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abstract: 'In this paper, we address the problem of unsupervised video summarization that automatically extracts key-shots from an input video. Specifically, we tackle two critical issues based on our empirical observations: (i) Ineffective feature learning due to flat distributions of output importance scores for each frame, and (ii) training difficulty when dealing with long-length video inputs. To alleviate the first problem, we propose a simple yet effective regularization loss term called variance loss. The proposed variance loss allows a network to predict output scores for each frame with high discrepancy which enables effective feature learning and significantly improves model performance. For the second problem, we design a novel two-stream network named Chunk and Stride Network (CSNet) that utilizes local (chunk) and global (stride) temporal view on the video features. Our CSNet gives better summarization results for long-length videos compared to the existing methods. In addition, we introduce an attention mechanism to handle the dynamic information in videos. We demonstrate the effectiveness of the proposed methods by conducting extensive ablation studies and show that our final model achieves new state-of-the-art results on two benchmark datasets.'
author:
- |
Yunjae Jung, Donghyeon Cho, Dahun Kim, Sanghyun Woo, In So Kweon\
Korea Advanced Institute of Science and Technology, Korea\
{yun9298a, cdh12242}@gmail.com, {mcahny, shwoo93, iskweon77}@kaist.ac.kr\
bibliography:
- 'egbib.bib'
title: Discriminative Feature Learning for Unsupervised Video Summarization
---
Introduction
============
Video has become a highly significant form of visual data, and the amount of video content uploaded to various online platforms has increased dramatically in recent years. In this regard, efficient ways of handling video have become increasingly important. One popular solution is to summarize videos into shorter ones without missing semantically important frames. Over the past few decades, many studies [@song2015tvsum; @ngo2003automatic; @lu2013story; @kim2014reconstructing; @khosla2013large] have attempted to solve this problem. Recently, Zhang [*et al.*]{} showed promising results using deep neural networks, and a lot of follow-up work has been conducted in areas of supervised [@zhang2016summary; @zhang2016video; @zhao2017hierarchical; @zhao2018hsa; @wei2018video] and unsupervised learning [@Mahasseni2017VAEGAN; @zhou2017deep].
Supervised learning methods [@zhang2016summary; @zhang2016video; @zhao2017hierarchical; @zhao2018hsa; @wei2018video] utilize ground truth labels that represent importance scores of each frame to train deep neural networks. Since human-annotated data is used, semantic features are faithfully learned. However, labeling for many video frames is expensive, and overfitting problems frequently occur when there is insufficient label data. These limitations can be mitigated by using the unsupervised learning method as in [@Mahasseni2017VAEGAN; @zhou2017deep]. However, since there is no human labeling in this method, a method for supervising the network needs to be appropriately designed.
Our baseline method [@Mahasseni2017VAEGAN] uses a variational autoencoder (VAE) [@kingma2013auto] and generative adversarial networks (GANs) [@goodfellow2014generative] to learn video summarization without human labels. The key idea is that a good summary should reconstruct original video seamlessly. Features of each input frame obtained by convolutional neural network (CNN) are multiplied with predicted importance scores. Then, these features are passed to a generator to restore the original features. The discriminator is trained to distinguish between the generated (restored) features and the original ones.
Although it is fair to say that a good summary can represent and restore original video well, original features can also be restored well with uniformly distributed frame level importance scores. This trivial solution leads to difficulties in learning discriminative features to find key-shots. Our approach works to overcome this problem. When output scores become more flattened, the variance of the scores tremendously decreases. From this mathematically obvious fact, we propose a simple yet powerful way to increase the variance of the scores. Variance loss is simply defined as a reciprocal of variance of the predicted scores.
In addition, to learn more discriminative features, we propose Chunk and Stride Network (CSNet) that simultaneously utilizes local (chunk) and global (stride) temporal views on the video. CSNet splits input features of a video into two streams (chunk and stride), then passes both split features to bidirectional long short-term memory (LSTM) and merges them back to estimate the final scores. Using chunk and stride, the difficulty of feature learning for long-length videos is overcome.
Finally, we develop an attention mechanism to capture dynamic scene transitions, which are highly related to key-shots. In order to implement this module, we use temporal difference between frame-level CNN features. If a scene changes only slightly, the CNN features of the adjacent frames will have similar values. In contrast, at scene transitions in videos, CNN features in the adjacent frames will differ a lot. The attention module is used in conjunction with CSNet as shown in [Fig. \[fig:overview\]]{}, and helps to learn discriminative features by considering information about dynamic scene transitions.
We evaluate our network by conducting extensive experiments on SumMe [@gygli2014creating] and TVSum [@song2015tvsum] datasets. YouTube and OVP [@de2011vsumm] datasets are used for the training process in augmented and transfer settings. We also conducted an ablation study to analyze the contribution of each component of our design. Quantitative results show the selected key-shots and demonstrate the validity of difference attention. Similar to previous methods, we randomly split the test set and the train set five times. To make the comparison fair, we exclude duplicated or skipped videos in the test set.
Our overall contributions are as follows. (i) We propose variance loss, which effectively solves the flat output problem experienced by some of the previous methods. This approach significantly improves performance, especially in unsupervised learning. (ii) We construct CSNet architecture to detect highlights in local (chunk) and global (stride) temporal view on the video. We also impose a difference attention approach to capture dynamic scene transitions which are highly related to key-shots. (iii) We analyze our methods with ablation studies and achieve the state-of-the-art performances on SumMe and TVSum datasets.
Related Work
============
Given an input video, video summarization aims to produce a shortened version that highlights the representative video frames. Various prior work has proposed solutions to this problem, including video time-lapse [@joshi2015real; @kopf2014first; @poleg2015egosampling], synopsis [@pritch2008nonchronological], montage [@kang2006space; @sun2014salient] and storyboards [@gong2014diverse; @gygli2014creating; @gygli2015video; @lee2012discovering; @liu2010hierarchical; @yang2015unsupervised; @gong2014diverse]. Our work is most closely related to storyboards, selecting some important pieces of information to summarize key events present in the entire video.
Early work on video summarization problems heavily relied on hand-crafted features and unsupervised learning. Such work defined various heuristics to represent the importance of the frames [@song2015tvsum; @ngo2003automatic; @lu2013story; @kim2014reconstructing; @khosla2013large] and to use the scores to select representative frames to build the summary video. Recent work has explored supervised learning approach for this problem, using training data consisting of videos and their ground-truth summaries generated by humans. These supervised learning methods outperform early work on unsupervised approach, since they can better learn the high-level semantic knowledge that is used by humans to generate summaries.
Recently, deep learning based methods [@zhang2016video; @Mahasseni2017VAEGAN; @sharghi2017query] have gained attention for video summarization tasks. The most recent studies adopt recurrent models such as LSTMs, based on the intuition that using LSTM enables the capture of long-range temporal dependencies among video frames which are critical for effective summary generation.
Zhang [*et al.*]{} [@zhang2016video] introduced two LSTMs to model the variable range dependency in video summarization. One LSTM was used for video frame sequences in the forward direction, while the other LSTM was used for the backward direction. In addition, a determinantal point process model [@gong2014diverse; @zhang2016summary] was adopted for further improvement of diversity in the subset selection. Mahasseni [*et al.*]{}. [@Mahasseni2017VAEGAN] proposed an unsupervised method that was based on a generative adversarial framework. The model consists of the summarizer and discriminator. The summarizer was a variational autoencoder LSTM, which first summarized video and then reconstructed the output. The discriminator was another LSTM that learned to distinguish between its reconstruction and the input video.
In this work, we focus on unsupervised video summarization, and adopt LSTM following previous work. However, we empirically worked out that these LSTM-based models have inherent limitations for unsupervised video summarization. In particular, two main issues exits: First, there is ineffective feature learning due to flat distribution of output importance scores and second, there is the training difficulty with long-length video inputs. To address these problems, we propose a simple yet effective regularization loss term called Variance Loss, and design a novel two-stream network named the Chunk and Stride Network. We experimentally verify that our final model considerably outperforms state-of-the-art unsupervised video summarization. The following section gives a detailed description of our method.
{width="85.00000%"}
Proposed Approach
=================
In this section, we introduce methods for unsupervised video summarization. Our methods are based on a variational autoencoder (VAE) and generative adversarial networks (GAN) as [@Mahasseni2017VAEGAN]. We firstly deal with discriminative feature learning under a VAE-GAN framework by using variance loss. Then, a chunk and stride network (CSNet) is proposed to overcome the limitation of most of the existing methods, which is the difficulty of learning for long-length videos. CSNet resolves this problem by taking a local (chunk) and a global (stride) view of input features. Finally, to consider which part of the video is important, we use the difference in CNN features between adjacent or wider spaced video frames as attention, assuming that dynamic plays a large role in selecting key-shots. [Fig. \[fig:overview\]]{} shows the overall structure of our proposed approach.
Baseline Architecture
---------------------
We adopt [@Mahasseni2017VAEGAN] as our baseline, using a variational autoencoder (VAE) and generative adversarial networks (GANs) to perform unsupervised video summarization. The key idea is that a good summary should reconstruct original video seamlessly and adopt a GAN framework to reconstruct the original video from summarized key-shots.
In the model, an input video is firstly forwarded through the backbone CNN (i.e., GoogleNet), Bi-LSTM, and FC layers (encoder LSTM) to output the importance scores of each frame. The scores are multiplied with input features to select key-frames. Original features are then reconstructed from those frames using the decoder LSTM. Finally, a discriminator distinguishes whether it is from an original input video or from reconstructed ones. By following Mahasseni [*et al.*]{}’s overall concept of VAE-GAN, we inherit the advantages, while developing our own ideas, significantly overcoming the existing limitations.
Variance Loss
-------------
The main assumption of our baseline [@Mahasseni2017VAEGAN] is “well-picked key-shots can reconstruct the original image well". However, for reconstructing the original image, it is better to keep all frames instead of selecting only a few key-shots. In other words, mode collapse occurs when the encoder LSTM attempts to keep all frames, which is a trivial solution. This results in flat importance output scores for each frame, which is undesirable. To prevent the output scores from being a flat distribution, we propose a variance loss as follows:
$$\begin{aligned}
{\pazocal{L}}_{V}(\textbf{\textit{p}}) = \frac{1}{\hat{V}(\textbf{\textit{p}}) + \textit{eps}},
\label{equ:var_loss}\end{aligned}$$
where $\textbf{\textit{p}}= \left \{ p_{t} : t = 1,..., T \right \}$, *eps* is epsilon, and $\hat{V}$$(\cdot)$ is the variance operator. $p_t$ is an output importance score at time $t$, and $T$ is the number of frames. By enforcing [Eq. (\[equ:var\_loss\])]{}, the network makes the difference in output scores per frames larger, then avoids a trivial solution (flat distribution).
In addition, in order to deal with outliers, we extend variance loss in [Eq. (\[equ:var\_loss\])]{} by utilizing the median value of scores. The variance is computed as follows: $$\begin{aligned}
\hat{V}_{median}((\textbf{\textit{p}})) = \frac{\sum \limits_{t=1}^{T} {|p_t - med(\textbf{\textit{p}})|^2}}{T},
\label{equ:var_loss_med}\end{aligned}$$ where $med(\cdot)$ is the median operator. As has been reported for many years [@Pratt1975medianfilter; @Huang1979median; @Zhang2014Wmedian], the median value is usually more robust to outliers than the mean value. We call this modified function variance loss for the rest of the paper, and use it for all experiments. 0 $$\begin{aligned}
\hat{V}_{median}(p_t) = \frac{\sum{|p_t - med(p_t)|^2}}{N}.
\label{equ:var_loss_med}\end{aligned}$$
Chunk and Stride Network
------------------------
To handle long-length videos, which are difficult for LSTM-based methods, our approach suggests a chunk and stride network (CSNet) as a way of jointly considering a local and a global view of input features. For each frame of the input video $\textbf{\textit{v}}= \left \{ v_{t} : t = 1,..., T \right \}$, we obtain the deep features $\textbf{\textit{x}}= \left \{ x_{t} : t = 1,..., T \right \}$ of the CNN which is GoogLeNet pool-5 layer.
As shown in [Fig. \[fig:overview\]]{} (a), CSNet takes a long video feature $\textbf{\textit{x}}$ as an input, and divides it into smaller sequences in two ways. The first way involves dividing $\textbf{\textit{x}}$ into successive frames, and the other way involves dividing it at a uniform interval. The streams are denoted as $\textbf{\textit{$c_{m}$}}$, and $\textbf{\textit{$s_{m}$}}$, where $\left \{m = 1,..., M \right \}$ and $M$ is the number of divisions. Specifically, $\textbf{\textit{$c_{m}$}}$ and $\textbf{\textit{$s_{m}$}}$ can be explained as follows: $$\begin{aligned}
c_{m} = \left \{ x_{i} : i = (m-1)\cdot(\frac{T}{M}) +1,..., m \cdot (\frac{T}{M}) \right \},\\
s_{m} = \left \{ x_{i} : i = m, m+k, m+2k, ...., m+T-M \right \},
\label{equ:csnet1}\end{aligned}$$ where $k$ is the interval such that $k=M$. Two different sequences, $c_{m}$ and $s_{m}$, pass through the chunk and stride stream separately. Each stream consists of bidirectional LSTM (Bi-LSTM) and a fully connected (FC) layer, which predicts importance scores at the end. Then, each of the outputs are reshaped into $\textbf{\textit{$c_{m}'$}}$ and $\textbf{\textit{$s_{m}'$}}$, enforcing the maintenance of the original frame order. Then, $\textbf{\textit{$c_{m}'$}}$ and $\textbf{\textit{$s_{m}'$}}$ are added with difference attention $d_t$. Details of the attentioning process are described in the next section. The combined features are then passed through sigmoid function to predict the final scores $p_t$ as follows: 0 $$\begin{aligned}
c_{t} = \sum\limits_{k=0}^{n-1} \Bigg( \sum\limits_{t=1 + k\lfloor{}T/n\rfloor{}}^{(k+1)\lfloor{}T/n\rfloor{}} {x_t} \Bigg),\\
s_{t} = \sum\limits_{k=0}^{n-1} \Bigg( \sum\limits_{t=1}^{\lfloor{}T/n\rfloor{}} {x_{n(t-1) + k+1}} \Bigg).
\label{equ:csnet2}\end{aligned}$$ $$\begin{aligned}
p^1_t = \textit{sigmoid}\Big(c'_{t} + d_t\Big),\\
p^2_t = \textit{sigmoid}\Big(s'_{t} + d_t\Big),\\
p_t = W[p^1_t + p^2_t].
\label{equ:csnet3}\end{aligned}$$ where $W$ is learnable parameters for weighted sum of $p^1_t$ and $p^2_t$, which allows for flexible fusion of local (chunk) and global (stride) view of input features.
Difference Attention
--------------------
In this section, we introduce the attention module, exploiting dynamic information as guidance for the video summarization. In practice, we use the differences in CNN features of adjacent frames. The feature difference softly encodes temporally different dynamic information which can be used as a signal for deciding whether a certain frame is relatively meaningful or not.
As shown in [Fig. \[fig:overview\]]{} (b), the differences $d^1_t$, $d^2_t$, $d^4_t$ between $x_{t+k}$, and $x_t$ pass through the FC layer ($d'^1_t$, $d'^2_t$, $d'^4_t$) and are merged to become $d_t$, then added to both $c_{m}$ and $s_{m}$. The proposed attention modules are represented as follows: $$\begin{aligned}
d_{1t} = |x_{t+1} - x_t|,\\
d_{2t} = |x_{t+2} - x_t|,\\
d_{4t} = |x_{t+4} - x_t|,\\
d_t = d'_{1t} + d'_{2t} + d'_{4t}.
\label{equ:diff}\end{aligned}$$
While the difference between the features of adjacent frames can model the simplest dynamic, the wider temporal stride can include a relatively global dynamic between the scenes.
------------------------------------------------ ------------------------------------------------
{width="0.48\linewidth"} {width="0.48\linewidth"}
[(a) Video 1]{} [(b) Video 15]{}
{width="0.48\linewidth"} {width="0.48\linewidth"}
[(c) Video 18]{} [(d) Video 41]{}
------------------------------------------------ ------------------------------------------------
Experiments
===========
Datasets
--------
We evaluate our approach on two benchmark datasets, SumMe [@gygli2014creating] and TVSum [@song2015tvsum]. SumMe contains 25 user videos with various events. The videos include both cases where the scene changes quickly or slowly. The length of the videos range from 1 minute to 6.5 minutes. Each video has an annotation of mostly 15 user annotations, with a maximum of 18 users. TVSum contains 50 videos with lengths ranging from 1.5 to 11 minutes. Each video in TVSum is annotated by 20 users. The annotations of SumMe and TVSum are frame-level importance scores, and we follow the evaluation method of [@zhang2016video]. OVP [@de2011vsumm] and YouTube [@de2011vsumm] datasets consist of 50 and 39 videos, respectively. We use OVP and YouTube datasets for transfer and augmented settings.
Evaluation Metric
-----------------
Similar to other methods, we use the F-score used in [@zhang2016video] as an evaluation metric. In all datasets, user annotation and prediction are changed from frame-level scores to key-shots using the KTS method in [@zhang2016video]. The precision, recall, and F-score are calculated as a measure of how much the key-shots overlap. Let “predicted" be the length of the predicted key-shots, “user annotated" be the length of the user annotated key-shots and “overlap" be the length of the overlapping key-shots in the following equations. $$\begin{aligned}
P=\frac{\text{overlap}}{\text{predicted}},
R=\frac{\text{overlap}}{\text{user annotated}},\\
\text{F-score}=\frac{2PR}{P+R} * 100\%.
\label{equ:metric}\end{aligned}$$
Evaluation Settings
-------------------
Our approach is evaluated using the Canonical (C), Augmented (A), and Transfer (T) settings shown in [Table \[tab:setting\]]{} in [@zhang2016video]. To divide the test set and the training set, we randomly extract the test set five times, 20% of the total. The remaining 80% of the videos is used for the training set. We use the final F-score, which is the average of the F-scores of the five tests. However, if a test set is randomly selected, there may be video that is not used in the test set or is used multiple times in duplicate, making it difficult to evaluate fairly. To avoid this problem, we evaluate all the videos in the datasets without duplication or exception.
Implementation Details
----------------------
For input features, we extract each frame by 2fps as in [@zhang2016video], and then obtain a feature with 1024 dimensions through GoogLeNet pool-5 [@szegedy2015going] trained on ImageNet [@russakovsky2015imagenet]. The LSTM input and hidden size is 256 reduced by FC (1024 to 256) for fast convergence, and the weight is shared with each chunk and stride input. The maximum epoch is 20, the learning rate is 1e-4, and 0.1 times after 10 epochs. The weights of the network are randomly initialized. M in CSNet is experimentally picked as 4. We implement our method using Pytorch.
#### Baseline
Our baseline [@Mahasseni2017VAEGAN] uses the VAE and GAN in the model of Mahasseni [*et al.*]{} We use their adversarial framework, which allows us unsupervised learning. Specifically, basic sparsity loss, reconstruction loss, and GAN loss are adopted. For supervised learning, we add binary cross entropy (BCE) loss between ground truth scores and predicted scores. We also put fake input, which has uniform distribution.
Quantitative Results
--------------------
In this section, we show the experimental results of our various approach proposed in the ablation study. Then, we compare our methods with the existing unsupervised and supervised methods and finally show the experimental results in canonical, augmented, and transfer settings. For fair comparison, we quote performances of previous research recorded in [@zhou2017deep].
------------------------------------------------ ----------------------------------------------
{width="0.48\linewidth"} {width="0.48\linewidth"}
[(a) CSNet 8]{} [(b) CSNet 2]{}
{width="0.48\linewidth"} {width="0.48\linewidth"}
[(c) CSNet 3]{} [(d) CSNet 4]{}
------------------------------------------------ ----------------------------------------------
#### Ablation study.
We have three proposed approaches: CSNet, difference attention and variance loss. When all three methods are applied, the highest performance can be obtained. The ablation study in [Table \[tab:ablation\]]{} shows the contribution of each proposed method to the performance by conducting experiments on the number of cases in which each method can be applied. We call these methods shown in exp. 1 to exp. 8 CSNet~1~ through CSNet~8~, respectively. If any of our proposed methods is not applied, we experiment with a version of the baseline in that we reproduce and modify some layers and hyper parameters. In this case, the lowest F-score is shown, and it is obvious that performance increases gradually when each method is applied.
Analyzing the contribution to each method, first of all, the performance improvement due to variance loss is immensely large, which proves that it is a way to solve the problem of our baseline precisely. CSNet~4~ is higher than CSNet~1~ by 4.1%, and CSNet~8~ is better than CSNet~5~ by 7.8%. The variance of output scores is less than 0.001 without variance loss, but as it is applied, the variance increases to around 0.1. Since we use a reciprocal of variance to increase variance, we can observe the loss of an extremely large value in the early stages of learning. Immediately after, the effect of the loss increases the variance as a faster rate, giving the output a much wider variety of values than before.
By comparing the performance with and without the difference attention, we can see that difference attention is well modeled in the relationship between static or dynamic scene changes and frame-level importance scores. By comparing CSNet~1~ to CSNet~3~, the F-score is increased by 1.2%. Similarly, CSNet~5~ and CSNet~7~ are higher than CSNet~2~ and CSNet~4~ by 1.5% and 2.0%. CSNet~8~ is greater than CSNet~6~ by 2.2%. These comparisons mean that the difference attention always contributes to these four cases.
We can see from our [Table \[tab:ablation\]]{} that CSNet also contributes to performance, and it is effective to design the concept of local and global features with chunk and stride while reducing input size of LSTM in temporal domain. Experiments on the number of cases where CSNet can be removed are as follow. CSNet~2~ is better than CSNet~1~ by 1.2%, and each CSNet~5~, CSNet~6~ outperform CSNet~3~, CSNet~4~ by 1.5%, 4.2%. Lastly, CSNet~8~ and CSNet~7~ have 4.4% difference.
Since each method improves performance as it is added, the three proposed approaches contribute individually to performance. With the combination of the proposed methods, CSNet~8~ achieves a higher performance improvement than the sum of each F-score increased by CSNet~2~, CSNet~3~ and CSNet~4~. In the rest of this section, we use CSNet~8~.
#### Comparison with unsupervised approaches.
[Table \[tab:unsupervised\]]{} shows the experimental results for SumMe and TVSum datasets using unsupervised learning in canonical settings. Since our approach mainly target unsupervised learning, CSNet outperforms both SumMe and TVSum over the existing methods [@elhamifar2012see; @khosla2013large; @de2011vsumm; @zhao2014quasi; @song2015tvsum; @zhou2017deep; @Mahasseni2017VAEGAN]. As a significant improvement in performance for the SumMe dataset, [Table \[tab:unsupervised\]]{} shows a F-score enhancement over 9.9% compared to the best of the existing methods [@zhou2017deep].
To the best of our knowledge, all existing methods are scored at less than 50% of the F-score in the SumMe dataset. Evaluation of the SumMe dataset is more challenging than the TVSum dataset in terms of performance. DR-DSN has already made a lot of progress for the TVSum dataset, but for the first time, we have achieved extreme advancement in the SumMe dataset which decreases the gap between SumMe and TVSum.
An interesting observation of supervised learning in video summarization is the non-optimal ground truth scores. Users who evaluated video for each data set are different, and every user does not make a consistent evaluation. In such cases, there may be a better summary than the ground truth which is a mean value of multiple user annotations. Surprisingly, during our experiments we observe that predictions for some videos receive better F-scores than in the results of ground truth. Unsupervised approaches do not use the ground truth, so it provides a step closer to the user annotation.
#### Comparison with supervised approaches.
We implemented CSNet~sup~ for supervised learning by simply adding binary cross entropy loss between prediction and ground truth to existing loss for CSNet. In [Table \[tab:supervised\]]{}, CSNet~sup~ obtains state-of-the-art results compared to existing methods [@gygli2014creating; @gygli2015video; @zhang2016summary; @zhang2016video; @zhou2017deep], but does not provide a better performance than CSNet. In general, supervision improves performance, but in our case, the point of view mentioned in the unsupervised approaches may fall out of step with using ground truth directly.
#### Comparison in augmented and transfer settings.
We compare our CSNet with other state-of-the-art literature with augmented and transfer settings in [Table \[tab:CAT\]]{}. We can make a fair comparison using the 256 hidden layer size of LSTM used by DR-DSN [@zhou2017deep], which is a previous state-of-the-art method. We obtain better performance in CSNet than CSNet~sup~, and our unsupervised CSNet performs better than the supervised method in any other approach except for GAN~sup~, which uses 1024 hidden size in TVSum dataset with augmented setting.
Qualitative Results
-------------------
#### Selected key-shots.
In this section, we visualize selected key-shots in two ways. First, in [Fig. \[fig:visual\]]{}, selected key-shots are visualized in bar graph form using various genre of videos. (a) - (d) show that many of our key-shots select peak points of labeled scores. In terms of the content of the video, the scenes selected by CSNet are mostly meaningful scenes by comparing colored bars with the images in [Fig. \[fig:visual\]]{}. Then, in [Fig. \[fig:visual-2\]]{}, we compare variants of our approach with a video 1 in TVSum. Although minor differences exist, each approach select peak points well.
#### Difference attention.
With a deeper analysis of difference attention, we visualize the difference attention in the TVSum dataset. Its motivation is to capture dynamic information between frames of video. We can verify our assumption that the dynamic scene should be more important than the static scene with this experiment. As shown in [Fig. \[fig:diff\]]{}, the plotted blue graph is in line with the selected key-shots, which highlight portions with high scores. The selected key-shots are of a motorcycle jump, which is a dynamic scene in the video. As a result, difference attention can effectively predict key-shots using dynamic information.
Conclusion
==========
In this paper, we propose discriminative feature learning for unsupervised video summarization with our approach. Variance loss tackles the temporal dependency problem, which causes a flat output problem in LSTM. CSNet designs a local and global scheme, which reduces temporal input size for LSTM. Difference attention highlights dynamic information, which is highly related to key-shots in a video. Extensive experiments on two benchmark datasets including ablation study show that our state-of-the-art unsupervised approach outperforms most of the supervised methods.
#### Acknowledgements
This research is supported by the Study on Deep Visual Understanding funded by the Samsung Electronics Co., Ltd (Samsung Research)
|
---
abstract: 'In this work we present a method for using Deep Q-Networks (DQNs) in multi-objective environments. Deep Q-Networks provide remarkable performance in single objective problems learning from high-level visual state representations. However, in many scenarios (e.g in robotics, games), the agent needs to pursue multiple objectives simultaneously. We propose an architecture in which separate DQNs are used to control the agent’s behaviour with respect to particular objectives. In this architecture we introduce *decision values* to improve the scalarization of multiple DQNs into a single action. Our architecture enables the decomposition of the agent’s behaviour into controllable and replaceable sub-behaviours learned by distinct modules. Moreover, it allows to change the priorities of particular objectives post-learning, while preserving the overall performance of the agent. To evaluate our solution we used a game-like simulator in which an agent - provided with high-level visual input - pursues multiple objectives in a 2D world.'
author:
- 'Tomasz Tajmajer [^1]'
bibliography:
- 'refs.bib'
title: 'Modular Multi-Objective Deep Reinforcement Learning with Decision Values'
---
Introduction
============
Many recent works on Reinforcement Learning focus on single-objective methods such as Deep Q-learning [@mnih-atari-2013; @Mnih2015]. As those methods provide great performance in task such as playing video games, many real-life problems require satisfying multiple objectives simultaneously. In single objective reinforcement learning the agent receives a single reward each time it performs an action. In multi-objective reinforcement learning (MORL) the agent receives multiple rewards - one for each objective. In particular, agents dealing with complex environments, such as autonomous robots or agents playing real-time video games, need to pursue multiple, often conflicting objectives.
To have a graspable example, lets consider an autonomous cleaning robot, which is able to clean floors, navigate through obstacles and autonomously return to charging station. The observable aggregated behaviour of such robot may be decomposed into three sub-behaviours: collision avoidance (ca), floor cleaning (fc) and recharging (rg). We may describe the objectives of the robot for each identified sub-behaviour in a multi-objective manner, or we can aggregate the sub-behaviours and define a single objective. In the former case, the robot-agent will receive a set of three rewards (\[$r_{ca}$, $r_{fc}$, $r_{rg}$\]) after each action. If the robot collides with a wall, it receives a negative reward related to collision avoidance ($r_{ca}$), yet the rewards related to floor cleaning and recharging do not depend on this event. However, in single-objective case, the robot will receive only one reward value (\[$r$\]) dependent on any of the three sub-behaviours. In case of collision, the the single-objective robot will receive a negative reward, but it will be indistinguishable from any negative reward provided with respect to other sub-behaviours such as depletion of batteries.
In single objective scenarios, we may find an optimal policy for which the sum of rewards collected by the agent is the highest possible. Methods such as Q-learning should converge to optimal policies [@Sutton_RL]. However, for multi-objective problems, many such optimal policies may exist, depending on the trade-offs between satisfying particular objectives [@MORL_overview].
Autonomous agents, such as our example cleaning robot, are not really independent - they usually have a purpose defined by another agent: human. This aspect is often neglected in the literature, but is significant when considering practical applications of intelligent agents in robotics, automation or even when designing AIs for video games (always winning AI is not the one that many humans would like to play against). Our cleaning robot may follow a policy for which collision avoidance has greater importance than floor cleaning - in such case the robot should focus on avoiding collisions even at the cost of worse performance at floor cleaning. It is however for the user of such robot to decide, what should be the proportion between carefulness and cleanliness. The user may even want to fully disable some functions (behaviours) of the robot. Yet, state of the art reinforcement learning methods, such as Deep Q-Learning, do not allow to modify the behaviour of the agent after it was trained.
We see that when considering practical applications it is desired to have a multi-objective reinforcement learning method with the following features available post-learning: 1) ability to select the sub-set of pursued objectives and 2) ability to change the impact of particular objectives on the overall policy of the agent. As we will show later, the method presented in this paper posses those features.
Multi-objective problems may be approached using *single-policy* or *multi-policy* methods. The simplest single-policy method uses a *scalarization function* [@scalarization], which converts multiple objectives into a single objective. Scalarization methods utilize a weight matrix to obtain a single score from multiple action-value functions. Some techniques assign linear priorities to objectives [@optimal_policies_with_multiple_criteria; @Vamplew2011]. This allows to obtain a single optimal policy with respect to objectives ordered by those priorities.
In contrast to single-policy methods, multi-policy MORL methods are used for find a set of policies. Their aim is to approximate the Pareto front of policies [@MORL_overview]. In multi-policy methods, the preference of objectives does not need to be set a priori as a Pareto optimal policy for any preference may be obtained at runtime [@MORL_pareto_dominating_p].
A natural approach in MORL is to use separate learning modules for each objective [@modular_sarsa]. Modularity allows to decompose the problem into components that are to some extent independent [@parallel_rl_multiple_rewards]; modularity may be required for providing features desired in practical applications that were listed earlier. Some works deal with transforming complex single-objective problems to many simpler objectives [@multiobjectivization]. Such methods may be used to benefit from modular approach while solving single-objective problems.
Although Deep Q-Networks gained much attention in recent years, not many works consider the use of DQNs in multi-objective problems. Recently authors of [@mo_drl] proposed a multi-policy learning framework that utilizes Deep Q-Networks.
Learning behaviours in embodied agents, such as robots, is a problem well fitted for reinforcement learning methods. In *embodied artificial intelligence*, the idea of *parallel, loosely coupled processes* [@understanding_intelligence] is proposed as a principle for designing embodied agents. It states, that the control logic for embodied agents should consists of many independent components dedicated for particular aspects of the agent’s behaviour. The aggregated behaviour of an agent emerges from cooperation or competence among those components.
In this work we will present a method for combining multiple Deep Q-Networks for solving multi-objective problems. We will introduce decision values used for more advanced scalarization of multiple Q-functions. Furthermore we will combine decision values with user define priorities, to have an architecture that can dynamically adapt its behaviour with respect to user’s preferences.
In section \[background\] we will briefly describe single- and multi- objective reinforcement learning. Next, in section \[main\_section\] we will describe how many separate DQNs may be used together and we will define decision values. In section \[eval\] we will present a simple 2D game - a virtual environment including an autonomous agent that has a local (situated) sensory inputs and may pursue different objectives. Finally in the last section we will evaluate our solution and present the results of our experiments.
Background
==========
Single Objective Reinforcement Learning
---------------------------------------
In the single-objective reinforcement learning an agent interacts with the environment by perceiving the state $s_t \in S$ and performing an action $a_t \in A$ for each step $t$. The actions are chosen by the agent according to some policy $\pi$. After performing an action, the agent receives a reward $r_t$. Then the agent observes the next state $s_{t+1}$ and the process repeats. The goal of the agent is to maximize the expected discounted reward $R_t = \sum^{\infty}_{k=0}{\gamma^k r_{t+k}}$, where $\gamma \in [0,1]$ is the discount factor.
In Q-learning actions are selected based on $Q(s,a)$, which represents the expected discounted reward for performing action $a$ in state $s$. For given state s, $a_t = \underset{a}{\arg\max} Q(s, a)$ is the optimal action. The policy of an agent, denoted by $\pi$, is the probability of selecting action $a$ in state $s$. If the agent always selects the optimal action, then we say that it follows an optimal policy $\pi_\star$. Knowing the $Q(s,a)$ allows to create an optimal policy simply by selecting the action with the highest Q-value. Deep Q-learning utilizes Deep Neural Networks for approximating $Q(s,a)$ values, thus enabling this method to be used in many real-world applications. Deep Q-Networks [@Mnih2015] may be used used with high-level visual inputs such as those provided by video games.
Multi-Objective Reinforcement Learning {#morl_section}
--------------------------------------
We may consider a more complex reinforcement learning scenario in which multiple objectives are pursued by the agent. Let $O$ be the set of objectives of an agent. We may assign a priority $p$ to each objective $o \in O$ such that $o_k$ will have lower priority than $o_j$ when $p(o_k) < p(o_j)$. For further analysis we will assume that $\forall_{o \in O} p(o) \geq 0$, so that priorities may be interpreted as weights.
The agent, instead of a single reward, receives a vector of rewards at each time-step $t$ with respect to each objective $o_i$, i.e: $\vec{r}_t = \left[r_{1,t}, r_{2,t}, \dots \, r_{n,t} \right]$, where $r_{i,t}$ corresponds to objective $o_i$. For each objective $o_i$ and step $t$ we may define the discounted return as: $$R_{i,t} = \sum^{\infty}_{k=0}{\gamma^k r_{i,t+k}}$$ Moreover, for each objective $o_i$ there is a Q-function $Q_i(s,a)$ that represents the expected discounted return $R_{i,t}$, i.e: $Q_i(s,a) = \mathbb{E}\left[ R_{i,t} \, | \, s_t = s, a_t = a\right]$. We may define a vector of Q-functions, which includes $Q(s,a)$ for each objective $o_i$: $$\vec{Q}(s,a) = \left[Q_1(s,a), Q_2(s,a), ..., Q_n(s,a)\right]$$ The function $Q_i(s,a)$ may be used by the agent to determine the optimal action with respect to objective $o_i$ at time-step $t$, given state $s_t$: $$\label{argmax_Q}
a_{i,t} = \underset{a}{\arg\max}\,Q_i(s_t, a)$$
The vector $\vec{a}_t = \left[a_{1,t}, a_{2,t}, ..., a_{n,t}\right]$ consists of actions optimal with respect to particular objectives at a given time-step $t$. Because at each step, the agent may perform only a single action, a method of reducing $\vec{a}_t$ to a single action is required.
A common method for selecting a single action is the scalarization [@scalarization] of $\vec{Q}(s,a)$ using some scalarization function and a weight vector $\vec{w}$. Typically a linear scalarization is applied, so that: $$SQ(s,a) = \sum^N_{i=1} {w_i Q_i(s,a)}$$ Then $SQ(s,a)$ may be used as in equation \[argmax\_Q\] to select an action. The weight wector in this case corresponds to priorities assigned to particular objectives.
In the further sections of this paper, we will show how to apply scalarization in Deep Q-Networks and we will introduce Decision Values to dynamically adjust the weights for improved performance of the agent. For simplicity, further in the text we will use the index $i$ to note that a particular value or function is defined for any objective $o_i$, and by $N$ we will define the number of objectives.
Using multiple DQNs {#main_section}
===================
We have considered an agent that have multiple objectives, receives rewards with respect to those objectives and has a separate Q-function for each objective. In this section we will describe how to merge q-values obtained from Deep Q-Networks for different objectives and how the impact of particular DQNs on the behaviours of the agent may be controlled by using Decison Values. Finally we will describe the learning process utilizing DQNs with Decision Values. We will refer to our method as to Multi-Objective Deep Q-Network with Decision Values (MODQN-DV).
Combining Q-values {#combining_dqns}
------------------
In case of multi-objective agent, we may use a separate DQN as an approximator for each $Q_i(s,a)$ in the $\vec{Q}(s,a)$ vector. Such agent would be controlled by multiple Deep Q-Networks working in parallel. Each DQN provides a list of q-values and we want to use q-values from all DQNs to select a single action $a$ that will be performed by the agent
Let us define a vector $\vec{q}_i$ that consists of q-values provided by $Q_i(s,a)$ for each possible action $a \in A$ and a single objective $o_i$, i.e.: $$\vec{q}_i = \left[Q_i(s, a_0), Q_i(s, a_1), ..., Q_i(s, a_j)\right]$$ In the single-objective case the optimal action $a$ would be equal to $a_j$ for such $j$ that $\vec{q}_{i,j} = \max \vec{q}_i$. For multi-objective case we can use scalarization to sum up all $\vec{q}$ vectors and then select the action corresponding to the maximal value of such scaled q-value vector. In this approach, q-values may be interpreted as votes of certain DQN, which are summed-up and the highest-voted action is selected. We need to stress here that simply adding the vectors does not produce a meaningful result yet. The q-values produced by different Q-functions are not scaled. In general q-values may be any real numbers. If we want them to represent votes for particular actions, each $\vec{q}_i$ vector needs to be rescaled to $[0,1]\subseteq\mathbb{R}$. Many approaches for scaling the vector may be applied. In our experiments we use the following scaling function for which $\min(\vec{q}_i)$ is mapped to $0$ and $\max(\vec{q}_i)$ to $1$:
$$scale(\vec{x}) = \frac{\vec{x} - \min(\vec{x})}{\max (\vec{x} - \min(\vec{x}))}$$
The scalarized q-vector is then defined as: $$\vec{q_s} = \sum_{i=1}^N w_i scale(\vec{q_i})$$
Now, using the rescaled $\vec{q}_i$ vectors we can sum them up and select one action with the highest total q-value. For example, let have actions ${a_1, a_2, a_3}$, weight vector $\vec{w} = [1, 1, 1]$, objectives $o_1$, $o_2$ and corresponding q-vectors $ \vec{q}_1 = [0, 0.6, 1]$ and $ \vec{q}_2 = [1, 0.5, 0]$. Adding them will result in vector $[1, 1.1, 1]$, for which the second element is the maximal, thus the corresponding action $a_2$ should be selected.
Decision Value {#decision_value}
--------------
The scalarization allows to combine outputs from multiple DQNs. However, such a combination does not guarantee a meaningful action selection. Let us return to previous to examples and consider a vacuum cleaner approaching a wall; actions ${a_1, a_2, a_3}$ correspond to turning left, going straight, and turning right respectively. If the vacuum cleaner perform the action proposed in $q_1$ it will turn right, alternatively if it uses $q_2$ then it will turn left. Using the sum will however lead to going straight forward and hitting the wall. So while both DQNs suggested a meaningful action, their sum is not meaningful at all. We see that using constant weights while summing q-values does not provide a solution for this problem.
To solve this issue, we would need to dynamically choose which q-value vectors are more important in a particular state. In other words, we would like to have a meta-policy for choosing the actual policy of the agent. However, as the agent pursues many objectives, it is hard to define this meta-policy with respect to all objectives. To overcome this problem we propose to indicate the *value* of the *decision* provided by each DQN with respect to corresponding objective pursued by the agent.
The proposed *decision values* may be indicated independently by each DQN based on the current state and used as additional weights while summing up q-value vectors. Going back to the previous example: let assume that $q_1$ is the output from DQN associated with collision avoidance and $q_2$ is the output from DQN associated with cleaning. As the robot approaches a wall, the decision regarding collision avoidance is clearly more important than the decision regarding cleaning. This is because if the robot does not make any decision, it will collide with the wall and receive a negative reward with respect to collision avoidance objective. However, not making the decision will not affect cleaning objective (assuming that the cleanliness of the floor in front of him is not different than in other places). Thus, at this particular state the value of $q_1$ is higher than the value of $q_2$ and $q_1$ should be summed with a higher weight.
We may define the decision value signal $d \in [0,1]\subseteq\mathbb{R}$, and by $d_i$ denote the decision value associated with $DQN_i$. Now the scalarized q-vector would use decision values instead of constant weights: $$\vec{q_d} = \sum_{i=1}^N d_i scale(\vec{q_i})$$
We may additionally include the external preferences indicated by values of priorities $p_i$ assigned to objectives as introduced in \[morl\_section\]. This way the q-values will be scaled both by dynamic decision values and static priorities. Moreover, for technical reasons, we need to add $\vec{\mu}$, which is a vector containing very small random values. This will ensure that in a rare cases when all decision values are equal to $0$, a random action will be chosen. Finally the scaled, decision value- and priority- weighted q-value vector denoted by $\vec{q_\sigma}$ is equal to:
$$\vec{q_\sigma} = \vec{\mu} + \sum_{i=1}^N d_i p_i scale(\vec{q_i})$$
Acquiring values of decisions
-----------------------------
Now, as we have a method of applying decision values in the scalarization of multiple objectives, let us explain in more details how decision values are defined and how they can be learned by reinforcement learning.
First we should consider how objectives of an agent are defined. Again let us refer to the vacuum cleaning robot example. If the agent had only two objectives: a) to seek dirt and b) to avoid colliding with obstacles, then we could define two reward/terminal states: state A - state in which dirt is collected, state B - state in which the robot is colliding with something. There is a notable difference between those two states. In the first case, the agent should be rewarded positively, but in the the latter case, it should be rewarded negatively. Moreover, if the agent is not in any of those states, it should be not rewarded at all. We can describe the first objective as being *attractive* (as it attracts the agent by positive rewards) and the second as being *repulsive* (as it repulses the agent by negative rewards). Many problems in robotics, games or other fields of AI may be presented using a set of attractive or repulsive objectives. In particular some problems may be decomposed into such set of objectives to promote more granular learning and control. Such decomposition is usually simpler and more intuitive compared to more advanced reward shaping techniques.
Let us consider an agent moving in a state-space with attractive and repulsive states. As the agent approaches one of those states, it becomes more critical to perform an action that will either move the agent towards such state or away from it. The value of the decision made with respect to an objective near a rewarding state rises as the distance to this state becomes shorter. This is a simple and intuitive heuristic: if an agent pursues multiple equally weighted objectives, then it probably should focus most on the objective that is already very close to being accomplished.
We can thus create a *decision reward* - the reward provided to the agent for performing a decision - which would be simply the absolute value of the reward provided with respect to an objective: $\rho_i = abs(r_i)$. Now we can define the decision value as a *state-value* function [@Sutton_RL], returning the value of the state $s$ under policy $\pi$, with respect to the decision rewards of a particular objective:
$$D_i(s) = \mathbb{E}_{\pi}\left[ \sum_{k=0}^{\infty} \gamma^k\,\rho_{i,t+k+1} \,\middle|\, s_t = s \right]$$
Such defined decision value will provide high values around rewarding states (either positive or negative) and low values in states which are far from rewarding states. In any state, the decision value will provide the importance of particular objective. The proposed decision value function will hence provide values representing the chances of achieving a rewarding state (with respect to some objective $o$) given the current state $s$ and following policy $\pi$. Where policy $\pi$ is the policy provided by the Q-function for a particular objective.
It is important to note, that the decision value, as defined, can not be directly used for scalarization, because its value may be any positive number. Moreover, the range of the values provided for different objectives may be very broad. To overcome this problem, the decision value needs to be scaled to be in range $[0, 1]$ as noted in section \[decision\_value\]. However, the unscaled decision value is needed during learning as it will be shown in the next section. We will therefore denote the unscaled decision value by $D_i$ and define the scaled decision value by $d_i$ as follows: $$d_i = \sigma\left (\frac{(D_i - \alpha_i)} {\beta_i} \right )$$
Where $\sigma$ is the sigmoid function; $\alpha_i$ and $\beta_i$ are derived during learning: $\alpha_i$ is an approximation of the mean value of $D_i$, while $\beta_i$ is an approximation of the standard deviation of $D_i$.
Learning
--------
Have defined decision values, we may move to the method of learning such values along with learning policies for particular objectives. We use Deep Q-Networks to approximate the values of Q-functions. Following the state of the art in this field a $DQN$ provides the approximated function $Q(s,a; \theta)$, where $\theta$ are the learnable parameters of the neural network. As in our model we use multiple DQNs, there is a function $Q_i(s, a; \theta_i)$ for a $DQN_i$ related to objective $o_i$. Each $DQN_i$ is optimised iteratively, using the following loss function for each iteration $j$: $$\begin{split}
L^{Q}_{i,j}(\theta_{i,j}) &= \mathbb{E}_{(s,a,r_i,s')\sim U(M_i)} [ ( r_i + \\
&+ \gamma \underset{a'}{\max}\,Q_i(s',a'; \theta_{i,j}^-) - Q_i(s,a; \theta_{i,j}) )^2 ]
\end{split}$$ As introduced in [@mnih-atari-2013], there are in fact two neural networks involved in the learning process of a single DQN. The *on-line network* $Q_i(s, a; \theta)$ is updated at each iteration, while the *target network* $Q_i(s', a';\theta^-)$ is updated only each $K$ iterations. Moreover *experience replay* is used to further improve the learning process. The agent stores experienced states, actions and rewards in a *replay memory* $M_i$ for each $DQN_i$ respectively. Then at each iteration, each $DQN_i$ is trained using a sample of past experiences selected uniformly at random from the corresponding replay memory $M_i$. Those samples are used as mini-batches for gradient descent optimization.
The Decision Value may be updated using TD-learning [@Sutton_RL] similarly as for any state value function, by using the following update rule: $$D_i(s_t) \leftarrow D_i(s_t) + \alpha \left[\rho_i + \gamma D_i(s_{t+1}) - D_i(s_t)\right]$$ As we use a neural network for approximating $D_i(s)$, we may define the loss function as follows: $$\begin{split}
L^{D}_{i,j}(\theta_{i,j}) &= \mathbb{E}_{(s, \rho_i, s')\sim U(M_i)} [ ( \rho_i +\\
&+ \gamma D_i(s'; \theta_{i,j}^-) - D_i(s; \theta_{i,j}) )^2 ]
\end{split}$$ The decision value is provided by an additional output of the DQN and the learning procedure is analogical to Q-function. Moreover the decision value requires scaling, for which the parameters $\alpha$ and $\beta$ need to be learned. If we include $\alpha$ and $\beta$ in the neural network parameters $\theta$, then the additional loss function for the decision value scaling would be defined as: $$\begin{split}
L^{d}_{i,j}(\theta_{i,j}) &= \mathbb{E}_{(s)\sim U(M_i)} [(0.5 - \sigma(D_i(s; \theta_{i,j})))^2 + \\
&+ (1 - \max_s(D_i(s; \theta_{i,j})) - \min_s(D_i(s; \theta_{i,j})))^2]
\end{split}$$ The neural network is optimized using a combined loss function for Q-values, decision values and scaling of the decision values: $$\begin{split}
L_{i,j}(\theta_{i,j}) &= L^{Q}_{i,j}(\theta_{i,j}) + L^{D}_{i,j}(\theta_{i,j}) + L^{d}_{i,j}(\theta_{i,j})
\end{split}$$
Evaluation {#eval}
==========
Cleaner - a 2D game-like virtual environment
--------------------------------------------
To evaluate the solution presented in this paper we created *Cleaner* - a simple game-like virtual environment, simulating the behaviour of an autonomous vacuum cleaner. The environment consists of an agent, walls, recharge areas and dirt consumable by the agent. Cleaner is presented in Figure \[game\]. The agent is a circular object that may move around the map by performing one of three actions: move forward, turn left and turn right. The map is a continuous space. The agent perceives the environment only by visual sense, i.e. a $W$ x $H$ pixel (width and height) rectangle situated in front of him. This visual input is converted to gray-scale (8bit). Agent’s world (white) is surrounded by walls and filled with obstacles (black rectangles) which agent can not pass. Agent may pick up dirt and recharge itself. Dirt is indicated by three small coaxial circles (black), while recharging field is indicated by a gray rectangle. Dirt re-spawns at random positions on the map after being consumed by the agent. The quantity of dirt, recharge fields and obstacles is constant during the episode. Eater is a simplified simulation of a mobile robot moving on a flat surface (e.g. floor) with a video camera attached at the top of the robot pointed towards the floor.
The agent has a battery level $E \leq E_{max}$, which is decreased at each time step by $E_{step}$. The battery level may be increased when the agent enters the recharging area by $(1 - E) \cdot 0.1$ each step. An episode ends when the agent’s energy level drops to $0$ or when $2000$ steps pass. The agent starts each game with initial battery level $E = E_{start}$. The position of dirt, recharge fields and obstacles as well as the initial position of the agent are chosen randomly at the start of the episode.
The agent has three objectives: (ca) collision avoidance, (fc) cleaning and (rg) recharging.
The rewards for particular objectives are as follows: objective (ca): $-1$ for collision, $0$ otherwise; objective (fc): $+1$ for collecting dirt, $0$ otherwise; objective (rg): $-1$ for for each step when $E < 0.1$, $(1 - E) \cdot 0.1$ while charging and $0$ otherwise.
In all experiments described in this chapter, the game options were as follows: $E_{start} = E_{max} = 1.0$, $E_{step} = 0.001$. The size of the agent sight rectangle is $W = 50$px, $H = 50$px. The quantity of food is 20. The numer of obstacles varies randomly from 1 to 5, and the number of charging areas varies randomly from 1 to 3.
MODQN-DV implementation {#implementation}
-----------------------
[ll]{} Parameter & Value\
learning steps & 1000000\
replay memory size & $10000$\
target network update rate & $1000$\
learning rate & $0.001$\
$\epsilon$ start value & $1$\
$\epsilon$ end value & $0.1$\
$\epsilon$ end step & 100000\
discount & $0.99$\
batch size & $32$\
optimizer & Adam\
Our implementation of the MODQN-DV was based on the baseline DQN implementation [@baselines] developed by OpenAI using TensorFlow[@tensorflow2015-whitepaper]. We expanded the standard DQN with additional decision value outputs and mechanism for scalarizing q-values from multiple DQNs. Each single DQN in a MODQN-DV consist of a convolution network with three convolution layers and no pooling layers, followed by a fully connected layer and the output layer. Dueling [@dueling] and double q-learning [@double_q_learning] were used. The additional decision value output is a single neuron linear layer connected to the state score layer used for dueling.
The parameters of the convolution network were kept default as provided in the baselines implementation. The size of the fully connected layer in our models is set to 128, and the size of the input image is our case is 50x50x1, thus the q-values are provided based only on an image input from a single state. The memory replay was modified to store rewards with respect to all objectives separately. The prioritized experience replay[@per] was not used in our implementation. The hyperparameters used for training DQNs during evaluation are presented in table \[dqn\_params\].
During training of the MODQN-DV, loss functions are used as specified in section \[learning\]. DQNs for all objectives are trained simultaneously and scaled decision values are used for scalarization during learning.
Experiments
-----------
[.5]{}
$p_{ca}$ $p_{fc}$ $p_{rg}$ $\Sigma r_{ca}$ $\Sigma r_{fc}$ $\Sigma r_{rg}$ $\Sigma\Sigma r_i$
--------------------- ---------- ---------- ----------------- ----------------- ----------------- -------------------- --
1 1 1 -88.4 47.6 -35.0 -75.9
$\Delta_{baseline}$ — — — —
**1** 0 0 **-51.9** 24.0 -46.2 -74.1
$\Delta_{baseline}$ **41.4%** -49.6% -32.0% 2.37%
0 **1** 0 -303.0 **50.0** -40.3 -293.3
$\Delta_{baseline}$ -242.7% **5.1%** -15.2% -286.74%
0 0 **1** -311.8 20.6 **-35.9** -327.2
$\Delta_{baseline}$ -252.6% -56.7% **-2.7%** -331.32%
**0.5** 0.3 0.2 **-45.7** 42.9 -39.2 -42.1
$\Delta_{baseline}$ **48.4%** -9.9% -12.2% 44.55%
**0.5** 0.2 0.3 **-68.4** 38.3 -39.5 -69.6
$\Delta_{baseline}$ **22.7%** -19.5% -12.9% 8.27%
0.2 **0.5** 0.3 -143.7 **51.3** -33.2 -125.6
$\Delta_{baseline}$ -62.5% **7.9%** 4.9% -65.63%
0.3 **0.5** 0.2 -90.0 **50.2** -34.7 -74.4
$\Delta_{baseline}$ -1.7% **5.6%** 0.9% 1.93%
0.2 0.3 **0.5** -140.6 45.2 **-34.7** -130.1
$\Delta_{baseline}$ -59.0% -4.9% **0.6%** -71.54%
0.3 0.2 **0.5** -123.1 42.4 **-33.8** -114.5
$\Delta_{baseline}$ -39.2% -10.9% **3.3%** -51.01%
-0.1in
[.5]{}
$p_{ca}$ $p_{fc}$ $p_{rg}$ $\Sigma r_{ca}$ $\Sigma r_{fc}$ $\Sigma r_{rg}$ $\Sigma\Sigma r_i$
--------------------- ---------- ---------- ----------------- ----------------- ----------------- -------------------- --
1 1 1 -61.0 51.3 -28.8 -38.5
$\Delta_{baseline}$ — — — —
**1** 0 0 **-77.6** 32.0 -45.0 -90.6
$\Delta_{baseline}$ **-27.2%** -37.6% -56.6% -135.26%
0 **1** 0 -518.2 **33.3** -58.5 -543.4
$\Delta_{baseline}$ -749.1% **-35.0%** -103.4% -1310.52%
0 0 **1** -126.9 31.4 **-27.8** -123.3
$\Delta_{baseline}$ 108.0% -38.7% **3.3%** -220.12%
**0.5** 0.3 0.2 **-35.7** 47.7 -35.7 -23.7
$\Delta_{baseline}$ **41.6%** -7.0% -24.1% 38.54%
**0.5** 0.2 0.3 **-40.3** 45.2 -32.6 -27.7
$\Delta_{baseline}$ **34.0%** -11.9% -13.4% 28.08%
0.2 **0.5** 0.3 -236.2 **49.8** -37.8 -224.2
$\Delta_{baseline}$ -287.0% **-2.8%** -31.5% -482.04%
0.3 **0.5** 0.2 -218.9 **50.3** -38.7 -207.3
$\Delta_{baseline}$ -258.7% **-1.8%** -34.5% -438.10%
0.2 0.3 **0.5** -86.7 41.9 **-29.4** -74.2
$\Delta_{baseline}$ -42.1% -18.3% **-2.1%** -92.71%
0.3 0.2 **0.5** -80.8 40.7 **-29.1** -69.3
$\Delta_{baseline}$ -32.4% -20.7% **-1.3%** -79.83%
-0.1in
To evaluate our method we conducted a series of experiments utilizing MODQN-DV and the cleaner environment. In particular we compared the performance of multiple Deep Q-Networks for case a) where decision values were enabled for scalarization and case b) where the decision values were disabled. This comparison gave us a clear indication of the impact of decision values on the performance. We will refer to case (a) as MODQN-DV (b) as to MODQN.
The experiment for both cases (a) and (b) were conducted as follows. First the DQNs were trained using the implementation and parameters as provided in section \[implementation\] and table \[dqn\_params\]. In case (a) the decision values were trained and used for scalarization. In case (b) decision values were disabled during training and their values were forcefully set to $1$. During training, the user defined priorities for objectives were set to $1$ in all cases (all objectives were weighted equally during scalarization). For each case the training procedure was repeated 6 times and all trained neural networks were saved. As show in figure \[sum\_of\_rewards\], the learning of MODQN-DV is stable over time.
Next, the trained MODQN-DV and MODQN networks were used for evaluation with 10 different sets of user defined priorities $(p_{ca}, p_{fc}, p_{rg})$ as provided in tables \[eval\_with\_dv\] and \[eval\_without\_dv\]. In a single evaluation, 100 episodes were played. The same sequence of randomly generated map layouts were used for each run. The sum of collected rewards were recorded for each run. For each set of priorities, 6 runs performed by 6 separately trained MODQN-DV and MODQN instances were averaged.
Results
-------
The results presented in the tables \[eval\_with\_dv\] and \[eval\_without\_dv\] are averaged sums of collected rewards with respect to each objective, namely: $\Sigma r_{ca}$ for objective (ca) - collision avoidance, $\Sigma r_{fc}$ for objective (fc) - cleaning and $\Sigma r_{rg}$ for objective (rg) - recharging. $\Sigma \Sigma r_i$ is the sum of the sums of rewards - it indicates the total performance of the agent. Priorities $(p_{ca}, p_{fc}, p_{rg})$ correspond to objectives (ca), (fc) and (cg).
The set of priorities: $(p_{ca}=1, p_{fc}=1, p_{rg}=1)$ was used as the baseline for evaluation (also those priories were used during training). For each row in the tables \[eval\_with\_dv\] and \[eval\_without\_dv\] there is an additional row marked as $\Delta_{baseline}$ with values showing the percentage of gain or loss of collected rewards with respect to the baseline value for each case. The green and red colours of the cells indicate if the reward gain for a particular set of priorities was better compared to the corresponding case in the second table.
The aim of the evaluation was to test how the overall performance of the agent changes when priorities are different from the initial values used during training. As we can see in table \[eval\_with\_dv\] on 7 of 9 cases, the use of MODQN-DV helped to preserve (or even increased) the overall performance of the agent in comparison to MODQN. Moreover, in almost all cases, the performance of the agent with respect to the objective with the highest priority (marked in bold in the tables) increased when decision values were used. However, it should be noted that in the baseline case, the overall performance of the agent using decision values was lower compared to the case without decision values. The results show that the proposed solution has a significant impact on the performance when priorities are modified post-training.
It is also worth noticing how the decision values change as the agent moves. As expected, the decision value for a particular objective rises when the agent approach a state where it could receive a reward. For instance, the value of collision avoidance rises significantly when the agent is very close to a wall or an obstacle. Moreover, the decision value drops when the agent is in a state far from receiving a reward. For example, if the agent does not perceive any walls or obstacles, then the collision avoidance decision value is lower than average. The agent thus usually selects the action, which is related to the most promising objective at a particular state.
Conclusions
===========
In this paper, we presented a method for using multiple Deep Q-Networks to approach multi-objective problems called Multi Objective Deep Q-Networks with Decision Values (MODQN-DV). We introduced decision values to DQNs in order to improve the scalarization of outputs from multiple DQNs. Our method requires only slight modification of existing DQN architectures, while it introduces a number of benefits: 1) it enables the decomposition of problems in to smaller sub-problems, for which independent DQNs may be trained simultaneously, 2) it provides a method for robust manipulation of priorities after the training, which also allows to completely disable DQNs responsible for particular behaviour/objective, 3) it allows to add new objectives to already trained agent without the need of retraining and to tune their impact on the behaviour of the agent.
In the experimental part, we shown that in most cases MODQN-DV improves the performance of the agent, that uses a different set of priorities compared to the training phase. The results are promising, however more tests should be performed using other benchmarks.
In this paper, we also introduce *cleaner* - a benchmark for multi-objective reinforcement learning problems that provides visual state representation. The authors are not aware of any other existing multi-objective benchmark that would be comparable to atari games benchmark or other provided by OpenAI.
In future work we want to improve the performance of MODQN-DV; one possible improvement is the use of common convolutional layers for all DQNs. It is particularly interesting to use MODQN-DV in very complex environments, such as video games. Recently published Starcraft 2 learning environment may be a good choice for further tests of MODQN-DV architecture, as strategy games may be perceived as multi-objective problems.
[^1]: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland; email: `t.tajmajer@mimuw.edu.pl`
|
---
abstract: 'We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle.'
address: |
Department of Mathematics\
Chalmers University of Technology\
S-412 96 GÖTEBORG\
SWEDEN
author:
- Bo Berndtsson
title: 'A comparison principle for Bergman kernels.'
---
[*Dedicated to the memory of Mikael Passare*]{}
Introduction.
=============
Let $\phi$ and $\psi$ be two plurisubharmonic functions in a complex manifold $X$, and let $\Omega$ be a relatively compact subdomain in $X$. Assume that on the boundary of $\Omega$, $\phi\leq\psi$, and that inside the domain the Monge-Ampere measures of $\phi$ and $\psi$ satisfy $$(dd^c\phi)^n\geq (dd^c\psi)^n.$$ Then the maximum principle for the Monge-Ampere equation asserts that the inequality $\phi\leq \psi$ holds inside the domain $\Omega$ too. (Here of course both the inequality between $\phi$ and $\psi$ on the boundary and the Monge-Ampere equation has to be given a precise meaning.) The maximum principle is easy to prove if the functions are sufficiently smooth, e g of class $\mathcal{C}^2$. For non regular functions the maximum principle can be derived from the so called comparison principle (see [@Kol]) of Bedford and Taylor, which also serves as a substitute for the maximum principle in some cases. The comparison principle states (again omitting precise assumptions) that $$\int_{\{\psi<\phi\}}(dd^c\phi)^n\leq \int_{\{\psi<\phi\}}(dd^c\psi)^n .$$
On the other hand it is well known that Monge-Ampère measures often can be approximated by measures defined by [*Bergman functions*]{}. Suppose that we have given on our manifold $X$ a positive measure, $\mu$, and consider the $L^2$-space of holomorphic functions $$A^2=A^2(X,\mu,\phi)=\{h\in H(X); \int |h|^2e^{-\phi} d\mu <\infty\},$$ or its closure in $L^2(X,\mu,\phi)$. We denote by $K_\phi(z,\zeta)$ the Bergman kernel for $A^2$ and let $$B_\phi(z)=K_\phi(z,z)e^{-\phi}$$ be the Bergman function, also known as the [*density of states function*]{}. It is a consequence of the asymptotic expansion formula of Tian-Catlin-Zelditch (see [@Zeld]) that we have $$\lim_{k\rightarrow \infty} k^{-n}B_{k\phi} d\mu =c_n (dd^c\phi)^n$$ if $\phi$ is plurisubharmonic and $\phi$ and $\mu$ are sufficiently regular. We can therefore think of $B_\phi d\mu$ as an approximation, or perhaps quantization, of the Monge-Ampere measure of $\phi$.
The main observation in this note is that a version of the comparison principle holds if we replace the Monge-Ampere operator by the density of states function, so that $$\int_{\psi<\phi} B_\phi d\mu\leq \int_{\psi<\phi} B_\psi d\mu.$$ As it turns out, this is an almost completely formal phenomenon, and it holds under very (but not completely) general circumstances. In particular, the plurisubharmonicity of $\phi$ and $\psi$ plays no role at all, and even the holomorphicity of functions in $A^2$ enters only in a very weak form, so similar results also hold in many other situations when we have a well behaved Bergman kernel, and also if we consider sections of line bundles instead av scalar valued functions. However, the setting of plurisubharmonic weights and holomorphic functions allows a slightly stronger statement with strict inequality, and in that context our main theorem is as follows.
Let $L$ be a holomorphic line bundle over a complex manifold $X$, and let $\phi$ and $\psi$ be two, possibly singular, metrics on $L$. Suppose that $dd^c\phi\geq -\omega$ and $dd^c\psi\geq -\omega$ for some smooth hermitean $(1,1)$-form $\omega$. Assume also that for some constant $C$, $\phi\leq \psi +C$ and that $\mu$ is given by a strictly positive continuous volume form. Then \_[<]{} B\_d\_[<]{} B\_d. Moreover, if $\emptyset\neq \{\psi<\phi\}\neq X$, strict inequality holds unless both sides are zero or infinity.
A few remarks are in order. The strict inequality is of less importance when we deal with Monge-Ampère measures, since one can often arrange that by an ad hoc small perturbation. For Bergman kernels this is less clear and that is the reason why we mention the (very weak) conditions for strict inequality. The condition that $\phi\leq \psi+C$ is sometimes phrased as ’$\psi$ is less singular than $\phi$’, and some condition like that is necessary. Indeed, if $\psi<\phi$ everywhere and we assume $X$ compact, the two integrals equal the dimensions of the space of sections of $L$ that are square integrable with respect to the respective metrics. If $\psi$ is more singular than $\phi$ it may well happen that the space of sections that have finite norm measured by $\psi$ is smaller than the space of sections that have finite norm measured by $\phi$, so the inequality cannot hold.
The abstract setting
====================
We will first deal with the abstract setting of general $L^2$-spaces with a Bergman kernel. Let $(X, \mu)$ be a measure space, let $e^{-\phi}$ be a measurable weight function on $X$, and let $\H_\phi$ be a Hilbert subspace of $L^2(e^{-\phi}d\mu)$. We assume that for any $z\in X$, point evaluation at $z$ is a bounded linear functional on $\H_\phi$. Then $\H_\phi$ has a Bergman kernel, $K_\phi(z,\zeta)$ and we denote $B_\phi(z)=K_\phi(z,z)e^{-\phi}$.
(Comparison principle for Bergman spaces.) Let $\phi$ and $\psi$ be two weight functions on $X$ such that for some constant $C$, $\phi\leq \psi +C$. Then \_[<]{} B\_d\_[<]{} B\_d.
To prove the comparison principle we need a, basically standard, lemma on derivatives of Bergman kernels.
Let $\phi_t$ be a differentiable family of weight functions with uniformly bounded derivative with respect to $t$ . Put $K_t=K_{\phi_t}$. Then $K_t$ is differentiable with respect to $t$ . Let $\dot{K_t}$ and $\dot{\phi_t}$ be the derivatives of $K_t$ and $\phi_t$ with respect to $t$. Then for $z$ and $\zeta$ fixed (z,)=\_X K\_t(z,w) K\_t(w,) e\^[-\_t(w)]{} d(w). Moreover, for the difference quotients we have, if $|\tau|\leq 1$, |(K\_[t+]{}(z,z)-K\_t(z,z))/|A K\_t(z,z) for some constant $A$ depending on the sup-norm of $\dot \phi$.
Note first that since $\dot \phi$ is bounded, $\phi_t-\phi_{t+\tau}$ is bounded for $|\tau|\leq 1$. Hence $e^{-\phi_t}$ and $e^{-\phi_{t+\tau}}$ are of the same magnitude and it follows from the extremal characterization of Bergman kernels that $K_t$ and $K_{t+\tau}$ are comparable as well. Let $\Delta(t,\tau)= e^{-\phi_t}-e^{-\phi_{t+\tau}}$. Since $$\Delta(t,\tau)=\int_0^\tau \dot\phi_s e^{-\phi_{t+s}}ds,$$ $|\Delta(t,\tau)|\leq A |\tau|e^{-\phi_t}
$ if $|\tau|\leq 1$. Next note that by the reproducing property of Bergman kernels (K\_[t+]{}-K\_t)(z,)=\_X K\_t(z,w)K\_[t+]{}(w,)(e\^[-\_t]{}-e\^[-\_[t+]{}]{})d(w). Hence for $|\tau|\leq 1$ $$|(K_{t+\tau}(z,z)-K_t(z,z))/\tau|\leq A\int_X |K_t(z,w)K_{t+\tau}(w,z)|e^{-\phi_t}d\mu(w).$$ Since $\phi_t-\phi_{t+\tau}$ is bounded this is less than $$A'\left( \int_X |K_t(z,w)|^2 e^{-\phi_t}d\mu(w)+ \int_X |K_{t+\tau}(z,w)|^2 e^{-\phi_{t+\tau}}d\mu(w)\right)\leq A'' K_t(z,z),$$ so we have proved (2.3). To prove (2.2) is very easy formally, just differentiating under the integral sign, but to prove that this is legitimate we have to work a bit more. We first multiply (2.4) by its conjugate and integrate with respect to $\zeta$. Letting $f(z,\zeta):=
(K_{t+\tau}-K_t)(z,\zeta)$ we get $$\int |f(z,\zeta|^2 e^{-\phi_{t+\tau}}d\mu(\zeta)=$$ $$\int K_t(z,w)K_t(w',z)\Delta(t,\tau)(w)\Delta(t,\tau)(w')\int K_{t+\tau}(\zeta,w')K_{t+\tau}(w,\zeta) e^{-\phi_{t+\tau}(\zeta)}d\mu(\zeta)d\mu(w)d\mu(w').$$ Using the reproducing property of Bergman kernels in the inner integral this is $$\int K_t(z,w)K_t(w',z) K_{t+\tau}(w,w')\Delta(t,\tau)(w)\Delta(t,\tau)(w')d\mu(w)d\mu(w').$$ Next we apply (2.4) to the integral with respect to $w'$ and get $$\int f(w,z) K_t(z,w)\Delta(t,\tau)(w) d\mu(w).$$ Then use that $|\Delta(t,\tau)|\leq |\tau| e^{-\phi_t}$ and apply Cauchy’s inequality to get
|f(z,)|\^2 e\^[-\_t]{}d()A || K\_t(z,z). We are now finally ready to prove (2.2). By (2.4) $$(K_{t+\tau}-K_t)(z,\zeta)/\tau=\int_X K_t(z,w)K_{t+\tau}(w,\zeta)(e^{-\phi_t}-e^{-\phi_{t+\tau}})/\tau d\mu(w).$$ By (2.5) we may replace $K_{t+\tau}$ by $K_t$ in the integral. After that we let $\tau$ tend to zero and get (2.2) by dominated convergence.
We now turn to the proof of the comparison principle Theorem 2.1. We first claim that we may assume that $\phi-\psi$ is bounded. To see this, let $u:= \psi-\phi$ so that $u\geq -C$. Put $u_0:= \min (u, 0)$, $\psi_0=\phi+u_0$. Then $\psi_0\leq \psi$ and $\psi_0-\phi$ is bounded. By the extremal characterization of Bergman kernels $K_{\psi_0}(z,z)\leq K_\psi(z,z)$. On the other hand, where $\psi<\phi$, $u<0$ so $u_0=u$. Hence $\psi_0=\psi$ and $B_{\psi_0}\leq B_\psi$. Moreover $\psi<\phi$ if and only if $u<0$ which is equivalent to $u_0<0$, so $\psi<\phi$ if and only if $\psi_0<\phi$. Hence it suffices to prove the theorem for $\psi_0$ since then $$\int_{\psi<\phi} B_{\phi} d\mu\leq\int_{\psi_0<\phi} B_{\psi_0} d\mu\leq \int_{\psi<\phi} B_{\psi} d\mu.$$
From now on we assume that $\phi-\psi$ is bounded and let $\rho$ be a measurable function on $X$ such that $$\int_X \rho(z)K_\phi(z,z) e^{-\phi} d\mu(z)<\infty.$$ The same integral with $\phi$ replaced by $\psi$ is then also bounded. Let $\phi_t=\phi+tu$, so that $\phi_0=\phi$ and $\phi_1=\psi$. Then we claim that by Proposition 2.2, if $$G(t):=\int_X \rho(z) B_{\phi_t} d\mu$$ then G’(t)=\_X -(z)(z)K\_t(z,z)e\^[-\_t]{} d+ \_X\_X (z) (w) K\_t(z,w)K\_t(w,z)e\^[-\_t(z)-\_t(w)]{}d(z)d(w). Again, this follows formally by the proposition and to justify the limit process we write $$(G(t+\tau)-G(t))= \int_X \rho K_t(e^{-\phi_{t+\tau}}-e^{-\phi_t})d\mu +
\int_X\rho(K_{t+\tau}-K_t)e^{-\phi_{t+\tau}}d\mu.$$ When we divide by $\tau$ and let $\tau\to 0$ we see that the first term converges to the first term of (2.6) by dominated convergence. For the second term we use (2.3) to conclude that we have dominated convergence in that integral as well.
In the first integral on the right hand side we insert the reproducing formula for the Bergman kernel $$K_t(z,z)=\int_X K_t(z,w)K_t(w,z)e^{-\phi_t(w)}d\mu(w)$$ which changes the right hand side to $$\int_X\int_X
\rho(z) (\dot{\phi_t}(w)-\dot{\phi_t}(z)) K_t(z,w)K_t(w,z)e^{-\phi_t(z)-\phi_t(w)}d\mu(z)d\mu(w).$$ We can write this more symmetrically as (1/2)\_X\_X ((z)-(w)) ((w)-(z)) |K\_t(z,w)|\^2 e\^[-\_t(z)-\_t(w)]{}d(z)d(w). Now recall that $\phi_t=\phi+tu$ so $\dot{\phi_t}=u$. Let $\rho$ be the characteristic function of the set where $ \psi-\phi=u<0$. Then (2.7) becomes $$(1/2)\int\int_{\{u(z)<0<u(w)\}}(u(w)-u(z))|K_t(z,w)|^2 e^{-\phi_t(z)-\phi_t(w)}d\mu(z)d\mu(w)-$$ $$-(1/2)\int\int_{\{u(w)<0<u(z)\}}(u(w)-u(z))|K_t(z,w)|^2 e^{-\phi_t(z)-\phi_t(w)}d\mu(z)d\mu(w).$$ Again using symmetry we get \_[u<0]{} B\_[\_t]{} d=\_[{u(z)<0<u(w)}]{}(u(w)-u(z))|K\_t(z,w)|\^2 e\^[-\_t(z)-\_t(w)]{}d(z)d(w). Clearly this expression is non negative, so we have proved Theorem 2.1 under the assumption that $$\int_X \rho(z)K_\phi(z,z) e^{-\phi} d\mu(z)<\infty,$$ i e that the left hand side of (2.1) is finite. If this does not hold, clearly the right hand side is infinite as well, so the (in)equality holds trivially.
[**Remark:**]{} Since the Bergman function $B_\phi(z)=K_\phi(z,z)e^{-\phi}$ does not change if we add a constant to $\phi$, we also have that \_[<+c]{} B\_d\_[<+c]{} B\_dfor any constant $c$. One may note that it follows already from (2.7) that the derivative is non negative if $\rho= k(u)$ is an increasing function of $u=\dot\phi$. This fact is however equivalent to (2.9) for all values of $c$. We have chosen to write the derivative in the form (2.8) since this makes it easy to see when we have strict inequality (see the next section).
The proof of Theorem 1.1
========================
It is now an easy matter to deduce Theorem 1.1 from Theorem 2.1. First we note that the setting of line bundles instead of scalar valued functions causes no extra difficulties. Indeed the proof goes through in the same way with only nominal changes. Alternatively, we could use that any line bundle has a discontinuous trivializing section and since continuity played no role in the proof, the line bundle case follows. It remains to prove that we have strict inequality if $B_\psi$ is non trivial and $\emptyset\neq\{\psi<\phi\}\neq X$. For this it suffices to show that the right hand side of (2.8) is strictly positive. But $$V:=\{(z,w); u(w)<0<u(z)\}$$ is by assumption non empty. Moreover, this set is open for the plurifine topology and therefore has positive Lebesgue measure, [@BT]. Hence it has positive $\mu$-measure if $\mu$ is given by a strictly positive continuous density. From this it follows that if $K_t(z,w)$ is not identically zero, it is different from zero almost everywhere on $V$, since it is holomorphic with respect to $z$ and $w$ (this is the only place we use holomorphicity). Hence the derivative of $G$ is strictly positive.
Finally we give a ’maximum principle’ for Bergman kernels which follows from Theorem 1.1 in the same way that the Monge-Ampère maximum principle follows from the classical comparison principle.
Under the same assumptions as in Theorem 1.1, let $\Omega\neq X$ be a subset of $X$ such that $B_\phi\geq B_\psi$ on $\Omega$. Assume that $\phi\leq\psi$ on $X\setminus \Omega$. Then $\phi\leq\psi$ everywhere.
Assume the set $U$ where $\psi<\phi$ is nonempty. Then $U$ is a subset of $\Omega$, and $\Omega$ is not equal to $X$. This contradicts Theorem 1.1.
\#1\#2\#3[[\#1]{}: [*\#2*]{}, \#3.]{}
[9999]{}
Bedford E. and Taylor, B. A: Fine topology, Shilov boundary and $(dd^c)^n$. J Func Anal 72 (1987), pp 225-251
Kolodziej, S: The Complex Monge-Ampère-equation and pluripotential theory. Mem Am Math Soc, 178 (2005)
Zelditch, S: Szegö kernels and a theorem of Tian. Internat Math Res Notices 1998, no 6, pp 317-331
|
EFI 08-19\
June 2008\
**$B$ decays dominated by $\omega$–$\phi$ mixing**
Michael Gronau[^1] and Jonathan L. Rosner
*Enrico Fermi Institute and Department of Physics, University of Chicago*
*Chicago, IL 60637, U.S.A.*
> Recently Belle has established the 90% confidence level (CL) upper limit $\b < 9.4 \times 10^{-7}$ for the branching ratio for $B^0\to J/\psi
> \phi$, a process expected to be suppressed by the Okubo-Zweig-Iizuka (OZI) rule disfavoring disconnected quark diagrams. We use information on $\omega$–$\phi$ mixing to establish likely lower bounds on this and related processes. We find that the Belle result is about a factor of five above our limit, while other decays such as $B^0 \to \bar D^0 \phi$ and $B^+
> \to \pi^+ \phi$, for which upper limits have been obtained by BaBar, could be observable with similar improvements in data. We argue that a significant enhancement of our predicted decay rates by rescattering is unlikely.
Certain $B$ meson decay processes are expected to be suppressed, as they involve either the participation of a spectator quark or, equivalently in terms of flavor topology, the rescattering of final states into one another. A number of such processes were suggested in Ref. [@Blok:1997yj]. Recently the Belle Collaboration has greatly improved upper limits on one such process, the decay $B^0 \to J/\psi \phi$, finding a 90% CL upper limit for the branching ratio of $\b < 9.4 \times 10^{-7}$ [@Liu:2008bt]. This process is expected to be suppressed by the Okubo-Zweig-Iizuka (OZI) rule disfavoring disconnected quark diagrams [@Okubo:1963fa]. In the present Letter we evaluate one potential mechanism for production of such a final state, the existence of $\omega$–$\phi$ mixing, and find that it leads to a predicted branching ratio about a factor of five below the present upper limit. We also evaluate the effect of this mixing upon several other processes and find that for $B^0 \to \bar D^0 \phi$ and $B^+
\to \pi^+ \phi$ a similar improvement in data should lead to observable signals.
Extensive discussions of $\omega$–$\phi$ mixing have been given in Refs.[@Benayoun:1999fv; @Benayoun:1999au; @Benayoun:2000ti; @Benayoun:2001qz; @Benayoun:2007cu]. We shall neglect isospin-violation and admixtures with the $\rho$. Then one can parametrize $\omega$–$\phi$ mixing in terms of an angle $\delta$ such that the physical $\omega$ and $\phi$ are related to the ideally mixed states $\omega^I \equiv (u \bar u + d \bar d)/\s$ and $\phi^I
\equiv s \bar s$ by \[mixing\] (
[c]{}\
) = (
[c c]{} &\
- &
) (
[c]{} \^I\
\^I
) A simplified analysis [@Benayoun:1999fv] implies a mixing angle of $\delta = - (3.34 \pm 0.17)^\circ$, while the most recent treatment [@Benayoun:2007cu] implies an energy-dependent mixing which varies from $-0.45^\circ$ at the $\omega$ mass to $-4.64^\circ$ at the $\phi$ mass.
![Quark diagrams for OZI-allowed $B^0\to J/\psi \omega^I$ and $B^0\to \bar D^0\omega^I$.[]{data-label="fig:csjpsi"}](csjpsi.ps){width="6.0cm"}
![Quark diagrams for OZI-suppressed $B^0\to J/\psi \phi^I$ and $B^0\to \bar D^0\phi^I$.[]{data-label="fig:jpsiphi"}](jpsiphi.ps){width="5.0cm"}
A systematic study of the effects of $\omega$–$\phi$ mixing on hadronic decays of non-strange $B$ mesons, $B^0\equiv \bar bd,
B^+\equiv \bar bu$, requires considering the three $\Delta S=0$ quark subprocesses, $\bar b \to \bar cu\bar d, \bar b\to \bar c c\bar d$ and $\bar b\to \bar u u\bar d$. Each one of these subprocesses leads to OZI-allowed decays involving $\omega^I$, while decays into final states with $\phi^I$ are OZI-suppressed. Quark diagrams describing two examples of OZI-allowed decays, $B^0\to J/\psi \omega^I$ and $B^0\to \bar D^0\omega^I$, and corresponding OZI-suppressed decays, $B^0\to J/\psi\phi^I$ and $B^0\to \bar D^0\phi^I$, are shown in Figs. \[fig:csjpsi\] and \[fig:jpsiphi\]. The first two processes are described by color-suppressed tree diagrams, while the other two processes involve $W$-exchange diagrams, to which an $s\bar s$ pair is attached through three gluons.
The situation in decays of $B_s\equiv \bar bs$ is the opposite relative to that in non-strange $B$ decays. That is, the $\omega$ and $\phi$ exchange roles. Here one considers the $\Delta S=1$ quark subprocess $\bar b\to \bar c c\bar s$ which leads to OZI-allowed decays involving $\phi^I$ and OZI-suppressed decays with $\omega^I$. (The quark subprocess $\bar b\to \bar c\bar us$ leads through W-exchange diagrams to OZI-allowed $B_s$ decays involving $\omega^I$ including $\bar D^0\omega^I$.) Examples of quark diagrams describing the OZI-allowed decay $B_s\to J/\psi \phi^I$ and the corresponding OZI-suppressed decay $B_s\to
J/\psi \omega^I$ are shown in Figs. \[fig:csbs\] and \[fig:bsjpsiom\]. As in the above examples of $B^0$ decays, the first process is governed by a color-suppressed tree amplitude, while the second decay is described by a W-exchange diagram, to which a $u\bar u$ or $d\bar d$ pair is attached by three gluons.
Neglecting contributions of OZI-suppressed amplitudes and small phase space differences between processes with $\omega$ or $\phi$ in the final state, Eq. (\[mixing\]) implies \[Xphi\] (B\^[0,+]{} X\^[0,+]{}) &=& \^2(B\^[+,0]{}X\^[+,0]{}) ,\
\[Xomega\] (B\_s X\^0) &=& \^2(B\_sX\^0) . The examples shown in Figs. \[fig:csjpsi\] and \[fig:csbs\] correspond to $X^0=J/\psi, \bar D^0$ in $B^0$ decays and $X^0=J/\psi$ in $B_s$ decays.
![Quark diagrams for OZI-allowed $B_s\to J/\psi \phi^I$.[]{data-label="fig:csbs"}](csbs.ps){width="6.0cm"}
![Quark diagrams for OZI-suppressed $B_s\to J/\psi \omega^I$.[]{data-label="fig:bsjpsiom"}](bsjpsiom.ps){width="5.0cm"}
In Table I we list OZI-allowed branching ratios of $B^0, B^+$ and $B_s$ decays, for which nonzero values have been measured, and upper limits on corresponding OZI-suppressed decays. The upper left part of the table, addressing non-strange $B$ mesons, includes also processes involving $\rho^0$, for which branching ratios are expected to be approximately equal to corresponding processes with $\omega$. (See Fig. \[fig:csjpsi\].) The approximately equal decay rates measured for $B^0\to\bar D^0\rho^0$ and $B^0\to \bar D^0\omega$ confirm this assumption. OZI-allowed branching ratios for $B_s$ decays involving $\phi$ are listed in the lower left part of the Table I.
Using Eqs. (\[Xphi\]) and (\[Xomega\]) with a universal value of $\delta = -4.64^\circ$ and the measured OZI-allowed branching ratios, we obtain predictions for the OZI-suppressed rates shown in the right-hand column of Table I. Values in parentheses, quoting predictions for OZI-suppressed $B_s$ decays involving $\omega$, are obtained for the small mixing angle $\delta = -0.45^\circ$ [@Benayoun:2007cu]. Predictions for $B^0$ and $B^+$ decays are compared with current upper bounds measured for these branching ratios. We note that the predictions for $\b(B^0\to \bar D^0\phi)$, $\b(B^0\to J/\psi\phi)$ and $\b(B^+\to \pi^+\phi)$ are about a factor five smaller than the current upper limits on these branching ratios.
---------------------------- --------------------------- ----------------------------------------- ------------------------- --------------------------- ------------------
Quark Decay ${\cal B}~(10^{-5})$ Decay
subprocess mode mode Upper limit Predicted
$\bar b\to\bar cu\bar d$ $B^0 \to \bar D^0 \rho^0$ 32$\pm$5 [@Kuzmin:2006mw] $\bar D^0 \phi$ $< 116$ [@Aubert:2007nw] 21$\pm$3
$\bar D^0 \omega$ 25.9$\pm$3.0 [@Aubert:2003sw] $\bar D^0 \phi$ $< 116$ [@Aubert:2007nw] 17$\pm$2
$\bar D^{*0} \omega$ 27$\pm$8 [@Aubert:2003sw] $\bar D^{*0} \phi$ – 18$\pm$5
$\bar b \to \bar cc\bar d$ $J/\psi \rho^0$ 2.7$\pm$0.4 [@Aubert:2007xw] $J/\psi \phi$ $< 9.4$ [@Liu:2008bt] 1.8$\pm$0.3
$\bar b\to \bar uu\bar d$ $B^+ \to\pi^+\omega$ 0.69$\pm$0.05 [@Jen:2006in] $B^+ \to \pi^+\phi$ $<2.4$ [@Aubert:2006nn] 0.45$\pm$0.03
$\rho^+ \omega$ $1.06^{+0.26}_{-0.23}$ [@Aubert:2006vt] $\rho^+ \phi$ $<160$ [@Bergfeld:1998ik] 0.7$\pm$0.2
$\bar b\to \bar cc\bar s$ $B_s \to J/\psi \phi$ 93$\pm$33 [@Abe:1996kc] $B_s \to J/\psi \omega$ – 61$\pm$22 (0.6)
$\psi(2S) \phi$ 48$\pm$22 [@Abulencia:2006jp] $\psi(2S) \omega$ – 32$\pm 15$ (0.3)
---------------------------- --------------------------- ----------------------------------------- ------------------------- --------------------------- ------------------
: Comparison of some OZI–allowed and OZI–suppressed branching ratios, in units of $10^{-5}$ and $10^{-7}$, respectively. Averages are taken from Ref. [@PDGup]. Upper limits are 90% CL. Predictions are based on an $\omega$-$\phi$ mixing angle $\delta= -4.64^\circ$. Parentheses denote predictions based on the very small admixture of $s \bar s$ expected in the $\omega$ in Ref.[@Benayoun:2007cu]. \[tab:comp\]
![Electroweak penguin diagram for OZI-suppressed $B^+\to \pi^+\phi^I$ and $B^+\to \rho^+\phi^I$.[]{data-label="fig:ewp"}](ewp.ps){width="6.0cm"}
While most OZI-allowed processes in Table I are described by color-suppressed tree diagrams as shown in Figs. \[fig:csjpsi\] and \[fig:csbs\], the CKM-suppressed charmless decays $B^+\to \pi^+\omega$ and $B^+\to
\rho^+\omega$ are dominated by color-allowed tree diagrams [@Gronau:1994rj; @Chiang:2003pm]. Contributions to these processes from color-suppressed tree diagrams are considerably smaller. This is demonstrated by 90$\%$ CL upper limits measured for corresponding color-suppressed branching ratios, $\b(B^0\to \pi^0\omega) < 0.12\times 10^{-5}$ [@Aubert:2004ih] and $\b(B^0\to \rho^0\omega) < 0.15\times 10^{-5}$ [@Aubert:2006vt], which are a factor six or seven below $\b(B^+\to \pi^+\omega)$ and $\b(B^+\to \rho^+\omega)$ given in Table I. The corresponding OZI-suppressed amplitudes for $B^+\to \pi^+\phi$ and $B^+\to \rho^+\phi$ each obtain an electroweak penguin contribution [@Gronau:1995hn] and a contribution from a singlet penguin diagram [@Dighe:1997wj], shown in Figs. \[fig:ewp\] and \[fig:p3g\], respectively. These amplitudes have been calculated in Ref. [@Beneke:2003zv] and [@Beneke:2006hg] within the framework of QCD factorization neglecting $\omega$-$\phi$ mixing. Branching ratios $\b(B^+\to \pi^+\phi)=(2-10)\times
10^{-9}$ and $\b(B^+\to \rho^+\phi)=(1-3)\times 10^{-8}$ were obtained, considerably smaller than the two corresponding predictions in Table I originating in $\omega$-$\phi$ mixing.
![Singlet penguin diagram for OZI-suppressed $B^+\to \pi^+\phi^I$ and $B^+\to \rho^+\phi^I$.[]{data-label="fig:p3g"}](p3g.ps){width="6.0cm"}
Assuming that small OZI-suppressed amplitudes do not interfere destructively with amplitudes due to $\omega$-$\phi$ mixing, the predictions presented in Table I for branching ratios of OZI-suppressed decays should be considered as likely lower bounds. In principle, these branching ratios may be enhanced by rescattering through intermediate states with larger decay rates. This possibility had been envisaged a few years before starting the operation of $e^+e^-$ $B$ factories [@Blok:1997yj]. We will now argue that experimental evidence obtained in certain experiments indicates that a significant enhancement by rescattering is unlikely in OZI-suppressed and other suppressed decays.
Consider the decay $B^0\to D^-_sK^+$, which is governed by a $W$-exchange amplitude represented by a quark subprocess $(\bar bd)\to (\bar cu)$, associated with a popping of an $s\bar s$ pair out of the vacuum. This exchange amplitude is expected to be suppressed by an order of magnitude ($\sim
\Lambda_{\rm QCD}/m_b$) relative to the corresponding color-favored tree amplitude for $B^0\to D^-\pi^+$ induced by the same quark subprocess, $\bar b\to \bar cu\bar d$ [@Gronau:1994rj; @Beneke:2000ry; @Mantry:2003uz]. This would imply $\b(B^0\to D^-_s K^+)/\b(B^0\to D^-\pi^+)\sim 10^{-2}$. Rescattering through dynamically favored intermediate states including $B^0\to D^-\pi^+\to D^-_sK^+$ and rescattering through other intermediate $C=-1, S=0$ states, with decay branching ratios at a level of a fraction of a percent, could enhance the branching ratio for $B^0\to D^-_sK^+$ relative to the above expectation. Experimentally, one finds [@Krokovny:2002pe] $\b(B^0\to D^-_sK^+)=(2.9\pm 0.5)\times 10^{-5}$, in comparison with [@PDGup] $\b(B^0\to D^-\pi^+)=(2.68\pm 0.13)\times10^{-3}$ which is two orders of magnitude larger. That is, rescattering effects do not enhance the rate for $B^0\to D^-_sK^+$ beyond the estimate based on an exchange amplitude.
One possible conclusion is that a significant enhancement of diagramatically suppressed decay rates by rescattering requires intermediate states with rates which are larger than the suppressed rates by [*more than two orders of magnitude*]{}. This requirement seems to follow from the multi-channel nature of the rescattering process occurring between the initial $B$ meson and the final state to which it decays. Examples for processes which have been shown to need an enhancement by rescattering are the decays $B\to K\pi$ [@Ciuchini:1997rj]. The short-distance loop-suppressed penguin amplitude dominating these processes is too small to account for the measured decay rates and requires an enhancement by long-distance rescattering [@Jain:2007dy]. The branching ratios of intermediate states including $B\to D^{(*)-}_s D^{(*)}$ are at a percent level, [*three orders of magnitude*]{} larger than the branching ratios calculated for $B\to K\pi$ using short short-distance physics. This is sufficient for a significant enhancement of the $B\to K\pi$ decay rates relative to this calculation.
A well-known charmless $B$ decay process dominated by a $W$-exchange amplitude is $B^0\to K^+K^-$ [@Blok:1997yj; @Gronau:1994rj]. This process receives rescattering contributions from tree-dominated intermediate states including $\pi^+\pi^-$ with a branching ratio [@PDGup] $\b(B^0\to\pi^+\pi^-) = (5.13\pm 0.24)\times 10^{-6}$. Assuming an order of magnitude suppression of the exchange amplitude relative to a tree amplitude as in $B^0\to D^-_sK^+$, and using the above criterion for no significant enhancement by rescattering, one expects $\b(B^0\to K^+K^-) \sim 5\times 10^{-8}$, almost an order of magnitude below the current 90% CL upper limit of [@PDGup] $4.1\times 10^{-7}$. Similarly, using [@PDGup] $\b(B^0\to\rho^+\rho^-) = (2.42\pm 0.31)\times 10^{-5}$, we predict $\b(B^0\to K^{*+}K^{*-})\sim 2\times 10^{-7}$. Very recently an upper limit at 90% CL has been measured [@Aubert:2008ap], $\b(B^0\to K^{*+}K^{*-}) <2.0\times 10^{-6}$, an order of magnitude above our prediction.
Consider now the OZI-suppressed decay $B^0\to \bar D^0\phi$. The quark diagram for this process shown in Fig. \[fig:jpsiphi\] describes an exchange amplitude $(\bar bd)\to (\bar cu)$ as in $B^0\to D^-_sK^+$, to which a pair of $s\bar s$ is attached by three gluons. The above argument for no significant rescattering effects in $B^0\to D^-_sK^+$ implies the absence of such effects also in $B^0\to \bar D^0\phi$. To demonstrate this explicitly, let us consider the two kinds of intermediate states through which rescattering into $\bar D^0\phi$ can occur:
1. States dominated by tree amplitudes such as $D^-\rho^+~[(\bar cd)(\bar du)]$. Rescattering through these intermediate states into $\bar D^0\phi$ $[(\bar cu)(\bar ss)]$ is OZI-suppressed \[see Fig. \[fig:OZI\](a)\] and is not expected to enhance the rate for $B^0\to\bar D^0\phi$.
2. States governed by exchange amplitudes including $D^-_sK^{*+}~[(\bar cs)(\bar su)]$. Rescattering through these states into $\bar D^0\phi~[(\bar cu)(\bar ss)]$ is OZI-allowed \[see Fig. \[fig:OZI\](b)\]. This rescattering is not expected to enhance the predicted branching ratio, $\b(B^0\to \bar D^0\phi)\simeq
2\times 10^{-6}$, because $\b(B^0\to D^-_sK^{*+})$ is expected to be only one order of magnitude larger, assuming that it does not differ much from $\b(B^0\to D^-_sK^+)=(2.9\pm 0.5)\times 10^{-5}$ .
![Quark diagrams for (a) OZI-suppressed rescattering $D^- \rho^+
\to \bar D^0 \phi$, (b) OZI-allowed rescattering $D_s^- K^{*+} \to \bar D^0
\phi$. \[fig:OZI\]](OZI.ps){width="98.00000%"}
A similar situation exists in $B^0\to J/\psi\phi$. Here OZI-suppressed rescattering occurs through intermediate tree-dominated states including $D^{(*)+}D^{(*)-}$, while OZI-allowed rescattering involves intermediate states such as $D^{(*)+}_sD^{(*)-}_s$. $B^0$ decays into the latter states are dominated by exchange amplitudes, which are expected to be suppressed by about an order of magnitude relative to the tree amplitudes in $B^0\to
D^{(*)+}D^{(*)-}$. Using the suppression measured for the exchange amplitude in $B^0\to D^-_sK^+$, this implies, for instance [@Gronau:2008ed], $\b(B^0\to D^+_sD^-_s) =
(4.0^{+1.8}_{-1.4})\times 10^{-6}$, an order of magnitude below the current upper limit on this branching ratio [@Zupanc:2007pu]. This branching ratio is only twenty times larger than the value predicted for $\b(B^0\to J/\psi\phi)$, which is expected to be insufficient for an enhancement of the latter branching ratio by rescattering through this class of intermediate states.
In the case of $B\to \pi^+\phi$ (or $B^+\to\rho^+\phi$) the situation is slightly different, but the condition for a significant enhancement by rescattering is also not met. The final state $\pi^+\phi$ may be reached by OZI-suppressed rescattering through intermediate states such as $\pi^+\rho^0$, or by OZI-allowed rescattering through states including $K^+\bar K^{*0}$. $B^+$ decay into the latter mode is dominated by a suppressed $\Delta S=0$ penguin amplitude [@Chiang:2003pm] implying a small branching ratio. The current 90$\%$ CL upper limit [@Aubert:2007ua], $\b(B^+\to K^+\bar K^{*0})<
1.1\times 10^{-6}$, shows that this branching ratio is at most twenty-five times larger than the predicted $\b(B^+\to\pi^+\phi)$. As discussed above, this is insufficient for enhancing the latter branching ratio by rescattering.
In conclusion, we have studied the consequences of $\omega$–$\phi$ mixing in OZI-suppressed hadronic decays of $B$ and $B_s$ mesons. We calculated branching ratios for $B$ decays involving $\phi$, which in the cases of $B^0\to \bar
D^0\phi$, $B^0\to J/\psi\phi$ and $B^+\to \pi^+\phi$ are each about a factor of five below the corresponding current upper limits. We used the observed suppression of branching ratios for decays dominated by $W$-exchange including $B^0\to D^-_sK^+$ to argue that a significant enhancement of these rates by rescattering is unlikely. Thus, the above three processes are predicted to be detectable with a factor of five increase in data. Effects of $\omega$–$\phi$ mixing in OZI-suppressed $B_s$ decays involving $\omega$ are much smaller than in nonstrange $B$ decays if one assumes a very small admixture of $s \bar s$ in the $\omega$ as suggested in Ref. [@Benayoun:2007cu]. The predicted branching ratios become a factor two smaller than in Table I for an energy-independent $\omega$-$\phi$ mixing angle of $\delta=-3.34^\circ$ [@Benayoun:1999fv].
M.G. would like to thank the Enrico Fermi Institute at the University of Chicago for its kind and generous hospitality. We thank Pavel Krokovny, Shunzo Kumano, and Yoshi Sakai for useful communications. This work was supported in part by the United States Department of Energy through Grant No. DE FG02 90ER40560.
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[^1]: On sabbatical leave from the Physics Department, Technion, Haifa 32000, Israel.
|
---
abstract: 'We model the intermediate-mass black hole HLX-1, using the [*Hubble Space Telescope*]{}, [*XMM-Newton*]{} and [*Swift*]{}. We quantify the relative contributions of a bluer component, function of X-ray irradiation, and a redder component, constant and likely coming from an old stellar population. We estimate a black hole mass $\approx (2^{+2}_{-1}) \times 10^4 M_{\odot}$, a spin parameter $a/M \approx 0.9$ for moderately face-on view, and a peak outburst luminosity $\approx 0.3$ times the Eddington luminosity. We discuss the discrepancy between the characteristic sizes inferred from the short X-ray timescale ($R \sim$ a few $10^{11}$ cm) and from the optical emitter ($R \sqrt{\cos \theta} \approx 2.2 \times 10^{13}$ cm). One possibility is that the optical emitter is a circumbinary disk; however, we disfavour this scenario because it would require a very small donor star. A more plausible scenario is that the disk is large but only the inner annuli are involved in the X-ray outburst. We propose that the recurrent outbursts are caused by an accretion-rate oscillation driven by wind instability in the inner disk. We argue that the system has a long-term-average accretion rate of a few percent Eddington, just below the upper limit of the low/hard state; a wind-driven oscillation can trigger transitions to the high/soft state, with a recurrence period $\sim$1 year (much longer than the binary period, which we estimate as $\sim$10 days). The oscillation that dominated the system in the last decade is now damped such that the accretion rate no longer reaches the level required to trigger a transition. Finally, we highlight similarities between disk winds in HLX-1 and in the Galactic black hole V404 Cyg.'
author:
- |
Roberto Soria,$^{1,2,3}$[^1] Aina Musaeva,$^{2}$ Kinwah Wu,$^{4}$ Luca Zampieri,$^{5}$ Sara Federle,$^{5}$ Ryan Urquhart,$^{1}$ Edwin van der Helm,$^{6}$ and Sean Farrell,$^{2}$\
$^{1}$International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia\
$^{2}$Sydney Institute for Astronomy, School of Physics A28, The University of Sydney, Sydney NSW 2006, Australia\
$^{3}$National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China\
$^{4}$Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK\
$^{5}$INAF, Astronomical Observatory of Padova, vicolo dell’Osservatorio 5, I-35122 Padova, Italy\
$^{6}$Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA, Leiden, the Netherlands
date: 'Accepted 2017 April 08. Received 2017 March 31; in original form 2017 November 15'
title: |
Outbursts of the intermediate-mass black hole HLX-1:\
a wind instability scenario
---
\[firstpage\]
black hole physics – X-rays: binaries – X-rays: individual: HLX-1
Introduction
============
The transient X-ray source HLX-1 [@farrell09] is arguably the most convincing off-nuclear intermediate-mass black hole (IMBH) candidate. The X-ray source is unambiguously associated with an H$\alpha$-emitting optical counterpart, providing a recession speed of $\approx$7,100 km s$^{-1}$ [@soria13b; @wiersema10]; this is consistent with the recession velocity of the surrounding galaxy cluster Abell 2877 [@malumuth92]. HLX-1 is projected in front of the halo of the S0 galaxy ESO243-49 (a member of Abell 2877), $\approx$7from its nucleus (Figure 1), which has a systemic recession speed of $\approx$6,700 km s$^{-1}$. For this reason, we assume the same luminosity distance $\approx$92 Mpc for both ESO243-49 and HLX-1 (NASA/IPAC Extragalactic Database). It is unclear whether HLX-1 is orbiting around ESO243-49 or it is simply a chance association within Abell 2877; in either case, HLX-1 is not a foreground Galactic object. If we assume isotropic emission, the peak X-ray luminosity in each of its six fully-monitored outbursts is $L_{\rm X} \approx (1$–$1.5) \times 10^{42}$ erg s$^{-1}$ [@yan15], corresponding to the Eddington luminosity of a $10^4 M_{\odot}$ BH. At outburst peak, the X-ray spectrum is well fitted with a radiatively efficient, multicolour disk model [@davis11; @godet12; @farrell12], typical of accreting BHs in the sub-Eddington high/soft state. The peak colour temperature ($kT_{\rm in} \approx 0.25$ keV) and the inner radius of the disk ($R_{\rm in} \sqrt{\cos \theta} \sim 5$–$10 \times 10^4$ km) are self-consistent with a $\sim$10$^4 M_{\odot}$ BH accreting just below its Eddington limit [@godet12]. The low rms variability is also consistent with an accretion disk in the high/soft state [@servillat11]. The outburst evolution after the peak is similar to that of sub-Eddington Galactic BHs: in the first few weeks, a power-law tail appears, strengthens and becomes flatter, while the disk component evolves to lower luminosities and temperatures along the characteristic $L \propto T_{\rm in}^4$ track [@servillat11; @godet12]. In each of its six fully-recorded outbursts (Figure 2), after $\sim$100–150 days and after it declined to a luminosity $\la 10^{41}$ erg s$^{-1}$, HLX-1 underwent a transition to the low/hard state, with a spectrum dominated by a power-law of photon index $\Gamma = 1.6 \pm 0.3$ [@servillat11; @yan15]. The detection of transient radio emission during an outburst [@webb12] and the presence of a jet in the low/hard state [@cseh15] also support the interpretation that HLX-1 cycles through the canonical states of an accreting BH, with radio flaring generally observed between the hard outburst rise and the transition to the thermal dominant state [@fender04].
Despite the success of this IMBH model based on canonical sub-Eddington accretion states, several questions remain unanswered; failure to solve such problems might even lead to a rejection of the canonical IMBH model, in favour for example of a beamed, highly super-Eddington stellar-mass BH [@lasota15]. The main outstanding problems are:
- what component of the optical/UV luminosity comes from the irradiated accretion disk (variable component) and what from a surrounding star cluster (constant component)?
- what is the size of the accretion disk and of the semi-major axis of the binary system? More specifically, can the accretion disk be small enough to explain the short outburst timescale but at the same time large enough to produce the observed optical luminosity?
- what causes the repeated outburst behaviour and determines the recurrence timescale: thermal-viscous instability, periastron passages, other types of mass transfer instabilities, oscillations induced by radiation pressure or outflows?
In this paper, we will discuss those sets of questions, re-examine published and unpublished observational results, and propose a new scenario that may be consistent with all the data. We will determine the BH mass and other system parameters consistent with this scenario.
\
\
\
\
------------ ----------------- ------------ -------- -------------------- --------------- ------------------ ------------------
Date Days after peak Instrument Filter Central Wavelength Exposure Time Brightness Brightness
(Å) (s) (Vegamag) (ABmag)
2010-09-13 6 ACS SBC F140LP 1528.0 2480 $22.13 \pm 0.15$ $24.30 \pm 0.15$
2010-09-23 16 WFC3 UVIS F300X 2814.8 1710 $22.60 \pm 0.05$ $24.07 \pm 0.05$
F390W 3922.9 712 $23.83 \pm 0.05$ $24.05 \pm 0.05$
F555W 5308.2 742 $24.04 \pm 0.05$ $24.02 \pm 0.05$
F775W 7647.6 740 $23.71 \pm 0.05$ $24.10 \pm 0.05$
WFC3 IR F160W 15369.2 806 $23.6 \pm 0.3$ $24.9 \pm 0.3$
2012-11-19 72 ACS SBC F140LP 1528.0 2428 $22.53 \pm 0.15$ $24.70 \pm 0.15$
WFC3 UVIS F300X 2814.8 1001 $23.2 \pm 0.1$ $24.7 \pm 0.1$
F336W 3354.8 983 $23.5 \pm 0.1$ $24.7 \pm 0.1$
F390W 3922.9 1068 $24.33 \pm 0.05$ $24.55 \pm 0.05$
F555W 5308.2 1045 $24.57 \pm 0.05$ $24.55 \pm 0.05$
F621M 6218.9 1065 $24.40 \pm 0.05$ $24.55 \pm 0.05$
F775W 7647.6 1040 $24.11 \pm 0.05$ $24.50 \pm 0.05$
WFC3 IR F105W 10551.0 1209 $24.1 \pm 0.2$ $24.7 \pm 0.2$
F160W 15369.2 1209 $23.7 \pm 0.3$ $25.0 \pm 0.3$
2013-07-05 300 ACS SBC F140LP 1528.0 2428 $23.83 \pm 0.15$ $26.00 \pm 0.15$
WFC3 UVIS F300X 2814.8 1004 $24.1 \pm 0.1$ $25.6 \pm 0.1$
F336W 3354.8 980 $24.2 \pm 0.1$ $25.4 \pm 0.1$
F390W 3922.9 1074 $25.2 \pm 0.2$ $25.4 \pm 0.1$
F555W 5308.2 1039 $25.0 \pm 0.1$ $25.0 \pm 0.1$
F621M 6218.9 1077 $24.85 \pm 0.05$ $25.00 \pm 0.15$
F775W 7647.6 1028 $24.51 \pm 0.15$ $24.90 \pm 0.15$
WFC3 IR F105W 10551.0 1209 $24.3 \pm 0.2$ $24.9 \pm 0.2$
F160W 15369.2 1209 $23.8 \pm 0.3$ $25.1 \pm 0.3$
------------ ----------------- ------------ -------- -------------------- --------------- ------------------ ------------------
Optical/UV and X-ray study
==========================
Setting the problem: distinguishing accretion disk and stellar contributions
----------------------------------------------------------------------------
The point-like optical source discovered at the position of HLX-1 [@soria10], in a region devoid of any other bright X-ray or optical point-like sources, might in principle be a chance coincidence, rather than being physically associated with the hyperluminous X-ray source. If it were a chance association, the detection of a redshifted H$\alpha$ emission line from the optical counterpart [@wiersema10; @soria13b] suggests that the optical source is at $\approx$100 Mpc while the X-ray source might be a lower-luminosity foreground X-ray binary. This is statistically implausible, and would also make the disk temperature and luminosity of HLX-1 no longer consistent with canonical BH accretion states [@farrell09; @yan15]. Further strong evidence of a physical connection between X-ray and optical emission comes from their variability properties, as discussed below.
The physical interpretation of the optical/UV emission has been a topic of intense debate. The source is too bright ($M_V \approx -11$ mag in outburst: @farrell12) to be an individual O star or even a group of few O stars. Two alternative possibilities were proposed [@farrell12; @soria12], based on combined fits of the X-ray and optical/UV spectrum in outburst: either the optical emission is dominated by a young, massive ($M_{\ast} \sim 10^6 M_{\odot}$) star cluster, or it comes mostly from the X-ray-irradiated accretion disk, supplemented by an older stellar population (age $> 10$ Gyr) to account for an observed near-infrared excess.
Different formation and evolution scenarios for the candidate IMBH are supported by either interpretation of the optical counterpart. If [**most of**]{} the near-UV and blue emission is from a young star cluster, a recent, localized episode of intense star formation is required. There is no other evidence of recent star formation in the halo of ESO243-49, but the young cluster could be the nucleus of a recently accreted gas-rich dwarf galaxy [@farrell12; @mapelli12; @mapelli13a; @mapelli13b]. A young, massive star cluster could be a suitable location for the formation and growth of an IMBH [@pz02; @gurkan04]. Instead, if the near-UV and blue light is mostly reprocessed thermal emission from an irradiated accretion disk, there is much less need for a massive supply of gas and substantial recent star formation in the star cluster. If the donor star is a red giant or asymptotic giant branch star, or a blue straggler, the cluster may consist entirely of an old population; alternatively, the cluster may contain a few young, massive stars near the centre (including the IMBH donor), perhaps formed from a small amount of gas swept up from the interstellar medium [@li16; @conroy11]. Moreover, if the near-UV and blue emission turn out to be from the irradiated disk, we can use that observed flux to obtain a characteristic size of the emitting region, and from that, constrain the size of the BH Roche lobe and the binary separation.
In the young cluster scenario, the optical/UV emission is not expected to show substantial variations on timescales shorter than the evolution or the dynamical timescale of the cluster. In the irradiated disk scenario, brightness variations in X-rays and optical are naturally expected, as a system could get into an outburst phase or retreat into quiescence. Very Large Telescope (VLT) observations taken just before and during the rise of the 2012 outburst showed an increase in the visual brightness of $\Delta V = 1.8 \pm 0.4$ mag between near-quiescence (365 days after the peak of the previous outburst) and outburst peak [@webb14]. In the $R$ band, a comparison of observations taken with the [*Hubble Space Telescope*]{} ([*HST*]{}), VLT, Gemini and Magellan at different times over 2009–2012 suggests a brightness change of $\Delta R = 0.9 \pm 0.4$ mag between outburst and quiescence [@farrell14; @webb14]; however, this result may be affected by systematic errors in the conversion between different filter bands. Moreover, the diffuse emission from the old (red) stellar population in the halo of ESO243-49 substantially reduces the detection significance of the HLX-1 counterpart from ground-based telescopes, especially towards quiescence.
[*HST*]{}, [*XMM-Newton*]{} and [*Swift*]{} data analysis
---------------------------------------------------------
In this work, we present and compare the photometric results from three sets of [*HST*]{} observations, taken with the Wide Field Camera 3 (UVIS and IR apertures), on 2010 September 23 ($\approx$16 days after outburst peak), on 2012 November 19 ($\approx$72 days after peak) and on 2013 July 5–6 ($\approx$300 days after peak) (Table 1). Ultraviolet observations were also taken on 2010 September 13, 2012 November 19 and 2013 July 5 with the Advanced Camera for Surveys Solar Blind Channel (ACS SBC). To model the optical/UV data, we fit them simultaneously with representative X-ray spectra taken as close as possible to the optical observations, and/or at a similar level of X-ray luminosity. For the 2012 [*HST*]{} data, we used a 54-ks [*XMM-Newton*]{} observation taken on November 23. For the 2013 [*HST*]{} data, we used a 141-ks [*XMM-Newton*]{} observation taken on July 3. For the 2010 [*HST*]{} data, no contemporaneous [*XMM-Newton*]{} observations are available, and individual [*Swift*]{} observations are too short to provide a meaningful constraint to the X-ray parameters. Therefore, we determined the [*Swift*]{} X-Ray Telescope (XRT) count rate at the time of the 2010 [*HST*]{} observations ([*Swift*]{} observation of 2010 September 23), and then stacked all [*Swift*]{} observations across the 6 observed outbursts, that have an observed count rate within a factor of 2 of the count rate at the [*HST*]{} epoch[^2].
\
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For the [*HST*]{} data, we downloaded the calibrated images (drizzled files) from NASA’s Mikulski Archive for Space Telescopes (MAST). HLX-1 is a well-isolated source, with no risk of confusion with other nearby point-like sources. Therefore we used aperture photometry to measure its brightness, with standard packages such as Version 7.4 and Version 2.16. Particularly in the redder filters, proper background subtraction is crucial because of the strong unresolved emission from the old stellar population in ESO243-49 and the gradient of such emission. For the source, we used a circular extraction region of radius 0”.2 (Figure 3). For the background, we used an elliptical annulus oriented at a parallactic angle (north through east) of 75$^{\circ}$, that is approximately parallel to the isophotes of ESO243-49 around the projected location of HLX-1. This was done to reduce the gradient of the unresolved emission in the background region. The semi-major axes of the outer background annulus were $1''$ and $0''.45$; those of the inner (exclusion) annulus were $0''.45$ and $0''.3$. For each filter and each epoch, we measured the background-subtracted count rate of HLX-1 within the $0''.2$ extraction region, and converted it to a $0''.4$ count rate. To do so, we selected isolated, brighter point-like sources in the same chip, and determined the ratio of the count rates from a $0''.2$ and $0''.4$ radius; as a safety check, we also compared our empirical values with the aperture correction values tabulated in the WFC3 and ACS online handbooks, and found them consistent. We then converted the $0''.4$ count rates to Vega and AB magnitudes, using the Zeropoint tables provided online by the Space Telescope Science Institute. Finally, we used the [FTOOLS]{} [@blackburn95] task [*flx2xsp*]{} to convert the observed [*HST*]{} flux densities into standard PHA files with their associated response files, which can be displayed and fitted in [XSPEC]{} [@arnaud96].
For [*XMM-Newton*]{}, we downloaded the data from NASA’s High Energy Astrophysics Science Archive Research Center (HEASARC) archive: we used ObsID 0693060401 (PI: S. Farrell) from 2012 November 23, and ObsID 0693060301 (PI: S. Farrell) from 2013 July 4–5. We reprocessed the European Photon Imaging Camera (EPIC) MOS and pn observation data files with the Science Analysis System (SAS) version 14.0.0. We checked for exposure intervals with high particle background, and removed them from the analysis; the good-time-interval was 49 ks for the 2012 dataset and 112 ks for the 2013 dataset. We extracted the source photons from a circular region with a radius of $30''$; the background photons were obtained from nearby regions located at similar distances from readout nodes, and avoiding chip gaps. We used the [SAS]{} task [*xmmselect*]{} to select single and double events (pattern 0–4 for pn and 0–12 for MOS1 and MOS2), and filtered them with the standard criteria FLAG $= 0$ & \#XMMEA\_EP for the pn and \#XMMEA\_EM for the MOS. We built response and ancillary response files with the [SAS]{} tasks [*rmfgen*]{} and [*arfgen*]{}. Finally, to increase the signal-to-noise ratio, we combined the pn and MOS spectra with [*epicspeccombine*]{}, creating an average EPIC spectrum for each of the two epochs. We grouped the two combined spectra to a minimum of 20 counts per bin; coincidentally, this gives us $\approx$3 bins per spectral resolution element, keeping in mind that the spectral resolution of a combined EPIC spectrum is $\approx$100 eV at $\approx$0.3–2 keV. Finally, we fitted the 2012 and 2013 EPIC spectra with [XSPEC]{} version 12.8.2 [@arnaud96], together with the corresponding [*HST*]{} spectra. For the [*Swift*]{}/XRT data, we used the online product generator [@evans09] to extract a lightcurve, determine the count rate on 2010 September 23, and build a stacked spectrum with all the XRT observations between 2009 and 2015 with a count rate within a factor of 2 of the reference one. The stacked spectrum comprises 101 snapshot observations for a total exposure time of 66 ks. A similar stacking technique was previously used by other authors ([*e.g.*]{}, @yan15 [@soria11]) to model the characteristic high/soft, intermediate and low/hard state spectra. We verified that a stacked spectrum including only data from the 2010 outburst (similar to what was done by @farrell12 for their broad-band modelling of X-ray and optical/UV data) produces similar results, but at lower signal-to-noise, which makes it harder to constrain the disk parameters.
Main results
------------
The immediate result of our comparison between the three [*HST*]{} observations is the sharp decline in the bluest part of the optical/UV spectrum going from the high/soft to the low/hard X-ray state; instead, there is only a small flux decrease in the redder bands (Table 1 and Figure 4). It was already known [@farrell12; @soria12; @farrell14] that a single irradiated disk is not sufficient to reproduce the 2010 [*HST*]{} optical spectrum: either an additional near-IR component (possibly an old stellar population), or an additional near-UV component (possibly a young stellar population), or both, are needed. For each of the three [*HST*]{} epochs, we applied a code developed by [@mucciarelli07] and [@patruno08], upgraded to calculate the magnitudes of an irradiated disk in the [*HST*]{} photometric bands (see also @mapelli13a): like for the 2010 data, we verified that at least one additional component is required also for the 2012 and 2013 data. By comparing the three datasets, we have now proved the correlation between near-UV emission and the level of X-ray irradiation. Therefore, the near-UV emission in outburst cannot be dominated by a young stellar population, because of its variability; the near-UV luminosity in the faintest observed epoch (2013) sets the upper limit to the constant contribution of young stars. On the other hand, the near-IR component can be dominated by constant emission from an old stellar population.
-------------------------- -------------------------------------------------- ---------------------------- ------------------------------- ----------------------------
Component Parameter
[*redden*]{}$_{\rm los}$ $E(B-V)_{\rm los}$ (mag) \[0.013\] \[0.013\] \[0.013\]
\[2pt\] $E(B-V)$ (mag) $< 0.12$ $<0.12$ $< 0.11$
\[2pt\] $_{\rm los}$ $N_{\rm H,los}$ ($10^{20}$ cm$^{-2}$) \[2.0\] \[2.0\] \[2.0\]
\[2pt\] $N_{\rm H}$ ($10^{20}$ cm$^{-2}$) $<1.3$ $< 1.4$ $<2.0$
\[2pt\] $kT_{\rm in}$ (keV) [$0.22_{-0.01}^{+0.01}$]{} [$0.083_{-0.040}^{+0.037}$]{} $< 0.073$
\[2pt\] $\Gamma$ [$1.8_{-0.6}^{+2.1}$]{} [$2.43_{-0.15}^{+0.15}$]{} [$2.04_{-0.20}^{+0.26}$]{}
\[2pt\] $kT_{\rm e}$ (keV) \[100\] \[100\] \[100\]
\[2pt\] $L_{\rm c}/L_{\rm d}$ [$0.12_{-0.09}^{+1.35}$]{} [$2.1_{-0.9}^{+\ast}$]{} [$16.8_{-15.4}^{+\ast}$]{}
\[2pt\] $f_{\rm in}$ \[0.1\] \[0.1\] \[0.1\]
\[2pt\] $r_{\rm irr}$ \[1.2\] \[1.2\] \[1.2\]
\[2pt\] $f_{\rm out}$ $(10^{-3})$ [$6.6_{-3.5}^{+1.5}$]{} [$43_{-16}^{+\ast}$]{} [$45_{-15}^{+23}$]{}
\[2pt\] log($R_{\rm out}$) [$3.68_{-0.06}^{+0.05}$]{} [$3.27_{-0.24}^{+0.30}$]{} $< 3.16$
\[2pt\] $K$ [$17.6_{-4.4}^{+12.2}$]{} [$39.5_{-28.5}^{+139}$]{} $> 3.0$
\[2pt\] $T_{\rm bb}$ (K) \[5270\] \[5270\] [$5270_{-400}^{+300}$]{}
\[2pt\] $N_{\rm bb}$ ($10^9$) \[3.9\] \[3.9\] [$3.9_{-0.9}^{+0.7}$]{}
\[2pt\] $kT_{\rm mk}$ (keV) \[0.45\] \[0.45\] [$0.45_{-0.13}^{+0.14}$]{}
\[2pt\] $N_{\rm mk}$ ($10^{-6}$) \[1.1\] \[1.1\] [$1.1_{-0.5}^{+0.7}$]{}
\[2pt\] $f_{\rm X}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) [$57_{-3}^{+3}$]{} [$7.2_{-0.2}^{+0.4}$]{} [$2.6_{-0.2}^{+0.2}$]{}
\[2pt\] $L_{\rm X} \cos \theta$ ($10^{40}$ erg s$^{-1}$) [$33.1_{-1.9}^{+1.8}$]{} [$4.1_{-0.2}^{+0.3}$]{} [$2.8_{-0.2}^{+0.4}$]{}
\[2pt\] $f_{\rm O}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) [$2.4_{-0.1}^{+0.1}$]{} [$1.4_{-0.1}^{+0.1}$]{} [$0.60_{-0.02}^{+0.02}$]{}
\[2pt\] $f_{\rm R}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) \[0.18\] \[0.18\] [$0.18_{-0.05}^{+0.04}$]{}
\[2pt\] $R_{\rm in} \, \sqrt{\cos \theta}$ ($10^8$ cm) [$46_{-6}^{+14}$]{} [$68_{-32}^{+61}$]{} $> 20$
\[2pt\] $\chi^2_{\nu}$ 1.29 (91.3/71) 0.88 (64.3/73) 0.75 (61.5/82)
\[2pt\]
-------------------------- -------------------------------------------------- ---------------------------- ------------------------------- ----------------------------
\[tab\_diskir\]
-------------------------- -------------------------------------------------- ----------------------------- ----------------------------- -----------------------------------
Component Parameter
[*redden*]{}$_{\rm los}$ $E(B-V)_{\rm los}$ (mag) \[0.013\] \[0.013\] \[0.013\]
\[2pt\] $E(B-V)$ (mag) $< 0.10$ $<0.15$ \[0.0\]
\[2pt\] $_{\rm los}$ $N_{\rm H,los}$ ($10^{20}$ cm$^{-2}$) \[2.0\] \[2.0\] \[2.0\]
\[2pt\] $N_{\rm H}$ ($10^{20}$ cm$^{-2}$) [$2.0_{-2.0}^{+2.8}$]{} $< 1.4$ $<28$
\[2pt\] $_{\rm \,X}$ $kT_{\rm bb}$ (keV) - - [$0.058_{-0.058}^{+0.059}$]{}
\[2pt\] $N_{\rm bb}$ - - [$37.7_{-37.7}^{+\ast}$]{}
\[2pt\] $kT_0$ (keV) - - \[[$0.058_{-0.058}^{+0.059}$]{}\]
\[2pt\] $kT_e$ (keV) - - \[100\]
\[2pt\] $\tau$ - - [$0.19_{-0.07}^{+0.15}$]{}
\[2pt\] $N_{\rm c}$ ($10^{-7}$) - - [$3.6_{-1.9}^{+\ast}$]{}
\[2pt\] $kT_{\rm in}$ (keV) [$0.21_{-0.03}^{+0.02}$]{} [$0.086_{-0.04}^{+0.03}$]{} -
\[2pt\] $\Gamma$ [$2.1_{-1.0}^{+\ast}$]{} [$2.45_{-0.21}^{+0.17}$]{} -
\[2pt\] $kT_{\rm e}$ (keV) \[100\] \[100\] -
\[2pt\] $L_{\rm c}/L_{\rm d}$ [$0.066_{-0.041}^{+9.5}$]{} [$2.0_{-0.6}^{+\ast}$]{} -
\[2pt\] $f_{\rm in}$ \[0.1\] \[0.1\] -
\[2pt\] $r_{\rm irr}$ \[1.2\] \[1.2\] -
\[2pt\] $f_{\rm out}$ $(10^{-3})$ [$3.6_{-3.1}^{+7.4}$]{} [$27_{-12}^{+\ast}$]{} -
\[2pt\] log($R_{\rm out}$) [$3.57_{-0.18}^{+0.13}$]{} [$3.16_{-\ast}^{+0.30}$]{} -
\[2pt\] $K$ [$27.5_{-11.1}^{+35.4}$]{} [$36.5_{-26.5}^{+89}$]{} -
\[2pt\] $_{\rm \,B}$ $T_{\rm bb}$ (K) \[21,170\] \[21,170\] [$21,170_{-2890}^{+3760}$]{}
\[2pt\] $N_{\rm bb}$ ($10^9$) \[3.6\] \[3.6\] [$3.6_{-1.4}^{+2.3}$]{}
\[2pt\] $_{\rm \,R}$ $T_{\rm bb}$ (K) \[5550\] \[5550\] [$5550_{-290}^{+250}$]{}
\[2pt\] $N_{\rm bb}$ ($10^9$) \[3.4\] \[3.4\] [$3.4_{-1.7}^{+3.0}$]{}
\[2pt\] $kT_{\rm mk}$ (keV) \[0.44\] \[0.44\] [$0.44_{-0.13}^{+0.14}$]{}
\[2pt\] $N_{\rm mk}$ ($10^{-6}$) \[1.1\] \[1.1\] [$1.1_{-0.5}^{+0.7}$]{}
\[2pt\] $f_{\rm X}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) [$57_{-2}^{+2}$]{} [$7.2_{-0.2}^{+0.4}$]{} [$2.6_{-0.2}^{+0.2}$]{}
\[2pt\] $L_{\rm X} \cos \theta$ ($10^{40}$ erg s$^{-1}$) [$38.1_{-4.9}^{+6.0}$]{} [$4.1_{-0.2}^{+0.3}$]{} [$2.8_{-0.2}^{+0.4}$]{}
\[2pt\] $f_{\rm O}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) [$1.9_{-0.1}^{+0.1}$]{} [$1.3_{-0.1}^{+0.1}$]{} [$0.56_{-0.04}^{+0.04}$]{}
\[2pt\] $f_{\rm B}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) \[0.37\] \[0.37\] [$0.37_{-0.01}^{+0.03}$]{}
\[2pt\] $f_{\rm R}$ ($10^{-14}$ erg cm$^{-2}$ s$^{-1}$) \[0.19\] \[0.19\] [$0.19_{-0.04}^{+0.02}$]{}
\[2pt\] $R_{\rm in} \, \sqrt{\cos \theta}$ ($10^8$ cm) [$57_{-13}^{+30}$]{} [$66_{-31}^{+57}$]{} -
\[2pt\] $\chi^2_{\nu}$ 1.25 (88.9/71) 0.89 (64.8/73) 0.73 (61.7/85)
\[2pt\]
-------------------------- -------------------------------------------------- ----------------------------- ----------------------------- -----------------------------------
\[tab\_diskir\]
\
\
\
\
Given the small number of optical/UV datapoints, and therefore the small number of free parameters we can introduce, we tried fitting the broad-band data with two simple models: one in which all of the near-UV emission comes from the irradiated disk at all epochs (“Model 1"), and one in which all of the near-UV emission in the faintest epoch (2013) comes from a young stellar population, and the additional irradiation component only appears in 2010 and 2012 (“Model 2"). Those are clearly extreme cases: in reality, the contribution from the young stellar population may be somewhere between the two cases.
Let us start with Model 1, which consists of an irradiated disk ([*diskir*]{} in [XSPEC]{}: @gierlinski08 [@gierlinski09]) plus a constant, cool blackbody component ([*bbodyrad*]{}). The parameters of the blackbody component are determined from the 2013 observation, when it dominates the optical spectrum. These parameters are then kept fixed for our fits to the 2010 and 2012 [*HST*]{} data, dominated by the bluer component. We find (Figure 5, Table 2) that we can reproduce the spectral energy distribution at all three epochs. The moderately high fit residuals in the 2010 dataset ($\chi^2 \approx 1.25$) are mostly due to the X-ray part of the spectrum (the stacked [*Swift*]{}/XRT data), and to a near-UV excess in the F140LP filter. The latter is probably caused by the fact that this band was observed 10 days before the other [*HST*]{} bands, closer to outburst peak, and the near-UV is the most sensitive colour to the effect of X-ray irradiation. More detailed analyses and discussions of the 2010 [*HST*]{} and broadband data have already been presented elsewhere [@farrell12; @soria12; @mapelli13a; @farrell14] and need not be repeated here. The new, interesting result of this work is that the irradiated disk plus blackbody model formally works also for the intermediate and low state observations. However, a very high reprocessing fraction $f_{\rm out} \approx 4\%$ is required in 2012 and 2013. In simple terms, the high value of $f_{\rm out}$ in the 2012 and 2013 fits is because the blue flux (modelled with irradiation) decreases more slowly than the soft X-ray flux in the three [*HST*]{} epochs used for our modelling, from 2010 to 2012 and 2013. In 2010, the observed 0.3–10 keV flux was $f_{\rm X} \approx 5.7 \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$, and the flux at $\lambda > 912$Å was $f_{\rm O} \approx 2.4 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, of which $f_{\rm R} \approx 1.8 \times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ modelled as a constant red component. That gives a ratio $f_{\rm X}/(f_{\rm O}-f_{\rm R}) \approx 26$ between the short- and long-wavelength emission of the accretion flow. In 2012, $f_{\rm X} \approx 7.2 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, and $f_{\rm O} \approx 1.4 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, so that $f_{\rm X}/(f_{\rm O}-f_{\rm R}) \approx 6$. In 2013, $f_{\rm X} \approx 2.6 \times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, and $f_{\rm O} \approx 6.2 \times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$, also for a ratio $f_{\rm X}/(f_{\rm O}-f_{\rm R}) \approx 6$.
At first sight, such high levels of optical reprocessing appear difficult to explain. They are an order of magnitude higher than predicted by standard thin-disk models [@dubus99; @king97; @dejong96; @vrtilek90], supported by observations of sub-Eddington Galactic X-ray binaries [@russell14; @gierlinski09; @hynes02]. In ultraluminous X-ray sources (ULXs), which in most cases are likely to be super-Eddington accretors, it is harder to distinguish between the emission from the donor star and from the irradiated disk [@heida14; @gladstone13; @grise12; @tao12; @tao11]; therefore, it is also more difficult to determine the disk reprocessing fraction. In one ULX where all optical emission was proved to be from the irradiated disk (M83 ULX), a reprocessing factor of $\approx$5 $\times 10^{-3}$ was inferred [@soria12b]; however, for other ULXs, broad-band emission models suggested reprocessing factors of a few $10^{-2}$ [@sutton14]. The geometric solid angle subtended by the disk is insufficient to explain such reprocessing factors only from direct X-ray illumination; however, if there is a strong outflow launched from the inner part of the disk, it was suggested [@sutton14; @narayan17] that some of the X-ray photons emitted along the polar funnel may be scattered isotropically and contribute to the illumination of the outer disk. In this paper, we argue that HLX-1 is a sub-Eddington source (IMBH accretor), so in that sense it would be more appropriate to compare its reprocessing fraction with those of Galactic X-ray binaries. On the other hand, we will also argue (Section 5.3) that it has a strong wind, which would be consistent with the photon scattering scenario proposed for ULXs and with a high reprocessing factor.
In fact, when we examine more carefully the physical meaning of $f_{\rm out}$ in 2012 and 2013, we propose other explanations for those high values. In 2012, optical and X-ray observations were not strictly simultaneous: the [*XMM-Newton*]{} observation happened 4 days after the [*HST*]{} observation. Those few days between the two observations are precisely the moment when HLX-1 started to switch from the high/soft to the low/hard state, with a drop in the [*Swift*]{}/XRT count rate by an order of magnitude (Figure 2, bottom panel). Thus, when the [*HST*]{} measurements were taken, the X-ray flux was almost certainly a few times higher than what was measured with [*XMM-Newton*]{} a few days later. We fitted the [*XMM-Newton*]{} and [*HST*]{} data together without accounting for the decrease in the X-ray flux: this means that we are almost certainly over-estimating by the same amount the true value of $f_{\rm out}$ needed to match the X-ray and UV portions of the SED. As for the high reprocessing fraction fitted to the 2013 data, we recall that our Model 1 represents the extreme case of no contribution from a young stellar population: in this sense, $f_{\rm out}$ here represents the hard upper limit to the reprocessing fraction in 2013. In the opposite case (Model 2), the same spectral energy distribution can be fitted with the other extreme case of $f_{\rm out} \rightarrow 0$. Let us consider now Model 2 (Table 3 and Figure 6). We start from the 2013 dataset, which gives us the constraint on the stellar contribution (no optical/UV disk emission). On the X-ray side, we replace the [*diskir*]{} model with a simple blackbody plus Comptonization model ([*bbodyrad*]{} $+$ [*comptt*]{}), which does not extend into the UV. On the optical/UV side, we fit the 2013 spectrum only with two blackbody components, one bluer and one redder, with no additional contribution from a disk. Having determined the old and young stellar contributions from the 2013 data (Table 3), we impose that the same two components are also present in the 2010 and 2012 spectra with the same (fixed) temperature and normalization, in addition to an irradiated disk component.
In summary, we find that Model 1 and Model 2 are statistically equivalent in all 3 epochs (compare Table 2 and Table 3). We cannot tell the difference between the scenarios of near-UV emission from the disk only, or from a young stellar population plus an irradiated disk. However, we can use the models to calculate the age and mass of the old stellar populations, and to put useful upper limits to the young stellar population. In Model 1, with only a red component, the best-fitting blackbody temperature corresponds to a dereddened $V = 25.56 \pm 0.10$ mag in the Vega system, with $V-I = 1.03 \pm 0.15$ mag, $B-V = 0.82 \pm 0.15$ mag. At a distance modulus of 34.8 mag (92 Mpc), the absolute brightness is $M_V \approx -9.2$ mag. We ran simulations of star cluster evolution with [Starburst99]{} Version 7.0.1 [@leitherer99; @leitherer14], for instantaneous star formation and metallicity $Z = 0.008$. We found that those optical colours and luminosities are consistent either with an intermediate-age star cluster with mass $M_{\ast} \approx 2 \times 10^5 M_{\odot}$ and age of $\approx$800–900 Myr, or with an old star cluster, in particular one with a mass $M_{\ast} \approx 3 \times 10^6 M_{\odot}$ and age of $\approx$6–8 Gyrs (Figure 7). We repeated the same analysis for Model 2, with two optical blackbody components (Table 3). The colder component has a dereddened $V = 25.41 \pm 0.10$ mag, $V-I = 0.93 \pm 0.15$ mag, $B-V = 0.76 \pm 0.15$ mag, similar to the red component in the first model. The hotter component has $V = 26.23 \pm 0.10$ mag, $V-I = -0.28 \pm 0.15$ mag, $B-V = -0.08 \pm 0.15$ mag. This is roughly consistent with the colours of a very young star cluster (age $\la 3$ Myr), with a mass $\la 10^4 M_{\odot}$, which can be taken as the firm upper limit to the young stellar component associated to HLX-1. More detailed spectral modelling, with proper spectral energy distributions for stellar populations of various ages and metallicities, in place of simple blackbody components, is left to follow-up work (C. Maraston, priv. comm.). The 2013 X-ray spectral fit is statistically improved (with F-test significance $> 99.9\%$) by the addition of a thermal-plasma component ([*mekal*]{} model in [XSPEC]{}), with solar abundance and fixed redshift $z = 0.0224$. We determined the temperature and normalization of the thermal plasma emission using the same strategy that we applied to the optical blackbody components: namely, we left those two [*mekal*]{} parameters free to vary in the 2013 spectrum, determined their best-fitting values, and then kept them frozen in the 2010 and 2012 spectra, assuming that the thermal plasma component does not vary on short timescales. (In the 2010 and 2012 X-ray spectra, the additional thermal plasma component also improves the fit but with $< 90\%$ significance, because of the comparatively stronger disk emission). We found a best-fitting [*mekal*]{} temperature of $\approx (0.5 \pm 0.1)$ keV, and a de-absorbed luminosity of $\approx (3.3 \pm 1.3) \times 10^{39}$ erg s$^{-1}$, almost identical in Model 1 and Model 2. This is also the same thermal-plasma temperature and luminosity that was found by [@servillat11] in an earlier [*XMM-Newton*]{} spectrum from 2010 May 14, also in the low/hard state (in-between the 2009 and 2010 outbursts). [@servillat11] used the higher spatial resolution of [*Chandra*]{}/ACIS-S to show (see their Sections 3.2–3.3) that the soft thermal component is likely to be diffuse emission from the bulge of ESO243-49 rather than from HLX-1. Such extended component cannot be resolved in [*XMM-Newton*]{} (the 30” source extraction region for HLX-1 includes also the nuclear region and most of the galaxy). Thus, the thermal plasma emission seen in the [*XMM-Newton*]{} spectra of HLX-1 may have a different physical interpretation than the thermal plasma emission seen in several ULXs [@middleton15; @sutton15; @pinto16; @urquhart16]. In the latter group of sources, the line residuals appear to come from massive outflows, directly associated with the accreting compact objects; they are interpreted as evidence that those sources are stellar-mass BHs or neutron stars accreting much above their Eddington limit. The lack of intrinsic X-ray thermal plasma features in the HLX-1 spectra, on the other hand, is consistent with the interpretation of this source as an IMBH accreting at an Eddington rate $\la$1. For these reasons, we did not include the flux and luminosity of the thermal plasma component in the values of $f_{\rm X}$ and $L_{\rm X}$ reported in Tables 2 and 3.
What is the mass of the black hole?
===================================
The normalization of the [*diskir*]{} model ([i.e.]{}, the fitted inner radius of the disk) in the high/soft and intermediate state (Tables 2 and 3) is consistent between Model 1 and Model 2. By taking a weighted average of the four best-fitting values from the high and intermediate state (two from Model 1 and two from Model 2) we obtain: $$R_{\rm in} \sqrt{\cos \theta} \approx 1.19 \, r_{\rm in} \sqrt{\cos \theta} \approx \left(49.3^{+12.4}_{-5.6}\right) \times 10^3 \ \ {\mathrm {km}},
$$ where the factor 1.19 takes into account the hardening factor and the inner-boundary condition, as discussed in [@kubota98]. (Notice that the four values used for this average are all consistent with each other within their errors.) Our best-fitting inner-disk radius is also consistent with those previously published in the literature [@farrell14; @farrell12; @soria12; @godet12; @soria11; @farrell09]. Instead, in the low/hard state, the size of the thermal seed component (disk-blackbody in the [*diskir*]{} model and blackbody in the [*comptt*]{} model) cannot be constrained: the data are also consistent with seed temperatures $\la 50$ eV (a range in which they can no longer be meaningfully constrained with the EPIC detectors), and a correspondingly larger normalization. The [*diskir*]{} model does suggest that $R_{\rm out}/R_{\rm in} \la 1400$ in the low/hard state (Table 2), a few times less than in the high state (when $R_{\rm out}/R_{\rm in} \approx 4000$–5000); if we assume that the outer disk radius remains approximately constant (as determined in Section 4.2), it means that the inner radius is moving further out (truncated disk), to a value $R_{\rm in} \ga 2 \times 10^5$ km, consistent with the canonical evolution of an accretion disk in BH transients at the end of an outburst.
Henceforth, we will use only the high-state and intermediate-state spectra for an estimate of the inner-disk radius and BH mass. The fact that in several disk-dominated X-ray spectra, measured with different instruments over different outbursts, the inner disk radius is consistently found to be $R_{\rm in} \, \sqrt{\cos \theta} \approx 50,000$–$100,000$ km (see also Table 3 in @farrell14) suggests that this parameter is physically meaningful, representing the innermost stable circular orbit $R_{\rm isco}$. We introduce a parameter $\alpha_{\rm s}$ to express the innermost stable orbit as function of the gravitational radius, such that $R_{\rm in} = R_{\rm isco} \equiv \alpha_{\rm s} \, GM/c^2$, where $1 < \alpha_{\rm s} \le 6$, depending on the BH spin. In the framework of the standard disk model, the BH mass is then $$M \approx \frac{(3.3^{+0.9}_{-0.4})}{\alpha_{\rm s} \, \sqrt{\cos \theta}} \times 10^4 M_{\odot}.$$ If we assume that the peak of each outburst corresponds to the Eddington luminosity (in the X-ray band), we obtain $M \approx 1 \times 10^4 M_{\odot}$ and $\alpha_{\rm s} \, \sqrt{\cos \theta} \approx 3.3$. However, there is no compelling reason for assuming Eddington-limited outbursts: many Galactic X-ray transients peak at only a fraction of Eddington [@remillard06; @fender04].
Fortunately, the [*XMM-Newton*]{} spectrum from 2012 November 23 provides new information on a particularly interesting stage of the outburst cycle, and an additional independent constraint to the BH mass. In the model-independent hardness-intensity diagram, based on the observed EPIC-pn count rates, we note (Figure 8) that the position of the source appears to fall in between the high/soft and the low/hard state. Using Galactic X-ray binary terminology, we could say that the system was on the lower/descending branch of the so called “Q diagram", which is often used to describe phenomenologically the evolution of stellar-mass BH transients [@fender04]. In fact, a full “Q" cycle has never been observed for HLX-1, perhaps because the outburst rise in the hard state and the transition to the soft state occur very quickly; based on [*Swift*]{} data, the hardness-intensity diagram for HLX-1 shows two clear states but little evidence of the track between the two [@yan15]. Regardless of the actual shape of the outburst cycle, the 2012 data suggest that HLX-1 was in the middle of the (fairly rapid) transition from the soft to the hard state. This scenario is confirmed by the sharp decline in the [*Swift*]{}/XRT count rate that occurs right at the time of the 2012 observation (Figure 2, bottom panel), over a timescale of less than a week. Our spectral modelling shows (Tables 2 and 3) that disk and Comptonized (power-law) components contribute at comparable levels; the power-law photon index is still relatively soft ($\Gamma \approx 2.2$), while the peak disk temperature $kT_{\rm in} \approx 120$ eV is a factor of 2 below the temperature at outburst peak.
Empirical evidence from Galactic X-ray transients shows that the soft-to-hard state transition at the end of an outburst occurs at a characteristic luminosity $L \approx L_{\rm X} \approx 0.01$–0.03 $L_{\rm Edd}$ [@maccarone03; @kalemci13]. We determined a de-absorbed luminosity $L_{\rm X} = (4.1^{+0.3}_{-0.2} \times 10^{40})/\cos \theta$ erg s$^{-1}$ for the 2012 [*XMM-Newton*]{} data, assuming a disk-like emission geometry. Taking $L_{\rm Edd} \approx 1.3 \times 10^{38} (M/M_{\odot})$ erg s$^{-1}$, we obtain: $$M \approx \frac{1.6^{+1.7}_{-0.6}}{\cos \theta}\, 10^4 \, M_{\odot}.$$ Combining Equations (2) and (3), we derive: $$\alpha_{\rm s} \approx \left(2.1^{+1.3}_{-1.1}\right) \, \sqrt{\cos \theta}.$$ Since $\cos \theta \le 1$, $R_{\rm isco} \la 3.4 GM/c^2$, which implies that the BH spin parameter $a/M \ga 0.7$. We do not have any direct measurements of $\cos \theta$; however, the low absorbing column density required by the X-ray spectral fits suggests that HLX-1 is more likely not seen at very high inclination. This is also consistent with the viewing angle $\theta \approx 30^{\circ}$ proposed by [@cseh15] to model the Doppler boosting of the radio jet in the low/hard state. The central value of our estimate (corresponding to a soft-to-hard transition luminosity of 0.02$L_{\rm Edd}$) is $\alpha_{\rm s} \approx 2.1$ for $\theta \approx 0^{\circ}$, $\alpha_{\rm s} \approx 2.0$ for $\theta \approx 30^{\circ}$, $\alpha_{\rm s} \approx 1.8$ for $\theta \approx 45^{\circ}$. This implies a spin parameter $a/M \approx 0.9$, almost independent of $\theta$, for $\theta \la 45^{\circ}$.
In conclusion, from the transition luminosity and the fitted inner disk radius, and for moderately face-on viewing angles, we estimate a BH mass $M \approx (2\pm 1) \times 10^{4} M_{\odot}$, a BH spin parameter $\ga 0.7$ for any viewing angle, and $\approx 0.9$ for face-on angles, a peak outburst luminosity $L_{\rm X} \approx (0.3 \pm 0.15)\,L_{\rm Edd}$. Our best-fitting values of mass and spin are very similar to those inferred by [@davis11] (see in particular their Fig. 5) from [*Swift*]{} data, in the case of $\theta \approx 0$. Our best-fitting BH mass is also similar to the value found by [@godet12], although the latter result was based on slim-disk models, which may not be relevant if the peak luminosity is always sub-Eddington. Note that our value of the peak outburst luminosity is about half the value usually reported in the literature ([*e.g.*]{}, @farrell09 [@farrell12; @yan15]), because we have defined luminosities as $(2\pi/\cos \theta)$ times the flux (as more suitable to disk-dominated emission) rather than $4\pi$ times the flux.
If we assume instead a perfectly isotropic emission during the 2012 intermediate state, the luminosity becomes $L_{\rm X} = (8.2^{+0.6}_{-0.4} \times 10^{40})$ erg s$^{-1}$ $\approx 0.01$–0.03 $L_{\rm Edd}$. Repeating the same derivation, we obtain $M \approx (3.3^{+3.4}_{-1.1}) \times 10^4 M_{\odot}$, and $\alpha_{\rm s} \, \sqrt{\cos \theta} = 1.1^{+0.7}_{-1.1}$. For a moderately face-on view, this implies an even higher BH spin, close to an extreme Kerr BH. The peak outburst luminosity would be $L_{\rm X} \approx (0.15 \pm 0.08)\,\cos \theta \, L_{\rm Edd}$.
How large is the disk?
======================
Following [@soria13a], we estimate the outer radius $R_{\rm out}$ of the accretion disk in two independent ways. The first method is to assume that the X-ray luminosity during the initial phase of outburst decline has an exponential-decay timescale of order of the viscous timescale at the outer edge of the disk. This can be understood in the framework of thin accretion disk models [[*e.g.*]{}, @frank02]; it also has an empirical analogy with the temporal brightness variations of Galactic X-ray binaries during their outbursts [@king98]. The second method assumes that the optical emission comes mostly from the irradiated disk (as discussed in Section 2.3): the characteristic size of the thermal emitter is derived from the observed optical luminosity and temperature.
Disk size from X-ray outburst properties
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During the 2009, 2010 and 2011 outbursts, the X-ray emission shows characteristic exponential-decay timescales $\tau \approx (3$–$6) \times 10^6$ s [@soria13a]. For the 2012, 2013 and 2015 outbursts, the $e$-folding decay timescales are shorter, $\tau \approx 2 \times 10^6$ s [@yan15], although this value is possibly affected by the earlier transition from the exponential to the linear decay phase (Fig. 6 in @yan15). Such differences between outbursts are not crucial for our attempt to provide an order-of-magnitude estimate of the disk size. Exponential decay timescales of a few weeks are near the upper limit but still within the range of those observed in Galactic X-ray binary transients [@yanyu15; @chen97], despite the much higher BH mass proposed by the IMBH scenario.
During the exponential decline of an outburst, when the rate of mass depletion in the disk (via accretion and/or outflows) is much higher than any ongoing mass transfer from the companion star, the decay timescale is related to the outer disk radius by the relation [@frank02]: $$\tau \approx R^2_{\rm out}/(3\nu),$$ where $\nu$ is the kinematic viscosity. In the Shakura-Sunyaev model [@ss73], $\nu = \alpha c_{\rm s} H$, where $\alpha \sim 0.1$–$0.5$ is a scaling parameter (not to be confused with the BH spin parameter used in Section 3), $c_{\rm s}$ is the mid-plane sound speed in the outer disk and $H$ is the disk scale-height. In the simplest empirical approximation, we can take $\alpha (H/R) \sim 0.01$ [@shahbaz98]. Then, $R_{\rm out} \sim 0.03 c_{\rm s} \tau$. For HLX-1, we expect characteristic mid-plane temperatures $T \sim$ a few $10^5$ K in the outer disk [@frank02]; hence, $c_{\rm s} \sim 40$–60 km s$^{-1}$ and $R_{\rm out} \sim 10^{12}$ cm. Using a more accurate expression of $\nu$ from the Shakura-Sunyaev disk solutions with Kramer’s opacity [@ss73; @frank02], for the observed X-ray properties of HLX-1, we re-obtain a characteristic disk radius $\la 10^{12}$ cm (Equation 10 in @soria13a) for the earlier outbursts. For the timescales measured in the most recent outbursts, a more likely estimate of the characteristic radius is $R_{\rm out} \sim$ a few $10^{11}$ cm.
When the disk is completely ionized (a condition expected to be easily satisfied in HLX-1: @lasota11), the mass content in the disk declines exponentially on the viscous timescale [@king98] after the peak of the outburst. This leads to a relation [@shahbaz98] between outer disk radius and peak luminosity $L_{\rm peak}$: $$R_{\rm out} \approx 1.0 \times 10^{12} \left(\frac{\eta}{0.1}\right)^{-1/3}
\left(\frac{L_{\rm peak}}{10^{42} {\rm erg~s}^{-1}}\right)^{1/3}
\left(\frac{\tau}{10^6 {\rm s}}\right)^{1/3} \ {\rm cm},$$ where $\eta \equiv L/(\dot{m}c^2)$ is the radiative efficiency. Again, for the observed decline timescales and peak luminosity, and efficiency $\eta \approx 0.3$ (suitable to a fast-spinning BH), we obtain that $R_{\rm out} \approx 10^{12}$ cm, consistent with our previous estimate.
Another (not entirely independent) estimate of disk size is based on the total fluence or energy released during an outburst. For HLX-1, the energy emitted as X-rays (which is a sufficiently good approximation to the total energy, for our order-of-magnitude estimate) declined from $\approx$10$^{49}$ erg to $\approx$4 $\times 10^{48}$ erg over the six recorded outbursts (Table 3 in @yan15), assuming isotropic emission, or a factor of 2 lower if we are looking at a face-on disk geometry. For a standard BH accretion efficiency $\eta \approx 0.1$, this energy corresponds to accreted masses between $\approx$10$^{29}$ g and $\approx$4 $\times 10^{28}$ g, in the earliest and latest outbursts, respectively. These values may be reduced by up to a factor of 2 in case of a face-one disk view, and another factor of 3 for a fast-spinning BH with $a/M \approx 0.9$. Considering the uncertainties, we can say that $10^{29}\,(0.1/\eta)$ g is a conservative upper limit to the mass accreted in each outburst. Surface densities profiles $\Sigma(R)$ for irradiated disks around a 10-$M_{\odot}$ BH, for various accretion rates, were calculated by [@dubus99]. The main effect of irradiation is to keep the outer disk in the hot state; in that regime, the surface density is well approximated by the Shakura-Sunyaev solution $$\Sigma \approx 5.2 \times 10^3 \, \alpha^{-4/5} \, \dot{M}_{21}^{7/10} \,
M_4^{1/4} \, R_{12}^{-3/4} \ {\mathrm {g~cm}}^{-2}$$ [@ss73; @frank02], where $M_4$ is the BH mass in units of $10^4 M_{\odot}$ and $\dot{M}_{21}$ the accretion rate in units of $10^{21}$ g s$^{-1}$ (suitable order of magnitude for the long-term-average accretion rate in HLX-1). The total mass $M_{\rm tot}$ in the disk up to a radius $R$ is given by $$\begin{aligned}
M_{\rm tot}(R) &\approx& 2 \pi \int_{0}^{R} \Sigma \, R \, dR \nonumber \\
&\approx& 2.6 \times 10^{28} \, \alpha^{-4/5} \, \dot{M}_{21}^{7/10} \, M_4^{1/4} \, R_{12}^{5/4} \ {\mathrm g}.\end{aligned}$$ Since the accreted mass during an outburst is $\la 10^{29} \,(0.1/\eta)$ g, and taking $\alpha \la 0.3$, $M_4 \approx 2$, from Equation (8) we conclude that a typical outburst can empty a disk (or a substantial fraction of the disk) of radius $R \la 10^{12}$ cm. If we also account for the factor-of-two reduction in the mass accreted in each outburst (in case of a face-on disk view), the disk size (or at least the portion of the disk) that can be emptied in each outburst is only $R \la 5 \times 10^{11}$ cm.
A radius $\sim$ a few $10^{11}$–$10^{12}$ cm is within the range of disk sizes in stellar-mass Galactic BH X-ray binaries (Fig. 1 in @remillard06); for example, it would be smaller than the disk in GRS 1915$+$105 ($R_{\rm out} \approx 5 \times 10^{12}$ cm: @rau03), similar to the disk radius in V404 Cyg ($R_{\rm out} \approx 9 \times 10^{11}$ cm: @munoz16) and slightly larger than the disk radii inferred for GROJ1655$-$40, GRO1744$-$28, GX339-4 [@shahbaz98; @homan05]. However, given the large difference in BH masses and therefore in the inner disk radius $R_{\rm in}$, for HLX-1 this would imply $R_{\rm out} \sim 100 R_{\rm in}$ while for Galactic stellar-mass BHs, $R_{\rm out} \sim 10^5 R_{\rm in}$.
Disc size from optical continuum fitting
----------------------------------------
As discussed in Section 2.3, the optical/UV continuum emission in the high and intermediate state must come mostly from an irradiated disk, with the addition of a red excess. Our [*diskir*]{} modelling of the 2010 [*HST*]{} data together with stacked high/soft [*Swift*]{} data (Tables 2 and 3) suggests an outer disk radius $R_{\rm out} \, \sqrt{\cos \theta} = (2.2^{+0.6}_{-0.4}) \times 10^{13}$ cm. This is consistent with the outer radii estimated by [@farrell12] and [@soria12], and is a factor of 3 smaller than the value fitted by [@farrell14]; a value $R_{\rm out} \, \sqrt{\cos \theta} \approx 3 \times 10^{13}$ cm was estimated by [@mapelli13a] with the irradiated disk model of [@patruno08]. The main reason why our optical radius estimate is on the lower side of the measurement distribution is that we have attributed all the red and near-IR emission in the low-state 2013 [*HST*]{} dataset to a stellar component rather than the disk (Section 3.2), and have then kept this model component constant in the spectra for the the intermediate and high states. This reduces the outer radius of the disk required to fit the optical data.
The effective temperature at the outer edge of the disk (where irradiation dominates over viscous heating) can be estimated from the relation $\sigma [T(R_{\rm out})]^4 \approx \left(f_{\rm out} \, L_{\rm X}\right)/\left(4\pi R_{\rm out}^2\right)$, where $f_{\rm out}$ is the reprocessing fraction. From our [*diskir*]{} fit for the 2010 [*HST*]{} dataset, we infer $T(R_{\rm out}) \approx 20,000$ K. An alternative way of estimating the outer disk temperature is to fit the 2010 [*HST*]{} datapoints with a single blackbody component, because the irradiated disk emission is dominated by the blackbody emission from the largest annulus. We verified that this method also suggests an outer disk temperature of $\approx$20,000 K (for line-of-sight extinction).
In conclusion, the disk radius derived from optical photometry is at least $\approx$20 times, and more likely 40 times, larger than the radius derived from the X-ray lightcurve. The two values cannot be reconciled even by taking into account systematic uncertainties in both methods. While the outburst timescale is within the range of Galactic X-ray binaries, the peak optical brightness of the disk ($M_V \approx -11$ mag) is several magnitudes brighter; the most luminous irradiated disks in Galactic X-ray binaries reach only $M_V \sim -5$ mag [@vanp94]. In Section 5, we will discuss two scenarios that address this discrepancy: either by attributing the optical emission to a circumbinary (CB) disk, or by confining the cycle of X-ray outbursts to the innermost rings of the disk.
Implications for the system parameters
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In the absence of dynamical measurements for HLX-1, determining the outer disk radius provides an important constraint on the binary period and orbital separation, and therefore also on the type of donor star and mass transfer mechanism. Analytical models and numerical simulations suggest that for $q \equiv M_2/M_1 \ll 1$ (where $M_1$ is the BH mass and $M_2$ the donor star masss), the disk can extend to the tidal truncation radius $R_{\rm T}$: $$R_{\rm out} \approx R_{\rm T} \approx 0.60\, a/(1+q),$$ [@warner95; @artymowicz94; @whitehurst88; @paczynski77; @papaloizou77]. For the range of mass ratios of interest here ($q \sim 10^{-5}$–$10^{-3}$), the radius of the donor star (assumed to be filling its Roche lobe) is well approximated by $$R_{2} \approx 0.49\, aq^{1/3},$$ [@eggleton83; @paczynski71]. Note also that for $q \ll 1$, the companion star fills its Roche lobe at a binary separation $a \approx 2 R_2 q^{-1/3}$, and gets tidally disrupted at a binary separation $a_{\rm td} \approx R_2 q^{-1/3}$ [@rees88]. The binary separation is related to the binary period via Kepler’s third law: $$\begin{aligned}
a &=& 1.50 \times 10^{13} \, (M_1 + M_2)^{1/3}\, P_{\rm {yr}}^{2/3} \ \ {\rm{cm}} \nonumber \\
&=& 0.63 \times 10^{13} \, (M_1 + M_2)_4^{1/3}\, P_{\rm {d}}^{2/3} \ \ {\rm{cm}},\end{aligned}$$ where the masses are in units of $M_{\odot}$ or $10^4 M_{\odot}$, respectively.
Here, we will briefly illustrate the characteristic system parameters corresponding to the two alternative cases of $R_{\rm out} \la 10^{12}$ cm and $R_{\rm out} \approx 2.5 \times 10^{13}$ cm (two representative values for the scenarios discussed in Section 4.1 and 4.2). We will also assume a BH mass of $\approx$$(2\pm1) \times 10^4 M_{\odot}$.
If it corresponds to the outer disk radius, the characteristic size derived from the X-ray lightcurve implies a binary separation $a \la 1.7 \times 10^{12}$ cm (Equation 9), and a binary period $P \la 2.3$ hr. The Roche lobe radius of the secondary is $R_2 \la 2.3 \times 10^{10}$ cm $\approx 0.33 R_{\odot}$ for $q = 2 \times 10^{-5}$, or $R_2 \la 3.7 \times 10^{10}$ cm $\approx 0.53 R_{\odot}$ for $q = 10^{-4}$. Thus, the inferred constraint on the secondary Roche lobe can only be satisfied by an M-dwarf main-sequence donor star with $M_2 \la 0.4 M_{\odot}$ [@boyajian12]. However, it is also possible that the donor star has a radius much lower than its initial radius on the main sequence because it has already lost most of its outer envelope as a result of the intense mass transfer. For a Roche-lobe-filling secondary star, and $0 < q \la 10^{-3}$, the period-density relation is well approximated by $P\rho^{1/2} \approx 0.41$ [@eggleton83], where $P$ is in days and $\rho$ in g cm$^{-3}$. In our compact scenario for HLX-1, a period of $\approx$2.3 hours corresponds to a density $\rho \approx 18$ g cm$^{-3}$. This is consistent with the density of an M3-M4 main-sequence star [@boyajian12].
As discussed in Section 4.2, a small disk of radius $R_{\rm out} \la 10^{12}$ cm inside the BH Roche lobe cannot explain the bright optical emission. Therefore, we propose that the latter component comes from an irradiated CB disk. Smoothed particle hydrodynamics models show [@artymowicz94] that the inner boundary of a CB disk is truncated by tidal forces at a radius $R_{\rm CB,in}$ varying between $\approx$1.8$a$ and $\approx$3$a$ depending on orbital eccentricity. The difference is not significant for our case, because the emission dip expected from this gap falls in the far-UV band, where we do not have any measurements. Chris Copperwheat (priv.comm.) used his code [@copperwheat05; @copperwheat07] to simulate irradiated disk spectra with or without a gap, and we verified that the difference is undetectable with the available X-ray and optical datapoints. The outer radius of the irradiation-dominated CB disk would have to extend to $R_{\rm CB,out} \ga 15a \approx 2.5 \times 10^{13}$ cm to reproduce the optical continuum; its continuum emission spectrum would look essentially identical to the spectrum of an irradiation-dominated accretion disk of the same size ($T(R) \propto R^{-1/2}$). See also [@farris14; @farris15; @yanlu15; @dorazio16; @dorazio13; @roedig14; @artymowicz96] for further modelling of CB disk structure and emission.
Observational evidence for mid-infrared emission from a dusty CB disk illuminated by the compact X-ray source has been found in several X-ray binaries, such as the Galactic BH candidates GRS1915$+$105 [@rahoui10], A0620$-$00 and XTEJ1118$+$480 [@muno06]. Among ULXs, a CB disk was proposed for Holmberg IX X-1 [@dudik16] and Holmberg II X-1 [@lau17]. A CB disk fed by outflows through the L2 point is also likely to be present in SS433, based on optical spectroscopic studies of its stationary H$\alpha$, Br$\gamma$, Paschen, and HeI emission lines [@blundell08; @perez09; @perez10; @bowler13]. From the rotational velocity shift of the red and blue line components, the characteristic emitting radius of the SS433 CB disk is $\sim$10$^{13}$ cm [@bowler13]. We have argued that in the CB disk scenario for HLX-1, its characteristic radius would also be $\sim$10$^{13}$ cm; however, its irradiating luminosity at outburst peak would be much higher than in SS433, and would keep the CB disk in the hot, optically-thick state with significant contributions to the UV continuum emission.
Let us now assume instead that the outer accretion disk radius is $R_{\rm out} \approx 2.5 \times 10^{13}$ cm, based on the optical continuum measurements. Following the same argument, for $q \ll 1$ and $M \approx 2 \times 10^4 M_{\odot}$, this disk radius implies a binary separation $a \approx 4 \times 10^{13}$ cm, and a binary period $P \approx 12$ d. The Roche lobe radius of the secondary is $R_2 \approx 5.5 \times 10^{11}$ cm $\approx 8 R_{\odot}$ for $q = 2 \times 10^{-5}$, or $R_2 \approx 9.5 \times 10^{11}$ cm $\approx 14 R_{\odot}$ for $q = 10^{-4}$, or $R_2 \approx 2.0 \times 10^{12}$ cm $\approx 29 R_{\odot}$ for $q = 10^{-3}$. From the period-density relation for $0 < q \la 10^{-3}$ we obtain a mean density $\rho \approx 1.2 \times 10^{-3}$ g cm$^{-3}$. This is consistent with several types of evolved stars ([*e.g.*]{}, blue supergiants, Hertzsprung gap stars, red giants). Massive stars in the last stages of stellar evolution are much more likely sources of continuously high mass-transfer rate (for timescales of a few $10^5$ yr) than a main-sequence M star. The only circumstance in which an M star can provide the required mass transfer rate is if it is currently undergoing tidal stripping of its envelope. Therefore, we conclude that the large disk scenario is much more consistent with the observations. This scenario would also imply that HLX-1 is fuelled in a way similar to ULXs, despite the likely large difference in the mass of the accretor. For completeness, in Section 5 we will outline possible outburst mechanisms both for the case of a smaller and of a larger orbital separation, keeping in mind that the latter case is more likely to apply to HLX-1.
What is the outburst mechanism?
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Summary of HLX-1 outburst models proposed in the literature
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The fast outburst rise and subsequent exponential decay, and the harder power-law spectrum in the low states, can be qualitatively explained with the disk instability model usually applied to X-ray binaries [@lasota08a; @lasota08b; @lasota01; @dubus01; @lasota00; @menou00; @dubus99; @hameury98; @hameury97]. In the model, the accretion disk extends to the innermost stable circular orbit $R_{\rm isco}$ when in the high state. Instead, in the low state, the disk is truncated at $R_{\rm tr} (\gg R_{\rm isco}$), within which the hot inflow is radiatively inefficient and geometrically thick. Moreover, in the low state, all disk annuli are on the cold branch of the density-temperature S-curve (characterised by neutral hydrogen and low viscosity), with inflowing matter piling gradually till the surface density exceeds the critical density. This occurs at $R$ slightly larger than $R_{\rm tr}$, from where a heating front propagates outwards (an inside-out outburst). The increased viscosity and accretion rate cause the inner disk to extend inwards towards $R_{\rm isco}$, and the system enters a high state, where the entire disk is on the hot branch of the S-curve. When the outburst subsides and the accretion rate diminishes, the outer disk cools down and the inner disk evaporates, creating a radiatively inefficient, optically thin, geometrically thick inflow.
However, it was soon realized that this model does not work well for HLX-1 [@lasota11]. The observed X-ray luminosity of $L_{\rm X} \approx 2 - 4 \times 10^{40}\;\!{\rm erg\;\!s}^{-1}$ during the low/hard state [@servillat11; @yan15] requires an accretion rate larger than a few $10^{20}\;\!{\rm g\;\!s}^{-1}$ in the hot, radiatively inefficient inner region, and hence $\dot{M}(R_{\rm tr}) \ga$ a few $10^{20}\;\! {\rm g\;\! s}^{-1}$. This implies that starting from a low-density disk in the cold state, the transition radius at which the disk becomes hot is $R_{\rm tr} \approx 2 \times 10^{13}\;\! {\rm cm}$ [@dubus01; @lasota08a]. If the inner disk was truncated at this large radius, it would take $\sim$10$^2$ yr to refill before it extends again to $R_{\rm isco}$. The high state would then be expected to last also for hundreds of years, which is contradicted by the observed rapid recurrence times and short ourbursts. The disk instability model predicts $\dot{M}(R) \propto R^{2.65}$ in the low state [@lasota01]. If we impose a smaller truncation radius in the low state, $R_{\rm tr} \la 10^{12}\;\! {\rm cm}$, to reconcile the refill time with the observed timescales, and $\dot{M}(R_{\rm {tr}}) \approx$ a few $10^{20}$ g s$^{-1}$, we obtain $\dot{M}(R_{\rm out}) \sim 10^{24}\;\!{\rm g\;\! s}^{-1}
\sim 10^{-2} M_{\odot}\;\! {\rm yr}^{-1}$, a value completely unphysical for a donor star. Moreover, the ratio between inter-outburst and outburst luminosity in HLX-1 is $\approx 3 \times 10^{-2}$, about five to six orders of magnitude higher than the ratios observed in transient BH X-ray binaries and predicted by the disk instability model. In summary, despite certain qualitative similarities in the temporal and spectral properties of HLX-1 and BH low-mass X-ray binaries, the disk instability model fails to account for the timescales of HLX-1’s outburst duty cycle and low/hard state luminosity of $L_{\rm X} \sim 10^{40}\;\! {\rm erg\;\! s}^{-1}$.
One may explain the repeat of the first few outbursts at almost the same time interval of $\approx 1$ year as successive bursts of mass transfer from a donor star each time it passes through periastron of a very eccentric orbit [@lasota11]. Taking the binary period as 1 year, the fitted BH mass of $\sim$10$^4 M_{\odot}$ then sets the orbital semimajor axis $a \approx 3 \times 10^{14}\;\!{\rm cm}$ ($\approx 20$ au). On the other hand, the small disk size required by the fast rise and decay (radius $\la 10^{12}$ cm) constrains the distance at the periastron, and hence the orbital eccentricity. The best solutions correspond to extremely high elliptical orbits (almost reaching the parabolic limit) [@lasota11; @soria13a]. Such systems would be short-lived and/or unstable, as the entire envelope of the donor star would be completely stripped after a few periastron passages. A less extreme alternative was proposed [@miller14], with an orbital eccentricity of “only” $\approx 0.9$, where the mass transfer occurs via wind capture, resulting in a lower specific angular momentum for the inflowing gas and hence a shorter accretion timescale. This scenario still cannot avoid the donor star to be tidally stripped.
Crucially, the recurrence time has increased substantially in the latest outbursts, up to 470 days between the rise of the last two outbursts in 2013–2015 [@yan15], and it has been a couple of years since the start of the last outburst in 2015 January. The periastron-passage scenario has to account for this development. Change in the orbital period can happen in highly eccentric, extreme mass-ratio systems undergoing mass loss and/or mass redistribution. The study of [@godet14] has shown that a donor star captured by an IMBH on an orbit with $e \ga 0.9998$ and periastron distance between $\approx$2 and 3 times the tidal disruption radius could reach a minimum period of about 1 year after a few dozen orbits. The star would lose a few percent of its mass at each passage, leading to a rapid increase in the orbital period, and eventually what is left of the star becomes unbound after only 2 or 3 more orbits. This is consistent with the observations and perhaps explains why there has been no outburst since 2015. Smoothed particle hydrodynamics simulations of a similar scenario (but with different assumptions on orbital eccentricity, mass loss rates and viscosity) by [@vanderhelm16] showed that more moderate eccentricities such as $e=0.7$ or even $e=0.95$ are not sufficient to reproduce the rapid change in the recurrence timescale.
An important feature of all models based on extremely high eccentricity and tidal stripping is that the accretion disk must be small, with a maximum size comparable to the periastron distance, in agreement with the X-ray timescales rather than with the optically-measured size. That means that such models require an additional optically thick, irradiated gas region to account for the optical/UV continuum. For a similar reason, we have suggested the presence of a CB disk in the case of a compact system (Section 4.3). However, the CB disk scenario would itself have serious troubles, in the case of a highly eccentric orbit. In that situation, the apastron distance would determine the size of the gap between inner disk and the CB disk; therefore, the inner radius (hottest part) of the CB disk would have to be located at a distance $\ga 3 \times 10^{14}$ cm, if the semimajor axis of the orbit is large enough to correspond to a binary period of $\approx$ 1 yr (from Kepler’s law). At those distances, the irradiated CB disk would be too cold to produce the observed UV/optical emission. Thus, a highly eccentric orbit would require an alternative, more complicated distribution of irradiated gas closer to the BH, possibly a transient combination of accretion streams, tidal streams and outflows around the stripped donor star. The high-eccentricity, periastron-passage mass transfer scenario also implies that our chance detection of HLX-1 as a hyperluminous X-ray source is an one-off event in the system’s evolutionary history, associated with a recent capture and rapid shredding of a donor star (on a timescale of a few years). HLX-1 should now head into quiescence.
Instead, here we want to consider whether HLX-1 can be a more stable system, capable of supporting high mass transfer for a much longer timescale ([*e.g.*]{}, $\sim$ 10$^5$ yr rather than $\sim$10 yr). For this, we need to discard the assumption that the outburst recurrence timescale is approximately the same as the orbital period. We will discuss and compare two alternative physical mechanisms that may produce outburst cycles, not based on the disk-instability scenario: one for the case of a small orbital separation, and one for a larger separation (consistent with the optical disk size measured in HLX-1). As we explained in Section 4.3, we will argue that the latter one is the more plausible.
A mass transfer instability scenario for the small disk case
------------------------------------------------------------
The first scenario assumes a small accretion disk with an outer radius $\la 10^{12}\;\! {\rm cm}$. The observed timescale for the luminosity decline corresponds to the time needed for the whole disk to drain, and the outburst is due to an intermittent or variable gas inflow. Without the aid of a highly eccentric orbit, an alternative mechanism is needed for the recurrent bursts of mass transfer from the donor star. Given the strong X-ray luminosity and small orbital separation, irradiation on the donor star is expected to be strong. Studies [@phillips02] have shown that the pressure exerted by a strong irradiation could press down the surface of the star, causing a temporary loss of contact with the critical Roche surface at the inner Lagrangian point (L1). When this happens, the star loses mass instead through the outer Lagrangian point (L2), feeding a CB disk. However, moderate irradiation by X-rays could have an opposite effect: irradiative heating may lead to an increase in the donor star’s atmospheric scale height, resulting in more mass flowing into the BH’s Roche lobe through L1 [@podsiadlowski91; @kovetz88]. The importance of irradiation feedback in BH X-ray binaries (especially those with a large accretion disk, which casts a shadow on the star) has been under debate [@vilhu94; @ritter00; @ritter08], but observations have shown that the process does operate in binary systems containing a white-dwarf accretor [@mroz16]. Persistent X-ray irradiation could drive cycles of high and low mass transfer rates in low-mass X-ray binaries, with peaks reaching $10^{-5} M_{\odot}\;\!{\rm yr}^{-1}$ or higher [@harpaz94], but the characteristic duty cycle of $> 10^4\;\!{\rm yr}$ implies that this mechanism is irrelevant for the outbursts seen in HLX-1.
With the effects of radiation pressure on the donor star surface in mind, we suggest the following scenario to account for the $\sim 100\;\!{\rm day}$ HLX-1 outburst cycles. At the peak of each outburst, the irradiating X-ray pressure becomes high enough to press down the donor star’s atmosphere, making the star detach from the critical Roche surface at L1. When the disk is not fed, the outburst subside on the viscous timescale. After most of the disk is consumed, the system enters a low/hard state, with a luminosity allowing the donor star to regain contact with the critical Roche surface and resume the mass transfer process. This sequence of events explain the presence of a CB disk (required to account for the optical luminosity), presumably fed through L2 at the moment that the star becomes detached at L1, in a natural manner. This scenario works when the X-ray luminosity $L_{\rm X} \ga 0.8 L_{\rm Edd,2}$ [@phillips02] (where $L_{\rm Edd,2}$ is the Eddington luminosity of the star). For HLX-1 $L_{\rm X} \approx 10^{42}$ erg s$^{-1}$ during the outburst peak. Such luminosity is $\gg L_{\rm Edd,2} \approx 10^{38}$ erg s$^{-1}$. The question is how much accretion luminosity reaches L1 in the presence of shadowing from the outer edge of the disk. There are several possibilities for a partial irradiation of L1 even in the presence of a disk shadow [@viallet08]. For instance, a warped disk may provide the amount of irradiation needed. Accretion disks in both the high/soft state and the super-Eddington regime could launch strong winds [@munoz16; @pinto16; @ponti12; @neilsen09], which can scatter a fraction of the X-ray emission down towards L1. Photons can also be scattered towards L1 by an extended hot corona above the inner disk or near the base of a jet. Note that for this to happen, the vertical extent of the X-ray scattering region would have to be $\ga 10^{11}\; \!{\rm cm}$, for a disk with a thickness $H/R \sim$ a few $10^{-2}$. From the observed temperatures and luminosities of outer disks in the Galactic X-ray binaries, [@viallet08] estimated that about $\sim 10^{-3}$ of the accretion luminosity could be scattered by the disk wind, outflow or corona to irradiate the region near L1. The condition $L_1 \ga 0.8 L_{\rm Edd,2}$ is thus easily satisfied, leading to the detachment of the donor star from the critical Roche surface at L1. As a comparison, the Galactic source SS433 is an example of an X-ray binary where the direct emission is partially occulted or eclipsed, but a residual hard X-ray component (multi-temperature bremsstrahlung and/or Compton scattering emission) is seen even when the disk is eclipsed by the companion star. The inferred emission region for this extended component is $\sim 10^{12} - 10^{13}\;\!{\rm cm}$ [@marshall12; @kubota10; @krivosheyev09; @cherepashchuk07; @kotani96], which is as large or larger than the size of the accretion disk ($R_{\rm out} \approx 2 \times 10^{12}$ cm: @gies02). The observed X-ray luminosity of SS433 ($L_{\rm X} \approx 10^{36}$ erg s$^{-1}$) is $\sim$10$^{-3}$–10$^{-4}$ times the true luminosity [@khabibullin16; @fabrika15; @medvedev10], because only a small fraction of X-rays are scattered at all angles by the outflow while most of the radiation is emitted inside the polar-axis funnel [@atapin15; @medvedev10]. If a similar fraction of the X-ray luminosity is scattered by a wind above the disk plane in HLX-1, the pressure exerted by the irradiation flux would satisfy the condition given in [@phillips02] and cause an intermittent mass transfer.
In conclusion, we stress that the scenario described in this Section is not the most likely explanation for HLX-1, but we have presented it for completeness, as an example of an at least theoretically possible cycle of variability in a highly luminous, compact system, with a donor star alternatively feeding the inner disk and the CB disk.
An oscillating wind scenario for the large disk case
----------------------------------------------------
The second scenario considers a system with an accretion disk extending to $R_{\rm out} \ga 10^{13}\;\! {\rm cm}$. The arguments outlined in Section 5.2, about irradiation suppressing mass transfer through L1 and inducing L2 overflow [@phillips02], are also applicable here. The larger binary separation in this case is compensated by a larger cross section for photon interception provided by the larger donor star. The larger orbital separation implies that the Roche lobe (Section 3.3.2) can accommodate a more massive donor, and hence permits a higher value for $L_{\rm Edd,2}$; in principle, this makes it harder for the irradiation condition to be satisfied. However, in the large-disk scenario, the condition is irrelevant and cannot explain the outburst cycle of HLX-1. Here is the reason. The amount of material stored in a steady-state standard disk with $R_{\rm out} \sim 10^{13}\;\! {\rm cm}$ is a few $10^{30}\;\!{\rm g}$ (Equation 4), but only a few $10^{28}\;\! {\rm g}$ are accreted into the BH during each outburst (Section 4.1 and @yan15). The large outer disk practically is a reservoir that tends to smooth out mass-transfer fluctuations from the donor star on timescales shorter than the fluid viscous timescale at the outer radius (which is $\sim$10$^3$ yr). Whether or not the donor star temporarily detaches from L1 during each outburst cycle is therefore unimportant, when the amount of mass involved in each outburst duty cycle is only a small fraction of the mass stored in the disk. The main feature of the large-disk scenario is that the outburst would terminate when a small part of the disk (presumably, the innermost part) is depleted. Thus, the observed alternation of high and low states must be driven by changes in the accretion rate through the disk at $R \la 0.1 R_{\rm out}$ rather than by variations in the mass transfer through L1.
A comparison of the outburst behaviours of HLX-1 and of the Galactic BH V404 Cyg [@kimura16] shows some interesting similarities. The binary parameters of V404 Cyg suggest that the accretion disk has a radius $R_{\rm out} \approx 9 \times 10^{11}\;\! {\rm cm}$. It is believed that during the outburst a massive thermal wind is launched, depleting the accretion disk; the wind is strong enough to deplete the mass inflow [@munoz16]. As a result, the inner disk region (within a radius $R \sim 0.1 R_{\rm out}$) is no longer fed (or is fed at much lower rates) by the mass inflow from larger radii. The outburst subsides on the viscous timescale of the inner part of the disk (inside the wind depletion radius), which is much shorter than the viscous timescale of the whole disk. An estimate of only $\approx 0.1$ percent of the disk mass is accreted during an outburst [@munoz16]. The optical continuum is mostly reprocessed emission from the irradiated surface of the full accretion disk. A consequence of the massive disk outflow is the formation of a large H$\alpha$-emitting nebula. These features of V404 Cyg are qualitatively similar to what we observe in HLX-1.
Another property that V404 Cyg and HLX-1 have in common is the relatively high luminosity in the inter-outburst periods. At $L_{\rm X} \approx 10^{33}$ erg s$^{-1}$, V404 Cyg is two orders of magnitude more luminous than typical BH X-ray binaries in quiescence [@munoz16; @rana16; @armas14]. Even so, the ratio of outburst to inter-outburst luminosities is still as high as $\sim$10$^5$ in V404 Cyg, while it is only $\approx$30 in HLX-1. The anomalously high luminosity in the inter-outburst periods can be explained by the continuing presence of low-level accretion from the already detached outer disk, which is not completely depleted after the outburst and is working as a reservoir [@munoz16]. In V404 Cyg, the double-peaked H$\alpha$ emission line in the inter-outburst period is strong evidence that the outer disk survives through quiescence [@munoz16]. In this scenario, the different ratio of outburst to inter-outburst luminosities suggests that the wind-driven disruption of the inner accretion flow is not as strong in HLX-1 as it is in V404 Cyg.
If the disk wind is strong enough to play a major role in regulating the outbursts of HLX-1, a damped disk-wind instability [@begelman83; @shields86] would be inevitable. This instability occurs when accretion, at a rate $\dot{M}_{\rm a}$ in the X-ray emitting region of the disk, triggers a massive wind in a larger annulus of the disk (at $R \equiv R_{\rm w} < R_{\rm out}$), with an outflow rate $\dot{M}_{\rm w}(t) \equiv C \dot{M}_{\rm a}(t)$, with $C \ga$ a few. For this to happen, the wind has to be directly induced by irradiation. The original model put forward by [@begelman83] and [@shields86] considered a thermal (Compton) wind, but the mechanism would also work with a radiatively driven wind. In a steady state, $\dot{M}_{\rm in} = \dot{M}_{\rm a} + \dot{M}_{\rm w} = (1+C)\, \dot{M}_{\rm a}$, where $\dot{M}_{\rm in}$ is the accretion rate in the outer region of the disk beyond the wind-dominated region. A small positive perturbation $\delta \dot{M}_{\rm a}$ will cause a prompt increase $\delta \dot{M}_{\rm w} = C \times \delta \dot{M}_{\rm a}$ of the outflow rate. As a consequence, $\dot{M}_{\rm in} < \dot{M}_{\rm a} + \dot{M}_{\rm w}$, and the disk is depleted inside $R_{\rm w}$. While the wind immediately reacts to the X-ray luminosity increase, it takes a viscous diffusion timescale for $\dot{M}_{\rm a}$ to feel the effect of the depleted inflow arriving from $R_{\rm w}$ and to re-adjust to a lower value. When that happens, the wind immediately decreases, so that $\dot{M}_{\rm in} > \dot{M}_{\rm a} + \dot{M}_{\rm w}$ and the inner disk builds up again. It is the difference in the response timescales of the radiation-induced wind and of the inner-disk accretion that creates and sustains the oscillations. If $C\ga 10$, oscillations are persistent; for $1 < C \la 10$, they are damped after a few cycles [@shields86]. The amplitude of the oscillations is $\dot{M}_{\rm a,max}/\dot{M}_{\rm a,min} \approx 10$ [@shields86]. Using the standard Shakura-Sunyaev viscosity parameterization, [@shields86] found an oscillation period $P_{\rm osc} \sim 0.1$–0.4 $\tau_{\rm w}$, where $\tau_{\rm w}$ is the viscous timescale in the wind region. The period increases when the wind efficiency parameter $C$ decreases, but is only a weak function of $C$.
We suggest that this model is applicable to HLX-1, which is expected to have strong disk winds in outburst. As a comparison, Galactic low-mass X-ray binaries in the high state, with $L_{\rm X} \sim 0.1 L_{\rm Edd}$, can have a mass outflow rate in a disk wind $\sim$1–20 times the accretion rate [@diaztrigo16], which is strong enough to allow the wind instability described above to develop. However, we need to estimate whether the wind instability predicts the correct outburst timescale.
For a thermal wind, the launching radius is $R_{\rm w} \sim 0.1 R_{\rm IC}$ [see @shields86], where $R_{\rm IC}$ is the Compton radius, where the isothermal sound speed of the gas equals the escape velocity from the disk. For an irradiating source spectrum peaking at energies $\approx 1 \;\! {\rm keV} (\approx 10^7\;\! {\rm K})$, the Compton radius $R_{\rm IC} \sim 10^{11} (M/M_{\odot})\;\!{\rm cm}$. Therefore, in HLX-1, thermal winds could only be launched if the disk extended to $\sim 10^{14}$ cm, which does not appear to be the case. However, when the effect of radiation pressure is also included, strong winds can be launched from smaller radii: in particular, [@proga02] showed that UV lines can drive powerful winds from regions of the disk that have a surface temperature $\la 50,000$ K. Based on the results of our broad-band modelling, we have estimated (Section 4.2) an effective temperature $T_{\rm eff} \approx 20,000$ K at $R \approx R_{\rm out} \approx (2 \times 10^{13})/\sqrt{\cos \theta}$ cm in the high state. In the irradiation-dominated part of the disk, $T_{\rm eff} \propto R^{-1/2}$ [@dubus99]. Thus, we expect a temperature $T_{\rm eff} \approx 50,000$ K at $R \approx R_{\rm w} \approx 3 \times 10^{12}$ cm $\approx 3 \times 10^{-3} R_{\rm IC}$. (As a comparison, for X-ray binaries $R_{\rm w} \sim 5 \times 10^{-2} R_{\rm IC}$, and for AGN, $R_{\rm w} \sim 10^{-4} R_{\rm IC}$: @proga02). Assuming a Shakura-Sunyaev disk [@ss73; @frank02] with Kramers opacity and $\dot{M} \sim 10^{21}$–$10^{22}\;\!{\rm g~s}^{-1}$ (spanning the range of plausible accretion rates for HLX-1 in the low and high state), we obtain a viscous timescale $\tau \sim $ a few years at $R \sim$ a few $10^{12}$ cm. (This can be verified simply by recalling that $\tau \sim \left[\alpha \, \Omega \, (H/R)^2\right]^{-1} \approx 6$ yr for $(H/R) \approx 0.01$, $\alpha \approx 0.3$, $M \approx 2 \times 10^4 M_{\odot}$.) Therefore, the wind oscillation period $P_{\rm osc}$ can be as short as $\approx$1 year [@shields86]. The observed outburst duty cycle in HLX-1 is consistent with this scenario. The increase in the outburst recurrence period, accompanied by a reduced fluence in each cycle [@yan15], is also consistent with damping of the oscillations, as the wind multiplication factor decreases.
Line-driven winds are effective when the UV luminosity $L_{\rm UV} > L_{\rm Edd}/M_{\rm max}$ [@murray95; @proga98; @proga02], where $M_{\rm max}$ is the limit of the force multiplier, which can be as high as $\approx$2000 for low-ionization gas. In the case of HLX-1, for the 2010 high/soft state spectrum we infer an unabsorbed luminosity in the 13.6–100 eV band $L_{\rm UV} \approx 0.15 L_{\rm X} \sim$ a few $10^{-2} L_{\rm Edd}$. Therefore, we can at least say that there are enough UV photons available in the system to drive a strong wind. However, the amount of mass carried in the wind depends on the ionization parameter of the gas, and hence on the amount of shielding from direct X-ray irradiation, which would over-ionize the wind and reduce the effectiveness of line driving. We do not have any empirical constraints on the wind parameters for HLX-1, perhaps because our line-of-sight is almost face-on, not intersecting the wind. Thus, a more detailed modelling of the energetics of the wind is beyond the scope of this work. However, we do note that optical spectra of HLX-1 have shown an H$\alpha$ emission line with a luminosity of a few $10^{37}$ erg s$^{-1}$ and full width at half maximum of $\approx$400 km s$^{-1}$ [@wiersema10; @soria13b]. The line is too narrow to be directly associated with a disk wind, but it suggests at least the presence of diffuse gas around the system.
A wind-driven sinusoidal oscillation in $\dot{M}_{\rm a}$ does not necessarily result in a sinusoidal oscillation in $L_{\rm X}$, because accreting BHs have distinct accretion states, and hysteresis in their cycle of states. If $\dot{M}_{\rm a}$ exceeds a few percent of $\dot{M}_{\rm Edd}$ at the peak of the oscillation, the system will switch to the high/soft state, before returning to the low/hard state during the low part of the oscillation. The fact that the time spent in the high/soft state was getting shorter from cycle to cycle [@yan15], and no new outbursts have been detected for a couple of years (since 2015 January), may both be a consequence of the damping of the oscillation. We suggest that the system is now settled in the low/hard state, where the long-term-average accretion rate is below the state-transition threshold of a few percent of the Eddington accretion rate, and the damped oscillation in $\dot{M}_{\rm a}$ does not reach the threshold for a hard-to-soft state transition any more. For example, we suggest a scenario in which the long-term-average accretion rate is $\approx 0.05 \dot{M}_{\rm Edd} \approx 10^{21}$ g s$^{-1}$ $\approx 2 \times 10^{-5} M_{\odot}$ yr$^{-1}$ (for an efficiency $\eta \approx 0.1$), just below the threshold for the collapse of the hot, radiatively inefficient corona and the resulting hard-to-soft transition. By analogy with Galactic X-ray transients, the radiative luminosity in this state is expected to be $\approx 0.02 L_{\rm Edd} \approx 4 \times 10^{40}$ erg s$^{-1}$ (consistent with the luminosity inferred from [*Swift*]{} since the end of the last outburst), with an additional similar contribution to the accretion power in the form of kinetic power of a jet[^3]. A positive perturbation in the accretion rate would push the system over the threshold for the high/soft state at $L \sim 0.1 L_{\rm Edd}$ (briefly peaking at $L \approx 0.3 L_{\rm Edd}$ for a few days after the transition). The subsequent wind-driven oscillation of the accretion rate below $\approx 0.02 \dot{M}_{\rm Edd}$ would bring the system back to the hard state.
An accretion rate $\dot{M} \approx 10^{21}$ g s$^{-1}$ is very high but comparable to the accretion rates proposed for the Galactic source SS433 and for the most powerful ULXs in nearby galaxies [@feng11; @sutton13; @fabrika15; @bachetti16], which are thought to be super-Eddington, stellar-mass accretors. For a 10-$M_{\odot}$ BH, an accretion rate $\dot{M} \approx 10^{21}$ g s$^{-1}$ corresponds to $\dot{m} \equiv \dot{M}/\dot{M}_{\rm Edd} \approx 100$ and a luminosity $L \approx (1 + 0.6 \ln \dot{m}) \approx 4 L_{\rm Edd} \approx 5 \times 10^{39}$ erg s$^{-1}$ [@poutanen07]. In HLX-1, for a BH mass $M \approx 2 \times 10^4 M_{\odot}$ and a binary separation $a \approx 4 \times 10^{13}$ cm (Section 3.3.2), the binary period is $P \approx 10$ d. A donor star of mass $M_2 \approx 15 M_{\odot}$ is able to fill its Roche lobe at a radius $R_2 \approx 25 R_{\odot}$ (typical of a blue supergiant). In summary, this scenario does not require that we are [*observing*]{} the system during a one-off, short-duration event, such as the tidal stripping of a recently-captured star over a few orbital timescales. The average mass transfer rates can instead be sustained for $\sim$10$^5$ yr (or $\sim$10$^4$ for a red-giant donor). It still does require peculiar [*formation*]{} conditions, such as a donor star in a close orbit around an IMBH, filling its Roche lobe as it evolves off the main sequence. The likelihood of such event is hard to estimate with the constraints at hand, and not knowing what fraction of compact stellar systems contains an IMBH in their core (@caputo17 estimate the existence of $\sim$10$^6$ IMBHs with masses $\sim$10$^4 M_{\odot}$ within 100 Mpc). On the other hand, from the observational point of view, HLX-1 is a unique system within at least 100 Mpc, and we should not discount one-off events. If our oscillation scenario is correct, we predict that the X-ray luminosity of HLX-1 over the next few years will remain at $\ga$10$^{40}$ erg s$^{-1}$; instead, if short-duration tidal stripping scenarios are correct, HLX-1 should soon fade into quiescence.
Conclusions
===========
We compared the optical/UV brightness of HLX-1 during three sets of [*HST*]{} observations taken at different times during the outburst cycle. We fitted the optical/UV data together with X-ray data taken at similar phases, to quantify how the optical/UV luminosity is a function of the X-ray flux. We showed that the optical/UV emission is well modelled with at least two components: a blue component, strongly dependent on the irradiating X-ray flux, and a constant red component. The latter is consistent with an underlying old stellar population, perhaps a globular cluster or the nucleus of a satellite dwarf. The bluer optical emission is dominated by an irradiated disk with an outer radius $R_{\rm out} \sqrt{\cos \theta} \approx 2 \times 10^{13}$ cm. The high level of irradiation of the outer accretion disk disfavours models based on strongly beamed X-ray emission (as proposed by @king14 if HLX-1 were a highly super-Eddington stellar-mass accretor), or at the very least it would require a peculiar wind structure in the polar funnel, in order to scatter a significant amount of photons onto the outer disk. The data are also consistent with an additional blue component coming from a young stellar population. The residual optical emission in the 2013 observations (when both the X-ray source and the optical counterpart were at their faintest level) clearly places an upper limit to the (constant) contribution of the young stellar population: we find that a single-population young stellar cluster must have a mass $\la 10^4 M_{\odot}$ to avoid exceeding the blue luminosity in the 2013 data.
From our analysis and comparison of [*XMM-Newton*]{} and [*Swift*]{} lightcurves and spectra, we argued that the 2012 November observations were fortuitously taken right at the time of the hard-to-soft transition at the end of an outburst cycle. As a result, based on the analogy with similar transitions in Galactic X-ray transients, we argued that the X-ray luminosity at that point was $\approx (2 \pm 1)\%$ of the Eddington luminosity. Combining this constraint with the independent constraint on the inner disk size from spectral fitting ($R_{\rm in} \sqrt{\cos \theta} \approx 50,000$ km), we estimated a BH mass $\approx (2^{+2}_{-1}) \times 10^4 M_{\odot}$, with a spin parameter $a/M \ga 0.9$ for a moderately face-on viewing angle. This is in agreement with the results of [@davis11] based only on detailed spectral modelling of the disk spectrum in the high state. For these system parameters, the peak observed luminosity in the high/soft state is $L_{\rm X} \approx 0.2$–$0.4 L_{\rm Edd}$ (similar to most Galactic BH transients), and the luminosity in the low/hard state is $L_{\rm X} \approx 0.01 L_{\rm Edd}$. With a different assumption of the geometry of emission in the intermediate state (isotropic rather than disk-like), we obtain a slightly different BH mass, $M \approx (3^{+3}_{-1}) \times 10^4 M_{\odot}$, but the qualitative result is the same: the system oscillates between sub-Eddington, canonical high/soft and low/hard states.
We then discussed the size of the accretion disk, based on two apparently contradicting pieces of information. The short timescale of the X-ray rise, decay and recurrence time points to a small disk, with $R_{\rm out} \la 10^{12}$ cm, and more likely $\sim$ a few $\times 10^{11}$ cm. Instead, the reprocessed UV/blue emission requires a surface with a radius $R_{\rm out} \ga 2 \times 10^{13}$ cm. To reconcile the two observations, we presented and compared two alternative scenarios, both based on a two-zone structure for the accretion flow: an inner region ($\la 10^{12}$ cm) that drives the X-ray outburst, and an outer region ($\ga 2 \times 10^{13}$ cm) that produces most of the optical/UV luminosity. In the first scenario, the inner region is the accretion disk inside the BH Roche lobe, and the outer region is a CB disk, with a gap in between due to the orbit of the donor star. The outburst may be triggered by a mass transfer instability due to the high irradiation seen by the donor star. The star would alternately feed the accretion disk (through the Lagrangian point L1) or the CB disk (through L2) depending on the irradiation condition. In the second scenario, both X-ray and optical/UV emission come from a large accretion disk. However, only the inner region of this inflow produces the X-ray outburst cycles on a short timescale; the outer disk acts as a long-term reservoir of matter. In this scenario, the mass accretion flow towards the inner region oscillates due to the feedback effect (wind instability) of radiatively-driven outflows, which deplete the middle region of the disk and regulate the flow that goes past a radius $\sim 10^{12}$ cm. The latter scenario is similar to the current interpretation of the outburst in the Galactic BH V404 Cyg. Based on the available optical/UV and X-ray luminosity alone, it is not possible directly to rule out either scenario. The spectral energy distribution expected from an irradiated CB disk is similar to the one expected from an irradiated outer disk of similar size (in both cases, $T \propto R^{-1/2}$). The only difference would be the presence and size of a gap at $R \approx 10^{12}$ cm in the case of a CB disk, due to the orbit of the donor star. The gap would affect disk annuli with a temperature $T_{\rm eff} \approx 10^5$ K, and would produce a small decrease in the far-UV flux, a spectral band that is not directly observable. However, the two scenarios have very different implications for the size and type of donor star and for the binary period. A binary separation of $\approx$1–$2 \times 10^{12}$ cm (small-disk scenario) implies a binary period of only $\approx$2–3 hr and a very small Roche lobe for the donor star, consistent only with the size and density of an M-dwarf close to its tidal disruption limit. A large-disk scenario implies a more plausible binary period $\sim$10 d, and the secondary Roche lobe can accommodate a blue supergiant or other types of post-main-sequence stars, able to provide a higher mass transfer rate over a longer timescale. If the donor star is a blue supergiant, this would confirm the presence of a small population of young stars around the IMBH, but the data do not rule out an older donor (ascending the red giant branch). Based on these arguments, we conclude that the large-disk scenario with wind-driven oscillations in the inner region is more consistent with the data and is the most plausible explanation for HLX-1.
More generally, the wind-driven oscillation scenario provides a relatively unexplored mechanism for the generation of super-orbital X-ray variability (in the case of HLX-1, with a super-orbital period $\approx$40 times longer than the binary period), in alternative to the inner-disk/jet precession scenario. We speculate that the same mechanism may explain other examples of super-orbital periodicities in luminous X-ray binaries [@farrell06; @farrell09b] and ULXs ([*e.g.*]{}, the 115-d periodicity in NGC5408 X-1: @strohmayer09 [@foster10; @grise13]).
According to our adopted scenario, the system is now in the low/hard state at $L_{\rm X} \sim 10^{-2} L_{\rm Edd}$, only a factor of a few below the accretion rate and luminosity at which we expect a transition to the high/soft state. We suggest that the oscillations in the accretion rate have damped to the point where even at its peak, $\dot{M}$ is no longer high enough to trigger the transition. We do not have empirical evidence to test the origin of the perturbation that caused the damped oscillatory behaviour over the last decade–it could be related to the evolution of the donor star, or to a super-orbital period, or just stochastic variability. In engineering terms, the radiatively-driven wind acts as a proportional controller on $\dot{M}_{\rm a}(t)$, with a lag time of order of the viscous timescale. The setpoint is a luminosity slightly below the threshold for the hard-to-soft transition. If the gain of the proportional controller is sufficiently high, the lag time inevitably causes oscillations. In earlier cycles, when the oscillating error was high enough, the system underwent a state transition.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Chris Copperwheat, Fabien Grisé, Jifeng Liu, Michela Mapelli, Claudia Maraston, James Miller-Jones, Manfred Pakull, Megumi Shidatsu, Vlad Tudor, Yoshihiro Ueda and the anonymous referee for useful suggestions and discussions. RS acknowledges support from a Curtin University Senior Research Fellowship. He is also grateful for support, discussions and hospitality at the Strasbourg Observatory and at the Aspen Center for Physics during part of this work. LZ acknowledges funding from the ASI-INAF contract [*NuSTAR*]{} I/037/12/0. EvdH acknowledges support from NWO (grants Number 643.200.503, 639.073.803 and 614.061.608) and NOVA as well as supervision from S. Portegies Zwart. This paper benefitted from discussions at the 2016 International Space Science Institute workshop “The extreme physics of Eddington and super-Eddington accretion on to Black Holes” in Bern, Switzerland (team PIs: Diego Altamirano & Omer Blaes). The quasi-simultaneous 2012 and 2013 [*XMM-Newton*]{} and [*HST*]{} data were obtained through the joint [*XMM-Newton*]{}/[*HST*]{} time allocation program. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We also used Leo C. Stein’s “Kerr ISCO Calculator” online.
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\[lastpage\]
[^1]: E-mail: roberto.soria@curtin.edu.au (RS)
[^2]: We can build this average spectrum because we also verified that all [*Swift*]{} observations in that range of count rates ($\approx$0.012–0.05 ct s$^{-1}$) have similar colours in a hardness-intensity diagram, consistent with the disk-dominated high/soft state; see also [@yan15].
[^3]: Here, we have assumed that the low/hard state is less radiatively efficient than the high/soft state: $\eta_{\rm lh} \approx 0.05$ at $\dot{M} \approx 0.05 \dot{M}_{\rm Edd}$, with a similar amount of power going into the jet.
|
---
author:
- |
[Yanhua Zhao, Xueyi Huang[^1][^2] and Huiqiu Lin]{}\
Department of Mathematics, East China University of Science and Technology,\
Shanghai 200237, P.R. China
title: '**The maximum spectral radius of wheel-free graphs**'
---
A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of a graph of order $n$ that does not contain subgraphs of particular kind. In this paper, we study the Brualdi-Solheid-Turán type problem for wheel-free graphs, and we determine the maximum (signless Laplacian) spectral radius of a wheel-free graph of order $n$. Furthermore, we characterize the extremal graphs.
**Keywords:** Wheel-free graph; Spectral radius; Extremal graph; Quotient matrix.
**AMS Classification:** 05C50
Introduction
============
Let $G$ be an undirected simple graph with vertex set $V(G)$ and edge set $E(G)$ (denote by $e(G)=|E(G)|$). For any $v\in V(G)$, let $N_k(v)$ denote the set of vertices at distance $k$ from $v$ in $G$. In particular, the vertex subset $N(v)=N_1(v)$ is called the *neighborhood* of $v$, and $d_v=|N(v)|$ is called the *degree* of $v$. The *adjacency matrix* of $G$ is defined as $A(G)=(a_{u,v})_{u,v\in V(G)}$, where $a_{u,v}=1$ if $uv\in E(G)$, and $a_{u,v}=0$ otherwise. Let $D(G)=\mathrm{diag}(d_v:v\in V(G))$ denote the diagonal matrix of vertex degrees of $G$. Then $Q(G)=D(G)+A(G)$ is called the *signless Laplacian matrix* of $G$. The (adjacency) *spectral radius* $\rho_A(G)$ and the *signless Laplacian spectral radius* $\rho_Q(G)$ of $G$ are the largest eigenvalues of $A(G)$ and $Q(G)$, respectively. In addition, for any $n\times n$ matrix $M$ with only real eigenvalues, we always arrange its eigenvalues in a non-increasing order: $\lambda_1(M)\geq \lambda_2(M)\geq \cdots \geq \lambda_n(M)$.
For any $S,T\subseteq V(G)$ with $S\cap T=\emptyset$, let $E(S,T)$ denote the set of edges between $S$ and $T$ in $G$ (denote by $e(S,T)=|E(S,T)|$), and let $G[S]$ denote the subgraph of $G$ induced by $S$. For any $e\in E(G)$, let $G-e$ denote the graph obtained by deleting $e$ from $G$. Given two graphs $G$ and $H$, let $G\nabla H$ denote the graph obtained from the disjoint union $G\cup H$ by adding all edges between $G$ and $H$. For any nonnegative integer $k$, let $kG$ denote the disjoint union of $k$ copies of $G$. As usual, we denote by $K_n$, $P_n$, $C_n$ and $W_n=K_1\nabla C_{n-1}$ the complete graph, the path, the cycle and the wheel graph on $n$ vertices, respectively. Also, we denote by $B_k$ the book graph with $k$-pages, $F_k$ the graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex, and $K_{s,t}$ the complete bipartite graph with two parts of size $s$ and $t$.
Let $\mathcal{H}$ be a family of graphs. A graph $G$ is called $\mathcal{H}$-*free* if it does not contain any graph of $\mathcal{H}$ as a subgraph. The *Turán number* of $\mathcal{H}$, denoted by $ex(n,\mathcal{H})$, is the maximum number of edges in an $\mathcal{H}$-free graph of order $n$. Let $Ex(n,\mathcal{H})$ denote the set of $\mathcal{H}$-free graphs of order $n$ with $ex(n,\mathcal{H})$ edges. To determine $ex(n,\mathcal{H})$ and characterize the graphs in $Ex(n,\mathcal{H})$ for various kinds of $\mathcal{H}$ is a basic problem in extremal graph theory (see [@CF; @NI5; @SI] for surveys). In particular, for $\mathcal{H}=\{W_{2k}\}$, the Simonovits’s theorem (see [@SIM Theorem 1, p. 285]) implies that $ex(n,\{W_{2k}\})=ex(n,\{K_4\})=\lfloor\frac{n^2}{3}\rfloor$ and $Ex(n,\{W_{2k}\})=\{T_{n,3}\}$ for sufficiently large $n$, where $T_{n,3}$ is the complete $3$-partite graph of order $n$ with part sizes as equal as possible. In 2013, Dzido [@DZ] improved this result to $n\geq 6k-10$ for $k\geq 3$. For $\mathcal{H}=\{W_{2k+1}\}$, Dzido and Jastrzȩbski [@DJ] proved that $ex(n,\{W_5\})=\lfloor\frac{n^2}{4}+\frac{n}{2}\rfloor$, $ex(n,\{W_7\})=\lfloor\frac{n^2}{4}+\frac{n}{2}+1\rfloor$, and $ex(n,\{W_{2k+1}\})\geq \lfloor\frac{n^2}{4}+\frac{n}{2}\rfloor$ for all values of $n$ and $k$. Very recently, Yuan [@YU] established that $ex(n,\{W_{2k+1}\})=\max\{n_0n_1+\lfloor\frac{(k-1)n_0}{2}\rfloor+2:n_0+n_1=n\}$ for $k\geq 3$ and sufficiently large $n$.
In spectral graph theory, the well-known Brualdi-Solheid problem (see [@BS]) asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of the extremal graphs. Up to now, this problem has been studied for many classes of graphs, and readers are referred to [@STE] for systematic results. As the blending of the Brualdi-Solheid problem and the general Turán type problem, Nikiforov [@NI3] proposed a Brualdi-Solheid-Turán type problem:
\[prob-2\] What is the maximum spectral radius of an $\mathcal{H}$-free graph of order $n$?
In the past few decades, much attention has been paid to Problem \[prob-2\] for various families of graphs $\mathcal{H}$ such as $\mathcal{H}=\{K_s\}$ [@NI1; @WI], $\{K_{s,t}\}$ [@BG; @NI1; @NI4], $\{{B_{k+1},K_{2,l+1}}\}$ [@SS], $\{F_k\}$ [@CFTZ] $\{P_s\}$ [@NI3], $\{C_{2k+1}\}$ [@NI2], $\{C_3,C_4\}$ [@LLT], $\{C_4\}$ [@NI1; @ZW], $\{C_5,C_6\}$ [@YWZ], $\{W_5,C_6\}$ [@ZWF], $\{C_6\}$ [@ZL], $\{\cup_{i=1}^kP_{s_i}\}$ [@CLZ], $\{C_{l}:l\geq 2k+1\}$ and $\{C_{l}:l\geq 2k+2\}$ [@GH]. For more results on extremal spectral graph theory, we refer the reader to [@NI5].
Motived by the general Turán type problem, it is natural to consider the Brualdi-Solheid-Turán type problem for $\{W_\ell\}$-free graphs, where $l$ is any fixed integer. However, it seems difficult to determine the maximum spectral radius of a $\{W_\ell\}$-free graph of order $n$. In this paper, we consider a closely related problem:
\[prob-3\] What is the maximal spectral radius of a wheel-free (i.e., $\{W_\ell:\ell\geq 4\}$-free) graph of order $n$?
Let $H_n$ be defined as $$\label{eq-1}
H_n=\left\{
\begin{array}{ll}
\frac{n-1}{4}K_2 \nabla \frac{n+1}{2}K_1 & \mbox{if $n\equiv 1~\mathrm{mod}~4$}, \\
\frac{n+1}{4}K_2 \nabla \frac{n-1}{2}K_1 & \mbox{if $n\equiv 3~\mathrm{mod}~4$}, \\
\frac{n}{4}K_2 \nabla \frac{n}{2}K_1 & \mbox{if $n\equiv 0~\mathrm{mod}~4$}, \\
(\frac{n-2}{4}K_2\cup K_1) \nabla \frac{n}{2}K_1 & \mbox{if $n\equiv 2~\mathrm{mod}~4$},
\end{array}
\right.$$ and let $F$ be the complement of $C_7$ shown in Figure \[fig-1\]. Notice that both $H_n$ and $F$ are wheel-free. As an answer to Problem \[prob-3\], we prove that
\[thm-1\] Let $G$ be a wheel-free graph of order $n\geq 4$. Then $$\rho_A(G) \leq \rho_A(H_n),$$ with equality holding if and only if $G=H_n$ for $n\neq7$ and $G=H_7$ or $F$ for $n=7$.
![The graph $F$.[]{data-label="fig-1"}](fig1.pdf){width="4cm"}
Furthermore, we consider the same problem for the signless Laplacian spectral radius of wheel-free graphs. Surprisingly, the following result shows that the extremal graphs are not the same as that of Theorem \[thm-1\].
\[thm-2\] Let $G$ be a wheel-free graph of order $n \geq 4$. Then $$\rho_Q(G) \leq \rho_Q(K_2\nabla (n-2)K_1)=\frac{n+2+\sqrt{(n+2)^2-16}}{2},$$ with equality holding if and only if $G=K_2\nabla (n-2)K_1$.
Some lemmas
===========
Let $M$ be a real symmetric matrix of order $n$, and let $[n]=\{1,2,\ldots,n\}$. Given a partition $\Pi:[n]=X_1\cup X_2\cup \cdots \cup X_k$, the matrix $M$ can be written as $$M=\begin{bmatrix}
M_{1,1}&M_{1,2}&\cdots &M_{1,k}\\
M_{2,1}&M_{2,2}&\cdots &M_{2,k}\\
\vdots& \vdots& \ddots& \vdots\\
M_{k,1}&M_{k,2}&\cdots &M_{k,k}\\
\end{bmatrix}.$$ If $M_{i,j}$ has constant row sum $b_{i,j}$ for all $i,j\in\{1,2,\ldots,k\}$, then $\Pi$ is called an *equitable partition* of $M$, and the matrix $B_\Pi=(b_{i,j})_{i,j=1}^k$ is called an *equitable quotient matrix* of $M$.
(Brouwer and Haemers [@BH p. 30]; Godsil and Royle [@GR pp. 196–198].)\[quotient\] Let $M$ be a real symmetric matrix, and let $B_\Pi$ be an equitable quotient matrix of $M$. Then the eigenvalues of $B_\Pi$ are also eigenvalues of $M$. Furthermore, if $M$ is nonnegative and irreducible, then $$\lambda_1(M) = \lambda_1(B_\Pi).$$
The following result is straightforward, and one can find a short proof in [@EZ].
\[row\_sum\] Let $M$ be a real symmetric matrix with row sums $R_1,R_2,\ldots,R_n$. Let $\lambda(M)$ be an eigenvalue of $M$ with an eigenvector $x$ all of whose entries are nonnegative. Then $$\min_{1\leq i \leq n} R_i \leq \lambda(M) \leq \max_{1\leq i\leq n} R_i.$$ Moreover, if all entries of $x$ are positive, then either of the equalities holds if and only if the row sums of $M$ are all equal.
\[extremal-graph\] Let $H_n$ be the graph defined in (\[eq-1\]) with $n\geq 4$. Let $\lambda_1$ denote the largest root of $x^3 - x^2 - \frac{n^2}{4}x + \frac{n}{2}=0$. Then $$\rho_A(H_n)=\left\{
\begin{array}{ll}
\frac{n+1}{2} & \mbox{if $n\equiv 1~\mathrm{mod}~4$}, \\
\frac{n+1}{2}& \mbox{if $n\equiv 3~\mathrm{mod}~4$}, \\
\frac{\sqrt{n^2+1}+1}{2} & \mbox{if $n\equiv 0~\mathrm{mod}~4$}, \\
\lambda_1>\frac{\sqrt{n^2 - 3}+1}{2} & \mbox{if $n\equiv 2~\mathrm{mod}~4$}.
\end{array}
\right.$$
If $n\equiv 1~\mathrm{mod}~4$, we see that $A(H_n)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
1& \frac{n+1}{2}\\
\frac{n-1}{2}& 0
\end{bmatrix}.$$ Then, by Lemma \[quotient\], we have $\rho_A(H_n)=\lambda_1(B_\Pi)= \frac{n+1}{2}$. Similarly, for $n\equiv 3~\mathrm{mod}~4$ and $n\equiv 0~\mathrm{mod}~4$, we have $\rho_A(H_n)= \frac{n+1}{2}$ and $\rho_A(H_n)= \frac{\sqrt{n^2+1}+1}{2}$, respectively. For $n\equiv 2~\mathrm{mod}~4$, observe that $A(H_n)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
1 & 0 & \frac{n}{2}\\
0 & 0 & \frac{n}{2}\\
\frac{n}{2}-1& 1 & 0\\
\end{bmatrix}.$$ By a simple calculation, the characteristic polynomial of $B_\Pi$ is equal to $$\varphi(B_\Pi,x)=x^3 - x^2 - \frac{n^2}{4}x + \frac{n}{2}.$$ Since $\varphi(B_\Pi,
\frac{\sqrt{n^2 - 3}+1}{2})=\frac{n-\sqrt{n^2 - 3}-1}{2}<0$ due to $n\geq 4$, we have $\lambda_1(B_\Pi)>\frac{\sqrt{n^2 - 3}+1}{2}$, which gives that $\rho_A(H_n)=\lambda_1(B_\Pi)>\frac{\sqrt{n^2 - 3}+1}{2}$ again by Lemma \[quotient\].
\[Q-extremal-graph\] For $n\geq 3$, we have $$\rho_Q(K_2\nabla (n-2)K_1)=\frac{n+2+\sqrt{(n+2)^2-16}}{2}.$$
Notice that $Q(K_2\nabla (n-2)K_1)$ has the equitable quotient matrix $$B_\Pi=\left(\begin{array}{ccccccc}
n & n-2\\
2& 2\\
\end{array}\right).$$ Then, by Lemma \[quotient\], we have $$\rho_Q(K_2\nabla (n-2)K_1)=\lambda_1(B_\Pi)=\frac{n+2+\sqrt{(n+2)^2-16}}{2},$$ as required.
Proofs of Theorems \[thm-1\] and \[thm-2\]
==========================================
For a wheel-free graph $G$, the following two facts are obvious.
\[fact-1\] For any $v \in V(G)$, $G[N(v)]$ is a forest.
\[fact-2\] For any two distinct $u, v\in V(G)$, $G[N(u)\cap N(v)]$ is $P_3$-free. Furthermore, if $uv\in E(G)$ then $G[N(u)\cap N(v)]$ is $K_2$-free.
First we shall give the proof of Theorem \[thm-1\].
Notice that both $H_n$ and $F$ are wheel-free. By using Sagemath v9.1[^3] [@ST], we find that, for $n\leq 10$ and $n\neq 7$, $H_n$ is the unique graph attaining the maximum spectral radius among all wheel-free graphs of order $n$, and for $n=7$, there is another extremal graph $F$ (see Figure \[fig-1\]), which satisfies $\rho_A(F)=\rho_A(H_7)=4$. From now on, we take $n\geq 11$ and assume that $G$ is a wheel-free graph of order $n$ with maximum spectral radius. We assert that $G$ is connected. If not, suppose that $G_1,\ldots,G_\omega$ are the components of $G$. Then we can add $\omega-1$ edges to $G$ so that the obtained graph $G^*$ is connected and wheel-free. By using the Rayleigh quotient and the Perron-Frobenius theorem, we can deduce that $\rho_A(G^*)>\rho_A(G)$, contrary to the assumption.
For any $v\in V(G)$, we denote by $\omega_v$ the number of components in $G[N(v)]$, $\bar{d}_v=|N_2(v)|$ the number of vertices at distance $2$ from $v$, and $R_v$ the row sum of $A(G)^2$ corresponding to $v$. Notice that $R_v$ is exactly the number of walks of length $2$ originating at $v$. Thus $$\label{eq-2}
R_v=d_v+2e(G[N(v)])+e(N(v),N_2(v))$$ for any $v\in V(G)$. Take $u\in V(G)$ such that $R_u=\max_{v\in V(G)}R_v$. We have the following three claims.
\[claim-1\] $R_u\geq \frac{(n+1)^2-1}{4}$.
By assumption, we find that $\rho_A(G)\geq \rho_A(H_n)$. If $n\not\equiv 2~\mathrm{mod}~4$, from the Perron-Frobenius theorem, Lemma \[row\_sum\] and Lemma \[extremal-graph\] we immediately deduce that $$R_u\geq \rho_A^2(G)\geq \rho_A^2(H_n)\geq \frac{(n+1)^2}{4}.$$ Similarly, for $n\equiv 2~\mathrm{mod}~4$, i.e., $n=4k+2$ with $k\in \mathbb{Z}$, we have $$R_u>\frac{(\sqrt{n^2 - 3}+1)^2}{4}>\frac{(n+1)^2}{4}-1=4k^2+6k+\frac{5}{4}.$$ Since $R_u$ is an integer, we conclude that $$R_u\geq 4k^2+6k+2=\frac{(n+1)^2-1}{4}.$$ This proves Claim \[claim-1\].
\[claim-2\] $\bar{d}_u=n-1-d_u$, or equivalently, $V(G)=\{u\}\cup N(u)\cup N_2(u)$.
Notice that $e(G[N(u)])=d_u-\omega_u$ by Fact \[fact-1\]. According to (\[eq-2\]), we have $$\begin{aligned}
R_u&\leq d_u+2(d_u-\omega_u)+d_u\bar{d}_u\leq d_u(\bar{d}_u+3)-2\leq \frac{(d_u+\bar{d}_u+3)^2}{4}-2.
\end{aligned}$$ Combining this with Claim \[claim-1\], we get $$(n+1)^2<(d_u+\bar{d}_u+3)^2\leq (n+2)^2,$$ which implies that $d_u+\bar{d}_u=n-1$ because $d_u+\bar{d}_u$ is an integer.
\[claim-3\] Let $p_u$ be the number of vertex-disjoint copies of $P_3$ in $G[N(u)]$. We have $p_u\leq 1.$
By Fact \[fact-2\], each vertex (if any) of $N_2(u)$ is adjacent to at most two vertices of any $P_3$ of $G[N(u)]$. Thus, if $p_u\geq 2$, we have $$\begin{aligned}
R_u&\leq d_u +2(d_u-\omega_u)+(d_u-p_u)\bar{d}_u\\
&\leq d_u +2(d_u-1)+(d_u-2)\bar{d}_u\\
&=-d_u^2+(n+4)d_u-2n
\end{aligned}$$ by (\[eq-2\]) and Claim \[claim-2\]. Combining the above inequality with Claim \[claim-1\] yields that $$d_u^2-(n+4)d_u+2n+\frac{(n+1)^2-1}{4}\leq 0,$$ which is impossible because $\Delta=(n+4)^2-4(2n+\frac{(n+1)^2-1}{4})=16-2n<0$. Thus we must have $p_u\leq 1$.
By Claim \[claim-3\], it suffices to consider the following two cases.
[**Case 1.**]{} $p_u=1.$
![The graph $G(a,b)$.[]{data-label="fig-2"}](fig2.pdf){width="6cm"}
In this case, we see that $G$ has the following six properties:
1. $d_u\in\{\frac{n+1}{2},\frac{n+3}{2},\frac{n+5}{2}\}$ when $n$ is odd, and $d_u\in\{\frac{n+2}{2},\frac{n+4}{2}\}$ when $n$ is even;
2. $G[N(u)]=G(a,b)$ (see Figure \[fig-2\]), where $a,b\geq 0$ and $a+2b+1=d_u\geq 6$;
3. $G[N_2(u)]=cK_2\cup dK_1$, where $c,d\geq 0$ and $2c+d=\bar{d}_u$;
4. $e(N(u),N_2(u))=d_u\bar{d}_u-\bar{d}_u$ or $d_u\bar{d}_u-\bar{d}_u-1$;
5. for any $v\in N_2(u)$ and $P_3\subseteq G[N(u)]$, we have $|N(v)\cap V(P_3)|=1$ or $2$, and for any fixed $P_3\subseteq G[N(u)]$, there is at most one $v\in N_2(u)$ such that $|N(v)\cap V(P_3)|=1$;
6. each vertex of $N_2(u)$ is not adjacent to $u_0$, where $u_0$ is the central vertex of $G[N(u)]=G(a,b)$ shown in Figure \[fig-2\].
For (P1), by Fact \[fact-1\], Fact \[fact-2\] and Claim \[claim-2\], we get $$\begin{aligned}
R_u&=d_u+2e(G[N(u)])+e(N(u),N_2(u))\\
&\leq d_u +2(d_u-1)+d_u\bar{d}_u-\bar{d}_u\\
&=-d_u^2+(n+3)d_u-n-1,
\end{aligned}$$ Combining this with Claim \[claim-1\], we obtain $\frac{n+3-\sqrt{5}}{2}\leq d_u\leq \frac{n+3+\sqrt{5}}{2}$. Considering that $d_u$ is an integer, we may conclude that $d_u\in\{\frac{n+1}{2},\frac{n+3}{2},\frac{n+5}{2}\}$ when $n$ is odd, and $d_u\in\{\frac{n+2}{2},\frac{n+4}{2}\}$ when $n$ is even. For (P2), if $G[N(u)]$ is disconnected, i.e., $\omega_u\geq 2$, we have $$\begin{aligned}
R_u&=d_u+2(d_u-\omega_u)+e(N(u),N_2(u))\\
&\leq d_u +2(d_u-2)+d_u\bar{d}_u-\bar{d}_u\\
&=-d_u^2+(n+3)d_u-n-3.
\end{aligned}$$ Combining this with Claim \[claim-1\] yields that $$d_u^2-(n+3)d_u+n+3+\frac{(n+1)^2-1}{4}\leq 0,$$ which is impossible because $\Delta=(n+3)^2-4(n+3+\frac{(n+1)^2-1}{4})=-3<0$. Thus $G[N(u)]$ is a tree by Fact \[fact-1\]. Furthermore, since $G[N(u)]$ has exactly one vertex-disjoint copy of $P_3$, we immediately deduce that $G[N(u)]=G(a,b)$ (see Figure \[fig-2\]), where $a,b$ are nonnegative integers such that $a+2b+1=d_u$. Notice that $d_u\geq 6$ follows from (P1) and $n\geq 11$. For (P3), it suffices to prove that $G[N_2(u)]$ is $P_3$-free. By contradiction, assume that there exists some copy of $P_3$ (say $v_1v_2v_3$) in $G[N_2(u)]$. As $p_u=1$, we also can take a copy of $P_3$ (say $u_1u_2u_3$) in $G[N(u)]$. By Fact \[fact-2\], each vertex of $N_2(u)$ is adjacent to at most two vertices of $\{u_1,u_2,u_3\}$, and there is at most one $w\in N(u)\setminus\{u_1,u_2,u_3\}$ such that $\{v_1,v_2,v_3\}\subseteq N(w)$. Thus we have $e(N(u),N_2(u))\leq d_u\bar{d}_u-\bar{d}_u-(d_u-4)$, and $$R_u\leq d_u +2(d_u-1)+d_u\bar{d}_u-\bar{d}_u-(d_u-4)=-d_u^2+(n+2)d_u-n+3.$$ by Fact \[fact-1\] and Claim \[claim-2\]. Combining this with Claim \[claim-1\] yields that $$d_u^2-(n+2)d_u+n-3+\frac{(n+1)^2-1}{4}\leq 0,$$ which is impossible because $\Delta=(n+2)^2-4(n-3+\frac{(n+1)^2-1}{4})=-2n+16<0$. This proves (P3). For (P4), if $e(N(u),N_2(u))\leq d_u\bar{d}_u-\bar{d}_u-2$, as in (P2), we also can deduce a contradiction. Thus the result follows because we have known that $e(N(u),N_2(u))\leq d_u\bar{d}_u-\bar{d}_u$ by Fact \[fact-2\]. For (P5), it is clear that $|N(v)\cap V(P_3)|\leq 2$ by Fact \[fact-2\]. Also, if $|N(v)\cap V(P_3)|=0$ or there are two vertices $v_1,v_2\in V(N_2(u))$ such that $|N(v_1)\cap V(P_3)|=|N(v_2)\cap V(P_3)|=1$, then $e(N(u),N_2(u))\leq d_u\bar{d}_u-\bar{d}_u-2$, which contradicts (P4). For (P6), suppose to the contrary that there exists some $v\in N_2(u)$ such that $u_0\in N(v)$. If $a=0$, then $b=\frac{d_u-1}{2}$, and we have $|N(v)\cap N(u)|\leq b+1=\frac{d_u+1}{2}$ by Fact \[fact-2\]. Hence, $$\begin{aligned}
e(N(u),N_2(u))&\leq d_u\bar{d}_u-(\bar{d}_u-1)-(d_u-|N(v)\cap N(u)|)\\
&\leq d_u\bar{d}_u-\bar{d}_u-\frac{1}{2}(d_u-3)\\
&<d_u\bar{d}_u-\bar{d}_u-1,
\end{aligned}$$ contrary to (P4). Similarly, if $a\geq 1$, then $|N(v)\cap N(u)|\leq b+2\leq \frac{d_u}{2}+1$ and $$e(N(u),N_2(u))\leq d_u\bar{d}_u-(\bar{d}_u-1)-(d_u-|N(v)\cap N(u)|)\leq d_u\bar{d}_u-\bar{d}_u-\frac{d_u}{2}+2.$$ Thus we have $$\begin{aligned}
R_u&\leq d_u +2(d_u-1)+d_u\bar{d}_u-\bar{d}_u-\frac{d_u}{2}+2=-d_u^2+\frac{2n+5}{2}d_u-n+1.
\end{aligned}$$ Combining this with Claim \[claim-1\], we obtain $$\begin{aligned}
d_u^2-\frac{2n+5}{2}d_u+n-1+\frac{(n+1)^2-1}{4}\leq 0,
\end{aligned}$$ which is impossible because $\Delta=(\frac{2n+5}{2})^2-4(n-1+\frac{(n+1)^2-1}{4})=\frac{41}{4}-n<0$ due to $n\geq 11$. This proves (P6).
According to (P2)–(P6), we see that $G$ must be of the form $G(a,b,c,d)$ or $G(a,b,c,d)-e$, where $G(a,b,c,d)$ is shown in Figure \[fig-3\], and $e$ is some edge between $N(u)$ and $N_2(u)$ in $G(a,b,c,d)$. Notice that $\rho_A(G(a,b,c,d))>\rho_A(G(a,b,c,d)-e)$. We consider the following three situations.
![The graph $G(a,b,c,d)$, where the thickest line represents the connection of all edges between $N(u)\setminus\{u_0\}$ and $N_2(u)$.[]{data-label="fig-3"}](fig3.pdf){width="12cm"}
[**Subcase 1.1.**]{} $b=0$.
In this situation, we have $G=G(d_u-1,0,c,d)$ or $G(d_u-1,0,c,d)-e$, which are both wheel-free. Thus we conclude that $G=G(d_u-1,0,c,d)$ by considering the fact that $G$ has the maximum spectral radius among all wheel-free graphs of order $n$. For the same reason, we assert that $G=G(d_u-1,0,\lfloor\frac{\bar{d}_u}{2}\rfloor,\bar{d}_u-2\lfloor\frac{\bar{d}_u}{2}\rfloor)=(\lfloor\frac{\bar{d}_u+2}{2}\rfloor K_2\cup (\bar{d}_u-2\lfloor\frac{\bar{d}_u}{2}\rfloor) K_1)\nabla (d_u-1)K_1$. Recall that $\bar{d}_u=n-1-d_u$. Then, by (P1), we can determine all possible forms of $G$, which are listed in Table \[tab-1\].
For $n\equiv 1~\mathrm{mod}~4$, we shall prove that $$\label{eq-3}
\left\{
\begin{aligned}
&\rho_A\Big(\Big(\frac{n-1}{4}K_2\cup K_1\Big)\nabla \frac{n-1}{2}K_1\Big)<\rho_A(H_n), \\&\rho_A\Big(\Big(\frac{n-5}{4}K_2\cup K_1\Big)\nabla \frac{n+3}{2}K_1\Big)<\rho_A(H_n).
\end{aligned}
\right.$$ Observe that $A((\frac{n-1}{4}K_2\cup K_1)\nabla \frac{n-1}{2}K_1)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
1 & 0 &\frac{n-1}{2}\\
0 & 0 & \frac{n-1}{2}\\
\frac{n-1}{2} &1 &0
\end{bmatrix},$$ of which the characteristic polynomial is equal to $$\varphi(B_\Pi,x)=x^3 - x^2 -\frac{n^2-1}{4}x + \frac{n-1}{2}.$$ As $\varphi(B_\Pi,\frac{n+1}{2})=\frac{n-1}{2}>0$, we have $\lambda_1(B_\Pi)<\frac{n+1}{2}$ or $\lambda_2(B_\Pi)>\frac{n+1}{2}$. We claim that the later case cannot occur, since otherwise we have $\lambda_3(B_\Pi)<-n$ by considering the trace of $B_\Pi$, which is impossible because $\varphi(B_\Pi,-n)=- \frac{3}{4}n^3 - n^2 +
\frac{1}{4}n - \frac{1}{2}<0$. It follows that $\rho_A((\frac{n-1}{4}K_2\cup K_1)\nabla \frac{n-1}{2}K_1)=\lambda_1(B_\Pi)<\frac{n+1}{2}=\rho_A(H_n)$ by Lemma \[extremal-graph\]. Similarly, one can verify that $\rho_A((\frac{n-5}{4}K_2\cup K_1)\nabla \frac{n+3}{2}K_1)<\rho_A(H_n)$. Thus (\[eq-3\]) holds, and $G=H_n$ by the maximality of $\rho_A(G)$. For $n\not\equiv 1~\mathrm{mod}~4$, by using a similar method, we find that $H_n$ always has the maximum spectral radius. Therefore, we conclude that $G=H_n$ in this situation.
\[tab-1\]
[**Subcase 1.2.**]{} $b=1$.
In this situation, we have $G=G(d_u-3,1,c,d)$ or $G(d_u-3,1,c,d)-e$, where $e$ is some edge between $N(u)$ and $N_2(u)$ in $G(d_u-3,1,c,d)$. Because $G$ is wheel-free, we must have $c\leq 1$.
If $c=0$, since both $G(d_u-3,1,c,d)$ and $G(d_u-3,1,c,d)-e$ are wheel-free, we conclude that $G=G(d_u-3,1,0,\bar{d}_u)=G(d_u-3,1,0,n-1-d_u)$ by the maximality of $\rho_A(G)$. Observe that $A(G)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
0 &1& d_u-3 &1 &1& 0\\
1 &0 &d_u-3 &1 &0 &0\\
1 &1& 0 &0& 0& n-1-d_u\\
1 &1 &0 &0 &1 &n-1-d_u\\
1 &0 &0& 1& 0 &n-1-d_u\\
0 &0 &d_u-3 &1 &1 &0
\end{bmatrix}
\begin{array}{l}
\{u\}\\
\{u_0\}\\
\{u_1,\ldots,u_{d_u-3}\}\\
\{v_1\}\\
\{w_1\}\\
\{z_1,\ldots,z_{n-1-d_u}\}
\end{array}$$ By a simple computation, the characteristic polynomial of $B_\Pi$ is equal to $$\begin{aligned}
\varphi(B_\Pi,x,d_u)&=x^6 + (d_u^2 - (n +2)d_u + n)x^4 + (4 - 2n)x^3 \\
&~~~- (3d_u^2 - (3n + 6)d_u + 6n + 3)x^2+ (2n - 8)x\\
&~~~+ d_u^2 - (n + 2)d_u + 3n - 3.\\
\end{aligned}$$ Notice that $$\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u\Big)=\alpha(n)\cdot d_u^2 + \beta(n)\cdot d_u +\gamma(n),$$ where $$\left\{
\begin{aligned}
\alpha(n)&=\frac{1}{16}n^4+ \frac{1}{8}n^3 - \frac{21}{32}n^2 - \frac{21}{32}n + \frac{209}{256},\\
\beta(n)&= - \frac{1}{16}n^5 - \frac{1}{4}n^4 + \frac{13}{32}n^3 + \frac{65}{32}n^2 + \frac{159}{256}n - \frac{209}{128},\\
\gamma(n)&=\frac{1}{64}n^6 + \frac{7}{64}n^5 - \frac{17}{256}n^4 - \frac{159}{128}n^3 - \frac{657}{1024}n^2 - \frac{1305}{1024}n - \frac{20991}{4096}.
\end{aligned}
\right.$$ Since $\alpha(n)>0$ due to $n\geq 11$, and $-\beta(n)/(2\alpha(n))=\frac{n+2}{2}$, we obtain $$\begin{aligned}
\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u\Big)&\geq \varphi\Big(B_\Pi,\frac{2n+1}{4},\frac{n+2}{2}\Big)\\
&=\frac{1}{64}n^5 - \frac{23}{256}n^4 - \frac{17}{32}n^3 + \frac{271}{512}n^2 - \frac{1405}{1024}n - \frac{24335}{4096}\\
&>0,
\end{aligned}$$ where the last inequality follows from $n\geq 11$. Thus we have $\lambda_1(B_\Pi)<\frac{2n+1}{4}$ or $\lambda_2(B_\Pi)>\frac{2n+1}{4}$. We shall prove that the later case cannot occur. Let $D=\mathrm{diag}(1,1,d_u-3,1,1,n-1-d_u)$. Then $$\begin{aligned}
\tilde{B}_\Pi&=D^{\frac{1}{2}}B_\Pi D^{-\frac{1}{2}}\\
&=
\left[\begin{smallmatrix}
0 &1 & \sqrt{d_u-3} & 1 & 1 & 0\\
1 &0 & \sqrt{d_u-3} & 1 & 0 & 0\\
\sqrt{d_u-3}& \sqrt{d_u-3} & 0 & 0 & 0 &\sqrt{(d_u-3)(n-1-d_u)}\\
1 &1 & 0 & 0 & 1 & \sqrt{n-1-d_u}\\
1 &0 & 0 & 1 & 0 & \sqrt{n-1-d_u}\\
0 &0& \sqrt{(d_u-3)(n-1-d_u)} &\sqrt{n-1-d_u}& \sqrt{n-1-d_u} & 0\\
\end{smallmatrix}\right]
\end{aligned}$$ is symmetric, and has the same eigenvalues as $B_\Pi$. Let $\tilde{B}_\Pi'$ be the matrix obtained by deleting the third row and column from $\tilde{B}_\Pi$. By the Cauchy interlacing theorem and Lemma \[row\_sum\], we have $$\begin{aligned}
\lambda_2(B_\Pi)=\lambda_2(\tilde{B}_\Pi)&\leq\lambda_1(\tilde{B}_\Pi')\\
&\leq \max\left\{\sqrt{n-1-d_u}+3, 2\sqrt{n-1-d_u}\right\}\\
&\leq \frac{2n+1}{4}
\end{aligned}$$ because $d_u\in \{\frac{n+1}{2},\frac{n+2}{2},\frac{n+3}{2},\frac{n+4}{2},\frac{n+5}{2}\}$ by (P1) and $n\geq 11$, as required. Therefore, we conclude that $\rho_A(G)=\lambda_1(B_\Pi)<\frac{2n+1}{4}<\frac{\sqrt{n^2-3}+1}{2}<\rho_A(H_n)$ by Lemma \[extremal-graph\], contrary to our assumption.
If $c=1$, since $G$ is wheel-free, we must have $G=G(d_u-3,1,1,\bar{d}_u-2)-e=G(d_u-3,1,1,n-3-d_u)-e$, where $e$ is an edge between $\{v_1,w_1\}$ and $\{x_1,y_1\}$ in $G(d_u-3,1,1,n-3-d_u)$. By symmetry, we may assume that $e=v_1x_1$ or $w_1y_1$. If $e=v_1x_1$, we see that $A(G)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
0 &1& d_u-3 &2& 0\\
1 &0 &d_u-3 &1 &0 \\
2 &2& 0 &0& n-3-d_u\\
2 &1 &0 &1 &n-3-d_u\\
0 &0 &d_u-3& 2& 0 \\
\end{bmatrix}
\begin{array}{l}
\{u,y_1\}\\
\{u_0,x_1\}\\
\{u_1,\ldots,u_{d_u-3}\}\\
\{v_1,w_1\}\\
\{z_1,\ldots,z_{n-3-d_u}\}
\end{array}$$ As above, we can deduce that $\lambda_1(B_\Pi)<\frac{2n+1}{4}$ or $\lambda_2(B_\Pi)>\frac{2n+1}{4}$. Again, we claim that the later case cannot occur. In fact, if $n=11$, we can directly verify the result because we have known that $d_u\in \{6,7,8\}$ by (P1). For $n\geq 12$, let $$\begin{aligned}
\tilde{B}_\Pi&=D^{\frac{1}{2}}B_\Pi D^{-\frac{1}{2}}\\
&=
\left[\begin{smallmatrix}
0 &1 & \sqrt{2(d_u-3)} & 2 & 0\\
1 &0& \sqrt{2(d_u-3)} & 1 & 0\\
\sqrt{2(d_u-3)} &\sqrt{2(d_u-3)} & 0 & 0 & \sqrt{(d_u-3)(n-3-d_u)}\\
2 &1 & 0 & 1 & \sqrt{2(n-3-d_u)}\\
0 &0 & \sqrt{(d_u-3)(n-3-d_u)} & \sqrt{2(n-3-d_u)} & 0\\
\end{smallmatrix}\right],
\end{aligned}$$ where $D=\mathrm{diag}(2,2,d_u-3,2,n-3-d_u)$, and let $\tilde{B}_\Pi'$ denote the matrix obtained by deleting the third row and column from $\tilde{B}_\Pi$. Then we have $$\begin{aligned}
\lambda_2(B_\Pi)=\lambda_2(\tilde{B}_\Pi)&\leq\lambda_1(\tilde{B}_\Pi')\\
&\leq \sqrt{2(n-3-d_u)}+4\\
&\leq \frac{2n+1}{4}
\end{aligned}$$ because $d_u\in \{\frac{n+1}{2},\frac{n+2}{2},\frac{n+3}{2},\frac{n+4}{2},\frac{n+5}{2}\}$ and $n\geq 12$, as required. Thus $\rho_A(G)=\lambda_1(B_\Pi)<\frac{2n+1}{4}<\frac{\sqrt{n^2-3}+1}{2}<\rho_A(H_n)$ by Lemma \[extremal-graph\], a contradiction. If $e=w_1y_1$, then $A(G)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
0 &1& d_u-3 &1& 1&0\\
1 &0 &d_u-3 &1 &0 &0 \\
2 &2& 0 &0& 0& n-3-d_u\\
2 &2 &0 &0& 1 &n-3-d_u\\
2 &0 &0 &1 &0 &n-3-d_u\\
0 &0 &d_u-3& 1& 1 &0 \\
\end{bmatrix}
\begin{array}{l}
\{u,x_1\}\\
\{u_0,y_1\}\\
\{u_1,\ldots,u_{d_u-3}\}\\
\{v_1\}\\
\{w_1\}\\
\{z_1,\ldots,z_{n-3-d_u}\}
\end{array}$$ By using a similar analysis as above, we can deduce that $\lambda_1(B_\Pi)<\frac{2n+1}{4}$, and so $\rho_A(G)=\lambda_1(B_\Pi)<\frac{\sqrt{n^2-3}+1}{2}<\rho_A(H_n)$, contrary to the assumption.
[**Subcase 1.3.**]{} $2\leq b \leq \frac{d_u-1}{2}$.
In this situation, we have $G=G(d_u-2b-1,b,c,d)$ or $G(d_u-2b-1,b,c,d)-e$, where $e$ is some edge between $N(u)$ and $N_2(u)$ in $G(d_u-2b-1,b,c,d)$. Since $G$ is wheel-free, we must have $c=0$. Notice that both $G=G(d_u-2b-1,b,0,d)$ and $G(d_u-2b-1,b,0,d)-e$ are wheel-free. Thus we conclude that $G=G(d_u-2b-1,b,0,d)=G(d_u-2b-1,b,0,\bar{d}_u)=G(d_u-2b-1,b,0,n-1-d_u)$.
If $b\neq \frac{d_u-1}{2}$, then $b\leq \frac{d_u-2}{2}$, and we see that $A(G)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
0 &1& d_u-2b-1 &b &b& 0\\
1 &0 &d_u-2b-1 &b &0 &0\\
1 &1& 0 &0& 0& n-1-d_u\\
1 &1 &0 &0 &1 &n-1-d_u\\
1 &0 &0& 1& 0 &n-1-d_u\\
0 &0 &d_u-2b-1 &b &b &0
\end{bmatrix}
\begin{array}{l}
\{u\}\\
\{u_0\}\\
\{u_1,\ldots,u_{d_u-2b-1}\}\\
\{v_1,\ldots, v_b\}\\
\{w_1,\ldots, w_b\}\\
\{z_1,\ldots,z_{n-1-d_u}\}
\end{array}$$ By a simple calculation, the characteristic polynomial of $B_\Pi$ is $$\begin{aligned}
\varphi(B_\Pi,x,d_u,b)&=x^6 + (d_u^2 - (n +2)d_u + n+b-1)x^4 + ((2b-2)d_u+2b+2 - 2bn)x^3 \\
&~~~- ((b+2)d_u^2 - (b^2+(b+2)n +3b+2)d_u + (b+1)(b+2)n +4b-1)x^2\\
&~~~+ ((2-2b)d_u+(2n - 6)b-2)x + d_u^2 - (n + 2b)d_u +(2b+1)(n-1).
\end{aligned}$$ We shall prove that $\lambda_1(B_\Pi)<\frac{2n+1}{4}$. It is easy to see that $\varphi\left(B_\Pi,\frac{2n+1}{4},d_u,b\right)$ can be expressed as $$\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,b\Big)=\alpha(d_u,n) \cdot b^2 +\beta(d_u,n) \cdot b +\gamma(d_u,n),$$ where $\alpha(d_u,n)=-\frac{1}{16}(2n + 1)^2(n-d_u)$, $\beta(d_u,n)$ and $\gamma(d_u,n)$ are the functions of $d_n$ and $n$. Since $\alpha(d_u,n)<0$, we must have $$\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,b\Big)\geq\min\left\{\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,2\Big),\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,\frac{d_u-2}{2}\Big)\right\}.$$ Recall that $d_u\in \{\frac{n+1}{2},\frac{n+2}{2},\frac{n+3}{2},\frac{n+4}{2},\frac{n+5}{2}\}$ by (P1). If $d_u=\frac{n+1}{2}$, by a simple computation, we obtain $$\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,2\Big)=\frac{1}{64}n^5 - \frac{19}{256}n^4 - \frac{7}{8}n^3 + \frac{195}{512}n^2 - \frac{4101}{1024}n -\frac{43067}{4096}>0$$ and $$\varphi\Big(B_\Pi,\frac{2n+1}{4},d_u,\frac{d_u-2}{2}\Big)=\frac{1}{128}n^5 - \frac{7}{256}n^4 - \frac{223}{512}n^3 - \frac{97}{256}n^2 + \frac{1005}{512}n -\frac{11361}{4096}>0$$ by considering that $n\geq 11$. For other values of $d_u$, one can verify that the above two inequalities also hold. Thus $\varphi(B_\Pi,\frac{2n+1}{4},d_u,b)>0$, which leads to $\lambda_1(B_\Pi)<\frac{2n+1}{4}$ or $\lambda_2(B_\Pi)>\frac{2n+1}{4}$. Again, the later case cannot occur. Let $$\begin{aligned}
\tilde{B}_\Pi&=D^{\frac{1}{2}}B_\Pi D^{-\frac{1}{2}}\\
&=
\left[\begin{smallmatrix}
0 &1 & \sqrt{d_u-2b-1} & \sqrt{b} & \sqrt{b}& 0\\
1 &0 & \sqrt{d_u-2b-1} & \sqrt{b} & 0 & 0\\
\sqrt{d_u-2b-1}& \sqrt{d_u-2b-1} & 0 & 0 & 0 &\sqrt{(d_u-2b-1)(n-1-d_u)}\\
\sqrt{b} &\sqrt{b} & 0 & 0 & 1 & \sqrt{b(n-1-d_u)}\\
\sqrt{b} &0 & 0 & 1 & 0 & \sqrt{b(n-1-d_u)}\\
0 &0& \sqrt{(d_u-2b-1)(n-1-d_u)} &\sqrt{b(n-1-d_u)}& \sqrt{b(n-1-d_u)} & 0\\
\end{smallmatrix}\right],
\end{aligned}$$ where $D=\mathrm{diag}(1,1,d_u-2b-1,b,b,n-1-d_u)$, and let $\tilde{B}_\Pi'$ be the matrix obtained by deleting the last row and column from $\tilde{B}_\Pi$. As above, we have $$\begin{aligned}
\lambda_2(B_\Pi)=\lambda_2(\tilde{B}_\Pi)\leq\lambda_1(\tilde{B}_\Pi')\leq \max\left\{\sqrt{d_u-2b-1}+2\sqrt{b}+1, 2\sqrt{d_u-2b-1}\right\}.
\end{aligned}$$ Notice that $$(\sqrt{d_u-2b-1}+2\sqrt{b})^2=d_u+2b-1+4\sqrt{(d_u-2b-1)b}\leq 3d_u-3.$$ Combining this with $d_u\in \{\frac{n+1}{2},\frac{n+2}{2},\frac{n+3}{2},\frac{n+4}{2},\frac{n+5}{2}\}$ and $n\geq 11$, we can deduce that $\sqrt{d_u-2b-1}+2\sqrt{b}+1\leq \frac{2n+1}{4}$. Also, one can verify that $2\sqrt{d_u-2b-1}\leq 2\sqrt{d_u-5}\leq \frac{2n+1}{4}$. Thus $\lambda_2(B_\Pi)\leq \frac{2n+1}{4}$, as required. Therefore, we have $\rho_A(G)=\lambda_1(B_\Pi)<\frac{2n+1}{4}<\frac{\sqrt{n^2-3}+1}{2}<\rho_A(H_n)$, which is impossible.
If $b=\frac{d_u-1}{2}$, then $G=G(0,\frac{d_u-1}{2},0,n-1-d_u)$, and $A(G)$ has the equitable quotient matrix $$B_\Pi=\begin{bmatrix}
0 &1& \frac{d_u-1}{2}&\frac{d_u-1}{2} & 0\\
1 &0 &\frac{d_u-1}{2} &0 &0\\
1 &1& 0 & 1& n-1-d_u\\
1 &0 &1 &0 &n-1-d_u\\
0 &0 &\frac{d_u-1}{2} &\frac{d_u-1}{2} &0
\end{bmatrix}
\begin{array}{l}
\{u\}\\
\{u_0\}\\
\{v_1,\ldots, v_{\frac{d_u-1}{2}}\}\\
\{w_1,\ldots, w_{\frac{d_u-1}{2}}\}\\
\{z_1,\ldots,z_{n-1-d_u}\}
\end{array}$$ By using a similar method, we also obtain $\rho_A(G)=\lambda_1(B_\Pi)<\frac{2n+1}{4}<\frac{\sqrt{n^2-3}+1}{2}<\rho_A(H_n)$, contrary to the assumption.
[**Case 2.**]{} $p_u=0$.
\[tab-2\]
In this case, since $G[N(u)]$ is $P_3$-free, we have $G[N(u)]=aK_2\cup bK_1$ with $a,b\geq 0$ and $2a+b=d_u$. First we shall prove that $a>0$. In fact, if $a=0$, i.e., $G[N(u)]$ is an empty graph, then $$R_u=d_u+2e(G[N(u)])+e(N(u),N_2(u))\leq d_u+d_u\bar{d}_u=-d_u^2+nd_u.$$ By Claim \[claim-1\], we deduce that $$d_u^2-nd_u+\frac{(n+1)^2-1}{4}\leq 0,$$ which is impossible because $\Delta=n^2-((n+1)^2-1)=-2n<0$. Next we claim that $G[N_2(u)]$ is also $P_3$-free. If not, since $G[N(u)]$ contains at least two vertices due to $a>0$, we have $e(N(u),N_2(u))\leq d_u\bar{d}_u-1$ by Fact \[fact-2\], and so $$\begin{aligned}
R_u&=d_u+2(d_u-\omega_u)+e(N(u),N_2(u))\\
&\leq d_u+2(d_u-\frac{d_u}{2})+d_u\bar{d}_u-1\\
&=-d_u^2+(n+1)d_u-1.
\end{aligned}$$ Combining this with Claim \[claim-1\] yields that $$d_u^2-(n+1)d_u+1+\frac{(n+1)^2-1}{4}\leq 0,$$ which is impossible because $\Delta=(n+1)^2-4(1+\frac{(n+1)^2-1}{4})=-3<0$. Hence, we can suppose $G[N_2(u)]=cK_2\cup dK_1$, where $c,d\geq 0$ and $2c+d=\bar{d}_u=n-1-d_u$. Then, again by Fact \[fact-2\], we have $e(N(u),N_2(u))\leq d_u\bar{d}_u-ac$, and so $$R_u\leq d_u+2\Big(d_u-\frac{d_u}{2}\Big)+d_u\bar{d}_u-ac=-d_u^2+(n+1)d_u-ac.$$ Combining this with Claim \[claim-1\], we obtain $$\label{eq-4}
d_u^2-(n+1)d_u+ac+\frac{(n+1)^2-1}{4}\leq 0,$$ which implies that $c=0$ because $a>0$ and $\Delta=1-4ac\geq 0$. Putting $c=0$ in (\[eq-4\]), we obtain $d_u=\frac{n}{2}$, $\frac{n+1}{2}$ or $\frac{n+2}{2}$. Furthermore, according to the above discussions, we must have $e(N(u),e(N_2(u)))=d_u\bar{d}_u$. Concluding these results, we obtain $G=(aK_2\cup bK_1)\nabla (\bar{d_u}+1)K_1=(aK_2\cup bK_1)\nabla (n-d_u)K_1$, where $a>0$ and $2a+b=d_u\in \{\frac{n}{2}, \frac{n+1}{2},\frac{n+2}{2}\}$. Notice that $(aK_2\cup bK_1)\nabla (n-d_u)K_1$ is always wheel-free. By considering the maximality of $\rho_A(G)$, we conclude that $G=(\lfloor\frac{d_u}{2}\rfloor K_2\cup (d_u-2\lfloor\frac{d_u}{2}\rfloor)K_1)\nabla (n-d_u)K_1$ with $d_u\in \{\frac{n}{2}, \frac{n+1}{2},\frac{n+2}{2}\}$. In Table \[tab-2\], we list all possible forms of $G$. If $n\equiv 1~\mathrm{mod}~4$, then $G=(\frac{n-1}{4}K_2\cup K_1)\nabla \frac{n-1}{2}K_1$. According to the analysis of Subcase 1.1, we have $\rho_A(G)=\rho_A((\frac{n-1}{4}K_2\cup K_1)\nabla \frac{n-1}{2}K_1)<\rho_A(H_n)$, contrary to our assumption. If $n\equiv 3~\mathrm{mod}~4$, then $G=H_n$, as required. For $n\equiv 0~\mathrm{mod}~4$ and $n\equiv 2~\mathrm{mod}~4$, as in Subcase 1.1, we can verify that the graph $H_n$ always has the maximum spectral radius.
We complete the proof.
Now we give the proof of Theorem \[thm-2\].
Assume that $G$ is a graph attaining the maximum signless Laplacian spectral radius among all wheel-free graphs of order $n$. As in the proof of Theorem \[thm-1\], we claim that $G$ is connected. Since $K_2 \nabla (n-2)K_1$ is wheel-free, we have $$\rho_Q(G)\geq \rho_Q(K_2 \nabla (n-2)K_1)=\frac{n+2+\sqrt{(n+2)^2-16}}{2}$$ by Lemma \[Q-extremal-graph\], which gives that $$\label{eq-5}
\rho_Q^2(G)-(n+2)\rho_Q(G)+4\geq 0.$$ Let $Q^*(G)=Q^2(G)-(n+2)Q(G)+4I_n$. Clearly, $\rho_Q^2(G)-(n+2)\rho_Q(G)+4$ is an eigenvalue of $Q^*(G)$ with an eigenvector all of whose entries are nonnegative. Let $R_v^*$ be the row sum of $Q^*(G)$ corresponding to $v\in V(G)$, and let $
R_u^*=\max_{v\in V(G)}R_v^*$. By Lemma \[row\_sum\] and (\[eq-5\]), we have $$\label{eq-6}
R_u^*\geq \rho_Q^2(G)-(n+2)\rho_Q(G)+4\geq 0.$$ On the other hand, we see that $$\label{eq-7}
\begin{aligned}
R_u^*&=2d_u^2+2[d_u+2e(G[N(u)])+e(N(u),N_2(u))]-2(n+2)d_u+4\\
&\leq 2d_u^2+2[d_u+2(d_u-\omega_u)+d_u\bar{d_u}]-2(n+2)d_u+4\\
&\leq 2d_u^2+2[d_u+2(d_u-1)+d_u(n-1-d_u)]-2(n+2)d_u+4\\
&=0,
\end{aligned}$$ where $\bar{d}_u=|N_2(u)|$ and $\omega_u$ are defined as in the proof of Theorem \[thm-1\]. Combining (\[eq-6\]) and (\[eq-7\]), we obtain $\omega_u=1$ (i.e., $G[N(u)]$ is a tree), $\bar{d}_u=n-1-d_u$ (i.e., $V(G)=\{u\}\cup N(u)\cup N_2(u)$) and $e(N(u),N_2(u))=d_u\bar{d}_u$ (that is, the edges between $N(u)$ and $N_2(u)$ form a complete bipartite graph). We consider the following two cases.
[**Case 1.**]{} $N_2(u)=\emptyset$.
In this situation, from the above arguments we obtain $G=K_1\nabla G[N(u)]$, where $G[N(u)]$ is a tree of order $n-1$. If $G(N[u])=K_{1,n-2}$, then $G=K_2\nabla (n-2)K_1$, as required. Now suppose $G(N[u])\neq K_{1,n-2}$. Let $\mathbf{x}$ be the unique unit positive eigenvector (or Perron vector) of $Q(G)$ corresponding to $\rho_Q(G)$, and let $\mathbf{x}_{v_0}=\max_{v\in N(u)} \mathbf{x}_{v}$. Let $\ell$ denote the diameter of $G[N(u)]$. Notice that $\ell\geq 3$ because $G(N[u])$ is a tree but $G(N[u])\neq K_{1,n-2}$. For $1\leq k\leq \ell$, let $N_k^*(v_0)$ denote the set of vertices at distance $k$ from $v_0$ in $G[N(u)]$. Since $G[N(u)]$ is a tree, we observe that $N_k^*(v_0)$ is an independent set of $G[N(u)]$, and each vertex of $N_k^*(v_0)$ has exactly one neighbor in $N_{k-1}^*(v_0)$. Let $G'$ be the graph obtained from $G$ by deleting those edges not incident with $v_0$ and connecting $v_0$ with all the resulting isolated vertices in $G[N(u)]$. It is clear that $G'=K_2\nabla (n-2)K_1$. Then we have $$\begin{aligned}
\rho_Q(G)&=\mathbf{x}^TQ(G)\mathbf{x}\\
&= \sum_{vw\in E(G)}(\mathbf{x}_v+\mathbf{x}_w)^2\\
&= \sum_{v\in N(u)}(\mathbf{x}_u+\mathbf{x}_v)^2+\sum_{vw\in E(G[N(u)])}(\mathbf{x}_v+\mathbf{x}_w)^2\\
&=\sum_{v\in N(u)}(\mathbf{x}_u+\mathbf{x}_v)^2+\sum_{w\in N(v_0)}(\mathbf{x}_{v_0}+\mathbf{x}_w)^2+\sum_{i=1}^{\ell-1}\sum_{v_iv_{i+1}\in E(N_i^*(v_0),N_{i+1}^*(v_0))}(\mathbf{x}_{v_i}+\mathbf{x}_{v_{i+1}})^2\\
&\leq \sum_{v\in N(u)}(\mathbf{x}_u+\mathbf{x}_v)^2+\sum_{w\in N(v_0)}(\mathbf{x}_{v_0}+\mathbf{x}_w)^2+\sum_{i=1}^{\ell-1}\sum_{v_iv_{i+1}\in E(N_i^*(v_0),N_{i+1}^*(v_0))}(\mathbf{x}_{v_0}+\mathbf{x}_{v_{i+1}})^2\\
&=\sum_{v\in N(u)}(\mathbf{x}_u+\mathbf{x}_v)^2+\sum_{w\in N(v_0)}(\mathbf{x}_{v_0}+\mathbf{x}_w)^2+\sum_{i=1}^{\ell-1}\sum_{v_{i+1}\in N_{i+1}^*(v_0)}(\mathbf{x}_{v_0}+\mathbf{x}_{v_{i+1}})^2\\
&= \sum_{v\in N(u)}(\mathbf{x}_u+\mathbf{x}_v)^2+\sum_{vw\in E(G'[N(u)])}(\mathbf{x}_v+\mathbf{x}_w)^2\\
&= \sum_{vw\in E(G')}(\mathbf{x}_v+\mathbf{x}_w)^2\\
&=\mathbf{x}^TQ(G')\mathbf{x}\\
&\leq \rho_Q(G').
\end{aligned}$$ We claim that $\rho_Q(G)<\rho_Q(G')=\rho_Q(K_2\nabla (n-2)K_1)$. In fact, if $\rho_Q(G)=\rho_Q(G')$, then from the above inequality and the Perron-Frobenius theorem we see that $\mathbf{x}$ is an eigenvector of $Q(G')$ corresponding to $\rho_Q(G')$ and $\mathbf{x}_v=\mathbf{x}_{v_0}$ for any non-pendant vertex $v$ of $G[N(u)]$. Let $v$ be a non-pendant neighbor of $v_0$ in $G[N(v)]$ (such a vertex exists because $\ell\geq 3$). Notice that $v$ is of degree $1$ in $G'[N(u)]$. By considering the eigenvalue-eigenvector equation of $\rho_Q(G')$ and $\mathbf{x}$ at $v_0$ and $v$, we have $$\left\{
\begin{aligned}
\rho_Q(G') \mathbf{x}_{v_0}&=(n-1) \mathbf{x}_{v_0}+\sum_{w\in N(u)\setminus\{v_0\}}\mathbf{x}_{w}+x_u,\\
\rho_Q(G') \mathbf{x}_{v}&=2 \mathbf{x}_{v}+\mathbf{x}_{v_0}+x_u,
\end{aligned}
\right.$$ which is impossible because $n\geq 4$, $v\in N(u)\setminus\{v_0\}$ and $\mathbf{x}_v=\mathbf{x}_{v_0}$. Therefore, in this situation, we conclude that $G=K_2\nabla (n-2)K_1$.
[**Case 2.**]{} $N_2(u)\neq \emptyset$.
In this situation, we see that $G[N(u)]$ is $P_3$-free because $G$ is wheel-free. Thus $G[N(u)]=K_1$ or $K_2$ by the above arguments. If $G[N(u)]=K_1$, then $d_u=1$. Let $v_0$ be the unique neighbor of $u$. Then $d_{v_0}=n-1$. Since $G[N(v_0)]$ must be a forest (with at least two components), according to the proof of Case 1, we conclude that $\rho_Q(G)<\rho_Q(K_2\nabla (n-2)K_1)$. If $G[N(u)]=K_2$, then $N_2(u)$ must be an independent set by Fact \[fact-2\], and so we obtain $G=K_2\nabla (n-2)K_1$.
We complete the proof.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are indebted to S.M. Cioabă, Zhiwen Wang and Zhenzhen Lou for many helpful suggestions. This research was supported by the National Natural Science Foundation of China (Nos. 11901540, 11671344 and 11771141).
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[^1]: Corresponding author.
[^2]: *Email address:* yhua030@163.com (Y. Zhao), huangxymath@163.com (X. Huang), huiqiulin@126.com (H. Lin).
[^3]: The code was uploaded to <https://github.com/XueyiHuang/Wheel-free-Graph.git>.
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---
abstract: 'Quantum gases in optical lattices offer an opportunity to experimentally realize and explore condensed matter models in a clean, tunable system. We investigate the Bose-Hubbard model on a microscopic level using single atom–single lattice site imaging; our technique enables space- and time-resolved characterization of the number statistics across the superfluid–Mott insulator quantum phase transition. Site-resolved probing of fluctuations provides us with a sensitive local thermometer, allows us to identify microscopic heterostructures of low entropy Mott domains, and enables us to measure local quantum dynamics, revealing surprisingly fast transition timescales. Our results may serve as a benchmark for theoretical studies of quantum dynamics, and may guide the engineering of low entropy phases in a lattice.'
author:
- 'Waseem S. Bakr'
- Amy Peng
- 'M. Eric Tai'
- Ruichao Ma
- Jonathan Simon
- 'Jonathon I. Gillen'
- Simon Fölling
- Lode Pollet
- Markus Greiner
bibliography:
- 'mott\_paper\_arXiv.bib'
title: Probing the Superfluid to Mott Insulator Transition at the Single Atom Level
---
Microscopic measurements can reveal properties of complex systems that are not accessible through statistical ensemble measurements. For example, scanning tunneling microscopy has allowed physicists to identify the importance of nanoscale spatial inhomogeneities in high temperature superconductivity [@lang_imaginggranular_2002], and single molecule microscopy [@moerner_single-molecule_1996] has enabled studies of local dynamics in chemical reactions revealing e.g. the importance of multiple reaction pathways [@zhuang_single-molecule_2000]. While previous ultracold quantum gas experiments have focused primarily on statistical ensemble measurements, the recently introduced single atom–-single lattice site imaging technique in a Quantum Gas Microscope (QGM) [@bakr_quantum_2009] opens the door for probing and controlling quantum gases on a microscopic level. Here we present a microscopic study of an atom-lattice system that realizes the bosonic Hubbard model and exhibits a quantum phase transition from a superfluid to a Mott insulator [@fisher_boson_1989; @jaksch_cold_1998; @greiner_quantum_2002]. In the weakly interacting superfluid regime, the many-body wavefunction factorizes into a product of states with well-defined phase on each lattice site, known as coherent states, with Poissonian number fluctuations. As the strength of the interaction increases, the number distribution is narrowed, resulting in a fixed atom number state on each site deep in the Mott insulator regime. We study this change in the number statistics across the transition; these microscopic studies are complementary to previous experiments that have focused on measuring ensemble properties such as long range phase coherence, excitation spectra or compressibility [@greiner_quantum_2002; @jordens_mott_2008; @schneider_metallic_2008]. Local properties such as onsite number statistics [@capogrosso-sansone_-site_2007] were accessible only indirectly [@greiner_collapse_2002; @gerbier_probing_2006; @jordens_mott_2008] and averaged over several shells of superfluid and Mott insulating domains in the inhomogeneous system, complicating quantitative interpretation. More recently, the shell structure was imaged through tomographic [@foelling_formation_2006], spectroscopic [@campbell_imagingmott_2006], and in-situ imaging techniques, coarse-grained over several lattice sites [@gemelke_in_2009].
![\[fig1\] Single-site imaging of atom number fluctuations across the superfluid-Mott insulator transition. ([**A**]{} – [**D**]{}) Images within each column are taken at the same final 2D lattice depth of ([**A**]{}) $6E_r$, ([**B**]{}) $10E_r$, ([**C**]{}) $12E_r$ and ([**D**]{}) $16E_r$. Top row: in-situ fluorescence images from a region of $10\times8$ lattice sites within the $n=1$ Mott shell that forms in a deep lattice. In the superfluid regime $({\bf A}, {\bf B})$, sites can be occupied with odd or even atom numbers, which appear as full or empty sites respectively in the images. In the Mott insulator, occupancies other than 1 are highly suppressed ([**D**]{}). Middle row: results of the atom detection algorithm [@som] for images in the top row. A full (empty) circle indicates the presence (absence) of an atom on a site. Bottom row: time of flight fluorescence images after 8ms expansion of the cloud in the 2D plane as a result of non-adiabatically turning off the lattice and the transverse confinement (averaged over 5 shots and binned over $5\times5$ lattice sites).](SqueezingVsDepth_Images.pdf){width="8.5cm"}
We start with a two-dimensional $^{87}$Rb Bose-Einstein condensate of a few thousand atoms confined in a single well of a standing wave, with a harmonic oscillator length of 130nm[@som]. The condensate resides $9\mu$m from an in-vacuum lens that is part of an imaging system with a resolution of $\sim600$nm. This high resolution system is used to project a square lattice potential onto the pancake cloud with a periodicity of $a=680$nm, as described in previous work [@bakr_quantum_2009]. The lattice depth is ramped exponentially with a time constant of 81ms up to a maximum depth of $16E_r$, where $E_r$ is the recoil energy of the effective lattice wavelength given by $h^2/8ma^2$, with $m$ being the mass of $^{87}$Rb and Planck’s constant $h$. In a homogeneous system in two dimensions, the transition to a Mott insulator with one atom per site occurs at a ratio of interaction energy to tunneling of $U/J=16.7$ [@kohl_superfluid_2005; @spielman_mott-insulator_2007; @capogrosso-sansone_monte_2008], corresponding to a lattice depth of $12.2E_r$. During this ramp, the initial transverse confinement of 9.5Hz is increased such that the cloud size remains approximately constant. After preparing the many-body state, we image the atoms by increasing the lattice depth several hundred-fold, and then illuminate the atoms with an optical molasses that serves to localize the atoms while fluorescence photons are collected by the high resolution optics. As a result of the imaging process, the many-body wavefunction is projected onto number states on each lattice site. In addition, light-assisted collisions immediately eject atoms in pairs from each lattice site, leaving behind an atom on a site only if its initial occupation was odd [@depue_unity_1999]. Remaining atoms scatter several thousand photons during the exposure time and can be detected with high fidelity. By preparing the sample repeatedly under the same conditions, we deduce the probability $p_{\mbox{\scriptsize \emph{odd}}}$ of having an odd number of atoms on a site before the measurement.
For a coherent state on a lattice site with mean atom number $\lambda$, $p_{\mbox{\scriptsize \emph{odd}}}$ is given by $1/2 (1-e^{-2\lambda})< 1/2$. In a Mott-insulating region in the zero temperature and zero tunneling limit, $p_{\mbox{\scriptsize \emph{odd}}}$ is 1 (0) for shells with an odd (even) atom number per site. Fig. \[fig1\] shows fluorescence images in a region of the cloud as the final depth of the lattice is increased. The initial superfluid density is chosen to obtain an insulator with two shells on the Mott side of the transition, and the region shown is in the outer shell containing one atom per site. For high filling fractions, the lattice sites in the images are barely resolved, but the known geometry of the lattice and imaging system point spread function obtained from images at sparser fillings allow reliable extraction of site occupations [@som].
![\[fig1E\] Measured value of $p_{\mbox{\scriptsize
\emph{odd}}}$ vs. the interaction to tunneling ratio ($U/J$). Data sets, with statistical error bars, are shown for regions that form part of the $n=1$ (squares) and $n=2$ (circles) Mott shells in a deep lattice. The lines are based on finite temperature Monte-Carlo simulations in a homogeneous system at constant temperature to interaction ratio ($T/U$) of 0.20 (dotted red), 0.15 (solid black) and 0.05 (dashed blue). The axis on the right is the corresponding odd-even variance given by $p_{\mbox{\scriptsize \emph{odd}}}(1- p_{\mbox{\scriptsize
\emph{odd}}})$.](SqueezingVsDepth.pdf){width="8.5cm"}
We determine $p_{\mbox{\scriptsize \emph{odd}}}$ for each site using 24 images at each final lattice depth. The transverse confining potential varies slowly compared to the lattice spacing and the system is to good approximation locally homogeneous. We make use of this to improve the error in our determination of $p_{\mbox{\scriptsize \emph{odd}}}$, by averaging over a group of lattice sites, in this case 51 (30) sites for regions in the first (second) shell (Fig. \[fig1E\]). In the $n=1$ shell, we detect an atom on a site with probability $94.9\pm 0.7\%$ at a lattice depth of $16 E_r$. We measure the lifetime of the gas in the imaging lattice and determine that $1.75\pm0.02\%$ of the occupied sites are detected as unoccupied due to atoms lost during the imaging exposure time of 1s because of background gas collisions. We correct for this effect in the given average occupation numbers and errorbars.
Measuring the defect density in the Mott insulator provides sensitive local thermometry deep in the Mott regime. Thermometry in the Mott state has been a long-standing experimental challenge [@weld_spin_2009; @trotzky_suppression_2009] and has acquired particular significance as experiments approach the regime of quantum magnetism [@duan_controlling_2003; @altman_phase_2003; @trotzky_time-resolved_2008] where the temperature scale should be on the order of the superexchange interaction energy. We directly image excitations of the $n=1$ Mott insulator, holes and doublons, as they both appear as missing atoms in the images. Similarly, for Mott insulators with higher fillings $n$, sites with excitations $(n+1, n-1)$ can be detected through their opposite parity signal. For finite tunneling rate $J$ much smaller than the interaction energy $U$, the admixture fraction of coherent hole-doublon pairs excitations is $\sim(J/U)^2$, whereas any other excitations are due to incoherent thermal fluctuations and are suppressed by a Boltzmann factor $e^{-U/T}$.
The theory curves presented in Fig. \[fig1E\] are the predicted $p_{\mbox{\scriptsize
\emph{odd}}}$ in the two shells for different values of $T/U$. The curves are obtained using a quantum Monte-Carlo “worm” algorithm [@prokofev_worm_1998; @pollet_engineering_2007], and the average temperature extracted using the data points at the three highest $U/J$ ratios is $T/U\sim0.16 \pm 0.03$. At the transition point for $n=1$, this corresponds to a temperature of 1.8nK. Assuming this value of $T/U$ to be the overall temperature, the thin layer between the Mott shells should be superfluid, and the transition to a normal gas is expected around a critical temperature of $zJ=2.8$nK, where $z$ is the number of nearest neighbors in the lattice [@gerbier_boson_2007].
Next we study the global structure of the Mott insulator. The high resolution images provide an atom-by-atom picture of the concentric shell structure, including the transition layers in between the insulating shells. In Fig. \[fig2\]A to D, the formation of the various shells, up to the fourth, is shown as the atom number in the trap is increased. Slowly varying optical potential disorder causes deviation from circular symmetry in the shells. The contour lines of the potential are directly seen in the images in Fig. \[fig2\]. In Fig. \[fig2\]E and F, we have compensated this disorder by projecting a light pattern generated using a digital micromirror device through the objective[@som], resulting in a nearly circular shell structure.
![\[fig2\] Single-site imaging of the shell structure in a Mott insulator. ([**A**]{} – [**D**]{}) The images show $p_{\mbox{\scriptsize
\emph{odd}}}$ on each site determined by averaging 20 analyzed fluorescence images. The lattice depth is $22E_r$ and the transverse confinement is 45Hz. As the atom number is increased from ([**A**]{}) to ([**D**]{}), the number of shells in the insulator increases from one to four. The value of $p_{\mbox{\scriptsize \emph{odd}}}$ for odd (even) numbered shells is close to one (zero). The atom numbers, determined by in-situ imaging of clouds expanded in the plane, are ([**A**]{}), $120\pm10$, ([**B**]{}), $460\pm20$, ([**C**]{}), $870\pm40$ and ([**D**]{}), $1350\pm70$. ([**E**]{}–[**F**]{}) Long wavelength disorder can be corrected by projecting an appropriate compensation light pattern onto the atoms, resulting in nearly circular shells. ([**E**]{}) shows $p_{\mbox{\scriptsize
\emph{odd}}}$ (average of 20 analyzed images) and ([**F**]{}) is a single shot raw image (arbitrary units).](ShellsVsDimpleVoltage.pdf){width="8.5cm"}
In a second series of experiments, we use on-site number statistics to probe the adiabaticity timescale for the transition, focusing on the local dynamics responsible for narrowing the number distribution. We start by increasing the lattice depth adiabatically to $11E_r$, still in the superfluid regime, using the same ramp described previously. Next the depth is ramped linearly to $16E_r$ where, for an adiabatic ramp, a Mott insulator should form. The ramp time is varied from 0.2ms to 20ms, and $p_{\mbox{\scriptsize \emph{odd}}}$ is measured in the first and second shells as before (Fig. \[fig3\]); we find that the data fits well to exponential curves that asymptote to the value of $p_{\mbox{\scriptsize \emph{odd}}}$ obtained in the adiabatic case. The fitted time constant in the first (second) shell is $3.5\pm0.5$ms ($3.9\pm1.3$ms).
![\[fig3\] Dynamics of on-site number statistics for a fast ramp from the superfluid regime to the Mott regime. $p_{\mbox{\scriptsize
\emph{odd}}}$ at the end of the ramp vs. ramp time is shown in the $n=1$ (squares) and $n=2$ (circles) shells, averaged over 19 datasets with statistical errorbars. Red lines are exponential fits. Inset: the two-part ramp used in this experiment. The first part is a fixed adiabatic exponential ramp ($t=81$ms) and the second is a linear ramp starting at $11E_r$ and ending at $16E_r$. The duration of the second ramp is varied in the experiment.](rampdata.pdf){width="8.5cm"}
Compared to the critical value of the tunneling time $h/J_c=68$ms for the first shell, the observed dynamics are counter-intuitively fast. This can be understood using a simple picture of two atoms in a double well. In this system, as the tunneling is varied, the minimal gap between the ground state and the first excited state is $U$, which sets the adiabaticity timescale. It is an open question whether this argument can be generalized to a lattice. In an infinite system, the appearance of Goldstone modes in the superfluid regime leads to a vanishing gap at the transition point, but the density of states is low for energies much less than $U$ [@knap_spectral_2010]. In fact, the $1/e$ timescale observed experimentally is comparable to $h/U_c=4.1$ms, where $U_c$ is the critical interaction energy for an $n=1$ insulator.
Although the local number statistics change on a fast timescale of $h/U$, entropy redistribution in the inhomogeneous potential should occur on a much slower timescale of $h/J$. Because superfluid and normal domains have a larger specific heat capacity than Mott domains, in an inhomogeneous system, entropy is expelled from the Mott domains and accumulates in the transition regions after crossing the phase transition if the system is in thermal equilibrium [@pollet_temperature_2008]. It was found, however, that in bulk Mott regions the insulating behavior makes entropy transport difficult, and global thermalization is slow on experimental timescales [@hung_slow_2010]. In our system, optical potential corrugations produce sizable potential gradients in some regions, leading to a heterostructure of almost one-dimensional Mott domains, about 1-2 lattice sites thick, surrounded by transition layers (Fig. \[fig4\]). We find remarkably low defect densities and sharp transitions between superfluid and Mott states in these regions. The measured defect probability per site in the domain shown is $0.8 \pm 0.8\%$. In these microscopic domains, each site of a Mott domain is in contact with a superfluid region. Such a configuration is likely to lead to fast thermalization, which would explain the low defect density we observe. This suggests that the lowest entropies in a Mott insulator might be obtained under conditions where the chemical potential is engineered so as to obtain alternating stripes (2D) or layers (3D) of insulating and superfluid regions [@popp_ground-state_2006; @capogrosso-sansone_monte_2008].
![\[fig4\] Low entropy Mott domains observed in a steep potential gradient. ([**A**]{}) Single shot in-situ image of a Mott insulator in a $16E_r$ deep lattice with 25Hz transverse confinement. The ring is an $n=1$ insulator enclosing an $n=2$ region. ([**B**]{}) Average $p_{\mbox{\scriptsize \emph{odd}}}$ over 24 images. Each pixel corresponds to a single lattice site. The red rectangle encloses a region containing a Mott insulator with $n=1$, a few lattice sites wide. ([**C**]{}) Column average of $p_{\mbox{\scriptsize \emph{odd}}}$ over the sites within the red rectangle in ([**B**]{}), with statistical errorbars.](sharpfeature.pdf){width="8.5cm"}
In addition to the number statistics studied in this work, single-site imaging could be applied to study spatial correlations in strongly correlated quantum gases[@kapit_even-odd_2010], and to directly measure entanglement in a quantum information context. The low defect Mott states we detect would provide an ideal starting point for quantum magnetism experiments; if the low entropy in the Mott domains can be carried over to spin models, it should be possible to realize magnetically ordered states such as antiferromagnets, which could be directly detected with single-site imaging.
Materials and Methods
=====================
Preparation of the two-dimensional condensate
---------------------------------------------
A nearly pure Bose-Einstein condensate of $5\times 10^4$ $^{87}$Rb atoms is prepared in the $F=1, m_f= -1$ state in a magnetic trap by radiofrequency (rf) evaporation. The Thomas-Fermi radii of the condensate are (3.1,3.1,27)$\mu$m. The condensate is transferred into a single well of a 1D standing wave with periodicity 9.2$\mu$m created by a beam reflected from the flat glass surface of an in-vacuum hemispheric lens. The light for this standing wave is centered at 755nm, has a 3nm spectral width and is incident at an angle of 2.3$^\circ$ relative to the surface. The condensate is loaded into the first nodal plane from the surface. The harmonic oscillator width of the condensate at full lattice depth along the direction perpendicular to the surface is 360nm. By increasing the bias field the confinement in the 2D plane is relaxed, resulting in an elliptic cloud with Thomas-Fermi radii (18, 36)$\mu$m in the 2D plane.
In order to obtain a suitable initial density for creating a Mott insulator, the atom number in the 2D plane must be reduced to a few thousand atoms in a reproducible way. For this, a red-detuned (840nm) beam with an 8$\mu$m waist is focused through the objective onto the center of the pancake and creates a “dimple” potential in the magnetically confined cloud. The magnetic confinement is then removed and the number of atoms remaining in the dimple trap is proportional to its depth, with a residual RMS fluctuation of 6%. A second collinear 840nm beam with a 27$\mu$m waist is then turned on, and the dimple is adiabatically ramped down to expand the cloud into the larger beam. The transverse confinement of the condensate provided by this beam is 9.5Hz. The 840nm light source used for creating these beams has a spectral width of 12nm. The short coherence length eliminates unwanted interferences which would corrugate the confining potential.
The interaction between the atoms is then further enhanced by increasing the axial trapping frequency by a factor of eight. This is achieved by turning on a second 1D standing wave at an angle of 14$^\circ$ to the glass surface, with a lattice spacing of 1.54$\mu$m. The maximum axial trapping frequency is 7.1kHz. The condensate resides in the sixth well from the surface. At this point, the 9.2$\mu$m standing wave is ramped down. The surface provides a reproducible way to overlap the nodes of these two standing waves. In addition, the proximity of the atoms to the glass surface enhances the resolution of the imaging system by the index of refraction of glass, resulting in a measured point spread function (PSF) with full width at half maximum of $\sim600$nm.
Preparation and imaging of the Mott insulator
---------------------------------------------
To bring the cloud into a strongly-correlated regime, a two-dimensional square lattice with 680nm periodicity is created in the plane by projecting a mask through the objective onto the atoms, as described in [@bakr_quantum_2009]. The lattice light, like the light used for producing the 1D standing waves, is centered at 755nm and has a spectral width of 3nm to reduce disorder in the potential.
The lattice depth is increased linearly to $0.4E_r$ in 50ms, and from there ramped exponentially to its final value ($16E_r$ for most experiments) with a time constant of 81ms. During the lattice ramp, the transverse confinement is increased so as to keep the size of the cloud constant, compensating for the increasing inter-atomic interaction and deconfinement due to the blue lattice. This allows for faster ramps while maintaining adiabaticity, because the density redistribution during the lattice ramp is minimized. Lattice depths are calibrated to 5% accuracy using Kapitza-Dirac scattering, and the tunneling matrix element and interaction energy at different depths are obtained from a band structure calculation. The dominating loss process in the lattice is three-body collisions in Mott shells with $n>2$, as observed in other experiments [@hung_slow_2010]. The rate for such losses is $\gamma
n(n-1)(n-2)$, with $\gamma=2\times10^{-3}$Hz for a lattice depth of $22E_r$ [@jack_signatures_2003]. Due to the relatively large lattice spacings, such losses are negligible in our lattice even for the fourth Mott shell ($<1$%).
For imaging the atoms, the same procedure as described in [@bakr_quantum_2009] is used. Briefly, the lattice depth is increased over three hundred-fold by changing the light source illuminating the mask to a monochromatic source detuned 50GHz to the blue of the D1 line. The frozen atom distribution, now projected onto number states in each well, is illuminated with a cooling molasses on the D2 line. Within the first 100$\mu$s, light assisted collisions eject atoms in pairs, leaving behind an atom only if the initial atom number on the site was odd. The remaining atoms are imaged in fluorescence by collecting the scattered molasses photons during a 1s exposure, resulting in $\sim2,000$ photons registered by the camera for each atom.
Atom numbers are measured by switching off the transverse confinement and letting the cloud expand in the 2D plane for 5ms before turning on the deep lattice used for fluorescence imaging. This ensures that the probability of two atoms being on the same site is negligible, avoiding photo-assisted losses for accurate atom number determination.
Image analysis
--------------
A sparse atom cloud image is used to extract the PSF. The geometry of the lattice is then extracted from such an image. First, the lattice spacing is obtained and then the region of interest is fitted in blocks of 10 by 10 lattice sites. The block centers are allowed to vary to extract any distortions of the lattice pattern due to imaging aberration over the field of view. A histogram of atom brightness is used to set a threshold that identifies the presence or absence of an atom on a site. The information about the PSF, lattice geometry and threshold obtained from these sparse images is then used to fit other images with much higher lattice filling, only allowing for a single global offset in the lattice phase determined by fitting atoms at the edges of the cloud.
During imaging, a small fraction of the atoms are lost due to background gas collisions. If this occurs before they scatter enough photons to surpass the detection threshold, they are not counted. The mean fraction of such uncounted atoms is $1.75 \pm 0.02\%$, determined from 15 movies (30 frames, 0.5s exposure per frame) of the atom population decay in the near-resonant lattice.
Correction of disorder in optical potentials
--------------------------------------------
Scatterers on optical surfaces produce ring-like patterns on the optical potentials used to trap the atoms.The spatial pattern of the disorder is static in time and for a lattice depth of $22E_r$, has an RMS gradient of $(0.13\pm0.01)U$ per lattice site and a characteristic length scale of 10 lattice sites. The contour lines of the potential are directly extracted from the shell structure boundaries in the Mott regime. Different contour lines are obtained by varying the atom number. The ability to observe these contour lines enables us to correct the potential by projecting an appropriate light pattern through the objective. This pattern is produced by illuminating a digital micromirror device (DLP Discovery 4100, Texas Instruments) with incoherent light of spectral width 1nm, centered at 840nm. A block of $14 \times 14$ mirrors maps onto a single lattice site in the plane of the atoms, allowing the creation of grayscale patterns, with the aperture of the objective providing Fourier filtering. An error diffusion algorithm [@liang_1.5_2009] is used to convert the desired grayscale image to a binary pattern. Potential corrections of either sign are possible by operating the micromirror device with a bias light level produced by flattening the profile of the Gaussian illumination beam.
We would like to thank G. Jotzu, E. Demler, D. Pekker, B. Wunsch, T. Kitagawa, E. Manousakis, and M. D. Lukin for stimulating discussions. This work was supported by a grant from the Army Research Office with funding from the DARPA OLE program, grants from AFOSR MURI, NSF, the Swiss National Science Foundation, and an Alfred P. Sloan Fellowship to M.G. The simulations were run on the Brutus cluster at ETH Zurich.
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---
abstract: 'Let $S$ be a set of $n$ points in $3$-dimensional space. A tetrahedralization $\mathcal{T}$ of $S$ is a set of interior disjoint tetrahedra with vertices on $S$, not containing points of $S$ in their interior, and such that their union is the convex hull of $S$. Given $\mathcal{T}$, $D_\mathcal{T}$ is defined as the graph having as vertex set the tetrahedra of $\mathcal{T}$, two of which are adjacent if they share a face. We say that $\mathcal{T}$ is Hamiltonian if $D_\mathcal{T}$ has a Hamiltonian path. Let $m$ be the number of convex hull vertices of $S$. We prove that by adding at most $\lfloor \frac{m-2}{2} \rfloor$ Steiner points to interior of the convex hull of $S$, we can obtain a point set that admits a Hamiltonian tetrahedralization. An $O(m^\frac{3}{2}) + O(n \log n)$ time algorithm to obtain these points is given. We also show that all point sets with at most $20$ convex hull points admit a Hamiltonian tetrahedralization without the addition of any Steiner points. Finally we exhibit a set of $84$ points that does not admit a Hamiltonian tetrahedralization in which all tetrahedra share a vertex.'
author:
- Francisco Escalona
- 'Ruy Fabila-Monroy'
- Jorge Urrutia
bibliography:
- 'hambib.bib'
title: Hamiltonian Tetrahedralizations with Steiner Points
---
Introduction
============
All point sets considered throughout this paper will be in general position in $\mathbb{R}^2$ and $\mathbb{R}^3$. This are point sets such that: in $\mathbb{R}^2$ not three of its elements are colinear and in $\mathbb{R}^3$ not four of its elements are coplanar.
Let $S$ be a set of $n$ points in $\mathbb{R}^3$. A tetrahedralization $\mathcal{T}$ of $S$ is a set of tetrahedra with vertices in $S$, such that :
1. Their union is the convex hull of $S$.
2. The tetrahedra only intersect at points, lines or faces.
3. The tetrahedra do not contain points of $S$ in their interior.
Given $\mathcal{T}$, we define the dual graph of $\mathcal{T}$, $D_\mathcal{T}$ to be the graph whose vertices are the elements of $\mathcal{T}$, two of which are adjacent if they share a common face.
In a similar way a triangulation of a point set of points $S$ in the plane is set of interior disjoint triangles with vertices on $S$, not containing points of $S$ and such that their union is the convex hull of $S$.
Again, the dual graph $D_\mathcal{T}$ of $\mathcal{T}$ is the graph having the elements of $\mathcal{T}$ as vertices, two of which are adjacent if they share and edge.
A Hamiltonian path (cycle) of a graph $G$ is a path (cycle) spanning all the vertices of $G$. If $D_\mathcal{T}$ contains a Hamiltonian path or cycle, we say that $\mathcal{T}$ is a Hamiltonian tetrahedralization (or triangulation, if $S$ is in the plane).
The problem of finding a Hamiltonian triangulation for a given point set in the plane, has been settled in both the existential and algorithmic sense: every set of $n$ points in the plane admits a hamiltonian triangulation and it can be computed in time $O(n\log n)$ [@skiena; @faurr].
The Hamiltonian Tetrahedralization Problem [@skiena], Problem 29 in [@open], has been a long standing problem in Computational Geometry. It is not known if every point set in general position in $\mathbb{R}^3$ admits a Hamiltonian tetrahedralization. It is conjectured in [@skiena], that the problem of finding such a tetrahedralization is $NP$-hard for arbitrary point sets.
Hamiltonian triangulations were initially studied, among other reasons, because they speed the rate at which a triangulation can be sent to a graphic processor for rendering ([@skiena; @spacetradeoffs]). A similar speed up applies for tetrahedralizations.
In this paper we study the problem of computing Hamiltonian tetrahedalizations by adding Steiner points.
We call the points in the interior of the convex hull of $S$, *interior points* and the points on the boundary of the convex hull, *exterior points* Let $m$ and $m'$ be the number of exterior and interior points of $S$ respectively; we denote the convex hull of $S$ by $Conv(S)$.
Our main result is thus:
\[teo:main\] Let $S$ be a set of $n$ points in general position with $m$ exterior points in $\mathbb{R}^3$. Then we need to add at most $\lfloor \frac{m-2}{2} \rfloor$ Steiner points to the interior of ${\mathit{Conv}}(S)$, so that the resulting point set admits a Hamiltonian tetrahedralization. Moreover this tetrahedralization can be found in time $O(m^{\frac{3}{2}})+O(n\log n)$
We note that the Steiner points are added as interior points of $S$, it is not hard to see that if this requirement is dropped a Hamiltonian tetrahedralization can be found by adding only two Steiner points.
The paper is organized as follows:
In Section \[join\] we present an algorithm that adds at most $\lfloor \frac{m-2}{2} \rfloor$ Steiner points, to the interior of ${\mathit{Conv}}(S)$. Our algorithm produces a Hamiltonian tetrahedralization. The overall complexity of the algorithm is $O(m^\frac{3}{2}) + O(n \log n)$. We consider its complexity and implementation issues in Section \[alg\]. In Section \[section:3ccp\] we study the dual graph of the convex hull of $S$. We show that all point sets with at most $20$ exterior points admit a Hamiltonian path tetrahedralization. In the same section a lower bound on the number of Steiner points our algorithm might add is given. With the same techniques we improve on the result of [@isora] and exhibit a point set of $84$ elements that does not admit a Hamiltonian pulling tetrahedralization (a pulling tetrahedralization is a tetrahedralization in which all tetrahedra share a point). Finally in Section \[section:conclusions\] a summary of results is given and new directions offered.
The algorithm {#join}
=============
In this section we sketch an algorithm that achieves Theorem \[teo:main\].
The main idea of our algorithm is to first add a point to $S$ to obtain a tetrahedralization such that its dual graph can be partitioned into cycles.
Steiner points are then inserted to join existing cycles. We continue this process until the cycle partition consists of just one cycle. This final cycle is a Hamiltonian cycle in the dual graph of the final tetrahedralization.
The first step is to remove the interior points and those exterior points of degree $3$ (that is, points adjacent to $3$ other points in the boundary of ${\mathit{Conv}}(S)$). We can do this in view of the following:
\[obsconv\] If the set of exterior points of $S$ admit a Hamiltonian tetrahedralization, so does $S$.
Consider an interior point $x$ of $S$ and suppose $S-\{x\}$ admits a Hamiltonian tetrahedralization $\mathcal{T}$. Let $\tau$ be the unique tetrahedron of $\mathcal{T}$ that contains $x$ in its interior. If we remove $\tau$ from $\mathcal{T}$ and add the four tetrahedra induced by the faces of $\tau$ with $x$, we obtain a tetrahedralization of $S$ and the Hamiltonian cycle of $D_{\mathcal{T}}$ can be extended to a Hamiltonian cycle of the new tetrahedralization. Applying this process recursively the result follows.
Assume thus that $S$ does not have interior points.
\[theo1\] Let $x$ be an exterior point of $S$ of degree $3$. If $S-\{x\}$ admits a Hamiltonian tetrahedralization, then so does $S$.
Suppose $S-\{x\}$ admits a Hamiltonian tetrahedralization $\mathcal{T}$. The three convex hull vertices of $S$ adjacent to $x$ form a face $F$ of the boundary of ${\mathit{Conv}}(S-\{x\})$. Let $\tau_1$ be the only tetrahedron of $\mathcal{T}$ that contains $F$ as a face and let $\tau_2$ be the tetrahedron induced by $x$ and $F$. Clearly $\tau_1 \cup \tau_2$ is convex. If we remove $\tau_1$ and $\tau_2$ from $\mathcal{T}$ and replace them with the three tetrahedra induced by the faces of $\tau_1$ (except $F$) and $x$, we obtain a tetrahedralization $\mathcal{T}'$ of $S$. The Hamiltonian cycle of $D_\mathcal{T}$ can now be extended to a Hamiltonian cycle of $D_{\mathcal{T}'}$.
Assume now that $S$ does not contains exterior points of degree $3$.
We insert a point $p_0$ in the interior of ${\mathit{Conv}}(S)$ and join every face of the boundary of ${\mathit{Conv}}(S)$ to $p_0$, forming a tetrahedralization $\mathcal{T}$ of $S \cup \{ p_0\} $.
Let $G$ be the graph induced by the $1$-skeleton of the boundary of ${\mathit{Conv}}(S)$, that is, the graph whose vertex set consists of the exterior points of $S$ and whose edges are the edges of the boundary of ${\mathit{Conv}}(S)$. It is easy to see that both $G$ and its dual graph are planar and $3$-connected. By construction, the dual graph of $G$ is isomorphic to $D_\mathcal{T}$. Since every face of $G$ is a triangle, $D_\mathcal{T}$ is a regular graph of degree $3$.
To obtain the initial partition, we use a theorem of Petersen [@petersen] that states that every $2$-connected cubic graph contains a perfect matching. Since $D_\mathcal{T}$ is $3$-connected, in particular it is $2$-connected and therefore contains a perfect matching $M$. If we remove the edges of $M$ from $D_\mathcal{T}$, we obtain a regular graph of degree $2$. This subgraph of $D_\mathcal{T}$ is the initial cycle partition.
\[steinerfig\]
{width="80.00000%"}
Joining cycles {#joincycles}
--------------
Consider two disjoint cycles, $C_1$ and $C_2$, in our cycle partition of $D_\mathcal{T}$, and suppose that there is an edge $e$ of $D_\mathcal{T}$ that has its end points $\tau_1$ and $\tau_2$ in $C_1$ and $C_2$ respectively. Since $\tau_1$ and $\tau_2$ are tetrahedra in $\mathcal{T}$, $e$ corresponds to a shared face $F$ of $\tau_1$ and $\tau_2$.
The join operation consists of adding a point $p$ to the interior of $\tau_1$ so that the line segment joining the point $q$ in $\tau_2$ opposite to $F$ in $\tau_2$ intersects $F$. We now remove $\tau_1$ and $\tau_2$ and replace them by the six tetrahedra induced by the faces of $\tau_1$, $\tau_2$ and $p$ (except $F$) as shown in Figure \[steinerfig\].
$C_1$ and $C_2$ are joined into a cycle passing the tetrahedra of $C_1 \cup C_2 - \{\tau_1, \tau_2\}$ plus the six new tetrahedra containing $p$ as a vertex (see Figure 2).
\[dualsteiner\]
\[\]\[\]\[1\][$C_1$]{} \[\]\[\]\[1\][$C_2$]{} \[\]\[\]\[1\][$C$]{} {width="50.00000%"}
We repeat this process until a single cycle is obtained. We will show in the next section that the number of Steiner points we need to insert before a Hamiltonian cycle is reached is at most $\lfloor \frac{m-2}{2} \rfloor$.
Complexity and implementation. {#alg}
==============================
In this section we will analyze the running time and implementation issues of the algorithm sketched in Section \[join\].
Suppose now that $S$ is a point set with $n$ points in $\mathbb{R}^3$ with $m$ exterior points and $m'$ interior points, $m+m'=n$. We first calculate the convex hull of $S$ in $O(n \log n)$, and then remove the interior points of $S$.
Next, we remove the exterior vertices of degree $3$. This can be done in $O(m)$ by using a priority queue with all exterior points of degree $3$. Each time one is removed, the degree of its neighbors is checked and if necessary they are added to the queue.
Adding the first Steiner point $p_0$ and tetrahedralizing as in the previous section takes time $O(m)$.
The complexity of finding the initial cycle partition described at the end of Section \[join\] is that of finding a perfect matching in $G$. In a graph with $|V|$ vertices and $|E|$ edges, a perfect matching can be found in time $O(|E| \sqrt{ |V|})$ [@micali]. Since we are dealing with a cubic graph, we have $|E|=\frac{3}{2} |V|$. Thus we can find the initial cycle cover in $O(\frac{3}{2} m \sqrt{m})=O(m^\frac{3}{2})$ time.
Once we have the initial cycle cover, we return the vertices that were removed. This is done before the join operations in order to take advantage of the structure of the tetrahedralization and return the exterior points of degree $3$ and interior points efficiently. Using the fact that $D_\mathcal{T}$ is a planar graph, the interior points and exterior points of degree $3$ can be added using point location at a cost of $O(\log m)$ per point. The exterior points of degree $3$ are added first and the interior points afterwards. As these points are returned, the initial cycle partition is updated as in Lemma \[obsconv\] and Theorem \[theo1\].
We have to be careful about the order in which the interior points are added. Suppose we have a tetrahedra $\tau$ which contains $k$ interior points that remain to be added, and that we return one of these points. When we retetrahedralize the point set, $\tau$ would be split into $4$ new tetrahedra. We have to guarantee that each of these tetrahedra receives a linear fraction of the points in $\tau$, for otherwise the iterative process could take as much as $O(k^2)$. That is, we need a splitter vertex (see [@avis]). Such a vertex can be found in time $O(k)$, thus ensuring a total of $O(m'\log m)$ running time.
Finally we proceed to merge the set of cycles obtained so far into a single cycle as in Subsection \[joincycles\]. Each time we join two cycles, we insert one Steiner point. Since $G$ has $m$ vertices, the number of faces of $G$ is $2m-4$, and since all the cycles obtained have at least four vertices, the initial cycle partition contains at most $\lfloor \frac{2m-4}{4} \rfloor$ elements. Thus the number of Steiner points required is at most $\lfloor \frac{m-2}{2} \rfloor$. Since there are $O(m)$ edges in $G$ the merging of the cycles can be done in time $O(m)$. The overall complexity of the algorithm is thus $O(m^\frac{3}{2}) + O(n \log n)$. From this and all previous observations we obtain Theorem \[teo:main\].
$3$-connected cubic planar graphs {#section:3ccp}
=================================
To conclude the paper we study tetrahedralizations of point sets in terms of the dual graph of the 1-skeleton of their convex hull. The $1$-skeleton of the convex hull of $S$ is the graph having as vertices the exterior points of $S$, two of which are adjacent if they are joined by an edge in the boundary of $Conv(S)$. In particular we prove that every point set of at most $20$ exterior points admits a Hamiltonian tetrahedralization. Also a set of $84$ points that does not contain a Hamiltonian pulling tetrahedralization is shown. Improving therefore on the result of [@isora]. The techniques employed, allow us to give a lower bound on the number of Steiner points our algorithm might add.
The convex hull of a point set in $\mathbb{R}^3$ is a convex polyhedron of triangular faces. It is known that the dual graph of such a polyhedron is a $3$-connected cubic planar graph ($3$CCP). The converse is also true, that is every $3$CCP graph can be realized as the dual graph of some polyhedron and therefore as the dual graph of the $1$-skeleton of the convex hull of a point set in convex position [@steinitz].
$3$CCP graphs are uniquely embeddable in the plane [@whitney]. This in particular means that faces of a $3$CPP graph are defined regardless of any particular embedding. Now, given a polyhedron $\mathcal{P}$, take any point $q$ in the interior of a face and do a stereo-graphic projection to a plane not containing $\mathcal{P}$ and such that any line segment joining a point of the plane and $q$ cuts $\mathcal{P}$ in its interior. This yields a planar embedding of the dual graph $D(\mathcal{P})$ of $\mathcal{P}$. $D(\mathcal{P})$ is the graph having as vertices the faces of $\mathcal{P}$ two of the adjacent if they share an edge. Note that in this embedding: faces of $D(\mathcal{P})$ correspond to all the faces of $\mathcal{P}$ containing a given point and all the faces of $\mathcal{P}$ containing a point correspond to faces of $D(\mathcal{P})$. Since all embeddings are essentially unique we may assume that this is always the case for any embedding of $D(\mathcal{P})$.
$3$CCP graphs were once conjectured to be Hamiltonian by Tait [@tait], until a $3$CCP non-Hamiltonian graph of $46$ vertices was found by Tutte [@tutte]. Nevertheless using an exhaustive computer search it has been shown that the smallest non-Hamiltonian $3$CCP graphs have $38$ vertices [@nonham], in other words all $3$CCP graphs of at most 36 vertices have a Hamiltonian cycle. We use this fact to show that all point sets of at most $20$ elements admit a Hamiltonian tetrahedralization.
At this point it should be stressed that we are making no assumption on the existence of interior points. Indeed if the dual graph of the convex hull of a point set contains a Hamiltonian cycle then joining all the vertices of the convex hull to an interior point would yield a tetrahedralization whose dual graph is isomorphic to the dual graph of the and thus would contain a Hamiltonian cycle.
\[eyeham\] Let $G$ be a $3$CCP Hamiltonian graph. There exists a face $F$ of $G$ so that $G-F$ contains a Hamiltonian path
Consider a planar embedding of $G$ and a Hamiltonian cycle $C$ of $G$. Define the distance of two vertices as the minimum length of the two paths joining them in $C$. Take any edge $e$ in $G$ not in $C$, joining vertices two vertices $x$ and $y$, whose distance in $C$ is minimum. The path $\Gamma$ in $C$ joining $x$ and $y$, realizing this distance together with $e$ forms a face $F$; since any other edge would join vertices at a strictly less distance. Now $C$-$\Gamma$ is a Hamiltonian path of $G-F$.
Take any vertex $p$ of $\mathcal{P}$ and consider the tetrahedralization $T_p$ formed by joining all other vertices of $\mathcal{P}$ to $p$. Such tetrahedralizations are known in the literature as “pulling” tetrahedralizations. Let $F_p$ be the corresponding face of $p$ in $D(\mathcal{P})$. $T_p$ is isomorphic to $D(\mathcal{P})-F_p$. Note that if $D(\mathcal{P})$ is Hamiltonian, Theorem \[eyeham\] implies the existence of a point $p$ such that $T_p$ is Hamiltonian.
By Euler’s formula, a $3$CCP graph on $n$ vertices has $\frac{n+4}{2}$ faces. Since all $3$CCP graphs of $36$ or less vertices are Hamiltonian we have:
\[corol\] Every point set in $\mathbb{R}^3$ in general position of at most $20$ points admits a Hamiltonian path (“pulling”) tetrahedralization.
Although it is a simple observation, Theorem \[eyeham\] serves as a bridge between $3$CCP graphs and point sets in space. For example it is known that $3$CCP graphs of at most $176$ vertices and face size at most $6$ are Hamiltonian. For point sets this implies that all sets of at most $90$ vertices and with vertices of degree at most $6$ in its convex hull admit a Hamiltonian tetrahedralizations. See [@mckay] for various similar results on $3$CCP graphs. Also a well known conjecture on $3$CCP graphs states that all bipartite $3$CCP graphs are Hamiltonian [@barnette].
Recently, point sets of $92$ with no Hamiltonian path pulling tetrahedralizations have been shown to exist [@isora]. We improve on this previous result and exhibit a set with less points ($84$) without a pulling Hamiltonian path tetrahedralizations. Our construction also enable us to find lower bounds on the number of Steiner points added by the algorithm presented in section \[join\] and \[alg\].
Blowing up vertices
-------------------
In this section we introduce an operation that will allow us to replace any vertex in a $3$CCP graph with another $3$CCP graph, so that the resulting graph is again a $3$CCP graph. Using this operation we will construct $3$CPP graphs with certain desired properties.
Let $G$ and $H$ be $3$CCP graphs and $v$ be any vertex of $H$. We may assume that $H$ is embedded in the plane so that $v$ is a vertex on the exterior face. Let $N_H(v)=\{v_1,v_2, v_3\}$ be the neighbourhood of $v$. Remove $v$ from $H$ and add a path of $4$ new vertices $(v_1',v_2',w,v_3')$. Join $v_i$ with $v_i'$; call the resulting graph $H'$, see Figure 3.
\[con3ccp\]
\[\]\[\]\[1\][$H$]{} \[\]\[\]\[1\][$H'$]{} \[\]\[\]\[1\][$v_1$]{} \[\]\[\]\[1\][$v_2$]{} \[\]\[\]\[1\][$v_3$]{} \[\]\[\]\[1\][$v_1'$]{} \[\]\[\]\[1\][$v_2'$]{} \[\]\[\]\[1\][$v_3'$]{} \[\]\[\]\[1\][$w$]{} \[\]\[\]\[1\][$v$]{} {width="80.00000%"}
Although $H'$ is no longer cubic, it can however be used to replace any vertex $u$ of $G$: Let $N_G(u)=\{u_1, u_2, u_3\}$ with its elements in a given order; Remove $u$ from $G$ and replace it with $H'$; join $v_i'$ with $u_i'$ The resulting graph $G'$ is a $3$CCP graph. By choosing an adequate order on the elements of $N_G(u)$, we can place $H$ in a particular face of $G$, see Figure 4.
\[fig:choice\]
We use this property to place a non Hamiltonian $3$CCP graph $H$ on each face of $K_4$ (which is a $3$CCP graph). Call the resulting graph $G$. The faces of $K_4$ isolate each non Hamiltonian graph. Assume that after a removal of a face of $G$ there exists a Hamiltonian path on $G$. Since there are $4$ copies of $H$ on $G$ two of them may contain an endpoint of the path, one more may contain the face that was removed. In the remaining copy of $H$ the Hamiltonian path enters and leaves $H$. This a contradiction since from this path we could derive a Hamiltonian cycle in $H$, see Figure 5.
\[fig:k4\]
{width="50.00000%"}
For $H$ in this construction we may use the smallest non Hamiltonian $3$CCP graph on $38$ vertices. After each insertion of $H$ into $K_4$, we remove a vertex and add $41$ new vertices. In total $G$ has $41 \times 4=164$ vertices. Therefore there exists a polyhedron of $\frac{164+4}{2}=84$ points in $\mathbb{R}^3$, whose dual graph is isomorphic to $G$. By the observations at the beginning of this section, there is no pulling Hamiltonian tetrahedralization of this polyhedron.
Using the same technique we can construct a $3$CCP graph that needs a linear number of disjoint cycles to cover its vertices. Take any $3$CCP graph and replace every vertex with a copy of the smallest non Hamiltonian graph $H$ of $38$ vertices. In total per vertex $41$ new vertices are added. It is easy to see that any partition of the vertices into cycles in this new graph would need at least a cycle per copy of $H$. Therefore since our algorithm adds one Steiner point per cycle, there are point sets for which our algorithm adds at least $\frac{n}{41}$ Steiner points.
This gives a lower bound on the number of Steiner points our algorithm might need to find a Hamiltonian tetrahedralization of a given point set.
Conclusions {#section:conclusions}
===========
In this paper we considered the problem of computing Hamiltonian Tetrahedralizations of point sets in $3$-space by adding Steiner points. An algorithm was detailed to do so for points sets with $n$ points and $m$ exterior points in time $O(m^\frac{3}{2}) + O(n \log n)$ . Our algorithm adds at most $\lfloor \frac{m-2}{2}\rfloor$ Steiner points.
It seems natural that there must be a compromise between number of Steiner points added and the running time of the algorithm employed to do so. A natural question would be to ask: What is the least number of points that need to be added while maintaining an efficient algorithm? We conjecture that a sublinear number of Steiner points cannot be added to obtain a Hamiltonian tetrahedralization in polynomial time.
Regarding the implementation of the algorithm, the algorithms we used as a subroutines are not straight forward to program. Simpler algorithms would also be desirable.
On the combinatorial side of the problem, it would be interesting to prove that a sublinear number of Steiner points suffice to obtain a Hamiltonian Tetrahedralization. Note that the current conjecture is that actually none are needed. This weakening of the conjecture, nevertheless is worth studying.
Finally we also showed that point sets with at most $20$ points always admit a Hamiltonian tetrahedralization. Actually this tetrahedralization is a pulling tetrahedralization. We also improved the previous upper bound of $92$ to $84$ on point sets without a Hamiltonian pulling tetrahedralization. This gap remains to be closed.
|
---
abstract: 'In this paper we study a class of nonlinearities for which a nonlocal parabolic equation with Neumann-Robin boundary conditions, for $p$-Laplacian, has finite time blow-up solutions.'
address:
- 'Department of Mathematics, University of Craiova, Craiova RO-200585, Romania'
- 'Department of Mathematics, University of Craiova, Craiova RO-200585, Romania'
author:
- 'Constantin P. Niculescu'
- Ionel Rovenţa
title: 'Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem'
---
[^1]
Introduction
============
It is a well known fact that convexity plays an important role in the different parts of mathematics, including the study of boundary value problems. The aim of our paper is to introduce a new class of generalized convex functions and to illustrate its usefulness in establishing a sufficient condition for the existence of finite time blow-up solutions for the evolutionary problem $$\left\{
\begin{array}
[c]{ccc}u_{t}-\Delta_{p}u=f(|u|)-\frac{1}{m(\Omega)}{\displaystyle\int\nolimits_{\Omega}}
f(|u|)\,dx & \text{in} & \Omega\\
& & \\
|\nabla u|^{p-2}\frac{\partial u}{\partial n}=0 & \text{on} & \partial
\Omega\,,
\end{array}
\right. \label{ft1}$$ with the initial conditions $$u(x,0)=u_{0}(x)\;\text{on}\;\Omega,\;\text{where}\;{\displaystyle\int\nolimits_{\Omega}}
u_{0}\,dx=0. \label{ft2}$$
Here $\Omega\subset\mathbb{R}^{N}$ is a bounded regular domain of class $C^{2}$, $f:[0,\infty)\mapsto\lbrack0,\infty)$ is a locally Lipschitz function, $m(\Omega)$ represents the Lebesgue measure of the domain $\Omega,$ and $\Delta_{p}=div(|\nabla u|^{p-2}\nabla u)$, for $p\geq2$, is the $p$-Laplacian operator.
The particular case where $p=2$ was recently considered by Soufi, Jazar and Monneau [@SJM], and Jazar and Kiwan [@JK] (under the assumption that $f$ is a power function of the form $f(u)=u^{\alpha},$ with $\alpha>1),$ and also by the present authors [@NR2010] (for $f$ belonging to a larger class of nonlinearities).
The problems of type (\[ft1\]) & (\[ft2\]) arise naturally in mechanics, biology and population dynamics. See [@JB], [@LAC], [@RSCC], [@YC] and [@JFMG]. For example, if we consider a couple or a mixture of two equations of the above type, the resulting problem describes the temperatures of two substances which constitute a combustible mixture, or represents a model for the behavior of densities of two diffusion biological species which interact each other.
Generalized convexity of order $\alpha$
=======================================
According to the classical Hermite-Hadamard inequality, the mean value of a continuous convex function $f:[a,b]\rightarrow\mathbb{R}$ lies between the value of $f$ at the midpoint of the interval $[a,b]$ and the arithmetic mean of the values of $f$ at the endpoints of this interval, that is, * * $$\,f\left( \frac{a+b}{2}\right) \leq\frac{1}{b-a}\,\int_{a}^{b}\,f(x)\,dx\leq\frac{f(a)+f(b)}{2}\,. \tag{$HH$}$$ Moreover, each side of this double inequality characterizes convexity in the sense that a real-valued continuous function $f$ defined on an interval $I$ is convex if its restriction to each compact subinterval $[a,b]\subset I$ verifies the left hand side of $(HH)$ (equivalently, the right hand side of $(HH))$. See [@BP] and [@NP2006] for details.
In what follows we will be interested in a class of generalized convex functions motivated by the right hand side of the Hermite-Hadamard inequality.
A real-valued function $f$ defined on an interval $[a,\infty)$ belongs to the class $GC_{\alpha}$ $($for some $\alpha>0)$, if it is continuous, nonnegative, and$$\frac{1}{\alpha+1}f(t)\geq\frac{1}{t-a}\int_{a}^{t}f(x)\,dx\text{\quad for
}t\text{ large enough.} \label{GCa}$$
Using calculus, one can see easily that the condition (\[GCa\]) is equivalent to the fact that the ratio$$\frac{\frac{1}{t-a}\int_{a}^{t}f(x)\,dx}{\left( t-a\right) ^{\alpha}}
\label{defC}$$ is nondecreasing for $t$ bigger than a suitable value $A\geq a.$ In turn, this implies that the mean value $\frac{1}{t-a}\int_{a}^{t}f(x)\,dx$ has a polynomial growth at infinity.
According to the Hermite-Hadamard inequality, every nonnegative, continuous and convex function $f:[a,\infty)\rightarrow\mathbb{R}$ with $f(a)=0$ belongs to the class $GC_{1}.$ The converse is not true because the membership of a function $f:[a,\infty)\rightarrow\mathbb{R}$ to the class $GC_{\alpha}$ yields only an asymptotic inequality of the form$$\frac{1}{\alpha+1}f(t)+\frac{\alpha}{\alpha+1}f(a)\geq\frac{1}{t-a}\int
_{a}^{t}f(x)\,dx\text{\quad for }t\text{ large enough.}$$
If $g\in C^{1}([0,\infty))$ and $g$ is nondecreasing, then the function $f(x)=g(x)(x-a)^{\alpha}$ belongs to the class $CG_{\alpha}\left(
[0,\infty)\right) ,$ whenever $\alpha>0$. In fact,$$\begin{aligned}
\frac{1}{t-a}\int_{a}^{t}f(x)dx & =\frac{(t-a)^{\alpha}}{\alpha+1}g(t)-\frac{1}{t-a}\int_{a}^{t}g^{\prime}(x)\frac{\left( x-a\right)
^{\alpha+1}}{\alpha+1}\;dx\\
& \leq\frac{1}{\alpha+1}f(t).\end{aligned}$$
As a consequence, $(x+\sin x)x$ provides an example of function of class $GC_{1}$ on $[0,\infty)$ which is not convex.
No positive constant can be a function of class $GC_{\alpha}$ for any $\alpha>0$.
Also, the restriction of a function $f:[a,\infty)\rightarrow\mathbb{R}$ of class $GC_{\alpha}$ to a subinterval $[b,\infty)$ is not necessarily a function of class $GC_{\alpha}.$
In the sequel we will describe some other classes of functions of class $GC_{\alpha}.$
The following concept of generalized convexity is due to S. Varosanec [@SV] and generalizes the usual convexity, $s$-convexity, the Godunova–Levin functions and $P$-functions.
Suppose that $h:[0,1]\rightarrow\mathbb{R}$ is a function such that $h(\lambda)+h(1-\lambda)\geq1$ for all $\lambda\in\lbrack0,1].$ A nonnegative function $f$ defined on an interval $I$ is called $h$-convex if $$f(\lambda x+(1-\lambda)y)\leq h(\lambda)f(x)+h(1-\lambda)f(y). \label{sconv1}$$ whenever $\lambda\in\lbrack0,1]$,$\,$ and $x,y\in I$.
\[PropV\]Suppose that $f$ is a nonnegative continuous function defined on an interval $[a,\infty)$ such that the following two conditions are fulfilled:
$i)$ $f(a)=0;$
$ii)$ $f$ is $h$-$convex$ with respect to a function $h$ with $\int_{0}^{1}h(\lambda)\;d\lambda\leq\frac{1}{\alpha+1}$, for some $\alpha>0.$
Then $f$ belongs to the class $GC_{\alpha}$.
In fact, $$\begin{aligned}
\frac{1}{t-a}\int_{a}^{t}f(x)\;dx & =\int_{0}^{1}f((1-\lambda)a+\lambda
t)\;d\lambda\\
& \leq f(t)\int_{0}^{1}h(\lambda)\;d\lambda+f(a)\int_{0}^{1}h(1-\lambda
)\;d\lambda\\
& \leq\frac{1}{\alpha+1}f(t).\end{aligned}$$
An important class of nonlinearities in partial differential operators theory is that of regularly varying functions, introduced by Karamata in [@K1930].
A positive measurable function $f$ defined on interval $[a,\infty)$ (with $a\geq0)$ is said to be regularly varying at infinity, of index $\sigma
\in\mathbb{R}$ *(*abbreviated, $f\in RV_{\infty}(\sigma)$*)*, provided that $$\lim_{x\rightarrow\infty}\frac{f(tx)}{f(x)}=t^{\sigma}\quad\text{for}\;\text{all}\;t>0.$$
All functions of index $\sigma$ are of the form $$f(x)=x^{\sigma}\exp\left( a(x)+\int_{0}^{x}\frac{\varepsilon(s)}{s}ds\right)
,$$ where $a(x)$ and $\varepsilon(x)$ are bounded and measurable, $a(x)\rightarrow
\alpha\in\mathbb{R}$ and $\varepsilon(x)\rightarrow0$ as $x\rightarrow\infty.$ In particular, so are $$x^{\sigma}\log x,\,\ x^{\sigma}\log\log x,~x^{\sigma}\exp\left( \frac{\log
x}{\log\log x}\right) ,\,\ x^{\sigma}\exp\left( \left( \log x\right)
^{1/3}\left( \cos\left( \log x\right) ^{1/3}\right) \right) .$$
See [@B] for details.
Semilinear problems with nonlinearities in the class of regularly varying functions have been studied by many people. See the paper by Cîrstea and Rădulescu [@CR2] and the references therein.
\[PropRV\]If $f\in RV_{\infty}(\sigma)$ with $\sigma>0,$ then$$\lim_{x\rightarrow\infty}\frac{F(x)}{xf(x)}=\frac{1}{\sigma+1},$$ where$$F(x):=\int_{0}^{x}f(s)\,ds. \label{primf}$$ As a consequence, if $f$ is also continuous, then $f$ is of class $GC_{\alpha
},$ whenever $\alpha\in(0,\sigma).$
To prove this, consider the change of variable $s=tx$ which yields $$F(x)=\int_{0}^{x}f(s)\,ds=\int_{0}^{1}xf(tx)\,dt.$$
The continuity of $f$ and the fact that $f\in RV_{\infty}(\sigma)$ assure the existence of a $\delta>0$ such that for every $x>\delta$ we have $$\frac{f(tx)}{f(x)}\leq t^{\sigma}+1,$$ whence the integrability of the function $t\rightarrow\frac{f(tx)}{f(x)}$ on $[0,1]$. Then$$\begin{aligned}
\lim_{x\rightarrow\infty}\frac{F(x)}{xf(x)} & =\lim_{x\rightarrow\infty}\int_{0}^{1}\frac{f(tx)}{f(x)}\,dt\\
& =\int_{0}^{1}\lim_{x\rightarrow\infty}\frac{f(tx)}{f(x)}\,dt=\int_{0}^{1}t^{\sigma}dt=\frac{1}{\sigma+1}.\end{aligned}$$ The commutation of the limit with the integral is motivated by the Lebesgue dominated convergence theorem.
Another important class of nonlinearities which appear in connection with the study of boundary blow-up problems for elliptic equations is the class of functions satisfying the Keller-Osserman condition. See [@Rad], [@DDGR], [@IR2008] and [@NR2010].
\[KO\] A nonnegative and nondecreasing function $f\in C^{1}([0,\infty))$ with $f(0)=0$ satisfies the generalized Keller-Osserman condition of order $p>1$ if $$\int_{1}^{\infty}\frac{1}{(F(t))^{1/p}}\;dt<\infty, \label{f2}$$ where $F$ is the primitive of $f$ given by the formula (\[primf\]).
If $f\in RV_{\infty}(\sigma+1)$ with $\sigma+2>p>1$ is a nondecreasing and continuous function, then $F\in RV_{\infty}(\sigma+2)$ and $F^{-1/p}\in
RV_{\infty}((-\sigma-2)/p)$. Since $(-\sigma-2)/p<-1$, we infer that $F^{-1/p}\in L^{1}([1,\infty))$ and thus $f$ satisfies the generalized Keller-Osserman condition.
It is worth to notice that the function $\exp(t)$ is not regularly varying at infinity though satisfies the generalized Keller-Osserman condition and belongs also to any class $GC_{\alpha}$ with $\alpha>0$.
Necessarily, if a function $f$ satisfies the generalized Keller-Osserman condition of order $p>1$, then$$\lim_{t\rightarrow\infty}\frac{F(t)}{t^{p}}=\infty, \label{growthF}$$ while $\frac{F(t)}{t^{p}}$ may be (or may be not) a monotonic function.
If $\frac{F(t)}{t^{p}}$ is nondecreasing for some $p>2$, then the function $f$ belongs to the class $GC_{p-1}$. In particular, this is the case of the function $f(t)=pt^{p-1}\log(t+1)+\frac{t^{p}}{t+1}$ (whose primitive is $F(t)=t^{p}\log(t+1)).$ Notice that this function does not satisfy the generalized Keller-Osserman condition of order $p.$
We end this section by discussing the connection Definition 1 with a class of functions due to W. Orlicz.
\[Orl\]An $\emph{N}$*-function* is any function $M:[0,\infty
)\rightarrow\mathbb{R}$ of the form$$M(x)=\int_{0}^{x}p(t)\,dt,$$ where $p$ is nondecreasing and right continuous, $p(0)=0,$ $p(t)>0$ for $t>0,$ and $\lim_{t\rightarrow\infty}p(t)=\infty.$
An $N$*-*function ** $M$ satisfies the $\Delta_{2}$-*condition* if there exist constants $k>0$ and $x_{0}\geq0$ such that$$M(2x)\leq kM(x)\text{\quad for all }x\geq x_{0}.$$
Any $N$*-*function ** $M$ is convex and plays the following properties:
$N1)$ $M(0)=0$ and $M(x)>0$ for $x>0;$
$N2)$ $\frac{M(x)}{x}\rightarrow0$ as $x\rightarrow0$ and $\frac{M(x)}{x}\rightarrow\infty$ as $x\rightarrow\infty.$
$\medskip$
Two examples of $N$-functions which satisfy the $\Delta_{2}$-condition are $\frac{x^{p}}{p}$ (for $p\geq1)$ and $t(\log t)^{+}$.
The $N$-functions which satisfy the $\Delta_{2}$-condition are instrumental in the theory of Orlicz spaces (which extend the $L^{p}(\mu)$ spaces). Their theory is available in many books, such as [@KR] and [@RR], and has important applications to interpolation theory [@BS] and Fourier analysis [@ZYG].
According to [@KR], page 23, the constant $k$ which appears in the formulation of $\Delta_{2}$-condition is always greater than or equal to 2.
Every $N$-function $M:[0,\infty)\rightarrow\mathbb{R}$ which satisfies the $\Delta_{2}$-condition belongs to the class $GC_{\alpha},$ whenever $\alpha
\in(0,2\log_{2}k)$.
Since $M$ is nondecreasing,$$M(tx)=M(2^{\log_{2}t}x)\leq M(2^{\left[ \log_{2}t\right] +1}x),$$ and taking into account the $\Delta_{2}$-condition we infer that $$\begin{aligned}
M(tx) & \leq M(x)k^{\left[ \log_{2}t\right] +1}\leq M(x)k^{\log_{2}t+1}\\
& \leq M(x)t^{2\log_{2}k},\end{aligned}$$ for $x\ $big enough and $t\geq2.$ Hence,$$\begin{aligned}
\int_{0}^{t}M(x)dx & =\int_{0}^{1}tM(ts)ds\\
& \leq\int_{0}^{1}tM(t)s^{2\log_{2}k}ds=\frac{1}{2\log_{2}k+1}tM(t)\end{aligned}$$ and the proof is done.
An application to the existence of finite time blow-up solutions
================================================================
This section is devoted to the existence of finite time blow-up solutions of the evolutionary $p$-Laplacian problem$$u_{t}-\Delta_{p}u=f(|u|)-\frac{1}{m(\Omega)}{\displaystyle\int\nolimits_{\Omega}}
f(|u|)\,dx\text{\quad in }\Omega\label{eveq}$$ with Neumann-Robin boundary values,$$|\nabla u|^{p-2}\frac{\partial u}{\partial n}=0\text{\quad on }\partial
\Omega\,, \label{ncond}$$ and the initial conditions$$u(x,0)=u_{0}(x)\;\text{on}\;\Omega,\;\text{where}\;{\displaystyle\int\nolimits_{\Omega}}
u_{0}\,dx=0. \label{inval}$$
As was mentioned in the introduction, we restrict ourselves to the case where $\Omega\subset\mathbb{R}^{N}$ is a bounded regular domain of class $C^{2}$, and $f:[0,\infty)\mapsto\lbrack0,\infty)$ is a locally Lipschitz function; $m(\Omega)$ represents the Lebesgue measure of the domain $\Omega,$ and $\Delta_{p}$, for $p\geq2$, is the $p$-Laplacian operator.
The purpose of this section, is to extend a natural energetic criterion for the blow-up in finite time of solutions of $(3.1)-(3.3).$ Our proof relies on the same idea used by Jazar and Kiwan [@JK] in the case where $p=2$ and $f$ is a power function.
We start by noticing that each solution $u$ of the problem above has the property $$\int_{\Omega}u\,dx=0$$ because the integral in the right hand side of (\[eveq\]) is $0$ and $$\begin{aligned}
\frac{d}{dt}\Big(\int_{\Omega}u\,dx\Big) & =\int_{\Omega}u_{t}\,dx=\int_{\Omega}\Delta_{p}u\,dx\\
& =\int_{\Omega}div(|\nabla u|^{p-2}\nabla u)\,dx=0.\end{aligned}$$ Hence, by the initial condition (\[inval\]), we have $\int_{\Omega}u\,dx=0$.
Next, it is easy to see that for $p>1$ the energy $$E(u(t))=\int_{\Omega}\Big(\frac{1}{p}|\nabla u|^{p}-\int_{0}^{u}f(|\tau|)\,d\tau\Big)dx,$$ of any solution $u$ of our evolutionary problem is nonincreasing in time. In fact,$$\begin{aligned}
\frac{dE(u(t))}{dt} & =\int_{\Omega}\left( |\nabla u|^{p-2}\nabla
u_{t}\nabla u-u_{t}f(|u|)\right) \,dx\\
& =\int_{\partial\Omega}\frac{\partial u}{\partial n}|\nabla u|^{p-2}u_{t}\,d\sigma-\int_{\Omega}u_{t}\Delta_{p}u\,dx-\int_{\Omega}u_{t}\,f(|u|)\,dx\\
& =-\int_{\Omega}u_{t}(\Delta_{p}u+f(|u|))\,dx=-\int_{\Omega}u_{t}^{2}\,dx,\end{aligned}$$ and by integrating both sides over $[0,t]$ we obtain the formula $$E(u(t))=E(u_{0})-\int_{0}^{t}\int_{\Omega}u_{t}^{2}\,dxdt,\quad\text{for\ all}\;t>0. \label{ft5}$$
According to this formula, if the initial energy $E(u_{0})$ is nonpositive, then $E(u(t))$ is nonpositive for all $t>0.$ In the case of generalized convex functions of order $\alpha$, with $\alpha>\frac{1}{p-1}$ we have$$C\int_{\Omega}uf(|u|)\,dx\geq\int_{\Omega}\int_{0}^{u}f(|t|)\,dtdx\geq\frac
{1}{p}\int_{\Omega}|\nabla u|^{p}, \label{ftinegmod}$$ where $C=\frac{1}{1+\alpha}\in(0,\frac{p-1}{p})$.
*(*The energetic criterion for blow-up in finite time, case **** $p\geq2$*)* \[ftthm1\] Assume that $f:[0,\infty)\mapsto
\lbrack0,\infty)$ is a locally Lipschitz function belonging to the class $GC_{a}$, with $\alpha>\frac{1}{p-1}$, and let $u$ be a solution of the problem $(3.1)-(3.3)$ corresponding to an initial data $u_{0}\in
C(\overline{\Omega})$, $u_{0}$ not identically zero.
If $E(u_{0})\leq0,$ then $u,$ as a function of $t,$ cannot be in $L^{\infty
}((0,T);L^{2}(\Omega))$ for all $T>0$. In other word, there is $T>0$ such that$$\underset{t\rightarrow T-}{\lim\sup}\left\Vert u(t)\right\Vert _{L^{2}}=\infty. \label{blowup}$$
Notice that the condition $E(u_{0})\leq0$ in Theorem \[ftthm1\] is also necessary for the blow-up in finite time (of the $L^{2}$ norm of $u(t)).$ In fact, (\[blowup\]) forces that $$\inf\left\{ E(u(t)):0<t<T\right\} =-\infty.$$ This can be argued by contradiction. If $E(u(t))\geq-C_{0}$, for some $C_{0}>0$, then the function$$h(t):=\frac{1}{2}\int_{\Omega}u^{2}(x,t)\,dx$$ verifies the condition $$\begin{aligned}
\frac{1}{2}h^{\prime}(t) & =\int_{\Omega}uu_{t}dx\leq\frac{1}{2}\int
_{\Omega}\left( u^{2}+u_{t}^{2}\right) dx\\
& =\frac{1}{2}(h(t)-E^{\prime}(u(t))),\end{aligned}$$ which yields $$(h(t)+E(u(t))+C_{0})^{\prime}\leq h(t)\leq h(t)+E(u(t))+C_{0}.$$ Therefore$$h(t)\leq h(t)+E(u(t))+C_{0}\leq(h(0)+E(u_{0})+C_{0})e^{t},\;\text{for all
}t\in(0,T),$$ and thus the $L^{2}$-norm of $u(t)$ is bounded.
The proof of Theorem \[ftthm1\] needs a preparation.
\[ftlem2\] Under the assumptions of Theorem \[ftthm1\] the two auxiliary functions$$h(t):=\frac{1}{2}\int_{\Omega}u^{2}(x,t)\,dx\quad\text{and\quad}H(t):=\int
_{0}^{t}h(s)\,ds$$ verify the following three conditions$:$$$\begin{gathered}
h^{\prime}(t)\geq\frac{1}{C}\int_{0}^{t}\int_{\Omega}u_{t}^{2}\,dt;
\label{ft6}\\
h^{\prime}(t)\geq2\Big(\frac{1}{Cp}-p+1\Big)\lambda h(t)\text{, for}\;\text{some }\lambda>0;\label{ft7}\\
\frac{1}{2C}\Big(H^{\prime}(t)-H^{\prime}(0)\Big)^{2}\leq H(t)H^{\prime\prime
}(t). \label{ft8}$$
In fact,$$\begin{aligned}
h^{\prime}(t) & =\int_{\Omega}u_{t}u\,dx=\int_{\Omega}u(\Delta
_{p}u+f(|u|))\,dx\\
& \geq\int_{\Omega}\Big(-(p-1)|\nabla u|^{p}+\frac{1}{C}\int_{0}^{u}f(|t|)\,dt\Big)dx\\
& =-\frac{1}{C}\int_{\Omega}\Big(\frac{1}{p}|\nabla u|^{p}-\int_{0}^{u}f(|t|)\,dt\Big)dx+\Big(\frac{1}{Cp}-p+1\Big)\int_{\Omega}|\nabla
u|^{p}\,dx.\end{aligned}$$
Hence,$$\begin{aligned}
h^{\prime}(t) & \geq-\frac{1}{C}E(u)+\Big(\frac{1}{Cp}-p+1\Big)\int_{\Omega
}|\nabla u|^{p}\,dx\\
& \geq-\frac{1}{C}E(u)\\
& =-\frac{1}{C}E(u_{0})+\frac{1}{C}\int_{0}^{t}\int_{\Omega}u_{t}^{2}\,dxdt\\
& \geq\frac{1}{C}\int_{0}^{t}\int_{\Omega}u_{t}^{2}\,dxdt.\end{aligned}$$
On the other hand, by the Poincaré inequality, we have $$\begin{aligned}
h^{\prime}(t) & \geq\Big(\frac{1}{Cp}-p+1\Big)\int_{\Omega}|\nabla
u|^{2}\,dx\\
& \geq\Big(\frac{1}{Cp}-p+1\Big)\lambda\int_{\Omega}u^{2}\,dx\\
& =2\Big(\frac{1}{Cp}-p+1\Big)\lambda h(t),\end{aligned}$$ where $\lambda$ is a suitable positive constant.
We pass now to the proof of (\[ft8\]). Since $$\begin{aligned}
H^{\prime}(t)-H^{\prime}(0) & =\int_{0}^{t}h^{\prime}(s)\,ds=\int_{0}^{t}\int_{\Omega}uu_{t}\,dxdt\\
& \leq\Big(\int_{0}^{t}\int_{\Omega}u^{2}\,dxdt\Big)^{1/2}\Big(\int_{0}^{t}\int_{\Omega}u_{t}^{2}\,dxdt\Big)^{1/2}\\
& \leq(2H(t))^{1/2}(Ch^{\prime}(t))^{1/2}=(2CH(t)H^{\prime\prime}(t))^{1/2},\end{aligned}$$ by (\[ft6\]) we infer that$$H^{\prime}(t)-H^{\prime}(0)=\int_{0}^{t}h^{\prime}(s)\,ds\geq0,$$ and thus$$\frac{1}{2C}\Big(H^{\prime}(t)-H^{\prime}(0)\Big)^{2}\leq H(t)H^{\prime\prime
}(t).$$
*Proof of Theorem \[ftthm1\]*. Suppose, by *reduction ad absurdum*, that the solution $u(x,\cdot)$ exists in$$L^{\infty}((0,T);L^{2}(\Omega))\,$$ for all $T>0$. By (\[ft7\]),$$\lim_{t\rightarrow\infty}H^{\prime}(t)=\lim_{t\rightarrow\infty}h(t)=\infty,
\label{ft10}$$ which yields, for each $\beta\in(0,1/C),$ the existence of a number $T_{0}>0$ such that for all $t>T_{0}$,$$\beta H^{\prime}(t)^{2}\leq\frac{1}{C}\Big(H^{\prime}(t)-H^{\prime
}(0)\Big)^{2}.$$
Now, by (\[ft8\]) we obtain$$\beta H^{\prime}(t)^{2}\leq2H(t)H^{\prime\prime}(t).$$
We will show, by considering the function $G(t)=H(t)^{-q}$, for a suitable $q>0,$ that the last inequality leads to a contradiction. In fact,$$\begin{aligned}
G^{\prime\prime}(t) & =qH(t)^{-q-2}\Big((q+1)(H^{\prime}(t))^{2}-H(t)H^{\prime\prime}(t)\Big)\\
& \leq qH(t)^{-q-2}\Big(\frac{2(q+1)}{\beta}-1\Big)H(t)H^{\prime\prime}(t),\end{aligned}$$ for all $t\geq T_{0},$ so that for $\beta\in\left( 0,1/C\right) $ and $q\in\left( 0,1/(2C)-1\right) $ with $2(q+1)<\beta<1/C,$ the corresponding function $G(t)$ is concave.
By (\[ft10\]), $\lim_{t\rightarrow\infty}H(t)=\infty$, whence $\lim
_{t\rightarrow\infty}G(t)=0$. Thus $G$ provides an example of a concave and strictly positive function which tends to $0$ at infinity, a fact which is not possible. Consequently $u,$ as a function of $t,$ cannot be in $L^{\infty
}((0,T);L^{2}(\Omega))$ for all $T>0$. The proof of Theorem \[ftthm1\] is done.
This research is supported by CNCSIS Grant $420/2008$.
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[^1]: Corresponding author: Rovenţa Ionel
|
---
abstract: 'The Rashba-Edelstein effect stems from the interaction between the electron’s spin and its momentum induced by spin-orbit interaction at an interface or a surface. It was shown that the inverse Rashba-Edelstein effect can be used to convert a spin- into a charge current. Here, we demonstrate that a Bi/Ag Rashba interface can even drive an adjacent ferromagnet to resonance. We employ a spin-torque ferromagnetic resonance excitation/detection scheme which was developed originally for a bulk spin-orbital effect, the spin Hall effect. In our experiment, the direct Rashba-Edelstein effect generates an oscillating spin current from an alternating charge current driving the magnetization precession in a neighboring permalloy (Py, Ni$_{80}$Fe$_{20}$) layer. Electrical detection of the magnetization dynamics is achieved by a rectification mechanism of the time dependent multilayer resistance arising from the anisotropic magnetoresistance.'
author:
- 'M. B. Jungfleisch'
- 'W. Zhang'
- 'J. Sklenar'
- 'W. Jiang'
- 'J. E. Pearson'
- 'J. B. Ketterson'
- 'A. Hoffmann'
title: 'Interface-driven spin-torque ferromagnetic resonance by Rashba coupling at the interface between non-magnetic materials'
---
Conventional spintronics relies on the exchange interaction between conduction electrons on one side and localized spins in magnetic materials on the other side [@Zutic]. Stimulated by the experimental demonstration of spin- to charge current conversion using bulk spin Hall effects (SHE), these kind of spin-orbital phenomena were actively investigated in the last decade and opened up the door to the research field of spin-orbitronics [@Dyakonov; @Hirsch; @Hoffmann; @Kajiwara]. SHEs can be investigated by means of spin-current injection from a ferromagnet (FM) into materials with large spin-orbit coupling, usually normal metals (NM) such as Pt or Pd [@Wei_PRB], and sensing the generated voltage generated by means of the inverse spin Hall effect (ISHE) [@Mosendz; @Mosendz_PRB; @Wei_PRL; @Saitoh; @Azevedo; @Wei_JAP; @Jungfleisch; @Jungfleisch_PRB]. Other interesting applications of SHEs are the effective magnetization switching of nanomagnets or the movement of domain walls [@STT; @Mesoscale_mag; @Wanjun_Science]. Furthermore, the ferromagnetic linewidth modulation as well as the excitation of spin waves and ferromagnetic resonance by SHE was demonstrated in ferromagnetic metals and insulators [@Liu_ST-FMR; @Joe_PRB; @Klein_PRL; @Wu; @Mellnik_ST-FMR]. The SHE is a bulk effect occurring within a certain volume of the NM determined by the spin-diffusion length. The conversion efficiency can be expressed by a material-specific parameter, the spin Hall angle $\gamma_\mathrm{SHE}$ [@Hoffmann].
Very recently, it has been shown that the inverse Rashba-Edelstein effect (IREE) can also be used for transformation of a spin- into a charge current [@Sanchez_Nat_com; @Wei_IREE; @Sangiao; @Ando_IREE]. The IREE is the inverse process to the Rashba-Edelstein effect (REE) [@Edelstein]. The REE originates from spin-orbit interaction in a 2D electron gas at interfaces or surfaces, which effectively produce a steady non-equilibrium spin polarization from a charge current driven by an electric field. The Hamiltonian of this interaction is given by [@Sanchez_Nat_com]: ${H_\mathrm{R}=\alpha_\mathrm{R}(k\times \hat{e}_\mathrm{z}) \cdot\sigma,}$ where $\alpha_\mathrm{R}$ is the Rashba coefficient, $\hat{e}_\mathrm{z}$ is the unit vector in $z$-direction \[see Fig. \[Fig1\](b,c)\] and $\sigma$ is the vector of Pauli matrices. As a result of this interaction the dispersion curves of the 2D electron gas are spin-split if $\alpha_\mathrm{R} \neq 0$, as illustrated in Fig. \[Fig1\](a). Analogous to the spin Hall angle, the spin- to charge current interconversion parameter can be defined as [@Sanchez_Nat_com]: $$\label{Rashba_parameter}
{\lambda_\mathrm{REE}=\alpha_\mathrm{R}\tau_\mathrm{S}/\hbar,}$$ where $\tau_\mathrm{S}$ is the effective relaxation time describing the ratio between spin injection and spin-momentum scattering and $\hbar$ is the reduced Planck constant. The spin-split 2D electron gas dispersions and Fermi contours of many Rashba surfaces and interfaces have been investigated spectroscopically [@Ast]. In general, large Rashba couplings occur at interfaces between heavy elements with strong spin-orbit interaction (such as Bi, Pb, and Sb) and other non-magnetic materials with small spin-orbit coupling such as Ag, Au, and Cu [@Ast; @Koroteev]. Even though, the interaction between a charge current and a non-zero spin density at a Rashba interface has been demonstrated by injection of a spin-pumping driven spin current at ferromagnetic resonance, the reverse process remains to be explored experimentally until now.
![\[Fig1\] (Color online) (a) Dispersion curves of a 2D electron gas are spin-split due to the REE. (b) Scheme of the ST-FMR experimental setup. (c) ST-FMR mechanism in Py/Ag/Bi multilayers. The alternating RF current drives an Oersted field $h_\mathrm{RF}$ exerting a field-like torque $\tau_\mathrm{\perp}$ on the magnetization $M$. At the same time a oscillatory transverse spin accumulation at the Py/Ag interface generated at the Ag/Bi interface by the REE exerts a damping-like torque $\tau_\mathrm{\vert\vert}$ on the magnetization.](Fig1.eps){width="1\columnwidth"}
Here, we demonstrate that a Bi/Ag Rashba interface can drive spin-torque ferromagnetic resonance (ST-FMR) in an adjacent ferromagnetic layer. We interpret our results in terms of an excitation by the direct REE, which drives an oscillating spin current from an alternating charge current that scatters of the Rashba interface (Ag/Bi). The generated spin current excites the magnetization precession in a neighboring permalloy (Py, Ni$_{80}$Fe$_{20}$) layer by the spin-transfer torque effect [@Liu_ST-FMR; @Slonczewski]. The precessional magnetization leads to resistance oscillations on account of the anisotropic magnetoresistance (AMR) of Py. The mixing between the applied alternating current and the oscillating resistance allows for a direct voltage detection of the induced magnetization dynamics [@Liu_ST-FMR; @Mellnik_ST-FMR]. Injecting an additional DC current to the sample results in an additional spin current generation due to the REE which enables to manipulate the ferromagnetic resonance linewidth by exerting a torque on the magnetization.
We fabricated the devices using magnetron sputtering and photolithography. The multilayers were prepared in the shape of $30\times5$ $\mu$m$^2$ stripes using lithography and lift-off on intrinsic Si substrates with $300$-nm thick thermally grown SiO$_2$. Four different types of multilayers were deposited using magnetron sputtering: Py, Py/Bi, Py/Ag and Py/Ag/Bi. In the case of the Py/Ag/Bi systems, the Ag thickness was $t_\mathrm{Ag}$ = 2, 4, 6, 10, 15 nm, the Py thickness $t_\mathrm{Py}= 9$ nm and the Bi thickness $t_\mathrm{Bi}= 4$ nm. The control samples feature a Py thickness of 7 nm, Ag thickness 6 nm and Bi thickness 4 nm. In a subsequent process step, the coplanar waveguide (CPW) was fabricated on top of the multilayers. Figure \[Fig1\](b) illustrates the experimental setup. A bias-T is used to apply a microwave signal and to detect the rectified DC voltage at the same time. The applied microwave power is kept constant at $+10$ dBm, unless otherwise mentioned. An in-plane magnetic field is applied at an angle of $\theta = 45^\circ$ \[see illustration in Fig. \[Fig1\](b,c)\]. While sweeping the external magnetic field the DC voltage is detected by a lock-in amplifier with an amplitude modulation at 3 kHz. All measurements were performed at room temperature.
![\[Fig2\] (Color online) Spectra of REE-driven ST-FMR measured at a frequency of 4 GHz and an applied microwave power of +10 dBm. Thickness in brackets given in nm. (a) Ag thickness dependence of the resonance signal. (b) Comparison between control samples and Py(9)/Ag(10)/Bi(4).](Fig2.eps){width="1\columnwidth"}
Figure \[Fig2\] shows typical spectra at an excitation frequency of $f=4$ GHz. Let’s first discuss the trilayers \[Fig. \[Fig2\](a)\]. In our experiment, magnetization dynamics is excited simultaneously by the Oersted field as well as by the REE which generates an oscillating spin current from the alternating charge current driving the magnetization precession in the neighboring permalloy layer when the condition of ferromagnetic resonance is fulfilled, $$\label{Kittel}
{f= \frac{\vert\gamma\vert}{2\pi} \sqrt{H_\mathrm{}(H_\mathrm{}+4\pi M_\mathrm{eff})}.}$$ Here, $M_\mathrm{eff}$ is the effective magnetization and $\vert\gamma\vert$ is the gyromagnetic ratio. Electrical detection of the magnetization dynamics is achieved by a rectification mechanism of the time dependent multilayer resistance arising from the AMR of Py. A rectification by spin pumping and IREE is a secondary effect in our experiment [@Liu_ST-FMR]. As apparent from Fig. \[Fig2\](a), the Py/Ag/Bi samples exhibit a superimposed symmetric and antisymmetric Lorentzian lineshape. The smallest Ag interlayer thickness of 2 nm shows the largest symmetric contribution, but the smallest absolute signal. With increasing $t_\mathrm{Ag}$ the signal tends to be more antisymmetric and the absolute value increases. The control samples are depicted in Fig. \[Fig2\](b). The pure Py sample features a small, antisymmetric Lorentzian signal due to a rectification by AMR. The Py/Bi sample exhibits a very small, mostly symmetric signal. Py/Ag features a reasonably large antisymmetric signal: The Ag layer is beneficial for the absolute voltage because a larger Oersted field is generated in the Py layer resulting in a higher AMR signal manifested in a substantial antisymmetric lineshape.
Figures \[Fig3\](a) and (c) illustrate how the resonance field and linewidth alter for different Ag interlayer thicknesses at various excitation frequencies. The excitation of ferromagnetic resonance is confirmed by a fit to Eq. (\[Kittel\]), see Fig. \[Fig3\](b). Furthermore, the data shown in Fig. \[Fig3\](d) is governed by a linear dependence between linewidth $\Delta H$ and the excitation frequency $f$: $$\label{damping}
\Delta H(f) = \Delta H_0 + 4\pi f \frac{\alpha}{\vert\gamma\vert},$$ where $\Delta H_0$ is the inhomogeneous linewidth broadening given by the zero-frequency intercept and $\alpha$ is the Gilbert damping parameter. This confirms the excitation of FMR in our samples.
The magnetization dynamics in a macrospin model is governed by a modified Landau-Lifshitz-Gilbert equation [@Mellnik_ST-FMR]: $$\label{ST-FMR}
\begin{split}
\frac{\mathrm{d}\hat{m}}{\mathrm{d}t}=-\vert\gamma\vert \hat{m}\times\vec{H}_\mathrm{eff} + & \alpha \hat{m}\times\frac{\mathrm{d}\hat{m}}{\mathrm{d}t} + \vert\gamma\vert \tau_{\parallel} \hat{m}\times(\hat{y} \times \hat{m})\\+ \vert\gamma\vert \tau_\perp \hat{y}\times \hat{m},
\end{split}$$ where $\hat{m}$ is the magnetization direction, $H_\mathrm{eff}$ is the effective magnetic field, $\tau_{||}$ and $\tau_{\perp}$ are the two acting torque components, and the coordinate system ($\hat{x}, \hat{y}, \hat{z}$) is defined as shown in Fig. \[Fig1\](b,c).
![\[Fig3\] (Color online) (a) Resonance at various excitation frequencies for different Ag thicknesses. (b) Dispersion measured for different Ag interlayer thicknesses, $t_\mathrm{Py}=9$ nm, $t_\mathrm{Bi}=4$ nm. A fit to Eq. (\[Kittel\]) confirms the excitation of ferromagnetic resonance; shown as solid lines. (c) Evolution of the FMR linewidth with $t_\mathrm{Ag}$ at different excitation frequencies. (d) Determination of Gilbert damping parameter $\alpha$. Solid lines show a fit to Eq. (\[damping\]).](Fig3.eps){width="1\columnwidth"}
![\[Fig4\] (Color online) (a) Deconvoluted symmetric and antisymmetric contribution to DC voltage amplitude for various Ag interlayer thicknesses. (b) Ratio $\mathrm{T}= V_\mathrm{antisymm}/(V_\mathrm{antisymm}+V_\mathrm{symm})$ as function of $t_\mathrm{Ag}$ for various frequencies.](Fig4.eps){width="1\columnwidth"}
The two vector components of the current-induced torque $\tau_{||}, \tau_{\perp}$ can be related to the amplitudes of the symmetric and antisymmetric components of the resonance lineshape [@Mellnik_ST-FMR]: (1) An in-plane component $\tau_\mathrm{\vert\vert} \sim \hat{m}\times(\hat{y}\times\hat{m})$ results in a symmetric contribution and (2) an out-of-plane component $\tau_\mathrm{\perp} \sim \hat{y}\times\hat{m}$ results in an antisymmetric contribution, see Fig. \[Fig1\](c) [@Mellnik_ST-FMR]. Figure \[Fig4\](a) illustrates the Ag thickness dependence of the amplitudes of both contributions, respectively, as a function of the driving RF frequency. We observe the following trend: The amplitudes increase with increasing $t_\mathrm{Ag}$ up to $t_\mathrm{Ag}\approx 7$ nm. At larger thicknesses, the antisymmetric contribution (dashed lines) remains constant up to $t_\mathrm{Ag}\approx 10$ nm before it decreases. The symmetric contribution, however, peaks at $t_\mathrm{Ag}\approx 7$ nm and reduces afterwards. In order to highlight this observation we plot a torque-ratio equivalent $\mathrm{T} = V_\mathrm{antisymm}/(V_\mathrm{antisymm}+V_\mathrm{symm})$ as a function of $t_\mathrm{Ag}$ in Fig. \[Fig4\](b). Clearly, the symmetric contribution to the lineshape is greatest at a lower Ag thickness and becomes negligible for larger $t_\mathrm{Ag}$. The reason for this trend is the larger Oersted field produced in samples with a thicker Ag layer and, thus, a larger out-of-plane torque contribution $\tau_\mathrm{\perp}$. As is apparent from Fig. \[Fig4\](b), this trend is independent on the excitation frequency. We also show the ratio of the control samples Py/Ag and Py in the same plot as a red dot and a green square, respectively.
We interpret our observations in the following way: If the observed increase of the symmetric component ($\sim \tau_\mathrm{\vert\vert}$) with respect to the antisymmetric component ($\sim \tau_\mathrm{\perp}$) was caused by the SHE in Ag, we should observe the same ratio for the control sample Py/Ag. As is apparent from Fig. \[Fig4\](b), this is not the case. Since Ag features a long spin-diffusion length of $\sim 300$ nm [@Ag-diffusion], it would also be possible that the SHE in Bi generates a spin current which diffuses through the Ag layer. However, since the generated voltage for the control sample Py/Bi is negligibly small, see Fig. \[Fig2\](b), this mechanism can also be ruled out. We conclude that the magnetization dynamics in our Py/Ag/Bi samples is driven by an interfacial charge-spin conversion due to the REE.
According to the spin-torque theory [@Slonczewski], an additional spin current injected into the FM layer will increase or decrease the effective magnetic damping, i.e., the linewidth, depending on its relative orientation with respect to the magnetization [@Liu_ST-FMR; @Wu]. Since Ag features a very small spin Hall angle [@Hailong] and our Bi layer is almost non-conducting [@Wei_IREE], the demonstration of the ferromagnetic linewidth manipulation by an additional DC current injection would be an independent manifestation of charge- to spin current conversion by the REE. Figure \[Fig5\] shows unambiguously that it is indeed possible to manipulate the resonance lineshape if an additional DC current is injected into the sample. For this purpose a rather small RF power of +2 dBm is chosen. Apparently, for a positive magnetic field polarity, a positive DC current leads to an enhanced linewidth, i.e., a damping enhancement. In contrast, a negative current leads to a decreased linewidth, i.e., a damping reduction. Reversing the field polarity results in an opposite trend. We find a relative linewidth change of $0.8\%$ mA$^{-1}$.
![\[Fig5\] (Color online) Manipulation of the FMR linewidth by a simultaneous injection of an electrical DC current. Py(15)/Ag(4)/Bi(4), $f = 4$ GHz, $P_\mathrm{RF} = +2$ dBm.](Fig5.eps){width="1\columnwidth"}
Although it isn’t physical to speak of a thickness in case of an interface effect, it is still possible to adapt a lineshape analysis approach which was presented originally in Ref. [@Liu_ST-FMR] to relate the spin Hall angle to the ratio symmetric/antisymmetric components of the resonance lineshape. We can estimate a spin Hall angle equivalent $\gamma^{*}$ if we hypothetically assume that the charge-spin conversion process was a bulk- rather than an interface-driven effect [@Liu_ST-FMR]: $$\label{SH_angle}
\gamma^{*} = \frac{S}{A}\frac{e \mu_0 M_\mathrm{S}t_\mathrm{Py} t_\mathrm{NM} }{\hbar} \sqrt{1+\frac{4\pi M_\mathrm{eff}}{H_\mathrm{}}}.
$$ Here, $t_\mathrm{NM}$ is the non-magnetic layer thickness. We find the spin Hall angle equivalent to be $\gamma^{*}\approx18$% for our Py/Ag/Bi samples, exceeding most paramagnetic metals. In our previous work we determined the REE conversion parameter $\lambda_\mathrm{REE}\approx 0.1$ nm [@Wei_IREE]. Using the relation $\lambda_\mathrm{REE}= 1/2 d \gamma^{*}$, where $d$ is the *interface layer* thickness [@Sanchez_Nat_com], we obtain $d\approx 1$ nm, which is a reasonable estimate.
In summary, we demonstrated the conversion of a charge- into a spin current by Rashba coupling of interface states by adapting a spin-torque ferromagnetic resonance excitation/detection technique. The Ag thickness dependence clearly demonstrates that the spin dynamics in the adjacent Py layer is driven by an interface-generated spin-polarized electron current that exerts a torque on the magnetization rather than a bulk effect such as the spin Hall effect. Our conclusions are further validated by a FMR linewidth modulation due to the spin current injection by applying an additional DC charge current to the sample stack. Our results will stimulate experimental and theoretical endeavors to explore novel interface- and surface-driven spin-orbital phenomena for the efficient excitation of magnetization dynamics.
We thank Roland Winkler for illuminating discussions. This work was supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division. Lithography was carried out at the Center for Nanoscale Materials, an Office of Science user facility, which is supported by DOE, Office of Science, Basic Energy Science under Contract No. DE-AC02-06CH11357.
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abstract: |
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$cycle covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$cycle. The minimum weight of $k-$cycle covering is the weighted covering number on $k-$cycle, denoted by $\tau_{k}(G_{w})$. In this paper, we design a $k-1/2$ approximation algorithm for the weighted covering number on $k-$cycle when $k$ is odd.
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$clique covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$clique. The minimum weight of $k-$clique covering is the weighted covering number on $k-$clique, denoted by $\widetilde{\tau_{k}}(G_{w})$. In this paper, we design a $(k^{2}-k-1)/2$ approximation algorithm for the weighted covering number on $k-$clique. Last, we discuss the relationship between $k-$clique covering and $k-$clique packing in complete graph $K_{n}$.
author:
- 'Zhongzheng Tang ${}^{b,c}$ Zhuo Diao ${}^{a}$[^1]'
bibliography:
- 'ref.bib'
date: |
${}^a$ School of Statistics and Mathematics, Central University of Finance and Economics Beijing 100081, China\
${}^b$ Academy of Mathematics and Systems Science, Chinese Academy of Sciences\
Beijing 100190, China\
${}^c$ School of Mathematical Sciences, University of Chinese Academy of Sciences\
Beijing 100049, China\
title: '**Approximation algorithms on $k-$ cycle covering and $k-$ clique covering**'
---
[**Keywords**]{}: [$k-$ cycle covering, $k-$ clique covering, $k-$ clique packing]{}
k-cycle covering
================
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$cycle covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$cycle. The problem of minimum weight of $k-$cycle covering can be described as follows: $$\begin{aligned}
\tau_{k}(G_{w})=\min\{\mathbf w^T\mathbf x: A\mathbf x\geq\mathbf 1,\mathbf x\in\mathbf \{0,~1\}\}\label{ILP1}\end{aligned}$$
Where $A$ is a $k$-cycle-edge adjacent matrix and $\mathbf w$ is the weight vector. $\mathbf x$ represents the characteristic vector of edge.We obtain the relaxed programming of as follows: $$\begin{aligned}
\min\{\mathbf w^T\mathbf x: A\mathbf x\geq\mathbf 1, 0 \leq\mathbf x\leq 1\}\label{LP1}\end{aligned}$$ We compute the optimal solution $\hat{\mathbf x}^*$ of in polynomial time, then we transfer $\hat{\mathbf x}^*$ to integral vector: $$\begin{aligned}
\label{eq1}
\mathbf x_e=
\begin{cases}
1 &\hspace{2cm} \hat{\mathbf x}^*_e\geq 1/k\\
0 &\hspace{2cm}o.w.
\end{cases}\end{aligned}$$ Obviously, $\mathbf x$ is a feasible solution of ILP and $\mathbf w^T\mathbf x\leq k \mathbf w^T\hat{\mathbf x}^*$, which implies a $k$-approximation algorithm of minimum $k$-cycle covering problem.
Solve LP and get the optimal solution $\hat{\mathbf x}^*$.
Compute $\mathbf x$ by equation.
The $(k-\frac{1}{2})$-approximation algorithm when $k$ is odd
=============================================================
We take advantage of specific strategy to reach better performance when $k$ is odd.
Set $E_{k_1}=\emptyset$.
Solve LP and get the optimal solution $\hat{\mathbf x}^*$.
**For** every $e\in E$
**If** $\hat{\mathbf x}^*_e\geq 2/(2k-1)$ **Then** $E_{k_1}=E_{k_1}\cup\{e\}$.
Suppose $E^\prime$ are these edges of all $k$-cycles in $G-E_{k_1}$ and let $G^\prime$ be a subgraph of $G$ induced by the edge set $E^\prime$.
Using the Greedy Algorithm or Random Algorithm, we can find an approximate solution of maximum weight bipartite graph
$B=(V_1,V_2,E_B)$, which satisfies $\mathbf W(E_B)\geq (1/2)\mathbf W(E^\prime)$. Set $E_{k_2}=E^\prime\setminus E_B$.
Output $E_k=E_{k_1}\cup E_{k_2}$.
The Algorithm \[alg2\] has $(k-\frac{1}{2})$ approximate ratio for the minimum $k$-cycle covering problem.
Suppose $\hat{\mathbf x}^*$ and $\mathbf x^*$ are the optimal solution of LP and ILP, respectively.\
Firstly, we indicate that $E_k$ is a $k$-cycle covering. Actually, for every $k$-cycle $C_{k}$ in $G$, if it doesn’t contain any edge in $E_{k_1}$, then it is a $k$-cycle in $G-E_{k_1}$, thus it is a $k$-cycle in $G'$. Because $G^\prime-E_{k_2}$ is a bipartite graph, of course, $G^\prime-E_{k_2}$ has no $k$-cycle (here $k$ is odd). Thus $C_{k}$ contains some edge in $E_{k_2}$. Above all, we prove that $E_k$ is a $k$-cycle covering of $G$.\
Additionally, we will show the approximate ratio.\
On one hand, according to the rounding regulation, we know that: $$\begin{aligned}
\label{ineq1}
\sum\limits_{e\in E_{k_1}}\mathbf w_e\leq(k-\frac{1}{2})\sum\limits_{e\in E_{k_1}}\mathbf w_e\hat{\mathbf x}^*_e.\end{aligned}$$ On the other hand, every $\hat{\mathbf x}^*_e$ related to $e\in E^\prime$ has the lower bound $1-2(k-1)/(2k-1)=1/(2k-1)$ on the grounds that there exists a $k$-cycle $C_k$ in $G^\prime$ containing $e$, satisfying $\hat{\mathbf x}^*_e\leq2/(2k-1)$ and $\sum_{e\in C_k}\hat{\mathbf x}^*_e\geq 1$. $$\begin{aligned}
\label{ineq2}
\sum\limits_{e\in E_{k_2}}\mathbf w_e\leq(1/2)\sum\limits_{e\in E^\prime}\mathbf w_e\leq(1/2)(2k-1)\sum\limits_{e\in E^\prime}\mathbf w_e \hat{\mathbf x}^*_e\leq (k-\frac{1}{2})\sum\limits_{e\in E\setminus E_{k_1}}\mathbf w_e\hat{\mathbf x}^*_e.\end{aligned}$$ Combine inequalities and : $$\begin{aligned}
\sum\limits_{e\in E_k}\mathbf w_e\leq(k-\frac{1}{2})\sum\limits_{e\in E}\mathbf w_e\hat{\mathbf x}^*_e\leq(k-\frac{1}{2})\sum\limits_{e\in E}\mathbf w_e \mathbf x^*_e.\end{aligned}$$ which completes the proof.
The hardness of $k$-cycle covering when $k$ is even
===================================================
According to Algorithm \[alg1\], we trivially derive the $k$ approximate ratio whatever $k$ is odd or even. In previous section, we have shown $(k-\frac{1}{2})$ approximate ratio when $k$ is odd, but unfortunately we can’t improve the approximate ratio when $k$ is even by using similar techniques. The following Theorem may tell us a possible reason and the hardness of the problem when $k$ is even.
(Paul Erd$\ddot{o}$s, Arthur Stone, 1946[@Erdos1946])\[thm:ex\] The extremal function $ex(n; H)$ is defined to be the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. $$\begin{aligned}
ex(n;H)=(\frac{r-2}{r-1}+o(1)){n\choose2}\end{aligned}$$ where $r$ is the color number of $H$.
It is known that, when $H$ is bipartite, $ex(n; H) = o(n^2)$. Consider the special case, $H$ is an even cycle $C_{k}$, $ex(n; C_{k}) = o(n^2)$ thus $\tau_{k}(K_{n})= {n\choose2}-o(n^2)$. We have:
$$\begin{aligned}
lim_{n\rightarrow \infty} \tau_{k}(K_{n})/{n\choose2}=1\end{aligned}$$
Thus there doesn’t exist constant $0<c<1$ such that for every graph $G(V,E)$, $\tau_{k}(G)\leq cm$ holds on which is a key quality in our Algorithm \[alg2\].
k-clique covering
=================
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$clique covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$clique. The problem of minimum weight of $k-$clique covering can be described as follows: $$\begin{aligned}
\widetilde{\tau_{k}}(G_{w})=\min\{\mathbf w^T\mathbf x: A\mathbf x\geq\mathbf 1,\mathbf x\in\mathbf \{0,~1\}\}\label{CILP1}\end{aligned}$$
Where $A$ is a $k$-clique-edge adjacent matrix and $\mathbf w$ is the weight vector. $\mathbf x$ represents the characteristic vector of edge.We obtain the relaxed programming of as follows: $$\begin{aligned}
\min\{\mathbf w^T\mathbf x: A\mathbf x\geq\mathbf 1, 0 \leq\mathbf x\leq 1\}\label{CLP1}\end{aligned}$$ We compute the optimal solution $\hat{\mathbf x}^*$ of in polynomial time, then we transfer $\hat{\mathbf x}^*$ to integral vector: $$\begin{aligned}
\label{ceq1}
\mathbf x_e=
\begin{cases}
1 &\hspace{2cm} \hat{\mathbf x}^*_e\geq 1/{k\choose2}\\
0 &\hspace{2cm}o.w.
\end{cases}\end{aligned}$$ Obviously, $\mathbf x$ is a feasible solution of ILP and $\mathbf w^T\mathbf x\leq {k\choose2} \mathbf w^T\hat{\mathbf x}^*$, which implies a ${k\choose2}$-approximation algorithm of minimum $k$-clique covering problem.
Solve LP and get the optimal solution $\hat{\mathbf x}^*$.
Compute $\mathbf x$ by equation.
The $(k^2-k-1)/2$-approximation algorithm for minimum $k$-clique covering problem
=================================================================================
Similarly with Algorithm \[alg2\], we have the following approximation algorithm for minimum $k$-clique covering problem.
Set $E_{k_1}=\emptyset$.
Solve LP and get the optimal solution $\hat{\mathbf x}^*$.
**For** every $e\in E$
**If** $\hat{\mathbf x}^*_e\geq 2/(2{k\choose2}-1)$ **Then** $E_{k_1}=E_{k_1}\cup\{e\}$.
Suppose $E^\prime$ are these edges of all $k$-cliques in $G-E_{k_1}$ and let $G^\prime$ be a subgraph of $G$ induced by the edge set $E^\prime$.
Using the Greedy Algorithm or Random Algorithm, we can find an approximate solution of maximum weight bipartite graph
$B=(V_1,V_2,E_B)$, which satisfies $\mathbf W(E_B)\geq (1/2)\mathbf W(E^\prime)$. Set $E_{k_2}=E^\prime\setminus E_B$.
Output $E_k=E_{k_1}\cup E_{k_2}$.
The Algorithm \[alg4\] has $(k^2-k-1)/2$ approximate ratio for the minimum $k$-clique covering problem.
Suppose $\hat{\mathbf x}^*$ and $\mathbf x^*$ are the optimal solution of LP and ILP, respectively.\
Firstly, we indicate that $E_k$ is a $k$-clique covering. Actually, for every $k$-clique $K_{k}$ in $G$, if it doesn’t contain any edge in $E_{k_1}$, then it is a $k$-clique in $G-E_{k_1}$, thus it is a $k$-clique in $G'$. Because $G^\prime-E_{k_2}$ is a bipartite graph, of course, $G^\prime-E_{k_2}$ has no $k$-clique ($G^\prime-E_{k_2}$ has no triangle). Thus $K_{k}$ contains some edge in $E_{k_2}$. Above all, we prove that $E_k$ is a $k$-clique covering of $G$.\
Additionally, we will show the approximate ratio.\
On one hand, according to the rounding regulation, we know that: $$\begin{aligned}
\label{ineq1}
\sum\limits_{e\in E_{k_1}}\mathbf w_e\leq({k\choose2}-\frac{1}{2})\sum\limits_{e\in E_{k_1}}\mathbf w_e\hat{\mathbf x}^*_e.\end{aligned}$$ On the other hand, every $\hat{\mathbf x}^*_e$ related to $e\in E^\prime$ has the lower bound $1-2({k\choose2}-1)/(2{k\choose2}-1)=1/(2{k\choose2}-1)$ on the grounds that there exists a $k$-clique $K_k$ in $G^\prime$ containing $e$, satisfying $\hat{\mathbf x}^*_e\leq2/(2{k\choose2}-1)$ and $\sum_{e\in K_k}\hat{\mathbf x}^*_e\geq 1$. $$\begin{aligned}
\label{ineq2}
\sum\limits_{e\in E_{k_2}}\mathbf w_e\leq(1/2)\sum\limits_{e\in E^\prime}\mathbf w_e\leq(1/2)(2{k\choose2}-1)\sum\limits_{e\in E^\prime}\mathbf w_e \hat{\mathbf x}^*_e\leq ({k\choose2}-\frac{1}{2})\sum\limits_{e\in E\setminus E_{k_1}}\mathbf w_e\hat{\mathbf x}^*_e.\end{aligned}$$ Combine inequalities and : $$\begin{aligned}
\sum\limits_{e\in E_k}\mathbf w_e\leq({k\choose2}-\frac{1}{2})\sum\limits_{e\in E}\mathbf w_e\hat{\mathbf x}^*_e\leq({k\choose2}-\frac{1}{2})\sum\limits_{e\in E}\mathbf w_e \mathbf x^*_e.\end{aligned}$$ which completes the proof.
$k$-clique covering and $k$-clique packing in $K_n$
===================================================
Given a graph $G(V,E)$, a $k-$clique packing is a set of edge-disjoint $k-$cliques in $G$. The problem of maximum number of $k-$clique packing can be described as follows:
$$\begin{aligned}
\widetilde{\nu_{k}}(G)=\max\{\mathbf 1^T\mathbf x: A^T\mathbf y\leq\mathbf 1,\mathbf y\in\mathbf \{0,~1\}\}\end{aligned}$$
It is easy to see for every graph $G$, $\widetilde{\nu_{k}}(G)\leq\widetilde{\tau_{k}}(G)\leq{k\choose2}\widetilde{\nu_{k}}(G)$ holds on.\
According to Theorem \[thm:ex\], the $k$-clique covering number of $K_n$ is $\widetilde{\tau_{k}}(K_{n})=(1/(k-1)-o(1)){n\choose2}$.\
As for the packing number, we need the classical results in Block Design Theory.\
A $2$-design (or BIBD, standing for balanced incomplete block design),denoted by $(v,k,\lambda)$-BIBD, is a family of $k-$ element subsets of $X$, called blocks, such that any pair of distinct points $x$ and $y$ in $X$ is contained in $\lambda$ blocks. Here $v$ is number of points, number of elements of $X$, $k$ is number of points in a block, $\lambda$ is number of blocks containing any two distinct points. We have next famous theorem:
([@Richard1975])\[thm:Block\] Given positive integers $k$ and $\lambda$, $(v,k,\lambda)$-BIBD exist for all sufficiently large integers $v$ for which the congruences $\lambda(v-1)\equiv0(\hspace{-2mm}\mod k-1)$ and $\lambda v(v-1)\equiv0(\hspace{-2mm}\mod k(k-1))$ are valid.
When $\lambda=1$, it is easy to see $(n,k,1)$-BIBD exists if and only if $K_n$ contains a perfect $k$-clique packing, which is a $k$-clique packing such that every edge belongs to a $k$-clique. Thus the above Theorem \[thm:Block\] is equivalent to the following Theorem:
\[thm:packing\] For all sufficiently large integers $n$ satisfying $n\equiv1,k~(\hspace{-2mm}\mod k(k-1))$, then $K_n$ contains a perfect $k$-clique packing.
For all sufficiently large integers $n$ satisfying $n\equiv1,k~(\hspace{-2mm}\mod k(k-1))$, $K_n$ contains $\frac{n(n-1)}{k(k-1)}$ edge-disjoint $k$-clique. So we have $\widetilde{\nu_{k}}(K_{n})\sim\sim\frac{n(n-1)}{k(k-1)}$ and $k$-clique covering number over $k$-clique packing number in $K_n$ is $k/2$ when $n\rightarrow\infty$, that is:
$$\begin{aligned}
lim_{n\rightarrow \infty} \widetilde{\tau_{k}}(K_{n})/\widetilde{\nu_{k}}(K_{n})= lim_{n\rightarrow \infty} (1/(k-1)-o(1)){n\choose2}/\frac{n(n-1)}{k(k-1)}=k/2\end{aligned}$$
Recall Tuza’s Conjecture, which is related to the ratio of triangle covering number and triangle packing number:
(Tuza, 1981[@tuza1981]) $\tau(G)\leq2\nu(G)$ holds for every graph $G$.
For the ratio of $k$-clique covering number and $k$-clique packing number in graph $G$, the trivial upper bound is ${k\choose2}$. We guess there exists a upper bound between $k/2$ and ${k\choose2}$ for every graph $G$.
[^1]: Corresponding author. E-mail: diaozhuo@amss.ac.cn
|
---
author:
- 'F. Cuisinier'
- 'J. Köppen & A. Acker'
- 'W.J. Maciel'
title: Stars and Planetary Nebulae in the Galactic Bulge
---
Introduction
============
Planetary Nebulae are interesting objects for the study of the Galactic Bulge because they concentrate the energy of their central stars in the emission lines of their spectra, and can therefore be observed relatively easily at this distance. Furthermore the masses of their progenitor stars varying from 0.8 to ${\rm 8 M_{\sun}}$, their ages span from 50 Myr to 25 Gyr, covering more than 95% of the possible ages in the Universe, and of course in the Bulge.
The only stars with a similar ages range, that are reasonabely observable in the Bulge, are the Red Giants. They are actually the direct precursors of the Planetary Nebulae. Some elements have their abundances unmodified by the stellar evolution in Red Giants as well as in Planetary Nebulae. These elements keep the fingerprints of the chemical composition of the ISM when the progenitor star was born, and because of the span of their ages, they allow to follow its evolution over a very wide time range.
One particular point of interest are the relative abundances of elements produced in type II and in type Ia supernovae. Type II supernovae explode very rapidely, after some Myr, e.g. quasi instantaneously on the Bulge evolution timescale, whereas type Ia supernovae explode after a period of the order of one Gyr. The relative abundances of type II and type Ia supernovae should thus allow to measure the timescale of the Bulge formation.
On the other hand, elements produced during the lifetimes of the progenitor stars should allow to determine their ages - at least statistically. In Planetary Nebulae, nitrogen is very easily detectable, and has its abundance modified in high mass progenitors, that are short lived. Nitrogen abundances in Planetary Nebulae should thus help to identify recent star formation.
Abundances in Stars and in Planetary Nebulae
============================================
We derived abundances for a sample of 30 PN, that we observed with high quality spectroscopy (Cuisinier et al. 2000). These abundances being of really better quality than others available in the literature, we will only consider these ones here.
Abundances for individual elements in stars are up to now only available for a sample of 11 red giants, from McWilliam & Rich (1994).
Unfortunately, a direct comparison of abundances is not possible, the elements detectable in stars with a good confidence being different from those detectable in Planetary Nebulae.
We compared therefore the distributions of O, S and Ar in Planetary Nebulae in the Bulge and in the Disk, these elements representing the pristine abundances of the ISM (Figure 1, left panel, for O). We found the abundances distributions to be quite similar, like the Fe abundances in the stars (Mc William & Rich 1994).
On the other hand, the N/O ratios comparison in the Bulge and the Disk (figure 1, left panel) show that the young progenitor, N-rich Planetary Nebulae, that are present in the Disk, are lacking in the Bulge. From the Planetary Nebulae, the Bulge does not seem to have formed stars recently.\
If the Red Giant and the Planetary Nebulae populations in the Bulge seem to be quite similar in the light of our study, the picture that arises from a comparison of the various elements originating from type II and type Ia supernovae that are detected in Planetary Nebulae and in Red Giants remains very puzzling: Mg and Ti, that are enhanced over Fe, seem to favor a quick evolution, whereas He, O, Si, S, Ar and Ca show normal abundances patterns, and favor a much slower evolution.
Cuisinier, F., Maciel, W.J., Köppen, J., Acker, A., Stenholm, B. 2000, , 353, 543
McWilliam, A., Rich, R.M., 1994, , 91, 749
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abstract: 'In this paper, we model nested polar code construction as a Markov decision process (MDP), and tackle it with advanced reinforcement learning (RL) techniques. First, an MDP environment with state, action, and reward is defined in the context of polar coding. Specifically, a state represents the construction of an $(N,K)$ polar code, an action specifies its reduction to an $(N,K-1)$ subcode, and reward is the decoding performance. A neural network architecture consisting of both policy and value networks is proposed to generate actions based on the observed states, aiming at maximizing the overall rewards. A loss function is defined to trade off between exploitation and exploration. To further improve learning efficiency and quality, an “integrated learning” paradigm is proposed. It first employs a genetic algorithm to generate a population of (sub-)optimal polar codes for each $(N,K)$, and then uses them as prior knowledge to refine the policy in RL. Such a paradigm is shown to accelerate the training process, and converge at better performances. Simulation results show that the proposed learning-based polar constructions achieve comparable, or even better, performances than the state of the art under successive cancellation list (SCL) decoders. Last but not least, this is achieved without exploiting any expert knowledge from polar coding theory in the learning algorithms.'
author:
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Email: {huanglingchen,zhanghuazi,lirongone.li,yiqun.ge,justin.wangjun}@huawei.com
title: Reinforcement Learning for Nested Polar Code Construction
---
Polar codes, Nested polar code construction, Markov decision process, Reinforcement learning, Integrated learning
Introduction {#section:intro}
============
In communication systems, the capacity of an AWGN channel is defined in theory [@Shannon]. Classic code construction methods are built upon coding theory, in which code performance can be theoretically modeled in terms of various types of code properties, e.g. minimum distance, decoding threshold, reliability, etc. However, it seems insufficient for us to rely on only these classic coding theory metrics in facing of such practical concerns and application-specific requirements as realistic channel types, decoding latency and complexity and so on.
Recently, artificial intelligence (AI) techniques have been applied to physical layer design. AI techniques can be a tool to design or optimize error correction codes [@AI:AI_coding], while leaving their legacy encoding and decoding architectures and implementations unchanged. Within a “constructor-evaluator” framework [@AI:AI_coding], AI algorithms such as policy gradient, genetic algorithm, and actor critic, are capable of constructing linear block codes and polar codes with as good performances as the state of the art. In [@AI:RL_LDPC], RL and Monte Carlo tree search (MCTS) are combined to guide edge growth in LDPC code construction. In [@AI:GeneAlg_Polar; @AI:GeneAlg_LDPC], genetic algorithms are used to design polar codes and LDPC codes. The main difference from [@AI:AI_coding] is that coding expert knowledge is utilized during the initialization to speed up the learning process.
In this paper, our motivation is to investigate the feasibility of using AI technologies to explore the design space for wireless systems. Channel code, especially polar code, is a good example for this endeavor. We propose novel RL algorithms for designing nested polar codes [@Polar:nested]. Because nested polar code construction (sequential information sub-channel selection) is inherently modeled as a Markov decision process (MDP), and RL algorithms can be applied to approach the optimum. To improve training efficiency and code performance, we propose an integrated learning paradigm and various parameter optimization techniques.
Preliminaries {#section:motivation}
=============
Polar code construction {#section:motivation:polar}
-----------------------
Polar codes [@Polar:Arikan] are the first class of capacity-achieving codes (under successive cancellation (SC) decoding). For polar codes, physical channels are synthesized to polarized subchannels, with the most reliable ones selected to carry information bits. As a result, an $(N,K)$ polar code is defined by the $K$ most reliable subchannel indices, namely information set $\cal I$. The remaining $(N-K)$ subchannel indices are defined as frozen set $\cal F$. For general binary-input memoryless channels, density evolution (DE) was applied to estimate subchannel reliability [@Polar:DE1_Mori; @Polar:DE2_Mori], and improved in [@Polar:DE3_Tal] and analyzed in [@Polar:DE4_Pedarsani] in terms of complexity. For AWGN channels, Gaussian approximation to density evolution (DE/GA) was proposed [@Polar:GA_Trifonov] to further reduce complexity with negligible performance loss.
To improve the performance of polar codes at finite length, enhanced decoding algorithms are proposed [@Polar:SCL; @Polar:SCS]. Among them, SC list (SCL) decoding achieves the best tradeoff among decoding latency, complexity and performance. However, to our best knowledge, for polar codes with SCL-based decoders, theoretically optimal code construction is still an open problem. Existing constructions either directly adopt DE/GA, which are designed for SC rather than SCL, or apply genetic algorithms for SCL decodings [@AI:AI_coding; @AI:GeneAlg_Polar].
Nested polar codes {#section:motivation:nested_polar}
------------------
In practical communication systems where code rate and length adaption is required, efficient code description is mandatory. For example, 5G enhanced mobile broadband (eMBB) [@Polar:212] supports thousands of polar codes with different $(N,K)$ combinations. It is impossible to store all code configurations separately, due to large overhead. It is much more convenient for description and implementation to impose a nested property [@Polar:nested], so that all polar codes of the same mother code length can be derived from a single nested sequence. Specifically, denote ${\cal F}_{N,K}$ as the frozen set of an $(N,K)$ polar code. ${\cal F}_{N,N-1}, {\cal F}_{N,N-2}, \cdots, {\cal F}_{N,0}$ can be constructed sequentially, on condition that ${\cal F}_{N,N-1} \subset {\cal F}_{N,N-2} \subset \cdots \subset {\cal F}_{N,0}$. As seen, a single nested sequence can be obtained as $\{{\cal F}_{N,N-1}, \textit{setdiff}( {\cal F}_{N,N-2}, {\cal F}_{N,N-1} ), \cdots, \textit{setdiff}( {\cal F}_{N,0}, {\cal F}_{N,1})\}$.
Nested polar codes are adopted by 5G in the form of a reliability sequence of length $N_{\max}=1024$ [@Polar:212]. To construct an $(N,K)$ polar code from the length-$N_{\max}$ nested sequence ($N=2^n \leq N_{\max}$),
1. First, a sequence $Seq_{N}$ of length $N$ is extracted from the length-$N_{\max}$ sequence (by taking all indices $\{i:i\in Seq_{N_{\max}},i<N\}$ while keeping the ordering).
2. Second, the last $K$ entries of $Seq_{N}$ are selected as the information set.
Reinforcement Learning for nested polar code construction {#section:learning}
=========================================================
In this section, we show that nested polar code construction is actually a Markov decision process (MDP) that can be tackled by reinforcement learning. We further discuss some applicable learning algorithms.
Constructing nested polar code with MDP {#section:learning:MDP}
---------------------------------------
Nested polar code construction can be modeled as an MDP for the following reasons:
- According to Markov property of nested polar code construction in Fig. \[fig:graphical\_models\], the construction of $(N,N-K-1)$ polar code $c_{K+1}$ and its performance $e_{K+1}$ depend only on that of $(N,N-K)$ polar code $c_{K}$ and a subchannel selection $b_K$;
- The goal is to optimize all $(N,K)$ polar codes for $K=1,2,\cdots,N-1$ through maximizing an overall performance metric $\sum_K{e_K}$.
To explicitly map the nested polar code construction into an MDP task, we define a base environment $\left( \mathcal {S}, \mathcal {A}, \mathcal {R} \right)$:
- A **state** is denoted by a length-$N$ binary vector $s_K^N \in \mathcal {S}=\{0,1\}^N$, whose support set is ${\cal F}_{N,N-K}$. The initial state $s_0^N$ is an all-zero vector corresponding to empty set (${\cal F}_{N,N}$).
- An **action** is denoted by an integer $a_K^N \in \mathcal {A}=\{0,1,...,N-1\}$, such that $a_K^N \notin {\cal F}_{N,N-K}$ and $a_K^N \cup {\cal F}_{N,N-K} = {\cal F}_{N,N-K-1}$.
- The **reward** value of state $s_K^N$ is $r_K^N \in \mathcal {R}$, representing the performance of the polar code defined by ${\cal F}_{N,N-K}$.
The state transfer process is deterministic, i.e., given $s_K^N$ and $a_K^N$, $s_{K+1}^N$ can be determined. The maximum length of an episode is $N$. A trajectory of the base environment, $(s_0^N$, $a_0^N$, $s_1^N$, $a_1^N$, $\cdots$, $s_{N-1}^N$, $a_{N-1}^N$, $s_{N}^N)$, corresponds to the nested polar code construction (ordered sequence) $\{a_0^N$, $a_1^N$, $\cdots$, $a_{N-1}^N\}$. In the following, the superscripts of $s_K^N$, $a_K^N$ and $r_K^N$ are omitted with some abuse of notation.
Following the “constructor-evaluator” framework [@AI:AI_coding], we propose to directly evaluate the rewards through decoding performance. Monte-Carlo (MC) simulations are conducted to output a block error rate (BLER) performance for each code construction. The evaluator implements SCL decoding algorithms, which generate a list of $L$ codewords. We name two types of SCL decoders based on final output selection:
- SCL-PM: select the first codeword, i.e, the most likely one with the smallest PM;
- SCL-Genie: select the correct codeword, as long as it is among the $L$ surviving ones.
Sufficient decoding error events are counted to obtain an accurate BLER estimation. Then, the reward value is defined as $r\triangleq-\log_{10}BLER$. [^1]
Reinforcement learning {#section:learning:RL_algs}
----------------------
Nested code construction is actually to search an optimal sequence in a large solution space. RL would help approach the optimum, dragged by a reward. The devised reward, through one real value metric, should represent the performances of all component codes.
Strictly speaking, an RL agent interacts with the MDP environment over discrete timesteps. At each timestep $t$, the agent observes a state $s_t$, chooses an action $a_t$ according to its policy $\pi(a_t|s_t)$ and obtains a reward $r_t$ from the environment. The goal of this agent is to optimize its policy in order to maximize the discounted return $R_t=\sum_{i=0}^{\infty} \gamma^{i} r_{t+i}$ at each timestep. Here discount factor $\gamma \in [0,1)$ is introduced to trade off the contribution of immediate and long term rewards to return value.
For the nested polar code construction task, the state space is $2^{N}$, the action space is $N$ and the solution space, i.e. trajectory space, is $N!$. Concerning the large solution space, it is necessary to have *sample efficient* RL algorithms. *Sample efficiency* is defined by number of samples used to solve the MDP task, where an MDP sample is a state-action-reward $(s,a,r)$ tuple. In literature, sample efficient RL algorithms include advantage actor critic (A2C), proximal policy optimization (PPO) [@RL:PPO] and actor critic using Kronecker-factored trust region (ACKTR) [@RL:ACKTR], etc.
We apply PPO [@RL:PPO] as it is by far the most advanced model-free algorithm. The PPO is an extension of A2C, where a Kullback-Leibler (KL) divergence constraint is imposed between the updated policy and the old policy, i.e. a trust region constraint [@RL:TRPO].
For the PPO, the policy loss function is defined, $$\label{Equ.actor}
Loss_{A} = \hat{A}(s_t,a_t) \cdot min\left(r_t(\theta), clip(r_t(\theta), 1-\epsilon, 1+\epsilon)\right),$$ where $\hat{A}(s_t,a_t)=R(s_t,a_t)-V(s_t)$ is the estimate of advantage function for taking action $a_t$ at state $s_t$; $r_t(\theta)=\frac{\pi_{\theta}(a_t|s_t)}{\pi_{\theta_{old}}(a_t|s_t)}$ is the probability ratio between the updated policy $\pi_{\theta}$ and the old policy $\pi_{\theta_{old}}$ for taking action $a_t$ at state $s_t$; $\pi_{\theta}(a_t|s_t)$ is the policy function parameterized by $\theta$; $\epsilon$ is a clipping ratio to constrain the probability ratio $r_t(\theta)$.
The value loss function is defined, $$\label{Equ.critic}
Loss_{C} = \left( \hat{A}(s_t,a_t) \right)^2.$$
For the advantage estimation $\hat{A}(s_t,a_t)$, a general advantage estimation method (GAE) [@RL:GAE] implements an exponential average among advantage estimations of different steps to trade off between the estimation bias and variance, $$\label{Equ.GAE}
\begin{aligned}
\hat{A}^{GAE}(s_t,a_t)& = (1-\lambda) \sum_{i=1}\left( \lambda^{i-1} \hat{A}^{i}(s_t,a_t) \right), \\
\hat{A}^{i}(s_t,a_t)& = \hat{R}^{i}(s_t,a_t) - V(s_t), \\
\hat{R}^{i}(s_t,a_t)& = \sum_{j=0}^{i-1}\left( \gamma^{j} r_{t+j} \right) + \gamma^{i} V^{\pi}(s_{t+i}),
\end{aligned}$$ where $\lambda$ is the exponential moving average parameter.
A policy function entropy regularization, defined in , can be considered in policy loss function to trade off between exploration and exploitation. $$\label{Equ.entropy}
H_{A}(s) = -\sum_{a}\pi(a|s)\log \pi(a|s).$$
Integrated learning for nested polar code construction {#section:int_learning}
======================================================
In the section, we propose an integrated learning method for nested polar code construction to improve the sample efficiency and code performance.
For reinforcement learning algorithms, policy function is initialized to explore all possible MDP trajectories with equal probability. However, for most trajectories in the trajectory space, the accumulated rewards are far worse than optimal one(s). Given prior knowledge about the distributions of actions with large rewards, the policy function can be pretrained to bias the exploration towards trajectories with larger accumulated rewards. Depending on the prior knowledge, this pretraining can significantly accelerate the learning process [@RL:Pretrain1; @RL:Pretrain2; @RL:Pretrain3].
In the context of polar code construction, we may rely on sub-optimal expert knowledge (e.g., DE/GA constructions) for pretraining, where direct state-action $(s,a)$ pairs (demonstrations) are available. However, genetic algorithm is the best choice to generate a large population of (sub-)optimal code constructions, corresponding to the distribution of states with large rewards. As the genetic algorithm converges, its population already contains code constructions with the best performances. Moreover, the genetic algorithm in [@AI:AI_coding] does not require any expert knowledge, which means the proposed method also learns everything from scratch.
An integrated learning is proposed in Alg. \[alg:integrated\_learning\]. Firstly, the polar code constructions are generated by genetic algorithm. These constructions are used to produce pretraining examples. Policy function is then pretrained in supervised learning manner. Nested polar code constructions are learned through reinforcement learning, as in section \[section:learning:RL\_algs\], with the pretrained policy function.
$polar\_codes\_constr$ = genetic\_algorithm() $(state, action)$ =\
example\_generation($polar\_codes\_constr$) $policy\_function\_pretrained$ =\
pretrain($policy\_function, state, action$) $nested\_polar\_codes\_constr$ =\
reinforcement\_learning($policy\_function\_pretrained$)
For each $(N,K)$ pair, we apply the genetic algorithm in [@AI:AI_coding] to generate a population of (sub-)optimal polar codes. In terms of MDP, these code constructions represent good states with large reward values. The remaining problem is how to design state-action $(s,a)$ pairs from these constructions. One characteristic of this MDP is that, the state is a collection (set) of history actions taken to reach it. Meanwhile, the order of actions is neither distinguishable from the state, nor relevant to reach the state. Therefore, the state-action $(s,a)$ pairs can be produced based on two intuitions,
1. Given a current state, if the agent can take one action to reach a good state, then this can be a valid state-action pair;
2. If the current state is a good state, a potentially good choice of action can be the ones that has not been taken to reach the current state, while is recorded by some other states with close information length.
The process to produce state-action $(s,a)$ pairs from good states is described in Alg. \[Alg:pretrain\].
example\_generation($polar\_codes\_constr$) $examples=[]$ \# Based on intuition-1): $data=state$ $data(n)=0$ $label=n$ $examples.append([data,label])$ \# Based on intuition-2): $data=state_s$ $label=n$ $examples.append([data,label])$ $examples$
Evaluation {#section:evaluation}
==========
In this section, we elaborate the model of the reinforcement learning algorithms, and evaluate the sample efficiency of various schemes.
Model {#section:evaluation:model}
-----
For the reinforcement learning algorithms, we use neural networks to represent the policy and value function. The same neural network architecture, shown in Fig. \[fig:NN\_architecture\], is deployed for all of the nested polar construction tasks. For an input state $s$, a feed-forward network was used for feature extraction, with two fulled connected layers, with $2N$ tanh units per layer. This feature layer was shared by policy and value function. For the output layer, the policy function used a linear layer to screen out previously selected subchannels (e.g., by subtracting a larger value from the corresponding entries), and followed by a softmax nonlinearity to generate probability mass function (PMF). The value function used a linear layer to output an estimated value for state $s$.
One synthesized loss function is used for simultaneous training of policy and value function, $$\label{Equ.Loss}
Loss=Loss_A + \beta_c Loss_C - \beta_e H_A$$ where $Loss_A$ and $Loss_C$ are the loss functions for actor and critic, and $H_A$ is the entropy for policy function, defined in section \[section:learning:RL\_algs\]. The policy function entropy is subtracted in loss function to encourage exploitation.
Reinforcement learning {#section:evaluation:RL}
----------------------
We conduct several experiments under SCL-Genie decoding. The default parameters are listed in Table \[Table.Params\].
Parameters values
----------------------------------- ----------------------
polar code length $N=256$
decoder SCL-Genie, SCL-PM
SC list size $L=8$
reward $r=-\log_{10}(BLER)$
BLER simulation error event count 1000
clipping ratio in policy loss $\epsilon=0.2$
critic loss weight $\beta_c=0.5$
entropy weight $\beta_e=0$
learning rate $3\cdot10^{-4}$
batch size $n_{batch}=64$
feature extraction network 1024,1024
discount factor $\gamma=0.2$
GAE factor $\lambda=0.95$
: Default parameters setting
\[Table.Params\]
Fig. \[fig:cmp\_alg\] shows the episode rewards of A2C, ACKTR and PPO for 100E3 timesteps. The number of timesteps for return estimation was optimized for each algorithm. PPO outperformed A2C and ACKTR in terms of sample efficiency by a significant margin, and was therefore adopted in the following experiments.
Fig. \[fig:cmp\_Ent\] shows the episode rewards for the amount of policy entropy evolved in loss function. Similar sample efficiency is observed for entropy weight $\beta_e \leq 0.01$. Nevertheless, entropy weight $\beta_e=0$ shows slightly better convergence performance.
Fig. \[fig:cmp\_gamma\] shows the episode rewards for discount factor selection. For a smaller discount factor, the sample efficiency is increased since the current return would be affected by shorter future actions. While for this learning task, the convergence performance is not compromised.
Integrated learning {#section:evaluation:pretrain}
-------------------
In this subsection, we evaluate the integrated learning to show its improved sample efficiency.
We first obtained a population of polar codes for each $(N,K)$ pair by genetic algorithm. Then we applied Alg. \[Alg:pretrain\] to generate examples of state-action $(s,a)$ pair. The same policy network architecture is used in integrated learning as in Fig. \[fig:NN\_architecture\]. The policy network was trained on randomly sampled examples with stochastic gradient descent to minimize the training loss function, $$\label{Equ.Loss_pre}
Loss_{pre}=Loss_{Apre} - \beta_{epre} H_{Apre}$$ where $Loss_{Apre}$ is the cross entropy between policy output and the (one-hot) action label, $H_{Apre}$ is the entropy value of policy function, with entropy weight $\beta_{epre}=1.0$. After 20 epoches of training, the policy function is saved for reinforcement learning as described in section \[section:learning:RL\_algs\].
Fig. \[fig:cmp\_pretrain\] shows the episode rewards for 100E3 timesteps for reinforcement learning and integrated learning. It is shown that integrated learning has better sample efficiency as well as larger episode reward values.
BLER performance {#section:results}
----------------
For nested polar code construction with code length of $256$, the error correction performance of the learned codes are compared with those constructed by DE/GA. It should be noted that the comparison is unfair with respect to description and implementation complexity, because the constructions by DE/GA are not necessarily nested.
We consider two MDP tasks with different decoders:
- SCL-Genie decoding under AWGN channel
- SCL-PM decoding under AWGN channel
The same learning method (parameters) are used for both MDP tasks.
For SCL-Genie decoding under AWGN channel, nested polar code constructions are learned by reinforcement learning and integrated learning with 1E6 training timesteps. Fig. \[fig:cmp\_SCLGenie\] shows the relative EsN0 value (at BLER of $10^{-2}$) for the three constructions. The nested polar constructions learned by both reinforcement learning and integrated learning outperform the case-by-case DE/GA constructions for a majority of cases. Integrated learning exhibits even better performance, with a maximum gain over DE/GA approaching 0.3dB.
For SCL-PM decoding under AWGN channel, nested polar code constructions are learned with 100E3 training timesteps. Fig. \[fig:cmp\_SCLPM\] shows that the learned nested polar constructions outperform the case-by-case DE/GA constructions for almost all information length. The maximum performance gain achieves as large as 1.2dB.
Conclusions {#section:conclusion}
===========
In this paper, the “constructor-evaluator” framework in [@AI:AI_coding] is adopted to construct polar codes. In particular, we show that constructing nested polar codes can be viewed as a Markov decision process. Thus, RL techniques are employed to iteratively optimize the “code construction” policy without expert intervention. To facilitate faster and better convergence, genetic algorithm is integrated in the RL algorithms to provide prior knowledge about (sub-)optimal code constructions. We carry out extensive experiments to compare the learning process under various settings. The polar code constructions for both SCL-PM and SCL-Genie decoders are obtained, which exhibit superior performance over classic constructions.
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[^1]: For SCL-Genie decoder, when $K \leq \log_{2}L$, the codeword would always be decoded correctly. For such cases, the reward value is set to $0$.
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****
**Ali H. Chamseddine$^{1,2}$ , Viatcheslav Mukhanov$^{3,4,5}$ **
[**Abstract**]{}
We present a theory of ghost-free massive gravity where the mass of the graviton is generated through the Brout-Englert-Higgs (BEH) mechanism and one of the four scalar fields used is that of mimetic gravity. The mass term is not of the Fierz-Pauli type and the constraint eliminates the Boulware-Deser ghost which is absent to all orders. We perform a detailed analysis using the methods of cosmological perturbation theory and consider quantum fluctuations of the degrees of freedom of massive graviton and mimetic matter. It is shown that for three of the degrees of freedom of the graviton of mass $m$ the nonlinear corrections become comparable to the linear terms already at a length scale of order $m^{-\frac{1}{2}}$. Thus, at smaller scales they become strongly coupled and the graviton remains with two transverse degrees of freedom which get strongly coupled only at Planck scale. The mimetic field behaves as cold particles of half of the graviton mass and could well explain the source of dark matter in our universe. In the weakly coupled domain mimetic matter is completely decoupled from the massive graviton.
Introduction
============
The problem of finding a consistent theory of massive gravity has recently attracted considerable interest. It is by now well established that the simplest way of giving mass to the graviton without explicit breaking of diffeomorphism invariance is by employing four scalar fields, which acquire vacuum expectation values [@tH; @CM]. As a result the scalar fields are absorbed making the graviton massive on Minkowski background. However, four scalar fields have generically four degrees of freedom and only three of them are needed to provide mass to the graviton. Out of four scalar fields, used to preserve global Lorentz invariance, one field must be taken with negative kinetic energy and if its perturbations around a spontaneously broken symmetry background propagates then there would appear a ghost mode leading to inconsistency of the theory. By adopting the Fierz-Pauli (FP) term for the mass [@FP] one can vanish the kinetic term for these perturbations, so that the dangerous mode disappears in the linear approximation. Generically it reappears again and starts to propagate on a nontrivial background and is known as nonlinear Boulware-Deser (BD) ghost [@BD]. There was recently an attempt to extend the FP-term to higher orders in such a way as to avoid the propagation of this field to all orders on a nontrivial background. The resulting theory which might avoid the non-linear BD-ghost is rather unambiguous and its action is given by an infinite expansion of a square root function [@Kab] which can be simplified and rewritten as a quadratic one by making use of auxiliary fields [@CMquad]. However, in some particular backgrounds, the ghost mode nevertheless gets excited, raising a question about consistency of the theory [@CMghost; @CMghost1].
One can wonder whether it is possible to have a ghost-free massive gravity which is not very restricted and well behaved to all higher orders? We will show in this paper that this can be easily achieved in mimetic gravity [@CMmim], with non Fierz-Pauli mass term, where we use the constrained scalar field imitating Dark Matter as one of the four fields needed to realize Brout-Englert-Higgs (BEH) mechanism for the graviton mass.
BEH-mechanism for massive gravity
=================================
In BEH-mechanism for massive gravity a central role is played by four *scalar* fields $\phi^{A}\left( x^{\mu}\right) ,$ $A=0,1,2,3,$ which in the broken symmetry phase, in Minkowski space-time, acquire vacuum expectation values $$\left\langle \phi^{A}\right\rangle =\delta_{\mu}^{A}x^{\mu}\equiv x^{A}.
\label{1}$$ Note that the vacuum state is degenerate and the different vacua are related by Poicnare transformations. From the auxiliary induced metric $H^{AB}=g^{\mu\nu}\phi_{,\mu}^{A}\phi_{,\nu}^{B},$ where $\phi_{,\mu}^{A}\equiv
\frac{\partial\phi^{A}}{\partial x^{\mu}},$ it is convenient to build the diffeomorphism invariant set of scalars$$\bar{h}^{AB}=H^{AB}-\eta^{AB}, \label{2}$$ and use them to give mass to the graviton. Here $\eta^{AB}=(1,-1,-1,-1)$ is the auxiliary Minkowski metric. Let us consider the small perturbations of the scalar fields around broken symmetry phase$$\phi^{A}=x^{A}+\chi^{A}, \label{3}$$ which induce small metric perturbations$$g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}. \label{4}$$ Then $$\bar{h}^{AB}=h^{AB}+\partial^{A}\chi^{B}+\partial^{B}\chi^{A}+h^{A\nu}\chi_{,\nu}^{B}+h^{B\nu}\chi_{,\nu}^{A}+\eta^{\mu\nu}\chi_{,\mu}^{A}\chi
_{,\nu}^{B}+h^{\mu\nu}\chi_{,\mu}^{A}\chi_{,\nu}^{B}, \label{5}$$ where $h^{AB}\equiv\delta_{\mu}^{A}\delta_{\nu}^{B}h^{\mu\nu},$ $\partial
^{A}\chi^{B}\equiv\eta^{A\nu}\chi_{,\nu}^{B},$ $\eta^{A\nu}\equiv\delta_{\mu
}^{A}\eta^{\mu\nu},$ $h^{A\nu}=\delta_{\mu}^{A}h^{\mu\nu}$ etc. The scalars $\bar{h}^{AB}$ are diffeomorphism invariant and by an appropriate choice of coordinate system $x^{\mu}$ we can impose four gauge conditions on fourteen functions $\chi^{A}$ and $h^{\mu\nu}$ out of which they are built. For instance, in so called *unitary* gauge where $\chi^{A}=0,$ $\bar{h}^{AB}$ coincides with the metric perturbations $h^{\mu\nu}.$ Thus, it is clear that $\overline{h}^{AB}$ can be used to construct, in a gauge invariant way, the mass term for the graviton via BEH-mechanism. For instance, a theory with the action$$S=\int d^{4}x\sqrt{-g}\left[ -\frac{1}{2}R+\frac{m^{2}}{8}\left( \bar{h}^{2}-\bar{h}^{AB}\bar{h}_{AB}\right) \right] , \label{6}$$ where $\bar{h}=\bar{h}_{A}^{A}$ and we use the units in which $8\pi G=1$, describes massive gravity with FP-mass term in broken symmetry phase. Note that contraction of capital indices guarantees the invariance of the theory with respect to a particular choice of the vacuum in a huge landscape of degenerate vacuum states related by Poincare transformations.
One can easily see why the Fierz-Pauli combination is so special by considering a decoupling limit when the metric perturbations $h^{\mu\nu}$ vanish. In this case the mass term $$\frac{m^{2}}{8}\left( \alpha\cdot\bar{h}^{2}-\bar{h}^{AB}\bar{h}_{AB}\right)
, \label{7}$$ where $\alpha$ is a numerical coefficient, becomes (up to total derivative)$$\frac{m^{2}}{2}\left( -\frac{1}{4}F^{AB}F_{AB}+\left( \alpha-1\right)
\left( \partial_{A}\chi^{A}\right) ^{2}+O\left( \chi^{3}\right) \right)
\label{8}$$ with $F_{AB}=\partial_{A}\chi_{B}-\partial_{B}\chi_{A}$. It is clear that unless $\alpha=1,$ the theory with Lagrangian (\[8\]) describes four fields one of which is inevitably a ghost. Only the choice of the mass combination $\bar{h}^{2}-\bar{h}^{AB}\bar{h}_{AB}$ insures that the term $\left( \bar
{h}_{00}\right) ^{2},$ that provides dynamics to the ghost field $\chi^{0},$ is absent in the Lagrangian and $\chi^{0}$ becomes similar to the gauge potential $A^{0}$ in Maxwell theory, which corresponds to the first term in (\[8\]). However, if one considers the propagation of the fields in a curved background (for instance induced by an external source) with metric $g_{\mu
\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, then the contribution $$\bar{h}^{2}=2\bar{h}_{i}^{i}\bar{h}_{0}^{0}+...=2h_{i}^{i}\left( \dot{\chi
}^{0}\right) ^{2}+..., \label{8a}$$ due to the higher order terms in (\[5\]) induces a propagator for $\chi^{0}$, thus resurrecting the nonlinear BD-ghost in non-trivial background even for FP mass term. One can try to get rid of the nonlinear ghost by adding to the action (\[6\]) higher order terms $\bar{h}^{3},\bar{h}\bar{h}^{AB}\bar
{h}_{AB},...$ and such strategy leads to a nearly unambiguous theory with action [@Kab]:$$S=\int d^{4}x\sqrt{g}\left[ -\frac{1}{2}R+\frac{m^{2}}{2}\left( S^{2}-S^{AB}S_{AB}\right) \right] , \label{8b}$$ where$$S_{AB}=\sqrt{\eta_{AB}+\bar{h}_{AB}}-\eta_{AB}. \label{8c}$$ To second order in $\overline{h}$ this theory reduces to FP theory. In higher orders it is represented by an infinite series in $\bar{h}$ and only with the help of auxiliary *vierbein type* fields the square root in (\[8c\]) can be given in finite form and the theory becomes quadratic in $S$ [@CMquad]. The theory (\[8b\]) looks promising from the point of view of keeping the field $\chi^{0}$ non-dynamical to all orders. However, it was shown in [@CMghost], [@CMquad] that the term $\bar{h}^{0i}\bar{h}_{0i},$ which is already present in FP-term, as part of the quadratic term $h^{A\nu}\chi_{,\nu}^{B}$ in (\[5\]), gives the following contribution to the action$$\bar{h}^{0i}\bar{h}_{0i}=h^{0i}h^{0i}\left( \dot{\chi}^{0}\right) ^{2}+...
\label{9}$$ which is not canceled by higher order terms. Thus, in some backgrounds the field $\chi^{0}$ starts to propagate again and there is no guarantee that this mode would not induce a ghost in an arbitrary background. A reader could wonder whether it is possible to choose a gauge $h^{0i}=0$ where the term (\[9\]) is absent$.$ It was shown in [@CMghost], [@CMghost1] that in this gauge the propagating $\chi^{0}$ mode reappears in a more complicated way. This is why the question about ghost free massive gravity remains yet open. In this paper we address the following questions:
- Is it in principle possible to construct ghost free theory which will describe massive graviton with five degrees of freedom in Minkowski background ?
- Must the FP mass term be necessarily always used in massive gravity?
Mimetic *non* Fierz-Pauli massive gravity
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As we have seen above the problem in massive gravity is that the field $\phi^{0}$ becomes dynamical by itself and thus inevitably leads to the appearance of linear or nonlinear ghost. The idea we exploit in this paper is to use in BEH-mechanism the mimetic field [@CMmim] as one of four fields responsible for the appearance of the graviton mass. This field $\phi^{0}$ always obeys the constraint $$g^{\mu\nu}\partial_{\mu}\phi^{0}\partial_{\nu}\phi^{0}=1,\label{10}$$ and in a synchronous coordinate system serves as time coordinate, so that the generic solution of (\[10\]) is$$\phi^{0}=t+A\label{11}$$ where $A$ is a constant of integration. Combined with the longitudinal mode of gravity it leads to the appearance of mimetic matter which can well imitate the observed Dark Matter in the Universe [@CMmim], [@CMVmim]. The remaining three scalar fields can provide three extra degree of freedom needed for the massive graviton. Thus, in such theory we are guaranteed to have no ghosts to any order in perturbation theory. It happens that in this case the mass term for the graviton is unambiguously fixed. It is not of the Fierz-Pauli type and the action of the corresponding theory is $$S={\displaystyle\int}
d^{4}x\sqrt{g}\left[ -\frac{1}{2}R+\frac{m^{2}}{8}\left( \frac{1}{2}\bar
{h}^{2}-\bar{h}^{AB}\bar{h}_{AB}\right) +\lambda\left( g^{\mu\nu}\partial_{\mu}\phi^{0}\partial_{\nu}\phi^{0}-1\right) \right] .\label{12}$$ The mass term has a relative coefficient of $-\frac{1}{2}$ between the $\bar{h}^{2}$ and $\bar{h}^{AB}\bar{h}_{AB}$ terms. The mimetic constraint term does not violate the Lorentz invariance of the ground state because it is obviously invariant with respect to the transformation $\phi^{0}\rightarrow\tilde{\phi}^{0}=\Lambda_{B}^{0}\phi^{B}$ for $\phi^{B}=\delta
_{\mu}^{B}x^{\mu}.$ Variation of this action with respect to the metric $g_{\mu\nu}$ gives the following modified Einstein equations $$\begin{aligned}
G_{\nu}^{\mu} & =2\lambda\partial^{\mu}\phi^{0}\partial_{\nu}\phi^{0}-\frac{m^{2}}{8}\left( \frac{1}{2}\bar{h}^{2}-\bar{h}^{AB}\bar{h}_{AB}\right) \delta_{\nu}^{\mu}\nonumber\\
& +\frac{m^{2}}{2}\left( \frac{1}{2}\bar{h}\partial^{\mu}\phi_{A}\partial_{\nu}\phi^{A}-\bar{h}_{B}^{A}\partial^{\mu}\phi_{A}\partial_{\nu}\phi^{B}\right) ,\label{13}$$ where $G_{\nu}^{\mu}=R_{\nu}^{\mu}-\frac{1}{2}R\delta_{\nu}^{\mu}$ is the Einstein tensor and we recall that capital indices are raised and lowered with the help of the auxiliary metric $\eta_{AB},$ while for Greek indices the space-time metric $g_{\mu\nu}$ is used. Variation with respect to the scalar fields $\phi_{A}$ leads to$$\partial_{\mu}\left( \sqrt{g}\left[ \frac{m^{2}}{2}\left( \frac{1}{2}\bar{h}\partial^{\mu}\phi^{A}-\bar{h}_{B}^{A}\partial^{\mu}\phi^{B}\right)
+2\lambda\delta_{0}^{A}\partial^{\mu}\phi^{0}\right] \right) =0.\label{phi}$$ Finally the constraint (\[10\]), which follows by varying with respect to $\lambda,$ takes the simple form $$\bar{h}_{0}^{0}=0.\label{14}$$ In the next section we will analyze linear perturbations in this model in component form, using the methods of cosmological perturbation theory [@vm]. An explicitly covariant study of these perturbations is given in a separate publication [@PRL].
Graviton on Minkowski background
================================
We first study the linearized theory considering small metric perturbations in Minkowski background, that is, assuming $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \label{15}$$ and $\left\vert h_{\mu\nu}\right\vert \ll1.$ Correspondingly the scalar fields deviate from their vacuum expectations values by $\chi^{A},$ that is,$$\phi^{A}=x^{A}+\chi^{A}. \label{16}$$ To explicitly reveal the true physical degrees of freedom of the massive graviton it is convenient to use the methods of cosmological perturbation theory and classify the perturbations with respect to irreducible representations of the spatial rotation group [@vm]. The metric component $h_{00}$ behaves as a scalar under rotations and it is convenient to denote it as$$h_{00}=2\phi, \label{17}$$ where $\phi$ is a 3-scalar. The space-time components $h_{0i}$ behaves as a 3-vector that can be written as a sum of a longitudinal and transverse parts:$$h_{0i}=B_{,i}+S_{i}, \label{18}$$ where $B_{,i}=\frac{\partial B}{\partial x^{i}}$ and $S_{i}$ has zero divergence, that is, $\partial^{i}S_{i}=0.$ Finally $h_{ij}$ can be decomposed as$$h_{ij}=2\psi\delta_{ij}+2E_{,ij}+F_{i,j}+F_{j,i}+\tilde{h}_{ij}, \label{19}$$ with $\partial^{i}F_{i}=0$ and the transverse traceless part $\tilde{h}_{ij}$ satisfies four conditions $\partial^{i}\tilde{h}_{ij}=0,$ $\tilde{h}_{i}^{i}=0,$ leaving us with two polarizations for the massless graviton in General Relativity$.$ Thus, the perturbations can be classified as *scalar* perturbations described by $\phi,$ $\psi,$ $B,$ and $E,$ *vector* perturbations corresponding to $S_{i}$ and $F_{i}$ and *tensor* perturbations $\tilde{h}_{ij}.$ In the linear approximation they are completely decoupled and thus can be studied separately$.$ In General Relativity in empty space the scalar and vector perturbations vanish and they are induced entirely by matter. In our theory, which can be treated as Einstein theory with extra scalar fields, these perturbations are due to the small perturbations $\chi^{A}$, which in turn can be decomposed as:$$\chi^{0}=\chi^{0},\text{ \qquad}\chi^{i}=\tilde{\chi}^{i}+\pi_{,i} \label{20}$$ where $\partial_{i}\tilde{\chi}^{i}=0.$ It is obvious that $\chi^{0}$ and $\pi$ excite scalar modes, while $\tilde{\chi}^{i}$ is responsible for vector perturbations. The remaining coordinate freedom$$x^{0}\rightarrow x^{0}+\xi^{0},\text{ \qquad\ }x^{i}\rightarrow x^{i}+\xi
^{i}=x^{i}+\zeta_{,i}+\tilde{\xi}^{i} \label{21}$$ with $\partial_{i}\tilde{\xi}^{i}=0,$ allows us to impose four gauge conditions. Two of them, which are due to the freedom in choosing $\xi^{0}$ and $\zeta$ refer to the scalar perturbations sector, where for instance two out of seven functions $\phi,\psi,B,$ $E,\chi^{0},\pi$ and $\lambda$ can be taken to vanish. Often it is very convenient to impose the Newtonian gauge conditions $B=E=0,$ which selects the coordinate system with explicit Newtonian limit of General Relativity. The coordinate freedom due to transverse $\tilde{\xi}^{i}$ leave the gauge freedom for the vector perturbations and we can set either $S_{i}$ or $F_{i}$ to be equal to zero. The transverse traceless part of the metric $\tilde{h}_{ij}$ is gauge invariant.
Keeping in equations (\[13\]) only those terms which contain the first order terms in perturbations, these equations are simplified to$$G_{0}^{0}\simeq2\lambda+\frac{m^{2}}{4}\bar{h}_{l}^{l}, \label{22a}$$$$G_{i}^{0}\simeq-\frac{m^{2}}{2}\bar{h}_{i}^{0}, \label{23}$$$$G_{k}^{i}\simeq\frac{m^{2}}{2}\left( \frac{1}{2}\bar{h}_{l}^{l}\delta_{k}^{i}-\bar{h}_{k}^{i}\right) , \label{24}$$ where we took into account that $\lambda,$ which is due to mimetic matter, is a first order quantity in perturbations and $\bar{h}_{0}^{0}=0$ thanks to constraint (\[14\]).
**Scalar perturbations.** We fix the gauge by taking $B=E=0.$ In this gauge the metric takes the form$$ds^{2}=(1+2\phi)dt^{2}-\left( 1-2\psi\right) \delta_{ik}dx^{i}dx^{k},
\label{25}$$ where $\phi$ is the Newtonian gravitational potential. As follows from (\[5\]) we have to first order in perturbations $$\bar{h}_{0}^{0}=-2\phi+2\dot{\chi}^{0},\qquad\text{ }\bar{h}_{i}^{0}=\chi
_{,i}^{0}-\dot{\pi}_{,i},\text{\qquad}\bar{h}_{k}^{i}=2\psi\delta_{k}^{i}+2\pi_{,ij}, \label{26}$$ where dot denotes derivative with respect to time $t.$ Taking into account that in this gauge (see, [@vm]): $$G_{0}^{0}=2\Delta\psi,\text{ }G_{i}^{0}=2\dot{\psi}_{,i},\text{ }G_{k}^{i}=\left( \phi-\psi\right) _{,ik}-(2\ddot{\psi}+\Delta\left( \phi
-\psi\right) )\delta_{ik}, \label{27}$$ and substituting the expressions above into (\[22a\])-(\[24\]) we obtain$$\Delta\psi=\lambda+\frac{m^{2}}{4}\left( 3\psi+\Delta\pi\right) , \label{28}$$$$\dot{\psi}=-\frac{m^{2}}{4}\left( \chi^{0}-\dot{\pi}\right) , \label{29}$$ from $0-0$ and $0-i$ equations. The off diagonal $i\neq k$ components of (\[24\]) give$$\phi-\psi=-m^{2}\pi, \label{30}$$ while taking the trace we obtain $$3\ddot{\psi}+\Delta\left( \phi-\psi\right) +\frac{m^{2}}{4}\left(
3\psi+\Delta\pi\right) =0. \label{31}$$ The constraint equation (\[14\]), when linearized, takes the form$$\phi=\dot{\chi}^{0}, \label{32}$$ and together with equations (\[28\])-(\[31\]) are enough to determine the five unknown variables $\phi,$ $\psi,$ $\lambda,$ $\chi^{0}$ and $\pi.$ Differentiating (\[29\]) and using (\[32\]) and (\[30\]) we have$$\ddot{\psi}=-\frac{m^{2}}{4}\left( \dot{\chi}^{0}-\ddot{\pi}\right)
=-\frac{m^{2}}{4}\left( \psi-m^{2}\pi-\ddot{\pi}\right) . \label{33}$$ Substituting this expression in (\[31\]) and taking into account (\[30\]) we derive the closed form equation for $\pi:$$$\ddot{\pi}-\Delta\pi+m^{2}\pi=0. \label{34}$$ Considering a plane-wave with wave-number $\vec{k}$ the solution of this equation is $$\pi_{k}=\left( A_{k}\sin\omega t+B_{k}\cos\omega t\right)
e^{i\overrightarrow{k}\cdot\overrightarrow{x}},\qquad\omega^{2}=k^{2}+m^{2},
\label{35}$$ where $A_{k}$ and $B_{k}$ are constants of integration. Using (\[34\]) we can rewrite (\[33\]) as$$\ddot{\psi}+\frac{m^{2}}{4}\psi=\frac{m^{2}}{4}\Delta\pi\label{36}$$ and substituting (\[35\]) in the right hand side of this equation we solve it to obtain$$\begin{aligned}
\psi & =C\left( x^{i}\right) \sin\frac{mt}{2}+D\left( x^{i}\right)
\cos\frac{mt}{2}\nonumber\\
& +\frac{m^{2}k^{2}}{4k^{2}+3m^{2}}\left( A_{k}\sin\omega t+B_{k}\cos\omega
t\right) e^{i\overrightarrow{k}\cdot\overrightarrow{x}}, \label{37}$$ where $C\left( x^{i}\right) $ and $D\left( x^{i}\right) $ are further space dependent constants of integration describing the contribution of mimetic matter to the gravitational potential $\psi.$ It then follows from (\[30\]) that$$\begin{aligned}
\phi & =\psi-m^{2}\pi=C\left( x^{i}\right) \sin\frac{mt}{2}+D\left(
x^{i}\right) \cos\frac{mt}{2}\nonumber\\
& -\frac{3m^{2}\omega^{2}}{4k^{2}+3m^{2}}\left( A_{k}\sin\omega t+B_{k}\cos\omega t\right) e^{i\overrightarrow{k}\cdot\overrightarrow{x}}.
\label{38}$$ From equation (\[29\]) one obtains$$\begin{aligned}
\chi^{0} & =\dot{\pi}-\frac{4}{m^{2}}\dot{\psi}=-\frac{2}{m}\left( C\left(
x^{i}\right) \cos\frac{mt}{2}-D\left( x^{i}\right) \sin\frac{mt}{2}\right)
\nonumber\\
& +\frac{3m^{2}\omega}{4k^{2}+3m^{2}}\left( A_{k}\cos\omega t-B_{k}\sin\omega t\right) e^{i\overrightarrow{k}\cdot\overrightarrow{x}}.
\label{39}$$ Finally we solve for $\lambda$ from equation (\[28\]) to get $$\lambda=\left( \triangle-\frac{3m^{2}}{4}\right) \left( C\left(
x^{i}\right) \sin\frac{mt}{2}+D\left( x^{i}\right) \cos\frac{mt}{2}\right)
\label{40}$$ Having determined that the scalar mode of massive graviton is represented by the field $\pi$ satisfying (\[34\]), we note that $\lambda$ is entirely decoupled from $\pi$ and obeys the equation $$\ddot{\lambda}+\frac{m^{2}}{4}\lambda=0, \label{41}$$ which describes massive mimetic matter. The contribution of this mimetic matter to the gravitational potentials is$$\phi=\psi=C\left( x^{i}\right) \sin\frac{mt}{2}+D\left( x^{i}\right)
\cos\frac{mt}{2}. \label{42}$$ To find the normalized independent quantization variables we need to calculate the action for the scalar perturbations. Let us first expand the action (\[12\]) to second order in scalar perturbations in the conformal Newtonian gauge, where $B=E=0,$ $$\begin{aligned}
S & ={\displaystyle\int}
d^{4}x\left[ -3\dot{\psi}^{2}-\psi\triangle\psi+2\phi\triangle\psi
+2\lambda\left( \dot{\chi}^{0}-\phi\right) \right. \nonumber\\
& -\frac{m^{2}}{4}\left( \dot{\chi}^{0}-\phi\right) ^{2}+\frac{m^{2}}{2}\left( \dot{\chi}^{0}-\phi\right) \left( 3\psi+2\triangle\pi\right)
\nonumber\\
& \left. +\frac{m^{2}}{4}\left( 3\psi^{2}+2\psi\triangle\pi-\triangle
\pi\triangle\pi\right) +\frac{m^{2}}{4}\left( \chi^{0}-\dot{\pi}\right)
_{,i}\left( \chi^{0}-\dot{\pi}\right) _{,i}\right] \label{44}$$ Variation of this action with respect to $\lambda$ gives the constraint $\phi=\dot{\chi}^{0}$, which when substituted in the action (\[44\]) reduces it to$$\begin{aligned}
S & ={\displaystyle\int}
d^{4}x\left[ -3\dot{\psi}^{2}-\psi\triangle\psi+2\dot{\chi}^{0}\triangle
\psi\right. \nonumber\\
& \left. +\frac{m^{2}}{4}\left( 3\psi^{2}+2\psi\triangle\pi-\triangle
\pi\triangle\pi+\left( \chi^{0}-\dot{\pi}\right) _{,i}\left( \chi^{0}-\dot{\pi}\right) _{,i}\right) \right] \label{45}$$ Next, as a result of variation with respect to $\chi^{0}$ we obtain the constraint (\[29\]), and this allows us to express $\chi^{0}$ and $\dot
{\chi}^{0}$ in terms of $\psi$ and $\pi,$$$S={\displaystyle\int}
d^{4}x\left[ \dot{\psi}\left( -3+\frac{4\triangle}{m^{2}}\right) \dot{\psi
}-\psi\triangle\psi-2\dot{\pi}\triangle\dot{\psi}+\frac{m^{2}}{4}\left(
3\psi^{2}+2\psi\triangle\pi-\triangle\pi\triangle\pi\right) \right]
\label{46}$$ The fields $\psi$ and $\pi$ have mixed propagators. To diagonalize this action we substitute $$\psi=\left( \triangle-\frac{3m^{2}}{4}\right) ^{-1}\left( \lambda
+\frac{m^{2}}{4}\triangle\pi\right) , \label{47}$$ that follows from $0-0$ equation (\[28\]). After some algebra one finds that the action separates in terms of $\pi$ and $\lambda$, $$\begin{aligned}
S & =-\frac{1}{2}{\displaystyle\int}
d^{4}x\triangle\pi\left( \frac{2m^{2}}{3m^{2}-4\triangle}\right) \left(
\partial_{0}^{2}-\triangle+m^{2}\right) \triangle\pi+\nonumber\\
& +\frac{1}{2}{\displaystyle\int}
d^{4}x\lambda\left( \frac{32}{m^{2}\left( 3m^{2}-4\triangle\right)
}\right) \left( \partial_{0}^{2}+\frac{m^{2}}{4}\right) \lambda, \label{48}$$ where one should understand $\triangle$ to be $-k^{2}$ for the plane-wave modes with the wave-number $\vec{k}.$ This shows that the correctly normalized modes of the corresponding quantum fields are $$\sqrt{\frac{2m^{2}}{3m^{2}+4k^{2}}}k^{2}\pi_{k},\text{ \ \qquad\ }\sqrt
{\frac{32}{m^{2}\left( 3m^{2}+4k^{2}\right) }}\lambda_{k}, \label{49}$$ for the graviton and mimetic matter. The typical amplitude of minimal quantum fluctuations of the properly normalized fields (\[49\]) in scales $L\sim\frac{1}{k}$ is of order $\left( \frac{k^{3}}{\omega_{k}}\right)
^{\frac{1}{2}}$ (see, for instance, formula (4.34) in [@MW]). Therefore, taking into account that $\omega_{k}=\sqrt{k^{2}+m^{2}}$ for $\pi$ mode and $\omega_{k}=\frac{1}{2}m$ for $\lambda$ mode we infer from (\[49\]) that the minimal level of quantum fluctuations in scales $L\ll m^{-1}$ for $\pi$ and $\lambda$ are $$\delta\pi_{L}\simeq m^{-1},\text{ \ \qquad}\delta\lambda_{L}\simeq\left(
\frac{m}{L^{5}}\right) ^{\frac{1}{2}}. \label{50}$$ As it follows from (\[47\]) and (\[30\]) they induce the corresponding metric perturbations of order$$\delta\phi_{L}\sim\delta\psi_{L}\simeq m^{2}\delta\pi_{L}\simeq m, \label{30a}$$ and$$\delta\phi_{L}=\delta\psi_{L}\simeq L^{2}\delta\lambda_{L}\simeq\left(
\frac{m}{L}\right) ^{\frac{1}{2}}, \label{30b}$$ for the scalar graviton mode and mimetic matter, respectively. Thus the metric perturbations due to the quantum fluctuations of the scalar modes always remain much smaller than unity up to the Planckian scales. Note that we work in Planck units where all constants are set to unity and in dimensional units the right hand side, for instance in (\[30b\]), is $\left( \frac
{m}{m_{\mathrm{Pl}}}\right) ^{\frac{1}{2}}\left( \frac{l_{\mathrm{Pl}}}{L}\right) ^{\frac{1}{2}}$, where $m_{\mathrm{Pl}}$ and $l_{\mathrm{Pl}}$ are the Planck mass and Planck length respectively.
The sign of kinetic energy for the field $\pi$ is positive and this field is not a ghost, while, as follows from (\[48\]), the apparent contribution to the *energy density* of mimetic field from the mode with $k\gg m,$ $$-\frac{4}{m^{2}k^{2}}\left( \dot{\lambda}^{2}+\frac{m^{2}}{4}\lambda
^{2}\right) , \label{51}$$ is negative and looks singular as $m^{2}\rightarrow0.$ However, a propagator for $\lambda$ does not not include a Laplacian. Therefore, $\dot{\lambda
}\propto m\lambda$, and hence the singularity $\frac{1}{m^{2}}$ in (\[51\]) is canceled$.$ Moreover, for the mimetic matter the main contribution to the energy density is linear in $\lambda,$ so that the total energy density is$$\varepsilon_{mim}\simeq\lambda-\frac{\lambda^{2}}{k^{2}}, \label{52}$$ where the second negative term just account for the negative contribution of gravitational self-interaction to the total energy density. This second term is smaller than the first term for $\lambda\ll k^{2}.$ When $\lambda$ becomes of order $k^{2}$ then we immediately see from (\[49\]) that $\psi$ becomes of order unity and linear perturbation theory breaks down. To clarify the situation further let us consider in Minkowski space a ball of radius $R$ filled by dust at rest with local energy density $\lambda.$ Then the gravitational potential far away from the ball is determined by the mass of the ball which in the leading order can be estimated as $M_{0}\simeq\lambda
R^{3}.$ However, in the next order, the negative gravitational energy of self interaction of matter within the ball, which is of order $-\frac{M_{0}^{2}}{R},$ also contributes to the mass observed far away from the ball. The resulting mass measured by a far away observer is$$M_{\mathrm{obs}}\simeq M_{0}-\frac{M_{0}^{2}}{R}. \label{53}$$ If the observer would about contribution of negative gravitational energy to the total mass he would conclude that the energy density within the ball is$$\varepsilon_{\mathrm{obs}}\simeq\frac{M_{\mathrm{obs}}}{R^{3}}\simeq
\lambda-R^{2}\lambda^{2}. \label{54}$$ Taking into account that $k\simeq R^{-1}$ this explains the origin of negative energy contribution to the mimetic matter energy density. When the second term in (\[54\]) becomes comparable with the first one we first get semi-closed worlds and finally when the the negative self-interaction energy exactly compensates the internal mass one obtains the closed universe with zero total energy with respect to an observer in asymptotically Minkowski space (see, for example, [@ZN] for details).
**Vector perturbations.** Let us now turn to vector perturbations taking the gauge $F_{i}=0$, so that the metric becomes$$ds^{2}=dt^{2}+2S_{i}dx^{i}dt-\delta_{ik}dx^{i}dx^{k}. \label{55}$$ As follows from (\[5\]), (\[19\]) and (\[20\]), we have, to linear order in perturbations,$$\bar{h}_{0}^{0}=0,\text{ \qquad}\bar{h}_{i}^{0}=-S_{i}+\dot{\tilde{\chi}}_{i},\text{ \qquad\ }\bar{h}_{k}^{i}=-\tilde{\chi}_{k,i}-\tilde{\chi}_{i,k},
\label{56}$$ where as we recall $\tilde{\chi}_{i}=-\tilde{\chi}^{i}$ and both $S_{i}$ and $\tilde{\chi}_{i}$ are transverse. Taking into account that in the gauge $F_{i}=0$: $$G_{0}^{0}=0,\text{ \qquad}G_{i}^{0}=\frac{1}{2}\Delta S_{i},\text{ \qquad
}G_{k}^{i}=-\frac{1}{2}(\dot{S}_{i,k}+\dot{S}_{k,i}), \label{57}$$ (see, [@vm]) equations (\[23\]) and (\[24\]) reduce to $$\Delta S_{i}=m^{2}\left( S_{i}-\dot{\tilde{\chi}}_{i}\right) , \label{58}$$$$\dot{S}_{i,k}+\dot{S}_{k,i}=-m^{2}\left( \tilde{\chi}_{i,k}+\tilde{\chi
}_{k,i}\right) , \label{59}$$ while equation (\[22a\]) is satisfied identically. As follows from (\[58\]) $$S_{i}=\frac{m^{2}}{m^{2}-\Delta}\dot{\tilde{\chi}}_{i}, \label{60}$$ and this, after being substituted in (\[59\]), gives $$\left( \ddot{\tilde{\chi}}_{i,k}+\ddot{\tilde{\chi}}_{k,i}\right)
-\Delta\left( \tilde{\chi}_{i,k}+\tilde{\chi}_{k,i}\right) +m^{2}\left(
\tilde{\chi}_{i,k}+\tilde{\chi}_{k,i}\right) =0, \label{61}$$ or, equivalently,$$\ddot{\tilde{\chi}}_{i}-\Delta\tilde{\chi}_{i}+m^{2}\tilde{\chi}_{i}=0.
\label{62}$$ This equation describes two vector modes of the massive graviton. The action for the vector modes can be obtained by expanding (\[12\]) to second order in perturbations:$$S=\frac{1}{4}\int d^{4}x\,\left[ S_{i,k}S_{i,k}+m^{2}\left( \left(
\dot{\tilde{\chi}}_{i}-S_{i}\right) \left( \dot{\tilde{\chi}}_{i}-S_{i}\right) -\tilde{\chi}_{i,k}\tilde{\chi}_{i,k}\right) \right] .
\label{63}$$ Variation of this action with respect to $S_{i}$ gives us constraint equation (\[58\]), and upon substituting (\[60\]), the action reduces to$$S=-\frac{1}{2}\int d^{4}x\,\left[ \tilde{\chi}_{i}\left( \frac{m^{2}\Delta
}{2\left( \Delta-m^{2}\right) }\right) \left( \partial_{0}^{2}-\Delta+m^{2}\right) \tilde{\chi}_{i}\right] . \label{64}$$ From this we deduce that the properly normalized modes for the vector perturbations are $$\sqrt{\frac{m^{2}\Delta}{2\left( \Delta-m^{2}\right) }}\tilde{\chi}_{i}.
\label{65}$$ Thus for plane-wave perturbations with momentum $k\simeq L^{-1}\gg m$ the typical minimal quantum fluctuations in scales $L$ is of order$$\delta\tilde{\chi}_{L}\simeq\frac{1}{mL}. \label{66}$$ Taking into account that $\delta\dot{\tilde{\chi}}_{L}\simeq\frac{\delta
\tilde{\chi}_{L}}{L}$ we infer from (\[60\]) that the corresponding metric perturbations are$$\delta S_{iL}\simeq m \label{67}$$
**Tensor perturbations.** The tensor perturbations $\tilde{h}_{ij}$ satisfy four extra conditions $\partial^{i}\tilde{h}_{ij}=0=\tilde{h}_{i}^{i}.$ They are gauge invariant and describe two degree of freedom of the graviton which have become massive. The equation for $\tilde{h}_{ij}$ immediately follows from (\[24\]), $$\left( \partial_{0}^{2}-\Delta+m^{2}\right) \tilde{h}_{ij}=0, \label{68}$$ and the corresponding action is$$S=-\frac{1}{8}\int d^{4}x\left[ \tilde{h}_{ij}\left( \partial_{0}^{2}-\Delta+m^{2}\right) \tilde{h}_{ij}\right] . \label{69}$$ The canonical quantization variable is $\tilde{h}_{ij}$ and hence the typical amplitude of quantum fluctuations for tensor modes in scales $L$ is of order$$\delta\tilde{h}_{ij}\simeq\frac{1}{L}, \label{70}$$ for $L\ll m^{-1}.$ They become of order unity at Planck scale $l_{\mathrm{Pl}}\simeq10^{-33}$ $\mathrm{cm}$ where they enter non-perturbative quantum gravity strong coupling regime and linearized theory ceases to be applicable. Although the amplitude of quantum scalar and vector metric perturbations is scale independent $\delta\psi_{L}\simeq\delta S_{L}\simeq m$ and remains small even at the Planck scale, these perturbations nevertheless enter the strong coupling regime at the energy scale which is much below the Planck scale, that is, well before the tensor modes (see next section).
Thus we have shown in this section that the theory with action (\[12\]) describes on a Minkowski background a massive graviton with five degrees of freedom. According to little (rotation) group representations they can be thought of as consisting of one scalar degree of freedom described by $\pi
,\ $two vector degrees of freedom $\tilde{\chi}_{i}$ and two tensor modes $\tilde{h}_{ij}.$ In addition we have a massive mimetic matter described by $\lambda$ which, in the linear approximation, completely decouples from the massive graviton. These results are valid only when the higher order corrections to the linearized equation are negligible. In the next section we determine the range of applicability of the linearized theory.
Strong coupling and massless limit
==================================
**Scalar modes.** To determine at which scales the nonlinear corrections to the linearized equations become important we need to calculate the next order terms in equations (\[13\]). Because the resulting expressions are too cumbersome even to second order, we will keep in the formulae only those terms which can become comparable to linear terms assuming that the gravitational potentials $\phi$ and $\psi$ are much smaller than unity and considering perturbations with $k^{2}\gg m^{2}.$ In the left hand side of equations (\[13\]) it is enough to keep only the linear terms because under the assumptions stated above all terms of order $\phi^{2},$ $\psi^{2}$ are small compared to the linear terms. Keeping in $\bar{h}_{0}^{0},$ defined in (\[5\]), only terms to second order in perturbations we have $$0=\bar{h}_{0}^{0}=-2\left( \phi-\dot{\chi}^{0}\right) +4\left( \phi
-\frac{1}{2}\dot{\chi}^{0}\right) ^{2}-\chi_{,i}^{0}\chi_{,i}^{0} \label{71}$$ It is clear that $O\left( \phi^{2}\right) $ terms are always much smaller than $\phi$ and can be skipped. The term $4\phi\dot{\chi}^{0}$ is much smaller than $\dot{\chi}^{0}$ for $\phi\ll1$ and hence can be neglected in $\bar
{h}_{0}^{0}.$ Therefore, keeping only relevant terms to second order in perturbations, the constraint $\bar{h}_{0}^{0}=0$ becomes$$-2\phi+2\dot{\chi}^{0}+\left( \dot{\chi}^{0}\right) ^{2}-\chi_{,i}^{0}\chi_{,i}^{0}\simeq0. \label{72}$$ For linear perturbations we can express $\phi$, $\psi$ and $\chi^{0}$ in terms of the independent fields $\pi$ and $\lambda$ using (\[35\])-(\[40\]):$$\phi=\dot{\chi}^{0}=-\frac{3m^{2}\omega^{2}\pi+4\lambda}{4k^{2}+3m^{2}}\simeq-\frac{3m^{2}}{4}\pi-\frac{\lambda}{k^{2}}, \label{73}$$$$\psi=\frac{m^{2}k^{2}\pi-4\lambda}{4k^{2}+3m^{2}}\simeq\frac{m^{2}}{4}\pi-\frac{\lambda}{k^{2}}, \label{74}$$$$\chi^{0}=\frac{3m^{4}\dot{\pi}+16\dot{\lambda}}{m^{2}\left( 4k^{2}+3m^{2}\right) }\simeq\frac{3m^{2}}{4k}\pi+\frac{2}{mk^{2}}\lambda,
\label{75}$$ where to simplify the formulae in the second equality we considered perturbations with scales $L\simeq k^{-1}\ll m^{-1}$ and therefore skipped all subleading corrections proportional to $\frac{m^{2}}{k^{2}}.$ Moreover in (\[75\]) we estimated the time derivatives as $\dot{\lambda}\simeq\frac
{m}{2}\lambda$ and $\dot{\pi}\simeq k\pi.$ It is clear that the $\left(
\dot{\chi}^{0}\right) ^{2}$ term in (\[72\]) is of order $\phi^{2}$ and can be neglected compared to the last term, which in the domain of applicability of the linear theory would signal us when the linearized equations fail. Namely, this happens at scales where $\chi_{,i}^{0}\chi_{,i}^{0}$ $\sim
\frac{\left( \chi^{0}\right) ^{2}}{L^{2}}$ becomes comparable with $\phi$ given in (\[73\]). The contributions of quantum fluctuations to the gravitational potential from the scalar mode of the graviton $\pi$ and mimetic matter $\lambda$ are given by (\[30a\]) and (\[30b\]) respectively. Therefore we can easily see that for $\pi$ mode the linearized approximation for constraint (\[72\]) remains always valid for these perturbations. For mimetic mode the last term in (\[72\]) starts to exceed the linear term at $L\leq m^{-\frac{3}{5}}$ when $\delta\lambda_{L}$ becomes of order $m^{2}$ and in case of mimetic matter the linear approximation fails for the constraint $\bar{h}_{0}^{0}=0$.
Now we calculate higher order corrections to equations (\[28\])-(\[31\]). Equation (\[28\]) becomes$$\begin{aligned}
\Delta\psi & =\lambda+\frac{m^{2}}{4}\left[ \left( 3\psi+\Delta\pi\right)
+\dot{\pi}_{,i}\dot{\pi}_{,i}+\frac{3}{2}\pi_{,ik}\pi_{,ik}-\frac{1}{2}\left(
\Delta\pi\right) ^{2}-\chi_{,i}^{0}\dot{\pi}_{,i}-\frac{1}{2}\chi_{,i}^{0}\chi_{,i}^{0}\right] \nonumber\\
& +O\left( \lambda\dot{\chi}^{0},\psi^{2},\psi\Delta\pi,...\right) ,
\label{77}$$ where we have assumed that the gravitational potentials $\psi,$ $\phi$ are much smaller than unity. We have neglected in (\[77\]) the higher order terms in metric dependence in both sides of Einstein equations as well as higher order terms such as $\psi\Delta\pi$ etc. because they are always smaller than the corresponding linear terms. Considering perturbations with wavelength $L\ll m^{-1}$ and taking into account (\[73\])-(\[75\]) we can estimate the various quadratic terms in (\[77\]) as $$\dot{\pi}_{,i}\dot{\pi}_{,i}\sim\pi_{,ik}\pi_{,ik}\sim\left( \Delta
\pi\right) ^{2}\sim\frac{\pi^{2}}{L^{4}};\text{ \qquad}\chi_{,i}^{0}\dot{\pi
}_{,i}\sim\frac{\lambda\pi}{mL};\text{ \qquad}\chi_{,i}^{0}\chi_{,i}^{0}\sim\frac{\lambda^{2}L^{2}}{m^{2}}, \label{78}$$ where we kept only the leading terms. For quantum fluctuations we have$$\frac{\pi^{2}}{L^{4}}\sim\frac{m^{2}}{\left( mL\right) ^{4}};\qquad\text{
}\frac{\lambda\pi}{mL}\sim\frac{m^{2}}{\left( mL\right) ^{7/2}};\text{
\qquad}\frac{\lambda^{2}L^{2}}{m^{2}}\sim\frac{m^{2}}{\left( mL\right) ^{3}}. \label{79}$$ These nonlinear terms have to be compared to the linear terms in the brackets on the right hand side of equation (\[77\])$$\Delta\pi\sim\frac{\pi}{L^{2}}\sim\frac{m}{\left( mL\right) ^{2}};\text{
\qquad}\psi\sim L^{2}\lambda\sim\frac{m}{\left( mL\right) ^{1/2}},
\label{80}$$ where the first term is entirely due to the contribution of $\pi$ mode of the graviton and the main contribution to the second term comes from mimetic matter. Because $mL\ll1,$ the term $\frac{\pi^{2}}{L^{4}}$ dominates among nonlinear corrections for quantum fluctuations and is entirely due to the scalar mode $\pi$ of the massive graviton. It becomes comparable to $\Delta
\pi$ at the scale$$L_{\pi}^{\mathrm{str}}\sim m^{-\frac{1}{2}}, \label{81}$$ and for $L\leq$ $L_{\pi}^{\mathrm{str}}$ the scalar mode of the graviton becomes strongly coupled and decouples from the two transverse degree of freedom which, in contrast, only becomes strongly coupled at Planck scale . Among the two contributions from mimetic matter the second term in (\[79\]) is obviously larger than the third term and becomes comparable with the linear term $\psi$ given in (\[79\]) at the scale$$L_{\mathrm{mim}}^{\mathrm{str}}\sim m^{-\frac{2}{3}}, \label{82}$$ where the quantized mimetic matter becomes strongly coupled. This scale is larger than the scale $\sim m^{-\frac{3}{5}}$ found above, where nonlinear corrections to the constraint $\bar{h}_{0}^{0}=0$ become important$.$ Hence the mimetic mode gets in the strongly coupled regime when $\delta\lambda_{L}$ is yet smaller than $m^{2}.$ Notice that at scale $\sim m^{-\frac{3}{5}}$ the smallest last term in (\[79\]) becomes of order $\psi$ given in (\[80\]). Equation (\[29\]) gets modified by the dominant nonlinear corrections as$$\dot{\psi}_{,i}=-\frac{m^{2}}{4}\left[ \left( \chi^{0}-\dot{\pi}\right)
_{,i}-\Delta\pi\chi_{,i}^{0}+2\chi_{,k}^{0}\pi_{,ik}+\Delta\pi\dot{\pi}_{,i}-3\pi_{,ik}\dot{\pi}_{,k}\right] +\lambda\chi_{,i}^{0}+... \label{83}$$ For quantum fluctuations the nonlinear terms inside the brackets become comparable with the linear ones at $L_{\pi}^{\mathrm{str}}\sim m^{-\frac{1}{2}}$ both for the scalar mode $\pi$ and quantum mimetic matter. The last term in this equation $\lambda\chi_{,i}^{0}$ becomes of order $m^{2}\chi_{,i}^{0}$ at scales $L\sim m^{-\frac{3}{5}}$ when the amplitude of quantum fluctuation of $\delta\lambda$ would reach the value $m^{2}.$ However, as we have seen above the linearized approximation for mimetic matter fails before, at $L_{\mathrm{mim}}^{\mathrm{str}}\sim m^{-\frac{2}{3}}.$
The higher order corrections to $i-k$ components of Einstein equations have the same structure as in (\[77\]) and hence do not lead to any further restrictions on the linear perturbation theory. Thus, the scalar mode of the graviton $\pi$ gets strongly coupled at length scales of order $m^{-\frac
{1}{2}}$ and at smaller scales the massive graviton loses this degree of freedom.
**Vector modes.** For the vector perturbations equations (\[58\]) and (\[59\]), taking into account the *relevant* quadratic terms, become$$\Delta S_{i}=m^{2}\left[ (S_{i}-\dot{\tilde{\chi}}_{i})+2\dot{\tilde{\chi}}_{k}\tilde{\chi}_{k,i}+\dot{\tilde{\chi}}_{k}\tilde{\chi}_{i,k}\right] ,
\label{84}$$$$\begin{aligned}
\dot{S}_{i,k}+\dot{S}_{k,i} & =-m^{2}\left[ \left( \tilde{\chi}_{i,k}+\tilde{\chi}_{k,i}\right) +\dot{\tilde{\chi}}_{i}\dot{\tilde{\chi}}_{k}-\delta_{ik}\dot{\tilde{\chi}}_{m}\dot{\tilde{\chi}}_{m}-\tilde{\chi
}_{i,m}\tilde{\chi}_{k,m}-2\tilde{\chi}_{m,i}\tilde{\chi}_{m,k}\right.
\nonumber\\
& \left. -\tilde{\chi}_{i,m}\tilde{\chi}_{m,k}-\tilde{\chi}_{k,m}\tilde
{\chi}_{m,i}+\delta_{ik}\left( \tilde{\chi}_{l,m}\tilde{\chi}_{l,m}+\frac
{1}{2}\tilde{\chi}_{m,l}\tilde{\chi}_{l,m}\right) \right] , \label{85}$$ where we have omitted the quadratic terms $S_{i}^{2},$ $S_{i}\dot{\tilde{\chi
}}_{i},$ $S_{m}\tilde{\chi}_{i,m}$ because the metric perturbations $S_{i}$ are always much smaller than unity. For plane-wave perturbations with momentum $k\simeq L^{-1}\gg m$ these quadratic terms in (\[84\]), (\[85\]) can be estimated as $\frac{\tilde{\chi}^{2}}{L^{2}}$ and these become comparable with the linear terms $\frac{\tilde{\chi}}{L}$ when $\frac{\tilde{\chi}}{L}\sim1$. For the minimal quantum fluctuations $\tilde{\chi}\sim\frac{1}{mL}$ (see (\[66\])) and hence vector modes of the massive graviton come in the strong coupling regime at the same scale as the scalar mode $\pi$$$L_{\tilde{\chi}}^{\mathrm{str}}\sim m^{-\frac{1}{2}}. \label{86}$$ Thus, for $L\ll m^{-\frac{1}{2}}$ the massive graviton loses three out of five degrees of freedom and the two remaining transverse degrees of freedom continue to propagate as if the graviton would be massless. One can easily check that these two degrees of freedom are not influenced much by scalar and vector degrees of freedom via nonlinear corrections because the metric perturbations they induce always remain much smaller than unity. The transverse degrees of freedom become strongly coupled only at the Planck scale. When mass of the graviton vanishes, the strong coupling scale for scalar and vector modes goes to infinity and the graviton has only two propagating degrees of freedom as it must be.
Static gravitational field by external source
=============================================
Let us consider the spherically symmetric field created by the mass $M$. In General Relativity the metric far away from the source (at distance $r$) can be written in the linear approximation as $$ds^{2}=(1+2\phi)dt^{2}-\left( 1-2\psi\right) \delta_{ik}dx^{i}dx^{k},
\label{87}$$ where $$\phi=\psi=\phi_{N}=-\frac{M}{8\pi r}, \label{88}$$ is the Newtonian gravitational potential (in units $8\pi G=1$).
In massive gravity this potential is modified at distances $r\gg m^{-1}$ where it decays exponentially fast. However, more nontrivial is that for the case of Fierz-Pauli mass term Bianchi identities inevitably enforce the scalar curvature perturbations to vanish in the linear order [@BD]:$$\delta R=2\Delta\left( 2\psi-\phi\right) =0, \label{89}$$ from which it follows that $\phi=2\psi.$ The bending of light measures the sum of two potentials $\phi+\psi,$ that must be equal to $2\phi_{N}.$ Hence, the gravitational potential $\phi$ determining for instance the motion of the planets must be in this case equal to $\frac{4}{3}\phi_{N},$ independently of the mass of the graviton, in contradiction with experiment. This is known as vDVZ discontinuity which survives in the linearized theory even in the limit when the mass of the graviton goes to zero [@vDVZ]. The resolution to this apparent contradiction was found by Vainshtein who showed that the scalar mode of the massive graviton which gives extra $\frac{1}{3}\phi_{N}$ contribution to the gravitational potential $\phi$ gets strongly coupled at the scale $R_{V},$ where the linearized approximation for this mode fails and the result of General Relativity is restored in the leading order for $r<R_{V}$ [@Vein]. Depending on the nonlinear extension of Fierz-Pauli mass term, the Vainshtein scale $R_{V}$ changes in the interval from $\left( \frac
{M}{m^{4}}\right) ^{\frac{1}{5}}$ to $\left( \frac{M}{m^{2}}\right)
^{\frac{1}{3}}$ (see, for example, [@ACM], and references there). When mass of the graviton goes to zero, Vainshtein scale grows to infinity and thus the range of scales with vDVZ discontinuity disappears.
In the theory considered here the mass term is not of FP type and vDVZ discontinuity does not arise even at linearized level. In the presence of an external source of mass $M,$ among equations (\[28\])-(\[32\]), describing the linearized scalar mode of massive graviton, only equation (\[28\]) gets modified as$$\Delta\psi=\frac{m^{2}}{4}\left( 3\psi+\Delta\pi\right) +\lambda+\frac{1}{2}T_{0}^{0}, \label{90}$$ where $T_{0}^{0}=M\delta^{3}\left( x\right) .$ In the consideration above this equation was used only to find $\lambda$ for a given $\psi$ and $\pi
.$ In turn the solution for $\psi$ and $\pi$ is given in (\[35\]) and (\[37\]) and remains unchanged. First of all let us note that $\pi$ completely decouples from mimetic matter and the gravitational field induced by a central source because $T_{0}^{0}$ comes only in combination with $\lambda$ and only in equation (\[28\]). Therefore we can set $\pi=0$ assuming that the constants of integration in (\[35\]) vanish. Then as follows from (\[37\]) and (\[38\])$$\begin{aligned}
\phi & =\psi=C\left( x^{i}\right) \sin\frac{mt}{2}+D\left( x^{i}\right)
\cos\frac{mt}{2}\nonumber\\
& =F\left( x^{i}\right) \cos\left( \frac{mt}{2}+\alpha\left(
x^{i}\right) \right) . \label{91}$$ It is interesting to note that in case of massive mimetic gravity the static solutions for the static source do not exist in the linearized version of the theory. The gravitational potential oscillates with a frequency proportional to the mass of the graviton. Of course on time scales $t\ll m^{-1}$ the potential does not change too much. Considering $t\ll m^{-1}$ and scales $r\ll
m^{-1}$ we can estimate the constant of integration in (\[91\]) in the presence of source $T_{0}^{0}$ from (\[90\]) to get$$\phi=\psi\simeq-\frac{1}{8\pi\cos\alpha}\left( \frac{M}{r}+\frac{2}{r}\int^{r}\lambda d^{3}x\right) \cos\left( \frac{mt}{2}+\alpha\right) ,
\label{92}$$ and as follows from (\[29\])$$\chi^{0}=-\frac{4}{m^{2}}\dot{\psi}\simeq-\frac{1}{4\pi m\cos\alpha}\left(
\frac{M}{r}+\frac{2}{r}\int^{r}\lambda d^{3}x\right) \sin\left( \frac{mt}{2}+\alpha\right) \label{93}$$ Clearly these solutions are approximate solutions valid only for $t\ll
m^{-1}.$ Keeping this in mind we will find the range of the scales where the linear approximation used to get them fails because of nonlinear corrections. From (\[72\]) and (\[77\]) it is clear that this happens when $\chi
_{,i}^{0}\chi_{,i}^{0}$ becomes of order $\phi=\psi.$ Neglecting the contribution of mimetic matter in (\[92\]), (\[93\]) and taking as a rough estimate$$\psi\sim\frac{M}{r},\text{ \ \ \ }\chi_{,i}^{0}\chi_{,i}^{0}\sim\frac{\left(
\chi^{0}\right) ^{2}}{r^{2}}\sim\frac{M^{2}}{m^{2}r^{4}}, \label{94}$$ we see that these terms become comparable at$$r_{\mathrm{str}}\sim\left( \frac{M}{m^{2}}\right) ^{\frac{1}{3}} \label{96}$$ and for $r<r_{\mathrm{str}}$ the nonlinear term $\chi_{,i}^{0}\chi_{,i}^{0}$ cannot be neglected. Thus the linearized approximation used to derive the non-static solutions (\[92\]), (\[93\]) is not valid for $r<r_{\mathrm{str}}$. Moreover as follows from (\[83\])$$\dot{\psi}_{,i}=\left( \lambda-\frac{m^{2}}{4}\right) \chi_{,i}^{0},
\label{97}$$ and hence when $\lambda$ becomes comparable with $m^{2}$ the linearized $0-i$ equations fail. One can see that this equation allows static solution for $\lambda=m^{2}/4.$ In this case the contribution of mimetic matter to the gravitational potential in (\[92\]) becomes comparable to the contribution of the source of mass $M$ at scales $r_{\mathrm{str}}\sim\left( \frac
{M}{m^{2}}\right) ^{\frac{1}{3}}.$
Thus, at scales $r<r_{\mathrm{str}}$ we find that the solution of the equation for central source is static and corresponds to$$\phi=\psi=\phi_{N}\left( 1+O\left( 1\right) \left( \frac{r}{r_{\mathrm{str}}}\right) ^{3}\right) , \label{98}$$ where the corrections due to the contribution of induced mimetic matter rapidly decrease towards the smaller scales. On large scales, for $m^{-1}>r>r_{\mathrm{str}}$ the static source produces the time dependent oscillating solution$$\phi=\psi\simeq O\left( 1\right) \phi_{N}\cos\left( \frac{mt}{2}+\alpha\right) . \label{99}$$ These time dependent oscillations are due to the induced mimetic matter which surrounds the static source.
Summary
=======
It is a rather interesting theoretical challenge to try to construct a theory of massive gravity free of obvious flaws in which, on Minkowski background, the graviton behaves like a massive particle. This is the main motivation for this work. Generically the five degrees of freedom of a massive graviton are accompanied by an extra ghost degree of freedom which propagates already at the linear level unless the mass term is taken to be in the Fierz-Pauli form [@FP]. For the FP term the dangerous mode remains non-dynamical in the linear approximation but generically reappears as nonlinear Boulware-Deser ghost [@BD] on non-trivial (different from Minkowski) backgrounds. There is a claim in the literature that there exists a nearly unambiguous higher order extension of the FP theory in which such ghost is absent on most of the non-trivial backgrounds [@Kab]. However, as we have shown in [@CMghost; @CMghost1], the ghost field can still reappear on some backgrounds . In this paper we have addressed the following questions:
- *Is it possible to have a rather simple theory of massive gravity, which remains ghost free irrespective of its nonlinear extension*?
- *Can we have non FP mass term and nevertheless avoid the appearance of a ghost?*
As we have shown in this paper the answer to both questions is positive. The only way to generate the graviton mass without explicitly spoiling diffeomorphism invariance is to use the BEH mechanism with four scalar fields. In this case, in the broken symmetry phase, the three scalar fields *donate* their degrees of freedom to the graviton which thus gets a total of five degrees of freedom and acquires mass. The fourth field, if unrestricted, becomes a ghost and even for the Fierz-Pauli mass form still propagates on some non-trivial backgrounds. We have proposed to use the constrained mimetic scalar instead of this dangerous field. This field becomes dynamical and can imitate dark matter when combined with the longitudinal mode of gravity. In mimetic massive gravity its behavior is slightly modified and mimetic matter still well imitates cold dark matter. Thus, the dangerous mode is avoided and instead we get a good candidate for dark matter irrespective of the nonlinear extension of the theory. The inevitable consequence of using mimetic field, which is always in *broken symmetry phase*, is the need to use non Fierz-Pauli type mass term, with a relative coefficient of $-\frac{1}{2}$ between the $\bar{h}^{2}$ and $\bar{h}_{B}^{A}\bar{h}_{A}^{B}$ terms instead of $-1.$ Only in this case we obtain a massive graviton in the broken symmetry phase. This graviton is completely decoupled from mimetic matter in the linear approximation and vDVZ discontinuity is completely avoided. Out of the five degrees of freedom, three degrees due to the scalar fields get in the strongly coupled regime at the same scale $L^{str}\sim
m^{-\frac{1}{2}}$ and at $L<L^{\mathrm{str}}$ the graviton has only two transverse degrees of freedom which become strongly coupled at Planck scale. This is quite different from the case of FP mass term where these scales are different for the scalar and vector modes. The other interesting feature of the theory is that the gravitational potential produced by a mass $M$ is static only at scales $L<$ $r_{\mathrm{str}}\sim\left( \frac{M}{m^{2}}\right) ^{\frac{1}{3}}$ while at larger distances it oscillates with frequency $\frac{m}{2}$ and thus vanishes, being averaged over the time scales $t>m^{-1}.$
****
The work of A. H. C is supported in part by the National Science Foundation Grant No. Phys-1518371. The work of V.M. is supported in part by *The Dark Universe* and the Cluster of Excellence EXC 153 *Origin and Structure of the Universe*. V.M. thanks ENS, where a part of this work is completed, for hospitality.
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abstract: 'Average consensus algorithms compute the global average of sensor data in a distributed fashion using local sensor nodes. Simple execution, decentralized philosophy make these algorithms suitable for WSN scenarios. Most of the researchers have studied the average consensus algorithms by modeling the network as an undirected graph. But, WSNs in practice consist of asymmetric links and the undirected graph cannot model the asymmetric links. Therefore, these studies fail to study the actual performance of consensus algorithms on WSNs. In this paper, we model the WSN as a directed graph and derive the explicit formulas of the ring, torus, $r$-nearest neighbor ring, and $m$-dimensional torus networks. Numerical results subsequently demonstrate the accuracy of directed graph modeling. Further, we study the effect of asymmetric links, the number of nodes, network dimension, and node overhead on the convergence rate of average consensus algorithms.'
author:
- 'Sateeshkrishna Dhuli and Y. N. Singh '
bibliography:
- 'References.bib'
title: Analysis of Average Consensus Algorithm for Asymmetric Regular Networks
---
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Average Consensus Algorithms, Regular Graphs, Asymmetric Links, Directed Graph, Convergence Rate, Wireless Sensor Networks.
Introduction
============
algorithms have been widely studied in the literature due to their decentralized philosophy and simple execution [@olfati2007], [@olshevsky2009], [@xiao2007]. These algorithms can be utilized when global network topology information is not known, and the network consists of power constrained nodes. In contrast to centralized algorithms, these algorithms compute the desired statistics at every node without the need of any fusion center. Hence, these algorithms are quite suitable for WSN scenarios. Consensus algorithms are iterative in nature, and their performance is measured by convergence rate[@olshevsky2011], [@cao2005]. Convergence rate of the consensus algorithms have been widely studied in the literature, most of the prior works have modeled the networks as an undirected graph due to the computational tractability. However, undirected graphs cannot model the applications which involve asymmetric links and may not characterize the actual network’s performance. In practice, wireless channels in low power wireless networks such as WSNs are known to be time-varying, unreliable, and asymmetric [@zhou2006], [@zamalloa2007], [@yang2006], [@kotz2003], [@ganesan2002], [@li2013]. Therefore, it is important to consider the WSN as a directed graph to accurately estimate the convergence rate. Convergence rate is characterized by the graph Laplacian eigenvalues[@asadi2017], [@schug2014]. Estimating the convergence rate for large-scale networks is a computationally challenging task. To evaluate the convergence rate, there are many algorithms available in the literature, such as *best constant* weights algorithm, *metropolis–-hastings* weights algorithm, *max-degree* weights algorithm [@xiao2007]. In this paper, we employ the *best constant* weights algorithm to derive the explicit expressions of the convergence rate. Recently, WSN has been modeled as an $r$-nearest neighbor network and explicit expressions of convergence time for average consensus algorithms have been derived in [@dhuli2015]. However, they considered the undirected graph modeling which cannot study the time-varying wireless channels of WSNs. In [@hao], authors modeled the WSN as a directed graph and proved that asymmetric weights improve the convergence rate of average consensus algorithms. The expected convergence rate of an asymmetric network has been examined in [@asadi2017]. In our work, we model the WSN as a directed graph and derive the explicit expressions of convergence rate for regular graphs.
Regular graph models are simple structures which allow the theoretical analysis that incorporates important parameters like connectivity, scalability, network size, node overhead, and network dimension [@vanka2010], [@lattice], [@dhuli2015]. These models represent the geographical proximity in the practical wireless sensor networks. In this paper, we model the WSN as a ring, torus, $r$-nearest neighbor network, $m$-dimensional torus networks and derive the explicit expressions for convergence rate of average consensus algorithms. The nearest neighbors can model node’s transmission radius or node overhead. We measure the absolute error to investigate the deviation of convergence rate results in directed graph modeling over undirected graph modeling. Our analytic expressions are extremely helpful in designing the optimization frameworks for controlling the performance of average consensus algorithms on WSNs. Our approach avoids the usage of huge computational resources to compute the convergence rate for large-scale WSNs.
Organization
------------
1)In section II, we give a brief review on consensus algorithms\
2)In section III, we model the WSN as a ring and derive the explicit expressions of convergence rate in terms of number of nodes and network overhead.\
(3)In section IV, we model the WSN as a torus network and $m$-dimensional torus network and derive the explicit expressions of convergence rate in terms of number of nodes and network dimension.\
(4)In section V, we model the WSN as a $r$-neighbor neighbor ring network and derive the explicit expressions of convergence rate in terms of nearest neighbors and number of nodes.\
(5)Finally, in section IV, we present the numerical results and study the effect of network parameters on the convergence rate.
Average Consensus Algorithm
===========================
Let $G= (V,E)$, be a directed graph with node set $V = \left\{ {1,2,......n} \right\}$ and an edge set $E \subseteq V \times V$. Let $A$ denotes the $n\times n$ adjacency matrix of graph $G$, where each entry of $A$ is represented by $a_{ij}$. The degree matrix $D$ is defined as the diagonal matrix whose entry is $d_{ii}$, where $d_{ii}=\sum_{j=1}^{n}a_{ij}=\sum_{j=1}^{n}a_{ji}$. The Laplacian matrix of a graph $G$ is expressed as $L=D-A$, whose entries are $$l_{ij} = l_{ji} = \left\{ \begin{array}{l}
\deg (v_i )\,\,\,if\,\,j = i ,\\
- a_{ij} \,\,\,\,\,\,\,\,\,if\,\,j \ne \,i .\\
\end{array} \right.
\label{1}$$ Let $x_i (0)$ denotes the real scalar variable of node i at $t=0$. Average consensus algorithm computes the average $x_{avg}=\frac{{\sum\nolimits_{i = 1}^n {x_i (0)} }}{n}$ at every node through a decentralized approach which does not require any sink node. At each step, node $i$ carries out its update based on its local state and communication with its direct neighbors. At time instant $t+1$, the real scalar variable at node $i$ is expressed as $$x_i (t + 1) = x_i (t) + h\sum\limits_{j \in N_i } {(x_j (t) - x_i (t))} ,\,\,\,i = 1,...,n,
\label{2}$$ where ‘*h*’ is a consensus parameter and $N_i$ denotes neighbor set of node ‘*i*’. This can be also expressed as a simple linear iteration as $$\textbf{x(t + 1)} = W\textbf{x(t)},\,\,\,\,t = 0,1,2...,
\label{3}$$ where ‘*W*’ denotes weight matrix, and $W_{ij}$ is a weight associated with the edge $(i,j)$. If we assign equal weight *h* to each link in the network, then optimal weight for a given topology is $$W_{ij}=\left\{\begin{matrix}
h & if ,\,\,\,\,(i,j) \in E,\\
1-hdeg(\nu_{i})&if,\,\,\,\,i=j,\\
0 & otherwise.
\end{matrix}\right.
\label{4}$$ and Weight matrix is given by $$W = I - hL.
\label{5}$$ where ‘*I*’ is a $n\times n$ identity matrix. Let $\lambda _n (W)$ be the $n^{th}$ eigenvalue of $W$, then $\lambda _n (W)=1-h\lambda _n (L)$ satisfies $$1=\lambda _{1}(W)> \lambda _{2}(W)> \lambda _{3}(W).........\lambda _{n}(W).
\label{6}$$ and let $\lambda _{n}(L)$ be the $n^{th}$ eigenvalue of Laplacian matrix satisfies $$0=\lambda _{1}(L)< \lambda _{2}(L)< \lambda _{3}(L).........\lambda _{n}(L).
\label{7}$$ Convergence rate of a average consensus algorithm can be measured by the spectral gap $\left| {1 - \lambda _2 (L)} \right|$[@hao], [@asadi2017], [@schug2014]. In this work, we employ the *best constant weights* algorithm to derive the closed-form expressions of convergence rate. *Best constant weights* algorithm gives the fastest convergence rate among the other uniform weight methods [@toulouse2015], [@2011local].
Best Constant Weights Algorithm
-------------------------------
Derive the generalized eigenvalue expression of Laplacian matrix and follow the below steps.\
(1)Compute the second smallest eigenvalue of Laplacian matrix ($\lambda _1 \left( L \right)$) and largest eigenvalue of Laplacian matrix ($\lambda _{n - 1} \left( L \right)$).\
(2)Obtain the consensus parameter ($h$) using $${\left| {1 - h\lambda _1 \left( L \right)} \right| = \left| {1 - h\lambda _{n - 1} \left( L \right)} \right|}
\label{8}$$\
(3)Substitute the ‘*h*’ in $\left| {1 - h\lambda _1 \left( L \right)} \right|$ and obtain the convergence parameter ($\gamma=h\lambda _1 \left( L \right)$).\
(4)Finally, evaluate the convergence rate ($R$) using $$R=1-\gamma.
\label{9}$$
Explicit Formulas of Convergence Rate for Ring Networks
=======================================================
In this section, we derive the explicit expressions of convergence rate for ring and $r$-nearest neighbor ring networks. Ring network with asymmetric links is as shown in the Fig. 1. We assume that forward link weight is $\frac{1-a}{2}$ and backward link weight is $\frac{1+a}{2}$, here ‘*a*’ denotes asymmetric link factor.
![Asymmetric Ring Network[]{data-label="fig:verticalcell"}](asym_cycle)
![Asymmetric Torus Network[]{data-label="fig:verticalcell"}](asym_torus)
\
$Definition$ $1$: The $(j+1)^{th}$ eigenvalue [@circulant] of a circulant matrix $circ\{a_1 ,a_2 ......a_n\}$ is defined as $$\lambda _j = a_1 + a_2 \omega ^{j} + .............. + a_n \omega ^{(n - 1)j},
\label{10}$$ where $\omega= e^\frac{2\pi i}{n}$ and $\{ a_l \} _{l = 1}^n $ are row entries of circulant matrix.\
$Theorem$ $1$: The $(j+1)^{th}$ eigenvalue of Laplacian matrix of a ring network for even number of nodes is expressed as $$\lambda _j (L) = 1 - \cos \frac{{2\pi j}}{n} + ai\sin \frac{{2\pi j}}{n}
\label{11}$$ $Proof$ $1$: Asymmetric ring network is as shown in the Fig. 1. Thus, Laplacian matrix can be written as $$L = circ\{ 1,\frac{{ - 1 + a}}{2},\underbrace {0,0,..,0}_{n - 3\,\,terms},\frac{{ - 1 - a}}{2}\}
\label{12}$$ Using (\[10\]), $(j+1)^{th}$ eigenvalue of a Laplacian matrix can be written as $$\lambda_j (L) = 1 - \cos \frac{{2\pi j}}{n} + ai\sin \frac{{2\pi j}}{n}
\label{13}$$\
$Theorem$ $2$: Convergence rate of a ring network for even number of nodes is expressed as $$R = \frac{{2 - 2a^2 - 2\cos \frac{{2\pi }}{n} + 2a^2 \cos \frac{{2\pi }}{n}}}{{3 - a^2 + ( - 1 + a^2 )\cos \frac{{2\pi }}{n}}}
\label{14}$$ $Proof$ $2$: For $n$=even, $\lambda _1 (L)$ is the second smallest value of Laplacian matrix and $\lambda _{\frac{n}{2}} (L)$ is the largest of a Laplacian matrix.\
Thus, rewrite the (\[8\]) as $$\left| {1 - h\lambda _1 (L)} \right| = \left| {1 - h\lambda _{\frac{n}{2}} (L)} \right|
\label{15}$$\
Substituting the expressions of $\lambda _1 (L)$ and $\lambda _{\frac{n}{2}} (L)$ in (\[15\]), gives $$h = \frac{{2 + 2\cos \frac{{2\pi }}{n}}}{{3 - \cos ^2 \frac{{2\pi }}{n} + 2\cos \frac{{2\pi }}{n} - a^2 \sin ^2 \frac{{2\pi }}{n}}}.
\label{16}$$ Thus, convergence factor ($\gamma$) is expressed as $$\gamma=\left |1-h \left(1-\cos \left(\frac{2 \pi }{n}\right)+i a \sin \left(\frac{2 \pi }{n}\right)\right) \right |
\label{17}$$ Substituting the $\gamma$ in (\[9\]) proves the $Theorem$ $2$.\
$Theorem$ $3$: Convergence rate of a ring network for odd number of nodes is expressed as $$\resizebox{0.9\hsize} {!} {$R = 1 - \frac{{\sqrt {2 + 4a^2 + 2a^4 - 2( - 1 + a^4 )\cos \frac{\pi }{n} + ( - 1 + a^2 )^2 \cos \frac{{2\pi }}{n} + 2\cos \frac{{3\pi }}{n} - 2a^4 \cos \frac{{3\pi }}{n} + \cos \frac{{4\pi }}{n} - 2a^2 \cos \frac{{4\pi }}{n} + a^4 \cos \frac{{4\pi }}{n}} }}{{\sqrt 2 \left( {2 - \left( { - 1 + a^2 } \right)\cos \frac{\pi }{n} + \left( { - 1 + a^2 } \right)\cos \frac{{2\pi }}{n}} \right)}}$}
\label{18}$$ $Proof$ $3$: For $n$=odd, $\lambda _1 (L)$ is the second smallest eigenvalue of Laplacian matrix and $\lambda _{\frac{n-1}{2}} (L)$ is the largest eigenvalue of a Laplacian matrix.\
Thus, rewrite the (\[8\]) as $$\left| {1 - h\lambda _1 (L)} \right| = \left| {1 -h\lambda _{\frac{{n - 1}}{2}} (L)} \right|
\label{19}$$ Substituting the $\lambda _1 (L) $ and $\lambda _{\frac{{n - 1}}{2}} (L)$ expressions in (\[19\]) results in $$\resizebox{0.9\hsize} {!} {$\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{n} + ia\sin \frac{{2\pi }}{n}} \right)} \right| = \left| {1 - h\left( {1 + \cos \frac{\pi }{n} + ia\sin \frac{\pi }{n}} \right)} \right|$}
\label{20}$$ Thus, we obtain $$\resizebox{0.9\hsize} {!} {$h = \frac{{2\left( {\cos \frac{\pi }{n} + \cos \frac{{2\pi }}{n}} \right)}}{{ - \cos ^2 \frac{{2\pi }}{n} + 2\cos \frac{{2\pi }}{n} - a^2 \sin ^2 \frac{{2\pi }}{n} + \cos ^2 \frac{\pi }{n} + a^2 \sin ^2 \frac{\pi }{n} + 2\cos \frac{\pi }{n}}}$}
\label{21}$$ Finally, we get $\gamma$ as $$\gamma=\left |1-h \left(1-\cos \left(\frac{2 \pi }{n}\right)+i a \sin \left(\frac{2 \pi }{n}\right)\right) \right |
\label{22}$$ Substituting the (\[22\]) value in (\[9\]) proves the $Theorem$ $3$.
Explicit Formulas of Convergence Rate for Torus Networks
========================================================
Torus network with asymmetric links is as shown in the Fig. 2. In this section, we derive the explicit expressions of convergence rate for a torus network and $m$-dimensional torus networks.\
$Theorem$ $4$: The eigenvalue of a torus network is expressed as $$\resizebox{1.0\hsize} {!} {$\lambda _{j_1 ,j_2 } (L) = 2 - \cos \frac{{2\pi j_1 }}{{k_1 }} - \cos \frac{{2\pi j_2 }}{{k_2 }} + ia\left( {\sin \frac{{2\pi j_1 }}{{k_1 }} + \sin \frac{{2\pi j_2 }}{{k_2 }}} \right)$}
\label{23}$$ $Proof$ $4$: Cartesian product of the two ring networks results in torus network. The eigenvalue of a torus network will be the addition of eigenvalues of the corresponding ring networks [@load]. $$\lambda _{j_1 ,j_2 } (L) = \lambda _{j_1 } (L)+\lambda _{j_2 } (L)
\label{24}$$ Here, we assume that the torus is formed by two ring networks with $k_1$ and $k_{2}$ nodes respectively. Then $(j_1+1)^{th}$ eigenvalue of a Laplacian matrix for a ring network can be expressed as $$\lambda _{j_1 } (L) = 1 - \cos \frac{{2\pi j_1 }}{{k_1 }} + ia\sin \frac{{2\pi j_1 }}{{k_1 }}
\label{25}$$ Similarly, $(j_2+1)^{th}$ eigenvalue of a Laplacian matrix for ring network is expressed as $$\lambda _{j_2 } (L) = 1 - \cos \frac{{2\pi j_2 }}{{k_2 }} + ia\sin \frac{{2\pi j_2 }}{{k_2 }}
\label{26}$$ Finally, using (\[24\]), (\[25\]), and (\[26\]) we obtain(\[23\]).\
$Theorem$ $5$: Convergence rate of a torus network for $k_1=even$ and $k_2=even$ is expressed as $$\frac{a^2 \sin ^2\left(\frac{2 \pi }{k_2}\right)+\cos ^2\left(\frac{2 \pi }{k_2}\right)+6 \cos \left(\frac{2 \pi }{k_2}\right)+9}{a^2 \sin ^2\left(\frac{2 \pi }{k_2}\right)+\cos ^2\left(\frac{2 \pi }{k_2}\right)-2 \cos \left(\frac{2 \pi }{k_2}\right)-15}
\label{27}$$ $Proof$ $5$: For $k_1=even$ and $k_2=even$ , $\lambda _{1,0} (L)$ is the second smallest eigenvalue and $\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2}} (L)$ is the largest eigenvalue of a Laplacian matrix. Thus, (\[8\]) can be rewritten as $$\left| {1 - h\lambda _{1,0} (L)} \right| = \left| {1 - h\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2}} (L)} \right|
\label{28}$$ Substituting the $\lambda _{1,0} (L)$ and $\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2}} (L)$ in (\[28\]) results in $$\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_2 }} + ia\sin \frac{{2\pi }}{{k_2 }}} \right)} \right| = \left| {1 - 4h} \right|
\label{29}$$ Thus, we obtain $$h = \frac{{6 + 2\cos \frac{{2\pi }}{{k_2 }}}}{{15 - \cos ^2 \frac{{2\pi }}{{k_2 }} + 2\cos \frac{{2\pi }}{{k_2 }} - a^2 \sin ^2 \frac{{2\pi }}{{k_2 }}}}
\label{30}$$ Finally, we obtain the convergence parameter $\gamma$ as $$\gamma=\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_2 }} + ia\sin \frac{{2\pi }}{{k_2 }}} \right)} \right|
\label{31}$$ Thus, substituting (\[31\]) in (\[9\]) proves the Theorem 5.\
$Theorem$ $6$: Convergence rate of a torus network for $k_1=odd$ and $k_2=odd$ is expressed as $$R = \sqrt {\frac{{a^2 p_1^2 \sin ^2 \frac{{2\pi }}{{k_2 }}}}{{q_1^2 }} + \left( {1 - \frac{{p_1 \sin ^2 \frac{\pi }{{k_2 }}}}{{q_1 }}} \right)^2 }
\label{32}$$ where $$\begin{aligned}
p_1 ={} & 4 \left(2 \cos \left(\frac{\pi }{k_1}\right)+\cos \left(\frac{\pi }{k_2}\right)+\cos \left(\frac{2 \pi }{k_2}\right)+1\right),
\end{aligned}$$
$$\begin{aligned}
q_1 ={} & -a^2 \sin ^2\left(\frac{2 \pi }{k_2}\right)+a^2 \left(\sin \left(\frac{\pi }{k_1}\right)+\sin \left(\frac{\pi }{k_2}\right)\right){}^2 \\
& {}-\cos ^2\left(\frac{2 \pi }{k_2}\right)+\left(2 \cos \left(\frac{\pi }{k_1}\right)+\cos \left(\frac{\pi
}{k_2}\right)\right){}^2\\
&+8 \cos \left(\frac{\pi }{k_1}\right)+4 \cos \left(\frac{\pi }{k_2}\right)+2 \cos \left(\frac{2 \pi }{k_2}\right)+3
\end{aligned}$$
$Proof$ $6$: For $k_1=odd$ and $k_2=odd$, $\lambda _{1,0} (L)$ is the second smallest eigenvalue and $\lambda _{\frac{{k_1-1 }}{2},\frac{{k_2-1}}{2}} (L)$ is the largest eigenvalue of a Laplacian matrix. Thus rewrite (\[8\]) as $$\left| {1 - h\lambda _{1,0} (L)} \right| = \left| {1 - h\lambda _{\frac{{k_1-1 }}{2},\frac{{k_2-1}}{2}} (L)} \right|
\label{33}$$ Substitute the expressions of $\lambda _{1,0} (L)$ and $\lambda _{\frac{{k_1-1 }}{2},\frac{{k_2-1}}{2}} (L)$ in (\[33\]) results in $$\resizebox{1.1\hsize} {!} {$\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_2 }} + ia\sin \frac{{2\pi }}{{k_2 }}} \right)} \right| = \left| {1 - h\left( {2 - \cos \frac{{\pi (k_1 - 1)}}{{k_1 }} - \cos \frac{{\pi (k_2 - 1)}}{{k_2 }} + ia\left( {\sin \frac{{\pi (k_1 - 1)}}{{k_1 }} + \sin \frac{{\pi (k_2 - 1)}}{{k_2 }}} \right)} \right)} \right|$}
\label{34}$$ Thus, we get $$\resizebox{1.1\hsize} {!} {$h=\frac{-2 \cos \left(\frac{2 \pi }{k_2}\right)+2 \left(2 \cos \left(\frac{\pi \left(k_1-1\right)}{k_1}\right)+\cos \left(\frac{\pi \left(k_2-1\right)}{k_2}\right)\right)-2}{0.16 \sin ^2\left(\frac{2 \pi }{k_2}\right)-0.16 \left(\sin \left(\frac{\pi
\left(k_1-1\right)}{k_1}\right)+\sin \left(\frac{\pi \left(k_2-1\right)}{k_2}\right)\right){}^2+\cos ^2\left(\frac{2 \pi }{k_2}\right)-2 \cos \left(\frac{2 \pi }{k_2}\right)-\left(2 \cos \left(\frac{\pi \left(k_1-1\right)}{k_1}\right)+\cos
\left(\frac{\pi \left(k_2-1\right)}{k_2}\right)\right){}^2+4 \left(2 \cos \left(\frac{\pi \left(k_1-1\right)}{k_1}\right)+\cos \left(\frac{\pi \left(k_2-1\right)}{k_2}\right)\right)-3}$}
\label{35}$$ Finally, we obtain the convergence parameter $\gamma$ as $$\gamma=\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_2 }} + ia\sin \frac{{2\pi }}{{k_2 }}} \right)} \right|
\label{36}$$ Substituting the (\[36\]) in (\[9\]) results in (\[32\]).\
$Theorem$ $7$: The eigenvalue of a $m$-dimensional torus network is expressed as $$\lambda _{j_1 ,j_2 .....j_m } (L) = m - \sum\limits_{l = 1}^m {\cos \frac{{2\pi j_l }}{{k_l }}} + ia\left( {\sum\limits_{l = 1}^m {\sin \frac{{2\pi j_l }}{{k_l }}} } \right)
\label{37}$$ $Proof$ $7$: Cartesian product of ‘*m*’ ring networks results in $m$-dimenional torus network. The eigenvalue of a torus network will be the addition of eigenvalues of corresponding $m$ ring networks [@load]. $$\lambda _{j_1 ,j_2 .....j_m } (L)= \lambda _{j_1 } (L)+\lambda _{j_2 }(L)+....+\lambda _{j_1 } (L)+\lambda _{j_m}(L)
\label{38}$$ Here, we assume that the torus is formed by cartesian product of ’*m*’ ring networks with $k_m$ nodes, $m = 1,2....$. Then $(j_1+1)^{th}$ eigenvalue of a Laplacian matrix for ring network with $k_1$ nodes can be expressed as $$\lambda _{j_1 } (L) = 1 - \cos \frac{{2\pi j_1 }}{{k_1 }} + ia\sin \frac{{2\pi j_1 }}{{k_1 }}
\label{39}$$ The $(j_2+1)^{th}$ eigenvalue of a Laplacian matrix for ring network with $k_2$ nodes is expressed as $$\lambda _{j_2 } (L) = 1 - \cos \frac{{2\pi j_2 }}{{k_2 }} + ia\sin \frac{{2\pi j_2 }}{{k_2 }}
\label{40}$$ Similarly, $(j_m+1)^{th}$ eigenvalue of a Laplacian matrix for ring network with $k_m$ nodes is expressed as $$\lambda _{j_m } (L) = 1 - \cos \frac{{2\pi j_m }}{{k_m }} + ia\sin \frac{{2\pi j_m }}{{k_m }}
\label{41}$$ Using (\[38\]), (\[39\]), (\[40\]), and (\[41\]) we can write the eigenvalue of a $m$-dimensional torus network as (\[37\]).\
$Theorem$ $8$: Convergence rate of a $m$-dimensional torus network for $k_1=k_2=...k_m=even$ is expressed as $$\resizebox{0.9\hsize} {!} {$R =\frac{a^2 \sin ^2\left(\frac{2 \pi }{k_1}\right)+(4 m-2) \cos \left(\frac{2 \pi }{k_1}\right)+\cos ^2\left(\frac{2 \pi }{k_1}\right)+(1-2 m)^2}{a^2 \sin ^2\left(\frac{2 \pi }{k_1}\right)+\cos ^2\left(\frac{2 \pi }{k_1}\right)-2 \cos \left(\frac{2 \pi
}{k_1}\right)-4 m^2+1}$}
\label{42}$$ $Proof$ $8$: For $k_1 = k_2 = ..... = k_m =even $, second smallest eigenvalue of a Laplacian matrix is $\lambda _{1,0,0.....,0} (L)$ and largest eigenvalue of Laplacian matrix is $\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2},.....\frac{{k_n }}{2}} (L)$. Thus, rewrite (\[9\]) as $$\left| {1 - h\lambda _{1,0,0.....,0} (L)} \right| = \left| {1 - h\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2},.....\frac{{k_n }}{2}} (L)} \right|
\label{43}$$ Substituting the $\lambda _{1,0,0.....,0} (L)$ and $\lambda _{\frac{{k_1 }}{2},\frac{{k_2 }}{2},.....\frac{{k_n }}{2}} (L)$ in (\[43\]) results in $$\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_1 }} + ia\sin \frac{{2\pi }}{{k_1 }}} \right)} \right| = \left| {1 - 2mh} \right|
\label{44}$$ Thus, we obtain $$h = \frac{{2 - 2\cos \frac{{2\pi }}{{k_1 }} - 4m}}{{1 - 4m^2 + \cos ^2 \frac{{2\pi }}{{k_1 }} - 2\cos \frac{{2\pi }}{{k_1 }} + a^2 \sin ^2 \frac{{2\pi }}{{k_1 }}}}
\label{45}$$ Substituting the $h$ in $\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_1 }} + ia\sin \frac{{2\pi }}{{k_1 }}} \right)} \right|$ gives $\gamma$. Finally, substituting $\gamma$ value in (\[9\]) results in (\[42\]).\
**Note**: We are unable to give the expression of Convergence rate of a $m$-dimensional torus network for $k_1=k_2=...k_m=odd$, because the expression is too long and unable to fit in this format.\
$Theorem$ $9$: Consensus parameter of a $m$-dimensional torus network for $k_1=k_2=...k_m=odd$ is expressed as\
$$\resizebox{1.0\hsize} {!} {$h = \frac{{2 - 2\cos \frac{{2\pi }}{{k_1 }} - 4m}}{{1 + \cos ^2 \frac{{2\pi }}{{k_1 }} - 2\cos \frac{{2\pi }}{{k_1 }} + a^2 \sin ^2 \frac{{2\pi }}{{k_1 }} - m^2 - \left( {\sum\limits_{i = 1}^m {\cos \frac{{\pi \left( {k_i - 1} \right)}}{{k_i }}} } \right)^2 - a^2 \left( {\sum\limits_{i = 1}^m {\sin \frac{{\pi \left( {k_i - 1} \right)}}{{k_i }}} } \right)^2 }}$}
\label{46}$$ $Proof$ $9$: For $n$=odd, second smallest eigenvalue of a Laplacian matrix is $\lambda _{1,0,0.....,0} (L)$ and largest eigenvalue of Laplacian matrix is $\lambda _{\frac{{k_1-1}}{2},\frac{{k_2-1}}{2},.....\frac{{k_n-1}}{2}} (L)$. Thus, (\[8\]) can be rewritten as $$\left| {1 - h\lambda _{1,0,0.....,0} (L)} \right| = \left| {1 - h\lambda _{\frac{{k_1 - 1}}{2},\frac{{k_2 - 1}}{2},.....\frac{{k_n - 1}}{2}} (L)} \right|
\label{47}$$ Substitute the $\lambda _{1,0,0.....,0} (L)$ and $\lambda _{\frac{{k_1-1}}{2},\frac{{k_2-1}}{2},.....\frac{{k_n-1}}{2}} (L)$ in (\[46\]) results in $$\resizebox{1.0\hsize} {!} {$\left| {1 - h\left( {1 - \cos \frac{{2\pi }}{{k_1 }} + ia\sin \frac{{2\pi }}{{k_1 }}} \right)} \right| = \left| {1 - h\left( {m - \sum\limits_{i = 1}^m {\cos \frac{{\pi \left( {k_i - 1} \right)}}{{k_i }} + ia\sum\limits_{i = 1}^m {\sin \frac{{\pi \left( {k_i - 1} \right)}}{{k_i }}} } } \right)} \right|$}
\label{48}$$ Simplifying (\[48\]) further results in (\[46\]).
Explicit Formulas of Convergence Rate for $r$-Nearest Neighbor Networks
=======================================================================
In this section, we derive the explicit expressions of convergence rate for $r$-nearest neighbor networks. In this network, nodes in the distance *r* gets connected. The variable *r* models the node’s transmission radius or node overhead in WSNs.\
$Theorem$ $10$: The $(j+1)^{th}$ eigenvalue of a $r$-nearest neighbor ring network is expressed as $$\lambda _j (L) = r - \sum\limits_{k = 1}^r {\cos \frac{{2\pi jk}}{n}} + ia\sum\limits_{k = 1}^r {\sin \frac{{2\pi jk}}{n}}
\label{49}$$ $Proof$ $10$: Laplacian matrix of a $r$-nearest neighbor ring network with $\textit{n}$ can be written as $$\resizebox{0.9\hsize} {!} {$L = circ\{ r\,\,\underbrace {\frac{{ - 1 + a}}{2}\,\,\frac{{ - 1 + a}}{2}\,\,.....\frac{{ - 1 + a}}{2}}_{r\,terms}.....\underbrace {\frac{{ - 1 - a}}{2}\,\,\frac{{ - 1 - a}}{2}....\frac{{ - 1 - a}}{2}}_{r\,\,terms}\}$}
\label{50}$$ Using (\[10\]) and (\[50\]), we obtain (\[49\]).\
$Theorem$ $11$: Convergence Rate of a $r$-nearest neighbor ring network for $n=even$ is expressed as $$R = 1 - \sqrt {\left( {\frac{{p_2 q_2 }}{{s_2 }}} \right)^2 + \frac{{r_2 q_2 }}{{s_2 }} + 1}
\label{51}$$, where $$\begin{aligned}
p_2 ={} & \frac{{a\sin \frac{\pi }{n}\cos \frac{{(2r + 1)\pi }}{n} - 0.5a\sin \frac{{2\pi }}{n}}}{{\cos \frac{{2\pi }}{n} - 1}}, \\
q_2 ={} & \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}} - \cos \pi r ,\\
r_2 ={} & \frac{{\sin \frac{\pi }{n}\sin \frac{{(2r + 1)\pi }}{n}}}{{\cos \frac{{2\pi }}{n} - 1}} + r + 0.5,
\end{aligned}$$ and
$$\begin{aligned}
s_2 ={} & \frac{0.25 a^2 \left(2 \sin \left(\frac{\pi }{n}\right) \cos \left(\frac{2 \pi r+\pi }{n}\right)-\sin \left(\frac{2 \pi }{n}\right)\right)^2}{\left(\cos \left(\frac{2 \pi }{n}\right)-1\right)^2} \\
& +\frac{\sin ^2\left(\frac{\pi }{n}\right) \sin
^2\left(\frac{2 \pi r+\pi }{n}\right)}{\left(\cos \left(\frac{2 \pi }{n}\right)-1\right)^2}\\
&+(r+0.5) \left(\cos (\pi r)-\csc \left(\frac{\pi }{n}\right) \sin \left(\frac{2 \pi r+\pi }{n}\right)\right)\\
&-0.25 \cos ^2(\pi r).
\end{aligned}$$
$Proof$ $11$: For $n$=even, $\lambda _1 (L)$ is the second smallest value of Laplacian matrix and $\lambda _{\frac{n}{2}} (L)$ is the largest eigenvalue of a Laplacian matrix. Thus, rewrite (\[9\]) as $$\left| {1 - h\lambda _1 (L)} \right| = \left| {1 - h\lambda _{\frac{n}{2}} (L)} \right|
\label{52}$$\
Substituting the $\lambda _1 (L)$ and $\lambda _{\frac{n}{2}} (L)$ in (\[52\]), results in $$\resizebox{1.0\hsize} {!} {$\left| {1 - h\left( {r + 0.5 - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{2\sin \frac{\pi }{n}}} + \frac{{ia}}{2}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)} \right)} \right| = \left| {1 - h\left( {r + 0.5 - 0.5\cos \pi r} \right)} \right|$}
\label{53}$$ Thus, we get $$\resizebox{0.9\hsize} {!} {$h = \frac{{\cos \pi r - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}}}{{\frac{1}{4}\left( {\frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)^2 - (r + 0.5)\left( {\frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}} - \cos \pi r} \right) - \frac{{\cos ^2 \pi r}}{4} + \frac{{e^2 }}{4}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)^2 }}$}
\label{54}$$ Thus, convergence factor is expressed as $$\resizebox{0.9\hsize} {!} {$\gamma=\left| {1 - h\left( {r + 0.5 - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{2\sin \frac{\pi }{n}}} + \frac{{ia}}{2}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)} \right)} \right|$}
\label{55}$$ Substitute (\[55\]) in (\[9\]) proves the $Theorem$ $11$.\
$Theorem$ $12$: Convergence Rate of a $r$-nearest neighbor ring network for $n=odd$ is expressed as $$R = 1 - \sqrt {\left( {\frac{{p_3 q_3 }}{{s_3 }}} \right)^2 + \frac{{r_3 q_3 }}{{s_3 }} + 1}
\label{56}$$, where
$$\begin{aligned}
p_3 ={} & \frac{{ - a\sin \frac{\pi }{n}\cos \frac{{(2r + 1)\pi }}{n} + 0.5a\sin \frac{{2\pi }}{n}}}{{\cos \frac{{2\pi }}{n} - 1}}, \\
q_3 ={} & - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}} + \frac{{\sin \frac{{\pi (n - 1)(2r + 1)}}{{2n}}}}{{\cos \frac{\pi }{{2n}}}} ,\\
r_3 ={} & - \frac{{\sin \frac{\pi }{n}\sin \frac{{(2r + 1)\pi }}{n}}}{{\cos \frac{{2\pi }}{n} - 1}} - r - 0.5,
\end{aligned}$$
and $$\begin{aligned}
s_3 ={} & -\frac{0.25 a^2 \left(2 \cos \left(\frac{\pi }{2 n}\right) \cos \left(\frac{\pi (n-1) (2 r+1)}{2 n}\right)-\sin \left(\frac{\pi }{n}\right)\right)^2}{\left(\cos \left(\frac{\pi }{n}\right)+1\right)^2} \\
& +\frac{0.25 a^2 \left(2 \sin \left(\frac{\pi}{n}\right) \cos \left(\frac{2 \pi r+\pi }{n}\right)-\sin \left(\frac{2 \pi }{n}\right)\right)^2}{\left(\cos \left(\frac{2 \pi }{n}\right)-1\right)^2} \\
& -\frac{\cos ^2\left(\frac{\pi }{2 n}\right) \sin ^2\left(\frac{\pi (n-1) (2 r+1)}{2n}\right)}{\left(\cos \left(\frac{\pi }{n}\right)+1\right)^2}+\frac{\sin ^2\left(\frac{\pi }{n}\right) \sin ^2\left(\frac{2 \pi r+\pi }{n}\right)}{\left(\cos \left(\frac{2 \pi }{n}\right)-1\right)^2} \\
&+(r + 0.5)\left( {\frac{{\sin \frac{{\pi (n - 1)(2r + 1)}}{{2n}}}}{{\cos \frac{\pi }{{2n}}}} - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right).
\end{aligned}$$ $Proof$ $12$: For $n$=odd, $\lambda _1 (L)$ is the second smallest eigenvalue of Laplacian matrix and $\lambda _{\frac{n-1}{2}} (L)$ is the largest eigenvalue of a Laplacian matrix.\
Thus, we rewrite the (\[8\]) as $$\left| {1 - h\lambda _1 (L)} \right| = \left| {1 - h\lambda _{\frac{n-1}{2}} (L)} \right|
\label{57}$$\
After substituting the $\lambda _1 (L)$ and $\lambda _{\frac{n-1}{2}} (L)$ expressions in (\[57\]), we obtain $$\resizebox{1.1\hsize} {!} {$\left| {1 - h\left( {r + 0.5 - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{2\sin \frac{\pi }{n}}} + \frac{{ia}}{2}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)} \right)} \right| = \left| {1 - h\left( {r + 0.5 - \frac{{\sin \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}} + \frac{{ia}}{2}\left( {\cot \frac{{\pi (n - 1)}}{{2n}} - \frac{{\cos \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}}} \right)} \right)} \right|$}
\label{58}$$ Thus, we obtain $$\resizebox{1.1\hsize} {!} {$h = \frac{{\frac{{\sin \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}} - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}}}{{\frac{1}{4}\left( {\frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)^2 + (r + 0.5)\left( { - \frac{{\sin \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}} + \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right) - \frac{1}{4}\left( {\frac{{\sin \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}}} \right)^2 + \frac{{e^2 }}{4}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)^2 - \frac{{e^2 }}{4}\left( {\cot \frac{{\pi (n - 1)}}{{2n}} - \frac{{\cos \frac{{(2r + 1)\pi (n - 1)}}{{2n}}}}{{\sin \frac{{\pi (n - 1)}}{{2n}}}}} \right)^2 }}$}
\label{59}$$ Finally, convergence factor $\gamma$ is expressed as $$\resizebox{1.05\hsize} {!} {$\gamma=\left| {1 - h\left( {r + 0.5 - \frac{{\sin \frac{{(2r + 1)\pi }}{n}}}{{2\sin \frac{\pi }{n}}} + \frac{{ia}}{2}\left( {\cot \frac{\pi }{n} - \frac{{\cos \frac{{(2r + 1)\pi }}{n}}}{{\sin \frac{\pi }{n}}}} \right)} \right)} \right|$}
\label{60}$$ Substituting the (\[60\]) in (\[9\]) proves the $Theorem$ $12$.
Numerical Results
=================
In this section, we present the numerical results to investigate the effect of asymmetric link factor, network dimension, number of nodes, and node overhead on the convergence rate of the average consensus algorithm. We have used the *Wolfram Mathematica* to solve the equations. Fig. 3 shows the comparison of convergence rates of asymmetric and symmetric ring networks. We have observed that the convergence rate decreases with both the number of nodes and asymmetric link weight. Fig. 4 shows the convergence rate versus $k_{1}$ and $k_{2}$. Here, convergence rate decreases with $k_{1}$ and $k_{2}$ exponentially. Fig. 5 shows the convergence rate versus asymmetric link factor for different values of $r$. We have noted that the convergence rate increases with the node overhead and decreases with asymmetric link factor. We noted that the convergence rate becomes ‘0’ at asymmetric factor $0.8$. To understand the effect of network dimension on the convergence rate, we plotted the Fig. 6. We have observed that the convergence rate decreases with the network dimension. To compute the error introduced by the symmetric network modeling, we compute the absolute error $R_{s}-R_{a}$, where $R_{s}$ and $R_{a}$ denote the convergence rates of symmetric and asymmetric networks respectively. Fig. 7 shows the Absolute Error versus Number of nodes. Here, the absolute error decreases with the number of nodes. Therefore, the effect of asymmetric link modeling on the convergence rate is high in small-scale networks. We have observed that the absolute error is significant for large values of asymmetric link factors.
![Comparison of convergence rates in asymmetric and symmetric ring networks.[]{data-label="fig:verticalcell"}](asymmetric1){width="8cm" height="5.5cm"}
![Convergence Rate versus $k_{1}$ versus $k_{2}$ of a torus network for $n$=odd.[]{data-label="fig:verticalcell"}](torusodd){width="6.5cm" height="5cm"}
![Convergence Rate versus Asymmetric Link Weight of a $r$-nearest neighbor network for $n$=400.[]{data-label="fig:verticalcell"}](rnearesteven){width="6.5cm" height="6.5cm"}
![Convergence Rate versus Network Dimension for $k_{1}=11$, $k_{2}=15$, $k_{3}=21$, $k_{4}=25$, $k_{5}=27$.[]{data-label="fig:verticalcell"}](m-dimensional_odd){width="6.5cm" height="5cm"}
![Absolute Error versus Number of nodes for $a=0.3$, $a=0.9$.[]{data-label="fig:verticalcell"}](AE){width="6.5cm" height="5cm"}
Conclusions
===========
In this work, we modeled the WSN as a directed graph and derived the explicit formulas for a ring, torus, $r$-nearest neighbor ring, and $m$-dimensional torus networks. Numerical results demonstrated that the convergence rate decreases significantly with asymmetrical link factor in small-scale WSNs. In large-scale WSNs, the effect of asymmetrical links on convergence rate decreases with the number of nodes. Further, we studied the impact of the number of nodes, network dimension, and node overhead on the convergence rate. We have observed that the convergence rate increases with the node overhead and decreases with node dimension. However, energy consumption rises with the node overhead. Since WSNs are energy constrained networks, it is essential to design an optimal framework to maximize the convergence rate without affecting the energy consumption. Our analysis avoids the use of sophisticated algorithms to study the convergence rate and also reduces the computational complexity drastically over the existing approaches.
|
---
abstract: 'We extend the Unified Radio Catalog, a catalog of sources detected by various (NVSS, FIRST, WENSS, GB6) radio surveys, and SDSS, to IR wavelengths by matching it to the IRAS Point and Faint Source catalogs. By fitting each NVSS-selected galaxy’s NUV-NIR spectral energy distribution (SED) with stellar population synthesis models we add to the catalog star formation rates, stellar masses, and attenuations. We further add information about optical emission line properties for NVSS-selected galaxies with available SDSS spectroscopy. Using an NVSS 20 cm ($F_\mathrm{1.4GHz}\gtrsim2.5$ mJy) selected sample, matched to the SDSS spectroscopic (“main” galaxy and quasar) catalogs and IRAS data ($0.04<z\lesssim0.2$) we perform an in depth analysis of the radio-FIR correlation for various types of galaxies, separated into i) quasars, ii) star forming, iii) composite, iv) Seyfert, v) LINER and vi) absorption line galaxies using the standard optical spectroscopic diagnostic tools. We utilize SED-based star formation rates to independently quantify the source of radio and FIR emission in our galaxies. Our results show that Seyfert galaxies have FIR/radio ratios lower than, but still within the scatter of, the canonical value due to an additional (likely AGN) contribution to their radio continuum emission. Furthermore, IR-detected absorption and LINER galaxies are on average strongly dominated by AGN activity in both their FIR and radio emission; however their average FIR/radio ratio is consistent with that expected for star forming galaxies. In summary, we find that most AGN-containing galaxies in our NVSS-IRAS-SDSS sample have FIR/radio flux ratios indistinguishable from those of the star-forming galaxies that define the radio-FIR correlation. Thus, attempts to separate AGNs from star-forming galaxies by their FIR/radio flux ratios alone can separate only a small fraction of the AGNs, such as the radio-loud quasars.'
author:
- 'I. Morić, V. Smolčić, A. Kimball, D. A. Riechers, Ž Ivezić, N. Scoville'
title: ' A closer view of the radio-FIR correlation: Disentangling the contributions of star formation and AGN activity'
---
Introduction {#sec:introduction}
============
The radio-FIR correlation is one of the tightest correlations in observational astrophysics (e.g. @helou85 [@condon92; @mauch07; @yun01; @bell03; @sargent10; @kovacs06; @murphy09; @appleton04]). The correspondence between the radiation in the (far-)infrared and that in the radio spans over nearly five orders of magnitude in various types of galaxies, ranging from dwarfs to ULIRGs. Given that the two observational windows, IR and radio, trace independent and different intrinsic physical mechanisms in galaxies – thermal vs. synchrotron radiation – the existence of such a tight correspondence is remarkable. It is generally believed that recent star formation in galaxies is the process that relates IR and radio emission.
The radio-FIR correlation has been extensively studied in the past both in the low [@helou85; @condon92; @garrett02; @mauch07; @yun01; @bell03] and high redshift universe [@sargent10; @kovacs06; @sajina08; @murphy09; @appleton04; @vlahakis08; @ibar08; @chapman05]. It has been shown that, out to redshifts of $z\sim3-4$, the FIR/radio ratios of various types of galaxies are essentially the same as those in the local universe. At higher redshifts radio quiet QSOs have been demonstrated to have FIR/radio ratios consistent with the local value, while the FIR/radio ratios of $z>4$ SMGs are found to be lower by a few factors. This is somewhat contrary to expectations, as the the FIR/radio ratio is expected to be rising with redshift (especially at $z\gtrsim3$) due to the increase of the cosmic microwave background (CMB) energy density ($U_\mathrm{CMB}$) with redshift, $U_\mathrm{CMB}\propto(1+z)^4$, which surpresses the non-thermal component of a galaxy’s radio continuum via inverse-Compton (IC) scattering (see Murphy 2009 for details). An explanation for this discrepancy can be provided by additional processes that add to a galaxy’s radio continuum, such as increased magnetic field strengths or AGN contribution, that may compensate for the radio continuum emission losses due to IC scattering.
The AGN contribution to the radio-FIR correlation has been studied in the past to some extent. Typically a low FIR/radio ratio, significantly offsetting a galaxy from the correlation, is thought to indicate a radio-loud AGN [e.g. @yun01; @condon02]. However, recent studies have shown that optically-selected AGN often follow the correlation, albeit with a slightly lower FIR/radio ratio. For example, based on SDSS-NVSS-IRAS data, @obric06 have demonstrated a tight correlation between radio and 60 $\mu$m fluxes for low-luminosity AGN (predominantly Seyferts and LINERs), which varies by only $\sim20\%$ relative to that of star forming galaxies. Utilizing 6dFGS-NVSS-IRAS data, @mauch07 inferred a lower average FIR/radio ratio for AGN-bearing galaxies (Seyferts, LINERs, and quasars), but still within the scatter of the correlation for star forming galaxies. Furthermore, studies of the correlation at higher redshifts have yielded a handful of interesting objects for which it has clearly been shown that a significant AGN contribution to IR and/or radio exists, yet their FIR/radio ratio is consistent with the canonical value for star forming galaxies (Riechers et al. 2009, Murphy et al. 2009).
In order to understand in more detail the contribution of AGN activity to the radio-FIR correlation, we perform an in-depth study of the radio-FIR correlation, with a large sample, as a function of galaxy type, and comparison with star formation rates for those individual samples. The various types of star forming and AGN-bearing galaxies have been drawn from the NVSS [@condon98], IRAS [@neugebauer84], and SDSS [@york00] sky surveys. In we present the data used in this paper. We present the correlation for various types of galaxies in . In we link the FIR and radio emission from galaxies in our sample to independently derived star formation rates, and in and we discuss and summarize our results, respectively. We adopt $H_0=70$, $\Omega_M=0.3$, $\Omega_\Lambda=0.7$, and define the radio synchrotron spectrum as $F_\nu\propto\nu^{-\alpha}$, assuming $\alpha=0.7$. Throughout the text we will often use the term ’quasar’ referring to both quasi-stellar radio sources and quasi-stellar objects.
Data and galaxy samples: Expanding the Unified Radio Catalog {#sec:data}
============================================================
Unified Radio Catalog
---------------------
@kim08 have constructed a catalog of radio sources detected by the GB6 (6 cm), FIRST [@becker95], NVSS (@condon98; 20 cm), and WENSS (92 cm) radio surveys, as well as the SDSS (DR6) optical survey [@york00]. This “Unified Radio Catalog” has been generated in such a way that it allows a broad range of 20 cm based sample selections and source analysis (see @kim08 for details). The 2.7 million entries are comprised of the closest three FIRST to NVSS matches (within $30''$) and vice-versa, as well as unmatched sources from each survey. All entries have been supplemented by data from the other radio and optical surveys, where available. Here we select from the Unified Radio Catalog (version 1.1) all 20 cm sources that have been detected by the NVSS radio survey (using $\mathrm{matchflag\_nvss}=-1$ and $\mathrm{matchflag\_first}\leq1$; see @kim08 for details). This selection yields a radio flux limited ($F_\mathrm{1.4GHz}\gtrsim2.5$ mJy) sample that contains 1,814,748 galaxies. In the following section we expand this catalog to IR wavelengths, and augment it with additional (spectroscopic and SED-based) information.
Expanding the Unified Radio Catalog
-----------------------------------
### IRAS
For the purpose of this paper, we have expanded the Unified Radio Catalog to IR wavelengths by cross-correlating it with the IRAS point-source and faint-source catalogs (hereafter PSC and FSC, resp.). The IRAS PSC contains 245,889 confirmed point sources detected at 12, 25, 60 and 100 $\mu$m, respectively @strauss90. The completeness of the catalog at these wavelengths reaches down to 0.4, 0.5, 0.6 and 1.0 Jy, respectively. The FSC was tuned to fainter levels based on the same IR data by point-source filtering the individual detector data streams and then coadding those using a trimmed-average algorithm (see @moshir92). The reliability of the FSC is slightly lower than that of the PSC ($\gtrsim94\%$ compared to 99.997$\%$); however its sensitivity is higher by a factor of $\sim2.5$. The FSC contains 173,044 point sources with flux densities typically greater than 0.2 Jy at 12, 25 and 60 $\mu$m and 0.5 Jy at 100 $\mu$m.
We used matching radius of $30"$, as optimized by @obric06, in cross-corelating the Unified Radio Catalog with the IR IRAS data. In we show the distribution of the distances between the IR and radio detections. The cumulative distribution displayed in shows that $\sim70\%$ of the positional matches are within an angular distance of $15''$.
Our NVSS-selected radio sample contains 18,313 galaxies with high quality IR photometry[^1] (see ). As the FSC and PSC have been generated based on the same data, most of the PSC sources are included in the FSC. In our entire NVSS-IRAS sample, 26% of the sources have a PSC detection but are not included in the FSC. This fraction, however, reduces to only 3% after an optical (SDSS) cross-match is performed. The 60 and 100 $\mu$m magnitudes reported in the PSC and FSC are in agreement for the union of the two IR samples. The biweighted mean of the flux difference (for a subsample with SDSS detections) is 0.02 and 0.03 Jy at 60 and 100 $\micron$, respectively. The root mean scatter of the 60 $\micron$ flux difference distribution is 0.06 Jy, while that of the 100 $\micron$ distribution is significantly larger, i.e.0.16 Jy. Therefore, in order to access the highest quality IR photometry, hereafter we use the values reported in either the FSC or PSC catalog corresponding to the higher photometric quality flag quoted in the catalogs. The distribution of the 60 $\mu$m and 20 cm flux densities are shown in .
### SDSS quasar and main galaxy sample catalogs
We have further matched the NVSS-selected sample from the Unified Radio Catalog with data drawn from i) the SDSS DR5 quasar sample [@schneider07], and ii) the DR4 “main” spectroscopic sample for which derivations of emission line fluxes from the SDSS spectra are available (see 2009 and references therein; note that the DR5 quasar and DR4 main galaxy catalogs were the most up-to-date versions available at the time). The latter was complemented with stellar masses, star formation rates, dust attenuations, ages, metallicities, and a variety of other parameters based on spectral energy distribution (SED) fitting of the SDSS ($ugriz$) photometry using the @bruzual03 stellar population synthesis models. The SED fitting was performed as described in detail in et al. (2008).
During the inspection of the validity of the final catalog, we have found about 1$\%$ of objects with different spectroscopic redshifts in various SDSS data releases ($\Delta z > 5\times10^{-4}$). We have excluded those from the sample. Furthermore, a small number ($\sim
0.2\%$) of duplicate objects was present in both the SDSS “main” galaxy sample and the SDSS Quasar Catalog. Visually inspecting their spectra yielded that most of these objects are better matched to the properties of the “main” galaxy sample (as no power-law continuum nor broad emission-lines were present in the spectrum), and we have excluded these from our quasar sample. A summary of the various radio-IR-optical samples is given in , and in and we show the radio (20 cm), optical ($r$ band), and far-IR luminosities as a function of redshift for the final NVSS-SDSS and NVSS-SDSS-IRAS samples (see eqs. \[for:lfir\] and \[for:lr\]). Note that the shallow IRAS sensitivity (compared to the NVSS and SDSS data) significantly reduces the number of objects, and biases the sample toward lower redshifts.
Radio-optical-IR samples
------------------------
### Star forming and AGN galaxy subsamples {#sec:bptselection}
We have used the optical spectroscopic information added to the NVSS selected sample to spectroscopically separate the galaxies present in the SDSS (DR4) “main” galaxy sample as absorption line, AGN (LINER/Seyfert), star forming, or composite galaxies.
We define emission-line galaxies as those where the relevant emission lines (H$\alpha$, H$\beta$, O\[III,$\lambda$5007\], N\[II,$\lambda$6584\], S\[II,$\lambda$$\lambda$6717,6731\]) have been detected at $\mathrm{S/N}\geq3$, and consider all galaxies with $\mathrm{S/N}<3$ in any of these lines as absorption line systems (see e.g.@best05 [@kewley06; @smo09]). As strong emission lines are not present in the spectra of the latter, yet they are luminous at 20 cm, they can be considered to be (low excitation) AGN (see e.g. @best05 [@smo08] for a more detailed discussion). Further, as illustrated in , using standard optical spectroscopic diagnostics [@bpt81; @kauffmann03b; @kewley01; @kewley06] we sort the emission-line galaxies into 1) star forming, 2) composite, 3) Seyfert, and 4) LINER galaxies. The last two classes have been selected ‘unambiguously’ by requiring combined criteria using three emission line flux ratios (see middle and right panels in ). A summary of the number of objects in each class is given in . It is noteworthy that the IR detection fraction is a strong function of spectral class. It is the lowest for absorption line (0.6%) and LINER (6.5%) galaxies, intermediate for Seyferts (22%) and the highest for composite (40%) and star forming (46%) galaxies. These results suggest lower amounts of dust (and gas; @solomon95) in the former or alternatively dominantly very cold dust that peaks at longer wavelenghts.
The redshift distribution of the various galaxy types with 20 cm NVSS and NVSS-IRAS detections is shown in the two top panels in . Note that the redshift distribution of 20 cm detected absorption line galaxies is biased toward higher-redshifts, compared to all other galaxy types (see top panel in ). However, this is not the case when an IRAS IR detection is required, as illustrated in the middle panel in . The IR detection fraction of the different galaxy classes is shown as a function of redshift in the bottom panel in . Except for the overall trend that absorption and LINER galaxies are detected less efficiently in the IR, there is no substantial difference between the detection fractions as a function of redshift for different types of spectroscopically selected galaxies.
Hereafter, we apply redshift range limits of 0.04 $<$ z $<$ 0.3 to our sample. The lower redshift limit is adopted from @kewley05. Kewley et al. explored effects of fixed-size aperture of the SDSS spectroscopic fibers on the spectral characteristics such as metallicity, star formation rate, and reddening. They concluded that a minimum aperture size covering $\approx20\%$ of spectral light was required to properly approximate global values. The SDSS fiber aperture of $3''$ diameter collects such a fraction of light for galaxies of average size, type, and luminosity at $z\gtrsim0.04$. The upper redshift limit of $z=0.3$ is equivalent to that of the SDSS “main” spectroscopic sample (note however that the majority of IR-detected galaxies are at $z<0.2$, see ). It is worth noting that, because of lower spectral signal-to-noise for fixed-luminosity galaxies at greater distances, galaxies with weak emission lines, such as LINERs, can get confused with absorption line galaxies at $z>0.1$ [@kewley05]. However, as LINER and absorption galaxies have similar physical properties, (e.g. 2009) we simply combine these two types of galaxies, and treat them hereafter as a single class.
### Quasar subsample
Matching the SDSS DR5 quasar catalog to the Unified radio catalog resulted in 4,490 matches (see ). The redshift range of our radio luminous quasars is $0.09-5.12$, with a median at $z=1.36$. Requiring IRAS detections biases the sample toward low redshifts ($0.12\leq z\leq1.15$), with a median redshift of $0.18$, and selects only $\sim0.5\%$ of the radio detected quasars. The radio ($\gtrsim10^{23}$ ) and FIR ($\gtrsim2\times10^{11}~\mathrm{L_\odot}$) luminosities (see eqs. \[for:lfir\] and \[for:lr\]) of our quasars are systematically higher than those of the SDSS “main” spectroscopic sample galaxies in our radio-optical-IR sample.
Quantifying the radio-FIR correlation for various source types {#sec:quant}
==============================================================
Parameterizing the radio-FIR correlation
----------------------------------------
The radio-FIR correlation is usually quantified by its slope via the $\textit{q}$ parameter [@helou85], defined as the logarithmic ratio of the far-infrared flux to radio flux density:
$$q = \log \left\{\frac{F_\mathrm{FIR}/(3.75 \times 10^{12} \,\mathrm{Hz})}{F_\mathrm{1.4GHz}}\right\}
\label{for:q}$$
where $F_\mathrm{1.4GHz}$ is the 1.4 GHz radio flux density in units of $\mathrm{W m^{-2} Hz^{-1}}$ and $F_\mathrm{FIR}$ is the far-infrared flux in units of $\mathrm{W m^{-2}}$. Following @sanders96, we define the latter as:
$$F_\mathrm{FIR} = 1.26 \times 10^{-14}\, ( 2.58 S_{60\micron} + S_{100\micron})
\label{for:fir}$$
where $S_{60\micron}$ and $S_{100\micron}$ are observed flux densities at 60 and 100$\micron$ (in Jy), respectively.
We compute the far-infrared luminosity as:
$$L_{FIR} = 4\pi D_{L}^{2} C F_{FIR}\left[L\odot\right]
\label{for:lfir}$$
where $D_{L}$ is the luminosity distance (in units of m) and $C$ is a scale factor used to correct for the extrapolated flux longward of the IRAS 100 $\mu$m filter. We use $C=1.6$ (see Tab. 1 in @sanders96). Note that this expression can also be utilized to compute the FIR luminosities for our IR-detected quasars, given their relatively low redshifts. The radio luminosity density is computed as:
$$L_\mathrm{1.4GHz} = \frac{4\pi D_{L}^{2}}{(1+z)^{1-\alpha}}F_\mathrm{1.4GHz}
\label{for:lr}$$
where $z$ is the redshift of the source, $F_\mathrm{1.4GHz}$ is its integrated flux density, and $\alpha$ the radio spectral index (assuming $F_\nu\propto\nu^{-\alpha}$). To compute the radio luminosities, we assumed a spectral index of $\alpha=0.7$.
Radio-FIR correlation for all sources
-------------------------------------
The radio-FIR correlation for the NVSS-SDSS-IRAS sample is summarized in . The radio and FIR flux densities (top left panel) and luminosities (top right panel) clearly show a tight correlation that holds over many orders of magnitude. In the middle panels we show the $q$ parameter, that characterizes the slope of the radio-FIR correlation (see eq. \[for:q\]), as a function of FIR and radio luminosities. The average $q$ is constant as a function of FIR luminosity (middle left panel), and it is decreasing with increasing radio power (middle right panel; see also below). In the bottom panels of we show the $q$ parameter as a function of redshift, as well as its distribution for all our NVSS-SDSS-IRAS sources (galaxies and quasars). We find that the average (biweighted mean) $q$ value for the entire NVSS-SDSS-IRAS sample is q=2.273$\pm$0.008, with a root-mean-square scatter of $\sigma=0.18$. This is in very good agreement with previous findings [@condon92; @yun01; @condon02; @bell03; @mauch07], and will be discussed in more detail in .
The quasars in our sample comprise the high luminosity end at both IR and radio wavelengths (they are also located at higher redshifts, compared to the IR and radio detected “main” galaxy sample). It is also obvious that there is a larger fraction of quasars that do not lie on the radio-IR correlation, compared to that for the “main” sample galaxies.
Radio-FIR correlation for different types of galaxies {#sec:gals}
-----------------------------------------------------
In we present the radio-FIR correlation for the SDSS “main” galaxy sample subdivided into different, spectroscopically selected galaxy types (absorption, LINER, Seyfert, composite and star forming galaxies; see and for details on the selection). The decrease in $q$ with increasing radio power for all types of galaxies (middle right panel) is consistent with various other observations (e.g. et al. 2008; Kartaltepe et al. 2010; Sargent et al. 2010, Ivison et al. 2010). Note that, given the definition of the FIR/radio ratio, for any sample in which $q$ does not vary with FIR luminosity, and has a non-zero dispersion, it is expected to decrease with increasing radio luminosity (see e.g. Condon 1984). To test whether the magnitude of the decrease is as expected from statistics or e.g. higher due to an additional effect (such as AGN contribution) we have computed the radio luminosity for each source based on its observed FIR luminosity and a FIR/radio ratio drawn from a Gaussian distribution with a dispersion of 0.18 and a mean of 2.27. We find that the decrease of $q$ with radio luminosity in the observed data is consitent with that in the simulated data, thus not requiring additional effects (such as increasing AGN contribution with increasing radio power) to explain this trend (at least in the radio luminosity range probed here).
A quantitative analysis of the radio-FIR correlation for different galaxy types is presented in Fig $\ref{fig:param}$. The spectroscopic selection of pure star forming galaxies allows us to quantify the radio-IR correlation in a rather unbiased manner. For our star forming galaxies we find an average $q$ value of $<q>=2.27\pm0.05$, with a small root-mean-square scatter of $\sigma=0.13$. It is interesting to note that as the AGN contribution rises in galaxies (as inferred based on optical spectroscopic diagnostics) the scatter in $q$ increases by $\sim50\%$ to $\sim150\%$. Interestingly, the scatter is the highest for Seyfert types of galaxies, for which we also find the lowest average $q$ value, $<q>=2.14\pm0.05$. These differences will further be discussed in .
Radio-FIR correlation for quasars {#sec:qso}
---------------------------------
In we quantify the radio-FIR correlation for the 21 IR-detected quasars in our sample. The distribution of the FIR/radio ratio cannot be well fit with a Gaussian distribution. The median $q$ value of the sample is 2.04, comparable to the average $q$ value we have found for Seyfert galaxies (2.14), and lower than that for star forming galaxies (2.27; see ). It is worth noting that the higher redshift quasars ($0.2\lesssim z\lesssim0.4$) appear to be biased toward more radio-loud AGN.
An independent view of the radio-FIR correlation: A link to star formation {#sec:lexcess}
==========================================================================
It is generally taken that recent star formation drives both the radio and FIR emission of galaxies that lie on the radio-FIR correlation [@condon92; @mauch07]. Therefore, a correlation is expected to be present between the SFRs and radio/FIR luminosities obtained from the fluxes of galaxies dominated by recent star formation. To shed light on the source of radio/FIR emission in our galaxies, in this section we investigate the correlation between their radio/FIR luminosities and star formation rates, independently determined based on fitting stellar population synthesis models to the NUV-NIR SED).
We have derived a star formation rate for every galaxy in our sample by fitting the @bruzual03 library of stellar population synthesis models to the SDSS $ugriz$ photometry (see ). In we show the radio and FIR luminosities of our sources as a function of our SED-based SFRs. As expected, a correlation is discernible between these two quantities. This is especially emphasized for star formation dominated galaxies (i.e. star forming and composite galaxies), the distribution of which agrees well with the commonly used radio/IR luminosity – SFR calibrations [@kennicutt98; @yun01]. Note that this is quite remarkable as the SFRs have been derived completely independently from the FIR or radio emissions in the galaxies.
From it is obvious that a large fraction of galaxies with significant AGN contribution (Seyfert, LINER and absorption galaxies) has an obvious excess of radio power and FIR luminosity compared to that expected from the galaxy’s SFR. The most obvious example of this are the LINER and absorption galaxies from both the NVSS-SDSS and NVSS-SDSS-IRAS samples.
To investigate whether star formation is the underlying source of radio/FIR emission in our galaxies, we further quantify the difference between the radio/FIR emission and that expected from star formation. We thus define an “excess” in radio/FIR emission relative to that expected from star formation ($\Delta \log{L_\mathrm{1.4Ghz}}$ and $\Delta \log{L_\mathrm{FIR}}$, resp.) as:
$$\Delta \log{L}= \log{L_\mathrm{data}} - \log{L_\mathrm{exp.}}
\label{for:delta}$$
where $\log{L_\mathrm{data}}$ is the logarithm of the 1.4 GHz or FIR luminosity derived based on NVSS or IRAS data (see ), and $\log{L_\mathrm{exp.}}$ is the luminosity (either at 1.4 GHz or FIR) expected based on the SED-derived SFR and the standard radio and FIR luminosity to SFR calibrations. To convert SFR to radio luminosity we use the calibration defined in @yun01: $\mathrm{SFR\,[M_\odot/yr]}=5.9\times10^{-22}\,L_\mathrm{1.4GHz}$ \[W/Hz\]. To convert SFR to FIR luminosity we use the standard conversion defined by Kennicutt (1998): $\mathrm{SFR\,[M_\odot/yr]}=4.5\times10^{-37}\,L_\mathrm{FIR}$ \[W\]. Prior to applying these conversions, derived using a Salpeter IMF, we have scaled our SED-based SFRs by -0.2 dex to convert from a Chabrier to a Salpeter IMF (we have additionally included a scaling factor of $\sim0.4$ dex to account for the star formation histories used in our models; see et al. 2008 and Walcher et al. 2008 for details).
shows the FIR ($\Delta \log\mathrm{L_{FIR}}$) versus 1.4 GHz ($\Delta \log\mathrm{L_{1.4GHz}}$) luminosity excess for different types of galaxies in the NVSS-SDSS-IRAS sample. As expected, star forming galaxies follow a normal distribution in both the FIR and radio luminosity excess, with a mean $\Delta \log{L}$ value of about zero ($<\Delta \log\mathrm{L_{FIR}}>=0.001$, $<\Delta
\log\mathrm{L_{1.4GHz}}>=0.06$). The root-mean-square scatter is $0.35$, and $0.32$ for the FIR and radio distributions, respectively. Assuming the validity of the SFR to radio/FIR luminosity calibrations, such a (normal) distribution is expected if FIR and radio emissions arise from star formation processes in the galaxies. From it is apparent that, as the AGN contribution (defined via optical spectroscopic emission line properties) rises in galaxies, the distribution of both the FIR and radio luminosity excess becomes highly skewed towards higher $\Delta
\log{L}$ values. Although for a fraction of optically selected AGN it is possible that the AGN contribution to the radio/FIR may be weak ($\Delta \log{L}\sim0$), and they may be overwhelmed by star formation (which results in the canonical FIR/radio ratio), the significant skewness of the $\Delta \log{L}$ distributions suggests an additional source of radio and FIR emission in AGN bearing galaxies (at least for $\Delta \log{L}>0$). Even more interesting is that galaxies with large luminosity excess in both FIR and radio emission ($\Delta \log{L}\gtrsim0.9$) predominantly have FIR/radio ratios consistent with the mean $q$ value for star forming galaxies (see Fig. 12). This will be discussed in more detail in .
Discussion {#sec:discussion}
==========
Comparison with previous results
--------------------------------
Extensive studies of the radio-FIR correlation [e.g. @helou85; @condon92; @yun01; @condon02; @obric06; @mauch07] have led to an average FIR/radio ratio in the local ($z<0.3$) universe of $q\sim2.3$, and lower for AGN-bearing galaxies (see Tab. 2 in @sargent10 for a summary). For example, using the IRAS 2 Jy galaxy sample ($F_\mathrm{60\micron}\geq2$ Jy; 1809 sources with optical counterparts and well determined redshifts) combined with NVSS data, @yun01 have found $<q>=2.34\pm0.01$. A lower average $q$ value is generally inferred when using radio selected samples, and reaching fainter in the IR (see @sargent10 for a detailed discussion of selection effects). Combining NVSS data with the optical Uppsala Galaxy Catalog (UGC) and the IRAS FSC and PSC, @condon02 have found $<q>=2.3$ and rms width $\sigma=0.18$. Furthermore, matching NVSS and 6dFGS survey data only with the IRAS FSC, @mauch07 inferred a mean $q$ value of 2.28 with a root-mean-square scatter of 0.22 for their entire sample. For a subset of radio-loud AGN (that would correspond to our Seyfert, LINER, absorption, and quasar classes combined) they found an even lower average value, $<q>=2.0$, and a sigificantly higher scatter in the FIR/radio ratio ($\sigma=0.5$).
In we have presented the distribution of $q$ for various types of our spectroscopically selected NVSS-SDSS-IRAS (PSC+FSC) galaxies. Our results yield that the dispersion is the tightest for star forming galaxies ($\sigma=0.13$), and rises by a factor of 1.5, 2.5, and 2.2 for composite, Seyfert and absorption/LINER galaxies, respectively. We find that the average FIR/radio ratio for all objects in our radio-optical-IR sample is $2.27\pm0.01$ with a dispersion of $0.2$. This is in very good agreement with the results from @mauch07. Furthermore, if we limit the 60 fluxes of our full sample to $\geq2$ Jy we obtain an average value of 2.34, consistent with that inferred by @yun01.
Our results yield a lower FIR/radio ratio ($<q>=2.14\pm0.05$) for Seyfert galaxies, and a significantly higher root-mean-square-scatter ($\sigma=0.3$), compared to that found for SF galaxies. It is interesting that the mean $q$ value for our IR-detected LINER and absorption line galaxies is comparable to that for star forming galaxies. However, the spread in $q$ for the former is significantly larger than for the latter (0.28 compared to 0.13, respectively). The average FIR/radio ratio for the 21 quasars in our sample is $q=2.04$, comparable to that inferred for Seyferts and lower than that for star forming galaxies. If we combine our AGN-bearing galaxies (quasars, Seyferts, LINERs, absorption galaxies) into one class in order to match the AGN sample of @mauch07, we infer an average $q$ of $2.16\pm 0.03$ (with a root-mean-square scatter of $\sigma=0.24$). This is in relatively good agreement with their results. In the next sections we will discuss the variation of $q$ with radio luminosity and the AGN contribution to the radio-FIR correlation.
AGN contribution to the radio-FIR correlation
-----------------------------------------------
A low $q$ value is often used to discriminate between star forming galaxies and AGN. For example @condon02 have classified radio-loud AGN as those having q $\leq$ 1.8. Assuming that the FIR emission arises solely from star formation, this criterion selects galaxies with more than 3 times the radio emission from galaxies on the FIR-radio correlation. @yun01 have used $q = 1.64$ as a star formation/AGN separator, identifying galaxies that emit in radio more than 5 times than predicted by the correlation. It is important to point out that these discriminating values are tuned to select only the most radio-loud AGN. Having i) separated our NVSS-SDSS-IRAS sample into various classes of AGN, and ii) independently estimated SFRs in their host galaxies, we can now analyze the physical source of FIR and radio emission in galaxies both following and offset from the radio-FIR correlation. Assuming that the additional source of FIR and radio emission (relative to that expected from star formation) observed in our composite, Seyfert, absorption and LINER galaxies (see ) arises from the central supermassive black hole, the distribution of our luminosity excess, $\Delta \log L$ defined in eq. \[for:delta\], allows us to constrain the [*average*]{} contribution of star formation and AGN activity to the total power output for a given galaxy population. Taking that star formation and AGN activity are the two dominant FIR/radio emission generators, i.e.$L_\mathrm{tot} = L_\mathrm{SF} + L_\mathrm{AGN}$, the average fractional contributions of these two sources ($<f_\mathrm{SF}>$, and $<f_\mathrm{AGN}>$) to the total power output can then be computed as $<f_\mathrm{SF}>=10^{-<\Delta \log L>}$, and $<f_\mathrm{AGN}>=1-<f_\mathrm{SF}>$, where $<\Delta \log L>$ denotes the average (median) of the $\Delta \log L$ distribution (see ).
The median $\Delta \log L$ values, and the fractional star formation/AGN contributions are summarized in . As expected, for star forming galaxies we infer that the average contribution to FIR and radio emission due to star formation is $100\%$. We find that composite objects are dominated by star formation at the $\sim80-90\%$ level. Further, the FIR emission from Seyfert galaxies arises predominantly from star formation ($\sim75\%$), while the AGN contribution to radio luminosity in Seyfert galaxies is about a factor of two higher in the radio than in the FIR (see ). The latter explains the lower average $q$ value (compared to the nominal value) for Seyfert galaxies inferred here, as well as in e.g. @obric06, @mauch07. Lastly, based on the above calculation IR detected absorption and LINER galaxies are on average strongly dominated by AGN activity ($\sim90\%$) in both their FIR and radio emission although their average FIR/radio ratio is consistent with that expected for star forming galaxies (see ).
One of the main results of this work is that, for the large majority of galaxies with radio and/or IR emission excess, we infer $q$ values consistent with the average FIR/radio ratio found for star forming galaxies (see ). Thus, although a significant AGN contribution is likely present in these galaxies (adding both to the FIR and radio emission), they would not be identified with a simple low-q discriminator value, as is commonly used to select radio-loud AGN. Our results indicate that the FIR/radio ratio is not particularly sensitive to AGN contribution and that the radio-FIR correlation is a poor discriminant of AGN activity, except for the most powerful AGN. This is consistent with observations of several AGN-bearing galaxies in the high-redshift universe.
Based on an SED analysis, Riechers et al. (2009) find that both the radio and FIR luminosity in the z=3.9 quasar APM08279+5255 are dominated by the central AGN, but that it has a $q$ value consistent with the local radio-FIR correlation. Furthermore, Murphy et al. (2009) have analyzed Spitzer-IRS spectra of a sample of 22 $0.6\lesssim z\lesssim
2.6 $ galaxies, composed of submillimeter galaxies, as well as X-ray and optically selected AGN in GOODS-N. Making use of their IRS spectra, they have performed a thorough starburst-AGN decomposition for each object which allowed them to estimate the fractional AGN contribution to the total IR luminosity output of each source. They demonstrate that the 4 galaxies having the largest mid-IR AGN fractions ($>60\%$) in their sample have $q$ values consistent with the canonical value. Furthermore, they find that the FIR/radio ratio shows no trend with the fractional contribution of AGN activity in the galaxies in the IR, consistent with our results.
Summary and conclusions {#sec:summary}
=======================
Based on a new radio-optical-IR catalog we have separated our radio (NVSS) and IR (IRAS) detected SDSS galaxies ($0.04<z\lesssim0.2$) into star forming, composite, Seyfert, LINER, absorption line galaxies and quasars, and we have performed an in-depth analysis of the radio-FIR correlation for various types of star forming and AGN-bearing galaxies. Utilizing our NUV-NIR SED based star formation rates in combination with FIR and radio luminosity (expected to directly trace star formation), we have statistically quantified the source of radio and FIR emission in the galaxies in our sample. We find that Seyfert galaxies and quasars have FIR/radio ratios lower than the canonical value for star forming galaxies. This is due to an additional contribution to their radio continuum emission, which likely arises from their AGN. We further show that FIR-detected absorption and LINER galaxies are on average strongly dominated by AGN activity in both their FIR and radio emission; however their average FIR/radio ratio is consistent with that expected for star forming galaxies. In summary, our results imply that most AGN-containing galaxies in our sample have FIR/radio flux ratios indistinguishable from those of the star-forming galaxies. Thus, attempts to separate AGNs from star-forming galaxies by their FIR/radio flux ratios alone is a poor discriminant of AGN activity, except for the most powerful radio-loud AGN.
We are grateful to the anonymous referee for helpful comments. The authors thank R. Beck, S. Charlot, O. Ilbert, K. K. Knudsen, M. Sargent, and J. Walcher for insightful discussions. IM thanks California Institute of Technology for generous hospitality. IM and DR acknowledge support from NASA through an award issued by JPL/Caltech. VS acknowledges support from the Owens Valley Radio Observatory, which is supported by the National Science Foundation through grant AST-0838260. VS & IM thank Unity through Knowledge Fund (www.ukf.hr) for collaboration support through the ’Homeland Visit’ grant. DR acknowledges support from from NASA through Hubble Fellowship grant HST-HF-51235.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. AK and ZI ackknowledge NSF grant AST-0507259 to the University of Washington. The research leading to these results has received funding from the European Union’s Seventh Framework programme under grant agreement 229517.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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[c c c c]{} & IRAS (FSC + PSC) & SDSS (MAIN + QUASAR) & IRAS - SDSS\
\
total radio sample & 18313 & 9591 & 524\
\
Quasars & – & 4490 & 21\
\
Absorption & – & 3072 & 16\
Composite & – & 654 & 203\
SF unambiguous & – & 621 & 216\
SF ambiguous & – & 9 & 0\
\
AGN unambiguous & – & 454 & 43\
AGN ambiguous & – & 291& 25\
Seyfert unambiguous & – & 200 & 37\
LINER unambiguous & – & 254 & 6\
\[table:class\]
------------ ------- ------- ------- -------
SF AGN SF AGN
SF 100% 0% 100% 0%
Composite 81.3% 18.7% 90.7% 9.3%
Seyfert 56.8% 43.2% 76.1% 23.9%
Abso+LINER 11.3% 88.7% 12.8% 87.2%
\[1ex\]
------------ ------- ------- ------- -------
: Fractions of star formation and AGN activity in radio and FIR regimes for various types of galaxies
\[table:fracs\]
![Distribution of distances between the radio and FIR detections for the NVSS-IRAS (full black line) and NVSS-SDSS-IRAS (dashed red line) samples. The cumulative distribution is shown in the inset.[]{data-label="fig:Distance_IRAS_NVSS"}](Fig1.eps){width="\columnwidth"}
![The distribution of flux density at 20 cm (top panel) and 60 $\mu$m (bottom panel) for various radio-selected samples indicated in the top right of the panels.[]{data-label="fig:LogFnvss"}](Fig2.eps){width="\columnwidth"}
[^1]: We take the IRAS quality indicator, reported in the FSC and PSC, to be $\geq2$ at 60 and 100 $\mu$m (the wavelength bands utilized here).
|
---
abstract: 'We introduce a new computational problem, the [[BackboneDiscovery]{}]{} problem, which encapsulates both [*functional*]{} and [*structural*]{} aspects of network analysis. While the topology of a typical road network has been available for a long time (e.g., through maps), it is only recently that fine-granularity functional (activity and usage) information about the network (like source-destination traffic information) is being collected and is readily available. The combination of functional and structural information provides an efficient way to explore and understand usage patterns of networks and aid in design and decision making. We propose efficient algorithms for the [[BackboneDiscovery]{}]{} problem including a novel use of edge centrality. We observe that for many real world networks, our algorithm produces a backbone with a small subset of the edges that support a large percentage of the network activity.'
author:
- |
Sanjay Chawla\
University of Sydney\
Sydney, Australia\
sanjay.chawla@sydney.edu.au\
- |
Venkata Rama Kiran Garimella\
Aalto University\
Helsinki, Finland\
kiran.garimella@aalto.fi
- |
Aristides Gionis\
Aalto University and HIIT\
Helsinki, Finland\
aristides.gionis@aalto.fi\
- |
Dominic Tsang\
University of Sydney\
Sydney, Australia\
dwktsang@yahoo.com
- 'Sanjay Chawla[^1]'
- 'Kiran Garimella[^2]'
- 'Aristides Gionis[^3]'
- 'Dominic Tsang[^4]'
bibliography:
- 'bibliography.bib'
title: Discovering the Network Backbone from Traffic Activity Data
---
Introduction
============
In this paper we study a novel problem, which combines [*structural*]{} and [*functional (activity)*]{} network data. In recent years there has been a large body of research related to exploiting structural information of networks. However, with the increasing availability of fine-grained functional information, it is now possible to obtain a detailed understanding of activities on a network. Such activities include source-destination traffic information in road and communication networks.
More specifically we study the problem of discovering the [*backbone*]{} of traffic networks. In our setting, we consider the topology of a network ${{\ensuremath{G}}}=({{\ensuremath{V}}},{{\ensuremath{E}}})$ and a traffic log ${{\ensuremath{\cal L}}}=\{({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$, recording the amount of traffic ${{\ensuremath{w}}}_i$ that incurs between source ${{\ensuremath{s}}}_i$ and destination ${{\ensuremath{t}}}_i$. We are also given a budget [[$B$]{}]{} that accounts for a total edge cost. The goal is to discover a sparse subnetwork [[$R$]{}]{} of [[$G$]{}]{}, of cost at most [[$B$]{}]{}, which summarizes as well as possible the recorded traffic [[$\cal L$]{}]{}.
{width="40.00000%"}{width="40.00000%"}
The problem we study has applications for both [*exploratory data analysis*]{} and [*network design*]{}. An example application of our algorithm is shown in Figure \[fig:londontube\]. Here, we consider a traffic log (Figure \[fig:londontube\], left), which consists of the most popular routes used on the London tube. The backbone produced by our algorithm takes into account this demand (based on the traffic log) and tries to summarize the underlying network, thus presenting us with insights about usage pattern of the London tube (Figure \[fig:londontube\], right). This representation of the ‘backbone’ of the network could be very useful to identify the important edges to upgrade or to keep better maintained in order to minimize the total traffic disruptions.
We only consider source-destination pairs in the traffic log, and not full trajectories, as source-destination information captures [*true mobility demand*]{} in a network. For example, data about the daily commute from home (source) to office (destination) is more resilient than trajectory information, which is often determined by local and transient constraints, like traffic conditions on the road, time of day, etc. Furthermore, in communication networks, only the source-ip and destination-ip information is encoded in TCP-IP packets. Similarly, in a city metro, check-in and check-out information is captured while the intervening movement is not logged.
The [[BackboneDiscovery]{}]{} problem is an amalgam of the $k$-[*spanner*]{} problem [@narasimhan2007geometric] and the [*Steiner-forest*]{} problem [@williamson2011design]. However, our problem formulation will have elements which are substantially distinct from both of these problems.
In the $k$-spanner problem the goal is to find a minimum-cost subnetwork [[$R$]{}]{} of [[$G$]{}]{}, such that for [*each pair*]{} of nodes $u$ and $v$, the shortest path between $u$ and $v$ on [[$R$]{}]{} is at most $k$ times longer than the shortest path between $u$ and $v$ on [[$G$]{}]{}. In our problem, we are not necessarily interested in preserving the $k$-factor distance between all nodes but for only a subset of them.
In the Steiner-forest problem we are given a set of pairs of terminals $\{({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)\}$ and the goal is to find a minimum-cost forest on which each source ${{\ensuremath{s}}}_i$ is connected to the corresponding destination ${{\ensuremath{t}}}_i$. Our problem is different from the Steiner-forest problem because we do not need all $\{({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)\}$ to be connected, and try to optimize a stretch factor so that the structural aspect of the network are also taken into account.
A novel aspect of our work is the use of edge-betweenness to guide the selection of the backbone [@Newman]. The intuition is as follows. An algorithm to solve the Steiner-forest problem will try and minimize the sum of cost of edges selected as long as as the set of terminal pairs $\{({{\ensuremath{s}}}_{i},{{\ensuremath{t}}}_{i})\}$ are connected and is agnostic to minimizing stretch factor. However, if the edge costs are inversely weighted with [*edge-betweenness*]{}, then edges that can contribute to reducing the stretch factor can be potentially included into the backbone.
To understand the differences of the proposed [[BackboneDiscovery]{}]{} problem with both the $k$-[*spanner*]{} and [*Steiner-forest*]{} formulations, consider the example shown in Figure \[counter\_example\]. In this example, there are four groups of nodes:
1. group $A$ consists of $n$ nodes, $a_1,\ldots,a_n$,
2. group $B$ consists of $n$ nodes, $b_1,\ldots,b_n$,
3. group $C$ consists of $2$ nodes, $c_1$ and $c_2$, and
4. group $D$ consists of $m$ nodes, $d_1,\ldots,d_m$.
Assume that $m$ is smaller then $n$, and thus much smaller than $n^2$. All edges shown in the figure have cost $1$, except the edges between $c_1$ and $c_2$, which has cost $2$. Further assume that there is one unit of traffic between each $a_i$ and each $b_j$, for $i,j=1,\ldots,n$, resulting in $n^2$ source-destination pairs (the majority of the traffic), and one unit of traffic between $d_i$ and $d_{i+1}$, for $i=1,\ldots,m-1$, resulting in $m-1$ source-destination pairs (some additional marginal traffic). The example abstracts a common layout found in many cities: a few busy centers (commercial, residential, entertainment, etc.) with some heavily-used links connecting them (group $C$), and some peripheral ways around, that serve additional traffic (group $D$).
Careful inspection of the above example highlights advantages of the backbone discovery problem:
- As opposed to the $k$-spanner problem, we do not need to guarantee short paths for all pairs of nodes, but only for those in our traffic log which makes our approach more general. In particular, based on the budget requirements a backbone could be designed for the most voluminous paths.
- Due to the budget constraint, it may not be possible to guarantee connectivity for all pairs in the traffic log. We thus need a way to decide which pairs to leave disconnected. Neither the $k$-spanner nor the Steiner-forest problems provision for disconnected pairs. In fact, it is possible that the optimal backbone may even contain cycles while leaving pairs disconnected. Again, allowing for a disconnected backbone, generalizes the Steiner-forest problem and may help provision for a tighter budget. In order to allow for a disconnected backbone, we employ the use of [*stretch factor*]{}, defined as a [*weighted harmonic mean*]{} over the source-destination pairs of the traffic log, which provides a principled objective to optimize connectivity while allowing to leave disconnected pairs, when there is insufficient budget.
- Certain high cost edges may be an essential part of the backbone that other problem formulations may leave out. For example, while the edge that connects the nodes in $C$ is a very important edge for the overall traffic (as it provides a short route between $A$ and $B$), the optimal Steiner-forest solution, shown in Figure \[counter\_example\_b\], prefers the long path along the nodes in $D$. Our algorithm includes the component $C$ (as seen in Figure \[counter\_example\_c\]) because it is an edge that has a high edge-betweenness.
The rest of the paper is organized as follows. In Section \[sec:problem\], we rigorously define the [[BackboneDiscovery]{}]{} problem. Section \[sec:algorithms\] introduces our algorithm based on the greedy approach, Section \[sec:experiments\] details our experimental evaluation, results and discussion. In Section \[sec:related\] we survey related work and distinguish our problem formulation with other relevant approaches. We conclude in Section \[sec:conclusions\] with a summary and potential directions for future work.
Problem definition {#sec:problem}
==================
Let ${{\ensuremath{G}}}=({{\ensuremath{V}}},{{\ensuremath{E}}})$ be a network, with $|{{\ensuremath{V}}}|={{\ensuremath{n}}}$ and $|{{\ensuremath{E}}}|={{\ensuremath{m}}}$. For each edge ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$ there is a cost ${{\ensuremath{c}}}({{\ensuremath{e}}})$. Additionally, we consider a traffic log [[[$\cal L$]{}]{}]{}, specified as a set of triples $({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)$, with ${{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i\in{{\ensuremath{V}}}$, and with $i=1,\ldots,{{\ensuremath{k}}}$. A triple $({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)$ indicates the fact that ${{\ensuremath{w}}}_i$ units of traffic have been recorded between nodes ${{\ensuremath{s}}}_i$ and ${{\ensuremath{t}}}_i$.
We aim at discovering the [*backbone*]{} of traffic networks. A backbone ${{\ensuremath{R}}}$ is a subset of the edges of the network ${{\ensuremath{G}}}$, that is, ${{\ensuremath{R}}}\subseteq{{\ensuremath{E}}}$ that provides a good summarization for the whole traffic in [[$\cal L$]{}]{}. In particular, we require that if the available traffic had used only edges in the backbone [[$R$]{}]{}, it should have been almost as efficient as using all the edges in the network. We formalize this intuition below.
Given two nodes ${{\ensuremath{s}}},{{\ensuremath{t}}}\in{{\ensuremath{V}}}$ and a subset of edges ${{\ensuremath{A}}}\subseteq{{\ensuremath{E}}}$, we consider the shortest path ${{\ensuremath{d}}}_{{\ensuremath{A}}}({{\ensuremath{s}}},{{\ensuremath{t}}})$ from [[$s$]{}]{} to [[$t$]{}]{}that uses only edges in the set [[$A$]{}]{}. In this shortest-path definition, edges are counted according to their cost [[$c$]{}]{}. If there is no path from ${{\ensuremath{s}}}$ to ${{\ensuremath{t}}}$ using only edges in [[$A$]{}]{}, we define ${{\ensuremath{d}}}_{{\ensuremath{A}}}({{\ensuremath{s}}},{{\ensuremath{t}}})=\infty$. Consequently, ${{\ensuremath{d}}}_{{\ensuremath{E}}}({{\ensuremath{s}}},{{\ensuremath{t}}})$ is the shortest path from ${{\ensuremath{s}}}$ to ${{\ensuremath{t}}}$ using all the edges in the network, and ${{\ensuremath{d}}}_{{\ensuremath{R}}}({{\ensuremath{s}}},{{\ensuremath{t}}})$ is the shortest path from ${{\ensuremath{s}}}$ to ${{\ensuremath{t}}}$ using only edges in the backbone [[$R$]{}]{}.
To measure the quality of a backbone [[$R$]{}]{}, with respect to some traffic log ${{\ensuremath{\cal L}}}=\{({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$ we use the concept of [*stretch factor*]{}. Intuitively, we want to consider shortest paths from ${{\ensuremath{s}}}_i$ to ${{\ensuremath{t}}}_i$, and evaluate how much longer are those paths on the backbone [[$R$]{}]{}, than on the original network. The idea of using stretch factor for evaluating the quality of a subgraph has been used extensively in the past in the context of spanner graphs [@narasimhan2007geometric].
In order to aggregate shortest-path information for all source–destination pairs in our log in a meaningful way, we need to address two issues. The first issue is that not all source–destination pairs have the same volume in the traffic log. This can be easily addressed by weighting the contribution of each pair $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ by its corresponding volume ${{\ensuremath{w}}}_i$.
The second issue is that since we aim at discovering very sparse backbones, many source–destination pairs in the log could be disconnected in the backbone. To address this problem we aggregate shortest-path distances using the [*harmonic mean*]{}. This idea, which has been proposed by Marchiori and Latora [@marchiori2000harmony] and has also been used by Boldi and Vigna [@boldi2013axioms] in measuring centrality in networks, provides a very clean way to deal with infinite distances: if a source–destination pair is not connected, their distance is infinity, so the harmonic mean accounts for this by just adding a zero term in the summation. Using the arithmetic mean is problematic, as we would need to add an infinite term with other finite numbers.
For a given ${{\ensuremath{A}}}\subset {{\ensuremath{E}}}$, and for each source destination pair $(s_{i},t_{i})_{i=1}^{k}$ we define, $$\lambda_{i} = \frac{d_{{\ensuremath{A}}}(s_{i},t_{i})}{d_{{\ensuremath{E}}}(s_{i},t_{i})}$$
Then we define two aggregate stretch factors.
1. The arithmetic mean (AM) stretch factor for a backbone $R$ for log ${{\ensuremath{\cal L}}}$ is given by $$\lambda_{{{\ensuremath{\cal L}}},AM}(R) = \frac{1}{k}\sum_{i=1}^{k}w_{i}\lambda_{i}$$
2. The harmonic mean (HM) stretch factor for a backbone $R$ is given by $$\lambda_{{{\ensuremath{\cal L}}},HM}(R) = \frac{k}{\sum_{i=1}^{k}\frac{w_{i}}{\lambda_{i}}}$$
The advantage of $\lambda_{{{\ensuremath{\cal L}}},HM}(R)$ is that can it can gracefully handle the case when $d_{R}(s_{i},t_{i}) =\infty$ as for that case $\frac{1}{\lambda_{i}} = 0$.\
Let $R^{\ast}_{am}$ be the optimal backbone for $\lambda_{{{\ensuremath{\cal L}}},AM}(R)$ and $R^{\ast}_{hm}$ be the optimal backbone for $\lambda_{{{\ensuremath{\cal L}}},HM}(R)$. Then $$\lambda_{{{\ensuremath{\cal L}}},HM}(R^{\ast}_{hm}) \leq \lambda_{{{\ensuremath{\cal L}}},AM}(R^{\ast}_{am})$$
By AM-HM inequality $$\lambda_{{{\ensuremath{\cal L}}},HM}(R^{\ast}_{am}) \leq \lambda_{{{\ensuremath{\cal L}}},AM}(R^{\ast}_{am})$$ The result then follows from the fact that $R^{\ast}_{hm}$ is optimal (wrt $\lambda_{{{\ensuremath{\cal L}}},HM}$) and therefore $$\lambda_{{{\ensuremath{\cal L}}},HM}(R^{\ast}_{hm}) \leq \lambda_{{{\ensuremath{\cal L}}},HM}(R^{\ast}_{am})$$
[**Fact 1:**]{} $\lambda_{{{\ensuremath{\cal L}}},HM}(R) \leq \lambda_{{{\ensuremath{\cal L}}},AM}(R) $ ( in the fully connected case..) ${{\ensuremath{\cal L}}}= \{({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$ by $$G_{{{\ensuremath{\cal L}}}}({{\ensuremath{A}}},F) =
F({{\ensuremath{d}}}_{{\ensuremath{A}}}({{\ensuremath{s}}}_1,{{\ensuremath{t}}}_1),\ldots,{{\ensuremath{d}}}_{{\ensuremath{A}}}({{\ensuremath{s}}}_k,{{\ensuremath{t}}}_k))$$ Here, $F$ is an abstract real-valued aggregate function whose exact form will be determined by the problem objective.
Overall, given a set of edges ${{\ensuremath{A}}}\subseteq{{\ensuremath{E}}}$, we measure the connectivity of the traffic log ${{\ensuremath{\cal L}}}= \{({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$, $|{{\ensuremath{\cal L}}}| = k$ by $${{\ensuremath{H}}}_{{\ensuremath{\cal L}}}({{\ensuremath{A}}}) =
\left( \sum_{i=1}^k {{\ensuremath{w}}}_i \right)
\left(
\sum_{i=1}^k \frac{{{\ensuremath{w}}}_i}{{{\ensuremath{d}}}_{{\ensuremath{A}}}({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)}
\right)^{-1}.$$ The [*stretch factor*]{} of a backbone [[$R$]{}]{} is then defined as $${{\ensuremath{\lambda}}}_{{\ensuremath{\cal L}}}({{\ensuremath{R}}})=
\frac{{{\ensuremath{H}}}_{{\ensuremath{\cal L}}}({{\ensuremath{R}}})}{{{\ensuremath{H}}}_{{\ensuremath{\cal L}}}({{\ensuremath{E}}})}.$$ The stretch factor is always greater or equal than 1. The closer it is to 1, the better the connectivity that it offers to the traffic log [[$\cal L$]{}]{}. [[**[Example.]{}**]{}]{} Consider the network $G$ in the figure below. Assume the log ${{\ensuremath{\cal L}}}$ is $\{(a,e,10),(c,d,12)\}$. Furthermore assume we are given a budget ${{\ensuremath{B}}}= 18$ to build the backbone. The shortest path from $a$ to $e$ is $a \rightarrow b \rightarrow d \rightarrow e$. Thus ${{\ensuremath{d}}}_{{\ensuremath{E}}}(a,e) = 14$. Similarly the shortest path from $c$ to $d$ is just $c \rightarrow d$ and thus ${{\ensuremath{d}}}_{{\ensuremath{E}}}(c,e) = 6$.
{width="0.5\columnwidth"}
If we take the two shortest paths as the backbone then the cost is $20$ which is above the budget. However, if the backbone ${{\ensuremath{R}}}$ consists of the edges $\{(a,c),(c,d),(d,e)\}$, then both $(a,e)$ and $(c,d)$ remain connected on the backbone and the resulting cost is $16$, which is under budget. Now ${{\ensuremath{d}}}_{{\ensuremath{R}}}(a,e) = 16$ and ${{\ensuremath{d}}}_{{\ensuremath{R}}}(c,d) = 6$. The resulting stretch factor is $$\lambda_{{{\ensuremath{\cal L}}}}({{\ensuremath{R}}}) = \frac{ \frac{10}{14}+\frac{12}{6}}{\frac{10}{16}+
\frac{12}{6}} = 1.034$$ Now suppose the budget ${{\ensuremath{B}}}$ is further reduced to $14$. One possible backbone ${{\ensuremath{R}}}$ is $\{(a,b),(b,d),(d,e)\}$. Notice that ${{\ensuremath{d}}}_{{\ensuremath{R}}}(c,d) = \infty$. The stretch factor is $\lambda_{{{\ensuremath{\cal L}}}}({{\ensuremath{R}}}) = 3.8$. The other option is to choose [[$R$]{}]{} as $\{(c,d)\}$. Now ${{\ensuremath{d}}}_{{\ensuremath{R}}}(a,e) = \infty$. Now even though the full budget is not consumed the stretch factor is reduced to $\lambda_{{{\ensuremath{\cal L}}}}({{\ensuremath{R}}}) = 1.357$. The example suggests that a potentially complex and non-linear relationship exists between the network topology, the travel log and a budget to build the backbone. $\Box$
We are now ready to formally define the problem of backbone discovery.
\[problem:optimal-backbone-design\] Consider a network ${{\ensuremath{G}}}=({{\ensuremath{V}}},{{\ensuremath{E}}})$ and a traffic log ${{\ensuremath{\cal L}}}=\{ ({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$. Consider also a cost budget [[$B$]{}]{}. The goal is to find a backbone network ${{\ensuremath{R}}}\subseteq{{\ensuremath{E}}}$ of total cost [[[$B$]{}]{}]{} that minimizes the stretch factor ${{\ensuremath{\lambda}}}_{{\ensuremath{\cal L}}}({{\ensuremath{R}}})$ or report that no such solution exists.
As one may suspect, [[BackboneDiscovery]{}]{} is an [[[[$\mathbf{NP}$]{}]{}-hard]{}]{} problem.
\[lemma:np-hardness\] The [[BackboneDiscovery]{}]{} problem, defined in Problem \[problem:optimal-backbone-design\], is [[[[$\mathbf{NP}$]{}]{}-hard]{}]{}.
We obtain a reduction from the [[SetCover]{}]{} problem: given a ground set ${{\ensuremath{U}}}=\{{{\ensuremath{u}}}_1,\ldots,{{\ensuremath{u}}}_{{\ensuremath{n}}}\}$, a collection ${{\ensuremath{\cal S}}}= \{{{\ensuremath{S}}}_1,\ldots,{{\ensuremath{S}}}_{{\ensuremath{m}}}\}$ of subsets of [[$U$]{}]{}, and an integer [[$k$]{}]{}, determine whether there are [[$k$]{}]{} sets in [[$\cal S$]{}]{}that cover all the elements of [[$U$]{}]{}.
Given an instance of the [[SetCover]{}]{} problem we form an instance of the [[BackboneDiscovery]{}]{} problem as follows (See Figure \[fig:np\_gadget\] for illustration). We create one node ${{\ensuremath{u}}}_i$ for each ${{\ensuremath{u}}}_i\in{{\ensuremath{U}}}$ and one node ${{\ensuremath{v}}}_j$ for each ${{\ensuremath{S}}}_j\in{{\ensuremath{\cal S}}}$. We also create a special node ${{\ensuremath{z}}}$. We create an edge $({{\ensuremath{u}}}_i,{{\ensuremath{v}}}_j)$ if and only if ${{\ensuremath{u}}}_i\in{{\ensuremath{S}}}_j$ and we assign to this edge cost 0. We also create an edge $({{\ensuremath{v}}}_j,{{\ensuremath{z}}})$ for all ${{\ensuremath{S}}}_j\in{{\ensuremath{\cal S}}}$ and we assign to this edge cost 1. As far as the traffic log is concerned, we set ${{\ensuremath{\cal L}}}=\{({{\ensuremath{u}}}_i,{{\ensuremath{z}}},1)\mid{{\ensuremath{u}}}_i\in{{\ensuremath{U}}}\}$, that is, we pair each ${{\ensuremath{u}}}_i\in{{\ensuremath{U}}}$ with the special node [[$z$]{}]{} with volume 1. Finally we set the budget ${{\ensuremath{B}}}={{\ensuremath{k}}}$. It is not difficult to see that the instance of the [[BackboneDiscovery]{}]{} problem constructed in this way has a solution with stretch factor 1 if and only if the given instance of the [[SetCover]{}]{} problem has a feasible solution.
![Reduction from Set Cover to [[BackboneDiscovery]{}]{} for the log ${{\ensuremath{\cal L}}}=\{(u_{i},z,1)|u_{i} \in U \}$[]{data-label="fig:np_gadget"}](np_gadget){width="50.00000%"}
Algorithm {#sec:algorithms}
=========
The algorithm we propose for the [[BackboneDiscovery]{}]{} problem is a [*greedy*]{} heuristic that connects one-by-one the source–destination pairs of the traffic log [[$\cal L$]{}]{}. A distinguishing feature of our algorithm is that it utilizes a notion of [*edge benefit*]{}. In particular, we assume that for each edge ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$ we have available a benefit measure ${{\ensuremath{b}}}({{\ensuremath{e}}})$. The higher is the measure ${{\ensuremath{b}}}({{\ensuremath{e}}})$ the more beneficial it is to include the edge [[$e$]{}]{} in the final solution. The benefit measure is computed using the traffic log [[$\cal L$]{}]{} and it takes into account the global structure of the network [[$G$]{}]{}.
The more central an edge is with respect to a traffic log, the more beneficial it is to include it in the solution, as it can be used to serve many source–destination pairs. In this paper we use [*edge-betweenness*]{} as a centrality measure, adapted to take into account the traffic log. We also experimented with [*commute-time centrality*]{}, but edge-betweenness was found to be more effective.
Our algorithm relies on the notion of [*effective distance*]{} ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})$, defined as ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})={{\ensuremath{c}}}({{\ensuremath{e}}})/{{\ensuremath{b}}}({{\ensuremath{e}}})$, where ${{\ensuremath{c}}}({{\ensuremath{e}}})$ is the cost of an edge ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$, and ${{\ensuremath{b}}}({{\ensuremath{e}}})$ is the edge-betweenness of [[$e$]{}]{}. The intuition is that by dividing the cost of each edge by its benefit, we are biasing the algorithm towards selecting edges with high benefit.
We now present our algorithm in more detail.
The greedy algorithm. {#section:greedy}
---------------------
As discussed above, our algorithm operates with effective distances ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})={{\ensuremath{c}}}({{\ensuremath{e}}})/{{\ensuremath{b}}}({{\ensuremath{e}}})$, where ${{\ensuremath{b}}}({{\ensuremath{e}}})$ is a benefit score for each edge [[$e$]{}]{}. The objective is to obtain a cost/benefit trade-off: edges with small cost and large benefit are favored to be included in the backbone. In the description of the greedy algorithm that follows, we assume that the effective distance ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})$ of each edge is given as input.
The algorithm works in an iterative fashion, maintaining and growing the backbone, starting from the empty set. In the $i$-th iteration the algorithm picks a source–destination pair $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ from the traffic log [[$\cal L$]{}]{}, and “serves” it. Serving a pair $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ means computing a shortest path ${{\ensuremath{p}}}_i$ from ${{\ensuremath{s}}}_i$ to ${{\ensuremath{t}}}_i$, and adding its edges in the current [[$R$]{}]{}, if they are not already there. For the shortest-path computation the algorithm uses the effective distances ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})$. When an edge is newly added to the backbone its cost is subtracted from the available budget. Here, the actual cost of the edge ${{\ensuremath{c}}}({{\ensuremath{e}}})$ (instead of the ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})$) is used. Also its effective distance is reset to zero, since it can be used for free in subsequent iterations of the algorithm. The source–destination pair that is chosen to be served in each iteration is the one that reduces the stretch factor the most at that iteration; and hence the greedy nature of the algorithm.
The algorithm proceeds until it exhausts all its budget or until the stretch factor becomes equal to 1 (which means that all pairs in the log are served via a shortest path). The pseudo-code for the greedy algorithm is shown as Algorithm \[algorithm:greedy\].
We are experimenting with two variants of this greedy scheme, depending on the benefit score we use. These are the following:
[[Greedy]{}]{}:
: We use uniform benefit scores (${{\ensuremath{b}}}({{\ensuremath{e}}})=1$).
[[GreedyEB]{}]{}:
: The benefit score of an edge is set equal to its [*edge-betweenness centrality*]{}.
Speeding up the greedy algorithm. {#sec:optimizations}
---------------------------------
As we show in the experimental section the greedy algorithm provides solutions of good quality, in particularly the variant with the edge-betweenness weighting scheme. However, the algorithm is computationally expensive, and thus in this section we discuss a number of optimizations. We start by analyzing the running time of the algorithm.
[[**[Running time.]{}**]{}]{} Assume that the benefit scores ${{\ensuremath{b}}}({{\ensuremath{e}}})$ are given for all edges ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$, and that the algorithm performs [[$I$]{}]{} iterations until it exhausts its budget. In each iteration we need to perform ${{\ensuremath{\cal O}}}({{\ensuremath{k}}}^2)$ shortest-path computations, where [[$k$]{}]{} is the size of the traffic log [[$\cal L$]{}]{}. A shortest path computation is ${{\ensuremath{\cal O}}}({{\ensuremath{m}}}+{{\ensuremath{n}}}\log {{\ensuremath{n}}})$, and thus the overall complexity of the algorithm is ${{\ensuremath{\cal O}}}({{\ensuremath{I}}}{{\ensuremath{k}}}^2({{\ensuremath{m}}}+{{\ensuremath{n}}}\log {{\ensuremath{n}}}))$. The number of iterations [[$I$]{}]{} depends on the available budget and in the worst case it can be as large as [[$k$]{}]{}. However, since we aim at finding sparse backbones, the number of iterations is typically smaller.
[[**[Optimizations with no approximation.]{}**]{}]{} We first show how to speed up the algorithm, while guaranteeing the same solution with the naïve implementation of the greedy. Since the most expensive component is the computation of shortest paths on the newly-formed network, we make sure that we compute the shortest path only when needed. Our optimizations consist of two parts.
First, during the execution of the algorithm we maintain the connected components that the backbone creates in the network. Then, we do not need to compute shortest paths for all $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ pairs, for which ${{\ensuremath{s}}}_i$ and ${{\ensuremath{t}}}_i$ belong to different connected components; we know that their distance is $\infty$. This optimization is effective at the early stages of the algorithm, when many terminals belong to different connected components.
Second, when computing the decrease in the stretch factor due to a candidate shortest path to be added in the backbone, for pairs for which we have to recompute a shortest-path distance, we first compute an optimistic lower bound, based on the shortest path on the whole network (which we compute once in a preprocessing step). If this optimistic lower bound is not better than the current best stretch factor then we can skip the computation of the shortest path on the backbone.
As shown in the empirical evaluation of our algorithms, depending on the dataset, these optimization heuristics lead to 20–35% improvement in performance.
[[**[Optimization based on landmarks.]{}**]{}]{} To further improve the running-time of the algorithm we compute shortest-path distances using landmarks [@DBLP:conf/wsdm/SarmaGNP10; @DBLP:conf/cikm/PotamiasBCG09], an effective technique to approximate distances on graphs. Here we use the approach of Potamias et al. [@DBLP:conf/cikm/PotamiasBCG09]: A set of [[$\ell$]{}]{} landmarks ${{\ensuremath{L}}}= \{{{\ensuremath{z}}}_1,\ldots,{{\ensuremath{z}}}_{{\ensuremath{\ell}}}\}$ is selected and for each vertex ${{\ensuremath{v}}}\in{{\ensuremath{V}}}$ the distances ${{\ensuremath{d}}}({{\ensuremath{v}}},{{\ensuremath{z}}}_i)$ to all landmarks are computed and stored as an ${{\ensuremath{\ell}}}$-dimensional vector representing vertex [[$v$]{}]{}. The distance between two vertices ${{\ensuremath{v}}}_1,{{\ensuremath{v}}}_2$ is then approximated as $\min_{i} \{ {{\ensuremath{d}}}({{\ensuremath{v}}}_1,{{\ensuremath{z}}}_i) +
{{\ensuremath{d}}}({{\ensuremath{v}}}_2,{{\ensuremath{z}}}_i) \}$, i.e., the tightest of the upper bounds induced by the set of landmarks [[$L$]{}]{}.
To select landmarks we use high-degree non-adjacent vertices in the graph, which is reported as one of the best performing methods by Potamias et al. [@DBLP:conf/cikm/PotamiasBCG09]. Note that the distances are now approximations to the true distances, and the method trades accuracy by efficiency via the number of landmarks selected. In practice a few hundreds of landmarks are sufficient to provide high-quality approximations even for graphs with millions of vertices [@DBLP:conf/cikm/PotamiasBCG09].
For the running-time analysis, note that in each iteration we need to compute the distance between all graph vertices and all landmarks. This can be done with [[$\ell$]{}]{} single-source shortest-path computations and the running time is ${{\ensuremath{\cal O}}}({{\ensuremath{\ell}}}({{\ensuremath{m}}}+{{\ensuremath{n}}}\log{{\ensuremath{n}}}))$. The landmarks can then be used to approximate shortest-path distances between all source-destination pairs in the traffic log [[$\cal L$]{}]{}, with running time ${{\ensuremath{\cal O}}}({{\ensuremath{k}}}{{\ensuremath{\ell}}})$. Thus, the overall complexity is ${{\ensuremath{\cal O}}}({{\ensuremath{I}}}{{\ensuremath{\ell}}}({{\ensuremath{k}}}+{{\ensuremath{m}}}+{{\ensuremath{n}}}\log{{\ensuremath{n}}}))$. Since [[$\ell$]{}]{} is expected to be much smaller than [[$k$]{}]{}, the method provides a significant improvement over the naïve implementation of the greedy. As shown in the experimental evaluation, using landmarks provides an improvement of up to 4 times in terms of runtime in practice.
We now provide an approximation guarantee related to the Greedy algorithm.
The greedy algorithm is an $O(k)$-approximation algorithm where $k$ is the number of source destination pairs.
Let $d^G$ be the total cost of the [[Greedy]{}]{}.
Let $d^G_i$ be the [*additional cost*]{} paid by the [[Greedy]{}]{} on its $i$-th iteration for the pair $(s_i,t_i)$. In other words, $d^G_i$ is the cost of the new edges added at the $i$-th step. Note that the total path used by the [[Greedy]{}]{} to connect $s_i$ and $t_i$ may use more edges, not accounted in $d^G_i$ but accounted in previous iterations. Note that $d^G = \sum_{i=1}^k d^G_i$.
Let $d^*_i$ be the [*shortest-path distance*]{} from $s_i$ to $t_i$ in the network.
Let $d^O$ be the [*total cost*]{} of the optimal solution of the [[BackboneDiscovery]{}]{} problem.
We have:
1. $d^G_i \le d^*_i$ (The shortest path was available for the [[Greedy]{}]{}. Since the [[Greedy]{}]{} picked another path, this path has smaller cost)
2. $d^*_i \le d^O$ (Any path connecting $s_i$ and $t_i$ has to be larger than the shortest path. So does the path on the optimal solution.)
Combining the above facts we get $$d^G = \sum_{i=1}^k d^G_i \le k d^O.$$
Algorithm based on primal–dual. {#section:pd}
-------------------------------
In our introduction, with the help of Figure \[counter\_example\], we argued that the [[BackboneDiscovery]{}]{} problem that we address in this paper has important differences with the [[$k$-[Spanner]{}]{}]{} and [[SteinerForest]{}]{} problems. Yet, existing approximation algorithms for the [[SteinerForest]{}]{}problem, which are based on the primal–dual method, offer an intuitive paradigm to think about sparsification of networks and connectivity of terminals. Thus, our second family of algorithms is a variation of the primal–dual scheme, inspired by an approximation algorithm for the [[SteinerForest]{}]{} problem [@williamson2011design] and adapted to our setting.
Recall that in the [[SteinerForest]{}]{} problem the objective is to find a minimum-cost tree that connects all pairs of terminals $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$. The primal–dual scheme is described in, e.g., the book of Williamson and Shmoys [@williamson2011design], but for completeness and in order to describe our algorithm better, we give a brief overview.
[[**[Basic primal–dual scheme.]{}**]{}]{} Given a source–destination pair $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$, denote by ${\cal S}_i$ the subsets of nodes separating ${{\ensuremath{s}}}_i$ and ${{\ensuremath{t}}}_i$, that is, ${{\ensuremath{{\cal S}}}}_i=\{ {{\ensuremath{S}}}\subseteq {{\ensuremath{V}}}\mbox{ such that } |{{\ensuremath{S}}}\cap\{{{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i\}|=1\}$. Also denote by $\delta({{\ensuremath{S}}})$ set of edges the have one endpoint in $S$ and the other not in $S$. Let $c_{{\ensuremath{e}}}$ denote the cost of the edge [[$e$]{}]{}, and let $x_{{\ensuremath{e}}}$ be a binary variable indicating whether the edge [[$e$]{}]{} is included in a solution. The Integer Program associated to the [[SteinerForest]{}]{} problem is the following: $$\begin{aligned}
\mbox{minimize } & \sum_{{{\ensuremath{e}}}\in{{\ensuremath{E}}}} c_{{\ensuremath{e}}}x_{{\ensuremath{e}}}& \\
\mbox{such that } & \sum_{{{\ensuremath{e}}}\in \delta({{\ensuremath{S}}})} x_{{\ensuremath{e}}}\ge 1 & \mbox{for
all } {{\ensuremath{S}}}\in{{\ensuremath{{\cal S}}}}_i \mbox{ for some } i.
$$ By relaxing the integrality constraint to $x_{{\ensuremath{e}}}\ge 0$ we obtain the dual of the resulting linear program: $$\begin{aligned}
\mbox{maximize } & \sum_{\exists i:{{\ensuremath{S}}}\in{{\ensuremath{{\cal S}}}}_i} y_{{\ensuremath{S}}}& \\
\mbox{such that } & \sum_{{{\ensuremath{S}}}\mid {{\ensuremath{e}}}\in \delta({{\ensuremath{S}}})} y_{{\ensuremath{S}}}\le
c_{{\ensuremath{e}}}& \mbox{for all } {{\ensuremath{e}}}\in{{\ensuremath{E}}}\\
\mbox{with } & y_{{\ensuremath{S}}}\ge 0 & \mbox{for all } {{\ensuremath{S}}}\in{{\ensuremath{{\cal S}}}}_i \mbox{ for some } i. \end{aligned}$$ Let [[[${\cal C}$]{}]{}]{} be the set of all the connected components [[[$C$]{}]{}]{} such that $|{{\ensuremath{C}}}\cap \{{{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i \} | = 1$. A solution set of edges [[$F$]{}]{} is initialized to the empty set. The dual variables $y_{{\ensuremath{C}}}$, for all ${{\ensuremath{C}}}\in {{\ensuremath{{\cal C}}}}$, are increased at the same rate until a dual inequality becomes tight for some edge ${{\ensuremath{e}}}\in\delta({{\ensuremath{C}}})$ for a set [[[$C$]{}]{}]{}, whose dual is increased. The edge [[[$e$]{}]{}]{} is added to the solution [[[$F$]{}]{}]{} and the process continues. Once a feasible solution [[[$F$]{}]{}]{} is obtained, i.e., all pairs $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ are connected in [[[$F$]{}]{}]{}, we go through the edges in the reverse order in which they were added and if an edge can be removed without affecting the feasibility of the solution, it is deleted.
Initially, the set [[[${\cal C}$]{}]{}]{} of connected components consists only of the $2{{\ensuremath{k}}}$ singleton sets $\{{{\ensuremath{s}}}_i\}$ and $\{{{\ensuremath{t}}}_i\}$, that is, ${{\ensuremath{{\cal C}}}}=
\bigcup_{i=1}^{{{\ensuremath{k}}}}\{{{\ensuremath{s}}}_i\}\,\cup\,\bigcup_{i=1}^{{{\ensuremath{k}}}}\{{{\ensuremath{t}}}_i\}$. As the algorithm proceeds, those connected components grow in size by adding more nodes to them, and by merging existing connecting components.
[[**[Adaptation for [[BackboneDiscovery]{}]{}.]{}**]{}]{} We first use the basic primal–dual scheme of growing and merging connecting components to obtain a tree of cost ${{\ensuremath{B}}}'$ in which all $({{\ensuremath{s}}}_i,{{\ensuremath{t}}}_i)$ pairs of the traffic log are connected. If the cost of the tree at that point is smaller than the available budget (${{\ensuremath{B}}}'<{{\ensuremath{B}}}$), we can add more edges and reduce further the stretch factor. We do this greedily: we add edges one-by-one and in each step we select the one that decreases the stretch factor the most. On the other hand, if the cost of the tree found by the expansion phase is larger than the available budget (${{\ensuremath{B}}}'>{{\ensuremath{B}}}$), we need to remove edges. Again we do this in a greedy fashion: we remove edges one-by-one and in each step we select the one that increases the stretch factor the least. As with the greedy, when computing the total cost of the solution and comparing with the total budget [[$B$]{}]{}, the actual costs ${{\ensuremath{c}}}({{\ensuremath{e}}})$ are used. Pseudocode for the approach is given as Algorithm \[algorithm:pd\].
[[**[Algorithm variants.]{}**]{}]{} Similar to the greedy algorithm, the primal–dual algorithm first computes edge benefits ${{\ensuremath{b}}}({{\ensuremath{e}}})$ and effective costs ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})={{\ensuremath{c}}}({{\ensuremath{e}}})/{{\ensuremath{b}}}({{\ensuremath{e}}})$. It then applies the method described above, with edge costs ${{\ensuremath{c}}}_{{\ensuremath{e}}}={{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}})$. Depending on the benefit scores used, uniform, edge-betweenness centrality, and commute-time centrality, we obtain three variants: , , and . We also use landmarks (described above) to approximate shortest-path computations.
[[**[Running time:]{}**]{}]{} Again we assume that the benefit scores ${{\ensuremath{b}}}({{\ensuremath{e}}})$ are given for all edges ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$. The overall running time of steps 1–7 in Algorithm \[algorithm:pd\] is ${{\ensuremath{\cal O}}}({{\ensuremath{m}}}^2\log {{\ensuremath{m}}})$, which is achieved by using the [*union-find*]{} data structure to update connected components in roughly linear time and a priority queue to update the edge dual variables. After obtaining a Steiner forest, we have to add or remove edges greedily in steps 8–13 in Algorithm \[algorithm:pd\] to minimize the stretch factor. In each iteration in this step we need to compute the shortest paths ${{\ensuremath{\cal O}}}({{\ensuremath{k}}}{{\ensuremath{m}}})$ times to select the best edge. Assuming that the total number of iterations is [[$I$]{}]{}, the total running time in the second part of the algorithm is ${{\ensuremath{\cal O}}}({{\ensuremath{I}}}{{\ensuremath{k}}}{{\ensuremath{m}}}({{\ensuremath{m}}}+{{\ensuremath{n}}}\log{{\ensuremath{n}}}))$. As with the greedy, in practice the number of iterations ([[$I$]{}]{}) is significantly smaller than [[$k$]{}]{}.
Edge-betweenness centrality. {#section:ebc}
----------------------------
As we already discussed in the previous sections, our greedy algorithm uses edge centrality measures for benefit scores ${{\ensuremath{b}}}({{\ensuremath{e}}})$. In this section we discuss edge betweenness, and in particular show how we adapt the measure to take into account the traffic log [[$\cal L$]{}]{}.
We first recall the standard definition of edge-betweennes. Given a network ${{\ensuremath{G}}}=({{\ensuremath{V}}},{{\ensuremath{E}}})$, we define ${{\ensuremath{{{\ensuremath{V}}}_2}}}=
{{{\ensuremath{V}}}\choose 2}$ to be the set of all pairs of nodes of ${{\ensuremath{G}}}$. Given a pair of nodes $({{\ensuremath{s}}}, {{\ensuremath{t}}})\in {{\ensuremath{{{\ensuremath{V}}}_2}}}$ and an edge ${{\ensuremath{e}}}\in{{\ensuremath{E}}}$, we define by ${{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}$ the total number of shortest paths from ${{\ensuremath{s}}}$ to ${{\ensuremath{t}}}$, and by ${{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}({{\ensuremath{e}}})$ the total number of shortest paths from ${{\ensuremath{s}}}$ to ${{\ensuremath{t}}}$ that pass though edge [[$e$]{}]{}.
The standard definition of edge-betweenness centrality ${{\ensuremath{EB}}}({{\ensuremath{e}}})$ of edge [[[$e$]{}]{}]{} is the following: $${{\ensuremath{EB}}}({{\ensuremath{e}}}) =
\sum_{({{\ensuremath{s}}},{{\ensuremath{t}}})\in{{\ensuremath{{{\ensuremath{V}}}_2}}}} \frac{{{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}({{\ensuremath{e}}})}{{{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}}.$$ In our problem setting we take into account the traffic log ${{\ensuremath{\cal L}}}=\{({{\ensuremath{s}}}_i, {{\ensuremath{t}}}_i, {{\ensuremath{w}}}_i)\}$, and we define the edge-betweenness ${{\ensuremath{EB_{{{\ensuremath{\cal L}}}}}}}({{\ensuremath{e}}})$ of an edge [[$e$]{}]{} with respect to log [[$\cal L$]{}]{}, as follows. $${{\ensuremath{EB_{{{\ensuremath{\cal L}}}}}}}({{\ensuremath{e}}}) =
\sum_{({{\ensuremath{s}}},{{\ensuremath{t}}},{{\ensuremath{w}}})\in{{\ensuremath{\cal L}}}} {{\ensuremath{w}}}\,\frac{{{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}({{\ensuremath{e}}})}{{{\ensuremath{\sigma}}}_{{{\ensuremath{s}}},{{\ensuremath{t}}}}}.$$ In this modified definition only the source–destination pairs of the log [[$\cal L$]{}]{} contribute to the centrality of the edge [[$e$]{}]{}, and the amount of contribution is proportional to the corresponding traffic. The adapted edge-betweenness can still be computed in ${{\ensuremath{\cal O}}}({{\ensuremath{n}}}{{\ensuremath{m}}})$ time [@Brandes2007Centrality].
Experimental evaluation {#sec:experiments}
=======================
The aim of the experimental section is to evaluate the performance of the proposed algorithm, the optimizations, and the edge-betweenness measure. We also compare our algorithm with other state-of-the-art methods which attempt to solve a similar problem.
[[**[Datasets.]{}**]{}]{} We experiment with six real-world datasets, four transportation networks, one web network and one internet-traffic network. For five of the datasets we also obtain real-world traffic, while for one we use synthetically-generated traffic. The characteristics of our datasets are provided in Table \[tab:datasets\], and a brief description follows.
[[**[[[LondonTube]{}]{}.]{}**]{}]{} The London Subway network consists of subway stops and links between them.[^5] We use the geographic distance between stations as a proxy for edge costs. We also obtain a traffic log ${{\ensuremath{\cal L}}}$ extracted from the Oyster card system.[^6] The log consists of aggregate trips made by passengers between pairs of stations during a one-month period (Nov-Dec 2009). We filter out source-destination pairs with traffic less than 100 and remove bi-directional pairs by selecting one of them at random and summing up their traffic.
[[**[[[USFlights]{}]{}]{}**]{}]{}. We obtain a large network of US airports, and the list of all flights within the US during 2009–2013, from the Bureau of Transportation Statistics.[^7] Flying distances between airports, obtained using Travelmath.com, are used as edge costs. The traffic volume is the number of flights between airports.
[[**[[[NYCTaxi]{}]{}]{}**]{}]{}. We obtain the complete road network of NYC using OpenStreetMap data.[^8] In this network each node corresponds to a road intersection and each link corresponds to a road segment. Edge costs are computed as the geographic distances between the junctions. Data on the pickup and drop-off locations of NYC taxis for 2013 was used to construct the traffic log.[^9] The $2\,000$ most frequently used source-destination pairs was used to create the traffic log.
[[**[[[Wikispeedia]{}]{}]{}**]{}]{}. Wikispeedia[^10] [@west2009wikispeedia] is an online crowd sourcing game designed to measure semantic distances between 2 wikipedia pages using the paths taken by humans to reach from one page to the other. This dataset contains a base network of hyperlinks between Wikipedia pages and the paths taken by users between two pages. We construct the traffic log using the unique (start, end) pages from this data.
[[**[[[UKRoad]{}]{}]{}**]{}]{}. Next we consider the UK road network.[^11] The network construction is similar to what was done with the [[NYCTaxi]{}]{} data. For simplicity we use only the largest connected component. Since we were not able to obtain real-world traffic data for this network, we generate synthetic traffic logs ${{\ensuremath{\cal L}}}$ simulating different scenarios. In particular we generate traffic logs according to four different distributions: ($i$) power-law traffic volume, power-law [[$s$]{}]{}-[[$t$]{}]{} pairs; ($ii$) power-law traffic volume, uniformly random [[$s$]{}]{}-[[$t$]{}]{} pairs; ($iii$) uniformly random traffic volume, power-law [[$s$]{}]{}-[[$t$]{}]{} pairs; and ($iv$) uniformly random traffic volume, uniformly random [[$s$]{}]{}-[[$t$]{}]{} pairs. These different distributions capture different traffic volume possibility and hence help in understanding the behavior of our algorithm with respect to the traffic log ${{\ensuremath{\cal L}}}$. [[**[[[Abeline]{}]{}]{}**]{}]{}. For a qualitative analysis we also consider the well known [[Abeline]{}]{} dataset consisting of a sample of the network traffic extracted from the Internet2 backbone[^12] and that carries the network traffic between major universities in the continental US. The network consists of twelve nodes and 15 high-capacity links. Associated with each physical link, we also have capacity of the link which serves as a proxy for the cost of the link. We obtain traffic logs from 2003 between all pairs of nodes. [[**[[[SJRoad]{}]{}]{}**]{}]{}. To study the scalability of our approach, we use the road network from the city of San Joaquin County.[^13] This road network is constructed in a similar way as the [[UKRoad]{}]{}, and synthetically generated traffic log is used, according to the same four distributions.
[[**[Baseline.]{}**]{}]{} To obtain better intuition for the performance of our methods we define a simple baseline, where a backbone is created by adding edges in increasing order of their effective distances ${{\ensuremath{\widehat{\ell}}}}({{\ensuremath{e}}}) =
{{\ensuremath{c}}}({{\ensuremath{e}}})/{{\ensuremath{b}}}({{\ensuremath{e}}})$, where ${{\ensuremath{b}}}({{\ensuremath{e}}})$ is edge-betweenness; this was the best-performing baseline among other baselines we tried, such as adding source–destination pairs one by one (i) randomly, (ii) in decreasing order of volume ($w_i$), (iii) in increasing order of effective distance defined using closeness centrality, etc.
Quantitative results.
---------------------
We focus our evaluation on three main criteria: (i) Comparison of the performance with and without the edge-betweenness measure; (ii) effect of the optimizations, in terms of quality and speedup; and (iii) effect of allocating more budget on the stretch factor.
[[**[Effect of edge-betweenness.]{}**]{}]{} We study the effect of using edge-betweenness in the Greedy algorithm. The results are presented in Figure \[fig:ebVSnormal\].
[[**[Effect of landmarks.]{}**]{}]{} Landmarks provide faster computation with a trade off for quality. Figure \[fig:landmarks\_timetaken\] shows the speedup achieved when using landmarks. In the figures, BasicGreedyEB indicates the greedy algorithm that doesn’t use any optimizations. GreedyEBCC makes use of the optimizations proposed in Section \[sec:optimizations\] which do not use approximation. GreedyEBLandmarks\* makes use of the landmarks optimatization and the \* indicates the number of landmarks we tried. Figure \[fig:landmarks\_performance\] shows the performance of [[GreedyEB]{}]{} algorithm with and without using landmarks. [[**[Budget vs. stretch factor.]{}**]{}]{} We examine the trade-off between budget and stretch factor for our algorithm and its variants. A lower stretch factor for the same budget indicates that the algorithm is able to pick better edges for the backbone. Figure \[fig:ebVSnormal\] shows the trade-off between budget and stretch factor for all our datasets. In all figures the budget used by the algorithms, shown in the $x$-axis, is expressed as a percentage of the total cost of all the edges in the network.
[[**[Key findings]{}**]{}]{}. From all the above results, we would like to highlight the following points.
[[**[1.]{}**]{}]{} The greedy algorithm and its variants performs much better than the baseline (See Figure \[fig:ebVSnormal\]). Note that baseline is not included in Figure \[fig:ebVSnormal\](g,h) because the edges in the baseline are added one-by-one and for a large interval of the cost, the stretch factor was very large or even infinity. This shows that the backbone produced by our greedy approach not only consists of edges with low benefit, but also tries to re-use a lot of edges, hence obtaining a lower stretch factor. [[**[2.]{}**]{}]{} The backbones discovered by our algorithms are sparse and summarize well the given traffic (Figures \[fig:ebVSnormal\], \[fig:landmarks\_performance\]). In all cases, with about 15% of the edge cost in the network it is possible to summarize the traffic with stretch factor close to 1. In some cases, even smaller budget (than 15%) is sufficient to reach a lower stretch-factor value.
[[**[3.]{}**]{}]{} For all variants of the greedy algorithm, the stretch factor decreases as the budget increases. This is expected as increase in budget implies the ability to include more edges in the backbone and thus leading to a decrease in the stretch factor.
[[**[3.]{}**]{}]{} Incorporating edge-betweenness as an edge-weighting scheme in the algorithm improves the performance, in certain cases there is an improvement of at least 50% (See Figure \[fig:ebVSnormal\]; in most cases, even though there is a significant improvement, the plot is overshadowed by a worse performing baseline). This is because, using edges of high centrality will make sure that these edges are included in many shortest paths, leading to re-using many edges. [[**[4.]{}**]{}]{} The optimizations we propose in Section \[sec:optimizations\] help in reducing the running time of our algorithm (See Figure \[fig:landmarks\_timetaken\]). For the optimizations not using landmarks, we see around 30% improvement in running time. Using landmarks substantially decreases the time taken by the algorithms (3–4 times). While there is a compromise in the quality of the solution, we can observe from Figures \[fig:landmarks\_performance\] that the performance drop is small in most cases and can be controlled by the choosing the number of landmarks accordingly. Our algorithms, using the various optimizations we propose, are able to scale for large, real-world networks with tens of thousands of nodes which is the typical size of a road/traffic network.
[[**[Budget and size of backbone.]{}**]{}]{} The greedy algorithm operates by connecting [[$s$]{}]{}-[[$t$]{}]{} pairs in each iteration and reducing stretch factor. On the other hand, the primal–dual algorithm works by adding or removing edges from the Steiner forest. One would expect that for a given cost, primal–dual will keep a larger percentage of [[$s$]{}]{}-[[$t$]{}]{} pairs connected than greedy. However, as we can see in Figure \[fig:cost\_st\_pairs\], we did not observe this behavior: greedy seems to be able to connect the same fraction of [[$s$]{}]{}-[[$t$]{}]{} pairs, for the same budget. This can be partially explained by the fact that the standard reverse deletion step of the primal–dual algorithm has been replaced by a greedy step to reduce stretch factor. A deeper analysis of the behavior of of the primal–dual algorithm for the problem is warranted to draw firmer conclusions.
Comparison to existing approaches {#sec:comparison}
---------------------------------
In this section, we compare the performance of [[BackboneDiscovery]{}]{} with other related work in literature. The comparison is done based on two factors (i) Stretch factor, (ii) Percentage of edges covered by the solution. Intuitively, a good backbone should try to minimize both, i.e. produce a sparse backbone, which also preserves the shortest paths between vertices as well as possible.
[[**[Comparison with Prize Collecting Steiner-forest (PCSF)]{}**]{}]{} - Prize Collecting Steiner-forest [@hajiaghayi2010prize] is a variant of the classic Steiner Forest problem, which allows for disconnected source–destination pairs, by paying a penalty. The goal is to minimize the total cost of the solution by ‘buying’ a set of edges (to connect the $s$–$t$ pairs) and paying the penalty for those pairs which are not connected. We compare the performance of our algorithm with PCSF, based on two factors (i) Stretch factor (Figure \[fig:pcsf\_comparison\]a), (ii) Percentage of edges covered by the solution (Figure \[fig:pcsf\_comparison\]b). We use the same ($s$,$t$) pairs that we use in our algorithm and set the traffic volume $w_i$ as the penalty score in PCSF. We first run PCSF on our datasets and compute the budget of the solution produced. Using the budget as input to our algorithm ([[GreedyEB]{}]{}), we compute our backbone.
We can see from Figure \[fig:pcsf\_comparison\]a that our algorithm produces a backbone with a much better stretch factor than PCSF. In most datasets, our algorithm produces a backbone which is at least 2 times better in terms of stretch factor.
Figure \[fig:pcsf\_comparison\]b compares the fraction of edges covered by our algorithm and PCSF. We observe that the fraction of edges covered by our algorithm is lower than that of PCSF. This could be because our algorithm re-uses edges belonging to multiple paths. Figures \[fig:pcsf\_comparison\](a,b) show that even though our solution is much better in terms of stretch factor, we produce sparse backbones (in terms of the percentage of edges covered).
[[**[Comparison with k-spanner]{}**]{}]{} - As described in Section \[sec:related\], our problem is similar to $k$-spanner [@narasimhan2007geometric] in the sense that we try to minimize the stretch factor. A $k$-spanner of a graph is a subgraph in which any two vertices are at most $k$ times far apart than on the original graph. One of the main advantages of our algorithm compared to spanners is that spanners can not handle disconnected vertices. We also propose and optimize a modified version of stretch factor in order to handle disconnected vertices. Similar to PCSF, we first compute a 2-spanner using a 2 approximation greedy algorithm and compute the budget used. We then run our algorithm for the same budget. Figures \[fig:spanner\_comparison\](a,b) show the performance of our algorithm in terms of stretch factor and percentage of edges covered. Our objective here is to compare the cost our algorithm pays in terms of stretch factor for allowing disconnected vertices. We can clearly observe that even though we allow for disconnected pairs, our algorithm performs slightly better in terms of stretch factor and also produces a significantly sparser backbone.
[[**[Comparison with Toivonen et al. [@ToivonenMZ10]]{}**]{}]{} - Next, we compare our algorithm with Toivonen, et al [@ToivonenMZ10]. Toivonen et al. propose a framework for path-oriented graph simplification, in which edges are pruned while keeping the original quality of the paths between all pairs of nodes. The objective here is to check how well we perform in terms of graph sparsification. Figures \[fig:toivonen\_comparison\](a,b) shows the comparison in terms of stretch factor and percentage of edges covered. Similar to the above approaches, we use the same budget as that used by Toivonen’s algorithm. We observe that for most of the datasets, their algorithm works poorly in terms of sparsification, pruing less than 20% of the edges (Figure \[fig:toivonen\_comparison\](b)). Our algorithm performs better both in terms of the stretch of the final solution as well as sparseness of the backbone.
The above results, comparing our work with the existing approaches showcase the power of our algoritm in finding a concise representation of the graph, at the same time maintaining a low stretch factor. In all the three cases, our algorithm performs considerably better than the related work.
[[**[Fairness]{}**]{}]{} - Though we claim that our approach performs better, we need to keep in mind that there might be differences between these algorithms. PCSF does not optimize for stretch factor. Spanners and Toivonen et al. do not have a traffic log (($s$,$t$) pairs). They also do not try to optimize stretch factor. For this section, we were just interested in contrasting the performance of our approach with existing state of the art methods and show how our approach is different and better at what we do.
Case study \#1: [[NYCTaxi]{}]{}.
--------------------------------
The backbone of the NYC taxi traffic, as discovered by our algorithms [[Greedy]{}]{} and [[GreedyEB]{}]{}, is shown in Figure \[fig:nycbackbone\]. We see that both backbones consist of many street stretches in the mid-town (around Times Square) while serving lower-town (Greenwich village and Soho) and up-town (Morningside heights). We also note that there are stretches to the major transportation centers, such as the LaGuardia airport, the World Financial Center Ferry Terminal, and the Grand Central Terminal, as well as to the Metropolitan museum. Comparing the [[Greedy]{}]{} and [[GreedyEB]{}]{} backbones, we see that [[GreedyEB]{}]{} emphasizes more on the traffic to lower-town, and ignores the northern stretch via Robert Kennedy bridge, as it is less likely to be included in many shortest paths. The case study reiterates the advantages of using edge-betweenness to guide the selection of the backbone to include edges which are likely to be used more and is consistent with the well established notion of Wardrop Equilibrium in Transportation Science that users (in a non-cooperative manner) seek to minimize their cost of transportation [@wardrop].
Case study \#2: [[Abeline]{}]{}. {#sec:abeline}
--------------------------------
We carry out a qualitative analysis on the Abilene dataset. The results of applying the [[Greedy]{}]{} algorithm are shown in Figure \[fig:abeline\].[^14] The results provide preliminary evidence that the backbone produced by our problem can be tightly integrated with software defined networks (SDN), an increasingly important area in communication networks [@KimF13]. The objective of SDN is to allow a software layer to control the routers and switches in the physical layers based on the profile and shape of the traffic. This is precisely what our solution is accomplishing in Figure \[fig:abeline\]. The design of data-driven logical networks will be an important operation implemented through an SDN and will help network designers manage traffic in real time.
Related work {#sec:related}
============
As already noted, [[BackboneDiscovery]{}]{} is related to the $k$-spanner and the Steiner-forest problem and the decision versions of both are known to be [[[[$\mathbf{NP}$]{}]{}-complete]{}]{} [@narasimhan2007geometric; @williamson2011design]. The $k$-spanner problem is designed to bound the stretch factor for [*all pairs*]{} of nodes and not just those from a specific set of $(s,t)$ pairs. The Steiner-forest problem on the other hand is designed to keep the $(s,t)$ pairs connected with a minimal number of edges and is agnostic about the stretch factor. Both these problems only consider structural information and completely ignore functional (activity) data that maybe available about the usage of the network. They also have strict limitations that all nodes need to be covered, which makes them restrictive.
The Prize collecting Steiner-forest problem (PCSF) [@hajiaghayi2010prize] is a version of the Steiner-forest problem that allows for disconnected source–destination pairs, by imposing a penalty for disconnected pairs. Even in this variant, there is no budget or stretch requirement and hence the optimization problem that PCSF solves is completely different from what we solve. We show how our algorithm fares in comparison to PCSF in Section \[sec:comparison\].
Given a network $G = (V,E)$, the $k$-spanner problem asks to find a subnetwork $H = (V,E')$ where the original distances between every pair of nodes are preserved within a factor $k$. Note that the node set of subgraph $G$ and $H$ are identical. The decision version of $k$-spanner problem is known to be [[[[$\mathbf{NP}$]{}]{}-complete]{}]{} [@narasimhan2007geometric]. The problem has been extensively studied in the computer science theory community and several heuristic and approximation results have been proposed. For example, a very popular greedy heuristic for networks embedded in a metric space consists of the following steps: ($i$) Start with the nodes $V$ of the original graph; ($ii$) Form a complete a network $G' = (V,C)$; ($iii$) Sort the edges in $C$ in increasing order; and ($iv$) Process each edge $e =(u,v)$ in the sorted order and issue a shortest-path query on the original network. If the length of the shortest path is greater than $k|(u,v)|$ then add the edge to $H$. It is known that the time complexity of the greedy algorithm is ${{\ensuremath{\cal O}}}(n^{2})$. However, this greedy algorithm does not extend to the case where the objective is to only preserve the stretch factor for a subset of terminals, $\{(s_{i},t_{i})\}$. The focus on terminals connects the [[BackboneDiscovery]{}]{} problem to the Steiner-forest problem.
The input of Steiner-forest problem consists of a weighted undirected network $G=(V,E,w)$ and $k$ pairs of nodes $(s_{i},t_{i})$. The objective is to find a subset of edges of $E$ of minimum cost such that every $(s_{i},t_{t})$ pair is connected. Again, the problem is [[[[$\mathbf{NP}$]{}]{}-hard]{}]{} [@williamson2011design]. However, as we discussed in the introduction, the objective of Steiner-forest problem is not designed to preserve the stretch factor of the paths between the $k$ terminals. Another enhancement in our work is to normalize edge costs with measures related to the structure of the network (like edge betweenness [@Brandes2007Centrality; @girvan2002community-structure; @measbetrandom05]) As we show in our experiments, this leads to finding solutions of better quality. Our work is different from trajectory mining [@giannotti2007trajectory; @zheng2009mining], which consider complete trajectories between source–destination pairs. We do not make use of the trajectories and are only interested in the amount of traffic flowing between a source and destination. Also, the type of questions we try to answer in this paper are different from that of trajectory mining. While trajectory mining tries to answer questions like “Which are the most used routes between A and B?”, our paper tries to use information about traffic from A to B in order to facilitate a sparse backbone of the underlying network which allows traffic to flow from A to B, also keeping global network characteristics in mind.
The [[BackboneDiscovery]{}]{} problem is also related to finding graph sparsifiers and simplifying graphs. For example, Toivonen et al. [@ToivonenMZ10] as well as Zhou et al [@ToivonenMZ10b], propose an approach based on pruning edges while keeping the quality of best paths between all pairs of nodes, where quality is defined on concepts such as shortest path or maximum flow. Misiolek and Chen [@flownetsimpl] propose an algorithm which prune edges while maintaining the source-to-sink flow for each pair of nodes. Mathioudakis et al. [@spine] and Bonchi et al. [@spine2] study the problem of discovering the backbone of a social network in the context of information propagation, which is a different type of activity than source–destination pairs, as considered here. In the work of Butenko et al. a heuristic algorithm for the minimum connected dominating subset of wireless networks was proposed [@heurminiwirelessspringer04]. There has been some work in social network research to extract a subgraph from larger subgraphs subject to constraints [@backdissocnetacm07; @netbackdisclusteraxviv12]. Other forms of network backbone-discovery have been explored in domains including biology, communication networks and the social sciences. The main focus of most of these approaches is on the trade-off between the level of network reduction and the amount of relevant information to be preserved either for visualization or community detection. While in this paper we try to also sparsify a graph, our objective and approach is completely different from the above because we cast the problem in a well-defined optimization framework where the [*structural*]{} aspects of the network are captured in the requirement to maintain a low stretch while the [*functional*]{} requirements are captured in maintaining connectedness between traffic terminals, which has not been done before.
In the computer network research community, the notion of software defined networks (SDN), which in principle decouples the network control layer from the physical routers and switches, has attracted a lot of attention [@CasadoFPLGMS09; @KimF13]. SDN (for example through OpenFlow) will essentially allow network administrators to remotely control routing tables. The [[BackboneDiscovery]{}]{} problem can essentially be considered as an abstraction of the SDN problem, and as we show in Section \[sec:abeline\], our approach can make use of traffic logs to help SDN’s make decisions on routing and switching in the physical layer.
Conclusions {#sec:conclusions}
===========
We introduced a new problem, [[BackboneDiscovery]{}]{}, to address a modern phenomenon: these days not only is the [*structural*]{} information of a network available but increasingly, highly granular [*functional (activity)*]{} information related to network usage is accessible. For example, the aggregate traffic usage of the London Subway between all stations is available from a public website. The [[BackboneDiscovery]{}]{}problem allowed us to efficiently combine structural and functional information to obtain a highly sophisticated understanding of how the Tube is used (See Figure \[fig:londontube\]). From a computational perspective, the [[BackboneDiscovery]{}]{} problem has elements of both the $k$-spanner and the Steiner-forest problem and thus requires new algorithms to maintain low stretch and connectedness between important nodes subject to a budget constraint. We compare our algorithm with other similar algorithms and show how our algorithm is different and performs better for our setting. Our case studies show the application of the proposed methods for a wide range of applications, including network and traffic planning.
Though our algorithm makes use of shortest paths, in practice, any other types of paths could be incorporated into our algorithm. We leave this generalization for future analysis. The use of harmonic mean not only allows us to handle disconnected (s,t)-pairs, but also makes our stretch factor measure more sensitive to outliers. For future work, we would also incorporate a deeper theoretical analysis of the algorithm and the stretch factor measure.
[^1]: University of Sydney
[^2]: Aalto University
[^3]: Aalto University
[^4]: University of Sydney
[^5]: [<http://bit.ly/1C9PbLT>]{}
[^6]: [<http://bit.ly/1qM2BYi>]{}
[^7]: [<http://1.usa.gov/1ypXYvL>]{}
[^8]: [<http://metro.teczno.com/#new-york>]{}
[^9]: [<http://chriswhong.com/open-data/foil_nyc_taxi/>]{}
[^10]: [<http://snap.stanford.edu/data/wikispeedia.html>]{}
[^11]: [<http://www.dft.gov.uk/traffic-counts/download.php>]{}
[^12]: [<http://www.internet2.edu>]{}
[^13]: [<http://www.cs.fsu.edu/~lifeifei/SpatialDataset.htm>]{}
[^14]: The two nodes in Atlanta have been merged.
|
---
abstract: 'For a graph $G$ let $L(G)$ and $l(G)$ denote the size of the largest and smallest maximum matching of a graph obtained from $G$ by removing a maximum matching of $G$. We show that $L(G)\leq 2l(G),$ and $L(G)\leq \frac{3}{2}l(G)$ provided that $G$ contains a perfect matching. We also characterize the class of graphs for with $L(G)=2l(G)$. Our characterization implies the existence of a polynomial algorithm for testing the property $L(G)=2l(G)$. Finally we show that it is $NP$-complete to test whether a graph $G$ containing a perfect matching satisfies $L(G)=\frac{3}{2}l(G)$.'
address:
- |
Department of Informatics and Applied Mathematics,\
Yerevan State University, Yerevan, 0025, Armenia
- |
Institute for Informatics and Automation Problems,\
National Academy of Sciences of Republic of Armenia, 0014, Armenia
author:
- 'Artur Khojabaghyan [^1] and Vahan V. Mkrtchyan [^2]'
title: On upper bounds for parameters related to construction of special maximum matchings
---
Introduction
============
In the paper graphs are assumed to be finite, undirected, without loops or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of a graph $G$, respectively. If $v\in V(G)$ and $e\in E(G)$, then $e$ is said to cover $v$ if $e$ is incident to $v$. For $V'\subseteq V(G)$ and $E'\subseteq E(G)$ let $G\backslash V'$ and $G\backslash E'$ denote the graphs obtained from $G$ by removing $V'$ and $E'$, respectively. Moreover, let $V(E')$ denote the set of vertices of $G$ that are covered by an edge from $E'$. A subgraph $H$ of $G$ is said to be spanning for $G$, if $V(E(H))=V(G)$.
The length of a path (cycle) is the number of its edges. A $k$-path ($k$-cycle) is a path (cycle) of length $k$. A $3$-cycle is called a triangle.
A set $V'\subseteq V(G)$ ($E'\subseteq E(G)$) is said to be independent, if $V'$ ($E'$) contains no adjacent vertices (edges). An independent set of edges is called matching. A matching of $G$ is called perfect, if it covers all vertices of $G$. Let $\nu (G)$ denote the cardinality of a largest matching of $G$. A matching of $G$ is maximum, if it contains $\nu (G)$ edges.
For a positive integer $k$ and a matching $M$ of $G$, a $(2k-1)$-path $P$ is called $M$-augmenting, if the $2^{nd}$, $4^{th}$, $6^{th}$,..., $(2k-2)^{th}$ edges of $P$ belong to $M$, while the endvertices of $P$ are not covered by an edge of $M$. The following theorem of Berge gives a sufficient and necessary condition for a matching to be maximum:
(Berge [@Harary]) A matching $M$ of $G$ is maximum, if $G$ contains no $M$-augmenting path.
For two matchings $M$ and $M'$ of $G$ consider the subgraph $H$ of $G$, where $V(H)=V(M\triangle M')$ and $E(H)=M\triangle M'$. The connected components of $H$ are called $M\triangle M'$-alternating components. Note that $M\triangle M'$ alternating components are always paths or cycles of even length. For a graph $G$ define:
$L(G)\equiv \max \{\nu (G\backslash F):F$ is a maximum matching of $G\},$
$l(G)\equiv \min \{\nu (G\backslash F):F$ is a maximum matching of $G\}.$
It is known that $L(G)$ and $l(G)$ are $NP$-hard calculable even for connected bipartite graphs $G$ with maximum degree three [@Complexity], though there are polynomial algorithms which construct a maximum matching $F$ of a tree $G$ such that $\nu (G\backslash F)=L(G)$ and $\nu (G\backslash
F)=l(G)$ (to be presented in [@Algorithm]).
In the same paper [@Algorithm] it is shown that $L(G)\leq 2l(G).$ In the present paper we re-prove this equality, and also show that $L(G)\leq \frac{3}{2}l(G)$ provided that $G$ contains a perfect matching.
A naturally arising question is the characterization of graphs $G$ with $L(G)= 2l(G)$ and the graphs $G$ with a perfect matching that satisfy $L(G)= \frac{3}{2}l(G)$. In this paper we solve these problems by giving a characterization of graphs $G$ with $L(G)= 2l(G)$ that implies the existence of a polynomial algorithm for testing this property, and by showing that it is $NP$-complete to test whether a bridgeless cubic graph $G$ satisfies $L(G)=\frac{3}{2}l(G)$. Recall that by Petersen theorem any bridgeless cubic graph contains a perfect matching.
Terms and concepts that we do not define can be found in [Diestel,Harary,Lov,West]{}.
Some auxiliarly results
=======================
We will need the following:
\[Ratios\] Let $G$ be a graph. Then:
1. for any two maximum matchings $F,F'$ of $G$, we have $\nu (G\backslash F')\leq 2\nu (G\backslash F)$;
2. $L(G)\leq 2l(G)$;
3. If $L(G)=2l(G)$, $F_{L},F_{l}$ are two maximum matchings of the graph $G$ with $\nu (G\backslash
F_{L})=L(G), $ $\nu (G\backslash F_{l})=l(G)$, and $H_{L}$ is **any** maximum matching of the graph $G\backslash F_{L}$, then:
1. $F_{l}\backslash F_{L}\subset H_{L};$
2. $H_{L}\backslash F_{l}$ is a maximum matching of $G\backslash
F_{l}$;
3. $F_{L}\backslash F_{l}$ is a maximum matching of $G\backslash
F_{l}$;
4. if $G$ contains a perfect matching, then $L(G)\leq \frac32l(G)$.
(a)Let $H'$ be any maximum matching in the graph $G\backslash F'$. Then:$$\begin{aligned}
\nu (G\backslash F') =|H'|=|H'\cap F|+|H'\backslash F|\leq |F\backslash F'|+\nu(G\backslash F)=
|F'\backslash F|+\nu (G\backslash F)\leq 2\nu (G\backslash F).\end{aligned}$$
\(b) follows from (a).
\(c) Consider the proof of (a) and take $F'=F_{L}$, $H'=H_L$ and $F=F_l$. Since $L(G)=2l(G)$, we must have equalities throughout, thus properties (c1)-(c3) should be true.
\(d) Let $F_{L},F_{l}$ be two perfect matchings of the graph $G$ with $\nu
(G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$, and assume $H_{L}$ to be a maximum matching of the graph $G\backslash F_{L}$. Define: $$\begin{aligned}
X =\{e=(u,v)\in F_{L}:u\text{ and }v\text{ are incident to an edge from }H_{L}\cap F_{l}\}, \\
x =|X|, k=|H_{L}\cap F_{l}|;\end{aligned}$$
Clearly, $(H_{L}\backslash F_{l})\cup X$ is a matching of the graph $G\backslash F_{l}$, therefore, taking into account that $(H_{L}\backslash F_{l})\cap X=\emptyset,$ we deduce $$\begin{aligned}
l(G) =\nu (G\backslash F_{l})\geq |H_{L}\backslash
F_{l}| +|X| =|H_{L}|
-|H_{L}\cap F_{l}| +|X| =L(G)-k+x.\end{aligned}$$
Since $F_{L}$ is a perfect matching, it covers the set $V(H_{L}\cap
F_{l})\backslash V(X)$, which contains$$|V(H_{L}\cap F_{l})\backslash V(X)| =2|
(H_{L}\cap F_{l})| -2|X| =2k-2x$$vertices. Define the set $E_{F_{L}}$ as follows: $$E_{F_{L}}=\{e\in F_{L}:e\text{ covers a vertex from }V(H_{L}\cap
F_{l})\backslash V(X)\}.$$ Clearly, $E_{F_{L}}$is a matching of $G\backslash F_{l}$, too, and therefore $$l(G)=\nu (G\backslash F_{l})\geq |E_{F_{L}}|=2k-2x.$$ Let us show that $$\max \{L(G)-k+x,2k-2x\}\geq \frac{2L(G)}{3}.$$Note that\
if $x\geq k-\frac{L(G)}{3}$ then $L(G)-k+x\geq L(G)-k+k-\frac{L(G)}{3}=\frac{2L(G)}{3}$;\
if $x\leq k-\frac{L(G)}{3}$ then $2k-2x\geq \frac{2L(G)}{3}$,\
thus in both cases we have $l(G)\geq \frac{2L(G)}{3}$ or $$\frac{L(G)}{l(G)}\leq \frac{3}{2}.$$
The proof of the theorem \[Ratios\] is completed. $\square$
\[1-1Case\] (Lemma 2.20, 2.41 of [@Algorithm]) Let $G$ be a graph, and assume that $u$ and $v$ are vertices of degree one sharing a neighbour $w\in V(G)$. Then: $$L(G)=L(G\backslash \{u,v,w\})+1,l(G)=l(G\backslash \{u,v,w\})+1.$$
\[L=2l1-1case\] Let $G$ be a graph with $L(G)=2l(G)$. Then there are no vertices $u,v$ of degree one, that are adjacent to the same vertex $w$.
Suppose not. Then lemma \[1-1Case\] and (b) of theorem \[Ratios\] imply $$\begin{aligned}
L(G) &=&1+L(G-\{u,v,w\})\leq 1+2l(G-\{u,v,w\})= \\
&=&1+2(l(G)-1)=2l(G)-1<2l(G)\end{aligned}$$a contradiction. $\square$
Characterization of graphs $G$ satisfying $L(G)=2l(G)$
======================================================
Let $T$ be the set of all triangles of $G$ that contain at least two vertices of degree two. Note that any vertex of degree two lies in at most one triangle from $T$. From each triangle $t\in T$ choose a vertex $v_t$ of degree two, and define $V_1(G)$ as follows: $$V_1(G)=\{v:d_G(v)=1\}\cup \{v_t:t\in T\}$$
\[Characterization\] Let $G$ be a connected graph with $|V(G)|\geq3$. Then $L(G)=2l(G)$ if and only if
(1)
: $G\backslash V_1(G)$ is a bipartite graph with a bipartition $(X,Y)$;
(2)
: $|V_1(G)|=|Y|$ and any $y\in Y$ has exactly one neighbour in $V_1(G)$;
(3)
: the graph $G\backslash V_{1}(G)$ contains $|X|$ vertex disjoint $2$-paths.
Sufficiency. Let $G$ be a connected graph with $|V(G)|\geq3$ satisfying the conditions (1)-(3). Let us show that $L(G)=2l(G)$.
For each vertex $v$ with $d(v)=1$ take the edge incident to it and define $F_1$ as the union of all these edges. For each vertex $v_t\in V_{1}(G)$ take the edge that connects $v_t$ to a vertex of degree two, and define $F_2$ as the union of all those edges. Set: $$F=F_1\cup F_2.$$
Note that $F$ is a matching with $|F|=|V_{1}(G)|=|Y|$. Moreover, since $G$ is bipartite and $|V_{1}(G)|=|Y|$, the definitions of $F_1$ and $F_2$ imply that there is no $F$-augmenting path in $G$. Thus, by Berge theorem, $F$ is a maximum matching of $G$, and $$\nu(G)=|F|=|V_{1}(G)|=|Y|.$$Observe that the graph $G\backslash F$ is a bipartite graph with $\nu(G\backslash F)\leq |X|$, thus $$l(G)\leq \nu(G\backslash F)\leq |X|.$$
Now, consider the $|X|$ vertex disjoint $2$-paths of the graph $G\backslash V_{1}(G)$ guaranteed by (3). (2) implies that these $2$-paths together with the $|F|=|V_{1}(G)|=|Y|$ edges of $F$ form $|X|$ vertex disjoint $4$-paths of the graph $G$.
Consider matchings $M_{1}$ and $M_{2}$ of $G$ obtained from these $4$-paths by adding the first and the third, the second and the fourth edges of these $4$-paths to $M_{1}$ and $M_{2}$, respectively. Define: $$F'=(F\backslash M_{2})\cup (M_{1}\backslash F).$$
Note that $F'$ is a matching of $G$ and $|F'|=|F|$, thus $F'$ is a maximum matching of $G$. Since $F'\cap M_2=\emptyset$, we have$$L(G)\geq \nu (G\backslash F')\geq |M_{2}|=2|X|\geq 2l(G).$$
\(b) of theorem \[Ratios\] implies that $L(G)=2l(G)$.
Necessity. Now, assume that $G$ is a connected graph with $|V(G)|\geq 3$ and $L(G)=2l(G)$. By proving a series of claims, we show that $G\backslash V_1(G)$ satisfies the conditions (1)-(3) of the theorem.
\[SpanningSubgraph\] For any maximum matchings $F_{L},F_{l}$ of the graph $G$ with $\nu (G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$, $F_{L}\cup F_{l}$ induces a spanning subgraph, that is $V(F_{L})\cup
V(F_{l})=V(G)$.
Suppose that there is a vertex $v\in V(G)$ that is covered neither by $F_{L}$ nor by $F_{l}$. Since $F_{L}$ and $F_{l}$ are maximum matchings of $G$, for each edge $e=(u,v)$ the vertex $u$ is incident to an edge from $F_{L}$ and to an edge from $F_{l}$.
Case 1: there is an edge $e=(u,v)$ such that $u$ is incident to an edge from $F_{L}\cap F_{l}$.
Note that $\{e\}\cup (F_{L}\backslash F_{l})$ is a matching of $G\backslash
F_{l}$ which contradicts (c3) of the theorem \[Ratios\].
Case 2: for each edge $e=(u,v)$ $u$ is incident to an edge $f_{L}\in
F_{L}\backslash F_{l}$ and to an edge $f_{l}\in F_{l}\backslash F_{L}$.
Let $H_{L}$ be any maximum matching of $G\backslash F_{L}$. Due to (c1) of theorem \[Ratios\] $f_{l}\in H_{L}$. Define: $$H'_{L}=(H_{L}\backslash \{f_{l}\})\cup \{e\}.$$ Note that $H'_{L}$ is a maximum matching of $G\backslash F_{L}$ such that $F_{l}\backslash F_{L}$ is not a subset of $H'_{L}$ contradicting (c1) of theorem \[Ratios\]. $\square$
\[AltComp2paths\] For any maximum matchings $F_{L},F_{l}$ of the graph $G
$ with $\nu (G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$, the alternating components $F_{L}\triangle F_{l}$ are $2$-paths.
It suffices to show that there is no edge $f_{L}\in F_{L}$ that is adjacent to two edges from $F_{l}$. Suppose that some edge $f_{L}\in F_{L}$ is adjacent to edges $f_{l}^{\prime }$ and $f_{l}^{\prime \prime }$ from $F_{l}$. Let $H_{L}$ be any maximum matching of $G\backslash F_{L}$. Due to (c1) of theorem \[Ratios\] $f'_{l},f''_{l}\in H_{L}$. This implies that $\{f_{L}\}\cup (H_{L}\backslash F_{l})$ is a matching of $G\backslash F_{l}$ which contradicts (c2) of theorem \[Ratios\]. $\square$
\[DegreeRequirements\]For any maximum matchings $F_{L},F_{l}$ of the graph $G$ with $\nu (G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$
1. if $u\in V(F_{l})\backslash V(F_{L})$ then $d(u)=1$ or $d(u)=2$. Moreover, in the latter case, if $v$ and $w$ denote the two neighbours of $u$, where $(u,w)\in F_l$, then $d(w)=2$ and $(v,w)\in F_L$.
2. if $u\in V(F_{L})\backslash V(F_{l})$ then $d(u)\geq 2.$
\(a) Assume that $u$ is covered by an edge $e_{l}\in F_{l}$ and $u\notin
V(F_{L})$. Suppose that $d(u)\geq 2$, and there is an edge $e=(u,v)$ such that $e\notin F_{l}$. Taking into account the claim \[SpanningSubgraph\], we need only to consider the following four cases:
Case 1: $v\in V(F_{l})\backslash V(F_{L})$.
This is impossible, since $F_{L}$ is a maximum matching.
Case 2: $v$ is covered by an edge $f\in F_{L}\cap F_{l}$;
Let $H_{L}$ be any maximum matching of $G\backslash F_{L}$. Due to (c1) of theorem \[Ratios\] $e_{l}\in H_{L}$, thus $e\notin H_{L}$.
Define: $$F_{L}^{\prime }=(F_{L}\backslash \{f\})\cup \{e\}.$$Note that $F_{L}^{\prime }$ is a maximum matching, and $H_{L}$ is a matching of $G\backslash F_{L}^{\prime }$. Moreover, $$\nu (G\backslash F_{L}^{\prime })\geq \left\vert H_{L}\right\vert =\nu
(G\backslash F_{L})=L(G),$$ thus $H_{L}$ is a maximum matching of $G\backslash F_{L}^{\prime }$ and $\nu
(G\backslash F_{L}^{\prime })=L(G)$. This is a contradiction because $F_{L}^{\prime }\triangle F_{l}$ contains a component which is not a $2$-path contradicting claim \[AltComp2paths\].
Case 3: $v$ is incident to an edge $f_{L}\in F_{L},$ $f_{l}\in F_{l}$ and $f_{L}\neq $ $f_{l}$.
Let $H_{L}$ be any maximum matching of $G\backslash F_{L}$. Due to (c1) of theorem \[Ratios\], $e_{l},f_{l}\in H_{L}$. Define: $$F_{L}^{\prime }=(F_{L}\backslash \{f_{L}\})\cup \{e\}.$$Note that $F_{L}^{\prime }$ is a maximum matching, and $H_{L}$ is a matching of $G\backslash F_{L}^{\prime }$. Moreover, $$\nu (G\backslash F_{L}^{\prime })\geq \left\vert H_{L}\right\vert =\nu
(G\backslash F_{L})=L(G),$$thus $H_{L}$ is a maximum matching of $G\backslash F_{L}^{\prime }$ and $\nu
(G\backslash F_{L}^{\prime })=L(G)$. This is a contradiction because $F_{L}^{\prime }\triangle F_{l}$ contains a component which is not a $2$-path contradicting claim \[AltComp2paths\].
Case 4: $v$ is covered by an edge $e_{L}\in F_{L}$ and $v\notin V(F_{l}).$
Note that if $e_{L}$ is not adjacent to $e_{l}$ then the edges $e,e_{L}$ and the edge $\tilde{e}\in F_{l}\backslash F_{L}$ that is adjacent to $e_{L}$ would form an augmenting $3$-path with respect to $F_{L}$, which would contradict the maximality of $F_{L}$.
Thus it remains to consider the case when $e_{L}$ is adjacent to $e_{l}$ and $d(u)=2$. Let $w$ be the vertex adjacent to both $e_{l}$ and $e_{L}$. Let us show that $d(w)=2$. Let $H_{L}$ be any maximum matching of $G\backslash F_{L}$. Due to (c1) of theorem \[Ratios\], $e_{l}\in H_{L}$. Define: $$F'_{L}=(F_{L}\backslash \{e_{L}\})\cup \{e\}.$$Note that $F'_{L}$ is a maximum matching, and $H_{L}$ is a matching of $G\backslash F'_{L}$. Moreover, $$\nu (G\backslash F'_{L})\geq |H_{L}| =\nu (G\backslash F_{L})=L(G),$$thus $H_{L}$ is a maximum matching of $G\backslash F'_{L}$ and $\nu
(G\backslash F'_{L})=L(G)$. Since $d(w)\geq 3$ there is a vertex $w'\neq u,v$ such that $(w,w')\in E(G)$ and $w'$ satisfies one of the conditions of cases 1,2 and 3 with respect to $F'_{L}$ and $F_{l}$. A contradiction. Thus $d(w)=2$.
Clearly, $(v,w)=e_L\in F_L$.
\(b) This follows from (a) of claim \[DegreeRequirements\] and corollary \[L=2l1-1case\]. $\square$
\[IntersectionEdges\] Let $F_{L},F_{l}$ be any maximum matchings of the graph $G$ with $\nu (G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$. Then for any maximum matching $H_{L}$ of the graph $G\backslash F_{L}$ there is no edge of $F_{L}\cap F_{l}$ which is adjacent to two edges from $H_{L}$.
Due to (c3) of theorem \[Ratios\] any edge from $H_{L}$ that is incident to a vertex covered by an edge of $F_{L}\cap F_{l}$ is also incident to a vertex from $V(F_{L})\backslash
V(F_{l})$. If there were an edge $e\in F_{L}\cap F_{l}$ which is adjacent to two edges $h_{L},h_{L}^{\prime }\in H_{L}$, then due to (c1) of theorem [Ratios]{} and (a) of claim \[DegreeRequirements\] we would have an augmenting $7$-path with respect to $F_{L}$, which would contradict the maximality of $F_{L}$. $\square$
\[ChoiceClaim\]
1. for any maximum matchings $F_{L},F_{l}$ of the graph $G$ with $\nu
(G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$, we have $(V(F_L)\backslash V(F_l))\cap V_1(G)=\emptyset$;
2. there is a maximum matching $F_{l}$ of $G$ with $\nu (G\backslash F_{l})=l(G)$ and a maximum matching $F_{L}$ of the graph $G$ with $\nu(G\backslash F_{L})=L(G),$ such that $V_1(G)\subseteq V(F_L\cap F_l)\cup (V(F_l) \backslash V(F_L))$.
\(1) On the opposite assumption, consider a vertex $x\in (V(F_L)\backslash V(F_l))\cap V_1(G)$. Since $x\in V_1(G)$ then $d(x)\leq 2$. On the other hand, (b) of claim \[DegreeRequirements\] implies that $d(x)\geq 2$, thus $d(x)=2$. Then there are vertices $y,z$ such that $(x,z)\in F_L$, $(z,y)\in F_l$. Note that due to (a) of claim \[DegreeRequirements\], we have $d(y)\leq 2$. Let us show that $d(y)=1$. Suppose that $d(y)=2$. Then due to (a) of claim \[DegreeRequirements\], we have that $d(z)=2$, thus $G$ is the triangle, which is a contradiction, since $G$ does not satisfy $L(G)=2l(G)$.
Thus $d(y)=1$. Since $x\in V_1(G)$, we imply that there is a vertex $w$ with $d(w)=2$ such that $w,x,z$ form a triangle. Note that $w$ is covered neither by $F_L$ nor by $F_l$, which contradicts claim \[SpanningSubgraph\].
\(2) Let $e_t$ be an edge of a triangle $t\in T$ connecting the vertex $v_t\in V_1(G)$ to a vertex of degree two. Let us show that there is a maximum matching $F_{l}$ of $G$ with $\nu (G\backslash F_{l})=l(G)$ such that $e_t\in F_l$ for each $t\in T$.
Choose a maximum matching $F_{l}$ of $G$ with $\nu (G\backslash F_{l})=l(G)$ that contains as many edges $e_t$ as possible. Let us show that $F_l$ contains all edges $e_t$. Suppose that there is $t_0\in T$ such that $e_{t_0}\notin F_l$. Define: $$F'_{l}=(F_{l}\backslash \{e\})\cup \{e_{t_0}\},$$where $e$ is the edge of $F_l$ that is adjacent to $e_{t_0}$. Note that $$\nu(G\backslash F'_l)\leq \nu(G\backslash F_l)=l(G),$$thus $F'_{l}$ is a maximum matching of $G$ with $\nu (G\backslash F_{l})=l(G)$. Note that $F'_{l}$ contains more edges $e_t$ than does $F_l$ which contradicts the choice of $F_{l}$.
Thus, there is a maximum matching $F_{l}$ of $G$ with $\nu (G\backslash F_{l})=l(G)$ such that $e_t\in F_l$ for all $t\in T$. Now, for this maximum matching $F_{l}$ of $G$ choose a maximum matching $F_{L}$ of the graph $G$ with $\nu(G\backslash F_{L})=L(G),$ such that $V(F_L\cap F_l)\cup (V(F_l) \backslash V(F_L))$ covers maximum number of vertices from $V_1(G)$. Let us show that $V_1(G)\subseteq V(F_L\cap F_l)\cup (V(F_l) \backslash V(F_L))$.
Suppose that there is a vertex $x\in V_1(G)$ such that $x\notin V(F_L\cap F_l)\cup (V(F_l) \backslash V(F_L))$. Note that due to claim \[SpanningSubgraph\] and (b) of claim \[DegreeRequirements\], any vertex of degree one is either incident to an edge from $F_L\cap F_l$ or to an edge $V(F_l) \backslash V(F_L)$. Thus due to definition of $V_1(G)$, $d(x)=2$ and if $y$ and $z$ denote the two neighbors of $x$, then $d(y)=2$ and $(y,z)\in E(G)$.
Since $x\notin V(F_L\cap F_l)$, we have that $(x,y)\notin F_L$, and since $x\notin (V(F_l) \backslash V(F_L))$, we have that $(y,z)\notin F_L$, thus $(x,z)\in F_L$, as $F_L$ is a maximum matching. Let $H_L$ be any maximum matching of $G\backslash F_L$. As $L(G)=2l(G)$, we have $(x,y)\in H_L$ ((c1) of theorem \[Ratios\]). Define: $$F'_{L}=(F_{L}\backslash \{(x,z)\})\cup \{(y,z)\}.$$Note that $F'_{L}$ is a maximum matching of $G$, $H_L$ is a matching of $G\backslash F_L$, thus $$\nu(G\backslash F'_L)\geq |H_L|=\nu(G\backslash F_L)=L(G).$$Therefore $F'_{L}$ is a maximum matching of $G$ with $\nu(G\backslash F'_L)=L(G)$. Now, observe that $V(F'_L\cap F_l)\cup (V(F_l) \backslash V(F'_L))$ covers more vertices than does $V(F_L\cap F_l)\cup (V(F_l) \backslash V(F_L))$ which contradicts the choice of $F_L$. The proof of the claim \[ChoiceClaim\] is completed. $\square$
\[IndependenceClaim\] For any maximum matchings $F_{L},F_{l}$ of the graph $G$ with $\nu(G\backslash F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$, we have
1. $V(F_L)\backslash V(F_l)$ is an independent set;
2. no edge of $G$ connects two vertices that are covered by both $F_L\backslash F_l$ and $F_l\backslash F_L$;
3. no edge of $G$ is adjacent to two different edges from $F_L\cap F_l$;
4. no edge of $G$ connects a vertex covered by $F_L\cap F_l$ to a vertex covered by both $F_L\backslash F_l$ and $F_l\backslash F_L$;
5. if $(u,v)\in F_L\cap F_l$ then either $u\in V_1(G)$ or $v\in V_1(G)$.
(1)There is no edge of $G$ connecting two vertices from $V(F_{L})\backslash V(F_{l})$ since $F_{l}$ is a maximum matching.
\(2) follows from (c1) and (c2) of theorem \[Ratios\].
\(3) follows from (c3) of theorem \[Ratios\].
\(4) Suppose that there is an edge $e=(y_{1},y_{2})$, such that $y_1$ is covered by $F_L\cap F_l$ and $y_2$ is covered by both $F_L\backslash F_l$ and $F_l\backslash F_L$. Consider a maximum matching $H_{L}$ of the graph $G\backslash F_{L}.$ Note that $y_{1}$ must be incident to an edge from $H_{L}$, as otherwise we could replace the edge of $H_{L}$ that is adjacent to $e$ and belongs also to $F_{l}\backslash F_{L}$ ((c1) of theorem \[Ratios\]) by the edge $e$ to obtain a new maximum matching $H_{L}^{\prime }$ of the graph $G\backslash F_{L}$ which would not satisfy (c1) of theorem \[Ratios\].
So let $y_{1}$ be incident to an edge $h_{L}\in H_{L}$, which connects $y_{1}$ with a vertex $x\in V(F_{L})\backslash V(F_{l})$. Note that due to claim \[IntersectionEdges\], (c1) of theorem \[Ratios\] and (a) of claim \[DegreeRequirements\], the edge $h_{L}$ lies on an $H_{L}-F_{L}$ alternating $4$-path $P$. Define: $$\begin{aligned}
F_{L}^{\prime } &=&(F_{L}\backslash E(P))\cup (H_{L}\cap E(P)), \\
H_{L}^{\prime } &=&(H_{L}\backslash E(P))\cup (F_{L}\cap E(P)).\end{aligned}$$Note that $F_{L}^{\prime }$ is a maximum matching of $G$, $H_{L}^{\prime }$ is a matching of $G\backslash F_{L}^{\prime }$ of cardinality $\left\vert
H_{L}\right\vert $, and $$\nu (G\backslash F_{L}^{\prime })\geq \left\vert H_{L}^{\prime }\right\vert
=\left\vert H_{L}\right\vert =\nu (G\backslash F_{L})=L(G),$$ thus $H_{L}^{\prime }$ is a maximum matching of $G\backslash F_{L}^{\prime }$ and $\nu (G\backslash F_{L}^{\prime })=L(G)$. This is a contradiction since the edge $e$ connects two vertices which are covered by $F_{L}^{\prime
}\backslash F_{l}$ and $F_{l}\backslash F_{L}^{\prime }$ ((2) of claim \[IndependenceClaim\]).
(5)Suppose that $e=(u,v)\in F_{L}\cap F_{l}$. Since $G$ is connected and $\left\vert V\right\vert \geq 3$, we, without loss of generality, may assume that $d(v)\geq 2$, and there is $w\in V(G),w\neq u$ such that $(w,v)\in E(G)$. Consider a maximum matching $H_{L}$ of the graph $G\backslash F_{L}.$ Note that $v$ must be incident to an edge from $H_{L}$, as otherwise we could replace the edge of $H_{L}$ that is incident to $w$ ($H_{L}$ is a maximum matching of $G\backslash F_{L}$) by the edge $(w,v)$ to obtain a new maximum matching $H_{L}^{\prime }$ of the graph $G\backslash
F_{L}$ such that $v$ is incident to an edge from $H_{L}^{\prime }$.
So we may assume that there is an edge $(v,q)\in H_{L}$, $q\neq u$. Note that due to claim \[IntersectionEdges\], (c1) of theorem [Ratios]{} and (a) of claim \[DegreeRequirements\] the edge $(q,w)$ lies on an $H_{L}-F_{L}$ alternating $4$-path $P$. Define: $$\begin{aligned}
F_{L}^{\prime } &=&(F_{L}\backslash E(P))\cup (H_{L}\cap E(P)), \\
H_{L}^{\prime } &=&(H_{L}\backslash E(P))\cup (F_{L}\cap E(P)).\end{aligned}$$Note that $F_{L}^{\prime }$ is a maximum matching of $G$, $H_{L}^{\prime }$ is a matching of $G\backslash F_{L}^{\prime }$ of cardinality $\left\vert
H_{L}\right\vert $, and $$\nu (G\backslash F_{L}^{\prime })\geq \left\vert H_{L}^{\prime }\right\vert
=\left\vert H_{L}\right\vert =\nu (G\backslash F_{L})=L(G),$$thus $H_{L}^{\prime }$ is a maximum matching of $G\backslash F_{L}^{\prime }$ and $\nu (G\backslash F_{L}^{\prime })=L(G)$. Since $u\in V(F_{l})\backslash
V(F_{L}^{\prime })$ (a) of claim \[DegreeRequirements\] implies that either $d(u)=1$ and therefore $u\in V_1(G)$, or $d(u)=d(v)=2$ and therefore either $u\in V_1(G)$ or $v\in V_1(G)$. Proof of the claim \[IndependenceClaim\] is completed. $\square$
We are ready to complete the proof of the theorem. Take any maximum matchings $F_{L},F_{l}$ of the graph $G$ guaranteed by the (2) of claim \[ChoiceClaim\] and consider the following partition of $V(G\backslash V_1(G))=V(G)\backslash V_1(G)$: $$\begin{aligned}
X =X(F_{L},F_{l})=V(F_{L})\backslash V(F_{l}), Y =Y(F_{L},F_{l})=V(G)\backslash (V_1(G)\cup X).\end{aligned}$$
Claim \[IndependenceClaim\] implies that $X$ and $Y$ are independent sets of vertices of $G\backslash V_1(G)$, thus $G\backslash V_1(G)$ is a bipartite graph with a bipartition $(X,Y)$.
The choice of maximum matchings $F_{L},F_{l}$, (a) of claim \[DegreeRequirements\], (5) of claim \[IndependenceClaim\] and the definition of the set $Y$ imply (2) of the theorem \[Characterization\].
Let us show that it satisfies (3), too.
Consider the alternating $2$-paths of $$(H_{L}\backslash F_{l})\triangle (F_{L}\backslash F_{l}).$$(c2), (c3) of theorem \[Ratios\] and the definition of the set $X$ imply that there are $|X|$ such $2$-paths. Moreover, these $2$-paths are in fact $2$-paths of the graph $G\backslash V_{1}(G) $. Thus $G$ satisfies (3) of the theorem. The proof of the theorem \[Characterization\] is completed. $\square$
\[PolynomialAlgorithm\] The property of a graph $L(G)=2l(G)$ can be tested in polynomial time.
First of all note that the property $L(G)=2l(G)$ is additive, that is, a graph satisfies this property if and only if all its connected components do. Thus we can concentrate only on connected graphs.
All connected graphs with $|V(G)|\leq 2$ satisfy the equality $L(G)=2l(G)$, thus we can assume that $|V(G)|\geq 3$.
Next, we construct a set $V_1(G)$, which can be done in linear time. Now, we need to check whether the graph $G\backslash V_1(G)$ satisfies the conditions (1)-(3) of the theorem \[Characterization\].
It is well-known that the properties (1) and (2) can be checked in polynomial time, so we will consider only the testing of (3).
From a graph $G\backslash V_{1}(G)$ with a bipartition $(X,Y)$ we construct a network $\vec{G}$ with new vertices $s$ and $t$. The arcs of $\vec{G}$ are defined as follows:
- connect $s$ to every vertex of $X$ with an arc of capacity $2$;
- connect every vertex of $Y$ to $t$ by an arc of capacity $1$;
- for every edge $(x,y)\in E(G)$, $x\in X$, $y\in Y$ add an arc connecting the vertex $x$ to the vertex $y$ which has capacity $1$.
Note that
- the value of the maximum $s-t$ flow in $\vec{G}$ is no more than $2\left\vert X\right\vert $ (the capacity of the cut $(S,\bar{S})$, where $S=\{s\}$, $\bar{S}=V(\vec{G})\backslash S$, is $2\left\vert
X\right\vert $);
- the value of the maximum $s-t$ flow in $\vec{G}$ is $2\left\vert
X\right\vert $ if and only if the graph $G\backslash V_{1}(G)$ contains $\left\vert X\right\vert $ vertex disjoint $2$-paths,
thus (3) also can be tested in polynomial time. $\square$
Recently Monnot and Toulouse in [@Monnot] proved that $2$-path partition problem remains $NP$-complete even for bipartite graphs of maximum degree three. Fortunately, in theorem \[Characterization\] we are dealing with a special case of this problem which enables us to present a polynomial algorithm in corollary \[PolynomialAlgorithm\].
$NP$-completeness of testing $L(G)=\frac{3}{2}l(G)$ in the class of bridgeless cubic graphs
===========================================================================================
The reader may think that a result analogous to corollary \[PolynomialAlgorithm\] can be proved for the property $L(G)=\frac{3}{2}l(G)$ in the class of graphs containing a perfect matching. Unfortunately this fails already in the class of bridgeless cubic graphs, which by the well-known theorem of Petersen are known to possess a perfect matching.
It is $NP$-complete to test the property $L(G)=\frac{3}{2}l(G)$ in the class of bridgeless cubic graphs.
Clearly, the problem of testing the property $L(G)=\frac{3}{2}l(G)$ for graphs containing a perfect matching is in $NP$, since if we are given perfect matchings $F_{L},F_{l}$ of the graph $G$ with $\nu (G\backslash
F_{L})=L(G),$ $\nu (G\backslash F_{l})=l(G)$ then we can calculate $L(G)$ and $l(G)$ in polynomial time.
We will use the well-known $3$-edge-coloring problem ([@Holyer]) to establish the NP-completeness of our problem.
Let $G$ be a bridgeless cubic graph. Consider a bridgeless cubic graph $G_{\bigtriangleup }$ obtained from $G$ by replacing every vertex of $G$ by a triangle. We claim that $G$ is $3$-edge-colorable if and only if $L(G_{\bigtriangleup })=\frac{3}{2}l(G_{\bigtriangleup })$.
Suppose that $G$ is $3$-edge-colorable. Then $G_{\bigtriangleup }$ is also $3 $-edge-colorable, which means that $G_{\bigtriangleup }$ contains two edge disjoint perfect matchings $F$ and $F^{\prime }$. This implies that $$L(G_{\bigtriangleup })\geq \nu (G_{\bigtriangleup }\backslash F)\geq
\left\vert F^{\prime }\right\vert =\frac{\left\vert V(G_{\bigtriangleup
})\right\vert }{2},$$On the other hand, the set $E(G)$ forms a perfect matching of $G_{\bigtriangleup }$, and $$l(G_{\bigtriangleup })\leq \nu (G_{\bigtriangleup }\backslash E(G))=\frac{\left\vert V(G_{\bigtriangleup })\right\vert }{3},$$since every component of $G_{\bigtriangleup }\backslash E(G)$ is a triangle. Thus: $$\frac{L(G_{\bigtriangleup })}{l(G_{\bigtriangleup })}\geq \frac{3}{2},$$(d) of theorem \[Ratios\] implies that $\frac{L(G_{\bigtriangleup })}{l(G_{\bigtriangleup })}=\frac{3}{2}$.
Now assume that $\frac{L(G_{\bigtriangleup })}{l(G_{\bigtriangleup })}=\frac{3}{2}$. Note that for every perfect matching $F$ of the graph $G_{\bigtriangleup }$ the graph $G_{\bigtriangleup }\backslash F$ is a 2-factor, therefore $$\begin{aligned}
L(G_{\bigtriangleup }) &=&\frac{\left\vert V(G_{\bigtriangleup })\right\vert
-w(G_{\bigtriangleup })}{2}, \\
l(G_{\bigtriangleup }) &=&\frac{\left\vert V(G_{\bigtriangleup })\right\vert
-W(G_{\bigtriangleup })}{2}\end{aligned}$$where $w(G_{\bigtriangleup })$ and $W(G_{\bigtriangleup })$ denote the minimum and maximum number of odd cycles in a $2$-factor of $G_{\bigtriangleup }$, respectively. Since $\frac{L(G_{\bigtriangleup })}{l(G_{\bigtriangleup })}=\frac{3}{2}$ we have $$W(G_{\bigtriangleup })=\frac{\left\vert V(G_{\bigtriangleup })\right\vert
+2w(G_{\bigtriangleup })}{3}.$$Taking into account that $W(G_{\bigtriangleup })\leq \frac{\left\vert
V(G_{\bigtriangleup })\right\vert }{3}$, we have: $$\begin{aligned}
W(G_{\bigtriangleup }) &=&\frac{\left\vert V(G_{\bigtriangleup })\right\vert
}{3}, \\
w(G_{\bigtriangleup }) &=&0.\end{aligned}$$Note that $w(G_{\bigtriangleup })=0$ means that $G_{\bigtriangleup }$ is $3$-edge-colorable, which in its turn implies that $G$ is $3$-edge-colorable. The proof of the theorem is completed. $\square$
[99]{}
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[^1]: email: arturkhojabaghyan@gmail.com
[^2]: email: vahanmkrtchyan2002@{ysu.am, ipia.sci.am, yahoo.com}
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abstract: 'The parametric variation of the eigenfrequencies of a chaotic plate is measured and compared to random matrix theory using recently calculated universal correlation functions. The sensitivity of the flexural modes of the plate to pressure is used to isolate this symmetry class of modes and simplify the data analysis. The size of the plate is used as the external parameter and the eigenvalues are observed to undergo one or two oscillations in the experimental window. The correlations of the eigenvalues are in good agreement with statistical measures such as the parametric number variance, the velocity autocorrelation, and the intralevel velocity autocorrelation derived for the Gaussian Orthogonal Ensemble of random matrix theory. Our results show that the theory can be also applied to wave systems other than quantum systems.'
address: |
$^1$ Department of Physics, Clark University, Worcester, MA 01610, U.S.A.\
$^2$ Center for Chaos and Turbulence Studies, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark.\
author:
- 'K. Schaadt$^{1,2}$ and A. Kudrolli$^1$'
title: Experimental investigation of universal parametric correlators using a vibrating plate
---
It has been widely recognized that the eigenvalues of a quantum system show universal features that depend only on the presence or absence of chaos in the corresponding classical or ray system [@Casati:1995; @Mehta:1990]. For example, it has been established that the eigenvalues of integrable systems display Poisson statistics, and chaotic systems with time-reversal symmetry show statistics which are similar to the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT) [@Bohigas:1984]. The universality has been confirmed using not only quantum systems, but also systems which obey an elastomechanical wave equation [@Weaver:1989; @Ellegaard:1995; @Ellegaard:1996]. The difference in the statistical properties has been recognized to be due to the presence of level repulsion which were demonstrated as avoided crossings as a system parameter was varied. However, it was postulated only recently that the resulting fluctuation of the energy levels also show universal properties which are independent of the nature of the parameter [@Simons:1993a; @Simons:1993b].
When a quantum system is subjected to a perturbation via an external parameter $X$, the eigenvalues change and oscillate as a function of $X$. Using supersymmetry techniques, Simons and Altshuler [@Simons:1993a; @Simons:1993b] were able to calculate the correlations as a function of external parameter for energy levels with Wigner-Dyson distributions of RMT. The agreement of their analytical results with numerical simulations of disordered metallic rings and a chaotic billiard led them to the remarkable conjecture that correlations in the eigenvalues show universal features which are [*independent*]{} of the nature of the perturbation after appropriate normalization. Here the proper rescaling required to compare across different systems is given by expressing the energy $E$ in units of the local mean level spacing $\Delta$, and the parameter in units of the square root of the local mean squared slope: $$\varepsilon = E/\Delta \;\;\;\;\;\;
x = \sqrt{\left< \left({d\varepsilon \over dX}\right)^2 \right>} X \,
\, ,
\label{x_def}$$ where $\varepsilon$ is the normalized energy, and $x$ is the rescaled external parameter.
The conjecture was tested further with numerical simulations of a Hydrogen atom in a magnetic field, where agreement was found over a certain parameter range, but systematic deviations were also found because the system is only partially chaotic [@Simons:1993c]. Although some of the correlations have been indirectly tested in the conductance fluctuations of electrons in ballistic cavities [@Sivan:1994; @Huibers:1998], and also in microwave cavities [@Barth:1998] and quartz blocks [@Ellegaard:1998] where bouncing ball-like modes complicates the analysis, there has been no report of a direct experimental test of their universality.
In this paper, we report direct experimental evidence for the universality of the above mentioned parametric correlators. A freely vibrating plate with the shape of a Sinai-Stadium [@Kudrolli:1994] is used and the smooth motion of the eigenfrequencies is measured as a function of the size of the plate. Two classes of uncoupled modes exist in an isotropic plate: [*flexural*]{}, for which the displacement is perpendicular to the plane of the plate, and [*in-plane*]{}, for which the displacement is in the plane of the plate [@Graff:1975; @Hugues:1997]. We are able to experimentally isolate the flexural modes and therefore can simplify the analysis by not having to consider problems associated with mixed symmetries. The flexural modes obey a scalar equation for the displacement $W$ perpendicular to the plate: $$(\nabla^2 - k^2)(\nabla^2 + k^2)W = 0 \;\;\;,
\label{flex_eq}$$ where $k$ denotes the wavenumber. The dispersion relation is given by $$f = {k^2 \over 2\pi} \sqrt{E_Y h^3 \over 12\rho(1-\nu^2)}\;\;\;,
\label{flex_disp_rel}$$ where $f$ is the frequency, $h$ is the thickness of the plate, $\rho$ is the density, $E_Y$ is Young’s modulus, and $\nu$ is Poisson’s ratio. Any solution $W$ of Eq. (\[flex\_eq\]) can be written as a superposition of two modes, $W_1$ and $W_2$, where $$(\nabla^2 + k^2)W_1 = 0 \;\;\;\;\, \mbox{and} \;\;\;\;
(\nabla^2 - k^2)W_2 = 0 \;\;.$$ $W_1$ is a solution to the Helmholtz equation with free boundary conditions. $W_2$ is an [*exponential*]{} mode or [*boundary*]{} mode. The boundary modes are responsible for only about one percent of the density of states [@Bogomolny:1998; @Hugues:1997] and do not appear to alter the universality of the eigenvalues. Equation (\[flex\_eq\]) is an approximation to the full elastomechanical wave equation in the limit where the wavelength is much larger than the thickness of the plate. The typical wavelength in our experiments is 8 mm, and the thickness of the plate is 2 mm. In this case, Eq. (\[flex\_eq\]) is a good approximation.
The eigenvalue statistics is first confirmed to agree very well with GOE statistics using traditional measures such as the spacing statistics $P(s)$ and the spectral rigidity $\Delta_{3}(L)$, demonstrating the quality of the data and the universality of the geometry. We then compare statistical observables of the eigenvalue motion as a function of the parameter to analytical calculations. In particular, we find that the data agrees with calculations of the parametric number variance $v(x)$ by Simons et al. [@Simons:1993c] and shows a linear behavior for small $x$ which is different from semiclassical calculations [@Goldberg:1991]. To investigate correlations in energy-parameter space, comparisons are made with the exact calculations for the intralevel velocity autocorrelation $\tilde
c(w, x)$ which describes the correlations between the rate of change of eigenvalues separated in energy by $w$ and in parameter by $x$ [@Simons:1993b]. Good agreement is observed for selected values of $w$ and over all $x$. Another statistical measure is the velocity autocorrelation $c(x)$ which correlates the rate of change of eigenvalue as a function of parametric separation $x$. For this quantity we find that the data is in good agreement at small and intermediate values of the parameter. Deviations are observed for larger $x$ where statistical sampling is poor. Combined, these results provide the first experimental evidence for the universality of a broad class of the statistical observables of parametric level motion that have been studied theoretically.
In the experiments we use an Aluminium plate of thickness 2.0 mm, machined in the shape of a quarter Sinai-Stadium with radii 40 mm and 70 mm (see Fig. \[plate\]). The plate rests on three piezoelectric transducers, of which one is a transmitter and two are receivers. We measure acoustic transmission spectra of the plate using a HP 4395A network analyzer. A sample of the transmission signal at different values of the parameter is shown in Fig. \[plate\]. The amplitude of the resonances depend on the location of the transducers but the eigenfrequencies are unchanged. The plate is kept in a temperature controlled oven held at 300 K to within 1 mK. A vacuum system ensures that the air pressure is below $10^{-1}$ Torr, which is low enough that air damping of the plate is insignificant compared to other damping mechanisms. Of the two classes of modes, flexural modes are more sensitive to the presence of air damping than in-plane modes because of the flexural out-of-plane oscillation. We find that going from vacuum to atmospheric pressure, the $Q$-factor of the flexural modes decreases by at least a factor of 3, whereas the $Q$-factor for the in-plane modes is unchanged.
We first measure the transmission spectrum of the plate, then decrease the size of the plate by sanding off material at the longest straight edge, as indicated in Fig. \[plate\]. The amount of material removed is determined by measuring the mass of the plate to within $5\times 10^{-5}$ grams. Approximately $5\times 10^{-2}$ grams is removed each time and in all 6% of the material is removed in 63 steps. The spectrum is measured in the interval between 100 kHz and 300 kHz. Periodically, the spectrum is also measured at 1 atmosphere to identify the flexural modes. After this separation, we find approximately 300 resonances, of which 25 drift out of the frequency window due to the overall increase in frequency when the size of the plate is decreased. A resonance frequency can be determined to within 0.5 Hz by fitting the resonance peak to a Breit-Wigner function. We are confident that all eigenfrequencies in the frequency window are detected, because it is impossible for the amplitude of a resonance peak to lie below our detection level for all 63 values of the parameter. The absence of interaction of the flexural modes with the in-plane modes is checked to within experimental accuracy by noting a lack of interaction at flexural-in-plane encounters (see Fig. \[plate\]).
In the data analysis, the implementation of the normalization or [*unfolding*]{} given by Eq. (\[x\_def\]) is of great importance. Since the cumulative level density or [*staircase function*]{} for a freely vibrating plate was recently calculated [@Bogomolny:1998], both the mean level spacing and the mean squared velocity are known analytically. This knowledge can be directly applied to our data, which makes the data analysis very clean from a theoretical viewpoint.
We start by presenting our result for the distribution of nearest neighbor spacings $P(s)$ and the spectral rigidity $\Delta_{3}(L)$ which are shown in Fig. \[spec\_stat\]. We find complete agreement with RMT for both observables. Fully chaotic systems are very rare and most chaotic geometries have regions in phase space which are integrable. The Sinai-Stadium geometry is no exception and is known to have small regions of integrability. However if these regions are very small, they can support an integrable level only at very high frequencies, and therefore complete agreement with GOE is expected and observed.
We now present the main results which is the correlations in the parametric variation of the eigenfrequencies. The parametric number variance $v(x)$ is defined as: $$v(x) = \langle (n(\varepsilon,x')-n(\varepsilon,x'+x))^2 \rangle \;\;,
\label{par_num_var_def}$$ where the average is over the parameter $x'$ and energy $\varepsilon$. Here, $n(\varepsilon,x)$ is the staircase function which counts the number of energy levels at fixed $x$ with energy lower than $\varepsilon$. The parameter $x$ has been normalized according to Eq. (\[x\_def\]), as explained above. The variance measures the difference in the number of eigenvalues which are below a fixed value of normalized energy $\varepsilon$. Therefore this quantity measures the collective motion of levels under parametric change [@Goldberg:1991]. The comparison of the data with the theory is shown in Fig. \[par\_num\_var\]. The $v(x)$ calculated from the data grows linearly from zero and has a slope of $0.8 \pm
0.01$ which is in excellent agreement with the calculated value of $\sqrt{2/\pi} \sim 0.797$ by Simons et al. [@Simons:1993c]. A saturation is expected at large values of $x$ and therefore the $v(x)$ becomes sub-linear at higher $x$.
However, $v(x)$ does not give an indication of the correlations in the oscillations of the eigenvalues with the parameter $x$. To investigate such correlations, a new set of measures are required that study the rate of change of eigenvalue as a function of parameter [@Simons:1993a]. One example is the [*intralevel velocity autocorrelation*]{} $\tilde
c(\omega,x)$, which correlates velocities which are separated by a distance $x$ in parameter space and by a distance $\omega$ in energy: $$\tilde c(\omega,x) = {{\sum_{n,m}\langle
\delta(\varepsilon_n(x')-\varepsilon_m(x'+x) - \omega)
{\partial \varepsilon_n(x') \over \partial x'}
{\partial \varepsilon_m(x'+x) \over \partial x'}
\rangle}
\over {\sum_{n,m}
\langle \delta(\varepsilon_n(x')-\varepsilon_m(x'+x) - \omega)\rangle}}$$ The average is over the parameter $x'$. Using the supersymmetric nonlinear $\sigma$ model developed by Efetov [@Efetov:1983], Simons and Altshuler derived an integral representation for the intralevel velocity autocorrelation. Another correlation is the [*velocity autocorrelation*]{} $c(x)$ which correlates velocities which belong to the same energy level: $$c(x) = \left < {\partial \varepsilon(x') \over \partial x'}
{\partial \varepsilon(x'+x) \over \partial x'}
\right >$$ The brackets denote an average over the parameter $x'$ and the energy $\varepsilon$. For this correlator no analytical results exist for intermediate values of $x$. Therefore we compare our result for $c(x)$ to a curve calculated by Mucciolo [@Mucciolo:1996] using large GOE matrices which agrees with the analytical results in the limit of large and small $x$.
We first present the result for the velocity autocorrelation $c(x)$ (see Fig. \[vel\_corr\]). For values of $x$ smaller than 1, we find good agreement with the numerical RMT curve [@Mucciolo:1996]. At larger values of $x$, however, we see a deviation which is outside the experimental error bars. The shape of the correlation function indicates that the slope ${\partial \varepsilon(x)/\partial x}$ changes smoothly and has opposite signs near $x = 0.5$ because the parameter $x$ has been normalized to correspond to approximately one oscillation for $x=1$. This behavior of the correlation functions indicates that, locally, there is a particular length scale over which eigenfrequencies oscillate. The distribution of velocities ${\partial
\varepsilon(x)/\partial x}$ of the eigenvalues should be a Gaussian with a mean value of zero. The data is shown in the inset to Fig. \[vel\_corr\]. The data is close to a Gaussian, but is slightly asymmetric with more velocities of small magnitude which are negative than positive. We emphasize that the mean slope is zero, indicating that this discrepancy does not originate in the normalization of the eigenfrequencies. We believe that the deviation is due to a finite data set. It appears that the correlations are very robust and give good agreement even if the velocity distribution is not exactly Gaussian.
To make a more stringent test of the correlations, we compare our data with the intralevel velocity autocorrelation $\tilde
c(\omega,x)$ for $\omega = 0.25$, $\omega =
0.50$, and $\omega = 1.0$ as shown in Fig. \[intralevel\_vel\_corr\]. We compare our data to a numerical evaluation of the integral representation of this correlator [@Simons:1993b]. In calculating these quantities we have averaged over a small energy window of $\delta \omega = 0.03$ which is also done in the theoretical calculations. The occurrence of the peaks in the correlation functions and the systematic increase of the value of $x$ where the peak occurs can be understood from the fact that near an avoided crossing, one has to go across by nearly as much along the normalized energy axis as along the parameter axis to encounter a similar slope (see Fig. \[plate\]). The comparison of the data in Fig. \[intralevel\_vel\_corr\] shows very good agreement for all three values of $\omega$, validating the theory.
In conclusion, we have investigated experimentally the parametric level motion of the flexural modes of a freely vibrating plate as a function of the size of the plate. We have used our data to calculate statistical quantities which probe the parametric motion of the levels, and found agreement with the universal predictions of RMT. The agreement with RMT suggests that the universal predictions for parametric level motion extends beyond quantum chaotic systems to a wider range of wave systems, including acoustical waves.
We thank B. Simons and E. Mucciolo for providing the theoretical data, and M. Oxborrow and J. Norton for technical assistance. We thank O. Brodier and H. Gould for useful discussions. This work was supported by the Danish National Research Council (K.S.), Research Corporation and by an Alfred P. Sloan Fellowship (A.K.). We thank Hewlett-Packard for a partial equipment grant.
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abstract: 'We show that if $M$ is a fibered, orientable 3-manifold, and if $\pi_1 M$ has a 1-relator presentation, then the presentation is induced by a Heegaard splitting of $M$. A corollary is that, for these manifolds, the rank of $\pi_1 M$ is equal to the (“restricted”) Heegaard genus of $M$. We also explore the analogy between 1-relator groups and Haken 3-manifolds, showing that every 1-relator group possesses a “1-relator hierarchy”.'
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[ ]{}
[Joseph D. Masters ]{}
Introduction
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Let $F$ be a closed, orientable surface. A *compression body* $W$ is a cobordism rel $\p$ between two surfaces $\p_+ W$ and $\p_-W$, such that $W \cong \p_+ W \times I \cup \textrm{ 2-handles} \cup \textrm{ 3-handles}$, and $\p_- W$ has no 2-sphere components. A *disk system* $\mathcal{D} = \{D_1, ..., D_n \}$ for $W$ is a set of compressing disks for $\p_+ W$, such that $W$ cut open along $\mathcal{D}$ is homeomorphic to $\p_- \times I$ if $\p_- \neq \emptyset$, or $B^3$ if $\p_- = \emptyset$.
Let $M$ be a compact, orientable 3-manifold. A *Heegaard splitting* of $M$ is a pair $(W, W^{\prime})$, where $W, W^{\prime}$ are compression bodies, such that $W \cup W^{\prime} = M$, and $W \cap W^{\prime} = \p_+ W = \p_+ W^{\prime}$. The surface $\p_+ W = F$ is called a *Heegaard surface* for $M$. The *genus* of the splitting $(W,W^{\prime})$ is the genus of $F$. The *Heegaard genus* of $M$, denoted $g(M)$, is the minimal genus among all Heegaard splittings of $M$.
In our setting, it will be convenient to restrict the class of Heegaard splittings. We shall define a *restricted Heegaard splitting* to be a Heegaard splitting $(W,W^{\prime})$, where $W$ is a handlebody. (This notion is closely related to the more standard notion of a “tunnel system”). The *restricted Heegaard genus*, denoted $g_0(M)$, is the minimal genus among all restricted Heegaard splittings. If $\p M$ has fewer than two components, then $g_0(M) = g(M)$.
Any restricted Heegaard splitting $(W, W^{\prime})$ determines a collection of finite presentations of $\pi_1 M$. Indeed, if we choose disk systems $\mathcal{D}$ and $\mathcal{E}$ for $W$ and $W^{\prime}$, respectively, we obtain a presentation for $\pi_1 M$, whose generators correspond to the elements of $\mathcal{D}$, and whose relators correspond to the attaching maps of the disks in $\mathcal{E}$. We say that the resulting presentation, $P$, is *induced by* $(W, W^{\prime})$. Note that different choices of $\mathcal{D}$ or $\mathcal{E}$ would lead to different presentations, all of which are said to be induced by $(W, W^{\prime})$. Any presentation obtained from $P$ by deleting trivial relators is also said to be induced from $(W, W^{\prime})$. We say that a presentation for a group is *geometric* if it is induced by a Heegaard splitting of some 3-manifold.
For example, if $M$ is the exterior of the trefoil knot, then $<x_1, x_2| x_1^2 x_2^{-3}>$ is induced by a genus 2 Heegaard splitting of $M$. Note that the 1-relator presentation $<x_1| id>$ is induced by Heegaard splittings of two different 3-manifolds; it is induced by the genus 1 Heegaard splitting of $S^2 \times S^1$, and also by the genus 1 Heegaard splitting of $D^2 \times S^1$, where, in the latter case, the 2-handle is attached along a trivial curve of the Heegaard torus.
Our main theorem is the following:
\[main\] If $M$ is a compact, orientable 3-manifold which fibers over $S^1$, then any 1-relator presentation for $\pi_1 M$ is induced by a Heegaard splitting of $M$.
If $G$ is any group, we let $rank(G)$ denote the minimal cardinality among all generating sets for $G$. We have the obvious inequality $rank \, \pi_1 M \leq g_0(M)$. There are examples where this inequality is sharp (see [@BZ]), however it is an open problem whether equality holds if $M$ is hyperbolic. [^1] Theorem \[main\] implies that equality does hold in the case where $M$ is fibered, and $G$ is a 1-relator group. This includes many hyperbolic examples.
Let $M$ be a compact, orientable 3-manifold which fibers over $S^1$, and suppose that $\pi_1 M$ has a 1-relator presentation. Then $rank(\pi_1 M) = g_0(M)$.
*Remark:* The hypotheses of the corollary force $rank(\pi_1 M) \leq 2$. See Theorem \[brown\].
Let $P = <x_1, .., x_n|w>$ be the given presentation for $\pi_1 M$. By Theorem \[main\], $P$ is induced by a Heegaard splitting of $M$.
First, suppose that $w = id$. There are now two possibilities. If $\p M = \emptyset$, then we must have $P = <x_1|id>$, and $M \cong S^2 \times S^1$. Thus $rank \, \pi_1 M = g_0(M) = 1$. If $\p M \neq \emptyset$, then $W$ is a solid torus, and the 2-handle is attached along a trivial curve in $\p W$. So in this case, $M \cong D^2 \times S^1$, and again $rank \, \pi_1 M = g_0(M) = 1$.
Suppose now that $w \neq id$. Recall that an element $w$ in a free group $F$ is *primitive* if it is contained in a free basis for $F$. If $w$ is non-primitive, then $\pi_1 M$ is non-free (see [@LS] Chapter II, Proposition 5.10), so $rank(\pi_1 M) = 2 = g_0(M)$. If $w$ is primitive, then $\pi_1 M \cong \mathbb{Z}$, and $M \cong D^2 \times S^1$ or $S^2 \times S^1$. In this case $rank(\pi_1 M) = g_0(M) = 1$.
Say that a word $w$ in a free group $F(n)$ is *geometric* if it can be represented by a simple closed curve on the boundary of a genus $n$ handlebody.
\[word\] If $w$ is a non-geometric word in $x_1,x_2$, and if $<x_1, x_2|w>$ has a map onto $\mathbb{Z}$ with finitely generated kernel, then $<x_1, x_2|w>$ is not the fundamental group of an orientable 3-manifold.
*Remark:* For any given word, these two conditions are straightforward to verify. The algorithm to determine if a word in $F(n)$ is geometric is given in [@Z], and an implementation, in the 2-generator case, is currently available at www.math.buffalo.edu/ jdmaster. The procedure to determine if $ker(\phi)$ is finitely generated is discussed in Section \[back\]. For example, Corollary \[word\] implies that the group $<x_1, x_2| x_1 x_2 x_1 x_2 x_1^{-1} x_2^{-2}>$ is not the fundamental group of an orientable 3-manifold.
(of Corollary \[word\]) Suppose the kernel is finitely generated, and $<x_1, x_2|w> = \pi_1 M$ for an orientable 3-manifold $M$. By Scott’s compact core theorem ([@Sc]), we may assume that $M$ is compact. By Stallings’ Theorem ([@S]), $M$ is fibered. Thus, by Theorem \[main\], the given 1-relator presentation for $\pi_1 M$ is geometric. Thus the word $w$ is geometric.
We expect that a version of Theorem \[main\] remains true without the hypothesis that $M$ is fibered.
\[cnj\] Let $M$ be a compact, orientable 3-manifold. Then\
A. any 1-relator presentation of $\pi_1 M$ is geometric; and\
B. if $M$ has no 2-sphere boundary components, then any 1-relator presentation of $\pi_1 M$ is induced by a Heegaard splitting of $M$.
Conjecture \[cnj\] A is equivalent to:
\[cnj2\] The group $<x_1, ..., x_n|w>$ is the fundamental group of an orientable 3-manifold if and only if $w$ is a geometric word.
Conjecture \[cnj\] B implies that rank equals restricted Heegaard genus for orientable 3-manifolds with 1-relator fundamental groups. In particular, Conjecture \[cnj\] B implies the Poincaré Conjecture.
Evidence for Conjecture \[cnj\] A is given by the manifolds in the SnapPea census. There are 4303 examples given there of 1-relator presentations of orientable-3-manifold groups. We have used a computer to verify that all of these presentations are in fact geometric.
Section \[back\] contains some background material. The proof of Theorem \[main\] is contained in Section \[proof\]. Section \[groups\] contains a general theorem about 1-relator groups, which is motivated by Conjecture \[cnj\]. The final section contains some general discussion about 1-relator groups and Conjecture \[cnj\].
We wish to thank Tao Li for a useful conversation, and Ilya Kapovich for help with the references.
Background {#back}
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**Brown’s criterion**\
Given a 1-relator group $G = <x_1, ..., x_n|w>$, where $w$ is cyclically reduced, and a surjection $\phi: G \ra \mathbb{Z}$. We shall describe a criterion, due to Brown, for determining if $ker(\phi)$ is finitely generated.
Suppose that the exponent sum on $x_1$ in $w$ is zero. This can always be arranged, by applying an appropriate free group automorphism to $w$ (see Lemma \[exp\]). If $n \geq 3$, then it is easy to see that $ker(\phi)$ is infinitely generated. If $n = 2$, then $ker(\phi)$ is generated by $\{ y_{i,j} = x_1^i x_j x_1^{-i} \}$, and we have $w = \Pi y_{\l_j,\m_j}$, for some finite sequence of integers $(\l_j)$. The following is proved in [@B].
\[brown\](Brown) $Ker(\phi)$ is finitely generated if and only if $n=2$, and $(\l_j)$ has a unique minimum and maximum.
This result follows from Theorem 4.2 (and the discussion imediately preceding it) in [@B]. The proof is an application of the Freiheitsatz (Theorem \[frei\] below).\
\
*Remark:* Theorem \[brown\] provides another method of showing that certain 1-relator groups are not 3-manifold groups. Suppose $G$ is as above, and that the sequence $(\l_j)$ has
i\. a unique maximum (resp. minimum) and
ii\. a repeated minimum (resp. maximum).\
Then $G$ is not the fundamental group of a compact, orientable 3-manifold.
Indeed, if $G = \pi_1 M$ for a compact, orientable 3-manifold $M$, then the cover $\widetilde{M}$ corresponding to $ker(\phi)$ has two ends. Assumption i. implies that one of the ends has a finitely generated group. Thus by an argument of Stallings (proof of Theorem 2 of [@S]), $M$ is fibered, and so $ker(\phi)$ is finitely generated. But, by assumption ii and Proposition \[brown\], $Ker(\phi)$ is not finitely generated, for a contradiction.
For example, this argument shows that a Baumslag-Solitar group of the type $<x_1, x_2|x_2 x_1 x_2^p x_1^{-1}>$ is not the fundamental group of a compact, orientable 3-manifold if $|p| >1$– a special case of a well-known result of Jaco [@J].\
\
**Magnus’s Freiheitsatz**\
One of the fundamental results about 1-relator groups is the following:
\[frei\] (Magnus’s Freiheitsatz) Let $G = <x_1, ..., x_n|w>$, where $w$ is cyclically reduced, and let $X \subset \{x_1, ..., x_n\}$. Suppose that $w$ involves some $x_i$ which is not in $X$. Then $X$ freely generates a free subgroup of $G$.
For a proof, see, for example, [@LS].\
\
**Uniqueness of the relator**\
The following result is Proposition II.5.8 in [@LS]:
(Magnus) \[uniqueword\] If two elements $w_1$ and $w_2$ have the same normal closure in a free group $F$, then $w_1$ is conjugate to $w_2$ or $w_2^{-1}$.
In particular, if one fixes a generating set for a 1-relator group, then the relator is determined, up to conjugacy and inverse.\
\
**The Moldavansky Splitting**\
Suppose that $G = <x_1, ..., x_n| w>$, where the exponent sum of $x_1$ on $w$ is zero. We shall now explain a method, due to Moldavansky, of representing $G$ as an HNN-extension, with a 1-relator vertex group. Further details of the construction are given in [@MS].
Let us represent $G$ as the fundamental group of a 2-complex $K$, consisting of a bouquet of $n$ circles, with a single 2-handle attached, according to the word $w$. Let $\phi: G \ra \mathbb{Z}$ be the homomorphism which sends any word to the exponent sum of $x_1$. Then $ker(\phi)$ corresponds to an infinite cyclic cover $\widetilde{K} \ra K$, as pictured in Fig. 1, and $\pi_1 {\wt K} = Ker(\phi)$ is generated by elements of the form $y_{i,j} = x_1^i x_j x_1^{-i}$, for $i \in \mathbb{Z}$ and $2 \leq j \leq n$. Let $\th: \wt K \ra \wt K$ be the covering transformation corresponding to the element $x_1$.
Let $\ell:S^1 \ra K$ be the attaching map of the 2-handle, and let ${\wt \ell}:S^1 \ra {\wt K}$ be a lift. Then $\wt \ell$ represents some word $\wt w$, in the generators $y_{i,j}$. We may choose the lift so that the $i$-coordinates of the $y_{i,j}$’s involved in $\wt w$ range from $0$ to $m$.
Consider the connected sub-complex $\overline{K}$ of $\wt K$ which contains the lifted 2-handle, and the 1-cells corresponding to $\{y_{i,j}: 0 \leq i \leq m, 2 \leq j \leq n\}$. Then $K = \overline{K}/(x \equiv \th x)$, and this gives $G$ the structure of an HNN-extension (see Fig. 1).
Proceeding algebraically now, consider the presentation: $$(*) \,\,\,\, <t, y_{i,j} \,\,\,\, 0 \leq i \leq m, \, 2 \leq j \leq n|
\wt w, t y_{i,j} t^{-1} = y_{i+1,j} \, \, \,\,
0 \leq i \leq m-1, 2 \leq j \leq n>.$$ Then, by mapping $t$ to $x_1$ and $y_{i,j}$ to $x_1^i x_j x_1^{-i}$, one may check that this is a presentation of $G$.
Now, let $G_v$ be the following 1-relator group: $$G_v = <y_{i,j} \,\,\,\, 0 \leq i \leq m, \, 2 \leq j \leq n
|{\wt w}>,$$ and let $G_e$ be a free group: $$\begin{aligned}
G_e = <y_{i,j} \,\,\,\, 0 \leq i \leq m-1, \,\, 2 \leq j \leq n>.\end{aligned}$$ (We have abused notation by re-using the symbol $y_{i,j}$.) Define maps $i^{\pm}:G_e \ra G_v$, by $i^+ y_{i,j} = y_{i,j}$, and $i^- y_{i,j} = y_{i+1, j}$. By Theorem \[frei\], these are both injections. Thus the presentation $(*)$ expresses $G$ as an HNN-extension with vertex group $G_v$, and edge group $G_e$.
For an illustration, suppose that $G = <x_1, x_2| x_2 x_1 x_2^2 x_1 x_2^3 x_1^{-2}>$. Then $G$ is an HNN-extension, with vertex group $G_v = <y_{0,1}, y_{1,1}, y_{2,1}| y_{0,1} y_{1,1}^2 y_{2,1}^3>$, edge group $G_e = <y_{0,1}, y_{1,1}>$, and inclusions $i^+:G_e \ra G_v$, sending $y_{i,j} \ra y_{i,j}$ and $i^-: G_e \ra G_v$, sending $y_{1,1} \ra y_{2,1}$ and $y_{2,1} \ra y_{3,1}$.\
\
**Heegaard splittings of bundles**\
Suppose $M$ is a fibered 3-manifold, with a fiber $F$, which is a compact surface with non-empty boundary, and monodromy $f:F \ra F$. Let $\{ x_1, ..., x_n \}$ be a free generating set for $\pi_1 F$, represented by a collection of embedded loops in $F$, meeting at a single point. Then $\pi_1 M$ has an HNN presentation: $$(**) \,\,\,\,\pi_1 M = <x_1, ..., x_n, t| x_1 = t (f_* x_1) t^{-1}, ...,
x_n = t (f_* x_n) t^{-1}>.$$ We have the following fact.
\[fiberedhs\] The presentation $(**)$ for $\pi_1 M$ is induced by a Heegaard splitting of $M$.
Let $F_1$ and $F_2$ be disjoint fibers in $M$. Let $\a$ be an arc, whose interior is disjoint from $F_1 \cup F_2$, with one endpoint in $F_1$ and the other endpoint in $F_2$. Let $S$ be the boundary of a regular neighborhood of $F_1 \cup F_2 \cup \a$. Then one may verify that $S$ is a Heegaard surface inducing the presentation $(*)$.
**Modifying Heegaard splittings**\
Suppose $(W, W^{\prime})$ is a Heegaard splitting of $M$. We may obtain a new Heegaard splitting for $M$ by attaching a trivial 1-handle to $W$, and drilling out the corresponding 1-handle from $W^{\prime}$. We say that the new Heegaard splitting is a *stabilization* of $(W, W^{\prime})$.
Suppose that there are essential disks $D \subset W$, and $D^{\prime} \subset W^{\prime}$, such that $|\p D \cap \p D^{\prime}| = 1$. Let $C \subset \p_+ W$ denote the boundary of a regular neighborhood of $\p D \cup \p D^{\prime}$. Then there is a 2-sphere $S \subset M$, such that $S \cap \p_+ W = C$. Let $E$ be the disk $S \cap W$. Then we may form a new Heegaard surface for $M$ by compressing $\p_+ W$ along $E$. We say that the resulting Heegaard splitting is a *destabilization* of $(W, W^{\prime})$.
There is a useful method for changing disk systems. Suppose $\mc{D}$ is a disk system for $W$, and that $D_1, D_2 \in \mc{D}$. Suppose that $\a$ is an arc in $\p W$, starting on $\p D_1$ and ending on $\p D_2$, with $int \, \a \cap D = \emptyset$ for all $D \in \mc{D}$. Let $\a \times I \subset \p W$ be an embedded neighborhood of $\a$, so that $(\a \times I) \cap \p D_i$ is a neighborhood of $\a \cap D_i$, and $(\a \times I) \cap \p D = \emptyset$ for $D \in \mc{D} - (D_1 \cup D_2)$. Then the *band sum* of $D_1$ and $D_2$ along $\a$ is the disk $E = D_1 \cup D_2 \cup N(\a)$. The set $\mc{E} = (\mc{D} - D_1) \cup E$ is then a new disk system for $W$.\
\
**Modifying group presentations**\
Suppose $P = <X|R>$ is a presentation for a group $G$, where $X = \{x_1, ..., x_m \}$ and $R = \{r_1, ..., r_n \}$. One may obtain a new generating set by applying a free group automorphism to the $x_i$’s. We say that the resulting presentations are *Nielsen equivalent*.
For example, if $r \neq s$, and $1 \leq r,s \leq m$, let $\t_{rs} :F(x_1, ..., x_m) \ra F(x_1, ..., x_m)$ be the automorphism satisfying $\t_{rs} x_r = x_r x_s$, and $\t_{rs} x_i = x_i$ if $i \neq r$.
Let $\bar{x}_i = \t_{rs} x_i$, and let $\bar{r}_i$ be the word obtained from $r_i$, by replacing $x_r$ with $\bar{x}_r \bar{x}_s^{-1}$ and $x_i$ with $\bar{x}_i$, for $i \neq r$. Then $\overline{P} = <\bar{x}_1, ..., \bar{x}_m|\bar{r}_1, ..., \bar{r}_n>$ is a presentation for $G$.
Let $\t_i: F(x_1, ..., x_m) \ra F(x_1, ..., x_m)$ which sends $x_i$ to $x_i^{-1}$, and stabilizes all the other generators. In this case, the new presentation for $G$ is obtained from $P$ by exchanging $x_i$ and $x_i^{-1}$ in all the relators $r_1, ..., r_n$.
The automorphisms $\t_{rs}$ and $\t_i$ are called *Nielsen transformations* of $F(x_1, ..., x_m)$. The operation which a Nielsen transformation induces on a group presentation is called a *Nielsen move*. For a proof of the following fact, see [@LS], Chapter I.
\[nielsen\] The Nielsen transformations generate $Aut(F(x_1, ..., x_m))$. In particular, if $\overline{P}$ is Nielsen equivalent to $P$, then $\overline{P}$ may be obtained from $P$ by a sequence of Nielsen moves.
We shall need the fact that Nielsen moves can be realized geometrically. The proof of the following proposition may be found in [@G].
\[Gr\] (Griffiths) Let $F(n) = \pi_1(W)$, for a handlebody $W$. Then every automorphism of $F(n)$ is induced by an automorphism of $W$.
In addition to the Nielsen moves, it will be convenient to define the following operations on $P$ (special examples of Tietze tranformations), which give new presentations of $\pi_1 M$:\
\
1. Add a new element $x^{\prime}$ to $X$, and also add $x^{\prime}$ to $R$. We say that $P$ has been **stabilized**.\
\
2. Suppose that there is a relation (not necessarily part of the presentation) in the group of the form $r = x^{-1} w$, where $x \in X$ and $w$ is a word which does not involve $x$. Then remove $x$ from $X$, remove $r$ from $R$ (if $r$ is in R), and, wherever $x$ appears in any relator $r^{\prime} \in R - r$, replace it with $w$. We say that $P$ has been **destabilized along $x$**.\
**Modifying presentations of 3-manifold groups**\
Suppose that $P$ is a presentation for $\pi_1 M$ induced by a Heegaard splitting $(W, W^{\prime})$ of $M$, where $W$ is a handlebody. By Proposition \[Gr\], any automorphism of $\pi_1 W$ is induced by a handlebody automorphism. Thus, if $\overline{P}$ is obtained from $P$ by a Nielsen move, then $\overline{P}$ is also induced by the splitting $(W, W^{\prime})$. It is easy to see that a stabilization of the splitting $(W, W^{\prime})$, results in a stabilization of $P$. Similarly, a destabilization of $(W, W^{\prime})$ corresponds to a destabilization of $P$, as described in the following lemma.
\[destab\] Let $(W, W^{\prime})$ be a Heegaard splitting of $M$, let $\mc{D}$ and $\mc{D}^{\prime}$ be disk systems for $W$ and $W^{\prime}$, respectively, and let $P = <x_1, ..., x_g|R_1, ..., R_s>$ be the induced presentation of $\pi_1 M$. Let $D_1 \in \mc{D}$ be the disk which is dual to the generator $x_1$, and suppose $D_1^{\prime} \in \mc{D}^{\prime}$, with $|D_1 \cap D_1^{\prime}| = 1$. Then $P$ can be destabilized along $x_1$ so that the resulting presentation is induced by a destabilization of $(W,W^{\prime})$.
Let $C \subset \p W$ be the boundary of a regular neighborhood of $\p D_1 \cup \p D_1^{\prime}$. Then $C$ bounds a disk in both $W$ and in $W^{\prime}$, so there is a sphere $S \subset M$ with $S \cap \p W = C$. Let $B$ be the ball bounded by $S$ with $D_1 \cup D_1^{\prime} \subset B$. Then $S$ is a reducing sphere for $(W, W^{\prime})$, resulting in a destabilized Heegaard splitting $(Z, Z^{\prime})$, where $Z = W \cup B$, and $Z^{\prime} = \overline{W^{\prime}- B}$.
We obtain a disk system for $Z$ as follows. Let $\{ p_1, ..., p_n \} = \bigcup_{D_j \in \mc{D}} D_j \cap D_1^{\prime}$, where $p_1 = D_1 \cap D_1^{\prime}$, and the $p_i$’s are ordered consecutively along the circle $\p D_1^{\prime}$. Suppose $n \geq 2$, and let $D_j \in \mc{D}$ be the disk containing $p_2$. Let $\a$ be an arc of $\p D_1^{\prime}$ with endpoints $p_1$ and $p_2$, which is disjoint from $p_i$ for $i \geq 3$. Then we replace $D_j$ with the band sum of $D_j$ and $D_1$ along $\a$. Let $\mc{E}$ be the resulting disk system for $W$, and then $|\bigcup_{E \in \mc{E}} E \cap D_1^{\prime}| = n-1$. The resulting presentation of $\pi_1 M$ is obtained from $P$ by applying a single Nielsen move– replacing the generator $x_1$ with $x_1 x_j^{\pm 1}$. Continuing, we obtain a disk system $\mc{E}$ for $W$, so that $E \cap D_1^{\prime} = \emptyset$ for all $E \in \mc{E} - D_1$. Then $\mc{E} - D_1$ is a system for $Z$. Let $P^*$ be the presentation of $\pi_1 M$ induced by the systems $\mc{E}$ and $\mc{D}^{\prime}$. Then $P^*$ is obtained from $P$ by applying a sequence of Nielsen moves, whose effect is to replace the generator $x_1$ with the relator corresponding to $\p D_1^{\prime}$. In $P^*$, the disk $D_1^{\prime}$ corresponds to the relation $x_1 = id$.
We obtain a disk system for $Z^{\prime}$ as follows. Let $\{p_1, ..., p_n\}
= \bigcup_{D_j^{\prime} \in \mc{D}^{\prime}} D_j^{\prime} \cap D_1$, where $p_1 = D_1^{\prime} \cap D_1$, and the $p_i$’s are ordered consecutively along the circle $\p D_1$. Suppose $n \geq 2$, and let $D_j^{\prime} \in \mc{D}^{\prime}$ be the disk containing $p_2$. Let $\a$ be an arc of $\p D_1$ with endpoints $p_1$ and $p_2$, which is disjoint from $p_i$ for $i \geq 3$. Then we replace $D_j^{\prime}$ with the band sum of $D_j^{\prime}$ and $D_1^{\prime}$ along $\a$. Let $\mc{E}^{\prime}$ be the resulting disk system for $W^{\prime}$, and then $|\bigcup_{E \in \mc{E}^{\prime}} E \cap D| = n-1$. The presentation of $\pi_1 M$ induced by $\mc{E}$ and $\mc{E}^{\prime}$ is obtained from $P^*$ by removing one occurence of $x_1$ from one relation. Continuing, we obtain a disk system $\mc{E}^{\prime}$, so that $D_1^{\prime} \in \mc{E}^{\prime}$, and $E \cap D_1= \emptyset$ for all $E \in \mc{E}^{\prime} - D_1^{\prime}$. Then $\mc{E}^{\prime} - D_1^{\prime}$ is a system for $Z^{\prime}$. Let $Q$ be the presentation of $\pi_1 M$ induced by the systems $\mc{E}-D_1$ and $\mc{E}^{\prime}-D_1^{\prime}$. Then $Q$ is obtained from $P^*$ by removing the generator $x_1$, the relator $x_1 = id$, and all occurrences of $x_1$ in the remaining relators of $P^*$. It follows that $Q$ is obtained from $P$ by destabilizing along $x_1$.
To prove Theorem \[main\], we require a technical version of Lemma \[destab\] that works for *immersed* disks.
\[immersed\] Let $(W, W^{\prime})$ be a Heegaard splitting of $M$, let $\mc{D}$ and $\mc{D}^{\prime}$ be disk systems for $W$ and $W^{\prime}$, respectively, and let $P = <x_1, ..., x_g|R_1, ..., R_s>$ be a presentation of $\pi_1 M$, where the generators are dual to the disks in $\mc{D}$, and each $R_i$ is represented by an immersed disk in $W^{\prime}$. Let $D_1 \in \mc{D}$ be the disk which is dual to the generator $x_1$. Suppose there is a compressing disk $D^{\prime}$ for $W^{\prime}$, so that $|D_1 \cap D^{\prime}| = 1$. Then $(W, W^{\prime})$ can be destabilized to a splitting $(Z, Z^{\prime})$, and $P$ can be destabilized along $x_1$ to a presentation $Q$, where the generators of $Q$ are dual to a disk system for $Z$, and the relations of $Q$ correspond to immersed disks in $Z^{\prime}$.
The proof is similar to the proof of Lemma \[destab\]. The disks $D_1, D^{\prime}$ determine a reducing sphere $S$, which bounds a ball $B$, giving a destabilized Heegaard splitting $(Z, Z^{\prime})$, where $Z = W \cup B$, and $Z^{\prime} = \overline{W^{\prime}- B}$. We obtain a disk system for $Z$ as above.
If $D_i^{\prime}$ is an immersed disk representing $R_i$, we can remove intersection points of $\p D_i^{\prime} \cap \p D_1$, by banding together $D_i^{\prime}$ and $D^{\prime}$, along a subarc of $\p D_1$. Unlike the situation in Lemma \[destab\], we also have to consider the possibility that $D_i^{\prime}$ meets $D^{\prime}$. However, these intersections can be removed by banding together $D_i^{\prime}$ and $D_1$, along a subarc of $\p D^{\prime}$. The result is an immersed disk $E_i^{\prime}$ in $Z^{\prime}$, and the word represented by $\p E_i^{\prime}$ is the same as the word represented by $\p D_i^{\prime}$. Repeating this procedure for each $E_i$, we obtain immersed disks in the handlebody $Z^{\prime}$, representing the relators in a presentation $Q$, obtained by destabilizing $P$ along $x_1$.
Proof of Theorem \[main\] {#proof}
=========================
Suppose $$P = <x_1, ..., x_n|w> \cong G = \pi_1 M,$$ for some compact, orientable, fibered 3-manifold $M$. The first step is to put the word $w$ into a standard form. We let $e_i(w)$ denote the exponent sum of the letter $x_i$ in the word $w$. We may assume that $n \geq 2$, for otherwise $G \cong \mathbb{Z}$ and $w = id$, so $M$ is either $S^2 \times S^1$ or $D^2 \times S^1$, and the presentation is induced by a genus 1 Heegaard splitting of $M$.
Since $M$ is fibered, there is a map $\phi: G \ra \mathbb{Z}$ with finitely generated kernel, and so, by Theorem \[brown\], we may assume that $n=2$, and the given presentation has the form: $$(1) \,\,\, \pi_1 M = <x_1, x_2|w>.$$
Suppose that $w$ is primitive. Then (by Proposition \[Gr\]), $w$ is geometric, and $\pi_1 M \cong \mathbb{Z}$. Then $M \cong S^2 \times S^1$ or $D^2 \times S^1$, and the presentation is induced by a Heegaard splitting of $M$. So assume, from now on, that $w$ is non-primitive.
\[exp\] There is an automorphism $\th$ of the free group on $x_1, x_2$, such that $\phi(\th(x_1)) = 0, \phi(\th x_2) = 1$.
Let $\phi x_1 = \a$ and $\phi x_2 = \b$. Let $\sigma$ be the Nielsen transformation which stabilizes $x_2$, and satisfies $\sigma x_{1} = x_{1}x_{2}$. Let $\tau$ be the Nielsen transformation which stabilizes $x_{1}$, and satisfies $\tau(x_{2}) = x_{2} x_{1}$. We have: $$\begin{aligned}
\phi(\sigma^{\pm 1} x_1) = \a,\\
\phi(\sigma^{\pm 1} x_2) = \b \pm \a,\\
\phi(\tau^{\pm 1} x_1) = \a \pm \b,\\
\phi(\tau^{\pm 1} x_2) = \b.\end{aligned}$$ Thus by applying the Euclidean algorithm to $(\a, \b)$, we may find an automorphism $\th$, such that $\phi(\th x_1) = 0$, and $\phi(\th x_2) = \pm 1$. Replacing $x_2$ with $x_2^{-1}$ if necessary, we have $\phi(\th x_2) = 1$.
We now return to the proof of Theorem \[main\]. Let $\th$ be the automorphism given by Lemma \[exp\]. By Proposition \[Gr\], the presentation $<x_1, x_2|w>$ is induced by a Heegaard splitting of $M$ if and only if the presentation $<x_1, x_2|\th(w)>$ is induced by a Heegaard splitting of $M$. Thus we may replace $w$ with $\th(w)$, and so assume that $\phi(x_1) = 0$ and $\phi(x_2) = 1$.
Let $y_i = x_2^i x_1 x_2^{-i}$. Since $\phi(w) = 0$, then we can represent the word $w$ in the free group $F(x_1, x_2)$ as a product of $y_i$’s: i.e. $w = y_{r_1} ... y_{r_k}$. Replacing $w$ by a conjugate, we may assume that $Min \{ r_1, .., r_k \} = 0$, and we let $n = Max \{ r_1, ..., r_k \}$. Then $w = w_2(y_0, ..., y_n)$, for some word $w_2$. Since $\phi$ has finitely generated kernel, then by Proposition \[brown\], the letters $y_0$ and $y_n$ appear only once each in the word $w_2$. In particular, the word $w_2$ is primitive. Since $w$ is non-primitive, we must have $n \geq 1$.
As explained in the previous section, the 1-relator group $G$ has a Moldavansky splitting. In our case, the splitting has the following structure: the vertex group is $\mathcal{G}_v = <y_0, ..., y_n|w_2> \subset G$; the stable letter is $t=x_2$; the edge group is $\mathcal{G}_e = <y_0, ..., y_{n-1}> \subset G$. The inclusion maps are: $$\begin{aligned}
i_+ :\mathcal{G}_e \ra \mathcal{G}_v\\
i_+(y_j) = y_j;\\
i_- :\mathcal{G}_e \ra \mathcal{G}_v,\\
i_-(y_j) = y_{j+1}.\end{aligned}$$ This splitting is represented by a triple $\mathcal{S} = (\mathcal{G}_v, \mathcal{G}_e, t)$. Since $y_n$ appears only once in $w_2$, then in fact $\mathcal{G}_v = \mathcal{G}_e$, and the splitting $\mathcal{S}$ decomposes $G$ as a mapping torus of a free group automorphism $\psi: \mathcal{G}_e \ra \mathcal{G}_e$. In particular, we have that $\mathcal{G}_e = Ker(\phi)$.
If $F \subset M$ is a non-separating incompressible surface, with dual loop $\ell$, we let $F^+$ and $F^-$ be the corresponding subsurfaces of $\p (\overline{M - F})$. Let $p = \ell \cap F$ be a base-point for $\pi_1 F$, and let $p^{\pm}$ be the pre-images of $p$ in $\p (\overline{M-F})$. Let $j_+: \pi_1(F^+,p^+) \ra \pi_1(\overline{M-F},p^+)$ be the map induced by inclusion. We say that the splitting $\mathcal{S}$ is *realized geometrically* if $F$ and $\ell$ can be chosen so that $i_*(\overline{M-F}, p^+) = \mathcal{G}_v \subset \pi_1 M$, $j_{+*}(\pi_1 F^+, p^+) = \mathcal{G}_e$, and $[\ell] = t$.\
\
*Claim:* The splitting $\mathcal{S}$ is realized geometrically.\
\
*Proof of claim:*
The map $\phi:\pi_1 M \ra \mathbb{Z}$ is dual to a fiber $F$, and the element $t= x_2$ is represented by a dual loop $\ell$ for $F$. Then $j_{+*} \pi_1 (F^+ ,p^+) = ker(\phi) = \mathcal{G}_e$. Also, $\overline{M-F} \cong F \times [0,1]$, and so $i_*:(\pi_1 \overline{M-F},p^+) = \mathcal{G}_e = \mathcal{G}_v$, as required.\
\
By the claim, the fiber of $M$ has free fundamental group, and therefore $M$ must have non-empty boundary. (In fact, by a result of Lyndon ([@L]), any compact, orientable, irreducible 3-manifold with a 1-relator fundamental group must have non-empty boundary.)
We now wish to write down an explicit presentation for $G$, corresponding to the splitting $\mathcal{S}$. First, we may conjugate and invert $w_2$, if necessary, so that it has the form $w_2 = y_n^{-1} w_3$, for some word $w_3$ which does not involve $y_n$. We may then eliminate the generator $y_n$ from our presentation for $G_v$, obtaining the following presentation for $\pi_1 M$: $$(2) \,\,\, \pi_1 M \cong
<y_0, ..., y_{n-1}, t| y_0 t y_1^{-1} t^{-1},
..., y_{n-2} t y_{n-1}^{-1} t^{-1}, y_{n-1} t w_3^{-1} t^{-1} >.$$
Our next task is to show that the presentation $(2)$ is induced by a Heegaard splitting. By the claim, there is a fiber surface $F$ in $M$, inducing the splitting $\mathcal{S}$, and hence inducing the presentation (2). The difficulty is that, *a priori*, the given generators $y_0, ..., y_{n-1}$ of the fiber group may not be representable by embedded loops in $F$.
Let $(W, W^{\prime})$ be the genus $n+1$ splitting for $M$ given by Lemma \[fiberedhs\]. Let $\mathcal{D} = \{D_0, ..., D_n\}$ and $\mathcal{E} = \{E_0, ..., E_{n-1}\}$ be disk systems for $W$ and $W^{\prime}$, respectively, inducing a presentation of the following form: $$(3) \,\,\, \pi_1 M = <z_0, ..., z_{n-1}, s|
s z_0 s^{-1} f_* z_0^{-1}, ..., s z_{n-1} s^{-1} f_* z_{n-1}^{-1}>.$$ (So $y_i$ is dual to $D_i$ for $i \leq n-1$, and $s$ is dual to $D_n$.) Since $s$ and $t$ are both represented by loops which are dual to $F$, then $s = t u$ for some $u \in i_* \pi_1 F$. It follows that $f_*$ is the composition of $\psi$ with an inner automorphism. Thus there is a fiber-preserving automorphism of $M$ which takes the presentation $(3)$ to:
$$(4) \,\,\, \pi_1 M = <z_0, ..., z_{n-1}, t|
t z_0 t^{-1} \psi z_0^{-1}, ..., t z_{n-1} t^{-1} \psi z_{n-1}^{-1}>.$$ Thus presentation (4) is induced by a Heegaard splitting, which we still refer to as $(W, W^{\prime})$. We also replace the disk systems $\{D_i\}$ and $\{ E_i \}$ with their images under the automorphism.
Since $y_0, ..., y_{n-1}$ and $z_0, ..., z_{n-1}$ are both free bases for $i_* \pi_1 F$, then by Proposition \[nielsen\] there is a sequence of Nielsen transformations taking $z_0, ..., z_{n-1}$ to $y_0, ..., y_{n-1}$. Suppose the first such move sends $z_i$ to $z_i z_j$, and stabilizes all the other generators. Then the corresponding Nielsen move, applied to presentation (4), gives: $$\pi_1 M = <\bar{z}_0, ..., \bar{z}_{n-1}, t|
t \bar{z}_0 t^{-1} \psi \bar{z}_0^{-1}, ...,
t \bar{z}_i \bar{z}_j^{-1} t^{-1}
\psi (\bar{z}_i \bar{z}_j^{-1})^{-1}, ...,
t \bar{z}_{n-1} t^{-1} \psi \bar{z}_{n-1}^{-1}>.$$
We now perform a Tietze transformation, multiplying (a conjugate of) the $i$th relation by (a conjugate of) the $j$th relation, to obtain the following presentation: $$\pi_1 M = <\bar{z}_0, ..., \bar{z}_{n-1}, t|
t \bar{z}_0 t^{-1} \psi \bar{z}_0^{-1}, ...,
t \bar{z}_i t^{-1} \psi \bar{z}_i^{-1}, ...,
t \bar{z}_{n-1} t^{-1} \psi \bar{z}_{n-1}^{-1}>.$$
To perform this move geometrically, we choose an (immersed) arc $\a \subset \p W$, with one endpoint on $\p E_i$ and the other on $\p E_j$, which is disjoint from all the disks $D_1, ..., D_n$. Then we attach a regular neighborhood of $\a$ to $E_i$ and $E_j$, obtaining an immersed disk whose boundary represents the relator $t \bar{z}_i t^{-1} \psi \bar{z}_i^{-1}$. Continuing, we may change the presentation $(4)$ to the presentation $(2)$, and we see that the relators $R_0, ..., R_{n-1}$ of $(2)$ are represented by *immersed* disks $E_0^{\dagger}, ..., E^{\dagger}_{n-1}$, in $W^{\prime}$.
By Dehn’s Lemma, there is a compressing disk $E_0^*$ for $W^{\prime}$, so that $\p E_0^* \subset \p E^{\dagger}_0$. Let $w^*$ be the word represented by $\p E_0^*$.\
\
**Claim:** The circle $\p E_0^*$ meets $\p D_0$ in a single point.\
\
Before proving the claim, we make a definition.\
\
**Definition:** For any word $u$ on a generating set $\mc{G}$, and any $g \in \mc{G}$, we define $n_g(u)$ to be the number of times the generator $g$, or its inverse, occurs in $u$. Note this is different from the exponent sum. For example, we have $n_t(R_0) = n_t(y_0 t y_1^{-1} t^{-1}) = 2$. Since $\p E_0^* \subset \p E^{\dagger}_0$, then $|\p E_0^* \cap D_i| \leq |\p E^{\dagger} \cap D_i|$ for each $i$. Therefore, for each generator $g$ in (2), we have $n_g(w^*) \leq n_g(R_0) = n_g(y_0 t y_1^{-1} t^{-1})$.\
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**Proof of claim:** We have $|\p E_0^* \cap \p D_0| \leq |\p E_0^{\dagger} \cap \p D_0| = 1$. Suppose $\p E_0^* \cap \p D_0 = \emptyset$. Thus $n_{y_0}(w^*) = 0$. Since $n_g(w^*) \leq n_g(y_0 t y_1^{-1} t^{-1})$ for each generator $g$, then $n_{y_1}(w^*) \leq 1$, $n_t(w^*) \leq 2$, and $n_g(w^*) = 0$ for all other generators $g$. Furthermore, since $w^*$ represents the trivial element in $\pi_1 M$, the exponent sum of $t$ in $w^*$ must be zero.
We see that the only possibilities are that $w^*$ is trivial, or else conjugate to $y_1^{\pm 1}$. The latter case is impossible, since the element $y_1$ is contained in a basis for the free group $\pi_1 F$. So $w^*$ is the trivial word. But then there is a reducing sphere $S$ for $(W, W^{\prime})$ which is disjoint from all the disks $D_0, ..., D_n$; therefore some non-empty subset of the generators $y_0, ..., y_{n-1}, t$ must be trivial in $\pi_1 M$. This is impossible, since $\{y_0, ..., y_{n-1}\}$ is a basis for the free group $\pi_1 F$. This proves the claim.\
By Lemma \[immersed\], we can destabilize $(W,W^{\prime})$ to a splitting $(Z,Z^{\prime})$, and we can destabilize the presentation $P$ along $y_1$ to obtain a presentation $Q$, where the relators of $Q$ are represented by immersed disks in $Z^{\prime}$. The presentation $Q$ has a relator of the form $y_2 t y_3^{-1} t^{-1}$. Since $y_2$ appears once in this relator, we can repeat the previous argument, to obtain a reducing sphere for the splitting $(Z, Z^{\prime})$, and a destabilization of the presentation $Q$ along $y_2$.
Continuing, we may reduce the Heegaard splitting $(W, W^{\prime})$ to a splitting $(Z, Z^{\prime})$, where $genus(Z) = 2$, and $\pi_1 Z$ is generated by $y_{n-1} = x_2^{n-1} x_1 x_2^{-(n-1)}$ and $t = x_2$. Applying an automorphism of the handlebody $Z$, we may take the generators of $\pi_1 Z$ to be $x_1, x_2$, and then the Heegaard splitting $(Z, Z^{\prime})$ determines a 1-relator presentation:
$$(6) \,\,\, \pi_1 M \cong <x_1, x_2| \overline{w}>,$$ where $\overline{w}$ is a geometric word. Comparing the presentation (6) with the original presentation $(1)$, and applying Theorem \[uniqueword\], we see that $w$ is conjugate to $\overline{w}$ or $\overline{w}^{-1}$ in the free group $<x_1, x_2>$. Therefore $w$ is geometric, and the given presentation $(1)$ is induced by a Heegaard splitting of $M$.
Hierarchies for 1-relator groups {#groups}
================================
It was observed in [@K] that 1-relator groups and Haken 3-manifolds share some common features. In this section, we shall make the analogy explicit, showing that every 1-relator group possesses a “1-relator hierarchy”.
Let $G$ be a 1-relator group. We define a *1-relator hierarchy* for $G$ to be a finite sequence of 1-relator groups $G_1, G_2, ..., G_n$, such that $G_1 \cong G$, $G_n$ is cyclic, and $G_i$ splits as an HNN-extension with vertex group $G_{i-1}$.
\[hier\] Let $G$ be a 1-relator group. Then $G$ has a 1-relator hierarchy.
The proof is by a “tower argument”, similar in spirit to the proof of Dehn’s Lemma. First we require some terminology. By a *loop* in a surface $S$, we understand an immersion from a circle into $S$. A loop in $S$ is *essential* if it cannot be homotoped into $\p S$. A loop in $S$ is said to *fill* $S$ if the complement of its image contains no essential loops.
Given a 1-relator group $G_1 = <x_{1,1}, ..., x_{1,n_1}|w_1>$, with $n_1 \geq 2$. We view the free group on $x_{1,1}, ..., x_{1,n_1}$ as $\pi_1 S_1$, for some compact surface $S_1$, where the $x_i$’s are represented by embedded loops meeting in a single point.
The word $w_1$ is represented by an immersed loop $\ell_1 \subset S_1$, and if $w_1$ is cyclically reduced, we may take $\ell_1$ to minimize the number of self-interections in its free homotopy class.\
\
Case 1: The loop $\ell_1$ does not fill $S_1$.\
\
In this case, there is an automorphism $\th:S_1 \ra S_1$, such that some generator $x_{1,i}$ does not appear in $w_1$. After re-labeling, we may assume that the missing generator is $x_{1,1}$. We choose such a $\th$, and we replace the presentation $<x_{1,1} ..., x_{1,n_1}|w_1>$ with the presentation $<x_{1,1}, ..., x_{1,n_1}| \th_*(w_1)>$, which defines an isomorphic group. We also replace $\ell_1$ with $\th(\ell_1)$.
The group $G_1$ then splits as a free product $G_1 \cong G_2 \ast <x_{1,1}>$. We relabel the ordered list $(x_{1,2}, ..., x_{1,n_1})$ by the ordered list $(x_{2,1}, ..., x_{2,n_2})$, and let $w_2$ be the result of re-labeling $\th(w_1)$. Then we have $G_2 \cong <x_{2,1}, ..., x_{2,n_2}| w_2>$, where $n_2 = n_1-1$. We view this free product as an HNN-extension, with trivial edge groups.
There is a sub-surface $S_2 \subset S_1$, such that $$i_* \pi_1 S_2 = <x_{2,1}, ..., x_{2,n_2}> \subset \pi_1 \widetilde{S}_1.$$ The word $w_2$ is represented by a loop $\ell_2 \subset S_2$. In fact, in this case, we may choose $\ell_2 = \ell_1$.\
\
Case 2: The loop $\ell_1$ fills $S_1$.\
\
Since $b_1(S_1) \geq 2$, there is a map $\phi:G_1 \ra \mathbb{Z}$, such that $w_1 \in Ker(\phi)$. By changing the generators of $\pi_1 S$, and re-writing $w_1$ in terms of the new generators, we may assume that $\phi$ is the exponent-sum function $e_{x_{1,1}}$.
Corresponding to the homomorphism $e_{x_{1,1}}: G_1 \ra \mathbb{Z}$, there is an infinite cyclic cover $\widetilde{S}_1 \ra S_1$. Let $\t: \widetilde{S}_1 \ra \widetilde{S}_1$ be the corresponding covering transformation. Let $D \subset \widetilde{S}_1$ be a compact fundamental domain for the action of $\t$. Let $k_1$ be minimal such that the surface $S_2 = D \cup t D \cup ... \cup t^{k_1} D$ contains a lift of $\ell_1$. We have: $$\pi_1 S_2 = <x_{1,2}, ..., x_{1,n_1},
x_{1,1} x_{1,2} x_{1,1}^{-1}, ...,
x_{1,1}^{k_1} x_{1,n_1} x_{1,1}^{-k_1}|>.$$ We may choose these generators so they are represented by embedded loops in $S_2$, meeting at a single point. We label the elements of this list consecutively as $(x_{2,1}, ..., x_{2,n_2})$. Then we see (as in Section 2) that $G_1$ is an HNN-extension with vertex group $$G_2 = <x_{2,1}, ..., x_{2,n_2} |w_2>,$$ for some word $w_2$, represented by a loop $\ell_2 \subset S_2$, which is a lift of $\ell_1$. The edge group is the free group $F(r_2)$, where $r_2 = (n_1-1)k_1$, and the images of the edge group are generated by $\{ x_{2,1}, ..., x_{2,r_2} \}$ and $\{x_{2,n_1}, ..., x_{2,n_2} \}$, respectively. By the Freiheitsatz, these are in fact free subgroups of $G_2$.\
\
Suppose that by the above procedure we have constructed a splitting of $G_1$, with vertex group $G_2 = <x_{2,1}, ..., x_{2,n_2}| w_2>$, where $G_2$ is represented by a loop $\ell_2$ in a surface $S_2$. If $n_2 \geq 2$, we may apply the above process to $G_2$. Continuing, we obtain a sequence of 1-relator groups $G_1, G_2, ...$, where $G_i = <x_{i,1}, ..., x_{i,n_i}|w_i>$, and $G_i$ is a vertex group for $G_{i-1}$. The sequence terminates at $G_k$ if and only if $n_k = 1$.
Topologically, for each $i$ we have surfaces $S_i$ and $\widetilde{S}_i$, (where possibly $\widetilde{S}_i = S_i$) so that $S_{i+1}$ is a subsurface of $\widetilde{S}_i$. We also have a loop $\ell_i \subset S_i$ representing the word $w_i$, such that $\ell_{i+1}$ is a homeomorphic lift of $\ell_i$.\
\
*Claim:* The sequence $G_1, G_2, ...$ terminates after finitely many steps.\
\
*Proof of claim:*
Note that when a curve is lifted to a finite cover, the number of self-intersections can never go up. Therefore the number of self-intersections of $\ell_{i+1}$ is less than or equal to the number of self-intersections of $\ell_i$ for all $i$. Thus if the sequence $G_1, G_2, ...$ is infinite, then the sequence $\ell_1, \ell_2, ... $ must contain infinitely many consecutive terms for which the number of self-intersections remains constant. We claim that this is impossible.
Indeed, suppose the number of self-intersections of $\ell_{i+1}$ is the same as the number of self-intersections of $\ell_i$, and suppose that $\ell_{i+1} \subset S_{i+1}$ is constructed as in Case 2. Then the lifts of $\ell_i$ to $\widetilde{S}_i$ are all disjoint. Let $C$ be the boundary of a regular neighborhood of one of these lifts in $\widetilde{S}_i$. Since each lift of $\ell_i$ is essential, and since no lift of $\ell_i$ fills $\widetilde{S}_i$, then some component of $C$ must be essential. Thus $C$ projects to a collection of loops in $S_i$, which are disjoint from $\ell_i$, at least one of which must be essential. Thus $\ell_i$ does not fill $S_i$. Therefore, $G_{i+1}$ is constructed from $G_i$ as in Case 1, and so $n_{i+1} = n_i-1$.
Similarly, if the number of self-intersections of $\ell_i, ..., \ell_{i+k}$ are all the same, then $n_{i+k} = n_i-k$. Thus if the sequence of $G_i$’s is infinite, we arrive at a contradiction.
Concluding remarks
==================
On 1-relator groups
-------------------
Often, facts about 1-relator groups are proved by induction on the length of the relator. Proofs break into two cases: if $G = <x_1, ..., x_n|w>$, and the word $w$ has zero exponent sum on one of its letters, then $G \cong G_2 \ast_t$, where $G_2$ is a 1-relator group, whose relator is shorter than the relator of $G$. One may then apply the induction hypothesis and the HNN structure to prove things about $G$.
If $w$ does not have this form, one shows that $G$ embeds nicely in a certain 1-relator group $\widehat{G}$, with $\widehat{G} \cong \widehat{G}_2 \ast_t$; where $\widehat{G}_2$ is a 1-relator group, whose relator is shorter than $w$. One may then apply the induction hypothesis to $\widehat{G}_2$, and attempt to transfer this information to $G$.
Theorem \[hier\] makes it possible to prove facts about 1-relator groups by inducting, instead, on the length of a hierarchy. This allows one to avoid the embedding step, and give proofs which are, to a 3-manifold topologist, more intuitive.
On Conjecture \[cnj\]
---------------------
Our main motivation for presenting Proposition \[hier\] is provided by Conjecture \[cnj\]. The standard method for 1-relator groups appears problematic in this case, since there is no guarantee that the embedding step can be performed geometrically. Proposition \[hier\] gives hope that the conjecture might be approachable by inducting, instead, on the length of a hierarchy.
However, the proof given for the fibered case does not directly generalize. Indeed, a key point in the proof of Theorem \[main\] is that if $M$ is fibered, then the Moldavansky splitting is realized geometrically. But, in general, it is *not* true that the Moldavansky splitting of a 1-relator 3-manifold group is geometric.
Let $M$ be the manifold $M017$ in the Snappea census. Then the presentation $$\pi_1 M = <x_1, x_2| x_1^2 x_2 x_1^3 x_2 x_1^2 x_2^{-2}>$$ is induced by a Heegaard splitting of $M$. Since $b_1(M) = 1$ and $\p M \cong T^2$, there is a unique (up to isotopy) longitude curve $\l \subset \p M$, whose image is trivial in $H_1(M, \mathbb{Q})$. One may compute that in $\pi_1 M$, the loop $\l$ represents the element $x_2^{-4} x_1^2 x_2 x_1^2 (x_1^3 x_2 x_1^2)^3$. Thus $[\l]$ has order 7 in $H_1(M, \mathbb{Z}) \cong \mathbb{Z} \times \mathbb{Z}/7$. It follows that any non-separating surface in $M$ must have at least seven boundary components, and thus have a fundamental group of rank at least 6. However, for this example, the Moldavansky edge group has rank 2.
[2]{} M. Abert and N. Nikolov “Rank gradient, cost of groups and the rank versus Heegard genus problem”, pre-print.
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W. Jaco, “Roots, relations and centralizers in three-manifold groups”, *Geometric topology (Proc. Conf., Park City, Utah, 1974*, pp. 283–309. Lecture Notes in Math., Vol. 438, Springer, Berlin, 1975.
I. Kapovich, “Howson property and one-relator groups”, *Comm. Algebra* **27**, no. 3 (1999), 1057–1072.
R. C. Lyndon, Cohomology theory of groups with a single defining relation, *Ann. of Math. (2)* **52** (1950), 650–665.
R. C. Lyndon and P. E. Schupp, *Combinatorial Group Theory*, Springer-Verlag, 1977.
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Scott, G. P. Compact submanifolds of $3$-manifolds, *J. London Math. Soc.* (2) **7** (1973), 246–250.
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Math Department\
SUNY at Buffalo\
jdmaster@buffalo.edu
[^1]: Very recently, Abert and Nikolov [@AN] have shown that are in fact hyperbolic examples for which inequality is sharp.
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abstract: 'Hate speech detection on Twitter is critical for applications like controversial event extraction, building AI chatterbots, content recommendation, and sentiment analysis. We define this task as being able to classify a tweet as racist, sexist or neither. The complexity of the natural language constructs makes this task very challenging. We perform extensive experiments with multiple deep learning architectures to learn semantic word embeddings to handle this complexity. Our experiments on a benchmark dataset of 16K annotated tweets show that such deep learning methods outperform state-of-the-art char/word n-gram methods by $\sim$18 F1 points.'
author:
- |
Pinkesh Badjatiya$^1$, Shashank Gupta$^1$, Manish Gupta$^{1,2}$, Vasudeva Varma$^1$\
\
title: Deep Learning for Hate Speech Detection in Tweets
---
Introduction
============
With the massive increase in social interactions on online social networks, there has also been an increase of hateful activities that exploit such infrastructure. On Twitter, hateful tweets are those that contain abusive speech targeting individuals (cyber-bullying, a politician, a celebrity, a product) or particular groups (a country, LGBT, a religion, gender, an organization, etc.). Detecting such hateful speech is important for analyzing public sentiment of a group of users towards another group, and for discouraging associated wrongful activities. It is also useful to filter tweets before content recommendation, or learning AI chatterbots from tweets[^1].
The manual way of filtering out hateful tweets is not scalable, motivating researchers to identify automated ways. In this work, we focus on the problem of classifying a tweet as racist, sexist or neither. The task is quite challenging due to the inherent complexity of the natural language constructs – different forms of hatred, different kinds of targets, different ways of representing the same meaning. Most of the earlier work revolves either around manual feature extraction [@waseem2016hateful] or use representation learning methods followed by a linear classifier [@djuric2015hate; @nobata2016abusive]. However, recently deep learning methods have shown accuracy improvements across a large number of complex problems in speech, vision and text applications. To the best of our knowledge, we are the first to experiment with deep learning architectures for the hate speech detection task.
In this paper, we experiment with multiple classifiers such as Logistic Regression, Random Forest, SVMs, Gradient Boosted Decision Trees (GBDTs) and Deep Neural Networks(DNNs). The feature spaces for these classifiers are in turn defined by task-specific embeddings learned using three deep learning architectures: FastText, Convolutional Neural Networks (CNNs), Long Short-Term Memory Networks (LSTMs). As baselines, we compare with feature spaces comprising of char n-grams [@waseem2016hateful], TF-IDF vectors, and Bag of Words vectors (BoWV).
Main contributions of our paper are as follows: (1) We investigate the application of deep learning methods for the task of hate speech detection. (2) We explore various tweet semantic embeddings like char n-grams, word Term Frequency-Inverse Document Frequency (TF-IDF) values, Bag of Words Vectors (BoWV) over Global Vectors for Word Representation (GloVe), and task-specific embeddings learned using FastText, CNNs and LSTMs. (3) Our methods beat state-of-the-art methods by a large margin ($\sim$18 F1 points better).
Proposed Approach
=================
We first discuss a few baseline methods and then discuss the proposed approach. In all these methods, an embedding is generated for a tweet and is used as its feature representation with a classifier.
: As baselines, we experiment with three broad representations. (1) Char n-grams: It is the state-of-the-art method [@waseem2016hateful] which uses character n-grams for hate speech detection. (2) TF-IDF: TF-IDF are typical features used for text classification. (3) BoWV: Bag of Words Vector approach uses the average of the word (GloVe) embeddings to represent a sentence. We experiment with multiple classifiers for both the TF-IDF and the BoWV approaches.
: We investigate three neural network architectures for the task, described as follows. For each of the three methods, we initialize the word embeddings with either random embeddings or GloVe embeddings. (1) CNN: Inspired by Kim et. al [@Kim14]’s work on using CNNs for sentiment classification, we leverage CNNs for hate speech detection. We use the same settings for the CNN as described in [@Kim14]. (2) LSTM: Unlike feed-forward neural networks, recurrent neural networks like LSTMs can use their internal memory to process arbitrary sequences of inputs. Hence, we use LSTMs to capture long range dependencies in tweets, which may play a role in hate speech detection. (3) FastText: FastText [@joulin2016bag] represents a document by average of word vectors similar to the BoWV model, but allows update of word vectors through Back-propagation during training as opposed to the static word representation in the BoWV model, allowing the model to fine-tune the word representations according to the task.
All of these networks are trained (fine-tuned) using labeled data with back-propagation. Once the network is learned, a new tweet is tested against the network which classifies it as racist, sexist or neither. Besides learning the network weights, these methods also learn task-specific word embeddings tuned towards the hate speech labels. Therefore, for each of the networks, we also experiment by using these embeddings as features and various other classifiers like SVMs and GBDTs as the learning method.
Experiments
===========
Dataset and Experimental Settings
---------------------------------
We experimented with a dataset of 16K annotated tweets made available by the authors of [@waseem2016hateful]. Of the 16K tweets, 3383 are labeled as sexist, 1972 as racist, and the remaining are marked as neither sexist nor racist. For the embedding based methods, we used the GloVe [@pennington2014glove] pre-trained word embeddings. GloVe embeddings[^2] have been trained on a large tweet corpus (2B tweets, 27B tokens, 1.2M vocab, uncased). We experimented with multiple word embedding sizes for our task. We observed similar results with different sizes, and hence due to lack of space we report results using embedding size=200. We performed 10-Fold Cross Validation and calculated weighted macro precision, recall and F1-scores.
We use ‘adam’ for CNN and LSTM, and ‘RMS-Prop’ for FastText as our optimizer. We perform training in batches of size 128 for CNN & LSTM and 64 for FastText. More details on the experimental setup can be found from our publicly available source code[^3].
\[tab:experiments\]
Method Prec Recall F1
-- ------------------------------------------------------ ----------- ----------- -----------
Char n-gram+Logistic Regression [@waseem2016hateful] 0.729 0.778 0.753
TF-IDF+Balanced SVM 0.816 0.816 0.816
TF-IDF+GBDT 0.819 0.807 0.813
BoWV+Balanced SVM 0.791 0.788 0.789
BoWV+GBDT 0.800 0.802 0.801
CNN+Random Embedding 0.813 0.816 0.814
CNN+GloVe 0.839 0.840 0.839
FastText+Random Embedding 0.824 0.827 0.825
FastText+GloVe 0.828 0.831 0.829
LSTM+Random Embedding 0.805 0.804 0.804
LSTM+GLoVe 0.807 0.809 0.808
CNN+GloVe+GBDT 0.864 0.864 0.864
CNN+Random Embedding+GBDT 0.864 0.864 0.864
FastText+GloVe+GBDT 0.853 0.854 0.853
FastText+Random Embedding+GBDT 0.886 0.887 0.886
LSTM+GloVe+GBDT 0.849 0.848 0.848
LSTM+Random Embedding+GBDT **0.930** **0.930** **0.930**
: Comparison of Various Methods (Embedding Size=200 for GloVe as well as for Random Embedding)
Results and Analysis
--------------------
Table \[tab:experiments\] shows the results of various methods on the hate speech detection task. Part A shows results for baseline methods. Parts B and C focus on the proposed methods where part B contains methods using neural networks only, while part C uses average of word embeddings learned by DNNs as features for GBDTs. We experimented with multiple classifiers but report results mostly for GBDTs only, due to lack of space.
As the table shows, our proposed methods in part B are significantly better than the baseline methods in part A. Among the baseline methods, the word TF-IDF method is better than the character n-gram method. Among part B methods, CNN performed better than LSTM which was better than FastText. Surprisingly, initialization with random embeddings is slightly better than initialization with GloVe embeddings when used along with GBDT. Finally, part C methods are better than part B methods. The best method is “LSTM + Random Embedding + GBDT” where tweet embeddings were initialized to random vectors, LSTM was trained using back-propagation, and then learned embeddings were used to train a GBDT classifier. Combinations of CNN, LSTM, FastText embeddings as features for GBDTs did not lead to better results. Also note that the standard deviation for all these methods varies from 0.01 to 0.025.
To verify the task-specific nature of the embeddings, we show top few similar words for a few chosen words in Table \[tab:caseStudies\] using the original GloVe embeddings and also embeddings learned using DNNs. The similar words obtained using deep neural network learned embeddings clearly show the “hatred” towards the target words, which is in general not visible at all in similar words obtained using GloVe.
\[tab:caseStudies\]
Target Word Similar words using GloVe Similar words using task-specific embeddings learned using DNNs
------------- ------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------------------
pakistan karachi, pakistani, lahore, india, taliban, punjab, islamabad mohammed, murderer, pedophile, religion, terrorism, islamic, muslim
female male, woman, females, women, girl, other, artist, girls, only, person sexist, feminists, feminism, bitch, feminist, blonde, bitches, dumb, equality, models, cunt
muslims christians, muslim, hindus, jews, terrorists, islam, sikhs, extremists, non-muslims, buddhists islam, prophet, quran, slave, jews, slavery, pedophile, terrorist, terrorism, hamas, murder
: Embeddings learned using DNNs clearly show the “racist” or “sexist” bias for various words.
Conclusions
===========
In this paper, we investigated the application of deep neural network architectures for the task of hate speech detection. We found them to significantly outperform the existing methods. Embeddings learned from deep neural network models when combined with gradient boosted decision trees led to best accuracy values. In the future, we plan to explore the importance of the user network features for the task.
[1]{}
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A. Joulin, E. Grave, P. Bojanowski, and T. Mikolov. . , 2016.
Y. Kim. . In [*EMNLP*]{}, pages 1746–1751, 2014.
C. Nobata, J. Tetreault, A. Thomas, Y. Mehdad, and Y. Chang. . In [*WWW*]{}, pages 145–153, 2016.
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[^1]: https://en.wikipedia.org/wiki/Tay\_(bot)
[^2]: http://nlp.stanford.edu/projects/glove/
[^3]: https://github.com/pinkeshbadjatiya/twitter-hatespeech
|
---
abstract: 'Bayesian model averaging (BMA) is the state of the art approach for overcoming model uncertainty. Yet, especially on small data sets, the results yielded by BMA might be sensitive to the prior over the models. Credal Model Averaging (CMA) addresses this problem by substituting the single prior over the models by a *set* of priors (credal set). Such approach solves the problem of how to choose the prior over the models and automates sensitivity analysis. We discuss various CMA algorithms for building an ensemble of logistic regressors characterized by different sets of covariates. We show how CMA can be appropriately tuned to the case in which one is prior-ignorant and to the case in which instead domain knowledge is available. CMA detects *prior-dependent* instances, namely instances in which a different class is more probable depending on the prior over the models. On such instances CMA suspends the judgment, returning multiple classes. We thoroughly compare different BMA and CMA variants on a real case study, predicting presence of Alpine marmot burrows in an Alpine valley. We find that BMA is almost a random guesser on the instances recognized as prior-dependent by CMA.'
address:
- |
Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)\
Galleria 2, 6928 Manno (Lugano), Switzerland
- |
Dipartimento di Elettronica, Informazione e Bioingegneria\
Politecnico di Milano, Italy\
author:
- Giorgio Corani
- Andrea Mignatti
bibliography:
- 'biblio.bib'
title: 'Credal Model Averaging for classification: representing prior ignorance and expert opinions.'
---
|
---
abstract: 'The main results in this paper provide upper bounds of the second order Dehn functions of three-dimensional groups Nil and Sol. These upper bounds are obtained by using the Varopoulos transport argument on dual graphs. The first step is to start with reduced handlebody diagrams of the three-dimensional balls either immersed or embedded in the universal covers of each group and then define dual graphs using the 0-handles as vertices, 1-handles as edges. The idea is to reduce the original isoperimetric problem involving volume of three-dimensional balls and areas of their boundary spheres to a problem involving Varopoulos’ notion of volume and boundary of finite domains in dual graphs.'
address: |
Dept. of Mathematics and Computer Science\
The Citadel,\
Charleston, SC 29409\
USA
author:
- Antara Mukherjee
bibliography:
- 'transport3.bib'
title: Isoperimetric Inequalities using Varopoulos Transport
---
Introduction
============
History of Filling Functions
----------------------------
####
The origin of the quest to find a link between topology and combinatorial group theory can be traced back to Belgian physicist Plateau’s (1873, [@Pla]) classical question whether every rectifiable Jordan loop in every 3-dimensional Euclidean space bounds a disc of minimal area. Since then geometers and topologists have been investigating various ways to obtain efficient fillings of spheres by minimal volume balls. Thanks to the efforts of Dehn [@Dehn] and Gromov [@Gr3] we now know that there is an intimate connection between this classical geometric problem and group theory. Various other results on Dehn functions can be found in papers by McCammond [@Cammond], Ol’shanski[ĭ]{} [@Ol] and Rips [@Rips1]. The most significant development in this area has been Gromov’s introduction of word hyperbolic groups.\
Important results in the area of Dehn functions using different techniques also appear in the pair of following papers, the first in 1997 (published in 2002) by Sapir, Birget and Rips ([@sapir]) and the second in 2002 by Birget, Ol’shanski[ĭ]{}, Yu, Rips and Sapir, ([@Ol1]). They showed that there exists a close connection between Dehn functions and complexity functions of Turing machines. One of their main results said that the Dehn function of a finitely presented group is equivalent to the time function of a two-tape Turing machine. More of this history and background on isoperimetric inequalities can be found in the paper by Bridson in [@Bri].
Since the 1990’s topologists have been interested in Dehn functions in higher dimensions. Gromov [@Gr2], Epstein et al.[@Ep], first introduced the higher order Dehn functions and Alonso et al. [@Al] and Bridson [@Bri1] produced the first few results in the context of these functions.
This paper would not have been possible without the support and guidance of my advisor, Dr. Noel Brady of the University of Oklahoma.
Goal of this research
---------------------
####
The main theorems of this paper provide upper bounds of the second order Dehn functions for 3-dimensional groups Nil and Sol.
\[A. Mukherjee\] \[th1\] The upper bound of the second order Dehn function (denoted by $\delta^{(2)}$) of the lattices in the Nil geometry is given by $\delta^{(2)}(n) {\preccurlyeq}n^{\frac{4}{3}} $.\
In other words, the upper bound of the second order Dehn function of the groups ${{\mathbb Z}}^2 \rtimes_\phi {{\mathbb Z}}$, where $\phi$ has eigenvalues $\pm 1$ and has infinite order, is given by, $\delta^{(2)}(n) {\preccurlyeq}n^{\frac{4}{3}} $.
\[A. Mukherjee\]\[th2\] The upper bound of the second order Dehn function (denoted by $\delta^{(2)}$) of the lattices of the 3-dimensional geometry Sol is given by $\delta^{(2)}(n) {\preccurlyeq}n\ln (n) $.\
In other words, that the second order Dehn functions for the groups ${{\mathbb Z}}^2 \rtimes_\phi {{\mathbb Z}}$, where the eigenvalues of $\phi$ are not $\pm 1$, $\delta^{(2)}(n) {\preccurlyeq}n\ln (n) $.
Overview of proof of the main theorems
======================================
####
The main goal of this research is to obtain upper bounds of second order Dehn function of the groups mentioned in the theorems above.
In order to obtain upper bounds, we start with a reduced, transverse diagram $f:(D^3,S^2)\rightarrow K$, where $D^3$ is a 3-ball, $S^2$ is its boundary sphere and $K$ is the 3-dimensional ambient space. We then define a dual Cayley graph $\Gamma$ in the ambient space $K$ where each vertex of $\Gamma$ is a 3-cell in $K$ and each edge is a 2-cell common to two adjacent 3-cells. Now, we consider a finite subset of vertices $D$ of $\Gamma$ corresponding to the 0-handles of the diagram mapped into $K$ and we define an integer-valued function $\phi_D:\Gamma^{(0)}\rightarrow {{\mathbb Z}}^+$ with finite support i.e, $\phi_D(\alpha)=\hspace{0.1cm}$number of pre-images of $\alpha$ in $(D^3,S^2)$ for all $\alpha\in D$, otherwise $\phi_D(\alpha)=0$. This leads us to the fact that the volume of the 3-ball $D^3$ and $||\phi_D||= \displaystyle\sum_{\sigma\in D}\phi_D(\sigma)$ are equal. The boundary of $D$ according to Varopoulos is $\partial_V D= \{\tau: \tau$ is a face of two 3-cells, $ \sigma_i, \sigma_j ;\phi_D(\sigma_i)\neq \phi_D(\sigma_j)\}$, next we define $\parallel \nabla \phi_D \parallel=
\displaystyle\sum_{\tau \in\partial_V D }| \phi_D(t(\tau))-\phi_D(i(\tau))|$, where $i,t$ are functions which determine the initial and terminal vertices of an edge in $\Gamma$. This function gives the number of edges in the boundary $\partial_V D$. In fact, we can show that $\parallel \nabla \phi_D \parallel {\leqslant}Vol^2(S^2)$. Therefore the problem of upper bound reduces to an inequality involving $||\phi_D||$ and $\parallel \nabla \phi_D \parallel$ provided $Vol^3(D^3)=||\phi_D||$ and $Vol^2(S^2){\geqslant}||\nabla \phi_D||$.
Finally, we show that $||\phi_D||{\leqslant}||\nabla \phi_D||^\frac{4}{3}$ for the lattices in the 3-dimensional geometry Nil and $||\phi_D||{\leqslant}||\nabla \phi_D||\ln(||\nabla \phi_D||)$ for the lattices in the 3-dimensional geometry Sol using a variation the Varopoulos transport argument.
Organization of the paper
-------------------------
####
This paper is organized as follows, in the third section we introduce ordinary Dehn functions as well as higher order Dehn functions and discuss results involving higher order Dehn functions.
In the fourth section we give a survey of generalized handle body diagrams in 2 and 3-dimensions which can be thought of as higher dimensional analogs of van Kampen diagrams. We use transverse maps for this and the main result here is to show that a reduced diagram can be obtained from an unreduced diagram without changing the map on the boundary. Reduced diagrams are a key to obtaining upper bounds for second order Dehn function.
The fifth section introduces the structure of the 3-manifolds which are torus bundles over the circle. We then describe the cell decomposition of the torus bundles and introduce the notion of dual graphs in the cell decomposition. Finally we focus on the main examples of this paper which are lattices in the 3-dimensional geometries Nil and Sol.
The main result in the sixth section is that the isoperimetric inequality involving $Vol^3(D^3)$ and $Vol^2(S^2)$ reduces to an inequality between $||\phi_D||$ and $||\nabla \phi_D||$. We do this by defining a dual graph in the ambient space.
In the last section we use the Varopoulos transport argument to obtain the upper bounds of second order Dehn functions in case of both Nil and Sol.
Basic Notions on Dehn Functions {#basic}
===============================
In this section we introduce some basic definitions on ordinary and higher dimensional Dehn functions. We also present a short survey of results involving higher dimensional later in the section. The definitions were primarily taken from [@Bri] and [@Bra].
Dehn Functions {#iso}
--------------
(*Dehn function*). Let $\mathcal{P}= \langle \mathcal{A} \mid \mathcal{R}\rangle$ be the finite presentation of a group $G$, where $\mathcal{A}$ denotes the set of generators and $\mathcal{R}$ denotes the set of all relators.\
We can define the *Dehn* *function* of $\mathcal{P}$ in the following way: ([@Bri])\
Given a word $w = 1$ in generators $\mathcal{A}^{\pm 1}$,\
$Area(w) = $ min$\{ N_w\in {{\mathbb N}}: \exists$ an equality $ w= \displaystyle\prod_{i=1}^{N_w} {x_ir_ix_i^{-1}}; x_i\in F(\mathcal{A})$ and $ r_i\in \mathcal{R} \}$, here $F(\mathcal{A})$ denotes the free group on the generating set $\mathcal{A}$.\
The *Dehn function* of $\mathcal{P}$ is $\delta_{\mathcal{P}}(n) \hspace{0.1cm} = \hspace{0.1cm}$ max$\{Area(w) : |w| {\leqslant}n \}$.
(*Equivalent Functions*).\[eq\] Two functions $f,g:[0,\infty)\rightarrow [0, \infty)$ are said to be $\sim$ equivalent if $f{\preccurlyeq}g$ and $g{\preccurlyeq}f$, where $f{\preccurlyeq}g$ means that there exists a constant $C>0 $ such that $f(x){\leqslant}Cg(Cx)+Cx$, for all $x{\geqslant}0$, (and modulo this equivalence relation it therefore makes sense to talk of “the” Dehn function of a finitely presented group). This equivalence is called *coarse Lipschitz equivalence*.
(*Isoperimetric Function of a Group*). A function $f:{{\mathbb N}}\rightarrow {{\mathbb N}}$ is an *isoperimetric function* for a group $G$ if the Dehn function $\delta_\mathcal{P} {\preccurlyeq}f$ for some (and hence any) finite presentation $\mathcal{P}$ of $G$.
Given a smooth, closed, Riemannian manifold $M$, in the rest of this section we shall describe the isoperimetric function of $M$ and discuss its relationship with the Dehn function of the fundamental group $\pi_1(M)$ of $M$.
Let $c:S^1\rightarrow M$ be a null-homotopic, rectifiable loop and define $FArea(c)$ to be the infimum of the areas of all Lipschitz maps $g:D^2\rightarrow X$ such that $g|_{\partial {D^2}}$ is a reparametrization of $c$.\
Note that the notion of area used here is the same as that of area in spaces introduced by Alexandrov [@Alex]. The basic idea is to define the area of a surface (or area of a map $g:D^2\rightarrow X$) to be the limiting area of approximating polyhedral surfaces built out of Euclidean triangles.
(*Isoperimetric or Filling function*) Let $M$ be a smooth, complete, Riemannian manifold. The genus zero, 2-dimensional, isoperimetric function of $M$ is the function $[0,\infty)\rightarrow [0,\infty)$ defined by, $\hspace{0.5cm}{Fill}^M_0(l):= sup\{FArea(c) \hspace{0.2cm}| \hspace{0.2cm}c:S^1 \rightarrow M \hspace{0.1cm}$ null-homotopic, $ length(c){\leqslant}l\} $.
The Filling Theorem provides an equivalence between Dehn function and the Filling function defined above.
The genus zero, 2-dimensional isoperimetric function $Fill^M_0$ of any smooth, closed, Riemannian manifold $M$ is $\sim$ equivalent to the Dehn function $\delta_{\pi_1 M}$ of the fundamental group of $M$.
Here are a few examples of manifolds and their Dehn functions.
1. The Dehn function of the fundamental group of a compact 2-manifold is linear except for the torus and the Klein bottle when it is quadratic.
2. The groups that interest us are fundamental groups of 3-manifolds and the Dehn functions of these groups can be characterized using the following theorem by Epstein and Thurston.\
*Let $M$ be a compact 3-manifold such that it satisfies Thurston’s geometrisation conjecture ([@Th]).\
The Dehn function of $\pi_1(M)$ is linear, quadratic, cubic or exponential. It is linear if and only if $\pi_1(M)$ does not contain ${{\mathbb Z}}^2$. It is quadratic if and only if $\pi_1(M)$ contains ${{\mathbb Z}}^2$ but does not contain a subgroup ${{\mathbb Z}}^2 \rtimes_\phi {{\mathbb Z}}$ with $\phi \in GL(2,{{\mathbb Z}})$ of infinite order. Subgroups ${{\mathbb Z}}^2 \rtimes_\phi {{\mathbb Z}}$ arise only if a finite-sheeted covering of $M$ has a connected summand that is a torus bundle over the circle, and the Dehn function of $\pi_1(M)$ is cubic only if each such summand is a quotient of the Heisenberg group.*
**Geometric Interpretation of the Dehn function**. {#geom}
--------------------------------------------------
####
The connection between maps of discs filling loops in CW complexes (or in other words a geometric interpretation of the Dehn function defined above) and the algebraic method of reducing words can be explained by any one of the following,
- van Kampen diagrams ([@short]),
- pictures ([@short])and,
- Handle body diagrams (Discussed in Section 3).
Higher dimensional Dehn functions
---------------------------------
####
Epstein et al. [@Ep] and Gromov [@Gr2] first introduced higher dimensional Dehn functions at about the same time. However later, Alonso *et al.* [@Al] and Bridson [@Bri1] provided equivalent definitions which were different from the two mentioned above. In the discussion on higher dimensional Dehn functions presented here we will be using Brady *et al*’s ([@Bra]) definition which is based on the prior definitions given by Bridson and Alonso *et al*. Before we introduce higher dimensional Dehn functions we note the definition of groups of type $\mathcal{F}_n$.
(*Eilenberg-MacLane complex*, [@Bri2]) The Eilenberg-MacLane complex (or classifying space) $K(\Gamma,1)$ for a group $\Gamma$ is a CW complex with fundamental group $\Gamma$ and contractible universal cover. Such a complex always exists and its homotopy type depends only on $\Gamma$.
(*Finiteness property $\mathcal{F}_n$*, [@wall]) A group $\Gamma$ is said to be of type $\mathcal{F}_n$ if it has an Eilenberg-MacLane complex $K(\Gamma,1)$ with finite $n$-skeleton. Clearly a group is of type $\mathcal{F}_1$ if and only if it is finitely generated and of type $\mathcal{F}_2$ if and only if it is finitely presented.
Intuitively, the $k$-*dimensional Dehn function*, $k{\geqslant}1$ , is the function $\delta^{(k)}: {{\mathbb N}}\rightarrow {{\mathbb N}}$ defined for any group $G$ which is of type $\mathcal{F}_{k+1}$ and $\delta^{(k)}(n)$ measures the number of $(k+1)$-cells that is needed to fill any singular $k$-sphere in the classifying space $K(G,1)$, comprised of at most $n \hspace{0.2cm} k$-cells. Up to equivalence the higher dimensional Dehn functions of groups are quasi-isometry invariants.
The following part of this section is devoted to the technical definition of higher dimensional Dehn function given by Brady *et al.*, ([@Bra]).
Henceforth we will denote an $n$-dimensional disc (or ball) by $D^{n}$ and an $n$-dimensional sphere by $S^{n}$.
(*Admissible maps*)\[adm\] Let $W$ be a compact $k$-dimensional manifold and $X$ a CW complex, an admissible map is a continuous map $f: W \rightarrow X^{(k)} \subset X$ such that $f^{-1}(X^{(k)}- X^{(k-1)})$ is a disjoint union of open $k$-dimensional balls, each mapped by $f$ homeomorphically onto a $k$-cell of $X$.
\[volume\] (*Volume of $f$*) If $f: W \rightarrow X$ is admissible we define the volume of $f$, denoted by $Vol^{k}(f)$, to be the number of open $k$-balls in $W$ mapping to $k$-cells of $X$.
Given a group $G$ of type $\mathcal{F}_{k+1}$, fix an aspherical CW complex $X$ with fundamental group $G$ and finite $(k+1)$-skeleton. Let $\widetilde{X}$ be the universal cover of $X$. If $f: S^k \rightarrow \widetilde{X}$ is an *admissible* map, define the *filling volume* of $f$ to be the minimal volume of an extension of $f$ to $D^{k+1}$ in the following way, FVol$(f) \hspace{0.2cm} = \hspace{0.2cm}$min$\{Vol^{k+1}(g)\hspace{0.1cm}|\hspace{0.1cm}g: D^{k+1}\rightarrow \widetilde{X}, g|_{\partial D^{k+1}} =f \}$, then, $k-$dimensional Dehn function of $X$ is $\delta^{(k)}(n) \hspace{0.2cm} = \hspace{0.2cm} $sup$\{$ FVol$(f)\hspace{0.1cm} |\hspace{0.1cm}f: S^{k}\rightarrow \widetilde{X}, Vol^{k}(f){\leqslant}n \}$.
Here are a few observations about higher dimensional Dehn functions,
1. Up to equivalence, $\delta^{(k)}(n)$ is a quasi-isometry invariant.
2. In the above definitions it is possible to use $X$ in place of $\widetilde{X}$ since $f: S^k \rightarrow {X}$ (or $f: D^{k+1} \rightarrow {X}$) and their lifts to $\widetilde{X}$ have the same volume.
All the groups discussed in this paper is at most 3-dimensional so we will restrict $k$ in the above definitions such that $k{\leqslant}2$.
The following are examples of second order Dehn functions.
(*Examples of groups and their second-order Dehn functions*):
1. By definition,the second order Dehn function of a 2-complex with contractible universal cover is linear.
2. The second order Dehn function of any group of every (word) hyperbolic group $H$ is linear and so is the direct product of $H$ with any finitely generated free group, both these results were established by Alonso *et al.* in [@Bog].
3. The second order Dehn function of any finitely generated abelian group with torsion-free rank greater that two is $\sim n^{3/2}$, e.g, ${{\mathbb Z}}^3 $ ([@Wang2]).
Transverse Maps, Handle Decompositions and Reduced Diagrams {#maps}
===========================================================
In this section we will discuss generalized handle decompositions which will help us compute upper bounds of higher dimensional Dehn functions in specific cases later in the paper.
Background on Handle Decompositions
-----------------------------------
####
Any compact, smooth or piecewise linear manifold, admits a handle decomposition ([@Mil], [@Rk]), also each handle decomposition can be made proper (see details in [@Rk]). In 1961 S. Smale [@Sm], established the existence of exact handle decompositions of simply connected and cobordisms of dimensionality $n{\geqslant}6$.
In this paper we will be using the generalized handle decomposition of manifolds, mainly due to Buoncristiano, Rourke, Sanderson, [@Bu]. This reference by Buoncristiano, Rourke, Sanderson ([@Bu]) is a lecture series on a geometric approach to homology theory.
Here they introduce the concept of transverse CW complexes. These complexes have all the same properties of ordinary cell complexes. The result from this article which we will be using in this paper is known as the Transversality Theorem, and using this theorem any continuous map may be homotoped to a transverse map (Definition \[transverse\]). Here is the statement of the Transversality theorem, this theorem is used to show the maps from the handle decompositions we construct to the ambient space are transverse.
\[Buoncristiano, Rourke and Sanderson, [@Bu]\] Suppose $X$ is a transverse CW complex (a CW complex is transverse if each attaching map is transverse to the skeleton to which it is mapped), and $f:M \rightarrow X$ is a map where $M$ is a compact piecewise linear manifold. Suppose $f\mid_{\partial M}$ is transverse, then there is a homotopy of $f\hspace{0.1cm} rel \hspace{0.1cm} \partial M$ to a transverse map.
In fact, if $M$ is a generalized handle decomposition i.e, it is constructed from another manifold with boundary $M_0$, by attaching finite number of generalized handles, then the map $f$ itself is homotopic to a transverse map.
Handlebody Diagrams {#transverse1}
-------------------
####
The following definitions and statement of Transversality theorem were taken from the lecture notes of a course [@For] taught by Max Forester at the University of Oklahoma.
\[handle\] (*Index i-Handle*) An index $i$-handle is written as $H^i=\Sigma^i \times D^{n-i}$, where $\Sigma^i$ is a connected $i$-manifold (we will consider $\Sigma^i = D^i$ in all our examples) and $D^{n-i}$ is a $(n-i)$ closed disk.
**Note**: The boundary of a $i$-handle is $\partial H^i = \partial \Sigma^i \times D^{n-i} \cup \Sigma^i \times \partial D^{n-i}$.\
Given an $n$-manifold $M_0$ with boundary and an $i$-handle $H^i$, let $\phi: \partial \Sigma^i \times D^{n-i} \rightarrow \partial M_0$ be an embedding. Form $M_0 \cup_\phi H^i$ a new manifold with boundary obtained from $M_0$ by attaching an $i$-handle in the following way, $(M_0 \amalg H^i)/(x \sim \phi(x), \forall x \in \partial \Sigma^i \times D^{n-i})$.
\[filtratiion\] (*Generalized Handle Decomposition*) A generalized handle decomposition of $M$ is a filtration: $\emptyset = M^{(-1)} \subset M^{(0)} \subset M^{(1)} \subset ......\subset M^{(n)} = M$ such that:
- Each $M^{(i)}$ is a codimension-zero submanifold of $M$. ($L \subset M$ is a codimension-zero submanifold if $L$ is an $n$-manifold with boundary and $\partial L$ is a submanifold of $M$.)
- $M^{(i)}$ is obtained from $M^{(i-1)}$ by attaching finitely many $i$-handles.
In case $M$ is a compact $n$-manifold with boundary denoted by, $\partial M$, then the generalized handle decomposition of $M$ is:
- A generalized handle decomposition of $\partial M$, namely: $\emptyset =N^{(-1)} \subset N^{(0)} \subset N^{(1)} \subset ......\subset N^{(n-1)} = \partial M$, where each $M^{(i)}$ is a codimension-zero submanifold of $\partial M$
- A filtration of $M$, $\emptyset = M^{(-1)} \subset M^{(0)} \subset M^{(1)} \subset ......\subset M^{(n)} = M$ where each $M^{(i)}$ is a codimension-zero submanifold of $M$ and $M^{(i)}$ is obtained from $M^{(i-1)}\cup N^{(i-1)}$ by attaching $i$-handles.
- Each $(i-1)$-handle of $N$ is a connected component of the intersection of $N$ with an $i$-handle of $M$ (this means that $N^{(i-1)}=\partial M\cap M^{(i)}$).
\[transverse\] (*Transverse Maps*) Let $M$ be a compact $n$-manifold and $X$ a cell-complex. A continuous map $f:M \rightarrow X $ is *transverse* if $M$ has a generalized handle decomposition such that for every handle $H^i=\Sigma^i \times D^{n-i}$ in $M$, the restriction $f\mid _{H^i} : \Sigma^i \times D^{n-i} \rightarrow X$ is given by $\phi \circ {pr}_2$ where ${pr}_2: \Sigma^i \times D^{n-i} \rightarrow D^{n-i}$ is a projection map to the second coordinate and $\phi$ is the characteristic map of an $(n-i)$-cell of $X$. We will refer to the generalized handle decomposition of $M$ as a *handle body diagram* or just a *diagram*.
**Note** An $i$-handle maps to a $(n-i)$-cell this implies, $f(M) \subset X^{(n)} =X$.
(*“good” CW complex*)\[good\] A CW complex is “good” if and only if, each attaching map is transverse to the skeleton to which it is mapped.
Next we have a version of the Transversality theorem which we will refer to later in this paper.
\[Ttheorem\] If $X$ is an $n$-dimensional, “good” CW complex and $M$ is a generalized handle decomposition of a compact $n$-manifold, then every continuous map $f: M \rightarrow X$ is homotopic to a transverse map $g$. Moreover, if $f \mid _{\partial M}$ is transverse, then there is a homotopy of $f$ rel $\partial M$ to a transverse map.
\[complex\] Every cell-complex is homotopy equivalent to a “good” cell-complex.
(*Unreduced Diagram*). \[reduced\] A diagram $f: (D^n,S^{n-1}) \rightarrow K$ is said to be *unreduced* if in the interior of $(D^n,S^{n-1})$ there exists two 0-handles $H^0_1$ and $H^0_2$ joined together by a 1-handle such that, $f(H^0_1)=f(H^0_2)$ is an open $n$-cell in $K$ and $({f|_{H^0_1}}^{-1}\circ f|_{H^0_2})$ is an orientation reversing map. Otherwise, the diagram is said to be *reduced*.
In other words, a diagram is *unreduced* if there exists another diagram with the same boundary length or area (in case of 2 or 3-dimensional cases respectively) but strictly smaller filling area or volume for 2 or 3-dimensional cases respectively. Under these circumstances we will eliminate these 0-handles along with the 1-handle connecting them but keeping the boundary of the diagram same and ensuring that we still have a disc. Hence, our intention is to get a *reduced* diagram from an *unreduced* one.
Next we will discuss how to obtain a reduced diagram from an unreduced one. This argument was given by Brady and Forester ([@For1]).\
\[annul\] Let $f: (D^n, S^{n-1}) \rightarrow K$ be an admissible map, and let $H^0_1$ and $H^0_2$ be 0-handles in $(D^n, S^{n-1})$ connected together with a 1-handle. Let $\alpha$ be a core curve in the 1-handle connecting $H^0_1$ and $H^0_2$ homeomorphic to an interval (Figure \[alpha\]). Suppose $f$ maps $\alpha$ to a point and maps $H^0_1$ and $H^0_2$ to the same $n$-cell, with opposite orientations. As $H^0_1$ and $H^0_2$ are 0-handles, there are homeomorphisms $h_i:(H^0_i, \partial H^0_i)\rightarrow (D^n,S^{n-1}) $ such that $f|_{H^0_i}= \phi \circ h_i$ for some characteristic map $\phi :(D^n,S^{n-1}) \rightarrow K$. We first consider the curve $\alpha$ along with a tubular neighborhood around it and collapse it to a point to get part $(ii)$ of Figure \[alpha\]. Next remove the interiors of $H^0_i$ from $(D^n, S^{n-1})$ and form a quotient $(D^n_1, S^{n-1}_1)$ by gluing boundaries via $h_0^{-1}\circ h_1$, an orientation reversing map. The new space maps to $K$ by $f$, and there is a homeomorphism $g:(D^n, S^{n-1}) \rightarrow (D^n_1, S^{n-1}_1)$. Now $f\circ g$ is an admissible map $(D^n, S^{n-1}) \rightarrow K$ with two fewer 0-handles. The map can be then be made transverse with the rest of the 0-handles unchanged. Figure \[alpha\] illustrates the method pictorially.
$H_2^0$ at 170 239 $H_1^0$ at 314 239 $\alpha$ at 262 230 $(i)$ at 240 181 $(ii)$ at 136 -6 $(iii)$ at 451 -6 $f$ at 141 192 $g$ at 293 91 $H_2^0$ at 80 44 $H_1^0$ at 162 46 ![$(i)$ two 0-handles joined by a 1-handle and core curve $\alpha$, $(ii)$ Picture of $(i)$ after $\alpha$ has been removed, and $(iii)$ Final Picture[]{data-label="alpha"}](alpha "fig:")
The connection between Linear Algebra and Cell Decomposition of Mapping Tori {#decomposition}
============================================================================
In this section we discuss the structure of the 3-dimensional manifolds that have the lattices of the Nil and Sol geometries as fundamental groups.
The 3-manifolds considered here are the mapping tori where the attaching maps corresponds to matrices in $SL_2({{\mathbb Z}})$. In other words, given a group of the form ${{\mathbb Z}}^2 \rtimes_{\psi_A} {{\mathbb Z}}$, where $\psi_A \in Aut({{\mathbb Z}}^2)$ and $A \in SL_2({{\mathbb Z}})$, the geometric realization of these groups are mapping tori where the attaching maps are the automorphisms of ${{\mathbb Z}}^2$. For example if $\psi_A$ is the identity map then, the corresponding space is ${{\mathbb Z}}^3 \subset {{\mathbb R}}^3$. Other specific examples we are interested in are the lattices in the 3-dimensional geometries Nil and Sol. In particular we will be looking at lattices corresponding to the matrix $\left(\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}\right)$ for Nil and $\left(\begin{array}{cc}
2 & 1 \\
1 & 1 \\
\end{array}\right)$ in case of Sol. Another way of looking at these are as torus bundles over the circle and they are described below.
Let us denote the mapping torus $\frac{T \times I}{(t,0)\sim
(\psi_A(t),1)} $ by $A_T$, where $\psi_A$ is the attaching map. Let $\psi_A$ be represented by the matrix $A\equiv$ $\left(\begin{array}{cc}
x & z\\
y & w \\
\end{array}\right)\in SL_2({{\mathbb Z}})$ . So, if the generating curves of the torus in $A_T$ are labeled $a,b$, then the presentation of the corresponding fundamental group is given by, $ \Gamma=\langle a,b, t \mid [a,b],
tat^{-1}=A(a)=a^xb^y,\hspace{0.15cm}tbt^{-1} = A(b)=a^zb^w \rangle$.\
Cell Decomposition of the Mapping Torus $A_T$
---------------------------------------------
####
We know that the mapping torus $A_T$ consists of two copies of the torus attached via the map $\psi_A$. Here we will demonstrate an effective way of triangulating the 2-cell spanned by the generators of the group $\Gamma$ and hence obtain a model space for $\Gamma$.
We subdivide the 2-cells of both copies of the torus in $A_T$ into either a number of triangular faces or a combination of triangular and quadrilateral faces. The following example illustrates this process in details.
\[lattice1\] Let $A \equiv \left(\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}\right)$, then the corresponding group is the 3-dimensional, integral Heisenberg group $\mathcal{H} = \langle a,b,t \hspace{0.15cm}\mid
\hspace{0.15cm} [a,b],\hspace{0.1cm} tat^{-1}=a,\hspace{0.1cm}
tbt^{-1}=ab \rangle$.
$a$ at 152 192 $b$ at 196 154 $a$ at 152 110 $b$ at 110 153 $a$ at 396 110 $b$ at 500 157 $a$ at 458 190 $b$ at 356 153 $\psi_A$ at 274 133 ![Sub-division of ${{\mathbb R}}^2$ under the action of the map $\psi_A$[]{data-label="grid"}](nilgrid1 "fig:")
$A$ at -5 -10 $B$ at -4 110 $C$ at 55 160 $D$ at 146 -5 $E$ at 206 51 $F$ at 206 160 $G$ at 145 105 $H$ at 58 58 $a$ at 81 24 $b$ at 122 66 $c$ at 28 31 $t$ at 215 105 $b$ at 65 -5 $b$ at 125 162 $a$ at 24 136 $a$ at 172 119 $c$ at 83 134 $c$ at 164 20 $b$ at 82 98 $t$ at -5 58 ![The mapping torus corresponding to the matrix $A$ above.[]{data-label="nil"}](nil12 "fig:")
This subdivision of the mapping tori below, (Figure \[nil\]) shows that the top has been divided into two triangular faces each of which can be mapped via $\psi_A$ to their exact replicas in base. The 3-cell in (Figure \[nil\]) also serves as the fundamental domain for the action of $\mathcal{H}$ on the corresponding universal cover. The base point is named $A$ and all other vertices of the cell are also labeled.
\[lattice2\] If we have the matrix $B \equiv \left(\begin{array}{cc}
2 & 1 \\
1 & 1 \\
\end{array}\right)$, then the corresponding group presentation is $\mathcal{S} =
\langle a,b,t \hspace{0.15cm}\mid \hspace{0.15cm}
[a,b],\hspace{0.1cm} tat^{-1}=a^2b,\hspace{0.1cm} tbt^{-1}=ab
\rangle$.\
$b$ at 120 81 $b$ at 45 81 $a$ at 75 43 $a$ at 81 114 $b$ at 386 73 $b$ at 383 133 $a$ at 401 99 $a$ at 347 101 $a$ at 458 101 $\psi_B$ at 240 117 ![Sub-division of two copies of ${{\mathbb R}}^2$ under the action of $\psi_B$ .[]{data-label="solgrid1"}](solgrid12 "fig:")
Again (Figure \[solgrid1\]) above shows that the subdivision is compatible to the relations in the group presentation $\mathcal{S}$.
$A$ at 4 -4 $B$ at -2 100 $C$ at 31 157 $D$ at 166 -2 $E$ at 195 51 $F$ at 197 157 $G$ at 173 97 $H$ at 26 51 $b_1$ at 141 34 $b_2$ at 44 24 $b_1$ at 2 115 $b_2$ at 19 146 $b_1$ at 180 110 $b_2$ at 190 140 $c_1$ at 148 116 $c_2$ at 63 141 $c_1$ at 66 58 $c_2$ at 144 58 $c_1$ at 34 -2 $c_2$ at 120 -2 $a$ at 95 19 $a$ at 99 155 $t$ at 172 71 $t$ at 200 91 $t$ at 0 69 $M_1$ at 185 125 $M_2$ at 5 131 $M_3$ at 100 57 $M_4$ at 81 -2 $d$ at 55 126 $d$ at 14 31 $d$ at 189 26 ![The mapping torus corresponding to the matrix $B$ above.[]{data-label="sol"}](sol12 "fig:")
This triangulation of the mapping tori below, (Figure \[sol\]) shows that the top has been divided into four triangular faces each of which can be mapped via $\psi_B$ to their exact replicas in base. This 3-cell serves as the fundamental domain for the action of $\mathcal{S}$ on the corresponding universal cover.
Upper Bounds- Reduction to Varopoulos Isoperimetric Inequality
==============================================================
Sections 6 and 7 are devoted to obtaining upper bounds for the second order Dehn functions of $\mathcal{H}$ and $\mathcal{S}$ using a variation of Varopoulos Transport argument.
In section 6 we reduce the original isoperimetric problem involving volume of 3-balls and areas of their boundary 2-spheres to a problem involving Varopoulos’ notion of volume and boundary of finite domains in dual graphs.\
Definitions
-----------
Since we will use barycentric subdivisions to obtain the dual graph, we will start this section with the following definitions. (These definitions and notations have been taken from[@Bri].)
(*Barycentric Subdivision of a convex polyhedral cell*) Let $C$ be a polyhedral cell in an $n$-dimensional polyhedral complex $K$. The barycentric subdivision of $C$ denoted by $C'$ is the simplicial complex defined as follows:\
There is one geodesic simplex in $C'$ corresponding to each strictly ascending sequence of faces $F_0 \subset F_1
\subset....\subset F_n$ of $C$; the simplex is the convex hull of barycenters of $F_i$. Note that the intersection in $C$ of two such simplices is again such a simplex. The natural map from the disjoint union of these geodesic simplices to $C$ imposes on $C$ the structure of a simplicial complex - this is $C'$.
(*Barycentric Subdivision of a polyhedral $n$-complex $K$*) Let $p: \coprod_\lambda C_\lambda \rightarrow K$ (where $C_\lambda$ are the polyhedral cells of $K$), be a projection. For each cell $C_\lambda$ we index the simplices of the barycentric subdivision $C'_{\lambda}$ by a set $I_\lambda$; so $C'_{\lambda}$ is the simplicial complex associated to $\coprod_{I_\lambda}S_i \rightarrow C_\lambda$ where $S_i$ denotes the simplices of $C'_\lambda$. Let $\lambda'= \coprod_\lambda
I_\lambda$. By composing the natural maps $\coprod_{I_\lambda}S_i
\rightarrow C_\lambda $ and $p: \coprod_\lambda C_\lambda
\rightarrow K$ we get a projection $p': \coprod_{i\in \lambda'}S_i
\rightarrow K$. Let $K'$ be the quotient of $\coprod_{i\in
\lambda'}S_i$ by the equivalence relation $[x\sim y$ iff $p'(x)=
p'(y)]$. $K'$ is the barycentric subdivision of $K$.
**Note** Given any complex, there is a poset $\mathcal{P}$ on the cells of the complex ordered by inclusion. Therefore for any ascending chain in $\mathcal{P}$ there is a simplex in the barycentric subdivision of the complex.
Dual Graphs {#dual}
-----------
####
The examples in the previous section gives us an idea of the cell decomposition of the spaces under consideration. The groups considered here are all finitely generated, so the groups act properly and cocompactly by isometries on their respective universal covers. In fact, the translates of the fundamental domain covers the universal cover $\widetilde{X}$ in each case.
It is essential to mention here that the only groups we are interested in are the 3-dimensional groups $\mathcal{H}$ and $\mathcal{S}$ from Section \[decomposition\] and we will use the letter $G$ to refer to them in general.
Next, we define the dual graph $\Gamma$ using Definition \[graph\]. The vertex set of $\Gamma$, $V_\Gamma=\{ \sigma: \sigma$ is a 3-cell of $\widetilde{X} \}$ while the edge set is, $E_\Gamma =\{ \tau: \tau$ is a codimension one face (2-cells) shared by two adjacent 3-cells of $K \}$.
There is a map that embeds the graph $\Gamma$ in $K$.
Consider the barycentric subdivision of both the graph $\Gamma$ and the universal cover $\widetilde{X}$, we denote these barycentric subdivisions by $\Gamma'$ and $\widetilde{X'}$ respectively. Next we map the vertices in $V_\Gamma$ to the barycenters of the 3-cells while we map the barycenter of an edge $\tau$, labeled by $\tau_m$ in $E_\Gamma$ to the barycenter of the codimension one face shared by the two 3-cells in $V_\Gamma$, serving as the initial and terminal vertices of $\tau$. Finally, if $\tau$ is an edge with initial and terminal vertices $\sigma_1$ and $\sigma_2$ respectively, then, the left half-edge of $\tau$ is mapped to the simplex in $K'$ corresponding to the ascending chain $\tau \subset \sigma_1$ in the poset $\mathcal{P}$ while, the right half-edge maps to the simplex in $K'$ corresponding to the ascending chain $\tau\subset \sigma_2$ in $\mathcal{P}$.
As there is a natural bijection between the barycentric subdivision of a space and the geometric realization of the space itself so, there is a map that embeds $\Gamma$ in $K$.
So, now we have a dual graph in $\widetilde{X}$ which is also a Cayley graph (with the same name $\Gamma$), with respect to a finite generating set which we will define subsequently. The aim of the remaining part of this section is to show that $\Gamma$ is quasi-isometric to $\widetilde{X}$ using the following lemma.
($\check{S}varc-Milnor$ Lemma,[@Bri]) Given a length space $X$. If a group $G$ acts properly and cocompactly by isometries on $X$, then $G$ is finitely generated and for any choice of basepoint $x_0 \in X$, the map $f: G \rightarrow X$, defined by $g \mapsto g.x_0$ is a quasi-isometry.
Let $C$ be the fundamental domain of $\widetilde{X}$ ( a compact subset of $\widetilde{X}$ such that its translates covers all of $\widetilde{X}$). We then define the generating set of the group $G$ in the following way, $\mathcal{A}= \{ g\in G \hspace{0.2cm}\mid \hspace{0.2cm} gC \cap C =codimension-one \hspace{0.1cm} face \}$.
In case of $\mathcal{H}$ the valence of a vertex is eight, while in the case of $\mathcal{S}$ the valence is twelve. Hence, the generating sets in these cases will contain four and six elements respectively. We will define the generating sets in detail for specific examples i.e, for the groups $\mathcal{H}$ and $\mathcal{S}$ in the following lemma.\
**Note**: In the following lemma, we shall denote the triangular faces of the cell decomposition obtained in the previous section as $\triangle XYZ$, where $X,Y,Z$ are the labels of vertices in the cell decomposition forming a triangle.
\[generator\] Given the cell decompositions for groups $\mathcal{H}$ and $\mathcal{S}$ in section \[decomposition\]:
1. $\mathcal{A}_0 = \{ b,c,t, tb\} $ is a finite generating set for $\mathcal{H}$, where $c=b^{-1}a$ (from Figure \[nil\]).
2. $\mathcal{A}_0 = \{ d,t,c_1c_2,td^{-1},tc_2^{-1}c_1^{-1},tb_1c_1^{-1}\} $ is a finite generating set for $\mathcal{S}$, where, $a_1a_2=a, d=ba=ab, c_1= ab_1, c_2=b_2a $ (from Figure \[sol\]).
(*of (1)*) We consider Figure \[nil\] for this part of the proof. The vertex $A$ is chosen as the base point of universal cover $\widetilde{X}$. The paths that take the base point to its images in copies of the fundamental domain (which are 3-cells sharing codimension one faces with the fundamental domain) represent the isometries that take the domain to its copies and hence they are the generators of the group with respect to the Cayley graph $\Gamma$. In case of $\mathcal{H}$, there are eight other 3-cells sharing codimension one faces with the fundamental domain or in other words, due to the cell decomposition shown in section \[decomposition\], any 3-cell in the universal cover shares a codimension one face with eight other 3-cells.
In the following lines we give a list of isometries and hence the words which generate translates of the fundamental domain that share a codimension one face with the domain.
The path from $A$ to $D$ represents the isometry $b$ taking the domain to the 3-cell to its right; path from $A$ to $H$ represents the word $c$ takes the domain to the cell behind itself; $A$ to $B$, the word $t$ takes the domain to the 3-cell on the face $\triangle BCG$; path from $A$ to $G$, the word $tb$ takes the domain to the 3-cell on the face $\triangle GFC$. The isometries that take the domain to the rest of the neighboring 3-cells, are inverses of the words already mentioned above. For example the isometry taking the domain to the 3-cell sharing the face $\triangle ADE$ is $t^{-1}$, while the one taking it to the 3-cell associated with the face $\triangle AHE$ is $b^{-1}t^{-1}$ etc. So it is clear that $\mathcal{A}_0 = \{ b,c,t, tb\} $ is a finite generating set for $\mathcal{H}$ and ${\mathcal{A}_0}^{-1} = \{b^{-1},c^{-1},t^{-1},b^{-1}t^{-1} \} $.
(*Proof. of (2)*) This can be shown in a similar way as above. In this case, the fundamental domain shares codimension one faces with twelve other 3-cells, (four cells each above and below, two on each side and the remaining two at the front and back). As before the translates $d$ and $t$ generate copies to the right and vertically above (and sharing the face $\triangle BFM_1$) the fundamental domain respectively. The translate $td^{-1}$ generates the copy sharing the face $\triangle BGM_1$, while $tc_2^{-1}c_1^{-1}$ generates the copy of the fundamental domain along the face $\triangle BM_2F$. Finally $tb_1c_1^{-1}$ is responsible for the copy of the domain sharing the face $\triangle M_2CF$ with the fundamental domain. So $\mathcal{A}_0 = \{ d,t,c_1c_2,td^{-1},tc_2^{-1}c_1^{-1},tb_1c_1^{-1}\} $. Also, it is easy to check that ${\mathcal{A}_0}^{-1} = \{d^{-1}, t^{-1}, dt^{-1},c_1c_2t^{-1},c_1b_1^{-1}t^{-1}\}$.
\[cay\] Cay($G, \mathcal{A}_0$), the Cayley graph of the group $G$ with respect to the generating sets $\mathcal{A}_0$ defined in Lemma \[generator\] is quasi-isometric to $\widetilde{X}$ .
Milnor’s Lemma says that the group $G$ is finitely generated and quasi-isometric to the ambient space $\widetilde{X}$. But the Cayley graph Cay$(G,\mathcal{A})$ with respect to any finite generating set $\mathcal{A}$ of the group $G$, is quasi-isometric to the group itself, this quasi-isometry can be seen as the natural inclusion $G \hookrightarrow Cay(G,\mathcal{A})$, defined by $g \mapsto g.1$ for all $g \in G$. This last quasi-isometry is also a simple illustration of Milnor’s Lemma.
Finally, two Cayley graphs associated to the same group but with different generating sets are quasi-isometric, this implies Cay$(G, \mathcal{A}_0)$ is quasi-isometric to $\widetilde{X}$.
Definitions and Notations {#var}
-------------------------
####
We start with the definition of a dual graph (Section \[dual\]).
\[graph\] Given an ambient $n$-dimensional space $K$, we define a graph $\Gamma$ with vertex set $V_\Gamma
\hspace{0.2cm}=\hspace{0.2cm}\{\sigma: \hspace{0.2cm}\sigma$ is a $n$-cell of $K\}$ and edge set $E_\Gamma\hspace{0.2cm}=\hspace{0.2cm}\{\tau: \hspace{0.2cm}\tau$ is a $(n-1)$-cell and $\tau$ is a face of exactly 2 $n$-cells of $K\}$.
Given a finitely presented group $G$, let $X$ be the corresponding $n$-dimensional cell-complex and let $\widetilde{X}$ be its universal cover. Let $f: (D^n, S^{n-1})\rightarrow \widetilde{X}$ be a reduced diagram (defined in Section \[transverse1\]) where $D^n$ and its boundary sphere $S^{n-1}$ are either embedded or immersed in $\widetilde{X}$. Note that the map $f$ considered here is transverse and hence admissible, so each $i$-handle in the diagram maps to an $(n-i)$-cell in $\widetilde{X}$. Next, we consider a finite subset $D$ of the vertex $V_\Gamma$ such that, $D=\{\sigma: \sigma $ is an $n$-cell in $\widetilde{X}$ such that $\sigma \in Im(f)\}$. Associated with $D$ is a function analogous to a characteristic map, given by, $\phi_D: V_\Gamma \rightarrow \mathbb{N}\cup \{0\}$ defined by, $\phi_D(\sigma)=$ number of pre-images of $\sigma$ under $f$.
\[phi1\] Let $||\phi_D||= \displaystyle\sum_{\sigma\in D}\phi_D(\sigma)$, this is the number of 0-handles in the diagram i.e, $||\phi_D||= Vol^n(D^n)$ where $Vol^n(D^n)$ denotes the volume of the $n$-ball $D^n$.
It is clear that if $f$ is an embedding in the above definition then $\phi_D$ is in fact the characteristic function of the set $D$.
\[vboundary\] The Varopoulos boundary of $D$ is defined to be the set of all $(n-1)$-cells $\tau \in E_\Gamma$ such that $\tau$ is a face of exactly two $n$-cells $\sigma_i,\sigma_j \in V_\Gamma$ such that $\phi_D(\sigma_i)\neq \phi_D(\sigma_j)$.\
**Notation**: The Varopoulos boundary will be denoted by, $\partial_V D$.
Next we define $\nabla \phi_D:E_\Gamma \rightarrow {{\mathbb N}}\cup \{0\}$ by, $\nabla \phi_D(\tau)= |\phi_D(t(\tau))-\phi_D(i(\tau))
|$, where $i$ and $t$ have the same definition as before.\
The cardinality of the Varopoulos boundary $|\partial_V D|$, in this case can be given by,\
$\hspace{2.5cm}\parallel \nabla \phi_D \parallel=
\displaystyle\sum_{\tau \in\partial_V D }| \phi_D(t(\tau))-\phi_D(i(\tau))
|$. Note that this definition says that $\tau \in E_\Gamma$ is a boundary edge of $D$ if $\phi_D(t(\tau)) \neq \phi_D(i(\tau))$.
Reducing to Varopoulos Isoperimetric Inequality
------------------------------------------------
####
In this section we show that our problem to obtain an upper bound for the second order Dehn functions can be reduced to finding an inequality between volume and boundary notions according to Varopoulos in case of $\mathcal{H}$ and $\mathcal{S}$. We start with the following lemma which works in general for dimensions 1 or more.
\[bound\] $\parallel \nabla \phi_D \parallel {\leqslant}| \partial
D^n |$, where $|\partial D^n |$ is the area or volume of the boundary sphere of the diagram $(D^n,S^{n-1})$ for $n>1$.
Let us consider the n-dimensional reduced diagram $g: (D^n,S^{n-1}) \rightarrow \widetilde{X}$ (Definition \[reduced\]). Let $\tau \in \widetilde{X}$ be the $(n-1)$-cell such that $i(\tau)= \sigma_1$ and $t(\tau)= \sigma_2$, for $\sigma_1,\sigma_2 \in D$. In terms of poset $\mathcal{P}$, $\tau \subset \sigma_1$ and $\tau \subset \sigma_2$ where $\sigma_1,\sigma_2$ are $n$-cells in $\widetilde{X}$ such that $\sigma_1,\sigma_2 \in D (\subset
V_\Gamma)$ and $\phi_D(\sigma_1)\neq \phi_D(\sigma_2)$.
at 85 116 at 133 116 at 178 116 at 314 107 at 130 53 at 135 13 at 275 108 at 314 107 ![2-dimensional example with pre-images $\sigma_1$ , $\sigma_2$ and 1-cell $\tau$ in $(D^2,S^1)$[]{data-label="cells"}](van1 "fig:")
By the definition of $\phi_D$, there are $\phi_D(\sigma_1)$ 0-handles in $(D^n,S^{n-1})$ that map onto $\sigma_1$ via $g$ and similarly there are $\phi_D(\sigma_2)$ 0-handles in $(D^n,S^{n-1})$ that map onto $\sigma_2$ via $g$. Next, since we have $\phi_D(\sigma_1)\neq \phi_D(\sigma_2)$, this implies $\tau$ is one of the $(n-1)$-cells forming the boundary $(n-1)$-sphere, i.e, $\tau \in \partial_V D$ and as both $n$-cells have more than one pre-images, thus, $\tau$ too has one or more pre-images in $(D^n,S^{n-1})$ associated with pre-images of both $\sigma_1$ and $\sigma_2$. The pre-images of $\sigma_1$ and $\sigma_2$ are either in the interior of $(D^n,S^{n-1})$ with pre-images of $\tau$ or they are at the boundary with $\tau$ as a boundary $(n-1)$-cell in some instances.
at 50 140 at 116 116 at 181 148 at 71 58 at 150 41 ![3-dimensional example with pre-images for cells $\sigma_1$ , $\sigma_2$ and 2-cell $\tau$ in $(D^3,S^2)$ []{data-label="cells1"}](van2 "fig:")
If all the pre-images of $\sigma_1$ and $\sigma_2$ are in the interior of $(D^n,S^{n-1})$ with all pre-images of $\tau$ in the interior, then this implies $\phi_D(\sigma_1) = \phi_D(\sigma_2)$, which is against our assumption. Without loss of generality let us assume that $\phi_D(\sigma_1) > \phi_D(\sigma_2)$. In this case if at most $\phi_D(\sigma_2)$ of the pre-images are in the interior of $(D^n,S^{n-1})$, then as we are considering handle decomposition of $n$-balls which are manifolds, the only way a pre-image of $\sigma_2$ appears in the interior is if it is accompanied with a pre-image of $\sigma_1$ and they share a pre-image of $\tau$ which is a 1-handle. Figures \[cells\] and \[cells1\] are illustrations of this in two and three dimensions respectively, where $\tau^i$ denotes a pre-image of $\tau$ while $\sigma_k^i$ etc. denotes the pre-images of $\sigma_k$ for $k=1,2$. In these figures, one pre-image of $\tau$, a 1-handle, is in the interior of $(D^n,S^{n-1})$ between pre-images of $\sigma_1$ and $\sigma_2$, while the other is at the boundary adjoined to the 0-handle which is another pre-image of $\sigma_1$ . This implies that at least $(\phi_D(\sigma_1) - \phi_D(\sigma_2)) = (\phi_D(i(\tau) - \phi_D(t(\tau))$ of the pre-images of $\sigma_1$ are at the boundary of $(D^n,S^{n-1})$ with $\tau$ as a boundary $(n-1)$-cell. Thus, $|\partial D^n | {\geqslant}\displaystyle \sum_{\tau\in \partial_V D} | \phi_D(i(\tau) - \phi_D(t(\tau)| $ which implies, $\parallel \nabla \phi_D \parallel {\leqslant}|\partial D^n|$.
**Note**: At this point, the problem involving the volume of the balls $Vol^n(D^n)$ and the area or volume of the boundary sphere $|\partial D^{n-1}|$, has reduced to one involving $||\phi_D||$ and $||\nabla\phi_D||$. In the next section we are going to use Varopoulos transport argument to prove the isoperimetric inequality involving $||\phi_D||$ and $||\nabla\phi_D||$. In case of the group $\mathcal{H}$ we will show that $||\phi_D||{\leqslant}const.||\nabla\phi_D||^{\frac{4}{3}} $ and in the case of $\mathcal{S}$, we will show that $||\phi_D||{\preccurlyeq}const.||\nabla\phi_D||\ln(||\nabla\phi_D||)$. These inequalities automatically provide upper bounds for the second order Dehn functions in both cases.
Upper Bounds- Varopoulos Transport Argument {#isop}
===========================================
####
In this section we are going to use Varopoulos transport to obtain isoperimetric inequalities in case of groups $\mathcal{H}$ and $\mathcal{S}$. We are going to consider reduced diagrams, since in case they are unreduced we can always use Proposition \[reduced\] from Section \[transverse1\] to obtain a reduced diagram. As before, we will denote the volume of an $n$-ball by $|D^n|$ and the volume of its boundary by $|\partial D^n|$, for any dimension $n$.
The Varopoulos isoperimetric inequality and Dehn functions have very little in common with each other. The only cases where they appear likely to agree are when the groups are fundamental groups of manifolds and also we are considering only top dimensional Dehn functions. So, in the cases we have here we can apply Varopoulos transport to obtain the isoperimetric inequality and hence the upper bounds of second order Dehn functions.
Intuition behind the Varopoulos argument {#intui}
----------------------------------------
####
In this section, we present the intuition behind the notion of transportation of mass from a finite-volume subset of a space. It is important to note here that all our examples are finitely presented groups and the space under consideration will be the universal covers associated to the groups.
The following argument is originally due to Varopoulos [@V]. It was used by Gromov in [@Gr] to demonstrate the transportation of mass (volume) in ${{\mathbb R}}^n$ and also that of a finite subset of group. This notion of transport was first described by Varopoulos in [@V], where he described transport in association with random walks. The same argument was further discussed by Gromov in [@Gr]. Gromov also used this argument in his paper on Carnot-Carathéodory spaces [@G]. The lemma here is appropriately called “Measure Moving lemma” and helps in the proof of isoperimetric inequalities of hypersurfaces in Carnot-Carathéodory manifolds. Before going into the technical details of the argument in Section \[comp\], we will sketch the idea behind the argument and the reason it works, in this section.
Given a graph $\Gamma$ let $D$ be a finite subset of the vertices of the graph transported by a path $\gamma$, then the amount of mass transported through the boundary of $D$ is obviously bounded above by $(|\gamma |
vol(\partial D))$. But we have to find a particular $\gamma$ to bound $vol(D)$ by $vol(\partial D)$, for this we compute average transport. Transport of $D$ corresponding to some $\gamma$ is defined as the mass of $D$ that is moved out of $D$ by the action of $\gamma$. In other words it is the number of vertices in the set $(D\gamma \setminus D)$, where $D\gamma \hspace{0.1cm}=\hspace{0.1cm} \{v\gamma \mid v \in D\} $.
at 80 160 at 48 74 at 152 81 ![Transport of $D$ []{data-label="D"}](D2 "fig:")
Next for the lower bound for the transport we have to show that it is possible to move a percentage of the set $D$ off it. It is always possible to choose the path $\gamma$ such that $l_\gamma$ is large enough that almost all of $D$ is transported off itself, but the key is to find a $\gamma$ in the graph such that it is small enough and moves at least half of $D$ off itself. Since the shape of $D$ maybe very unpredictable Figure \[avg\], therefore transport via a path $\alpha$ maybe very small compared to the mass of $D$ again for another path $\gamma$ the transport maybe very large. In order to solve this problem we bound the length of the path by considering a ball of radius $R$ in $\Gamma$, denoted by $B(R)$ such that $|B(R)| \approx 2|D|$ and taking the average transport over all $\gamma \in B(R)$. Once we show that the average transport is at least half of $D$, we know that there is at least one path $\gamma_0$ such that the transport of $D$ via $\gamma_0$ is at least half of the mass of $D$. This inequality in turn leads to the respective isoperimetric inequalities of the groups we discuss in this context.
at 61 23 at 48 74 at 109 160 ![Transport of $D$ with $\alpha$ is small compared to the mass of $D$ while that with respect to $\gamma$ is large compared to $D$ []{data-label="avg"}](avg1 "fig:")
The Transport Computation {#comp}
-------------------------
####
Given a finitely presented group $G$, let the $\Gamma$ be the dual Cayley graph (defined in Section \[var\]) corresponding to the universal cover of $n$-complex $X$ corresponding to $G$. This graph is infinite but it is locally finite. The edges are directed and labeled, also there is only one outgoing (incoming) edge with a given label at any vertex. $\Gamma$ is a Cayley graph with respect to the presentation of the groups defined in Section \[dual\]. Also the graph is endowed with the path metric and each edge is isomorphic to the unit interval $[0,1]$.
As defined in the previous section, in the following discussion the vertex set of $\Gamma$ will be denoted by $V_\Gamma$ and edge set by $E_\Gamma$. Next, consider the subset $D$ in $V_\Gamma$ corresponding to the $n$-cells in the image of $f:(D^n,S^{n-1})\rightarrow X$. Let us consider the case when $f$ is an embedding. Then we denote the map $\phi_D$ by the characteristic function $\chi_D : V_\Gamma \rightarrow
\{0,1\}$ defined by $\chi_D(\sigma)= 1$ when $\sigma \in D$, otherwise $\chi_D(\sigma)=0$. In this case $||\chi_D||=|D|$, where $|D|$ denotes the number of vertices in $D$.
Next, $\nabla\chi_D : \partial_V D \rightarrow \{0,1\}$ is defined in the following way,\
$\nabla\chi_D(\tau) = |\chi_D(t(\tau)) - \chi_D(i(\tau))|$, where $i,t: E_\Gamma \rightarrow V_\Gamma$ gives the initial and terminal vertices respectively of any edge in $E_\Gamma$.
Therefore, $| \partial_{V} D |\hspace{0.1cm}=\hspace{0.1cm}
\displaystyle\sum_{\tau \in \partial_{V} D} \hspace{0.2cm}| \chi_D(t(\tau)) - \chi_D(i(\tau))
|$.
Let $\gamma \in {B}(r)\subset \Gamma$, where ${B}(r)$ represents a ball of radius $r$ in the graph. We choose $r$ large enough such that $|{B}(r)| {\geqslant}2| D |> |{B}(r-1)|$.\
Varopoulos Transport $T_D^\gamma
\hspace{0.2cm}=\hspace{0.2cm}|D\gamma \setminus D|$\
Average Transport $\widehat{T_D^\gamma}\hspace{0.2cm}=\hspace{0.2cm}\frac{1}{|{B}(r)|}\displaystyle\sum_{\gamma\in
{B}(r)} \hspace{0.2cm} T_D^\gamma$.\
The following is a variation of an argument given by Varopoulos, [@V].
\[ave\]$\hspace{0.5cm}\widehat{T_D^\gamma}\hspace{0.2cm}{\geqslant}\hspace{0.2cm}\frac{1}{2}|D|$.
$\hspace{0.3cm}\widehat{T_D^\gamma}\hspace{0.2cm}=\hspace{0.2cm}\frac{1}{|{B}(r)|}\displaystyle\sum_{\sigma,\gamma}|
\{(\sigma,\gamma)\hspace{0.2cm}|
\hspace{0.2cm}\sigma \in D , \hspace{0.2cm}\sigma\gamma \in (V_\Gamma
\setminus D),\hspace{0.2cm}\gamma \in B(r)\}\hspace{0.2cm}|$\
$\hspace{2.0cm}= \hspace{0.2cm}
\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in B(r)}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\left(\chi_D(\sigma) - \chi_D(\sigma\gamma)\right)$\
$\hspace{2.0cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\frac{1}{| B(r)|}\displaystyle\sum_{\gamma \in
B(r)}\hspace{0.2cm}\left(\chi_D(\sigma)-\chi_D(\sigma\gamma)\right)$\
$\hspace{2.0cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma\in
D}\hspace{0.2cm}\left(\hspace{0.2cm}\frac{| B(r)|}{|
B(r)|}\hspace{0.2cm}\chi_D(\sigma)\hspace{0.2cm} -\hspace{0.2cm}
\frac{\displaystyle\sum_\gamma \hspace{0.2cm} \chi_D(\sigma\gamma)}{|
B(r)|}\hspace{0.2cm}\right)$\
$\hspace{2.0cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma\in D}\hspace{0.2cm} \left(
1 \hspace{0.2cm}- \hspace{0.2cm}\frac{|
B_\sigma(r)\cap D |}{| B(r)|}\right)$, where $B_\sigma(r)$ is a ball of radius $r$ at vertex $\sigma$.\
But since we assumed that $|{B}(r)| \hspace{0.2cm}>
2|D|$, so we have,\
$\hspace{2.65cm}\widehat{T_D^\gamma}\hspace{0.2cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}( 1 - \frac{1}{2})$,\
$\hspace{1.6cm}$ or, $\hspace{0.2cm}\widehat{T_D^\gamma}\hspace{0.2cm}{\geqslant}\hspace{0.2cm}\frac{1}{2}|D|$\
So there is ${\gamma_0} \in B_1(r)$ such that $T_D^{\gamma_0} {\geqslant}\frac{|D|}{2} $.
Next we obtain an upper bound for the transport $T_D^\gamma$ in the following proposition.
\[lemma 1\] $T_D^\gamma\hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_\gamma \hspace{0.2cm}| \partial_{V} D|$; where $l_\gamma$ is the length of $\gamma$.
The path corresponding to the word $\gamma$ can be expressed as a sequence of the generators in $\Gamma$, namely, $a_1a_2a_3....a_{l_\gamma}$ where $a_i= \alpha^{\pm1}$ or $=\beta^{\pm1}$ for $1{\leqslant}i {\leqslant}l_\gamma$ .\
***Notation*:** Let $a_1a_2...a_k = \alpha_k$ for $1{\leqslant}k {\leqslant}l_\gamma$ and $\alpha_0 $ is the identity of the group.\
The Varopoulos Transport as defined before is,
$\hspace{1.0cm}T_D^\gamma \hspace{0.2cm}=\hspace{0.2cm}|D\gamma \setminus D|$
$\hspace{0.5cm} \therefore
T_D^\gamma\hspace{0.2cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma\in D}|
\{(\sigma,\gamma)\hspace{0.2cm}|
\hspace{0.2cm}y \in D , \hspace{0.2cm}\sigma\gamma \in (V_\Gamma
\setminus D) ,\hspace{0.2cm}\gamma \in
B_1(r)\}\hspace{0.2cm}|$
$\hspace{1.65cm}= \hspace{0.2cm} \displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}| \chi_D(\sigma) - \chi_D(\sigma\gamma)|$
Now using the sequence and notation defined above, we can write,
$\hspace{1.0cm}T_D^\gamma
\hspace{0.2cm}{\leqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\left(\displaystyle\sum_{i=1}^{l_\gamma}|\chi_D(\sigma\alpha_i)
- \chi_D(\sigma \alpha_{i-1})|\right)$\
So in the inner sum, in the expression above, the terms have value either $0$ or $1$, the terms which have value $1$, represent boundary edges.
In order to establish the upper bound for the transport of $D$ by $\gamma \in \Gamma$, we will show that each of the boundary edge mentioned above appears at most $l_\gamma$ times in the sum. So, we start with the transport of a vertex $\sigma_i \in D$ via the path $\gamma$. Let us denote the edge between the vertices $\sigma_i\alpha_{j-1}$ and $\sigma_i\alpha_j$ by $\tau$ where $1{\leqslant}j {\leqslant}l_\gamma$. Now let us express the path $\gamma\hspace{0.2cm}$ as the sequence $\gamma_1\tau\gamma_2$; where $\gamma_1,\gamma_2$ are two sub-paths of $\gamma$ such that the initial vertex of $\gamma_1$ is $\sigma_i$ while the terminal vertex of $\gamma_2$ is $\sigma_i\gamma$, and $\tau$ is the label of the $j^{th}$ edge of $\gamma$. Then, by uniqueness of path liftings in a Cayley graph, it is known that $\gamma_1$ and $\gamma_2$ are both unique with respect to initial vertex $\sigma_i$. In other words, the paths corresponding to $\gamma$ originating from vertices of $D$ other than $\sigma_i$, do not have $\tau$ as the $j^{th}$ edge. So if the path $\gamma$ originating from vertex $\sigma_j \in D$, (where $\sigma_j\neq \sigma_i$), can be expressed as $\gamma_3
\tau\gamma_4$, then, here $\tau$ is the label for say the $k^{th}, (k\neq j)$ edge of this path while $\gamma_3$ and $\gamma_4$ are both unique sub-paths with respect to $\sigma_j$. So a particular edge in path $\gamma$ can appear at most $l_\gamma$ times.
Therefore, $T_D^\gamma \hspace{0.2cm}{\leqslant}\hspace{0.2cm} l_\gamma
\displaystyle\sum_{\sigma_i,\sigma_j \in V_\Gamma}\hspace{0.2cm}|
\chi_D(\sigma_i)-\chi_D(\sigma_j)|$.\
So, $T_D^\gamma \hspace{0.2cm}{\leqslant}\hspace{0.2cm} l_\gamma
\hspace{0.2cm} |\partial_{V} D | $.
Next we will consider $f:(D^n,S^{n-1})\rightarrow X$ to be an immersion, and so instead of a characteristic function we consider a non-negative, integer-valued function $\phi_D$ (Section \[var\]) and show that the Varopoulos argument works in this case too. Assume as before, that $\gamma \in {B_1}(r)\subset G$, where ${B_1}(r)$ represents a ball of radius $r$ centered at the identity in the graph. We choose $r$ large enough such that $|{B_1}(r)| {\geqslant}2\parallel \phi_D \parallel > |{B_1}(r-1)|$.
Varopoulos Transport $T_D^\gamma = \displaystyle\sum_{\sigma \in D} |
\phi_D(\sigma)-\phi_D(\sigma \gamma) | $.
$\therefore$ Average Varopoulos Transport is given by,
$\hspace{2.5cm}\widehat{T_D^\gamma}\hspace{0.2cm}=\hspace{0.2cm}\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in
{B_1}(r)} \hspace{0.2cm} T_D^\gamma$ .\
**Note**: The definitions of $||\phi_D||$, $||\nabla\phi_D||$ used below can be found as Remark \[phi1\] and Definition \[vboundary\] respectively in Section \[var\].\
The following result is a variation of an argument given by Coulhon and Saloff-Coste, [@Co].\
\[prop1\] $\widehat{T_D^\gamma} {\geqslant}\frac{1}{2}||\phi_D||$
$\hspace{3.0cm}
\widehat{T_D^\gamma}=\hspace{0.2cm}\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in
{B_1}(r)}\displaystyle\sum_{\sigma \in D} | \phi_D(\sigma)-\phi_D(\sigma \gamma) | $\
$\hspace{4.5cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in
{B_1}(r)} | \phi_D(\sigma)-\phi_D(\sigma \gamma) | $\
$\hspace{4.5cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in {B_1}(r)} |
\phi_D(\sigma)|- | \phi_D(\sigma \gamma) |$\
$\hspace{4.5cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\frac{1}{|{B_1}(r)|}\hspace{0.2cm} \left(|
B_1(r)| \phi_D(\sigma)- \displaystyle\sum_{\gamma\in
{B_1}(r)}\phi_D(\sigma \gamma)\right)$\
$\hspace{4.5cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\left( \phi_D(\sigma) -
\frac{1}{|{B_1}(r)|}\displaystyle\sum_{\gamma\in
{B_1}(r)}\phi_D(\sigma\gamma)\right) $\
Since, $\displaystyle\sum_{\gamma\in {B_1}(r)}\phi_D(\sigma\gamma)
\hspace{0.2cm}
{\leqslant}\hspace{0.2cm}\parallel \phi_D \parallel $, we have,\
$\hspace{3.5cm} \widehat{T_D^\gamma}\hspace{0.2cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\left( \phi_D(\sigma) - \frac{\parallel
\phi_D \parallel}{| B_1(r) |}\right)$\
According to our initial assumption, $\frac{\parallel \phi_D
\parallel}{| B_1(r) |} {\leqslant}\frac{1}{2}$, and that implies, $\frac{\parallel \phi_D \parallel}{| B_1(r) |} {\leqslant}\frac{\phi_D(\sigma)}{2}$, for any particular $\sigma \in D$.\
$\hspace{3.5cm}\therefore \widehat{T_D^\gamma}\hspace{0.2cm}{\geqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in D} \frac{\phi_D(\sigma)}{2} =
\frac{1}{2}\parallel \phi_D \parallel$\
In particular, $\exists \gamma_0 \in B_1(r)$ such that, $T_D^{\gamma_0}
{\geqslant}\frac{\parallel \phi_D \parallel}{2}$.
\[prop2\] $T_D^\gamma {\leqslant}l_\gamma||\nabla\phi_D||$, where $l_\gamma$ denotes the length of the path/word $\gamma$.
We will use the same argument as in proof of Lemma \[lemma 1\] to show this. The path corresponding to the word $\gamma$ can be expressed as before by a sequence of the generators in $\Gamma$, namely, $a_1a_2a_3....a_{l_\gamma}$ where $a_i= \alpha^{\pm1}$ or $=\beta^{\pm1}$ for $1{\leqslant}i {\leqslant}l_\gamma$ .\
***Notation*:** Let $a_1a_2...a_k = \alpha_k$ for $1{\leqslant}k {\leqslant}l_\gamma$ and $\alpha_0 $ is the identity of the group.\
The Varopoulos Transport as defined before is,
$\hspace{2.0cm}T_D^\gamma \hspace{0.2cm}=\hspace{0.2cm}|D\gamma \setminus D|$
$\hspace{1.7cm} \therefore
T_D^\gamma\hspace{0.2cm}=\hspace{0.2cm}\displaystyle\sum_{\sigma\in D}\hspace{0.2cm}| \phi_D(\sigma) - \phi_D(\sigma\gamma)|$
As before,
$\hspace{2.0cm}T_D^\gamma
\hspace{0.2cm}{\leqslant}\hspace{0.2cm}\displaystyle\sum_{\sigma \in
D}\hspace{0.2cm}\left(\displaystyle\sum_{i=1}^{l_\gamma}|\phi_D(\sigma\alpha_i)
- \phi_D(\sigma \alpha_{i-1})|\right)$\
The terms in the inner sum are either zero or a natural number. In the case when they are non-zero, they represent boundary edges in the Varopoulos sense.
So as in the proof of Lemma \[lemma 1\], each of these afore-mentioned boundary edges appear in the sum at most $l_\gamma$ times. Therefore,
$\hspace{4.0cm}T_D^\gamma \hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_\gamma \displaystyle\sum_{\sigma_i\sigma_j \in V_\Gamma}|
\phi_D(\sigma_i)-\phi_D(\sigma_j)| $,
which means, $T_D^\gamma \hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_\gamma \parallel \nabla\phi_D\parallel$.
Isoperimetric Inequalities for groups of Polynomial growth
----------------------------------------------------------
### **A 2-dimensional Example**
####
Here we will discuss the 2-dimensional example ${{\mathbb Z}}^2$. Let us consider the presentation $\langle a,b \hspace{0.1cm}|\hspace{0.1cm}[a,b]\rangle$ for ${{\mathbb Z}}^2$. Let $\widetilde{X}$ be the universal cover of the 2-complex $X$ corresponding to the presentation given above for ${{\mathbb Z}}^2$. As before let us denote a ball of radius $r$ centered at the identity in $\widetilde{X}$ by $B_1(r)$.
Let us choose $r$ such that $ | B_1(r) |{\geqslant}2 ||\phi_D||
> | B_1(r-1) |$. Also, $| B_1(r) | \hspace{0.2cm}\sim \hspace{0.2cm}\mathcal{O}(r^2)$.
From the propositions above, we already know that:\
$\frac{1}{2}||\phi_D||\hspace{0.2cm} {\leqslant}T_D^{ \gamma_0}
\hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_{\gamma_0} \hspace{0.2cm}
|\partial_{V} D |$ for some $\gamma_0 \in B_1(r)$.
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\leqslant}\hspace{0.2cm} 2
l_{\gamma_0 } \hspace{0.2cm} ||\nabla\phi_D||$
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm}||\phi_D||^{\frac{1}{2}}\hspace{0.2cm} |||\nabla\phi_D||$ ; since $l_{\gamma_0} {\leqslant}\hspace{0.2cm} r$
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm}\hspace{0.2cm}||\nabla\phi_D||^{2}$\
When the 2-disc along with its boundary circle is embedded in $\widetilde{X}$ via the transverse map $f: (D^2,S^1) \rightarrow \widetilde{X}$, $||\nabla\phi_D||\hspace{0.2cm}= \hspace{0.2cm}
|\partial D^2|$. On the other hand, in the case when we have a reduced diagram $f: (D^2,S^1) \rightarrow \widetilde{X}$, such that the disc and its boundary are not embedded, then by Lemma \[bound\] $||\nabla\phi_D||{\leqslant}|\partial D^2|$. Hence we have the following isoperimetric inequality.
$\hspace{2cm}\therefore ||\phi_D|| {\preccurlyeq}\hspace{0.05cm}
||\nabla\phi_D||^{2}{\leqslant}(2Const.)^{2}
|\partial D^2|^{2}$.
$\hspace{2cm}\therefore Vol^2(D^2) {\preccurlyeq}|\partial D^2|^{2}$.
### **A 3-dimensional Example**
####
In this section we present the an upper bound for the second-order Dehn functions of the 3-dimensional group $\mathcal{H}$ and consequently all cocompact lattices in the Nil geometry. In other words we complete the proof of Theorem \[th1\] here. Let $M$ be the 3-manifold corresponding to the lattice $\mathcal{H}$ in the Nil geometry mentioned above in Example \[lattice1\] (along with the triangulation shown). Let $\widetilde{X}$ be its universal cover. So, one can find numerous copies of $M$ inside $\widetilde{X}$. Let $B_n$ represent a ball of radius $n$ in $\widetilde{X}$.\
$||\phi_D|| \hspace{0.05cm}{\preccurlyeq}\hspace{0.05cm}||\nabla\phi_D||^\frac{4}{3}$
Let $\Gamma$ denote the dual Cayley graph embedded in $\widetilde{X}$ corresponding to the generating set $\mathcal{A}_0$ defined in Lemma \[generator\] part $(i)$ where $\widetilde{X}$ is the universal cover of the 3-complex corresponding to $\mathcal{H}$. Let us consider the reduced 3-dimensional diagram $f:(D^3,S^2)\rightarrow \widetilde{X}$ (defined in Section \[transverse1\]). Let $D$ be the finite set of vertices in $\Gamma$ dual to the 0-handles present in the diagram mentioned above. Next, let us choose a ball of radius $r$ in the graph $\Gamma$ such that $ | B_1(r) |{\geqslant}2 ||\phi_D||>| B_1(r-1) |$ , where $r>2$ is real and $r$ is sufficiently large. Also, $| B_1(r) | \hspace{0.2cm}\sim \hspace{0.2cm}\mathcal{O}(r^4)$, ([@Gr5],[@Bass]). From Section \[comp\], we already know that, $\frac{1}{2}||\phi_D||\hspace{0.2cm} {\leqslant}T_D^{ \gamma_0}
\hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_{\gamma_0} \hspace{0.2cm}
|\partial_{V} D|$ for some $\gamma_0 \in B_1(r)$. Also as $l_{\gamma_0} {\leqslant}\hspace{0.2cm} r$ and $ r-1{\leqslant}( 2||\phi_D||)^\frac{1}{4}\Rightarrow r{\preccurlyeq}( ||\phi_D||)^\frac{1}{4}$ and we have the following,
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\leqslant}\hspace{0.2cm} 2
l_{\gamma_0 } \hspace{0.2cm} ||\nabla\phi_D||$
$\hspace{2cm}\therefore||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} 2
||\phi_D||^{\frac{1}{4}}\hspace{0.2cm} ||\nabla\phi_D||$
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm}||\nabla\phi_D||^\frac{4}{3}$.
(Proof of Theorem \[th1\]) Given a reduced diagram $f:(D^3,S^2)\rightarrow \widetilde{X}$, if the 3-ball and its boundary sphere are embedded in $\widetilde{X}$, then $||\nabla\phi_D||=|\partial D^3|$. If they are not embedded then by Lemma \[bound\], $||\nabla\phi_D||{\leqslant}|\partial D^3|$. Hence we have the following inequality.\
$\hspace{0.8cm}\therefore Vol^3(D^3){\preccurlyeq}|\partial D^3 |^\frac{4}{3}$, where $|\partial D^3 |$ is the volume of the boundary sphere. Therefore, by the definition of $\delta^{(2)}$, if $x$ is the maximum number of $3$-cells in the boundary sphere, then $\delta^{(2)}(x) {\preccurlyeq}x^\frac{4}{3}$.
Isoperimetric Inequalities for groups of Exponential growth
-----------------------------------------------------------
####
In this section we present the upper bound for the second-order Dehn functions of $\mathcal{S}$ and consequently all cocompact lattices in the Sol geometry. In other words, the proof of Theorem \[th2\] will be completed here.\
$||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} \ln (||\nabla\phi_D||)\hspace{0.2cm} ||\nabla\phi_D||$.
We start with a reduced 3-dimensional diagram $(D^3,S^2)$, corresponding to a finitely presented group $G$. In this sub-section, we have a 3-dimensional example with exponential growth namely, the solvable group $\mathcal{S}$.
Let us choose $r$ such that $ |B_1(r)| {\geqslant}2 ||\phi_D||> |B_1(r-1)| $, $|B_1(r)| \hspace{0.1cm}\sim \hspace{0.1cm}Ce^{\ln(k)r}$, $k,C$ are both positive constants, ([@Wolf],[@Mil2]). Therefore we have,\
$\hspace{2.2cm} ||\phi_D|| {\geqslant}\hspace{0.1cm}C e^{\ln(k)r}$.
$\hspace{2cm}\therefore r {\preccurlyeq}\ln(||\phi_D||) $
Next, from Section \[comp\], we already know that, $\frac{1}{2}||\phi_D||\hspace{0.2cm} {\leqslant}T_D^{ \gamma_0}
\hspace{0.2cm} {\leqslant}\hspace{0.2cm}l_{\gamma_0} \hspace{0.2cm}
||\nabla\phi_D||$ for some $\gamma_0 \in B_1(r)$.
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\leqslant}\hspace{0.2cm} 2
l_{\gamma_0 } \hspace{0.2cm} ||\nabla\phi_D||$
$\hspace{2cm}\therefore ||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} \ln(||\phi_D||)\hspace{0.2cm} ||\nabla\phi_D||$ ; since $l_{\gamma_0} {\leqslant}\hspace{0.2cm} r\hspace{2cm} (*)$
As in the case of $\mathcal{H}$, we can say the in the embedded case $||\nabla\phi_D||=| \partial
D^3 |$, while in the immersed case we have $|| \nabla \phi_D || {\leqslant}| \partial D^3 |$, using the Lemma \[bound\] above. Hence we have the following isoperimetric inequality,
Taking natural logarithm, $\ln$, on either side of $(*)$ we get,
$\hspace{2.4cm} \ln( ||\phi_D||) \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} \ln( \ln(||\phi_D||
)\hspace{0.2cm} ||\nabla\phi_D||)$,
$\hspace{2cm} \therefore \ln( ||\phi_D||) \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} \ln( \ln(||\phi_D||)\hspace{0.2cm}+ \hspace{0.2cm}\ln (||\nabla\phi_D||)$,
Now from $(*)$,for large values of $||\phi_D||$,
$\hspace{2.4cm}\ln(||\phi_D||){\leqslant}\frac{||\phi_D||}{\ln(||\phi_D||)}{\preccurlyeq}||\nabla\phi_D||$,
$\hspace{2cm} \therefore \ln(||\phi_D||) \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm} \ln (||\nabla\phi_D||)$
Again from $(*)$,
$\hspace{2.45cm} ||\phi_D|| \hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm}\ln (||\nabla\phi_D||)\hspace{0.2cm} ||\nabla\phi_D||$.
(Proof of Theorem \[th2\]) From the lemma above we have, $ Vol^3(D^3)\hspace{0.2cm}{\preccurlyeq}\hspace{0.2cm}\ln (|\partial D^3|)\hspace{0.2cm}|\partial D^3| $, where $|\partial D^3 |$ is the volume of the boundary sphere. Therefore, by the definition of $\delta^{(2)}$, if $x$ is the maximum number of $3$-cells in the boundary spheres then $\delta^{(2)}(x) {\preccurlyeq}x\ln(x)$.
|
---
abstract: |
Arpanet, Internet, Internet of Services, Internet of Things, Internet of Skills. What next? We conjecture that in 15-20 years from now we will have the Internet of Neurons, a new Internet paradigm in which humans will be able to connect bi-directionally to the net using only their brain. The Internet of Neurons will provide new, tremendous opportunities thanks to constant access to unlimited information. It will empower all those outside of the technical industry, actually it will empower all human beings, to access and use technological products and services as everybody will be able to connect, even without possessing a laptop, a tablet or a smartphone. The Internet of Neurons will thus ultimately complete the currently still immature democratization of knowledge and technology. But it will also bring along several enormous challenges, especially concerning security (as well as privacy and trust).
In this paper we speculate on the worldwide deployment of the Internet of Neurons by 2038 and brainstorm about its disruptive impact, discussing the main technological (and neurological) breakthroughs required to enable it, the new opportunities it provides and the security challenges it raises. We also elaborate on the novel system models, threat models and security properties that are required to reason about privacy, security and trust in the Internet of Neurons.
author:
-
-
title: |
Privacy, Security and Trust\
in the Internet of Neurons
---
New Internet paradigm; Brainwaves; Human computer; Security; Privacy; Trust
Introduction: from the human computer to... the human computer
==============================================================
We all carry around a computer, regardless of who we are, how old we are, where we live, what job we do, what education we received. No, we are not talking about your laptop, your tablet or your smartphone. We are talking about your *brain*.
In fact, the term “computer” has been in use from the early $17^\mathrm{th}$ century, way before electronic computers became available. It was introduced simply to mean “one who computes”, namely a person whose job is to perform complex mathematical calculations. In that sense, people often speak of “human computer” to make this distinction clear.[^1] Throughout the centuries, human computers, working alone or in teams, have provided significant contributions to groundbreaking scientific discoveries, ranging from trigonometry to astronomy, to the dawn of nuclear energy and nuclear weapons (e.g., the complex computations crucially related to nuclear fission in the Manhattan Project) and to the space race [@Shetterly16].
When electronic computers became available in the second half of the $20^\mathrm{th}$ century, human computers became useless, and “human computer” is nowadays mainly used to refer to individuals with prodigious powers of mental arithmetic who display their abilities in theaters or TV shows. Electronic computers also brought along a revolution that has transformed the economic, social, educational, and political landscape in a profound and indelible manner: the *net*.
The technical foundations of the Internet were laid by the Advanced Research Projects Agency Network *ARPANET* [@ARPANET] towards the end of the 1960s. Soon after, new overseas nodes of the network were created and the definition of the standard TCP/IP officially launched the *Internet* as a set of interconnected networks through these packet switching protocols. Advances in hardware and software at the end of the $20^\mathrm{th}$ century enabled mobile connectivity to billions of laptops and (smart)phones. This *Mobile Internet* gave rise to the *Internet of Services (IoS)* [@schroth2007web; @mandula2015mobile], with the flourishing of e-commerce, health-care portals, booking services, streaming websites and, last but not least, social networks. This redefined entire segments of the economy in the first decade of the $21^\mathrm{st}$ century, and was soon followed by the *Internet of Things (IoT)*, a network of physical devices, vehicles, home appliances and other items embedded with electronics, software, sensors, actuators, and connectivity which enables these objects to connect and exchange data [@gubbi2013internet; @Al-Fuqaha2015; @osisanwo2015internet; @mandula2015mobile].
The next, and $5^\mathrm{th}$, evolution of the Internet is expected to be the *Tactile Internet*, which has been defined by the International Telecommunication Union (ITU) as a network that is based on 5G and combines ultra-low latency with extremely high availability, reliability and security [@Simsek2016; @TactileInternet]. The Tactile Internet will encompass human-to-machine and machine-to-machine interaction, enabling tactile and haptic sensations and the control of the IoT in real time. It will unleash the full potential of the fourth industrial revolution (a.k.a. Industry 4.0), and revolutionize the way we learn and work through the *Internet of Skills* (a.k.a. Human 4.0, [@Dohler2017]). Although 5G has long passed the embryonic stage, and the testing phase is now underway, extra works must to be done to cover 5G security challenges [@ahmad2018overview; @schneider2015towards] in order to consider 5G a fully adoptable technology. However, capitalizing on 5G and ultra-low delay networking as well as on AI and robotics, the Internet of Skills will enable the real-time delivery of skills in digital form remotely and globally.
After this brief overview of the past, and the near future, of the Internet, it is time to ask what will came next. We conjecture the return of the human computer, but in a different guise. We predict the coming of the next, and maybe ultimate, phase of the Internet evolution: the *Internet of Neurons* will rest upon a novel paradigm in which humans are able to connect bi-directionally to the net using only their brain. The Internet of Neurons will provide new, tremendous opportunities thanks to constant access to unlimited information. It will empower all those outside of the technical industry, actually it will empower all human beings, to access and use technological products and services as everybody will be able to connect, even without possessing a laptop, a tablet or a smartphone. The Internet of Neurons will thus ultimately complete the currently still immature democratization of knowledge and technology. But it will also bring along several enormous challenges, especially concerning security (as well as privacy and trust).
In the rest of this paper we speculate on the worldwide deployment of the Internet of Neurons by 2038 and brainstorm about its disruptive impact, discussing the technological (and neurological) breakthroughs required to enable it, the new opportunities it provides and the security challenges it raises. We also elaborate on the novel system models, threat models and security properties that are required to reason about privacy, security and trust in the Internet of Neurons.
We proceed as follows. In Section \[sec:IoN\], we introduce the Internet of Neurons. In Section \[sec:PST\], we discuss privacy, security and trust issues in the Internet of Neurons. In Section \[sec:concl\], we draw conclusions.
The Internet of Neurons: From brainwaves to packets, and vice versa {#sec:IoN}
===================================================================
In a 2014 interview [@HawkingUSAToday], Stephen Hawking said
> *We are all now connected by the Internet, like neurons in a giant brain.*
Although Hawking is famous for his predictions (as well as for his scientific results, of course), in this case he was not prophesying the advent of what we call the Internet of Neurons. However, it is interesting to note that he used the same keywords (we found this quote when we googled “Internet of Neurons” to see if somebody had already had the idea) and that, in a brain, like in the Internet, it is actually all a matter of connectivity.
How would connectivity work in the Internet of Neurons? At the root of all our thoughts, emotions and behaviors is the communication between neurons within our brains. *Brainwaves* are produced by synchronized electrical pulses from masses of neurons communicating with each other. Hence, to realize the *brain-net*, which is one of the frontiers of *brain-computer interaction* and thus of human-computer interaction, we need to interface brainwaves with the packets that are received and sent by computers or other external devices.[^2]
Some approaches have already been proposed, and prototypical devices and software built, for the realization of *brain-computer interfaces* [@Graimann]. We can summarize the methodology behind brain-computer interaction, through a brain-computer interface, as the following sequence of steps:
1. Collect brainwaves by recording activity directly from the brain (invasively or non-invasively) in real-time.
2. Convert the complex waveforms of brainwaves into data.
3. Encode the parsed information and issue action instructions.
4. Feed back the externally perceived information in real-time in the form of signals that the brain can read (possibly through a stimulating device).
Note that the system must rely on intentional control, i.e., users must choose to perform a mental task whenever they want to accomplish a goal with the brain-computer interface.
Nowadays, it is already possible to detect and process brainwaves (e.g., using EEG sensors placed on the scalp) and a number of solutions have been proposed to provide a form of uni-directional communication and thus address at least steps 1) and 2) of this methodology. Let us consider three interesting examples. The neurotechnology company “Neuralink” was founded in 2016 by Elon Musk and others with the aim of developing an ultra-high-bandwidth implantable brain-computer interface to connect humans and computers [@winkler2017elon]. While Neuralink is still in early stages, the “Brainternet” project [@linesbrain] has developed an apparently more rudimentary but effective technology that streams brainwaves onto the Internet (by converting brainwaves into signals and streaming them to an online server using a Rasperry Pi computer). In 2018, the startup Neurable will release the VR game “Awakening” in which the gamer’s brain essentially acts as mouse thanks to a brain-scanning headband paired with software that interprets the neural signals, thus allowing for hands-free control [@Strickland]. Other application areas that brain-computer interfaces are currently being developed for are, for instance, education (e.g., for monitoring of students’ attention in real time) and medical care (e.g., for monitoring and treatment of Parkinson’s and other serious brain diseases, with the eventual goal of human enhancement as aspired by Neuralink and other projects).
These technologies are promising, but they are still far from addressing steps 3) and 4) in a satisfactory way. The Internet of Neurons will require more than a uni-directional information flow; it will require a bi-directional information flow, in which
- brainwaves are translated into data and
- data is translated into signals that the brain can parse.
Some exploratory research is being carried out that attempts to bridge neuroscience with computer science and telecommunications, but brain-computer bi-directional information flow is still largely unchartered territory.
Nonetheless, we conjecture that by 2023, in five years from now[^3], advances in neurology and in brain-computer interaction, combined with technological innovations, will have led to the creation of a device able to connect the human brain to the Internet bi-directionally, and without resorting to any invasive surgical operations. This device won’t be bulky; it will be portable, light and chargeable inductively so that we will be able to connect to the Internet anywhere anytime. It could take the form of a lightweight headphone like in Fig. \[fig:pod-notrust\], or more likely simply be a button-like pod that we will attach to our temples. Or it could even be a tiny implant, although non-invasive procedures are typically to be preferred.
The device will communicate bidirectionally with the brain via brainwaves (as illustrated by the brainwave symbol on the forehead of the human in Fig. \[fig:pod-notrust\]) and with the Internet via wireless communication (as illustrated by the standard symbol) to and from appropriate routers. The device must thus be capable of reading the brainwaves in real-time, more or less like EEG readers are capable of doing now, but it must also be capable of interpreting the brainwaves and transform them into their digital version, sending the coded version to the Internet. The device must also be capable of receiving incoming data, convert it into brainwaves (Step (3)) and send them to the brain (Step (4)).
Being able to convert data into brainwaves and vice versa is necessary in this phase. Progress in Machine Learning, AI and Big Data have made it possible to interpret brainwaves [@neuralencoding] mapping them with words or pictures creating a valid and applicable brainwaves-to-digital and digital-to-brainwaves codification. Feeding back the converted data into the brain requires techniques capable of stimulating the brain with signals. *Electroconvulsive therapy (ECT)*, *rapid transcranial magnetic stimulation (rTMS)* and *magnetic seizure therapy* are techniques able to deliver stimulation pulses through the tissue directly to the brain, even wirelessly [@chen2015wireless; @grossman2017noninvasive].
![Configuration 1 — Bi-directional brain-Internet connection by means of a wearable device[]{data-label="fig:pod-notrust"}](newBraing)
![Configuration 2 — Bi-directional brain-Internet direct connection[]{data-label="fig:podless-notrust"}](newBraing2)
We also conjecture that advances in software and hardware will make sure that in 2038 there will be no more need for any wearable device to connect: as depicted in Fig. \[fig:podless-notrust\], humans will be able to connect to the Internet directly with their brainwaves, possibly through routers that “read” brainwaves remotely (say from a distance of a few meters like wireless routers do now with wireless signals), and transform the brainwaves into data and vice versa (i.e., brains downloading and uploading information from the network). This may sound like the killing argument of “tin-foil-hat conspiracy theorists”, who wear hats made from one or more sheets of aluminum foil in the belief that the hat will shield the brain from threats such as electromagnetic fields, mind control, and mind reading. However, this too is not really science fiction: research is ongoing on developing sensors that can be used to monitor the human electroencephalogram without electrical or physical contact with the body [@Harland02; @Prance08; @Rendon17]. There is still a long way to go until these sensors are actually able to do more than just monitor but actually allow for the full realization of the four steps that we described above, but several foundation stones for the Internet of Neurons have been, or are in the process of being, laid so it is necessary that we start thinking about the privacy, security and trust challenges that will plague the Internet of Neurons. Some of these challenges will mirror the challenges that are plaguing Internet as we know it today, but other challenges will be novel and even more intriguing.
Privacy, security and trust {#sec:PST}
===========================
The potential offered by the technological revolution underlying the Internet of Neurons will be as varied as the problems related to privacy, security and trust that it will cause. In order to reason about these problems, we will need to provide suitable definitions, where a security definition is typically provided by combining a system model with a threat model and with one or more security properties that the system should guarantee even in the presence of an attacker. In the following, we discuss the main features of such models and properties for the Internet of Neurons. In our analysis, we thus take into account the two configurations suggested in the previous section, where the connection is made with or without a device, pointing out analogies with, and differences from, current research and technologies.
System model
------------
To provide a model of the system means to give a clear, and preferably formal, definition that provides enough detail to be able to understand and specify how the system behaves, encompassing both when it behaves correctly and securely, and when it behaves in unexpected and insecure ways.
In the security literature, security models have been formulated in a number of different ways. For instance, encryption and decryption operators are typically described by means of mathematical formulas along with some algebraic structure to capture the operators’ properties; security protocols are typically described by means of state transition systems that specify how the knowledge of the protocol agents evolves over time; firewalls are typically described by means of sets of rules regulating how packets are filtered; access control systems are typically described by means of security policies, requests and permissions; software systems are typically described directly by their source code (or by the specification that can be learned or inferred by interacting with the code) or by dataflow and/or control flow specifications. These are just some examples, but all of them have in common the need to represent the infrastructure and how information flows among the system’s agents (a.k.a. principals or entities).
For example, for Configuration 1 (Fig. \[fig:pod-notrust\]), we can identify the following agents:
- the human being,
- the device,
- the router(s),
- the Internet,
connected by the following communication channels:
- a short-range channel between human being and device,
- a medium-range channel between device and router,
- a long-range (and possibly wired) channel between router and Internet.
Different protocols will be used to transmit information over these channels. The channel between the device and the router and the channel between the router and the Internet might actually employ protocols similar to the wireless protocols that we are already using today — in fact, if we are interested in a formal analysis of the system, we could even abstract away the channel between the router and the Internet and simply consider a medium–to-long-range channel between device and Internet. The channel between the human being and the device will, however, require new protocols able to translate between brainwaves and data packets, as the technologies that we discussed in the previous section are attempting to do.
For Configuration 2 (Fig. \[fig:podless-notrust\]), we can identify the following agents:
- the human being,
- the router(s),
- the Internet,
connected by the following communication channels:
- a medium-range channel between human being and router,
- a long-range (and possibly wired) channel between router and Internet.
As before, different protocols will be used to transmit information over these channels. We expect that it will be possible to generalize to this configuration the protocols developed for the short-range brain-device communication in Configuration 1.
In both cases, the model of the configuration will need to be extended with models of the agents (including their actions and their states), of the security protocols used (including routing protocols), of the messages being sent, of the cryptography used and so on. We expect that many of the modeling languages and techniques that are in use today will be applicable with reasonable extensions, except of course for the translation brainwave-data, which will require considerable work. A starting point could be the formalization of this translation as a new cryptographic operator that encodes brainwaves into data along with the inverse operator that decodes data into brainwaves; identifying and formalizing the properties of these operators won’t be easy though.
Threat model
------------
A number of questions need to be answered in order to provide a threat model:
- *Who is the attacker?* Is he an outsider or an insider? Is he an agent (a human or a machine) trying to attack the communication between the human and the Internet? Is he perhaps the router, or even the human itself? What if the human behaves honestly but makes mistakes, or thinks “wrong thoughts” (whatever they may be) that make the system vulnerable? How would social engineering look like in this case?
![Possible attacker locations in Configuration 1[]{data-label="fig:threatdevice"}](threatdevice)
![Possible attacker locations in Configuration 2[]{data-label="fig:threatbrainwaves"}](threatbrainwaves)
![Trusted area in Configuration 1[]{data-label="fig:trustthreat"}](trustthreat)
- *Where is the attacker?* For instance, can the attacker attack all communication channels in the two configurations as in Fig.s \[fig:threatdevice\] and \[fig:threatbrainwaves\]? Or can we assume that the system contains a trusted network area? For example, Fig. \[fig:trustthreat\] assumes that the short-range channel between brain and device cannot be attacked, perhaps supposing that the device itself is able to provide a kind of shield creating some “noise” that isolates the human brain and prevents remote reading (and writing) of brainwaves, like noise-cancellation headphones do with the urban noise. Another approach could be to establish some kind of “encryption” between brain and device, mapping device signals to a specific person’s individual brainwaves. Alternatively, a more radical way would be to “implant” the device preventing possible substitutions with tampered devices. Other approaches could be possible. This situation is similar to the assumptions that are currently often made when reasoning about the security of complex security protocols (such as those built by composing subprotocols) [@AlmousaMMV15; @ModersheimV14; @ModersheimV09] or of cyber-physical systems [@LanotteMMV17], where the attacker can only tamper with some, but not all, channels and devices. We thus expect that these recent works will be particularly useful.
- *What is the power of the attacker?* What are his computational resources? Does he possess a certain amount of computation time to devote to his attack? Does he possess, or control, devices that allow him to access the different channels and the messages sent on them? Or perhaps should we assume that the attacker can inject some malicious code in the device or the router? In that way, he could not only do harm to the system or even spoof a router to gain access to the human brain, but perhaps also physical harm to the human, by tampering with the device that has direct access to the brain. The attacker could also spoof another human to gain access to a router. We will return to this when we discuss security properties in the next subsection. In fact, we must also answer the question: *What is the attacker trying to achieve?* What can he do on the different channels? Read, replace, modify, intercept messages and perhaps even brainwaves? To that end, we need to consider the security properties that the system is trying to achieve.
Properties
----------
Let us now discuss the main security properties that we could ask the Internet of Neurons to guarantee. Note that although we focus on the traditional security properties, it is obvious that the categorical imperative of the Internet of Neurons is actually the *safety* of the human being, i.e.,
no harm should occur to the (brain of the) human being.
The Internet is already putting human safety at risk in several ways nowadays [@kiley2002does; @kuss2013internet; @lee2013substance], but in the Internet of Neurons failure to guarantee one or more security properties (e.g., consequences of the Internet “tampering” directly the human brain) might actually expose, directly or indirectly, humans to novel and much more dangerous risks.
### Privacy, Confidentiality and Authentication
*Information privacy* (a.k.a. *data privacy*) is the relationship between the collection and dissemination of data, technology, the public expectation of privacy, and the legal and political issues surrounding them. *Internet privacy* is a subset of information privacy that concerns the storing, repurposing, provision to third parties, and displaying of information pertaining to oneself by means of the Internet. In the Internet of Neurons, our “persona” is using one of the most private information we have: our thoughts, represented by brainwaves.
Thoughts and emotions are intrinsically and intricately related. In psychology, emotions are described as unconscious feelings that are the result of mostly unconscious thoughts [@pettinelli2011psychology]. A number of works have been published on how to extract human emotions from brainwaves using electroencephalography (EEG) [@ismail2016human; @kao2015brainwaves; @lahane2015approach; @chai2010classification]. What would happen if the attacker were able to extract our emotions from the brainwaves that we are sending in the Internet of Neurons? How can we protect them from being stolen?
In Configuration 1 (as shown in Fig. \[fig:threatdevice\]), the attacker could intercept the brainwaves received by the device before they are coded and transmitted to the router and then the network. A study carried out in 2011 demonstrated technologies able to reconstruct images from brainwaves [@berkley], so that, also thanks to some spoofing techniques, the attacker could intercept our communication, reverse it into brainwaves and thus obtain the raw data of our thoughts, even in their binary version. This hypothesis becomes even stronger if we consider a device-less configuration (as shown in Fig. \[fig:threatbrainwaves\]) where there is no encoding of brainwaves and they are broadcast over the air to the Internet. This is reminiscent of the attacks that can be carried out by eavesdropping from a distance on the sound emanated by different keyboard keys [@Asonov04; @Zhuang2009] or by eavesdropping from a distance on the data that is displayed on a computer screen [@BackesDU08]. In these two kinds of attacks, the attacker learns how to recognize and reconstruct the sound or image generated. We expect that advances in machine learning, coupled with those in neuroscience and brainwave-data translation, will make brainwave eavesdropping and reconstruction possible with affordable attacking devices.
Another major issue concerns *location privacy*. Several indoor and outdoor location techniques can be used to trace our position [@gpsproc; @werner2014basic], which can have positive or negative consequences. For instance, in 2011, the Chinese government announced that it would track people’s movements through their cell phones for better traffic control [@guardianchina; @landwehr2012privacy], while a study of the Haitian population after the 2010 earthquake showed that similar tracking is extremely useful in informing where people are and where relief aid should go [@bengtsson2011improved]. The Internet of Neurons won’t be exempt from mass surveillance issues, allowing attackers, including governments or Internet providers, to violate the users’ location privacy.
We could assume that every brainwave-data device will have a unique identifier like most of the devices have, such as a uuid [@leach2005universally] or a global identifier that is created when the device accesses some services [@jones2012creating]. Tracking these identifiers will be possible, e.g., along the lines of [@koneru1999tracking]. Removing the device (and its identifier) as is done in Configuration 2, will help mitigate these problems, but still it won’t guarantee location privacy. Recent studies [@armstrong2015brainprint; @kumari2014brainwave; @ruiz2016cerebre] have namely shown that it is possible to create brainwave patterns to identify users, and thus use *brainprinting* as a biometric *authentication* factor.[^4] In both of the configurations that we considered, with or without a device, the attacker could then track a specific user relying just on her brainprint. To that end, the attacker would, of course, need to know the user’s brainprint, but, mimicking how authentication is done today, we could imagine a sort of brainprint certificate issued by a certification authority of a public-brainprint infrastructure[^5], or we could simply consider the Internet or the Internet provider as the attacker able to track the movement of its users. In this case, in order to attempt to achieve location privacy, users should try to change their brainprint. One way to alter one’s thought pattern would be to learn to think differently than usual, e.g., thinking “happy thoughts” that obfuscate the normal pattern. This sounds a bit “mystical”, but maybe one could indeed learn to confuse one’s own brainwaves while still functioning normally as a human being. Alcohol and drugs might help here (although it might then be difficult to remember one’s password [@drunk]) or also physical exercise, workout, fatigue, hunger and stress, which all have been shown to alter one’s EEG [@chuangpassthoughts].
Another solution for privacy and location privacy, as well as for confidentiality, would be to encrypt. However, while we could use standard encryption algorithms (such as RSA, Triple DES or AES) to encrypt the wireless communication from device to router and from router to Internet, it is at best unclear how to encrypt the actual brainwaves, which are transmitted from brain to device in Configuration 1 or broadcast over the air in Configuration 2. But maybe one day somebody will devise an algorithm that allows humans to carry out mental encryption much in the same way as one can learn how to carry out mental calculations.
The device of Configuration 1 could raise other privacy questions. For instance, it could determine health-related issues while it is reading the user’s brainwaves and provide, or sell, such information to health-insurance companies or the government. Could it also determine the user’s emotions and thoughts? Will the user trust the device? How could we protect information that we know (e.g., passwords or other confidential data) from being read and distributed by the device? One could, similar to “happy thoughts” above, try to suppress one’s thoughts about such confidential information when wearing the device, but this will be difficult if not impossible.[^6] Or one could learn to store some thoughts in *private mental drawers*, like some mentalists are (supposedly) able to do. In any case, to ensure that users will trust the device, it will at the very least be necessary to carry out a strict procedure of testing and certification of the device before it is deployed. Similar comments apply also for Configuration 2, but referring to the router rather than to the wearable device.
### Integrity
What does integrity mean in the Internet of Neurons? How can we protect thoughts and brainwaves? The attacker will attempt to tamper with all communication channels, the digital and the mental ones. In the case of digital channels (from device to router or from router to the network), we will likely be able to use integrity-preserving solutions similar to the ones that are available now (cryptographic checksums, hash functions, message authentication codes, digital signatures, and so on).[^7] There is of course also the question of the integrity of the human mind itself, i.e., protecting the brain from “malicious brainwaves” generated from malicious data from the network. In this case, we will need techniques for mental firewalls, input sanitization, sandboxing or Chinese-walling, thereby ensuring the security of the information contained in the other parts of the brain.
### Availability
Besides for malfunctioning of the device and the router, and of jamming of the wireless signals, availability in the Internet of Neurons can be threatened by a *Distributed Denial of Service (DDoS)* attack when the brain is overwhelmed by the amount of incoming information, thus putting the human at risk. Filtering mechanisms will be necessary to control the flow of data. On the other hand, the Internet of Neurons will enable opportunities that are unthinkable now. For instance, studies about sleep-learning [@simon1955learning; @rudoy2009strengthening; @antony2012cued; @arzi2012humans] have shown that our mind is able to learn if it is stimulated during the night under certain conditions. The Internet of Neurons would enable us to learn while we are sleeping thanks to the direct connection of our brain to the Internet. Actually, we could be learning in every waking moment, committing part of our brain to learning and leaving the remaining part untouched for everyday operations, i.e., for our brain’s normal daily activity. We could even commit part of our brain as a CPU, e.g., for mining and other cryptographic calculations, as we have imagined in [@MMM].
### Anonymity
One way to achieve at least some degree of anonymity in today’s Internet is to use an anonymizing service (such as Mixes, I2P or TOR) that addresses the issue of IP tracking [@zantout2011i2p; @reed1998anonymous] by encrypting packets within multiple layers of encryption. Anonymity is achievable because, as the packet follows a predetermined route through the anonymizing network, each router sees the previous router as the origin and the next router as the destination, and no router knows both the true origin and the true destination of the packet.
In Configuration 1 of the Internet of Neurons, some of the nodes of the network are actually other users with their devices, whereas other nodes are classic nodes like routers, computers and so on. In this case, the device could negotiate a preemptive path passing through a number of other devices creating a sort of onion routing. However, this kind of solution might not be applicable in Configuration 2 because it is unclear who would actually negotiate a route and apply multiple layers of encryption, unless we assume that brains are able to connect directly with each other, which is something that we will discuss in a bit more detail as we draw our conclusions.
Conclusions {#sec:concl}
===========
The premise of this paper is that in 2038, in 20 years from now, the human brain will be at the center of a new Internet paradigm that we call Internet of Neurons. Some parts of our paper are deliberately science fiction (almost in the style of the Black Mirror TV series or other futuristic series and movies), but actually, as we have shown by means of the many ongoing works that we discussed, the seeds of the Internet of Neurons are already present in several of the technologies that are being used today or are under development. The opportunities will be prodigious, but repercussions for privacy, security and trust will be enormous and, frankly, tremendously scary. We have thus tried to dissect some of those challenges that researchers will have to face once this is all real (and trust us, it will become real in one form or the other). However, we have only skimmed the surface.
More work is needed to fully understand and reason about system and threat models and security properties, specifying the ones we discussed above in more detail but also considering other properties that could be relevant for the Internet of Neurons. Moreover, we have made the quite strong assumption that brainwaves will need to be translated to data (and vice versa) as the Internet will still transmit packets. But by, say, 2050, it could well be that the network will follow a radically different model, perhaps thanks to advances in quantum computing or in “brainwave computing” (a discipline that we just invented), allowing the network to directly process brainwaves as shown in Fig. \[fig:brainwavesinternet\]. But why stop here? If brainwave transmission protocols are possible, then it means that the network is able to read the brainwaves that a brain is emanating, but also that the brain is able to receive brainwaves in input. How long will it then take before we find a way for brains to connect not only to the network but also to each other? Some research in this direction is already ongoing [@grau2014conscious; @0111332] and the ultimate Internet of Neurons might then simply be based on direct brain-brain connections as the one in Fig. \[fig:brain-brain\].
![Bi-directional brain-Internet connection by means of brainwaves[]{data-label="fig:brainwavesinternet"}](brainwavesinternet)
![Bi-directional brain-brain direct connection[]{data-label="fig:brain-brain"}](esp)
Finally, there is an elephant in the room that we have not addressed in this paper. In addition to technological and neurological questions, some of which we discussed above, there are a huge number of economical, political and ethical issues that we don’t really feel competent to address, but that will have to be tackled before we open our mind to the Internet. Who will pay for the Internet of Neurons? Will all citizens be taxed? Will governments or perhaps corporations provide it for free? Given that nothing is actually free, what will they want in return? In the wake of the recent scandals on data collection (such as the Facebook–Cambridge Analytica data scandal that involved the collection of personally identifiable information of up to 87 million Facebook users), we are skeptical that the Internet of Neurons will be exempt from massive personal data collection and mining, possibly opening up the possibility for big-brother scenarios in which citizens are always observed and tracked in order to control and influence their thoughts, opinions, votes, in brief, their whole life.
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[^1]: In his famous 1950 paper [@Turing1950], Alan Turing wrote: “The human computer is supposed to be following fixed rules; he has no authority to deviate from them in any detail.”
[^2]: Note that we are here assuming that the “normal” network will still be operating through packets, although by then advances in quantum computing (i.e., computing using quantum-mechanical phenomena, such as superposition and entanglement) might have provided for new modes of data transmission. But this is a topic for another paper.
[^3]: Actually, it is not really important whether it will be in 10, 20 or 30 years, but rather that this will happen for sure, in one form or the other. And this time we should do it right, considering security from the start, unlike what happened when Internet was first designed as pointed out Danny Hills in [@LoAndBehold]:
> *Because the internet was designed for a community that trusted each other, it didn’t have a lot of protections in it. We didn’t worry about spying on each other, for example. We didn’t worry about somebody sending out spam, or bad emails, or viruses, because such a person would have been banned from the community.*
[^4]: Other studies [@ThorpeOS05; @MerrillCC17] have investigated *pass-thought authentication*, which allows users to submit both a knowledge factor (i.e., a secret thought) and an inherence factor (i.e., the unique way that thought is expressed) in a single step, by performing a single mental task.
[^5]: The process behind the brainwave authentication methods that have been proposed requires the registration of a brainwave pattern: a sequence of images or a sequence of words are shown to a user and her brainwaves are stored as her brainprint. This process has to be done in exactly the same way for each user in order to obtain an impartial brainprint. Through this brainprint, an authentication system is able to recognize a user and then, if desired, to authenticate her requests. Note that in Configuration 1 we will also need to authenticate, and protect, the pairing of brain and device.
[^6]: This is reminiscent of the *paradox of thought suppression* [@Wegner94], which originates from a challenge that Fyodor Dostoevsky posed in his 1863 essay “Winter Notes on Summer Impressions”: *Try to pose for yourself this task: not to think of a polar bear, and you will see that the cursed thing will come to mind every minute*.
[^7]: In the case of analog channels and signals (from the device to the brain or from the brain to the router), integrity of analog brainwaves could be evaluated in the same way in which we recognize a friend’s voice: first by recognition of familiar analog speech sounds, then by recognition of familiar linguistic patterns, and eventually by recognition of familiar behavioral cues and, if needed, through private shared history.
|
---
abstract: |
Decentralised optimisation tasks are important components of multi-agent systems. These tasks can be interpreted as n-player potential games: therefore game-theoretic learning algorithms can be used to solve decentralised optimisation tasks. Fictitious play is the canonical example of these algorithms. Nevertheless fictitious play implicitly assumes that players have stationary strategies. We present a novel variant of fictitious play where players predict their opponents’ strategies using Extended Kalman filters and use their predictions to update their strategies.
We show that in 2 by 2 games with at least one pure Nash equilibrium and in potential games where players have two available actions, the proposed algorithm converges to the pure Nash equilibrium. The performance of the proposed algorithm was empirically tested, in two strategic form games and an ad-hoc sensor network surveillance problem. The proposed algorithm performs better than the classic fictitious play algorithm in these games and therefore improves the performance of game-theoretical learning in decentralised optimisation.\
[**[Keywords:]{}**]{} Multi-agent learning, game theory, fictitious play, decentralised optimisation, learning in games, Extended Kalman filter.
author:
- 'Michalis Smyrnakis[^1]'
bibliography:
- 'ekf.bib'
title: 'Multi-agent learning using Fictitious Play and Extended Kalman Filter'
---
Introduction
============
Recent advance in technology render decentralised optimisation a crucial component of many applications of multi agent systems and decentralised control. Sensor networks [@sn], traffic control [@tf] and scheduling problems [@sc] are some of the tasks where decentralised optimisation can be used. These tasks share common characteristics such as large scale, high computational complexity and communication constraints that make a centralised solution intractable. It is well known that many decentralised optimisation tasks can be cast as potential games [@wlu; @autonomous], and the search of an optimal solution can be seen as the task of finding Nash equilibria in a game. Thus it is feasible to use iterative learning algorithms from game-theoretic literature to solve decentralised optimisation problems.
A game theoretic learning algorithm with proof of convergence in certain kinds of games is fictitious play [@learning_in_games; @fp4]. It is a learning process where players choose an action that maximises their expected rewards according to the beliefs they maintain about their opponents’ strategies.The players update their beliefs about their opponents’ strategies after observing their actions. Even though fictitious play converges to Nash equilibrium, this convergence can be very slow. This is because it implicitly assumes that other players use a fixed strategy in the whole game. @cj addressed this problem by representing the fictitious play process as a state space model and by using particle filters to predict opponents’ strategies. The drawback of this approach is the computational cost of the particle filters that render difficult the application of this method in real time applications.
The alternative that we propose in this article is to use instead of particle filters, extended Kalman filters (EKF) to predict opponents’ strategies. Therefore the proposed algorithm has smaller computational cost than the particle filter variant of fictitious play algorithm that proposed by @cj. We show that the EKF fictitious play algorithm converges to a pure Nash equilibrium, in 2 by 2 games with at least one pure Nash equilibrium and in potential games where players have two available actions. We also empirically observe, in a range of games, that the proposed algorithm needs less iterations than the classic fictitious play to converge to a solution. Moreover in our simulations, the proposed algorithm converged to a solution with higher reward than the classic fictitious play algorithm.
The remainder of this paper is organised as follows. We start with a brief description of game theory, fictitious play and extended Kalman filters. Section \[new\_algorithm\] introduces the proposed algorithm that combines fictitious play and extended Kalman filters. The convergence results we obtained are presented in Section \[theory\]. In Section \[parameters\] we propose some indicative values for the EKF algorithm parameters. Section \[simulation\] presents the simulation results of EKF fictitious play in a 2$\times$2 coordination game, a three player climbing hill game and an ad-hoc sensor network surveillance problem. In the final section we present our conclusions.
Background
==========
In this section we introduce some definition from game theory that we will use in the rest of this article and the relation between potential games and decentralised optimisation. We also briefly present the classic fictitious play algorithm and the extended Kalman filter algorithm.
Game theory definitions
-----------------------
We consider a game $\Gamma$ with $\mathbb{I}$ players, where each player $i$,$i=1,2,\ldots,\mathbb{I}$, choose his action, $s^{i}$, from a finite discrete set $S^{i}$. We then can define the joint action that is played in a game as the set product $S=\times_{i=1}^{i=\mathbb{I}}S^{i}$. Each Player $i$ receive a reward, $u^{i}$, after choosing an action . The reward is a map from the joint action space to the real numbers, $u^{i}:S \rightarrow R$. We will often write $s=(s^{i},s^{-i})$, where $s^{i}$ is the action of Player $i$ and $s^{-i}$ is the joint action of Player $i$’s opponents. When players select their actions using a probability distribution they use mixed strategies. The mixed strategy of a player $i$, $\sigma^{i}$, is an element of the set $\Delta^{i}$, where $\Delta^{i}$ is the set of all the probability distributions over the action space $S^{i}$. The joint mixed strategy, $\sigma$, is then an element of $\Delta=\times_{i=1}^{i=\mathbb{I}}\Delta^{i}$. Analogously to the joint actions we will write $\sigma=(\sigma^{i},\sigma^{-i})$. In the special case where the players choose an action with probabiity one we will say that players choose their actions using pure strategies. The expected utility a player $i$ will gain if he chooses a strategy $\sigma^{i}$ (resp. $s^{i}$), when his opponents choose the joint strategy $\sigma^{-i}$ is $u^{i}(\sigma^{i},\sigma^{-i})$ (resp. $u^{i}(s^{i},\sigma^{-i})$).
A common decision rule in game theory is best response (BR). The best response is defined as the action that maximizes players’ expected utility given their opponents’ strategies. Thus for a specific opponents’ strategy $\sigma^{-i}$ we evaluate the best response as: $$BR^{i}(\sigma^{-i})= \mathop{\rm argmax}_{s^{i} \in S} \quad
u^{i}(s^{i},\sigma^{-i})
\label{eq:br}$$
@nash showed that every game has at least one equilibrium, which is a fixed point of the best response correspondence, $\sigma^{i} \in BR(\sigma^{-i})$. Thus when a joint mixed strategy $\hat{\sigma}$ is a Nash equilibrium then: $$u^{i}(\hat{\sigma}^{i},\hat{\sigma}^{-i})\geq u^{i}(s^{i},\hat{\sigma}^{-i}) \qquad \textrm{for all } s^{i} \in S^{i}
\label{eq:nashutil}$$
Equation \[eq:nashutil\] implies that if a strategy $\hat{\sigma}$ is a Nash equilibrium then it is not possible for a player to increase his utility by unilaterally changing his strategy. When all the players in a game select their actions using pure strategies then the equilibrium actions are referred as pure strategy Nash equilibria. A pure equilibrium is strict if each player has a unique best response to his opponents actions.
Decentralised optimisation tasks as potential games
---------------------------------------------------
A class of games that are of particular interest in multi agent systems and decentralised optimisation tasks are potential games, because of their utility structure. In particular in order to be able to solve an optimisation task decentrally the local functions should have similar characteristics with the global function that we want to optimise. This suggests that an action which improves or reduces the utility of an individual should respectively increase or reduce the global utility. Potential games have this property, since the potential function (global function) depict the changes in the players’ payoffs (local functions) when they unilaterally change their actions. More formally we can write $$u^{i}(s^{i},s^{-i})-u^{i}(\tilde{s^{i}},s^{-i})=
\phi(s^{i},s^{-i})-\phi(\tilde{s^{i}},s^{-i})$$ where $\phi$ is a potential function and the above equality stands for every player $i$, for every action $s^{-i}\in S^{-i}$, and for every pair of actions $s^{i}$, $\tilde{s^{i}} \in S^{i}$, where $S^{i}$ and $S^{-i}$ represent the set of all available actions for Player $i$ and his opponents respectively. Moreover potential games has at least one pure Nash equilibrium, hence there is at least one joint action $s$ where no player can increase their reward, therefore the potential function, through a unilateral deviation.
It is feasible to choose an appropriate form of the agents’ utility function in order for the global utility to act as a potential of the system. Wonderful life utility is a utility function that introduced by @wlu and applied by @autonomous to formulate distributed optimisation tasks as potential games. Player $i$’s utility, when wonderful life utility is used, can be defined as the difference between the global utility $u_{g}$ and the utility of the system when a reference action is used as player’s $i$ action. More formally when player $i$ chooses an action $s^{i}$ we write $$u^{i}(s^{i})=u_{g}(s^{i},s^{-i})-u_{g}(s^{i}_{0},s^{-i})
\label{eq:wlu}$$ where $s^{i}_{0}$ denotes the reference action of player $i$. Hence the decentralised optimisation problem can be cast as a potential game and any algorithm that is proved to converge to a Nash equilibrium of a potential game, which is a local or the global optimum of the optimisation problem, will converge to a joint action from which no player can increase the global reward through unilateral deviation.
Fictitious play
---------------
Fictitious play [@brown_fict], is a widely used learning technique in game theory. In fictitious play each player chooses his action according to the best response to his beliefs about his opponents’ joint mixed strategy $\sigma^{-i}$.
Initially each player has some prior beliefs about the strategy that each of his opponents uses to choose an action based on a weight function $\kappa_{t}$. The players, after each iteration, update the weight function and therefore their beliefs about their opponents’ strategies and play again the best response according to their beliefs. More formally in the beginning of a game Player $i$ maintains some arbitrary non-negative initial weight functions $\kappa_{0}^{j}$, $\forall j\in[1, \mathbb{I}] \backslash \{i\}$, that are updated using the formula: $$\kappa_{t}^{j}(s^{j}) = \kappa_{t-1}^{j}(s^{j})+\mathfrak{I}_{s^j_t=s^j}
\label{eq:kappa}$$ for each $j$, where $\mathfrak{I}_{s^j_t=s^j}=\left\{\begin{array}{cl}1&\mbox{if
$s^j_t=s^j$}\\0&\mbox{otherwise.}\end{array}\right.$.\
The mixed strategy of opponent $j$ is estimated from the following formula:
$$\sigma_{t}^{j}(s^{j})=\frac{\kappa^j_{t}(s^j)}{\sum_{s' \in S^{j}}\kappa^j_{t}(s')}.
\label{eq:fp1p}$$
Player $i$ based on his beliefs about his opponents’ strategies, chooses the action which maximises his expected payoffs. When player $i$ uses equation (\[eq:fp1p\]) to update the beliefs about his opponents’ strategies he treats the environment of the game as stationary and implicitly assumes that the actions of the players are sampled from a fixed probability distribution. Therefore the recent observations have the same weight as the initial ones. This approach leads to poor adaptation when the other players choose to change their strategies.
Fictitious play as a state space model
--------------------------------------
We follow @cj and we will represent fictitious play process as a state-space model. According to this state space model each player has a propensity $Q_{t}^{i}(s^{i})$ to play each of his available actions $s^{i} \in S^{i}$, and then he forms his strategy based on these propensities. Finally he chooses his actions based on his strategy and the best response decision rule. Because players have no information about the evolution of their opponents’ propensities, and under the assumption that the changes in propensities are small from one iteration of the game to another, we model propensities using a Gaussian autoregressive prior on all propensities. We set $Q_{0}\sim N(0,I)$ and recursively update the value of $Q_{t}$ according to the value of $Q_{t-1}$ as follows: $$Q(s_{t})=Q(s_{t-1})+\eta_{t}
\label{eq:propen}$$ where $\eta_{t}\sim N(0,\chi^{2}I)$. The action of a player then is related to his propensity by the following sigmoid equation for every $s^{i} \in S^{i}$ $$s^{i}=\frac{e^{(Q^{i}(s^{i})/\tau)}}{\sum_{\tilde{s} \in S^{i}}e^{(Q_{t}(\tilde{s})/\tau)}}.$$ Therefore players will assume that at every iteration $t$ their opponents have a different strategy $\sigma_{t}$.
Kalman filters and Extended Kalman filters
------------------------------------------
Our objective is to estimate player $i$’s opponent propensity and thus to estimate the marginal probability $p(Q_{t},s_{1:t})$. This objective can be represented as a Hidden Markov Model (HMM). HMMs are used to predict the value of an unobserved variable $x_{t}$, the hidden state, using the observations of another variable $z_{1:t}$. There are two main assumptions in the HMM representation. The former one is that the probability of being at any state $x_{t}$ at time $t$ depends only at the state of time $t-1$, $x_{t-1}$. The latter one is that an observation at time $t$ depends only on the current state $x_{t}$. One of the most common methods to estimate $p(x_{1:t},z_{1:t})$ is Kalman filters and its variations. Kalman filter [@kalman] is based on two assumptions, the first is that the state variable is Gaussian. The second is that the observations are the result of a linear combination of the state variable. Hence Kalman filters can be used in cases which are represented as the following state space model: $$\label{eq:kf_state}
\begin{split}
x_t= & Ax_{t-1}+\xi_{t-1} \textrm{ hidden layer} \\
y_{t}= & Bx_{t}+\zeta_{t} \textrm{ observations}
\end{split}$$ where $\xi_t$ and $\zeta_t$ follow a zero mean normal distribution with covariance matrices $\Xi=q_{t}I$ and $Z=r_{t}I$ respectively, and $A$, $B$ are linear transformation matrices. When the distribution of the state variable $x_t$ is Gaussian then $p(x_{t}|y_{1:t})$ is also a Gaussian distribution, since $y_{t}$ is a linear combination of $x_{t}$. Therefore it is enough to estimate its mean and variance to fully characterise $p(x_{t}|y_{1:t})$.
Nevertheless in the state space model we want to implement, the relation between Player $i$’s opponent propensity and his actions is not linear. Thus we should use a more general form of state space model such as: $$\label{eq:ekf_state}
\begin{split}
x_t&=f(x_{t-1})+\xi_{t} \\
y_{t}&=h(x_{t})+ \zeta_{t}
\end{split}$$ where $\xi_{t}$ and $\zeta_{t}$ are the hidden and observation state noise respectively, with zero mean and covariance matrices $\Xi=q_{t}I$ and $Z=r_{t}I$ respectively. The distribution of $p(x_{t}|y_{1:t})$ is not a Gaussian distribution because $f(\cdot)$ and $h(\cdot)$ are non-linear functions. A simple method to overcome this shortcoming is to use a first order Taylor expansion to approximate the distributions of the sate space model in (\[eq:ekf\_state\]). In particular we let $x_{t}=m_{t-1}+\epsilon$, where $m_{t}$ denotes the mean of $x_{t}$ and $\epsilon \sim N(0,P)$. We can rewrite (\[eq:ekf\_state\]) as: $$\label{eq:ekf_taylor}
\begin{split}
x_t&=f(m_{t-1}+\epsilon)+w_{t-1}=f(m_{t-1})+F_{x}(m_{t-1})\epsilon +\xi_{t-1}\\
y_{t}&=h(m_{t}+\epsilon)+\zeta_{t}=h(m_{t})+H_{x}(m_{t})\epsilon+\zeta_{t}
\end{split}$$ where $F_{x}(m_{t-1})$ and $H_{x}(m_{t})$ is the Jacobian matrix of $f$ and $h$ evaluated at $m_{t-1}$ and $m_{t}$, respectively. If we use the transformations in (\[eq:ekf\_taylor\]) then $p(x_{t}|y_{1:t})$ is a Gaussian distribution.
Since $p(x_{t}|y_{1:t})$ is a Gaussian distribution to fully characterise it we need to evaluate its mean and its variance. The EKF process [@ekf1; @ekf2] estimates this mean and variance in two steps the prediction and the update step. In the prediction step at any iteration $t$ the distribution of the state variable is estimated based on all the observations until time $t-1$, $p(x_{t}|y_{1:t-1})$. The distribution of $p(x_{t}|y_{1:t-1})$ is Gaussian and we will denote its mean and variance as $m_{t}^{-}$ and $P_{t}^{-}$ respectively. During the update step the estimation of the prediction step is corrected in the light of the new observation at time $t$, so we estimate $p(x_{t}|y_{1:t})$. This is also a Gaussian distribution and we will denote its mean and variance as $m_{t}$ and $P_{t}$ respectively.
The prediction and the update steps of the EKF process [@ekf1; @ekf2] to estimate the mean and the variance of $p(x_{t}|y_{1:t-1})$ and $p(x_{t}|y_{1:t})$ respectively are the following:\
**Prediction Step** $$\begin{aligned}
\label{eq:pdstep}
m_{t}^{-}= &f(m_{t-1}) \nonumber \\
P_{t}^{-}=&F(m_{t-1})P_{t-1}F(m_{t-1})+\Xi_{t-1} \nonumber\end{aligned}$$ where the $j,j'$ element of $F(m_{t})$ is defined as $$[F(m_{t}^{-})]_{j,j'}=\frac{\partial f(x_{j},r)}{\partial x_{j'}}\arrowvert_{x=m_{t}^{-}, q=0}$$ **Update Step** $$\begin{aligned}
\label{eq:updatestep}
v_t&=&z_{t}-h(m_{t}^{-}) \nonumber \\
S_{t}&=&H(m_{t}^{-})P_{t}^{-}H^{T}(m_{t}^{-})+Z \nonumber \\
K_{t}&=&P_{t}^{-}H^{T}(m_{t}^{-})S_{t}^{-1} \nonumber \\
m_{t}&=&m_{t}^{-}+K_{t}v_{t}\nonumber \\
P_{t}&=&P_{t}^{-}-K_{t}S_{t}K_{t}^{T} \nonumber\end{aligned}$$ where $z_{t}$ is the observation vector (with 1 in the entry of the observed action and 0 everywhere else) and the $j,j'$ element of $H(m_{t})$ is defined as: $$[H(m_{t}^{-})]_{j,j'}=\frac{\partial h(x_{j},r)}{\partial x_{j'}}\arrowvert_{x=m_{t}^{-}, r=0}$$
Fictitious play and EKF {#new_algorithm}
=======================
For the rest of this paper we will only consider inference over a single opponent mixed strategy in fictitious play. Separate estimates will be formed identically and independently for each opponent. We therefore consider only one opponent, and we drop all dependence on player $i$, and write $s_{t}$, $\sigma_{t}$ and $Q_{t}$ for Player $i$’s opponent’s action, strategy and propensity respectively. Moreover for any vector $x$, $x[j]$ will denote the $j_{th}$ element of the vector and for any matrix $y$, $y[i,j]$ will denote the $(i,j)_{th}$ element of the matrix.
We can use the following state space model to describe the fictitious play process: $$\begin{aligned}
\label{eq:fpekf}
Q_{t}&=Q_{t-1}+\xi_{t-1} \nonumber \\
s_{t}&= h(Q_{t})+\zeta_{t} \nonumber\end{aligned}$$ where $\xi_{t-1} \sim N(0,\Xi)$, is the noise of the state process and $\zeta_{t}$ is is the error of the observation state with zero mean and covariance matrix $Z$, which occurs because we approximate a discrete process like best responses, equation (\[eq:br\]), using a continuous function $h(\cdot)$. Hence we can combine the EKF with fictitious play as follows. At time $t-1$ Player $i$ has an estimation of his opponent’s propensity using a Gaussian distribution with mean $m_{t-1}$ and variance $P_{t-1}$, and has observed an action $s_{t-1}$. Then at time $t$ he uses EKF prediction step to estimate his opponent’s propensity. The mean and variance of $p(Q_{t}|s_{1:t-1})$ of the opponent’s propensity approximation are: $$\begin{aligned}
%\label{eq:fppred}
m^{-}_{t}=m_{t-1} \nonumber \\
P_{t}^{-}=P_{t-1}+\Xi \nonumber\end{aligned}$$ Player $i$ then evaluates his opponents strategies using his estimations as: $$\sigma_{t}(s_{t})=\frac{exp(m_{t}^{-}[s_{t}] / \tau)}{\sum_{\tilde{s} \in S}exp(m_{t}^{-}[\tilde{s}]/\tau)}.
\label{eq:strategies}$$ where $m_{t}^{-}[s_{t}]$ is the mean of Player $i$’s estimation about the propensity of his opponent to play action $s_{t}$. Player $i$ then uses the estimation of his opponent strategy , equation (\[eq:strategies\]), and best responses, equation (\[eq:br\]), to choose an action. After observing the opponent’s action $s_{t}$, Player $i$ correct his estimations about his opponent’s propensity using the update equations of EKF process. The update equations are:
$$\begin{aligned}
v_t&=&z_{t}-h(m_{t}^{-}) \nonumber \\
S_{t}&=&H(m_{t}^{-})P_{t}^{-}H^{T}(m_{t}^{-})+Z \nonumber \\
K_{t}&=&P_{t}^{-}H^{T}(m_{t}^{-})S_{t}^{-1} \nonumber \\
m_{t}&=&m_{t}^{-}+K_{t}v_{t} \nonumber \\
P_{t}&=&P_{t}^{-}-K_{t}S_{t}K_{t}^{T} \nonumber
\label{eq:fpupdate}\end{aligned}$$
where $h=\frac{exp(Q_{t}[s^{'}]/\tau)}{\sum_{\tilde{s} \in S} exp(Q_{t}[\tilde{s}]/\tau)}$, and $\tau$ is a temperature parameter. The Jacobian matrix $H(m_{t}^{-})$ is defined as\
$[H(m_{t}^{-})]_{j,j'}=\left\{\begin{array}{cl} \frac{\sum_{j \neq j'}\exp(m_{t}^{-}[j])\exp(m_{t}^{-}[j'])}{(\sum_{j}\exp(m_{t}^{-}[j]))^2}&\mbox{if
$j=j'$}\\- \frac{\exp(m_{t}^{-}[j])\exp(m_{t}^{-}[j'])}{(\sum_{j}\exp(m_{t}^{-}[j]))^2} &\mbox{if $j \neq$ j'}\end{array}\right.$.
Table \[skata\] summarises the fictitious play algorithm when EKF is used to predict opponents strategies.
[p[12cm]{}]{} At time $t$
1. Player $i$ maintains some estimations about his opponents propensity up to time $t-1$, $p(Q_{t-1}|s{1:t-1})$. Thus he has an estimation of the mean $m_{t-1}$ and the covariance $P_{t-1}$ of this distribution.
2. Then Player $i$ is updating his estimations about his opponents propensities $p(Q_{t}|s{1:t-1})$ using equations, $m_{t}^{-}=m_{t-1}$, $P_{t}^{-}=P_{t-1}+W_{t-1}$.
3. Based on the weights of step 1 each player updates his beliefs about his opponents strategies using $\sigma_{t}^{j}(s^{j})=\frac{exp(m_{t}^{-}(j)/\tau)}{\sum_{j'}exp(m_{t}^{-}(j)/\tau)}$.
4. Choose an action based on the beliefs of step 3 according to best response.
5. Observe opponent’s action $s_{t}$.
6. Update the propensities estimates using $ m_{t}=m_{t}^{-}+K_{t}v_{t}$ and .
7. set t=t+1
\
Theoretical Results {#theory}
===================
In this section we present the convergence results we obtained for games with at least one pure Nash equilibrium and players who have 2 available actions, $s=(1,2)$. We will denote as $-s$ the action that a player does not choose, for example if Player $i$’s opponent chooses action 1, $s=1$ and hence $-s=2$. Also we will denote as $m[1]$ and $m[2]$ the estimated means of opponent’s propensity of action 1 and 2 respectively. Similarly $P[1,1]$ and $P[2,2]$ will represent the variance of the propensity’s estimation of action 1 and 2 respectively, and $P[1,2],P[2,1]$ their covariance.
The proposed algorithm has the following two properties:
\[prop1\] If at iteration $t$ of the EKF fictitious play algorithm, action $s$ is played from Player $i$’s opponent, then the estimation of his opponent propensity to play action $s$ increases, $m_{t-1}[s]<m_{t}[s]$. Also the estimation of his opponent propensity to play action $-s$ decreases, $m_{t-1}[-s]>m_{t}[-s]$
The proof of Proposition \[prop1\] is on Appendix \[append1\].
Proposition \[prop1\] implies that players, when they use EKF fictitious play, learn their opponent’s strategy and eventually they will choose the action that will maximise their reward base on their estimation. Nevertheless there are cases where players may change their action simultaneously and trapped in a cycle instead of converging in a pure Nash equilibrium. As an example we consider the game that is depicted in Table \[tab:simcoord\].
L R
--- ----- -----
U 1,1 0,0
D 0,0 1,1
: Simple coordination game[]{data-label="tab:simcoord"}
This is a simple coordination game with two pure Nash equilibria the joint actions $(U,L)$ and $(D,R)$. In the case were the two players start from joint action $(U,R)$ or $(D,L)$ and they always change their action simultaneously then they will never reach one of the two pure Nash equilibria of the game.
\[prop2\] In a $2 \times 2$ game where the players use EKF fictitious play process to choose their actions, and the variance of the observation state is set to $Z=rI+\epsilon I$, with high probability the two players will not change their action simultaneously infinitely often. We define $\epsilon$ as a random number from normal distribution with zero mean and arbitrarily small covariance matrix, $I$ is the identity matrix.
The proof of Proposition \[prop2\] is on Appendix \[append2\].
We should mention here that the reason we set $Z=rI+\epsilon I$ is in order to break any symmetries that occurred because the initialisation of the EKF fictitious play algorithm. Based on Proposition \[prop1\] and \[prop2\] we can infer the following propositions and theorems.
\[prop3\] (a) In a game where players have two available actions if $s$ is a Nash equilibrium, and $s$ is played at date $t$ in the process of EKF fictitious play, $s$ is played at all subsequent dates. That is, strict Nash equilibria are absorbing for the process of EKF fictitious play. (b) Any pure strategy steady state of EKF fictitious play must be a Nash equilibrium.
Consider the case where players beliefs $\hat{\sigma}_{t}$, are such that their optimal choices correspond to a strict Nash equilibrium $\hat{s}$. In EKF fictitious play process players’ beliefs are formed identically and independently for each opponent based on equation (\[eq:strategies\]). By Proposition \[prop1\] we know that players’ estimations about their opponents’ propensities and therefore their strategies, that each player maintains for the other players, will increase for the actions that are included in $\hat{s}$ and will be reduced otherwise. Thus the best response to their beliefs $\hat{\sigma}_{t+1}$ will be again $\hat{s}$ and since $\hat{s}$ is a Nash equilibrium they will not deviate from it. Conversely, if a player remains at a pure strategy profile, then eventually the assessments will become concentrated at that profile, because of Proposition \[prop1\], hence if the profile is not a Nash equilibrium, one of the players would eventually want to deviate.
\[prop4\] Under EKF fictitious play, if the beliefs over each player’s choices converge, the strategy profile corresponding to the product of these distributions is a Nash equilibrium.
Suppose that the beliefs of the players at time t, $\sigma_{t}$, converges to some profile $\hat{\sigma}$. If $\hat{\sigma}$ were not a Nash equilibrium, some player would eventually want to deviate and the beliefs would also deviate since based on Proposition \[prop1\] players eventually learn their opponents actions.
\[theo1\] The EKF fictitious play process converges to the Nash equilibrium in $2\times2$ games with at least one pure Nash equilibrium, when the covariance matrix of the observation space error, $Z$, is defined as in Proposition \[prop2\], $Z=rI+\epsilon I$.
We can distinct two possible initial states in the game. In the first players’ initial beliefs of the players actions are such that their initial joint action $s_{0}$ is a Nash equilibrium. From Proposition \[prop3\] and equation (\[eq:strategies\]) we know that they will play the joint action which is a Nash equilibrium for all the iterations of the game.
The second case where the initial beliefs of the players are such that their initial joint action $s_{0}$ is not a Nash equilibrium is divided in 2 subcategories. The first include $2 \times 2$ games with only one pure Nash equilibrium. In this case, one of the two players has a dominant action, thus for all the iterations of the game he will choose the dominant action. This action maximises his expected payoff regardless the other player’s strategy and thus he will select this action in every iteration of the game. Therefore because of Proposition \[prop1\] the other player will learn his opponent’s strategy and players will choose the joint action which is the pure Nash equilibrium.
The second category includes $2\times2$ games with 2 pure Nash equilibria, like the simple coordination game that is depicted in Table \[tab:simcoord\]. In this case players initial joint action $s_{0}=(s^{1},s^{2})$ is not a Nash equilibrium. Then the players will learn their opponent’s strategy, Proposition \[prop1\] and Equation (\[eq:strategies\]), and they will change their action. We know from Proposition \[prop2\] that in a finite time with high probability the players will not change their actions simultaneously, and hence they will end up in a joint action that will be one of the two pure Nash equilibria of the game.
We can extend the results of Theorem \[theo1\] in $n \times 2$ games with a better reply path. A game with a better reply path can be represented as a graph were its edges are the join actions of the game $s$ and there is a vertex that connects $s$ with $s'$ iff only one player $i$ can increasing his payoff by changing his action [@payton_young]. Potential games have a better reply path.
\[theo2\] The EKF fictitious play process converges to the Nash equilibrium in $n\times 2$ games with a better reply path when the covariance matrix of the observations space error, $Z$, is $Z=r+\epsilon I$.
Similarly to the $2 \times 2$ games if the initial beliefs of the players are such that their initial joint action $s_{0}$ is a Nash equilibrium, from Proposition \[prop3\] and equation (\[eq:strategies\]), we know that they will play the joint action which is a Nash equilibrium for the rest of the game.
Moreover in the case of the initial beliefs of the players are such that their initial joint action $s_{0}$ is not a Nash equilibrium based on Proposition \[prop1\] and Proposition \[prop2\] after a finite number of iterations because the game has a better reply path the only player that can improve his payoff by changing his actions will choose a new action which will result in a new joint action $s$. If this action is not the a Nash equilibrium then again after finite number of iterations the player who can improve his payoff will change action and a new joint action $s'$ will be played. Thus after the search of the vertices of a finite graph, and thus after a finite number of iterations, players will choose a joint action which is a Nash equilibrium.
Simulations to define algorithm parameters $\Xi$ and $Z$. {#parameters}
=========================================================
The covariance matrix of the state space error $\Xi=qI$ and the measurement error $Z=rI$ are two parameters that we should define in the beginning of the EKF fictitious play algorithm and they affect its performance. Our aim is to find values, or range of values, of $q$ and $r$ that can efficiently track opponents’ strategy when it smoothly or abruptly change, instead of choosing $q$ and $r$ heuristically for each opponent when we use the EKF algorithm. Nevertheless it is possible that for some games the results of the EKF algorithm will be improved for other combinations of $q$ and $r$ than the ones that we propose in this section.
We examine the impact of EKF fictitious play algorithm parameters in its performance in the following two tracking scenarios. In the first one a single opponent chooses his actions using a mixed strategy which changes smoothly and has a sinusoidal form over the iterations of the tracking scenario. In particular for $t=1, 2, \ldots, 100$ iterations of the game: $\sigma_{t}(1)=\frac{cos\frac{{2\pi t}}{n}+1}{2}=1-\sigma_t(2)$, where $n=100$. In the second toy example Player $i$’s opponent change his strategy abruptly and chooses action 1 with probability $\sigma_{t}^{2}(1)=1$ during the first 25 and the last 25 iterations of the game and for the rest iterations of the game $\sigma_{t}^{2}(1)=0$. The probability of the second action is calculated as: $\sigma_{t}^{2}(2)=1 - \sigma_{t}^{2}(1)$.
We tested the performance of the proposed algorithm for the following range of parameters $10^{-4} \leq q \leq 1$ and $10^{-4} \leq r \leq 1$. We repeated both examples 100 times for each of the combinations of $q$ and $r$. Each time we measured the absolute error of the estimated strategy against the real one. The combined average absolute error when both examples are considered is depicted on Figure \[fig:mseqr\]. The darkest areas of the contour plot represent the areas where the average absolute error is minimised.
![Combined absolute error for both tracking scenarios. The range of both parameters, $q$ and $r$ is between $10^{-4}$ and $1$.[]{data-label="fig:mseqr"}](mseb1.png)
The average absolute error is minimised for a range of values of $q$ and $r$, that form two distinct areas. In the first area, the wide dark area of Figure \[fig:mseqr\], the range of $q$ and $r$ were $0.08 \leq q \leq 0.4$ and $0.2 \leq r \leq 1$ respectively. In the second area, the narrow dark area of Figure \[fig:mseqr\], the range of $q$ and $r$ were $0.001 \leq q \leq 0.025$ and $0.08 \leq r \leq 0.13$ respectively. The minimum error which we observed in our simulations was in the narrow area and in particular when $\Xi=0.01I$ and $Z=0.1I$, where $I$ is the identical matrix.
Simulation results {#simulation}
==================
This section is divided in two parts. The first part contains results of our simulations in two strategic form games and the second part contains the results we obtained in an ad-hoc sensor network surveillance problem. In all the simulations of this section we set the covariance matrix of the hidden and the observations state to $\Xi=0.01I$ and $Z=(0.1+\epsilon)I$ respectively, where $\epsilon \sim N(0,10^{-5})$ and $I$ is the identical matrix.
Simulations results in strategic form games
-------------------------------------------
In this section we compare the results of our algorithm with those of fictitious play in two coordination games. These games are depicted in Tables \[tab:simcoord\] and \[tab:3pclh\]. The game that is depicted in Table \[tab:simcoord\], as it was described in Section \[theory\] , is a simple coordination game with two pure Nash equilibria, its diagonal elements. Table \[tab:3pclh\] presents an extreme version of the climbing hill game [@clh] in which three players must climb up a utility function in order to reach the Nash equilibrium where their reward is maximised.
[c|c|c|c]{}
---
U
M
D
---
&
U M D
--- ---- ----
0 0 0
0 50 40
0 0 30
U
&
U M D
------ ---- ----
-300 70 80
-300 60 0
0 0 0
M
&
------------------------------
U M D
---------------------- --- ---
**[100]{} & -300&90\
0& 0& 0\
0&0&0\
&D&**
------------------------------
We present the results of 50 replications of a learning episode of 50 iterations for each game. As it is depicted in Figures \[fig:res1\] and \[fig:res2\] the proposed algorithm performs better than fictitious play in both cases. In the simple coordination game that is shown in Table \[tab:simcoord\], the EKF fictitious play algorithm converges to one of the pure equilibria after a few iterations. On the other hand fictitious play is trapped in a limit cycle in all the replications where the initial joint action was not one of the two pure Nash equilibria. For that reason the players’ payoff for all the iterations of the game was either 1 utility unit or 0 utility units depending to the initial joint action. In the climbing hill game, Table \[tab:3pclh\] the proposed algorithm converges to the Nash equilibrium after $35$ iterations when fictitious play algorithm do not converge even after 50 iterations.
![Results of EKF and classic fictitious play in the simple coordination game of Table \[tab:simcoord\][]{data-label="fig:res1"}](res1.png)
![Probability of playing the (U,U,D) equilibrium for the EKF fictitious play (solid line) and fictitious play (dash line) for the three player climbing hill game[]{data-label="fig:res2"}](res2.png)
Ad-hoc sensor network surveillance problem.
-------------------------------------------
We compared the results of our algorithm against those of fictitious play in a coordination task of a power constrained sensor network, where sensors can be either in a sense or sleep mode [@faraneli; @fict_ql]. When the sensors are in sense mode they can observe the events that occur in their range. During their sleep mode the sensors harvest the energy they need in order to be able function when they are in the sense mode. The sensors then should coordinate and choose their sense/sleep schedule in order to maximise the coverage of the events. This optimisation task can be cast as a potential game. In particular we consider the case where $\mathbb{I}$ sensors are deployed in an area where $E$ events occur. If an event $e$, $e \in E$, is observed from the sensors then it produce some utility $V_e$. Each of the sensors $i=1,\ldots,\mathbb{I}$ should choose an action $s^{i}=j$, from one of the $j=1,\ldots,J$ time intervals which they can be in sense mode. Each sensor $i$ when it is in sense mode can observe an event $e$, if it is in its sense range, with probability $p_{ie}=\frac{1}{d_{ie}}$, where $d_{ie}$ is the distance between the sensor $i$ and the event $e$. We assume that the probability each sensor has to observe an event is independent from the other sensors. If we denote as $i_{in}$ the sensors that are in sense mode when the event $e$ occurs and $e$ is in their sensing range, then we can write the probability an event $e$ to be observed from the sensors, $i_{in}$ as $$1- \prod_{i \in i_{in}}{(1-p_{ie})}$$ The expected utility that is produced from the event $e$ is the product of its utility $V_e$ and the probability it has to be observed by the sensors, $i_{in}$ that are in sense mode when the event $e$ occurs and $e$ is in their sensing range. More formally we can express the utility that is produced from an event $e$ as: $$U_{e}(s)=V_{e}(1- \prod_{i \in i_{in}}{(1-p_{ie}}))
\label{eq:target_utility}$$ The global utility is then the sum of the utilities that all events, $e \in E$, produce $$U_{global}(s)= \sum_{e}{U_{e}(s)}. \label{eq:global_utility}$$
Each sensor after each iteration of the game receives some utility which is based on the sensors and the events that are inside his communication and sense range respectively. For a sensor $i$ we denote $\tilde{e}$ the events that are in its sensing range and $\tilde{s}^{-i}$ the joint action of the sensors that are inside his communication range. The utility that sensor $i$ will receive if his sense mode is $j$ will be $$U_{i}(s^{i}=j,\tilde{s}^{-i})= \sum_{\tilde{e}}{U_{\tilde{e}}(s^{i}=j,\tilde{s}^{-i})} \label{eq:individual_utility}.$$
We compared the performance of the two algorithms in 2 instances of the above scenario one with 20 and one with 50 sensors that are deployed in a unit square. In both instances sensors had to choose one time interval of the day that they will be in sense mode and use the rest time intervals to harvest energy. We consider cases where sensors had to choose their sense mode between 2, 3 and 4 available time intervals. Sensors are able to communicate with other sensors that are at most 0.6 distance units away, and can only observe events that are at most 0.3 distance units away. Moreover in both instances we assumed that 20 events took place in the unite square area. Those events were uniformly distributed in space and time, so an event could evenly appear in any point of the unit square area and it could occur at any time with the same probability. The duration of each event was uniformly chosen between (0-6\] hours and each event had a value $V_{e} \in (0-1]$. Figures \[fig:res3\] and \[fig:res4\] depict the average results of 50 replications of the game for the two algorithms. For each instance, both algorithms run for 50 iterations. To be able to average across the 50 replications we normalise the utility of a replication by the global utility that the sensors will gain if they were only in sense mode during the whole day.
As we observe in Figures \[fig:res3\] and \[fig:res4\] EKF fictitious play converges to a stable joint action faster than the fictitious play algorithm. In particular on average the EKF fictitious play algorithm needed 10 “negotiation” steps between the sensors in order to reach a stable joint action, when fictitious ply needed more than 25. Moreover the classic fictitious play algorithm was always resulted in joint actions with smaller reward than the proposed algorithm.
Conclusion
==========
We have introduced a variation of fictitious play that uses Extended Kalman filters to predict opponents’ strategies. This variation of fictitious play addresses the implicit assumption of the classic algorithm that opponents use the same strategy in every iteration of the game.
We showed that, for $2 \times 2$ games with at least one pure Nash equilibrium, EKF fictitious play converges in the pure Nash equilibrium of the game. More over the proposed algorithm converges in games with a better reply path, like potential games, and $n$ players that have 2 available actions.
EKF fictitious play performed better than the classic algorithm algorithm in the strategic form games and the ad-hoc sensor network surveillance problem we simulated. Our empirical observations indicate that EKF fictitious play converges to a solution that is better than the classic algorithm and needs only a few iterations to reach that solution. Hence by slightly increasing the computational intensity of fictitious play less communication is required between agents to quickly coordinate on a desired solution.
Acknowledgements
================
This work is supported by The Engineering and Physical Sciences Research Council EPSRC (grant number EP/I005765/1).
Proof of Proposition 1 {#append1}
======================
We will base the proof of Proposition \[prop1\] on the properties of EKF when they used to estimate opponent’s strategy with two available actions. If player $i$’s opponent has two available actions $1$ and $2$, then we can assume that at time $t-1$ Player $i$ maintains beliefs about his opponent’s propensity, with mean $m_{t-1}$ and variance $P_{t-1}$. Moreover based on these estimations he chooses his strategy $\sigma_{t-1}$. At the prediction step of this process he uses the following equations to predict his opponent’s propensity and choose an action using best response.
$$m_{t}^{-} = \left(
\begin{array}{c}
m^{-}_{t-1}[1] \\
m^{-}_{t-1}[2] \\
\end{array} \right)$$
$$P_{t}^{-} = \left(
\begin{array}{cc}
P^{-}_{t-1}[1,1] &P^{-}_{t-1}[1,2] \\
P^{-}_{t-1}[2,1]& P^{-}_{t-1}[2,2] \\
\end{array} \right)+ qI$$
without loss of generality we can assume that his opponent in iteration $t$ chooses action 2. Then the update step will be : $$v_t=z_{t}-h(m_{t}^{-})$$ since Players $i$’s opponent played action 2 and $h=\frac{exp(Q_{t}[s^{'}]/\tau)}{\sum_{\tilde{s} \in S} exp(Q_{t}[\tilde{s}]/\tau)}$ we can write $v_{t}$ and $H_{t}(m_{t}^{-})$ as: $$\begin{aligned}
v_{t} = &\left(
\begin{array}{c}
0 \\
1 \\
\end{array} \right)-
\left(
\begin{array}{c}
\sigma_{t-1}(1) \\
1- \sigma_{t-1}(1) \\
\end{array} \right) \nonumber \\
= &\left(
\begin{array}{c}
- \sigma_{t-1}(1) \\
\sigma_{t-1}(1) \\
\end{array} \nonumber \right)\end{aligned}$$
$$H_{t}(m_{t}^{-}) = \left(
\begin{array}{cc}
a_{t} & -a_{t} \\
-a_{t}& a_{t} \\
\end{array} \right)$$
where $a_{t}$ is defined $a_{t}=\sigma_{t-1}(1)\sigma_{t-1}(2)$. The estimation of will be: $$S_{t} = a^{2}\left(
\begin{array}{cc}
b & -b \\
-b& b
\end{array} \right)+Z$$ where $b=P^{-}_{t}[1,1]+P^{-}_{t}[2,2]-2P^{-}_{t}[1,2]$. The Kalaman gain, $K_{t}=P_{t}^{-}H^{T}(m_{t}^{-})S_{t}^{-1}$ can be written as
$$K_{t}= \frac{1}{2r b+r^2}\left(
\begin{array}{cc}
P_{t}^{-}[1,1] & k \\
k& P_{t}^{-}[2,2] \\
\end{array} \right) \left(
\begin{array}{cc}
a_{t}& -a_{t} \\
-a_{t}&a_{t} \\
\end{array} \right)\left(
\begin{array}{cc}
b+r & b\\
b& b+r \\
\end{array} \right)$$
up to a multiplicative constant we can write $$K_{1} \sim \left(
\begin{array}{cc}
c & -c\\
-d& d \\
\end{array} \right)$$ where $c=P_{t}^{-}[1,1]-P_{t}^{-}[1,2]$ and $d=P_{t}^{-}[2,2]-P_{t}^{-}[1,2]$. The updates then for the mean and variance are:
$$\begin{aligned}
m_{t}=&m_{t}^{-}+K_{t}v_{t} \nonumber \\
P_{t}=&P_{t}^{-}-K_{t}S_{t}K_{t}^{T} \nonumber
%\label{31}\end{aligned}$$
The mean of the Gaussian distribution that is used to estimate opponent’s propensities is: $$\begin{aligned}
\label{eq:mupd}
m_{t} = &\left(
\begin{array}{c}
m_{t}[1] \\
m_{t}[2] \\
\end{array} \right)
=\left(
\begin{array}{c}
m_{t}^{-}[1]-2\sigma(1)\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)}\\
m_{t}^{-}[2]+2\sigma(1)\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)}\\
\end{array} \right)\end{aligned}$$
Based on the above we observe that $m_{t}(1)<m_{t-1}(1)$ and $m_{t}(2)>m_{t-1}(2)$ which completes the proof.
Proof of Proposition 2 {#append2}
======================
We consider $2 \times 2$ games with at least one pure Nash equilibrium. In the case that only one Nash equilibrium exists, a dominant strategy exists and thus one of the players will not deviate from this action. Hence we are interested in in $2\times2$ games with two pure Nash equilibria. Without loss of generality we consider a game with similar structure to the simple coordination game that is depicted in Table \[tab:simcoord\]. with two equilibria, the joint actions in the diagonal of the payoff matrix, $(U,L)$ and $(D,R)$. We will present calculations for Player 1,but the same results hold also for Player 2. We define $\lambda$ as the necessary confidence level that Player 1’s estimation of $\sigma_{t}(L)$ should reach in order to choose action $U$. Hence we Player 1 will choose $D$ if: $$\begin{aligned}
\sigma_{t}(1) &>\lambda \Leftrightarrow \nonumber \\
\frac{exp(m_{t}^{-}[1])}{exp(m_{t}^{-}[1])+exp(m_{t}^{-}[2])} & > \lambda \Leftrightarrow \nonumber \\
m_{t}^{-}[1] &> \ln(\frac{\lambda}{1-\lambda}) +m_{t}^{-}[2] \Leftrightarrow \nonumber \\
m_{t-1}[1] &> \ln(\frac{\lambda}{1-\lambda}) +m_{t-1}[2] \nonumber\end{aligned}$$
In order to prove Proposition \[prop2\], we need to show that when a player changes his action his opponent will change his action at the same iteration with probability less than 1. In the case where at time $t-1$ the joint action of the players is $U,R$ then Player $1$ believes that his opponent will play $L$, while he observing him playing $R$. Assume that Player 2’s beliefs about Player 1’s strategies has reached the necessary confident level about Players 1’s strategy and at iteration $t$ he will change his action from $R$ to $L$. Player 1 will also change his action at the same time if
$$m_{t-1}[2] > \ln(\frac{1-\lambda}{\lambda}) +m_{t-1}[1]$$
We want to show that players will not change actions simultaneously with probability 1. Hence it is enough to show that
$$Prob( m_{t-1}[1] > \ln(\frac{\lambda}{1-\lambda}) +m_{t-1}[2] )>0
\label{pithan1212}$$
We can replace $m_{t-1}[1]$ and $m_{t-1}[2]$ with their equivalent from (\[eq:mupd\]) and write:
$$\begin{aligned}
m_{t}^{-}[1]-2\sigma(1)\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)} & > &\ln( \frac{\lambda}{1-\lambda})+ m_{t}^{-}[2]+2\sigma(1)\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)} \Leftrightarrow \nonumber \end{aligned}$$
$$\begin{aligned}
-4\sigma(1)\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)} &> & \ln(\frac{\lambda}{1-\lambda})+ m_{t}^{-}[2] - m_{t}^{-}[1] \Leftrightarrow \nonumber \\
\frac{a(b-k)}{4a^{2}(b-k)+(r+\epsilon)} &<&\frac{\ln(\frac{\lambda}{1-\lambda})+ m_{t}^{-}[2] - m_{t}^{-}[1]}{-4\sigma(1)} \nonumber\end{aligned}$$
Solving this with respect to $\epsilon$ we have
$$\epsilon >\frac{a(b-k)\sigma(1)}{\ln(\frac{\lambda}{1-\lambda})+ m_{t}^{-}[2] - m_{t}^{-}[1]}-a^{2}(b-k) -r$$
Thus we can write (\[pithan1212\]) as: $$Prob(\epsilon >\frac{a(b-k)\sigma(1)}{\ln(\frac{\lambda}{1-\lambda})+ m_{t}^{-}[2] - m_{t}^{-}[1]}-a^{2}(b-k) -r )>0
\label{pithan1111}$$ Since $\epsilon$ is a Gaussian white noise (\[pithan1111\]) is always true.
We also consider the case where at time $t-1$ the joint action of the players is $D,L$ then Player 1 believes that his opponent will play $R$, while he observing him playing $L$. Assume that Player 2’s beliefs about Player 1’s strategies has reached the necessary confident level and at $t$ he will change his action from $L$ to $R$. Player 1 will also change his action at the same time if
$$m_{t-1}[1] > \ln(\frac{\lambda}{1-\lambda}) +m_{t-1}[2]$$
We want to show that Players will not change actions simultaneously with probability 1. Hence it is enough to show that
$$Prob(m_{t-1}[2] > \ln(\frac{1-\lambda}{\lambda}) +m_{t-1}[1])>0
\label{pithan}$$
We can rewrite (\[pithan\]) using the results we obtained for $m_{t-1}[1]$ and $m_{t-1}[2]$ in (\[eq:mupd\]) again as
$$Prob(\epsilon >\frac{a(b-k)\sigma(1)}{\ln(\frac{\lambda}{1-\lambda})+ m_{t}^{-}[2] - m_{t}^{-}[1]}-a^{2}(b-k) -r )>0
\label{pithan1}$$
Since $\epsilon$ is a Gaussian white noise (\[pithan1\]) is always true.
If we define $\xi_{t}$ the event that both players change their action at time $t$ simultaneously, and assume that the two players have change their actions simultaneously at the following iterations $t_{1}, t_{2}, \ldots, t_{t}$, then the probability that they will also change their action simultaneously at time $t_{T+1}$, $P(\xi_{t_1},\xi_{t_2}, \ldots, \xi_{t_T}, \xi_{t_T+1})$ is almost zero for large but finite $T$.
[^1]: michalis.smyrnakis@manchester.ac.uk
|
---
abstract: 'The Sc galaxy NGC 6181 was observed at the 6m telescope of SAO RAS with the scanning Perot-Fabry interferometer in the H$_\alpha$ emission line and at the 1m telescope of SAO RAS in $BVRI$ broadband filters with CCD. Subtraction of the mean circular rotation curve from the two-dimensional velocity field has revealed a ring-like zone with a diameter about of 2 kpc where strong radial gas motions are present. The form of the ring is almost perfectly circular in the plane of the galaxy. It is located closer to the center than the beginning of the well-defined spiral structure, but outside of the central bulge-dominated region. The detected radial velocity reduced to the plane of the galaxy is about 100 $km\cdot s^{-1}$ and probably is azimuthally dependent. The very inner region of the galaxy, $r<3\arcsec$ or 0.5 kpc, shows a turn of the dynamical major axis by about $30^o$. Central continuum isophotes are also twisted which suggests the presence of small nuclear bar .'
author:
- 'O. K. Sil’chenko'
- 'A. V. Zasov'
- 'A. N. Burenkov'
- 'J. Boulesteix'
title: 'A Ring-like Zone of Strong Radial Gas Motions in the Disk of NGC 6181 [^1]'
---
Introduction
============
NGC 6181 is an isolated late-type giant spiral galaxy whose high surface brightness of gas emission and inclination of about $60^o$ are favorable for a detailed kinematical study. Global parameters of the galaxy are listed in Table 1 being taken basically from LEDA (Lyon-Meudon Extragalactic Database). The first kinematical investigation of NGC 6181 was undertaken nearly 30 years ago by Burbidge et al. (1965). They obtained three spectral long-slit cross-sections of the galaxy and found line-of-sight velocity distributions to be quite asymmetrical. They concluded that NGC 6181 is not an axisymmetric galaxy being intermediate between a barred and a normal spiral. The form of thin dust lanes in the center of NGC 6181 gave some support to this hypothesis. In addition, they noted that “the structural center of the galaxy is not the center of the velocity distribution”, because the systemic velocities determined from the outer and the inner parts of line-of-sight velocity curves disagreed.
Our group (Afanasiev et al. 1992) has repeated a long-slit kinematical study of NGC 6181 at the 6m telescope of the Special Astrophysical Observatory (SAO RAN) and fully confirmed the unusual asymmetrical character of the one-dimensional line-of-sight velocity distributions. It has been proposed that it is the southern part of the galaxy which reveals strong non-circular gas motions up to distances of about 20 from the center. But it became evident that the obtaining of the two-dimensional velocity field is necessary to clarify the situation.
----------------------------- ----------------------------------------------------
Hubble type SAB(rs)c
$R_{25}$ 12.3 kpc
$B_T^0$ 11.7
$M_B$ –20.93
$V_r(radio)$ 2374 $km\cdot s^{-1}$
$V_{gal.std-of-rest}$ 2492 $km\cdot s^{-1}$
Distance 33.6 Mpc ($H_0$=75 $km\cdot s^{-1}\cdot Mpc^{-1}$)
Inclination $66.6^o$
*[MA]{}$_{phot}$ & $175^o$\
*
----------------------------- ----------------------------------------------------
: Global parameters of NGC 6181
Observations
============
The two-dimensional velocity field of NGC 6181 was obtained at the 6m telescope on September 24, 1993. The scanning Perot-Fabry interferometer was installed inside of the pupil plane of a focal reducer which was attached to the F/4 prime focus of the telescope. An intensified photon counting system (IPCS) $512\times 512$ was used as the detector. Instrument parameters are given in Table 2.
------------------------------------ ----------------------------
Input telescope beam F/4
Output camera beam F/2.4
Number of pixels $512\times 512$
Pixel scale 0.35
Field $3\arcmin \times 3\arcmin$
Filter wavelength $\lambda_f$=6620 Å
Filter FWHM 10 Å
Galaxy H$_\alpha$ wavelength $\lambda_g$=6615 Å
Filter transmission at $\lambda_g$ 35 %
Etalon interference order 501 at 6562.8Å
Free spectral range at $\lambda_g$ 603 $km\cdot s^{-1}$
Finesse at $\lambda_g$ 18
Number of scanned channels 32
Channel step 18.84 $km\cdot s^{-1}$
Number of scanned cycles 7
Basic exposure time 20 s
Total exposure time 4480 s
Calibration line Neon 6598.98 Å
Seeing ${< 1.5\arcsec}$
------------------------------------ ----------------------------
: Scanning Perot-Fabry observations parameters
The “Image-interferometry method” was described in detail earlier (Boulesteix et al. 1983, Amram et al. 1991). Observational data (galaxy and wavelength calibration data) were converted into cubes of 32 images ($256\times256$), with the linear scale being 0.70 arcsec/px and a spectral resolution of about 2–2.5 channels (40–50 $km\cdot s^{-1}$). A reduction of observational data (correction for phase shifting, subtraction of night-sky emission spectrum, construction of velocity map, a.s.o.) was done by using standard methods, and the Perot-Fabry reducing software ADHOC developed at Marseille Observatory (Boulesteix 1993) was used.
The continuum subtracted H$_\alpha$ image of NGC 6181 (Fig. 1) reveals that the H$_\alpha$ emission is rather strong over the whole galactic disk. So the high quality of the two-dimensional velocity distribution obtained allows to get more detailed information about gas motions in the disk of this galaxy. A sub-cube ($210 \,px\times 185 \,px \times 32 \,channels$), containing the main fraction of H$_\alpha$ emission from the galaxy, was extracted from a reduced data cube. This smaller cube was used for the analysis of the velocity field.
Date Filter Exposure time Zenithal distance
------------- -------- --------------- -------------------
7/8.06.94 $I$ 600 s $49^o$
7/8.06.94 $I$ 100 s $52^o$
8/9.06.94 $I$ 600 s $50^o$
8/9.06.94 $R$ 200 s $52^o$
8/9.06.94 $R$ 200 s $53^o$
8/9.06.94 $R$ 200 s $54^o$
10/11.07.94 $B$ 300 s $57^o$
10/11.07.94 $V$ 300 s $59^o$
10/11.07.94 $B$ 300 s $60^o$
10/11.07.94 $V$ 300 s $61^o$
10/11.07.94 $I$ 300 s $63^o$
: Direct image observations
Direct images of NGC 6181 were obtained at the telescope Zeiss–1000 of SAO RAN. Eleven frames of the galaxy have been derived with a CCD camera through the $B$, $V$, $R$ and $I$ filters of Johnson’s system (the log of the observations is given in Table 3). The seeing quality ranged from 2.1 ($R$ images) to 2.8 ($V$ and $I$ images).
Gray-scaled sky-subtracted $I$ and $B$ images are presented in Fig. 2a and 2b; flux-calibrated isophotes in $B$ (21.8 $mag/arcsec^2$ and 25.3 $mag/arcsec^2$ for the innermost and the outermost isophotes respectively with the step of 0.5 $mag/arcsec^2$) are shown in Fig.2c. The $B$ and $V$ frames were calibrated by using 14 aperture photoelectric measurements of NGC 6181 from Burstein et al. (1987). The range of aperture radii is from 16 to 48. Zero-point magnitudes are obtained with accuracy better than 0.01 mag, color terms are found to be negligible. The sky brightnesses are estimated as 21.26 $mag/arcsec^2$ in $V$ and 22.12 $mag/arcsec^2$ in $B$; these values coincide with mean sky brightnesses measured fifteen years ago in the Special Astrophysical Observatory at $z=60^o$ by Neizvestny (1981). To check our $BV$ calibration, we compared multi-aperture photoelectric data for NGC 6181 taken from the catalogue of Longo & Vaucouleurs (1983, 1985) (18 entities excluding old data of PET-54 and BIG-51) with values simulated for the same apertures from our CCD frames. In Fig. 3 one can see a rather good consistency with the photoelectric data, even for large apertures. The calibration of $R$ and $I$ frames was indirect and less precise because standard stars were not observed. Instead $B-V$ colors for five faint stars in the field of the galaxy were measured, and adopting them to be dwarfs we ascribe them mean $V-R$ and $V-I$ colors in accordance with their spectral types (Straizys 1977). The formal accuracy of the calibration constants determined in such a way is 0.12 mag, but we admit a possible systematic shift of our $R$ and $I$ magnitudes by up to 0.3 mag.
Photometric properties of NGC 6181
==================================
A monochromatic image was constructed from the Perot-Fabry observational data in order to see the morphology of the inner part of NGC 6181. The H$_\alpha$ map of the central part of NGC 6181 (Fig. 1) reveals two bright central sources none of which coincides with the center of the isophotes in the continuum. Two faint tails of H$_\alpha$ emission embrace the central continuum source which is located in the area of very weak emission. Bearing in mind the absence of radio emission from the NGC 6181 nucleus, one may conclude that the nucleus of this galaxy is very quiescent.
The direct images obtained with the 1m telescope were used first of all to find the precise position of the center of the galaxy in the continuum. It was determined with respect to five nearby stars. Location of the center was compared with the H$_\alpha$ distribution and with the dynamical center position (see the next section). In addition we tried to derive some surface brightness distribution characteristics. Fig.2 demonstrates rather smooth image in $I$, with a weak bar-like disk elongation in the inner part, and a more clumpy image in $B$; prominent spiral arms extend up to the outermost radii in all passbands, confirming the grand-design classification of arms made by Elmegreen & Elmegreen (1984).
Fig. 4 presents $B$ and $V$ azimuthally averaged radial surface brightness profiles assuming $\it{PA}$ (line of nodes)=$173^o$ and inclination $56^o$ in accordance with the velocity field analysis (see below). It shows that this galaxy possesses a very compact bulge which does not affect light distributions beyond the radius 7. In the range 12–25 a brightness excess is noticeble over the simple exponential law extrapolated from the outer parts; this excess is reproduced in all four filters being the largest (0.15 mag) in the $B$ passband. It seems that the radius of 25 is a boundary between two disk subsystems. The disk scale measured in the range 25–45 for all four passbands slightly decreases from blue to red (Fig. 5) being in general accordance with earlier results of Elmegreen & Elmegreen (1984) and Roth (1994).
The azimuthally averaged color tends to be bluer up to $r\approx
36\arcsec$ and then some reddening occurs, but the slopes of these trends are quite different for different colors (Fig. 6 and 7). In Fig. 7 we try to compare radial color variations in NGC 6181 with models for old stellar populations showing pure metallicity trend (Worthey 1994) and with a mean observational sequence of galactic colors (Buta & Williams 1995), which is known to be defined mainly by different present-time star formation rate. Color excesses expected due to interstellar reddening in the Galaxy are also shown. The reddening in $B-V$ at $r\approx 5\arcsec$ has an azimuthally non-homogeneous character: it is a distinct, very red spot to the west from the nucleus obviously related to a local dust concentration; the optical characteristics of the dust may be unusual because the spot is absent in $V-R$ and $V-I$ colors. The other color variations seem to be rather azimuthally homogeneous. Comparison of the observed and the expected color trends shows that between the radii 7 and 25, i. e. in the inner disk distinguished by some brightness excess, the observed color variations may be satisfactorily explained by variations of star formation intensity, because the point grid is roughly parallel to the observational sequence of integrated galactic colors. However the observed color variations in the outer disk are more complicated and rather unusual – especially for $r > 36\arcsec$, where the reddening of $B-V$ and $V-R$ occurs under the constant $V-I$. This looks quite inexplicable in the frames of simple effects which influence the color.
The isophote form analysis (pure-ellipse fitting) was carried out to check a possible deviation from axial symmetry. The ellipticity between 1 and 7 from the center gradually increases from 0.10 (bulge) to 0.35 – a behavior which is quite normal for a galaxy whose inclination is about $60^o$. The radial dependence of $\it{PA}_0$ is presented in Fig. 8. We see an unambiguous turn of isophotes in the very center of NGC 6181. Measurements in all passbands show that at the radius of 2–3 the position angle of the major axis is $+3^o$ with an uncertainty less than $1^o$, which differs by $\approx 5^o$ from the orientation of the outermost isophotes ($175^o$, Nilson 1973; $178.4^o\pm0.5^o$, our measurement of the SKYVIEW isophote at the $r=65\arcsec$). In the radius range 10–20 isophotes are also twisted by $10^o-13^o$, but in the opposite sense with respect to the innermost region. Only beyond $r\approx 25\arcsec$, where the radial brightness distribution follows a pure exponential law, the isophote major axis becomes aligned with the line of nodes.
Velocity field of the ionized gas in NGC 6181
=============================================
Fig. 9 presents the observed velocity field of NGC 6181. It looks quite regular, with prominent signs of rotation. However in the center of the galaxy a twist of the zero-velocity line is seen which gives evidence for non-circular gas motions in this area in a good agreement with the photometrical data (see below).
A fit of the circular rotation model was made for the full velocity map, with dimensions of $210\times185$ pixels ($147\arcsec\times130\arcsec$), or within a radius of about 75(12.2 kpc) from the center. Special codes were written for data processing. The galactic disk was supposed to be thin and flat, that is it does not have any tilt of warp within the optical radius; so, we looked for inclination $i$ and position angle of the line of nodes $\it{MA}$ for the full radius range. As a first step, we determined the position of the dynamical center and a systemic velocity suggesting a central symmetry of the velocity field. The dynamical center appears to coincide with the center of broadband isophotes with an accuracy of one pixel. The systemic velocity is found to be 2375 $km\cdot s^{-1}$ which agrees with earlier determinations (Table 1). The dispersion of systemic velocity values determined over all pairs of symmetrically taken points of the galaxy is 16 $km\cdot s^{-1}$ which is close to our accuracy of individual velocity determinations. Then we verified if the whole line-of-sight velocity field can be fitted by a pure circular rotation. The mean line-of-sight velocity residuals (r.m.s.) were calculated for $30^o$ ranges of $i$ and $\it{MA}$; the minimum of the velocity residual calculated over the total velocity field reveals the true values of these parameters. This approach assumes that any velocity field distortions, if they exist, are of local nature.
The agreement is found the best for the following orientation parameters: $i_0=56^o$, $\it{MA}=173^o$, the $\it{MA}$ being more strictly limited and $i_0$ having less accuracy. If one compares these values with the photometric parameters of NGC 6181 – for example, $i=61^o$ (Bottinelli et al. 1984) and $\it{MA}=175^o$ (Nilson 1973) or with the data from the Table 1 – it becomes clear that the bulk of the gas in the galaxy rotates circularly.
Fig. 10 presents the azimuthally averaged rotation curve. For $r>25\arcsec$ it is nicely flat at the level of 210 $km\cdot s^{-1}$ with an r.m.s. error of individual points not worse than 5-7 $km\cdot s^{-1}$. The maximal rotation velocity estimated from the width of the HI line at 21 cm, $W50$, is 222 $km\cdot s^{-1}$ (LEDA Consultation), so our rotation curve for NGC 6181 is in accordance with previously known data.
Despite the generally good accordance between the observed velocity field and the circular rotation model, there are three ranges of radial distances where essential systemic deviations from a pure circular rotation are detected with the regions of maximal deviations at $r = 30\arcsec - 40\arcsec$ , $r \approx 12 \arcsec$ and $r< 3
\arcsec$. The first area is located near to the dynamical major axis and coincides neither with spiral arms nor with bright HII regions. However, the color profiles discussed in the previous section demonstrate a turnover of color radial trends at this radius. In the southern half of the galaxy this region is distinguished by the excess of azimuthal velocity of order of 30–50 $km\cdot s^{-1}$, and in the northern half of the galaxy there is a similar velocity depression of about the same value. As these two areas are located symmetrically with respect to the galactic center, this anomaly may be considered rather as a kind of regular wave distortion of the velocity field than as a local velocity anomaly.
The model velocity field calculated in the frame of pure circular rotation with the parameters mentioned above has been subtracted from the observed velocity field. The central part of the residual velocity field is presented in Fig. 11. Here we see two halves of the ring-like region where deviations from circular rotation model locally exceed 50 $km\cdot s^{-1}$. Being deprojected onto the plane of the galaxy, this region looks like three quarters of a perfect circular ring with a mean radius of 11 (about 1.8 kpc); the eastern half of the ring has positive residual velocities up to 55 $km\cdot s^{-1}$, the western part has negative ones, from –30 to –40 $km\cdot s^{-1}$. The width of the ring is at least 5 pixels, which corresponds to 0.8 kpc. The fact that the switching of residual velocity sign takes place near the line of nodes implies that the residual velocities here are mostly radial ones (here and below, we admit that the gas motion is in the plane of the disk). Together with a circular deprojected shape of the ring, it also gives evidences that the line of nodes of this structure is close to the $\it{MA}$ of the global galactic disk and that the ring lies exactly in the galactic plane being purely an internal feature of the galactic gaseous subsystem. Stemming from the slightly asymmetric minor-axis surface brightness profile of the bulge one may conclude that the western half of the galactic disk is the nearest one to us. In this case we may conclude that NGC 6181 possesses a trailing spiral pattern and radially expanding gas motions in the ring.
It is worth noting that the ring of radially moving gas lies closer to the center than the beginning of the well-defined spiral arms and does not reveal itself in a brightness distribution. Nevertheless ionized gas in the ring shows systemic velocity residuals of much higher amplitude than in the bright spiral arms.
In the very center of NGC 6181, within the region where the major axis of continuum isophotes is twisted ($r<5\arcsec$), a clear sign of elliptical gas rotation is seen in the two-dimensional velocity field. Analysing an azimuthal dependence of central velocity gradient, we find that the maximum of the cosine curve computed by the least square approximation\
dv$_r$/dr = \[44.2 cos($\it PA$ – 325.5$^o$) – 4.7\] $km\cdot s^{-1}\cdot arcsec^{-1}$\
is shifted by about $30^o$ relative to the line of nodes $\it{MA}=173^o$ (Fig. 12). Hence, circumnuclear gas rotation in NGC 6181 parallel with the isophote major axis twist gives strong evidence for the presence of a small bar in the very center of the galaxy. So NGC 6181 may be applied to a small number of known galaxies where nuclear bar reveals itself both from photometric and kinematic data.
A physical connection between the central mini-bar and the ring-like zone of gas expansion may be suspected. It follows from radial velocities of gas in the ring deprojected onto the plane of the galaxy, assuming that these motions are purely radial. Parameters of the galactic plane orientation used for deprojection were taken from the best fit model of circular rotation: $i_0=56^o$, $\it{MA}=173^o$. It appeared that the radial velocity of the ring expansion varies along the eastern half of the ring from 50 to 120 $km\cdot s^{-1}$ (Fig. 13), and the position angle of the maximum expansion velocity roughly coincides with the position angle of the minimum of cosine curve describing the azimuthal dependence of the central velocity gradient. It gives some evidence that the position angle of the largest radial velocities is related to the orientation of nuclear bar.
Kinematically distinct, ring-like inner regions of systemic radial motions of gas, similar to what we observe in NGC 6181, were not known yet. The only analogy, which can be mentioned, is the famous “3 kpc arm” in our Galaxy: it possesses radial velocities of about 100–150 $km\cdot s^{-1}$ and is probably connected with a triaxial structure of the Galactic center. The other radially expanding gaseous ring, found in NGC 4725 (Buta 1988), has a radius of 10–13 kpc being a structure of a quite different scale.
Note, that morphologically distinguished nuclear rings, which reveal themselves as zones of brightness, not of velocity, excess, often accompany nuclear bars (Buta & Crocker 1993). Numerical simulations confirm that gaseous subsystems may give ring-like response to a triaxial potential form (Combes & Gerin 1985). So what we found may be considered as a kinematical counterpart of such structures as nuclear rings, related to general disk structure. In this case, as in the case of our Galaxy, it is not necessary to interpret radial gas velocities in the ring-like zone as an evidence of its real expansion due to some explosion event: there is no hint of the presence of a shock front or enhanced star formation in front of the ring or in the ring itself. A more realistic explanation is that we observe here an unusually large amplitude of hydrodynamical oscillations of gas velocities associated with the density waves which penetrate deep into the inner part of the disk (Fridman et al., in preparation).
The other feature of the velocity field of NGC 6181 is a multicomponent structure of emission line profiles in some HII regions of the disk. We performed Gauss analysis of two-component emission line profiles for the central part of the galaxy $64\times64$ pixels, or $45\arcsec\times45\arcsec$. The primary – more strong and everywhere narrow – component reveals a velocity field which excellently agrees with the field obtained in the previous analysis: a general circular rotation, elliptical gas motions in the center and ring-like zone of radial gas motions. The secondary, more weak and broad component (with gas velocity dispersion up to 200 $km\cdot s^{-1}$) appears only in the four brightest HII regions; it is absolutely absent in the ring-like zone of the radial gas motions. The difference between the “first” and the “second” velocity component averaged over the total region of the galaxy is zero, but for the bright HII region, nearest to the dynamical center, there exists a switch of velocity difference sign between the northern and the southern halves. It allows to suspect that we deal with a proper rotation of a giant star formation site.
Concluding remarks
==================
Measurements of the line-of-sight velocity field for gas emission in the giant Sc galaxy NGC 6181 have allowed us to reveal some rare phenomena. The most unusual of them is the presence of a ring-like region with striking systemic deviations from the general circular rotation which may be interpreted as a strong gas radial motion. More elaborate interpretation which connects this motion with the presence of hydrodynamical 3D oscillations of gas velocities related to density wave phenomena will be given elsewhere (Fridman et al., in preparation). A small nuclear bar is also found in the galaxy both by kinematic and photometric methods.
We are very grateful to the observers of the Special Astrophysical Observatory RAS – S.N. Dodonov, S.V. Drabek, and V.V. Vlasiuk assisting us at the 6 m telescope, and V.O. Chavushyan and S.K. Balayan which obtained the data at the 1 m telescope. During the data analysis we have used the Lyon-Meudon Extragalactic Database (LEDA) supplied by the LEDA team at the CRAL-Observatoire de Lyon (France) and of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We acknowledge also an anonymous referee for the very useful comments and advices.
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[^1]: Based on observations collected with the 6m telescope at the Special Astrophysical Observatory (SAO) of the Russian Academy of Sciences (RAS).
|
---
abstract: 'We study the Schrödinger functional coupling for lattice Yang-Mills theory coupled to an improved bosonic spinor field, which corresponds to QCD with minus two light flavors. This theory serves as a less costly testcase than QCD for the scaling of the coupling.'
address: 'Institut für Physik, Humboldt-Universität zu Berlin, Invalidenstr. 110, D-10115 Berlin, Germany'
author:
- 'Bernd Gehrmann, Stefan Kurth, Juri Rolf and Ulli Wolff'
title: 'Schrödinger functional at $N_{\rm f}=-2$[^1]'
---
Introduction
============
The non-perturbative computation of the strong coupling constant $\alpha_s$ from hadronic scales to perturbative high energy scales is one of the main interests of the ALPHA collaboration. For this computation, a finite volume renormalization scheme is used in which the coupling runs with the space-time volume [@Luscher:1991wu]. The scale evolution of the coupling is computed recursively with the help of the step scaling function, which is an integrated variant of the beta function for finite changes of the scale. Furthermore, Schrödinger functional boundary conditions and ${\rm O}(a)$ improvement are employed. An important property of the step scaling function is that it can be calculated by simulating pairs of lattices with different sizes $L$ for the same $a$ and taking the continuum limit.
After this programme had been successfully used in the quenched approximation [@heplat9309005], first results for full QCD with two flavors are now available [@Frezzotti:2000rp]. Unfortunately, full QCD is notoriously costly to simulate, and thus the data for the step scaling function do not reach very close to the continuum limit. In order to learn more about the extrapolation to the continuum and about the effect of ${\rm O}(a)$ improvement, we have studied the step scaling function in a Yang-Mills theory coupled to a bosonic spinor field, which has a local interaction and is therefore much cheaper to simulate. It corresponds to setting the number of flavors $N_{\rm f}=-2$ in the partition function, and is therefore closely related to full QCD in perturbation theory. In the literature, this model is known as *bermion model* [@deDivitiis:1995yz]. Our focus here however is not on extrapolating results from negative to positive flavor number.
Bermion model
=============
For a detailed discussion of the lattice setup, boundary conditions and the algorithm we have used, we refer to [@heplat0106025] and references therein.
The Schrödinger functional, as the partition function of the system, is an integral over all gauge and quark fields which fulfill the given boundary conditions. After integrating out the quark fields, it is $$\begin{aligned}
Z = e^{-\Gamma} &=& \int D[U] D[\bar\psi] D[\psi]
e^{-S[U,\bar\psi,\psi]} \nonumber\\
&=& \int D[U] e^{-S_g} \det(M^\dagger M)^{N_{\rm f}/2}\end{aligned}$$ with a gauge action $S_g$. For $N_{\rm f}=-2$, the determinant can be written as $$Z = \int D[U] D[\phi^+] D[\phi] e^{-S_g-S_b},$$ with a now local bosonic action $$S_b[U,\phi] = a^4 \sum_x |(M\phi)(x)|^2.$$ The fields $\phi(x)$ carry color and Dirac indices.
We have chosen the bulk improvement coefficient $c_{\rm sw}$ by extrapolating the non-perturbative data for $N_{\rm f}=0$ [@heplat9609035] and $N_{\rm f}=2$ [@heplat9803017] in $N_{\rm f}$. An explicit calculation of $c_{\rm sw}$ along the lines of these references for the most critical parameters in our simulations has proven a good accuracy of this extrapolation. The boundary improvement coefficients $c_t$ and $\tilde c_t$ have been set to their perturbative values.
As the improved bosonic action depends quadratically on each link $U_{x\mu}$ through the clover term, it is difficult to employ finite step size algorithms such as the combination of heatbath and overrelaxation steps used in [@heplat9907007]. Because of the large additional cost of hybrid Monte Carlo algorithms, we have decided to use global heatbath steps for the bosonic fields and local overrelaxation steps with respect to the unimproved action for the gauge fields. The clover term is taken into account by an acceptance step. The acceptance rate has turned out to be high enough (about 70 %) for such an approach. Nevertheless, there is a significant overhead due to the calculation of the action difference necessary for the acceptance step, resulting in a cost factor of about $12$ compared to unimproved Wilson bermions. On the other hand, comparing with data from [@Frezzotti:2000rp], improved bermions are about a factor of $10$ cheaper than dynamical fermions, with a better scaling in $a$. In figure \[fig:perf\] we see a scaling of the cost with $a^{-2.5}$.
Results
=======
We have computed the step scaling function $\Sigma(u,a/L)$ for the couplings $u=\bar g^2=0.9793$ and $u=1.5145$ and lattice sizes $L=4,5,6,8$. Most simulations were done on APE100 parallel computers with up to 256 nodes with 50 MFlops each. For the simulation of the largest lattice $L/a=16$ at $u=1.5145$ we have also used a crate of APEmille.
Figure \[fig:bermimp\] shows the results for the step scaling function plotted against $(a/L)^2$. Within the error bars, no linear dependence an $a/L$ is visible, and therefore an extrapolation linear in $(a/L)^2$ is justified. The extrapolated values are consistent with perturbation theory, and their error bars are of the same size as the perturbative 3-loop contribution.
In [@heplat9907007], the step scaling function was already computed in the unimproved bermion theory for $u=0.9793$. Figure \[fig:bermcombined\] shows these results together with the data after implementing improvement. The linear cutoff effects for this observable are of the order of a few percent. In the plot, both data sets are fitted under the constraint of universality, i.e. that their continuum limit agrees. This fit is linear plus quadratic in $a/L$ for the Wilson bermion data and quadratic in $a/L$ in the improved data. Although the additional input from the Wilson data is included, the joint continuum limit $\sigma_{\rm combined}(0.9793)=1.1059(43)$ is almost the same as the value $\sigma_{\rm improved}(0.9793)=1.1063(46)$. A linear plus quadratic fit in $a/L$ of the unimproved data alone would have given the continuum result $\sigma_{\rm unimproved}(0.9793) = 1.103(12)$.
This indicates a success of the improvement programme. The main contribution to the cost for the calculation of $\sigma(u)$ comes from the largest lattice. When using an improved action, the lattice size needed for a reliable extrapolation to the continuum is smaller than without improvement. Even in the model used here, where the algorithmic implementation implies a large overhead for the inclusion of the clover term, this leads to the improved case being more cost effective. In simulations with dynamical fermions and algorithms like Hybrid Monte Carlo, the implementation of a clover term is possible with much lower overhead, and the advantage should therefore be higher.
[**Acknowledgements.**]{} We would like to thank Peter Weisz for essential checks on our perturbative calculations and Rainer Sommer for helpful discussions. DESY provided us with the necessary computing resources and the APE group contributed their permanent assistance. This work is supported by the Deutsche Forschungsgemeinschaft under Graduiertenkolleg GK 271 and by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00145.
[99]{} Martin [Lüscher]{}, Peter Weisz and Ulli Wolff, [*Nucl. Phys.*]{}, B359:221–243, 1991.
Martin [Lüscher]{}, Rainer Sommer, Peter Weisz and Ulli Wolff, [*Nucl. Phys.*]{}, B413:481–502, 1994.
Roberto Frezzotti, Martin Hasenbusch, Ulli Wolff, Jochen Heitger and Karl Jansen, [*Comput. Phys. Commun.*]{}, 136:1–13, 2001.
Giulia de Divitiis et al., [*Nucl. Phys.*]{}, B437:447–470, 1995.
Bernd Gehrmann, Stefan Kurth, Juri Rolf and Ulli Wolff, [*Nucl.Phys.*]{}, B612:3–24, 2001
Juri Rolf and Ulli Wolff, [*Nucl. Phys. Proc. Suppl.*]{}, 83:899–901, 2000.
Martin [Lüscher]{}, Stefan Sint, Rainer Sommer, Peter Weisz, Ulli Wolff, [*Nucl.Phys.*]{}, B491:323–343, 1997
Karl Jansen and Rainer Sommer, [*Nucl.Phys.*]{}, B530:185–203, 1998
[^1]: Talk given by B. Gehrmann at Lattice 2001, Berlin
|
---
abstract: 'The color evaporation model simply states that charmonium production is described by the same dynamics as $D \bar D$ production, [ *i.e.*]{}, by the formation of a colored $c \bar c$ pair. Its color happens to be bleached by soft final-state interactions. We show that the model gives a complete picture of charmonium production including low-energy production by proton, photon and antiproton beams, and high-energy production at the Tevatron and HERA. Our analysis includes the first next-to-leading-order calculation in the color evaporation model.'
address: 'Department of Physics, University of Wisconsin, Madison, WI 53706'
author:
- 'J. F. Amundson[^1], O. J. P. Éboli[^2], E. M. Gregores[^3], F. Halzen[^4]'
title: |
Quantitative Tests of Color Evaporation:\
Charmonium Production
---
Introduction
============
In a recent paper [@amundson] we pointed out that an unorthodox prescription for the production of rapidity gaps in deep inelastic scattering, proposed by Buchmüller and Hebecker [@bh], suggests a description of the production of heavy quark bound states which is in agreement with the data. The prescription represents a reincarnation of the “duality” or “color evaporation” model. It is very important to study the validity of this approach as it questions the conventional treatment of the color quantum number in perturbative QCD.
The conventional treatment of color, [*i.e.*]{}, the color singlet model, has run into serious problems describing the data on the production of charmonium [@review]. While attempts to rectify the situation exist [@bbl], the color singlet model has remained the standard by which other approaches are measured. Here we show that the color evaporation approach, which actually predates the color singlet approach, describes the available data well. After a discussion of the model itself, we show how the ratio of production by antiprotons to production by protons forms a quantitative test of the model. The model passes the test extraordinarily well. We then show how further quantitative tests can be performed using the the data on photoproduction and hadroproduction total cross sections. We finally show that the model successfully describes $p_T$ distributions at the Tevatron.
The Model
=========
Color evaporation represents a fundamental departure from the way color singlet states are treated in perturbation theory. In fact, color is “ignored”. Rather than explicitly imposing that the system is in a color singlet state in the short-distance perturbative diagrams, the appearance of color singlet asymptotic states depends solely on the outcome of large-distance fluctuations of quarks and gluons. These large-distance fluctuations are probably complex enough for the occupation of different color states to respect statistical counting. In other words, color is a nonperturbative phenomenon. In Fig. \[fig:csm\] we show typical diagrams for the production of $\psi$-particles using the competing treatments of the color quantum number. In the diagram of Fig. \[fig:csm\]a, the color singlet approach, the $\psi$ is produced in gluon-gluon interactions in association with a final state gluon which is required by color conservation. This diagram is related by crossing to the hadronic decay $\psi \rightarrow 3$ gluons.
In the color evaporation approach, the color singlet property of the $\psi$ is initially ignored. For instance, the $\psi$ can be produced to leading order by $q\bar q$-annihilation into $c\bar c$, which is the color-equivalent of the Drell-Yan process. This diagram is calculated perturbatively; its dynamics are dictated by short-distance interactions of range $\Delta x \simeq m_{\psi}^{-1}$. It does not seem logical to enforce the color singlet property of the $\psi$ at short distances, given that there is an infinite time for soft gluons to readjust the color of the $c \bar c$ pair before it appears as an asymptotic $\psi$ or, alternatively, $D \bar D$ state. It is indeed hard to imagine that a color singlet state formed at a range $m_{\psi}^{-1}$, automatically survives to form a $\psi$. This formalism was, in fact, proposed almost twenty years ago [@cem; @fh:1a; @fh:1b; @gor] and subsequently abandoned for no good reason.
------ --- -------
=0.3 =0.45
(a) (b)
------ --- -------
\[fig:cem\]
The color evaporation approach to color leads to a similar description of bound and open charm production. In the color evaporation model $$\sigma_{\rm onium} = \frac{1}{9} \int_{2 m_c}^{2 m_D} dm~
\frac{d \sigma_{c \bar{c}}}{dm} \; ,
\label{sig:on}$$ and $$\begin{aligned}
\sigma_{\rm open} &=& \frac{8}{9} \int_{2 m_c}^{2 m_D} dm~
\frac{d \sigma_{c \bar{c}}}{dm}
+ \int_{2 m_D} dm~\frac{d \sigma_{c \bar{c}}}{dm}
\label{sig:op}\end{aligned}$$ where the cross section for producing heavy quarks, $\sigma_{c \bar
c}$, is computed perturbatively. Diagrams are included order-by-order irrespective of the color of the $c \bar c$ pair. The coefficients $\frac{1}{9}$ and $\frac{8}{9}$ represent the statistical probabilities that the $3\times\bar3$ charm pair is asymptotically in a singlet or octet state.
The model also predicts that the sum of the cross sections of all onium states is given by Eq. (\[sig:on\]). This relation is, unfortunately, difficult to test experimentally, since it requires measuring cross sections for [*all*]{} of the bound states at a given energy.
Other approaches similar in spirit can be found in Refs. [@bbl] and [@hoyer]. The color evaporation approach differs from Ref. [@bbl], the formalism of Bodwin, Braaten and Lepage (BBL), in the way that the $c \bar c$ pair exchanges color with the underlying event. In the BBL formalism, multiple gluon interactions with the $c\bar c$ pair are suppressed by powers of $v$, the relative velocity of the heavy quarks within the $\psi$. The color evaporation model assumes that these low-energy interactions can take place through multiple soft-gluon interactions; this implies a statistical treatment of color.
The color evaporation model assumes a factorization of the production of the $c\bar{c}$ pair, which is perturbative and process dependent, and the materialization of this pair into a charmonium state by a mechanism that is nonperturbative and process independent. This assumption is reasonable since the characteristic time scales of the two processes are very different: the time scale for the production of the pair is the inverse of the heavy quark mass, while the formation of the bound state is associated to the time scale $1/\Lambda_{\rm
QCD}$. Therefore, comparison with the $\psi$ data requires knowledge of the fraction $\rho_\psi$ of produced onium states that materialize as $\psi$’s, [*i.e.,*]{} $$\sigma_\psi = \rho_\psi \sigma_{\rm onium} \; ,
\label{frac}$$ where $\rho_\psi$ is assumed to be a constant. This assumption is in agreement with the low energy data [@gksssv; @schuler]. We demonstrated in Ref. [@amundson] that simple statistical counting estimates of $\rho_{\psi}$, $\rho_{\chi}$, etc., accommodate all charmonia data to better than a factor of 2.
The Tests
=========
We discussed several qualitative tests of the color evaporation picture in Ref. [@amundson]. One such test that we did not mention is the polarization of produced charmonium. In the framework of the color evaporation model the multiple soft gluon exchange destroys the initial polarization of the heavy quark pair [@mirkes]. This fact is in agreement with the measurements of the $\psi$ polarization made in fixed-target $\pi^-N$ [@pin] and $\bar{p}N$ [@pbarn] reactions. The color singlet model fails to describe this feature of charmonium production [@tang] since it predicts that $\psi$’s are produced transversely polarized. The predictions in the literature for polarization in the BBL formalism have been somewhat controversial[@braatenchen].
One of the most striking features of color evaporation is that the production of charmonium at low energies is dominated by the conversion of a colored gluon into a $\psi$, as in Fig. \[fig:cem\]b. In the conventional treatment, where color singlet states are formed at the perturbative level, 3 gluons (or 2 gluons and a photon) are required to produce a $\psi$. The result is that in the color evaporation model $\psi$’s are hadroproduced not only by gluon-gluon initial states, but also via quark-antiquark fusion. In the color singlet approach such diagrams only appear at higher orders of perturbation theory and their contribution is small.
We can distinguish between the two pictures experimentally by comparing the production of charmonium by proton beams with that from antiproton beams. The color evaporation model predicts an enhanced $\psi$ cross section for antiproton beams, while the color singlet model predicts the same cross section for the production of $\psi$’s, whether we use proton or antiproton beams. The prediction of an enhanced $\bar p$ yield compared to $p$ yield at low energies is obviously correct: the ratio of antiproton and proton production of $\psi$’s exceeds a factor 5 close to threshold. (See Fig. \[fig:part-anti\].) In fact, it has been known for some time that $\psi$’s are predominantly produced by $q\bar q$ states [@fh:1a; @fh:1b; @gor]. Nonetheless, we should note that for sufficiently high energies, gluon initial states will eventually dominate because they represent the bulk of soft partons. This can be seen in Fig. \[fig:part-anti\] where the ratio gets close to unity for center-of-mass energies as low as 25 GeV.
The merits of the color evaporation approach can be first appreciated by studying the data in calculation-independent ways. The factorization of the production of the $c\bar{c}$ pairs and the formation of onium states implies that the energy dependence and kinematic distributions of the measured cross section for the different bound states should be equivalent. Moreover, in the approximation $m_{c}\approx m_{D}$ this equivalence extends to the production of open $D \bar D$ pairs. In Fig. \[fig:justdata\] we display charm photoproduction data for both open charm and bound state production with common normalization in order to show their identical energy behavior. In Fig. \[fig:hadro-data\] we display charm hadroproduction data in a similar fashion.
Further quantitative tests of color evaporation are made possible by the fact that the factor $\rho_\psi$ is the same in hadro- and photoproduction. Once $\rho_\psi$ has been empirically determined for one initial state, the cross section is predicted without free parameters for the other. We show next that color evaporation passes this test, quantitatively accommodating all measurements, including the high energy Tevatron and HERA data, which have represented a considerable challenge for the color singlet model.
In Fig. \[fig:photopro\] we compare the photoproduction data with theory, using the NLO perturbative QCD calculation of charm pair production from Ref. [@nlo]. The solid line is the NLO prediction for open charm production using the GRV HO distribution function and the scale $\mu = m_c = 1.45$ GeV. The dashed line is the prediction using the MRS A distribution function with the scale $\mu = 2 m_c =
2.86$ GeV. We obtained these charm quark masses from the best fit to the data for each structure function. For the $J/\psi$ production data we employed the parameters used for describing open charm, and determined the fragmentation factor $\rho_\psi$ to be 0.50 using GRV HO, or 0.43 using MRS A. Note that the factor $\rho_\psi$ possesses a substantial theoretical uncertainty due to the choice of scales and parton distribution functions. We conclude the photoproduction of $J/\psi$ and $D\bar{D}$ is well described by the color evaporation model. This reaction determines the only free parameter, $\rho_\psi\approx 0.5$.
At this point the predictions of the color evaporation model for hadroproduction of $\psi$ are completely determined, up to higher order QCD corrections. These can be determined by fitting the hadroproduction cross section of $D \bar{D}$ pairs with a global $K$ factor. This factor is subsequently used to correct the $\psi$ prediction. In Fig. \[fig:hadro-fit\] we compare the color evaporation model predictions with the data. In order to fit the $D\bar{D}$ cross section with the NLO result for the production of $c\bar{c}$ pairs we introduced a factor $K=1.27$ (1.71) for the GRV HO (MRS A) distribution function with the same scale $\mu$ and charm quark mass determined by photoproduction. Inserting this $K$ factor and the constant $\rho_\psi$ obtained from photoproduction into Eq. (\[sig:on\]), we predict $\psi$ hadroproduction without any free parameters. We conclude from Fig. \[fig:hadro-fit\] that the color evaporation model describes the hadroproduction of $\psi$ very accurately.
As a final test, we turn to $p_{T}$ distributions at the Tevatron. The CDF collaboration has accumulated large samples of data on the production of prompt $\psi$, $\chi_{c{\tiny J}}$, and $\psi^\prime$ [@exp:pt]. This data set allows a detailed study of the $p_T$ distribution of the produced charmonium states. Since all the charmonium states share the same production dynamics in the color evaporation model, their $p_T$ distributions should be the same, up to a multiplicative constant. This prediction is confirmed by the CDF data, as we can see in Fig. \[fig:cdf-pt\].
In order to obtain the theoretical prediction, we computed the processes $g + g \to [c\bar{c}] + g$, $q + \bar{q} \to [c\bar{c}] +
g$, and $g + q \to [c\bar{c}] + q$ at tree level using the package MADGRAPH [@tim]. This calculation should give a reliable estimate of the $p_T$ distribution at large values of $p_T$. We require that the $c\bar{c}$ pair satisfy the invariant mass constraint on Eq. (\[sig:on\]). Our results are shown in Fig. \[fig:cdf-pt\], which required a $K$ factor of $2.2$ in order to fit the total $\psi$ production. As we can see from this figure, the color evaporation model describes the general features of the $p_T$ distribution of the different charmonium states. Furthermore, we should keep in mind that the factor $K$ is, in general, $p_T$ dependent and that it is usually larger at low $p_T$ [@vogt]. Higher order corrections such as soft-gluon resummation are expected to tilt our lowest order prediction, bringing it to a closer agreement with the data [@sean].
Conclusions
===========
The color evaporation model died an untimely death. In its current reincarnation its qualitative validity can be proven directly from the available experimental data by taking ratios of production by different beams and production of different particles. Moreover, we have shown that its validity extends to the quantitative regime once we use next-to-leading-order QCD calculations for the photo- and hadroproduction of $\psi$ and $D\bar{D}$ pairs.
The color evaporation model explains all available data on $p_T$ distribution and energy-dependence of the cross section for the production of $\psi$ in all the available energy range. Moreover, it sheds some light in the relation between the production of charmonium states and $D\bar{D}$ pairs.
We would like to thank P. Nason and R. K. Ellis for providing us the code containing the next-to-leading-order QCD cross sections. We would also like to thank S. Fleming for his insight. This research was supported in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation, by the U.S. Department of Energy under grant DE-FG02-95ER40896, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).
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[^1]: Email: amundson@phenom.physics.wisc.edu
[^2]: Permanent address: Instituto de Física, Universidade de São Paulo, C.P. 66318, CEP 05389-970 São Paulo, Brazil. E-mail: eboli@phenom.physics.wisc.edu
[^3]: Email: gregores@phenos.physics.wisc.edu
[^4]: Email: halzen@phenxh.physics.wisc.edu
|
---
abstract: 'A numerical model for tsunami simulations Cliffs is exercised with the complete set of NTHMP-selected benchmark problems focused on inundation, such as simulating runup of a non-breaking solitary wave onto a sloping beach, runup on a conical island, a lab experiment with a scaled model of Monai Valley, and the 1993 Hokkiado tsunami and inundation of the Okushiri Island.'
author:
- |
Elena Tolkova [^1]\
NorthWest Research Associates, Redmond, WA, USA
title: Cliffs Benchmarking
---
Model Overview
==============
Cliffs computes tsunami propagation and land inundation under the framework of the non-linear shallow-water theory. Cliffs uses modified finite-difference scheme VTCS-2 [@titov1995] for numerical integration of the 1D shallow-water equations, dimensional splitting for solving in two spacial dimensions, and a custom land-water interface. Cliffs inundation algorithm is based on a staircase representation of topography and treats a moving shoreline as a moving vertical wall (a moving cliff), which gave the model its name.
Cliffs performs computations in a single grid given initial deformation of the free surface or the sea floor, and/or initial velocity field, and/or boundary forcing. It also computes boundary time-series input into any number of enclosed grids, to allow further refinement of the solution with one-way nesting. The flow of computations and input/output data types are similar to that in MOST version 4 (not documented; benchmarked for NTHMP in 2011 [@tolk-nthmp]), which was developed as an adaptation of a curvilinear version of the MOST model [@curvimost] to spherical coordinate systems arbitrary rotated on the Globe. Cliffs’ computational flow has been optimized to focus specifically on geophysical and Cartesian coordinate systems, and to allow easy switch between the coordinate systems, as well as 1D and 2D configurations. Cliffs is coded in Fortran-95 and parallelized using OpenMP. NetCDF format is used for all input and output data files.
Cliffs is developed, documented, and maintained in a GitHub repository[^2] by E. Tolkova. Cliffs code was written in 03/2013-09/2014, and has been occasionally revisited thereafter. The model’s description is kept current in a detailed User Manual ([<http://arxiv.org/abs/1410.0753>]{}), which comments on deviations from the description given in the model’s birth certificate [@cliffs]. Cliffs code is copyrighted under the terms of FreeBSD license.
#### The developer’s web site
<http://elena.tolkova.com/Cliffs.htm> contains links to the model-related resources, including the code repository, ready-to-try modeling examples, user manual and other related publications. Results of this (and previous) benchmarking with supplementary animations can also be browsed at <http://elena.tolkova.com/Cliffs_benchmarking.htm>.
Numerics
========
VTCS-2 finite-difference approximation was introduced by VT and CS in 1995 for numerically integrating the non-linear shallow-water equations [@titov1995]. VTCS-2 difference scheme is described in detail in [@titov1995; @titov1998], and presented in a coding-friendly form in [@cliffs]. Burwell et al (2007) analyzed diffusive and dispersive properties of the VTCS-2 solutions. The present version of Cliffs uses a modification of the VTCS-2 scheme, described in \[DS\].
A 1D algorithm can be efficiently applied to solving 2D shallow-water equations using well-known dimensional splitting method [@strang; @yan; @leveque2002]. In this way, the VTCS-2 scheme was extended to handle 2D problems in Cartesian coordinates [@titov1998], geophysical spherical coordinates [@titov97] (the first mentioning of the MOST model), and in an arbitrary orthogonal curvilinear coordinates [@curvimost].
However, the dimensional splitting results in specific sensitivity to the boundary conditions. Cliffs development started with a seemingly minor change to the reflective (vertical wall) boundary condition in VTCS-2/MOST models, which allowed to reduce numerical dissipation on reflection [@cliffsA]. The modification was extended to include runup computations, by treating a moving shoreline as a moving vertical wall [@cliffs]. Cliffs inundation algorithm is more compact and approximately 10% more efficient computationally than the present MOST algorithm. Below, Cliffs numerics is described in more details, with an emphasize on how Cliffs is different from VTCS-2 / MOST.
Difference Scheme {#DS}
-----------------
The shallow-water equations (SWE) solved in Cliffs are given below in matrix notation in the Cartesian coordinates: $$W_t=A(W)W_x+B(W)W_y+C(W)
\label{eq1}$$ where subscript denotes partial derivatives; $W=\begin{pmatrix} h & u & v \end{pmatrix}^T$ is a vector of state variables; $h$ is height of the water column; $u, \ v$ are particle velocity components in $x$ and $y$ directions; $$A=-\begin{pmatrix} u & h & 0 \\ g & u & 0 \\ 0 & 0 & u \end{pmatrix}, \ B=-\begin{pmatrix} v & 0 & h \\ 0 & v & 0 \\ g & 0 & v \end{pmatrix}, \ C=\begin{pmatrix} 0 \\ gd_x-\alpha^x \\ gd_y-\alpha^y \end{pmatrix} ;
\label{eqABC}$$ $g$ is acceleration due to gravity; $\begin{pmatrix} \alpha^x & \alpha^y \end{pmatrix}$ are the components of acceleration due to friction (Manning friction formulation is used); $d$ is undisturbed water depth, or vertical coordinate of sea bottom measured down from the mean sea level (MSL). Negative values of $d$ correspond to dry land and give the land elevation relative to MSL.\
Cliffs computes tsunami propagation using a numerical method by Titov and Synolakis (1998), which breaks the original SWE into separate problems of reduced complexity, to be solved sequentially. First, by making use of dimensional splitting, the original 2D problem becomes a sequence of 1D problems for the same state variables, to be solved row-wise and column-wise in an alternate manner:
$$W_t=A(W)W_x+C_1(W)
\label{eq2A}$$
$$W_t=B(W)W_y+C_2(W)
\label{eq2B}$$
\[eq2AB\]
where $C_1=\begin{pmatrix} 0 & gd_x-\alpha^x & 0 \end{pmatrix}^T$, $C_2=\begin{pmatrix} 0 & 0 & gd_y-\alpha^y \end{pmatrix}^T$. Next, by transitioning to Riemann invariants $$p=u+2 \sqrt{gh}, \ \ \ q=u-2 \sqrt{gh}$$ with corresponding eigenvalues $ \lambda_{p,q}=u \pm \sqrt{gh}$, each 1D problem becomes three independent convection problems for a single variable each. The resulting problem set originating with a system follows:
$$p_t =-(\lambda_1\cdot p_x - g d_x) -\alpha^x
\label{pq1}$$
$$q_t=-( \lambda_2\cdot q_x - g d_x) -\alpha^x
\label{pq2}$$
$$v_t = - u \cdot v_x
\label{pq3}$$
\[pq\_most\]
#### Two possible finite difference approximations
for (for analogically) are: $$p^{n+1}_j=p^n_j - \Delta t \cdot Q(j+1,j-1)+ \lambda_j \Delta t^2 \cdot \frac{ \hat{Q}_j- \hat{Q}_{j-1}}{\Delta x_{j-1}+\Delta x_j}-\alpha_j{\Delta t}
\label{solver1}$$ or $$p^{n+1}_j=p^n_j - \frac{\Delta t}{2} \cdot \left( \hat{Q}_{j-1} + \hat{Q}_j \right)+ \lambda_j \Delta t^2 \cdot \frac{ \hat{Q}_j- \hat{Q}_{j-1}}{\Delta x_{j-1}+\Delta x_j}-\alpha_j{\Delta t}
\label{solver2}$$ where $$Q(k,j)=\frac{1}{2} (\lambda_k+\lambda_j) \frac{p_k-p_j}{x_k-x_j}-g\frac{d_k-d_j}{x_k-x_j}, \ \ \ \ \hat{Q}_j=Q(j+1,j)
\label{eqQ}$$ with $\Delta x_j=x_{j+1}-x_j$ being a space increment. In the steady state, $Q=0$, which ensures automatic preservation of the steady state.
The two stencils differ in that to approximate a spatial derivative at point $j$, uses a difference across two cells $Q(j-1,j+1)$, whereas uses an average of the left and right one-cell differences $\hat{Q}_{j-1}$ and $\hat{Q}_j$. Since term $Q$ is nonlinear with respect to depth and wave variables $$Q(j-1,j+1) \ne \frac{1}{2} \left( \hat{Q}_{j-1} + \hat{Q}_j \right) ,$$ the two approximations yield noticeably different results for strongly nonlinear problems and/or over large depth variations. Both solvers and are first-order accurate in time and second-order accurate in space for a uniform grid spacing. Should the spacing $\Delta x_j$ vary, an added numerical error of order $\Delta x_j - \Delta x_{j-1}$ would arise. In a basin with constant depth, each scheme is stable under the known limit on the Courant number: $|\lambda| \Delta t/ \Delta x \le 1$.
Scheme is the original VTCS-2 scheme utilized in MOST; scheme is used in Cliffs. One example where the MOST and Cliffs numerical schemes operate substantially different is given below.
### Dam Break Problem {#DBP}
{width="70.00000%"} \[dam\]
Correct application of the VTCS-2 stencil requires the characteristics of the same family not to change their direction in three successive nodes - a condition which cannot incorporate rarefaction waves [@leveque2002]. Consequently, VTCS-2 scheme fails to correctly simulate a classical Dam Break problem[^3] [@stoker; @leveque2002]. As seen in Fig. \[dam\], a few modifications to VTCS-2 difference scheme implemented in Cliffs has enabled the model to handle the Dam Break problem. Figure \[dam\] displays simulated profiles of water volume 36 s after instantaneous removal of a 2.5-m-high dam vs. the analytical solution. At $t=0$, 2.5-m-deep water occupied half-space $x>0$. The solutions to the problem are computed by Cliffs with its modified VTCS-2 solver, and by Cliffs with plugged-in original VTCS-2. The moving shoreline algorithm is that of Cliffs model in both cases. For this problem, Cliffs solver yields results coinciding with the theoretical solution, whereas VTCS-2 / MOST solver computes an unrealistic discontinuous waveform.
Land-water interface: vertical wall
-----------------------------------
According to VTCS-2 algorithm, the sea-going Riemann invariant in the last wet node next to a reflective boundary (a vertical wall) is assigned a value opposite to that of the wall-going invariant (Titov and Synolakis, 1995, 1998). This treatment sets the reflective wall immediately next to the last wet node. However, as discussed below, this treatment does not mix well with the splitting technique.
Unless a straight boundary coincides with either x or y axis, it can not be treated by a splitted scheme as straight, but rather as a step-like. Consider, for example, a diagonal channel shown in Figure \[dots\]. When the splitted VTCS-2 solver performs computations in x-direction, it “sees" this boundary as a set of steps whose vertical segments are aligned with the edge wet nodes, as shown on the north-west channel boundary with a black line. When computations are performed in y-direction, the solver “sees" a different set of steps. Now the horizontal segments of the steps are aligned with the edge nodes, as shown by a gray line on the north-west boundary. This displacement of the reflective boundary within a splitted cycle results in increased dissipation, occasionally causing unrealistically low later waves (reflected from coastlines) in simulations of historical tsunami events with MOST.
In Cliffs, the reflective boundary conditions are formulated by introducing a mirror ghost node coinciding with the first dry node. This treatment sets the reflecting wall exactly in the middle between the neighboring wet and dry nodes, where it remains steady for the entire splitted cycle (see Fig. \[dots\]). This modification has resulted in better representation of the later waves in simulations of the real-world tsunamis [@cliffs].
{width="30.00000%"} \[dots\]
Land-water interface: moving shoreline
--------------------------------------
Unlike the VTCS-2 approach, Cliffs boundary conditions are obtained without making any assumptions about the direction of the characteristics, which allows to apply them on a moving boundary as well. Hence Cliffs inundation algorithm is based on a staircase representation of topography, and treats a vertical interface between wet and dry cells (a shoreline) as a vertical wall. Reflective boundary conditions applied on an instant shoreline are formulated using a mirror ghost node coinciding with a shoreline dry node.
To enable wetting/drying, a common procedure of comparing the flow depth with an empirical threshold $h_{min}$ is used. The wet area expands when a flooding depth in a shoreline wet node exceeds a threshold (runup), and shrinks when a flow depth in a cell is below the threshold (rundown). On the wet area expansion, dry cells to be flooded are pre-filled with water to $h_{min}$ depth. Hence the runup occurs on an inserted cushion $h_{min}$ high which is removed on rundown.
Testing Cliffs with NTHMP Benchmark Problems
============================================
Cliffs is seeking to join a list of ten (on 2016) tsunami models endorsed by the National Tsunami Hazard Mitigation Program (NTHMP) after passing the inundation benchmarks, which were selected according to NOAA’s standards and criteria [@synolak2007; @nthmp]. Detailed descriptions of the benchmarks, as well as topography and laboratory or survey data when applicable, can be found in a repository of benchmark problems [https:// github.com/ rjleveque/ nthmp-benchmark-problems](https:// github.com/ rjleveque/ nthmp-benchmark-problems) for NTHMP, or in the NOAA/PMEL repository [http:// nctr.pmel.noaa.gov/ benchmark/](http:// nctr.pmel.noaa.gov/ benchmark/). Cliffs performance with these benchmark problems (BPs) is presented below. Wherever applicable, the results are displayed and the errors are evaluated with the NTHMP provided scripts. Current Cliffs benchmarking results with complementary animations can also be found at <http://elena.tolkova.com/Cliffs_benchmarking.htm>.
BP1 - Solitary wave on a simple beach (nonbreaking - analytic)
--------------------------------------------------------------
[![Water surface profiles for an initial solitary wave $0.0185 d$ high climbing up a 1:19.85 beach at $t (g/d)^{1/2}=35,40,45,50,55,60,65$ (shown in a plot), black - Cliffs numerical solution, red dots - analytical solution. Plotted with NTHMP provided script. []{data-label="Sprf185"}](cliffs_bp1_profiles-eps-converted-to "fig:")]{}
In this BP, a non-breaking solitary wave of initial height $H=0.0185d$ over depth $d$ normally approaches and climbs onto a plane sloping beach. The Objective for this BP is to model the surface elevation in space and time within 5% of the calculated value from the analytical solution.
The geometry of the beach and the wave-profile are described in many articles ([@synolak1987; @titov1995; @synolak2007]) as well as in the Internet repositories mentioned above. The 1D bathymetry consists of a flat segment of depth $d$ connected to a beach with a slope 1:19.85. The x coordinate increases seaward, $x=0$ is the initial shoreline position, and the toe of the beach is located at $x=19.85d$. The runup simulation of a non-breaking solitary wave with $H/d=0.0185$ was performed with Cliffs using a 384-node grid, which encompassed a $45d$-long segment of constant depth $d$ connected to $50d$-long 1:19.85 slope. In simulations, the depth of the flat part of the basin was $d=1$ m. The grid spacing was set to 1 m (depth) over the flat segment, and then varied as $\sqrt{d}$, but not less than $0.1d$. Time increment $\Delta t=0.03$ s yields Courant number $\Delta t {\sqrt{gd}} / \Delta x =0.1$, thus providing for emulating physical dispersion with numerical dispersion [@burwell; @tolk-nthmp]. Friction coefficient was set to zero. Depth threshold $h_{min}$ was set to 2 mm. Results with dimension of length are expressed in units of $d$, time is expressed in units of $\tau=\sqrt{d/g}$.
Figure \[Sprf185\] shows water surface profiles computed with Cliffs vs. an analytical solution provided by NTHMP. Figure \[Stmsers\] shows surface elevation time histories computed with Cliffs vs. the analytical solution. Cliffs solution approximated the analytical solution well within the limits of the no-more-than-5% error. Namely, the mean normalized standard deviation and max wave amplitude errors, as defined by the NTHMP [@nthmp], are 3% and 2% respectively for the computed surface profiles, 2% and 1% for the surface elevation time-history at the first location, and 3% and 2% for the surface elevation time-history at the second location.
Maximal computed runup height is $0.087d$.
BP4 - Solitary wave on a simple beach (nonbreaking - lab)
---------------------------------------------------------
[![Water surface profiles for an initial solitary wave $0.0185 d$ high climbing up a 1:19.85 beach at $t (g/d)^{1/2}=30,40,50,60,70$ (shown in a plot), black - Cliffs numerical solution, red dots - lab data. Plotted with NTHMP provided script. []{data-label="Slab185"}](cliffs_bp4_profiles00185-eps-converted-to "fig:")]{}
The Objective of this BP is to approximate lab measurements of a solitary wave attacking a plane sloping beach, with the same geometry as in BP1. Simulation set-up was the same as for BP1, but friction with Manning roughness coefficient $n=0.015$ was included in the simulation aiming to reproduce the lab experiment. Figure \[Slab185\] shows water surface profiles computed with Cliffs vs. lab data provided by NTHMP. Cliffs’ errors – mean normalized standard deviation (N.St.D.) and max wave amplitude error – are 6% and 4% respectively. For comparison, among all NTHMP-approved tsunami inundation models, the mean N.St.D. errors range is 7-11%, and mean MaxWave errors range is 2-10% [@nthmp]. Maximal computed runup height is $0.077d$ (lower runup value is due to friction).
BP6: Solitary wave on a Conical Island {#sec_island}
--------------------------------------
This BP requires to numerically simulate a wave-tank experiment where a plane solitary wave attacks and inundates a conically shaped island [@briggs; @liu1995]. A model of the conical island with 1:4 slope and 7.2 m diameter at the base was constructed near the center of a flat $29.3 \times 30$ m basin filled with water $d=32$ cm deep. In three separate experiments, three different incident solitary waves were generated: $0.05d$, $0.1d$, and $0.2d$ high, labeled cases A, B, and C correspondently. In each case, time histories at several locations around the island, and the angular distribution of runup were recorded. The Objective for this BP is to predict the runup measurements around the island with no more that 20% mean errors as calculated by the NTHMP provided scripts.
The problem was simulated using two nested grids. The outer grid at $30$ cm spacing enclosed the entire basin area starting at the wave-maker. Simulations were forced through the western boundary, for the duration of the direct pulse. Boundary velocity was computed according to the paddle trajectories and complemented with the surface elevation as in a purely incident wave. Run-up onto the island was simulated with a finer 10 m x 10 m grid spaced at $5$ cm; the roughness coefficient was $n=0.015$; $h_{min}=5$ mm. The inundated area was obtained using Cliffs’ “maxwave" output. The inundation boundary was drawn through interfaces between always dry nodes (NaNs in the maxwave output) and the nodes which get wet at any time during the simulation (have valid maxwave values).
The computed inundated area around the island and the runup height distribution in each of the three cases vs. measurements are shown in Figures \[conical1\] - \[conical2\]. Simulated time histories at four gages closest to the island (gage 6 at depth 31.7 cm, gage 9 at 8.2 cm, gage 16 at 7.9 cm, and gage 22 on the lee side at depth 8.3 cm) vs. laboratory measurements for the three cases are shown in Figures \[conical3A\]-\[conical3C\]. As seen in Table \[tblcon\], all errors are below the 20% threshold.
[![Inundation around the island in cases A, B, and C: computed with Cliffs (green), laboratory measurements (black circles), original coastline (black); wave attacks from the west. []{data-label="conical1"}](conical_inund_ABC-eps-converted-to "fig:")]{}
------------ ---- ------- ---- ------ -------- -----
Errors Runup Gauges
A B C A B C
N.St.D., % 14 18 11 8 8 7.5
Max., % 12 4 7 18.5 19 11
------------ ---- ------- ---- ------ -------- -----
: Relative errors of Cliffs computations of the runup heights around the island, and of the water level time-histories at the gages.[]{data-label="tblcon"}
BP7: Runup onto a lab model of the Monai Valley
-----------------------------------------------
[![Five frames extracted from the video record of the lab experiment with 0.5 sec interval[]{data-label="monai2"}](Monai_videoframes-eps-converted-to "fig:")]{}
[![Five snapshots of simulated water height distribution with 0.5 s interval. []{data-label="monai3"}](Monai_frames-eps-converted-to "fig:")]{}
{width="50.00000%"} \[monai1\]
The 1993 Hokkaido tsunami (see BP9) caused extreme runup at the tip of a narrow gully at Monai, Okushiri island, Japan. This event was reproduced in a wave tank experiment with a 1:400 scale version of the Monai valley [@liu2008; @synolak2007]. The Objective of this BP is demonstrating a tsunami model’s ability to capture shoreline dynamics with an extreme runup and rundown over a complex topography.
The domain of computations represents a $5.5 \times 3.4$ m portion of the tank next to the shoreline. Water level dynamics in this region were recorded on video and at three gages along the shoreline. Five frames extracted from the video record of the lab experiment with 0.5 sec interval [@nicolsky2011] are shown in Figure \[monai2\]. The first frame occurred approximately at 15.3 s of the lab event time.
The lab experiment has been modeled with a provided grid at 1.4 cm spacing, friction coefficient $n=0.012$, $h_{min}=1$ mm. Simulations were forced with an incident wave through the sea-side boundary of the domain prescribed for the first 23 s. Five sea surface snapshots 0.5 s apart are shown in Figure \[monai3\]. The first snapshot was taken at 15.6 s. By visual comparison, the simulations agree closely with the recorded shoreline position in space and in time. Close fit in the direct wave is also observed between modeled and recorded water level time histories at the gages, in Figure \[monai1\]. Maximal computed runup height is 8.6 cm.
BP9: Okushiri Island tsunami (field)
------------------------------------
[![Left: computational domain used to simulate the 1993 Hokkaido tsunami with contours of the two nested grids; initial sea surface deformation with contour lines at -1, -0.5, 1, 2, 3, 4 m levels, subsidence contours are shown with dashed lines, uplift with solid lines. The deformation area near Okushiri is zoomed-in in the bottom left corner. Right: 3-d nesting level grid around Okushiri island with contours of the 4-th level grids around Monai and Aonae.[]{data-label="OKsetup"}](bp9_setup-eps-converted-to "fig:")]{}
{width="50.00000%"} \[OKsurvey\]
On July 12, 1993, the Mw 7.8 earthquake west of Okushiri island, Japan, generated a tsunami that has become a test case for tsunami modeling efforts [@takahashi; @synolak2007; @nthmp]. Detailed runup measurements around Okushiri island were conducted by the Hokkaido tsunami survey group which reported up to 31.7 m runup near Monai village. Also, high-resolution bathymetric surveys were performed before and after the EQ, which allowed to evaluate the deformation of the ocean bottom due to the quake. The conventional practice in modeling geophysical tsunamis is to assume that the tsunami originates with the initial deformation of the sea surface following that of the ocean bottom.
The computational domain used to simulate the 1993 Okushiri tsunami and the initial sea surface deformation are shown in the left pane in Figure \[OKsetup\]. The computations at a resolution of 30 sec of the Great arc were refined with nested grids spaced at 10 arc-sec, 3 arc-sec (circling Okushiri island), 15 m (enclosing Aonae area), and 6 m (enclosing Monai valley). The grids contours are pictured in Figure \[OKsetup\]. In the two outer level grids, a vertical wall was imposed in 1 m deep water. In the 3 arc-sec grid around Okushiri, waves higher than $h_{min}=0.5$ m were permitted to inundate the next dry cell, that is, to advance by another 90 m inland. In the two finest grids, a wave was permitted to inundate once it was higher than $h_{min}=0.1$ m. Friction coefficient was set to $n=0.025$ in the two outer grids, $n=0.04$ in Okushiri-sized grid, and $n=0.03$ in the two finest grids.
This BP has several objectives. The fist Objective is to model the inundation around Okushiri Island and match field measurements of runup heights with no less than 20% accuracy. In the NTHMP-provided script, runup error is evaluated by a sophisticated comparison between computed and measured sets of minimal, maximal, and mean runup values in unspecified surroundings of prescribed reference points. Figure \[OKsurvey\] shows locations of the NTHMP’s reference points (blue circles) and field measurements (red dots). Considering that (1) the reference point surroundings should better contain at least one field measurement, but preferably a few; and (2) uncertainty in geo referencing is about 0.011 deg = 1.2 km; the range of computed runup values was taken from a box with 1.5 km side (as shown in the figure) around each reference point in the 3 arc-s Okushiri grid; and from a box with 120 m side in both Aonae and Monai grids. As evaluated by NTHMP’s script, Cliffs reproduced the observed runup with an error of 15%.
Figures \[AOrunup\] and \[MBrunup\] display computed maximal runup heights in the high-resolution grids focused on Aonae and Monai areas. Maximum modeled runup height obtained at Hamatsumae was 13 m (reported up to 13.2 m), on Aonae peninsula 12.5 m (reported 12.4 m ), and in Monai valley 21 m (reported up to 31.7 m). It’s common among numerical models to under-estimate runup at Monai valley for the reasons discussed in [@nicolsky2011].
The second Objective of this BP is to reproduce wave dynamics around Aonae peninsular, namely to simulate arrival of the first wave to Aonae 5 min after the earthquake coming from the west, and the second wave coming from the east a few mins later. Snapshots of computed sea surface in the Aonae grid 5 and 9 min after the EQ shown in Figure \[waves\] clearly display the wanted waves.
Lastly, Figure \[gages\] presents modeled and observed sea levels at Iwanai and Esashi tide gages. The comparison between the model and the observations might be little informative, given that available bathymetry at these sites has too coarse resolution for a coastal location (30 arc-s and 10 arc-s), and that observations at Iwanai are scarsely sampled.
{width="70.00000%"} \[AOrunup\]
{width="50.00000%"} \[MBrunup\]
{width="70.00000%"} \[gages\]
[99]{}
Briggs, M. J., Synolakis, C. E., Harkins, G. S., and Green, D. R. (1995), Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys., 144, 3/4, 569Ð593.
Burwell, D., Tolkova, E., and Chawla, A. (2007), Diffusion and Dispersion Characterization of a Numerical Tsunami Model. Ocean Modelling, 19 (1-2), 10-30. doi:10.1016/j.ocemod.2007.05.003
LeVeque, R.J., Finite volume methods for hyperbolic problems. (Cambridge University Press, UK 2002).
Li, Y., and Raichlen, F. (2002), Non-breaking and breaking solitary wave run-up. J. Fluid Mech., 456, 295Ð318.
Liu, P.L.-F., Cho, Y.-S., Briggs, M., Kanoglu, U., and Synolakis, C. (1995), Runup of solitary waves on a circular island. Jornal of Fluid Mech. 302, 259-285.
Liu, P.L.-F., Yeh, H., and Synolakis C. (2008), Advanced Numerical Models for Simulating Tsunami Waves and Runup. Advances in Coastal and Ocean Engineering, 10, 223-230.
Nicolsky, D.J., Suleimani, E.N., and Hansen, R.A. (2011), Validation and verification of a numerical model for tsunami propagation and runup. Pure Appl. Geophys. 168, 1199-1222.
\[NTHMP\] National Tsunami Hazard Mitigation Program, July 2012. Proceedings and Results of the 2011 NTHMP Model Benchmarking Workshop. Boulder: U.S. Department of Commerce/NOAA/NTHMP; NOAA Special Report. 436 p.
Shi, F., Kirby, J.T., Harris, J.C., Geiman, J.D., Grilli, S.T. (2012), A high-order adaptive time-stepping TVD solver for Boussinesq modeling of breaking waves and coastal inundation. Ocean Modelling, 43-44, 36-51.
Stoker, J.J. (1957), Water Waves (Interscience Pub., Inc., New York, NY, USA).
Strang, G. (1968), On the construction and comparison of difference schemes. SIAM Journal on Numerical Analysis, 5(3), 506-517.
Synolakis, C.E. (1987), The runup of solitary waves. J. Fluid Mech., 185, 523-545.
Synolakis, C.E., Bernard, E.N., Titov, V.V, Kanoglu, U., and Gonzalez, F.I. (2007), Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. NOAA Tech. Memo. OAR PMEL-135, NTIS: PB2007-109601, NOAA/Pacific Marine Environmental Laboratory, Seattle, WA, 55 pp.
Takahashi, T. (1996), Benchmark problem 4: the 1993 Okushiri tsunami - Data, conditions and phenomena. In Long-Wave Runup Models, World Scientific, 384-403.
Titov, V. V., and Synolakis, C. E. (1995), Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2, J. Waterw., Port, Coastal, Ocean Eng., 121(6), 308Ð316.
Titov, V., and Gonzalez F.I. (1997), Implementation and testing of the Method of Splitting Tsunami (MOST) model. NOAA Tech. Memo. ERL PMEL-112 (PB98-122773), NOAA/Pacific Marine Environmental Laboratory, Seattle, WA, 11 pp.
Titov, V. V., and Synolakis, C. E. (1998), Numerical modeling of tidal wave runup, J. Waterw., Port, Coastal, Ocean Eng., 124(4), 157Ð171.
Tolkova, E. (2008) Curvilinear MOST and its first application: Regional Forecast version 2. In: Burwell, D., Tolkova, E. Curvilinear version of the MOST model with application to the coast-wide tsunami forecast, Part II. NOAA Tech. Memo. OAR PMEL-142, 28 pp.
Tolkova, E. (2012), MOST (Method of Splitting Tsunamis) Numerical Model. In: \[NTHMP\] National Tsunami Hazard Mitigation Program. Proceedings and Results of the 2011 NTHMP Model Benchmarking Workshop. Boulder: U.S. Department of Commerce/NOAA/NTHMP; NOAA Special Report. 436 p.
Tolkova, E. (2014), LandÐWater Boundary Treatment for a Tsunami Model With Dimensional Splitting. Pure and Applied Geophysics, 171( 9), 2289-2314. DOI 10.1007/s00024-014-0825-8
Tolkova, E. (2014a), Comparative simulations of the 2011 Tohoku tsunami with MOST and Cliffs. http://arxiv.org/abs/1401.2700
Yanenko N.N. (1971). The method of Fractional Steps. Translated from Russian by M.Holt. Springer, New York, Berlin, Heidelberg.
[^1]: e.tolkova@gmail.com; elena@nwra.com
[^2]: <https://github.com/Delta-function/cliffs-src>
[^3]: Dam break problem is a frequent textbook example of the shallow-water flow with known analytical solution. In spite of its popularity, this problem is omitted in PMEL-135 Report setting benchmarking criteria [@synolak2007], and consequently, it is not included with the NTHMP’s set of inundation benchmarks.
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**VERTICES AND THE CJT EFFECTIVE POTENTIAL**
Bijan Haeri, Jr.
*Department of Physics*
*Purdue University*
*West Lafayette, Indiana 47907-1396*
1.5cm
ABSTRACT
The Cornwall-Jackiw-Tomboulis effective potential is modified to include a functional dependence on the fermion-gauge particle vertex, and applied to a quark confining model of chiral symmetry breaking.
The study of dynamical chiral symmetry breaking has benefited greatly from the Cornwall-Jackiw-Tomboulis (CJT) effective potential [@cjt]. The CJT potential is an effective potential of composite operators, specifically the propagators of a theory. Minimizing the CJT potential with respect to a propagator gives the Schwinger-Dyson (SD) equation for that propagator. The SD equation can have more than one solution. These solutions must be inserted in the CJT potential to find the solution corresponding to a stable minimum in the CJT potential, thus determining the physical solution. In its simplest form the CJT potential as a functional of propagators is of practical use only in solving propagator SD equations, which have a severe truncation of their vertex SD equations, as is the case for the ladder approximation, where the full vertex is replaced by the bare vertex. The usual CJT potential does not have a functional dependence on the vertices of a theory, as a result the SD equations of the vertices cannot be found by minimizing the CJT potential. There exist aternative means of constructing full vertices, such as solving the Ward identity of the vertex (the gauge technique ), to arrive at a full vertex as a functional of the propagators, which is valid in the infrared regime. Putting the gauge technique vertex in the propagtor SD equation by hand is inconsistent with minimizing the CJT potential, which is not a functional of the gauge technique vertex. Therefore testing the stability of the CJT potential with solutions found from propagator SD equations containing the gauge technique vertex can be unreliable (as will be demonstrated in section 3).
We will extend the CJT formalism to write an effective potential for QED which has a functional dependence on the fermion-fermion-gauge particle vertex in addition to the propagators, and apply it to a confining model of dynamical chiral symmetry breaking. Interestingly, it will be found that the chiral symmetry breaking solution of the effective quark propagator SD equation minimizes the modified CJT potential. Section 2 contains a derivation of the modified CJT potential for QED, while section 3 applies the results to a specific model.
Consider massless QED with its gauge particle and fermion propagators, and fermion-fermion-gauge particle vertex denoted as $\Delta_{\mu \nu}(x,y)$, $S(x,y)$, and $\Gamma_{\mu}(x,y,z)$ respectively. The CJT effective action is given by $$\begin{aligned}
\Gamma(\Delta_{\mu \nu},S)=\Gamma_{0}(\Delta_{\mu \nu},S)+
\Gamma_{2}(\Delta_{\mu \nu},S),\end{aligned}$$ where $$\begin{aligned}
\Gamma_{0}(\Delta_{\mu \nu},S)=
&{{-i\hbar}\over{2}}Tr\ ln(
{{\Delta_{0}}_{\mu \alpha}}^{-1}\Delta_{\alpha \nu})
+{{i\hbar}\over{2}}Tr({{\Delta_{0}}_{\mu \alpha}}^{-1}
\Delta_{\alpha \nu}-g_{\mu \nu}) \nonumber \\
&+i\hbar Tr\ ln(S_{0}^{-1}S)-i\hbar Tr(S_{0}^{-1}S-1),\end{aligned}$$ while $\Gamma_{2}(\Delta_{\mu \nu},S)$ is the infinite sum over all possible 2 point-irreducible (2PI) Feynman graphs involving $\Delta_{\mu \nu}$ and $S$, and ${\Delta_{0}}_{\mu \nu}$ and $S_{0}$ are the bare gauge particle and fermion propagators respectively. Including a vertex source in the generator of connected Green’s function, $W$, gives $W=W(G_{\mu \nu},K,\rho_{\mu})$, where $G_{\mu\nu}$, $K$, and $\rho_{\mu}$ are the sources for $\Delta_{\mu \nu}$, $S$, and $\Gamma_{\mu}$ respectively. Perfoming a triple Legendre transformation, we find $$\begin{aligned}
\Gamma (\Delta_{\mu \nu},S,\Gamma_{\mu})=
&W-i\hbar\int d^4xd^4y\left({1\over2}G_{\mu \nu}(x,y)\Delta^{\mu \nu}
(y,x)-K(x,y)S(y,x)\right)\nonumber \\
&-i\hbar\int d^4xd^4yd^4z \rho^{\mu}(x,y,z)\Gamma_{\mu}(z,y,x),\end{aligned}$$ corresponding to $$\begin{aligned}
\Gamma(\Delta_{\mu \nu},S,\Gamma_{\mu})=\Gamma_{0}(\Delta_{\mu \nu},S)+
\Gamma_{2}(\Delta_{\mu \nu},S,\Gamma_{\mu}),\end{aligned}$$ where $\Gamma_{2}(\Delta_{\mu \nu},S,\Gamma_{\mu})$ is an infinite sum over all 2PI Feynman graphs containing all possible interactions of full propagators and full vertices, as well as all the Feynman graphs that make up $\Gamma_{2}(\Delta_{\mu \nu},S)$. Evaluating the effective action at a stationary point gives $$\begin{aligned}
{{\delta \Gamma}\over{\delta
\Delta_{\mu \nu}}}=-{{i\hbar}\over{2}}G_{\mu \nu}|_{G_{\mu \nu}=0},\end{aligned}$$ $$\begin{aligned}
{{\delta \Gamma}\over{\delta S}}=i\hbar K|_{S=0},\end{aligned}$$ and $$\begin{aligned}
{{\delta \Gamma}\over{\delta \Gamma_{\mu}}}=-i\hbar
\rho_{\mu}\vert_{\rho_{\mu}=0},\end{aligned}$$ where Eq.’s 5-7 are the SD equations for $\Delta_{\mu \nu}(x-y)$, $S(x-y)$, and $\Gamma_{\mu}(x-y,y-z)$ respectively, where the propagators(vertex) are(is) a funtion of 1(2) variables, rather than 2(3) before the sources were set to zero. At the stationary point the effective action can be written as $$\begin{aligned}
\Gamma(\Delta_{\mu \nu},S,\Gamma_{\mu})=-V(\Delta_{\mu \nu},S,\Gamma_{\mu})
\int d^4x,\end{aligned}$$ where the effective potential, $V(\Delta_{\mu \nu},S,\Gamma_{\mu})$, is given by [@cnwl] $$\begin{aligned}
V(\Delta_{\mu \nu},S,\Gamma_{\mu})=V_0(\Delta_{\mu \nu},S,\Gamma_{\mu})+
V_{2}(\Delta_{\mu \nu},S,\Gamma_{\mu}),\end{aligned}$$ $$\begin{aligned}
V_0(\Delta_{\mu \nu},S,\Gamma_{\mu})=&-i\hbar\int {{d^4p}\over{2{\pi}^4}}
Tr\Biggl(-{1\over2}ln({{\Delta_{0}}_{\mu\alpha}}^{-1}(p)
\Delta_{\alpha \nu}(p)) \nonumber \\
&+{1\over{2}}({{\Delta_{0}}_{\mu \alpha}(p)}^{-1}
\Delta_{\alpha \nu}(p)-g_{\mu \nu}) \nonumber \\
&+ln(S_{0}(p)^{-1}S(p))-(S_{0}(p)^{-1}S(p)-1)\Biggr),\end{aligned}$$ where $V_{2}(\Delta_{\mu \nu},S,\Gamma_{\mu})$ is the infinite sum over vacuum graphs shown in Fig. 1. Eqs. 5-7 can alternatively be written as $$\begin{aligned}
{{\delta V}\over{\delta
\Delta_{\mu \nu}(p)}}=0,\end{aligned}$$ $$\begin{aligned}
{\delta V\over{\delta S(p)}}=0,\end{aligned}$$ and $$\begin{aligned}
{\delta V\over{\delta \Gamma_{\mu}(p,q)}}=0,\end{aligned}$$ which are displayed in Figs. 2-4 respectively. The coffecients of the graphs on the right hand side of Fig. 1 are determined by first examining the vertex SD equation shown in Fig. 4, and then either of the propagator SD equations (Figs. 2 and 3); using Fig. 4 it can be seen that $V_{2}(S,\Gamma_{\mu})=V_{2}(S)$ as we expect.
The vertex SD equation is not solvable as the infinite series shown in Fig. 4. Truncating the series beyond the first graph on the right hand side of Fig. 4, and keeping only the first graph on the right hand side of Fig. 2 (choosing the Landau gauge) is a common approximation (ladder approximation) used to solve the fermion SD equation. Another approximation scheme of determining the vertex known as the modified gauge technique is to formulate an ansatz for the vertex in terms of the fermion propagator that solves the Ward identity for the vertex and preserves the multiplicative renormalizability and gauge covariance of the fermion SD equation [@jk; @bh2]. But since the vertex ansatz depends on the fermion propagator, the fermion SD equation is modified to give
$$\begin{aligned}
{{\delta V}\over{\delta S}}={{\delta V}\over{\delta S}}\Biggr\vert_
{\Gamma_{\mu}}+{{\delta \Gamma_{\mu}}\over{\delta S}} {{\delta V}
\over{\delta \Gamma_{\mu}}}\Biggr\vert_{S}=0.\end{aligned}$$
The first term in the middle corresponds to the usual fermion SD equation (Eq. 12) when the second term in the middle vanishes, that is the vertex SD equation (Eq. 13) is satisfied, otherwise there will be extra terms in the fermion SD equation (besides those appearing on the right hand side of Fig. 3).
Quark confinement in QCD has long been modeled by replacing the gluon propagator appearing in the effective quark propagator SD equation with a $1\over{k^4}$ potential, representing a linearly rising potential in coordinate space [@cc]. The $1\over{k^4}$ potential has an infrared divergence, which is commonly regulated to give a $\delta^4(k)$ potential [@pagels]. Here we will follow this example and replace the gluon propagator with: $$\begin{aligned}
\Delta_{\mu \nu}(k)=ag_{\mu \nu}{{16\pi^4}\over{C_{f}}}\delta^4(k),\end{aligned}$$ where $C_f={{N^2-1}\over{2N}}$ is the Casimir eigenvalue of the of the fundamental representation of $SU(N)$, and $\sqrt{a}$ is a scale associated with confinement. It is usual at this point to completely truncate the quark-quark-gluon (qqg) vertex by replacing the full qqg vertex $\Gamma_{\mu}$, by a bare vertex $\gamma_{\mu}$ in the quark propagator SD equation, and thus leaving one SD equation to solve [@mn]. We will alternatively solve the quark propagator SD equation with the full qqg vertex intact. The actual qqg vertex we will use is the modified qqg vertex found by resumming Feynman diagrams to redefine the n-point Green’s functions in terms of modified Green’s functions [@bh1; @hh]. In this case the modified gluon propagator $\hat{\Delta}_{\mu \nu}(k)$ is replaced by Eq. 15, while the full modified qqg vertex, $\hat{\Gamma}_{\mu}(p,k)$ satisfies a simple Ward identity $$\begin{aligned}
\hat{\Gamma}_{\mu}(p,p)={{\partial S^{-1}(p)}\over{\partial p_{\mu}}}.\end{aligned}$$ Using Eq. 16, $\hat{\Gamma}_{\mu}(p,p)$ can be solved exactly in terms of $S(p)$ without the need to solve the vertex SD equation of Fig. 4. Writing $S(p)$ in terms of scalar functions $$\begin{aligned}
S(p)={1\over{\alpha \rlap /p -\beta (p)}},\end{aligned}$$ we find that $$\begin{aligned}
\hat{\Gamma}_{\mu}(p,p)=\alpha(p)\gamma_{\mu}+
2p_{\mu}\rlap /p {{\partial \alpha(p)}\over{\partial p^2}}-
\{ \gamma_{\mu},\rlap /p\} {{\partial \beta(p)}\over{\partial p^2}}\end{aligned}$$ satisfies Eq. 16. It is useful to multiply Eq. 18 on the left and right by $S(p)$, so that it becomes $$\begin{aligned}
S(p)\hat{\Gamma}_{\mu}(p,p)S(p)=-{{\partial S(p)}\over{\partial p_{\mu}}},\end{aligned}$$ and write $S(p)$ as $$\begin{aligned}
S(p)=A(p)\rlap /p+B(p),\end{aligned}$$ giving $$\begin{aligned}
S(p)\hat{\Gamma}_{\mu}(p,p)S(p)=-A(p)\gamma_{\mu}-
2p_{\mu}\rlap /p {{\partial A(p)}\over{\partial p^2}}-
\{ \gamma_{\mu},\rlap /p\} {{\partial B(p)}\over{\partial p^2}},\end{aligned}$$ where $\alpha(p)$ and $\beta(p)$ are related to $A(p)$ and $B(p)$ by $$\begin{aligned}
\alpha(p)={{A(p)}\over{p^2A^2(p)-B^2(p)}}\end{aligned}$$ $$\begin{aligned}
\beta(p)={{B(p)}\over{p^2A^2(p)-B^2(p)}}\end{aligned}$$ The SD equation for the effective quark propagator is $$\begin{aligned}
S(p)=S_0(p)+ig^2C_fS_0(p)\int {{d^4k}\over{16\pi^4}} \gamma_{\mu}S(k)
\hat{\Gamma}_{\nu}(k,p)S(p)\hat{\Delta}^{\mu \nu}(p-k),\end{aligned}$$ where $$\begin{aligned}
S_0(p)={1\over{\rlap /p}}.\end{aligned}$$ Substituting Eq. 15, and Eq. 21 into Eq. 24, we find $$\begin{aligned}
p^2{{\partial A(p)}\over{\partial p^2}}=
{1\over{2a}}\left((p^2-4a)A(p)-1\right)\end{aligned}$$ $${{\partial B(p)}\over{\partial p^2}}={{B(p)}\over{2a}}$$ Eqs. 26, 27, 22, and 23 have been solved in Euclidean space (i.e. $p^2=-{p_E}^2$) by Burden, Roberts, and Williams [@brw], who find for the case of chiral symmetry breaking, $$A(x)=-{1\over{2a}}{{x-1+e^{-x}}\over{x^2}},$$ $$B(x)={1\over{2\sqrt{a}}}e^{-x},$$ $$\alpha_b(x)={{2x(e^{-x}+x-1)}\over{x^3e^{-2x}+2(e^{-x}+x-1)^2}},$$ $$\beta(x)=-\sqrt{a}{{2x^3}\over{x^3e^{-x}+2e^x(e^{-x}+x-1)^2}},$$ where $x={{{p_E}^2}\over{2a}}$. For the chiral symmetry preserving case $A(x)$ has the same solution as in Eq. 28, while $$\alpha_s(x)={{x}\over{e^{-x}+x-1}},$$ $$\beta(x)=B(x)=0.$$ $\alpha_s(x)$, and $\alpha_b(x)$ are the solutions of $\alpha(x)$ corresponding to the chiral symmetric and chiral symmetry breaking cases respectively. To determine which solution is prefered, we need to evalute both sets of solutions in Eq. 9, namely we must calculate $$\begin{aligned}
\Delta V\equiv V_{s}-V_{b},\end{aligned}$$ where $V_{s}$ is the modified CJT effective potential evaluated with the chiral symmetric solution to $S(p)$, and $\Gamma_{\mu}(p,p)$ (Eq. 28, 32, and 33), while $V_{b}$ uses the chiral symmetry breaking solution (Eq. 28-31). Defining $$\tilde{V}\equiv {1\over{n_fNa^2}}V(S,\Gamma_{\mu}),$$ where $n_f$ is the number of fermions, we find as in Ref. 10 that, $$\tilde{V_0}_{s}-\tilde{V_0}_{b}=-{1\over{2\pi^2}}\int xdx\
ln\Biggl(1+{{\bar{B}^2}\over{2x\bar{A}^2}}\Biggr)=-.04,$$ where $\bar{A}\equiv aA$, and $\bar{B}\equiv aB$. $V_2(S)$ in Fig. 1b does not contribute to Eq. 34. The second and third terms in the series on the right hand side of Fig. 1a have contributions to Eq. 34 given by, $$\begin{aligned}
\Delta\tilde{V}|_{2}=&-{1\over{\pi^2}}\int xdx
\Biggl(x(\bar{A}+2x\bar{A}')(\alpha_s'-\alpha_b')
+(2\bar{A}+x\bar{A}')
(\alpha_s-\alpha_b) \nonumber \\
&-x\beta'B'\Biggl),\end{aligned}$$ and $$\begin{aligned}
\Delta\tilde{V}|_{3}=&-{2\over{\pi^2}}\int xdx
\Biggl([x\bar{A}^2-2x^2\bar{A}'(\bar{A}+x\bar{A}')]
(\alpha_s\alpha_s'-\alpha_b\alpha_b'
+x(\alpha_s'^2-\alpha_b'^2)) \nonumber \\
&+(\bar{A}^2+x \bar{A}\bar{A}'+x^2\bar{A}'^2)
(\alpha_s^2-\alpha_b^2-x\bar{\beta}^2) \nonumber \\
&+x({1\over{2}}\alpha_b \bar{A}+x\alpha_b
\bar{A}'+x\alpha_b' \bar{A}
+2x^2\alpha_b'\bar{A}')\beta'B'\nonumber \\
&-x(\alpha_b^2+x\alpha_b\alpha_b'+x^2\alpha_
b'^2)\bar{B}'^2+{1\over 2}x^2\beta'^2B'^2\Biggr),\end{aligned}$$ where $\bar{\beta}\equiv a\beta$, and prime denotes partial deifferentiation with respect to x. Eq. 38 has an infrared singularity in $\tilde{V_2}_{s}|_{3}$, which originates from a mismatch in infrared singularities between the left and right sides of Fig. 4. As x approaches zero we find that $\alpha_S={2\over{\epsilon}}$, where $\epsilon$ is infinitesmal; therefore in the chiral symmetric case, Fig. 4 will have a singularity structure that goes as $${b\over{\epsilon}}={c\over{\epsilon^2}}+{d\over{\epsilon^3}}+...,$$ where b, c, and d are constants. Presumably the right hand side will sum up to give a power of $1\over\epsilon$, which can be achieved by replacing $\delta^4(k)$ in Eq. 15 with $\epsilon\delta^4(k)$ in the limit of small x to obtain a power of $1\over\epsilon$ for each term on the right hand side of Fig. 4. But such a resummation is only valid in the infrared regime, and we are interested in performing an integral over all momenta. We notice instead that the infrared singularity in Eq. 38 is isolated to the term involving $\alpha_{s}^2\bar{A}^2$ (other terms are also infrared singular, but the singularity cancels among them); in fact in graphs involving more than four vertices the singularity is also dominate in the term containing a power of $\alpha_{s}\bar{A}$. We find that terms invoving $\alpha_{s}\bar{A}$ and $\alpha_{b}\bar{A}$ resum into a logarithm for the graphs shown in Fig. 1a (and the graph involving 6 vertices, we have not checked 8 and higher vertex graphs). Replacing the sum of the terms on the right hand sides of Eqs. 37 and 38 involving $\bar{A}(\alpha_{s}-\alpha_{b})$, and ${\bar{A}}^2({\alpha_{s}}^2-{\alpha_{b}}^2)$ respectively by, $$\begin{aligned}
-{1\over{\pi^2}}\int xdx\Biggl(4(\alpha_{s}-\alpha_{b})\bar{A}
+ln\Biggl({{1-2\alpha_{s}\bar{A}}\over{1-2\alpha_{b}\bar{A}}}\Biggr)\Biggr),\end{aligned}$$ and then adding Eqs. 36, 37, and 38, we find that $$\begin{aligned}
{\tilde{V}_{s}}-{\tilde{V}_{b}}=.1>0.\end{aligned}$$ To accurately determine ${\tilde{V}_{s}}-{\tilde{V}_{b}}$, we need to sum the infinite terms on the right hand side of Eq. 1a for both the symmetric and symmetry breaking cases, which may be tractable, but will not be pursued here. By keeping 3 terms in the series in Fig. 1a, we have demonstrated the possibility that the chiral symmetry breaking solution is the physical solution to Eq. 24 unlike the conclusion reached in Ref. 11. The author is grateful for helpful discussions with J.M. Cornwall. The author also benefited from discussions with S.T. Love and C.D. Roberts. This work was supported by the US Department of Energy under contract DE-AC02-76ER01428 (Task B).
[**Figure 1.**]{} The interaction part of the effective potential.
[**Figure 2.**]{} The gauge particle propagator SD equation.
[**Figure 3.**]{} The fermion propagator SD equation.
[**Figure 4.**]{} The vertex SD equation.
[11]{} J.M. Cornwall, R. Jackiw, E.T. Tomboulis, Phys. Rev. D[**10**]{}, 2428 (1974). J.M. Cornwall, private communication. J.E. King, Phys. Rev. D[**27**]{}, 1821 (1983). B. Haeri, Phys. Rev. D[**43**]{}, 2701 (1991). J.M. Cornwall, Phys. Rev. D[**22**]{}, 1452 (1980). H. Pagels, Phys. Rev. D[**15**]{}, 2991 (1977). H.J. Munczek, and A.M. Nemirovsky Phys. Rev. D[**28**]{}, 181 (1983). B. Haeri, Phys. Rev. D[**38**]{}, 3799 (1988). B. Haeri, and M.B. Haeri, Phys. Rev. D[**43**]{}, 3732 (1991). C.J. Burden, C.D. Roberts, and A.G. Williams, Phys. Lett.B[**285**]{}, 347 (1992). The middle graphs in Fig. 1b contain symmetry numbers that do not appear in a pertubative summation of the vertex, thus naively inserting a closed expression for the full vertex in the left hand side of Fig. 1b as done in H. Munczek, Phys. Lett. B[**175**]{}, 215 (1986), and K. Stam, Phys. Lett. B[**152**]{}, 238 (1985), is not equivalent to Fig 1a, and therefore is not an exact expression.
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abstract: 'The ratio of the proton’s electric to magnetic form factor, $G_E/G_M$, can be extracted in elastic electron-proton scattering by measuring either cross sections, beam-target asymmetry or recoil polarization. Separate determinations of $G_E/G_M$ by cross sections and recoil polarization observables disagree for $Q^2 > 1$ (GeV/c)$^2$. Measurement by a third technique might uncover an unknown systematic error in either of the previous measurements. The beam-target asymmetry has been measured for elastic electron-proton scattering at $Q^2$ = 1.51 (GeV/c)$^2$ for target spin orientation aligned perpendicular to the beam momentum direction. This is the largest $Q^2$ at which $G_E/G_M$ has been determined by a beam-target asymmetry experiment. The result, $\mu G_E/G_M$ $= 0.884 \pm 0.027 \pm 0.029$, is compared to previous world data.'
author:
- 'M. K. Jones'
- 'A. Aghalaryan'
- 'A. Ahmidouch'
- 'R. Asaturyan'
- 'F. Bloch'
- 'W. Boeglin'
- 'P. Bosted'
- 'C. Carasco'
- 'R. Carlini'
- 'J. Cha'
- 'J.P. Chen'
- 'M.E. Christy'
- 'L. Cole'
- 'L. Coman'
- 'D. Crabb'
- 'S. Danagoulian'
- 'D. Day'
- 'J. Dunne'
- 'M. Elaasar'
- 'R. Ent'
- 'H. Fenker'
- 'E. Frlez'
- 'D. Gaskell'
- 'L. Gan'
- 'J. Gomez'
- 'B. Hu'
- 'J. Jourdan'
- 'C. Keith'
- 'C.E. Keppel'
- 'M. Khandaker'
- 'A. Klein'
- 'L. Kramer'
- 'Y. Liang'
- 'J. Lichtenstadt'
- 'R. Lindgren'
- 'D. Mack'
- 'P. McKee'
- 'D. McNulty'
- 'D. Meekins'
- 'H. Mkrtchyan'
- 'R. Nasseripour'
- 'I. Niculescu'
- 'K. Normand'
- 'B. Norum'
- 'D. Pocanic'
- 'Y. Prok'
- 'B. Raue'
- 'J. Reinhold'
- 'J. Roche'
- 'D. Rohe'
- 'O.A. Rondón'
- 'N. Savvinov'
- 'B. Sawatzky'
- 'M. Seely'
- 'I. Sick'
- 'K. Slifer'
- 'C. Smith'
- 'G. Smith'
- 'S. Stepanyan'
- 'L. Tang'
- 'S. Tajima'
- 'G. Testa'
- 'W. Vulcan'
- 'K. Wang'
- 'G. Warren'
- 'F.R. Wesselmann'
- 'S. Wood'
- 'C. Yan'
- 'L. Yuan'
- 'J. Yun'
- 'M. Zeier'
- 'H. Zhu'
title: ' Proton $G_E$/$G_M$ from beam-target asymmetry '
---
Introduction
============
Understanding the structure of the nucleon has long been a goal of nuclear physics and elastic electron-nucleon scattering has been an important tool in this quest. In the one-photon exchange (Born) approximation, the structure of the nucleon can be characterized in terms of the electric and magnetic form factors, $G_E$ and $G_M$, which depend only on the four-momentum transfer squared, $Q^2 = -t$. At $Q^2 = 0$, the proton form factors are defined as $G_E = 1$ and $G_M = \mu$, where $\mu = 2.7928$ is the proton’s magnetic moment. The proton form factors can be extracted individually in elastic electron-proton scattering by measuring cross sections at the same $Q^2$ but different beam energies (Rosenbluth technique). In addition, spin observables in elastic electron-proton scattering are sensitive to the ratio of $G_E$ to $G_M$.
Historically, the Rosenbluth technique was used to measure $G_E$ and $G_M$ with elastic scattering identified by detection of the scattered electron. The cross section can be written as: $$\begin{aligned}
\frac{d\sigma}{d\Omega} &=& \frac{\alpha^2 E^{\prime} \cos^2\frac{\theta_e}{2}}{4(1+\tau)E^{3}\sin^4\frac{\theta_e}{2}}
\left[G^{2}_{E} + \frac{\tau}{\epsilon}G^{2}_{M}\right] \label{eq:xn} \\
\tau &=& \frac{Q^2}{4M^2} \,\,\,\,\,\,\,\, Q^2 = 2EE^{\prime}(1-\cos\theta_e) \nonumber\\
\epsilon &=& \left[1+2(1+\tau)\tan^2\frac{\theta_e}{2}\right]^{-1} \nonumber\end{aligned}$$ where $E$ and $E^{\prime}$ are the incoming and outgoing electron energies, $M$ is the proton mass and $\theta_e$ is the outgoing electron’s scattering angle. $G_M$ in Eq. \[eq:xn\] is multiplied by $Q^2$ and dominates the cross sections at large $Q^2$ at all $\epsilon$. For example, at $Q^2 = 6$ (GeV/c)$^2$, the contribution of $G_E$ to the elastic cross section is 7% at $\epsilon = 0.9$, assuming $\mu G_E/G_M = 1$.
At SLAC, $G_E/G_M$ was measured to $Q^2 = 8.8$ (GeV/c)$^2$ using the Rosenbluth technique [@An94]. A recent JLab Hall C experiment [@Ch04] in the same $Q^2$ range agrees with the SLAC data. These data were combined together with other cross sections measurements for a global analysis by Arrington [@Ar04d]. The $\mu G_E/G_M$ extracted from the global analysis is plotted in Fig. \[fig:rat1\] and labeled “World xn”. The dashed line in Fig. \[fig:rat1\] is $\mu G_E/G_M$ from a fit by Arrington to that data with a polynomial parametrization of $G_E$ and $G_M$.
Previous cross sections measurements detected electrons to identify an elastic event. A recent JLab experiment [@Qa05] in Hall A identified an elastic scattering event by detection of the scattered proton. This experimental approach has different systematic errors compared to electron detection and has many advantages in terms of reducing the systematic error. The $\mu G_E/G_M$ are plotted in Fig. \[fig:rat1\] and labeled as “JLab05”. The new data agree well with the recent fit to previous world data which demonstrates that the systematic errors in the Rosenbluth technique are understood.
Early on, it was proposed [@Ak68; @Do69; @Ak74] that measuring polarization observables in elastic electron-proton scattering would be an alternative method to extract the electric form factor given that the dominant magnetic form factor is determined by cross section data. In 1976, an experiment [@Al76] measured the beam-target asymmetry for elastic $ep$ scattering at $Q^2 = 0.76$ (GeV/c)$^2$. But given that the experiment used a longitudinally polarized target, the asymmetry was extremely insensitive to $G_E/G_M$ and could only restrict the relative sign between $G_E$ and $G_M$.
With the advent of high duty factor, high current, and highly polarized electron beam accelerators such as Jefferson Lab and the Mainz Microtron, experiments which measure the proton and neutron electro-magnetic form factors have reached a new level of precision over a larger $Q^2$ range by measuring polarization observables in elastic electron-nucleon scattering (see Ref. [@Hy04] for a review of the recent experiments). The proton $G_E/G_M$ ratios have been extracted from measurement of the recoil polarization components of the scattered protons in elastic scattering of polarized electrons from an unpolarized proton target. Both the transverse, $P_x$, and longitudinal, $P_z$, components of scattered proton’s recoil polarization are dependent on $G_E/G_M$. By simultaneously measuring both components, one can extract $G_E/G_M$ from the ratio of polarization components, $P_x/P_z$, which cancels systematic errors from the beam polarization and the analyzing power.
The first measurements of $G_E/G_M$ using the polarization transfer technique were done at MIT-Bates [@Mi98] in the 1990’s at $Q^2 = 0.38$ and 0.5 (GeV/c)$^2$ and are plotted in Fig. \[fig:rat1\]. The results agree with $G_E/G_M$ from the Rosenbluth technique. The polarization transfer technique was used in Hall A at Jefferson Lab [@Jo00; @Ga02] to measure $G_E/G_M$ to $Q^2 =$5.6 (GeV/c)$^2$ and the data are plotted in Fig. \[fig:rat1\]. A linear fall-off with $Q^2$ is seen which is in sharp contrast to the nearly flat $Q^2$ dependence of $G_E/G_M$ measured with the Rosenbluth technique. The absolute systematic error on the polarization transfer technique is given by the solid band at the bottom of Fig. \[fig:rat1\]. Reconciling the $G_E/G_M$ results from the two techniques is impossible given the systematic error quoted for both techniques. A recent result [@Ma06] using the polarization transfer technique in Hall C at Jefferson Lab for $G_E/G_M$ at $Q^2 =$1.13 (GeV/c)$^2$ is plotted in Fig. \[fig:rat1\] with the error bar that is dominated by statistics.
![ \[fig:rat1\] (Color online) Ratio $\mu G_E/G_M$ plotted as a function of $Q^2$. “World xn” and JLAB05 [@Qa05] used the Rosenbluth technique. Results using the recoil polarization technique are from MIT-Bates [@Mi98], JLAB00 [@Jo00], JLAB02 [@Ga02] and JLAB06 [@Ma06]. The band at the bottom is the systematic error on the data from JLAB00 and JLAB02. The dashed curve is a recent fit [@Ar04d] to the world cross section data. ](gepgmp_recoil_pol_comp_xn_fig1.eps){width="\columnwidth"}
One possible solution that reconciles the different $G_E/G_M$ from the two experimental techniques is inclusion of two-photon exchange mechanisms which are not part of the standard radiative correction procedure which reduces the raw cross section data to the Born cross sections needed in Eq. \[eq:xn\] for determination of $G_E$ and $G_M$. The Coulomb distortion effect is one type of two-photon exchange mechanism (exchange of one hard and one soft photon) which has been neglected in $ep$ experiments. Calculations [@Ar04c] which include Coulomb distortion effects when extracting the form factors from the cross sections find that $\mu G_E/G_M$ is reduced by about 0.05 for $Q^2 > 1$ (GeV/c)$^2$ while the effect on $\mu G_E/G_M$ is gradually reduced at smaller $Q^2$.
More general calculations [@Bl05; @Af05] of the contribution of two-photon exchange mechanisms in elastic electron-proton scattering have been done. The two calculations take different approaches to the model of the nucleon which is needed as part of the two-photon exchange calculation. The approach of Ref. [@Bl05] is applicable to lower $Q^2$ then that of Ref. [@Af05]. In both calculations, the contribution of the two-photon exchange amplitude has an $\epsilon$-dependence which has the same sign as the $G_E$ contribution to the cross section and is large enough to effect the extracted value of $G_E$. Therefore, the extracted $G_E/G_M$ for the Rosenbluth technique is reduced.
In addition to a linear $\epsilon$-dependence, both calculations have a nonlinear $\epsilon$-dependence in the two-photon contribution to the cross section. A global analysis [@Tvaskis:2005ex] of the $\epsilon$-dependence of elastic and inelastic cross sections found that the elastic ( inelastic) data was consistent with a maximum deviation from a linear fit of $\leq 0.4\%$ ($0.7\%$). But this level of precision is obtained by averaging over $0.2 < Q^2 < 5.2$ (GeV/c)$^2$ range. Since the amount of nonlinearity can change with $Q^2$, more precise data is needed for comparison to theory. An approved JLab experiment [@Ar04] is an extensive study of non-linearity in the $\epsilon$-dependence of the elastic electron-proton cross section at fixed $Q^2$ for a number of different $Q^2$.
The effect of two-photon exchange amplitude on the polarization components is small, though the size of the contribution changes with $\epsilon$. The recoil polarization measurements at JLab ran at $\epsilon$ between 0.45 and 0.77. From Ref. [@Bl05], the measured $P_x/P_z$ at $\epsilon = 0.5$ should be reduced by factors of 0.9975 and 0.97 at $Q^2 = 1$ and 6 (GeV/c)$^2$, respectively. At $Q^2 = 3$ (GeV/c)$^2$, Ref. [@Bl05] predicts that $P_x/P_z$ will be 4% larger at $\epsilon = 0.05$ compared to $\epsilon = 1$ due to contributions from two-photon amplitudes. Interestingly, the calculation of Ref. [@Af05] predicts that $\epsilon$-dependence of $P_x/P_z$ will have a slope of the opposite sign. Complementing the approved JLab cross section experiment, an upcoming JLab experiment [@Gi04] will measure the $\epsilon$-dependence of $P_x/P_z$ at $Q^2= 2.6$ (GeV/c)$^2$.
The two-photon models need to be tested by comparing predictions of additional observables to data. Experiments planned at Jefferson Lab [@Af04] and proposed at VEPP-3 in Novosibirsk [@Ar04b] would precisely measure the $\epsilon$-dependence of the ratio of cross sections, $R_{e+e-}$, for elastic electron-proton scattering to positron-proton scattering at a fixed $Q^2$. In absence of two-photon mechanisms, the ratio would be one and independent of beam energy. The present data set for $R_{e+e-}$ is limited with most measurements at $\epsilon > 0.6$. Previous experimental data was re-examined [@Arrington:2003ck] and found that combining all data for $Q^2 < 2$ (GeV/c)$^2$ gives a slope of -5.7 $\pm$ 1.8 % for the $\epsilon$-dependence of $R_{e+e-}$. Indeed, the calculation of Ref. [@Bl05] predicts an $\epsilon$-dependence which is consistent with the large error bars of the existing data.
Checking on the possibility of an unknown systematic error in the Rosenbluth or recoil polarization technique is also important. Measurement of the beam-target asymmetry in elastic electron-proton scattering offers an independent technique of determining $G_E/G_M$. The systematic errors are different when compared to either the Rosenbluth technique or the polarization transfer technique. For elastic scattering, the recoil polarization of scattered proton is directly related to the beam-target asymmetry by time reversal invariance. Therefore, sensitivity of the beam-target asymmetry to two-photon effects is the same as in the recoil polarization technique. By measuring $G_E/G_M$ by a third technique and comparing to previous results, the discovery of unknown or underestimated systematic errors in the previous measurements is possible.
Experimental Set-up
===================
The experiment was performed in Hall C at the Thomas Jefferson National Accelerator Facility (Jefferson Lab). The main purpose of the experiment was a measurement of the inclusive parallel and perpendicular spin asymmetries in the resonance region for electron scattering on polarized proton and deuterium targets. This report presents a subset of the data which measured the perpendicular beam-target asymmetry for elastic electron-proton scattering.
Polarized electrons with 5.755 GeV/c momentum were scattered from polarized frozen ammonia ($^{15}$NH$_3$) with the spin of the polarized target aligned perpendicular to the beam. The scattered electrons were detected at 13.15$^{\circ}$ in the High Momentum Spectrometer (HMS) which was set at a central momentum of 4.73 GeV/c. Electron particle identification was done by a combination of a gas Cerenkov detector and lead-glass calorimeter. A cut was placed to use a momentum range of $\pm$8%.
The frozen ammonia target [@Cr95] is polarized by dynamic nuclear polarization and operated at 1 K in a 5 T magnetic field. The magnetic field is created by a pair of superconducting Helmholtz coils which produces a uniform magnetic field that selects the spin direction of the protons. The refrigerator is a $^4$He evaporation type which is installed vertically along the center of the magnet. The coils can be rotated independently of the refrigerator so that the target spin can be aligned to any angle relative to the beam. The angle of the coils relative to the beam was measured to a precision of 0.1$^{\circ}$.
To make the target, frozen ammonia is pulverized into small fragments which are sifted to get fragments of the same size. The fragments are stored in sample bottles in liquid nitrogen dewars. For use in the experiment, the ammonia fragments are placed in a cylindrical container which is 3 cm long with a diameter of 2.5 cm. Inside the container is a coil for measuring the NMR signal. The container is placed on an insert ladder so that the beam passes through the container lengthwise. The insert ladder can be rotated independently of the magnet coils and refrigerator so that the beam enters the container perpendicular to its face. To check the orientation of the insert ladder, a target was placed on the insert which consisted of L-shaped rods of tungsten separated by 3cm. From reconstruction of the rods, the insert ladder was determined to be rotated 6$^{\circ}$ relative to the beam direction. The insert ladder held two frozen ammonia containers which were designated as TOP and BOTTOM. Additional targets on the insert were a 6.9-mm-thick $^{12}$C disk and an empty container. The targets are in a bath of liquid helium that is cooled by the refrigerator.
To maintain reasonable target polarization, the beam current was limited to 100 nA and was uniformly rastered. The uniformity of the raster was obtained by independently and simultaneously rastering at a fast frequency (17.9 kHz in vertical direction and 24.2 kHz in the horizontal direction) over 1 mm square spot and slow frequency (30 Hz) over 0.9 cm maximum radius spiral pattern. The slow raster frequency was the same frequency as the flipping of the beam helicity. Each of the rasters could independently be turned on or off and the raster size changed. The beam position was measured on an event-by-event basis using an array of secondary emission monitors [@St00] located upstream of the target.
At thermal equilibrium at 5 T and 1 K, the protons have a small polarization of 0.51% and the electrons have a large polarization of 99.8%. By applying a microwave radiation to the target material at a frequency near the electron spin-flip resonance frequency, the electron polarization is transferred to the proton. The protons have a slow relaxation time compared to the electrons and slowly the polarization of the protons builds up. The spin vector of the polarized protons is aligned parallel or anti-parallel to the field direction by changing the frequency of the microwaves and measurements were done at both microwave frequencies. For this data set, the target field was aligned at 90$^{\circ}$ to the beam direction with positive target polarization defined as the target field pointing toward beam left. The target polarization slowly decreased with exposure to the beam. When it became too small, the target was retracted from the beam to be annealed and repolarized.
The target polarization, $P_T$, was measured by the NMR technique. To extract absolute polarization, the NMR signal was calibrated by a known polarization at thermal equilibrium with no microwave radiation and no beam. Under these conditions, the proton polarization can be accurately calculated and used to determine the calibration constant, $C_{TE}$, of the NMR signal. $C_{TE}$ was determined separately for the bottom and top target, since each target has an individual NMR setup. The normalization was taken from the weighted average of a series of thermal equilibrium (TE) measurements which gives a small statistical error on $C_{TE}$. To determine the systematic error on $C_{TE}$, three separate series of TE measurements were done for one target at different times and the standard deviation was found to be 2.9%. This was used as the relative systematic error on the target polarization for both targets.
The accelerator at Jefferson Lab produces highly polarized beam that can be simultaneously delivered to all three experimental halls. The polarized beam was produced by photo-emission from a semiconductor cathode using polarized laser light from a pulsed diode laser. Each hall had its own diode laser which produces a narrow pulse, but a small continuous noise was also present. This produced a leakage current from the other hall’s laser underneath the main beam pulse for that hall. The leakage current was measured in Hall C by an intrusive method. The rate in the HMS was measured with the Hall C laser turned on (normal conditions) and turned off (only leakage current). The ratio of the two rates is a measure of the leakage current. Throughout the experiment the leakage current was measured every 12 hours and on average the leakage current was found to be 2% of the total current. This is the leakage current from both Halls A and B.
The polarization of electrons produced at the cathode depends on the laser wavelength. At the time of this experiment, Hall A wanted high current and was not interested in polarized beam, while Halls B and C wanted low current and polarized beam. The wavelength of the laser chosen for the Hall A system produced a high current beam with $\approx$ 35% polarization which is about half the beam polarization for Hall C. A 2% leakage current from Hall A dilutes the Hall C beam polarization by about 1%. Since both Hall B and C were at the same laser wavelength, the beam polarization at the injector is the same for Halls B and C. This means that Hall B leakage current does not effect the polarization of beam to Hall C, but changes to the relative amount from Halls A and B to the total leakage current in Hall C does change the Hall C beam polarization. There was no measurement of the relative amount of leakage current from Halls A and B in the total measured leakage current. In addition to dilution from leakage current, the longitudinal beam polarization at the Hall C target depends on the energy per pass, the number of passes and the setting the spin rotator in the injector which was set to maximize the product of longitudinal beam polarization in Halls B and C. Therefore, to accurately know the beam polarization in Hall C, a measurement must be made near the Hall C target.
The beam polarization, $P_B$, was measured in Hall C using the Møller polarimeter [@Ha99]. Møller measurements were taken when the target was retracted from the beam for annealing. The Møller measurements were done at beam currents of 100 and 200 nA. The measurements were taken throughout the run period and are plotted in Fig. \[fig:moller\] as a function of run number. The average $P_B$ was $65.6 \pm 0.38$% and was used to determine the elastic asymmetry in Eq. \[eq:asymexp\]. The beam polarization was assumed to be constant throughout the perpendicular target field running and no time dependent nor run-by-run adjustment to the beam polarization was done. The relative systematic error for the Møller measurement is 0.7%. The beam polarization could be different during the Møller measurements and the actual running due to changing leakage currents in Halls A and B. If the leakage current was mainly from Hall B then there would be no dependence of the Hall C beam polarization on leakage current. The worse case would be assuming that the leakage current is dominantly from Hall A. With that condition and assuming that leakage varies from 0% to 4% then an estimate of the relative systematic error on the beam polarization from changes in the leakage current is 1%. Combining these errors in quadrature gives a relative systematic error of 1.3% on the beam polarization.
![ \[fig:moller\] Beam polarization, $P_B$, as a function of run number. The line is the weighted average of $P_B$. ](moller_run.eps){width="\columnwidth"}
When the target field is perpendicular to the beam direction, the incoming electrons are bent downward before the target by the magnetic field. Two chicane magnets before the target bend the incoming beam up so that, when combined with the target field, the beam is incident horizontally on the target. The electrons scattered toward the HMS are bent downward and have an average out-of-plane angle of 3.4$^{\circ}$.
Normally, the position, angle and momentum of the scattered electron are determined by measuring HMS focal plane position and angles of the electron and then reconstructing the target quantities using an optics matrix. In addition, the HMS optics matrix takes into account the vertical position of the beam at the target. The calculation of momentum and out-of-plane angle are sensitive to the vertical position. The HMS optics matrix has been determined without the target field. The effect of the target field can be mimicked by using a effective vertical position at the target with the known HMS optics matrix in an iterative procedure. The reconstructed angles and momentum of the electron are determined using the known HMS optics matrix and an assumed effective vertical position at the target. The electron is tracked from the entrance of the HMS back through the target field to the center of the target using a tabulated map of the target field and the reconstructed electron momentum and angle. The difference is taken between this tracked vertical position at the target center and the vertical position of the beam measured by the SEM. If the difference is larger than 1 mm, then a new effective vertical position is assumed and the procedure is iterated until the difference between the tracked and measured vertical position is less than 1 mm.
To check the angle reconstruction, data were taken with the sieve collimator which has a 9x9 grid of holes. The pattern of sieve holes were properly reconstructed by the algorithm described above. The momentum reconstruction was checked by looking at the reconstructed final state mass, $W = \sqrt{M^2 + 2(E-E^{\prime})M - Q^2}$. The peak position of $W$ was plotted as a function of different target variables. The $W$ peak position had a slight dependence on the out-of-plane angle and no dependence on the other target variables. An azimuthal angle dependence was added to the map of the target field used in the calculation of the electron’s track which changed the electron’s reconstructed momentum and eliminated the dependence of $W$ on the out-of-plane angle.
Experimental Results
====================
From Ref. [@Do86], the beam-target asymmetry, $A_p$, for elastic electron-proton scattering is related to the ratio of the proton’s electric to magnetic form factors, $r = G_E/G_M$, by the formula: $$A_p = \frac{-br\sin\theta^{\star}\cos\phi^{\star} - a\cos\theta^{\star}}{r^2+c}
\label{eq:asym}$$ in which $\theta^{\star}$ and $\phi^{\star}$ are the polar and azimuthal angles between the momentum-transfer vector, $\vec{q}$, and the proton’s spin vector. $a,b,c$ are kinematic factors: $$\begin{aligned}
a &=& 2\tau\tan\frac{\theta_e}{2}\sqrt{1+\tau+(1+\tau)^2\tan^2\frac{\theta_e}{2}} \label{eq:a}\\
b &=& 2\tan\frac{\theta_e}{2}\sqrt{\tau(1+\tau)} \label{eq:b} \\
c &=& \tau + 2\tau(1+\tau)\tan^2\frac{\theta_e}{2} \label{eq:c} \end{aligned}$$ The measured asymmetry, $A_{m}$, is defined as $(N^{+}-N^{-})/(N^{+}+N^{-})$ where $N^{+}$ and $N^{-}$ are the raw counts normalized for deadtime and charge for opposite beam helicities. The elastic asymmetry for the perpendicular target field is $$A_{p} = \frac{A_{m}}{f P_B P_T} + N_{c} \label{eq:asymexp}$$ where the measured asymmetry is normalized by $P_T$, $P_B$ and the dilution factor, $f$. The dilution factor is the ratio of the yield from scattering off free protons to that from the entire target. $N_{c}$ is correction to the measured asymmetry which eliminates the contribution from quasi-elastic $^{15}$N scattering under the elastic peak.
In Fig. \[fig:df\]a, the yield, $Y_{tot}$, for scattering off the entire BOTTOM target is plotted versus W. The peak at $W$ $\approx$ 938 MeV for elastic scattering off free protons is evident on top of the background from quasi-elastic scattering from other target material. The width of the elastic peak is $\sigma = 14$ MeV and is determined by the resolution in the scattered electron’s momentum and angle. The width is consistent with a combination of 1.5 mr resolution in $\theta_e$ and $1.5 \times 10^{-3}$ resolution in $E^{\prime}$. These resolutions are about 50% larger than the typical resolutions found with no target field and smaller raster size.
![ \[fig:df\] a) The yield, $Y_{tot}$, for scattering from the entire BOTTOM target is plotted as open circles versus W. The error is smaller than the circle size. The solid line is $Y_{back}$, the $^{12}$C+He yield which has been normalized to $Y_{tot}$ in the region of 0.6 $<$ $W$ $<$ 0.85 GeV.\
b) The dilution factor, $f$, for the BOTTOM target versus W. The dotted line indicates zero to guide the eye. ](df_perp.eps){width="\columnwidth"}
To determine the shape of the quasi-elastic background under the elastic peak, data were taken with a $^{12}$C disk (immersed in the liquid helium bath) of areal density comparable to the ammonia in the target. The solid line in Fig. \[fig:df\]a is the yield, $Y_{back}$, from the $^{12}$C+He data which has been normalized to the BOTTOM target yield in the region 0.6 $<$ $W$ $<$ 0.85 GeV. The normalization factor was $1.212 \pm 0.007$ for the BOTTOM target and $1.235 \pm 0.007$ TOP target. One can see that the $^{12}$C+He matches the shape of $^{15}$N+He in the region 0.6 $<$ $W$ $<$ 0.85 GeV. The assumption that the shape of the $^{12}$C+He is similar to the $^{15}$NH$_3$+He in the $W$ region under the elastic peak was tested by a Monte Carlo simulation using realistic cross section models and including radiative corrections. The Monte Carlo predicts that normalization factor is 1.19 (1.22) for BOTTOM (TOP) target at $W$ = 0.77 GeV and has slight $W$ dependence of 0.04 every $\Delta W$ = 0.1 GeV. The difference in normalization factor between the BOTTOM and TOP targets is caused by different packing fractions (the ratio of NH$_3$ to helium in the target).
The dilution factor, $f$, is $1-Y_{back}/Y_{tot}$ and $f$ for the BOTTOM target is plotted in Fig. \[fig:df\]b. In the calculation of the dilution factor, the $W$ dependence of the normalization factor was not taken into account. For W$ < 0.85$ GeV, $f$ is zero and flat indicating that the shape of the $^{12}$C+He data is well matched to the shape of the $^{15}$N+He background with a constant normalization at all W$ < 0.85$ GeV. Near $W$ = 0.938 GeV, $f$ reaches a peak of about 0.66 and drops off to near constant value of 0.10 for the $W$ region of the elastic radiative tail up to pion production threshold ($W = 1.075$ GeV). By combining the statistical error on the normalization factor and the error due to assuming a flat $W$ dependence to the normalization factor, the relative systematic error of 1.1% on the dilution factor was calculated.
Typically, data taking was divided into runs of one hour duration and $P_T$ changed during the run. $P_T$ was continuously measured and recorded during the experiment every 20 seconds by an automated procedure. The average proton polarization for all runs was 66% (71%) when running with the BOTTOM (TOP) target. The charge-weighted average target polarization, $P^{ave}_{T}$, and $A_{m}$ were measured for each run. In Fig. \[fig:asymw\], the weighted average of $A_{m}/P^{ave}_T$ for all runs is plotted as a function of $W$ for BOTTOM and TOP targets.
![ (Color online) \[fig:asymw\] The asymmetry, $A_{m}/P^{ave}_T$, as a function of W. For the BOTTOM (TOP) target, the asymmetry is plotted as a solid (open) circle. Each data set is slightly shifted in $W$ for clarity. ](am_w.eps){width="\columnwidth"}
The protons in $^{15}$N are polarized and contribute to $A_m$. The contribution is characterized in terms of the correction term, $N_c$, in Eq. \[eq:asymexp\]. $N_c$ is equal to $f_{N}/f\times P_{N}/P{_T}\times {A_N}$ in which $f_N$, $P_N$ and $A_N$ are the dilution factor, polarization and asymmetry for the proton in $^{15}$N. $A_N$ can be estimated from models [@Ro99]. From the angular momentum decomposition of the $p_{1/2}$ level that is populated by the unpaired proton in the single particle shell model, one expects $A_N = -A_p/3$. The polarization of the proton in $^{15}$N relative to $P_T$ has been measured in separate experiments [@Cr95; @Br90]. The data was fitted by the formula: $$P_N = -0.01\times(0.312+5.831|P_T|+8.935|P_T|^2 + 8.685|P_T|^3) \nonumber$$ For $P_T = 71$% one gets $P_N = -12$%. The dilution factor, $f_N$, is the ratio of the yield for scattering from the polarized proton in $^{15}$N to the yield from scattering from the entire target. $f_N$ is like $f$ in that it varies with $W$ and $f_N = 0.03$ at $W = 940$ MeV. The asymmetry is corrected for $N_c$ at each $W$ and, to give a flavor of the size of the correction, $N_c = -0.0002$ at $W = 940$ MeV which is a 0.2% correction to $A_p$.
$A_p$ is plotted as a function of $W$ for both the BOTTOM and TOP targets in Fig. \[fig:apw\]. For W$ < 0.9$ GeV, $f$ is very small with relatively large error, so the error on $A_p$ becomes larger than the scale of the y-axis. In the region $0.9 <$ $W$ $ < 1.0$ GeV, $A_p$ is constant, and the error bars are small due to the large magnitudes of $A_m/P_t$ and $f$. For W$ > 1.0$ GeV, in the region of the elastic radiative tail, $A_{p}$ is still constant, but the error bars are larger. For the region $0.9 <$ $W$ $ < 1.0$ GeV, the average $A_{p}$ is $-0.1004 \pm 0.0042$ ($-0.0994 \pm 0.0044$) for BOTTOM (TOP) target. Radiative corrections to $A_{p}$ were calculated using the MASCARAD code of Ref. [@Af01] and shift $A_{p}$ by $-0.0004$. Including the radiative correction, the average $A_{p}$ from both targets is $-0.1003 \pm 0.0031$.
![ (Color online) \[fig:apw\] The asymmetry, $A_{p}$, as a function of W. For the BOTTOM (TOP) target, the asymmetry is plotted as a solid (open) circle. Each data set is slightly shifted in $W$ for clarity. The solid line is the weighted average of $A_p$ combining both targets. ](asym_corr_w.eps){width="\columnwidth"}
Using Eq. \[eq:asym\], $G_E/G_M$ can be determined directly from $A_p$ using the formula: $$\begin{aligned}
\frac{G_E}{G_M} = &-&\frac{b}{2A_p}\sin\theta^{\star}\cos\phi^{\star} \nonumber \\
&+&\sqrt{\frac{b^2}{4A^2_p}\sin^2\theta^{\star}\cos^2\phi^{\star}-\frac{a}{A_p}\cos\theta^{\star}-c}
\label{eq:gegm}\end{aligned}$$ in which $a,b$ and $c$ are the kinematic factors given in Eqns. \[eq:a\]-\[eq:c\]. The average $\theta_e$ is 13.22$^{\circ}$ and the average $Q^2$ is 1.509 (GeV/c)$^2$. The lab coordinate system is defined by the incoming and scattered electron’s momentum vectors, $k$ and $k^{\prime}$, as positive $z$-direction along $\hat{k}$, $\hat{y} = \hat{k} \times \hat{k}^{\prime}$ and $\hat{x} = \hat{y} \times \hat{z}$ with $+\phi$ rotation from $+\hat{x}$ to $+\hat{y}$. Since the scattered electron is bent downwards by the target’s magnet field, the average azimuthal angle, $\phi_e$, is out-of-plane with a value of 348.8$^{\circ}$. The $\vec{q}$ points at the angles $\theta_q$ = 50.43$^{\circ}$ and $\phi_q$ = 168.8$^{\circ}$. For Eq. \[eq:gegm\], one needs the polar and azimuthal angles, $\theta^{\star}$ and $\phi^{\star}$, between the $\vec{q}$ and the proton’s spin vector. Specifically, when the proton’s spin vector is pointing at $\theta_s$ =90$^{\circ}$ and $\phi_s$ =180$^{\circ}$, $\theta^{\star}$ and $\phi^{\star}$ can be calculated by the formulas: $$\begin{aligned}
\theta^{\star} &=& \arccos(\sin\theta_q\cos\phi_e) \nonumber \\ \nonumber
\phi^{\star} &=& 180 + \arctan\left[\frac{\tan\phi_e}{-\cos\theta_q}\right]\end{aligned}$$ For the present kinematics, $\theta^{\star}$ = 40.87$^{\circ}$ and $\phi^{\star}$ = 197.26$^{\circ}$. With these kinematic factors and the radiatively corrected average $A_p$, $\mu G_E/G_M$ $= 0.884 \pm 0.027$. The solution to Eq. \[eq:asym\] for $G_E/G_M$ is double-valued. The positive value of the square root was chosen, since the negative solution gives an unreasonable value of $\mu G_E/G_M = -4.05$. For this kinematic point, the systematic error on $\Delta (G_E/G_M)/(G_E/G_M) = 0.97\times\Delta A_p/A_p$. The total relative systematic error on $\mu G_E/G_M$ is 3.3%. A break down of the systematic errors is given in Table \[tab:sys\]. The beam and target polarization are the dominant contributions systematic contributions.
Variable Error $\Delta r$/$r$
------------------ --------------- ----------------
$\theta_e$ 0.5 mr 0.2%
$\theta^{\star}$ 0.1$^{\circ}$ 0.1%
$\phi^{\star}$ 1.0$^{\circ}$ 0.45%
$E$ 0.003 GeV 0.005%
$E^{\prime}$ 0.005 GeV 0.01%
$f$ 1.1% 1.1%
$P_T$ 2.9% 2.8%
$P_B$ 1.3% 1.3%
Total 3.3%
: Relative systematic errors on $r = G_E/G_M$. []{data-label="tab:sys"}
Conclusion
==========
In Fig. \[fig:ratio\], the ratio $\mu G_E/G_M$ from this experiment is compared to previous measurements. A recent global fit of $G_E$ and $G_M$ to the world cross section data has been done [@Ar04d] and the result for $\mu G_E/G_M$ is plotted by a dashed line in Fig. \[fig:ratio\]. The solid line is $\mu G_E/G_M$ from a fit to all nucleon form factors by Lomon [@Lo02] which only uses proton $G_E/G_M$ from the polarization transfer technique at large $Q^2$. The difference between the two curves is 12% at $Q^2$ = 1.509 (GeV/c)$^2$. The statistical error and systematic error for this measurement are comparable to previous $\mu G_E/G_M$ values from cross-section and recoil polarization experiments. The data point is midway between the two curves so it is about 2$\sigma$ away from either curve. Unfortunately, the new measurement does not help to determine whether the discrepancy between $\mu G_E/G_M$ from the Rosenbluth technique and the polarization transfer technique is due to unknown systematic errors in either technique.At this $Q^2$, inclusion the Coulomb distortion effects [@Ar04c] in the Rosenbluth technique would reduce $\mu G_E/G_M$ by 0.05 which would make it overlap with the present data point and bring measurements from all three techniques into reasonable agreement.
![ \[fig:ratio\] (Color online) Ratio $\mu G_E/G_M$ plotted as a function of $Q^2$. The ratio $\mu G_E/G_M$ from this experiment is plotted as a filled circle with the error bar being the statistical and systematic error combined in quadrature. The solid line is a fit [@Lo02] to all form factor data, which only included proton $G_{E}/G_{M}$ from Ref. [@Jo00] and [@Ga02] for large $Q^2$. Other symbols are same as in Fig. \[fig:rat1\]. ](gepgmp_recoil_pol_comp_xn_fig2.eps){width="\columnwidth"}
The inclusion of two-photon exchange mechanisms in the extraction of the Born cross section will reduce $\mu G_E/G_M$ and bring it closer to $\mu G_E/G_M$ determined by this measurement and previous measurements using the polarization transfer technique. A calculation [@Bl05] including all two-photon exchange mechanisms would reduce $\mu G_E/G_M$ by about 0.08 compared to the dashed line in Fig. \[fig:ratio\]. This beam-asymmetry measurement is at $\epsilon = 0.963$ which minimizes the contribution from two-photon exchange mechanisms and, from Ref. [@Bl05], $\mu G_E/G_M$ would be reduced by roughly a factor of 0.995 by accounting for the two-photon amplitude mechanisms.
This experiment is the first to measure $G_E/G_M$ using beam-target asymmetry in elastic $ep$ scattering. To definitively distinguish between experimental techniques at this $Q^2$, a beam-target asymmetry experiment needs to reduce both the statistical and systematic error. The systematic error which is hardest to reduce is the error on the target polarization. One approach would be to simultaneously measure the beam-target asymmetry at a given $Q^2$ with two separate spectrometers which are at the same electron scattering angle but opposite sides of the beam. By taking the ratio of the two asymmetry measurements, the beam and target polarization will cancel and $G_E/G_M$ can be extracted with no systematic from either polarization measurement. Another approach would be to measure at higher $Q^2$ where the percentage difference between $G_E/G_M$ extracted from the two experimental techniques is larger, since the systematic error on the beam and target polarization is independent of $Q^2$. To compensate for the falling cross-section, the experiment either has to run longer or use large acceptance detectors to keep the statistical error from growing too large. Dedicated experiments have been proposed [@Wa01; @Zh04] at Jefferson Lab to measure $G_E/G_M$ by beam-target asymmetries using both these experimental approaches.
We would like to thank the Hall C technical staff and the accelerator operators for their efforts and dedication. This work was supported by Schweizerische Nationalfonds, the Department of Energy contract DE-FG02-96ER40950 and by the Institute of Nuclear and Particle Physics of the University of Virginia. The Southern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility for the United States Department of Energy under contract DE-AC05-84ER40150.
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abstract: 'On 2010 August 1, a global solar event was launched involving almost the entire Earth$-$facing side of the Sun. This event mainly consisted of a C3.2 flare, a polar crown filament eruption and two Earth-directed coronal mass ejections (CMEs). The observations from the Solar Dynamics Observatory (SDO) and the Solar Terrestrial Relations Observatory (STEREO) showed that all the activities were coupled together, suggesting a global character of the magnetic eruption. We reconstruct the three-dimensional geometry of the polar crown filament using observations from three different viewpoints (STEREO A, B and SDO) for the first time. The filament undergoes two eruption processes. Firstly, the main body of the filament rises up, while it also moves towards the low-latitude region with a change in inclination by $\sim$ 48$\degr$ and expands only in the altitudinal and latitudinal direction in the field of view of Atmospheric Imaging Assembly. We investigate the true velocities and accelerations of different locations along the filament, and find that the highest location always has the largest acceleration during this eruption process. During the late phase of the first eruption, part of the filament material separates from the eastern leg. This material displays a projectile motion and moves towards the west at a constant velocity of 141.8 km s$^{-1}$. This may imply that the polar crown filament consists of at least two groups of magnetic systems.'
author:
- 'Ting Li, Jun Zhang, Yuzong Zhang, Shuhong Yang'
title: 'Three-Dimensional Reconstruction of an Erupting Filament with SDO and STEREO Observations'
---
Introduction
============
The twin Solar Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008; Howard et al. 2008) spacecraft provide simultaneous observations from two different points of view, which enables us to reconstruct the three-dimensional (3D) geometry of coronal structures. After the launch of the STEREO mission, there have been many studies about the 3D reconstructions using data from the pair of Extreme UltraViolet Imagers (EUVI; see Wuelser et al. 2004). Feng et al. (2007) presented the first reconstruction of the 3D shape of coronal loops in an active region based on a triangulation technique. Aschwanden et al. (2008a, b, 2009) discussed the 3D geometry of coronal loops and determined the electron density and temperature of loops. Besides the coronal loops, the STEREO data have been used to reconstruct the active regions (Rodriguez et al. 2009; Wiegelmann & Inhester 2006), polar plumes (Curdt et al. 2008; Feng et al. 2009), polar coronal jets (Patsourakos et al. 2008) and coronal mass ejections (CMEs; Mierla et al. 2008; Timothy & Tappin 2008; Wood et al. 2009; Frazin et al. 2009).¡¡
Until now, there have been many studies about the nature of filament eruption and its role in disrupting the coronal magnetic fields (e.g., Zhang & Wang 2001; Zhang et al. 2001; Liu et al. 2007; Jiang et al. 2007; Liu et al. 2010; Guo et al. 2010a; Guo et al. 2010b). The dynamics of erupting filaments are very important to understand the associated physical mechanisms. However, filament velocity derived using traditional method has a disadvantage because it is only a projected velocity. Only recently, the true process of filament eruption could be properly judged by using the STEREO data (Li et al. 2010; Xu et al. 2010; Zapiór & Rudawy 2010; Gosain et al. 2009; Liewer et al. 2009; Bemporad 2011; Panasenco et al. 2011; Thompson 2011). Gissot et al. (2008) presented a fully automated optical-flow algorithm and estimated the height of an erupting filament from simultaneously obtained EUVI images. Thompson (2008) traced out a filament in 3D space and found a rotation of $\sim$ 140$\degr$ from the original filament orientation. Bemporad (2009) found that the early filament expansion was anisotropic and occurred mainly on a plane parallel to the plane of the sky.
It is well known that the inner core of CME is made up of filament material (e.g., House et al. 1981). The slow rise of eruptive filaments linked to the streamer swelling is considered as the precursor of CMEs (Filippov & Koutchmy 2008). Thus an accurate measurement of filament motion is very useful in predicting the occurrence of a CME, as well as to constrain or construct the CME triggering mechanisms (Gopalswamy et al. 2006). Nevertheless, only several filaments have been reconstructed so far about their 3D shape and evolution using the STEREO data. The 3D physical picture of filament eruption is far from being completely understood. It is disadvantageous to reconstruct a 3D configuration of solar features by only using STEREO data because of the large separation angle between the two satellites. With the launch of the Solar Dynamics Observatory (SDO; Schwer et al. 2002), this would be much improved by using data of SDO and STEREO. In this work, we make the 3D reconstruction of a polar crown filament using observations from the three different viewpoints, and use a visualization method to display the reconstructed filament. Section 2 describes the instrumentation and observations. The 3D reconstruction technique and results are presented in Section 3. The eruptions of the polar crown filament are described in Sections 4, and the conclusions and discussion are given in Section 5.
Instrumentation and observations
================================
The Atmospheric Imaging Assembly (AIA; Lemen et al. 2011) on the SDO has already provided a rich database since its launch in February 2010. AIA images are taken in 10 different wavelength bands, including one visible line, two ultraviolet and seven extreme ultraviolet (EUV) channels. It provides full-disk images, covering a wide range of temperatures, with high cadence (up to 12$-$24 s) and spatial resolution (0.6 arcsec pixel$^{-1}$). The full-disk longitudinal magnetic field from the Helioseismic and Magnetic Imager (HMI; Schou et al. 2011) aboard SDO is available now, with high cadence ($\sim$ 45 s) and spatial resolution (0.5 arcsec pixel$^{-1}$).
The Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) imaging package on each spacecraft of the STEREO consists of the following five telescopes: the EUVI imager, inner (COR1) and outer (COR2) coronagraphs, and inner (HI1) and outer (HI2) heliospheric imagers. EUVI images are taken at four wavelengths centered at 304 [Å]{}, 171 [Å]{}, 195 [Å]{}, and 284 [Å]{} with time cadences of 10, 2.5, 10, and 20 minutes, respectively, for the data used here. COR1 has a field of view (FOV) from 1.4 to 4 $R_{sun}$ and COR2 from 2.5 to 15 $R_{sun}$.
On 2010 August 1, a spectacular solar event was launched involving almost the entire Earth$-$facing side of the Sun. At about 05:00 UT, a large-scale ($\sim$ 750,000 km) polar crown filament stretching across the solar northern hemisphere began to rise slowly. Then a C3.2 flare occurred at 07:24 UT in active region 11092, with a distance of $\sim$ 430,000 km from the long filament. At about 07:50 UT, an Earth-directed CME (CME front “a" in Figure 1) was observed by the COR1 aboard STEREO A. The average velocity of the first CME in the FOV of COR1 was about 624 km s$^{-1}$, and the angular width was $\sim$ 144$\degr$. At about 09:00 UT, another Earth-directed CME occurred and the eruptive filament was the bright core of the second CME (right panels in Figure 1). The second CME had an average velocity of 260 km s$^{-1}$ in the FOV of COR1, and its angular width was about 83$\degr$. Based on the Michelson Doppler Imager (MDI) magnetograms, we carried out the global reconstruction of 3D magnetic-field structure under the potential-field assumption. The boundary element method (BEM) was used in the extrapolation (Yan & Sakurai 2000; Wang et al. 2002; Zhang et al. 2007). We found that there were two groups of magnetic structures overlying the filament (the large-scale magnetic structures denoted by character “a" and the smaller magnetic structures beneath the large-scale ones denoted by “b" in top panels of Figure 1).
Here, we concentrate on the eruption process of the large polar crown filament. The filament eruption was recorded in all AIA channels, and the 304 [Å]{} images with a time cadence of 12 s were used. The eruption was also observed as a limb event at 304 [Å]{} by the two EUVIs aboard STEREO. The pixel size of the EUVI image on the solar disk was 1.6 arcsec, and the time cadence for 304 [Å]{} images was 10 minutes.
Reconstruction technique and results
====================================
For the first time, we combine data from SDO and STEREO to reconstruct the erupting filament. The angle of separation between STEREO A and B during our observations is around 149$\degr$.6. It is difficult to define the same feature in EUVI A and B images because of the large separation angle. The separation angle between STEREO A and SDO is 78$\degr$.8, and that between STEREO B and SDO 70$\degr$.8. So observations from the three different viewpoints are used here to derive the 3D erupting process of the polar crown filament. The western part of the filament was observed clearly by STEREO A and SDO, and thus we reconstruct the western part by using data from STEREO A and SDO (Figures 2 and 3). Similarly, the eastern part is reconstructed by using observations of STEREO B and SDO (Figures 2 and 4).
In order to reconstruct the 3D shape of the eruptive filament, we use a routine called SCC$_{-}$MEASURE (developed by W. Thompson), which is available in the STEREO package of the Solar Software library. The routine uses triangulation to determine the 3D coordinate of the tiepointed feature. It is a widget based application that allows the user to locate (and select with a cursor) the same feature in two images from different views. This routine makes use of an approximate “epipolar constraint" in locating the same feature in both images (Inhester 2006). The two observer positions and any object point to be reconstructed exactly define a plane, which is known as an epipolar plane. By definition, epipolar planes are projected on both observer¡¯s images as lines and these lines are called the epipolar lines. Once we identify a feature in one image, it is possible to determine the projection of the epipolar plane (i.e., epipolar line) passing the same feature in the second image. After selecting the same feature in both images, the 3D coordinates are then determined as longitude, latitude, and radial distance from the center of the Sun.
It must be mentioned that there exists a limiting factor in reconstructing filaments, because the reconstruction technique relies on the ability to identify the same feature in two images from different views. In fact, much of the corona is optically thin, so the emission at each pixel may be the result of a line-of-sight integration effect. Moreover, features may appear different from different angles due to the projection effect. In order to reduce the uncertainty, we examine the movies of filament eruption carefully and determine the features according to their evolution. We also use the bright or dark patterns in adjacent areas to identify the features. As seen in Figure 7, the measurement is repeated by two researchers, and each researcher measures twice. The standard deviations in the four sets of measurements are considered as the realistic uncertainties. The average values in the four sets of measurements are thought to be the real values.
About four hours after the onset of the rise, the filament lifted up to a high altitude above the solar surface and it became blurry because of the steep radial intensity gradient. So we reconstruct the 3D shape and evolution of the eruptive filament between 05:06 and 09:26 UT. We choose the top edge, the main-body line and the bottom edge as three baselines of the filament. For the western part of the filament, 20 pairs of points are placed along each baseline in the images of STEREO A and SDO. For the eastern part, 16 pairs of points are placed along each baseline in the images of STEREO B and SDO. Therefore, 36 pairs of points are placed along each baseline for the entire filament. We interpolate 500 points by using a cubic spline along each baseline among the selected points to smooth the baseline. Then we simply fill in the enclosed region between any two baselines with 1000 triangular elements to produce the extended regions. Using this visualization method, we can display the reconstructed filament seen from any visual angle.
The reconstructed filament seen from the SDO viewpoint at three different times is shown in Figure 2. The filament began to gradually lift up at 05:06 UT. It spanned a longitude range of about 42$\degr$.3 (from 3$\degr$.3 E to 39$\degr$.0 W measured from the central meridian) and a latitude range of about 26$\degr$.0 (from 20$\degr$.6 N to 46$\degr$.6 N). The filament had the initial length of 1.07 $R_{sun}$, and the highest part at that time was approximately at 1.15 $R_{sun}$. Then it moved towards the north pole as seen from the SDO, and rose to a height of 1.31 $R_{sun}$ at 07:36 UT. At about 09:00 UT, the western part of the filament broke up and then it was ejected from the Sun (bottom panels).
From the STEREO A perspective (Figure 3), the evolution of the western part is displayed clearly. The filament has two main endpoints at its west, composing a reverse$-$Y shape. The distance between the two ends of the reverse$-$Y shape is 197.4 Mm. The distance between the northern (southern) end and the intersection is 109.2 Mm (135.8 Mm). The northern endpoint could not be seen clearly before the rapid ascent of the filament (top panels). With the ascent of the filament, the reverse$-$Y structure is obviously seen and the endpoint brightenings are observed simultaneously (bottom panels). Strong and relatively concentrated brightenings occur at the northern endpoint, while weaker and more dispersed EUV enhancements are seen along the multiple, curved threads at its southern endpoint. At 08:36 UT, the strands of the filament anchoring the two ends are separated from each other. Then the strands anchoring the northern end is disconnected, and the area of brightenings at the northern endpoint expanded towards the northeast. The barbs of the filament could be observed clearly from the STEREO B perspective (denoted by white circles in Figure 4). At 05:06 UT, all the barbs are tightly anchored in the photosphere. Then the barbs are elongated and finally cut off from the solar surface in sequence due to the rise of the eastern part.
We reconstruct the eastern part of the filament by using data of STEREO B and SDO, and reconstruct the western part using data of STEREO A and SDO. For the eastern part, the STEREO A viewpoint is the third view to confirm the result. The eastern part of the reconstructed filament seen from STEREO A is similar to the observed eastern part (Figure 3). Similarly, the STEREO B viewpoint is the third view to confirm the result for the western part. The reconstructed western part seen from STEREO B is similar to the observed western part (Figure 4). These imply that our reconstruction results are reliable to some extent.
We display the reconstructed filament seen from three different points of view in Figure 5. The left panels show the results seen from the top of the filament center, which is the central location of the main body at 05:06 UT on the solar surface (denoted by plus signs in panels (a) and (e); with a longitude of 12$\degr$.1 W and a latitude of 39$\degr$.4 N). With the radial distance from the center of the Sun increasing, the latitude of the filament decreases gradually. The longitude of the filament remains approximately invariable. This implies that the eruption is non-radial. The angle between the radial direction (lines in panels (d)$-$(f); connecting the filament center at 05:06 UT with the solar center) and the line connecting the highest part with the filament center at 05:06 UT is considered as the inclination of the filament (panel (e)). The filament is initially inclined northward with an inclination of $\sim$ 40$\degr$. Then it moves towards the low-latitude region and is inclined southward with an inclination of $\sim$ 8$\degr$ at 09:26 UT. Seen from the north pole, the velocity of the filament has a large Earth$-$directed component (panels (g)$-$(i)). This is consistent with the Earth$-$directed CME associated with the filament eruption (right panels in Figure 1).
Eruptions of the filament
=========================
First eruption of the filament
------------------------------
In order to analyze the filament eruption in detail, we investigate the true velocities and accelerations of different sites along the filament. Eleven points (points “1" to “11" in Figure 6) are selected along the filament body at each time. Points “1" to “4" are defined using data of SDO and STEREO B, and points “5" to “11" are selected in SDO and STEREO A images. Due to motions of the material, it is difficult to define the same feature during a period. So we examine the movies of the filament eruption carefully, and select four specific points (“1", “5", “10" and “11") according to their continuous evolution. Points “1" and “11" are near the two ends of the filament. Points “5" and “10" are placed at the top and bottom corners, respectively. Then we place other points equally between each two specific neighboring points according to SDO/AIA images. Thus all the 11 points along the main body are defined at each time.
Because the velocity-location plots of the 11 points at different times are similar, we only display the plots at three times (Figures 7(a) and (b)). Point “5" at the top corner of the filament is always the highest among the 11 points. It also has the largest velocity and acceleration at any time. Moreover, the true heights, velocities and accelerations of the highest point between 04:56 UT and 09:36 UT are presented (Figure 7(c)). The height of the filament increased by 0.72 $R_{sun}$ over four hours. There is some evidence for a slight, slow rise before the fast eruption. Before 07:06 UT, the filament lifted up gradually; the velocity and acceleration were less than 13.8 km s$^{-1}$ and 3.0 m s$^{-2}$, respectively. Then it ascended more rapidly and reached a velocity of 144.8 km s$^{-1}$ at 09:26 UT. Arrows in Figure 7(c) show the time of the C3.2 flare start (07:24 UT) and peak (08:56 UT). About 30 minutes before the flare onset, the acceleration of the highest point began to increase and reached a value of 9.2 m s$^{-2}$ at 07:36 UT. Then it decreased to 5.2 m s$^{-2}$ at 07:56 UT and increased to 41.2 m s$^{-2}$ at 09:16 UT afterwards. When the velocity of the filament continuously increased to about 90 km s$^{-1}$ at 09:06 UT, the associated CME was clearly observed in STEREO A/COR1 images. At this time, the velocity of the CME was 163 km s$^{-1}$ (Figure 7(d)), which is more than the velocity of the filament by a factor as large as 1.8. Then the velocity of the second CME increased to 762 km s$^{-1}$ at 13:18 UT in the FOV of COR2 (Figure 7(d)).
We also investigate the variations of the longitudes and latitudes for the reconstructed features during the fast-eruptive phase (between 07:06 and 09:26 UT). As seen in Figure 8, the longitudes of points “2" to “4" at the eastern leg show a decrease of $\sim$ 7$\degr$, while points “6" to “9" at the western leg show an increase in longitude of about 12$\degr$. The latitudes of all the points except “9" are seen to decrease by $\sim$ 5$\degr$. The measurement is repeated four times, and the corresponding errors in longitude and latitude are about 0$\degr$.6 and 0$\degr$.3, respectively. Similar to the analysis of Joshi & Srivastava (2011), our analysis indicates that the filament experiences a twist in a counterclockwise direction.
In the process of the filament eruption, the EUV enhancements at the endpoints are observed clearly (top panel in Figure 9). At about 07:20 UT, the initial brightenings occurred at the eastern endpoints. At 08:20 UT, the brightenings at the western endpoints were observed. Comparison with the magnetograms shows that the western (eastern) ends of the filament are rooted in negative- (positive-) polarity fields (Figure 9). The magnetic polarity of the ends determines the direction of the axial field component of the filament. According to the empirical rule of Martin (1998), if the barbs point forward and to the right (left), then the axial field points to the right (left) when viewed from the positive-polarity side of the polarity inversion line, and the filament has “left-handed" (“right-handed") chirality. Therefore, the filament is left-handed.
Second eruption of the filament
-------------------------------
At the late phase of the first eruption ($\sim$ 09:30 UT), the eastern leg of the erupting filament obviously started to expand. Then partial material from the filament separated from the eastern leg (see rectangle “1" in Figure 10(b)). The separated material gradually diffused towards the west and the orientation of the separated material became mainly northeast-southwest at 10:16 UT from the longitudinal direction at 09:51 UT (Figure 10(b)). This material displayed a projectile motion and moved along the filament channel in AIA images (Figure 10(a)). The material became invisible at 11:25 UT when it moved to the west limb.
In order to follow the motion of the material in detail, we select the slice (line A-B in Figure 10) along which the material moves from the running ratio 304 [Å]{} images. As seen in the running ratio stack plot (Figure 10(c)), the material moves at a nearly constant speed. The velocities of two clear structures are about 145.2 km s$^{-1}$ and 138.3 km s$^{-1}$. In the first eruption, the main body of the filament lifted up and was accelerated (Figure 7(c)). In the second eruption, partial material of the eastern leg displayed a projectile motion toward the west and the velocity remained constant.
Conclusions and discussion
==========================
On 2010 August 1, a C3.2 flare, a polar crown filament eruption and two Earth-directed CMEs were coupled together, indicating the global character of solar activity (Wang et al. 2007; Zhukov & Veselovsky 2007; Zhou et al. 2007). Schrijver & Title (2011) analyzed a series of events occurring on 2010 August 1-2, and concluded that all substantial coronal activities were initiated from a connected network of large-scale separators, separatrices, and quasi-separatrix layers. Also they found that the magnetic field lines emanating from the flare region slid over a quasi-separatrix layer connecting the polar crown filament. For the first time, we reconstruct the polar crown filament using observations from three different viewpoints (STEREO A, B and SDO). The initial length of the filament was 1.07 $R_{sun}$, and the highest part was approximately at 1.15 $R_{sun}$. Three hours after the initial ascent, it rose to a height of 1.47 $R_{sun}$ and the length increased to 1.75 $R_{sun}$, about 1.6 times as long as the initial length. Two western endpoints and a partial spine of the filament compose a reverse$-$Y structure. During the eruption, the barbs are elongated and finally cut off from the solar surface in sequence. AIA observations show that the slow-rise phase of the filament lasts about two hours, and the velocity and acceleration at the highest location are less than 13.8 km s$^{-1}$ and 3.0 m s$^{-2}$, respectively. Then it ascends more rapidly and reaches a velocity of 144.8 km s$^{-1}$ at 09:26 UT and an acceleration of 41.2 m s$^{-2}$ at 09:16 UT. The highest location always has the largest acceleration, implying that the location bears the largest force during the eruption process. This result is different from a former study (Li et al. 2010) that the location bearing the largest force varies during a filament eruption process.
The reconstructed filament is processed with a visualization method and we can view it from any visual angle. The reconstructed western part of the filament gained with STEREO A and SDO data seen from the third viewpoint (STEREO B) is similar to the observations. This indicates that our reconstruction results are reliable to some extent. Seen from the north pole, the velocity of the filament has a large Earth$-$directed component. This is consistent with the Earth$-$directed CME accompanying the eruptive filament. The filament moves towards the low-latitude region, with an inclination change of $\sim$ 48$\degr$ with respect to the radial direction in the FOV of AIA. The filament expands only in altitude and in the latitudinal direction, similar to the conclusion made by Bemporad (2009), who found that the radial and latitudinal expansions of the filament are much larger than the longitudinal ones and suggested that the filament can be approximated as a 2D “ribbon-like" feature, instead of a 3D twisted flux tube.
In addition, at the late phase of the first eruption, partial material separates from the eastern leg. This material displays a projectile motion, and moves towards the west at a constant velocity of 141.8 km s$^{-1}$. To our knowledge, this phenomenon has never been reported before. The velocity of the second eruption remains approximately constant, while the velocity of the first eruption increases with time, which indicates that their eruption mechanisms are different. Hirose et al. (2001) concluded that the filament with helical fields accelerate upward in a round-shaped expanding field structure forming a CME containing the eruptive filament. The first eruption corresponds to the rise of helical magnetic fields of the filament (Jing et al. 2004; Sterling et al. 2007; DeVore & Antiochos 2008), and the second eruption is the motion of filament material along the magnetic system due to unequal magnetic pressure. The observations may imply that the polar crown filament consists of at least two groups of magnetic systems.
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![Extrapolated coronal magnetic field lines seen from the SDO (top left) and STEREO A (top right) perspectives at 00:03 UT on 2010 August 1 and composite images (middle and bottom panels) of STEREO-A COR1 and EUVI 304 [Å]{}. The backgrounds in the top panels are the MDI line-of-sight magnetogram at 00:03 UT and the STEREO A 195 [Å]{} image at 00:05 UT, respectively. Characters “a" and “b" denote the large-scale and small magnetic structures. Red and green curves represent high and low magnetic lines. Arrows “I" indicate the filament material. The plus signs denote the flare position.\[fig\]](fig1.eps)
![Reconstructed filament seen from the SDO viewpoint. Left: a series of SDO/AIA 304 [Å]{} observed images showing the evolution of the filament. Right: a series of reconstructed images seen from the SDO viewpoint showing the eruption process of the filament. The black circles denote the western endpoints of the filament.\[fig\]](fig2.eps)
![Comparison of the observed filament (left panels) and the reconstructed filament (right panels) seen from the STEREO A view. The black circles denote the western endpoints of the filament.\[fig\]](fig3.eps)
![Comparison of the observed filament (left panels) and the reconstructed filament (right panels) seen from the STEREO B view. The white circles denote the barbs of the filament.\[fig\]](fig4.eps)
![Evolution of the reconstructed filament seen from the top view (left panels), the side view (middle panels) and the north polar view (right panels). In the heliographic coordinate system, the top view of the filament has a longitude of 12$\degr$ and a latitude of 39$\degr$, and the side view has a longitude of 102$\degr$ and a latitude of 0$\degr$. \[fig\]](fig5.eps)
![Locations of 11 points selected in SDO and STEREO images. The circles denote the 11 points along the filament. \[fig\]](fig6.eps)
![(a): True height-location plots of the 11 points (points “1" to “11" in Figure 6) at three different times. (b): Velocity-location and acceleration-location plots of the 11 points. (c): True height-time, velocity-time and acceleration-time plots of the highest point between 04:56 UT and 09:36 UT. (d): Projected height-time and velocity-time plots of the associated CME measured in STEREO A/COR1 and COR2 images. \[fig\]](fig7.eps)
![Longitude-time and latitude-time plots of different points of the filament. Points “2" to “4" are from the eastern leg, and points “6" to “9" are from the western leg (Figure 6). \[fig\]](fig8.eps)
![AIA 171 [Å]{} image (upper panel) and HMI line-of-sight magnetogram (lower panel) at the filament location. The white contours are the filament at 05:06 UT. The red circles denote the multiple endpoints of the filament. The red rectangle is the 171 [Å]{} image at 07:40 UT. \[fig\]](fig9.eps)
![Appearance of the second eruption of filament material seen in AIA 304 [Å]{} (panel (a)) and running-difference images (panel (b)) and the running ratio stack plot obtained along slice A-B (panel (c)). The black contour denotes the filament material along line A-B.\[fig\]](fig10.eps)
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abstract: 'We have shown previously that a merger of marginally outer trapped surfaces (MOTSs) occurs in a binary black hole merger and that there is a continuous sequence of MOTSs which connects the initial two black holes to the final one. In this paper, we confirm this scenario numerically and we detail further improvements in the numerical methods for locating MOTSs. With these improvements, we confirm the merger scenario and demonstrate the existence of self-intersecting MOTSs formed in the immediate aftermath of the merger. These results will allow us to track physical quantities across the non-linear merger process and to potentially infer properties of the merger from gravitational wave observations.'
author:
- 'Daniel Pook-Kolb'
- Ofek Birnholtz
- Badri Krishnan
- Erik Schnetter
bibliography:
- 'mots\_merger.bib'
date: '2019-06-28'
title: 'Self-intersecting marginally outer trapped surfaces'
---
Introduction {#sec:intro}
============
Numerous binary black hole merger events have now been observed by gravitational wave detectors [@Abbott:2016blz; @TheLIGOScientific:2016pea; @LIGOScientific:2018mvr; @Green:2017voq; @Zackay:2019tzo; @Nitz:2018imz; @Venumadhav:2019lyq]. The general features of the gravitational wave signal from such events are now well known. The first is the inspiral regime where the signal is a chirp of increasing amplitude and frequency, and the system is effectively modeled as two point particles orbiting around each other and emitting gravitational waves as the orbit decays. As the two black holes approach each other and coalesce to form a final common black hole, the inspiral description is no longer valid, and non-perturbative aspects of general relativity become important; this is the merger regime. Eventually, as the final black hole reaches equilibrium, the gravitational wave signal can be well modeled as a superposition of damped sinusoids (and, in principle, much weaker power-law tails). Corresponding to this behavior of the gravitational wave signals, one visualizes the black holes themselves separately in the three different regimes. The inspiral regime consists of two disjoint black hole horizons slightly distorted by each other’s gravitational field. The merger is visualized as two horizons very close to each other and merging to form a single horizon which is initially very distorted. Finally, the ringdown is modeled as a perturbed Kerr horizon settling down to a final equilibrium Kerr black hole.
These features of the waveform must be correlated in some way with properties of the gravitational field in the strong field region. In particular, the three regimes must correspond in some way to properties of the black hole horizons. The details of the correlations between the different portions of the gravitational wave signal and the behavior of the horizons, and the precise demarcations between the three regimes are yet to be fully quantified. A full understanding of these correlations is obviously necessary to have a complete picture of a binary black hole merger (see e.g. [@Gupta:2018znn; @Jaramillo:2011rf; @Jaramillo:2012rr; @Kamaretsos:2012bs; @Kamaretsos:2011um; @Bhagwat:2017tkm]). It is also of interest to understand further quantitative features of the merger, such as the evolution of physical quantities across the merger. This includes, among other things, the fluxes of energy and angular momentum, and the evolution of higher order multipoles during the merger. These might be correlated with interesting features of the radiative multipoles found in [@Borhanian:2019kxt]. Numerical simulations are capable of solving the Einstein equations with high accuracy for binary black hole mergers (see e.g. [@Pretorius:2005gq; @Campanelli:2005dd; @Baker:2005vv; @Szilagyi:2009qz]). Such simulations provide an obvious avenue for exploring such questions.
To understand the correlations between the gravitational wave signal and the black hole horizons, we need to first decide precisely what geometrical quantities on the horizon should be considered. In fact, we need to go a further step backwards and decide what kind of horizons should be considered. There are two different ways of visualizing horizons using either event horizons or marginally trapped surfaces. Both of these descriptions are in good agreement in the inspiral and ringdown regimes, but differ substantially during the merger where non-linear and non-perturbative effects of general relativity are especially important. Consider first the event horizon description. An event horizon is the boundary of the region which is causally disconnected from an asymptotically far away observer. It is clear that locating an event horizon requires knowledge of the global properties of the spacetime infinitely far into the future. It is possible, though not trivial, to locate event horizons in numerical binary black hole simulations [@Hughes:1994ea; @Diener:2003jc; @PhysRevLett.74.630; @Thornburg:2006zb], and this yields the well known “pair of pants” picture [@Matzner:1995ib]. The cross-sections of the “pair of pants” corresponds with the expectations described above. At early times, the cross-section of the event horizon consist of two disjoint surfaces corresponding to the two separate black holes, and a single spherical surface at the end. There are several interesting features of the event horizon in the merger, including the existence of a toroidal phase early in the merger and the non-differentiability of the event horizon [@PhysRevD.60.084019]; the non-differentiability is in fact a general feature of event horizons [@Chrusciel:2000gj; @Chrusciel:1996tw].
The “pair of pants” picture is intuitively appealing and moreover it seems to provide a complete picture of the black hole merger in accordance with our physical expectations. In reality however, this picture is not so useful, both as a matter of principle and therefore also for any detailed quantitative studies. The problems can be traced back to the global and teleological nature of event horizons: to locate them, one needs to know what happens in the spacetime far in the future. In perturbative situations and when the end-state is known or assumed, it is indeed possible to obtain expressions for the fluxes of energy and angular momentum through the event horizon [@Hawking:1972hy]. In general dynamical situations however, this is not true. There are simple examples, even in spherical symmetry, when the area of the event horizon grows without any corresponding flux of energy [@Ashtekar:2004cn]. Due to these teleological properties, there is no possible local expression of general validity for, say, the fluxes of energy and angular momentum through event horizons. It is thus not clear how to carry out the program of understanding the merger and relating it to gravitational wave observations outlined at the beginning of the previous paragraph. As a side remark, the teleological property also makes it difficult to locate event horizons in numerical simulations in real time, but in any case, it is certainly possible to locate them once the simulations are complete.
There is an alternate way of visualizing a binary black hole merger which, for both conceptual and practical reasons, is of much greater importance in numerical simulations. The starting point is an unusual property of certain surfaces in the black hole region, first pointed out by Penrose [@Penrose:1964wq]. This requires the notion of the expansion $\Theta$ of a congruence of light rays; $\Theta$ is the logarithmic rate of change of an infinitesimal cross-section transverse to the null geodesics. A round sphere in flat space has $\Theta>0$ for the outgoing light rays and $\Theta< 0$ for the ingoing ones. In the black hole region, there exist spheres (the trapped surfaces) for which both sets of light rays have negative expansion. The outermost such sphere at any given time has vanishing outgoing expansion; these are the marginally trapped surfaces. In stationary situations such as for a Schwarzschild or Kerr black hole, cross-sections of the event horizon are also marginally trapped surfaces, but this correspondence is not true in non-stationary situations. Thus, cross-sections of the event horizon are marginally trapped surfaces very early in the inspiral regime or at very late times. At intermediate times, especially near the merger, the two notions are very different. Furthermore, unlike event horizons, marginal surfaces are not teleological and can be located at any given time without reference to any future properties of spacetime. It is possible to define physical quantities such as mass, angular momentum, multipole moments, and fluxes of energy and angular momentum quasi-locally, i.e. on the marginal surfaces. For this reason, marginal surfaces are widely used in numerical simulations when referring to the properties of black holes. There is a large literature on these quasi-local definitions and their applications to various problems in classical and quantum black hole physics (see [@Ashtekar:2004cn; @Booth:2005qc; @Faraoni:2015pmn; @Krishnan:2013saa] for reviews).
Despite this progress, there is still a missing ingredient, namely a unified treatment of inspiral, merger and ringdown. Thus far, all studies of binary black hole coalescence using marginal surfaces have considered the pre- and post-merger regimes separately. The reason for this is that, until recently, it was not known how marginal surfaces behave across the merger; near the merger the marginal surfaces are extremely distorted and previous numerical methods were not successful in tracking such highly distorted surfaces. Using improved numerical methods [@Pook-Kolb:2018igu], we have recently shown the first evidence for the existence of a continuous sequence of marginal surfaces which interpolates between the two disjoint initial black holes and the single final remnant black hole [@Pook-Kolb:2019iao]. This is the analog of the “pair of pants” picture for event horizons. In the present work, with further improvements in numerical methods for locating marginal surfaces, we shall provide further unambiguous evidence for this scenario. We shall also show the existence of marginal surfaces with self-intersections. In a companion paper we shall study physical characteristics of the world-tube of marginal surfaces, which is the other important ingredient for physical applications.
The scenario we obtain for the merger is summarized in Fig. \[fig:merger1\]. The details showing how these results are obtained will be explained in the next sections. The figure shows four snapshots of the MOTSs at various times[^1] in a head-on binary black hole merger starting with Brill-Lindquist initial data. We initially have only the two individual MOTSs without a common horizon. As the black holes get closer, a common MOTS is formed which immediately bifurcates into outer and inner portions visible in the second snapshot. The outer portion loses its distortions as it approaches its equilibrium state, while the inner MOTS becomes increasingly distorted. At some point, just shortly after the third snapshot, the two individual MOTSs touch each other exactly at the time when they merge with the inner common MOTS. After this merger, the two individual MOTSs go through each other. Surprisingly, it turns out that the inner common MOTS continues to exist after the merger and now has self-intersections as shown in the last snapshot. The remainder of this paper will be devoted to explaining how we arrive at this result. A detailed study of the physical aspects of this scenario will be presented elsewhere.
{width="80.00000%"}
Sec. \[sec:motsdefn\] summarizes the basic definitions and concepts that we shall need for this paper. The improved numerical algorithm for locating marginal surfaces is described in Sec. \[sec:motsfinder\] and Sec. \[sec:validation\] shows various numerical tests to validate the method. Sec. \[sec:numerics\] discusses our modifications to the numerical methods used to evolve Cauchy data using the Einstein equations. These modifications allow us to reach the required numerical accuracy and convergence, and to carry out our simulations more efficiently. Sec. \[sec:selfintersect\] puts together all these ingredients and presents our main results. For a particular initial configuration (the head on collision of comparable mass non-spinning black holes), the merger of marginally trapped surfaces is demonstrated with high numerical accuracy. The merger involves the formation of a marginally trapped surface with self-intersections, showing topology change in a binary black hole merger.
Marginally outer trapped surfaces {#sec:motsdefn}
=================================
Let $\ell^a$ be a congruence of future directed null geodesics, and let $n^a$ be another such congruence satisfying $\ell^a n_a = -1$. Let $q_{ab}$ be the Riemannian metric in the 2-dimensional space transverse to both $\ell^a$ and $n^a$. The divergence of $\ell^a$ and $n^a$ are respectively $$\Theta_{(\ell)} = q^{ab}\nabla_a\ell_b\,,\quad \Theta_{(n)} = q^{ab}\nabla_an_b\,.$$ Let ${\ensuremath{\mathcal{S}}}$ be a closed spacelike 2-surface with null normal fields $\ell^a$ and $n^a$ respectively. We assume that it is possible to assign outgoing and ingoing directions on ${\ensuremath{\mathcal{S}}}$, and by convention, $\ell^a$ and $n^a$ are the outgoing and ingoing null normals respectively. The classification of ${\ensuremath{\mathcal{S}}}$ based on conditions on the expansions are the following:
- Trapped: $\Theta_{(n)}< 0$, $\Theta_{(\ell)} < 0$
- Un-trapped: $\Theta_{(n)}< 0$, $\Theta_{(\ell)} > 0$
- Marginally trapped: $\Theta_{(n)}< 0$, $\Theta_{(\ell)} = 0$
- Marginally outer-trapped: $\Theta_{(\ell)} = 0$ (no condition on $\Theta_{(n)}$)
All of these refer to future-directed $\ell^a$. Thus we should say future-trapped rather than just trapped, but we shall only consider future directed cases. The most important case for us is the marginally outer trapped surface (MOTS) lying within a spatial slice $\Sigma$.
As mentioned in the introduction, there is a large literature on the application of MOTSs to study black holes in various contexts (see e.g. [@Ashtekar:2004cn; @Booth:2005qc; @Gourgoulhon:2005ng; @Hayward:2004fz; @Jaramillo:2011zw; @Krishnan:2013saa; @Krishnan:2007va]). They are regularly used in numerical relativity simulations to compute physical quantities [@Dreyer:2002mx; @Schnetter:2006yt; @Gupta:2018znn], and this formalism leads naturally to various versions of quasi-local black hole horizons.
While we shall not delve into the mathematical and physical characteristics of MOTSs here, it shall be useful to understand the stability operator for a MOTS and its relevance for time evolution. For a given MOTS ${\ensuremath{\mathcal{S}}}$ consider a smooth one-parameter family of closed spherical surfaces ${\ensuremath{\mathcal{S}}}_\lambda$ which are *variations* of ${\ensuremath{\mathcal{S}}}$ in the normal direction [@Newman1987] within the spatial hypersurface $\Sigma$.
On each ${\ensuremath{\mathcal{S}}}_\lambda$, just as for ${\ensuremath{\mathcal{S}}}$, we can define the null normals and calculate the expansion $\Theta_{(\ell)}(\lambda)$, which will of course generally not vanish. The differentiation of $\Theta_{(\ell)}(\lambda)$ leads to an operator $L$ on ${\ensuremath{\mathcal{S}}}$: $$\delta_{fr}\Theta_{(\ell)} =: Lf\,.$$ Here $r^a$ refers to the unit outward pointing spacelike normal to ${\ensuremath{\mathcal{S}}}$ (within $\Sigma$) and $f$ is a scalar function on ${\ensuremath{\mathcal{S}}}$. Along the 1-parameter family ${\ensuremath{\mathcal{S}}}_\lambda$, every point on ${\ensuremath{\mathcal{S}}}$ traces out a curve with tangent vector $fr^a$. The variation of the expansion, i.e. the left hand side of the above equation, is the derivative of the expansion along these curves. This procedure defines an elliptic operator $L$ on a MOTS and the precise expression for $L$ can be worked out. Generically it is of the form $$\label{eq:stability_operator}
Lf = -\Delta f + \gamma^a \partial_a f + \beta f\,,$$ Here $\Delta$ is the Laplace-Beltrami operator on ${\ensuremath{\mathcal{S}}}$ compatible with $q_{ab}$, $\gamma^a$ is a vector field on $\mathcal{S}$ related to black hole spin, and $\beta$ is a scalar related to the intrinsic (two-dimensional) Ricci scalar of ${\ensuremath{\mathcal{S}}}$. Thus, $L$ is not necessarily a self-adjoint operator due to the presence of $\gamma^a$, and its eigenvalues are not necessarily real. Nevertheless, its principal eigenvalue $\Lambda_0$, i.e. the eigenvalue with the smallest real part is indeed real. In this paper we shall restrict ourselves to non-spinning black holes with vanishing $\gamma^a$ so that all eigenvalues are real.
The primary utility of $L$ is that it determines the behavior of ${\ensuremath{\mathcal{S}}}$ under time evolution. It was shown that if the principal eigenvalue is positive, then the MOTS evolves smoothly in time [@Andersson:2005gq; @Andersson:2007fh; @Andersson:2008up]. This stability condition is equivalent to saying that an outward deformation of $\mathcal{S}$ makes it untrapped which is what we expect to happen for the apparent horizon. While not emphasized in [@Andersson:2005gq; @Andersson:2007fh; @Andersson:2008up], the condition for the existence of ${\ensuremath{\mathcal{S}}}$ under time evolution is the invertibility of $L$. Thus, if 0 is not in the spectrum of $L$, then ${\ensuremath{\mathcal{S}}}$ continues to evolve smoothly. In the case when $\Lambda_0<0$ (which will happen in our case), we must consider the next eigenvalue $\Lambda_1$ and check that it does not vanish. See e.g. [@Booth:2017fob; @Sherif:2018scu; @Mach:2017peu] as examples of studies which consider this notion of stability in specific examples.
Numerical methods for locating highly distorted MOTSs {#sec:motsfinder}
=====================================================
Consider a Cauchy surface $\Sigma$ on which we wish to locate a MOTS ${\ensuremath{\mathcal{S}}}$. Let $\Sigma$ be equipped with a Riemannian metric $h_{ij}$ with the associated Levi-Civita connection $D_a$, and let the extrinsic curvature of $\Sigma$ be $K_{ij}$. Let $r^a$ be the unit-spacelike normal to ${\ensuremath{\mathcal{S}}}$ within $\Sigma$ and let $\tau^a$ be the unit-timelike normal to $\Sigma$. Then, a suitable choice of null-normals to ${\ensuremath{\mathcal{S}}}$ is $$\ell^a = \frac{1}{\sqrt{2}} \left(\tau^a+ r^a\right)\,,\quad n^a = \frac{1}{\sqrt{2}}\left(\tau^a - r^a\right)\,.$$ The condition $\Theta_{(\ell)}=0$ is rewritten as $$\label{eq:motsequation}
D_ar^a + K_{ab}r^ar^b - K = 0\,.$$ This is the equation that we must solve to find ${\ensuremath{\mathcal{S}}}$. The conventional approach [@Thornburg:2006zb; @Thornburg:2003sf] assumes that the surface is defined by a level-set function $$\label{eq:starshaped}
F(r,\theta,\phi) = r - h(\theta,\phi)\,,$$ where $(r,\theta,\phi)$ are spherical coordinates on $\Sigma$. This assumes that ${\ensuremath{\mathcal{S}}}$ is *star-shaped* with respect to the origin in the chosen coordinate system. In other words, any ray drawn from the origin must intersect the surface only once. This assumption will not hold for the surfaces of interest for us. A variant of this method was proposed in [@Pook-Kolb:2018igu] and shown to be capable of locating extremely distorted surfaces. This new method is based on using a *reference* surface $\sigma_R$, and representing ${\ensuremath{\mathcal{S}}}$ in terms of distances $h(\lambda, \mu)$ from $\sigma_R$, where $\lambda, \mu$ parameterize $\sigma_R$. As long as the reference surface is chosen appropriately, the method can be used to locate almost arbitrarily distorted surfaces. For example, in a numerical evolution, one could choose $\sigma_R$ to be the MOTS located in the previous time step. The problem of locating ${\ensuremath{\mathcal{S}}}$ then translates to solving a nonlinear partial differential equation for the horizon function $h$. This can be done e.g. via a pseudospectral method, which is what we chose.
For our present application, we have implemented two additional features compared to what was used in [@Pook-Kolb:2018igu]. These features are meant to deal with two additional complications that we must necessarily deal with: i) surfaces which have a very narrow “neck” (almost like a figure-eight), and in some instances have features like cusps and self intersections. For this purpose, motivated by the methods used in [@Jaramillo:2009zz], we employ bi-spherical coordinates [@Ansorg:2005bp]. ii) Unlike in [@Pook-Kolb:2018igu] where the MOTS finder was applied to analytical initial data, we now have to deal with numerically generated data on a finite mesh. This requires the use of interpolation schemes some of which were already used in [@Pook-Kolb:2019iao]. We now describe in turn both of these additional features. We shall still be restricted to axisymmetry in this work, reducing the task of finding the horizon function $h$ to a one-dimensional problem. However, no in-principle difficulties are foreseen for general non-axisymmetric cases.
Bi-spherical coordinates {#subsec:bipolar}
------------------------
![ Visualizations of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in bipolar coordinates at different simulation times $T$. The left column shows the MOTS and lines of constant $s$ and $t$ in the $(x,z)$ plane while the right column contains ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in the $(t,s)$ plane. Note that only positive values of $s$ are shown, though the full MOTS is of course symmetric about $s=0$. The first row shows a slightly distorted MOTS in both representations. At $T=5.5{\ensuremath{\,\mathcal{M}}}$ (second row), ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ is highly distorted in the $(x,z)$ plane and only slightly distorted in the bi-spherical coordinates. The last row shows a case of a self-intersecting ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$. The dot marks the location of the “neck” in all cases. []{data-label="fig:bipolar1"}](figs/bipolar-t3_0 "fig:"){width="48.00000%"} ![ Visualizations of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in bipolar coordinates at different simulation times $T$. The left column shows the MOTS and lines of constant $s$ and $t$ in the $(x,z)$ plane while the right column contains ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in the $(t,s)$ plane. Note that only positive values of $s$ are shown, though the full MOTS is of course symmetric about $s=0$. The first row shows a slightly distorted MOTS in both representations. At $T=5.5{\ensuremath{\,\mathcal{M}}}$ (second row), ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ is highly distorted in the $(x,z)$ plane and only slightly distorted in the bi-spherical coordinates. The last row shows a case of a self-intersecting ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$. The dot marks the location of the “neck” in all cases. []{data-label="fig:bipolar1"}](figs/bipolar-t5_5 "fig:"){width="48.00000%"} ![ Visualizations of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in bipolar coordinates at different simulation times $T$. The left column shows the MOTS and lines of constant $s$ and $t$ in the $(x,z)$ plane while the right column contains ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in the $(t,s)$ plane. Note that only positive values of $s$ are shown, though the full MOTS is of course symmetric about $s=0$. The first row shows a slightly distorted MOTS in both representations. At $T=5.5{\ensuremath{\,\mathcal{M}}}$ (second row), ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ is highly distorted in the $(x,z)$ plane and only slightly distorted in the bi-spherical coordinates. The last row shows a case of a self-intersecting ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$. The dot marks the location of the “neck” in all cases. []{data-label="fig:bipolar1"}](figs/bipolar-t6_5 "fig:"){width="48.00000%"}
For axisymmetric surfaces, choosing the symmetry axis to be the $z$ axis, we can restrict ourselves to the $(x,z)$ plane and it is often convenient to characterize any point using polar coordinates, i.e. using the distance from the origin and the angle of the position vector with the $z$ axis. However these coordinates are not optimal for describing surfaces with a very narrow neck connecting two spherical portions, i.e. close to a figure-eight in shape. We use instead the bipolar coordinates $(s,t)$ which are based on two foci located at $x=0$, $z=c \pm a$: $$\label{eq:bipolar}
x = \frac{a\sin s}{\cosh t - \cos s}\,,\quad
z = \frac{a\sinh t}{\cosh t - \cos s} + c\,.$$ The $(s,t)$ coordinates make the highly distorted inner common MOTS ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ much easier to parameterize.
Examples demonstrating the effect of this coordinate transformation for three different simulation times are shown in Fig. \[fig:bipolar1\]. The three snapshots are at times i) $T=3{\ensuremath{\,\mathcal{M}}}$ which is a bit after the top right panel of Fig. \[fig:merger1\] and ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ does not have extreme distortions; ii) $T=5.5{\ensuremath{\,\mathcal{M}}}$, shortly before the bottom left panel in Fig. \[fig:merger1\] where ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ has a very narrow neck, and finally iii) $T=6.5{\ensuremath{\,\mathcal{M}}}$, a little bit before the bottom right panel of Fig. \[fig:merger1\], and ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ has self-intersections.
The bi-spherical coordinates are employed only for ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$; none of the other horizons have the narrow neck and these coordinates are unnecessary to locate them. To determine the value of $c$ in , we first find the two individual MOTSs ${{\ensuremath{\mathcal{S}}}_{1}}$ and ${{\ensuremath{\mathcal{S}}}_{2}}$ and choose $c$ to lie in the coordinate center between the lowest point of ${{\ensuremath{\mathcal{S}}}_{1}}$ and the upper-most point of ${{\ensuremath{\mathcal{S}}}_{2}}$. As detailed below, we find the various MOTSs in a series of time slices produced by the numerical simulation. During this [*tracking*]{} of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$, we numerically approximate the optimal value for $a$ as a post-processing step once the MOTS is located. In practice, this is done by representing ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in bi-spherical coordinates and expressing the coordinate functions $s(\lambda), t(\lambda)$ as a truncated series of sines and cosines, respectively, which have the correct symmetry for the problem. We use a slightly lower number of basis functions than necessary to obtain convergence and check the residual expansion of the now imperfect representation. Varying the parameter $a$, we repeat this process to find the value resulting in the lowest residual. The value for $a$ determined this way is then used for finding the MOTS in the next slice, assuming the optimal parameter varies slowly with simulation time.
A further optimization is to re-parameterize the reference surface $\sigma_R$ prior to finding the MOTS. A natural choice of parameterization would be the proper length or proper length in coordinate space, the latter obviously being better suited for our numerical representation of the surface. If the curve representing $\sigma_R$ in coordinate space is $\lambda \mapsto \gamma_R(\lambda)$, this would mean that $\Vert\gamma_R'(\lambda)\Vert_2 \equiv \text{const}$. However, we obtained faster convergence by taking a non-constant speed function such that $\Vert\gamma_R'(\lambda)\Vert_2$ is roughly[^2] proportional to $1/k_{AB}k^{AB}$, where $k_{AB}$ is the second fundamental form of $\sigma_R$ embedded in coordinate space.
Utilization of bi-spherical coordinates together with the above re-parameterization has led to convergent solutions ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ with about one order of magnitude fewer collocation points compared to the previous method.
Interpolating numerical data {#subsec:interpolation}
----------------------------
In each time step, our axisymmetric numerical simulations produce data on a 2-dimensional grid of points lying equidistant in the $(x,z)$ coordinate plane. However, for the nonlinear search for a MOTS ${\ensuremath{\mathcal{S}}}$, the expansion $\Theta_{(\ell)}$ and its derivatives have to be computed on a set of points $x_n\in\mathbb{R}^2$ along trial surfaces ${\ensuremath{\mathcal{S}}}^{i}$, c.f. [@Pook-Kolb:2018igu], Section III.B. This requires evaluating the components of the metric $h_{ij}$, its first and second spatial derivatives, the extrinsic curvature $K_{ij}$ and its first spatial derivatives at the points $x_n$ which generally do not coincide with any of the grid points of the simulation.
In [@Pook-Kolb:2019iao] we used $4$th order accurate $5$-point Lagrange interpolation. Derivatives were obtained by evaluating $4$th order accurate finite differencing derivatives using the data on the grid and then interpolating the results using $5$-point Lagrange interpolation. For the present paper, however, we switched to quintic Hermite interpolation, which allows us to control the values along with first and second derivatives of the interpolant at the grid points. These derivatives are evaluated using $6$th order accurate finite differencing. Derivatives between the grid points are then computed by analytically differentiating the interpolating polynomial. The advantage is that now first and second derivatives are continuous throughout, which is not the case with Lagrange interpolation.
Interpolation of discrete data will be more accurate with increased grid resolution. However, it will never be exact and even floating point accuracy cannot be neglected, especially near the punctures at computationally feasible resolutions. These additional inaccuracies may limit the numerical convergence as they move the plateau we see below in Fig. \[fig:spectral\_convergence\] up—for example when moving closer to the punctures or reducing the grid resolution—or down. To account for this effect while tracking a MOTS through simulation time, we compute the expansion between the collocation points each time the expansion drops below a pre-set tolerance [*at*]{} the collocation points. After this, we increase the spectral resolution and continue until the tolerance is met at the now larger set of collocation points. This is repeated until the expansion between the collocation points no longer improves, signaling that we have reached the plateau.
A second criterion for stopping to increase the spectral resolution is derived from the absolute values of the coefficients $a_n$ of the spectral representation of the horizon function $h$. In a pseudospectral method using a basis of cosines, one expects these coefficients to fall off exponentially for large $n$ if the solution exists. We hence stop increasing the resolution if sub-exponential fall-off of the $a_n$ is found following a region of exponential convergence. This prevents our code from overfitting ${\ensuremath{\mathcal{S}}}$ to features introduced by the interpolation method, which happens especially for lower resolution simulations.
Validating the MOTS finder {#sec:validation}
==========================
With the addition of numerical simulations, the task for our MOTS finder has become more general compared to the purely time-symmetric cases considered in [@Pook-Kolb:2018igu]. Therefore, and in light of the surprising result of a self-intersecting MOTS, it is important to validate the method and test it for correctness in an analytic case where the result is known. We shall later present convergence results for further validation.
For this purpose we construct a non-time-symmetric slice with analytically known horizon shape. We choose a slice of the Schwarzschild spacetime in Kerr-Schild coordinates [@Matzner:1998pt], i.e. $$\begin{aligned}
\label{eq:schwarzschildKS}
h_{ij} &=& \delta_{ij} + \frac{2m}{r} \frac{x_i x_j}{r^2}\,,\\
\label{eq:schwarzschildKS_curv}
K_{ij} &=& \frac{2m}{r^4} \frac{1}{\sqrt{1+2m/r}}
\left[ r^2 \delta_{ij} - \left( 2 + \frac{m}{r} \right) x_i x_j \right]
\,,\end{aligned}$$ where $\delta_{ij}$ is the flat metric, $x_i$ are the standard Cartesian coordinates for the flat metric, and we shall often use $(x,y,z)$ instead of $x_i$ when no confusion can arise. For Schwarzschild, the radial coordinate is just $r^2= x^2+y^2+z^2$. These data have nontrivial extrinsic curvature with the horizon being located at $r=2m$.
To make the horizon non-star-shaped and thus the task more difficult (but still axisymmetric), we transform the coordinates $(x,z) \rightarrow (\bar{x},\bar{z})$ via $$\begin{aligned}
\label{eq:schwarzschildCoordTransform}
\bar x = x \left(1 - \frac{\beta}{\cosh((z-z_0)/\gamma)}\right)
\,,\quad
\bar z = z\,.\end{aligned}$$ These equations are used to sample $h_{ij}$ and $K_{ij}$ on grids of various resolutions from $1/h = 30$ to $1/h = 1920$. We choose a reference shape that is close but not identical to the horizon. The MOTS ${\ensuremath{\mathcal{S}}}$ and the reference shape $\sigma_R$ are shown in the first panel of Fig. \[fig:schwarzschildKW-plot\]. For this test we compute the area $A$ of ${\ensuremath{\mathcal{S}}}$ and compare it to the exact area $A_{\rm exact} = 16\pi m^2$, where $m=1$. We also compute the maximum coordinate distance $\Vert{\ensuremath{\mathcal{S}}}-{\ensuremath{\mathcal{S}}}_{\rm exact}\Vert_\infty$ of the numerical solutions to the exact horizon. The second panel demonstrates that our numerical solutions converge to the expected solutions as the resolution of the numerical grid is increased.
![ [*Top*]{}: Horizon ${\ensuremath{\mathcal{S}}}$ and reference shape $\sigma_R$ for the transformed slice of Schwarzschild spacetime. The parameters for the transformation via are $\beta = 0.97$, $\gamma = 0.7{\ensuremath{\,\mathcal{M}}}$ and $z_0 = 0.8{\ensuremath{\,\mathcal{M}}}$. [*Bottom*]{}: Convergence of the area (dashed) and surface coordinate shape (solid) with increased grid resolution. In each case, the spectral resolution was chosen such that a further increase does not result in a lower residual expansion (see section \[subsec:convergence\]). This thus shows the error introduced by the spatial discretization and interpolation. []{data-label="fig:schwarzschildKW-plot"}](figs/SchwarzschildTransformed-plot "fig:"){width="40.00000%"} ![ [*Top*]{}: Horizon ${\ensuremath{\mathcal{S}}}$ and reference shape $\sigma_R$ for the transformed slice of Schwarzschild spacetime. The parameters for the transformation via are $\beta = 0.97$, $\gamma = 0.7{\ensuremath{\,\mathcal{M}}}$ and $z_0 = 0.8{\ensuremath{\,\mathcal{M}}}$. [*Bottom*]{}: Convergence of the area (dashed) and surface coordinate shape (solid) with increased grid resolution. In each case, the spectral resolution was chosen such that a further increase does not result in a lower residual expansion (see section \[subsec:convergence\]). This thus shows the error introduced by the spatial discretization and interpolation. []{data-label="fig:schwarzschildKW-plot"}](figs/SchwarzschildTransformed-convergence "fig:"){width="45.00000%"}
The numerical evolutions {#sec:numerics}
========================
Formulations, Discretization, and Implementation
------------------------------------------------
We set up initial conditions for the spacetime geometry as two puncture black hole using the method of Brill and Lindquist [@PhysRev.131.471]. To evolve the geometry, we use the BSSN formulation of the Einstein equations with a $1+\log$ slicing and a $\Gamma$-driver shift condition [@Alcubierre:2000xu; @Alcubierre:2002kk]. We also impose axisymmetry throughout the calculation.
For our setup (see below), we choose a domain with $x \in [0;10]$, $z \in [-10;10]$, and $T \in [0;7]$. (Due to axisymmetry, we only consider the hyperplane $y = 0$.) For simplicity, we use Dirichlet boundary conditions to set all time derivatives to zero at the outer boundary. We check that the errors introduced by the artificial boundary conditions do not affect the geometry near the MOTSs.
We choose a Cartesian basis for the tangent space, i.e. we represent vectors and tensors via their $x, y, z$ components. Although axisymmetry requires that certain components or linear combinations of components must vanish, we do not explicitly impose such conditions. Instead, we only impose axisymmetry on spatial derivatives: We require that the Lie derivatives of all quantities in the $\phi$ direction be zero, and we use this to remove all $y$ derivatives. ($y$ derivatives are then either $0$, or are replaced by combinations of various $x$ derivatives.) We use l’Hôpital’s rule to regularize these expressions on the axis. This closely follows the approach described in [@Pretorius_2005], extended to handle second derivatives as well. The set of expressions for handling first and second $y$ derivatives for all tensor ranks appearing in the BSSN formulation is lengthy, and is available in a Mathematica script as part of Kranc [@Husa:2004ip; @Kranc:web].
In our discretization, we also require a small region “on the other side” of the axis (where $x<0$), which we calculate by rotating the region with $x>0$ by $\pi$.
We also experimented with the *Cartoon* method [@Alcubierre:1999ab] to impose axisymmetry. Cartoon uses a spatial rotation in the $\phi$ direction and then spatial interpolation to populate points away from the $y$ axis, so that $y$ derivatives can be calculated in the standard manner. We found that the Cartoon method does not work well with higher order (higher than $4$th) finite differencing: The result of a Lagrange interpolation is not continuous, which leads to large oscillations when derivatives are taken near the axis where the Cartoon rotation angle is large.
In our setup, the punctures are located on the $z$ axis and are initially at $z_\pm = \pm 0.65$. The puncture masses are $m_+ = 0.5$ and $m_- = 0.8$ (i.e. the “upper” black hole is smaller). The punctures have no linear or angular momentum.
Details of initial and gauge conditions are described in [@wardell_barry_2016_155394]. Our exact parameter settings are available in the parameter files in the repository [@pook_kolb_daniel_2019_3260171].
We use $6$th order finite differencing to discretize space. We also add a $6$th order Kreiss-Oliger artificial dissipation, which reduces our spatial accuracy to $5$th order. We use a $6$th order accurate Runge-Kutta time integrator. Our discretization is globally $5$th order accurate, as we demonstrate below in section \[sec:convergence\]. We do not use mesh refinement nor multiple grid patches as these would not be beneficial for our calculations that span only a short time and a small region of space, compared to systems of orbiting binary black holes.
Compared to $4$th and $8$th order discretizations, $6$th order is most efficient for us. $4$th order calculations require significantly higher resolutions, and $8$th order calculations are significantly slower since they use larger stencils and require more integrator substeps. $8th$ order calculations also require higher resolutions before their error falls below that of $6$th order calculations.
We perform our calculation via the *Einstein Toolkit* [@Loffler:2011ay; @EinsteinToolkit:web]. We use *TwoPunctures* [@Ansorg:2004ds] to set up initial conditions and an axisymmetric version of *McLachlan* [@Brown:2008sb] to solve the Einstein equations, which uses *Kranc* [@Husa:2004ip; @Kranc:web] to generate efficient C++ code.
Accuracy, Convergence {#sec:convergence}
---------------------
To demonstrate the accuracy of our discretization, we plot in Fig. \[fig:constr\] the Hamiltonian constraint $$\label{eq:ham}
\mathcal{H} = K_{ab} K^{ab} - K - R$$ on grid points close to the inner common MOTS at two different times for different grid resolutions. Here, $R$ is the Ricci scalar of the slice $\Sigma$. There is no significant difference between the two times. Note that in coordinate space, ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ lies closer to the punctures in its upper than in its lower half, compare also Fig. \[fig:merger1\]. In terms of the curve’s proper length parameter $\bar\lambda$ (scaled to $\bar\lambda\in[0,\pi]$), this corresponds to $\bar\lambda \lesssim \pi/2$ and $\bar\lambda \gtrsim \pi/2$, respectively, where our representation only covers half of the plotted MOTS (say for positive $x$ values) due to axisymmetry.
The results have been scaled to account for $5$th order convergence. We indeed find $5$th order convergence for $1/h \geq 240$ closer to the punctures and for $1/h \geq 120$ further away from the punctures. In that latter region, the highest resolution results with $1/h = 960$ show slightly larger errors than expected from $5$th order accuracy.
This is caused by round-off errors starting to dominate the finite difference derivatives, as is demonstrated in Fig. \[fig:differentiation\_roundoff\]. Here, the different curves represent the results obtained using stencils of $3$ to $9$ points for the derivatives of the metric components, corresponding to $2$nd to $8$th order accuracy. We see the typical behavior of convergence up to the resolution at which the round-off error becomes dominant. This happens at lower resolutions for the higher order methods as these reach the round-off limit earlier. Note that the optimal resolution depends on the function being approximated and in our case becomes larger the closer we get to the puncture. This explains the different behavior in the first and second half of the plots in Fig. \[fig:constr\].
![ Convergence of the Hamiltonian constraint for increasing resolutions $1/h = 60$, $120$, $240$, $480$, $960$ at one time step before (upper panel) and after (lower panel) the individual horizons touch. The constraint is computed at grid points close to ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ and plotted over the proper length (scaled to $[0,\pi]$) of the curve representing ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in the $(x,z)$ plane. []{data-label="fig:constr"}](figs/ham_inner_5_35_proper_mult "fig:"){width="48.00000%"} ![ Convergence of the Hamiltonian constraint for increasing resolutions $1/h = 60$, $120$, $240$, $480$, $960$ at one time step before (upper panel) and after (lower panel) the individual horizons touch. The constraint is computed at grid points close to ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ and plotted over the proper length (scaled to $[0,\pi]$) of the curve representing ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ in the $(x,z)$ plane. []{data-label="fig:constr"}](figs/ham_inner_5_73_proper_mult "fig:"){width="48.00000%"}
![ Hamiltonian constraint computed at one point of a slice of the Schwarzschild spacetime in Kerr-Schild coordinates as defined in , for grid resolutions $1/h = 20$ to $1/h = 10^5$. Since this is an exact solution of the Einstein equations, we expect $\mathcal{H} \equiv 0$, and this figure thus shows the discretization error. The constraint is evaluated at a coordinate distance of $r \approx 0.24\,m$ from the puncture. []{data-label="fig:differentiation_roundoff"}](figs/ham_over_res){width="48.00000%"}
The existence of self-intersecting MOTSs {#sec:selfintersect}
========================================
With the technical improvements at hand, we now turn to the main result of this paper, namely the merger of the inner MOTS with the two individual horizons, and the occurrence of self intersecting MOTSs just after this merger (see Fig. \[fig:merger1\]). We will study a single configuration with high resolution. We focus primarily on numerical accuracy and convergence to confirm the merger scenario and the existence of self-intersecting MOTSs. There are obviously numerous physical and geometrical properties of great interest. First however, we need to prove this scenario numerically beyond any reasonable doubt, which is what we shall do here. A detailed discussion of the interesting physical and geometrical properties of the world tube of MOTSs will be postponed to a forthcoming paper. Similarly, we shall not discuss here the various extensions to non-time symmetric and non-axisymmetric data. As mentioned previously, we start with Brill-Lindquist initial data with the bare masses $m_+ = 0.5$ and $m_- = 0.8$. The initial coordinate separation between the punctures is $1.3{\ensuremath{\,\mathcal{M}}}$ (i.e. $1$ in units of the total ADM mass $M_{\rm ADM} = m_+ + m_-$). Simulations are performed at various grid resolutions: $1/h = 60$, $120$, $240$, $480$, $960$. We have already shown in the previous section that the numerical solution of the Einstein equations for the given initial data is sufficiently accurate and all constraint violations converge at the expected rate when $h$ is varied. Given this numerical spacetime, we can use our horizon finder to locate the various MOTSs. It remains to be shown now that the surfaces thus found are indeed MOTSs.
Before proceeding further, it might be useful to clarify the nature of the MOTS with self-intersections shown in the bottom right panel of Fig. \[fig:merger1\]. Viewed as a submanifold of the 3-dimensional Riemannian spatial slice $\Sigma$, this manifold might appear to be non-differentiable at the point of self-intersection and one might be concerned that there is no well defined normal to the manifold at that point (and hence no well defined expansion either). This is however incorrect, and formally the curve is simply understood as an *immersion* instead of an embedding. In the present case, because of axisymmetry, we can restrict ourselves to a two-dimensional section (say the $x$-$z$ plane as we have been using so far). Then the horizon is simply a parameterized curve, i.e. a mapping of the circle $S^1$ into $\Sigma$, $f:S^1\rightarrow \Sigma$ (this is precisely how this curve is defined numerically). Using the map $f$, we can push forward tangent vectors to $\Sigma$ and thus we have well defined normals depending on which direction one traverses the point of self-intersection (see Fig. \[fig:knot\]). The relevant topological property of the curve is the winding number, i.e. the number of rotations that a tangent vector undergoes when we go all the way around the curve; each loop adds +1 to the winding number. Curves with different winding numbers cannot be smoothly deformed into each other [@CM_1937__4__276_0]. This is why in order to get the self-intersections, it is necessary to go through the cusp at the merger. In non-axisymmetric situations, we have to necessarily deal with mappings of $S^2$ into the 3-manifold $\Sigma$ (which are in fact simpler [@10.2307/1993205]), but we shall not discuss this here.
![ Tangent vectors at a regular crossing-point of a curve. As we traverse the curve following the arrows from the top-right, we push-forward tangent vectors in the usual way. Thus, the first time the self-intersection is crossed, the tangent vector is $V$. The second time, i.e. after traversing the loop in the clockwise direction, the tangent vector is $W$. Normal vectors are also well defined along the curve and uniquely specified once an outward direction is specified at any point. In our specific example, we say that at the north pole, the outward direction is the $+z$ direction. []{data-label="fig:knot"}](figs/normals_self_crossing.png){width="0.5\columnwidth"}
Convergence {#subsec:convergence}
-----------
Except for the modifications introduced earlier in Sec. \[sec:motsfinder\], we employ the same basic Newton-Kantorovich search as in [@Pook-Kolb:2018igu] with each step being performed using a pseudospectral method. If the nonlinear search converges, we expect the exponential convergence of the individual pseudospectral steps to carry over to the solution of the full nonlinear problem.
![ Convergence of the residual expansion of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ at one time before (upper panel) and after (lower panel) the individual horizons touch. Note that the inner common MOTS has self-intersections in the latter case. We plot the maximum absolute residual expansion between the collocation points over the pseudospectral resolution used to find the MOTS. This is independent of the grid resolution ${\rm res} = 1/h$ of the simulation. Exponential convergence is clearly visible up to reaching the plateau in the various cases. The plots also show that the plateau moves downward with increased grid resolution and that at lower resolution, we can identify a nonzero negative slope within the plateau, indicating the overfitting effect mentioned in the text. []{data-label="fig:spectral_convergence"}](figs/spectral_convergence_T5_35 "fig:"){width="48.00000%"} ![ Convergence of the residual expansion of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ at one time before (upper panel) and after (lower panel) the individual horizons touch. Note that the inner common MOTS has self-intersections in the latter case. We plot the maximum absolute residual expansion between the collocation points over the pseudospectral resolution used to find the MOTS. This is independent of the grid resolution ${\rm res} = 1/h$ of the simulation. Exponential convergence is clearly visible up to reaching the plateau in the various cases. The plots also show that the plateau moves downward with increased grid resolution and that at lower resolution, we can identify a nonzero negative slope within the plateau, indicating the overfitting effect mentioned in the text. []{data-label="fig:spectral_convergence"}](figs/spectral_convergence_T5_73 "fig:"){width="48.00000%"}
This is indeed the case, as can be seen in Fig. \[fig:spectral\_convergence\]. It shows the maximum residual expansion between the collocation points for ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ at two different times of the simulation: one at $T=5.35{\ensuremath{\,\mathcal{M}}}$, where the MOTS is already highly distorted, and one at $T\approx5.7333{\ensuremath{\,\mathcal{M}}}$. This second case is [*after*]{} the individual MOTSs touch. At this stage, ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ lies in the inside of ${{\ensuremath{\mathcal{S}}}_{1}}\cup {{\ensuremath{\mathcal{S}}}_{2}}$ and intersects itself. There is no qualitative difference in convergence and the plateau is approximately at the same level for the same grid resolution.
We also see in Fig. \[fig:spectral\_convergence\] that the negative slope continues into the plateau region. This effect is more pronounced for lower grid resolutions and not noticeable for $1/h=960$. It is caused by fitting the horizon to features introduced by the interpolation. We avoid this unphysical effect in practice by limiting the pseudospectral resolution as described at the end of Sec. \[subsec:interpolation\].
Instead of varying the pseudospectral resolution, we can test convergence for different grid resolutions $1/h$ of the simulation. The quantity we use here is the convergence of the coordinate shapes of the curves representing the MOTSs. Fig. \[fig:curve\_distances\] shows that we indeed find convergence of the shapes.
![ Convergence of the coordinate shapes of the MOTSs for increasing numerical resolutions $1/h = 60$, $120$, $240$, $480$, $960$. Shown is the maximum coordinate distance of the horizons found in lower resolution simulations to the respective horizon found for $1/h = 960$. []{data-label="fig:curve_distances"}](figs/curve_distances_inner){width="48.00000%"}
We show in Fig. \[fig:residual-expansion\], as a function of time, the residual expansion of the various MOTSs for the highest resolution that we have considered, namely $1/h = 960$. The residual expansion is one of the key ingredients which gives us confidence that the surfaces we find are indeed MOTSs. Note first that for all the “easy” cases, namely for the two individual MOTSs ${\ensuremath{\mathcal{S}}}_{1,2}$ and for the apparent horizon, the residual expansion is no more than $\mathcal{O}(10^{-11})$. These horizons do not have any portions with extreme curvatures and there is no difficulty in locating them. In fact, the residual expansion is largest for the smaller horizon, and is $\mathcal{O}(10^{-12})$ for the larger horizon and the apparent horizon. The difficult case is of course the inner common horizon, which required the various technical improvements detailed earlier. The most difficult cases are those which have the narrow neck and correspondingly highly curved portions. There is a small duration of time near ${\ensuremath{T_{\rm touch}}}$ where we are not able to locate ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$. At all the other times shown in the plot, the residual expansion is no more than $\mathcal{O}(10^{-9})$. In fact, away from ${\ensuremath{T_{\rm touch}}}$, the residual expansion is as good as for the other MOTSs. In particular, this is true after $T \sim 5.7{\ensuremath{\,\mathcal{M}}}$. At these times ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ has developed self-intersections. Thus, our confidence in the existence of self-intersecting MOTSs is the same as our confidence in the existence of the other MOTSs, which of course are already well established.
{width="\textwidth"}
Area and stability {#subsec:area}
------------------
Some quantitative numbers for this evolution are:
- The common horizon forms at ${\ensuremath{T_{\rm common}}}\approx 1.37460222{\ensuremath{\,\mathcal{M}}}$.
- The two individual horizons touch at ${\ensuremath{T_{\rm touch}}}\approx 5.5378176{\ensuremath{\,\mathcal{M}}}$.
- The area of the inner horizon reaches a minimum at ${\ensuremath{T_{\rm min}}}\approx 5.50592{\ensuremath{\,\mathcal{M}}}$, i.e. just a little bit before ${\ensuremath{T_{\rm touch}}}$. This behavior of ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ was previously noted in [@Pook-Kolb:2019iao].
These values were computed at the various resolutions up to $1/h = 960$ and converge up to the shown number of decimal places; compare also Fig. \[fig:t\_convergence\].
![ Convergence of the various characteristic times with increased grid resolution. Shown is the difference between the value found at the finest resolution $1/h = 960$ and the respective lower resolution result. ${\ensuremath{T_{\rm common}}}$ is the time when the common horizon forms, ${{\ensuremath{\mathcal{S}}}_{1}}$ and ${{\ensuremath{\mathcal{S}}}_{2}}$ touch at ${\ensuremath{T_{\rm touch}}}$, and the inner common horizon has a local minimal area at ${\ensuremath{T_{\rm min}}}$. []{data-label="fig:t_convergence"}](figs/time_convergence){width="48.00000%"}
The areas of the various horizons are plotted as functions of time in Fig. \[fig:area-bl5\]. The bottom-right panel presents a useful picture of the merger process. It shows the areas of the apparent horizon, the inner common horizon and the sum of the areas of the individual horizons. It shows the formation and bifurcation of the apparent horizon and it also shows the merger, i.e. the crossing of the curves for the inner horizon and the individual horizons.
{width="\textwidth"}
The principal eigenvalue of the stability operator for the various horizons is shown in Fig. \[fig:stability-bl5\]. We see that $\Lambda_0$ is always positive for ${\ensuremath{\mathcal{S}}}_{1,2}$ and for the apparent horizon, and that it is not strongly varying. ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ is more interesting. When it is initially born, it coincides with the apparent horizon and has $\Lambda_0=0$. At all subsequent times, ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$ has $\Lambda_0<0$; to understand its stability we need to consider the next eigenvalue $\Lambda_1$. But already from Fig. \[fig:stability-bl5\], we see interesting behavior of $\Lambda_0$ for ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$, namely a cusp at ${\ensuremath{T_{\rm touch}}}$. Fig. \[fig:stability1-inner-bl5\] shows $\Lambda_1$ for the inner horizon, and it is seen to be positive thus demonstrating stability. Again, we see a cusp-like behavior near ${\ensuremath{T_{\rm touch}}}$.
![ First stability parameters $\Lambda_0$ (i.e. the principal eigenvalue of the stability operator) for the various horizons. $\Lambda_0$ is positive for all horizons except ${{\ensuremath{\mathcal{S}}}_{\insubscript}}$, for which we instead plot $-\Lambda_0$. A cusp is clearly seen for the inner horizon. All the other horizons show unremarkable behavior in this respect; they remain stable as far as they can be reliably tracked. []{data-label="fig:stability-bl5"}](figs/stability-bl5-res960){width="\columnwidth"}
![ The stability parameter for the inner horizon (upper panel) and zoom in around ${\ensuremath{T_{\rm touch}}}$ (lower panel). We lose numerical precision very close to the merger time as the pseudospectral resolution becomes very large thereby increasing the condition number of the matrix of the discretized problem. []{data-label="fig:stability1-inner-bl5"}](figs/stability1-inner-bl5-res960){width="\columnwidth"}
Conclusions {#sec:conclusions}
===========
In this paper we examined in detail the scenario for the merger of MOTSs outlined previously in [@Pook-Kolb:2019iao]. We have done this by evolving a particular Brill-Lindquist setup and finding all MOTSs at various times. We have tracked the inner common horizon with high accuracy. In particular, we present strong numerical evidence that the inner horizon merges with the two individual horizons precisely at the time when they touch. Moreover, we find that the inner horizon develops self-intersections just after the merger. This provides then a connected sequence of MOTSs taking us from the two disjoint initial horizons to the final apparent horizon. We have also studied some basic properties of the MOTSs including their area and stability. There are numerous other interesting physical and geometric properties of the world tube of MOTSs which shall be studied in detail in forthcoming work.
We thank Abhay Ashtekar, Bernd Brugmann, Luis Lehner, and Andrey Shoom for valuable discussions and comments. We are especially grateful to Jose-Luis Jaramillo for extensive discussions and for suggesting the use of bipolar coordinates.
The MOTS finder [@pook_kolb_daniel_2019_3260171] used in this research is developed in Python with *SimulationIO* [@erik_schnetter_2019_3258858] being used for reading the numerical simulation data. The libraries *SciPy* [@Jones_SciPy], *NumPy* [@van_der_Walt_NumPy], *mpmath* [@mpmath], *SymPy* [@meurer2017sympy] and *Matplotlib* [@Hunter:2007; @michael_droettboom_2018_1202077] were used for certain numerical, validation and plotting tasks.
O.B. acknowledges the National Science Foundation (NSF) for financial support from Grant No. PHY-1607520. This research was also supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. This research was enabled in part by support provided by SciNet (www.scinethpc.ca) and Compute Canada (www.computecanada.ca). Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium [@Loken_2010]. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund – Research Excellence; and the University of Toronto.
[^1]: We define the factor ${\ensuremath{\mathcal{M}}} := M_{\rm ADM} / 1.3$ to be able to state our coordinate quantities in terms of the ADM mass, which in our simulations was chosen to be $1.3$.
[^2]: We smoothen the speed function along the MOTS by exponentially damping the coefficients of a cosine series representation. This reduces higher frequencies in the density of collocation points along ${\ensuremath{\mathcal{S}}}$.
|
---
abstract: 'We explore the growth of super-massive black holes and host galaxy bulges in the galaxy population using the Millennium Run $\Lambda$CDM simulation coupled with a model of galaxy formation. We find that, if galaxy mergers are the primary drivers for both bulge and black hole growth, then in the simplest picture one should expect the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation to evolve with redshift, with a larger black hole mass associated with a given bulge mass at earlier times relative to the present day. This result is independent of an evolving cold gas fraction in the galaxy population. The evolution arises from the disruption of galactic disks during mergers that make a larger fractional mass contribution to bulges at low redshift than at earlier epochs. There is no comparable growth mode for the black hole population. Thus, this effect produces evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation that is driven by bulge mass growth and not by black holes.'
author:
- |
\
$^1$Department of Astronomy, University of California, Berkeley, CA, 94720, USA\
$^2$Max-Planck-Institut für Astrophysik, D-85740 Garching, Germany
bibliography:
- '../../../paper.bib'
date: 'Accepted —. Received —;in original form —'
title: 'Evolution in the black hole mass–bulge mass relation: a theoretical perspective'
---
cosmology: theory, galaxies: evolution, galaxies: active, black hole physics
Introduction {#intro}
============
Super-massive black hole masses are strongly correlated with their host bulge stellar mass, the so-called [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation [@Magorrian1998; @Marconi2003; @Haring2004]. This is at least true in the local universe, but also expected to extend out to higher redshifts. This correlation suggests a common mechanism linking the growth of these two galactic components, with evidence proposing galaxy mergers as the most likely candidate. If true, and given that the global galaxy merger rate in a $\Lambda$CDM universe evolves strongly with time, one may ask if we should expect to see the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation also evolve.
Support for the idea that bulges and black holes grow through mergers arises primarily from the success of numerical simulations and galaxy formation models in reproducing many observed galaxy scaling relations. Such works illustrate that much of the bulge mass of a galaxy can be accounted for by the disruption of disk stars from the merger progenitors, and merger triggered starbursts in the cold gas disk [@Barnes1992; @Mihos1994; @Mihos1996; @Cox2004]. As a growth mechanism for black holes, merger induced perturbations of the gas close to the central massive object can drive gas inward, fueling what is observed to be a ‘quasar’ period in a galaxy’s history [see e.g. @Kauffmann2000; @DiMatteo2005].
In this simple picture the amount of cold gas present in a merging system plays a large part in how rapidly the black hole and bulge can grow. If the growth dependence for each is a simple constant scaling with gas mass, as is commonly assumed in many models of galaxy formation, then their mass ratio will, on average, be approximately independent of any evolution in the global cold gas fraction. This is because both bulges and black holes then co-evolve at a similar pace (drawing their new mass from the same gas reservoir). Furthermore, during a merger bulges will add to bulges, and black holes may coalesce. Thus, from this alone, one should expect little evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation.
In this paper we explore an additional growth channel through which bulges gain mass that black holes do not have. This is the disruption of merged satellite *disks*, and in the event of a major merger, the disruption of the central galaxy *disk*. The stellar mass in such disks will have previously never contributed to the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation. If the bulge growth rate from such disrupted disks is not constant with time, then evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation can occur.
We investigate this behavior using the Millennium Run $\Lambda$CDM simulation [@Springel2005] and a model for galaxy formation [@Croton2006]. This model follows the growth of galaxies (including their individual disk, bulge and black hole components) from high redshift to the present day, and provides a solid framework within which to undertake our analysis. The results we find, however, are not be unique to our particular implementation of the galaxy formation physics but arise from the simple assumptions described above regarding black hole and bulge growth in galaxies. Our aim in using this particular model is to illustrate what one may expect to see if these underlying growth mechanisms turn out to be true.
This paper is organized as follows. In Section \[method\] we briefly describe the Millennium Run $\Lambda$CDM dark matter simulation and our model of galaxy formation, including the simple implementation of bulge and black hole growth. In Section \[results\] we will use this model to investigate how black hole and bulges co-evolve together in the galaxy population from high redshift to the present. We finish in Section \[discussion\] with a discussion of the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation in light of these results.
Galaxy Formation in a Cosmological Context {#method}
==========================================
The galaxy formation model we use to study the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation is identical to that described in [@Croton2006] (including parameter choices), with the exception of one non-essential detail, discussed below. This model of galaxy formation is implemented on top of the Millennium Run $\Lambda$CDM dark matter simulation [@Springel2005]. Below we briefly outline the relevant aspects of the simulation and model to our current work, and refer the interested reader to the above references for further information.
The Millennium Run follows the dynamical evolution of $10^{10}$ dark matter particles in a periodic box of side-length $500\,h^{-1}$Mpc with a mass resolution per particle of $8.6\times 10^8\,h^{-1}{\rm
M}_{\odot}$. We adopt cosmological parameter values consistent with a combined analysis of the 2dFGRS [@Colless2001] and first year WMAP data [@Spergel2003; @Seljak2005]: $\Omega_\Lambda=0.75$, $\Omega_{\rm m}= \Omega_{\rm dm}+\Omega_{\rm b}=0.25$, $\Omega_{\rm
b}=0.045$, $h=0.73$, and $\sigma_8=0.9$. Friends-of-friends (FOF) halos are identified in the simulation using a linking length of 0.2 the mean particle separation, while substructure *within* each FOF halo is found with an improved and extended version of the [SUBFIND]{} algorithm of @Springel2001. Having determined all halos and subhalos at all output snapshots we then build the hierarchical merging trees that describe in detail how structures grow as the universe evolves. These trees form the backbone onto which we couple our model of galaxy formation.
Inside each tree, virialised dark matter halos at each redshift are assumed to attract ambient gas from the surrounding medium, from which galaxies form and evolve. Our model effectively tracks a wide range of galaxy formation physics in each halo, including reionization of the inter-galactic medium at high redshift, radiative cooling of hot gas and the formation of cooling flows, star formation in the cold disk and the resulting supernova feedback, black hole growth and active galactic nuclei (AGN) feedback through the ‘quasar’ and ‘radio’ epochs of AGN evolution, metal enrichment of the inter-galactic and intra-cluster medium, and galaxy morphology shaped through mergers and merger induced starbursts. As galaxy mergers and the resulting growth of bulges and black holes are central to the questions at hand, we will now describe these in more detail.
A simple picture of black hole and bulge growth {#model}
-----------------------------------------------
### The “static” model {#static}
A satellite galaxy orbiting within a larger halo will feel dynamical friction [@Binney1987] and eventually spiral inward to merge with the central galaxy of the system. Mergers are believed to trigger galactic starbursts, where some (perhaps large) fraction of the cold disk gas is converted into stars on a timescale much shorter than that typically found in quiescent star forming disks. To model this event, when a merger occurs we assume the following mass of stars are formed in a burst from the combined cold gas mass of the progenitor galaxies, as found in the SPH simulations of [@Cox2004]: $$\Delta m_{\rm starburst} =
0.56\ m_{\rm R}^{0.7} \ m_{\rm cold}~,
\label{starburst}$$ where $m_{\rm R} = m_{\rm sat} / m_{\rm central}$ is the merger mass ratio of the merging galaxies, and $m_{\rm cold}$ the total mass of cold gas present during the merger. These stars contribute to the spheroid of the final galaxy. For the results presented in this paper, the typical mass of stars in a bulge formed through starbursts is $\sim\!10\%$, which, from Eq. \[starburst\], indicates an average gas fraction in the merging progenitors of $\simgt\!30\%$. This is consistent with smoothed particle hydrodynamic simulations of merging galaxies which suggest that such a gas fraction is required to explain the local Fundamental Plane [e.g. @Hernquist1993; @Robertson2005].
Mergers also perturb the cold gas disk, and this can trigger the accretion of gas onto the central super-massive black hole. [@Croton2006] showed that, under reasonable assumptions, merger triggered ‘quasar’ mode events are sufficient to reproduce the local [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation as well as the observed local black hole mass density of the universe. To include such events, we apply an empirical relation similar to that described in [@Kauffmann2000] and assume that during a merger the gas accreted onto the black hole is proportional to the cold gas present, but in a way that is less efficient in lower mass halos: $$\Delta m_{\rm BH} = 0.03 \ m_{\rm R} \
\Big[ 1 + (280\,\rm{km\,s^{-1}}/V_{\rm vir})^2 \Big]^{-1}
\ m_{\rm cold}~,
\label{quasar}$$ where $V_{\rm vir}$ is the virial velocity of the system, and $m_{\rm
R}$ and $m_{\rm cold}$ are defined above. Here, the coefficient $0.03$ normalizes the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation to match that observed locally. It is important to note that in this picture the ratio $\Delta m_{\rm BH}
/ \Delta m_{\rm starburst}$, i.e. the relative growth rate of black holes and bulges due to converted cold gas, is expected to be essentially constant and independent of redshift, even if the gas fraction itself changes with redshift (note that the virial velocity of a system is only weakly dependent on time).
In addition to starbursts, bulges also grow from the stellar remnants of merged satellites. In our implementation any existing satellite disk is permanently disrupted during a merger and its stars, whose orbits will be heavily randomized from strong tidal forces, are added to the final galaxy bulge. Furthermore, we assume that if the baryonic mass ratio of the merging galaxies is large enough a major merger has occurred. Major mergers are sufficiently energetic that the disk of the central galaxy is also destroyed and its stars added to the bulge: we trigger such events when $m_{\rm R}\!>\!0.3$.
To summarize, aside from the benign contribution during mergers of bulges to bulges and black holes to black holes, bulges in our model grow through both starbursts and disrupted disks, whereas black holes grow only by accretion. Importantly, black holes have no comparable growth mode from disrupted disks, and in Section \[results\] we will explicitly show the significance of this effect. Finally, given that we have made no implicit assumption regarding evolution in the growth of either bulges or black holes, we hereafter refer to this model as the *static* model.
### The “dynamic” model {#dynamic}
The above model is not unique in its ability to reproduce the local black hole and bulge populations. In the next section we will find it useful to consider a variation to this model, which we call the *dynamic* model, in order to explore the sensitivity of our results to the input physics. This change is applied to Eq. \[quasar\] and assumes that gas disks are more centrally concentrated at higher redshift [@mo1998] and are thus more efficient at feeding the black hole during a merger. We incorporate this idea in the simplest possible way through a transformation of the feeding efficiency coefficient: $0.03\,\rightarrow\,0.01\,(1+z)$ (note that the change in coefficient renormalizes our result to remain on the observed local [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation). We point out that now, by construction, we have introduced an evolution to the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation, and this evolution is will be dependent on an evolving gas fraction. Using both static and dynamic implementations of black hole growth our interest is to measure the strength of the change in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation under the above physical assumptions to gain a sense for the range of possibilities that may seen observationally.
A disclaimer
------------
To keep our model as simple as possible we do not consider other processes through which bulges or black holes can grow. This includes the tracking of disk instabilities which contribute to the bulge, as used by [@Croton2006] (this is our only variation from their model). Importantly, we do not claim that other growth modes are not important to the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation. Instead, we assume that mergers, as described above, are the *primary* mechanism that determines the mass history of the bulge and black hole components of a galaxy. This allows us to explore the degree to which merger triggered disrupted stellar disks are able to drive evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation. We do not rule out the possibility that more complicated processes cancel out this effect.
Results
=======
In Fig. \[fig1\] we plot the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation of our semi-analytic model galaxy population at four epochs, $z\!=\!0,1,3,6$. The filled circles in each panel represent the static model described in Section \[static\], while the open squares show galaxies where evolution in the black hole feeding efficiency has been assumed, the dynamic model described in Section \[dynamic\]. For reference, the solid line gives the best fit through the observations of [@Haring2004] for a sample of $30$ galaxies in the nearby universe with well measured bulge and super-massive black hole masses. The local population of both models have been normalized to that found by @Haring2004 and thus match it reasonably well. Contrasting this to the $z\!=\!6$ galaxy population we find clear evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation, where even for the static case a change in amplitude is observed. For both both static and dynamic models this indicates that the characteristic mass of a black hole residing in a bulge of given mass is larger at high redshift than at low, with the difference between models coming only from the degree of evolution found. To quantify this evolution we perform a simple $\chi^2$ power-law fit with unity slope to each result at each epoch, which we illustrate in each panel with long and short dashed lines for the static and dynamic models, respectively. These fits are limited to galaxies with $m_{\rm bulge}\!>\!10^9 M_{\odot}$. We emphasize that, for both dynamic *and* static models, the amplitude of the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation *decreases* with time, finally settling on the observed relation by the present day. This justifies the high normalization chosen in Eq. \[quasar\]. Interestingly, both models show no significant change in slope with redshift.
To understand the origin of the evolution found in Fig. \[fig1\] we separate the mass growth to model black holes and bulges into their respective components. This is measured at each redshift by first independently summing the total mass contributed from each growth channel (i.e. starbursts or disrupted disks for bulge growth, merger driven accretion for black hole growth) to all galaxies with bulges having $m_{\rm bulge}\!>\!10^9 M_{\odot}$. We then consider the ratio of total bulge to black hole growth from these channels (which is also the mean relative growth rate) to quantify which dominates and when. We do this first for the simplest case, the static model, now shown in Fig. \[fig2\]. In the top panel we plot the ratio of growth rates, $\dot{m}_{\rm BH}$/$\dot{m}_{\rm bulge}$, for bulge growth through either starbursts (dashed line, Eq. \[starburst\]) or disrupted disks (solid line), both relative to the single black hole growth mode of gas accretion (Eq. \[quasar\]). (Note that we do not concern ourselves with growth from black hole–black hole or bulge–bulge merging, since, by definition, this does not change the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation.) We find that the relative growth of bulges and black holes from the existing cold gas supply present during the merger is approximately constant with time (dashed line). This is expected (see Section \[static\]) and simply reflects the fact that, although individually their growth rates can be strongly affected by an evolving cold gas fraction, when taken as a ratio this evolution cancels. In contrast, the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} growth ratio from disrupted disks is a strong function of redshift, as demonstrated by the solid line, with a change of almost an order-of-magnitude between $z\!=9$ and the present day. This increase is driven by bulge growth that arises from both merged satellite disks and major merger disruption of central galaxy disks.
The bottom panel in Fig. \[fig2\] illustrates how this translates into an evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation itself. Here we show the previous $\chi^2$ power-law fits from Fig. \[fig1\] as a function of redshift (solid line) and $1 \sigma$ scatter around the mean (dashed lines). The clear decrease in the amplitude of the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation by a factor of approximately $3$ tracks closely the increase in the growth of bulges from disrupted disks. This demonstrates the simple idea we set out in Section \[intro\], that if mergers are the primary mechanism which shape both black hole and bulge growth in the galaxy population, then a larger fractional contribution to the bulge from disrupted disks should result in an inevitable evolution of the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation. This holds true even when no explicit evolution the the growth modes of black holes and bulges is assumed.
In Fig. \[fig3\] we redo the analysis of Fig \[fig2\] now using the dynamical model described in Section \[dynamic\]. This allows us to explore the sensitivity of [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} evolution when some evolution in the black hole growth rate has been assumed. The top panel of Fig. \[fig3\] clearly shows a much stronger effect than that in the previous figure, with the relative growth of black holes and bulges from both the cold gas reservoir (dashed line) and disk disruption (solid line) changing with time with an additional factor of approximately $1+z$. This results in a significant boost to the previous evolution seen in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation, as shown in the bottom panel, with approximately an order-of-magnitude difference now predicted between high and low redshift. When one restricts the comparison to between redshift $z\!=\!1$ and the present day, the difference in amplitude is still a factor of $\sim\!2$, which may statistically be an observable quantity in the near future.
Discussion
==========
The rapid increase of bulge growth at late times in our static model is a consequence of two well understood effects. The first is the steady rise in the star formation rate density of the universe from high redshift to approximately $z\!=\!1\!-\!2$. If one accepts, as a general rule, the conventional wisdom that the bulk of this star formation occurs in stellar disks, then the outcome is a strongly increasing growth of disk mass across the galaxy population with time. As disks grow the second effect then becomes increasingly important. This effect stems from the hierarchical nature of a CDM universe, where mergers become more frequent as the universe ages, assembling structure from the bottom up. As we discussed in Section \[model\], mergers also transform disks into bulges. Thus, at late times, a larger fraction of the total stellar mass in the universe becomes locked up in the spheroid component of the population relative to earlier epochs. This results in the accelerated bulge growth seen in Fig. \[fig2\], and which drives the evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation shown in Fig. \[fig1\].
Observationally it is difficult to measure black hole and bulge masses. In the local universe [@Magorrian1998] estimate $m_{\rm
BH}\!\sim 0.006\,m_{\rm bulge}$ from a sample 32 galaxies, while both [@Marconi2003] and [@Haring2004] independently find $m_{\rm
BH}\!\sim 0.002\,m_{\rm bulge}$ from improved measurements of $\sim\!30$ galaxies. Although the statistics are still poor and the uncertainty large, locally at least all observations appear to be converging to a consistent result. At higher redshifts the picture is much less clear. For example, [@Shields2003] claim little evolution in the relation can be inferred out to $z\!\sim
3$[^1], and [@Adelberger2005] measure the quasar–galaxy cross-correlation function and find consistency with the local [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} ratio from a sample of $79$ $z\!\sim\!2.5$ quasars. On the other hand, [@Treu2004] see variations in the $m_{\rm BH}$–$\sigma$ relation at $z\!=\!0.37$, while [@Mclure2005] measure some evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation using the 3CRR sample of radio galaxies. Similarly, [@Rix2001] use gravitational lensing to find that quasar host galaxies at $z\!\sim\!2$ are much fainter than their low redshift counterparts containing quasars of similar luminosity, and [@Walter2004] find a significant deviation from the local [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation for a $z\!=\!6.4$ quasar host galaxy. Future observations will need to clarify the exact nature of both the high redshift black hole and host galaxy populations.
Recent theoretical work to understand the cosmological assembly of stars and super-massive black holes have led to interesting results. [@Wyithe2003] present a model for super-massive black hole growth that successfully matches many local and high redshift AGN related observations. Their work results in a $m_{\rm BH}$–$\sigma$ relation constant with redshift while predicting the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation evolves as $m_{\rm BH}/m_{\rm bulge} \propto \xi(z)^{1/2}(1+z)^{3/2} \sim
(1+z)^{1.15}$, where the final approximation is valid when $z\!<\!2$, and $\xi(z)$ depends only on the cosmological parameters and is a weak function of redshift. An evolution of this kind would be consistent with the scaling assumed in our dynamical model (Section \[dynamic\]).
Similarly, [@Merloni2004b] constrain phenomenologically the joint evolution of super massive black holes and their host spheroids by fitting simultaneously the total stellar mass and star formation rate densities as a function of redshift, as well as the hard X-ray selected quasar luminosity function. With the latter they assume that black holes grow exclusively through accretion. Assuming a present day disc to spheroid ratio of $0.5$ [@Tasca2005], their work favors a model in which the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation evolves as $\sim\!(1+z)^{1/2}$. This is a weaker effect than found by @Wyithe2003, however demonstrates both the range of evolution that may be expected, and most importantly, that such non-zero evolution can arise naturally from simple studies of black hole and bulge growth.
As discussed in Section \[intro\], in a $\Lambda$CDM universe the effect described in this paper will be present in any model of black hole and bulge assembly driven by mergers. Indeed, this has already been seen in the semi-analytic model of [@Cattaneo2005] who find similar [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} evolution to that found here (compare their Fig. 6 with our Fig. \[fig1\]). Unfortunately they do not discuss the origin of this behavior, but instead choose to focus on the disruption of galactic discs in relation to the scatter and slope of the relation. @Cattaneo2005 grow bulges both as we do *and* from disk instabilities, which interestingly produces a bi-modal [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}distribution at high redshift. For simplicity we have removed bulge growth through disk instabilities [as originally used in @Croton2006], although when included we also see such bi-modality. This bi-modal prediction of the high redshift [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation provides a novel test of the mechanisms through which bulge growth may occur.
Theoretical arguments and numerical work have demonstrated that galaxy mergers are capable of simultaneously triggering growth in both bulges and black holes in a way so as to jointly reproduce many of their properties currently observed in the local universe. If mergers are the primary drivers of black hole and bulge growth in the galaxy population, then we have shown one should expect to see an evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation which arises from an increasing contribution of disrupted disks to bulges as the universe ages. In this picture, *evolution in the growth of bulges drives an evolution in the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{} relation*, distinct from the growth rate of black holes. At the very least, even if the physics governing bulge and black hole growth turns out to be much more complex and cannot be expressed in a simplified manner (as is currently assumed by most models of galaxy formation), in a $\Lambda$CDM universe this effect should still be present and must be included in any interpretation of the [$m_{\rm BH}\,$–$\,m_{\rm bulge}$]{}relation measured at different redshifts. We await future high redshift observations, e.g. the Galaxy Evolution from Morphological Studies (GEMS) project [@Rix2004], to clarify the situation further.
Acknowledgments {#acknowledgements .unnumbered}
===============
This work was supported in part by NSF grant AST00-71048 and from the International Max Planck Research School in Astrophysics Ph.D. fellowship. Thanks to Simon White, Andrea Merloni and Eliot Quataert. Special thanks to Eric Bell and Hans-Walter Rix for an initial very motivating discussion. The Millennium Run simulation used in this paper was carried out by the Virgo Supercomputing Consortium at the Computing Centre of the Max-Planck Society in Garching. Semi-analytic galaxy catalogs from the simulation are publicly available at http://www.mpa-garching.mpg.de/galform/agnpaper.
\[lastpage\]
[^1]: However see their most recent work [@Shields2005]
|
---
abstract: 'For a big class of smooth dagger spaces — dagger spaces are ’rigid spaces with overconvergent structure sheaf’ — we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of Berthelot’s rigid cohomology also in the non-smooth case. We need a careful study of de Rham cohomology in situations of semi-stable reduction.'
author:
- '<span style="font-variant:small-caps;">Elmar Grosse-Klönne</span>'
title: Finiteness of de Rham cohomology in rigid analysis
---
[**Introduction**]{}
Let $(R,\pi,k={\mbox{\rm Frac}}(R),\bar{k}=R/(\pi))$ be a complete discrete valuation ring of mixed characteristic. It is a simple observation that the de Rham cohomology $H_{dR}^*(X)$ of a positive dimensional smooth affinoid $k$-rigid space $X$ computed with respect to its (usual) structure sheaf is not finite dimensional. The idea of instead using an overconvergent structure sheaf arises naturally from the paper [@mw] of Monsky and Washnitzer: The Monsky-Washnitzer cohomology of a smooth affine $\bar{k}$-scheme ${\mbox{\rm Spec}}(A)$ is the de Rham cohomology of $\tilde{A}^{\dagger}\otimes_Rk$, where $\tilde{A}^{\dagger}$ is a weakly complete formal lift of $A$. Monsky-Washnitzer cohomology has recently been shown to be finite dimensional (independently by Berthelot [@berfi] and Mebkhout [@mebend]). The algebra $\tilde{A}^{\dagger}\otimes_Rk$ can be geometrically interpreted as a $k$-algebra of overconvergent functions on the rigid space ${\mbox{\rm Sp}}(\tilde{A}\otimes_Rk)$, where $\tilde{A}$ is a lifting of $A$ to a formally smooth $\pi$-adically complete $R$-algebra. In [@en1dag] we introduce a category of ’$k$-rigid spaces with overconvergent structure sheaf’, which we call $k$-dagger spaces, and study a functor $X\mapsto X'$ from this category to the category of $k$-rigid spaces which is not far from being an equivalence. For example, $X$ and $X'$ have the same underlying $G$-topological space and the same stalks of structure sheaves. Finiteness of Monsky-Washnitzer cohomology implies finiteness of de Rham cohomology for affinoid $k$-dagger spaces with good reduction — in the above notation, the algebra $\tilde{A}^{\dagger}$ gives rise to the affinoid $k$-dagger space $X$ with $\Gamma(X,{\cal O}_X)=\tilde{A}^{\dagger}\otimes_Rk$. Our main result generalizes this as follows:\
[**Theorem A:**]{} (=Corollary \[ende\] $+$ Theorem \[kokomend\]) [*Let $X$ be a quasi-compact smooth $k$-dagger space, $U\subset X$ a quasi-compact open subset, $Z\to X$ a closed immersion. Then $T=X-(U\cup Z)$ has finite dimensional de Rham cohomology $H_{dR}^*(T)$.*]{}\
By [@en1dag], 3.2, this implies finiteness of de Rham cohomology also for certain smooth $k$-rigid spaces $Y$: For example, if $Y$ admits a closed immersion $i$ into a polydisk without boundary (at least if $i$ extends to a closed immersion with bigger radius), or if $Y$ is the complement of a quasi-compact open subspace in a smooth proper $k$-rigid space. But our main corollary is of course:\
[**Corollary B:**]{} (=Corollary \[rifi\]) [*For a ${\bar k}$-scheme $X$ of finite type the $k$-vector spaces $H_{rig}^*(X/k)$ (see [@berfi]) are finite dimensional.*]{}\
We do not reprove finiteness of Monsky-Washnitzer cohomology, rather we reduce our Theorem A to it. A big part is devoted to the study of de Rham cohomology in situations of semi-stable reduction. We need and prove:\
[**Theorem C:**]{} (=Theorem \[monovor\]) [*Let ${\cal X}$ be a strictly semi-stable formal $R$-scheme, let ${\cal X}_{\bar{k}}=\cup_{i\in I}Y_i$ be the decomposition of the closed fibre into irreducible components. For $K\subset I$ set $Y_K=\cap_{i\in K}Y_i$. Let $X^{\dagger}$ be a $k$-dagger space such that its associated rigid space is identified with ${\cal X}_k$. For a subscheme $Y\subset{\cal X}_{\bar{k}}$ let $]Y[_{\cal X}^{\dagger}$ be the open dagger subspace of $X^{\dagger}$ corresponding to the open rigid subspace $]Y[_{\cal X}$ of ${\cal X}_k$. Then for any $\emptyset\ne J\subset I$ the canonical map $H_{dR}^*(]Y_J[_{\cal X}^{\dagger})\to H_{dR}^*(]Y_J-(Y_J\cap(\cup_{i\in I-J}Y_i))[^{\dagger}_{\cal X})$ is bijective.*]{}\
Another important tool is de Jong’s theorem on alterations by strictly semi-stable pairs, in its strongest sense.\
We proceed as follows. After recalling some facts on dagger spaces in section 0, we formulate in section 1 some basic concepts about ${\cal D}$-modules on rigid and dagger spaces. This follows the complex analytic case, see for example [@mebdmod]. Instead of reproducing well known arguments, we focus only on what is specific to the non-archimedean case. Then we construct a long exact sequence for de Rham cohomology with supports in blowing-up situations. As in [@hadr] (for algebraic $k$-schemes), it results from the existence of certain trace maps for proper morphisms; we define such trace maps, based on constructions from [@chia], [@bey], [@vdpse]. Finally we prove the important technical fact that the de Rham cohomology $H_{dR}^*(X)$ of a smooth dagger space $X$ depends only on its associated rigid space $X'$; hence knowledge of $X'$ (for example a decomposition into a fibre product) gives information about $H_{dR}^*(X)$. In the section 2, we begin to look at $R$-models of the associated rigid spaces; more specifically, we consider the case of semi-stable reduction. Its main result is Theorem C. It enables us to reduce Theorem A, in the case where $U=\emptyset$ and $X$ has semi-stable reduction, to the finiteness of Monsky-Washnitzer cohomology. In section 3 we first prove Theorem A in case $U=\emptyset=Z$: After reduction to the case where $X$ is affinoid and defined by polynomials, we apply de Jong’s theorem to an $R$-model of a projective compactification of $X$ to reduce to the finiteness result of section 2. The case of general $Z$ is handled by a resolution of singularities ([@bier]). Then we treat the case of general $U$ by another application of de Jong’s theorem. The formal appearence of these last arguments bears similarity with the finiteness proofs in [@berfi], [@hadr]. But there are also distinctive features: in the simultanuous control of special and generic fibre; and in particular in our second application of de Jong’s theorem: We apply it to a certain closed immersion of $R$-schemes $\bar{X}_{\bar{k}}\cup\bar{Y}\to\bar{X}$, where the space $X-U$ we are interested in is realized in (the tube $]\bar{Y}[$ of) [*the compactifying divisor, not*]{} in its open complement.\
Dagger spaces
=============
Let $k$ be field of characteristic $0$, complete with respect to a non-archimedean valuation $|.|$, with algebraic closure $k_a$, and let $\Gamma^*=|k_a^*|=|k^*|\otimes\mathbb{Q}$.\
We gather some facts from [@en1dag]. For $\rho\in\Gamma^*$ the $k$-affinoid algebra $T_n(\rho)$ consists of all series $\sum a_{\nu}X^{\nu}\in
k[[X_1,\ldots,X_n]]$ such that $|a_{\nu}|\rho^{|\nu|}$ tends to zero if $|\nu|\to\infty$. The algebra $W_n$ is defined to be $W_n=\cup_{\stackrel{\rho>1}{\rho\in\Gamma^*}}T_n(\rho)$ [^1]. A $k$-dagger algebra $A$ is a quotient of some $W_n$; a surjection $W_n\to A$ endows it with a norm which is the quotient of the Gauss norm on $W_n$. All $k$-algebra morphisms between $k$-dagger algebras are continuous with respect to these norms, and the completion of a $k$-dagger algebra $A$ is a $k$-affinoid algebra $A'$ in the sense of [@bgr]. There is a tensor product $\otimes^{\dagger}_k$ in the category of $k$-dagger algebras. As for $k$-affinoid algebras, one has for the set ${\mbox{\rm Sp}}(A)$ of maximal ideals of $A$ the notions of rational and affinoid subdomains, and for these the analogue of Tate’s acyclicity theorem ([@bgr],8.2.1) holds. The natural map ${\mbox{\rm Sp}}(A')\to{\mbox{\rm Sp}}(A)$ of sets is bijective, and via this map the affinoid subdomains of ${\mbox{\rm Sp}}(A)$ form a basis for the strong $G$-topology on ${\mbox{\rm Sp}}(A')$ from [@bgr]. Imposing this $G$-topology on ${\mbox{\rm Sp}}(A)$ one gets a locally $G$-ringed space, an affinoid $k$-dagger space. (Global) $k$-dagger spaces are built from affinoid ones precisely as in [@bgr]. The fundamental concepts and properties from [@bgr] translate to $k$-dagger spaces.\
There is a faithful functor from the category of $k$-dagger spaces to the category of $k$-rigid spaces, assigning to a $k$-dagger space $X$ a $k$-rigid space $X'$ (to which we will refer as the associated rigid space; but we will use the notation $(?)'$ not only for this functor). $X$ and $X'$ have the same underlying $G$-topological space and the same stalks of structure sheaf. A smooth $k$-rigid space $Y$ admits an admissible open affinoid covering $Y=\cup V_i$ such that $V_i=U_i'$ for uniquely determined (up to non canonical isomorphisms) affinoid $k$-dagger spaces $U_i$. Furthermore, this functor induces an equivalence between the respective subcategories formed by partially proper spaces (see below). In particular, there is an analytification functor from $k$-schemes of finite type to $k$-dagger spaces. For a smooth partially proper $k$-dagger space $X$ with associated $k$-rigid space $X'$, the canonical map $H_{dR}^*(X)\to H_{dR}^*(X')$ between the de Rham cohomology groups is an isomorphism: this follows from applying [@en1dag], 3.2 to the morphism between the respective Hodge-de Rham spectral sequences. By a dagger space not specified otherwise, we will mean a $k$-dagger space, and similarly for dagger algebras, rigid spaces etc..\
In the sequel, all dagger spaces and rigid spaces are assumed to be quasi-separated. We denote by ${\bf D}=\{x\in k; |x|\le1\}$ (resp. ${\bf D}^0=\{x\in k; |x|<1\}$) the unitdisk with (resp. without) boundary, with its canonical structure of $k$-dagger or $k$-rigid space, depending on the context. For $\epsilon\in\Gamma^*$, the ring of global functions on the polydisk $\{x\in k^n;\mbox{ all }|x_i|\le\epsilon\}$, endowed with its canonical structure of $k$-dagger space, will be denoted by $k<\epsilon^{-1}.X_1,\ldots,\epsilon^{-1}.X_n>^{\dagger}$. The dimension $\dim(X)$ of a dagger space $X$ is the maximum of all $\dim({\cal O}_{X,x})$ for $x\in X$. We say $X$ is pure dimensional if $\dim(X)=\dim({\cal O}_{X,x})$ for all $x\in X$.\
A morphism $f:X\to Y$ of rigid or dagger spaces is called partially proper (cf. [@huet p.59]), if $f$ is separated, there is an admissible open affinoid covering $Y=\cup_i Y_i$, and for all $i$ admissible open affinoid coverings $f^{-1}(Y_i)=\cup_{j\in J_i} X_{ij}=\cup_{j\in J_i} X_{ij}'$ with $X_{ij}\subset\subset_{Y_i}X_{ij}'$ for every $j\in J_i$ (where $\subset\subset_{Y_i}$ is defined as in [@bgr]).\
\[hironaka\] Let $Z\to X$ be a closed immersion into an affinoid smooth dagger space. There is a proper surjective morphism $g:\tilde{X}\to X$ with $\tilde{X}$ smooth, $g^{-1}(Z)$ a divisor with normal crossings on $\tilde{X}$, and $g^{-1}(X-Z)\to(X-Z)$ an isomorphism.
[Proof:]{} Write $X={\mbox{\rm Sp}}(W_n/I), Z={\mbox{\rm Sp}}(W_n/J)$ with ideals $I\subset J\subset W_n$. Since these ideals are finitely generated, there is a $\rho>1$ and ideals $I_{\rho}\subset J_{\rho}\subset T_n(\rho)$ such that $I=I_{\rho}.W_n$ and $J=J_{\rho}.W_n$, and such that the rigid space $X_{\rho}={\mbox{\rm Sp}}(T_n(\rho)/I_{\rho})$ is smooth. Apply [@bier 1.10] to the closed immersion $Z_{\rho}={\mbox{\rm Sp}}(T_n(\rho)/J_{\rho})\to X_{\rho}$ to get a morphism of rigid spaces $\tilde{X}_{\rho}\to X_{\rho}$ with the desired properties. Its restriction to the partially proper open subspace $\cup_{\rho'<\rho}{\mbox{\rm Sp}}(T_n(\rho')/(I_{\rho}))\subset X_{\rho}$ is a morphism of partially proper spaces (compositions of partially proper morphisms are partially proper, [@huet]), hence by [@en1dag],2.27 is equivalent with a morphism of dagger spaces. The restriction of the latter to $X$ does the job.\
${\cal D}$-modules
==================
For a smooth dagger (or rigid) space $X$ let $I={\mbox{\rm Ker}}({\cal O}_{X\times X}\to{\cal O}_X, f\otimes g\mapsto fg)$. For $n\in\mathbb{N}$ we view ${\cal P}_X^n={\cal O}_{X\times X}/I^{n+1}$ via $d_{1,n}:{\cal O}_X\to{\cal P}_X^n,\quad f\mapsto f\otimes1$ as ${\cal O}_X$-algebra and define ${\cal D}_X^n={\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal P}_X^n,{\cal O}_X)$. Then ${\cal D}_X=\lim_{\stackrel{\to}{n}}{\cal D}_X^n$ comes in a natural way with the structure of a sheaf of rings on $X$ (as in [@EGA],IV,16.8.10). By modules over ${\cal D}_X$ which are not explicitly declared as ${\cal D}_X$-right-modules we always mean ${\cal D}_X$-left-modules. ${\cal O}_X$ becomes a ${\cal D}_X$-module setting $P.f=P(d_{2,n}(f))$ for $P\in{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal P}_X^n,{\cal O}_X), f\in{\cal O}_X$ and $d_{2,n}:{\cal O}_X\to{\cal P}_X^n, f\mapsto 1\otimes f$. For a morphism $X\to Y$ of smooth spaces with $J={\mbox{\rm Ker}}({\cal O}_{X\times_YX}\to{\cal O}_X, f\otimes g\mapsto fg)$ we define $\Omega_{X/Y}^1=J/J^2$. By means of $d:{\cal O}_X\to\Omega^1_{X/Y},\quad a\mapsto(1\otimes a-a\otimes1)$ we form in the usual way the complex $\Omega_{X/Y}^{\bullet}=(\bigwedge^{\bullet}\Omega_{X/Y}^1,d)$. If $Y={\mbox{\rm Sp}}(k)$ we write $\Omega_X^{\bullet}$. If $X$ is of pure dimension $r$ we write $\omega_X=\Omega_X^r$, and if $f:X\to Y$ is a morphism of smooth and pure dimensional spaces, we write $\omega_{X/Y}=\omega_X\otimes_{{\cal O}_X}f^*\omega_Y^{\otimes(-1)}$.\
Suppose $X={\mbox{\rm Sp}}(A)$ with a regular $k$-dagger algebra (resp. $k$-affinoid algebra) $A$. Then $d:A\to\Omega^1_X(X)=\Omega^1_A$ is the universal $k$-derivation of $A$ into finite $A$-modules. A dagger space $X$ is smooth if and only if its associated rigid space $X'$ is smooth, and $\Omega_{X'}^1$ is canonically identified with the coherent ${\cal O}_{X'}$-module obtained by completing on affinoid open pieces the coherent ${\cal O}_X$-module $\Omega_X^1$.\
${\cal D}_X$ is a coherent ${\cal D}_X$-module (see [@doktor], 5.5).\
If $X={\mbox{\rm Sp}}(A)$, then ${\cal D}_X(X)=D_A$ is a both-sided noetherian ring, and $\dim(A)={\mbox{\rm coh.dim}}(D_A)$ (see [@mebdmod],p.42-55).\
Let $X$ be smooth of pure dimension. Then $\omega_X$ can be equipped with a canonical structure of ${\cal D}_X$-right-module, and there is an equivalence between the category of ${\cal D}_X$-right-modules and the category of ${\cal D}_X$-(left-)modules, where a ${\cal D}_X$-right-module ${\cal E}$ is sent to ${\underline{\mbox{\rm Hom}}}_{{\cal O}_X}(\omega_X,{\cal E})$, and a ${\cal D}_X$-(left-)module ${\cal M}$ is sent to $\omega_X\otimes_{{\cal O}_X}{\cal M}$. This can be seen as over the complex numbers, cf. [@mebdmod].\
For a morphism $f:Z\to Y$ of smooth pure dimensional dagger (resp. rigid) spaces we define $${\cal D}_{Z\to Y}={\cal O}_Z\otimes_{f^{-1}{\cal O}_Y}f^{-1}{\cal D}_Y,$$ which is a $({\cal D}_Z, f^{-1}{\cal D}_Y)$-bimodule, and $${\cal D}_{Y\leftarrow Z}=\omega_Z\otimes_{f^{-1}{\cal O}_Y} f^{-1}({\underline{\mbox{\rm Hom}}}_{{\cal O}_Y}(\omega_Y,{\cal D}_Y))=\omega_{Z/Y}\otimes_{{\cal O}_Z}{\cal D}_{Z\to Y},$$ which is a $(f^{-1}{\cal D}_Y,{\cal D}_Z)$-bimodule, cf.[@mebdmod].\
In particular, if ${\cal M}$ is a ${\cal D}_Z$-module, ${\cal D}_{Y\leftarrow Z}\otimes_{{\cal D}_Z}{\cal M}$ becomes a $f^{-1}{\cal D}_Y$-module, and we get a left-derived functor $${\cal D}_{Y\leftarrow Z}\otimes_{{\cal D}_Z}^{\bf L}(.):{\mbox{\rm D}}^-({\cal D}_Z)\to {\mbox{\rm D}}^-(f^{-1}{\cal D}_Y).$$
As in the complex case, we have a canonical ${\cal D}_X$-linear projective resolution $$\begin{gathered}
0\to{\cal D}_X\otimes_{{\cal O}_X}\bigwedge^n{\cal T}_X\to\ldots\to{\cal D}_X\otimes_{{\cal O}_X}{\cal T}_X\to{\cal D}_X\tag{$*$}\end{gathered}$$ of ${\cal O}_X$, with ${\cal T}_X={\underline{\mbox{\rm Hom}}}_{{\cal O}_X}(\Omega^1_X,{\cal O}_X)$. Therefore the definition ${\mbox{\rm DR}}({\cal M})=R{\underline{\mbox{\rm Hom}}}_{{\cal D}_X}({\cal O}_X,{\cal M})$ makes sense for any ${\cal M}\in{\mbox{\rm D}}({\cal D}_X)$. For $r\in\mathbb{N}$ we define the $k$-vector spaces $H_{dR}^r(X)=H^r(X,{\mbox{\rm DR}}({\cal O}_X))$.\
Application of $\omega_X\otimes_{{\cal O}_X}(.)$ to $(*)$ yields ${\mbox{\rm DR}}({\cal M})\cong\omega_X\otimes_{{\cal D}_X}^{\bf L}{\cal M}[-\dim(X)]$, and if $Y$ is another smooth space, $Z=X\times Y\to Y$ the canonical projection, there is a canonical isomorphism $\Omega^{\bullet}_{Z/Y}\cong{\cal D}_{Y\leftarrow Z}\otimes_{{\cal D}_Z}^{\bf L}{\cal O}_Z[-\dim(X)]$.\
We recall a definition from [@huet],5.6. Let $f:Z\to Y$ be partially proper, let ${\cal F}$ be an abelian sheaf on $Z$. If $Y$ is quasi-compact, let $$\Gamma_c(Z/Y,{\cal F})=\{s\in\Gamma(Z,{\cal F}) |\mbox{ there is a quasi-compact}$$$$\mbox{admissible open } U\subset Z \mbox{ such that } s\in{\mbox{\rm Ker}}(\Gamma(Z,{\cal F})\to\Gamma(Z-U,{\cal F}))\}.$$If $Y$ is arbitrary, let$$\Gamma_c(Z/Y,{\cal F})=\{s\in\Gamma(Z,{\cal F}) |\mbox{ for all quasi-compact}$$$$\mbox{admissible open } Y'\subset Y \mbox{ we have } s|_{f^{-1}(Y')}\in\Gamma_c(f^{-1}(Y')/Y',{\cal F}) \}.$$ Then $V\mapsto\Gamma_c(f^{-1}(V)/V,{\cal F})$ defines a sheaf $f_!{\cal F}$ on $Y$. We denote by $Rf_!:{\mbox{\rm D}}^+(Z)\to {\mbox{\rm D}}^+(Y)$ the functor induced by the left exact functor $f_!(-)$. In the following we always assume tacitly that $Rf_!:{\mbox{\rm D}}^+(Z)\to {\mbox{\rm D}}^+(Y)$ extends to $Rf_!:{\mbox{\rm D}}(Z)\to {\mbox{\rm D}}(Y)$ and $Rf_!:{\mbox{\rm D}}^-(Z)\to {\mbox{\rm D}}^-(Y)$ (this will be true in the cases relevant for us; namely, if $Z=X\times Y$ with $X$ finitely admissibly covered by open subspaces which are Zariski closed in some $({\bf D}^0)^m$, and $f$ is the projection — as in \[spur\] — then there is an $n\in\mathbb{N}$ such that $R^if_!{\cal F}=0$ for all $i>n$, all ${\cal F}$. See [@doktor], p.47).\
If $Z$ and $Y$ are smooth and pure dimensional and if ${\cal M}\in{\mbox{\rm D}}({\cal D}_Z)$, we define $$f_+({\cal M})=Rf_!({\cal D}_{Y\leftarrow Z}\otimes_{{\cal D}_Z}^{\bf L}{\cal M})\in{\mbox{\rm D}}({\cal D}_Y).$$ If ${\cal M}\in{\mbox{\rm D}}^-({\cal D}_X)$ we get $f_+({\cal M})\in{\mbox{\rm D}}^-({\cal D}_Y)$.\
\[kohkomp\] Let $X\stackrel{f}{\to}Y\stackrel{g}{\to}Z$ be partially proper morphisms between smooth pure dimensional dagger (resp. rigid) spaces. We assume that $g$ is a projection or a closed immersion. For ${\cal M}\in{\mbox{\rm D}}^-({\cal D}_X)$ there is a canonical isomorphism $g_+(f_+{\cal M})\cong(g\circ f)_+{\cal M}$ in ${\mbox{\rm D}}^-({\cal D}_Z)$.
[Proof:]{} We have to show the projection formula$${\cal D}_{Z\leftarrow Y}\otimes_{{\cal D}_Y}^{\bf L}Rf_!({\cal D}_{Y\leftarrow X}\otimes_{{\cal D}_X}^{\bf L}{\cal M})\cong Rf_!(f^{-1}{\cal D}_{Z\leftarrow Y}\otimes_{f^{-1}{\cal D}_Y}^{\bf L}{\cal D}_{Y\leftarrow X}\otimes_{{\cal D}_X}^{\bf L}{\cal M}).$$If $g$ is a projection $Y=W\times Z\to Z$, we see that ${\cal D}_{Z\leftarrow Y}=\omega_{Y/Z}\otimes_{{\cal O}_Z}{\cal D}_Z$ is a coherent ${\cal D}_Y={\cal D}_W\otimes_k{\cal D}_Z$-right-module (because $\omega_{Y/Z}=\omega_W\otimes_k{\cal O}_Z$ is a coherent ${\cal D}_W\otimes_k{\cal O}_Z$-right-module). Therefore we obtain the above projection formula using the way-out-lemma ([@rd],I,7), since for finite free ${\cal D}_Y$-right-modules (instead of ${\cal D}_{Z\leftarrow Y}$) the projection formula is evident. If $g$ is a closed immersion, ${\cal D}_{Z\leftarrow Y}$ is a locally free ${\cal D}_Y$-right-module. This can be shown as in the complex analytic case using the fact that, locally for an admissible covering of $Z$, there are isomorphisms $Y\times{\bf D}^{{\mbox{\rm codim}}_Z(Y)}\cong Z$ such that $g$ corresponds to the zero section — this is [@kidr],Theorem 1.18, in the rigid case, but holds true also in the dagger case as one observes by examining the proof in loc.cit.. By the commutation of $Rf_!$ with pseudo-filtered limits (cf. [@huet],5.3.7, or [@doktor], 4.8) again one reduces the proof of the projection formula to the case of finite free ${\cal D}_Y$-right-modules.\
Let $Y\to X$ be a closed immersion into a smooth dagger (resp. rigid) space, defined by the coherent ideal ${\cal J}\subset{\cal
O}_X$. Then $${\underline
\Gamma}_{*Y}({\cal E})=\lim_{\stackrel{\to}{n}}{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/{{\cal
J}^n},{\cal E})$$ and $${\cal E}(*Y)=\lim_{\stackrel{\to}{n}}{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({{\cal
J}^n},{\cal E})$$ for a ${\cal D}_X$-module ${\cal E}$ are again ${\cal D}_X$-modules. We get right-derived functors $R{\underline
\Gamma}_{*Y}(-)$ and $R(-)(*Y)$ as functors ${\mbox{\rm D}}^+({\cal D}_X)\to{\mbox{\rm D}}^+({\cal D}_X)$, but also as functors ${\mbox{\rm D}}({\cal D}_X)\to{\mbox{\rm D}}({\cal D}_X)$ and ${\mbox{\rm D}}^-({\cal D}_X)\to{\mbox{\rm D}}^-({\cal D}_X)$. We have distinguished triangles $$R{\underline \Gamma}_{*Y}({\cal
K})\to{\cal K}\to R{\cal K}(*Y)\stackrel{+1}{\to}$$ for all ${\cal K}\in{\mbox{\rm D}}({\cal D}_X)$.\
We list some properties of the above functors. The proofs are similar to those in [@mebdmod], the projection formulas needed can be justified as in \[kohkomp\].\
(a) If $f:Z\to Y$ is a partially proper morphism between smooth pure dimensional dagger (resp. rigid) spaces and if ${\cal M}\in{\mbox{\rm D}}^-({\cal D}_Z)$, there is a canonical isomorphism ${\mbox{\rm DR}}(f_+{\cal M})[\dim(Y)]\cong Rf_!{\mbox{\rm DR}}({\cal M})[\dim(Z)]$.\
(b) If in addition $T\to Y$ is a closed immersion and if $T_Z=T\times_YZ$, there is a canonical isomorphism $R{\underline
\Gamma}_{*T}(p_+{\cal M})\cong p_+(R{\underline\Gamma}_{*T_Z}{\cal M})$.\
(c) Let $X$ be smooth, let $Y_i\to X$ be closed immersions ($i=1,2$) and let ${\cal M}\in{\mbox{\rm D}}({\cal D}_X)$. There is a canonical isomorphism $$R{\underline\Gamma}_{*Y_1}(R{\underline\Gamma}_{*Y_2}({\cal M}))\cong R{\underline\Gamma}_{*(Y_1\cap Y_2)}({\cal M})$$ and a distinguished triangle $$R{\underline\Gamma}_{*(Y_1\cap Y_2)}({\cal M})\to R{\underline\Gamma}_{*Y_1}({\cal M})\oplus R{\underline\Gamma}_{*Y_2}({\cal M})\to R{\underline\Gamma}_{*(Y_1\cup Y_2)}({\cal M})\stackrel{+1}{\to}.$$(d) Let $X$ be smooth, let $Y\to X$ be a closed immersion and let ${\cal M}\in{\mbox{\rm D}}({\cal D}_X)$. There is a canonical isomorphism $R{\underline\Gamma}_{*Y}({\cal O}_X)\otimes_{{\cal O}_X}^{\bf L}{\cal M}\cong R{\underline\Gamma}_{*Y}({\cal M})$.\
(e) Let $X$ be smooth and affinoid, ${\mbox{\rm Sp}}(B)=Y\to X$ a closed immersion of pure codimension $d$ such that all local rings $B_x$ (for $x\in{\mbox{\rm Sp}}(B)=Y$) are locally complete intersections, and let ${\cal F}$ be a ${\cal D}_X$-module which is locally free as ${\cal O}_X$-module. Then $R^i{\underline\Gamma}_{*Y}({\cal F})=0$ for all $i\neq d$ (a well known algebraic fact!).\
\[dmodgys\] Let $Z\to Y\stackrel{s}{\to}X$ be a chain of closed immersions, where $Y$ and $X$ are smooth and pure dimensional. If $d={\mbox{\rm codim}}(s)$, there is a canonical isomorphism $s_+R{\underline\Gamma}_{*Z}{\cal O}_Y\cong R{\underline\Gamma}_{*Z}{\cal O}_X[d]$.
[Proof:]{} (compare with [@mebend],3.3-1) First, .(b),(c) allows us to assume $Z=Y$. Denote by $I\subset{\cal O}_X$ the ideal of $Y$ in $X$. Observe ${\cal D}_{Y\to X}={\cal D}_X/I.{\cal D}_X$ and that this, as well as ${\cal D}_{X\leftarrow Y}$, is locally free over ${\cal D}_Y$, cf. the proof of \[kohkomp\]. We claim that there is a canonical map of ${\cal D}_X$-right-modules$${\underline{\mbox{\rm Ext}}}^d_{{\cal O}_X}({\cal O}_X/I,\omega_X)\otimes_{{\cal D}_Y}{\cal D}_{Y\to X}\to\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Ext}}}^d_{{\cal O}_X}({\cal O}_X/I^k,\omega_X).$$Indeed, choose an injective resolution $J^{\bullet}$ of the ${\cal D}_X$-right-module $\omega_X$ and define the morphism of complexes$${\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/I,J^{\bullet})\otimes_{{\cal D}_Y}{\cal D}_{Y\to X}\to\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/I^k,J^{\bullet})$$as follows: If $g$ is a local section of ${\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/I,J^{m})$ and if $P$ is a local section of ${\cal D}_{Y\to X}$, represented by the local section $\tilde{P}$ of ${\cal D}_X$, then $g\otimes P$ is sent to the following local section of $\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/I^k,J^{m})$: The ${\cal O}_X$-linear map ${\cal O}_X\to J^{m}$ which sends $1_{{\cal O}_X}$ to $g(1_{{\cal O}_X}).\tilde{P}$ actually induces an element of $\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Hom}}}_{{\cal O}_X}({\cal O}_X/I^k,J^{m})$: Indeed, if $\tilde{P}$ is of order $n$, then $g(1_{{\cal O}_X}).\tilde{P}$ is annihilated by $I^{n+1}$. We obtain the promised map. Now since we have as usual $\omega_Y\cong{\underline{\mbox{\rm Ext}}}^d_{{\cal O}_X}({\cal O}_X/I,\omega_X)$, we get a map $$\omega_Y\otimes_{{\cal D}_Y}{\cal D}_{Y\to X}\to\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Ext}}}^d_{{\cal O}_X}({\cal O}_X/I^k,\omega_X)$$ of ${\cal D}_X$-right-modules. Applying ${\underline{\mbox{\rm Hom}}}_{{\cal O}_X}(\omega_X,.)$ (cf. .) it becomes the map $$\begin{gathered}
{\cal D}_{X\leftarrow Y}\otimes_{{\cal D}_Y}{\cal O}_Y\to\lim_{\stackrel{\to}{k}}{\underline{\mbox{\rm Ext}}}^d_{{\cal O}_X}({\cal O}_X/I^k,{\cal O}_X)=R^d{\underline\Gamma}_{*Y}{\cal O}_X\tag{$*$}\end{gathered}$$ of ${\cal D}_X$-left-modules. We claim that $(*)$ is an isomorphism. Indeed, if $x_1,\ldots, x_n$ are local coordinates on $X$ such that $Y$ is defined by $x_1,\ldots, x_d$ and if $\delta_1,\ldots,\delta_n$ is the basis of ${\underline{\mbox{\rm Hom}}}_{{\cal O}_X}(\Omega_X,{\cal O}_X)$ dual to $dx_1,\ldots,dx_n$, one verifies that both sides in $(*)$ are identified with ${\cal D}_X/(x_1,\ldots,x_d,\delta_{d+1},\ldots,\delta_n)$. Since the right hand side in $(*)$ is already all of $R{\underline\Gamma}_{*Y}{\cal O}_X[d]$ (due to .(e)), and since on the left hand side we may write $\otimes^{\bf L}$ instead of $\otimes$, we are done.\
\[spur\]\[kovhom\]\[funktkov\] Let $X$ be smooth, proper and of pure dimension $n$, let $Y$ be smooth and pure dimensional and let $Z=X\times Y\stackrel{p}{\to}Y$ be the canonical projection. Then there is a canonical trace map $p_+{\cal O}_Z[n]\to{\cal O}_Y$. If furthermore $T\to Y$ and $S\to Z\times_YT=X\times T$ are closed immersions, there are canonical trace maps $p_+R{\underline\Gamma}_{*S}{\cal O}_Z[n]\to R{\underline\Gamma}_{*T}{\cal O}_Y$ and $$R\Gamma(Z,{\mbox{\rm DR}}(R{\underline\Gamma}_{*S}{\cal O}_Z))[2n]\to R\Gamma(Y,{\mbox{\rm DR}}(R{\underline\Gamma}_{*T}{\cal O}_Y)),$$ which are isomorphisms if the composition $S\to X\times T\to T$ is an isomorphism.
[Proof:]{} We begin with the rigid case. In [@bey] it is described a finite admissible open covering $X=\cup_iU_i$ such that all $U_J=\cap_{i\in J}U_i$ (for $J\subset I$) have the following properties: there is a closed immersion $U_J\to({\bf D}^0)^{n_J}$ for some $n_J\in\mathbb{N}$, and for all affinoid $\hat{Y}$, all coherent ${\cal O}_{U_J\times\hat{Y}}$-modules ${\cal F}$ and all $j>n$ we have $R^j(p_{J,\hat{Y}})_!{\cal F}=0$, where $p_{J,\hat{Y}}:U_J\times\hat{Y}\to\hat{Y}$ denotes the projection. By means of Mayer-Vietoris sequences we get $R^jp_!{\cal F}=0$ for all coherent ${\cal O}_Z$-modules ${\cal F}$, all $j>n$, hence $R^mp_!\Omega^{\bullet}_{Z/Y}=0$ for $m>2n$ and $$R^{2n}p_!\Omega^{\bullet}_{Z/Y}=\frac{R^np_!\Omega^n_{Z/Y}}{{\mbox{\rm im}}(R^np_!\Omega^{n-1}_{Z/Y}\to R^np_!\Omega^n_{Z/Y})}.$$ In view of $p_+{\cal O}_Z[n]\cong Rp_!\Omega^{\bullet}_{Z/Y}$ (cf. .), to give $p_+{\cal O}_Z[n]\to{\cal O}_Y$ it is therefore enough to give a map $t_p:R^np_!\omega_{Z/Y}\to{\cal O}_Y$ vanishing on ${\mbox{\rm im}}(R^np_!\Omega^{n-1}_{Z/Y}\to R^np_!\Omega^n_{Z/Y})$. We take the following map (compare with [@vdpse], [@bey]): For $$U_J\to D_J=({\bf D}^0)^{n_J}\subset{\bf D}^{n_J}={\mbox{\rm Sp}}(k<T_1,\ldots,T_{n_J}>)$$ as above and projection $p_{D_J,\hat{Y}}:D_J\times\hat{Y}\to\hat{Y}$, there is a canonical identification$$R^{n_J}(p_{D_J,\hat{Y}})_!\omega_{D_J\times\hat{Y}/\hat{Y}}(\hat{Y})=\{\omega=\sum_{\stackrel{\mu\in\mathbb{Z}^{n_J}}{\mu<0}}a_{\mu}T^{\mu}dT_1\wedge\ldots\wedge dT_{n_J}; a_{\mu}\in{\cal O}_{\hat{Y}}(\hat{Y})$$$$\mbox{ and }\omega\mbox{ converges on }\{t\in D_J;\mbox{ all }|t_i|<\epsilon\}\times\hat{Y}\mbox{ for some }0<\epsilon<1\}.$$For elements $\omega$ of this module set $t(\omega)=a_{(-1,\ldots,-1)}$. On the other hand, we have the Gysin map $g$:$$R^n(p_{J,\hat{Y}})_!\omega_{U_J\times\hat{Y}/\hat{Y}}(\hat{Y})\cong R^n(p_{J,\hat{Y}})_!\underline{Ext}^{n_J-n}({\cal O}_{U_J\times\hat{Y}},\omega _{D_J\times\hat{Y}/\hat{Y}})(\hat{Y})$$$$\to R^{n_J}(p_{D_J,\hat{Y}})_!\omega_{D_J\times\hat{Y}/\hat{Y}}(\hat{Y}).$$Locally it can be described as $\eta\mapsto \tilde{\eta}\wedge dX_{n+1}/X_{n+1}\wedge\ldots\wedge dX_{n_J}/X_{n_J}$ where $\tilde{\eta}$ is a lift of $\eta$ and $X_{n+1},\ldots, X_{n_J}$ are local equations for $U_J$ in $D_J$.\
We get $t\circ g:R^n(p_{J,\hat{Y}})_!\omega_{U_J\times\hat{Y}/\hat{Y}}(\hat{Y})\to{\cal O}_{\hat{Y}}(\hat{Y})$. This is seen to be independent of $n_J$ and of the chosen embedding $U_J\to({\bf D}^0)^{n_J}$, hence glues, for varying $J$, to give the desired map $t_p$. By construction, it vanishes on ${\mbox{\rm im}}(R^np_!\Omega^{n-1}_{Z/Y}\to R^np_!\Omega^n_{Z/Y})$.\
In the dagger case, we argue by comparison with the associated morphism $p':Z'\to Y'$ of rigid spaces: Due to [@en1dag], 3.5, we have again $R^jp_!{\cal F}=R^jp_*{\cal F}=0$ for all coherent ${\cal O}_Z$-modules ${\cal F}$, all $j>n$, and the composition of the canonical map $R^np_!\omega_{Z/Y}\to R^np'_!\omega_{Z'/Y'}$ with $t_{p'}:R^np'_!\omega_{Z'/Y'}\to{\cal O}_{Y'}$ has its image in ${\cal O}_Y\subset {\cal O}_{Y'}$, hence we obtain a map $t_p:R^np_!\omega_{Z/Y}\to{\cal O}_Y$ (this can be checked locally on $Y$; if $Y$ is affinoid, then this $t_p$ is the direct limit of the maps $t_{p_{\epsilon}}$ for the morphisms of rigid spaces $p_{\epsilon}:X'\times Y_{\epsilon}'\to Y_{\epsilon}'$, for appropriate extensions $Y'\subset Y_{\epsilon}'$).\
If now in addition $T$ and $S$ are given, we can derive from $t_p$ the other promised maps using the isomorphisms from . (note that $$R\Gamma(Z,{\mbox{\rm DR}}(R{\underline\Gamma}_{*S}{\cal O}_Z))\cong R\Gamma(Y,Rp_!{\mbox{\rm DR}}(R{\underline\Gamma}_{*S}{\cal O}_Z))$$ because $S\to T$ is quasi-compact). Finally, suppose $S\to T$ is an isomorphism. Our additional statement in this situation is seen to be local on $X$. By the definition of $t_p:R^np_!\omega_{Z/Y}\to{\cal O}_Y$ we may substitute our $X$ by $X=({\bf D}^0)^n$ (dropping the assumption on properness). Passing to an admissible covering of $Y$, we may assume that there is a section $s:Y\to Z$ of $p:Z\to Y$ inducing the inverse of $S\to T$. It comes with an isomorphism$$R{\underline\Gamma}_{*T}{\cal O}_Y\cong R{\underline\Gamma}_{*T}p_+R{\underline\Gamma}_{*Y}{\cal O}_Z[n]\cong p_+R{\underline\Gamma}_{*S}{\cal O}_Z[n]$$by .,\[dmodgys\]. It is enough to show that its composition with the map in question $p_+R{\underline\Gamma}_{*S}{\cal O}_Z[n]\to R{\underline\Gamma}_{*T}{\cal O}_Y$ is an isomorphism. Of course this will follow once we know that the underlying map $${\cal O}_Y\cong p_+s_+{\cal O}_Y\cong p_+R{\underline\Gamma}_{*Y}{\cal O}_Z[n]\to p_+{\cal O}_Z[n]\to{\cal O}_Y$$is the identity. Since ${\cal O}_Y(Y)$ is jacobson, we may assume $Y={\mbox{\rm Sp}}(k)$ for this. The definition of $t_p$ as above does not depend on the choice of the closed embedding into some $({\bf D}^0)^n$: This tells us that the map $t_{id_Y}$ we get for $X=Y={\mbox{\rm Sp}}(k)$, computed by means of the embedding $s$, is the identity; but on the other hand, it is precisely the map we are interested in, by the compatibility of the Gysin map in the definition of $t_p$ with the Gysin map in \[dmodgys\].\
\[homdefbkt\] Let $g_i:Z\to Y_i$ ($i=1,2$) be closed immersions into smooth affinoid dagger spaces $Y_i$ of pure dimension $n_i$. Then we have $$R\Gamma(Y_1,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_{Y_1}))[2n_1]\cong R\Gamma(Y_2,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_{Y_2}))[2n_2].$$
[Proof:]{} Because of .,\[dmodgys\] applied to closed immersions $Y_i\to{\bf D}^{n_i}$ we may assume $Y_i={\bf D}^{n_i}$. If $l:Z\to X=Y_1\times Y_2$ is the diagonal embedding it is enough to give isomorphisms $$\quad R\Gamma(Y_i,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_{Y_i}))\cong R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_X))[2n_{3-i}].$$ Let $i=1$. The open embedding $j:X\to W=Y_1\times{\bf P}^{n_2}_k$ induced by the open embedding into projective space $Y_2={\bf D}^{n_2}\to{\bf P}^{n_2}_k$ induces a closed immersion $(j\circ l):Z\to W$, and we have $R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_X))\cong R\Gamma(W,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_W))$. Now apply \[funktkov\].\
Let $Z$ be an affinoid $k$-dagger space, $q\in\mathbb{N}$. The definition $$h^{dR}_q(Z,k)=h^{dR}_q(Z)=\dim_k(H^{2n-q}(Y,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_Y))),$$ where $Z\to Y$ is a closed embedding into a smooth affinoid $k$-dagger space $Y$, is justified by \[homdefbkt\]. For a finite field extension $k\subset k_1$ let $(?)_1=(?)\times_{{\mbox{\rm Sp}}(k)}{{\mbox{\rm Sp}}(k_1)}$. Then $h^{dR}_q(Z,k)=h^{dR}_q(Z_1,k_1)$. Indeed, clearly $\dim_k(H_{dR}^*(X/k))=\dim_{k_1}(H_{dR}^*(X_1/k_1))$ for any smooth $k$-dagger space $X$, hence $$\dim_k(H^*(X,{\mbox{\rm DR}}(R\underline{\Gamma}_Z{\cal O}_X)))=\dim_{k_1}(H^*(X_1,{\mbox{\rm DR}}(R\underline{\Gamma}_{Z_1}{\cal O}_{X_1})))$$ for closed subspaces $Z$ of smooth $k$-dagger spaces $X$. Now use \[algana\] below.\
Let $f:X\to Y$ be a finite étale morphism of smooth dagger (or rigid) spaces, $Y$ irreducible, $X=\cup X_i$ the decomposition into connected components. Assume all maps $f|_{X_i}:X_i\to Y$ to be surjective. Then there is an $l=\deg(f)\in\mathbb{N}$ and a trace map $$t:f_*\Omega^{\bullet}_X\to\Omega^{\bullet}_Y$$ such that the composition $\Omega_Y^{\bullet}\to f_*\Omega^{\bullet}_X\stackrel{t}{\to}\Omega^{\bullet}_Y$ is multiplication by $l$. In particular, $H^i_{dR}(Y)\to H^i_{dR}(X)$ is injective for all $i\in\mathbb{N}$.\
Indeed, for admissible open connected $U={\mbox{\rm Sp}}(A)\subset Y$ with decomposition $f^{-1}(U)=\cup_j{\mbox{\rm Sp}}(B_j)$ such that each $B_j$ is free over $A$, let $t_j:B_j\to A$ be the trace map, and for $q\in\mathbb{Z}$ let $f_*\Omega^q_X(X)\to\Omega^q_Y(Y)$ be the $A$-linear map $$f_*\Omega^q_X(X)=\Omega_{B_j}^q=\Omega_A^q\otimes_AB_j\to\Omega_A^q=\Omega^q_Y(Y)$$ which sends $\omega\otimes b$ to $t_j(b).\omega$. By the same computation as in [@hadr], p.35, we see that for varying $q$ it commutes with the differentials. Clearly it glues for varying $U$, and the number $l=\sum_jl_j$, where $l_j$ denotes the rank of $B_j$ over $A$, is independent of $U$ and fulfills our requirement.\
(In fact, the étaleness of $f$ is not really needed: $f$ is flat in any case, by regularity of $X$ and $Y$. If $U$ is as above, let $L_j={\mbox{\rm Frac}}(B_j)$, $K={\mbox{\rm Frac}}(A)$; then $f$ induces finite separable field extensions $K\subset L_j$; let $l=\sum_j[L_j:K]$. The trace maps $\sigma_j:L_j\to K$ give rise to $\Omega_{L_j/k}^q=\Omega_{K/k}^q\otimes_KL_j\to\Omega_{K/k}^q, \omega\otimes f\mapsto \sigma_j(f).\omega$, restricting to $\sigma_j:\Omega_{B_j}^q\to\Omega_A^q$. Compare with the discussion in [@mw], thm.8.3.. We do not need this.)\
Let $X$ be a smooth dagger (or rigid) space, $j:U\to X$ an open immersion with complement $Y=X-j(U)$. We do not put a structure of dagger (or rigid) space on $Y$. By $R{\underline\Gamma}_Y(.):{\mbox{\rm D}}^+({\cal D}_X)\to{\mbox{\rm D}}^+({\cal D}_X)$ we denote the right-derived functor of the left exact functor $${\cal F}\mapsto {\mbox{\rm Ker}}({\cal F}\to j_*j^{-1}{\cal F})$$on abelian sheaves on $X$, and by $Rj_*:{\mbox{\rm D}}^+({\cal D}_U)\to{\mbox{\rm D}}^+({\cal D}_X)$ we denote the right-derived functor of $j_*$. Note that $Rj_*{\mbox{\rm DR}}({\cal L})\cong{\mbox{\rm DR}}(Rj_*{\cal L})$ for ${\cal L}\in{\mbox{\rm D}}^+({\cal D}_U)$. If $j':U'\to X$ is another open immersion with complement $Y'=X-j'(U')$ such that $U'\cup U$ is an admissible covering of an admissible open subset of $X$, there is a distinguished triangle $$R{\underline \Gamma}_{Y\cap Y'}({\cal K})\to
R{\underline \Gamma}_Y({\cal K})\oplus R{\underline
\Gamma}_{Y'}({\cal K})\to R{\underline \Gamma}_{Y\cup Y'}({\cal K})\stackrel{+1}{\to}.$$
\[algana\](a) Let $Y\to X$ be a closed immersion into a smooth dagger (or rigid) space $X$. The canonical map $$\begin{gathered}
R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))\to R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_Y{\cal O}_X))\tag{$*$}\end{gathered}$$ is an isomorphism.\
(b) Let $Z\to Y$ be another closed immersion. There is a long exact sequence$$\ldots\to H^i(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_X))\to H^i(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))$$$$\to H^i(X-Z,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y-Z}{\cal O}_{X-Z}))\to H^{i+1}(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Z}{\cal O}_X))\to\ldots.$$
[Proof:]{} [**(A):**]{} The rigid case. First assume that $Y$ is locally defined by a single equation. Then the inclusion $j:U=(X-Y)\to X$ of the complement is a quasi-Stein morphism, hence ([@kiaub]) acyclic for coherent ${\cal O}_U$-modules (since we do not know if the analogue in the dagger case holds, we are forced to distinguish). It follows ${\mbox{\rm DR}}(Rj_*{\cal O}_U)=j_*\Omega^{\bullet}_U$. On the other hand $R{\cal O}_X(*Y)={\cal O}_X(*Y)$, and by [@kidr],Thm.2.3, the canonical map $DR({\cal O}_X(*Y))\to j_*\Omega^{\bullet}_U$ is an isomorphism. We get (a) for this type of $Y$. For general $Y$ assertion (a) is now deduced by an induction on the number of defining local equations, using the Mayer Vietoris sequences from . and .. Assertion (b) follows from (a) and the fact that for every sheaf ${\cal F}$ on $X$, we have a natural distinguished triangle $$R{\underline\Gamma}_Z{\cal F}\to R{\underline\Gamma}_Y{\cal F}\to Rj_*R{\underline\Gamma}_{Y-Z}j^{-1}{\cal F}\stackrel{+1}{\to}$$where $j:(X-Z)\to X$ is the open immersion: Take an injective resolution $I^{\bullet}$ of ${\cal F}$, then $j^{-1}I^{\bullet}$ is an injective resolution of $j^{-1}{\cal F}$, and $$0\to {\underline\Gamma}_Z I^{\bullet}\to {\underline\Gamma}_Y I^{\bullet}\to j_*{\underline\Gamma}_{Y-Z}j^{-1}I^{\bullet}\to 0$$ is exact.\
[**(B):**]{} The dagger case. Again (b) follows from (a). For (a), first assume that $Y$ is also smooth. As in the proof of \[kohkomp\], we find an affinoid admissible open covering $X=\cup_{i\in I}U_i$ such that for each $i\in I$ either $U_i\cap Y$ is empty or there exists an isomorphism $\phi_i:U_i\cong{\bf D}^m\times(U_i\cap Y)$ such that $U_i\cap Y\to U_i$ is the zero section. By a Cech argument one sees that it is enough to prove that for all finite and non-empty subsets $J$ of $I$, if we set $U_J=\cap_{i\in J}U_i$, the canonical map$$R\Gamma(U_J,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))\to R\Gamma(U_J,{\mbox{\rm DR}}(R{\underline\Gamma}_{Y}{\cal O}_X))$$is an isomorphism. If $U_J\cap Y$ is empty this is trivial, so we assume $U_J\cap Y$ is non-empty. Choose one $j\in J$. For $\epsilon\in|k^*|\cap]0,1]$ let ${\bf D}(\epsilon)$ be the closed disk of radius $\epsilon$ (with its dagger structure) and $$U_{j,J,\epsilon}=\phi_j^{-1}({\bf D}^m(\epsilon)\times(U_J\cap Y)).$$The set of the open subspaces $U_{j,J,\epsilon}$ is cofinal in the set of all open neighbourhoods of $U_J\cap Y$ in the affinoid space $U_{j,J,1}$. Since $U_J\cap U_{j,J,1}$ is such a neighbourhood, we find an $\epsilon_0$ such that $U_{j,J,\epsilon_0}\subset U_J$. Now observe that the canonical restriction maps$$R\Gamma(U_J,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))\to R\Gamma(U_{j,J,\epsilon_0},{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))$$$$R\Gamma(U_J,{\mbox{\rm DR}}(R{\underline\Gamma}_{Y}{\cal O}_X))\to R\Gamma(U_{j,J,\epsilon_0},{\mbox{\rm DR}}(R{\underline\Gamma}_{Y}{\cal O}_X))$$are isomorphisms. Therefore we need to show that $$R\Gamma(U_{j,J,\epsilon_0},{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))\to R\Gamma(U_{j,J,\epsilon_0},{\mbox{\rm DR}}(R{\underline\Gamma}_{Y}{\cal O}_X))$$is an isomorphism. In other words, we may assume from the beginning that $X={\bf D}^m\times Y$ and $Y\to X$ is the zero section. Let $D={\bf D}^m\subset{\bf P}^m_k=(\mbox{projective space})=P$, let $0$ be its origin, let $W=P\times Y$, $V=P-\{0\}$ and think of $Y=\{0\}\times Y$ as embedded into $W$. Since the natural restriction maps $$R\Gamma(W,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_W))\to R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_X))$$ $$R\Gamma(W,{\mbox{\rm DR}}(R{\underline\Gamma}_Y{\cal O}_W))\to R\Gamma(X,{\mbox{\rm DR}}(R{\underline\Gamma}_Y{\cal O}_X))$$are isomorphisms, it suffices to show that $$R\Gamma(W,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_W))\to R\Gamma(W,{\mbox{\rm DR}}(R{\underline\Gamma}_Y{\cal O}_W))$$ is an isomorphism. The dagger spaces $\{0\}, P$ and $V$ are partially proper, therefore [**(A)**]{}(b) applies to give us the long exact Gysin sequence $$\ldots\to H_{dR}^{i-2m}(\{0\})\to
H_{dR}^i(P)\to H_{dR}^i(V)\to H_{dR}^{i-2m+1}(\{0\})\to\ldots.$$ By the Künneth formulas (in this case easily derived from \[semtub\] below), we thus obtain the long exact sequence $$\ldots\to H_{dR}^{i-2m}(Y)\to
H_{dR}^i(W)\to H_{dR}^i(W-Y)\to H_{dR}^{i-2m+1}(Y)\to\ldots.$$Because of $H_{dR}^{*-2m}(Y)\cong H^*(W,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_W))$, this implies what we want.\
Now for arbitrary $Y$, we may as in [**(A)**]{} suppose that $Y$ is defined by a single equation and that $X$ is affinoid. Then we can reduce to the case where $Y$ is a divisor with normal crossings as in [@grodr], considering a proper surjective morphism $g:X'\to X$ with $X'$ smooth, $U'=g^{-1}(U)\to U$ an isomorphism and $g^{-1}(Y)$ a divisor with normal crossings on $X'$ (such a $g$ exists by \[hironaka\]). But in view of the Mayer Vietoris sequences from . and ., the normal crossings divisor case is equivalent with the case where $Y$ is smooth, which has been treated above.\
\[blasexa\] Let $f:X'\to X$ be a proper morphism of dagger (or rigid) spaces, $Y\to X$ a closed immersion such that $f|_{X'-Y'}:(X'-Y')\to (X-Y)$ is an isomorphism, where $Y'=X'\times_XY$. Let $b:X\to Z$ be a closed immersion into a smooth space and $a:X'\to W'$ a locally closed immersion into a smooth proper space of pure dimension $n$. Then $(a,b\circ f):X'\to W'\times Z=Z'$ is a closed embedding, and there is a long exact sequence$$\ldots\to H^i(Z',{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y'}{\cal O}_{Z'}))\to H^{i-2n}(Z,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y}{\cal O}_Z))\oplus H^i(Z',{\mbox{\rm DR}}(R{\underline\Gamma}_{*X'}{\cal O}_{Z'}))$$ $$\to H^{i-2n}(Z,{\mbox{\rm DR}}(R{\underline\Gamma}_{*X}{\cal O}_Z))\to H^{i+1}(Z',{\mbox{\rm DR}}(R{\underline\Gamma}_{*Y'}{\cal O}_{Z'}))\to\ldots.$$
[Proof:]{} By \[spur\], there is a morphism between the acyclic complexes which we get when we apply \[algana\](b) to $Y'\to X'\to Z'$ and to $Y\to X\to Z$; observe $$H^i(Z'-Y',{\mbox{\rm DR}}(R{\underline\Gamma}_{*X'}{\cal O}_{Z'}))\cong H^i(W'\times(Z-Y),{\mbox{\rm DR}}(R{\underline\Gamma}_{*X'}{\cal O}_{Z'}))$$for this, and that by the construction of the trace map in \[spur\], it is indeed a morphism of complexes, i.e. the resulting diagrams commute. Every third rung of this morphism of complexes is bijective (also by \[spur\]), therefore we can perform a diagram chase according to the pattern [@hadr],p.44.\
\[rigdrdf\] Let $X_1$ and $X_2$ be smooth dagger spaces, let $\phi:X_1'\to X_2'$ be an isomorphism of the associated rigid spaces. Then $\phi$ gives rise to an isomorphism $\phi^{\dagger}:H^*_{dR}(X_2)\cong H^*_{dR}(X_1)$.
[Proof:]{} Set $X=X_1\times X_2$, $X'=X_1'\times X_2'$ and $\tilde{\phi}=(id,\phi):X_1'\to X'$, and let $\Delta={\mbox{\rm im}}(\tilde{\phi})$, a Zariski closed subspace of $X'$ isomorphic to $X_1'$. The canonical projections $X_1\leftarrow X\to X_2$ induce maps $$H^*_{dR}(X_1)\stackrel{a_1}{\longrightarrow}H^*(X,\lim_{\stackrel{\to}{V}}j_{V*}\Omega^{\bullet}_V)\stackrel{a_2}{\longleftarrow}H^*_{dR}(X_2),$$where in the middle term $V$ runs through the open immersions $j_V:V\to X$ of dagger spaces with $\Delta\subset V'$, where $V'$ is the rigid space associated with $V$, regarded as an open subspace of $X'$. We claim that the $a_i$ are isomorphisms. The claim is local, so we may assume $X_1, X_2$ are affinoid and connected, there are elements $t_1,\ldots, t_m\in{\cal O}_{X_1}(X_1)=A_1$ such that $dt_1,\ldots, dt_m$ is a basis of $\Omega_{X_1}^1(X_1)$ over $A_1$, an open affinoid subspace $U\subset X$ such that $\Delta\subset U'$, where $U'\subset X'$ is the associated rigid space, an element $\delta\in\Gamma^*$, and an isomorphism of rigid spaces$$\rho:U'\to{\mbox{\rm Sp}}(k<\delta^{-1}.Z_1,\ldots,\delta^{-1}.Z_m>)\times\Delta$$ where $\delta^{-1}.Z_i$ is sent to $\delta^{-1}.(t_i\otimes1-1\otimes(\phi^*)^{-1}(t_i))\in {\cal O}_{U'}(U')$, compare [@kidr],1.18. For $0<\epsilon\le\delta$ let $$U_{\epsilon}'=\rho^{-1}({\mbox{\rm Sp}}(k<\epsilon^{-1}.Z_1,\ldots,\epsilon^{-1}.Z_m>)\times\Delta),$$ an open subspace of $X'$, and let $U_{\epsilon}$ be the corresponding open subspace of $X$. Since $U_{\epsilon}'$ is a Weierstrass domain in $X'$, the same is true for $U_{\epsilon}$ in $X$ (if necessary, modify the defining functions slightly to get overconvergent ones); in particular, $U_{\epsilon}$ is affinoid, so $j_{{U_{\epsilon}}*}\Omega^{\bullet}_{U_{\epsilon}}=Rj_{{U_{\epsilon}}*}\Omega^{\bullet}_{U_{\epsilon}}$. Since $X$ is quasi-compact, $\lim_{\stackrel{\to}{V}}$ commutes with formation of cohomology, and since the ${U_{\epsilon}}$ are cofinal in $\{V\}$, it is now enough to show that for arbitrary $0<\epsilon\le\delta$ the maps $$b_{i,\epsilon}:H_{dR}^*(X_i)\to H_{dR}^*(U_{\epsilon})$$ are isomorphisms ($i=1,2$). By [@boartin], we can find an isomorphism $\sigma:X_1\to X_2$ such that the induced map $\sigma':X_1'\to X_2'$ is close to $\phi$; in particular so close that for $\tilde{\sigma}=(id,\sigma):X_1\to X$ we have ${\mbox{\rm im}}(\tilde{\sigma})\subset U_{\epsilon/2}$. Similarly, we can approximate the map ${\cal O}_{X_1'}(X_1')\cong{\cal O}_{\Delta}(\Delta)\stackrel{\rho^*}{\to}{\cal O}_{U'}(U')\to{\cal O}_{U_{\epsilon}'}(U_{\epsilon}')$ by a map ${\cal O}_{X_1}(X_1)\to{\cal O}_{U_{\epsilon}}(U_{\epsilon})$. Its extension to the map$${\cal O}_{X_1}(X_1)\otimes_k^{\dagger}k<\epsilon^{-1}.Z_1,\ldots,\epsilon^{-1}.Z_m>^{\dagger}\to{\cal O}_{U_{\epsilon}}(U_{\epsilon})$$which sends $\epsilon^{-1}.Z_i$ to $\epsilon^{-1}.(t_i\otimes1-1\otimes(\sigma^*)^{-1}(t_i))$ is an isomorphism since its completion is close to the isomorphism obtained from $\rho$. So we have an isomorphism $$U_{\epsilon}\cong{\mbox{\rm Sp}}(k<\epsilon^{-1}.Z_1,\ldots,\epsilon^{-1}.Z_m>^{\dagger})\times X_1$$ where the closed immersion $\tilde{\sigma}:X_1\to U_{\epsilon}$ corresponds to the zero section. Hence the maps $H_{dR}^*(U_{\epsilon})\to H_{dR}^*(X_1)$ induced by $\tilde{\sigma}$ are isomorphisms, by \[semtub\]. Since $\tilde{\sigma}$ is a section for the canonical map $U_{\epsilon}\to X_1$ which gives rise to $b_{1,\epsilon}$, we derive the bijectivity of $b_{1,\epsilon}$. That $b_{2,\epsilon}$ is bijective is seen symmertrically. Now we define $\phi^{\dagger}=a_1^{-1}\circ a_2$.\
One can show that this construction is compatible with compositions: If $X_3$ is a third dagger space with associated rigid space $X_3'$, and if $\gamma:X_2'\to X_3'$ is an isomorphism, than $\phi^{\dagger}\circ\gamma^{\dagger}=(\gamma\circ\phi)^{\dagger}$, see [@en3dag]. We do not need this here.\
De Rham cohomology of tubes of a semi-stable reduction
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From now on let $R$ be a complete discrete valuation ring of mixed characteristic $(0,p)$, let $\pi\in R$ be a uniformizer, $k$ its fraction field, and $\bar{k}=R/(\pi)$ its residue field.
\[semtub\] (a) Let $r, n\in\mathbb{Z}$, $0\le r\le n$, let $\mu$ and $\gamma_i$ for $1\le i\le r$, and $\delta_i$ for $1\le i\le n$ be elements of $\Gamma^*$ such that $\gamma_i\le\delta_i$ for all $1\le i\le r$, and $\prod_{i=1}^r\delta_i\ge \mu$. Define the open dagger subspace $V$ of the dagger affine space ${\bf A}_k^n$ by$$V=\{(x_1,\ldots,x_n)\in{\bf A}_k^n|$$$$\prod_{i=1}^r|x_i|\ge\mu, |x_i|\le\delta_i \mbox{ for all }1\le i\le n,\mbox{ and }|x_i|\ge\gamma_i \mbox{ for all }1\le i\le r\}.$$Let $X_1,\ldots, X_r$ be the first $r$ standard coordinates on ${\bf A}_k^n$. Then $H_{dR}^q(V)$ is the $k$-vector space generated by the classes of the $q$-forms $$dX_{i_1}/X_{i_1}\wedge\ldots\wedge dX_{i_q}/X_{i_q}$$ with $1\le i_1<\ldots<i_q\le r$. In particular, if $r=0$ we have $H_{dR}^q(V)=0$ for all $q>0$. If $X$ is another smooth dagger space, the canonical maps$$\oplus_{q_1+q_2=q}H_{dR}^{q_1}(X)\otimes_kH_{dR}^{q_2}(V)\stackrel{\beta}{\to}H_{dR}^q(X\times V)$$ are bijective.\
(b) Suppose even $\gamma_i<\delta_i$ for all $1\le i\le r$, and $\prod_{i=1}^r\delta_i<\mu$. Define the open dagger (resp. rigid) subspace $V$ of the dagger (resp. rigid) affine space ${\bf A}_k^n$ by$$V=\{(x_1,\ldots,x_n)\in{\bf A}_k^n|$$$$\prod_{i=1}^r|x_i|>\mu, |x_i|<\delta_i \mbox{ for all }1\le i\le n,\mbox{ and }|x_i|>\gamma_i \mbox{ for all }1\le i\le r\}.$$ Then the same assertions as in (a) hold (of course, if $V$ is the dagger (resp. rigid) space, then $X$ should be a dagger (resp. rigid) space, too).
[Proof:]{} (a) Note that $V$ is affinoid. We may assume that $X$ is also affinoid and connected, $X={\mbox{\rm Sp}}(B)$. After a finite extension of $k$ we may assume there are $\underline{\delta}_i, \underline{\gamma}_i$ and $\underline{\mu}$ in $k$ such that $|\underline{\delta}_i|=\delta_i$, $|\underline{\gamma}_i|=\gamma_i$ and $|\underline{\mu}|=\mu$. We regard ${\cal O}_{X\times V}(X\times V)$ as a subring of $$B<\underline{\delta}_1^{-1}.X_1,(\underline{\delta}_1^{-1}.X_1)^{-1},\ldots,\underline{\delta}_n^{-1}.X_n,(\underline{\delta}_n^{-1}.X_n)^{-1}>$$(to compute in this ring was suggested by the referee).\
(i) We begin with the following observation. Let $$a=\sum_{v\in\mathbb{Z}^n}\theta_v\prod_{j=1}^nX_j^{v_j}\in{\cal O}_{X\times V}(X\times V),$$$\theta_v\in B$, $\theta_v=0$ whenever there is a $r< j\le n$ with $v_j<0$. Fix $1\le l\le n$. We claim$$b=\sum_{\stackrel{v\in\mathbb{Z}^n}{v_l\ne0}}v_l^{-1}\theta_v\prod_{j=1}^nX_j^{v_j}\in{\cal O}_{X\times V}(X\times V),$$i.e. also this sum converges in ${\cal O}_{X\times V}(X\times V)$. Indeed, consider the surjection of dagger algebras $$D=B\otimes_k^{\dagger}k<X_1,\ldots,X_n,Y_1,\ldots,Y_r,Z>^{\dagger}\stackrel{\tau}{\longrightarrow}{\cal O}_{X\times V}(X\times V),$$$$X_i\mapsto\underline{\delta}_i^{-1}.X_i,\quad\quad Y_i\mapsto(\underline{\gamma}_i^{-1}.X_i)^{-1},\quad\quad Z\mapsto(\underline{\mu}^{-1}.\prod_{i=1}^rX_i)^{-1}.$$By definition of $\otimes_k^{\dagger}$, we have $$D=\lim_{\stackrel{\to}{X_{\epsilon},\delta}}{\cal O}_{X_{\epsilon}}(X_{\epsilon})\hat{\otimes}_kk<\delta^{-1}.X_1,\ldots,\delta^{-1}.X_n,\delta^{-1}.Y_1,\ldots,\delta^{-1}.Y_r,\delta^{-1}.Z>$$where the $X_{\epsilon}$ run through the strict neighbourhoods of $X'$, the rigid space associated with $X$, in an appropriate affinoid rigid space which contains $X'$ as a relatively compact open subset, and where $\delta$ runs through all $\delta>1$. Hence, a given $$c=\sum_{w\in\mathbb{Z}_{\ge0}^{n+m+1}}\beta_w(\prod_{j=1}^nX_j^{w_j})(\prod_{j=1}^rY_j^{w_{j+n}})Z^{w_{n+r+1}}\in D$$$(\beta_w\in B)$ is an element of $${\cal O}_{X_{\epsilon}}(X_{\epsilon})\hat{\otimes}_kk<\delta^{-1}.X_1,\ldots,\delta^{-1}.X_n,\delta^{-1}.Y_1,\ldots,\delta^{-1}.Y_r,\delta^{-1}.Z>$$ for some $X_{\epsilon},\delta$, and one easily sees that $$d=\sum_{\stackrel{w\in\mathbb{Z}_{\ge0}^{n+m+1}}{w_l-w_{l+n}-w_{n+r+1}\ne 0}}(w_l-w_{l+n}-w_{n+r+1})^{-1}\beta_w(\prod_{j=1}^nX_j^{w_j})(\prod_{j=1}^rY_j^{w_{j+n}})Z^{w_{n+r+1}}$$ is then an element of $${\cal O}_{X_{\epsilon}}(X_{\epsilon})\hat{\otimes}_kk<\delta_1^{-1}.X_1,\ldots,\delta_1^{-1}.X_n,\delta_1^{-1}.Y_1,\ldots,\delta_1^{-1}.Y_r,\delta_1^{-1}.Z>$$ for any $1<\delta_1<\delta$; in particular $d\in D$, too. Clearly, if $\tau(c)=a$, then $\tau(d)=b$, and the claim follows.\
(ii) For $1\le i_1<\ldots<i_t\le n$ we write $$dX_{i}/X_i=dX_{i_1}/X_{i_1}\wedge\ldots\wedge dX_{i_t}/X_{i_t}$$ $$(dX_{i}/X_i)^{\hat{i_t}}=dX_{i_1}/X_{i_1}\wedge\ldots\wedge dX_{i_{t-1}}/X_{i_{t-1}}.$$ Every $\omega\in\Omega^q_{X\times V}(X\times V)$ can uniquely be written as a convergent series $$\begin{gathered}
\omega=\sum_{0\le t\le q}\sum_{1\le i_1<\ldots<i_t\le n}\sum_{v\in\mathbb{Z}^n}\sigma_{t,i,v}(\prod_{j=1}^nX_j^{v_j})dX_{i}/X_i\tag{$*$}\end{gathered}$$ with $\sigma_{t,i,v}\in\Omega^{q-t}_X(X)$ and $\sigma_{t,i,v}=0$ whenever there is a $j$ with $r<j\le n$ and $v_j\le 0$. For $0\le l\le n$ let $\Omega_l^q\subset \Omega^q_{X\times V}(X\times V)$ be defined by the additional condition: $\sigma_{t,i,v}=0$ whenever $l<i_t$ or $v_j\ne0$ for some $j$ with $l<j\le n$; this condition means, that no $X_j$, no $dX_j/X_j$ for $l<j$ occurs in $\omega$. We claim that, if $l>0$, every $\omega\in\Omega^q_l$ with $d\omega=0$ can be written modulo exact forms as $\omega=\omega_0+\omega_1dX_l/X_l$ with $\omega_0\in\Omega^q_{l-1}$ and $\omega_1\in\Omega^{q-1}_{l-1}$ and $d\omega_0=0$ and $d\omega_1=0$, and even $\omega_1=0$ if $l>r$. Indeed, if $\omega$ is represented as in $(*)$, then by (i) the series$$\eta=\sum_{0\le t\le q}\sum_{1\le i_1<\ldots<i_t=l}\sum_{\stackrel{v\in\mathbb{Z}^n}{v_l\ne0}}v_l^{-1}\sigma_{t,i,v}(\prod_{j=1}^nX_j^{v_j})(dX_{i}/X_i)^{\hat{i_t}}$$converges in $\Omega^{q-1}_l$, and subtracting $d\eta$ from $\omega$ we see that we may suppose $\sigma_{t,i,v}=0$ whenever $i_t=l$ and $v_l\ne0$. But $d\omega=0$ implies $\sigma_{t,i,v}=0$ whenever $v_l\ne0$ and $i_t<l$, so in any case we have $\sigma_{t,i,v}=0$ if only $v_l\ne0$. From this the claim follows.\
(iii) Let $\omega\in\Omega^q_{X\times V}(X\times V)=\Omega_n^q$ be such that $d\omega=0$. Repeated application of (ii) shows that modulo exact forms $\omega$ can be written as $$\omega=\sum_{0\le t\le q}\sum_{1\le i_1<\ldots<i_t\le r}\sigma_{t,i}dX_{i}/X_i$$with uniquely determined $\sigma_{t,i}\in\Omega^{q-t}_X(X)$ such that $d\sigma_{t,i}=0$. This provides us with an inverse map for $\beta$ and proves the assertion on $H_{dR}^q(V)$.\
(b) Again we may assume $X$ is affinoid. In the dagger case, we then exhaust $V$ by affinoid dagger spaces of the type considered in (a) and conclude by using our result in (a) (passing to the limit). In the rigid case, $X\times V$ is quasi-Stein, hence acyclic for coherent modules ([@kiaub]). Therefore, in this case, we can argue literally as in (a) — since here $V$ is defined by strict inequalities, we do not need overconvergence.\
We call a closed immersion ${\cal Z}\to{\cal X}$ of noetherian $\pi$-adic formal $R$-schemes a strictly semi-stable formal pair $({\cal X},{\cal Z})$ over $R$, if there is an $n\in\mathbb{N}$, a Zariski open covering ${\cal X}=\cup_i{\cal U}_i$ and for all $i$ a pair $s(i), r(i)\in\mathbb{N}$ with $n\ge s(i)\ge r(i)\ge 1$ and an étale morphism $$q_i:{\cal U}_i\to{\mbox{\rm Spf}}(R<X_1,\ldots,X_n>/(X_1\ldots X_{r(i)}-\pi))$$such that ${\cal Z}|_{{\cal U}_i}=\cup_{j=r(i)+1,\ldots,s(i)}V(q_i^*X_j)$. We call ${\cal X}$ a strictly semi-stable formal $R$-scheme if $({\cal X},\emptyset)$ is a strictly semi-stable formal pair over $R$.\
For a $\pi$-adic topologically finite type (tf) formal $R$-scheme ${\cal X}$ with generic fibre ([@bolu]) the rigid space ${\cal X}_k$, there is a specialization map $s:{\cal X}_k\to{\cal X}$, and if $Y\to{\cal X}_{\bar{k}}$ is an immersion into the closed fibre, then $]Y[_{\cal X}=s^{-1}(X)$ is an admissible open subspace of ${\cal X}_k$, the tube of $Y$.
\[monovor\] Let ${\cal X}$ be a strictly semi-stable formal $R$-scheme, let ${\cal X}_{\bar{k}}=\cup_{i\in I}Y_i$ be the decomposition of the closed fibre into irreducible components. For $K\subset I$ set $Y_K=\cap_{i\in K}Y_i$. Let $X^{\dagger}$ be a $k$-dagger space such that its associated rigid space is identified with ${\cal X}_k$. For a subscheme $Y\subset{\cal X}_{\bar{k}}$ let $]Y[_{\cal X}^{\dagger}$ be the open dagger subspace of $X^{\dagger}$ corresponding to the open rigid subspace $]Y[_{\cal X}$ of ${\cal X}_k$. Then for any $\emptyset\ne J\subset I$ the canonical map $$H_{dR}^*(]Y_J[_{\cal X}^{\dagger})\to H_{dR}^*(]Y_J-(Y_J\cap(\cup_{i\in I-J}Y_i))[^{\dagger}_{\cal X})$$ is bijective.
[Proof:]{} (i) Suppose $I\ne J$. For $L\subset(I-J)$ set $$G_L=]Y_J-(Y_J\cap(\cup_{i\in L}Y_i))[^{\dagger}_{\cal X}.$$ For $m\in\mathbb{N}$ let $P_m(I-J)$ be the set of subsets of $I-J$ with $m$ elements, and $G^m=\cup_{L\in P_m(I-J)}G_L$. Let $G^0=]Y_J[^{\dagger}_{\cal X}$. Then $G^{m+1}\subset G^m$ for all $m\ge0$, and $$G^{|I-J|}=]Y_J-(Y_J\cap(\cup_{i\in I-J}Y_i))[^{\dagger}_{\cal X}.$$ It is enough to show that $$\begin{gathered}
H_{dR}^*(G^m)\to H_{dR}^*(G^{m+1})\tag*{$(*)_m$}\end{gathered}$$ is bijective for all $m\le|I-J|-1$. Since $G^m=\cup_{L\in P_m(I-J)}G_L$ is an admissible covering, it is enough to show that $$\begin{gathered}
H_{dR}^*(\cap_{L\in Q}G_L)\to H_{dR}^*(\cap_{L\in Q}G_L\cap G^{m+1})\tag*{$(**)_m$}\end{gathered}$$ is bijective for all $Q\subset P_m(I-J)$. But $\cap_{L\in Q}G_L=G_M$ for $M=\cup_{L\in Q}L$, and $$\cap_{L\in Q}G_L\cap G^{m+1}=\cup_{i\in(I-(M\cup J))}(G_M-(G_M\cap]Y_i[^{\dagger}_{\cal X})).$$ If now ${\cal X}$ is replaced by its open formal subscheme which on the underlying topological space is the complement of $\cup_{i\in M}Y_i$ in ${\cal X}_{\bar {k}}$, then this means the replacement of $]Y_J[^{\dagger}_{\cal X}$ by $G_M$ and of $I$ by $I-M$ (but $J$ stays the same). In this way $(**)_m$ takes the form $(*)_0$, therefore it suffices to prove $(*)_0$. Note $G^1=]Y_J[^{\dagger}_{\cal X}-]Y_I[^{\dagger}_{\cal X}$, i.e. we must prove that $$H_{dR}^*(]Y_J[_{\cal X}^{\dagger})\to H_{dR}^*(]Y_J-Y_I[^{\dagger}_{\cal X})$$ is bijective.\
(ii) Let ${\cal X}=\cup_{s\in S}{\cal U}_s$ be an open covering of ${\cal X}$ as in the definition of strict semi-stability. For a finite non-empty subset $T$ of $S$ let ${\cal U}_T=\cap_{s\in T}{\cal U}_s$. It is enough to show that for each such $T$ the map $$H_{dR}^*(]Y_J\cap{\cal U}_T[_{{\cal U}_T}^{\dagger})\to H_{dR}^*(](Y_J-Y_I)\cap{\cal U}_T[_{{\cal U}_T}^{\dagger})$$is bijective. This is trivial if $Y_I\cap{\cal U}_T$ is empty. If $Y_I\cap{\cal U}_T$ is non-empty, the irreducible components of the reduction $({\cal U}_T)_{\bar{k}}$ of ${\cal U}_T$ correspond bijectively to those of ${\cal X}_{\bar{k}}$, so we can replace ${\cal X}$ by ${\cal U}_T$. In other words, it is enough to prove the statement in (i) in the following case: ${\cal X}={\mbox{\rm Spf}}(A)$ is affine, and there is an étale morphism ${\cal X}={\mbox{\rm Spf}}(A)\stackrel{\phi}{\to}{\mbox{\rm Spf}}(R<X_1,\ldots,X_n>/(X_1\ldots X_r-\pi))$. Let $f_i=\phi^*(X_i)\in A$, inducing $\bar{f}_i\in A/(\pi)$. Passing to an open covering of ${\cal X}$ we may suppose that each $V(\bar{f_i})$ is irreducible (and non-empty), so we derive an identification $\{1,\ldots,r\}=I$. For $\lambda\in\Gamma^*$ with $\lambda<1$ set $$F_{\lambda}=\{x\in G^0|\quad|f_j(x)|\le\lambda\mbox{ for all }j\in J\}$$ and $E_{\lambda}=F_{\lambda}\cap G^1$. Then $G^0=\cup_{\lambda<1}F_{\lambda}$ is an admissible covering, and to prove $(*)_0$ it suffices to show the bijectivity of $H_{dR}^*(F_{\lambda})\to H_{dR}^*(E_{\lambda})$ for all such $\lambda$.\
(iii) For $\beta\in\Gamma^*$ with $\beta<1$ and $i\in I-J$ set $$F_{\lambda,\beta}^i=\{x\in F_{\lambda}|\quad|f_i(x)|\ge\beta\}\quad\mbox{ and }\quad F_{\lambda,\beta}=\cup_{i\in I-J}F_{\lambda,\beta}^i.$$ We have $E_{\lambda}\subset F_{\lambda,\beta}\subset F_{\lambda}$, and it suffices now to prove that the following maps are bijective:$$\begin{gathered}
\quad\lim_{\stackrel{\to}{\beta\to 1}}H^*_{dR}(F_{\lambda,\beta})\to H^*_{dR}(E_{\lambda})\tag{1}\\\quad H^*_{dR}(F_{\lambda})\to H^*_{dR}(F_{\lambda,\beta})\quad\mbox{ for }\beta<1.\tag{2}\end{gathered}$$ Note $G_{\{i\}}=]Y_J-(Y_J\cap Y_i)[^{\dagger}_{\cal X}=\{x\in]Y_J[^{\dagger}_{\cal X}; |f_i(x)|=1\}$ for $i\in I-J$. We compare the admissible covering $E_{\lambda}=\cup_{i\in I-J}(G_{\{i\}}\cap F_{\lambda})$ with the admissible covering $F_{\lambda,\beta}=\cup_{i\in I-J}F_{\lambda,\beta}^i$: since the direct limit is exact, to prove the bijectivity of (1), it is enough to prove the bijectivity of $$\begin{gathered}
\quad\lim_{\stackrel{\to}{\beta\to 1}}H^*_{dR}(\cap_{i\in K}F^i_{\lambda,\beta})\to H^*_{dR}(\cap_{i\in K}G_{\{i\}}\cap F_{\lambda})\tag{3}\end{gathered}$$for all $\emptyset\ne K\subset(I-J)$. But the affinoid dagger space $\cap_{i\in K}G_{\{i\}}\cap F_{\lambda}$ is the inverse limit of the affinoid dagger spaces $\cap_{i\in K}F^i_{\lambda,\beta}$, in particular $$\Gamma(\cap_{i\in K}G_{\{i\}}\cap F_{\lambda},\Omega^{\bullet})=\lim_{\stackrel{\to}{\beta\to 1}}\Gamma(\cap_{i\in K}F^i_{\lambda,\beta},\Omega^{\bullet}),$$ so the bijectivity of (3) follows from the exactness of direct limits.\
(iv) It remains to prove the bijectivity of the maps (2). Set $$S_{\lambda,\beta}=\{x\in F_{\lambda}|\quad|f_i(x)|\le\beta\mbox{ for all }i\in I-J\}.$$ Then $F_{\lambda}=S_{\lambda,\beta}\cup F_{\lambda,\beta}$ is an admissible covering, and the bijectivity of (2) is equivalent to that of $$\begin{gathered}
\quad H^*_{dR}(S_{\lambda,\beta})\to H^*_{dR}(S_{\lambda,\beta}\cap F_{\lambda,\beta}).\tag{4}\end{gathered}$$ (v) We claim that there is a $\pi$-adic affine formally smooth tf formal ${\mbox{\rm Spf}}(R)$-scheme ${\mbox{\rm Spf}}(D)$ and an isomorphism of rigid spaces
$$]Y_I[_{{\cal X}}=]V((\bar{f}_i)_{i=1,\ldots,r})[_{{\cal X}}\cong$$$${\mbox{\rm Sp}}(D\otimes_Rk)\times\{x\in{\mbox{\rm Sp}}(k<T_1,\ldots,T_r>/(T_1\ldots T_r-\pi))|\quad|T_i(x)|<1\mbox{ for all }i\}$$such that $f_i\in A$ corresponds to $T_i$. This is constructed as follows: Let $\hat{A}$ be the $(f_1,\ldots,f_r)$-adic completion of $A$. Since $\hat{A}/(\pi,f_1,\ldots,f_r)=A/(\pi,f_1,\ldots,f_r)=\bar{D}$ is a smooth $\bar{k}$-algebra ($\phi$ is étale), there is a section $\bar{D}\stackrel{\bar{s}}{\to}\hat{A}/(\pi)$ for the canonical surjection $\hat{A}/(\pi)\to\bar{D}$, a lift of $\bar{D}$ to a smooth $R$-algebra $\tilde{D}$ (see [@elk]) and a lift of $\bar{s}$ to a morphism $D\stackrel{s}{\to}\hat{A}$, where $D$ is the $\pi$-adic completion of $\tilde{D}$. The extension $D[[T_1,\ldots,T_r]]\to\hat{A},\quad T_i\mapsto f_i$ of $s$ induces an isomorphism $$D[[T_1,\ldots,T_r]]/(T_1\ldots T_r-\pi)\cong\hat{A}.$$ This gives the desired isomorphism of rigid spaces (compare [@berco], 0.2.7, for the computation of tubes).\
(vi) To prove the bijectivity of (4), we may now, in view of \[rigdrdf\], assume that there is a smooth $k$-dagger algebra $B$ and an isomorphism of dagger spaces $$]Y_I[^{\dagger}_{{\cal X}}=]V((\bar{f}_i)_{i=1,\ldots,r})[^{\dagger}_{{\cal X}}\cong$$$${\mbox{\rm Sp}}(B)\times\{x\in{\mbox{\rm Sp}}(k<T_1,\ldots,T_r>^{\dagger}/(T_1\ldots T_r-\pi))|\quad|T_i(x)|<1\mbox{ for all }i\}$$such that $f_i\in A$ corresponds to $T_i$. Let $$N=\{x\in S_{\lambda,\beta}|\quad|f_i(x)|=\beta\mbox{ for all }i\in I-J\}.$$ It suffices to show the bijectivity of the two maps $$\begin{gathered}
\quad H^*_{dR}(S_{\lambda,\beta})\to H^*_{dR}(N)\tag{5}\\\quad H^*_{dR}(S_{\lambda,\beta}\cap F_{\lambda,\beta})\to H^*_{dR}(N).\tag{6}\end{gathered}$$The bijectivity of $(5)$ follows immediately from \[semtub\]. Finally, consider the admissible covering $S_{\lambda,\beta}\cap F_{\lambda,\beta}=\cup_{i\in I-J}(S_{\lambda,\beta}\cap F_{\lambda,\beta}^i)$. To prove the bijectivity of $(6)$, it is enough to prove that of $$\begin{gathered}
H^*_{dR}(\cap_{i\in K}S_{\lambda,\beta}\cap F^i_{\lambda,\beta})\to H^*_{dR}(N)\tag{7}\end{gathered}$$for all $\emptyset\ne K\subset(I-J)$, which again can be done using \[semtub\].
\[maxschni\] In \[monovor\], suppose in addition that ${\cal X}$ is quasi-compact. Then for every $J\subset I$ and $q\in\mathbb{N}$ one has $\dim_k(H_{dR}^q(]Y_J[^{\dagger}_{\cal X}))<\infty$, and also $\dim_k(H^q_{dR}(X^{\dagger}))<\infty$.
[Proof:]{} Since $X^{\dagger}=\cup_{i\in I}]Y_i[^{\dagger}_{\cal X}$ is a finite admissible covering, the second claim follows from the first. For the first we may, due to \[monovor\], assume that $J=I$, shrinking ${\cal X}$ if necessary. Passing to a finite covering, we may assume that ${\cal X}$ is affine, and that there is an étale morphism ${\cal X}\stackrel{\phi}{\to}{\mbox{\rm Spf}}(R<X_1,\ldots,X_n>/(X_1\ldots X_r-\pi))$ such that $Y_I=\cap_{i\in I}Y_i=\cap_{i=1}^rV(f_i)$ with $f_i=\phi^*(X_i)$. From the proof of \[monovor\] we see that we may assume, again due to \[rigdrdf\], that there is a smooth $R$-algebra $\tilde{D}$, with weak completion $D^{\dagger}$, such that, if $B$ denotes the $k$-dagger algebra $D^{\dagger}\otimes_Rk$, we have an isomorphism of dagger spaces $$]Y_I[^{\dagger}_{\cal X}=]V((\bar{f}_i)_{i=1,\ldots,r})[^{\dagger}_{\cal X}\cong$$$${\mbox{\rm Sp}}(B)\times\{x\in{\mbox{\rm Sp}}(k<T_1,\ldots,T_r>^{\dagger}/(T_1\ldots T_r-\pi))|\quad|T_i(x)|<1\mbox{ for all }i\}$$such that $f_i$ corresponds to $T_i$. By \[semtub\] this yields isomorphisms $$H_{dR}^q(]Y_I[^{\dagger}_{\cal X})\cong H_{dR}^q({\mbox{\rm Sp}}(B)\times V)\cong \oplus_{q=q_1+q_2}H_{dR}^{q_1}({\mbox{\rm Sp}}(B))\otimes_kH_{dR}^{q_2}(V)$$where we write $$V=\{(x_1,\ldots,x_{r-1})\in{\bf A}_k^{r-1}|\quad\prod_{i=1}^{r-1}|x_i|>|\pi|, |x_i|<1 \mbox{ for all }1\le i\le r-1\}.$$Now since $Y_I={\mbox{\rm Spec}}(\tilde{D}/(\pi))$ we have $H_{dR}^*({\mbox{\rm Sp}}(B))=H_{MW}^*(Y_I)$, and the latter is known to be finite dimensional by [@berfi],[@mebend]. But by \[semtub\] also $H_{dR}^*(V)$ is finite dimensional. We are done.
\[faser\] Let $n\ge s\ge r\ge 1$, let $$q:{\mbox{\rm Spf}}(A)\to{\mbox{\rm Spf}}(R<X_1,\ldots,X_n>/(X_1\ldots X_r-\pi))$$ be an étale morphism and let $B=A/(q^*X_j)_{j=r+1,\ldots,s}$. There is an isomorphism$$]{\mbox{\rm Spec}}(B/(\pi))[_{{\mbox{\rm Spf}}(A)}\cong {\mbox{\rm Sp}}(B\otimes_Rk)\times({\bf D}^0)^{s-r}$$such that the standard coordinates on $({\bf D}^0)^{s-r}$ correspond to $q^*X_{r+1},\ldots,q^*X_s$.
[Proof:]{} A strictly semi-stable formal $R$-scheme ${\cal X}$ carries a canonical log. structure: The log. structure ${\cal M}_{\cal X}$ associated with the divisor $X$, the reduction modulo $(\pi)$ of ${\cal X}$. In particular, ${\mbox{\rm Spf}}(R)$ gives rise to a formal log. scheme $S$, and $({\cal X},{\cal M}_{\cal X})\to S$ is a log. smooth morphism of formal log. schemes. In our situation, ${\mbox{\rm Spf}}(A)$ and ${\mbox{\rm Spf}}(B)$ are strictly semi-stable formal $R$-schemes, and the embedding $({\cal U},{\cal M}_{\cal U})\to ({\cal V},{\cal M}_{\cal V})$ is an exact closed immersion of formal $S$-schemes.\
Now we prove \[faser\], using the above log. structures. Because of [@EGA],IV, 18.3.2.1, we may work over the truncations mod $(\pi^n)$. Due to the extension property of morphisms from exact nilimmersions to log. smooth objects provided by [@kalo],3.11, as analogous to the classical extension property of morphisms from nilimmersions to smooth objects, one easily verifies the following transposition of [@EGA],0,19.5.4 to the log. context: Let $({\mbox{\rm Spec}}(B),M')\stackrel{g}{\to}({\mbox{\rm Spec}}(A),M)\stackrel{f}{\to}({\mbox{\rm Spec}}(C),N)$ be morphisms of affine log. schemes such that $g$ is an exact closed immersion defined by the ideal $I\subset A$ and such that $f$ and $f\circ g$ are log. smooth. Then $I/I^2$ is a projective $B$-module, and if $\hat{A}$ resp. $\hat{\cal S}$ are the respective $I$-adic completions of $A$ resp. ${\mbox{\rm Sym}}_B(I/I^2)$, then there is a section $B\to\hat{A}$ together with an isomorphism of $B$-algebras $\hat{\cal S}\cong\hat{A}$. The lemma follows, because by the method [@berco],0.2.7 to compute tubes, all we have to do is to construct an $R$-isomorphism $B[[T_1,\ldots,T_{s-r}]]\cong\hat{A}$ with $T_i\mapsto q^*X_{i+r}$, where $\hat{A}$ is the ${\mbox{\rm Ker}}(A\to B)$-adic completion of $A$.\
The finiteness theorem
======================
We recall the terminology from [@dejo]. An $R$-variety is an integral separated flat $R$-scheme of finite type. Let $g:X\to{\mbox{\rm Spec}}(R)$ be an $R$-variety and $X_i, i\in I$, the irreducible components of the closed fibre $X_{\bar {k}}$. For $J\subset I$ set $X_J=\cap_{i\in J}X_i$. Then $X$ is called strictly semi-stable over $R$ if the following (1)-(4) are fulfilled: (1) The generic fibre $X_k$ is smooth over $k$. (2) $X_{\bar {k}}$ is reduced, i.e. $X_{\bar {k}}=\cup_{i\in I}X_i$ scheme theoretically. (3) $X_i$ is a divisor on $X$ for all $i\in I$. (4) $X_J$ is smooth over $\bar{k}$ of codimension $|J|$ for all $J\subset I$, $J\ne\emptyset$.\
Let $Z\subset X$ be Zariski closed with its reduced structure, such that $X_{\bar {k}}\subset Z$. Then $(X,Z)$ is called a strictly semi-stable pair over $R$ if the following (1)-(3) are fulfilled: (1) $X$ is strictly semi-stable over $R$. (2) $Z$ is a divisor with normal crossings on $X$. (3) Decompose $Z=Z_f\cup X_{\bar {k}}$ with $Z_f\to{\mbox{\rm Spec}}(R)$ flat and let $Z_f=\cup_{i\in K}Z_i$ be the decomposition into irreducible components. Then $Z_L=\cap_{i\in L}Z_i$ is a union of strictly semi-stable $R$-varieties for all $L\subset K$, $L\ne\emptyset$.\
Note that for a strictly semi-stable pair $(X,Z)$ over $R$, the $\pi$-adic completion of $Z_f\to X$ is a strictly semi-stable formal pair over $R$ in the sense of ..\
We call a dagger space $H$ quasi-algebraic, if there is an admissible covering of $H$ by dagger spaces $U$, which admit an open embedding into the dagger analytification of a $k$-scheme of finite type. If $H$ is quasi-algebraic, there is even an admissible covering of $H$ by open affinoids $U$ such that for each $U$ there are $n, r\in\mathbb{N}$, polynomials $f_j\in k[X_1,\ldots,X_n]$ and isomorphisms $U\cong{\mbox{\rm Sp}}(k<X_1,\ldots,X_n>^{\dagger}/(f_1,\ldots,f_r))$ (see [@en1dag], 2.18).
\[dejong\]([@dejo],6.5) If $Y$ is a proper $R$-variety and $Z\subset Y$ is a proper Zariski closed subset with $Y_{\bar {k}}\subset Z$, then there is a finite extension $R\to R'$ of complete discrete valuation rings, an $R'$-variety $X$, a proper surjective morphism of $R$-schemes $f:X\to Y$ and an open dense subscheme $U\subset Y$ such that $f^{-1}(U)\to U$ is finite and $(X,f^{-1}(Z)_{red})$ is a strictly semi-stable pair over $R'$.\
\[endlich\] For an affinoid quasi-algebraic dagger space $H$, the numbers $h_q^{dR}(H)$ are finite for all $q\ge0$.
[Proof:]{} Induction on $m=\dim(H)$. We may suppose $$H={\mbox{\rm Sp}}(k<X_1,\ldots,X_n>^{\dagger}/(f_1,\ldots,f_r))$$ with polynomials $f_j\in R[X_1,\ldots,X_n]$, and set $V={\mbox{\rm Spec}}(R[X_1,\ldots,X_n]/(f_1,\ldots,f_r))$. We regard $H$ as an open subspace of the dagger analytification of the generic fibre $V_k$ of $V$. The decomposition of $V$ into irreducible components induces a decomposition of $H$ into Zariski closed quasi-algebraic subspaces. Therefore we can reduce our claim by means of .,(c) and the induction hypothesis to the case where $V$ is irreducible, $m=\dim(V_k)$. Since $h_q^{dR}(H)$ depends only on the reduced structure of $H$, we may assume $V$ is reduced, hence integral. Choose an open immersion $V\to Y$ into a projective $R$-variety $Y$. For the pair $(Y,Y_{\bar {k}})$ choose $R'$ and $f:X\to Y$ and $U\subset Y$ as in \[dejong\] — since $h_q^{dR}(H)$ is not effected by finite extensions of $k$, we may suppose $R=R'$. Let $X_k\stackrel{g_k}{\to}X'_k\stackrel{\pi_k}{\to}Y_k$ be the Stein factorization of the map $f_k:X_k\to Y_k$ of generic fibres; note that $X'_k$ is integral. From [@EGA],III, 4.4.1 it follows that there is a closed subscheme $T'\subset X'_k$, $\dim(T')<m$, such that for $T=g_k^{-1}(T')$ one has: $g_k|_{X_k-T}:(X_k-T)\to (X_k'-T')$ is an isomorphism. Since the locus of smoothness over $k$ is dense in $X'_k$ and in $Y_k$, and since $\pi_k$ is finite, we find a closed subscheme $S\subset Y_k$, $\dim(S)<m$, such that for $S'=X'_k\times_{Y_k}S$ one has: $Y_k-S$ and $X'_k-S'$ are smooth over $k$, and $(X'_k-S')\to(Y_k-S)$ is étale. For a $k$-scheme $L$ of finite type, we denote by $L^{\dagger}$ its analytification as a dagger space. We regard $H$ as an open subspace of $Y_k^{\dagger}$ and set\
$H_X=H\times_{Y_k^{\dagger}}X_k^{\dagger},\quad\quad\quad\quad\quad H_{X'}=H\times_{Y_k^{\dagger}}X_k'^{\dagger},\quad\quad\quad H_S=H\times_{Y_k^{\dagger}}S^{\dagger},$\
$H_{X',S'}=H_{X'}\times_{X_k'^{\dagger}}S'^{\dagger},\quad\quad H_{X,T}=H_X\times_{X_k^{\dagger}}T^{\dagger},\quad\quad H_{X',T'}=H_{X'}\times_{X_k'^{\dagger}}T'^{\dagger}$.\
As auxiliary data we choose closed embeddings $H\to N$ resp. $H_{X'}\to M$ into smooth dagger spaces of pure dimension $n$ resp. $h$.\
We have $\dim_k(H^i(N,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H}{\cal O}_N)))=h_{2n-i}^{dR}(H)$. By \[algana\] there is a long exact sequence$$\ldots\to H^i(N,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_S}{\cal O}_N))\to H^i(N,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H}{\cal O}_N))$$$$\to H^i(N-H_S,{\mbox{\rm DR}}(R{\underline\Gamma}_{*(H-H_S)}{\cal O}_{N-H_S}))\to H^{i+1}(\ldots.$$ The numbers $\dim_k(H^i(N,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_S}{\cal O}_N)))=h_{2n-i}^{dR}(H_S)$ are finite by induction hypothesis, so it is enough to show $\dim_k(H^i(N-H_S,{\mbox{\rm DR}}(R{\underline\Gamma}_{*(H-H_S)}{\cal O}_{N-H_S})))<\infty$, which by smoothness of $H-H_S$ is equivalent with $\dim_k(H^{i-2l}_{dR}(H-H_S))<\infty$ (where $l={\mbox{\rm codim}}_N(H)$). The canonical maps $H^j_{dR}(H-H_S)\to H^j_{dR}(H_{X'}-H_{X',S'})$ are injective (.), so it is enough to show $\dim_k(H^j_{dR}(H_{X'}-H_{X',S'}))<\infty$. Since $H_{X'}-H_{X',S'}$ is smooth one has $$H^j_{dR}(H_{X'}-H_{X',S'})\cong H^{j+2t}(M-H_{X',S'},{\mbox{\rm DR}}(R{\underline\Gamma}_{*(H_{X'}-H_{X',S'})}{\cal O}_{M-H_{X',S'}}))$$ (where $t={\mbox{\rm codim}}_M(H_{X'})$). By \[algana\] there is a long exact sequence $$\ldots\to H^i(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X',S'}}{\cal O}_M))\to H^i(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X'}}{\cal O}_M))$$ $$\to H^i(M-H_{X',S'},{\mbox{\rm DR}}(R{\underline\Gamma}_{*(H_{X'}-H_{X',S'})}{\cal O}_{M-H_{X',S'}}))\to H^{i+1}(\ldots.$$Again $\dim_k(H^i(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X',S'}}{\cal O}_M)))=h_{2h-i}^{dR}(H_{X',S'})$, a finite number by induction hypothesis. It remains to show $\dim_k(H^i(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X'}}{\cal O}_M)))<\infty$.\
Set $P=X_k^{\dagger}\times M$ and consider the closed immersion $H_X\to P$. From \[blasexa\] we get (with $b=\dim(X_k)$) an exact sequence$$\ldots\to H^i(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_P))\to H^{i-2b}(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X',T'}}{\cal O}_M))\oplus H^i(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_X}{\cal O}_P))$$$$\to H^{i-2b}(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X'}}{\cal O}_M))\to H^{i+1}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_P))\to\ldots.$$ Here $\dim_k(H^{2h-j}(M,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X',T'}}{\cal O}_M)))=h_j^{dR}(H_{X',T'})$ is finite by induction hypothesis. Furthermore $$H^{2h+i}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_P))=H^i(H_X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_{H_X}))$$since $H_X$ is smooth, and for all open affinoid $U\subset H_X$ also $$\dim_k(H^i(U,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_{H_X})))=h_{2b-i}^{dR}(U\cap H_{X,T}),$$ a finite number by induction hypothesis. Since $H_X$ is quasi-compact it follows that $\dim_k(H^i(H_X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_{X,T}}{\cal O}_{H_X})))<\infty$. So we are left with showing that $\dim_k(H^{2h+i}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H_X}{\cal O}_P)))=\dim_k(H_{dR}^i(H_X))$ is finite. By construction, the rigid space associated with $H$ is the generic fibre of an open formal subscheme of the $\pi$-adic completion of $Y$. It follows that the rigid space associated with $H_X$ is the generic fibre of an open formal subscheme of the $\pi$-adic completion of $X$. Thus $\dim_k(H_{dR}^i(H_X))<\infty$ follows from \[maxschni\].
\[ende\] If $X$ is a smooth quasi-compact dagger space and if $i:Z\to X$ is a closed immersion, then the $k$-vector spaces $H_{dR}^i(X-Z)$ are finite dimensional.
[Proof:]{} We may assume $X$ is affinoid. Choose a proper surjective morphism $f:\tilde{X}\to X$ such that $\tilde{X}$ is smooth, $\tilde{Z}=f^{-1}(Z)$ is a divisor with normal crossings on $\tilde{X}$ and $(\tilde{X}-\tilde{Z})\to(X-Z)$ is an isomorphism (\[hironaka\]). Passing to an appropriate finite affinoid admissible open covering of $\tilde{X}$ we see that we may assume from the beginning: $X$ is affinoid and $Z$ is a normal crossings divisor on $X$ such that all its irreducible components are smooth. Now note that if $W=X$ or if $W$ is the intersection of some irreducible components of $Z$, then $W$ is quasi-algebraic: Indeed, since it it smooth and affinoid, it follows from [@elk], th. 7, p. 582 that the associated rigid space $W'$ can be defined by polynomials,. In particular $W'$ is the rigid space associated to a quasi-algebraic affinoid dagger space $W_1$, and by [@en1dag], 1.15 there exists a (non-canonical) isomorphism $W\cong W_1$. Now \[endlich\] says that $H^q(X,{\mbox{\rm DR}}(R{\underline\Gamma}_{*W}{\cal O}_{X}))$ is finite dimensional for all $q$ and all such $W$. Repeated application of \[algana\] gives the Corollary.\
\[kokomend\] Let $X$ be a quasi-compact smooth dagger space, $U\subset X$ a quasi-compact admissible open subset. Then $H_{dR}^q(X-U)$ is finite dimensional for all $q\in\mathbb{N}$.
[Proof:]{} We prove by induction on $n\in\mathbb{N}$:\
${\bf (a_n)}$ For every quasi-compact smooth dagger space $X$ with $\dim(X)\le n$, every quasi-compact open $U\subset X$ and every $q\in\mathbb{N}$, we have $\dim_k(H^q_{dR}(X-U))<\infty$.\
${\bf (b_n)}$ For every quasi-compact smooth dagger space $Y$, every closed immersion $T\hookrightarrow Y$ with $\dim(T)\le n$, every quasi-compact open $V\subset Y$ and every $q\in\mathbb{N}$, we have $\dim_k(H^q(Y-V,{\mbox{\rm DR}}(R{\underline\Gamma}_{*T}{\cal O}_Y)))<\infty.$\
Here ${\bf (a_0)}$ is evident, and so is ${\bf (b_0)}$ because of \[dmodgys\].\
${\bf (b_{n-1})\Rightarrow(a_n):}$ We may suppose $\dim(X)=n$ and $X$ is affinoid and connected. Since $U$ is the union of finitely many rational subdomains of $X$, Mayer-Vietoris sequences allow us to reduce to the case where $U$ is a rational subdomain of $X$. Choose an affine formal $R$-scheme ${\cal X}$ such that its generic fibre ${\cal X}_k$ is the rigid space associated with $X$. For subschemes $Z\subset{\cal X}_{\bar {k}}$ we denote by $]Z[_{\cal X}^{\dagger}$ the open dagger subspace of $X$ corresponding to $]Z[_{\cal X}\subset {\cal X}_k$. By [@bolu], after an admissible blowing up and further localization we may suppose: There is a closed subscheme $Z\subset{\cal X}_{\bar {k}}$ such that $]Z[_{\cal X}^{\dagger}=X-U$. From [@elk], th. 7, p. 582 it follows that ${\cal X}$ is locally defined by polynomials, i.e. we may suppose $${\cal X}={\mbox{\rm Spf}}(R<X_1,\ldots,X_m>/(f_1,\ldots,f_r))$$ with $f_i\in R[X_1,\ldots,X_m]$. We view $X$ as an open subspace of the dagger analytification of the generic fibre $T_k$ of $T={\mbox{\rm Spec}}(R[X_1,\ldots,X_m]/(f_1,\ldots,f_r))$. Since $X$ is smooth, it is contained in the dagger analytification of a single irreducible component of $T_k$. This component is the generic fibre of a closed subscheme of $T$; dividing out the $\pi$-torsion and the nilpotent elements of its coordinate ring, we see that we may suppose that $T$ is integral. We may also suppose that $Z$ is defined by a single equation in $T_{\bar{k}}={\cal X}_{\bar {k}}$. We choose a closed subscheme $Y\subset T$ defined by a single equation such that $Y_{\bar {k}}=Z$. Furthermore we choose an open embedding $T\to\bar{X}$ into a projective $R$-variety $\bar{X}$, and define $\bar{Y}$ to be the schematic closure of $Y$ in $\bar{X}$. Since ${\bf (a_n)}$ is proven in case $U=\emptyset$ by \[endlich\], we may suppose $\bar{Y}\cup\bar{X}_{\bar {k}}$ is a proper subset of $\bar{X}$. Therefore we can apply \[dejong\] to $(\bar{X},\bar{Y}\cup\bar{X}_{\bar {k}})$: Performing a base change with a finite extension of $R$, we may assume that there is a surjective proper morphism $\bar{\phi}:\bar{V}\to\bar{X}$ of $R$-varieties, an open dense subscheme of $\bar{X}$ over which $\bar{\phi}$ is finite, and such that $(\bar{V},\bar{\phi}^{-1}(\bar{Y}\cup\bar{X}_{\bar{k}})_{red})$ is a strictly semi-stable pair over $R$. Let $\bar{\cal V}$, resp. $\bar{\cal W}$, resp. $\bar{\cal X}$ be the $\pi$-adic formal completion of $\bar{V}$, resp. $\bar{\phi}^{-1}(\bar{Y})_{red}$, resp. $\bar{X}$, and set ${\cal V}=\bar{\cal V}\times_{\bar{\cal X}}{\cal X}$ and ${\cal W}=\bar{\cal W}\times_{\bar{\cal V}}{\cal V}$. Let $\bar{V}^{\dagger}_k$ be the dagger analytification of the generic fibre of $\bar{V}$, and let $V$ be its open dagger subspace whose associated rigid space is ${\cal V}_k$. As before, for subschemes $S\subset{\cal V}_{\bar{k}}$, we denote by $]S[^{\dagger}_{\cal V}$ the open dagger subspace of $V$ corresponding to $]S[_{\cal V}\subset {\cal V}_k$. Now $\bar{\phi}$ induces a morphism $\phi_k:V\to X$ of dagger spaces, and if $F=X-U$, we have $]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}=\phi_k^{-1}(F)$.\
Claim: It is enough to show $\dim_k(H_{dR}^*(]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}))<\infty$ for all $q\in\mathbb{N}$.\
Using the induction hypothesis ${\bf (b_{n-1})}$, this can be shown similarly as in the proof of \[endlich\]: Let $V\stackrel{f}{\to}D\stackrel{g}{\to}X$ be the Stein factorization of $\phi_k$ (for example obtained from the Stein factorization of the algebraic morphism $\bar{\phi}$). Let $Q=g^{-1}(F)$, let $D\to P'$ be a closed immersion into a smooth affinoid dagger space $P'$ and let $P\subset P'$ be an open subspace such that $P'-P$ is quasi-compact and open in $P'$ and such that $Q\to P'$ factorizes through a closed immersion $Q\to P$. Let $T\to F$ be a closed immersion with $\dim(T)<n$, such that for $L=g^{-1}(T)$ we have: $Q-L$ is smooth and $(Q-L)\to(F-T)$ is étale. By \[algana\] there are long exact sequences $$\ldots\to H^i(F,{\mbox{\rm DR}}(R{\underline\Gamma}_{*T}{\cal O}_F))\to H_{dR}^i(F)$$$$\to H_{dR}^i(F-T)\to H^{i+1}(F,{\mbox{\rm DR}}(R{\underline\Gamma}_{*T}{\cal O}_F))\to\ldots$$ and$$\ldots\to H^i(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*L}{\cal O}_P))\to H^i(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Q}{\cal O}_P))$$$$\to H^i(P-L,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Q-L}{\cal O}_{P-L}))\to H^{i+1}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*L}{\cal O}_P))\to\ldots.$$Now $H^i(F,{\mbox{\rm DR}}(R{\underline\Gamma}_{*T}{\cal O}_F))$ and $H^i(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*L}{\cal O}_P))$ are finite dimensional by induction hypothesis ${\bf (b_{n-1})}$. Since $Q-L$ is smooth we have $$H^i(P-L,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Q-L}{\cal O}_{P-L}))\cong H_{dR}^{i-2l}(Q-L)$$ (where $l={\mbox{\rm codim}}_P(Q)$), and by . the canonical maps $H^q_{dR}(F-T)\to H_{dR}^q(Q-L)$ are injective. Together we obtain that to prove $\dim_k(H^q_{dR}(F))<\infty$, it is enough to prove $\dim_k(H^q(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Q}{\cal O}_P)))<\infty$ for all $q\in\mathbb{N}$. Now choose a closed immersion $G\to Q$ with $\dim(G)<n$ such that for $H=G\times_Q]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}$ we have: $(]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}-H)\to(Q-G)$ is an isomorphism. Let $E=\bar{V}_k^{\dagger}\times P$ with $]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}$ diagonally embedded. By \[blasexa\] there is a long exact sequence$$\ldots\to H^i(E,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H}{\cal O}_E))\to H^{i-2n}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*G}{\cal O}_P))\oplus H^i(E,{\mbox{\rm DR}}(R{\underline\Gamma}_{*]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}}{\cal O}_E))$$ $$\to H^{i-2n}(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*Q}{\cal O}_P))\to H^{i+1}(E,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H}{\cal O}_E))\to\ldots.$$We have $\dim_k(H^i(E,{\mbox{\rm DR}}(R{\underline\Gamma}_{*H}{\cal O}_E)))<\infty$ and $\dim_k(H^j(P,{\mbox{\rm DR}}(R{\underline\Gamma}_{*G}{\cal O}_P)))<\infty$ by induction hypothesis ${\bf (b_{n-1})}$. On the other hand $$H^i(E,{\mbox{\rm DR}}(R{\underline\Gamma}_{*]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}}{\cal O}_E))=H_{dR}^{i-2n}(]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}),$$ and altogether the claim follows.\
Now we prove $\dim_k(H_{dR}^q(]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}))<\infty$. Note that ${\cal W}$ is the $\pi$-adic formal completion of $\bar{\phi}^{-1}(Y)_{red}$. Let ${\cal W}={\cal W}_f\cup{\cal W}_0$ be the decomposition into the $R$-flat part ${\cal W}_f$ and the $\pi$-torsion part ${\cal W}_0$. Since $Y$ is defined by a single equation in $T$, the same is true for $\bar{\phi}^{-1}(Y)_{red}$ in $\bar{\phi}^{-1}(T)_{red}$. Since $(\bar{V},\bar{\phi}^{-1}(\bar{Y}\cup\bar{X}_{\bar {k}})_{red})$ is a strictly semi-stable pair over $R$, this means that $\bar{\phi}^{-1}(Y)_{red}$ is the union of some irreducible components of $\bar{\phi}^{-1}(Y\cup T_{\bar {k}})_{red}$. After passing to a finite Zariski open covering of ${\cal V}$ we may therefore suppose: There is an étale morphism $${\cal V}\stackrel{\psi}{\to}{\mbox{\rm Spf}}(R<X_1,\ldots,X_m>/(X_1\ldots X_r-\pi))$$ for some $m\ge r\ge1$, and if we set ${\cal W}^i=V(\psi^*X_i)$ for $i\le m$, there are subsets $J\subset\{r+1,\ldots,m\}$ and $N\subset\{1,\ldots,r\}$ such that ${\cal W}_f=\cup_{i\in J}{\cal W}^i$ and ${\cal W}_0=\cup_{i\in N}{\cal W}^i$. The covering $]{\cal W}_{\bar {k}}[^{\dagger}_{\cal V}=\cup_{i\in J\cup N}]{\cal W}^i_{\bar {k}}[^{\dagger}_{\cal V}$ is admissible, so it is enough to show: For all $I\subset J$, all $M\subset N$, all $q\in\mathbb{N}$ we have $$\dim_k(H_{dR}^q(\cap_{i\in I\cup M}]{\cal W}^i_{\bar {k}}[^{\dagger}_{\cal V}))<\infty.$$ Set ${\cal R}=\cap_{i\in I}{\cal W}^i$, a strictly semi-stable formal $R$-scheme. By construction, the equations defining the closed immersion ${\cal R}\to{\cal V}$ are contained in ${\cal O}_V$; they define a Zariski closed dagger subspace of $V$ whose associated rigid space is the generic fibre of ${\cal R}$. Similar as before we define its open subspaces $]S[^{\dagger}_{\cal R}$ for $S\subset {\cal R}_{\bar {k}}$. Let $C={\cal R}_{\bar {k}}\cap(\cap_{i\in M}{\cal W}^i)$. We claim $$H_{dR}^q(\cap_{i\in I\cup M}]{\cal W}^i_{\bar {k}}[^{\dagger}_{\cal V})=H_{dR}^q(]C[^{\dagger}_{\cal V})\cong H_{dR}^q(]C[^{\dagger}_{\cal R}).$$ Indeed, from \[faser\] we derive an isomorphism $]C[_{\cal R}\times({\bf D}^0)^{|I|}\cong]C[_{\cal V}$ of rigid spaces. Because of \[rigdrdf\], to prove our claim we may therefore assume that there is an isomorphism of dagger spaces$$]C[^{\dagger}_{\cal R}\times({\bf D}^0)^{|I|}\cong]C[^{\dagger}_{\cal V}$$ and then the claim is obvious. But $H_{dR}^q(]C[^{\dagger}_{\cal R})$ is finite dimensional by \[maxschni\], because $C$ is the intersection of some irreducible components of ${\cal R}_{\bar {k}}$.\
${\bf (a_n)+(b_{n-1})\Rightarrow(b_n):}$ Passing to a finite affinoid covering of $Y$ we may suppose (\[hironaka\]): There is a smooth $P$ and a proper surjective $P\stackrel{f}{\to}Y$ such that $S=f^{-1}(T)$ is a divisor with normal crossings on $P$ and $(P-S)\to(Y-T)$ is an isomorphism. As in ${\bf (b_{n-1})\Rightarrow(a_n)}$ one shows that it suffices to prove $\dim_k(H^q(f^{-1}(Y-V),{\mbox{\rm DR}}(R{\underline\Gamma}_{*S}{\cal O}_P)))<\infty$; so we suppose $T$ is a divisor with normal crossings on $Y$. But then, in view of .(c), the problem is equivalent with the one where $T$ is smooth (of arbitrary codimension). Thus it is reduced by means of ., \[dmodgys\] to the induction hypothesis ${\bf (a_n)}$.\
As an application of \[kokomend\], we will show in [@en3dag] that the de Rham cohomology groups of smooth rigid Stein spaces are topologically separated for their canonical topology, hence are Fréchet spaces. We also define reasonable de Rham cohomology groups for arbitrary rigid spaces (the underlying idea is that of \[rigdrdf\]), and derive from \[kokomend\] Künneth and duality formulas for them.
\[rifi\] For a $\bar{k}$-scheme $Y$ of finite type, the $k$-vector spaces $H_{rig}^q(Y/k)$ are finite dimensional for all $q\in\mathbb{N}$.
[Proof:]{} Let ${\cal X}$ be a proper smooth $\pi$-adic tf formal ${\mbox{\rm Spf}}(R)$-scheme, $Y\to{\cal X}_{\bar {k}}$ an immersion with schematic closure $j:Y\to\bar{Y}$ in ${\cal X}_{\bar {k}}$. Then $]\bar{Y}[_{\cal X}$ is a partially proper rigid space, therefore equivalent with a dagger space $Q$. Let $X$ be the open subspace of $Q$ whose underlying set is identified with $]Y[_{\cal X}$. From [@en1dag], 5.1 we get an isomorphism $H_{rig}^q(Y/k)\cong H^q_{dR}(X)$, but $H^q_{dR}(X)$ is finite dimensional by \[kokomend\].
[abcdefgh]{} , Cohomologie rigide et cohomologie rigide à supports propres, Première partie, Prépublication IRMAR 96-03, Université de Rennes (1996) , Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. [**128**]{} (1997), 329-377 , On Serre duality for coherent sheaves on rigid-analytic spaces, Manuscripta Mathematica [**93**]{}, Fasc 2 (1997), 219-245 , Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. [**128**]{} (1997), 207-302 , A rigid analytic version of M.Artin’s Theorem on analytic equations, Math. Ann. [**255**]{} (1981), 395-404 , Non-Archimedean Analysis, Grundl. d. math. Wissensch. [**261**]{} (1984), Springer-Verlag , Formal and rigid geometry, Math. Ann. [**295**]{} (1993), 291-317 , Duality in Rigid Analysis, Trento 1989, LNM 1454, 191-214, Springer-Verlag (1990) , Smoothness, semistability and alterations, Publ. Math. I.H.E.S. [**83**]{} (1996), 51-93 , Solutions d’équations à coefficients dans un anneau hensélien, Ann. sci. Éc. Norm. Sup. [**6**]{} (1973), no.4, 553-604 , DeRham-Kohomologie in der rigiden Analysis, Preprintreihe SFB 478, Universität Münster, Heft 39 (1999) , Rigid analytic spaces with overconvergent structure sheaf, J. reine und angew. Math. [**519**]{} (2000), 73-95 , De Rham cohomology of rigid spaces, in preparation , Eléments de géométrie algébrique, Publ. Math. I.H.E.S. [**4**]{}, [**8**]{}, [**11**]{}, [**17**]{}, [**20**]{}, [**24**]{}, [**28**]{}, [**32**]{} (1960-67) , On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.E.S. [**29**]{} (1966), 95-103 , On the de Rham cohomology of algebraic varieties, Publ. Math. I.H.E.S. [**45**]{} (1976), 5-99 , Residues and Duality, Lectures Notes in Mathematics, Springer-Verlag, [**20**]{} (1966) , Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Vieweg (1996) , Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, Geometry and Number Theory, J. Hopkins Univ. Press (1989), 191-224 , Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Invent. Math. [**2**]{} (1967), 256-273 , Die de Rham Kohomologie algebraischer Mannigfaltigkeiten über einem bewerteten Körper, Publ. Math. I.H.E.S. [**33**]{} (1967), 5-20 , Le formalisme des six opérations de Grothendieck pour les $D_X$-modules cohérents, Travaux en cours [**35**]{} (1989), Hermann, Paris , Sur le théorème de finitude de la cohomologie $p$-adique d’une variété affine non singulière, American Journal of Mathematics [**119**]{} (1997), no.5, 1027-1081 , Formal cohomology: I, Annals of Math. [**88**]{} (1968), 181-217 , Serre duality for rigid analytic spaces, Indag. math, N.S. [**3(2)**]{} (1992), 219-235
[^1]: The notation $W_n$ is taken from \[[*U. Günzer*]{}, Modellringe in der nichtarchimedischen Funktionentheorie, Indag. math. [**29**]{} (1967), 334-342\]. There the author assigns only the name of Washnitzer to this algebra. However, the referee pointed out that the name Monsky-Washnitzer algebra is the usual one.
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abstract: 'We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function and then transforming the k-bit coupling ($k\geq 3$) terms to quadratic terms using ancillary variables. Our resource-efficient method uses $\mathcal{O}(\text{log}^2(N))$ binary variables (qubits) for finding the factors of an integer $N$. We present how to factorize 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits, respectively. This method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. The method is general and could be used to factor larger integers as the number of available qubits increases.'
author:
- Shuxian Jiang
- 'Keith A. Britt'
- 'Alexander J. McCaskey'
- 'Travis S. Humble[[^1]]{}'
- 'Sabre Kais[[^2]]{}'
title: Quantum Annealing for Prime Factorization
---
Introduction
============
The integer factorization problem reduces an integer $N$ to its prime factors $p$ and $q$. This problem is fundamental in number theory with broad implications for cryptographic data storage and communications [@wagstaff2013joy]. While finding factors of an integer is a computational hard problem, it is not believe to belong to the class of NP-hard problems. In a practical sense though, all known classical factoring algorithms which are deterministic and don’t have unproved hypotheses require time exponential in $\log N$. Thus, the integer factorization problem is used as the basic hardness assumption for many encryption methods including the widely deployed RSA cryptographic system. The fastest, known classical algorithm for integer factorization is the general number field sieve method [@lenstra1990number], which scales exponentially in the number of operations required with respect to the integer $N$.
Quantum computing theory has shown the potential to reduce the number of operations required for solving certain problems, and many efforts have been undertaken to develop a quantum computer that can solve the integer factorization problem. The quantum methods for solving factorization problem could be regarded as probabilistic methods, compared to the classical deterministic methods. Shor’s algorithm is perhaps the most well-known quantum algorithm for integer factorization. To factor an integer $N$, Shor’s algorithm requires a polynomial number of operations [@shor1999polynomial], thus providing an exponential speedup over the general number sieve. Shor’s algorithm works by reducing the factorization problem to the order-finding problem. Many attempts have been made to implement Shor’s algorithm on quantum computing hardware. Vandersypen et al. [@vandersypen2001experimental] used seven spin-1/2 nuclei in a molecule as qubits to factor $N=15$, while Lanyon et al. [@lanyon2007experimental], Lu et al. [@lu2007demonstration], and Politi et al. [@politi2009shor] have implemented compiled versions of Shor’s algorithm using photonic systems for factoring $N=15$. Mart[í]{}n-L[ó]{}pez et al. [@martin2012experimental] factored $21$ using qubit recycling, and Lucero et al. [@lucero2012computing] used superconducting qubits to factor 15. Geller et al.[@geller2013factoring] used a simplified version of Shor’s algorithm for factoring products of the Fermat primes 3, 5, 17, 257, and 65537. Shor’s algorithm is not only useful for solving integer factorization problem, but can be also used to solve other order-finding problems. Very recently, Grosshans et al.[@grosshans2015factoring] proposed factoring safe semi-primes using the quantum order finding algorithm which reduced the failure probability.
Shor’s quantum algorithm for order-finding is presented within the circuit model of quantum computation. But another, equally powerful model of quantum computing (up to a polynomial reduction) is the quantum adiabatic computing model [@farhi2001quantum](QAC), which can also be used to solve the integer factorization problem. Peng et al. [@peng2008quantum] have developed methods for factoring 21 using QAC and they implemented this algorithm in a three-qubit NMR quantum processor, while Xu et al. [@XZLZPD12] factored 143 using similar NMR technology. A novel approach by Schaller et al. [@schaller2007role] used multiplication tables to produce a set of equations and used those to produce a quadratic cost function. Dridi et al. [@dridi2017prime] further optimized this method using Gröbner bases, reducing the number of auxiliary variables and equations required.
In this contribution, we introduce a new procedure for solving the integer factorization problem using a variant of QAC known as quantum annealing. Our approach is based on a direct mathematical transformation of the problem to an Ising Hamiltonian, which can be realized using currently available quantum processors. Next, in order to account for hardware constraints, we introduce a modified multiplication method that reduces the range of the coefficients in the cost function without increasing the number of qubits required. The method is general, resource-efficient, and does not rely on ad-hoc calculations.
Quantum annealing solves optimization problems using quantum fluctuations [@kadowaki1998quantum]. Quantum adiabatic computation (QAC), as developed by Farhi et al. [@farhi2001quantum], approaches the same task given a complex Hamiltonian whose ground state encodes the solution to the optimization problem. This computation begins in the ground state of a simple, well-characterized Hamiltonian, which is then adiabatically evolved to the complex, problem Hamiltonian. According to the adiabatic theorem[@M62], the system state will also evolve the ground state of the problem Hamiltonian provided the evolution is sufficiently slow to prevent excitations to any higher-lying state. At the end of the annealing process, the measured qubits will encode the optimal solution to the problem within a bounded degree of certainty (due to noise within the closed system).
The time-dependent Hamiltonian of the quantum system is given by combining the initial Hamiltonian and the final Hamiltonian [@RevModPhys.90.015002] $$H(t) = (1- \frac{t}{T})H_B + \frac{t}{T}H_P.$$ Here $H_B$ is the initial Hamiltonian with a well-known and easily constructed ground state, which we consider to have the general form $$\label{eq:Hb}
H_B=-\sum \sigma_x^{(i)}$$ with Pauli operator $\sigma_x$ defining the $x$-basis. The Hamiltonian $H_P$ is the final Hamiltonian whose ground state encodes the solution to a given instance of the optimization problem. Finding the prime factors of an integer will be mapped to the final Ising Hamiltonian of the general form $$H_P=\sum{{h_i}{\sigma_z^{(i)}}} + \sum{{J_{ij}} }{\sigma_z^{(i)} }{\sigma_z^{(j)}}$$ where $\sigma_z$ defines the $z$-basis and the local fields ${h_i}$ and the couplings ${J_{ij}}$ define the factorization problem instance. $H_P$ gives the total energy of the system.
The time-dependent Hamiltonian $H(t)$ of the physical system evolves according to Schrödinger equation $$\label{eq:Schr}
i\frac{d}{dt}\ket{\psi(t)}=H(t)\ket{\psi(t)}$$ where $|\psi(t)\rangle$ is the state of the system at any time $t\in[0,T]$. Let $\ket{\phi_i(t)}$ be the $i$-th instantaneous eigenstate of $H(t)$, that is, $H(t)\ket{\phi_i(t)}=E_i(t)\ket{\phi_i(t)}$ holds through the entire evolution. If the system is initialized in the ground state $|\phi_0(t=0)\rangle$, then the evolution proceeds slow enough to avoid exciting to the higher-lying eigenstates, e.g., $|\phi_1(t)\rangle$. Ultimately, the system will be prepared in the instantaneous ground eigenstate $|\phi_0(t=T)\rangle$.
The direct method to factor $N = pq$, where $p$ and $q$ are prime numbers is to let $l_1 = \lfloor \log_2(p) \rfloor, l_2 = \lfloor \log_2(q) \rfloor$ and, without loss of generality, we take $p = (x_{l_1-1}x_{l_1-2}...x_1 1)_2, q = (x_{l_1+l_2-2}x_{l_1+l_2-3}...x_{l_1} 1)_2$, and $l_1>l_2$ where $x_i$ are binary numbers. So $p = \sum_{i=1}^{l_1-1}2^{i}x_i+1$ and $q=\sum_{j=l_1}^{l_1+l_2-2}2^{j}x_j+1$. We can define the cost function $f(x_1,x_2,x_3,x_4,...,x_{l_1+l_2-2})= (N-pq)^2$. To reduce the order of the cost function to quadratic, we need $\binom{l_1}{2} + \binom{l_2}{2} = \frac{l_1(l_1-1)}{2}+ \frac{l_2(l_2-1)}{2}$ auxiliary variables. If $l_1=l_2=l$, the number of auxiliary variables is $l \times (l-1)$. Plus the variables to denote the factors, we used $ 2\times (l-1) + l \times (l-1) = (l+2) \times (l-1)= \mathcal{O}(l^2)= \mathcal{O}(\text{log}^2(N))$ binary variables in total. We could also let $p = (1 x_{l_1-2}...x_1 1)_2, q = (1 x_{l_1+l_2-4}...x_{l_1-1} 1)_2$ when lengths of $p$ and $q$ are fixed.
we illustrate this direct method through the factorization of $N = 15$. We have $\log_2(p)\leq 2 <n_2=\log_2(q)<4$, which means $p$ is at most 2 bits, $q$ is at most 3 bits, then we define $p=(x_1 1)_2=x_1\times 2+1, q=(x_2x_3 1)_2=x_2\times 2^2+x_3\times 2+1, x_i \in \{0,1\})$, because $p$ and $q$ are prime numbers. The objective function $f(x_1,x_2,x_3)= (N-pq)^2$ to be minimized has the following form: $$\begin{aligned}
f=128x_1x_2x_3-56x_1x_2-48x_1x_3+16x_2x_3-52x_1-52x_2-96x_3+196.\end{aligned}$$ Now, we reduced the 3-local term to 2-local term as follows [@BH01]: for $x,y,z \in \{0,1\}$, $xy = z \ \text{iff} \ xy-2xz-2yz+3z = 0$, and $
xy \neq \ z \ \text{iff} \ xy-2xz-2yz+3z > 0$. It is also easy to check that $x_1x_2x_3= x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4)$ if $x_4 = x_1 x_2$, and $x_1x_2x_3 < x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4)$ if $ \ x_4 \neq x_1 x_2$. Thus, the $x_1x_2x_3$ term could be transformed to quadratic form by replacing $x_1x_2$ with $x_4$, plus a constrained condition as the penalty term: $$\begin{aligned}
\label{eq:replace}
\min(x_1x_2x_3) = \min(x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4)\end{aligned}$$ Here we transformed the 3-local term to 2-local by introducing a new variable and replacing the constrained condition with a penalty term into the original function. We obtain (see Appendix A.1 for more details) an Ising function to be optimized with the local fields $h^T$ and couplings $J$ written as
-- --
-- --
Results {#sec:results}
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We use the D-Wave 2000Q quantum annealer to demonstrate the method. In order to solve problems on the D-Wave hardware we need to embed the problem Hamiltonian onto the *chimera* hardware graph while maintaining the energy minimization objective function [@Humble2014]. This process requires users of the D-Wave system to solve two problems: minor embedding[@choi2008minor; @Klymko2014] and parameter setting[@choi2011minor].
A *minor embedding* of $G(V,E)$ in $G'(V',E')$ is defined by a mapping $\phi:G\mapsto G'$ such that each vertex $v\in G$ is mapped to a connected subtree $T_v$ of $G'$ and if $(u,v)\in E$ then there exist $i_u,i_v\in G'$ such that $i_u\in T_u$, $i_v\in T_v$ and $(i_u,i_v)\in E'$. If such a mapping $\phi$ exists between $G$ and $G'$, we say $G$ is a *minor* of $G'$ and we use $G\le_m G'$ to denote such relationship.
In parameter setting, we assign each node and each edge in the minor embedding graph such that: (1) for each node in the tree $T_i$ expanded by the same vertex $i$, its value $h'_{i_k}$ satisfies $\sum h'_{i_k} = h_i$, (2) for each edge in the tree $T_i$ expanded by the same vertex $i$, the value $J_{i_{k},i_{k'}}$ needs to be large enough to make sure all physical qubits that correspond to the same logical qubit to be of the same value and (3) for each edge in the minor embedding graph which is in the original graph, we could use the same $J_{ij}$ value.
The experimental results of factoring 15 and 21 are shown in Figure \[fig:experiment15\]. These plots show the decoded solutions in order of lowest energy to highest energy (left to right). In some cases, the observed bits were decoded as the correct factors. For example, there are several (3, 7) solutions for $N=21$. Only the first (leftmost) corresponds to the lowest energy state. The others were always higher energy solutions.
------------------------------------------- ----------------------------------------------
(a)Factoring $15$ with annealing time 200 \(b) Factoring $21$ with annealing time 2000
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In order to factor larger numbers and perform the quantum annealing on the D-wave machine with reasonable mitigation of control hardware bits of precision, we introduce the modified multiplication table method. The modified multiplication table method allows us to reduce the range of Ising parameter values used as coefficients to the qubits and couplers, thereby reducing the bits of precision required by control hardware to satisfy the final Hamiltonian.
The modified multiplication table method uses local minimizations of the product of individual bits in the bit strings of $p$ and $q$ to reduce the number of variables needed in the global minimization of the final Hamiltonian. In addition to reducing the number of logical qubits needed to describe the final Hamiltonian, this method also shrinks the range of coefficient values needed to describe the final Hamiltonian. A detailed analysis of the range of coefficients is shown in Appendix A.5. Of note is that the modified multiplication table method did not eliminate the need for 4-body and 3-body terms to be reduced to quadratic terms, but it did reduce the number of these higher body terms making it possible for us to embed them on the D-Wave hardware. We used approximately $\text{log}(N)$ binary variables to denote the factors and about $\text{log}(N)$ binary variables to denote the carries where $N$ is the number to be factored, plus $\text{log}^2(N)/4$ auxiliary binary variables in this scheme. In total, we need roughly $\text{log}^2(N)/4$ binary variables (qubits).
For an illustrative example, we consider factoring $143 = 11 \times 13$. We have the multiplication table[@dattani2014quantum].
$2^{7}$ $2^{6}$ $2^{5}$ $2^{4}$ $2^{3}$ $2^{2}$ $2^{1}$ $2^{0}$
------------------- --------- --------- --------- --------- --------- -------------- --------- ---------
p $1$ $p_{2}$ $p_{1}$ $1$
q $1$ $q_{2}$ $q_{1}$ $1$
$p_{2}$ $p_{1}$
$p_{1}q_{1}$ $q_{1}$
$q_{2}$ $q_{2}$
1 $p_{2}$
carries $c_{4}$ $c_{3}$
$p\times q = 143$ 1 0 0 1 1
: Multiplication table for $11\times13=143$ in binary.[]{data-label="tab:table1"}
Instead of using carries in every bits[@dattani2014quantum] [@dridi2017prime],[@pal2016hybrid], we only use carries twice (let them to be $c_1,c_2,c_3,c_4$ with $(c_2 c_1)_2=c_2\times 2+c_1$, $(c_4 c_3)_2=c_4\times 2+c_3, c_i = 0,1$) determined by the divided columns. This method skipped the step to calculate the system of equations for each column, thus greatly reduced the burden of computation. To determine how to divide the columns we need to balance the number of unknown variables (carries) and the range of coefficients in the problem Hamiltonian. Instead of making the sum of each column equal to every bits of the number to be factored as in a conventional multiplication table, we make each block of the multiplication table equal to the corresponding block of the number to be factored, thus greatly reducing the number of carry qubits used while keeping their coefficients in a reasonable range (given the hardware platform). The equations for these blocks give the following cost function $f(p_1,p_2,q_1,q_2,c_1,c_2,c_3,c_4)$ to be optimized (more details in Appendix A.2): $$\begin{aligned}
f & = & (2p_2+2p_1 q_1 +2q_2-8c_2-4c_1+p_1+q_1-3)^2+(2q_1+2p_2 q_2 +2p_1+2c_2-8c_4-4c_3+ p_2 q_1+p_1 q_2\\
& & +c_1+1)^2+(q_2+p_2+c_3+2c_4-2)^2 .\end{aligned}$$ This function could be expanded and further simplified using the property $x^2=x \text{ for } x=0,1$. But there will still be cubic terms like $c_1p_1q_1$ and quartic terms like $p_1p_2q_1q_2$. In order to convert it to Ising Hamiltonian, we need to reduce these high order terms to two order terms as explained in the Appendix A.2. Replacing $p_1q_1$ with $t_1$, $p_1q_2$ with $t_2$, $p_2q_2$ with $t_3$, and $p_2q_1$ with $t_4$, we used $2\times 2 =4$ auxiliary variables. After doing a variable conversion from $p_1$ to $s_1$, $p_2$ to $s_2$, $q_1$ to $s_3$, $q_2$ to $s_4$, ..., $p_2q_1$ to $s_{12}$ as the final step, we transfer the cost function to an Ising type Hamiltonian as shown in Appendix A.2.
Next we embed the problem to D-wave machine using the following method (note: for larger $N$ factoring experiments which can’t be embedded directly using this method due to the limitation of the Chimera graph, we relied on a D-Wave provided heuristic embedding algorithm). If $n$ qubits are needed in the Hamiltonian, we divide $n$ into $\lceil \frac{n}{4} \rceil$ groups. For each group, we use 4 copies of the nodes with each $h'_{i_k} = \frac{1}{4} h_i$. We assign each edge in the tree $T_i$ the negative number with largest absolute value to make it a penalty term. This method guarantees the nodes correspond to the same original qubit have the same value. We assign each edge corresponds to the original edge in the problem graph the same $J_{ij}$ value. The embedded graph to D-Wave machine is in Figure \[emb\_143\].
\[fig: chimera\_12\] ![Embedding the factoring instance $N=143$ to Chimera graph. The nodes with the same color denote the same original qubit, with their connected lines corresponding to strong couplings. The left footnotes refer to which spin the node was embedded.[]{data-label="emb_143"}](143.pdf "fig:"){width="0.6\linewidth"}
The results graph are shown in Figure \[fig:results 143\]. The final state of the system will be $\ket{1 -1 -1 \ 1}$ or $\ket{-1 \ 1 \ 1 -1}$ with high probability, which corresponds to solutions $p=(1 p_2 p_1 1)=(1 1 0 1)_2 = 13$, $q=(1 q_2 q_1 1) = (1 0 1 1)_2 = 11$ or $p = 11, q= 13$.
---------------------------- ----------------------------
\(a) Result $13 \times 11$ \(b) Result $11 \times 13$
---------------------------- ----------------------------
Next, we factorize larger numbers such as $N = 59989 = pq$, where $p>q$ are prime numbers. We start by fixing the length of the factors by setting the binary representation of the factors to $p = (1 p_6 p_5 p_4 p_3 p_2 p_1 1)_2,q = (1 q_6 q_5 q_4 q_3 q_2 q_1 1)_2$, $p_i, q_i\in \{0,1\}$. We predefine how to divide the columns and the number of carries given the multiplication table shown in Appendix A.3. Using the method described for factoring 143, we write the equations and the corresponding cost function of the factorization problem then convert this cost function to the Ising Hamiltonian. We need $6\times 6 = 36$ auxiliary variables, based on the same variable replacement rules stated in the previous section, plus $6 + 6= 12$ variables to denote the factors and 11 variables to denote the carries. This gives a total number of 59 variables. Sometimes the carries in the multiplication table will overlap, as is the case for factoring $376289 = 659\times 571$ shown in the multiplication table of Appendix A.4. In such circumstances, we add these carries in the table and then use the same method as before to find the corresponding Ising Hamiltonian. This Hamiltonian has $8+8+14+8\times8 =94$ qubits. As a point of reference, applying this method to the current factoring record for RSA-768 would require approximately 147,456 qubits.
Conclusions
===========
In this paper we have presented two general methods for factoring integers using quantum annealing by converting the problem to an Ising Hamiltonian. Both methods use $\mathcal{O}(\text{log}^2(N))$ qubits in total, where $N$ is the number to be factored. The novelty of our demonstration of quantum annealing for prime factorization is based on the reduction in quantum resources required to execute factoring and the experimental verification of the algorithmic accuracy using currently available hardware. As a proof-of-concept, we have demonstrated these methods by factoring integers using the D-Wave 2000Q quantum annealing hardware, but these methods may be used on any other quantum annealing system with a similar number of qubits, qubit degree of connectivity, and hardware parameter precision. Assuming that quantum annealing hardware systems will continue to grow both in the number of qubits and bits of precision capabilities, our methods offer a promising path toward factor much larger numbers in the future.
Finally, we note that while our demonstrations of factoring have made use of currently available quantum annealers, there is an outstanding question regarding the asymptotic complexity for this approach. It is well known that algorithmic complexity within the QAC model depends on the minimum spectral gap between the ground and first-excited states of the underlying time-dependent Hamiltonian. Attempts to classify the complexity of the spectral gap with respect to system size have not yet succeed and, indeed, Cubitt, Perez-Garcia, and Wolf have proven that the problem of claiming a Hamiltonian has a gap is undecidable in general [@cubitt2015undecidability]. Nonetheless, there is hope that our resource-efficient algorithms may find use in pre-processing potential factors for noisy factorization algorithms, e.g., as suggested by Patterson et al. within the context of RSA [@Patterson2012].
[**[Methods]{}**]{} We calculated the factors $p$ and $q$ of a coprime integer $N$ using a implementation of the algorithms described in Sec. \[sec:results\]. The programmed implementation was written in C/C++ or Python using the XACC programming framework [@mccaskey2017extreme]. XACC enables integration of the D-Wave solver application programming interface (SAPI) using a directive-based programming model. Pre-processing of the input $N$ generated the Ising parameters for a logical Hamiltonian that was then embedded into the hardware graph structure. For the 2000Q processor, the hardware graph was a complete 16-by-16 Chimera structure over 2048 qubits using the SAPI version 3.0 sapi\_findembedding method, which is based on the Cai, Macready and Roy randomized algorithm [@cai2014practical]. Access to these methods were managed using the XACC dwsapi-embedding plugin [@mccaskey2017extreme]. The corresponding biases and couplings for the embedded problem were generated using the logical Ising parameters. The output of the embeddeding was a program implementation of the physical Ising model that was submitted for execution on the D-Wave processor. Additional parameters for the execution included the number of samples $S$ and the annealing duration $T$. The default annealing schedule for the 2000Q was used for all executions. The output from each of the $S$ executions was a measured binary string designating $\pm1$ values for each spin variable. The number of samples was $S = 10,000$. Each returned string was then classified according to the corresponding energy for the physical Ising model and subsequently decoded according to the algorithm in Sec. \[sec:results\] into the factors $p$ and $q$. A histogram of all solutions returned for a specific annealing time was recorded.
[**Acknowledgments**]{} Access to the D-Wave 2000Q was provided by the Quantum Computing Institute at Oak Ridge National Laboratory and Google Quantum Artificial Intelligence Lab, USRA-Purdue. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Appendix
========
Factoring $N = 15 = 5 \times 3$
-------------------------------
Define $p=(x_1 1)_2=x_1*2+1, q=(x_2x_3 1)_2=x_2*2^2+x_3*2+1, x_i \in \{0,1\})$, $p$ and $q$ are prime numbers.\
The cost function is $$\begin{aligned}
& & f(x_1,x_2,x_3) \\
& = & (N-pq)^2 \\
& = & [15-(x_1*2+1)(x_3*2^2+x_2*2+1)]^2 \\
& = & 128x_1x_2x_3-56x_1x_2-48x_1x_3+16x_2x_3-52x_1-52x_2-96x_3+196.\end{aligned}$$ Use the replacement in Eq.\[eq:replace\], we got $$\begin{aligned}
& & f'(x_1,x_2,x_3,x_4) \\
& = & 128(x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4))-56x_1x_2-48x_1x_3+16x_2x_3\\
& & -52x_1-52x_2-96x_3+196 \\
& = &200x_1x_2 - 48x_1x_3 - 512x_1x_4 + 16x_2x_3 - 512x_2x_4 + 128x_3x_4 \\
& & - 52x_1- 52x_2 - 96x_3+768x_4+ 196.\end{aligned}$$ with $$\underset{x_1 x_2=x_4} {\min} f(x_1,x_2,x_3,x_4) = {\min}f'(x_1,x_2,x_3,x_4)$$ because the coefficient of $x_1x_2x_3$ term is positive.
Then we do variable replacement using $x_i= \frac{1-s_i}{2}, i = 1,2,3,4$ $$\begin{aligned}
& & f'(x_1,x_2,x_3,x_4) \\
& = & 200\frac{1-s_1}{2}\frac{1-s_2}{2} -48\frac{1-s_1}{2}\frac{1-s_3}{2}-512\frac{1-s_1}{2}\frac{1-s_4}{2}+16\frac{1-s_2}{2}\frac{1-s_3}{2}\\
& & -512\frac{1-s_2}{2}\frac{1-s_4}{2}+128\frac{1-s_3}{2}\frac{1-s_4}{2}-52\frac{1-s_1}{2}-52\frac{1-s_2}{2}\\
& & -96\frac{1-s_3}{2}+768\frac{1-s_4}{2}+196\\
& = & 116s_1 + 100s_2 + 24s_3 - 160s_4 + 50s_1s_2 -12s_1s_3 - 128s_1s_4 + 4s_2s_3 -128s_2s_4 +32s_3s_4 + 298\\
& = & 2g(s_1,s_2,s_3,s_4)\end{aligned}$$ $g(s_1,s_2,s_3,s_4)$ is the energy function of $$\begin{aligned}
& & H_P (\sigma_z^{(1)},\sigma_z^{(2)},\sigma_z^{(3)},\sigma_z^{(4)})\\
& =&58\sigma_z^{(1)}+50\sigma_z^{(2)}+12\sigma_z^{(3)}-80\sigma_z^{(4)}+25\sigma_z^{(1)}\sigma_z^{(2)}-6\sigma_z^{(1)}\sigma_z^{(3)}-64 \sigma_z^{(1)}\sigma_z^{(4)}+2\sigma_z^{(2)}\sigma_z^{(3)}-64\sigma_z^{(2)}\sigma_z^{(4)}\\
& & +16\sigma_z^{(3)}\sigma_z^{(4)}+149I. \end{aligned}$$
Factoring $N = 143 = 13 \times 11$
----------------------------------
From multiplication table \[tab:table1\], we could get the equations for each blocks $$\begin{aligned}
(p_2+p_1 q_1 +q_2-(c_2\times 4 + c_1\times 2))\times 2+(p_1+q_1) &=& (1 1)_2 = 3 \\
(q_1+p_2 q_2 +p_1 + c_2-(c_4\times 4 +c_3\times 2))\times 2 + (1+p_2 q_1+p_1 q_2 +1+ c_1)& = & (0 1)_2=1\\
(1+c_4)\times 2+ (q_2+p_2+c_3)& = & (1 0 0)_2=4\end{aligned}$$ They could be further simplified as $$\begin{aligned}
2p_2+2p_1 q_1 +2q_2-8c_2-4c_1+p_1+q_1-3 &=& 0 \\
2q_1+2p_2 q_2 +2p_1+2c_2-8c_4-4c_3+ p_2 q_1+p_1 q_2+c_1+1& = & 0 \\
q_2+p_2+c_3+2c_4-2& = & 0\end{aligned}$$ We define the cost function to be squares of the left of equations. That is $$\begin{aligned}
& & f(p_1,p_2,q_1,q_2,c_1,c_2,c_3,c_4) \\
& = & (2p_2+2p_1 q_1 +2q_2-8c_2-4c_1+p_1+q_1-3)^2+(2q_1+2p_2 q_2 +2p_1+2c_2-8c_4-4c_3+ p_2 q_1+p_1 q_2+c_1+1)^2\\
& & +(q_2+p_2+c_3+2c_4-2)^2
$$ Expand and simplify the function using the property $x^2=x \text{ for } x=0,1$. Then reduce the higher order terms to two order terms according to the following rule noticing that there will be negative high order terms: $$\begin{cases}
x_1x_2x_3 = x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4) \ \ \text{if} \ x_4 = x_1 x_2\\
x_1x_2x_3 < x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4) \ \ \text{if} \ x_4 \neq x_1 x_2
\end{cases}$$ $$and \begin{cases}
-x_1x_2x_3 = -x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4) \ \ \text{if} \ x_4 = x_1 x_2\\
-x_1x_2x_3 < -x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4) \ \ \text{if} \ x_4 \neq x_1 x_2
\end{cases}$$ So the negative term $-x_1x_2x_3$ could be transformed to quadratic term in the same way as the positive term $x_1x_2x_3$.
The cost function could be minimized as long as the transformed one is minimized $$\begin{aligned}
\label{eq:replace}
\min(x_1x_2x_3) & = & \min(x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4) \\
\min(-x_1x_2x_3) & = & \min(-x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4)\end{aligned}$$
Replace $p_1q_1$ with $t_1$,$p_1q_2$ with $t_2$, $p_2q_2$ with $t_3$, $p_2q_1$ with $t_4$, using the variable replacement rule if the coefficient of the term is positive or negative respectively. The cost function becomes $$\begin{aligned}
& & f(p_1,p_2,q_1,q_2,c_1,c_2,c_3,c_4,t_1,t_2,t_3,t_4) \\
& = & 43 c_1 + 120 c_2 + 5 c_3 + 44 c_4 + 3 p_1 - 11 p_2 + 3 q_1 - 11 q_2 + 444 t_1 + 252 t_2 + 372 t_3 + 252 t_4 + 68 c_1 c_2 - 8 c_1 c_3 \\
& & - 16 c_1 c_4 - 16 c_2 c_3 - 32 c_2 c_4 + 68 c_3 c_4 - 4 c_1 p_1 - 16 c_1 p_2 - 8 c_2 p_1 - 32 c_2 p_2 - 16 c_3 p_1 + 2 c_3 p_2 - 32 c_4 p_1 + 4 c_4 p_2 \\
& & - 4 c_1 q_1 - 16 c_1 q_2 - 8 c_2 q_1 - 32 c_2 q_2 - 16 c_3 q_1 + 2 c_3 q_2 - 32 c_4 q_1 + 4 c_4 q_2 - 16 c_1 t_1 + 2 c_1 t_2 - 32 c_2 t_1 + 4 c_1 t_3\\
& & + 4 c_2 t_2 + 2 c_1 t_4 + 8 c_2 t_3 - 8 c_3 t_2 + 4 c_2 t_4 - 16 c_3 t_3 - 16 c_4 t_2 - 8 c_3 t_4 - 32 c_4 t_3 - 16 c_4 t_4 + 4 p_1 p_2 + 158 p_1 q_1 \\
& & + 95 p_1 q_2 + 95 p_2 q_1 + 142 p_2 q_2 + 4 q_1 q_2 - 296 p_1 t_1 - 168 p_1 t_2 + 12 p_2 t_1 + 12 p_2 t_2 - 248 p_2 t_3 - 168 p_2 t_4 - 296 q_1 t_1 \\
& & + 12 q_2 t_1 - 168 q_2 t_2 - 168 q_1 t_4 - 248 q_2 t_3 + 12 q_2 t_4 + 2 t_1 t_3 + 14\end{aligned}$$ Then we do a variable transformation to make the variable in the domain {-1,1} using $x_i=\frac{1-s_i}{2}$ if we let $x_1=p_1, x_2=p_2,..., x_{12}=t_4$.
-- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
$p_1$ $p_2$ $q_1$ $q_2$ $c_1$ $c_2$ $c_3$ $c_4$ $t_1$ $t_2$ $t_3$ $t_4$
$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
$s_1$ $s_2$ $s_3$ $s_4$ $s_5$ $s_6$ $s_7$ $s_8$ $s_9$ $s_{10}$ $s_{11}$ $s_{12}$
-- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- -------------- --------------
$$\begin{aligned}
& & f'(p_1,p_2,q_1,q_2,c_1,c_2,c_3,c_4,t_1,t_2,t_3,t_4) \\
& = & 2f(p_1,p_2,q_1,q_2,c_1,c_2,c_3,c_4,t_1,t_2,t_3,t_4) \\
& = & (261s_1)/2 + (215s_2)/2 + (261s_3)/2 + (215s_4)/2 - 41s_5 - 82s_6 + 3s_7 + 6s_8 - 137s_9 - 81s_{10} - 107s_{11} - 81s_{12} \\
& & + 2s_1s_2 + 79s_1s_3 + (95s_1s_4)/2 + (95s_2s_3)/2 - 2s_1s_5 + 71s_2s_4 - 4s_1s_6 - 8s_2s_5 + 2s_3s_4 - 8s_1s_7 - 16s_2s_6 \\
& & - 2s_3s_5 - 16s_1s_8 + s_2s_7 - 4s_3s_6 - 8s_4s_5 - 148s_1s_9 + 2s_2s_8 - 8s_3s_7 - 16s_4s_6 - 84s_1s_{10} + 6s_2s_9 - 16s_3s_8 \\
& & + s_4s_7 + 34s_5s_6 + 6s_2s_{10} - 148s_3s_9 + 2s_4s_8 - 4s_5s_7 - 124s_2s_{11} + 6s_4s_9 - 8s_5s_8 - 8s_6s_7 - 84s_2s_{12} - 84s_4s_{10} \\
& & - 8s_5s_9 - 16s_6s_8 - 84s_3s_{12} - 124s_4s_{11} + s_5s_{10} - 16s_6s_9 + 34s_7s_8 + 6s_4s_{12} + 2s_5s_{11} + 2s_6s_{10} + s_5s_{12} + 4s_6s_{11} \\
& & - 4s_7s_{10} + 2s_6s_{12} - 8s_7s_{11} - 8s_8s_{10} - 4s_7s_{12} - 16s_8s_{11} - 8s_8s_{12} + s_9s_{11} + 794\end{aligned}$$
This corresponds to Ising Hamiltonian with local fields $${\bf h}^T = \bordermatrix{
~ & \sigma_z^{(1)} & \sigma_z^{(2)} & \sigma_z^{(3)} & \sigma_z^{(4)} & \sigma_z^{(5)} & \sigma_z^{(6)} & \sigma_z^{(7)} & \sigma_z^{(8)} & \sigma_z^{(9)} & \sigma_z^{(10)} & \sigma_z^{(11)} & \sigma_z^{(12)} \cr
~ & 130.5 & 107.5 & 130.5 &107.5&-41&-82&3&6&-137&-81&-107&-81}$$ and coupling terms: $$\label{eq:J}
\begin{array}{ccl}
{\bf J} & = & \displaystyle
\bordermatrix{
~ & \sigma_z^{(1)} & \sigma_z^{(2)} & \sigma_z^{(3)} & \sigma_z^{(4)} & \sigma_z^{(5)} & \sigma_z^{(6)}& \sigma_z^{(7)}& \sigma_z^{(8)}& \sigma_z^{(9)}& \sigma_z^{(10)}& \sigma_z^{(11)}& \sigma_z^{(12)} \\[0.02in]
\sigma_z^{(1)} & &2&79&47.5&-2&-4& -8 & -16 &-148 &-84 &0 & 0\\[0.02in]
\sigma_z^{(2)} & & &47.5&71& -8 & -16 & 1 & 2 & 6 &6 &-124&-84 \\[0.02in]
\sigma_z^{(3)} & & & & 2 &-2&-4& -8 & -16 &-148&0 &0 & -84\\[0.02in]
\sigma_z^{(4)} & & & & & -8 & -16 & 1 & 2 & 6& -84 & -124 & 6\\[0.02in]
\sigma_z^{(5)} & & & & & & 34 & -4 & -8 & -8 & 1& 2& 1 \\[0.02in]
\sigma_z^{(6)} & & & & & & & -8 & -16 & -16 & 2& 4& 2\\[0.02in]
\sigma_z^{(7)} & & & & & & & & 34 & 0 & -4 &-8 & -4 \\[0.02in]
\sigma_z^{(8)} & & & & & & & & & 0& -8 & -16 & -8 \\[0.02in]
\sigma_z^{(9)} & & & & & & & & & &0 & 1 & 0 \\[0.02in]
\sigma_z^{(10)}& & & & & & & & & & & 0 & 0 \\[0.02in]
\sigma_z^{(11)}& & & & & & & & & & & & 0 \\[0.02in]
\sigma_z^{(12)}& & & & & & & & & & & & \\[0.02in]}
\end{array}$$
Factoring $ N = 59989 = 251\times 239$
---------------------------------------
The following table shows how to divide the columns to blocks. The lengths of each carries ($c_{11}c_{10}c_9$, $c_8c_7c_6$, $c_5c_4c_3$, $c_2c_1$) are determined by what is the largest carry for the numbers in right next block are(assuming each variable to be 1, then add them up). For example, the maximum carry for the right-most block (except the least significant bit) is 3 which is 11 in binary, so the length of the carry for this block is 2. This carry is represented as $c_2c_1$.
Factoring $N = 376289 = 659\times 571$
--------------------------------------
The following table shows the carries $c_{14}c_{13}$, $c_{12}c_{11}c_{10}$, $c_{9}c_{8}c_7c_6$, $c_5c_4c_3$, $c_2c_1$ for corresponding blocks. There are overlaps in the column $2^{14}$.
Range of Coefficients
---------------------
Define the lengths of $p$ and $q$ as $l_1$ and $l_2$, respectively. Let $l_1 = \frac{log(N)}{2} = O(log(N))$ and $l_2 = \frac{log(N)}{2} = O(log(N))$. Suppose each block contains 3 columns as in Table 2 for factoring 59989. Then the sum for one block is not larger than $(l_2+1)+2(l_2+1)+4(l_2+1)=7l_2+7$, assuming all unknown bits in $p$ and $q$ are 1’s and all carries from the block on the right hand side are 1’s. Thus, the length of the sum for the current block is at most $log(7l_2+7) = O(log(log(N)))$. Therefore, the length of the carry from the sum of current block is at most $log(7l_2+7)-3 = O(log(log(N)))$. Because the length of the carry plus the width of the block (which is 3 in this case) determines the range of the coefficients in the cost function, the maximum coefficient in the cost function corresponding to one block is $(2^{O(log(log(N)))})^2=O((log(N))^2)$. This square comes from transforming the equation for each block to a square that makes the equation hold. (See appendix A.2 for examples of these equations.) There are approximately $\frac{log(N)}{3}$ blocks in total for this example, such that the coefficient in the combined cost function containing all cost functions for each block is no larger than $\frac{log(N)}{3} * O((log(N))^2) = O((log(N))^3)$. Note that for the majority of cases, this becomes $O((log(N))^2)$ because most of terms in different blocks are different. Since the variable replacement only effects the scale of the range of the coefficients linearly, the coefficients of the final quadratic cost function are polynomially large with regard to the size of $N$, the number to be factored.
[^1]: humblets@ornl.gov
[^2]: kais@purdue.edu
|
---
abstract: 'We have computed wind models with time-dependent dust formation and grain-size dependent opacities, where (1) the problem is simplified by assuming a fixed dust-grain size, and where (2) the radiation pressure efficiency is approximated using grain sizes based on various means of the actual grain size distribution. It is shown that in critical cases, the effect of grain sizes can be significant. For well-developed winds, however, the effects on the mass-loss rate and the wind speed are small.'
author:
- 'Lars Mattsson, and Susanne Höfner'
title: 'Relaxing the Small Particle Approximation for Dust-grain opacities in Carbon-star Wind Models'
---
[**Introduction.**]{} In a new grid of C-star wind models Mattsson et al. (2010, A&A, 509, 14) used the so-called [*small-particle limit*]{} (SPL) of Mie theory, i.e., grains are assumed to be small compared to the photon wavelengths when calculating their optical properties. In that limit the wavelength- and grain size-dependence of the opacity can be greatly simplified. Fig. 1 shows the distribution of actual grain sizes found in a representative subsample of models taken from the model grid (presented in Mattsson et al. 2010, A&A, 509, 14). The maximum grain size $a_{\rm d}$ (the “dividing radius” between the SPL and non-SPL regemes) where deviations in opacities from the small particle limit are less than 10% is marked by the vertical dashed line. The vast majority of the grid models did not meet this criterion (see Fig. 1).
[**Models.**]{} We have re-computated a selection of models adopting different relaxations of the small-particle approximation (SPA) and picked models such that two categories emerge: those with strong well-developed winds and those with slow critical winds (Mattsson & Höfner 2010, A&A, submitted). We use an “optimised” (OPT) constant grain radius $a_{\rm gr} = 3.55\cdot 10^{-5}$ cm for all grains (the peak in $Q_{\rm rp}$, see Fig.1). We also consider mean grain radii derived from one of the moments $K_i$ ($i = 1,2,3$) of the actual grain-size distribution as effective grain sizes used in the raditive transfer calculations.
[**Results and Conclusions.**]{} In the critical-wind cases, the effect of grain sizes can be significant. Mass-loss rates may increase by a factor of two, or more, and wind speeds by as much as an order of magnitude (see Fig. 2, open symbols). Furthermore, the corresponding models with grain-size dependent opacities that have resultant winds tend to have much lower degrees of dust condensation, compared to their SPA counterparts (again, see Fig. 2). Consequently, the “dust-loss rates” are much lower in these new models. In contrast, for well-developed dust-driven winds (Fig. 2, filled symbols), where the dust formation has saturated, the effect of grain sizes on mass-loss rate, wind speed and dust-to-gas ratio is almost negligible.
This work was partly supported by the Swedish Research Council (Vetenskapsrådet).
{width="35mm"} {width="35mm"}
![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2a "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2b "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2c "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2d "fig:"){width="28mm"}\
![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2e "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2f "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2g "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2h "fig:"){width="28mm"}\
![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2i "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2j "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2k "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2l "fig:"){width="28mm"}\
![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2m "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2n "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2o "fig:"){width="28mm"} ![From left to right: Mass-loss rates, wind speeds, mean degrees of dust condensation and dust-to-gas ratios for “$Q_{\rm rp}$-optimised” and K1, K2, K3 models versus the corresponding SPA models. The dashed lines show the case of equal values. \[agr\_spa\_Kopt\] ](mattsson_f2p "fig:"){width="28mm"}
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abstract: 'Consider a continuous surjective self map of the open annulus with degree $d>1$. It is proved that the number of Nielsen classes of periodic points is maximum possible whenever $f$ has a completely invariant essential continuum. The same result is obtained in negative degree $|d|>1$ and for just forward invariant essential continua, provided that the continuum is locally connected. We also deal with the problem of wether there is a representative of each Nielsen class in the filled set of the invariant continuum. Moreover, if the map extends continuously to the boundary of the annulus and both boundary components are either attracting or repelling, the hypothesis on the existence of the invariant continuum is no longer needed for obtaining all the periodic points in the interior of the annulus.'
address:
- 'J. Iglesias, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay'
- 'A. Portela, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
- 'A. Rovella, Universidad de La República. Facultad de Ciencias. Centro de Matemática. Iguá 4225. C.P. 11400. Montevideo, Uruguay'
- 'J. Xavier, Universidad de La República. Facultad de Ingenieria. IMERL. Julio Herrera y Reissig 565. C.P. 11300. Montevideo, Uruguay '
author:
- 'J.Iglesias, A.Portela, A.Rovella and J.Xavier'
title: 'Existence of periodic points for self maps of the annulus.'
---
Introduction.
=============
This paper deals with the existence of periodic points for annulus maps. A map is any continuous endomorphism of the open annulus $A$. The [*degree*]{} of an annulus map $f$ is the integer $d$ such that $f_*(n)=d.n$, where $f_*$ is the endomorphism of the fundamental group $\Pi_1(A)\sim {{\mathbb{Z}}}$ induced by $f$. We are particularly interested in the case where the degree of the map has modulus greater than one.
We will consider two different ways of saying that a map has abundance of periodic orbits. The first one is completeness, and is related to Nielsen equivalence. The second one is the growth rate inequality which will be discussed shortly. A map of the annulus is complete if for every $n$ the number of Nielsen classes of fixed points of $f^n$ is equal to $|d^n-1|$. This is the maximum possible for a map of degree $d$. Roughly speaking, a continuous map $f:A\to A$ of degree $d$, $|d|>1,$ is said to be complete if it has as many different Nielsen classes of periodic points as the map $z^d$ acting on the circle. Nielsen equivalence and completeness are explained in the next section.
This notion of completeness is related to a great deal of problems on existence of periodic orbits in surface dynamics. For example, a complete map always satisfies the growth rate inequality: $$\limsup_{n\to \infty}\frac{1}{n}\ln( \#\{{{\mbox{Fix}}}(f^n)\}) \geq \ln (d).$$
It is an open problem if the rate inequality holds for differentiable maps of degree $d$ acting on the sphere $S^2$ (Problem 3 posed in [@shub2]).
Other results related to the existence of periodic points for $C^0$ maps, (not necessarily homeomorphisms) were obtained by Hagopian in [@h], where some conditions are imposed on plane continua in order to obtain the existence of fixed points for any $C^0$ map defined on it.
Existence of recurrence is mostly present in maps with $|d|>1$ because $|d|>1$ implies an ’expansion’ on the direction of a simple nontrivial closed curve. This idea will be used frequently throughout the article.
The covering map $z\to z^d$ defined in the annulus $\{z\in {{\mathbb{C}}}: 0<|z|<1\}$ is a trivial example of a map without periodic points in the open annulus, and so the mere existence of periodic orbits becomes of interest. As covering maps lift to homeomorphisms of the plane, the problem also relates to surface homeomorphisms dynamics, in particular with what is called Brouwer Theory. In [@iprx] completeness of annulus coverings of degree $d$, $|d|>1$, was proved under the assumption that an essential continuum is preserved.
In this paper we study the same problem for general continuous maps (not necesarily coverings). Some definitions are needed before stating the results.
We say that a map $f: A \to A$ is complete on $K\subset A$ if $f$ is complete and every Nielsen class has a representative in $K$. Whenever $K$ is a compact subset of the (open) annulus, denote by ${{\mbox{Fill}}}(K)$ the union of $K$ with the bounded (relatively compact) components of its complement.
\[t1\] Let $f: A \to A$ be a surjective degree $d$ map of the annulus, and $K\subset A$ an essential continuum such that $f^{-1}(K) = K$. If $d>1$, then $f$ is complete on ${{\mbox{Fill}}}(K)$.
The ingredientes for the proof of this Theorem are Lefschetz Theory (to obtain fixed points enclosed by curves with nonzero index), Caratheodory Theory (constructions of crosscuts with determined properties) and Nielsen Theory (existence of fixed points in every Nielsen class is assured by proving that every lift of $f$ to its universal covering has a fixed point).
The proof obtained fails for $d<-1$ and it is easy to construct counterexamples of the proof. It is an open problem if the assertion of the Theorem is true also for negative values of $d$.
The hypothesis $f^{-1}(K)=K$ is also important in the proof. Assuming that $K$ is locally connected, this assumption and the surjectivity of $f$ can be dropped, and a satisfactory conclusion is obtained.
\[t2\] Let $f: A \to A$ be a degree $d$ map of the annulus, and $K\subset A$ a locally connected essential continuum such that $f(K) \subset K$. If $|d|>1$, then $f$ is complete on ${{\mbox{Fill}}}(K)$.
These statements imply the existence of periodic points in the set ${{\mbox{Fill}}}(K)$ and there are examples where the periodic points are not contained in $K$. Posing some conditions on the ’behaviour at infinity’ of the map $f$, completeness is obtained without any other assumption. To state these conditions, note that the annulus can be compactified with two points, called ends of $A$.
\[t3\] Let $f: A \to A$ be a degree $d$ map of the annulus, where $|d|>1$. Assume that $f$ extends continuously to the ends of $A$. Each one of the following conditions imply that $f$ is complete.
1. Both ends of $A$ are attracting.
2. Both ends of $A$ are repelling.
See sections 4,5 and 6 where completeness is obtained for maps of the annulus under different sets of hypothesis. In Section 2 the notion of Nielsen class is explained, in Section 3 the index and Lefschetz fixed point Theorem are introduced without proofs, and some of the lemmata needed in the proofs of the theorems are included. Section 4 is devoted to the proof of Theorem \[t1\]. In Section 5 the proofs for locally connected $K$ are given, and the last Section contains two simple applications of these results to maps of the sphere.
Nielsen Theory.
===============
Two fixed points $p$ and $q$ of a map $f:A\to A$ are said [*Nielsen equivalent*]{} if there exists a curve $\gamma$ joining $p$ to $q$ such that $f\gamma$ is homotopic to $\gamma$ relative to its endpoints. This is an equivalent relation whose classes are called Nielsen classes. Two fixed points that are very close are equivalent. Of course the existence of a continuum of fixed points gives just one class of Nielsen equivalent fixed points. If $|d|>1$, then the map $p_d(z)=z^d$ on the annulus $A={{\mathbb{C}}}\setminus\{0\}$ has exactly $|d-1|$ fixed points, and also $|d-1|$ Nielsen classes.
It is not difficult to see that a degree $d$ map of the annulus cannot have more than $|d-1|$ classes of fixed points. A map $f$ is [*complete*]{} if for every $n$ the number of Nielsen classes of fixed points of $f^n$ is equal to that of the map $p_d$. So, asking for a map $f$ to be complete is to ask that is has the maximum possible (non artificial) periodic orbits. The following proposition is a standard result in Nielsen theory (see Jiang’s book [@jiang] for further information on the subject) and will be used to prove all the results throughout this paper. Alternatively, you can see Corollary 1 in [@iprx].
\[comp\] A degree $d$ map of the annulus has $|d-1|$ Nielsen classes of fixed points if and only if any lift of $f$ to the universal covering of the annulus has a fixed point.
Note that in order to prove completeness of a map $f$ satisfying the hypothesis of Theorem \[t1\], Theorem \[t2\] or Theorem \[t3\], it suffices to show that any lift of $f$ has a fixed point. Indeed, note that in any case, every iterate $f^k$ of $f$ satisfies the same hypothesis as $f$. It follows that $f^k$ has $|d^k-1|$ classes of fixed points.
The index. {#3}
==========
Let $\Pi:{{\mathbb{R}}}^2\to {{\mathbb{R}}}^2\setminus\{0\}$, defined by $\Pi(z)=\exp(2\pi i z)$ denote the universal covering of the punctured plane. The index of a plane closed curve $\gamma$ defined in an interval $I=[a,b]$ such that $\gamma(t)\neq 0$ for every $t$, is defined as the first coordinate of $\gamma'(b)-\gamma'(a)$, where $\gamma'$ is any lift of $\gamma$ under $\Pi$: that is, $\gamma': [a,b]\to {{\mathbb{R}}}^2$ is a curve such that $\Pi(\gamma'(t))=\gamma(t)$ for every $t$. This does not depend on the choice of the lift. Note that the index is an integer.
We abuse notation and make no difference between the curve and its trace in the plane.
\[index\] Let $\gamma$ be a closed curve and $f$ a continuous map defined on $\gamma$ and without fixed points on $\gamma$. Define the Lefschetz index of $f$ in $\gamma$ as the index of the closed curve $f\circ\gamma-\gamma$. This index will be denoted by $I_f(\gamma)$.
Let $\gamma$ be a simple closed curve in the plane, so that its complement has two connected components, one is bounded and denoted ${{\mbox{int}}}(\gamma)$ and the other unbounded, denoted ${{\mbox{ext}}}(\gamma)$.
The curve $\gamma$ is positively oriented if the index of $t\to \gamma(t)-p_0$ is $1$ whenever $p_0$ is a point that belongs to ${{\mbox{int}}}(\gamma)$. If $\gamma$ is positively oriented, then ${{\mbox{int}}}(\gamma)$ is located to the left of $\gamma$.
[**Lefschetz fixed point Theorem.**]{}
*Let $f$ be a continuous self-map of the plane.*
1. If $\gamma$ is a simple closed curve such that $I_f(\gamma)\neq 0$, then $f$ has a fixed point in the bounded component of ${{\mathbb{R}}}^2\setminus\gamma$.
2. Let $A$ be an annulus in the plane whose boundary components are positively oriented curves $\gamma_0$ and $\gamma_1$, and assume that $I_f(\gamma_0)\neq I_f(\gamma_1)$, then $f$ has a fixed point in $A$.
It is our prupose to use the above result to prove existence of periodic points. We will need some techniques for calculating indexes of curves. The next assertion follows by continuity of $I_f(\gamma)$ on $f$ and $\gamma$ whenever the index is defined.
\[l1\] Assume that $\gamma$ is a closed curve and that $f_t$ is a homotopy. If no $f_t$ has a fixed point in $\gamma$, then $t\to I_{f_t}(\gamma)$ is constant.
Moreover, if $\gamma_t$ is a free homotopy and $f$ has no fixed points on any $\gamma_t$, then $t\to I_f(\gamma_t)$ is constant.
The following is an immediate consequence:
\[l3\] Let $f$ be fixed point free on a simple closed curve $\gamma$. If $P$ is a finite subset of $\gamma$, then there exist $\epsilon>0$ and $\delta>0$ such that the index of $f'$ on $\gamma'$ is equal to the index of $f$ in $\gamma$ whenever the following conditions hold:
1. $\gamma'$ is a simple closed curve such that $(\gamma'\setminus\gamma) \cup (\gamma\setminus\gamma')$ is contained in $V_\delta(P)$, the $\delta$ neighborhood of $P$,
2. $f'$ is a continuous map such that $f'=f$ outside $V_\delta(P)$,
3. the image under $f'$ of $V_\delta(P)$ is contained in $V_\epsilon(f(P))$.
The proof of this result follows from the fact that if $\epsilon$ and $\delta$ are sufficiently small, then both $f$ and $f'$ and $\gamma$ and $\gamma'$ are homotopic. This previous corollary and the following two criteria is all that is needed to calculate indexes in the sequel.
Assume that $f$ is constant, equal to $p$ and that $\gamma$ is a positevely oriented simple closed curve. The index $I_f(\gamma)$ is equal to $1$ if $p$ belongs to ${{\mbox{int}}}(\gamma)$, equal to $0$ if $p$ belongs to ${{\mbox{ext}}}(\gamma)$, and not defined when $p\in\gamma$.
We proceed to generalize this statement.
\[l2\]Let $f$ be defined on a positively oriented simple closed curve $\gamma:[0,1]\to {{\mathbb{R}}}^2$. Let $s$ be an open arc of $\gamma$ and assume that the image of $s$ is compactly contained in ${{\mbox{int}}}(\gamma)$. Assume also that $f$ has no fixed points on $\gamma$. Then, there exists a homotopy $\{f_t : 0\leq t\leq 1\}$ beginning at $f=f_0$, and a time $t'\in (0,1)$ such that the following conditions hold:
1. $f_t(s)\subset {{\mbox{int}}}(\gamma)$ for every $t<t'$.
2. $f_t(s)\subset{{\mbox{ext}}}(\gamma)$ for every $t>t'$
3. $f_{t'}(s)\subset s$.
4. For every $t$ it holds that $f_t=f$ in $\gamma\setminus s'$ where $s'$ is an open arc compactly containing $s$, and $f_t(s')\cap\gamma\subset s$.
In addition, for a homotopy satisfying all these properties, it holds that $I_{f_0}(\gamma)=I_{f_1}(\gamma)+1$.
Let $h_t:{{\mathbb{R}}}^2\to{{\mathbb{R}}}^2$ be a homotopy satisfying that $h_0=id$, $h_1$ carries $f(s)$ to a point $p$ contained in ${{\mbox{int}}}(\gamma)$, and for every $t$ it holds that $h_t({{\mbox{int}}}(\gamma))\subset{{\mbox{int}}}(\gamma)$ and $h_t$ is the identity in a neighborhood $U$ of the closure of ${{\mbox{ext}}}(\gamma)$. Then $h_t\circ f$ is a homotopy begining at $f$, having no fixed points on $\gamma$, and such that the final map has $h_1(f(s))=p$.\
So it can be assumed that $f$ is constant in $s$, say $f(s)=p_0\in {{\mbox{int}}}(\gamma)$. Next, let $p(t)$ be a curve defined in $[0,1]$ such that $p(0)=p_0$, $p(t)\in {{\mbox{int}}}(\gamma)$ if $t<1/2$, and $p(t)\in {{\mbox{ext}}}(\gamma)$ if $t>1/2$ and $p(1/2) \in s$. Take an arc $s'$ containing $s$ in its interior, so that $s'\setminus s$ has two connected components whose images under $f$ are contained in the complement of $U$. Let $f_t$ be a homotopy beginning at $f$, such that following three conditions hold: $f_t(s)=p(t)$, $f_t(s'\setminus s)\cap (\gamma\setminus s)=\emptyset$, and $f_t=f$ in $\gamma\setminus s'$. Then the conditions of the lemma hold with $t'=1/2$. This proves the first assertion of the lemma.\
It remains to prove that $I_{f_0}(\gamma)=I_{f_1}(\gamma)+1$, the second assertion of the lemma. It was explained above that $f_t$ can be assumed to be constant in $s$ for every $t$. So the problem reduces to estimate the jump of $I_{f_t}(\gamma)$ when $t$ passes through $t'$. The construction made implies that no $f_t$ has fixed points in $\gamma\setminus s$ so that every $f_t$ can be assumed to be equal to $f$ in $\gamma\setminus s'$, where $s'$ is a small neighborhood of $s$ whose image under $f_t$ is contained in ${{\mbox{int}}}(\gamma)$ for every $t<t'$.
Then the jump can be calculated as if $s$ is the segment $(-1,1)$ contained in the real axis, oriented from right to left, $s'$ is the arc $(-2,2)$ and the image of $s$ changes from the lower half plane to the upper one passing through the origin, while the intersection of $f_t(s')$ with the real axis is contained in $s$, and $f_1=f_0$ in the rest of $\gamma$. Moreover, it may also be assumed that $f_t (s')$ is piecewise linear. See Figure \[ind\]. In this case, the index change over $s'$ can be calculated directly from the definition.
![[]{data-label="ind"}](figura17.eps)
Analogously, we have:
\[meter\] Let $f$ be defined on a positively oriented simple closed curve $\gamma:[0,1]\to {{\mathbb{R}}}^2$. Let $s$ be an open arc of $\gamma$ and assume that the image of $s$ is compactly contained in ${{\mbox{int}}}(\gamma)$. Assume also that $f$ has no fixed points on $\gamma$. Then, there exists a homotopy $\{f_t : 0\leq t\leq 1\}$ beginning at $f=f_0$, and a time $t'\in (0,1)$ such that the following conditions hold:
1. $f_t(s)\subset {{\mbox{ext}}}(\gamma)$ for every $t<t'$.
2. $f_t(s)\subset{{\mbox{int}}}(\gamma)$ for every $t>t'$
3. $f_{t'}(s)\subset s$.
4. For every $t$ it holds that $f_t=f$ in $\gamma\setminus s'$ where $s'$ is an open arc compactly containing $s$, and $f_t(s')\cap\gamma\subset s$.
In addition, for a homotopy satisfying all these properties, it holds that $I_{f_0}(\gamma)=I_{f_1}(\gamma)-1$.
\[rectangulo\] Let $R\subset {{\mathbb{R}}}^2$ be the square $[-1,1]^2$. Let $f$ be defined on $\partial R$ such that:
- $f(\{y=1\})\subset \{y<1\}$,
- $f(\{y=-1\})\subset \{y>-1\}$,
- $f(\{x=1\})\subset \{x>1\}$,
- $f(\{x=-1\})\subset \{x<-1\}$.
Then, $I_f (\gamma) = -1$, where $\gamma$ is $\partial R$ with the positive orientation.
Let $h_t(x,y)=(x,(1-t)y)$, so that $h_0$ is the identity and $h_1(x,y)=(x,0)$. Let $f_t=h_t\circ f$. Then $f_0=f$ and the hypotheses imply that no $f_t$ has fixed points in $\partial R$ (for example, take any $(x,1)\in \partial R$ and let $f(x,1)=(u,v)$, then $v<1$ implies that $f_t(x,1)=(u, (1-t)v)$ and $(1-t)v<1$ for every $t\in (0,1)$). Note that the image of $\partial R$ under $f_1$ is contained in the line $y=0$, and that the image of $x=1$ is contained in $x>1$ and the image of $x=-1$ contained in $x<-1$. It follows from Corollary \[meter\] that one can obtain a map $f_2$ such that $f_2(\partial R)$ is contained in the interior of $R$, so that $1=I_{f_2}(\partial R)=I_{f_1}(\partial R)+2= I_{f}(\gamma)+2$.
Of course, the fact that $R$ is a square is not essential in the hypothesis, and we will use the lemma and remark above to calculate index of more general curves:
\[torcido\] Let $\alpha$ and $\beta$ be disjoint simple proper lines in the plane, each one of which separate the plane. Let $\gamma$ and $\delta$ be another pair of disjoint curves separating the plane. Assume also that each $\gamma$ and $\delta$ intersect $\alpha$ in one point and $\beta$ in one point. Now let $\Gamma$ be the simple closed curve determined by the four arcs of the curves delimited by the intersection points, with the positive orientation. Now let $f$ be a map defined on $\Gamma$ such that $f(\Gamma\cap\alpha)$ is contained in the component of the complement of $\alpha$ that contains $\beta$, $f(\Gamma\cap\beta)$ is contained in the component of the complement of $\beta$ that contains $\alpha$, that $f(\Gamma\cap\delta)$ is contained in the component of the complement of $\delta$ that does not contain $\gamma$ and that $f(\Gamma\cap\gamma)$ is contained in the component of the complement of $\gamma$ that does not contain $\delta$. Then the index of $f$ in $\Gamma$ is equal to $-1$.
Note that there exists a homeomorphism $H$ isotopic to the identity such that $H(\alpha)=\{y=1\}$, $H(\beta)=\{y=-1\}$, $H(\gamma)=\{x=-1\}$ and $H(\delta)=\{x=1\}$. Then the map $H\circ f$ satisfies the hypothesis of Lemma \[rectangulo\] so $-1 = I_{H\circ f}(\partial R)=I_f(\Gamma)$.
\[rk1\] If the last two items are changed to $f(\{x=1\})\subset \{x<-1\}$ and $f(\{x=-1\})\subset \{x>1\}$, then the conclusion is $I_f(\partial R)=1$. To prove that apply the same homotopy and then observe that there exists $f_2$ homotopic to $f_1$ such that $f_2(\partial R)$ is contained in the interior of $R$, but now the homotopy from $f_1$ to $f_2$ has no fixed points in $\partial R$. The conclusion follows immediately.
As before, one obtains:
\[torcido2\] Let $\alpha$ and $\beta$ be disjoint simple proper lines in the plane, each one of which separate the plane. Let $\gamma$ and $\delta$ be another pair of disjoint curves separating the plane. Assume also that each $\gamma$ and $\delta$ intersect $\alpha$ in one point and $\beta$ in one point. Now let $\Gamma$ be the simple closed curve determined by the four arcs of the curves delimited by the intersection points, with the positive orientation. Now let $f$ be a map defined on $\Gamma$ such that $f(\Gamma\cap\alpha)$ is contained in the component of the complement of $\alpha$ that contains $\beta$, $f(\Gamma\cap\beta)$ is contained in the component of the complement of $\beta$ that contains $\alpha$, that $f(\Gamma\cap\delta)$ is contained in the component of the complement of $\gamma$ that does not contain $\delta$ and that $f(\Gamma\cap\gamma)$ is contained in the component of the complement of $\delta$ that does not contain $\gamma$. Then the index of $f$ in $\Gamma$ is equal to $1$.
Proof of Theorem 1.
===================
[**Theorem \[t1\].**]{} [*Let $f: A \to A$ be a surjective map with degree $d$, $K\subset A $ an essential continuum such that $f^{-1}(K)=K$. If $d>1$, then $f$ is complete on ${{\mbox{Fill}}}(K)$.*]{}
As was already explained, it is sufficient to prove that every lift of $f$ has a fixed point on the preimage of ${{\mbox{Fill}}}(K)$ under the covering projection .
The proof will use indexes of curves, so it is convenient to interpret the annulus as a subset of the plane. The map $f$ will be considered as a map from $A:={{\mathbb{R}}}^2\setminus \{S\}$ into itself, where $S$ is the origin (called South; the other end of the annulus is $\infty$, called North). The map $f$ is assumed to have degree $d$, which is equivalent to impose that the index of the closed curve $t\in[0,1]\to f(\exp(2\pi i t))$ is equal to $d$.
As $K$ is essential, the complement of $K$ in the annulus $A={{\mathbb{R}}}^2\setminus \{S\}$ has two distinguished connected components. The unbounded one is denoted $A_N$, while $A_S$ stands for the connected component of $A\setminus K$ that accumulates on $S$. Note that $A_S\cup \{S\}$ is simply connected, thus Riemann’s Theorem implies that there exists a conformal bijection $h$ from $A_S\cup \{S\}$ to the unit disc ${{\mathbb{D}}}$ that carries $S$ to the origin.
\[ends\] Note that as $K$ is connected, then $A_N$ and $A_S$ are the unique components of $A\setminus K$ that are essential in $A$, while the others are simply connected. As the degree of $f$ is nonzero, the image under $f$ of an essential open set is essential, and as $f^ {-1}(K)\subset K$ then $f(A_N)$ is a subset of $A_S$ or a subset of $A_N$. The same can be said about $A_S$. On the other hand, as $f$ was assumed to be surjective, only two possibilities remain: either $f(A_N)\subset A_N$ and $f(A_S)\subset A_S$ or $f(A_N)\subset A_S$ and $f(A_S)\subset A_N$. We will consider the two cases separately.
We will assume first that each $A_S$ and $A_N$ are $f$-invariant.
Constructions in the annulus.
-----------------------------
We begin with some constructions of curves in the annulus, this will not give the desired conclusions, not before lifting the curves to the universal covering and calculating their index under a given lift $F$ of $f$.
We will prove first that $f$ has fixed points in the annulus. Assume by contradiction that $f$ has no fixed point.
It will be convenient to recall some results from Caratheodory’s theory. Let $\Omega$ be a simply connected region in the plane with a conformal bijection $h:\Omega\to {{\mathbb{D}}}$. A crosscut in $\Omega$ is a simple curve $c$ contained in $\Omega$ except for its extreme points which belong to the boundary of $\Omega$. If $h(c(t))\neq 0$ for every $t$, then the complement of $c$ in $\Omega$ has two connected components, one of which contains $h^{-1}(0)$; the other one is denoted $N(c)$. A point $a$ in the boundary of $\Omega$ is accesible if there exists an arc that is contained in $\Omega$ except for one extreme point in $a$. It is well known that accesible points are dense in the boundary of $\Omega$. Moreover, given two accesible points there exists a crosscut in $ \Omega$ whose extremes are the given points. Assume that $\overline\Omega$ is compact, (which in our case is obvious since $A_S\cup \{S\}$ will stand for $\Omega$). Under this assumption, given a positive number $\delta$, there exists a finite set of accesible points, each one of them can be joined with another by a crosscut of diameter less than $\delta$. These crosscuts will be used to construct an essential simple closed curve.
Let $h$ be a conformal bijection from $A_S\cup\{S\}$ to the unit disc carrying $S$ to the origin $0$. It is claimed that for every sufficiently small $\delta>0$, there exist a finite sequence of crosscuts $\{c_i:1\leq i\leq k\}$ in $A_S$ such that
1. $N(c_i)\cap N(c_j)=\emptyset$ whenever $i\neq j$.
2. $\gamma=c_1.c_2\ldots c_k$ is an essential positively oriented simple closed curve ,
3. each $c_i$ has diameter less than $\delta$,
4. $f(c_i)\cap c_i=\emptyset$.
(See figure 2 a).
The proofs of the first, second and third properties follow from the comments above. To prove the last one, note that the contrary assumption implies that for every sufficiently small positive $\delta_n$ there exists a point $x_n$ in some crosscut $c_n$ of diameter less than $\delta_n$ such that $f(x_n)\in c_n$, which implies that the distance from $f(x_n)$ to $x_n$ is less than $\delta_n$. An accumulation point of the sequence $x_n$ must belong to the boundary of $A_S$, that is contained in $K$, and must be a fixed point of $f$, contrary to our assumption.
Except for the extreme points of the crosscuts, the remaining points of $\gamma$ are contained in the region $A_S$. By assumption in Remark \[ends\], $f$ carries $A_S$ to itself. Thus the curve $f(\gamma)$ is contained in $A_S$ excepting for the images of the extreme points of the crosscuts, that are also contained in $K$. Note that the image of a crosscut is not necessarily a crosscut.
Let $P$ be the intersection of $\gamma$ with $K$. $P$ is finite and by Corollary \[l3\] there exists $\epsilon >0$ and $\delta >0$ such that one can modify $f$ in the $\delta$ neighborhood of $P$ arbitrarily, whenever the image of the perturbation is still contained in the $\epsilon$ neighborhood of $f(P)$. Using the fact that $A_S\cup \{S\}$ is conformally equivalent to the unit disc, it is possible to construct a curve in $A_S$ that is as close to the boundary of $A_S$ as wished. With this in mind, let $\alpha$ be a simple closed curve contained in $A_S$ that is close to the boundary of $A_S$ in such a way that the intersection of $\alpha$ with $\gamma$ only occurs in $V_\delta(P)$ and that the intersection of $\alpha$ with $f(\gamma)$ is contained in $V_\epsilon(f(P))$. In order to calculate the index of $f$ in $\gamma$, we apply Corollary \[l3\] and we may assume that $\gamma$ and $f(\gamma)$ are both contained in the closure of $A^{\alpha}_S$, the connected component of the complement of $\alpha$ that contains $S$. Therefore it can be assumed (for the purposes of index calculation) that the boundary of $A_S$ is a simple closed curve $\alpha$, of course, essential in the annulus $A$.
The next step consists in changing $f$ to a homotopic map $f'$ in order to obtain that $f'(\gamma)\cap\gamma=\emptyset.$ In the process, the index will change, but by means of Lemma \[l2\] and Lemma \[rectangulo\] the difference between $I_f(\gamma)$ and $I_{f'}(\gamma)$ will be easily calculated.\
Using Property (C4), there are two possibilities remaining for each $c_i$: either $f(c_i)\subset N(c_i)$, or $f(c_i)\cap N(c_i)=\emptyset$.
Assume that $f(c_i)\subset N(c_i)$. The image under $f$ of the arc $c_i$ is contained in ${{\mbox{ext}}}(\gamma)$; a direct application of Lemma \[l2\] implies that the map $f$ is homotopic to a map $f_i'$ satisfying that $f'_i(c_i)\subset {{\mbox{int}}}(\gamma)$ and $I_{f_i'}(\gamma)=I_f(\gamma)+1$ as $ \gamma$ is positively oriented.\
If there were exactly $J$ of the crosscuts $c_i$ satisfying $f(c_i)\subset N(c_i)$, then change $f$ by a map $f'$ such that $f'(c_i)\cap N(c_i)=\emptyset$ and $I_{f'}(\gamma)=I_f(\gamma)+J$. Moreover $f'$ is equal to $f$ outside a small neighborhood of each $c_i$.\
Now use $f'$ instead of $f$. Note that, even if $f'(c_j)\cap N(c_j)=\emptyset$ for every $j$, it may very well happen that $f'(c_j)\cap\gamma\neq\emptyset$.
Let $A^{\gamma}_S$ be the connected component of $A\backslash \gamma$ containing $S$. Let $R_t$ be a deformation retract of $A^{\alpha}_S$ to $\gamma_S$ be such that a point in $N(c_i)$ is sent under application of $R_1$ to a point in the interior of the arc $c_i$ and such that $R_t(\overline{N(c_i)})\subset \overline{N(c_i)}$ for all $t$. Note that $R_t\circ f'$ cannot have fixed points in $\gamma$ because for a point $x$ in $c_j$ it is known that $f'(x)$ does not belong to the closure of $N(c_j)$. It follows that the index of $R_0\circ f'=f'$ and the index of $R_1\circ f'$ with respect to the curve $\gamma$ are equal. Note that the image of $\gamma$ under $R_1\circ f'$ is contained in the closure of ${{\mbox{int}}}(\gamma)$, so a small $C^0$ perturbation $f''$ of $R_1\circ f'$ does not change the index and satisfies $f''(\gamma)\subset {{\mbox{int}}}(\gamma)$ and $I_{f''}(\gamma)=1$.
In conclusion, the following assertion was proved.
\[c1\] There exists a map $f'$ homotopic to $f$ such that $f'=f$ in $K$ and all the components of the complement of $K$ other than $A_S$. Moreover, it holds that $$I_f(\gamma)+J=I_{f'}(\gamma)=1,$$where $J$ is the number of crosscuts $c_i$ satisfying $f(c_i)\subset N(c_i)$.
Now, we want to explain briefly what happens if the construction like that of $\gamma$ and the map $f'$ is done in $A_N$, the unbounded component of the complement of $K$ containing $N$. Now $A_N\cup \{N\}$ gives a simply connected region and there exists a Riemann map carrying $A_N\cup N$ to the unit disc and $N$ to $0$. The construction of a curve $\gamma_N$ by concatenation of crosscuts of $A_N$ satisfying the properties (C1) to (C4) is the same as above. Now the component $N(c_i)$ of a crosscut $c_i$ is the bounded component of the complement of $c_i$ in $A_N$. Observe in addition that if the curve $\gamma$ is positively oriented, then a point in $N(c_i)$ is located in ${{\mbox{int}}}(\gamma)$, therefore by Lemma \[l2\] $I_f(c_i)=I_{f'}(c_i)+1$ whenever $c_i$ is a crosscut such that $f(c_i)\subset N(c_i)$ and $f'$ is homotopic to $f$ and constructed exactly as in the previous part. The construction of the curve $\alpha$ is analogous in this case. The only difference appears when calculating $I_{f'}(\gamma)$:
\[c2\] There exists a map $f'$ homotopic to $f$ such that $f'=f$ in $K$ and all the components of the complement of $K$ other than $A_N$. Moreover, it holds that $$I_f(\gamma)-J'=I_{f'}(\gamma)=d,$$ where $J'$ is the number of crosscuts $c_i$ such that $f(c_i)$ is contained in $N(c_i)$ and $\gamma = c_1.c_2.\ldots c_n$ with the positive orientation.
It remains to see why $I_{f'}(\gamma)=d$, but this is a simple calculation based on the fact that $\gamma$ is a simple closed curve nontrivial in the annulus, that $f'(\gamma)$ has index $d$ and is contained in the unbounded component of $\gamma$.
Assuming that $d>0$, the conclusion is the following:
\[c3\] Let $f$ be a map of the annulus of positive degree $d>1$. If $K$ is an essential compact subset of the annulus such that $f^{-1}(K)=K$ and both unbounded components of the complement of $K$ are invariant under $f$, then $f$ has a fixed point in ${{\mbox{Fill}}}(K)$.
Note first that the curves $\gamma_{N}$ and $\gamma_{S}$ (the notation makes reference to the curve $\gamma$ constructed in $A_N$, resp. $A_S$) are as close to the set $K$ as wished. Both curves are positively oriented, and their indexes differ, because $$I_{f}(\gamma_{N})\geq d>1\geq I_f(\gamma_{S}).$$
Now Lefschetz fixed point Theorem (see Section \[3\]) shows that $f$ has a fixed point in the annulus bounded by $\gamma_{N}$ and $\gamma_{S}$. As these curves can be constructed arbitrarily close to the boundary of $A_S$ and $A_N$, it comes that $f$ has a fixed point in ${{\mbox{Fill}}}(K)$.
This is not sufficient to conclude that $f$ is complete. It just proves that $f$ has a fixed point, also the same argument shows that every iterate of $f$ has fixed points, but this contains no additional information. We need to prove that every lift of $f$ has fixed points.
Adapting the proof of Corollary 3 a well known Theorem due to Cartwright and Littlewood follows: [*If an orientation preserving homeomorphism $f$ of the plane leaves invariant a continuum $K$ not separating the plane, then $f$ has a fixed point in $K$.*]{}\
Indeed, take a curve like $\gamma$ in the complement of $K$ and the arguments above show that the index $I_f(\gamma)$ is equal to $d+J$, where $d=1$ (as $f$ is a homeomorphism) and $J\geq 0$.
Lifting the construction.
-------------------------
We will lift the construction in the annulus and consider an arbitrary lift $F$ of $f$. We will prove that $F$ has a fixed point (in order to prove completeness by Proposition 1). Assume by contradiction that $F$ is fixed point free. It is not straightforward to lift the construction. The difficulty comes at first sight, when the preimage of $K$ under the covering projection is not compact. But there are others, as will become clear while calculating the index of a curve.
The universal covering of the annulus is given by the map $\Pi:{{\mathbb{C}}}\to {{\mathbb{C}}}\setminus\{0\}$, written in complex coordinates as $\Pi(z)=\exp(2\pi iz)$. Note that $\Pi(z+1)=\Pi(z)$ so the group of deck transformations is generated by the translation $z\to z+1$.
Let $K'$ be the preimage of $K$ under the covering projection. Note that $K'$ is not a compact set but is a closed subset of some horizontal strip $-T\leq y\leq T$; we will assume $T=1$ to simplify notation. Moreover, $K'$ separates the plane in at least two components, one contains $\{y\leq -1\}$, denoted $A'_S$ and the other contains $\{y\geq 1\}$, denoted $A'_N$. Denote as usual $z=(x,y)$ for the compex number $z=x+iy$.
Let $\gamma$ and $\alpha$ the curves in $A_S$ constructed above and denote by $\gamma ^S$ and $\alpha ^S$ the preimages of $\gamma$ and $\alpha$ under the covering projection. These are embedded lines, each one of which separates the plane in two components.
Choose $q_0^S\in \alpha^S\cap \gamma ^S$ and let $\gamma_0 ^S$ be the lift of $\gamma$ that begins at $q_0^S$. For $m\in {{\mathbb{Z}}}$, let $$\Gamma_m^S= \bigcup_{k=-m}^m (\gamma_0^S + (k,0)).$$
We define analogously $\gamma^N, \alpha ^N, q_0^N$ and $\Gamma _m ^N$. The closed strip contained between the lines $\alpha^N$ and $\alpha^S$ is denoted by $A^0$. Let $V_0$ be a simple curve from $q_0^S$ to $q_0^N$ contained in $A^0$, and let $V_m = V_0 + (m,0)$. Let $V_0'\supset V_0$ an embedded line that separates the plane leaving $\gamma_0^S \cup \gamma_0^N$ in the connected component of ${{\mathbb{R}}}^2\backslash V_0'$ to the right of $V_0'$. Let $V'_m = V'_0 + (m,0)$. Clearly $V_0$ separates the horizontal strip $A^0$. Note that as $F(x+1,y)=F(x,y)+(d,0)$, and $d>1$, then there exists $m>0$ such that the following conditions hold:
1. The image of $V_m$ under $F$ is contained in the connected component of ${{\mathbb{R}}}^2\setminus V'_m$ to the right of $V'_m$.
2. The image of $V_{-m}$ under $F$ is contained in the connected component of ${{\mathbb{R}}}^2\setminus V'_{-m}$ to the left of $V'_{-m}$.
3. $F(V_{\pm m})\cap \Gamma_m^S = \emptyset$ and $F(V_{\pm m})\cap \Gamma_m^N = \emptyset$
Finally, let $\beta$ be the simple closed curve $$\beta=\Gamma_m^N. V_m.(\Gamma_m^S). (V_{-m})$$ positively oriented.
Recall we have assumed that $F$ has no fixed points. The lift of $\gamma$ is a concatenation of crosscuts, each one satisfiyng $F(c'_i)\cap c'_i=\emptyset$, where $c'_i$ lifts $c_i$. There exists $F'$, homotopic to $F$, such that $F'(\Gamma_m^S)\cap\Gamma_m^S=\emptyset$. Let $J^S$ be the number of crosscuts contained in $\Gamma_m^S$ such that $F(c'_i)\subset N(c'_i)$. Define $J^N$ analogously. Moreover, note that $F'(\Gamma_m^S)$ is contained in the region $A'_S$, as well as $F'(\Gamma_m^N)\subset A'_N$. By application of Lemma \[l2\] it follows that the $I_F(\beta)=I_{F'}(\beta)+J^S+J^N$.
Note that $F'(\beta)\cap\beta=\emptyset$. The fact that $F'(\Gamma_m^N)$ is contained in $A'_N$ and that $F'(\Gamma_m^S)\subset A'_S$ and that $F'=F$ in $V_m\cup V_{-m}$ and the choice of $m$ imply that $I_{F'}(\beta)=1$ as $\beta$ is contained in ${{\mbox{int}}}F(\beta)$.
This implies that the index of $I_F(\beta)\neq 0$ and this is absurd since we have assumed that $F$ was fixed point free. It follows that every lift of $f$ has fixed points.
![[]{data-label="levantado"}](figura30.eps)
Recall that to finish the proof of this theorem, it remains to consider the case where the unbounded components of the complement of $K$ are not invariant, but two-periodic (Remark \[ends\]). That is:\
[*Assume that $f(A_S)=A_N$ and $f(A_N)=A_S$*]{}.\
Let $k$ be an even number. Then $f^k$ satisfies the assumptions made above, (that $A_S$ and $A_N$ are $f^k$-invariant), therefore any lift $G$ of $f^k$ has a fixed point. On the other hand, let $k$ be an odd number and $G$ a lift of $f^k$. Note first that there exists a positive number $R$ such that $G(\{y=R\})$ is contained in $\{y\leq 0\}$ and $G(\{y=-R\})$ is contained in $\{y\geq 0\}$. By assumption, $d>1$, which implies that there exists a number $m$ such that $G( m,y)\cap\{x<m\}=\emptyset$ and $G(- m,y)\cap\{x>-m\}=\emptyset$ whenever $-R\leq y\leq R$. Let $\Gamma$ be the simple closed curve whose trace is the boundary of the rectangle with vertices $x=\pm m$, $y=\pm R$; give $\Gamma$ the positive orientation. Then $\Gamma$ satisfies the hypothesis of Lemma \[rectangulo\]. Thus the index of $G$ in $\Gamma$ is equal to $-1$. It follows that $G$ has fixed points. This finishes the proof of Theorem \[t1\].
\[cambia\]
1. It was assumed that $f$ is surjective. It can be seen from the proofs above that it is sufficient to assume that the invariant continuum $K$ is contained in the interior of $f(A)$.
2. Assume that $f(A_S)=A_N$ and $f(A_N)=A_S$, then the following assertion is also true:\
[*Let $f$ be a map of the annulus that interchange the ends (meaning that $f$ can be continuously extended to $A\cup\{S,N\}$ in such a way that $\{S,N\}$ is two-periodic). If the degree of $f$ has absolute value greater than $1$, then for every odd $k$, the map $f^k$ has at least $|d^k-1|$ fixed points.*]{} For $d>1$ use the arguments above and for $d<-1$, the conclusion follows from Remark \[rk1\]. In particular, the growth rate inequality $\limsup_{n\to \infty}\frac{1}{n}\ln( \#\{{{\mbox{Fix}}}(f^n)\}) \geq \ln (d)$ holds for $f$.
Locally connected boundary.
===========================
Let $K\subset A$ be an essential locally connected continuum. We will use the following properties:
1. There exists a simple essential closed curve contained in $K$. See Theorem 43, p.193 in [@mo].
2. $ K'=\Pi^{-1}(K)$ is connected and locally connected.
3. Let $U$ be an unbounded connected component of $A\backslash K$. For all $\epsilon >0$ there exists a neighborhood $W$ of $K$ in $\overline U$ such that the radial retraction $r$ of $W$ onto $K$ satisfies $d(x,r(x))<\epsilon$ for all $x\in W$ (here ’radial’ stands for the induced retraction on $W$ by the Riemann map). This is true because $K$ is locally connected and therefore the Riemann map extends continuously to $\overline {{\mathbb{D}}}$.
[**Theorem \[t2\].**]{} [*Let $f: A \to A$, $K\subset A $ an essential continuum such that $f(K)\subset K$. If $K$ is locally connected and $|d|>1$, then $f$ is complete on ${{\mbox{Fill}}}(K)$.*]{}
Notations: $\Pi:A'\to A$ is the universal covering of the annulus $A$, the component of the complement of $K$ that contains $N$ is called $A_N$ and that containing $S$ is denoted $A_S$. Moreover, $A'$ is identified with the product ${{\mathbb{R}}}\times (-1,1)$, and define $A'_N=\Pi^{-1}(A_N)$ and $A'_S=\Pi^{-1}(A_S)$.
Note that for all $\epsilon>0$ there exists a simple closed curve $\alpha \subset A_N$ such that $d(z,K)<\epsilon$ for every $z\in \alpha \cup f (\alpha)$. Analogously, there exists a simple closed curve $\beta\subset A_S$ satisfying the same properties. Let $\alpha '$ and $\beta '$ be the preimages of $\alpha$ and $\beta$ by the covering projection. Note that both $\alpha'$ and $\beta'$ separate $A'$ and as they do not intersect, then there is also a component of $A'\setminus(\alpha'\cup\beta')$ whose boundary is $\alpha'\cup\beta'$, and this component, denoted $A'_m$, contains $K'$.
Let $F$ be any lift of $f$. Suppose that ${{\mbox{Fix}}}(F)\cap {{\mbox{Fill}}}K' = \emptyset$ (otherwise we are done). So, there exists $\epsilon >0$ such that $d(x,F(x))>\epsilon$ for all $x\in {{\mbox{Fill}}}K'$. For this $\epsilon$, choose $\alpha$ and $\beta$ as above. By Property (3) of locally connected sets at the beginning of this section, there exists a deformation retract $r:W\to K'$ such that $d(r, id)<\epsilon/2$, where $W$ is a neighborhood of $K'$. So, $r\circ F(\alpha')$ is contained in $K'$. Analogously, $r\circ F(\beta')$ is contained in $K'$. Besides, the set of fixed points of $r\circ F$ in ${{\mbox{Fill}}}K'$ are the same as those of $F$. If $d>1$, we may choose real numbers $x_0< x_1$ such that the intersection of the vertical segment $(x_1,y)$ with $A'_m$ is mapped under $r\circ F$ strictly to its right, and the intersection of the vertical segment $(x_0,y)$ with $A'_m$ is mapped under $r\circ F$ strictly to its left.
Let $\gamma$ be the simple closed curve constructed with the segments of $\alpha'$ and $\beta'$ between $x_0$ and $x_1$ and the subarcs of the vertical segments $(x_0,y)$ and $(x_1,y)$ that are contained in $A'_m$.
Note that if $\gamma$ is positively oriented then Corollary \[torcido\] (of Lemma \[rectangulo\]) gives $I_{r\circ F}=-1$. This was assuming $d>1$. If $d<-1$, then the vertical segments at $x_0$ and $x_1$ can be taken so that the vertical at $x_0$ is mapped to the right of $x_1$ and the vertical at $x_1$ is mapped to the left of $x_0$. Now apply Corollary \[torcido2\] to obtain that the index is equal to $1$. So, ${{\mbox{Fix}}}(F)\cap {{\mbox{Fill}}}(K')\neq \emptyset$.
[**Example.**]{} We will show that if $d\leq -1$ it may happen that $f$ is fixed point free on $K$. Let $K$ be as in Figure 1 (a). We will construct a map $F: A' \to A'$ such that $F(x+1) = F(x)-1$. Take a fundamental domain $D$ of $K'$ as in Figure 1 (b) and we proceed to define $F$ on $D$. Let $j:[0,1]\to D $ a parametrization such that: $j(x)\neq j(y)$ for all $\{x,y\}\neq \{a,b\}$, $x\neq y$, and such that $j(a)= j(b)$. First define $F_0: [0,1]\to [0,1]$ (discontinuous) as in Figure \[figura3\] (c). Note that there exists $F: D\to D$ such that $jF_0(t)= F j(t)$. Moreover $F$ is continuous on $D$, as the disconitnuity of $F_0$ can be avoided on $j([0,1])$ because $j(a)=j(b)$. Clearly, $F_0$ is fixed point free. If $F$ has a fixed point $x=j(t)$, then $$x= F(x)= Fj(t)= jF_0(t).$$ So, $jF_0(t) = j(t)$ implying $t= F_0(t)$ because $t$ and $F_0(t)$ are different from $a$ and $b$.
Note that $F$ factors to a map $f: A \to A$ of degree $-1$ and that it is fixed point free by construction. We have then proved that with $d= -1$ one can avoid having a fixed point on $K$. Note that with exactly the same technique one can build a map $f$, not complete in $K$, of any given degree $d<-1$.\
To assure the existence of periodic points in $K$, and not just ${{\mbox{Fill}}}(K)$, we assume in the next Theorem that $K$ is the boundary of one of the unbounded components of its complement and that $d>1$.
\[t4\] Let $f: A \to A$ be a map of degree $d$, $K\subset A$ a locally connected essential continuum such that $f(K)\subset K$. Assume that $K$ is the boundary of one of the unbounded connected components of its complement. If $d>1$, then $f$ is complete on $K$.
By hypothesis $K'$ is the boundary of a simply connected region $R'= \Pi ^{-1}(R) $.\
The preimage under $\Pi$ of the curve given by property (1) at the beginning of this section, gives a continuous $1$-periodic function $i:{{\mathbb{R}}}\to K'$ with the following properties:
1. $i$ is a homeomorphism from ${{\mathbb{R}}}$ onto $i({{\mathbb{R}}})$.
2. $i({{\mathbb{R}}})$ separates the plane.
[**Claim.**]{} Let $C$ be a connected component of $ K'\setminus i({{\mathbb{R}}})$. Then $\overline C$, the closure of $C$, intersects $i({{\mathbb{R}}})$ in a one point set.\
To prove this claim note first that the intersection is not empty because $ K'$ is connected as $K$ is locally connected. Now assume that there are points $x<y$ such that $i(x)$ and $i(y)$ belong to $\overline C$. As $i({{\mathbb{R}}})$ is contained in $ K'$ and $K'$ is the boundary of an unbounded region $ R'$, for all $z\in (x,y)$ there exists a simple embedded line $r\subset R'$ landing at $i(z)$ and separating $R'$ in two connected components $A$ and $B$. As $C$ must have points in $A$ and in $B$, this contradicts the fact that $C$ is connected.\
From now on the unique point of $i({{\mathbb{R}}})\cap\overline C$ is denoted by $p(C)$.
Define a map $\psi: K'\to i({{\mathbb{R}}})$ that is the identity on $i({{\mathbb{R}}})$ and to each point $x$ in a component $C$ of $ K'\setminus i(R)$ assigns the point $p(C)$. We claim that as $ K'$ is locally connected, $\psi$ is continuous. To see this, note that if $z\in i({{\mathbb{R}}})$, then for all neighborhood $V$ of $z$ there exists a neighborhood $U$ of $z$ such that if $p(C)\in U$ and $p(C)\neq z$ then $C\subset V$. Otherwise, there exist a sequence $C_n$ of different components accumulating both on $z$ and in a point $z'$ in $\partial V\backslash i({{\mathbb{R}}})$. This contradicts local connectivity. Continuity of $\psi$ now follows.
Define $F'=i^{-1}\psi F i$. This is a continuous function from ${{\mathbb{R}}}$ to ${{\mathbb{R}}}$. The assumption $d>1$ implies that there is a point $x_0$ such that $F'(x)<x$ for every $x<x_0$ and a point $x_1$ such that $F'(x)>x$ for every $x>x_1$. Let $y_0$ be the supremum of the set of points $y$ such that $F'(y)<y$. As $F'$ is continuous, $F'(y_0)=y_0$ and by definition of $y_0$, $F'(z)\geq z$ for every $z>y_0$. Then either $i(y_0)$ is fixed for $F$ or $F(i(y_0))$ belongs to a component $C$ of $K'\setminus i({{\mathbb{R}}})$ such that $p(C)=i(y_0)$. In the latter case, let $V\subset C$ be a neighborhood of $Fi(y_0)$ in $K'$ such that $V\cap i({{\mathbb{R}}})=\emptyset$. Now let $z>y_0$ be close to $y_0$ in such a way that $Fi(z)\in V$. So $F(i(z))\in C$ and it follows that $F'(z)=y_0$. This is a contradiction, so the unique conclusion left is $F(y_0)=y_0$.
[**Theorem \[t3\]**]{}. [*Let $f: A \to A$ be a degree $d$ map of the annulus, where $|d|>1$. Each one of the following conditions imply that $f$ is complete.*]{}
1. Both ends of $A$ are attracting.
2. Both ends of $A$ are repelling.
The hypothesis of the first assertion implies that defining $f(N)=N$ and $f(S)=S$ gives a continuous extension of $f$ to $A^*$, the compactification of the annulus with two points, in such a way that $N$ and $S$ become topological attractors. Then there exists a simple closed essential curve $\alpha$ whose image under $f$ is contained in $\alpha_N$, the connected component of $A^*$ that contains $N$. Similarly, there is a simple closed essential curve $\beta$ whose image under $f$ is contained in $\beta_S$. Now let $F$ be a lift of $f$, $\alpha'$ and $\beta'$ the preimages of $\alpha$ and $\beta$ under the covering projection, so that there is a region $L\subset A'$ whose boundary is the union of $\alpha'$ and $\beta'$. Let $V_0$ be a simple arc having one point in $\alpha'$ another point in $\beta'$ and whose interior is contained in $L$, and let $V_m=V_0+(0,m)$ for $m\in{{\mathbb{Z}}}$. Let $\Gamma_m$ be the simple closed curve formed with $V_m$, $V_{-m}$ and parts of $\alpha'$ and $\beta'$. The fact that $|d|>1$ implies that there exists $m>0$ such that $F(\Gamma_m)$ does not intersect $\Gamma_m$ and that $I_{F}(\Gamma_m)=\pm 1$, depending on $d>0$ or $d<0$. It follows that $F$ has a fixed point. As every iterate of $f$ satisfies the same hypothesis, then $f$ is complete.
For the case of repelling ends, there exists a curve $\alpha$, close to the end $N$ of the annulus, such that $f(\alpha)$ is contained in the component of the complement of $\alpha$ that does not contain $N$. There will be as well a curve $\beta$, close to $S$, whose image under $f$ is contained in the component of the complement of $\beta$ that contains $N$. Next consider a lift $F$ of $f$, take the preimages of $\alpha$ and $\beta$ in the universal covering and cut it with vertical arcs, in such a way to obtain a curve $\gamma$ whose sides are $\alpha'$, $\beta'$ and vertical segments, and satisfies the properties of the curve $\gamma$ of Corollary \[torcido\] if $d>1$. The Corollary implies that the index of $F$ in $\gamma$ is equal to $-1$. On the other hand, if $d<-1$, then the comments succeding Corollary \[torcido\] imply that $I_F(\gamma)=1$. This implies that every lift of $f$ has a fixed point. As every $f^k$ satisfies the same hypothesis, it follows that $f$ is complete.
Applications
============
The first result is a consequence of Hagopian’s theorem [@h] which states that an arcwise connected plane continuum $K$ that does not separate the plane has the fixed point property (i.e. every continuous map from $K$ to $K$ has a fixed point in $K$). We will use this theorem on the universal covering, so we need that the lift of K is arcwise connected. This may not happen even if the set K is arcwise connected (see Figure 6). However, note that in this example the set $K$ is simply connected. If one assumes that $K$ has a nontrivial closed loop, this guarantees that the lift is arcwise connected.\
\[hago\] Let $f: A \to A$, $K\subset A $ a continuum such that $f(K)\subset K$. Assume that $K$ is arcwise-connected, and $i_* : \pi_1 (K)\to \pi_1 (A)$ is surjective.
If $d<-1$ and $k$ is odd, then every lift $F$ of $f^k$ has a fixed point. It follows that $f$ satisfies the growth rate inequality.
If $d= -1$, then $f$ has a fixed point in ${{\mbox{Fill}}}(K)$.
Compare with Theorem \[t1\] where it was proved that $f^{-1}(K) = K$ and $d>1$ implies completeness.
By hypothesis, $K'$ and ${{\mbox{Fill}}}(K')$ are arcwise connected. Take $F$ any lift of $f^k$. As $k$ is odd, then $f^k$ has negative degree. One can modify $F$ outside ${{\mbox{Fill}}}(K')$ so that it can be extended to ${{\mathbb{R}}}\times [0,1]$. Let $D$ be the compactification of ${{\mathbb{R}}}\times [0,1]$ with two points $-\infty, +\infty$, which is a topological closed disk. As $d<0$, $F$ interchanges these two points and ${{\mbox{Fill}}}(K')\subset D$ is compact. By Hagopian’s Theorem [@h], $F$ must have a fixed point in ${{\mbox{Fill}}}(K')$, and this point does not belong to $\{-\infty, +\infty\}$. It follows that $F$ has a fixed point in ${{\mbox{Fill}}}(K')$. Concluding, if $d<-1$ then every lift of $f^k$ with odd $k$ has a fixed point. This implies that $f^k$ has at least $|d^k-1|$ fixed points, so $f$ satisfies the growth rate inequality. For $d=-1$, just the existence of a fixed point in ${{\mbox{Fill}}}(K)$ can be assured.
As an application, we obtain a forcing result for $f: S^2\to S^2$: a completely invariant orbit of period two implies that the growth inequality holds for $f$.
Let $f: S^2\to S^2$ be a map of degree $d$, $|d|>1$. Suppose that there exists $p,q \in S^2$ such that $f^{-1}(p) = \{q\}$ and $f^{-1} (q) = \{p\}$. Then, $f^k$ has at least $|d^k-1|$ fixed points for every odd $k$.
The restriction of $f$ to $S^2\setminus\{p,q\}$ is a degree $d$ map of the annulus satisfying the hypothesis of the second assertion in Remark \[cambia\] after the proof of Theorem \[t1\].
Another result for the sphere:
Let $f: S^2\to S^2$ be a map of degree $d>1$. Assume that there exists two disjoint simply connected nonempty open sets $U_1$ and $U_2$ such that $f^{-1}(U_i)= U_i$ for $i=1, 2$. Then, $f$ has at least the same number of periodic points as $z^d$ on $S^1$.
Let $K$ be the complement of $U_1\cup U_2$ in $S^2$. Then $K$ is a completely invariant continuum. For $i=1, 2$ choose points $p_i\in U_i$. Let $f'$ be a map homotopic to $f$ such that $f=f'$ in a neighborhood of $K$ and such that $(f')^{-1}(p_i)=\{p_i\}$ for $i=1, 2$. Then $f'$ is a degree $d$ map of the annulus $S^2\setminus\{p_1,\ p_2\}$ and has a completely invariant essential continuum $K$. By Theorem \[t1\] $f'$ is complete on $K={{\mbox{Fill}}}(K)$, so it has at least the same number of periodic points as $z^d$ on $S^1$. The conclusion follows because $f=f'$ on $K$.
[Poinc]{}
(2006), 99–120.
|
---
abstract: 'The novel “Volume-Enclosing Surface exTraction Algorith” (VESTA) generates triangular isosurfaces from computed tomography volumetric images and/or three-dimensional ($3$D) simulation data. Here, we present various benchmarks for GPU-based code implementations of both VESTA and the current state-of-the-art Marching Cubes Algorithm (MCA). One major result of this study is that VESTA runs significantly faster than the MCA.'
author:
- 'B. R. Schlei[^1]'
title: 'GPU Programming - Speeding Up the 3D Surface Generator VESTA'
---
Introduction
============
NVIDIAs toolkits (*cf., e.g.,* Ref. [@NVIDIA]) for the development of CUDA-based software contain, among many other things, example code for an extended version [@BOUR94] of the original MCA [@LORE87]. Here, we compare the performance of this code with our CUDA- and ANSI-C-based implementation of VESTA [@BRS12] on a Linux-based (*i.e.,* openSUSE $13.1$) PC with a GeForce GTX $750$ Ti graphics card.
In particular, the times that we have measured (*cf.,* Table 1) are averages over $1000$ runs each. The measurements start after the data sets have been loaded into texture memory, and they stop after all point coordinates and triplets of point IDs (*i.e.,* triangles) have been computed on the GPU.
---------------------- ----------------- ----------------- -----------------
**Technique** Extended MCA
**Mode** **DCED / L** **DCED / L** **Mixed / H**
**(a)$\:\:$ Points** $ 19,218$ $ 12,814$ $\: 15,292$
**Triangles** $ 6406$ $ 6406$ $\: 11,362$
**Time (ms)** [$ 1.43(5)$]{} [$ 1.28(5)$]{} $\: 1.37(4)$
**(b)$\:\:$ Points** $ 6,128,724$ $ 4,085,840$ $\: 4,852,644$
**Triangles** $ 2,042,908$ $ 2,042,908$ $\: 3,576,516$
**Time (ms)** [$ 98.5(1)$]{} [$ 71.3(1)$]{} $\: 75.9(4)$
**(c)$\:\:$ Points** $ 5,566,998$ $ 3,699,086$ $\: 4,346,120$
**Triangles** $ 1,855,666$ $ 1,855,666$ $\: 3,147,604$
**Time (ms)** [$ 23.0(1)$]{} [$ 18.7(1)$]{} $\: 22.4(1)$
**(d)$\:\:$ Points** $ 33,240$ $ 22,208$ $\: 25,894$
**Triangles** $ 11,080$ $ 11,080$ $\: 18,350$
**Time (ms)** [$ 0.82(4)$]{} [$ 0.63(4)$]{} $\: 0.74(4)$
**(e)$\:\:$ Points** $ 13,859,304$ $ 9,267,824$ $\: 11,178,649$
**Triangles** $ 4,619,768$ $ 4,619,768$ $\: 8,441,610$
**Time (ms)** [$ 111.2(1)$]{} [$ 84.4(1)$]{} $\: 94.6(6)$
---------------------- ----------------- ----------------- -----------------
: Benchmarks for various processed tomographic data sets: for (a) – (c), *cf.,* Ref. [@BRS12] and Ref.s therein, (d) Bucky.raw data is a portion of [@NVIDIA], and (e) Happy Buddha VRI file [@HAPPY]. For the selected isovalues, *cf.,* Fig. 1. []{data-label="tab01"}
![VESTA high resolution “mixed” mode (Mixed/ H) isosurface renderings, where the isovalues equal to (a) $139$, (b) $150$, (c) $135$, (d) $128$, and (e) $150$, respectively.[]{data-label="fig01"}](figure01){width="82mm"}
Results
=======
For the here considered data sets [@NVIDIA; @BRS12; @HAPPY], the extended MCA is about (a) $12\%$, (b) $38\%$, (c) $23\%$, (d) $30\%$, and (e) $32\%$, *slower* than the marching variant of VESTA [@BRS12], when the latter is executed in its low resolution “disconnect” mode (DCED/L). Furthermore, VESTA is also faster even if higher resolution isosurfaces are computed (*cf.,* Fig. 1), which have about twice the number of triangles (*cf.,* Table 1).
Note that the current code implementation of VESTA does *not* yet use parallel streaming, *nor* it does call device kernels from within kernels. As a consequence, further GPU-based code optimisations may result in an even faster VESTA code.
[9]{} NVIDIA CUDA Toolkit 6.5; for more detail, *cf.,* https:// developer.nvidia.com/cuda-toolkit/
P. Bourke, “Polygonising a scalar field”, May 1994; for more detail, *cf.,* http://paulbourke.net/geometry/polygonise/
W. E. Lorenzen and H. E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm”, Comput. Graph. 21 (1987), p. 163.
B. R. Schlei, “Volume-Enclosing Surface Extraction”, Computers & Graphics 36 (2012) p. 111, doi: 10.1016/j.cag.2011. 12.008.
http://graphics.stanford.edu/software/volfill/
[^1]: b.schlei@gsi.de
|
---
author:
- 'B. Charlier [^1]'
- 'G. Nardi, A. Trouvé [^2]'
bibliography:
- 'bibliography.bib'
title: 'The matching problem between functional shapes via a $BV$-penalty term: a $\Gamma$-convergence result'
---
[^1]: IMAG (UMR 5149), Université Montpellier, Campus du Triolet (CC051), Place Eugène Bataillon, 34095 Montpellier, France. Email: [benjamin.charlier@umontpellier.fr]{}
[^2]: CMLA, UMR 8536, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94230 Cachan, France. Emails: [nardi,trouve@cmla.ens-cachan.fr]{}
|
---
abstract: |
The extended second order cones were introduced by S. Z. Németh and G. Zhang in \[S. Z. Németh and G. Zhang. [*Extended Lorentz cones and variational inequalities on cylinders*]{}. J. Optim. Theory Appl., 168(3):756-768, 2016\] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in \[R. Sznajder. [*The Lyapunov rank of extended second order cones*]{}. Journal of Global Optimization, 66(3):585-593, 2016\] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. Németh and G. Zhang in \[S.Z. Németh and G. Zhang. [*Positive operators of Extended Lorentz cones*]{}. arXiv:1608.07455v2,2016\] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. In this note we give formulas for projecting onto the extended second order cones. In the most general case the formula depends on a piecewise linear equation for one real variable which is solved by using numerical methods.
[**Keywords:**]{} [ Semi-smooth equation, extended second order cone, metric projection, piecewise linear Newton method]{}
author:
- 'O. P. Ferreira[^1]'
- 'S. Z. Németh [^2]'
title: How to project onto extended second order cones
---
Introduction
============
The Lorentz cone is an important object in theoretical physics. In recent times it has been rebranded as second order cone and used for various application in optimization. Some robust optimization, plant location and investment portfolio manangement problems were formulated as as a second order cone program [@AG2003]. Another good survey paper with a wide range of applications of second order cone programming is [@MR1655138]. More recent connections of second order cone programming and second order cone complementarity problem with physics, mechanics, economics, game theory, robotics, optimization and neural networks were considered in [@MR3158056; @MR2377196; @MR2116450; @MR3010551; @MR2925039; @MR2568432; @MR2522815; @KCY2011; @MR2179239]. The importance of the second order cone is nowadays notorious not only in theoretical physics, but in optimization as well.
Thus far, there is no closed-form expression for metric (orthogonal) projection onto a general closed convex cone. A nice property of the second order cone is that it admits an explicit representation of the projection mapping onto it (see [@MR1885570 Proposition 3.3]). The original motivation for extending the second order cone was inspired by using iterative methods for solving complementarity problems and variational inequalities [@NZ20151; @NZ2016a]. These iterative methods are based on the property that the projection onto the closed convex set defining the problem is isotone with respect to the order defined out by a cone. Usually this is a very restrictive condition. However, cylinders and in particular cylinders with cone base admit isotone projections onto them with respect to the extended second order cones. Therefore, variational inequalities on cylinders and mixed complementarity problems can be solved by using such iterative techniques based on monotone convergence [@NZ20151; @NZ2016b].
Later it turned out that many of these cones could be even more useful because the bilinearity rank (or Lyapunov rank) [@RudolfNoyanPappAlizadeh2011; @GowdaTao2014; @GowdaTrott2014; @Trott2014; @OrlitzkyGowda2016] of them is higher than the dimension of the underlying space and therefore they have good numerical properties. More specifically, for $p>1$ this is true whenever $q^2-3q+2>2p$ [@RS2016], where $p$, $q$ are from the definition of the extended second order cone (see Definition \[def-ext-lor\]). Such cones are “numerically good” cones when solving complementarity problems defined on them. The extended second order cones are also irreducible [@RS2016]. But to be really usable from optimization point of view we need easy ways of projecting onto them. In this paper we show that projecting onto an extended second order cone it is “almost possible” by using closed-form expressions. We present a set of formulas for projecting onto an extended second order cone which is subject to solving a piecewise linear equation with one real variable only. The method of finding these expressions is based on the special form of the complementarity set of the extended second order cone and Moreau’s decomposition theorem [@MR0139919] for projecting onto cones. The latter problem of projecting onto the extended second order cone is a particular conic optimization problem with respect to this cone. Although, the problem of projecting the point $(x,u)\in\mathbb R^p\times\mathbb R^q$ into the extended second order cone $L$ (see Definition \[def-ext-lor\]) can be transformed into the second order conic optimization problem $$\min \left\{ \|y-x\|^2+\|v-u\|^2~:~ (y,v)\in\mathbb R^p\times\mathbb R^q, ~ ~ \ell_i(y,v)\in\mathcal{L},\quad i=1,\ldots,p\right\},$$ where $\mathcal{L}=\{(t,u)\in\mathbb R\times\mathbb R^q:t\ge\|u\|\}$ is the second order cone in $\mathbb R^{q+1}\equiv\mathbb R\times\mathbb R^q$ and $\ell_i:\mathbb R^p\times\mathbb R^q\to\mathbb R\times\mathbb R^q$ are the linear mappings defined by $\ell_i(y,v)=(y_i,v)$, the complexity of our method is much simpler than solving the reformulated problem, because apart from closed-form expressions, it contains only one piecewise linear equation. By considering such a reformulation one would lose the useful special structure of the cone, which is the cornerstone for the simplicity of our method.
Certainly, the explicit representation of the projection mapping onto the second order cone (see ) should not be handled as a conic optimization problem and the need to solve a simple piecewise linear equation for $p>1$ makes our method just slightly more complex. The above observation about why one shouldn’t reformulate the projection onto the extended second order cone into a second order conic optimization problem, together with the irreducibility of the second order cone, clearly shows that this cone “deserves a closer look”.
The structure of the paper is as follows: In Section 2 we fix the notation and the terminology used throughout the paper. In Section 3 we present the formulas for projecting onto the extended second order cone. In Section 4 we solve the piecewise linear equation involved in these formulas by using the semi-smooth Newton’s method and a method based on Picard’s iteration. Finally, we make some remarks in the last section.
Preliminaries
=============
Let $\ell,m,p,q$ be positive integers such that $m=p+q$. We identify the the vectors of ${\mathbb R}^\ell$ with $\ell\times 1$ matrices with real entries. The scalar product in ${\mathbb R}^\ell$ is defined by the mapping $${\mathbb R}^\ell\times{\mathbb R}^\ell\ni (x,y)\mapsto\langle x,y \rangle:=x^\top y\in{\mathbb R}$$ and the corresponding norm by $${\mathbb R}^\ell\ni x\mapsto\|x\|:=\sqrt{\langle x,x\rangle}\in{\mathbb R}.$$ For $x,y\in{\mathbb R}^\ell$ denote $x\perp y$ if $\langle x,y\rangle=0$. We identify the elements of ${\mathbb R}^p\times{\mathbb R}^q$ with the elements of ${\mathbb R}^m$ through the correspondence $${\mathbb R}^p\times{\mathbb R}^q\ni(x,y)\mapsto (x^\top,y^\top)^\top.$$ Through this identification the scalar product in ${\mathbb R}^p\times{\mathbb R}^q$ is defined by $$\langle (x,y),(u,v)\rangle:=\langle (x^\top,y^\top)^\top,(u^\top,v^\top)^\top\rangle=\langle x,u\rangle+\langle y,v\rangle.$$ A closed set $K\subset{\mathbb R}^\ell$ with nonempty interior is called a *proper cone* if $K+K\subset K$, $K\cap(-K)=\{0\}$ and $\lambda K\subset K$, for any $\lambda$ positive real number. The *dual cone* of a proper cone $K\subset{\mathbb R}^\ell$ is a proper cone defined by $$K^*:=\{x\in{\mathbb R}^\ell~:~\langle x,y\rangle\ge0,\mbox{ }\forall y\in K\}.$$ A proper cone $K\subset{\mathbb R}^\ell$ is called subdual if $K\subset K^*$, superdual if $K^*\subset K$ and self-dual if $K^*=K$. If $K,D\subset{\mathbb R}^\ell$ are proper cones such that $D=K^*$, then $D^*=K$ and the cones $K$, $D$ are called *mutually dual*.
For a proper cone $K\in{\mathbb R}^\ell$ denote $$C(K):=\left\{(x,y)\in K\times K^*~:~x\perp y\right\}$$ the *complementarity set* of $K$.
Let $C\in{\mathbb R}^\ell$ be a closed convex set. The projection mapping $P_C\colon{\mathbb R}^\ell\to{\mathbb R}^\ell$ onto $C$ is the mapping defined by $$P_C(x):=\mbox{argmin}\{\|x-y\|:y\in C\}.$$ We recall here Moreau’s decomposition Theorem [@MR0139919] (stated here for proper cones only):
\[th:mt\] Let $K\subset{\mathbb R}^\ell$ be a proper cone, $K^*$ its dual cone and $z\in{\mathbb R}^\ell$. Then, the following two statements are equivalent:
1. $z=x-y$ and $(x,y)\in C(K)$,
2. $x=P_K(z)$ and $y=P_{K^*}(-z)$.
Theorem \[th:mt\] implies $$z=P_K(z)-P_{K^*}(-z),$$ with $P_K(z)\perp P_{K^*}(-z)$.
For $z\in{\mathbb R}^\ell$ we denote $z=(z_1,\dots,z_\ell)^\top$. Let $\ge$ denote the component-wise order in ${\mathbb R}^\ell$, that is, the order defined by ${\mathbb R}^\ell\ni x\ge y\in{\mathbb R}^\ell$ if and only if $x_i\ge y_i$ for $i=1,\dots,\ell$. Denote by $0$ the vector in ${\mathbb R}^\ell$ or a scalar zero (it will not lead to any confusion), by $e$ the vector of ones in ${\mathbb R}^\ell$ and by ${\mathbb R}^\ell_+=\{x\in{\mathbb R}^\ell~:~ x\ge0\}$ the nonnegative orthant. The proper cone ${\mathbb R}^\ell_+$ is self-dual. For a real number $\alpha\in{\mathbb R}$ denote $\alpha^+:=\max(\alpha,0)$ and $\alpha^-:=\max(-\alpha,0)$. For a vector $z\in{\mathbb R}^\ell$ denote $z^+:=(z_1^+,\dots,z_\ell^+)$, $z^-:=(z_1^-,\dots,z_\ell^-)$, $|z|:=(|z_1|,\dots,|z_\ell|)$, $\mbox{sgn}(z):=(\mbox{sgn}(z_1),\dots,\mbox{sgn}(z_\ell))$ and $\mbox{diag}(z)$ the $\ell\times\ell$ diagonal matrix with entries $\mbox{diag}(z)_{ij}:=\delta_{ij}z_i$, where $i,j\in\{1,\dots,\ell\}$. It is known that $z^+=P_{{\mathbb R}^\ell_+}(z)$ and $z^-=P_{{\mathbb R}^\ell_+}(-z)$.
We recall from [@NZ20151] the following definition of a pair of mutually dual extended second order cones $L$, $M$:
\[def-ext-lor\] $$\begin{aligned}
L&:=\left\{(x,u)\in{\mathbb R}^p\times{\mathbb R}^q:x\ge\|u\|e\right\}, \\
M&:=\left\{(x,u)\in{\mathbb R}^p\times{\mathbb R}^q:\langle x,e\rangle\ge\|u\|,x\ge0\right\}.\end{aligned}$$ where $\ge$ denotes the component-wise order.
It is known that both $L$ and $M$ are proper cones, $L$ is subdual $M$ is superdual and if $p=1$, then both cones reduce to the second order cone. The cones $L$ and $M$ are polyhedral if and only if $q=1$. If we allow $q=0$ as well, then the cones $L$ and $M$ reduce to the nonnegative orthant. More properties of the extended second order cones can be found in [@NZ20151; @RS2016; @NZ2016b].
Projection formulas for extended second order cones
===================================================
In this section we give formulas for projecting onto the pair of mutually dual extended second order cones. Before presenting our main theorem, we need some preliminary results for these cones. Let $p,q$ be positive integers.
\[pm\] Let $x,y\in{\mathbb R}^p$ and $u,v\in{\mathbb R}^q\setminus\{0\}$. We have that $(x,u,y,v):=((x,u),(y,v))\in C(L)$ if and only if there exists a $\lambda>0$ such that $v=-\lambda u$, $\langle y,e\rangle=\|v\|$ and $(x-\|u\|e,y)\in C({\mathbb R}^p_+)$.
Suppose first that there exists $\lambda>0$ such that $v=-\lambda u$, $\langle y,e\rangle=\|v\|$ and $(x-\|u\|e,y)\in C({\mathbb R}^p_+)$. Hence, $(x,u)\in L$ and $(y,v)\in M$. Moreover, $$\langle (x,u),(y,v)\rangle=\langle x,y\rangle+\langle u,v\rangle=\|u\|\langle e,y\rangle-\lambda\|u\|^2=\|u\|\|v\|-\lambda\|u\|^2=0.$$ Thus, $(x,u,y,v)\in
C(L)$. Conversely, suppose that $(x,u,y,v)\in C(L)$. Then, $(x,u)\in L$, $(y,v)\in M$ and $$0=\langle (x,u),(y,v)\rangle=\langle x,y\rangle+\langle u,v\rangle\ge\langle\|u\|e,y\rangle+\langle u,v\rangle\ge\|u\|\|v\|+\langle u,v\rangle\ge0.$$ Hence, there exists $\lambda>0$ such that $v=-\lambda u$, $\langle e,y\rangle=\|v\|$ and $\langle x-\|u\|e,y\rangle=0$. It follows that $(x-\|u\|e,y)\in C({\mathbb R}^p_+)$.
Before presenting the main result of this section we introduce a piecewise linear function and establish some important properties of it. This function will play an important role in the sequel, namely, the formulas for the projection will depend on its single positive zero. The piecewise linear function $ \psi\colon[0, +\infty) \to {\mathbb R}$ is defined by $$\label{eq:psy}
\psi(\lambda):=-\lambda \|w\|+\left\langle e,[(\lambda+1)z-\|w\|e]^-\right\rangle.$$ For stating the next proposition we need to define the following diagonal matrix, which we will see is related to the subdifferential $\partial \psi$ of $\psi$: $$\label{def:N(x)}
N(\lambda):=\mbox{diag}\left(-\mbox{sgn}\left( [(\lambda+1)z-\|w\|e]^-\right)\right), \qquad \lambda\in [0, +\infty).$$
\[pr:psy\] The function $\psi$ is convex. Moreover, if $$z^+\not\ge\|w\|e, \qquad \langle z^-,e\rangle<\|w\|,$$ then we have:
1. $-\|w\|+\left\langle e, N(\lambda)z\right\rangle \in \partial \psi(\lambda)$ and $-\|w\|+\left\langle e, N(\lambda)z\right\rangle <0$, for all $\lambda \geq 0$;
2. $\psi$ has a unique zero $\lambda_* >0$.
We first note that the function $\psi$ can be equivalently given by $$\label{def:edpsi}
\psi(\lambda):=-\lambda \|w\|+\sum_{i=1}^{p} \psi_i(\lambda), \qquad \psi_i(\lambda):= [(\lambda+1)z_i-\|w\|]^-, \qquad \lambda \geq 0.$$ Since the sum and the maximum of two convex functions is convex, it follows that the function $\psi_i(\lambda)=\max\{-(\lambda+1)z_i+\|w\|,~0\}$ is convex for all $i=1, \ldots p$. Hence, the result of the first part follows.
1. The definitions of $ \psi$ and $\psi_i$ in imply that $\partial \psi(\lambda)= -\|w\| + \sum_{i=1}^{p} \partial \psi_i(\lambda)$. Moreover, considering that $ \psi_i(\lambda)=\max\{-(\lambda+1)z_i+\|w\|,~0\}$, we have $ -\mbox{sgn}\left(
[(\lambda+1)z_i-\|w\|]^-\right)z_i\in \partial \psi_i(\lambda)$, for all $i=1, \ldots p$. Therefore, using , the inclusion follows. To prove the inequality, note that implies that the entries of $N(\lambda)$ are equal to $0$ or $-1$, for all $\lambda \geq 0$. Thus, from the assumption $\langle z^-,e\rangle<\|w\|$ we have $-\|w\| +\left\langle e, N(\lambda) z\right\rangle <
0$, for all $\lambda \geq 0$.
2. First, we show that has a positive zero. Note that $z\not\geq \|w\|e$, otherwise it would follow that $z^+=z\geq \|w\|e$, which contradicts our assumptions. Then, there exists $ i_0\in \{1, \ldots, p\}$ such that $z_{i_0} <\|w\|$. Hence, from we have $\psi(0)> \|w\|- z_{i_0}> 0$. If $\lambda>0$ is sufficiently large, then $\mbox{sgn}[(\lambda+1)z_i-\|w\|]= \mbox{sgn} z_i$ and consequently $[(\lambda+1)z_i-\|w\|]^-\leq (\lambda+1)z_i^-+\|w\|$. By using the last inequality, and the assumption $\langle z^-,e\rangle<\|w\|$, we conclude that for $\lambda>0$ sufficiently large, it is true that $$\begin{gathered}
\psi(\lambda)\leq -\lambda\|w\|+\left\langle e,(\lambda+1)z^-+\|w\|e\right\rangle= \\\left[-\|w\|+\left\langle
z^-,e\right\rangle\right]\lambda+ \|w\| +\left\langle e,z^-\right\rangle<0.
\end{gathered}$$ Since $\psi$ is continuous, there is a $\lambda_*>0$ such that $\psi(\lambda_*)=0$. By contradiction we assume that $\psi$ has two positive zeroes $\bar{\lambda}$ and $\hat{\lambda}$. Let $0<\hat{\lambda}< \bar{\lambda}$. Since $\psi$ is convex and $-\|w\|+ \left\langle e, N(\lambda)z\right\rangle \in \partial \psi(\lambda)$, we have $ \psi(\hat{\lambda})\geq \psi(\bar{\lambda}) +[-\|w\|+ \left\langle e,
N(\bar{\lambda})z\right\rangle][\hat{\lambda}-\bar{\lambda}] $. Due to $\psi(\hat{\lambda})=\psi(\bar{\lambda})=0$ and considering that $0<\hat{\lambda}<\bar{\lambda}$, the last inequity implies that $- \|w\|+ \left\langle e,
N(\lambda)z\right\rangle\geq 0$, which contradicts the second part of item 1. Therefore, $\psi$ has a unique positive zero.
Now we ready to state and prove the main result of the paper.
\[th:pelc\] Let $(z,w)\in{\mathbb R}^p\times{\mathbb R}^q$. Then, we have
1. If $z^+\ge\|w\|e$, then $P_L(z,w)=(z^+,w)$ and $P_M(-z,-w)=(z^-,0)$.
2. If $\langle z^-,e\rangle\ge\|w\|$, then $P_L(z,w)=(z^+,0)$ and $P_M(-z,-w)=(z^-,-w)$.
3. If $z^+\not\ge\|w\|e$ and $\langle z^-,e\rangle<\|w\|$, then the piecewise linear equation $$\label{ess}
\lambda \|w\|=\left\langle e,[(\lambda+1)z-\|w\|e]^-\right\rangle.$$ has a unique positive solution $\lambda>0$, $$\label{epl}
P_L(z,w)=\left(\left[z-\frac{1}{\lambda +1}\|w\|e\right]^++\frac{1}{\lambda+1}\|w\|e,~\frac{1}{\lambda+1}w\right)$$ and $$\label{epm}
P_M(-z,-w)=\left(\left[z-\frac{1}{\lambda +1}\|w\|e\right]^-,~-\frac{\lambda}{\lambda+1}w\right)$$
We will use Moreau’s decomposition theorem for $L$ for proving all three items. In this case this theorem states that, $P_L(z,w)=(x,u)$ and $P_M(-z,-w)=(y,v)$ if and only if $(z,w)=(x,u)-(y,v)$ and $(x,u,y,v)\in C(L)$.
1. This is exactly the case when $v=0$.
Indeed, $v=0$ implies $P_L(z,w)=(x,u)$ and $P_M(-z,-w)=(y,0)$. Hence, $z=x-y$, $w=u$, $x\ge\|u\|e$, $y\ge0$ and $\langle x,y\rangle=0$. By using Moreau’s decomposition theorem for ${\mathbb R}^p_+$, we have that $z=x-y$, $x\ge0$, $y\ge0$ and $\langle x,y\rangle=0$ implies $x=z^+$ and $y=z^-$. Since, $w=u$ and $x\ge\|u\|e$, we get $z^+\ge\|w\|e$.
Conversely, suppose that $z^+\ge\|w\|e$. Then $(z^+,w,z^-,0)\in C(L)$. Hence, by Moreau’s decomposition Theorem for $L$, we get $P_L(z,w)=(z^+,w)$ and $P_M(-z,-w)=(z^-,0)$. Thus, $v=0$.
2. This is exactly the case when $u=0$.
Indeed, $u=0$ implies $P_L(z,w)=(x,0)$ and $P_M(-z,-w)=(y,v)$. Hence, $z=x-y$, $w=-v$, $x\ge0$, $\langle y,e\rangle\ge\|v\|$, $y\ge0$ and $\langle x,y\rangle=0$. By using Moreau’s decomposition theorem for ${\mathbb R}^p_+$, we have that $z=x-y$, $x\ge0$, $y\ge0$ and $\langle x,y\rangle=0$ implies $x=z^+$ and $y=z^-$. Since $w=-v$ and $\langle y,e\rangle\ge\|v\|$, we get $\langle z^-,e\rangle\ge\|w\|$.
Conversely, suppose that $\langle z^-,e\rangle\ge\|w\|$. Then, it is easy to check that $(z^+,0,z^-,-w)\in C(L)$. Then, by Moreau’s decomposition Theorem for $L$, we get $P_L(z,w)=(z^+,0)$ and $P_M(-z,-w)=(z^-,-w)$. Thus, $u=0$.
3. This is exactly the case when $u\ne 0$ and $v\ne 0$.
From Proposition \[pm\] it follows that $(z,w)=(x,u)-(y,v)$ and $(x,u,y,v)\in C(L)$ is equivalent to $z=x-y$, $w=u-v$ and the existence of a $\lambda>0$ such that $v=-\lambda u$, $\langle y,e\rangle=\|v\|$ and $(x-\|u\|e,y)\in C({\mathbb R}^p_+)$. On the other hand, by Moreau’s decomposition theorem for ${\mathbb R}^p_+$, $(x-\|u\|e,y)\in C({\mathbb R}^p_+)$ is equivalent to $x-\|u\|e=[x-\|u\|e-y]^+$ and $y=[x-\|u\|e-y]^-$. Hence, $$\label{epli}
P_L(z,w)=\left(x,\frac{1}{\lambda+1}w\right)$$ and $$\label{epmi}
P_M(-z,-w)=\left(y,-\frac{\lambda}{\lambda+1}w\right)$$ if and only if $z=x-y$ and $\lambda>0$ is such that $$\label{ebm}
\langle y,e\rangle=\frac{\lambda}{\lambda+1}\|w\|,$$ $$\label{ex}
x=\left[z-\frac{1}{\lambda+1}\|w\|e\right]^++\frac{1}{1+\lambda}\|w\|e$$ and $$\label{ey}
y=\left[z-\frac{1}{\lambda+1}\|w\|e\right]^-.$$ From equations and follows equation and from equations and follows equation , where $\lambda>0$ is given by equation , which is a combination of equations and . The uniqueness of $\lambda>0$ which satisfies follows from the uniqueness of $P_L(z,w)$ and $P_M(z,w)$.
The next remark will recover the well known formulas for projecting onto the second order cone (see for example [@MR1885570 Proposition 3.3]).
\[eq:fplc\] Let $(z,w) \in {\mathbb R}\times{\mathbb R}^q$ and $L$ be the second order cone. Then, letting $u:= [z- \|w\|]^+$ and $v:= [z+ \|w\|]^+$ we conclude that Theorem \[th:pelc\] implies that $$\label{eq:projef}
P_{L}(z,w)= \begin{cases}
\frac{1}{2} \left( u + v ,\, \left[v -u\right] \displaystyle \frac {w}{\|w\|}\right), & w \neq 0, \\
\\
\ \left(z^+,\, 0\right), & w= 0.
\end{cases}$$ Indeed, for $p=1$, the conditions in item 3 in Theorem hold if and only if $0\leq |z|< \|w\|$ and equation becomes $\lambda \|w\|= [(\lambda+1)z-\|w\|]^-$, which obviously can have only nonnegative solutions, because the right hand side of the equation is nonnegative. Moreover, $\lambda=0$ cannot be a solution because that would imply $|z|-\|w\|\ge
z-\|w\|>0$. Hence, the conditions in item 3 hold if and only if becomes $\lambda\|w\|=(\|w\|-(\lambda+1)z)$. This latter equation has the unique positive solution $$\label{eq:lambda}
\lambda=\frac{\|w\|-z}{\|w\|+z}.$$ By using equation and , it is just a matter of algebraic manipulations to check that holds for this case. The cases described by items 1 and 2 can be similarly checked.
Numerical methods for projecting
================================
In this section we present three well known numerical methods to find the unique zero of the piecewise linear equation , in order to project onto the extended second order cones. We note that $(z,w)\in{\mathbb R}^p\times{\mathbb R}^q$ satisfies the two conditions in item 3 of Theorem \[th:pelc\] if and only if $$\label{eq: item3}
\exists ~ i_0\in \{1, \ldots, p\}; \quad 0\leq z_{i_0}^+<\|w\|, \qquad 0\leq \sum_{i=1}^p z_i^{-} < \|w\|. \qquad$$ [*Throughout this section we will assume that $(z,w)\in{\mathbb R}^p\times{\mathbb R}^q$ satisfies* ]{}.
Semi-smooth Newton method
-------------------------
In order to study , we consider the piecewise linear function $\psi$ defined by . It follows from Proposition \[pr:psy\] that $\psi$ is convex and its unique zero, namely $\lambda_*>0$, is the solution of . The [*semi-smooth Newton method*]{} for finding the zero of $\psi$, with a starting point $\lambda_{0}\in (0, +\infty)$, it is formally defined by $$\label{eq:nmg}
\psi(\lambda_k)+ s_k\left(\lambda_{k+1}-\lambda_{k}\right)=0, \qquad s_k \in \partial \psi(\lambda_k), \qquad k=0,1,\ldots,$$ where $ s_k$ is any subgradient in $ \partial \psi(\lambda_k)$. Let $N(\lambda)$ be defined by equation . Item 1 of Proposition \[pr:psy\] implies that $-\|w\|+ \left\langle e, N(\lambda)z\right\rangle
\in \partial \psi(\lambda)$. Since $N(\lambda) [(\lambda+1)z-\|w\|e]= [(\lambda+1)z-\|w\|e]^-$, by setting $s_k= -\|w\| + \left\langle e,
N_kz\right\rangle$ with $$\label{def:Nk}
N_k:= N(\lambda_k),$$ equation implies $$-\lambda_k \|w\|+\langle e, N_k \left[(\lambda_k+1)z-\|w\|e\right]\rangle+ \left[ -\|w\| + \left\langle e, N_kz\right\rangle\right]\left[\lambda_{k+1}-\lambda_{k}\right]=0.$$ After simplification, we get $$\label{eq:nm}
\left[- \|w\| +\left\langle e, N_kz\right\rangle\right] \lambda_{k+1}= - \left\langle e, N_k\left[z-\|w\|e\right]\right\rangle, \qquad k=0,1,\ldots,$$ which formally defines the [*semi-smooth Newton sequence*]{} $\{\lambda_{k}\}$ for solving .
For $p=1$, the conditions in hold if and only if $0\leq |z|< \|w\|$. Thus, if $z\leq 0$, then $N_k\equiv -1$ and $\lambda_{k+1}=[\|w\|-z]/[\|w\|+z]$ for all $k=0,1,\ldots$. Now, if $z > 0$ then letting $ 0<\lambda_0 < [\|w\|-z]/z$, we have $N_0\equiv
-1$ and $\lambda_{1}=[\|w\|-z]/[\|w\|+z]$. Therefore, from Remark \[eq:fplc\], we conclude that the semi-smooth Newton sequence solves equation for $p=1$ with only one iteration.
The proof of the next proposition is based on ideas similar to some arguments in [@MR3464994].
\[teo-finite\] For any $\lambda_0>0$ the sequence $\{\lambda_k\}$ defined in is well defined and converges after at most $2^p$ steps to the unique solution $\lambda_*>0$ of .
Proposition \[pr:psy\] implies that $\psi$ is convex and $-\|w\|+ \left\langle e, N(\lambda)z\right\rangle \in \partial \psi(\lambda)$. Thus, we have $$\label{eq-chave}
\psi(\mu )- \psi(\lambda) -[-\|w\|+ \left\langle e, N(\lambda)z\right\rangle](\mu-\lambda) \geq 0, \qquad \mu, \lambda \in [0, +\infty).$$ On the other hand, it follows from and that the sequence $\{\lambda_k\}$ is equivalently defined as follows $$\label{eq:nmg2}
\psi(\lambda_k)+ \left[- \|w\| +\left\langle e, N_kz\right\rangle\right]\left(\lambda_{k+1}-\lambda_{k}\right)=0, \qquad k=0,1,\ldots.$$ By combining the above equality with the definition in and the equality in , we can conclude that $$\label{Fnegativa}
\psi(\lambda_{k+1} )\geq \psi(\lambda_{k}) +[-\|w\| + \left\langle e, N_{k}z\right\rangle](\lambda_{k+1}-\lambda_{k})=0, \qquad \, k=0,1,\ldots.$$ By letting $\mu=\lambda^{*}$ and $\lambda=\lambda_k$ in inequality and by using again the definition in , we obtain that $$\label{13insolution}
0=\psi(\lambda_* )\geq \psi(\lambda_k) + [-\|w\|+ \left\langle e, N_kz\right\rangle](\lambda_*-\lambda_k), \qquad \, k=0,1,\ldots.$$ Proposition \[pr:psy\] implies that $-\|w\| +\left\langle e, N_kz\right\rangle <0$, for all $k=0,1,\ldots$. Then, by dividing both sides of by $-\|w\| +\left\langle e, N_kz\right\rangle$ and by using , after some algebras we obtain $$\label{I+}
\lambda_{k+1}= \lambda_k-[-\|w\| +\left\langle e, N_kz\right\rangle]^{-1}\psi(\lambda_k)\le \lambda_*, \qquad \, k=0, 1, \ldots.$$ On the other hand, $\psi(\lambda_{k} )\geq 0 $, for all $k=0,1,\ldots$. Thus, after dividing both sides of the equality in by $\|w\| -\left\langle e, N_kz\right\rangle$ and some algebraic manipulations, we conclude $$\label{increasing}
0<\lambda_{k}\le \lambda_k-[-\|w\| +\left\langle e, N_kz\right\rangle]^{-1}\psi(\lambda_k)= \lambda_{k+1}, \qquad \, k=0, 1, \ldots.$$ Hence, by combining with , we conclude that $ 0<\lambda_{k}\le \lambda_{k+1}\le \lambda_*$ , for all $k=0,1,\ldots$. Hence, $\{\lambda_{k}\}$ converges to some $\bar{\lambda}>0$. By using again and that the entries of $N_k$ are equal to $0$ or $-1$, we have $$\begin{aligned}
|\psi(\bar{\lambda})|=\lim_{k\to\infty}|\psi(\lambda_k)|=\lim_{k\to\infty}|\left[- \|w\| +\left\langle e,
N_kz\right\rangle\right]\left(\lambda_{k+1}-\lambda_{k}\right)|\\\le \left[ \|w\| +\left\langle e,
|z|\right\rangle\right]\lim_{k\to\infty}|\lambda_{k+1}-\lambda_{k}|=0.\end{aligned}$$ Hence, $\{\lambda_{k}\}$ converges to $\bar{\lambda}=\lambda_*$ the unique zero of $\psi$, which is the solution of .
Finally, we establish the finite termination of the sequence $\{\lambda_{k}\}$ at $\lambda_{*}$, the unique solution of . Since the entries of ${N}(\lambda)$ are equal to $0$ or $-1$, $N(\lambda)$ has at most $2^p$ different possible configurations. Then, there exist $j, \ell \in {\mathbb N}$ with $1\leq j<2^p$ and $1\leq \ell < 2^p$ such that $N(\lambda_j)=N(\lambda_{j+\ell})$. Hence, from we have $$\begin{aligned}
\lambda_{j+1}=-\left[ -\|w\| +\left\langle e, N_jz\right\rangle\right]^{-1}\left\langle e, N_j\left[z-\|w\|e\right]\right\rangle\\=-\left[- \|w\| +\left\langle e, N_{j+\ell}z\right\rangle\right]^{-1}\left\langle e, N_{j+\ell}\left[z-\|w\|e\right]\right\rangle=\lambda_{j+\ell+1}.\end{aligned}$$ By applying this argument inductively, $\lambda_{j+1}=\lambda_{j+\ell+1}$, $\lambda_{j+2}=\lambda_{j+\ell+2}$, $\ldots$, $\lambda_{j+\ell}=\lambda_{j+2\ell}$, $\lambda_{j+\ell+1}=\lambda_{j+2\ell+1}=\lambda_{j+1}.$ Thus, by using and the last equality, we conclude that $$\lambda_{j+1}\leq \lambda_{j+2} \leq \dots \leq \lambda_{j+\ell+1}\leq \lambda_{j+1}.$$ Hence, $\lambda_{j+1}= \lambda_{j+2}$ and in view of we conclude that $\psi(\lambda_{j+1})=0$ and $\lambda_{j+1}$ is the solution of , i.e., $\lambda_{j+1}=\lambda_{*}$.
The next proposition shows that under a further restriction on the point which is projected the convergence of the semi-smooth Newton sequence is linear.
Assume that $0<\alpha<1$ and $\left\langle e, |z|\right\rangle<\alpha(1+\alpha)^{-1}\|w\|$. Then, for any $\lambda_0>0$ , the sequence $\{\lambda_k\}$ in is well defined and converges linearly to the unique solution $\lambda_*$ of : $$\label{eq:cr}
|\lambda_* -\lambda_{k+1}|\leq \alpha |\lambda_*-\lambda_k|, \qquad k=0,1,\ldots.$$
Proposition \[pr:psy\] and imply $-\|w\| +\left\langle e, N_kz\right\rangle < 0$ for all $k=0,1,\ldots$, which implies that the sequence $\{\lambda_k\}$ is well defined. Proposition \[pr:psy\] also implies that has a zero $\lambda_*\in (0, +\infty)$. Hence, by using , and the definition of $\psi$, after some algebra we obtain that $$\begin{gathered}
\lambda_* -\lambda_{k+1}= \left[ -\|w\| +\left\langle e, N_kz\right\rangle\right]^{-1} \big{[} \lambda_* \|w\|-\langle e, [(\lambda_*+1)z-\|w\|e]^-\rangle \\
-\lambda_k \|w\|+\langle e, [(\lambda_k+1)z-\|w\|e]^-\rangle+ \left[ -\|w\| +\left\langle e, N_kz\right\rangle\right]\left[ \lambda_* -\lambda_{k}\right]\big{]}, \end{gathered}$$ for all $k=0,1,\ldots.$ On the other hand, since $N(\lambda) [(\lambda+1)z-\|w\|e]= [(\lambda+1)z-\|w\|e]^-$, after some calculations we have $$\begin{gathered}
\lambda_* \|w\|-\langle e, [(\lambda_*+1)z-\|w\|e]^-\rangle - \\ \lambda_k \|w\|+ \langle e, [(\lambda_k+1)z-\|w\|e]^-\rangle+ \left[ -\|w\| +\left\langle e, N_kz\right\rangle\right]\left[ \lambda_* -\lambda_{k}\right]= \\
-\langle e, N_*[(\lambda_*+1)z-\|w\|e]\rangle +\langle e, N_k[(\lambda_k+1)z-\|w\|e]\rangle + \left\langle e, N_kz\right\rangle \left[ \lambda_* -\lambda_{k}\right], \end{gathered}$$ for all $k=0,1,\ldots$, where $N_*:= N(\lambda_*)$. By combining the above two equalities, we obtain that $$\begin{gathered}
\label{eq:taylor1}
\lambda_* -\lambda_{k+1}= \left[ -\|w\| +\left\langle e, N_kz\right\rangle\right]^{-1} \big{[}- \langle e, N_*[(\lambda_*+1)z-\|w\|e]\rangle +\\ \langle e, N_k[(\lambda_k+1)z-\|w\|e]\rangle + \left\langle e, N_kz\right\rangle \left[ \lambda_* -\lambda_{k}\right] \big{]} .\end{gathered}$$ Define the auxiliary piecewise linear convex function $\zeta(\lambda) := \langle e, N(\lambda)[(\lambda+1)z-\|w\|e]\rangle$. Thus, except possibly at $p$ points, $\zeta$ is differentiable and there holds $$\zeta(\lambda_*)= \zeta(\lambda_k) + \int_{0}^{1} \left\langle e, N(\lambda_k + t(\lambda_*-\lambda_k))z\right\rangle[\lambda_*-\lambda_k]dt,$$ due to $\left\langle e, N(\lambda)z\right\rangle \in \partial \zeta (\lambda)$; see [@HiriartUrrutyLemarechal1993 Remark 4.2.5, pag. 26]. Hence, by simple combination of the two latter equalities, we have $$\begin{gathered}
\lambda_* -\lambda_{k+1}=\\ -\left[- \|w\| +\left\langle e, N_kz\right\rangle\right]^{-1} \int_{0}^{1} \left\langle e, \left[N(\lambda_k +t(\lambda_*-\lambda_k))-N_k\right]z\right\rangle dt [\lambda_*-\lambda_k],\end{gathered}$$ for all $k=0,1,\ldots.$ Since implies that the entries of the matrix $N$ are equal to $0$ or $-1$, we obtain $$| \left\langle e, \left[N(\lambda_k + t(\lambda_*-\lambda_k))-N_k\right]z\right\rangle|\leq \sum_{j=1}^p|z_j|=\left\langle e, |z|\right\rangle.$$ Thus, combining above equality with last inequality, we obtain that $$|\lambda_* -\lambda_{k+1}|\leq |\|w\| -\left\langle e, N_kz\right\rangle|^{-1} \left\langle e, |z|\right\rangle |\lambda_*-\lambda_k|, \qquad k=0,1,\ldots.$$ Therefore, as we are under the assumption $\left\langle e, |z|\right\rangle<\alpha(1+\alpha)^{-1}\|w\|$, we have $ \left\langle e, |z|\right\rangle/[\|w\| -\left\langle e,N_kz\right\rangle] <\alpha<1$, holds and the sequence $\{\lambda_k\}$ converges to $\lambda_*$, which concludes the proof.
Picard’s method
---------------
In this section we present a method based on Picard’s iteration for solving equation under a further restriction on the point which is projected. The statement of the result is as follows:
If $\left\langle e, |z|\right\rangle<\|w\|$, then for all $\lambda_0>0$ the sequence given by the iteration $$\label{eq:pm}
\lambda_{k+1} =\frac{1}{\|w\|}\left\langle e, [(\lambda_k+1)z-\|w\|e]^-\right\rangle, \qquad k=1, \ldots,$$ converges to the unique solution of the semi-smooth equation .
It is sufficient to prove that $\varphi \colon [0, +\infty) \to {\mathbb R} $ defined by $$\varphi(\lambda)=\frac{1}{\|w\|}\left\langle e, [(\lambda+1)z-\|w\|e]^-\right\rangle$$ is a contraction. Indeed, the definition of $\varphi$ implies $$\begin{aligned}
|\varphi(\lambda) - \varphi(\mu)|&\leq \frac{1}{\|w\|} \sum_{i=1}\left|[(\lambda+1)z_i-\|w\|]^- - [(\mu+1)z_i-\|w\|]^-\right| \\
&\leq \frac{1}{\|w\|} \sum_{i=1}\left|z_i (\lambda -\mu)\right| =\frac{\left\langle e, |z|\right\rangle}{\|w\|}\left|\lambda -\mu\right|, \qquad \lambda, \mu \in [0, +\infty). \end{aligned}$$ Since we are under the assumption $\left\langle e, |z|\right\rangle<\|w\|$, the last inequality implies that $\varphi$ is a contraction and the result follows.
Final remarks {#final-remarks .unnumbered}
=============
The [*extended second order cones*]{} (ESOCs) are likely the most natural extensions of the second order cones. Also, the complementarity problems defined on them often have nice computational properties as remarked in the introduction. Finally, we found almost closed-form formulas for projecting onto them. The formulas depend only on a piecewise linear equation for a real parameter. Not so much of the ESOCs is known, nevertheless, we stipulate that they will become an important class of cones in optimization.
For a given point in the ambient space the projection can be obtained easily in at most $2^p$ steps, by assigning signs to the components of the second vector in the scalar product on the right hand side of the piecewise linear equation , solving for $\lambda$, and if there is a solution, then checking that the solution corresponds to the a priori assumed signs. However, this method is computationally unviable for larger $p$. Therefore, we developed numerical methods for solving based on the semismooth Newton method and Picards iterations. Although the semismooth Newton method always converges in at most $2^p$ steps, it needs some restriction on the point which is projected to prove that is globally linearly convergent. A similar type of restriction is needed for Picard’s method to prove that it is globally convergent.
The complexity of our projection method is considerably lower than the complexity of solving the reformulation of the projection problem into a second order conic optimization problem. It is expected that there are other conic optimization problems with respect to the extended second order cone which are easier to solve than transforming them into second order conic optimization problems. We plan to solve conic optimization and complementarity problems on the extended second order cone (similarly to the second order cone in [@BCFNP2017]) and to find practical examples which can be modeled by such problems. Early studies of Lianghai Xiao (PhD student of the second author) suggest that the extended second order cones could be useful for portfolio selection, see [@MR0103768; @MR2925782] and signal processing problems, see [@MR3618179; @MR3674429].
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[^1]: IME/UFG, Avenida Esperança, s/n, Campus Samambaia, Goiânia, GO, 74690-900, Brazil (e-mail:[orizon@ufg.br]{}). The author was supported in part by FAPEG, CNPq Grant 305158/2014-7 and PRONEX–Optimization(FAPERJ/CNPq).
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---
abstract: 'We propose a unified framework for neural net normalization, regularization and optimization, which includes Path-SGD and Batch-Normalization and interpolates between them across two different dimensions. Through this framework we investigate the issue of invariance of the optimization, data dependence and the connection with natural gradients.'
author:
- |
Behnam Neyshabur\
Toyota Technological Institute at Chicago\
Chicago, IL 60637, USA\
`bneyshabur@ttic.edu` Ryota Tomioka\
Microsoft Research\
Cambridge, UK\
`ryoto@microsoft.com` Ruslan Salakhutdinov\
Department of Computer Science\
University of Toronto, Canada\
`rsalakhu@cs.toronto.edu`\
Nathan Srebro\
Toyota Technological Institute at Chicago\
Chicago, IL 60637, USA\
`nati@ttic.edu`\
bibliography:
- 'iclr2016.bib'
title: |
Data-Dependent Path Normalization\
in Neural Networks
---
Introduction
============
The choice of optimization method for non-convex, over-parametrized models such as feed-forward neural networks is crucial to the success of learning—not only does it affect the runtime until convergence, but it also effects which minimum (or potentially local minimum) we will converge to, and thus the generalization ability of the resulting model. Optimization methods are inherently tied to a choice of geometry over parameter space, which in turns induces a geometry over model space, which plays an important role in regularization and generalization [@neyshabur15b].
In this paper, we focus on two efficient alternative optimization approaches proposed recently for feed-forward neural networks that are based on intuitions about parametrization, normalization and the geometry of parameter space: [**Path-SGD**]{} [@NeySalSre15] was derived as steepest descent algorithm with respect to particular regularizer (the $\ell_2$-path regularizer, i.e. the sum over all paths in the network of the squared product over all weights in the path [@NeyTomSre15]) and is invariant to weight reparametrization. [**Batch-normalization**]{} [@IofSze15] was derived by adding normalization layers in the network as a way of controlling the variance of the input each unit receives in a data-dependent fashion. In this paper, we propose a unified framework which includes both approaches, and allows us to obtain additional methods which interpolate between them. Using our unified framework, we can also tease apart and combine two different aspects of these two approaches: data-dependence and invariance to weight reparametrization.
Our unified framework is based on first choosing a per-node complexity measure we refer to as $\gamma_v$ (defined in Section \[sec:unified\]). The choice of complexity measure is parametrized by a choice of “normalization matrix” $R$, and different choices for this matrix incorporate different amounts of data dependencies: for path-SGD, $R$ is a non-data-dependent diagonal matrix, while for batch normalization it is a data-dependent covariance matrix, and we can interpolate between the two extremes. Once $\gamma_v$ is defined, and for any choice of $R$, we identify two different optimization approaches: one relying on a normalized re-parameterization at each layer, as in batch normalization (Section \[sec:bn\]), and the other an approximate steepest descent as in path-SGD, which we refer to as DDP-SGD (Data Dependent Path SGD) and can be implemented efficiently via forward and backward propagation on the network (Section \[sec:ddp-sgd\]). We can now mix and match between the choice of $R$ (i.e. the extent of data dependency) and the choice of optimization approach.
One particular advantage of the approximate steepest descent approach (DDP-SGD) over the normalization approach is that it is invariant to weight rebalancing (discussed in Section \[sec:node-rescaling\]). This is true regardless of the amount of data-dependence used. That is, it operates more directly on the model (the function defined by the weights) rather than the parametrization (the values of the weights themselves). This brings us to a more general discussion of parametrization invariance in feedforward networks (Section \[sec:rescaling\]).
Our unified framework and study of in invariances also allows us to relate the different optimization approaches to Natural Gradients [@Ama98]. In particular, we show that DDP-SGD with full data-dependence can be seen as an efficient approximation of the natural gradient using only the diagonal of the Fisher information matrix (Section \[sec:natural\]).
Related Works {#related-works .unnumbered}
-------------
There has been an ongoing effort for better understanding of the optimization in deep networks and several heuristics have been suggested to improve the training [@lecun-98x; @larochelle2009exploring; @difficulty; @sutskever2013importance]. Natural gradient algorithm [@Ama98] is known to have a very strong invariance property; it is not only invariant to reparametrization, but also to the choice of network architecture. However it is known to be computationally demanding and thus many approximations have been proposed [@grosse2015scaling; @martens2015optimizing; @desjardins2015natural]. However, such approximations make the algorithms less invariant than the original natural gradient algorithm. [@pascanu2013revisiting] also discuss the connections between Natural Gradients and some of the other proposed methods for training neural networks, namely Hessian-Free Optimization [@martens2010deep], Krylov Subspace Descent [@vinyals2011krylov] and TONGA [@roux2008topmoumoute].
[@ollivier2015riemannian] also recently studied the issue of invariance and proposed computationally efficient approximations and alternatives to natural gradient. They study invariances as different mappings from parameter space to the same function space while we look at the invariances as transformations (inside a fixed parameter space) to which the function is invariant in the model space (see Section \[sec:rescaling\]). Unit-wise algorithms suggested in Olivier’s work are based on block-diagonal approximations of Natural Gradient in which blocks correspond to non-input units. The computational cost of the these unit-wise algorithms is quadratic in the number of incoming weights. To alleviate this cost, [@ollivier2015riemannian] also proposed quasi-diagonal approximations which avoid the quadratic dependence but they are only invariant to affine transformations of activation functions. The quasi-diagonal approximations are more similar to DDP-SGD in terms of computational complexity and invariances (see Section \[sec:node-rescaling\]). In particular, ignoring the non-diagonal terms related to the biases in quasi-diagonal natural gradient suggested in [@ollivier2015riemannian], it is then equivalent to diagonal Natural Gradient which is itself equivalent to special case of DDP-SGD when $R_v$ is the second moment (see Table \[tab:framework\] and the discussion on relation to the Natural Gradient in Section \[sec:ddp-sgd\]).
Feedforward Neural Nets {#sec:ff}
=======================
We briefly review our formalization and notation of feedforward neural nets. We view feedforward neural networks as a parametric class of functions mapping input vectors to output vectors, where parameters correspond to [*weights*]{} on connections between different [ *units*]{}. We focus specifically on networks of ReLUs (Rectified Linear Units). Rather than explicitly discussing units arranged in layers, it will be easier for us (and more general) to refer to the connection graph as a directed acyclic graph $G(V,E)$ over the set of node $V$, corresponding to units $v\in V$ in the network. $V$ includes the inputs nodes $V_{\rm in}$ (which do not have any incoming edges), the output nodes $V_{\rm out}$ (which do not have any outgoing edges) and additional internal nodes (possibly arranged in multiple layers). Each directed edge $(u\rightarrow v)\in E$ (i.e. each connection between units) is associated with a weight $w_{u
\rightarrow v}$. Given weight settings ${\mathbf{w}}$ for each edge, the network implements a function $f_{{\mathbf{w}}}:{\mathbb{R}}^{\abs{V_{\rm
in}}}\rightarrow {\mathbb{R}}^{\abs{V_{\rm out}}}$ as follows, for any input ${\mathbf{x}}\in{\mathbb{R}}^{\abs{V_{\rm in}}}$:
- For the input nodes $v\in V_{\rm in}$, their output $h_v$ is the corresponding coordinate of the input ${\mathbf{x}}$.
- For each internal node $v$ we define recursively $z_v=\sum_{(u\rightarrow v)\in E} w_{u\rightarrow v}\cdot h_u$ and $h_v=[z_v]_+$ where $[z]_+=\max(z,0)$ is the ReLU activation function and the summation is over all edges incoming into $v$.
- For output nodes $v\in V_{\rm out}$ we also have $z_v=\sum_{(u\rightarrow v)\in E} w_{u\rightarrow v}\cdot h_u$, and the corresponding coordinate of the output $f_{\mathbf{w}}({\mathbf{x}})$ is given by $z_v$. No non-linearity is applied at the output nodes, and the interpretation of how the real-valued output corresponds to the desired label is left to the loss function (see below).
- In order to also allow for a “bias” at each unit, we can include an additional special node $v_{\rm bias}$ that is connected to all internal and output nodes, where $h_{v_{\rm bias}}=1$ always ($v_{\rm bias}$ can thus be viewed as an additional input node whose value is always $1$).
We denote $N^{{\text{in}}}(v)=\left\{ u \middle| (u\rightarrow v)\in
E\right\}$ and $N^{{\text{out}}}(v)=\left\{ u \middle| (v\rightarrow u)\in
E\right\}$, the sets of nodes feeding into $v$ and that $v$ feeds into. We can then write ${\mathbf{h}}_{N^{\text{in}}(v)}\in{\mathbb{R}}^{\left|
N^{\text{in}}(v)\right|}$ for the vector of outputs feeding into $v$, and ${\mathbf{w}}_{\rightarrow v}\in{\mathbb{R}}^{\left| N^{\text{in}}(v)\right|}$ for the vector of weights of unit $v$, so that $z_v=\inner{{\mathbf{w}}_{\rightarrow
v}}{{\mathbf{h}}_{N^{\text{in}}(v)}}$.
We do not restrict to layered networks, nor do we ever need to explicitly discuss layers, and can instead focus on a single node at a time (we view this as the main advantage of the graph notation). But to help those more comfortable with layered networks understand the notation, let us consider a layered fully-connected network: The nodes are partitioned into layers $V=V_0 \cup V_1 \cup \ldots
V_d$, with $V_{\rm in}=V_0$, $V_{\rm out}=V_d$. For all nodes $v\in
V_i$ on layer $i$, $N^{\text{in}}(v)$ is the same and equal to $N^{\text{in}}(v)=V_{i-1}$, and so ${\mathbf{h}}_{N^{\text{in}}(v)}={\mathbf{h}}_{V_{i-1}}$ consists of all outputs from the previous layer and we recover the layered recursive formula $h_v=[\inner{{\mathbf{w}}_{\rightarrow
v}}{{\mathbf{h}}_{V_{i-1}}}]_+$ and ${\mathbf{h}}_{V_{i}}=[\mathbf{W}_i
{\mathbf{h}}_{V_{i-1}}]_+$, where $\mathbf{W}\in
{\mathbb{R}}^{\left|V_i\right|\times\left|V_{i-1}\right|}$ is a matrix with entries $w_{u\rightarrow v}$, for each $u\in V_{i-1}, v\in V_i$. This description ignores the bias term, which could be modeled as a direct connection from $v_{\rm bias}$ into every node on every layer, or by introducing a bias unit (with output fixed to 1) at each layer. Please see Figure \[fig:notation\] for an example of a layered feedforward network and a summary of notation used in the paper.
We consider supervised training tasks, where each input ${\mathbf{x}}$ is associated with a desired label $y$ and how well the network captures this label is quantified by a loss function $\ell(f_{{\mathbf{w}}}({\mathbf{x}}),y)$. For example, in a classification problem $y$ is one of $|V_{\rm out}|$ classes and a cross-entropy (soft-max) loss might be used. We also refer to a source distribution $\D_{(\mathbf{x},\mathbf{y})}$ over input-label pairs, where the goal is to minimize the expected loss: $$L_{\D}({\mathbf{w}})=\E_{(\mathbf{x},y) \sim \D}
\left[ \ell(f_{{\mathbf{w}}}(\mathbf{x}),y) \right]$$ All expectations, unless specified otherwise, refer to expectation w.r.t. this source distribution.
Invariances and Node-wise Rescaling {#invariances-and-node-wise-rescaling .unnumbered}
-----------------------------------
Once the architecture (graph $G$) is fixed, every choice of weight ${\mathbf{w}}$ defines a function $f_{\mathbf{w}}$. But this parameterization is not unique–the same function $f$ could be parameterized by two different weight settings (i.e. we could have $f_{\mathbf{w}}=f_{{\mathbf{w}}'}$ even though ${\mathbf{w}}\not={\mathbf{w}}'$). Ideally, we’d like to work as directly as possibly on the functions rather then the parameterization. It is therefor important to understand different “invariances”, i.e. different transformations that can be applied to the weights without changing the function. We note that the notion of invariance we define here is tied to a fixed network architecture $G$: we do not consider transformation that changes the network architecture, such as insertion of a linear layer [as in, e.g. @ollivier2015riemannian].
We say that network $G$ is invariant to an invariant transformation $T({\mathbf{w}})$, iff for any weight setting ${\mathbf{w}}$, $f_{{\mathbf{w}}}=f_{T({\mathbf{w}})}$. We say that an update rule $\mathcal{A}$ (e.g. a rule for obtaining the next iterate from the current iterate in an optimization procedure) is invariant to transformation $T$ iff for any weight setting ${\mathbf{w}}$, $f_{\mathcal{A}({\mathbf{w}})}=f_{\mathcal{A}(T({\mathbf{w}}))}$. That is, whether we start an iterative optimization procedure using updates $\mathcal{A}$ at ${\mathbf{w}}$ or the at the equivalent ${\mathbf{w}}'=T({\mathbf{w}})$, we would always be working on the same function (only with a different parameterization).
An important invariance in feedforward ReLU network is [*node-wise rescaling*]{} (or rebalancing). For any positive scalar $\rho$ and for any internal node $v$ ($v\notin V_{\rm in}$ and $v\notin V_{\rm
out}$), we can scale all the incoming weights into $v$ by $\rho$ and all the outgoing weights by $1/\rho$ without changing the computation in the networks. That is, the following transformation ${\mathbf{w}}'=T({\mathbf{w}})$ satisfies $f_{{\mathbf{w}}}=f_{{\mathbf{w}}'}$: $$\label{eq:node-wise}
\begin{aligned}
w'_{v\rightarrow u} &= \rho w_{v\rightarrow u}\quad (\forall u\in
N^{{\text{out}}}(u)),\\
w'_{u\rightarrow v} &=\rho^{-1} w_{u\rightarrow v}\quad
(\forall u\in N^{\text{in}}(v))\\
w'_{u\rightarrow u'} &=w_{u\rightarrow u'} \quad (\textrm{otherwise})
\end{aligned}$$ We can combined multiple such rescalings to push the scaling up or down the network without changing the computation. One of our goals is obtaining optimization algorithms that are invariant to such transformations.
$R_v$ Normalized reparametrization Diagonal steepest descent
----------------------------------------------------------------------------------------------- --------------- ------------------------------ --------------------------------
$D=\diag\left({\boldsymbol{\gamma}}^2_{N^{{\text{in}}(v)}}\right)$ Path-Norm Unit-wise Path-Normalization [**Path-SGD**]{}
$C={\text{Cov}}\left({\mathbf{h}}_{N^{{\text{in}}(v)})}\right)$ Variance [**Batch-Normalization**]{}
$M=\E\left[{\mathbf{h}}_{N^{{\text{in}}(v)})} {\mathbf{h}}_{N^{{\text{in}}(v)})}^\top\right]$ Second Moment [**Diag. Natural Gradient**]{}
DDP-Norm DDP-Normalization DDP-SGD
Node-wise Rescaling Invariant Yes No Yes
: Some of the choices for $R_v$ in the proposed unified framework.[]{data-label="tab:framework"}
A Unified Framework {#sec:unified}
===================
We define a complexity measure on each node as follows: $$\label{eq:R}
\gamma_v({\mathbf{w}}) = \sqrt{{\mathbf{w}}_{\rightarrow v}^\top R_v{\mathbf{w}}_{\rightarrow v} }$$ where $R_v$ is a positive semidefinite matrix that could depend on the computations feeding into $v$, and captures both the complexity of the nodes feeding into $v$ and possibly their interactions. We consider several possibilities for $R_v$, summarized also in Table 1.
A first possibility is to set $R_v=\diag\left({\boldsymbol{\gamma}}^2_{N^{{\text{in}}(v)}}\right)$ to a diagonal matrix consisting of the complexities of the incoming units. This choice does not depend on the source distribution (i.e. the data), and also ignores the effect of activations (since the activation pattern depends on the input distribution) and of dependencies between different paths in the network. Intuitively, with this choice of $R_v$, the measure $\gamma_v({\mathbf{w}})$ captures the “potential” (data independent) variability or instability at the node.
Another possibility is to set $R_v$ to either the covariance (centralized second moment) or to the (uncentralized) second moment matrix of the outputs feeding into $v$. In this case, $\gamma^2_v({\mathbf{w}})$ would evaluate to the variance or (uncentralized) second moment of $z_v$. We could also linearly combined the data independent measure, which measures inherent instability, with one of these the data-dependent measure to obtain: $$\label{eq:gamma}
\gamma^2_v({\mathbf{w}}) = \alpha S(z_v) + (1-\alpha)\sum_{u \in
N^{{\text{in}}} (v)}\gamma^2_u({\mathbf{w}}) w^2_{u\rightarrow v}\quad (v\notin V_{\rm
in}),$$ where $S(z_v)$ is either the variance or uncentralized second moment, and $\alpha$ is a parameter.
The complexity measure above is defined for each node of the network separately, and propagates through the network. To get an overall measure of complexity we sum over the output units and define the following complexity measure for the function $f_{{\mathbf{w}}}$ as represented by the network: $$\gamma^2_{\rm net}({\mathbf{w}}) = \sum_{v\in V_{\rm out}}\gamma^2_v({\mathbf{w}}).$$
For $R_v=\diag\left({\boldsymbol{\gamma}}^2_{N^{{\text{in}}(v)}}\right)$, this complexity measure agrees with the $\ell_2$-Path-regularizer as introduced by [@NeyTomSre15]. This is the sum over all paths in the network of the squared product of weights along the path. The path-regularizer is also equivalent to looking at the minimum over all “node rescalings” of ${\mathbf{w}}$ (i.e. all possibly rebalancing of weights yielding the same function $f_{\mathbf{w}}$) of the $\max_v \norm{{\mathbf{w}}_{\rightarrow v}}$. But, unlike this max-norm measure, the path-regularizer does [*not*]{} depend on the rebalancing and is invariant to node rescalings [@NeyTomSre15].
For data-dependent choices of $R_v$, we also get a similar invariance property. We refer to the resulting complexity measure, $\gamma^2_{\rm net}({\mathbf{w}})$, as the Data-Dependent-Path (DDP) regularizer.
After choosing $R_v$, we will think of $\gamma_v$ as specifying the basic “geometry” and bias (for both optimization and learning) over weights. In terms of learning, we will (implicitly) prefer weights with smaller $\gamma_v$ measure, and correspondingly in terms of optimization we will bias toward smaller $\gamma_v$ “balls” (i.e. search over the part of the space where $\gamma_v$ is smaller). We will consider two basic ways of doing this: In Section \[sec:bn\] we will consider methods that explicitly try to keep $\gamma_v$ small for all internal nodes in the network, that is explicitly search over simpler weights. Any scaling is pushed to the output units, and this scaling hopefully does not grow too much due. In Section \[sec:ddp-sgd\] we will consider (approximate) steepest descent methods with respect to the overall $\gamma_{\rm net}$, i.e. updates that aim at improving the training objective while being small in terms of their effect on $\gamma_{\rm net}$.
DDP-Normalization: A Batch-Normalization Approach {#sec:bn}
=================================================
In this Section, we discuss an optimization approach based on ensuring $\gamma_v$ for all internal nodes $v$ are fixed and equal to one—that is, the complexity of all internal nodes is “normalized”, and any scaling happens only at the output nodes. We show that with a choice of $R_v={\text{Cov}}\left({\mathbf{h}}_{N^{{\text{in}}(v)})}\right)$, this is essentially equivalent to Batch Normalization [@IofSze15].
Batch-Normalization [@IofSze15] was suggested as an alternate architecture, with special “normalization” layers, that ensure the variance of node outputs are normalized throughout training. Considering a feed-forward network as a graph, for each node $v$, the Batch-Normalization architecture has as parameters an (un-normalized) incoming weight vector $\tilde{{\mathbf{w}}}$ and two additional scalars $c_v,b_v\in\R$ specifying scaling and shift respectively. The function computed by the network is then given by a forward propagation similar to standard feed-forward ReLU networks as described in Section \[sec:ff\], except that for each node an un-normalized activation is first computed: $$\label{eq:ztilde}
\tilde{z}_v = \inner{\tilde{{\mathbf{w}}}_{\rightarrow v}}{{\mathbf{h}}_{N^{{\text{in}}}(v)}}$$ Then, this activation is normalized to obtain the normalized activation, which is also scaled and shifted, and the output of the unit is the output of the activation function for this activation value: $$\label{eq:bn}
\begin{gathered}
z_v=c_v
\frac{\tilde{z}_v-\E[\tilde{z_v}]}{\sqrt{{\text{Var}}(\tilde{z}_v)}}+b_v \\
h_v=[z_v]_+
\end{gathered}$$ The variance and expectation are actually calculated on a “mini-batch” of training examples, giving the method its name. Batch-normalization then proceeds by training the architecture specified in and through mini-batch stochastic gradient descent, with each gradient mini-batch also used for estimating the variance and expectation in for all points in the mini-batch.
Instead of viewing batch-normalization as modifying the architecture, or forward propagation, we can view it as a re-parameterization, or change of variables, of the weights in standard feed-forward networks as specified in Section \[sec:ff\]. In particular, instead of specifying the weights directly through ${\mathbf{w}}$, we specify them through $\tilde{{\mathbf{w}}},{{\mathbf{b}}}$ and ${{\mathbf{c}}}$, with the mapping: $$\begin{aligned}
&\tilde{\gamma}^2_v = \tilde{{\mathbf{w}}}_{\rightarrow v}^\top
R_v\tilde{{\mathbf{w}}}_{\rightarrow v} \quad\quad R_v={\text{Cov}}(h_{N^{{\text{in}}}(v)})
\label{eq:tildegamma}\\
&w_{u \rightarrow v} =
\begin{cases}
c \frac{\tilde{w}_{u \rightarrow v}}{\tilde{\gamma}_v} & u\neq
v_{\text{bias}}\\
b-c \frac{\E\left[\inner{\tilde{{\mathbf{w}}}_{\rightarrow v}}{{\mathbf{h}}_{N^{{\text{in}}}(v)}}\right]}{\tilde{\gamma}_v}& u=v_{\text{bias}}\\
\end{cases}\label{eq:reparam}\end{aligned}$$ The model class of functions used by Batch-Normalization is thus exactly the same model class corresponding to standard feed-forward network, just the parameterization is different. However, the change of variables from ${\mathbf{w}}$ to $\tilde{{\mathbf{w}}},{{\mathbf{b}}},{{\mathbf{c}}}$ changes the geometry implied by the parameter space, and consequently the trajectory (in model space) of gradient updates—effectively transforming the gradient direction by the Jacobian between the two parameterizations. Batch-Normalization can thus be viewed as an alternate optimization on the same model class as standard feed-forward networks, but with a different geometry. The reparametrization ensures that $\gamma_v({\mathbf{w}})=c_v$ for all nodes—that is, the complexity is explicit in the parameterization and thus gets implicitly regularized through the implicit regularization inherent in stochastic gradient updates.
The re-parameterization is redundant and includes more parameters than the original parameterization ${\mathbf{w}}$—in addition to one parameter per edge, it includes also two additional parameters per node, namely the shift $b_v$ and scaling $c_v$. The scaling parameters at internal nodes can be avoided and removed by noting that in ReLU networks, due to the node-rescaling property, all scaling can be done at the output nodes. That is, fixing $c_v=1$ for all internal $v$ does not actually change the model class (all functions realizable by the model can be realized this way). Similarly, we can also avoid the additional shift parameter $b_v$ and rely only on bias units and bias weights $\tilde{w}_{v_{\rm
bias}\rightarrow v}$ that get renormalized together with weights. The bias term $\tilde{w}_{v_{\rm bias}\rightarrow v}$ does [*not*]{} affect normalization (since it is deterministic and so has no effect on the variance), it just gets rescaled with the other weights.
We thus propose using a simpler reparametrization (change of variables), with the same number of parameters, using only $\tilde{{\mathbf{w}}}$ and defining for each internal unit: $$\label{eq:ddp-reparam}
w_{u\rightarrow v}=\frac{\tilde{w}_{u\rightarrow v} }{\tilde{\gamma}_v}$$ with $\tilde{\gamma}_v$ as in , and with the output nodes un-normalized: ${\mathbf{w}}_{\rightarrow V_{\rm out}} =
\tilde{{\mathbf{w}}}_{\rightarrow V_{\rm out}}$. This ensures that for all internal nodes $\gamma_v({\mathbf{w}})=1$.
Going beyond Batch-Normalization, we can also use the same approach with other choices of $R_v$, including all those in Table 1: We work with a reparametrization $\tilde{{\mathbf{w}}}$, defined through and but with different choices of $R_v$, and take gradient (or stochastic gradient) steps with respect to $\tilde{{\mathbf{w}}}$. Expectations in the definition of $R_v$ can be estimated on the stochastic gradient descent mini-batch as in Batch-Normalization, or on independent samples of labeled or unlabeled examples. We refer to such methods as “DDP-Normalized” optimization. Gradients in DDP-Normalization can be calculated implemented very efficiently similar to Batch-Normalization (see Appendix \[sec:BN-imp\]).
When using this type of DDP-Normalization, we ensure that for any internal node $\gamma_v({\mathbf{w}})=1$ (the value of $\tilde{\gamma}_v$ can be very different from $1$, but what is fixed is the value of $\gamma_v$ as defined in in terms of the weights ${\mathbf{w}}$, which in turn can be derived from $\tilde{{\mathbf{w}}}$ through ), and so the overall complexity $\gamma_{\rm
net}({\mathbf{w}})$ depends only on the scaling at the output layer.
Another interesting property of DDP-Normalization updates is that for any internal node $v$, the updates direction of $\tilde{w}_{\rightarrow v}$ is exactly orthogonal to the weights:
\[thm:orthogonal\] For any weight $\tilde{{\mathbf{w}}}$ in DDP-Normalization and any non-input node $v\notin V_{{\text{in}}}$ $$\inner{\tilde{{\mathbf{w}}}_{\rightarrow v}}{ \frac{ \partial L }{\partial \tilde{{\mathbf{w}}}_{\rightarrow v} } }=0$$
The fact that the gradient is orthogonal to the parameters means weight updates in DDP-Normalization are done in a way that it prevents the norm of weights to change considerably after each updates (the proof is given in Appendix \[sec:proofs\]).
DDP-SGD {#sec:ddp-sgd}
=======
We now turn to a more direct approach of using our complexity measure for optimization. To do so, let us first recall the strong connection between geometry, regularization and optimization through the specific example of gradient descent.
Gradient descent can be thought of as steepest descent with respect to the Euclidean norm—that is, it takes a step in a direction that maximizes improvement in the objective while also being small in terms of the Euclidean norm of the step. The step can also be viewed as a regularized optimization of the linear approximation given by the gradient, where the regularizer is squared Euclidean norm. Gradient Descent is then inherently linked to the Euclidean norm—runtime of optimization is controlled by the Euclidean norm of the optimum and stochastic gradient descent yields implicit Euclidean norm regularization. A change in norm or regularizer, which we think of as a change of geometry, would then yield different optimization procedure linked to that norm.
What we would like is to use the DDP-regularizer $\gamma_{\rm {{\rm net}}}({\mathbf{w}})$ to define our geometry, and for that we need a distance (or divergence) measure corresponding to it by which we can measure the “size” of each step, and require steps to be small under this measure. We cannot quite do this, but instead we use a diagonal quadratic approximation of $\gamma_{\rm net}({\mathbf{w}})$ about our current iterate, and then take a steepest descent step w.r.t. the quadratic norm defined by this approximation.
Specifically, given a choice of $R_v$ and so complexity measure $\gamma_{\rm net}({\mathbf{w}})$, for the current iterate ${\mathbf{w}}^{(t)}$ we define the following quadratic approximation: $$\hat{\gamma}^2_{\rm net}({\mathbf{w}}^{(t)}+\Delta {\mathbf{w}}) = \gamma^2_{\rm net}({\mathbf{w}}^{(t)})+
\inner{\nabla \gamma^2_{\rm net}({\mathbf{w}}^{(t)})}{\Delta {\mathbf{w}}} +
\frac{1}{2} \Delta {\mathbf{w}}^\top
\diag\left(\nabla^2 \gamma^2_{\rm net}({\mathbf{w}}^{(t)})\right) \Delta {\mathbf{w}}$$ and the corresponding quadratic norm: $$\norm{{\mathbf{w}}'-{\mathbf{w}}}^2_{\hat{\gamma}^2_{\rm net} } =
\norm{{\mathbf{w}}'-{\mathbf{w}}}^2_{\diag(\frac{1}{2}\nabla^2 \gamma^2_{\rm
net}({\mathbf{w}}^{(t)}))}=\sum_{(u\rightarrow v)\in G}
\frac{1}{2}\frac{\partial^2 \gamma^2_{\rm net}}{\partial {\mathbf{w}}^2_{u\rightarrow v}} ({\mathbf{w}}'_{u\rightarrow v}-{\mathbf{w}}_{u\rightarrow v})^2.$$ We can now define the DDP-update as: $$\label{eq:ddp}
{\mathbf{w}}^{(t+1)}=\min_{{\mathbf{w}}} \eta \inner{\nabla L(w)}{{\mathbf{w}}-{\mathbf{w}}^{(t)}} +
\frac{1}{2} \norm{{\mathbf{w}}'-{\mathbf{w}}}^2_{\hat{\gamma}^2_{\rm net} }.$$ Another way of viewing the above approximation is as taking a diagonal quadratic approximation of the Bergman divergence of the regularizer. Solving yields the update: $$w^{(t+1)}_{u\rightarrow v} = w_{u\rightarrow v} - \frac{\eta}{\kappa_{u\rightarrow v}({\mathbf{w}})} \frac{\partial
L}{\partial w_{u\rightarrow v}}({\mathbf{w}}^{(t)}) \quad\quad\textrm{where: } \kappa_{u\rightarrow v}({\mathbf{w}})=\frac{1}{2}\frac{\partial^2 \gamma^2_{\rm
net}}{\partial w^2_{u\rightarrow v}}.$$ Instead of using the full gradient, we can also use a limited number of training examples to obtain stochastic estimates of $\frac{\partial
L}{\partial w_{u\rightarrow v}}({\mathbf{w}}^{(t)})$—we refer to the resulting updates as DDP-SGD.
For the choice $R_v=\diag(\gamma^2_{N^{{\text{in}}}(v)})$, we have that $\gamma^2_{\rm net}$ is the Path-norm and we recover Path-SGD [@NeySalSre15]. As was shown there, the Path-SGD updates can be calculated efficiently using a forward and backward propagation on the network, similar to classical back-prop. In Appendix \[sec:ddp-imp\] we show how this type of computation can be done more generally also for other choices of $R_v$ in Table 1.
Relation to the Natural Gradient {#sec:natural .unnumbered}
--------------------------------
The DDP updates are similar in some ways to Natural Gradient updates, and it is interesting to understand this connection. Like the DDP, the Natural Gradients direction is a steepest descent direction, but it is based on a divergence measure calculated directly on the function $f_{{\mathbf{w}}}$, and not the parameterization ${\mathbf{w}}$, and as such is invariant to reparametrizations. The natural gradient is defined as a steepest descent direction with respect to the KL-divergence between probability distributions, and so to refer to it we must refer to some probabilistic model. In our case, this will be a conditional probability model for labels ${\mathbf{y}}$ conditioned on the inputs ${\mathbf{x}}$, taking expectation with respect to the true marginal data distribution over ${\mathbf{x}}$.
What we will show that for the choice $R_v=\E[{\mathbf{h}}_{N^{{\text{in}}}(v)}{\mathbf{h}}_{N^{{\text{in}}}(v)}^\top]$, the DDP update can also be viewed as an approximate Natural Gradient update. More specifically, it is a diagonal approximation of the Natural Gradient for a conditional probability model $q({\mathbf{y}}| {\mathbf{x}};{\mathbf{w}})$ (of the labels ${\mathbf{y}}$ given an input ${\mathbf{x}}$) parametrized by ${\mathbf{w}}$ and specified by adding spherical Gaussian noise to the outputs of the network: ${\mathbf{y}}|{\mathbf{x}}\sim \mathcal{N}(f_{\mathbf{w}}({\mathbf{x}}),I_{|V_{\rm out}|})$.
Given the conditional probability distribution $q({\mathbf{y}}|{{{\mathbf{x}}}};{\mathbf{w}})$, we can calculate the expected Fisher information matrix. This is a matrix indexed by parameters of the model, in our case edges $e=(u\rightarrow
v)$ on the graph and their corresponding weights $w_e$, with entries defined as follows: $$\begin{aligned}
\label{eq:fisher-information-m}
F({{{\mathbf{w}}}})[e,e'] = {\mathbb{E}}_{{{{\mathbf{x}}}}\sim p({\mathbf{x}})}{\mathbb{E}}_{{\mathbf{y}}\sim q({\mathbf{y}}|{{{\mathbf{x}}}};{\mathbf{w}})}\left[
\frac{\partial \log q({\mathbf{y}}|{{{\mathbf{x}}}};{\mathbf{w}})}{\partial w_e}
\frac{\partial \log q({\mathbf{y}}|{{{\mathbf{x}}}};{\mathbf{w}})}{\partial w_{e'}}
\right],
\end{aligned}$$ where $x\sim p({\mathbf{x}})$ refers to the marginal source distribution (the data distribution). That is, we use the true marginal distributing over ${\mathbf{x}}$, and the model conditional distribution ${\mathbf{y}}|{\mathbf{x}}$, ignoring the true labels. The Natural Gradient updates can then be written as(see appendix \[sec:ng\] for more information): $$\label{eq:ng-m}
{\mathbf{w}}^{(t+1)} = {\mathbf{w}}^{(t)} - \eta F({\mathbf{w}}^{(t)})^{-1} \nabla_{\mathbf{w}}L({\mathbf{w}}^{(t)}).$$
If we approximate the Fisher information matrix with its diagonal elements, the update step normalizes each dimension of the gradient with the corresponding element on the diagonal of the Fisher information matrix: $$\label{eq:dng}
w^{(t+1)}_{e} = w^{(t)}_{e} - \frac{\eta}{F({\mathbf{w}})[e,e]} \frac{\partial L}{\partial
w_{e}}({\mathbf{w}}^{(t)}).$$
Using diagonal approximation of Fisher information matrix to normalize the gradient values has been suggested before as a computationally tractable alternative to the full Natural Gradient [@lecun1998neural; @schaul2013no]. [@ollivier2015riemannian] also suggested a “quasi-diagonal" approximations that includes, in addition to the diagonal, also some non-diagonal terms corresponding to the relationship between the bias term and every other incoming weight into a unit.
For our Gaussian probability model, where $\log
q({\mathbf{y}}|{\mathbf{x}})=\frac{1}{2}\norm{{\mathbf{y}}-f_{\mathbf{w}}({\mathbf{x}})}^2+{\rm
const}$, the diagonal can be calculated as: $$\label{eq:diag}
F({\mathbf{w}})[e,e] = \E_{{\mathbf{x}}\sim
p({\mathbf{x}})}\left[\sum_{v'\in V_{\text{out}}} \left(\frac{\partial
f_{\mathbf{w}}({\mathbf{x}})[v']}{\partial w_e}\right)^2\right],$$ using . We next prove that this update is equivalent to DDP-SGD for a specific choice of $R_v$, namely the second moment.
The Diagonal Natural Gradient indicated in equations and is equivalent to DDP-SGD for $R_v=\E\left[{\mathbf{h}}_{N^{{\text{in}}}(v)}{\mathbf{h}}_{N^{{\text{in}}}(v)}^\top\right]$.
We calculate the scaling factor $\kappa_{u\rightarrow v}({\mathbf{w}})$ for DDP-SGD as follows: $$\begin{aligned}
\kappa_{u\rightarrow v}({\mathbf{w}})&=\frac{1}{2}\frac{\partial^2 \gamma^2_{\rm net}}{\partial w_{u\rightarrow v}^2}=\frac{1}{2}\sum_{v'\in V_{{\text{out}}}}\frac{\partial^2 \E[z^2_{v'}]}{\partial w_{u\rightarrow v}^2} = \sum_{v'\in V_{{\text{out}}}} \frac{\partial}{\partial w_{u\rightarrow v}}\left(\frac{1}{2}\frac{\partial \E[z^2_{v'}]}{\partial w_{u\rightarrow v}}\right)\\
&= \sum_{v'\in V_{{\text{out}}}}\frac{\partial}{\partial w_{u\rightarrow v}}\left(\E\left[ z_{v'} \frac{\partial z_{v'}}{\partial w_{u\rightarrow v}}\right]\right)
= \sum_{v'\in V_{{\text{out}}}}\frac{\partial}{\partial w_{u\rightarrow v}}\left(\E\left[ z_{v'} h_u\frac{\partial z_{v'}}{\partial z_{v}}\right]\right)\\
&= \sum_{v'\in V_{{\text{out}}}}\E\left[ h^2_u\left(\frac{\partial z_{v'}}{\partial z_{v}}\right)^2\right] = \E\left[h^2_u \sum_{v'\in V_{{\text{out}}}} \left(\frac{\partial z_{v'}}{\partial z_{v}}\right)^2\right]\\
&=\E\left[\sum_{v'\in V_{\text{out}}} \left(\frac{\partial
f_{\mathbf{w}}({\mathbf{x}})[v']}{\partial w_e}\right)^2\right] = F({\mathbf{w}})[u\rightarrow v,u\rightarrow v]\\\end{aligned}$$ Therefore, the scaling factors in DDP-SGD with $R_v=\E\left[{\mathbf{h}}_{N^{{\text{in}}}(v)}{\mathbf{h}}_{N^{{\text{in}}}(v)}^\top\right]$ are exactly the diagonal elements of the Fisher Information matrix used in the Natural Gradient updates.
Node-wise invariance {#sec:node-rescaling}
====================
In this section, we show that DDP-SGD is invariant to node-wise rescalings (see Section \[sec:ff\]), while DDP-Normalization does not have favorable invariance properties.
DDP-SGD on feedforward networks
-------------------------------
In Section \[sec:ff\], we observed that feedforward ReLU networks are invariant to node-wise rescaling. To see if DDP-SGD is also invariant to such rescaling, consider a rescaled ${\mathbf{w}}'=T({\mathbf{w}})$, where $T$ is a rescaling by $\rho$ at node $v$ as in . Let ${\mathbf{w}}^+$ denote the weights after a step of DDP-SGD. To establish invariance to node-rescaling we need to show that ${\mathbf{w}}'^+=T({\mathbf{w}}^+)$. For the outgoing weights from $v$ we have: $$\begin{aligned}
w'^{+}_{v\rightarrow j} &= \rho w_{v\rightarrow j} -
\frac{\rho^2 \eta}{\kappa_{v\rightarrow j}({\mathbf{w}})}\frac{\partial
L}{\rho \partial w_{v\rightarrow j}}({\mathbf{w}})\\
&=\rho\left(w_{v\rightarrow j}- \frac{\eta}{\kappa_{v\rightarrow
j}({\mathbf{w}})}\frac{\partial L}{\partial w_{v\rightarrow j}}({\mathbf{w}})\right)
= \rho w^{+}_{v\rightarrow j}\\\end{aligned}$$ Similar calculations can be done for incoming weights to the node $v$. The only difference is that $\rho$ will be substituted by $1/\rho$. Moreover, note that due to non-negative homogeneity of ReLU activation function, the updates for the rest of the weights remain exactly the same. Therefore, DDP-SGD is node-wise rescaling invariant.
SGD on DDP-Normalized networks
------------------------------
Since DDP-Normalized networks are reparametrization of feedforward networks, their invariances are different. Since the operations in DDP-Normalized networks are based on $\tilde{w}$, we should study the invariances for $\tilde{w}$. The invariances in this case are given by rescaling of incoming weights into a node, i.e. for an internal node $v$ and scaling $\rho>0$: $$T(\tilde{w})_{k\rightarrow v} =\rho \tilde{w}_{k\rightarrow v}\quad
(\forall k\in N^{\text{in}}(v))\notag$$ while all other weights are unchanged. The DDP-Normalized networks are invariant to the above transformation because the output of each node is normalized. The SGD update rule is however not invariant to this transformation: $$\begin{aligned}
T(\tilde{w})^{+}_{k\rightarrow v} &= \rho \tilde{w}_{k\rightarrow
v}-\eta\frac{\partial L}{\rho \partial \tilde{w}_{k\rightarrow
v}}(\tilde{{\mathbf{w}}})
\neq \rho\left(\tilde{w}_{k\rightarrow v}-\eta \frac{\partial L}{\partial
\tilde{w}_{k\rightarrow v}}(\tilde{{\mathbf{w}}})\right) = \rho
\tilde{w}^{+}_{k\rightarrow v}\\\end{aligned}$$
Understanding Invariances {#sec:rescaling}
=========================
The goal of this section is to discuss whether being invariant to node-wise rescaling transformations is sufficient or not.
Ideally we would like our algorithm to be at least invariant to all the transformations to which the model $G$ is invariant. Note that this is different than the invariances studied in [@ollivier2015riemannian], in that they study algorithms that are invariant to reparametrizations of the same model but we look at transformations within the the parameter space that preserve the function in the model. This will eliminate the need for non-trivial initialization. Thus our goal is to characterize the whole variety of transformations to which the model is invariant and check if the algorithm is invariant to all of them.
We first need to note that invariance can be composed. If a network $G$ is invariant to transformations $T_1$ and $T_2$, it is also invariant to their composition $T_1\circ T_2$. This is also true for an algorithm. If an algorithm is invariant to transformations $T_1$ and $T_2$, it is also invariant to their composition. This is because $f_{T_2\circ T_1\circ
\mathcal{A}({{{\mathbf{w}}}})}=f_{T_2\circ \mathcal{A}(T_1\circ
{{{\mathbf{w}}}})}=f_{\mathcal{A}(T_2\circ T_1({{{\mathbf{w}}}}))}$.
Then it is natural to talk about the [*basis*]{} of invariances. The intuition is that although there are infinitely many transformations to which the model (or an algorithm) is invariant, they could be generated as compositions of finite number of transformations.
In fact, in the infinitesimal limit the directions of infinitesimal changes in the parameters to which the function $f_{{{{\mathbf{w}}}}}$ is insensitive form a subspace. This is because for a fixed input ${{{\mathbf{x}}}}$, we have $$\begin{aligned}
\label{eq:taylor-expansion}
f_{{{{\mathbf{w}}}}+{\boldsymbol{\Delta}}}(x) = f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}}) + \sum\nolimits_{e\in E}\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}{\partial w_{e}}\cdot \Delta_{e} + O(\|{\boldsymbol{\Delta}}\|^2),\end{aligned}$$ where $E$ is the set of edges, due to a Taylor expansion around ${{{\mathbf{w}}}}$. Thus the function $f_{{{{\mathbf{w}}}}}$ is insensitive (up to $O(\|{\boldsymbol{\Delta}}\|^2)$) to any change in the direction ${\boldsymbol{\Delta}}$ that lies in the (right) null space of the Jacobian matrix $\partial
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})/\partial {{{\mathbf{w}}}}$ for all input ${{{\mathbf{x}}}}$ simultaneously. More formally, the subspace can be defined as $$\begin{aligned}
\label{eq:defN}
N({{{\mathbf{w}}}}) = \bigcap\nolimits_{{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|}}\textrm{Null}\left(\frac{\partial
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}{\partial {{{\mathbf{w}}}}}\right).\end{aligned}$$ Again, any change to ${{{\mathbf{w}}}}$ in the direction ${\boldsymbol{\Delta}}$ that lies in $N({{{\mathbf{w}}}})$ leaves the function $f_{{{{\mathbf{w}}}}}$ unchanged (up to $O(\|{\boldsymbol{\Delta}}\|^2)$) at [*every*]{} input $x$. Therefore, if we can calculate the dimension of $N({{{\mathbf{w}}}})$ and if we have ${\rm dim}N({{{\mathbf{w}}}})= |V_{\rm internal}|$, where we denote the number of internal nodes by $|V_{\rm internal}|$, then we can conclude that all infinitesimal transformations to which the model is invariant can be spanned by infinitesimal node-wise rescaling transformations.
Note that the null space $N({{{\mathbf{w}}}})$ and its dimension is a function of ${{{\mathbf{w}}}}$. Therefore, there are some points in the parameter space that have more invariances than other points. For example, suppose that $v$ is an internal node with ReLU activation that receives connections only from other ReLU units (or any unit whose output is nonnegative). If all the incoming weights to $v$ are negative including the bias, the output of node $v$ will be zero regardless of the input, and the function $f_{{{{\mathbf{w}}}}}$ will be insensitive to any transformation to the outgoing weights of $v$. Nevertheless we conjecture that as the network size grows, the chance of being in such a degenerate configuration during training will diminish exponentially.
When we study the dimension of $N({{{\mathbf{w}}}})$, it is convenient to analyze the dimension of the span of the row vectors of the Jacobian matrix $\partial
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})/\partial {{{\mathbf{w}}}}$ instead. We define the degrees of freedom of model $G$ at ${{{\mathbf{w}}}}$ as $$\begin{aligned}
\label{eq:dof}
d_G({{{\mathbf{w}}}}) = {\rm dim}\left(\bigcup\nolimits_{{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|}}{\rm Span}\left(
\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}{\partial {{{\mathbf{w}}}}}[v,:] : v\in V_{\rm out}
\right)\right),
\end{aligned}$$ where $\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v,:]/\partial {{{\mathbf{w}}}}$ denotes the $v$th row vector of the Jacobian matrix and ${{{\mathbf{x}}}}$ runs over all possible input ${{{\mathbf{x}}}}$. Intuitively, $d_{G}({{{\mathbf{w}}}})$ is the dimension of the set of directions that changes $f_{{{{\mathbf{w}}}}}(x)$ for at least one input $x$.
Due to the rank nullity theorem $d_G({{{\mathbf{w}}}})$ and the dimension of $N({{{\mathbf{w}}}})$ are related as follows: $$\begin{aligned}
d_G({{{\mathbf{w}}}}) + {\rm dim}\left(N({{{\mathbf{w}}}})\right)=|E| ,\end{aligned}$$ where $|E|$ is the number of parameters. Therefore, again if $d_G({{{\mathbf{w}}}})=|E| -
|V_{\rm internal}|$, then we can conclude that infinitesimally speaking, all transformations to which the model is invariant can be spanned by node-wise rescaling transformations.
Considering only invariances that hold uniformly over all input ${{{\mathbf{x}}}}$ could give an under-estimate of the class of invariances, i.e., there might be some invariances that hold for many input ${{{\mathbf{x}}}}$ but not all. An alternative approach for characterizing invariances is to define a measure of distance between functions that the neural network model represents based on the input distribution, and infinitesimally study the subspace of directions to which the distance is insensitive. We can define distance between two functions $f$ and $g$ as $$\begin{aligned}
D(f,g) = {\mathbb{E}}_{{{{\mathbf{x}}}}\sim \mathcal{D}}\left[m(f({{{\mathbf{x}}}}),g({{{\mathbf{x}}}}))\right],\end{aligned}$$ where $m:{\mathbb{R}}^{|V_{\rm out}|\times |V_{\rm out}|}\rightarrow {\mathbb{R}}$ is a (possibly asymmetric) distance measure between two vectors ${{{\mathbf{z}}}},{{{\mathbf{z}}}}'\in{\mathbb{R}}^{|V_{\rm out}|}$, which we require that $m({{{\mathbf{z}}}},{{{\mathbf{z}}}})=0$ and $\partial m/\partial {{{\mathbf{z}}}}'_{{{{\mathbf{z}}}}={{{\mathbf{z}}}}'}=0$. For example, $m({{{\mathbf{z}}}},{{{\mathbf{z}}}}')=\|{{{\mathbf{z}}}}-{{{\mathbf{z}}}}'\|^2$.
The second-order Taylor expansion of the distance $D$ can be written as $$\begin{aligned}
D(f_{{{{\mathbf{w}}}}}\| f_{{{{\mathbf{w}}}}+{\boldsymbol{\Delta}}}) &=\frac{1}{2}
{\boldsymbol{\Delta}}{^\top}\cdot F({{{\mathbf{w}}}})\cdot
{\boldsymbol{\Delta}}+o(\|{\boldsymbol{\Delta}}\|^2),\end{aligned}$$ where $$\begin{aligned}
F({{{\mathbf{w}}}})&={\mathbb{E}}_{{{{\mathbf{x}}}}\sim \mathcal{D}}\left[\left(\frac{\partial
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}{\partial {{{\mathbf{w}}}}}\right){^\top}\cdot\left.\frac{\partial^2
m({{{\mathbf{z}}}},{{{\mathbf{z}}}}')}{\partial {{{\mathbf{z}}}}'^2}\right|_{{{{\mathbf{z}}}}={{{\mathbf{z}}}}'=f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}
\cdot\left(\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}{\partial {{{\mathbf{w}}}}}\right)\right]
\end{aligned}$$ and $\partial^2 m({{{\mathbf{z}}}},{{{\mathbf{z}}}}')/\partial {{{\mathbf{z}}}}'^2|_{{{{\mathbf{z}}}}={{{\mathbf{z}}}}'=f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})}$ is the Hessian of the distance measure $m$ at ${{{\mathbf{z}}}}={{{\mathbf{z}}}}'=f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})$.
Using the above expression, we can define the input distribution dependent version of $N({{{\mathbf{w}}}})$ and $d_{G}({{{\mathbf{w}}}})$ as $$\begin{aligned}
N_{\mathcal{D}}({{{\mathbf{w}}}}) = {\rm Null} F({{{\mathbf{w}}}}),\qquad
d_{G,\mathcal{D}}({{{\mathbf{w}}}}) = {\rm rank} F({{{\mathbf{w}}}}).
\end{aligned}$$ Again due to the rank-nullity theorem we have $d_{G,\mathcal{D}}({{{\mathbf{w}}}})+{\rm dim}(N_{\mathcal{D}}({{{\mathbf{w}}}}))=|E|$.
As a special case, we obtain the Kullback-Leibler divergence $D_{\rm KL}$, which is commonly considered as [*the*]{} way to study invariances, by choosing $m$ as the conditional Kullback-Leibler divergence of output $y$ given the network output as $$\begin{aligned}
m({{{\mathbf{z}}}},{{{\mathbf{z}}}}') = {\mathbb{E}}_{y\sim q(y|{{{\mathbf{z}}}})}\left[\log\frac{q(y|{{{\mathbf{z}}}})}{q(y|{{{\mathbf{z}}}}')}\right],\end{aligned}$$ where $q(y|{{{\mathbf{z}}}})$ is a link function, which can be, e.g., the soft-max $q(y|{{{\mathbf{z}}}})=e^{z_y}/\sum_{y'=1}^{|V_{\rm out}|}e^{z_{y'}}$. However, note that the invariances in terms of $D_{\rm KL}$ depends not only on the input distribution but also on the choice of the link function $q(y|{{{\mathbf{z}}}})$.
Path-based characterization of the network {#sec:path-network}
------------------------------------------
A major challenge in studying the degrees of freedom is the fact that the Jacobian $\partial f_{{{{\mathbf{w}}}}}(x)/\partial {{{\mathbf{w}}}}$ depends on both parameter $w$ and input $x$. In this section, we first tease apart the two dependencies by rewriting $f_{{{{\mathbf{w}}}}}(x)$ as the sum over all directed paths from every input node to each output node as follows: $$\begin{aligned}
\label{eq:f-as-sum-over-paths}
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v] &= \sum\nolimits_{p\in\Pi(v)}g_{p}({{{\mathbf{x}}}})\cdot\pi_p({{{\mathbf{w}}}})\cdot x[{\rm
head}(p)],
\end{aligned}$$ where $\Pi(v)$ is the set of all directed path from any input node to $v$, ${\rm head}(p)$ is the first node of path $p$, $g_{p}({{{\mathbf{x}}}})$ takes 1 if all the rectified linear units along path $p$ is active and zero otherwise, and $\pi_p({{{\mathbf{w}}}})=\prod_{e\in E(p)} w(e)$ is the product of the weights along path $p$; $E(p)$ denotes the set of edges that appear along path $p$.
Let $\Pi=\cup_{v\in V_{\rm out}}\Pi(v)$ be the set of all directed paths. We define the path-Jacobian matrix $J({{{\mathbf{w}}}})\in{\mathbb{R}}^{|\Pi|\times |E|}$ as $J({{{\mathbf{w}}}})=(\partial \pi_p({{{\mathbf{w}}}})/\partial w_e)_{p\in \Pi, e\in E}$. In addition, we define ${\boldsymbol{\phi}}({{{\mathbf{x}}}})$ as a $|\Pi|$ dimensional vector with $g_p({{{\mathbf{x}}}})\cdot x[{\rm
head}(p)]$ in the corresponding entry. The Jacobian of the network $f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})$ can now be expressed as $$\begin{aligned}
\label{eq:jacobian}
\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v]}{\partial {{{\mathbf{w}}}}}&= J_v({{{\mathbf{w}}}}){^\top}{\boldsymbol{\phi}}_v({{{\mathbf{x}}}}),
\end{aligned}$$ where where $J_v({{{\mathbf{w}}}})$ and ${\boldsymbol{\phi}}_v({{{\mathbf{x}}}})$ are the submatrix (or subvector) of $J({{{\mathbf{w}}}})$ and ${\boldsymbol{\phi}}({{{\mathbf{x}}}}) $ that corresponds to output node $v$, respectively[^1]. Expression clearly separates the dependence to the parameters ${{{\mathbf{w}}}}$ and input ${{{\mathbf{x}}}}$.
Now we have the following statement (the proof is given in Appendix \[sec:proofs\]).
\[thm:dof\] The degrees-of-freedom $d_{G}({{{\mathbf{w}}}})$ of neural network model $G$ is at most the rank of the path Jacobian matrix $J({{{\mathbf{w}}}})$. The equality holds if ${\rm dim}\left({\rm
Span}({\boldsymbol{\phi}}({{{\mathbf{x}}}}):{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|})\right)=|\Pi|$; i.e. when the dimension of the space spanned by ${\boldsymbol{\phi}}({{{\mathbf{x}}}})$ equals the total number of paths $|\Pi|$.
An analogous statement holds for the input distribution dependent degrees of freedom $d_{G,\mathcal{D}}({{{\mathbf{w}}}})$, namely, $d_{G,\mathcal{D}}({{{\mathbf{w}}}})\leq {\rm rank} J({{{\mathbf{w}}}})$ and the equality holds if the rank of the $|\Pi|\times |\Pi|$ path covariance matrix $({\mathbb{E}}_{{{{\mathbf{x}}}}\sim\mathcal{D}}\left[
\partial^2 m({{{\mathbf{z}}}},{{{\mathbf{z}}}}')/\partial z'_v\partial z'_{v'}\phi_{p}({{{\mathbf{x}}}})\phi_{p'}({{{\mathbf{x}}}})
\right])_{p,p'\in \Pi}$ is full, where $v$ and $v'$ are the end nodes of paths $p$ and $p'$, respectively.
It remains to be understood when the dimension of the span of the path vectors ${\boldsymbol{\phi}}({{{\mathbf{x}}}})$ become full. The answer depends on ${{{\mathbf{w}}}}$. Unfortunately, there is no typical behavior as we know from the example of an internal ReLU unit connected to ReLU units by negative weights. In fact, we can choose any number of internal units in the network to be in this degenerate state creating different degrees of degeneracy. Another way to introduce degeneracy is to insert a linear layer in the network. This will superficially increase the number of paths but will not increase the dimension of the span of ${\boldsymbol{\phi}}({{{\mathbf{x}}}})$. For example, consider a linear classifier $z_{\rm out}=\inner{{{{\mathbf{w}}}}}{{{{\mathbf{x}}}}}$ with $|V_{\rm in}|$ inputs. If the whole input space is spanned by ${{{\mathbf{x}}}}$, the dimension of the span of ${\boldsymbol{\phi}}({{{\mathbf{x}}}})$ is $|V_{\rm in}|$, which agrees with the number of paths. Now let’s insert a linear layer with units $V_1$ in between the input and the output layers. The number of paths has increased from $|V_{\rm in}|$ to $|V_{\rm
in}|\cdot|V_1|$. However the dimension of the span of ${\boldsymbol{\phi}}({{{\mathbf{x}}}})={{\mathbf{1}}}_{|V_1|}\otimes {{{\mathbf{x}}}}$ is still $|V_{\rm in}|$, because the linear units are always active. Nevertheless we conjecture that there is a configuration ${{{\mathbf{w}}}}$ such that ${\rm dim}\left({\rm
Span}({\boldsymbol{\phi}}({{{\mathbf{x}}}}):{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|})\right)=|\Pi|$ and the set of such ${{{\mathbf{w}}}}$ grows as the network becomes larger.
Combinatorial characterization of the rank of path Jacobian
-----------------------------------------------------------
Finally, we show that the rank of the path-Jacobian matrix $J({{{\mathbf{w}}}})$ is determined purely combinatorially by the graph $G$ except a subset of the parameter space with zero Lebesgue measure. The proof is given in Appendix \[sec:proofs\].
\[thm:rank-path-jacobian\] The rank of the path Jacobian matrix $J({{{\mathbf{w}}}})$ is generically (excluding set of parameters with zero Lebesgue measure) equal to the number of parameters $|E|$ minus the number of internal nodes of the network.
Note that the dimension of the space spanned by node-wise rescaling equals the number of internal nodes. Therefore, node-wise rescaling is the [*only*]{} type of invariance for a ReLU network with fixed architecture $G$, if ${\rm dim}\left({\rm
Span}(\phi({{{\mathbf{x}}}}):{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|})\right)=|\Pi|$ at parameter ${{{\mathbf{w}}}}$.
As an example, let us consider a simple 3 layer network with 2 nodes in each layer except for the output layer, which has only 1 node (see Figure \[fig:net2221\]). The network has 10 parameters (4, 4, and 2 in each layer respectively) and 8 paths. The Jacobian $(\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})/\partial {{{\mathbf{w}}}})$ can be written as $(\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})/\partial {{{\mathbf{w}}}})
= J({{{\mathbf{w}}}}){^\top}\cdot {\boldsymbol{\phi}}({{{\mathbf{x}}}})$, where $$\begin{aligned}
\label{eq:J-2221}
J({{{\mathbf{w}}}}) &=
\left[
\begin{array}{c|c|c}
\begin{array}{cccc}
w_5 w_9 & & & \\
& w_5w_9 & & \\
& & w_6w_9 & \\
& & & w_6w_9 \\
\hline
w_7w_{10} & & & \\
& w_7w_{10} & & \\
& & w_8w_{10} & \\
& & & w_8w_{10} \\
\end{array}
&
\begin{array}{cccc}
w_9w_1 & & & \\
w_9w_2 & & & \\
& w_9w_3 & & \\
& w_9w_4 & &\\
\hline
& & w_{10}w_1 & \\
& & w_{10}w_2 & \\
& & & w_{10}w_3 \\
& & & w_{10}w_4
\end{array}
& \begin{array}{cc}
w_5w_1 & \\
w_5w_2 & \\
w_6w_3 & \\
w_6w_4 & \\
\hline
& w_7w_1 \\
& w_7w_2 \\
& w_8w_3 \\
& w_8w_4
\end{array}
\end{array}
\right]
\intertext{and}
\phi({{{\mathbf{x}}}})^\top&=
\begin{bmatrix}
g_1({{{\mathbf{x}}}})x[1] & g_2({{{\mathbf{x}}}})x[2] & g_3({{{\mathbf{x}}}})x[1] & g_4({{{\mathbf{x}}}})x[2] & g_5({{{\mathbf{x}}}})x[1] &
g_6({{{\mathbf{x}}}})x[2]& g_7({{{\mathbf{x}}}})x[1] & g_8({{{\mathbf{x}}}})x[2]
\end{bmatrix}.\notag\end{aligned}$$ The rank of $J({{{\mathbf{w}}}})$ in is (generically) equal to $10-4=6$, which is smaller than both the number of parameters and the number of paths.
![A 3 layer network with 10 parameters and 8 paths.[]{data-label="fig:net2221"}](net2221.pdf){width=".28\textwidth"}
Conclusion and Future Work
==========================
We proposed a unified framework as a complexity measure or regularizer for neural networks and discussed normalization and optimization with respect to this regularizer. We further showed how this measure interpolates between data-dependent and data-independent regularizers and discussed how Path-SGD and Batch-Normalization are special cases of optimization with respect to this measure. We also looked at the issue of invariances and brought new insights to this area.
Implementation
==============
DDP-Normalization {#sec:BN-imp}
-----------------
Given any batch of $n$ data points to estimate mean, variance and the gradient, the stochastic gradients for the weight $\tilde{{\mathbf{w}}}$ (weights in the DDP-Normalized network) can then be calculated through the chain rule: $$\begin{aligned}
\label{eq:dl}
\frac{\partial L}{\partial \tilde{{\mathbf{w}}}_{\rightarrow v}}
&=\frac{1}{n\tilde{\gamma}_v}\sum_{i=1}^{n} \frac{\partial L}{\partial
z_{v}^{(i)}}\left[ {{\mathbf{h}}}_{N^{{\text{in}}}(v)}^{(i)} - \frac{1}{n}
\sum_{j=1}^n {{\mathbf{h}}}_{N^{{\text{in}}}(v)}^{(j)} -\frac{
\hat{z}^{(i)}_v }{
2\tilde{\gamma}_v^2}\frac{\partial \tilde{\gamma}^2_v}{\partial \tilde{{\mathbf{w}}}_{\rightarrow
v}} \right]\\
\frac{\partial L}{\partial z^{(i)}_{u}} &= \frac{1}{\tilde{\gamma}_v}
\left[\sum_{v \in N^{{\text{out}}}(u)} \tilde{w}_{u\rightarrow
v}\left(\frac{\partial L}{\partial z_{v}^{(i)}} -
\frac{1}{n}\sum_{j=1}^n\frac{\partial L}{\partial
z_{v}^{(j)}}\left(1-\alpha\frac{\hat{z}^{(i)}_v\hat{z}^{(j)}_v}{\tilde{\gamma}^2_v}\right)\right)\right]_{z^{(i)}_{u}\geq 0}\\ \notag\end{aligned}$$ where $\hat{z}^{(i)}_{v} = \tilde{z}^{(i)}_{v} - \frac{1}{n}
\sum_{j=1}^n \tilde{z}^{(j)}_{v}$ and we have: $$\label{eq:dgam}
\frac{\partial \tilde{\gamma}^2_v}{\partial \tilde{{\mathbf{w}}}_{\rightarrow
v}}=2(1-\alpha)\tilde{{\mathbf{w}}}_{\rightarrow v} +
\frac{2\alpha}{n}\sum_{i=1}^n \hat{z}^{(i)}_v \left({{\mathbf{h}}}_{N^{{\text{in}}}(v)}^{(i)} - \frac{1}{n}
\sum_{j=1}^n {{\mathbf{h}}}_{N^{{\text{in}}}(v)}^{(j)}\right)$$ Similar to Batch-Normalization, all the above calculations can be efficiently carried out as vector operations with negligible extra memory and computations.
DDP-SGD {#sec:ddp-imp}
-------
In order to compute the second derivatives $\kappa_e({\mathbf{w}})=\frac{\partial^2 \gamma^2_{\rm net}}{\partial w^2_e}$, we first calculate the first derivative. The backpropagation can be done through $\gamma^2_u$ and $z^{(i)}_{u}$ but this makes it difficult to find the second derivatives. Instead we propagate the loss through $\gamma^2_u$ and the second order terms of the form $z^{(i)}_{u_1}z^{(i)}_{u_2}$: $$\begin{aligned}
\frac{\partial \gamma^2_{{\rm net}}}{\partial \gamma^2_u} &=
(1-\alpha)\sum_{v\in N^{\text{out}}(u)} \frac{\partial \gamma^2_{{\rm
net}}}{\partial \gamma^2_v} w^2_{u\rightarrow v}\end{aligned}$$ $$\frac{\partial \gamma^2_{{\rm net}}}{\partial
(z_{u_1}^{(i)}z_{u_2}^{(i)})} = \alpha \left[\frac{ \partial
\gamma^2_{\rm net}}{\partial \gamma^2_{u_1}}\right]_{u_1=u_2} +
\left[\sum_{(v_1,v_2)\in \left(N^{\text{out}}(u_1)\right)^2} \frac{\partial
\gamma^2_{\rm net}}{\partial
(z_{v_1}^{(i)}z_{v_2}^{(i)})}w_{u_1\rightarrow v_1} w_{u_2
\rightarrow v_2} \right]_{z^{(i)}_{u_1}>0, z^{(i)}_{u_2}>0}$$ Now we can calculate the partials for $w_{u\rightarrow v}$ as follows: $$\frac{\partial \gamma^2_{\rm net}}{\partial w_{u\rightarrow v}} =
2(1-\alpha)\frac{\partial \gamma^2_{\rm net}}{\partial \gamma^2_v}
\gamma^2_u w_{u\rightarrow v} + 2\sum_{i=1}^n\sum_{v'\in N^{\text{out}}(u)}
\frac{\partial \gamma^2_{\rm net}}{\partial (z^{(i)}_v z^{(i)}_{v'})}
h_{u}^{(i)}z_{v'}^{(i)}$$ Since the partials $\frac{\partial \gamma^2_{{\rm net}}}{\partial
\gamma^2_u}$ and $\frac{\partial \gamma^2_{{\rm net}}}{\partial
(z_{u_1}^{(i)}z_{u_2}^{(i)})}$ do not depend on $w_{u\rightarrow v}$, the second order derivative can be calculated directly: $$\kappa_{u\rightarrow v}({\mathbf{w}})=\frac{1}{2}\frac{\partial^2 \gamma^2_{\rm
net}}{\partial w_{u\rightarrow v}^2} = (1-\alpha)\frac{\partial
\gamma^2_{\rm net}}{\partial \gamma^2_v} \gamma^2_u +
\sum_{i=1}^n\frac{\partial \gamma^2_{\rm net}}{\partial
\left({z^{(i)}_v}^2\right)}\left(h^{(i)}_u\right)^2$$
Natural Gradient {#sec:ng}
================
The natural gradient algorithm [@Ama98] achieves invariance by applying the inverse of the Fisher information matrix $F({{{\mathbf{w}}}}^{(t)})$ at the current parameter ${{{\mathbf{w}}}}^{(t)}$ to the negative gradient direction as follows: $$\begin{aligned}
{{{\mathbf{w}}}}^{(t+1)} &= {{{\mathbf{w}}}}^{(t)} + \eta{\boldsymbol{\Delta}}^{(\rm natural)},\notag
\intertext{where}
\label{eq:ng-argmin}
\Delta^{(\rm natural)} &=
\argmin{\Delta\in{\mathbb{R}}^{|E|}}
\inner{-\frac{\partial L}{\partial w}({{{\mathbf{w}}}}^{(t)})}{{\boldsymbol{\Delta}}}
,\quad {\rm s.t.}\quad {\boldsymbol{\Delta}}{^\top}F({{{\mathbf{w}}}}^{(t)}){\boldsymbol{\Delta}}\leq \delta^2\\
\label{eq:ng}
&=-F^{-1}({{{\mathbf{w}}}}^{(t)})\frac{\partial L}{\partial w}({{{\mathbf{w}}}}^{(t)}).
\end{aligned}$$ Here $F({{{\mathbf{w}}}})$ is the Fisher information matrix at point ${{{\mathbf{w}}}}$ and is defined with respect to the probabilistic view of the feedforward neural network model, which we describe in more detail below.
Suppose that we are solving a classification problem and the final layer of the network is fed into a softmax layer that determines the probability of candidate classes given the input $x$. Then the neural network with the softmax layer can be viewed as a conditional probability distribution $$\begin{aligned}
\label{eq:cond-prob}
q(y|{{{\mathbf{x}}}})= \frac{\exp(f_{{\mathbf{w}}}({{{\mathbf{x}}}})[v_y])}{\sum_{v\in V_{\rm
out}}\exp(f_{{\mathbf{w}}}({{{\mathbf{x}}}})[v])},
\end{aligned}$$ where $v_y$ is the output node corresponding to class $y$. If we are solving a regression problem a Gaussian distribution is probably more appropriate for $q(y|{{{\mathbf{x}}}})$.
Given the conditional probability distribution $q(y|{{{\mathbf{x}}}})$, the Fisher information matrix can be defined as follows: $$\begin{aligned}
\label{eq:fisher-information}
F({{{\mathbf{w}}}})[e,e'] = {\mathbb{E}}_{{{{\mathbf{x}}}}\sim p({\mathbf{x}})}{\mathbb{E}}_{y\sim q(y|{{{\mathbf{x}}}})}\left[
\frac{\partial \log q(y|{{{\mathbf{x}}}})}{\partial w_e}
\frac{\partial \log q(y|{{{\mathbf{x}}}})}{\partial w_{e'}}
\right],
\end{aligned}$$ where $p(x)$ is the marginal distribution of the data.
Since we have $$\begin{aligned}
\label{eq:partial-logq}
\frac{\partial \log q(y|{{{\mathbf{x}}}})}{\partial w_{u\rightarrow v}}
=\frac{\partial \log q(y|{{{\mathbf{x}}}})}{\partial z_v}\cdot h_u
=\sum_{v'\in V_{\rm out}}
\frac{\partial \log q(y|{{{\mathbf{x}}}})}{\partial z_{v'}}\cdot
\frac{\partial z_{v'}}{\partial z_{v}}\cdot h_u
\end{aligned}$$ using the chain rule, each entry of the Fisher information matrix can be computed efficiently by forward and backward propagations on a minibatch.
Proofs {#sec:proofs}
======
First note that we can calculate the following inner product using equation : $$\begin{aligned}
\inner{\tilde{{\mathbf{w}}}_{\rightarrow v}}{\frac{ \partial \tilde{\gamma}_v^2}{ \partial \tilde{{\mathbf{w}}}_{\rightarrow v}}}
&= 2(1-\alpha)\norm{ \tilde{{\mathbf{w}}}_{\rightarrow v} }_2^2 +
\frac{2\alpha}{n}\sum_{i=1}^n (\hat{z}^{(i)})^2 \\
&= 2(1-\alpha)\norm{ \tilde{{\mathbf{w}}}_{\rightarrow v} }_2^2 +
2\alpha {\text{Var}}(\tilde{z}_v) = 2\tilde{\gamma}_v^2\end{aligned}$$ Next, by equation we get: $$\begin{aligned}
\inner{ \tilde{{\mathbf{w}}}_{\rightarrow v}}{ \frac{\partial L}{\partial \tilde{{\mathbf{w}}}_{\rightarrow v}} }
&=\frac{1}{n\tilde{\gamma}_v}\sum_{i=1}^{n} \frac{\partial L}{\partial
z_{v}^{(i)}}\left[ \hat{z}^{(i)}_v -\frac{
\hat{z}^{(i)}_v }{
2\tilde{\gamma}_v^2}\inner{\tilde{{\mathbf{w}}}_{\rightarrow v} }{ \frac{\partial \tilde{\gamma}^2_v}{\partial \tilde{{\mathbf{w}}}_{\rightarrow
v}}} \right]\\
&= \frac{1}{n\tilde{\gamma}_v}\sum_{i=1}^{n} \frac{\partial L}{\partial
z_{v}^{(i)}}\left[ \hat{z}^{(i)}_v -\frac{
\hat{z}^{(i)}_v }{
2\tilde{\gamma}_v^2} 2 \tilde{\gamma}^2_v \right] = 0\end{aligned}$$
First we see that is true because $$\begin{aligned}
\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v]}{\partial {{{\mathbf{w}}}}} =\Bigl(\sum_{p\in\Pi(v)}
\frac{\partial \pi_p({{{\mathbf{w}}}})}{\partial w_e}
\cdot g_{p}({{{\mathbf{x}}}})\cdot x[{\rm head}(p)]\Bigr)_{e\in E}
= J_{v}({{{\mathbf{w}}}}){^\top}\cdot \phi_{v}({{{\mathbf{x}}}}).\end{aligned}$$ Therefore, $$\begin{aligned}
\bigcup_{{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|}} {\rm Span}\left(
\frac{\partial f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v]}{\partial {{{\mathbf{w}}}}}: v\in V_{\rm out}
\right)
&= \bigcup_{{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|}}{\rm Span}\left(
J_{v}({{{\mathbf{w}}}}){^\top}\cdot \phi_{v}({{{\mathbf{x}}}}): v\in V_{\rm out}
\right)\notag\\
\label{eq:span}
&=J({{{\mathbf{w}}}}){^\top}\cdot {\rm Span}\left(\phi({{{\mathbf{x}}}}): {{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|}\right).\end{aligned}$$ Consequently, any vector of the form $(\frac{\partial
f_{{{{\mathbf{w}}}}}({{{\mathbf{x}}}})[v]}{\partial w_e})_{e\in E}$ for a fixed input ${{{\mathbf{x}}}}$ lies in the span of the row vectors of the path Jacobian $J({{{\mathbf{x}}}})$.
The second part says $d_{G}({{{\mathbf{w}}}})={\rm rank}J({{{\mathbf{w}}}})$ if ${\rm
dim}\left({\rm Span}(\phi({{{\mathbf{x}}}}):{{{\mathbf{x}}}}\in{\mathbb{R}}^{|V_{\rm in}|})\right)=|\Pi|$, which is the number of rows of $J({{{\mathbf{w}}}})$. We can see that this is true from expression .
First, $J({{{\mathbf{w}}}})$ can be written as an Hadamard product between path incidence matrix $M$ and a rank-one matrix as follows: $$\begin{aligned}
J({{{\mathbf{w}}}}) &= M \circ \left({{{\mathbf{w}}}}^{-1} \cdot {\boldsymbol{\pi}}^\top({{{\mathbf{w}}}})\right),\end{aligned}$$ where $M$ is the path incidence matrix whose $i,j$ entry is one if the $i$th edge is part of the $j$th path, ${{{\mathbf{w}}}}^{-1}$ is an entry-wise inverse of the parameter vector ${{{\mathbf{w}}}}$, ${\boldsymbol{\pi}}({{{\mathbf{w}}}})=(\pi_p({{{\mathbf{w}}}}))$ is a vector containing the product along each path in each entry, and $\top$ denotes transpose.
Since we can rewrite $$\begin{aligned}
J({{{\mathbf{w}}}}) &= {\rm diag}({{{\mathbf{w}}}}^{-1})\cdot M \cdot{\rm diag}({\boldsymbol{\pi}}({{{\mathbf{w}}}})),\end{aligned}$$ we see that (generically) the rank of $J({{{\mathbf{w}}}})$ is equal to the rank of zero-one matrix $M$.
Note that the rank of $M$ is equal to the number of linearly independent columns of $M$, in other words, the number of linearly independent paths. In general, most paths are not independent. For example, in Figure \[fig:net2221\], we can see that the column corresponding to the path $w_2w_7w_{10}$ can be produced by combining 3 columns corresponding to paths $w_1w_5w_9$, $w_1w_7w_{10}$, and $w_2w_5w_9$.
In order to count the number of independent paths, we use mathematical induction. For simplicity, consider a layered graph with $d$ layers. All the edges from the $(d-1)$th layer nodes to the output layer nodes are linearly independent, because they correspond to different parameters. So far we have $n_dn_{d-1}$ independent paths.
Next, take one node $u_0$ (e.g., the leftmost node) from the $(d-2)$th layer. All the paths starting from this node through the layers above are linearly independent. However, other nodes in this layer only contributes linearly to the number of independent paths. This is the case because we can take an edge $(u,v)$, where $u$ is one of the remaining $n_{d-2}-1$ vertices in the $(d-2)$th layer and $v$ is one of the $n_{d-1}$ nodes in the $(d-1)$th layer, and we can take any path (say $p_0$) from there to the top layer. Then this is the only independent path that uses the edge $(u,v)$, because any other combination of edge $(u,v)$ and path $p$ from $v$ to the top layer can be produced as follows (see Figure \[fig:dependence\]): $$\begin{aligned}
(u,v)\rightarrow p = (u,v)\rightarrow p_0 - (u_0,v)\rightarrow p_0 +
(u_0,v)\rightarrow p.
\end{aligned}$$ Therefore after considering all nodes in the $d-2$th layer, we have $$\begin{aligned}
n_{d}n_{d-1} + n_{d-1}(n_{d-2}-1)
\end{aligned}$$ independent paths. Doing this calculation inductively, we have $$\begin{aligned}
n_{d}n_{d-1} + n_{d-1}(n_{d-2}-1) + \cdots + n_{1}(n_0-1)
\end{aligned}$$ independent paths, where $n_0$ is the number of input units. This number is clearly equal to the number of parameters ($n_dn_{d-1}+\cdots+ n_{1}n_0$) minus the number of internal nodes ($n_{d-1}+\cdots+n_1$).
![Schematic illustration of the linear dependence of the four paths $(u_0,v)\rightarrow p_0$, $(u_0,v)\rightarrow p$, $(u,v)\rightarrow p_0$, and $(u,v)\rightarrow p$. Because of this dependence, any additional edge $(u,v)$ only contributes one additional independent path.[]{data-label="fig:dependence"}](dependence.pdf){width="70.00000%"}
[^1]: Note that although path activation $g_p({{{\mathbf{x}}}})$ is a function of ${{{\mathbf{w}}}}$, it is insensitive to an infinitesimal change in the parameter, unless the input to one of the rectified linear activation functions along path $p$ is at exactly zero, which happens with probability zero. Thus we treat $g_p({{{\mathbf{x}}}})$ as constant here.
|
---
abstract: 'We present the results of targeted observations and a survey of 1612-, 1665-, and 1667-MHz circumstellar OH maser emission from asymptotic giant branch (AGB) stars and red supergiants (RSGs) in the Small Magellanic Cloud (SMC), using the Parkes and Australia Telescope Compact Array radio telescopes. No clear OH maser emission has been detected in any of our observations targeting luminous, long-period, large-amplitude variable stars, which have been confirmed spectroscopically and photometrically to be mid- to late-M spectral type. These observations have probed $3-4$ times deeper than any OH maser survey in the SMC. Using a bootstrapping method with LMC and Galactic OH/IR star samples and our SMC observation upper limits, we have calculated the likelihood of not detecting maser emission in any of the two sources considered to be the top maser candidates to be less than 0.05%, assuming a similar pumping mechanism as the LMC and Galactic OH/IR sources. We have performed a population comparison of the Magellanic Clouds and used *Spitzer* IRAC and MIPS photometry to confirm that we have observed all high luminosity SMC sources that are expected to exhibit maser emission. We suspect that, compared to the OH/IR stars in the Galaxy and LMC, the reduction in metallicity may curtail the dusty wind phase at the end of the evolution of the most massive cool stars. We also suspect that the conditions in the circumstellar envelope change beyond a simple scaling of abundances and wind speed with metallicity.'
author:
- |
Steven R. Goldman$^{1}$[^1], Jacco Th. van Loon$^{1}$, José F. Gómez$^{2}$, James A. Green$^{3,4}$, Albert A. Zijlstra$^{5}$, Ambra Nanni$^{6}$, Hiroshi Imai$^{7}$, Patricia A. Whitelock$^{8,9}$, Martin A. T. Groenewegen$^{10}$ and Joana M. Oliveira$^{1}$\
$^{1}$Astrophysics Group, Lennard-Jones Laboratories, Keele University, ST5 5BG, UK\
$^{2}$Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la Astronomía s/n, 18008 Granada, Spain\
$^{3}$SKA Organisation, Jodrell Bank Observatory, Lower Withington, Macclesfield, Cheshire, SK11 9DL, UK\
$^{4}$CSIRO Astronomy and Space Science, Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia\
$^{5}$Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester,\
Oxford Road, Manchester, M13 9PL, UK\
$^{6}$Dipartimento di Fisica e Astronomia Galileo Galilei, vicolo dell’Osservatorio 3, 35122 Padova PD, Italy\
$^{7}$Department of Physics and Astronomy, Kagoshima University, 1-21-35 Korimoto, Kagoshima 890-0065, Japan\
$^{8}$South African Astronomical Observatory (SAAO), PO Box 9, 7935 Observatory, South Africa\
$^{9}$Astronomy Department, University of Cape Town, 7701 Rondebosch, South Africa\
$^{10}$Koninklijke Sterrenwacht van België, Ringlaan 3, B-1180 Brussels, Belgium
bibliography:
- 'references.bib'
date: 'Accepted 2017 XXXXXXX XX. Received 2016 XXXXXXX XX'
title: 'A dearth of OH/IR stars in the Small Magellanic Cloud'
---
\[firstpage\]
masers – stars: AGB and post-AGB – supergiants – stars: mass-loss – stars: winds, outflows – Magellanic Clouds
Introduction
============
The Magellanic Clouds have considerably different Asymptotic Giant Branch (AGB) populations resulting from different star formation histories (SFHs). The Small Magellanic Cloud (SMC), with a stellar mass of $3.7 \times 10^8$ M$_{\odot}$ [@2012ApJ...761...42S], is nearly a quarter of the stellar mass of the Large Magellanic Cloud (LMC) at $1.7 \times 10^9$ M$_{\odot}$ [@2004AJ....127.1531H; @2009AJ....138.1243H]. Within the LMC, a peak in star formation around 100 Myr ago [@2009AJ....138.1243H] has resulted in favourable conditions for bright maser emitting evolved stars. Moreover, according to the recent work by @2017MNRAS.465.4817S based on the stellar evolutionary tracks computed with the <span style="font-variant:small-caps;">parsec</span> [@2012MNRAS.427..127B] and <span style="font-variant:small-caps;">colibri</span> [@2017ApJ...835...77M] codes, the minimum mass of a Hot-Bottom Burning (HBB) star, which provides an oxygen-rich environment favorable for maser-emission, is around 4.2 M$_{\odot}$ for Z=0.006; the exact value of the mass will depend on the metallicity and on the underlying stellar model. As a consequence, given the 100 Myr evolution time of $5-6$ M$_{\odot}$ stars, this has resulted in a disproportionally large population of sources undergoing HBB [@2015MNRAS.454.4235D; @2017MNRAS.465.4817S]. Conversely, the SFH of the SMC derived by @2004AJ....127.1531H shows a minimum in the star formation around 100 Myr ago, which has likely resulted in a smaller population of massive oxygen-rich AGB stars within the SMC. Even if the minimum in the star formation rate (SFR) is not confirmed by the most recent SFH derived by @2015MNRAS.449..639R, the SFR by @2004AJ....127.1531H is in good agreement with the one derived by @2015MNRAS.449..639R about 100 Myr ago. The more recent SFR is difficult to constrain using Hubble Space Telescope data as it requires a wide survey area [@2013MNRAS.431..364W].
AGB stars of masses 1 M~$\odot$~ $\lesssim$ M $\lesssim 8$ M~$\odot$~, and some red supergiants (RSGs) with masses M $\gtrsim$ 8 M~$\odot$~, go through periods of intense mass-loss. Near the end of their lifetimes, AGB stars will lose $50 - 80\%$ of their initial mass, at a rate of up to $10^{-4}$ M$_{\odot}$ yr$^{-1}$ , and contribute a significant amount of chemically enriched material to the interstellar medium. They may also be the largest contributors of dust to the universe [@1989IAUS..135..445G], but this remains controversial as the effects of metallicity and luminosity on the dust contribution from these stars are still quantitatively unclear, as are the contributions from other major sources, novae and supernovae. The mass loss has an important effect on the evolution of the star itself truncating its evolution as a cool (super)giant and, in the case of red supergiants, affecting the conditions for the unavoidable core-collapse supernova.
Towards the end of their lives, AGB and RSG stars will become obscured by circumstellar dust. This dust allows the stars to lose mass in the form of a stellar “superwind” [@Iben1983]. This wind is driven by radiation pressure on the dust grains that form at several stellar radii, though how the outflow is launched from the stellar photosphere is still not clear. As the dust grains move outward, they drag along gas and create winds that reach up to 30 km s$^{-1}$. The dust is composed of either carbonaceous or silicate dust grains depending on the ratio of carbon-to-oxygen within the stellar envelope. The chemical composition of the dust and envelope can be significantly altered by internal processes like HBB and/or third dredge-up events.
Evolved oxygen-rich AGB stars are known for exhibiting OH maser emission at 1612 MHz. While carbon-rich evolved AGB or “carbon stars” tend to be the more-obscured than their oxygen-rich counterparts, maser emission does not occur within them and thus we will exclude them from this study. Evolved AGB stars are significantly obscured by dust, which leaves them opaque at visible wavelengths. The stellar light from these stars is re-radiated by the dust grains making them bright in the infrared (IR). From these unique characteristics these stars have been designated as OH/IR stars. OH/IR stars are generally brightest in their OH maser emission at 1612 MHz, but also exhibit H~2~O and SiO masers. The 1612-MHz maser transition occurs at several hundred stellar radii where the outer-most H~2~O is photodissociated by interstellar ultraviolet radiation. As the transition probes the final outward wind speeds propagating towards and away from us, we observe a double-horned maser profile that is created from the Doppler-shifted 1612-MHz transition. By taking half the peak separation of the maser profile, we get the expansion velocity of the envelope of the star. This expansion velocity can be used to test and refine dust-driven wind theory. The expansion velocity has been shown to follow a relationship of: $v$~exp~ $\propto \psi^{1/2}L^{1/4}$, where $L$ is the luminosity, and $\psi$ is the dust-to-gas ratio [@2017MNRAS.465..403G]. The dust-to-gas ratio has shown strong empirical evidence to scale approximately with the metal content and thus we expect the expansion velocity should show a dependence on both metallicity and luminosity, a phenomenon that has been seen observationally in the LMC and Galactic samples [@1992ApJ...397..552W; @2004MNRAS.355.1348M; @2017MNRAS.465..403G].
Previous searches for oxygen-rich dusty stars in the SMC
--------------------------------------------------------
Two OH maser studies have targeted SMC circumstellar OH maser emission in the past (discussed further in Appendix A). The deepest observation resulted in a noise level $\sim$8 mJy. The Infrared Astronomical Satellite [*IRAS*; @1984ApJ...278L...1N], the Midcourse Space Experiment [*MSX*; @2001AJ....121.2819P], the Two-Micron All-Sky Survey (*2MASS*) and more recently the *Spitzer Space Telescope* [@2004ApJS..154....1W] have revealed large samples of AGB and RSG stars in the SMC [@1992ApJ...397..552W; @2011AJ....142..103B; @2017ApJ...834..185K]. However the search for deeply embedded sources continues. Based on MSX and 2MASS data, the IR-bright sample from the MSX survey was expected to be composed of one-third carbon stars and one-half oxygen-rich stars in the SMC. Instead, two-thirds of the sample were found to be carbon stars, and no heavily obscured oxygen-rich stars were detected, with the remaining third composed of RSGs, post-AGB stars, YSOs, PN, and oxygen-rich stars without significant dust.
Other recent searches have discovered a few candidate OH/IR stars, some of which have been observed in the IR [@2015ApJ...811..145J; @2017ApJ...834..185K], and some at radio frequency, but only a few deeply embedded evolved stars like those within the LMC or our galaxy, either carbon- or oxygen-rich, have been found in the SMC . It is not a requirement for OH/IR stars to have a high self-extinction, but as the maser luminosity is dependent on the dust, the most luminous maser sources are expected to.
As was noted by @2016MNRAS.457.1456V, the most obscured carbon stars in the LMC and SMC have initial masses of $2.5-3 M_{\odot}$ and $\sim1.5 M_{\odot}$, respectively, as a result of different SFHs. This has lead to a difference in initial stellar masses in their Thermally-Pulsing AGB (TPAGB) phase in the Magellanic Clouds.
Moving to the higher mass stars, previous searches have found a small sample of dusty RSGs in the SMC, but they are not obscured to the same degree as the OH maser-emitting LMC RSGs [@2010AJ....140..416B]. They are both warmer and less dusty than the massive O-rich OH maser-emitting AGB stars and RSGs (called OH/IR stars) in the LMC and the Galaxy. While these stars are few, the results of the Surveying the Agents of Galaxy Evolution in the tidally stripped, low metallicity Small Magellanic Cloud (SAGE-SMC) survey [@2011AJ....142..102G] suggest that the small known sample of dusty RSGs within the Magellanic Clouds contributes more dust than the entire sample of optically bright RSGs [@2010AJ....140..416B]. Hence it is critically important that we obtain measurements of the expansion velocities of these sources to constrain current wind driving models.
[ l c c c c c c]{} Object & $L$ & Pulsation period & Pulsation amplitude & $J-K$ & $F_{24\mu m}$ & Spectral\
name & ($10^{3}$ L$_{\odot}$) & (d) & (mag) & (mag) & (mJy) & type\
*Parkes targets*\
IRAS00483$-$7347 & 8 & 859& 1.7 & 2.8 & 60.0 & M8\
IRASF00486$-$7308 & 1 & 062& 2.7 & 1.6 & 91.1 & M4\
IRAS00591$-$7307 & 8 & 092 & 1.9 & 1.3 & 95.4 & M5\
IRAS01074$-$7140 & 4 & 523& 2.8 & 1.2 & 396 & M5\
\
*ATCA sources*\
OGLE SMC-LPV-15504 & 5 & 543 & 0.8& 1.3 & 31.6 & M3$-$4\
2MASS J01033142$-$7214072 & 7 & 471& 1.8 & 3.7 & 19.2 &
\
HV 12149 & 8 & 769 & 2.3 & 1.4 & 12.7 & M8\
OGLE SMC-LPV-14322 & 1 & 483 & 1.9 & 3.5 & 19.0 &
\
2MASS J00592646$-$7223417 & 8 & & & 4.4 & 18.9 &
\
BMB-B75 & 8 & 761 & 2.1 & 1.3 & 92.0 & M6\
OGLE J004942.72$-$730220.4 & 8 & 563& 0.4 & 3.9 & 27.8 &
\
MSX SMC 018 & 0 & 897 & 2.5& 2.5 & 62.0 &\
\[smc\_maser\_candidates\]
Description of the Sample
=========================
Past observations have succeeded in detecting interstellar OH maser emission in the SMC [@2013MNRAS.432.1382B], but have failed to detect circumstellar OH maser emission [@2012arXiv1210.0983V]. We have targeted luminous evolved stars to investigate and scrutinize the most promising candidates for circumstellar maser emission to date (Table \[smc\_maser\_candidates\]). Our targets were chosen based on results from , as well as pulsation periods newly derived from the Optical Gravitational Lensing Experiment (OGLE) [@2011AcA....61..217S].
Our observations include four sources targeted for being dusty evolved oxygen-rich sources with the Parkes radio telescope as well as an ATCA survey with four fields spread across the SMC. With Parkes we have targeted four of the best candidates for OH maser emission in the SMC. With the ATCA’s FWHM primary beam at 1612 MHz of 29 arcminutes, we have surveyed a significant portion of the SMC bar region (Fig. \[smc\_fields\]). We have obtained spectra for seven sources that are known to be mid$-$late M-type oxygen-rich sources, one source that is likely to be oxygen-rich (MSX SMC 018), and four sources that are likely carbon stars.
We expect to detect maser emission in several of our observed sources. The 1612-MHz maser is pumped by IR radiation at 35 $\mu$m with a typical efficiency of 23% (discussed further in Section 4.4). Simplistically assuming this efficiency, the maser flux of our targeted source IRAS 00483$-$7347 should be over 19 times the noise level of the observation.
![The *Spitzer* MIPS 24 $\mu$m image of the SMC overlaid with the full width at half-maximum (FWHM) of the primary beam at 1612 MHz of our ATCA interferometric observations of 29 arcminutes (shown in blue) and those of the Parkes single-dish observations at 14 arcminutes (shown in red).[]{data-label="smc_fields"}](engels_smc_fields.pdf){width="1\columnwidth"}
Observations
------------
Radial velocities mentioned in this paper are in a barycentric reference frame. We adopt a systemic radial velocity of $\approx 146$ km s$^{-1}$ for the SMC but radial velocities of stars and gas within the SMC vary by about $\pm 50$ km s$^{-1}$ around this value, stemming from a combination of dispersion and gradient on the sky .
### Parkes observations
We present archival (reprocessed) Parkes observations taken in 2003 from August 18 to 20 [@2004MNRAS.355.1348M], and new observations taken in 2005 from July 7 to 13 and August 13 to 19, observing the 1612-MHz OH satellite line. With the multibeam receiver and multibeam correlator, we used a dual polarization setup with 8-MHz bandwidth (1489 km s$^{-1}$) and 8192 channels centered at 117 km s$^{-1}$ yielding a 0.18 km s^$-1$^ channel^$-1$^ velocity resolution; the observations used a frequency-switching calibration technique with a switching frequency of 0.5 MHz (93 km s$^{-1}$ at 1612 MHz) and a 14 arcminute beam. IRAS 00591$-$7307 was observed in 2003, IRAS F00486$-$7308 and IRAS 01074$-$7140 in 2005, and IRAS 00483$-$7347 in both epochs.
[lcccc]{} & & & On-source &\
Target name & & & Integration &\
& & & time (h) &\
*Parkes observations*\
IRAS00483$-$7347 & 00h 50m 09s & $-73^{\circ} \, 31^{\prime} \, 29^{\prime \prime}$ & 58.8 & 3.7\
IRASF00486$-$7308 & 00h 50m 31s & $-72^{\circ} \, 51^{\prime} \, 30^{\prime \prime}$ & 31.1 & 4.5\
IRAS00591$-$7307 & 01h 00m 48s & $-72^{\circ} \, 51^{\prime} \, 02^{\prime \prime}$ & 10.0 & 7.7\
IRAS01074$-$7140 & 01h 09m 02s & $-71^{\circ} \, 24^{\prime} \, 10^{\prime \prime}$ & 61.9 & 3.4\
\
*ATCA observations*\
SMC North & 01h 03m 60s & $-72^{\circ} \, 01^{\prime} \, 00^{\prime \prime}$ & 1.4 & 8\
SMC Centre & 00h 59m 00s & $-72^{\circ} \, 10^{\prime} \, 60^{\prime \prime}$ & 0.0 & 9\
BMB-B75 & 00h 52m 13s & $-73^{\circ} \, 08^{\prime} \, 53^{\prime \prime}$ & 1.2 & 0\
SMC South & 00h 48m 60s & $-73^{\circ} \, 25^{\prime} \, 60^{\prime \prime}$ & 8.4 & 9\
\[smc\_obs\]
### ATCA observations
The data were taken in 2012 between January 1 and January 5, and in 2015 on June 12. Four observations of varying integration times between 8 and 11 hours were done for each targeted region (displayed in Table \[smc\_obs\]). The observations were done with the CFB 1M-0.5k correlator configuration with zoom-bands [@2011MNRAS.416..832W] and the 6A array configuration, and observed all four OH maser transitions at 18 cm (1612, 1665, 1667, and 1720 MHz). The bandwidth used was 2.5 MHz with 5121 channels, corresponding to zoom-band velocity widths of $\sim$ 465 km s$^{-1}$ and a velocity resolution of 0.09 km s$^{-1}$ centred at $\sim50$ km s$^{-1}$. The FWHM of the primary beam of ATCA at 1612 MHz is 29 arcminutes. Three of the observations were blind maser searches that targeted two star formation regions in the north and another in the south (Fig. \[smc\_fields\]). These observations used PKS 1934-638 and PKS 0823-500 as flux and bandpass calibrators and PKS 2353-686 and PKS 0230-790 as secondary calibrators. The fourth observation targeted the region covering the evolved star BMB-B75 toward the southern SMC field and used PKS 1934-638 as a bandpass and a flux calibrator and PKS J0047-7530 as a secondary calibrator.
Data reduction
--------------
### Parkes data reduction
Using the <span style="font-variant:small-caps;">casa asap</span> toolkit [@2007ASPC..376..127M], spectra were extracted for each of our four targeted sources. Employing a frequency-switching calibration, out-of-band scans were subtracted to ensure a flat baseline. The spectra of IRAS 00483$-$7347 synthesised from a combination of both epochs and those of the remaining targets are displayed and discussed in Section 3. The resulting spectra were resampled to a resolution of 0.5 km s$^{-1}$. This was done to mitigate large spikes from RFI while preserving adequate spectral resolution.
### ATCA data reduction
The ATCA data were inspected, flagged, and calibrated using <span style="font-variant:small-caps;">miriad</span> [@1995ASPC...77..433S]. The visibility data were then transformed into three-dimensional data cubes, weighting the visibilities naturally. The source-finding package <span style="font-variant:small-caps;">duchamp</span> [@2012MNRAS.421.3242W] was used to search for maser sources within the FWHM of each of the fields. For the resulting peaks, our intended targets, and any other potential targets from @2011AJ....142..103B, @2012ApJ...753...71R, or SIMBAD, a spectrum was extracted for a region the size of the synthesized beam ($\sim 7^{\prime \prime}$). All overlapping fields were mosaicked, and <span style="font-variant:small-caps;">duchamp</span> was used to search within the ATCA’s FWHM primary beam, centred between the mosaicked fields. The resulting spectra were resampled to a resolution of 0.5 km s$^{-1}$, as was done with the Parkes spectra.
Results
=======
Our targeted and blind maser searches within the SMC have yielded no clear OH maser emission. This is the case for the 1612-MHz OH observations (Fig. \[smc\_masers\]), as well as the 1665-, 1667-, and 1720-MHz OH maser observations; the 1665-MHz OH mainline observations are shown in Appendix C. The average sensitivities of the 1612-MHz observations are listed in Table \[smc\_obs\]. While seemingly unsuccessful, these observations have in fact given us valuable upper limits. In the following sections we will employ our new limits to draw conclusions about the absence of an OH/IR population in the SMC, and what we can predict for future maser searches.
The H[i]{} column density has shown to be a good indicator of the likely systemic velocities of our sources. In addition to the Parkes and ATCA observations, we have also included 1420-MHz H[i]{} data from the combined ATCA and Parkes multibeam H[i]{} maps [@1999MNRAS.302..417S]. The survey covers an area of 20 square degrees with a velocity resolution of 1.6 km s$^{-1}$, a velocity range from $\sim 90 - 215$ km s$^{-1}$, and an angular resolution of 98 arcseconds (FWHM). For each source in our SMC sample, a spectrum was extracted using a 1-arcminute region centred on the sources. The resulting spectra have been plotted below each maser spectra. While some of our sources lie outside these H[i]{} regions, they provide supporting evidence for sources which lie in higher density regions of the SMC. Within the observations we find no indication of maser emission.
{width="8.7cm"} {width="8.7cm"}
[lcccccc]{} Object & $F_{24\mu \textrm{m}}$ & $F_{35\mu \textrm{m}, \textrm{predicted}}$ & $F_{\textrm{OH,\,predicted}}$& $\sigma$ & Maximum& Likelihood\
name & (mJy) & (mJy) & (mJy) & (mJy) & efficiency & of detection\
*Parkes targets*\
IRAS00483$-$7347 & 60.0 & 7 & 1 & 3.7 & 3.6% & 7.7%\
IRASF00486$-$7308 & 91.1 & 9 & 9 & 4.5 & 4.5% & 0.7%\
IRAS00591$-$7307 & 95.4 & 4 & 0 & 7.7 & 2.1% & 3.3%\
IRAS01074$-$7140 & 96.0 & 5 & 4 & 3.4 & 4.3% & 7.7%\
\
*ATCA sources*\
OGLE SMC-LPV-15504 & 31.6 & 3 & 3 & 7.1 & 8.9% & 5.8%\
2MASS J01033142$-$7214072 & 19.2 & 8 & 1 & 6.3 & &\
HV 12149 & 12.7 & 7 & 2 & 6.5 & 8.0% & 1.2%\
OGLE SMC-LPV-14322 & 19.0 & 1 & 3 & 7.6 & &\
2MASS J00592646$-$7223417 & 18.9 & 3 & 3 & 6.5 & &\
BMB-B75 & 92.0 & 3 & 0 & 9.5 & 6.5% & 9.7%\
OGLE J004942.72$-$730220.4 & 27.8 & 4 & 3 & 0.0 & &\
MSX SMC 018 & 62.0 & 6 & 7 & 7.2 & 8.7% & 7.5%\
\[limits\]
### Parkes targets {#parkes-targets .unnumbered}
Our four Parkes targets are IRAS 00483$-$7347, IRAS F00486$-$7308, IRAS 00591$-$7307, and IRAS 01074$-$7140. All of these sources have infrared characteristics typical of OH maser emitting AGB stars and RSGs (Table 1) and show 10 $\mu$m silicate features in emission, confirming they are oxygen-rich [@1995ApJ...449L.119G; @2011ApJS..196....8L]. IRAS 00483$-$7347 is our best candidate for OH maser emission with an expected mass of $6-7$ M$_{\odot}$ [@2015MNRAS.454.4235D], and has been found to be the most massive HBB AGB star in the SMC from its large Rb enhancement [@2009ApJ...705L..31G]. It has been suggested to be a strong candidate for a super-AGB star , which should not affect the maser emission, but may explain its peculiarity among the other SMC stars. IRAS F00486$-$7308 has shown strong molecular bands around 3 $\mu$m, indicative of an advanced evolutionary stage where the star is cooler with stronger pulsations capable of extending the molecular atmosphere [@vanLoon:2008iv]. IRAS 01074$-$7140 is very bright in the IR ($F_{24}$ = 0.4 Jy) but does not have a particularly reddened colour or long pulsation period. This source has also shown to be Li-enhanced, indicative of HBB [@1989MNRAS.238..769W; @1995ApJ...441..735S].
### ATCA sources {#atca-sources .unnumbered}
With our ATCA survey of the SMC we have obtained spectra for eight sources that we have considered good maser candidates. Four of these sources (OGLE SMC-LPV-15504, HV 12149, BMB-B75, and MSX SMC 018) have shown 10 $\mu$m silicate features in emission. OGLE SMC-LPV-15504 has a known systemic velocity of $176.8\pm0.3$ km s$^{-1}$ [@2003AJ....126.2867M] from spectroscopic observations of the Ca [ii]{} triplet, which greatly simplifies our search in velocity space. HV 12149 has an initial mass of around $4-5$ M$_{\odot}$, and is likely to have just initiated HBB . This source is unique with its late spectral type yet bright $K$-band flux of $M_{K}$= 8.6 mag [@2003yCat.2246....0C], typical of a RSG. After observing BMB-B75, @2015ApJ...811..145J found that the far-IR emission detected by *Herschel* was due to a spatially coincident galaxy. The mid-IR IRS spectrum of the source shows red-shifted (z $\sim$ $0.16$) emission peaks of Polycyclic Aromatic Hydrocarbons (PAHs) at 6.3 and 11.3 $\mu$m and Ne <span style="font-variant:small-caps;">iii</span> at 12.8 and 15.5 $\mu$m from the background galaxy [@2011ApJS..196....8L; @2017ApJ...834..185K], but the silicate emission detected from the source appears unaffected. This source has also been found to have 6 cm continuum [@2012SerAJ.184...93W], emission likely emanating from the background galaxy. MSX SMC 018 was categorised as an M-type star by @vanLoon:2008iv who used 3$-$4 $\mu$m spectra for classification, after which a spectral type of M7 was determined by . The source shows deep TiO bands, characteristic of oxygen-rich evolved AGB (as opposed to RSG) stars, and weak crystalline silicate features [@2017ApJ...834..185K]. The remaining four sources, 2MASS J01033142$-$7214072, OGLE SMC-LPV-14322, 2MASS J00592646$-$7223417, and OGLE J004942.72$-$730220.4, are evolved AGB stars within our observed fields that (as we will discuss in Section 4.5) are likely to be carbon stars.
Discussion
==========
A dearth of OH masers in the SMC
--------------------------------
In trying to understand the lack of maser emission in any of our SMC candidates we will first look at our most promising SMC sources for maser emission, IRAS 00483$-$7347. This observation had a 59-hour integration and targets our most luminous candidate. It is possible that the source is uniquely void of maser emission. However, assuming the source is a maser-emitting source, we should be able to detect maser emission with a $F_{35}$-to-F~OH~ pumping efficiency down to 3.6%, given the source’s predicted $F_{35}$ (Calculated from the best-fit <span style="font-variant:small-caps;">dusty</span> model; see Section 4.5), and the noise level of the observation. Within the rest of our sample of targeted SMC sources, we would expect a number of them to also show OH maser emission. The sample includes oxygen-rich sources slightly less luminous, reddened, and cool, compared to the LMC OH masing sources. We show in Figure \[smc\_L\_vs\_J-K\] that the SMC sources are typically less reddened at a given luminosity as opposed to the LMC sources, with IRAS 00483$-$7347 again being the exception. Within the figure we have included <span style="font-variant:small-caps;">parsec-colibri</span> stellar isochrones [@2017ApJ...835...77M] for different stellar ages that correspond to 2, 4, and 5 solar mass sources for both Magellanic Cloud metallicities. The point within each isochrone where the carbon-to-oxygen ratio becomes greater than 1 is also marked. We see that several of our SMC sources lie upon the 5 M$_{\odot}$ SMC track, where the C/O ratio is still less than 1. These sources have also all been spectroscopically confirmed to be oxygen-rich. We see that the four lower-luminosity sources lie between 2 and 4 M$_{\odot}$ at the SMC metallicity indicating that they are likely carbon stars. We have only plotted isochrones for the lower mass regime as the paths of higher mass isochrones on the HR diagram are more difficult to model and constrain.
![The luminosity as a function of $J-K$ colour for our samples in the SMC and LMC. Also shown are <span style="font-variant:small-caps;">parsec-colibri</span> stellar isochrones [@2017ApJ...835...77M] corresponding to 2, 4, and 5 solar mass stars in their TPAGB phase for both the LMC and SMC metallicities; also marked on the isochrones are the points at which the carbon-to-oxygen ratio is greater than one.[]{data-label="smc_L_vs_J-K"}](smc_mdot_and_L3.pdf){width="\columnwidth"}
Using the maser efficiency distribution from past maser detections and the noise levels of each targeted SMC observation, we have used a bootstrapping method to predict the likelihood of achieving a maser-pumping efficiency that would result in a $3 \sigma$ level detection (Table \[limits\]). By replacing each SMC source with a randomly selected maser efficiency from 86 sources in the Galactic and LMC samples from @2017MNRAS.465..403G with one million iterations, we have calculated the collective probability of not achieving this level in any of our four targeted observation as 0.11%. This suggests that there is likely an underlying reason for the lack of maser emission in the SMC.
![The cumulative expected maser peak flux distribution of our SMC sample using the detected LMC maser peak sample from @2017MNRAS.465..403G and scaling fluxes down by 40% for distance and another 40% for metallicity. Also shown are the typical sensitivities at different significance levels of a 10 hour integration with ATCA ($1\sigma\simeq 8.8$ mJy).[]{data-label="maser_distribution"}](maser_distribution1.pdf){width="\columnwidth"}
{width="49.00000%"} {width="49.00000%"}
We have also plotted the maser flux peak distribution expected from the SMC sample (Fig. \[maser\_distribution\]). We have resampled the LMC maser spectra to 0.5 km s$^{-1}$ resolution to match the SMC spectra. The cumulative density of the peak fluxes of the resulting spectra have been used to estimate the SMC peak maser fluxes. We have scaled the fluxes by 40% for distance, assuming distances of 50 and 60 kpc to the LMC [@2013pss5.book..829F] and SMC [@2009AJ....138.1661S], respectively. Assuming the sources are optically thick, the maser emission should depend only on the $F_{35}$ as this value takes metallicity into account. If this is not the case, then the emission will depend on the OH abundance which should scale with metallicity. To compensate for this potential effect we have scaled all LMC peak fluxes down by an additional 40% for the difference in metallicity. The plot shows that we would have expected 9 sources with peak maser fluxes above the typical noise level of our ATCA observations and 6 sources with peak fluxes above 3$\sigma$. However, this does not account for the difference in star formation history in the Magellanic Clouds. Using values for the star formation rate for the evolution time of $5-6$ M$_{\odot}$ stars (100 Myr) of 3 M$_{\odot}$yr$^{-1}$ and 0.75 M$_{\odot}$yr$^{-1}$ for the LMC and SMC, respectively [@2004AJ....127.1531H; @2009AJ....138.1243H], we would expect to have detected one fourth of the sources found in the LMC. This gives us an expected detectable maser population of $1-2$ sources above 3$\sigma$, whereas we detect none.
{width="49.00000%"} {width="49.00000%"}
OH/IR populations in the Magellanic Clouds
------------------------------------------
### Carbon/Oxygen abundance
The SMC hosts a smaller population of evolved AGB stars than the LMC, and has a higher fraction of carbon-rich evolved AGB stars [@2011AJ....142..103B]. It has also been observed that evolved AGB stars and RSGs in the SMC have less dust than their LMC counterparts . There are a number of possible explanations for the smaller population of oxygen-rich evolved stars in the SMC, four of which have been proposed by @vanLoon:2008iv. The first possibility is that oxygen-rich dust grains are more transparent at near-IR wavelengths, which coupled with a smaller dust content in the SMC, have left few obscured oxygen-rich sources. The second possibility is that higher luminosity oxygen-rich sources, which have undergone Hot-Bottom Burning (HBB), are extended, resulting in more diffuse dust. A third possibility is that the size of the SMC, the initial mass function, and the rapid evolution of higher-mass sources, have left few in this evolutionary phase. A final possibility is that as carbon stars form at lower masses within the Magellanic Clouds , this has resulted in fewer lower-mass oxygen-rich sources.
### Warmer evolved red supergiants
For RSGs, studies have shown that while values for $T_{\textrm{eff}}$ and $M_{\textrm{bol}}$ show agreement between Galactic and LMC sources, sources within the SMC show a much larger spread in $T_{\textrm{eff}}$ for a given $M_{\textrm{bol}}$ [@2006ApJ...645.1102L]. This is expected as a result of the enhanced effects of rotational mixing at lower metallicity . In addition to a larger spread in $T_{\textrm{eff}}$, lower metallicity RSGs have been shown to have earlier spectral types [@1985ApJS...57...91E; @2003AJ....126.2867M]. The expectation is that as spectral types are dependent on Titanium oxide (TiO) bands, and lower metallicity stars will have a lower abundance of TiO, this would result in lower metallicity sources being categorised as earlier spectral types. At the same time, lower metallicity RSGs have also been found to have warmer median $T_{\textrm{eff}}$ [@2005ApJ...628..973L; @2006ApJ...645.1102L; @2009AJ....137.4744L; @2007ApJ...660..301M]. As a decrease in the abundance of heavy elements results in a decrease in opacity, and an increase in a star’s surface brightness, this results in warmer RSGs at lower metallicity [@2013EAS....60..269L]. This would suggest fewer late-type RSGs exist within lower metallicity environments. Yet, observations have shown small populations of late-type RSGs in these lower metallicity environments .
Isolating the OH/IR sample
--------------------------
In order to determine the number of sources within the Magellanic Clouds that are capable of OH maser emission, we have considered photometry from the *Spitzer* SAGE-SMC catalogue [@2011AJ....142..102G; @2011AJ....142..103B]; we have compared these to the SAGE-LMC photometry [@2006AJ....132.2034B; @2006AJ....132.2268M; @2009AJ....138.1003B; @2010AJ....140..416B; @2009AJ....137.4810S; @2010AJ....139.1553V].
The sources within the @2017MNRAS.465..403G LMC OH/IR sample were primarily categorized as Far Infrared Objects (FIR) by @2011AJ....142..103B except for IRAS 05280$-$6910 which was designated as an extreme or x-AGB star (however showed that the source is bright at far-IR wavelengths), and IRAS 04553$-$6825, IRAS 05003$-$6712, and IRAS 05558$-$7000, which were not included in the catalogue. In the past, colour-magnitude diagrams (CMDs) have been used to separate different stellar populations within the Magellanic Clouds [@2011MNRAS.411.1597W]. Plotting mid-IR colours against mid-IR flux densities, the red supergiants and massive AGB stars create a distinct branch. Looking at the *Spitzer* IRAC \[3.6\]$-$\[8.0\] colour against the \[24\] flux density for the LMC OH/IR sample, the sources occupy a similar high position (or “masing zone”) on one of these branches (Fig. \[in\_LMC\]). We have plotted two CMDs, one with SAGE-LMC sources that fall within our ATCA and Parkes fields (*left*) and one with sources that do not (*right*). This is done to get an overall idea of the fraction of potential maser sources that have not been covered by our maser survey. We have done this also for our sources in the SMC (Fig. \[in\_SMC\]).
Within the LMC sample, there are two sources in our fields that have been identified as AGB stars and lie within the LMC masing zone, but do not exhibit circumstellar maser emission. Within our field are the sources MSX LMC 1210 and MSX LMC 1207, sources that do not exhibit maser emission. The first source, MSX LMC 1210, with a $J-K$ colour of 2.94 mag, is bright at mid-IR ($F_{24}=387$ mJy), and has a pulsation period of 1050 d [@2009AcA....59..239S]. The second source, IRAS 05287$-$6910, is also bright in the mid-IR ($F_{24}$=1.1 Jy), exhibits silicate in emission [@2011ApJS..196....8L], and lies in the cluster NGC 1994 . The cluster has an age of 11.5 Myr [@1988AJ.....96.1383E] and a metallicity of 0.58 Z$_{\odot}$.
Within the masing zone of sources not within the @2017MNRAS.465..403G LMC fields are three blue supergiants, three young stellar objects, four emission line stars, two carbon stars, a planetary nebula, and two sources that would be good targets for OH maser emission. The first source, IRAS 04523$-$7043, has a 24 $\mu$m flux density of 3.59 mag, and a pulsation period of 890 d [@2009AcA....59..239S]. The second source, 2MASS J05241334$-$6829587, has a 24 $\mu$m flux density of 2.58, a $J-K$ colour of 3.2 mag, a pulsation period of 899 days [@2011yCat..35360060S].
Within our SMC fields, there are three target sources that lie within the scaled LMC masing zone in the CMD: IRAS 00483$-$7347, IRAS 01074$-$7140, and MSX SMC 018 (Fig. \[in\_SMC\]). To the right of the region are several young stellar objects and emission line stars [@2013ApJ...778...15S]. There are also two sources that do not lie in any of our observed fields, IRAS 00350$-$7436 and IRAS 00337$-$7409 (HD 3407) within this same masing region. IRAS 00350$-$7436 has been found to be the most luminous carbon-rich object in the SMC [@1989MNRAS.238..769W; @Srinivasan:2016jp]. In the past the source has been categorized as a carbon star [@2015MNRAS.451.3504R], a post-AGB star and an interacting binary star [@1989MNRAS.238..769W]. The second source, IRAS 00337$-$7409, is a foreground star with a radial velocity of 58 km s$^{-1}$ [@1972POStr...2....1F]. No other sources within the SAGE-SMC survey lie within the masing zone as defined by the LMC OH/IR sources. This gives us confidence that we have surveyed all SMC sources that we would expect to exhibit circumstellar OH maser emission at our current detection thresholds. Also shown in the figure is the Thorne-Żytkow Object candidate star HV 2112 [@2014MNRAS.443L..94L; @2014MNRAS.445L..36T]. These stars, while yet to be confirmed observationally, are thought to be red giants or supergiants with neutron stars at their cores. Alternatively, it has been suggested it could be a super-AGB star [@1983ApJ...272...99W; @1990ApJ...361L..69S; @2014MNRAS.445L..36T], or even a Galactic foreground S-type star [@2016MNRAS.458L...1M], though its SMC membership seems secure on the basis of the radial velocity of $v_{\rm rad}=150$ km s$^{-1}$ and proper motion [@2016MNRAS.459L..31W]. The position of this source, with respect to the other SMC sources, suggests it is not particularly reddened or luminous at 24 $\mu$m.
Maser strength
--------------
Now that we have established the expectation of maser detections, we will aim to understand their absence. Several conditions must be met in order to produce circumstellar OH maser emission. First, as the transition is a population inversion, a dense population of OH with number densities typically $10^{12}-10^{16}$m$^{-3}$ is required [@2006evn..confE..42R]. The masing environment must have a sufficiently long path length to provide the high column density, needed to amplify the maser [@2005MNRAS.364..783G]. Masers can also only occur in a certain radius range of the parent star. If masers occur inside a radius of around several hundred au from the star, densities are so high that collisional pumping rapidly increases and “quenches” the maser. Outside a few thousand au, lower densities are not enough to sustain the pump rate [@1992ASSL..170.....E; @Vlemmings2006]. In addition to having a suitable masing environment, significant IR flux must be present to pump the maser. It is possible that circumstellar environments in the SMC do not adhere to these conditions.
### OH abundance
Within oxygen-rich evolved stars, the abundant oxygen allows for the formation of a variety of minerals, CO, and a significant amount of water. A smaller fraction of water has been found around carbon stars [@2001Natur.412..160M; @2010Natur.467...64D] likely a result of sublimating comets or the Fischer-Tropsch catalysis mechanism which converts CO and H$_{2}$ into hydrocarbons and water using iron as a catalyst [@2004ApJ...600L..87W]. In oxygen-rich stars water molecules are transported by stellar pulsation shocks, and the dust-driven wind, out to typical distances of $20-80$ au [e.g. @2002PASJ...54..757S and references therein]. This water is then photodissociated by interstellar UV radiation, creating OH. The abundance of OH will depend on the initial abundance of water but also the efficiency of converting water into OH, i.e. the penetrating strength of the UV radiation within the circumstellar envelope .
The observations and modeling of the conditions in metal-poor dwarf galaxies and nearby globular clusters indicate a much stronger radiation field and more porous molecular gas in lower metallicity environments . With more ionized hydrogen and less interstellar dust, the expectation is that the lower opacity will have an impact on chemical abundances in circumstellar environments. This may also be the case with our SMC sources, yet several LMC sources that also lie within clusters (e.g. IRAS 05280$-$6910 and IRAS 05298$-$6957) seem unaffected by their cluster environment .
The source of OH in evolved stars is water, and an understanding of the abundance of water can inform our understanding of the abundance of OH. Past surveys have targeted 22-GHz water maser emission in the SMC [@2013MNRAS.432.1382B], and while interstellar maser emission has been detected in star forming regions, no circumstellar maser emission has been confirmed. All of our brightest SMC sources (IRAS 00483$-$7347, IRAS F00486$-$7308, IRAS 00591$-$7307, IRAS 01074$-$7140, BMB-B75, and MSX SMC 018) have shown water in absorption at $\sim$ 3 $\mu$m [@vanLoon:2008iv]. It was suggested that this may be indicative of an advanced evolutionary stage, yet they also note that the comparison LMC sample does not show strong absorption from these molecular bands. They go on to suggest that it may be possible that less of the water molecules are being locked up in dust or that the population of water molecules may be closer to the stellar surface, increasing the column density as well as the excitation temperature. This result may hint at a potential difference in the circumstellar environments of our LMC and SMC OH/IR samples. However, the lack of OH maser emission in these sources is not due to a lack of water within their circumstellar environments. A deep survey for water maser emission in our top maser candidates may provide a better opportunity to observe circumstellar maser emission in the SMC.
It is highly possible that a denser circumstellar wind will result in a higher water column density. This higher density would create a more extended water masing region and result in a higher water maser flux density. With water closer to the star, it is less susceptible to external forces like interstellar UV radiation that convert water to OH. Also unlike 1612-MHz OH maser emission, water masers are pumped by both radiation and collisions with H$_2$ molecules . It is possible that the relative strengths of the maser pumping mechanisms may differ at lower metallicity. With less dust, and as a result lower mid-IR flux, collisional pumping may be more effective at lower metallicity. These facts have led us to the conclusion that the chances of detection in the SMC may favour water over OH masers.
### Maser pumping
The 1612-MHz maser is a product of a population inversion that results from the pumping by infrared photons of a transition at 35 $\mu$m. As a result, the strength of maser emission will depend on the dominant source of IR emission, circumstellar dust. The theoretical maximum pumping efficiency for the 1612-MHz maser transition is 25% [@1992ASSL..170.....E]. This assumes the maser is saturated and the maser emission is isotropic. However, results from have shown stars with pump rates as low as 0.05, 1/5 of this value. We see a large spread in pumping efficiency within the LMC and Galactic samples that extends far above this theoretical maximum with a median maser efficiency of 23% (Fig. \[maser\_efficiency\]). We expect that as the radio and IR measurements were taken at different times, and these sources are highly variable, this has contributed to the large scatter. We have calculated median maser efficiencies for the Galactic Centre, Galactic bulge and LMC OH/IR samples from @2017MNRAS.465..403G of 34.8%, 10.5%, 38.5%, respectively (see below). It should be noted that these are derived from spectra spanning a range of velocity resolutions between $\sim 0.5-1.5$ km s$^{-1}$, which will introduce some uncertainty. The much lower maser efficiency in the Galactic Bulge sample is surprising as SED fitting has found typically lower gas-to-dust ratio, and dramatically higher optical depth in dust emission than in the Galactic Centre and LMC samples [@2017MNRAS.465..403G]. This implies a larger supply of IR photons per OH molecule in the Galactic Bulge sources, yet we see a lower maser efficiency.
Figure \[maser\_efficiency\] shows the relationship of maser efficiencies and luminosities of the Galactic Centre, Galactic Bulge and LMC OH/IR samples from @2017MNRAS.465..403G. We have used predicted 35 $\mu$m flux densities from the best-fit <span style="font-variant:small-caps;">dusty</span> model of the sources (further explained in Section 4.5). The OH maser flux densities ($F$~OH~) are the peak flux density of both maser peaks combined for the Galactic samples from , and LMC sample from @2017MNRAS.465..403G. Also shown in this figure, are the upper limits of the maser efficiency of the SMC non-detections, as well as the two sources that we consider our best maser candidates IRAS 00483$-$7347 and IRAS 01074$-$7104, calculated using the ratio of the predicted 35 $\mu$m flux densities from the best-fit <span style="font-variant:small-caps;">dusty</span> model of the sources (further explained in Section 4.5) and three times the noise level. Our two best maser candidates are clearly below the expected maser efficiencies of all other sources of comparable luminosities except for the unique LMC RSG IRAS 05280$-$6910, which is suspected to exhibit a unique geometry [@2017MNRAS.465..403G]. This again suggests that, while not the case in the LMC, the lower metallicity of the SMC may be a critical factor in determining the presence of maser emission. These pumping efficiencies are comparable to those found in , yet we have expressed our values in terms of the peak intensity and not photon flux density, and our comparative analysis between the galaxy and the Magellanic Clouds still points to the expectation of detection within our SMC sample.
![Maser efficiency of converting 35 $\mu$m photons to 1612-MHz maser photons for the Galactic Centre (GC), Galactic bulge (GB), and LMC samples from @2017MNRAS.465..403G. The dotted line represents the median efficiency for the three samples of $23\%$. Also plotted are the 3$\sigma$ upper limits for the maser efficiencies of the SMC sample, our best two candidates: IRAS 01074$-$7104 and IRAS 00483$-$7308 (SMC best), and the SMC sources likely to be carbon stars (SMC carbon), taken from the ratio of the predicted 35 $\mu$m flux to three times the observation noise level.[]{data-label="maser_efficiency"}](maser_efficiency1.pdf){width="\columnwidth"}
[ l c c c c c c c]{} Object & Grain & $L$ & v~exp,Goldman~ & $T$~eff,DUSTY~ & $T$~inner~ & $\tau$ & $\dot{M}$\
name & type & ($10^3$ L~$\odot$~) & (km s$^{-1})$ & (K)& (K)& & (M~$\odot$~ yr$^{-1}$)\
*Parkes targets*\
IRAS00483$-$7347 & O & 8 & 5.4 & 3100 & 200 & 1.3 & 1.8 $\times 10^{-4}$\
IRASF00486$-$7308 & O & 1 & 3.3 & 3700 & 000 & 0.2 & 3.9 $\times 10^{-5}$\
IRAS00591$-$7307 & O & 8 & 4.3 & 2800 & 600 & 0.1 & 3.4 $\times 10^{-5}$\
IRAS01074$-$7140& O & 4 & 3.1 & 3100 & 350 & 0.4 & 5.9 $\times 10^{-5}$\
\
*ATCA sources*\
OGLE SMC-LPV-15504 & O & 5 & 3.4 & 2700 & 000 & 0.1 & 1.5 $\times 10^{-5}$\
2MASS J01033142$-$7214072 & C & 4 & 1.3 & 3700 & 000 & 0.2 & 1.2 $\times 10^{-5}$\
& O & 7 & 1.6 & 3000 & 400 & 1.4 & 1.8 $\times 10^{-5}$\
HV 12149 & O & 8 & 3.5 & 3000 & 400 & 0.1 & 1.0 $\times 10^{-5}$\
OGLE SMC-LPV-14322 & C & 1 & 2.0 & 3400 & 400 & 0.2 & 2.0 $\times 10^{-5}$\
& O & 1 & 2.0 & 3000 & 400 & 1.0 & 2.0 $\times 10^{-5}$\
2MASS J00592646$-$7223417 & C & 5 & 1.5 & 3500 & 000 & 0.2 & 1.8 $\times 10^{-5}$\
& O & 8 & 1.7 & 3000 & 400 & 1.3 & 1.8 $\times 10^{-5}$\
BMB-B75 & O & 8 & 3.8 & 3400 & 600 & 0.1 & 3.5 $\times 10^{-5}$\
OGLE J004942.72$-$730220.4 & C & 6 & 1.5 & 2800 & 800 & 0.1 & 1.6 $\times 10^{-5}$\
& O & 8 & 1.7 & 3000 & 400 & 1.8 & 2.2 $\times 10^{-5}$\
MSX SMC 018 & O & 0 & 3.8 & 2700 & 600 & 0.3 & 7.0 $\times 10^{-5}$\
\[table\_smc\_sed\_results\]
\
\
\
\
\
### OH column density
The minimum OH column density required to pump the 1612-MHz OH maser is around 1.3 $\times$ $10^{19}$ cm$^{-2}$ [@1976ApJ...205..384E]. Using our derived mass-loss rate (discussed in Section 4.5) and making several assumptions, we have calculated the expected OH column density of IRAS 00483$-$7347. Assuming the outflowing material is exclusively H$_2$ in an r$^{-2}$ distribution and using the inner OH shell radius determined for NML Cyg and VY CMa of 3.0 $\times$ 10$^{15}$ cm [@1979ApJ...233..119B], this yields a value for the H$_2$ column density. While the radius of NML Cyg has been confirmed using mainline OH maser emission [@2004MNRAS.348...34E], this value has been found to be larger ($\sim$ 300 au) for the more unique source VY CMa [@1982ApJ...253..199B]. NML Cyg and VY CMa have derived luminosities and mass-loss rates similar to those of IRAS 00483$-$7347 listed in Section 4.5 , and should have a similar inner OH shell radius. Assuming a ratio of 5000:1 for the ratio of H$_2$ to OH [@2005IAUS..231..499O], and scaling the abundance by 1/5 for metallicity, the fastest expansion velocity that would yield a value above the minimum column density for OH maser emission is 2.8 km s$^{-1}$. While slow for solar metallicity sources, it is still unclear if this value is realistic for the SMC’s lower metallicity sample. From the @2017MNRAS.465..403G expansion velocity relation, the expected value, given the source’s luminosity and metallicity is 5.4 km s$^{-1}$. This may suggest that the column density of the source has dropped below the masing threshold. The ratio of OH-to-H$_2$ remains the largest uncertainty in this calculation, as @2005IAUS..231..499O describes estimates more as order-of-magnitude approximations. Whether the abundance also scales linearly with the metallicity is also unclear. Regardless, we are observing sources very near the threshold for maser emission. The determining factor in whether maser emission is seen in the Galactic and LMC maser samples is the supply of IR photons to the system. Within the SMC, it seems that this becomes a two-body problem, where the maser strength drops with the metallicity squared, from a dependence on both the dust emission and the lower abundance of OH with respect to H$_2$.
Mass-Loss and evolutionary phase
--------------------------------
It is expected that metal-poor AGB stars need longer to evolve towards cool temperatures and strong pulsations [@1991ApJ...375L..53B]. This would result in more massive white dwarfs as the core has more time to grow. Yet, observations have shown that the white dwarf mass functions between the SMC, LMC, and the Galaxy are similar [@2004ApJ...614..716V; @2007ApJ...656..831V; @2007ASPC..372...85W]. This would mean that the onset and strength of the dust-driven wind phase is similar within the Galaxy and the Magellanic Clouds [@vanLoon:2008iv]. However this is only the case for lower-mass AGB stars. The white dwarf mass function does not constrain the strength or duration of the superwind phase in either massive AGB stars or RSGs, which dominate our OH/IR samples in the Magellanic Clouds. With a shorter superwind phase, we would expect fewer sources exhibiting circumstellar maser emission.
{width="49.00000%"} {width="49.00000%"}
Using the same SED fitting method as we applied to our LMC sample (see Goldman et al. 2017), we have fit the SEDs of our SMC sources (Fig. \[smc\_seds\]) and calculated luminosities and mass-loss rates for our SMC sample assuming a gas-to-dust ratio of 1000 . The results show mass-loss rates $\sim 10^{-5}$ M~$\odot$~ yr$^{-1}$, and luminosities typically $\sim 50,000$ L$_{\odot}$, similar to the LMC sample (Table \[table\_smc\_sed\_results\]). Due to the assumptions within the modeling of spherical symmetry, distance, optical properties, zero drift speed, and the fact that the model is an idealised system, it would be misleading to quote formal errors for these values. The luminosities are mostly affected by geometry and variability, but are generally accurate to within a factor two. The mass-loss rates are much less certain, and carry a systematic uncertainty of an order of magnitude due to our poor knowledge of the optical properties of the circumstellar grains . The table also shows expansion velocities calculated using the relation from @2017MNRAS.465..403G. While this relation has only been tested in metallicity environments between one half and twice solar, we expect the relation to give a good approximation within this metallicity regime. It is possible that at lower metallicity, a different set of dust optical constants may be better suited to reproduce the observed silicate features. However, implementing these changes would compromise the integrity of the comparison of samples, and thus we have used the same set of <span style="font-variant:small-caps;">dusty</span> models and dust optical constants as in the LMC analysis.
{width="49.00000%"} {width="49.00000%"}
While there are only three sources (IRAS 00483$-$7347, IRAS 01074$-$7140, and MSX SMC 018) with comparable absolute flux densities at 24 $\mu$m to the LMC sample, there are several sources with comparable luminosities (Fig. \[lum\_hist\]). Our most luminous source IRAS 00483$-$7347 has a long pulsation period, late spectral type, a large Rb enhancement, typical of an AGB star that has undergone HBB. Yet the source has a high luminosity of 138,000 L$_{\odot}$ suggestive of an RSG, which is more likely to indicate that the source is likely a super-AGB star.
When comparing the mass-loss rates and luminosities for the SMC sample to the LMC OH/IR sample (Fig. \[lum\_hist\], \[smc\_M\_vs\_L\]), we see that the higher luminosity SMC sources ($ > 2\times 10^{4}$ L$_{\odot}$) tend to have a lower mass-loss rate at a given luminosity (except for IRAS 00483$-$7347). This is not surprising given the SMC sources are not as optically thick as the LMC sources. We have also included the SMC mass-loss rates calculated using the @2017MNRAS.465..403G mass loss prescription (SMC prescription). Using the prescription we assume a gas-to-dust ratio of 1000, yet this parameter does not significantly affect the calculated mass-loss rate. The values we calculate are typically higher than what we calculate from SED fitting, except for several sources that have SEDs that do not conform well to any of the models within our grids (2MASS J01033142$-$7214072, OGLE SMC-LPV-14322, 2MASS J00592646$-$7223417, OGLE J004942.72$-$730220.4). We have modelled these four sources using a similar grid of <span style="font-variant:small-caps;">dusty</span> models that use the carbon dust grains from @1988ioch.rept...22H. We have over-layed the best fit carbon-SED for these sources and find that they better reproduce the observed SED. This along with their position with respect to the <span style="font-variant:small-caps;">parsec-colibri</span> isochrones (Fig. \[smc\_L\_vs\_J-K\]) and their low luminosities may signify that they are in fact less evolved non-maser-emitting carbon stars. Within the \[3.6\]$-$\[8.0\] vs. absolute flux density at 24 $\mu$m CMD (Fig. \[in\_SMC\]), these less luminous sources occupy an area typical of carbon stars, providing further evidence for this conclusion.
Looking at mass-loss rates versus pulsation period (Fig. \[smc\_M\_vs\_L\]) we see a spread of values for the SMC that span our other samples. We also see an increase in the mass-loss rate with pulsation period for the SMC sample. There are several sources with comparable luminosities, pulsation periods and mass-loss rates, suggesting that the SMC sources are in a similar evolutionary stage within the superwind phase. An example being the SMC source BMB-B75 and the LMC source IRAS 05402$-$6956, with luminosities of 58,000 and 51,000 L$_{\odot}$, pulsation periods of 1453 and 1393 d, and mass-loss rates of 3.5 and 8.4 $\times$ 10$^{-5}$ M$_{\odot}$ yr$^{-1}$, respectively.
From our new relation for the circumstellar expansion velocity [@2017MNRAS.465..403G], we have calculated the expected expansion velocities of our SMC sample (shown in Table \[table\_smc\_sed\_results\]) assuming a metallicity of one fifth solar. Typical expansion velocities for these sources are expected around 3 km s$^{-1}$, yet it is possible that as the strength of radiation pressure wanes, other wind-driving forces take over, resulting in underestimated predictions for the wind speed. We have also modelled SMC sources with <span style="font-variant:small-caps;">dusty</span> models calculated with a simple inverse square density distribution (as opposed to our SEDs calculated using a full hydrodynamical solution) and found negligible difference between the results. This may indicate that the material is blown out without the need for pulsation-enhancement. With a slower wind we also expect more efficient stimulated emission as a result of higher density. This higher rate of collisions may then make it easier to quench the maser. Another scenario worth mentioning is the possibility of binarity in the SMC sample. While we have no evidence to suggest this, symbiotic Miras tend to have weaker maser emission as a result of the decreased coherence within the OH shell [@1995MNRAS.276..867S]. This may provide another possible explanation for a decrease in masing sources within the SMC. We can thus conclude that there is no strong evidence for AGB stars and RSGs in the metal-poor SMC, at $\sim0.2$ Z$_\odot$, to experience weaker mass loss than those in the more metal-rich systems, the LMC ($\sim0.5$ Z$_\odot$) and Galactic Bulge and Centre ($\sim0.5$–2 Z$_\odot$).
Future prospects
----------------
The Galactic Australian Square Kilometre Array Pathfinder (GASKAP) Spectral Line Survey is an upcoming high spectral resolution (0.2 km s$^{-1}$) ASKAP survey of the 21-cm HI, and 18-cm OH lines, in the Milky Way galactic plane, the LMC, the SMC, and the Magellanic Bridge and Stream [@2013PASA...30....3D]. GASKAP will survey these regions with sensitivities better than that of Parkes (flux density sensitivity around 0.5 mJy with a 200 hour integration). As the entire SMC will fit within the field of view of ASKAP, this is feasible and there is therefore a good chance that GASKAP will detect OH maser emission from the most luminous candidates wherever they reside in the SMC. It was predicted by @2015aska.confE.125E, based on the relation between 1612 MHz OH maser emission and mass-loss rate [@1996MNRAS.279...32Z], that around two dozen OH maser sources should be detectable in the SMC at a 3-$\sigma$ sensitivity of 0.1 mJy. Following the same approach, for the range in mass loss parameters listed in Table 4 for our SMC sample we would predict $3-29$ mJy. We are therefore somewhat more optimistic about the prospects for GASKAP to detect more than a few OH masers in the SMC, or for it to confirm that the maser mechanism breaks down at the metallicity of the SMC. This survey will also be a precursor to much larger surveys using the Square Kilometre Array, which will reveal the full extent of the maser samples in these regions or show that the maser mechanism breaks down at the metallicity of the SMC.
Conclusions
===========
Using the properties of past OH maser host stars, we have identified and observed the brightest SMC sources likely to exhibit circumstellar maser emission. Our new, deep 1612-MHz OH maser observations with single and interferometric instruments have not resulted in any clear maser emission from our targeted or in-field sources, yet a number of maser-emitting sources may still lie below our observation detection thresholds. Assuming the masing mechanism is the same within the Galaxy and Magellanic Clouds, the top two candidates for maser emission have upper limits for maser efficiency (converting $F_{35}$ to $F_{\textrm{OH}}$) of 3.6 and 4.3%. These values are dramatically lower than the typical maser efficiency around 23%. This may point to differences within the circumstellar environments of evolved stars at lower metallicity that may include a low OH abundance or strong interstellar radiation. We cannot adequately quantify the effects of metallicity on the strength of circumstellar OH maser emission. We suspect that some of these sources are near the OH column density threshold for maser emission and also suspect that maser emission may still lie below our detection threshold. It is also possible that the OH masing phase does not last as long in the SMC as at higher metallicity, at least for massive AGB and RSGs. This could have implications for the contribution from these stars to the total dust budget in the galaxy, and for the contribution in which the RSGs undergo core collapse. Future observation with the GASKAP survey [@2013PASA...30....3D] will be able to add clarity to our understanding of circumstellar maser emission and the effects of metallicity.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the referee, Dieter Engels, for his helpful and constructive comments. We would like to thank the support staff at CSIRO for their help with the observations and Michele Costa for his help with the bootstrapping code, and Leen Decin for her helpful discussion on circumstellar environments. And we would also like to thank Phillip Edwards for generously allocating directors discretionary time to the project. AN and JvL acknowledge an award from the Sir John Mason Academic Trust in support of theoretical investigations of dusty winds and evolved star populations. JvL, JO and HI acknowledge awards from the Daiwa Foundation, 11595/12380, and the Great Britain Sasakawa Foundation, 5219, in support of their population studies of astrophysical masers. PAW acknowledges a research grant from the South African National Research Foundation (NRF). AN acknowledges the support from the project STARKEY funded by the ERC Consolidator Grant, G.A. No. 615604. JFG is supported by MINECO (Spain) grant AYA2014-57369-C3-3 (co-funded by FEDER). This paper makes use of the SIMBAD database of the CDS.
Past SMC maser targets
======================
There have been two previous 1612-MHz maser surveys in the SMC conducted by @2004MNRAS.355.1348M and @1992ApJ...397..552W. The observations by @2004MNRAS.355.1348M targeted two sources, IRAS 00483$-$7347, which we have re-observed with a higher sensitivity, and IRAS 00591$-$7307 which we have reprocessed to present a complete and uniform dataset. The observations by @1992ApJ...397..552W focused on 15 sources targeted for having high IRAS 25 $\mu$m flux densities and that fit the criteria: $0.0 < $ log $ (F_{25}/F_{12}) < 0.5$. These observations achieved noise levels typically $\sim$ 40 mJy and did not result in any clear maser detections in the SMC. The 15 targets are composed of an evolved AGB star, a Seyfert 2 Galaxy, a star forming region, 2 foreground stars, 2 YSOs, 3 H[ii]{} regions, and 5 sources whose identity is still unclear (Table A1). We have presented new observations for the re-observed AGB star, HV 12149. The foreground sources are the red giant IRAS 00542$-$7334 (CM Tuc) , and the F star IRAS 00490$-$7125 (HD 5028) [@2006AstL...32..759G]. The unknown source IRAS 00435$-$7339 lies near a star-forming region. The Seyfert 2 galaxy that we observed is the well studied peculiar source IRAS 00521$-$7054 . None of the unknown sources, or the source IRAS 00477$-$7322 have 2MASS counterparts.
Target Type Reference
------------------- --------------------- ----------------------
HV 12149 AGB (SpT=M8.5)
IRAS 00430$-$7326 YSO @2010AJ....139.1553V
IRAS 00435$-$7339 ?
IRAS 00466$-$7322 H[ii]{} region @2012ApJ...755...40P
IRAS 00477$-$7322 ?
IRAS 00477$-$7343 H[ii]{} region @2004AJ....128.2206I
IRAS 00490$-$7125 F-type star @2006AJ....132..161G
IRAS 00517$-$7240 ?
IRAS 00521$-$7054 Seyfert 2 galaxy @1987NASCP2466..241E
IRAS 00542$-$7334 Foreground AGB
IRAS 00553$-$7317 ?
IRAS 01039$-$7305 YSO @vanLoon:2008iv
IRAS 01126$-$7332 Star-forming region @2013MNRAS.432.1382B
IRAS 01208$-$7532 ?
IRAS 01228$-$7324 H[ii]{} region
YSO OH maser spectra
====================
After observing, it was found that two sources within our maser survey fields were in fact YSOs. These sources were suggested as YSO candidates by @vanLoon:2008iv and then confirmed by @2013MNRAS.428.3001O. We present the 1612-MHz spectra for the sources IRAS 01039$-$7305 and IRAS 01042$-$7215. They were observed for 7.4 and 6.3 hours, respectively. YSO’s are expected to have their strongest emission in the mainlines transitions, but will also exhibit maser emission at 1612 MHz. We do not see any clear maser emission in either source but do see several large narrow spikes, likely a result of RFI.
![OH 1612 MHz maser observation of IRAS 01039$-$7305 and IRAS 01042$-$7215. As in Figure 2, the <span style="font-variant:small-caps;">HI</span> spectra are presented for velocity reference.[]{data-label="yso spectra"}](01039-7305.pdf "fig:"){height="8"} ![OH 1612 MHz maser observation of IRAS 01039$-$7305 and IRAS 01042$-$7215. As in Figure 2, the <span style="font-variant:small-caps;">HI</span> spectra are presented for velocity reference.[]{data-label="yso spectra"}](01042-7215.pdf "fig:"){height="8"}
Mainline ATCA observations
==========================
Each of our ATCA observations observed multiple OH maser lines. In addition to the 1612-MHz OH satellite line, the observations also targeted the 1665- and 1667-MHz OH mainlines, which probe regions closer to the star, within the 1612-MHz maser profile. The 1612-MHz maser dominates at longer path lengths [@1991MNRAS.252...30G], where the 1665-MHz mainline emission can come from the OH being located in a high density region (close to the star). High densities ($\gtrsim 10^7$ cm$^{-3}$) suppress the 1667-MHz emission compared to the 1665-MHz emission. The 1665-MHz spectra for each of the sources observed with the ATCA have been resampled to a velocity resolution of 0.5 km s$^{-1}$ and are displayed below the 1612-MHz spectra in the figure below. The much more sensitive Parkes observation of IRAS 00483$-$7347 is shown for the 1612-MHz spectrum.
\[lastpage\]
![The 1612- and 1665-MHz OH maser spectra of SMC sources observed with the ATCA. []{data-label="mainlines"}](SMC_FIG_all_mains1.pdf){width="99.00000%" height="0.99\textheight"}
[^1]: E-mail: s.r.goldman@keele.ac.uk
|
---
abstract: 'If dark matter has a finite size that is larger than its Compton wavelength, the corresponding self-interaction cross section decreases with the velocity. We investigate the implications of this Puffy Dark Matter for addressing the small-scale problems of the $\Lambda$CDM model. In particular, we show that the way the non-relativistic cross section varies with the velocity is largely independent of the dark matter internal structure when the range of the mediating force is very short. We present an explicit example in the context of a QCD-like theory of dark matter and show that low-threshold direct detection experiments have the potential to probe Puffy Dark Matter.'
author:
- Xiaoyong Chu
- 'Camilo Garcia-Cely'
- Hitoshi Murayama
bibliography:
- 'ref.bib'
title: Puffy Dark Matter
---
[**Introduction.**]{} While the standard cosmology as described by the $\Lambda$CDM model has been well established thanks to the overwhelming amount of observational data, the particle nature of dark matter (DM) still eludes us. In this regard, the distribution of DM in astrophysical objects may provide a very important clue [@Spergel:1999mh]. Actually, a handful of studies of the inner region of small-scale astrophysical halos claim that DM self-scatters with a cross section per unit of mass of $\sigma/m \gtrsim \unit[1]{cm^2/g}$ [@Vogelsberger:2012ku; @Rocha:2012jg; @Peter:2012jh]. This is known as self-interacting dark matter (SIDM) and provides an appealing explanation to the seemingly mass deficit observed in objects such as dwarf galaxies when compared to the predictions of collisionless DM. See [@Tulin:2017ara; @Bullock:2017xww] for recent reviews of these shortcomings of the $\Lambda$CDM model as well as for a discussion of alternative solutions such as those invoking other exotic DM candidates (e.g.[@Doroshkevich:1989bf; @Colombi:1995ze; @Cen:2000xv; @Hogan:2000bv; @Bode:2000gq; @AvilaReese:2000hg; @Hu:2000ke; @Peebles:2000yy]) or baryonic effects (e.g.[@Navarro:1996bv; @Gelato:1998hb; @Binney:2000zt; @Gnedin:2001ec; @ElZant:2001re; @Weinberg:2001gm; @Ahn:2004xt; @Tonini:2006gwz; @Martizzi:2011aa; @Read:2018fxs]).
Yet, the aforementioned cross sections in small-scale objects are marginally consistent with observations of clusters of galaxies, in which DM moves relatively faster (see e.g. [@Robertson:2016xjh; @Randall:2007ph; @Kaplinghat:2015aga]). Thus, barring uncertainties, a velocity dependence of $\sigma/m$ is preferred, with lower values at higher velocities. Due to this, point-like DM particles that self-scatter by means of a short-range interaction are often said to be disfavored because the corresponding cross section is nearly constant. In this context, mechanisms for obtaining a velocity-dependent $\sigma/m$ include light mediators inducing a long-range interaction [@Feng:2009hw; @Tulin:2013teo], resonant SIDM [@Chu:2018fzy], and processes involving inelastic scatterings [@McDermott:2017vyk; @Vogelsberger:2018bok; @Chu:2018nki].
In this work, we point out that, supposing that the DM particle has a finite size, $r_\text{DM}$, the self-scattering cross section typically decreases for DM velocities larger than $(mr_\text{DM})^{-1}$, even if the interaction associated with the scattering has a very short range. As is shown in Fig. \[fig:fi2\], a momentum transfer much smaller than $r_\text{DM}^{-1}$ is too small to measure the internal structure of the DM, so the latter acts as a point-like particle. On the other hand, when the momentum transfer becomes larger than $r_\text{DM}^{-1}$, the internal structure of the particle is probed. As specified below, this can happen in such a way that the phase difference among the scattered waves leads to a suppression in the scattering cross section. This is indeed the desired velocity dependence of DM self-scattering. We will refer to this scenario as Puffy DM. We would like to remark that, beside the self-scattering effects, the fact that DM has a finite size leads to a very rich phenomenology, as has been explored for several concrete DM candidates (e.g.[@Coskuner:2018are; @Nussinov:1985xr; @Chivukula:1989qb; @Kaplan:2009de; @Feldstein:2009tr; @Kumar:2011iy; @Cline:2013zca; @Wise:2014jva; @Krnjaic:2014xza; @Wise:2014ola; @Detmold:2014qqa; @Hardy:2014mqa; @Hardy:2015boa; @Mitridate:2017oky; @Gresham:2018anj; @Francis:2018xjd; @Contino:2018crt; @Braaten:2018xuw]).
This manuscript is organized as follows. In Sec. II we elaborate further on the elastic scattering of finite-size DM particles. In Sec. III we discuss the implications in DM halos. In the following section we present a QCD-like model of Puffy DM. Finally we conclude in Sec. V.
![Form factors as a function of momentum-transfer $q$ in units of the inverse root-mean-square radius $r_\text{DM}$. Solid, dashed and dotted lines correspond to the dipole, the tophat and the Gaussian distributions (see Table \[table:Fq\]). []{data-label="fig:fi2"}](FF_scatter.pdf){width="47.00000%"}
[**II. Scattering of finite-size DM particles.**]{} Let us first consider the scattering of two finite-size objects, which –for simplicity– will be modeled as a collection of point-like constituents that coherently scatter by means of a spin-independent Yukawa interaction. The corresponding charge density, $\rho(\vec{\bf r})$, characterizes the finite shape of the scattering object. We will also assume that the contribution of the binding force to the scattering rate is negligible. This is the case e.g. if such a force leads to a momentum-suppressed scattering amplitude. Then, the interaction Hamiltonian for two objects described by the density profiles $\rho_{1}(\vec{\bf x})$ and $\rho_{2}(\vec{\bf y})$ is $$\begin{aligned}
H_{\it int} &=&
\int d \vec{\bf x} d \vec{\bf y} \rho_{1}(\vec{\bf x})
\frac{ \alpha e^{-|\vec{\bf x}-\vec{\bf y}|/\lambda }}{ |\vec{\bf x}-\vec{\bf y}|} \rho_{2}(\vec{\bf y})\nonumber\\
&=& \int \frac{d \vec{\bf q}}{(2\pi)^{3}} F_{1} (\vec{\bf q})
\frac{ 4\pi \alpha}{\vec{\bf q}^{2}+\lambda^{-2}} F_{2} (-\vec{\bf q})\,.
\label{eq:Yuk}\end{aligned}$$ where $\lambda$ is the range of the interaction, $\alpha$ is a coupling constant, and we have introduced the form factor $F_i(\vec{\bf q}) \equiv \int {d \vec { \bf r} \, e^{i \vec {\bf q} \cdot \vec { \bf r}} } \rho_i(\vec{\bf r})$. Hence, the center-of-mass differential cross section in the Born approximation is $$\frac{d\sigma}{d\Omega} = S \left|
F_{1} (\vec{\bf q}) \frac{2 \mu \alpha}{\vec{\bf q}^{2} +\lambda^{-2}} F_{2} (-\vec{\bf q}) \,\pm\, (\vec{\bf q}\to-\vec{\bf q}) \right|^{2},
\label{eq:scattering1}$$ where $\mu$ is the reduced mass and $\vec{\bf q}$ is the momentum transfer. For identical (non-identical) particles, the second term must (not) be included and $S=1/2\, (1)$.
Shape $\rho(r)$ $r_\text{DM}$ $F(q)$
---------- ------------------------------------------------------ ----------------- ---------------------------------------------------------
tophat $ \frac{3}{4\pi r_0^3} \theta(r_0-r)$ $2\sqrt{3}r_0$ $\frac{3 (\sin (r_0 q)-r_0 q \cos (r_0 q))}{r_0^3 q^3}$
dipole $\frac{e^{-r/r_0}}{8\pi r_0^3} $ $\sqrt{3/5}r_0$ $\frac{1}{\left(1+r_0^2 q^2\right)^{2}}$
Gaussian $ \, \frac{1}{8 r_0^3 \pi^{3/2}} e^{-r^2/(4r_0^2) }$ $\sqrt{6} r_0$ $e^{-r_0^2q^2}$
: Form factors for different density distributions. []{data-label="table:Fq"}
An illustrative example is the electron scattering off larger objects. This is determined by a Coulomb interaction ($\lambda \to \infty$) with $F_e(\vec{\bf q}) =1$. In this case, Eq. gives the well-known Rutherford scattering formula, which can be used to infer the shape of finite-size objects. When applied to the proton, one finds a density distribution decreasing exponentially with a characteristic scale $r_0^{-2} = 0.71$GeV$^2$ [@Perdrisat:2006hj]. The latter is the dipole distribution (see Table \[table:Fq\]), generally expected from wave-function solutions to various potential wells [@Landau:1991wop].
![Transfer cross section as a function of the force range, $\lambda$, and the DM size, $r_\text{DM}$, both in units of $1/mv$. Here $\sigma_0$ of Eq. is assumed to be constant. []{data-label="fig:fourier"}](SIDMvsPuffy.pdf){width="45.00000%"}
We apply now Eq. to non-relativistic DM. Assuming that the DM particle is spherical, i.e. $F(\vec{\bf q})=F(q)$, the $S$-wave differential cross section reads $$\label{eq:ansatz}
\frac{d \sigma}{d \Omega}= \frac{\sigma_{0} }{8\pi}
\left[ \frac{F(q)^2}{1+\lambda^2q^2} + \left( \theta \to \pi-\theta \right)\right]_{q= m v\sin\theta/2}^2 \,,$$ where $\sigma_0= 4 \pi (m \alpha \lambda^2)^2 $. Here $\theta$ and $v$ are respectively the scattering angle and the relative velocity in the center-of-mass frame. While the exact form of $\rho(r)$ –and hence $F(q)$ in Eq. – needs to be determined by solving for the wave function from the Schr[ö]{}dinger equation of the composite state, the differential cross section is not sensitive to the details of $\rho(r)$ as long as it is always positive (no screening) and it goes to zero sufficiently fast at large radii. In that case, the DM size –or more precisely– the root-mean-square radius $$r^2_\text{DM}\equiv \int d \vec{\bf r}\, \rho(r) r^2 =-6 \,\frac{d^2 F(q)}{dq^2} \Bigg|_{q=0} \,
\label{eq:size}$$ is positive. Thus, $F(q)$ decreases for small momenta from $F(0)=\int {d \vec{\bf r}} \rho (r)$, which can be normalized to $1$ without loss of generality. Fig. \[fig:fi2\] illustrates this for the three representative distributions as listed in Table \[table:Fq\]. Together with Eq. \[eq:ansatz\], all this implies that the cross section is constant at low velocities and eventually approaches zero, even if the range of the interaction is extremely short.
[**III. DM scattering in astrophysical halos.**]{} Because of the form factor, for low velocities we expect isotropic scattering, whereas for larger velocities forward scattering is more probable. Due to this, the transfer cross section, $\sigma_T \equiv \int d\Omega (1-|\cos\theta|)d\sigma/d\Omega$, captures the self-interaction effects in DM halos better than $\sigma$ (see e.g. [@Kummer:2017bhr]), and will be adopted below.
Fig. \[fig:fourier\] illustrates the dependence of $\sigma_T$ on the interaction range $\lambda$ and the particle size $r_\text{DM}$. As apparent from the plot, $\sigma_T$ is largely independent of the exact expression for the form factor and therefore of $\rho(r)$. Furthermore, roughly speaking, the transfer cross section is constant for $ mv\ll \text{min}\{\lambda^{-1}, r_\text{DM}^{-1}\}$, starts decreasing at $ mv \sim \text{min}\{\lambda^{-1}, r_\text{DM}^{-1}\} $, and approximately scales as $1/v^4$ for $mv\gg \text{min}\{\lambda^{-1}, r_\text{DM}^{-1}\}$. See Appendix \[appendix:sigma\] for details. Interestingly, when the range of the Yukawa force is much larger than the size of the DM, this precisely describes the Born regime of SIDM with a light mediator [@Tulin:2013teo]. Furthermore, the figure shows that there is a one-to-one correspondence between the latter and the self-scattering of finite-size DM by a short-range force, both giving the same transfer cross section.
![Velocity dependence of the transfer cross section of Puffy DM. Best-fit curves to data [@Kaplinghat:2015aga] for the dipole (solid), tophat (dashed) and the Gaussian (dotted) distributions in Table \[table:Fq\]. The inset shows the 95% C.L. contours together with the corresponding parameter sets of the main figure. []{data-label="fig:fit"}](fit.pdf){width="49.00000%"}
The DM relative velocity in astrophysical halos typically follows a Maxwell-Boltzmann distribution truncated at the corresponding escape velocity, $v_\text{max}$. The velocity-averaged transfer cross section is then [^1] $$\langle \sigma_T v \rangle = \int^{v_\text{max}}_0 f(v) \sigma_T v dv \,,\quad f (v ) = \frac{32 v^2e^{-{4v^2}/{\pi \langle v \rangle ^2}}}{ \pi^2 \langle v \rangle^3} \,.
\label{eq:sigmaTv}$$ In the context of SIDM as a solution to the small-scale structure problems, a semi-analytical method has been proposed in [@Kaplinghat:2015aga] to infer, from observational data, the value of $\langle \sigma_T v\rangle/m$ for a given DM halo (see also [@Valli:2017ktb]). This method was applied to five clusters from [@Newman:2012nw], seven low-surface-brightness spiral galaxies in [@KuziodeNaray:2007qi] and six dwarf galaxies of the THINGS sample [@Oh:2010ea]. Fig. \[fig:fit\] shows these results respectively in green, blue and red. While these values should be taken with caution due to large uncertainties in the SIDM modeling of astrophysical objects (see e.g.[@Sokolenko:2018noz]), the set of points is in agreement with observations from the Bullet Cluster giving $\sigma_T/m\lesssim\unit[1.3]{cm^2/g}$ at cluster scales [@Randall:2007ph; @Robertson:2016xjh], which is one of the strongest constraints on DM self-interactions. Postulating a DM finite size much larger than the range of the Yukawa force, i.e. $\lambda\ll r_\text{DM} $, provides an excellent fit to the velocity-dependent cross section preferred by the galactic and cluster systems. The corresponding best-fit of Eq. to the data above is shown in Fig. \[fig:fit\] for the dipole, the tophat and the Gaussian distributions, separately. As expected from the aforementioned remarks, there is almost no dependence on details of the form factors even though they correspond to substantially different density distributions. The figure also shows that, in order to have the right velocity dependence, the DM size needs to be hundreds of times larger than the Compton wavelength. This explains the name Puffy DM.
If the Yukawa force is associated to a mediator $\rho$, requiring $\lambda= 1/m_\rho\ll r_\text{DM}$ implies $ m_\rho \gg 10^{-3} m$. This shows that the mediator can be lighter than the DM and still the velocity dependence is determined by the DM size. Likewise, if we impose $\alpha \lesssim m_\rho/m$ as required in the Born expansion, $\sigma_0/m= 4 \pi (m \alpha \lambda^2)^2/m \sim \unit[1]{cm^2/g} $ leads to $m\lesssim \unit[20]{GeV}$.
[**IV. A model of Puffy DM.**]{} Here we only sketch a possible realization of Puffy DM while details will be discussed elsewhere. It is a QCD-like confining theory with $N_c$ colors and two flavors of quarks: one “charm quark” much heavier than the confining scale $\Lambda$ and one nearly massless “down quark”. They respectively have charges $+2/3$ and $-1/3$ under a dark $U(1)_D$ gauge group. This is associated with a massive “dark photon” $\gamma_{D}$, which can act as a portal to the Standard Model (SM) by means of the kinetic mixing between the $U(1)_D$ group and the SM hypercharge. There are no dark weak interactions. We assume there is an asymmetry so that anti-charm quarks are annihilated while the remaining charm quarks end up in the baryonic $\Sigma_{c} (cdd)$ state. The latter interacts by exchanging the pseudo-scalar $\eta (d\bar{d})$ and the vector $\rho(d\bar{d})$, which lead to attractive and repulsive forces respectively.
On the one hand, it is likely that the $\eta$-exchange dominates binding $\Sigma_c$ baryons into nuclei because its range is larger given that the $\eta$ mass is due to the anomaly and hence suppressed as $m_{\eta} \sim \Lambda / \sqrt{N_{c}}$, as opposed to the $\rho$ mesons for which $m_{\rho} \sim \Lambda$. In view of this, in the following we assume the typical mass number is $10 \lesssim A \lesssim 100$. On the other hand, the nucleus-nucleus scattering is dominated by the exchange of $\rho$ mesons because the latter are essentially massive gauge bosons coupled to $d$-number ($A/2$) giving rise to coherent spin-independent scattering, while the $\eta$-exchange induces a spin-dependent momentum-suppressed scattering. Therefore, the range of the scattering force $\Lambda^{-1}$ is shorter than the size of the nuclei $r_\text{DM}\sim A^{1/3} m_{\eta}^{-1} \sim A^{1/3} \Lambda^{-1} \sqrt{N_{c}}$. As a result, this model is a realization of Puffy DM.
For instance, parameters such as $N_c=3$, $A\sim10$, $m_c\sim m_{\Sigma_{c}} \sim \unit[1]{GeV}$, $r_\text{DM}^{-1}\sim \unit[15]{MeV}$, $m_\eta\sim \unit[20]{MeV}$, $\Lambda \sim m_\rho\sim\unit[30]{MeV}$ and $\alpha \sim m_\rho/m$ realize the desired self-scattering cross section and its velocity dependence. We take $m_{\gamma_{D}} < m_{\eta}/2$ so that $\eta$ decays into $\gamma_{D} \gamma_{D}$ from the anomaly [^2]. Then the size of the kinetic mixing is either (A) $10^{-5} \lesssim \epsilon \lesssim 10^{-3}$ or (B) $\epsilon \ll 10^{-10}$ to satisfy beam-dump experimental data and supernova observations [@Hardy:2016kme; @Chang:2016ntp]. In the cosmological history, presumably much of the entropy in this sector ends up in a thermally populated gas of $\eta$ mesons. These decay via $\eta \rightarrow \gamma_{D} \gamma_{D} \rightarrow 2(e^{+} e^{-})$ before Big-Bang Nucleosynthesis (BBN) for the range (A) if $g_{D} \gtrsim 10^{-9}$. On the other hand, the direct detection forces $g_{D} \lesssim 10^{-6}$ for a 10 GeV DM particle.
Even though the nucleus is $U(1)_{D}$ neutral, it has a finite charge radius similar to the neutron, which approximately equals the DM size $r_\text{DM}$. Therefore the kinetic mixing between our photon and the dark photon leads to nuclear recoils. We estimate the corresponding current direct-detection limits by implementing such a recoil spectrum in DDCalc [@Athron:2018hpc; @Workgroup:2017lvb]. The results are shown in Fig. \[fig:DDbound\] for various choices of the charge radius and $\unit[1]{MeV}\ll m_{\gamma_D} \ll m_Z$.
![Direct detection bounds on our QCD-like theory of Puffy DM from Xenon1T [@Aprile:2017iyp], CMDSlite[@Agnese:2015nto] and CRESST II [@Angloher:2015ewa]. See text for details. []{data-label="fig:DDbound"}](DirectDetection.pdf){width="49.00000%"}
For the parameter range (B), the dark sector decouples from the Standard Model early, and hence it may be much cooler and the additional entropy ejection is limited. This case needs to be studied separately.
[**V. Conclusions.**]{} We have shown that if DM is a composite state with a size hundreds of times larger than its Compton wavelength, the corresponding self-interaction cross section varies with velocity in a way that is largely independent of its internal structure. For cross sections larger than $\unit[1]{cm^2/g}$ at $v\to 0$, this provides a solution to the problems of the $\Lambda$CDM model in small-scale astrophysical objects while still being in agreement with cluster observations. An important aspect of this scenario is that it does not require a long-range force mediating DM self-scatterings. A QCD-like theory where DM is a dark nucleon has been used to illustrate our results, which are nevertheless general and can be applied to a broader range of theories. For this reason, we believe Puffy DM opens up a new avenue for SIDM model-building.
[**Acknowledgments.**]{} We thank Fady Bishara, Bob Cahn, Ranjan Laha, Kai Schmidt-Hoberg and Sebastian Wild for enlightening discussions. X.C. is supported by the ‘New Frontiers’ program of the Austrian Academy of Sciences. C.G.C. is supported by the ERC Starting Grant NewAve (638528). H.M. thanks the Alexander von Humboldt Foundation for support while this work was completed. H.M. was supported by the NSF grant PHY-1638509, by the U.S. DOE Contract DE-AC02-05CH11231, by the JSPS Grant-in-Aid for Scientific Research (C) (17K05409), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (15H05887, 15K21733), and by WPI, MEXT, Japan.
Appendix: The transfer cross section {#appendix:sigma}
====================================
The transfer cross section for DM scattering is
$$\sigma_T = \frac{\sigma_0}{8\pi}\int d\Omega\,(1-|\cos\theta|)
\left[ \frac{F(q)^2}{1+\lambda^2q^2} + \left( \theta \to \pi-\theta \right)\right]^2
= \int^{\frac{(mv)^2}{2}}_0
\left[ \frac{ \,F(q)^2}{1+\lambda^2q^2} + \left(q^2 \to (mv)^2-q^2 \right) \right]^2\frac{2\sigma_0q^2dq^2 }{(mv)^4}. \label{app:eqsT}$$
Here we focus on the Puffy DM, where $r_\text{DM}^{-1} \ll \lambda^{-1}$. On the one hand, taking the low velocity limit, $m v \ll r_\text{DM}^{-1} $, the factor in the square bracket approaches $2$ and thus $\sigma_T \to \sigma_0$ at $v\to 0$. On the other hand, for $m v \gg r_\text{DM}^{-1}$, $F(q)$ is suppressed at $q \gg r_\text{DM}^{-1}$ so that the result of the integral is insensitive to its upper limit. The integral is not sensitive to $\lambda$ either, because for any $q \gtrsim \lambda^{-1}$ there is always $q \gg r_\text{DM}^{-1}$. Taking this into account allows us to approximate $\sigma_T$ by $$\begin{aligned}
\sigma_T \simeq \frac{2 \sigma_{0} }{m^4v^4}\, \int^\infty_0 dq^2 q^2 F(q)^4 \simeq {\sigma_{0} \over (c\, {m v } r _\text{DM})^4 } \,,\end{aligned}$$ with $c= 0.23,\,3.9,\,0.97 $ for the tophat, the dipole, and the Gaussian distributions, respectively. Therefore, $\sigma_T $ scales as $1/v^4$ at $m v \gg r_\text{DM}^{-1}$. The behavior derived here is different from that of effective range theories [@Bethe:1949yr], since the latter applies to each partial wave of the scattering cross section, while our result applies to the total transfer cross section. Note that at very large $mv$ incoherent scattering starts playing a role. Nevertheless, its contribution is much smaller than $\sigma_T$, and is therefore neglected for simplicity.
[^1]: Here we will assume that $v_\text{max} \to \infty$ since the integral converges quite fast due to the Boltzmann factor.
[^2]: For slightly heavier dark photons the process $\eta \eta \rightarrow \gamma_{D} \gamma_{D}$ can sufficiently reduce the abundance of $\eta$ mesons. If a dark axion, $a$, exists, the annihilation $\eta \eta \to \eta a$ can play the same role.
|
---
author:
- 'Gary R. Lawlor'
title: 'Double bubbles for immiscible fluids in $\RRR n$'
---
We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem [@hhs], [@hmrr], [@br].
As part of the proof we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.
Introduction
============
The double bubble problem in $\RRR 3$ and its variants have been a focus of research since about 1989. The first published proof for the minimization property of a multiple bubble was done by an undergraduate group advised by Frank Morgan and headed by Joel Foisy [@fo]. They proved that a standard double bubble in the plane (two overlapping disks separated by a circular arc or a line segment, with all bounding arcs meeting at $120^\circ$ angles) has the least perimeter required to separately enclose two given amounts of area.
Hass, Hutchings and Schlafly [@hhs] proved in 1995 that the least surface area required to separately enclose two equal amounts of volume is achieved by the standard double bubble.
A beautiful combination of symmetry and variational arguments, including ingeniously crafted variations, culminated in the triumph of Hutchings, Morgan, Ritoré and Ros [@hmrr] as they proved that in $\RRR 3$, standard double bubbles of unequal volumes also minimize surface area. This required analyzing all equilibrium double bubble surfaces of revolution and eliminating those that could not be minimizers because of instability of the equilibrium or because a fragment of a bubble was too small.
Students of Morgan soon extended the result to higher dimensions, an enterprise that culminated in Reichardt’s [@br] proof of the double bubble theorem in all dimensions.
The paper [@hmrr] also addresses the question of immiscible fluids, and proves that for certain volumes and for nearly-unit weights on surface area, the minimizers are standard. This is the result that we extend in the present paper to all volume pairs and all weights in $n$ dimensions:
0.1in [**Theorem 8.1.**]{} *Standard weighted double bubbles in $\RRR n$ all minimize weighted surface area among piecewise smooth boundaries of pairs of open regions with prescribed volume. These minimizers are unique.* 0.22in
Acknowledgments
===============
The author gained valuable insight and momentum at the Workshop on isoperimetric problems, space-filling, and soap bubble geometry in Edinburgh, Scotland, in March of 2012, and would like to express gratitude to the hosting International Centre for Mathematical Sciences and to the workshop organizers.
The author would also like to thank the referee for many helpful suggestions.
A new symmetry argument
=======================
We will give a new argument (Proposition \[sym\]) for why a minimizing double bubble must be a surface of revolution. This argument is more robust than the previously known proof given in [@hmrr] in that it requires less knowledge of regularity of the minimizer.
We first demonstrate the idea by giving a new proof of the regular isoperimetric theorem in $\RRR n$.
\[isoper\] The unique minimizer of surface area for enclosing a given volume in $\RRR n$ is a round ball.
The proof involves existence, bisection, angular stretch, and reflection. The heart of the argument is the fact that an angular stretch multiplies volume by a factor at least as large as it multiplies surface area, and the inequality is strict except on a surface of revolution.
We will first do the proof in $\RRR 2$. Given a quantity of area to be enclosed, there exists a region $C$ of minimum perimeter enclosing that area.
Choose any orthonormal basis for $\RRR 2$. Translate a line perpendicular to the first basis vector until it bisects the area of $C$. Call the bisecting line the $x$ axis.
Both halves of $C$ must have precisely half the original perimeter; otherwise we could replace the larger-perimeter half with the reflection of the smaller, contradicting minimality of $C$. Note also that none of the length of the perimeter can be contained in the bisecting line, since this length could be discarded before reflecting.
Choose one half of $C$, and translate a line perpendicular to the other basis vector until it bisects the area of that half of $C$. Call the bisecting line the $y$ axis.
Choose one of the resulting quarters of $C$ and call it $C'$. Then $C'$ has a quarter of the area and, necessarily, a quarter of the original perimeter of $C$.
Certainly $C'$ is connected; otherwise we could choose a component of $C'$ whose percentage of the perimeter of $C$ is no larger than its percentage of the area of $C$. Dilation stretches area by a larger factor than perimeter, so we could obtain a better piece than $C'$, and reflect it twice to contradict the minimality of $C$.
Now here is the main point. Suppose $C'$ is not a quarter circle centered at the origin. Simply do an angular stretch, mapping $\theta$ to $2\theta$, as in Figure 1.
\[ellipse\]  \[stretch\]
This doubles the area, but stretches the perimeter by *less* than 2, since it is not all lined up with the $\theta$ direction. Reflecting the stretched image across the $x$ axis then completes an enclosure that contradicts the minimality of $C$.
So $C'$ is a quarter circle. The same argument can be applied to the other quarter of $C$ above the $x$ axis to show that it, too, is a quarter circle. Of course, these quarter circles must line up since otherwise some of the perimeter would lie on the axis, which would lead to a contradiction as commented above.
Similarly the half of $C$ below the $x$ axis must be a half circle. A priori, the $y$ axis selected to bisect the bottom half of $C$ might not be the same as the one that bisects the top half, but in the end, the upper and lower half circles must meet. So $C$ is, indeed, a circle.
Now move to $\RRR n$. The argument is similar; bisect a minimizer $M$ with a hyperplane, choose a half and bisect it with a perpendicular hyperplane, and continue until obtaining a piece $M'$ of $M$ in an orthant of $\RRR n$ having $1/2^n$ of the original volume and surface area. One by one, do angular stretches (always by a factor of 2) on $M'$ until obtaining a region lying in a half space of $\RRR n$ and having half the volume of $M$. Now unite this region with its reflection. The result is an improvement on the original $M$ unless $M'$ was a piece of a round ball. Similarly, each of $2^n$ pieces of $M$ must be spherical pieces and must line up, so that the original $M$ had to be a round ball.
Unification
===========
We introduce here another key idea that we call *unification*, in which we combine a family of problems, with their differing constraints, into a single minimization problem, with all surfaces competing together. This placing of an optimization problem into a broader field makes it harder for a non-optimal competitor to have first variation zero. Indeed, in the double bubble case we will show that there are *no* competitors with unified first variation zero except the minimizers themselves: the standard weighted double bubbles. This will eliminate the need for handling the more difficult questions of instability of equilibria as in [@hmrr].
In order to allow constrained quantities to vary without losing the nature of the original question, we transfer the constraints over to the measurement scheme. We do this by dividing the measure of a competitor by the expected minimum measure for whatever constraints it satisfies. We call this quotient the *relative area* of the competitor surface.
This levels the playing field and unifies whole families of minimization problems — each with its conjectured minimizer — into the single problem of seeing whether the relative area can ever be less than 1. Constraints such as fixed volumes no longer keep the optimization problems segregated; when volumes change, this simply changes the denominator for the relative area calculation.
It is often helpful to reduce the set of competitors before unifying; in the present paper, for example, we will unify only after restricting our attention to surfaces of revolution.
\[uspace\] (Unification space and relative area) Let $\{(Q_\alpha,T_\alpha,M_\alpha)\}$ be a collection of minimization problems. For each $\alpha$, $T_\alpha$ denotes a set of competing objects (generically, “surfaces”) that vie to minimize the quantity $Q_\alpha$ (such as weighted surface area), and $M_\alpha\in T_\alpha$ denotes a *conjectured* minimizer.
We select for each $\alpha$ a subset $$\chi_\alpha\subseteq T_\alpha$$ containing $M_\alpha,$ and let $$\chi=\{(Q_\alpha,\chi_\alpha,M_\alpha)\}.$$
Then for any competitor $S\in \cup \chi_\alpha$, we define the *relative area* $\mu(S)$ by finding the class $\chi_\alpha$ that $S$ belongs to and letting $$\mu(S)=\frac{Q_\alpha(S)}{Q_\alpha(M_\alpha)}.$$
More generally, in the absence of conjectured minimizers for some constraint values, one might replace the denominator with a conjectured lower bound on the measurement of competitors satisfying those constraints.
Outline of the weighted double bubble proof
===========================================
Unification and the new symmetry argument will pave the way for the double bubble theorem to follow from an application of the Gauss map to the exterior of bubble clusters, amounting in essence to a calibration via the Gauss map. Gauss map calibration was introduced by Kleiner [@kl] for proving isoperimetric inequalities in manifolds; see also [@hhm] for another application of this idea.
Kleiner’s paper [@kl] notes (p. 38) an observation that he traces back to Almgren and others, that relates mean curvature to the derivative of surface area as volume changes in the isoperimetric profile for single bubbles. Unification can be seen as applying this observation to multiple bubbles and extending it beyond mean curvature to include other quantities such as weighted area.
Unification appears promising for a wide variety of minimization problems, and can be used in connection with a number of other methods. When applied to multiple bubble problems, one benefit of unification is to change average inequalities into piecewise inequalities. A unified equilibrium surface that did better than the expected minimum would not only have smaller *total* surface area than the corresponding standard double bubble, it must have smaller surface area for *each piece* (the exterior of each bubble and the interface). In addition, the mean curvature on each piece must be smaller.
These inequalities on corresponding pieces of a competitor versus the proposed minimizer open the way for the following simple argument.
First, smaller mean curvature for a comparison surface means smaller Gauss curvature (or in higher dimensions, product of principal curvatures — that is, the Jacobian of the Gauss map, also called Gauss-Kronecker curvature). This is because the standard double bubbles consist of spherical caps, whose equal principal curvatures already give the best possible conversion factor between the sum and the product of mean curvatures.
Second, the image of the Gauss map (applied to the exterior surfaces of a competitor) has overlap(s) because of the outward bending at the singular meeting(s) of the two bubbles. A nonstandard unified minimizer would, because of its disconnected singular set, have to have more overlap area than a standard double bubble; this is closely related to the isoperimetric theorem *within* the sphere; indeed, we use the latter in our proof.
The greater overlap would require such a minimizer to generate more total Gauss image area in order to cover the sphere. But it would have to achieve this with smaller-area domains and smaller Jacobians, which is impossible.
Double bubbles in $\RRR n$
==========================
\[swdb\] A *standard weighted double bubble* is made up of three distinct caps of $(n-1)$-spheres, all meeting along their common boundary $(n-2)$-sphere; the middle cap may be a flat $(n-1)$-disk. The angles between caps are related to the weights as in Figure 2; a triangle with sides perpendicular to the tangent planes at the junction must have side lengths proportional to the weights. This is equivalent to the condition that the three conormal vectors (i.e., tangent to the surface and normal to the boundary) with lengths equal to the weights should sum to zero.
We allow three degenerate cases for standard weighted double bubbles. The first involves disjoint spheres. This corresponds to an interface weight being at least as large as the sum of the two exterior weights. The second occurs when one prescribed volume is zero. The third case involves one sphere inside the other, and occurs when the exterior weight of the enclosed sphere is at least as large as the sum of the other exterior weight and the interface.
\[exists\] For any pair of nonnegative volumes (not both zero) and nonnegative weights (not all zero) there is exactly one standard weighted double bubble with those volumes and weights. Further, the radii and distance between centers in this bubble depend in a locally Lipschitz fashion upon the volumes and weights.
The degenerate cases, as described in Definition \[swdb\], are clear. Assume, then, that the weights satisfy the strict triangle inequality. In particular, all three are positive; label them $w_0$ for the interface and $w_1,w_2$ for the respective exteriors.
Our proof builds on that of Proposition 2.1 in [@hmrr].
Consider a unit sphere through the origin as in Figure 3. Form a small triangle with side lengths $w_0, w_1, w_2$, with the side of length $w_1$ being perpendicular to the tangent plane to the sphere at the origin and the side of length $w_2$ being exterior to the sphere.
Take another sphere intersecting the first at the origin (and elsewhere) at the correct angle so that the triangle’s side of length $w_2$ is perpendicular to the tangent plane to the second sphere at the origin.
Find the unique third sphere that contains the intersection of the first two spheres and whose tangent plane at the origin is perpendicular to the triangle side of length $w_0$.
Form a standard weighted double bubble from the caps of these spheres, with the bubble’s interface being a subset of the third sphere.
Now we will vary this figure. With the angles held fixed, reducing the size of the second sphere increases the curvatures of both the second sphere and the interface, causing the volume of bubble 1 to increase and the volume of bubble 2 to decrease. Increasing the size of the second sphere has the opposite effect. Thus, we can adjust the second sphere until the ratio of volumes is correct, after which we dilate the picture to obtain the correct volume pair. This gives existence and uniqueness.
For the Lipschitz dependence, suppose we have a standard weighted double bubble and we change slightly the prescribed volumes and/or weights. First adjust the angles at which the spheres meet, to accommodate the new weights; this changes the volumes slightly. To arrive at the new target volumes, as before adjust one radius until the volume ratio is correct, then dilate.
It is geometrically clear that in the step where one radius is held fixed, the partial derivatives by the other radius of both volumes are nonzero. The same is true in the dilation step. One volume or the other will either increase on both steps or decrease on both steps, which means that if either of the radius adjustments were large, then at least one volume change would have been large. This contradiction completes the proof of the local Lipschitz dependence.
We are now ready to define our unification space.
\[un\] For any pair of nonnegative volumes $(V_1,V_2)$ (not both zero) and nonnegative weights $w_0, w_1, w_2$ satisfying the triangle inequality, let $\alpha=(V_1,V_2,w_0,w_1,w_2)$ and define $\chi_\alpha$ to be the family of all double bubble competitors that are piecewise-smooth surfaces of revolution (unions of $(n-2)$-spheres centered on a common line), enclosing volumes $V_1,V_2$, with connected exterior region. Let $\chi$ be the union of all such $\chi_\alpha$.
Let $Q_\alpha(S)$ measure weighted surface area of a competitor $S$, with weights $w_1,w_2\in[0,1]$ on the exteriors of bubbles 1 and 2, respectively, and $w_0$ on the interface between them.
Our conjectured minimizers $M_\alpha$ are the standard weighted double bubbles of Definition \[swdb\].
To justify the assumption that the region of $\RRR n$ exterior to both bubbles is connected, suppose instead that some hollow were left unfilled by the surrounding bubbles. The boundary of the hollow consists of interface with bubble 1 and interface with bubble 2; whichever has larger (unit-weight) surface area can be deleted, thus filling the hollow with the appropriate bubble. This may increase the weight on the remaining interface, but because of the triangle inequality on the weights, the total weighted surface area will not increase. Since this process would increase a volume, the proposed minimum weighted surface area would also increase, and relative area would decrease.
0.1in
We need to discuss existence, symmetry and regularity of minimizers of $\mu$.
There exists a minimizer of relative area in the unification space $\chi$.
Of course, if the infimum of $\mu$ is 1, then every proposed minimizer realizes the minimum relative area. So suppose that the infimum of $\mu$ is $\mu_0<1$.
By Morgan’s argument in [@begin], sections 13.4 and 16.2, for any fixed volumes and weights satisfying the strict triangle inequality, there exists a double bubble that minimizes weighted surface area. A priori this is an integral current but might not be piecewise smooth, but we will soon see that it is, in fact, piecewise smooth.
On the other hand, if the weights only satisfy the non-strict triangle inequality, then any piece of the expensive surface may be thought of as two membranes superimposed, representing the two cheaper surfaces. Thus, by the regular isoperimetric inequality, we cannot do better than two separate spheres or one inside another, depending on whether the interface or one of the exteriors is the expensive surface.
Scaling of volume pairs or of weight triples does not affect relative area, so we may restrict attention to the classes in which the larger volume equals 1 and the largest weight equals 1. Then we get a minimum relative area function $f$ on a compact space of volume pairs and weights; if $f$ is continuous then there exists a minimizer of relative area. To verify continuity, note first that for any two nearby sets of volume pairs and weights, the proposed minima of weighted areas are nearly equal. Second, a small perturbation of the actual minimizer in either class gives a candidate for minimization in the other class; thus, the actual minimum is less than or nearly equal to the minimum in the other class. But this works in both directions, so the actual minima are nearly equal. Dividing by the nearly equal proposed minima, we obtain nearly equal relative areas, and we see that $f$ is continuous and there exists a $\mu$ minimizer $C_0$.
Now the weights that go with $C_0$ must satisfy the strict triangle inequality; otherwise $\mu(C_0)$ would equal 1. From this we can deduce that $C_0$ has finitely many bubble components; otherwise a tiny component of volume $\epsilon$ could be either eliminated or merged with the other bubble. One of these choices would remove at least half of the weighted surface area of the tiny component, which by the regular isoperimetric inequality is at least a constant times $\epsilon^{2/3}$. But the corresponding adjustment in the proposed minimum surface area would only amount to $O(\epsilon)$, so that $\mu$ would be reduced, yielding a contradiction.
Now by the symmetry argument below, $C_0$ must in fact be a surface of revolution. Each of the finitely many segments in the generating network of curves for $C_0$ must be a generalized Delaunay curve and thus smooth. So $C_0$ is, indeed, piecewise smooth. (Note also that this last argument also works for the minimizers within each class, so that all are piecewise smooth as claimed above.)
For symmetry, we begin with a version of the ham sandwich theorem for double bubbles.
\[bisectors\] Given two regions in $\RRR n$, every 2-plane in $\RRR n$ contains a vector normal to some hyperplane that divides the volumes of both regions in half.
Let $V_1$, $V_2$ be regions in $\RRR n$ and $P$ a 2-plane. Choose a point $\ppp\in P$. For each vector $\vvv\in P$ there is a hyperplane normal to $\vvv$ that bisects the volume of $V_1$. If $V_1$ is not connected this hyperplane may not be unique; make it unique by choosing it to pass as near to $\ppp$ as possible. Call the hyperplane $H(\vvv)$, and do the same for all vector directions $\vvv\in P$.
Let $f(\vvv)$ be the difference between the volume of $V_2$ lying on the positive side of $H(\vvv)$ (with respect to the direction of $\vvv$) and the volume of $V_2$ on the negative side.
Because of the uniqueness of $H(\vvv)$ it will follow that $f$ varies continuously with $\vvv$. Also, $f(-\vvv)=-f(\vvv)$ for all $\vvv$, so by the intermediate value theorem, $f$ equals zero for some $\vvv\in P$.
\[sym\] In $\RRR n$ with $n>2$, an integral current that is a minimizer of weighted surface area enclosing two fixed volumes must be a surface of revolution, that is, a union of $(n-2)$-spheres centered at points of a fixed axis line $L$. It follows that a minimizer of $\mu$ is also a surface of revolution.
We follow the ideas of the proof of the isoperimetric Theorem \[isoper\].
Let $M$ be a double bubble competitor that minimizes weighted surface area for its given enclosed volumes. Using Lemma \[bisectors\], find a hyperplane $H_1$ that bisects both volumes, and a second hyperplane perpendicular to $H_2$ that bisects both half-volumes on one side of $H_1$, and so forth until obtaining a piece $M'$ of $M$ bounded by $n-1$ hyperplanes and having $1/2^{n-1}$ of each original volume (and, necessarily, the same fraction of the original surface area) of $M$.
Let $L$ be the line formed by the intersection of the $n-1$ hyperplanes.
Iteratively apply angle-doubling stretches until matching half the volumes of $M$ on one side of a hyperplane. Reflect across the last hyperplane to complete a figure matching the original volumes of $M$. Now unless $M'$ was already a piece of a surface of revolution, this stretching will have multiplied surface area by less than it did volume, contradicting the minimality of $M$.
Each time when we chose one half or the other of a bisected pair of volumes, the choice was arbitrary. So all of $M$ must be a union of $2^{n-1}$ pieces of surfaces of revolution. If they did not match up, there would be surface area on the bisecting planes, leading to a contradiction as in the isoperimetric proof \[isoper\]. In the end, all of $M$ must be a surface of revolution. 0.1in
Of course, any minimizer of $\mu$ is also a minimizer of weighted surface area for its given enclosed volumes, so it, too, is a surface of revolution, as required.
Now that we know a minimizer is a surface of revolution, its regularity also follows:
\[regu\] The planar generating network of a minimizer of $\mu$ has finitely many junction points around which the exterior region and the two bubble regions meet. The junction points are connected by smooth curve segments.
If the triangle inequality on the surface weights is not strict, the isoperimetric inequality guarantees that a minimizer must consist of two spheres, either nested or disjoint.
Suppose then that the weights satisfy the strict triangle inequality. Then there is a lower bound on the volume of a component of a minimizer, since a microscopic component could either be deleted (merged with the exterior) or merged with the opposite bubble, depending on which option reduced the (unweighted) surface area the most. The isoperimetric inequality applied to the tiny component, together with the strictness of the triangle inequality on the weights, would then guarantee a sufficient decrease in weighted surface area to more than pay the tiny cost of adjusting the volumes in the proposed minimizer.
A single component could meet the exterior in many fingers, separated by little components of the opposite bubble. So we might still worry about the finiteness of the entire structure. But if we imagine building up the planar picture from scratch by adding one connected bubble component at a time, since the exterior region must be connected, each addition of a bubble component adds at most two new junction points. There are finitely many components, so there are finitely many junction points.
Between junction points, segments of curve are smooth, being governed by a differential equation arising from variational calculus.
\[existreg\] The smooth pieces of a minimizer of relative area meet in threes along $(n-2)$-spheres at angles matching those of the corresponding standard weighted double bubbles.
Let $S$ be a minimizer of $\mu$, which by Proposition \[sym\] is a surface of revolution. Let $M$ the standard double bubble with the same volumes and weights as $S$. Let $\Sigma_S$ be the network of planar curves that generate $S$, and $\Sigma_M$ the generating network for $M$.
It is a standard fact that these curves never meet in fours, since then two of the four separated regions would be pieces of the same region (or exterior) and the junction could either separate or pinch together and reduce the surface area they generate.
A consideration of forces pulling on the triple junction shows that the tangent vectors to the three curves meeting at a junction point, taken with lengths equal to the weights on the corresponding surface areas, must sum to zero. This matches the condition in Definition \[swdb\].
Critical points of $\mu$
========================
An analysis of critical points in a unification space includes the behavior in both the (abstract) interior and boundary of that space.
A point of the unification space boundary is, by definition, a set of prescribed volumes and weights that cannot be varied in all directions. For example, if the weights satisfy only the non-strict triangle inequality, then there are only certain ways in which we can vary the weights from there and still maintain the required triangle inequality.
The two types of boundary points are:
1. one of the prescribed volumes is zero, or
2. the weights only satisfy the non-strict triangle inequality.
In all such cases the regular isoperimetric inequality shows that the minimizers have relative area 1.
Requirements for an interior critical point of $\mu$
----------------------------------------------------
A critical point must, of course, be in equilibrium with respect to the standard variational principles that apply to *fixed* volumes and weights. Thus, a critical point will consist of pieces of constant mean curvature that meet at angles prescribed according to the weights (see Definition \[swdb\]).
\[lessarea\] Let $\mu_0$ be the minimum value of the relative area $\mu$ on $\mathcal{K}$, and let $w_0, w_1, w_2$ and $V_1,V_2$ be weights and volumes of some minimizer $C_0$ that achieves relative area $\mu_0$. Supposing that $\mu_0<1$, the surface $C_0$ will have to be a nonstandard constant mean curvature weighted double bubble of revolution. Let $M_0$ be the associated standard double bubble with the same weights and volumes.
Then the two exterior surface areas of $C_0$ must be less than or equal to $\mu_0$ times the corresponding surface areas of $M_0$.
In degenerate cases, $\mu_0=1$ and $C_0=M_0$.
Suppose now that the theorem is false for a nondegenerate case; say the surface area of $C_0$’s bubble 1 exterior is greater than $\mu_0$ times that of $M_0$. Now begin to reduce, at unit speed, the weight $w_1$. At the same time, move $M_0$ continuously through the space of proposed minimizers to continue to match the prescribed values of volume and weight. The important point here is that $M_0$ has first variation zero in weighted area $Q$ with respect to changes that preserve volumes, so the initial rate of change of $Q(M_0)$ is due entirely to the change in $w_i$ and not in the shape of $M_0$.
We see that the initial rates of change of $Q(M_0)$ and $Q(C_0)$ are equal to (minus) the respective areas of the exterior of bubble 1 in each. This causes $\mu$ to dip below $\mu_0$, contradicting the assumption that we already had the minimizer of $\mu$.
\[lesscurvature\] With hypotheses as in the previous proposition, the (constant) mean curvature on each exterior piece of $C_0$ is less than or equal to $\mu_0$ times the corresponding mean curvature on $M_0$.
The proof is similar to that of the previous proposition. Supposing the proposition false, push inward slightly on any exterior having larger mean curvature than specified. To first order this would change the surface areas of $M_0$ and $C_0$ by amounts that would enable a decrease in $\mu$, which was already at its minimum.
\[bigsleeve\] With the same hypotheses as above, the Gauss image of the exterior of either bubble of $C_0$ has less area than the Gauss image of the corresponding bubble exterior for $M_0$.
For fixed mean curvature, Gauss-Kronecker curvature (the product of principal curvatures) is largest when principal curvatures are equal, so that by Proposition \[lesscurvature\], the Gauss-Kronecker curvature at any regular point of $C_0$ is at most $\mu_0$ times the constant mean curvature on the corresponding spherical piece of $M_0$. But the Gauss-Kronecker curvature is also the Jacobian of the Gauss map. By Proposition \[lessarea\], the exterior pieces of $C_0$ have less area than the corresponding pieces of $M_0$, and with a smaller Jacobian, the areas of their Gauss images will also be smaller than for $M_0$.
Size of the Gauss map overlap due to the singular set
=====================================================
The singularities in the exterior of an equilibrium double bubble cause an overlap in the union of images of the Gauss map applied to the smooth portions of the exterior. We need to know that there is always more total overlap area for a nonstandard competitor than for a standard one.
An *antenna* of a competing surface of revolution is a vector bisecting the angle at a singularity, as in Figure 6. In the example of the figure, there are one *leftward-pointing* antenna and two *rightward-pointing* antennae. A vertical vector is considered as both leftward and rightward pointing.
The most important characteristic of an antenna will be its steepness, which we will also refer to as its height (independent of the position of the singular point where the vector starts).
A *sleeve* will refer to the Gauss image of a smooth exterior component of a competitor surface of revolution; an *end sleeve* is a spherical cap.
A *cuff* will refer to the overlap annulus formed as the intersection between two consecutive sleeves. By Proposition \[existreg\], cuffs have the same width, which is $\pi/3$ for unit surface weights.
The *inner perimeter* of a cuff is the smaller of the surface areas of the two $(n-2)$-dimensional boundary spheres of the cuff.
We will call one cuff higher than another if the corresponding antenna is steeper; equivalently, the higher of two cuffs is the one with a larger inner perimeter.
\[antennae\] 
Notice that a standard double bubble has two end sleeves that share one cuff. All other serious competitors will have at least one extra double-cuffed sleeve in the middle.
We need the following technical lemma.
\[tech\] Let $\beta>0$, $c<d$, and $f(t)>0$ for $t\in(c,d+\beta)$. Suppose that $f'(t)/f(t)$ is a decreasing function on $(c,d)$. Then the function $$h(t)=\frac{\int_{t}^{t+\beta}f(\tau)d\tau}{f(t)}$$ is also decreasing on $(c,d)$.
Differentiating $h(t)$ we get $$\frac{d h(t)}{dt}=\frac{f(t)\bigl(f(t+\beta)-f(t)\bigr)-f'(t)\int_{t}^{t+\beta}f(t)dt}{f(t)^2}.$$ Consider the numerator of the above expression. To show that it is negative, first note that it would be zero if $\beta=0$. Differentiating by $\beta$, we get $$f(t)f'(t+\beta)-f'(t)f(t+\beta)=f(t+\beta)f(t)\Bigl(\frac{f'(t+\beta)}{f(t+\beta)}-\frac{f'(t)}{f(t)}\Bigr).$$ If $f'/f$ is decreasing, the expression will be negative, as desired.
\[higherlower\] A higher cuff has greater surface area, while a lower cuff has greater surface area *per unit* of inner perimeter.
Set up spherical coordinates horizontally so as to agree with Figure 6, so that in spherical coordinates two cuffs are defined by $t_1 \le \phi \le u_1$ and $t_2\le \phi \le u_2$. By reflecting each cuff, if necessary, we may assume that $t_i\le \pi-u_i$ for $i=1,2$. Since the cuffs have the same width, we can define $\beta=u_1-t_1=u_2-t_2>0$.
Let $\alpha_k$ denote the $k$-dimensional volume of the unit ball in $\RRR k$, and set $$f(t)=(n-1)\alpha_{n-1}\sin^{n-1}(t).$$ Then the inner perimeters of the cuffs are $f(t_1)$ and $f(t_2)$, and the areas of the cuffs are $$\int_{t_1}^{u_1} f(t)\,dt \mbox{ and }\int_{t_2}^{u_2} f(t)\,dt.$$ To see that the higher cuff has the larger surface area, note that $$\frac{d}{ds}\int_s^{s+\beta} \sin^{n-1}(t)\,dt\,\,=\,\, \sin^{n-1}(s+\beta)-\sin^{n-1}(s),$$ which is positive until $\pi-(s+\beta)=s$.
Lemma \[tech\] shows that the lower cuff has greater surface area per unit of inner perimeter.
We now prove the main proposition of this section, showing that a hypothetical minimizer that did better than standard would have to have more overlap area than the standard.
\[moreover\] Suppose that $C_0$ is a minimizer of relative area (and thus a surface of revolution, by Proposition \[sym\]) and that $C_0$ has less weighted surface area than $M_0$, the corresponding standard weighted double bubble. Then $C_0$ must have a double-cuffed sleeve whose cuffs together have more area than the cuff of $M_0$.
Let $Y$ be the larger of the two sleeves of $M_0$, and $K_M$ its cuff. We will divide into two cases.
*Case 1:* All antennae of $C_0$ point to the same side — all leftward or all rightward. Since the largest sleeve of $C_0$ is an end sleeve and by Corollary 6.3 must be smaller than $Y$, its cuff must be higher than $K_M$, so by Proposition \[higherlower\], we are finished.
*Case 2:* Antennae point both ways. Consider any sleeve $\mathcal{S}_C$ having a leftward and a rightward antenna. As before, if either cuff of $\mathcal{S}_C$ is at least as high as $K_M$, then it has at least as much surface area, and together with the other cuff we have strictly more area, as desired.
On the other hand, suppose that both cuffs of the sleeve are lower than $K_M$. If we can show that the two cuffs together have more inner perimeter than does $K_M$, then by Proposition \[higherlower\], the total area of the two cuffs will be greater than the area of $K_M$.
By Proposition \[bigsleeve\], the sleeve $\mathcal{S}_C$ is smaller than $Y$, so its complement $\mathcal{S}_C'$ is larger than the complement $Y'$ on the unit sphere. Further, since there is a great sphere disjoint from $\mathcal{S}_C'$, as we reduce the area of $\mathcal{S}_C'$ to match that of $Y'$ we also reduce its total perimeter. Since $Y'$ is a spherical cap, by the isoperimetric theorem on the sphere, its perimeter is uniquely least for its given area. We deduce that the perimeter of $Y'$ (and thus of $Y$) is less than that of $\mathcal{S}_C'$ (and thus of $\mathcal{S_C}$). Now the perimeter of $\mathcal{S}_C$ is the total inner perimeter of its two cuffs, while the perimeter of $Y$ is the inner perimeter of its cuff $K_M$. By Proposition \[higherlower\], the cuffs have more area than $K_M$, as required.
The main theorem
================
\[mainthm\] Standard weighted double bubbles in $\RRR n$ all minimize weighted surface area among piecewise smooth boundaries of pairs of open regions with prescribed volume. These minimizers are unique.
Suppose some standard weighted double bubble (Definition \[swdb\]) is not a minimizer. By Morgan’s argument in [@begin], sections 13.4 and 16.2, and by Propositions \[sym\] and \[regu\], there must exist a piecewise smooth surface of revolution having the same volumes and smaller surface area than the standard. Restrict attention, then, to piecewise smooth surfaces of revolution, and expand the space of competitors to $\chi$ (see Definitions \[uspace\] and \[un\]), thus allowing volumes and surface weights to vary.
By Proposition \[existreg\], within $\chi$ there exists a minimizer $C_0$ of relative area $\mu$ with $\mu(C_0)=\mu_0$ for some $\mu_0<1$. Let $M_0$ be the standard weighted double bubble whose volumes and surface weights match those of $C_0$.
Now $C_0$ must be a critical point in the space $\chi$. Boundary critical points have $\mu=1$, so $C_0$ is an interior critical point.
By Propositions \[lessarea\] and \[lesscurvature\], the exterior surface of each bubble of $C_0$ has surface area and mean curvature at most $\mu_0$ times the corresponding value for $M_0$.
By Corollary 6.3, the Gauss image of the exterior of each bubble of $C_0$ has less area than the Gauss image of the corresponding bubble exterior for $M_0$. Moreover, by Proposition \[moreover\] there is more overlap in the Gauss image of $C_0$ than in the image of $M_0$. Taken together, these facts would prevent the Gauss image of the exterior of $C_0$ from covering the sphere, a contradiction.
To prove uniqueness, now suppose that besides a standard minimizer $M_1$ with certain volumes and weights, there exists another minimizer $M_2$ with the same volumes and weights. The surface areas and mean curvatures on exterior pieces of $M_2$ cannot exceed the corresponding quantities for $M_1$; otherwise a slight reduction in volume or weight would reduce $\mu$ below 1, which we now know cannot happen. The singular set on the exterior cannot consist of more than a single $(n-2)$-sphere or by Proposition \[moreover\] there would be too much Gauss image overlap, and the surface areas and mean curvatures on the exterior of $M_2$ must, in fact, equal those of $M_1$ or the Gauss map on the exterior of $M_2$ could not cover the sphere.
The exterior of $M_2$ must now be identical to that of $M_1$. Then the interface surface area and curvature must also be the same in $M_2$ as $M_1$, so that $M_2$ is forced to use the most economical way to connect its singular sphere to the axis, namely the same way that $M_1$ does so.
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---
abstract: |
In the classical contamination models, such as the gross-error (Huber and Tukey contamination model or Case-wise Contamination), observations are considered as the units to be identified as outliers or not, this model is very useful when the number of considered variables is moderately small. @Alqallaf2009 shows the limits of this approach for a larger number of variables and introduced the Independent contamination model (Cell-wise Contamination) where now the cells are the units to be identified as outliers or not. One approach to deal, at the same time, with both type of contaminations is filter out the contaminated cells from the data set and then apply a robust procedure able to handle case-wise outliers and missing values. Here we develop a general framework to build filters in any dimension based on statistical data depth functions. We show that previous approaches, e.g., @Agostinelli2015a and @Zamar2017 are special cases. We illustrate our method by using the half-space depth.
**Key Words**: Case-wise Contamination, Cell-wise Contamination, Filters, Robust Statistics, Statistical Data Depth Functions.
**Mathematics Subject Classification** 62G35 $\cdot$ 62G05
author:
- Giovanni Saraceno
- Claudio Agostinelli
title: Robust Multivariate Estimation Based On Statistical Data Depth Filters
---
Introduction {#sec:introduction}
============
One of most common problem in real data is the presence of outliers, i.e. observations that are well separated from the bulk of data, that may be errors that affect the data analysis or can suggest unexpected information. According to the classical Tukey-Huber Contamination Model (THCM), a small fraction of rows can be contaminated and these are the units considered as outliers. Since the $1960$’s many methods have been developed in order to be less sensitive to such outlying observations. A complete introduction and explanation of the developments in robust statistics is given in the book by @Maronna2006.
In some application, e.g. in modern high-dimensional data sets, the entries of an observation (or cells) can be independently contaminated. @Alqallaf2009 first formulated the Independent Contamination Model (ICM), taking into consideration this cell-wise contamination scheme. According to this paradigm, given a fraction $\epsilon$ of contaminated cells, the expected fraction of contaminated rows is $$1 - (1-\epsilon)^p$$ which exceeds the $50\%$ breakdown point for increasing value of the contaminatin level $\epsilon$ and the dimension $p$. Traditional robust estimators may fail in this situation. Furthermore, @Agostinelli2015b shows that both type of outliers, case-wise and cell-wise, can occur simultaneously.
@Gervini2002 introduced the idea of an adaptive univariate filter, identifying the proportion of outliers in the sample measuring the difference between the empirical distribution and a reference distribution. Then, it is used to compute an adaptive cutoff value, and finally a robust and efficient weighted least squares estimator is defined. Starting from this concept of outlier detection, @Agostinelli2015a introduced a two-step procedure: in the first step large cell-wise outliers are flagged by the univariate filter and replaced by NA’s values [a technique called snipping in @Farcomeni2014]; in the second step a Generalized S-Estimator [@Danilov2012] is applied to deal with case-wise outliers. The choice of using GSE is due to the fact that it has been specifically designed to cope with missing values in multivariate data. @Zamar2017 improved this procedure proposing the following modifications:
- They combined the univariate filter with a bivariate filter to take into account the correlations among variables.
- In order to handle also moderate cell-wise outliers, they proposed a filter as intersection between the univariate-bivariate filter and *Detect Deviating Cells* (DDC), a filter procedure introduced by @Rousseeuw2018.
- Finally, they constructed a Generalized Rocke S-estimator (GRE) replacing the GSE, to face the lost of robustness in case of high-dimensional case-wise outliers.
Here, we want to define a new filter in general dimension $d$, with $1 \le d \le p$, based on the statistical data depth functions and it will be used in combination with the GSE. Note that if $d=1$ we filter the cell-wise outliers considering the variables independent. Section \[sec:filters\] introduces the main idea on how to construct filters based on statistical depth functions, in subsection \[sec:halfspace\] we illustrate the procedure by using the half-space depth function while in subsections \[sec:pvariate1\] and \[sec:pvariate2\] we introduce two different strategies to mark observations/cells as outliers. Section \[sec:gerviniyohai\] shows how the approaches in @Agostinelli2015a and @Zamar2017 are special cases of our framework and we introduce a statistical data depth function namely Gervini-Yohai depth function. Section \[sec:example\] illustrates the features of our approach using a real data set while Section \[sec:simulation\] reports the results of a Monte Carlo experiment. Appendix \[sm:statisticaldatadepth\] discusses general properties a statistical data depth function should have, Appendix \[sm:gerviniyohaidepth\] studies the Gervini-Yohai depth properties and Appendix \[sm:simulation\] contains full results of the Monte Carlo experiment.
Filters based on Statistical Data Depth Function {#sec:filters}
================================================
Let ${\ensuremath{\boldsymbol{X}}}$ be a $\mathbb{R}^p$-valued random variable with distribution function $F$. For a point ${\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p$, we consider the statistical data depth of ${\ensuremath{\boldsymbol{x}}}$ with respect to $F$ be $d({\ensuremath{\boldsymbol{x}}};F)$ such that $d$ satisfies the four properties given in @Liu1990 and @Zuo2000a and reported in Appendix \[sm:statisticaldatadepth\] of the Supplementary Material. Given an independent and identically distributed sample ${\ensuremath{\boldsymbol{X}}}_1, \ldots, {\ensuremath{\boldsymbol{X}}}_n$ of size $n$, we denote $\hat{F}_n(\cdot)$ its empirical distribution function and by $d({\ensuremath{\boldsymbol{x}}}; \hat{F}_n)$ the sample depth. We assume that, $d({\ensuremath{\boldsymbol{x}}}; \hat{F}_n)$ is a uniform consistent estimator of $d({\ensuremath{\boldsymbol{x}}}; F)$, that is, $$\sup_{{\ensuremath{\boldsymbol{x}}}}| d({\ensuremath{\boldsymbol{x}}}; \hat{F}_n) - d({\ensuremath{\boldsymbol{x}}}; F) | \stackrel{a.s.}{\rightarrow} 0 \qquad n \rightarrow \infty ,$$ a property enjoined by many statistical data depth functions, e.g., among others simplicial depth [@Liu1990], half-space depth [@Tukey1975]. One important feature of the depth functions is the $\alpha$-depth trimmed region given by $R_\alpha(F) = \{ {\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p: d({\ensuremath{\boldsymbol{x}}}; F) \ge \alpha\}$; for any $\beta \in [0,1]$, we will denote $R^\beta(F)$ the smallest region $R_\alpha(F)$ that has probability larger that or equal to $\beta$ according to $F$. Throughout, subscripts and superscripts for depth regions are used for depth levels and probability contents, respectively. Let $C^\beta(F)$ be the complement in $\mathbb{R}^p$ of the set $R^\beta(F)$. Let $m = \max_{{\ensuremath{\boldsymbol{x}}}} d({\ensuremath{\boldsymbol{x}}};F)$, be the maximum of the depth, for simplicial depth $m \le 2^{-p}$, for half-space depth $m \le 1/2$.
Given a high order quantile $\beta$, we define a filter of dimension $p$ based on $$\label{general_definition}
d_n = \sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F)} \{ d({\ensuremath{\boldsymbol{x}}}; \hat{F}_n) - d({\ensuremath{\boldsymbol{x}}}; F) \}^+ ,$$ where $\{a\}^+$ represents the positive part of $a$, and we mark as outliers all the $\lfloor n d_n/m \rfloor$ observations with the smallest population depth (where $\lfloor a \rfloor$ is the largest integer less then or equal to $a$). This define a filter in the general dimension $p$.
We have the following result, with obvious proof.
\[generalconsistency\] If $\sup_{{\ensuremath{\boldsymbol{x}}}}| d({\ensuremath{\boldsymbol{x}}}; \hat{F}_n) - d({\ensuremath{\boldsymbol{x}}}; F) | = o(n)$ (a.s.) then $n d_n \rightarrow 0$ as $n \rightarrow \infty$.
If the above result holds, then the filter would be consistent. In the next subsection we are going to illustrate this approach using the half-space depth.
Filters based on Half-space Depth {#sec:halfspace}
---------------------------------
Let ${\ensuremath{\boldsymbol{X}}}$ be a $\mathbb{R}^p$-valued random variable with distribution function $F$. For a point ${\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p$, the half-space depth of ${\ensuremath{\boldsymbol{x}}}$ with respect to $F$ is defined as the minimum probability of all closed half-spaces including ${\ensuremath{\boldsymbol{x}}}$: $$d_{HS} ({\ensuremath{\boldsymbol{x}}};F) = \min_{H \in \mathcal{H}({\ensuremath{\boldsymbol{x}}})} P_F({\ensuremath{\boldsymbol{X}}} \in H).$$ where $\mathcal{H}({\ensuremath{\boldsymbol{x}}})$ indicates the set of all half-spaces in $\mathbb{R}^p$ containing ${\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p$.
A random vector ${\ensuremath{\boldsymbol{X}}} \in \mathbb{R}^p$ is said elliptically symmetric distributed, denoted by ${\ensuremath{\boldsymbol{X}}} \sim E_p(h, {\ensuremath{\boldsymbol{\mu}}}, {\ensuremath{\boldsymbol{\Sigma}}})$, if it has a density function given by $$f_0({\ensuremath{\boldsymbol{x}}}) \propto |{\ensuremath{\boldsymbol{\Sigma}}}^{-1/2}| h(({\ensuremath{\boldsymbol{x}}}-{\ensuremath{\boldsymbol{\mu}}})^\top{\ensuremath{\boldsymbol{\Sigma}}}^{-1}({\ensuremath{\boldsymbol{x}}}-{\ensuremath{\boldsymbol{\mu}}})) .$$ where $h$ is a non-negative scalar function, ${\ensuremath{\boldsymbol{\mu}}}$ is the location parameter and ${\ensuremath{\boldsymbol{\Sigma}}}$ is a $p \times p$ positive definite matrix. Denote by $F_0$ the corresponding distribution function and by $\Delta_{{\ensuremath{\boldsymbol{x}}}} = ({\ensuremath{\boldsymbol{x}}} - {\ensuremath{\boldsymbol{\mu}}})^\top {\ensuremath{\boldsymbol{\Sigma}}}^{-1} ({\ensuremath{\boldsymbol{x}}} - {\ensuremath{\boldsymbol{\mu}}})$ the squared Mahalanobis distance of a $p$-dimensional point ${\ensuremath{\boldsymbol{x}}}$. By Theorem 3.3 of @Zuo2000b if a depth is affine equivariant (\[sm:PropDepthAffineInvariance\]) and has maximum at ${\ensuremath{\boldsymbol{\mu}}}$ (\[sm:PropDepthMaximalityAtCenter\]) (see Appendix \[sm:statisticaldatadepth\]) then a depth is such that $d({\ensuremath{\boldsymbol{x}}}; F_0) = g(\Delta_{{\ensuremath{\boldsymbol{x}}}})$ for some non increasing function $g$ and we can restrict ourselves without loss of generality, to the case ${\ensuremath{\boldsymbol{\mu}}} = {\ensuremath{\boldsymbol{0}}}$ and ${\ensuremath{\boldsymbol{\Sigma}}} = {\ensuremath{\boldsymbol{I}}}$ where ${\ensuremath{\boldsymbol{I}}}$ is the identity matrix of dimension $p$. Under this setting, it is easy to see that the half-space depth of a given point ${\ensuremath{\boldsymbol{x}}}$ is given by $d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0) = 1 - F_{0,1}(\sqrt{\Delta_{{\ensuremath{\boldsymbol{x}}}}})$, where $F_{0,1}$ is a marginal distribution of ${\ensuremath{\boldsymbol{X}}}$.
If the function $h$ is such that $$\frac{\exp(-\frac{1}{2} \Delta)}{h(\Delta)} \rightarrow 0 , \qquad \Delta \rightarrow \infty ,$$ then, there exists a $\Delta^\ast$ such that for all ${\ensuremath{\boldsymbol{x}}}$ so that $\Delta_{{\ensuremath{\boldsymbol{x}}}} > \Delta^\ast$, $d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0) \ge d_{HS}({\ensuremath{\boldsymbol{x}}}; \Phi)$, where $\Phi$ is the distribution function of the standard normal. Hence, $$\sup_{\{{\ensuremath{\boldsymbol{x}}}: \Delta_{{\ensuremath{\boldsymbol{x}}}} > \Delta^\ast\}} [d_{HS}({\ensuremath{\boldsymbol{x}}}; \Phi) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0)] < 0$$ and therefore, for all $\beta > 1 - 2 F_{0,1}(-\sqrt(\Delta^{\ast}))$, $$\sup_{C^\beta(F_0)} [d_{HS}({\ensuremath{\boldsymbol{x}}}; \Phi) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0)] < 0 \ .$$
Given an independent and identically distributed sample ${\ensuremath{\boldsymbol{X}}}_1, \ldots, {\ensuremath{\boldsymbol{X}}}_n$, we define the filter in general dimension $p$ introduced previously, where here we use the half-space depth $$d_n = \sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F)} \{ d_{HS}({\ensuremath{\boldsymbol{x}}}; \hat{F}_n) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n})) \}^+ ,$$ where $\beta$ is a high order quantile, $\hat{F}_n(\cdot)$ is the empirical distribution function and $F({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n})$ is a chosen reference distribution which depends on a pair of initial location and dispersion estimators, ${\ensuremath{\boldsymbol{T}}}_{0n}$ and ${\ensuremath{\boldsymbol{C}}}_{0n}$. Hereafter, we are going to use the normal distribution $F = N({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n})$. For ${\ensuremath{\boldsymbol{T}}}_{0n}$ and ${\ensuremath{\boldsymbol{C}}}_{0n}$ one might use, e.g., the coordinate-wise median and the coordinate-wise MAD for a univariate filter as in @Zamar2017. In order to compute the value $d_n$, we have to identify the set $C^\beta(F) = \{ {\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p | d_{HS}({\ensuremath{\boldsymbol{x}}},F) \le d_{HS}(\eta_\beta,F) \}$ where $\eta_\beta$ is a large quantile of $F$. By Corollary 4.3 in @Zuo2000b, and denoting with $\Delta_x = ({\ensuremath{\boldsymbol{x}}} - {\ensuremath{\boldsymbol{T}}}_{0n})^\top {\ensuremath{\boldsymbol{C}}}_{0n}^{-1}({\ensuremath{\boldsymbol{x}}} - {\ensuremath{\boldsymbol{T}}}_{0n})$ the squared Mahalanobis distance of ${\ensuremath{\boldsymbol{x}}}$ using the initial location and dispersion estimates, the set can be rewritten as $C^\beta(F) = \{ {\ensuremath{\boldsymbol{x}}} \in \mathbb{R}^p | \Delta_x > (\chi^2_p)^{-1}(\beta) \}$, where $(\chi^2_p)^{-1}(\beta)$ is a large quantile of a chi-squared distribution with $p$ degrees of freedom.
Now we want to show that the result given by Proposition \[generalconsistency\] holds for this particular case.
Consider a random vector $({\ensuremath{\boldsymbol{X}}}_1, \ldots , {\ensuremath{\boldsymbol{X}}}_n) \sim F_0({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)$ and suppose that $F_0$ is an elliptically symmetric distribution. Also consider a pair of location and dispersion estimators ${\ensuremath{\boldsymbol{T}}}_{0n}$ and ${\ensuremath{\boldsymbol{C}}}_{0n}$ such that ${\ensuremath{\boldsymbol{T}}}_{0n} \rightarrow {\ensuremath{\boldsymbol{\mu}}}_0$ and ${\ensuremath{\boldsymbol{C}}}_{0n} \rightarrow {\ensuremath{\boldsymbol{\Sigma}}}_0$ a.s.. Let $F$ be a chosen reference distribution and $\hat{F}_n$ the empirical distribution function. If the reference distribution satisfies $$\sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F_0)} [d_{HS}({\ensuremath{\boldsymbol{x}}}; F) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0)] < 0$$ where $\beta$ is some large quantile of $F_0$, then $$n d_n \rightarrow 0 \mbox{ as } n \rightarrow \infty$$
In @Donoho1992, it is proved that for i.i.d. ${\ensuremath{\boldsymbol{X}}}_1, {\ensuremath{\boldsymbol{X}}}_2, ... , {\ensuremath{\boldsymbol{X}}}_n$ with distribution $F_0$, as $n \rightarrow \infty$ $$\sup_{{\ensuremath{\boldsymbol{t}}} \in \mathbb{R}^d} |d_{HS}({\ensuremath{\boldsymbol{t}}},F_0) - d_{HS}({\ensuremath{\boldsymbol{t}}},\hat{F}_n)| \rightarrow 0 \mbox{ a.s.}$$ Note that, by the continuity of $F$, $F({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n}) \rightarrow F({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)$ a.s.. Hence, for each $\varepsilon > 0$ there exists $n_0$ such that for all $n > n_0$ we have $$\begin{aligned}
\sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F_0)} \{ d_{HS}({\ensuremath{\boldsymbol{x}}}; & \hat{F}_n) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n})) \} \le \\
& \sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F_0)} \{ d_{HS}({\ensuremath{\boldsymbol{x}}}; \hat{F}_n) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)) \} + \\
& \sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F_0)} \{ d_{HS}({\ensuremath{\boldsymbol{x}}}; F_0({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)) \} + \\
& \sup_{{\ensuremath{\boldsymbol{x}}} \in C^\beta(F_0)} \{ d_{HS}({\ensuremath{\boldsymbol{x}}}; F({\ensuremath{\boldsymbol{\mu}}}_0, {\ensuremath{\boldsymbol{\Sigma}}}_0)) - d_{HS}({\ensuremath{\boldsymbol{x}}}; F({\ensuremath{\boldsymbol{T}}}_{0n}, {\ensuremath{\boldsymbol{C}}}_{0n})) \} \\
\le & \frac{\varepsilon}{2} + 0 + \frac{\varepsilon}{2} = \varepsilon\end{aligned}$$
In the next example, we illustrate a univariate filter based on half-space depth that controls independently the left and the right tail of the distribution.
In the univariate case, given a point $x$ there exist only two halfspaces including it, hence the half-space depth assumes the explicit form $$\begin{aligned}
d_{HS}(x;F) & = \min (P_F((-\infty,x]),P_F([x,\infty))) \\
& = \min (F(x) , 1 - F(x) + P_F(X = x)),\end{aligned}$$ and considering the empirical distribution function $\hat{F}_n(\cdot)$, the halfspace depth will be $$d_{HS}(x,\hat{F}_n) = \min \biggl( \frac{1}{n} \sum_{i=1}^n I(X_i \leq x) , \frac{1}{n} \sum_{i=1}^n I(X_i \geq x) \biggr).$$
Consider ${\ensuremath{\boldsymbol{T}}}_{0n} = (T_{0n,1}, \ldots, T_{0n,p})$ and ${\ensuremath{\boldsymbol{S}}}_{0n} = (S_{0n,1}, \ldots, S_{0n,p})$, a pair of initial location and dispersion estimators. Here we choose for ${\ensuremath{\boldsymbol{T}}}_{0n}$ and ${\ensuremath{\boldsymbol{S}}}_{0n}$ respectively the coordinate-wise median and the median absolute deviation (MAD). For each variable $(X_{1j}, X_{2j}, \ldots, X_{nj})$ ($j=1, \ldots, p$), we denote the standardized version of $X_{ij}$ by $Z_{ij}= \frac{X_{ij} - T_{0n,j}}{S_{0n,j}}$. Let $F_j$ a chosen reference distribution for $Z_{ij}$; here we use the standard normal distribution, i.e., $F_j = \Phi$. Let $\hat{F}_{n,j}$ be the empirical distribution for the standardized values, that is $$\hat{F}_{n,j}(t) = \frac{1}{n} \sum_{i=1}^{n} I(Z_{ij} \leq t) \qquad j = 1, \ldots, p.$$ We define the proportion of flagged outliers by $$d_{n,j} = \max \biggl( \sup_{t \leq -\eta_{\beta,j}} \{d_{HS}(t,\hat{F}_{n,j}) - d_{HS}(t, F_j)\}^+ ; \sup_{t \geq \eta_{\beta,j}} \{d_{HS}(t,\hat{F}_{n,j}) - d_{HS}(t, F_j)\}^+ \biggr),$$ where $\eta_{\beta,j} = F_j^{-1}(\beta)$ is a large quantile of $F_j$. Note that, according to (\[general\_definition\]), we are considering the set $C^\beta(F_j) = \{x \in \mathbb{R} : d_{HS}(x,F_j) < d_{HS}(\eta_{\beta,j})\}$, which results in the simpler form written above considering the definition of the half-space depth in the univariate case. Here, if we consider the order statistics $Z_{(i),j}$, define $i_- = \min \{ i : Z_{(i),j} > -\eta_{\beta,j} \}$ and $i_+ = \max \{ i : Z_{(i),j} < \eta_{\beta,j} \}$. Using the definition of half-space depth function in the univariate case, presented above, the previous expression can be written as $$d_{n,j} = \max \biggl( \sup_{i < i_-} \{\frac{i}{n} - F_j(Z_{(i),j})\}^+ , \sup_{i > i_+} \{ F_j(Z_{(i),j}) - \frac{i-1}{n}\}^+ \biggr).$$ Then, we flag $\lfloor nd_{n,j} \rfloor$ observations with the smallest depth value as cell-wise outliers and replace them by NA’s.
A consistent univariate, bivariate and $p$-variate filter {#sec:pvariate1}
---------------------------------------------------------
Given a sample ${\ensuremath{\boldsymbol{X}}}_1, \ldots, {\ensuremath{\boldsymbol{X}}}_n$ where ${\ensuremath{\boldsymbol{X}}}_i \in \mathbb{R}^p, i = 1, \ldots, n$, we first apply the univariate filter described in the previous example to each variable separately. Filtered data are indicated through an auxiliary matrix ${\ensuremath{\boldsymbol{U}}}$ of zeros and ones, with zero corresponding to a NA value. Next we want to identify the bivariate outliers by iterating the filter over all possible pairs of variables. Consider a pair of variables ${\ensuremath{\boldsymbol{X}}}^{(jk)} = \{ (X_{ij},X_{ik}) \}, i = 1, \ldots, n$. The initial location and dispersion estimators are, respectively, the coordinate-wise median and the $2 \times 2$ sub-matrix $S^{(jk)}$ of the estimate $S$ computed by the generalized S-estimator on non-filtered data. Note that, this ensure the positive definiteness property for $S$ and each $d \times d$ sub-matrix corresponding to a subset of $d$ variables. For bivariate points with no flagged components by the univariate filter we compute the squared Mahalanobis distance $\Delta^{(jk)}_i$ and hence apply the bivariate filter, for all $1 < j < k < p$. At the end we want to identify the cells $(i,j)$ which have to be flagged as cell-wise outliers. The procedure used for this purpose is described in @Zamar2017 and reported here. Let $$J = \{ (i,j,k) : \Delta_i^{(jk)} \mbox{ is flagged as bivariate outlier} \}$$ be the set of triplets which identifies the pairs of cells flagged by the bivariate filter in rows $i = 1, \ldots, n$. For each cell $(i,j)$ in the data, we count the number of flagged pairs in the $i$-th row in which the considered cell is involved: $$m_{ij} = \#\{ k : (i,j,k) \in J\}.$$ In absence of contamination, $m_{ij}$ follows approximately a binomial distribution $Bin(\sum_{k \not = j}{\ensuremath{\boldsymbol{U}}}_{jk},\delta)$ where $\delta$ represents the overall proportion of cell-wise outliers undetected by the univariate filter. Hence, we flag the cell $(i,j)$ if $m_{ij} > c_{ij}$, where $c_{ij}$ is the $0.99$-quantile of $Bin(\sum_{k \not = j}{\ensuremath{\boldsymbol{U}}}_{jk},0.1)$. Finally, we perform the $p$-variate filter as described in subsection \[sec:halfspace\] to the full data matrix. Detected observations (rows) are directly flagged as $p$-variate (case-wise) outliers. We denote the procedure based on univariate, bivariate and $p$-variate filters, HS-UBPF.
A sequencing filtering procedure {#sec:pvariate2}
--------------------------------
Suppose we would like to apply a sequence of $k$ filters with different dimension $1 \le d_1 \le d_2 \le \ldots \le d_k \le p$. For each $d_i$, $i = 1, \ldots, k$, the filter updates the data matrix adding NA values to the $d_i$-tuples identified as $d_i$-variate outliers. In this way, each filter applies only those $d_i$-tuples that have not been flagged as outliers by the filters with lower dimension.
Initial values for each procedures rather than $d_1$ would be obtained by applying the GSE to the actual filtered values.
This procedure aims to be a valid alternative to that used in the presented HS-UBPF filter to perform a sequence of filters with different dimensions. However, this is a preliminary idea, indeed it has not been implemented yet.
Gervini-Yohai $d$-variate filter {#sec:gerviniyohai}
================================
In this Section we are going to show that the filters introduced in [@Agostinelli2015a] are a special case of our approach, using the following Gervini-Yohai depth $$d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G) = 1 - G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) ,$$ where $G$ is a continuous distribution function, ${\ensuremath{\boldsymbol{\mu}}}(F)$ and ${\ensuremath{\boldsymbol{\Sigma}}}(F)$ are the location and scatter matrix functionals and $\Delta(t, F) = \Delta({\ensuremath{\boldsymbol{t}}}, {\ensuremath{\boldsymbol{\mu}}}(F), {\ensuremath{\boldsymbol{\Sigma}}}(F)) = ({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F))^\top {\ensuremath{\boldsymbol{\Sigma}}}(F)^{-1} ({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F))$ is the squared Mahalanobis distance. Appendix \[sm:gerviniyohaidepth\] shows that this is a statistical data depth function. Let $\{ G_n \}_{n=1}^\infty$ be a sequence of discrete distribution functions that might depends on $\hat{F}_n$ and such that $\sup_{t} |G_n(t) - G(t)| \stackrel{a.s.}{\rightarrow} 0$, we might define the finite sample version of the Gervini-Yohai depth as $$d_{GY}({\ensuremath{\boldsymbol{t}}}, \hat{F}_n, G_n) = 1 - G_n(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(\hat{F}_n),{\ensuremath{\boldsymbol{\Sigma}}}(\hat{F}_n))) \ ,$$ however for filtering purpose we will use two alternative definitions later on. The use of $G_n$, that might depend on the data, instead of $G$ makes this sample depth semiparametric. We notice that the Mahalanobis depth, which is completely parametric, cannot be used for the purpose of defining a filter in a similar fashion.
Let $1 \le d \le p$, $j_1, \ldots, j_d$ be an $d$-tuple of the integer numbers $1, \ldots, p$ and, for easy of presentation, let ${\ensuremath{\boldsymbol{Y}}}_i = (X_{ij_1}, \ldots , X_{ij_d})$ be a subvector of dimension $d$ of ${\ensuremath{\boldsymbol{X}}}_i$. Consider a pair of initial location and scatter estimators $${\ensuremath{\boldsymbol{T}}}_{0n}^{(d)} = \left (
\begin{array}{ll}
T_{0n,j_1} \\
\ldots \\
T_{0n,j_d}
\end{array}
\right )
\quad \mbox{ and } \quad
{\ensuremath{\boldsymbol{C}}}_{0n}^{(d)} = \left (
\begin{array}{lll}
C_{0n,j_1j_1} & \ldots & C_{0n,j_1j_d} \\
\ldots &\ldots & \ldots \\
C_{0n,j_dj_1}& \ldots & C_{0n,j_dj_d}
\end{array}
\right ) \ .$$ Now, define the squared Mahalanobis distance for a data point ${\ensuremath{\boldsymbol{Y}}}_i$ by $\Delta_i = \Delta({\ensuremath{\boldsymbol{Y}}}_i, \hat{F}_n) = \Delta({\ensuremath{\boldsymbol{Y}}}_i, {\ensuremath{\boldsymbol{T}}}_{0n}^{(d)}, {\ensuremath{\boldsymbol{C}}}_{0n}^{(d)})$. Consider $G$ the distribution function of a $\chi_d^2$, $H$ the distribution function of $\Delta = \Delta(\cdot, F)$ and let $\hat{H}_n$ be the empirical distribution function of $\Delta_i$ ($1 \le i \le n$). We consider two finite sample version of the Gervini-Yohai depth, i.e., $$d_{GY}({\ensuremath{\boldsymbol{t}}}, \hat{F}_n, G) = 1 - G(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n)) ,$$ and $$d_{GY}({\ensuremath{\boldsymbol{t}}}, \hat{F}_n, \hat{H}_n) = 1 - \hat{H}_n(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n)) .$$ The proportion of flagged $d$-variate outliers is defined by $$d_n = \sup_{{\ensuremath{\boldsymbol{t}}} \in A} \{ d_{GY}({\ensuremath{\boldsymbol{t}}}, \hat{F}_n, \hat{H}_n) - d_{GY}({\ensuremath{\boldsymbol{t}}}, \hat{F}_n, G) \}^+ .$$ Here $A = \{ {\ensuremath{\boldsymbol{t}}} \in \mathbb{R}^d: d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G) \leq d_{GY}({\ensuremath{\boldsymbol{\zeta}}}, F, G) \}$, where ${\ensuremath{\boldsymbol{\zeta}}}$ is any point in $\mathbb{R}^d$ such that $\Delta({\ensuremath{\boldsymbol{\zeta}}}, F) = \eta$ and $\eta = G^{-1}(\alpha)$ is a large quantile of $G$. Then, we flag $\lfloor nd_{n} \rfloor$ observations. It is easy to see that, $$\begin{aligned}
d_n & = \sup_{{\ensuremath{\boldsymbol{t}}} \in A} \{ [1 - \hat{H}_n(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n))] - [1 - G(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n))] \}^+ \\
& = \sup_{{\ensuremath{\boldsymbol{t}}} \in A} \{ G(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n)) - \hat{H}_n(\Delta({\ensuremath{\boldsymbol{t}}}, \hat{F}_n)) \}^+ \\
& = \sup_{\Delta \ge \eta} \{ G(\Delta) - \hat{H}_n(\Delta) \}^+\end{aligned}$$ since $d_{GY}$ is a non increasing function of the squared Mahalanobis distance of the point ${\ensuremath{\boldsymbol{t}}}$.
We can rephrase Proposition 2. in @Zamar2017, that states the consistency property of the filter as follows.
Consider a random vector ${\ensuremath{\boldsymbol{Y}}} = (X_{1}, \ldots, X_{d}) \sim F_0$ and a pair of location and scatter estimators ${\ensuremath{\boldsymbol{T}}}_{0n}$ and ${\ensuremath{\boldsymbol{C}}}_{0n}$ such that ${\ensuremath{\boldsymbol{T}}}_{0n} \rightarrow {\ensuremath{\boldsymbol{\mu}}}_0 = {\ensuremath{\boldsymbol{\mu}}}(F_0) \in \mathbb{R}^d$ and ${\ensuremath{\boldsymbol{C}}}_{0n} \rightarrow {\ensuremath{\boldsymbol{\Sigma}}}_0 = {\ensuremath{\boldsymbol{\Sigma}}}(F_0)$ a.s.. Consider any continuous distribution function $G$ and let $\hat{H}_n$ be the empirical distribution function of $\Delta_i$ and $H_0(t) = \Pr (({\ensuremath{\boldsymbol{Y}}} - {\ensuremath{\boldsymbol{\mu}}}_0)^t {\ensuremath{\boldsymbol{\Sigma}}}_0^{-1}({\ensuremath{\boldsymbol{Y}}} - {\ensuremath{\boldsymbol{\mu}}}_0) \le t )$. If the distribution $G$ satisfies: $$\label{equ:prop2}
\max_{{\ensuremath{\boldsymbol{t}}} \in A} \{ d_{GY}({\ensuremath{\boldsymbol{t}}},F_0,H_0) - d_{GY}({\ensuremath{\boldsymbol{t}}},F_0,G) \} \le 0 ,$$ where $A = \{ {\ensuremath{\boldsymbol{t}}} \in \mathbb{R}^d: d_{GY}({\ensuremath{\boldsymbol{t}}}, F_0, G) \leq d_{GY}({\ensuremath{\boldsymbol{\zeta}}}, F_0, G) \}$, where ${\ensuremath{\boldsymbol{\zeta}}}$ is any point in $\mathbb{R}^d$ such that $\Delta({\ensuremath{\boldsymbol{\zeta}}}, F_0) = \eta$ and $\eta = G^{-1}(\alpha)$ is a large quantile of $G$, then $$\frac{n_0}{n} \rightarrow 0 \qquad \text{a.s.}$$ where $$n_0 = \lfloor nd_{n} \rfloor .$$
Note that $$d_{GY}({\ensuremath{\boldsymbol{t}}},\hat{F}_n,\hat{H}_n) - d_{GY}({\ensuremath{\boldsymbol{t}}},\hat{F}_n, G) = G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{T}}}_{0n},{\ensuremath{\boldsymbol{C}}}_{0n})) - \hat{H}_n(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{T}}}_{0n},{\ensuremath{\boldsymbol{C}}}_{0n}))$$ and condition in equation (\[equ:prop2\]) is equivalent to $$\max_{\Delta \ge \eta} \{G(\Delta) - H_0(\Delta) \} \le 0 ,$$ The rest of the proof is the same as in Proposition 2. of @Zamar2017.
Example {#sec:example}
=======
We consider the weekly returns from $01/08/2008$ to $12/28/2010$ for a portfolio of 20 small-cap stocks used in @Zamar2017.
With this example we want to compare the filter introduced in @Agostinelli2015a (indicated as GY-UF in case of univariate filter and GY-UBF for univariate and bivariate filter) and the same filter with the improvements proposed in @Zamar2017 (indicated here as GY-UBF-DDC-C) to the presented filter based on statistical data depth functions, using the halfspace depth (HS-UF for the univariate filter, HS-UBF for the univariate-bivariate filter, HS-UBPF for the univariate-bivariate-$p$-variate filter and HS-UBPF-DDC-C for the combination of the HS-UBPF with the modifications in @Zamar2017).
![Small-cap stock returns. QQ-plots of the variables, green: observations marked as outliers.[]{data-label="fig:scsr-qqplots"}](qqplot-small-cap){width="\textwidth"}
Figure \[fig:scsr-qqplots\] shows the normal QQ-plots of the 20 variables. The returns in all stocks seem to roughly follow a normal distribution, but with the presence of large outliers. The returns in each stock that lie 3 MAD’s away from the coordinate-wise median are displayed in green in the figure. In total, the $4.4\%$ of cells are outside; if these are cell-wise outliers then they propagate to $37.6\%$ of the cases.
![Squared Mahalanobis distances of the weekly returns based on the MLE, the GY filters (GY-UF, GY-UBF) and the filters based on half-space depth (HS-UF, HS-UBF, HS-UBPF). Observations with one or more cells flagged as outliers are displayed in green.[]{data-label="fig:md"}](mahalanobis-distance){width="\textwidth"}
Figure \[fig:md\] shows the squared Mahalanobis distances (MDs) of the weekly returns based on the estimates given by the MLE, the GY-UF, the GY-UBF, the HS-UF, the HS-UBF and the HS-UBPF. Observations with one or more cells flagged as outliers are displayed in green. We say that the estimate identifies an outlier correctly if the MD exceeds the $99.99\%$ quantile of a chi-squared distribution with 20 degrees of freedom. We see that the MLE estimate does a very poor job recognizing only 8 of the 59 cases. The GY-UF, HS-UF, HS-UBF and HS-UBPF show a quite similar behavior, doing better then the MLE but they miss about one third of the cases. The GY-UBF identifies all but seven of the cases.
![Squared Mahalanobis distances of the weekly returns based on the GY-UBF-DDC-C and the corresponding filter based on halfaspace depth, HS-UBPF-DDC-C). Observations with one or more cells flagged as outliers are displayed in green.[]{data-label="fig:md-ddc"}](md-ddc-c){width="\textwidth"}
Figure \[fig:md-ddc\] shows the Mahlanobis distances produced by GY-UBF-DDC-C and HS-UBPF-DDC-C. Here we can see that the GY-UBF-DDC-C misses 13 of 59 cases while the HS-UBPF-DDC-C has missed 15 cases. Although they seem not to do a better job, these two filters are able to flag some observations, not identified before, as case-wise outliers. These outliers are more clearly highlighted by HS-UBPF-DDC-C.
![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](gy-wts-htld "fig:"){width="32.00000%"} ![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](gy-htld-wsbc "fig:"){width="32.00000%"} ![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](gy-wsbc-sur "fig:"){width="32.00000%"}\
![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](halfspace-wts-htld "fig:"){width="32.00000%"} ![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](halfspace-htld-wsbc "fig:"){width="32.00000%"} ![Bivariate scatter plot of small-cap stock returns. In the first row the Gervini-Yohai depth is used. Blue: outliers detected by the GY-UF univariate filter; orange: outliers detected by the bivariate step of GY-UBF. In the second row the Half-space depth is uded. Blue: outliers detected by the HS-UF univariate filter; orange: outliers detected by the bivariate step of HS-UBF.[]{data-label="fig:scsr"}](halfspace-wsbc-sur "fig:"){width="32.00000%"}
Figure \[fig:scsr\] shows the bivariate scatter plot of WTS versus HTLD, HTLD versus WSBC and WSBC versus SUR where the GY-UBF and HS-UBF filters are applied, respectively. The bivariate observations with at least one component flagged as outlier are in blue, and outliers detected by the bivariate filter are in orange. We see that the HS-UBF identifies less outliers with respect to the GY-UBF.
Monte Carlo results {#sec:simulation}
===================
We performed a Monte Carlo simulation to assess the performance of the proposed filter based on halfspace depth. After the filter flags the outlying observations, the generalized S-estimator is applied to the data with added missing values. Our simulation study is based on the same setup described in @Zamar2017 to compare significantly the performance of our filter with respect to the filter introduced in their work.
We considered samples from a $N_p({\ensuremath{\boldsymbol{0}}}, {\ensuremath{\boldsymbol{\Sigma}}}_0)$, where all values in $diag({\ensuremath{\boldsymbol{\Sigma}}}_0)$ are equal to $1$, $p = 10, 20, 30, 40, 50$ and the sample size is $n = 10p$. We consider the following scenarios:
- Clean data: data without changes.
- Cell-Wise contamination: a proportion $\epsilon$ of cells in the data is replaced by $X_{ij} \sim N(k,0.1^2)$, where $k = 1, \ldots, 10$.
- Case-Wise contamination: a proportion $\epsilon$ of cases in the data matrix is replaced by ${\ensuremath{\boldsymbol{X}}}_i \sim 0.5N(c{\ensuremath{\boldsymbol{v}}},0.1^2{\ensuremath{\boldsymbol{I}}}) + 0.5N(-c{\ensuremath{\boldsymbol{v}}},0.1^2{\ensuremath{\boldsymbol{I}}})$, where $c = \sqrt{k(\chi^2_p)^{-1}(0.99)}$, $k = 1, 2, \ldots,20$ and ${\ensuremath{\boldsymbol{v}}}$ is the eigenvector corresponding to the smallest eigenvalue of ${\ensuremath{\boldsymbol{\Sigma}}}_0$ with length such that $({\ensuremath{\boldsymbol{v}}}-{\ensuremath{\boldsymbol{\mu}}}_0)^\top{\ensuremath{\boldsymbol{\Sigma}}}_0^{-1}({\ensuremath{\boldsymbol{v}}}-{\ensuremath{\boldsymbol{\mu}}}_0) = 1$.
The proportions of contaminated rows chosen for case-wise contamination are $\epsilon = 0.1, 0.2$, and $\epsilon = 0.02,0.05$ for cell-wise contamination. The number of replicates in our simulation study is $N=200$.
We measure the performance of a given pair of location and scatter estimators $\hat{{\ensuremath{\boldsymbol{\mu}}}}$ and $\hat{{\ensuremath{\boldsymbol{\Sigma}}}}$ using the mean squared error (MSE) and the likelihood ratio test distance (LRT), as in @Zamar2017: $$\begin{aligned}
& MSE = \frac{1}{N}\sum_{i=1}^N (\hat{{\ensuremath{\boldsymbol{\mu}}}}_i - {\ensuremath{\boldsymbol{\mu}}}_0)^\top (\hat{{\ensuremath{\boldsymbol{\mu}}}}_i - {\ensuremath{\boldsymbol{\mu}}}_0) \\
& LRT(\hat{{\ensuremath{\boldsymbol{\Sigma}}}},{\ensuremath{\boldsymbol{\Sigma}}}_0) = \frac{1}{N}\sum_{i=1}^N D(\hat{{\ensuremath{\boldsymbol{\Sigma}}}}_i,{\ensuremath{\boldsymbol{\Sigma}}}_0)\end{aligned}$$ where $\hat{{\ensuremath{\boldsymbol{\Sigma}}}}_i$ is the estimate of the $i$-th replication and $D({\ensuremath{\boldsymbol{\Sigma}}},{\ensuremath{\boldsymbol{\Sigma}}}_0)$ is the Kullback-Leibler divergence between two Gaussian distributions with the same mean and variances ${\ensuremath{\boldsymbol{\Sigma}}}$ and ${\ensuremath{\boldsymbol{\Sigma}}}_0$. Finally, we computed the maximum average LRT distances considering all contamination values $k$.
----- ------------ ------ ------ ------ ------ --------- -------- ---------
$p$ $\epsilon$ GY HS GY HS HS-UBPF GY-UBF HS-UBPF
10 0 0.8 0.8 0.9 0.9 0.9 1.0 1.0
0.02 1.2 1.3 1.3 1.3 1.3 1.1 1.1
0.05 4.6 4.8 4.6 4.5 4.3 2.4 2.5
20 0 1.3 1.4 1.4 1.5 1.5 1.8 1.8
0.02 3.9 4.4 4.2 4.5 4.4 2.5 2.5
0.05 11.0 12.2 11.3 11.9 11.8 8.2 8.3
30 0 1.9 1.9 2.0 2.0 2.0 3.4 3.4
0.02 6 6.7 6.5 6.8 6.6 5.0 5.1
0.05 14.5 16.9 15.1 16.8 16.6 13.4 13.9
40 0 2.4 2.5 2.6 2.6 2.6 5.8 5.8
0.02 7.5 8.5 8.2 8.6 8.5 9.2 9.3
0.05 17.4 20.8 18.1 20.7 20.5 20.0 20.0
50 0 2.9 3.0 3.1 3.1 3.2 5.1 5.1
0.02 8.8 10.0 9.7 10.2 10.0 12.2 12.5
0.05 19.9 24.5 20.8 24.3 24.1 24.5 24.7
----- ------------ ------ ------ ------ ------ --------- -------- ---------
: Maximum average LRT distance under cell-wise contamination.[]{data-label="tab:max_lrt_cell"}
Table \[tab:max\_lrt\_cell\] shows the average LRT distances under cell-wise contamination. We see that the univarite and univariate-bivariate filters have more problems in filtering moderate cell-wise outliers (for example $k=2$), while show a constant and optimal behavior for increasing contamination values of $k$. GY-UBF-DDC-C and HS-UBPF-DDC-C have lower maximum average LRT distances, but are higher for large $k$. This behavior is shown in Figure \[fig:k-30-icm-0.05\] (top) where the average LRT distances versus different contamination values are displayed, with $0.05$ of cell-wise contamination level and $p=30$.
----- ------------ ------- ------- ------- ------- --------- -------- ---------
$p$ $\epsilon$ GY HS GY HS HS-UBPF GY-UBF HS-UBPF
10 0 0.8 0.8 0.9 0.9 0.9 1.0 1.0
0.1 10.5 12.6 14.9 13.7 7.8 3.6 3.8
0.2 93.0 104.5 125.3 107.9 50.5 18.7 18.3
20 0 1.3 1.4 1.4 1.5 1.5 1.8 1.8
0.1 26.7 33.6 39.3 37.8 15.3 7.1 7.0
0.2 111.7 110.1 125.1 114.8 110.2 19.6 19.7
30 0 1.9 1.9 2.0 2.0 2.0 3.4 3.4
0.1 50.5 49.6 59.2 57.0 22.3 9.0 9.2
0.2 111.0 108.3 119.1 114.1 114.6 17.1 17.0
40 0 2.4 2.5 2.6 2.6 2.6 5.8 5.9
0.1 57.4 59.8 63.6 61.6 29.3 16.2 16.5
0.2 109.7 106.7 114.6 113.4 113.8 19.3 19.0
50 0 2.9 3.0 3.1 3.1 3.2 5.1 5.0
0.1 61.5 61.3 65.1 63.4 38.4 30.9 31.5
0.2 108.8 105.6 112.1 113.2 113.4 20.6 19.2
----- ------------ ------- ------- ------- ------- --------- -------- ---------
: Maximum average LRT distance under case-wise contamination.[]{data-label="tab:max_lrt_case"}
Table \[tab:max\_lrt\_case\] shows the maximum average LRT distances under case-wise contamination. Overall, the GY-UBP-DDC-C and HS-UBPF-DDC-C outperform all the other filters obtaining better results. Excluding these two, we see that the HS-UBPF is competitive in case of moderate case-wise contamination. An illustration of their behavior is given in Figure \[fig:k-30-thcm-0.1\] (top) which shows the average LRT distances for different values of $k$, with $0.1$ of case-wise contamination level and $p=30$.
----- ------------ ---- ---- ---- ---- --------- -------- ---------
$p$ $\epsilon$ GY HS GY HS HS-UBPF GY-UBF HS-UBPF
10 0 11 11 11 11 11 13 13
0.02 13 13 13 13 13 15 15
0.05 19 20 20 20 20 20 20
20 0 5 5 5 5 5 7 7
0.02 7 7 7 7 7 8 8
0.05 15 16 15 15 15 16 16
30 0 3 4 4 3 4 6 6
0.02 5 5 5 5 5 7 7
0.05 13 14 13 14 14 15 15
40 0 3 3 3 3 3 6 6
0.02 4 5 5 5 4 7 7
0.05 13 14 13 14 14 15 16
50 0 2 2 2 2 2 4 4
0.02 4 4 4 4 4 6 6
0.05 12 14 12 14 14 14 15
----- ------------ ---- ---- ---- ---- --------- -------- ---------
: Maximum average MSE distance under cell-wise contamination.[]{data-label="tab:max_mse_cell"}
----- ------------ ---- ----- ----- ----- --------- -------- ---------
$p$ $\epsilon$ GY HS GY HS HS-UBPF GY-UBF HS-UBPF
10 0 11 11 11 11 11 13 13
0.1 15 17 17 17 14 17 16
0.2 94 112 137 123 76 25 25
20 0 5 5 5 5 5 7 7
0.1 11 13 14 13 8 8 8
0.2 65 70 92 77 73 13 13
30 0 3 4 4 4 4 6 6
0.1 10 10 12 11 7 6 6
0.2 49 52 71 57 57 8 8
40 0 3 3 3 3 3 6 6
0.1 8 9 10 9 6 5 5
0.2 40 43 60 46 46 7 7
50 0 2 2 2 2 2 4 3
0.1 7 8 8 8 6 5 5
0.2 34 36 52 39 39 5 5
----- ------------ ---- ----- ----- ----- --------- -------- ---------
: Maximum average MSE distance under case-wise contamination.[]{data-label="tab:max_mse_case"}
Table \[tab:max\_mse\_cell\] and Table \[tab:max\_mse\_case\] show the maximum average MSE under cell-wise and case-wise contamination, respectively. The values in the tables are the MSE values multiplied by 1000 for a better visualization and model comparison. Under case-wise contamination, the GY-UBF-DDC-C and HS-UBPF-DDC-C outperform the other filters, and have also competitive results for cell-wise contamination. In Figure \[fig:k-30-icm-0.05\] (bottom) and Figure \[fig:k-30-thcm-0.1\] (bottom) the average MSE versus different contamination values $k$ are displayed, with $p=30$ and $0.05$ of cell-wise contamination and $0.1$ of case-wise contamination respectively. We highlight the nice redescending performance of the HS-UBPF for both LRT and MSE, not shared by the other filters.
![Average LRT (top) and average MSE (bottom) in $0.05$ cell-wise contamination level versus the contamination value $k$, $p=30$.[]{data-label="fig:k-30-icm-0.05"}](k-lrt-30-icm-0-05 "fig:"){width="80.00000%"} ![Average LRT (top) and average MSE (bottom) in $0.05$ cell-wise contamination level versus the contamination value $k$, $p=30$.[]{data-label="fig:k-30-icm-0.05"}](k-mse-30-icm-0-05 "fig:"){width="80.00000%"}
![Average LRT (top) and average MSE (bottom) in $0.1$ case-wise contamination level versus the contamination value $k$, $p=30$.[]{data-label="fig:k-30-thcm-0.1"}](k-lrt-30-thcm-0-1 "fig:"){width="80.00000%"} ![Average LRT (top) and average MSE (bottom) in $0.1$ case-wise contamination level versus the contamination value $k$, $p=30$.[]{data-label="fig:k-30-thcm-0.1"}](k-mse-30-thcm-0-1 "fig:"){width="80.00000%"}
Conclusions {#sec:conclusions}
===========
Considering the two-step procedure introduced in @Agostinelli2015a and improved by @Zamar2017, we present a new filter based on statistical data depth functions that can be used in place of the previous filters, intended as a generalization of such filters. Furthermore, we also combine the depth filter HS-UBPF and DDC, as suggested by @Zamar2017. As shown in the example, the filter HS-UBPF is able to identify large outlying observations and removes less cells than the GY-UBF. In addition, it also detects the case-wise outliers, which are clearly highlighted.
If we consider the performance of the entire procedure, our simulations show that using HS-UBPF we obtain the best estimates in case of moderate proportion of contamination, but it is still competitive for higher percentage of contamination, also for high-dimensional dataset, under both types of contamination models. Generally, the GY-UBF and HS-UBPF combined with DDC outperform the other filters. Differences in performance of these two estimators are not clearly visible. However the HS-UBPF has shown, especially under the case-wise contamination an interesting behaviour for moderate contamination level.
Further research on this filter could be needed to explore the performance of the estimator in different types of data and how it can vary with respect to the dimensions $p$ and $n$, for example in flat datasets (e.g., $n \approx 2p$). In addition different statistical data depth functions could be used in place of the half-space depth.
\[sec:appendix\]
Statistical data depth properties {#sm:statisticaldatadepth}
=================================
A **depth function** $d(\cdot; F)$ measures the centrality of a point w.r.t. a probability distribution $F$. $$d = \mathbb{R}^{p} \rightarrow \mathbb{R}^+ \cup \{ 0 \}, \qquad {\ensuremath{\boldsymbol{x}}} \rightarrow d({\ensuremath{\boldsymbol{x}}}; F)$$
A statistical depth function should satisfy the following Properties [@Liu1990; @Zuo2000a]
1. \[sm:PropDepthAffineInvariance\] Affine invariance: $d({\ensuremath{\boldsymbol{x}}}; F)=d({\ensuremath{\boldsymbol{A}}} {\ensuremath{\boldsymbol{x}}}+ {\ensuremath{\boldsymbol{b}}}; F_{{\ensuremath{\boldsymbol{A}}},{\ensuremath{\boldsymbol{b}}}})$;
2. \[sm:PropDepthMaximalityAtCenter\] Maximality at center: if $F$ is “symmetric” around ${\ensuremath{\boldsymbol{\mu}}}$ then $d({\ensuremath{\boldsymbol{x}}}; F) \leq d({\ensuremath{\boldsymbol{\mu}}}; F)$ for all ${\ensuremath{\boldsymbol{x}}}$; for a more detailed discussion on symmetry see @Serfling2006.
3. \[sm:PropDepthMonotonicity\] Monotonicity: if (\[sm:PropDepthMaximalityAtCenter\]) holds, then $$d({\ensuremath{\boldsymbol{x}}}; F) \le d({\ensuremath{\boldsymbol{\mu}}} + \alpha ({\ensuremath{\boldsymbol{x}}} - {\ensuremath{\boldsymbol{\mu}}}); F) \qquad \alpha \in [0,1] \ ;$$
4. \[sm:PropDepthZero\] Approaching zero: $\parallel {\ensuremath{\boldsymbol{x}}} \parallel \rightarrow \infty \Rightarrow d({\ensuremath{\boldsymbol{x}}}; F) \rightarrow 0$.
Gervini-Yohai depth {#sm:gerviniyohaidepth}
===================
Here we want to show that the Gervini-Yohai depth, defined as $d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G) = 1 - G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F)))$, is a proper statistical depth function, i.e., it satisfies the four properties introduced above.
1. Affine invariance: it follows directly from the affine invariance property of the Mahalanobis distance;
2. Maximality at center: if $F$ is elliptically symmetric around ${\ensuremath{\boldsymbol{\mu}}}(F)$, $$d_{GY}({\ensuremath{\boldsymbol{\mu}}}(F), F, G) = 1 - G(\Delta({\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) = 1 - G(0) .$$ For any ${\ensuremath{\boldsymbol{t}}} \not= {\ensuremath{\boldsymbol{\mu}}}(F)$ we have $$\begin{aligned}
\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F)) & > 0 \\
G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) & \ge G(0) \\
1-G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) & \le 1- G(0) \\
d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G) & \le d_{GY}({\ensuremath{\boldsymbol{\mu}}}(F), F, G) , \end{aligned}$$ when $G$ is strictly monotone then strict inequality holds, and ${\ensuremath{\boldsymbol{\mu}}}(F)$ is the unique maximizer of the Gervini-Yohai depth.
3. Monotonicity: $$\begin{aligned}
\Delta({\ensuremath{\boldsymbol{\mu}}}(F) + \alpha({\ensuremath{\boldsymbol{t}}}-{\ensuremath{\boldsymbol{\mu}}}(F)),{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F)) & = (\alpha({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F)))^\top {\ensuremath{\boldsymbol{\Sigma}}}(F)^{-1}(\alpha({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F))) \\
& = \alpha^2 ({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F))^\top {\ensuremath{\boldsymbol{\Sigma}}}(F)^{-1}({\ensuremath{\boldsymbol{t}}} - {\ensuremath{\boldsymbol{\mu}}}(F)) \\
& = \alpha^2 \Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F)) \\
& \le \Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))\end{aligned}$$ Then $d_{GY}({\ensuremath{\boldsymbol{\mu}}}(F) + \alpha({\ensuremath{\boldsymbol{t}}}-{\ensuremath{\boldsymbol{\mu}}}(F)), F, G) \ge d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G)$.
4. Approaching zero: if $\parallel {\ensuremath{\boldsymbol{t}}} \parallel \rightarrow \infty $ we have that $\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F)) \rightarrow \infty$ and consequently $G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) \rightarrow 1$. Then $$d_{GY}({\ensuremath{\boldsymbol{t}}}, F, G) = 1 - G(\Delta({\ensuremath{\boldsymbol{t}}},{\ensuremath{\boldsymbol{\mu}}}(F),{\ensuremath{\boldsymbol{\Sigma}}}(F))) \rightarrow 0$$
Monte Carlo experiment {#sm:simulation}
======================
Results for all combinations of the model parameters explored in the Monte Carlo simulation are reported in this section.
![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-10-icm-0-02 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-10-icm-0-05 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-20-icm-0-02 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-20-icm-0-05 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-30-icm-0-02 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-icm-1"}](k-lrt-30-icm-0-05 "fig:"){width="45.00000%"}
![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-icm-2"}](k-lrt-40-icm-0-02 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-icm-2"}](k-lrt-40-icm-0-05 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-icm-2"}](k-lrt-50-icm-0-02 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-icm-2"}](k-lrt-50-icm-0-05 "fig:"){width="45.00000%"}
![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-10-icm-0-02 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-10-icm-0-05 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-20-icm-0-02 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-20-icm-0-05 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-30-icm-0-02 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-icm-1"}](k-mse-30-icm-0-05 "fig:"){width="45.00000%"}
![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-mse-icm-2"}](k-mse-40-icm-0-02 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-mse-icm-2"}](k-mse-40-icm-0-05 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-mse-icm-2"}](k-mse-50-icm-0-02 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in cell-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-mse-icm-2"}](k-mse-50-icm-0-05 "fig:"){width="45.00000%"}
In Figures \[fig:k-lrt-icm-1\], \[fig:k-lrt-icm-2\] and Figures \[fig:k-mse-icm-1\], \[fig:k-mse-icm-2\] the average LRT and average MSE versus different contamination values $k$ are displayed, respectively.
![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-10-thcm-0-1 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-10-thcm-0-2 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-20-thcm-0-1 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-20-thcm-0-2 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-30-thcm-0-1 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-lrt-thcm-1"}](k-lrt-30-thcm-0-2 "fig:"){width="45.00000%"}
![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-thcm-2"}](k-lrt-40-thcm-0-1 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-thcm-2"}](k-lrt-40-thcm-0-2 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-thcm-2"}](k-lrt-50-thcm-0-1 "fig:"){width="45.00000%"} ![Average LRT versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (40,50).[]{data-label="fig:k-lrt-thcm-2"}](k-lrt-50-thcm-0-2 "fig:"){width="45.00000%"}
![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-10-thcm-0-1 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-10-thcm-0-2 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-20-thcm-0-1 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-20-thcm-0-2 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-30-thcm-0-1 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$ (10,20,30).[]{data-label="fig:k-mse-thcm-1"}](k-mse-30-thcm-0-2 "fig:"){width="45.00000%"}
![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$.[]{data-label="fig:k-mse-thcm-2"}](k-mse-40-thcm-0-1 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$.[]{data-label="fig:k-mse-thcm-2"}](k-mse-40-thcm-0-2 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$.[]{data-label="fig:k-mse-thcm-2"}](k-mse-50-thcm-0-1 "fig:"){width="45.00000%"} ![Average MSE versus the contamination value $k$ in case-wise contamination, considering all combinations of contamination level $\epsilon$ and number of variables $p$.[]{data-label="fig:k-mse-thcm-2"}](k-mse-50-thcm-0-2 "fig:"){width="45.00000%"}
Figures \[fig:k-lrt-thcm-1\], \[fig:k-lrt-thcm-2\] and Figures \[fig:k-mse-thcm-1\], \[fig:k-mse-thcm-2\] show the average LRT and average MSE versus different contamination values $k$, respectively.
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---
abstract: 'The transverse momentum dependent gluon density obtained with CCFM evolution is determined from a fit to the latest combined HERA structure function measurements.'
author:
- |
[*Hannes Jung$^{1,2}$, Francesco Hautmann$^3$*]{}\
$^1$DESY, Notkestra[ß]{}e 85, 22607 Hamburg, Germany\
$^2$CERN, 1211 Genève 23, Switzerland\
$^3$Theoretical Physics Department, University of Oxford, Oxford OX1 3NP, GB
title: Determination of transverse momentum dependent gluon density from HERA structure function measurements
---
Introduction
============
The combined measurements of the structure function at HERA [@Aaron:2009aa] allow the determination of parton distribution functions to be carried out to high precision. While these data have been used to determine the collinear parton densities, the transverse momentum distributions (TMD) or unintegrated gluon distributions were only based on older and much less precise measurements [@Jung:2002wn; @Hansson:2003xz].
In high energy factorization [@Catani:1990eg] the cross section is written as a convolution of the partonic cross section $\hat{\sigma}(É \kt)$ which depends on the transverse momentum $\kt$ of the incoming parton with the $\kt$-dependent parton density function ${\cal \tilde A}\left(x,\kt,{p}\right)$: $$\sigma = \int
\frac{dz}{z} d^2k_t \hat{\sigma}(\frac{x}{z},k_t) {\cal \tilde A}\left(x,\kt,{p}\right)\label{kt-factorisation}$$
[r]{}[0.28]{} -0.7cm {width="15.00000%"} -0.3cm
where ${p}$ is the factorization scale. The evolution of ${\cal \tilde A}\left(x,\kt,{p}\right)$ can proceed via the BFKL, DGLAP or via the CCFM evolution equations. Here, an extension of the CCFM evolution is applied (to be also used in the parton shower Monte Carlo event generator CASCADE [@Jung:2010si]) which includes the use of two loop $\alpha_s$ as well as applying a consistency constraint [@Kwiecinski:1996td; @Ciafaloni:1987ur; @Andersson:1995ju] in the $g\to gg$ splitting function [@Andersson:2002cf]: $$P_{gg}(z,{p},\kt ) = {{\bar \alpha}_\mathrm{s}}\left(\kt^2 \right)
\left( \frac{(1-z)}{z} + \frac{z(1-z)}{2}\right) \Delta_{ns}
+ {{\bar \alpha}_\mathrm{s}}({p}^2) \left(\frac{z}{1-z} +
\frac{z(1-z)}{2}\right) ,
\label{fullsplitt}$$ with $ \Delta_{ns} $ being the non-Sudakov form factor. The consistency constraint is given by [@Kwiecinski:1996td] (see Fig. \[Fig:branching\]): $$q_t^2 < \frac{(1-z) \kt^2}{ z} \label{consistency-constraint}
$$
Evolution
=========
Since the CCFM evolution cannot be easily written in an analytic closed form, a Monte Carlo method, based on [@Marchesini:1990zy; @Marchesini:1992jw], is used. However, the Monte Carlo solution is time consuming, and cannot be used in a straightforward way in a fit program. For a realistic solution, first a kernel $ {\cal \tilde A}\left(x'',\kt,{p}\right) $ is determined from the MC solution of the CCFM evolution equation, and then is folded with the non-perturbative starting distribution ${\cal A}_0 (x)$: $$\begin{aligned}
x {\cal A}(x,\kt,{p}) &= &x\int dx' \int dx'' {\cal A}_0 (x) {\cal \tilde A}\left(x'',\kt,{p}\right) \delta(x' \cdot x'' - x) \\
&= &\int dx' \int dx'' {\cal A}_0 (x) {\cal \tilde A}\left(x'',\kt,{p}\right) \frac{x}{x'} \delta(x'' - \frac{x}{x'}) \\
& = & \int dx' {{\cal A}_0 (x') }
\cdot \frac{x}{x'}{ {\cal \tilde A}\left(\frac{x}{x'},\kt,{p}\right) } \end{aligned}$$ The kernel ${\cal \tilde A}$ includes all the dynamics of the evolution, Sudakov form factors and splitting functions and is determined in a grid of $50\otimes50\otimes50$ bins in $x,\kt,{p}$.
The calculation of the cross section according to eq.(\[kt-factorisation\]) involves a multidimensional Monte Carlo integration which is time consuming and suffers from numerical fluctuations, and cannot be used directly in a fit procedure involving the calculation of numerical derivates in the search for the minimum. Instead the following procedure is applied: $$\begin{aligned}
\sigma_r(x,Q^2) & = & \int_x^1 d x_g {\cal A}(x_g,\kt,{p}) \hat{ \sigma}(x,x_g,Q^2) \\
& = & \int d x_g\; dx'\; dx'' {\cal A}_0 (x') {\cal \tilde A}(x'',\kt,{p})\cdot \hat{ \sigma}(x,x_g,Q^2) \cdot \delta(x' \,x'' -x_g) \\
& = & \int dx'\; dx'' {\cal A}_0 (x') \cdot {\cal \tilde A}(x'',\kt,{p}) \cdot \hat{ \sigma}(x,x'\,x'',Q^2) \\
& = & \int_x^1 dx' {\cal A}_0 (x') \cdot \int_{x/x'}^1 dx'' {\cal \tilde A}(x'',\kt,{p}) \cdot \hat{ \sigma}(x,x'\,x'',Q^2) \\
& = & \int_x^1 dx' {\cal A}_0 (x') \cdot \tilde{ \sigma}(x/x',Q^2) \label{final-convolution}
\end{aligned}$$ Here, first $ \tilde{ \sigma}(x',Q^2)$ is calculated numerically with a Monte Carlo integration on a grid in $x$ for the values of $Q^2$ used in the fit. Then the last step (i.e. eq.(\[final-convolution\])) is performed with a fast numerical gauss integration, which can be used in standard fit procedures.
The fit to the HERA structure function measurements is performed applying the `herafitter` program to determine the parameters of the starting distribution ${\cal A}_0$ at the starting scale $Q_0$: $$\begin{aligned}
x{\cal A}_0(x,\kt) &=& N x^{-B_g} \cdot (1 -x)^{C_g}\left( 1 -D_g x\right)
\label{a0}\end{aligned}$$
Fit to HERA structure function
==============================
The parameters $N,B_g,C_g,D_g$ in eq.(\[a0\]) are determined from a fit to the combined structure function measurement [@Aaron:2009aa] in the range $x<0.01$ and $Q^2>5$ GeV. In addition to the gluon induced process $\gamma^* g^* \to q\bar{q}$ the contribution from valence quarks is included via $\gamma^* q \to q$ using a CCFM evolution of valence quarks as described in [@Deak:2010gk]. The results presented here are obtained with the `herafitter` package, treating the correlated systematic uncertainties separately from the uncorrelated statistical and systematic uncertainties. To obtain a reasonable fit to the structure function data, the starting scale $Q_0$ as well as $\Lambda_{qcd}$ has been varied. An acceptable $\chi^2/ndf $ could only be achieved when applying the consistency constraint eq.(\[consistency-constraint\]): without consistency constraint the best $\chi^2/ndf \sim 14 - 28$, depending on which form of the splitting function is used. With consistency constraint and the splitting function eq.(\[fullsplitt\]) the best fit gives $\chi^2/ndf \sim 1.5$ for $Q_0=1.8$ GeV and $\Lambda_{qcd} = 0.17$ GeV at $n_f=4$ flavours. It has been checked, that the $\chi^2/ndf$ does not change significantly when using 3 instead of 4 parameters for the initial starting distribution ${\cal A}_0$.
In fig.\[Fig:updf\] the resulting unintegrated gluon density [**JH-set0**]{} is shown for 2 values of ${p}^2$ compared to [**set A0**]{} [@Jung:2004gs].
0.6cm
The uncertainties of the pdf are obtained within the `herafitter` package from a variation of the individual parameter uncertainties following the procedure described in [@Pumplin:2001ct] applying $\Delta \chi^2=1$. The uncertainties on the gluon are small (much smaller than obtained in standard fits), since only the gluon density is fitted. The uncertainty bands for the gluon density are shown in fig. \[Fig:updf-uncertainty\](left).
![(left): Uncertainties of the uPDF at ${p}^2= 25$ GeV$^2$. (right): Cross section of $b$-jet production as a function of $p_t$ for different bins in $y$ as measured by CMS [@Chatrchyan:2012dk] compared to predictions from [@Jung:2010si] using the unintegrated gluon density described here[]{data-label="Fig:updf-uncertainty"}](jung-hannes-fig5.pdf){width="35.00000%"}
In fig. \[Fig:updf-uncertainty\](right) the prediction for $b$-jet cross section as calculated from [@Jung:2010si] using the gluon density described here (labeled as [**set0**]{}) is shown together with a prediction using an older set (labeled as [**setA 0**]{} [@Jung:2004gs]) in comparison with a measurement from CMS [@Chatrchyan:2012dk].
0.5 cm
[**Acknowledgments**]{}. We thank the conveners for the invitation and excellent organization of the meeting.
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F. Aaron [*et al.*]{} [JHEP [ **1001**]{} (2010) 109](http://dx.doi.org/10.1007/JHEP01(2010)109), [[ arXiv:0911.0884 \[hep-ex\]]{}](http://arxiv.org/abs/0911.0884). 61 pages, 21 figures. H. Jung. Acta Phys. Polon. [**B33**]{} (2002) 2995–3000, [[arXiv:hep-ph/0207239]{}](http://arxiv.org/abs/hep-ph/0207239). M. Hansson and H. Jung. [[arXiv:hep-ph/0309009]{}](http://arxiv.org/abs/hep-ph/0309009). S. Catani, M. Ciafaloni, and F. Hautmann. [Nucl. Phys. [ **B366**]{} (1991) 135](http://dx.doi.org/10.1016/0550-3213(91)90055-3). H. Jung, S. Baranov, M. Deak, A. Grebenyuk, F. Hautmann, [*et al.*]{} [Eur.Phys.J. [**C70**]{} (2010) 1237](http://dx.doi.org/10.1140/epjc/s10052-010-1507-z), [[arXiv:1008.0152 \[hep-ph\]]{}](http://arxiv.org/abs/1008.0152). J. Kwiecinski, A. D. Martin, and P. Sutton. [Z.Phys. [ **C71**]{} (1996) 585](http://dx.doi.org/10.1007/s002880050206), [[ arXiv:hep-ph/9602320 \[hep-ph\]]{}](http://arxiv.org/abs/hep-ph/9602320).
M. Ciafaloni. [Nucl. Phys. [ **B296**]{} (1988) 49](http://dx.doi.org/10.1016/0550-3213(88)90380-X). B. Andersson, G. Gustafson, and J. Samuelsson. [Nucl.Phys. [**B467**]{} (1996) 443](http://dx.doi.org/10.1016/0550-3213(96)00114-9). Revised version. B. Andersson [*et al.*]{} [Eur. Phys. J. [**C25**]{} (2002) 771](http://dx.doi.org/10.1007/s10052-002-0998-7), [[arXiv:hep-ph/0204115]{}](http://arxiv.org/abs/hep-ph/0204115). G. Marchesini and B. R. Webber. [Nucl. Phys. [ **B349**]{} (1991) 617](http://dx.doi.org/10.1016/0550-3213(91)90389-F). G. Marchesini and B. R. Webber. [Nucl. Phys. [ **B386**]{} (1992) 215](http://dx.doi.org/10.1016/0550-3213(92)90181-A). F. Aaron [*et al.*]{} Eur.Phys.J. [**C64**]{} (2009) 561, [[arXiv:0904.3513 \[hep-ex\]]{}](http://arxiv.org/abs/0904.3513). 35 pages, 10 figures. “”, 2012. <http://herafitter.hepforge.org/>.
M. Deak, F. Hautmann, H. Jung, and K. Kutak. [[arXiv:1012.6037 \[hep-ph\]]{}](http://arxiv.org/abs/1012.6037). H. Jung. [[arXiv:hep-ph/0411287]{}](http://arxiv.org/abs/hep-ph/0411287). J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston, [*et al.*]{} [Phys.Rev. [**D65**]{} (2001) 014013](http://dx.doi.org/10.1103/PhysRevD.65.014013), [[arXiv:hep-ph/0101032 \[hep-ph\]]{}](http://arxiv.org/abs/hep-ph/0101032). S. Chatrchyan [*et al.*]{} JHEP [**1204**]{} (2012) 084, [[arXiv:1202.4617 \[hep-ex\]]{}](http://arxiv.org/abs/1202.4617).
|
---
abstract: 'We consider the ground-state properties of the two-component spin-orbit coupled ultracold bosons subject to a rotationally symmetric in-plane gradient magnetic field. In the non-interacting case, the ground state supports giant-vortices carrying large angular momenta without rotating the trap. The vorticity is highly tunable by varying the amplitudes and orientations of the magnetic field. Interactions drive the system from a giant-vortex state to various configurations of vortex lattice states along a ring. Vortices exhibit ellipse-shaped envelops with the major and minor axes determined by the spin-orbit coupling and healing lengths, respectively. Phase diagrams of vortex lattice configurations are constructed and their stabilities are analyzed.'
author:
- 'Xiang-Fa Zhou'
- 'Zheng-Wei Zhou'
- Congjun Wu
- 'Guang-Can Guo'
title: 'The in-plane gradient magnetic field induced vortex lattices in spin-orbit coupled Bose-Einstein condensations'
---
introduction
============
Spin-orbit (SO) coupling plays an important role in contemporary condensed matter physics, which is linked with many important effects ranging from atomic structures, spintronics, to topological insulators [@zutic2004; @Hasan2010; @Qi2011]. It also provides a new opportunity to search for novel states with ultracold atom gases which cannot be easily realized in condensed matter systems. In usual bosonic systems, the ground state condensate wavefunctions are positive-definite known as the “no-node” theorem [@feynman1972; @wu2009]. However, the appearance of SO coupling invalidates this theorem [@wu2011]. The ground state configurations of SO coupled Bose-Einstein condensations (BEC) have been extensively investigated and a rich structure exotic phases are obtained including the ferromagnetic and spin spiral condensations [@wu2011; @stanescu2008; @wang2010; @ho2011], spin textures of the skyrmion type [@wu2011; @hu2012; @sinha2011; @li2012; @kawakami2012], and quantum quasi-crystals [@gopalakrishnan2013], etc. On the experiment side, since the pioneering work in the NIST group [@lin2009], it has received a great deal of attention, and various further progresses have been achieved [@zhang2012v1; @wang2012; @cheuk2012; @qu2013; @olson2014]. Searching for novel quantum phases in this highly tunable system is still an on-going work both theoretically and practically [@wilson2013; @achilleos2013; @kartashov2013; @ozawa2013; @deng2012; @zhang2012; @lobanov2014; @luo2014], which has been reviewed in [@zhou2013; @dalibard2011; @zhai2014; @galistki2013; @goldman2013].
On the other hand, effective gradient magnetic fields have been studied in various neutral atomic systems recently. For instance, it has been shown in Ref. [@anderson2013; @xu2013] that SO coupling can be simulated by applying a sequence of gradient magnetic field pulses without involving complex atom-laser coupling. In optical lattices, theoretic and experimental progresses show that SO coupling and spin Hall physics can be implemented without spin-flip process by employing gradient magnetic field [@kennedy2013; @aidelsburger2013]. This represents the cornerstone of exploring rich many-body physics using neutral ultracold atoms. Additionally, introducing gradient magnetic fields has also been employed to create various topological defects including Dirac monopoles [@pietila2009] and knot solitons [@kawaguchi2008]. It would be very attractive to investigate the exotic physics by combining both SO coupling and the gradient magnetic field together in ultracold quantum gases.
In this work, we consider the SO coupled BECs subject to an in-plane gradient magnetic field in a $2$D geometry. Our calculation shows that this system support a variety of interesting phases. The main features are summarized as follows. First, the single-particle ground states exhibit giant vortex states carrying large angular momenta. It is very different from the usual fast-rotating BEC system, in which the giant vortex state appears only as meta-stable states [@schweikhard2004; @mueller2002]. Second, increasing the interaction strength causes the phase transition into the vortex lattice state along a ring plus a giant core. The corresponding distribution in momentum space changes from a symmetric structure at small interaction strengths to an asymmetric one as the interaction becomes strong. Finally, the size of a single vortex is determined by two different length scales, namely, the SO coupling strength together with the healing length. Therefore, the vortex exhibits an ellipse-shaped envelope with the principle axes determined by these two scales. This is different from the usual vortex in rotating BECs [@fetter2009; @zhou2011; @xu2011; @radic2011; @aftalion2013; @fetter2014], where an axial symmetric density profile is always favored.
The rest of this article is organized as follows. In Sect. \[sect:ham\], the model Hamiltonian is introduced. The single particle wavefunctions are described in Sect. \[sect:single\]. The phase transitions among different vortex lattice configurations are investigated in Sect. \[sect:interaction\]. The possible experimental realizations are discussed in Sect. \[sect:experiment\]. Conclusions are presented in Sect. \[sect:conclusions\].
The model Hamiltonian {#sect:ham}
=====================
We consider a quasi-$2$D SO coupled BEC subject to a spatially dependent magnetic field with the following Hamiltonian as $$\begin{aligned}
H=\int d \vec{r}^2 && \hspace{-.45cm} \hat{\psi}(\vec{r})^{\dag}
\Big\{ \frac{\vec{p}^2}{2m} + \Lambda r \left ( \cos\theta \hat{r}
+\sin\theta \hat{\varphi} \right ) \cdot \vec{\sigma} \nn \\
&+& \frac{1}{2}m\omega^2r^2 \Big \}
\hat{\psi}(\vec{r})+H_{soc}+H_{int},
\label{eq:ham_1}\end{aligned}$$ where $\hat{r}=\vec{r}/r$ with $\vec{r}=(x, y)$; $\vec{\sigma}=(\sigma_x, \sigma_y)$ are the usual Pauli matrices; $m$ is the atom mass; $\omega$ is the trapping frequency; $\Lambda$ is the strength of the magnetic field, and $\theta$ denotes the relative angle between the magnetic field and the radial direction $\hat r$. Physically, this quasi-2D system can be implemented by imposing a highly anisotropic harmonic trap potential $V_H=\frac{1}{2}m(\omega^2 r^2+\omega^2_zz^2)$. When $\omega_z \gg \omega$, atoms are mostly confined in the $xy$-plane, and the wavefunction along $z$ axis is determined as a harmonic ground state with the characteristic length $a_z=\sqrt{\hbar/(m\omega_z)}$.
For simplicity, the SO coupling employed below has the following symmetric form as $$\begin{aligned}
H_{soc}&=&\int d \vec{r}^2 \hat{\psi}(\vec{r})^{\dag}
\Big[ \frac{\lambda}{m} (p_x \sigma_x +p_y \sigma_y)
\Big]\hat{\psi}(\vec{r}) \nn\end{aligned}$$ with $\lambda$ the SO coupling strength. We note that due to this term, the magnetic fields which couples to spin can be employed as a useful method to control the orbit degree of freedom of the cloud. The interaction energy is written as $$\begin{aligned}
H_{int}=\frac{g_{2D}}{2} \int d \vec{r}^2 \hat{\psi}(\vec{r})^{\dag}\hat{\psi}(\vec{r})^{\dag}\hat{\psi}(\vec{r})\hat{\psi}(\vec{r}).\end{aligned}$$ Here the contact interaction between atoms in bulk is $g=4\pi\hbar^2 a_s/m$, where $a_s$ is the scattering length. For the quasi-2D geometry that we focus on, the effective interaction strength is modified as $g_{2D}=g_{3D}/(\sqrt{2\pi}a_z)$.
single-particle properties {#sect:single}
==========================
[![The single-particle dispersion of the Hamiltonian Eq. (\[reducedsingleparticleH\]) with lower energy branch as a function of the reduced magnetic fields $\beta$ for fixed $\alpha=10$ and different values of $\theta=0$ (a), and $\frac{1}{4}\pi$ (b). The inset in (b) shows that the ground states crossing for certain values of $\beta$ at $\theta=\pi/4\neq0$, while there is no crossing in (a) at $\theta=0$. \[spectrum\]](fig1a.eps "fig:"){width="0.8\linewidth"} ![The single-particle dispersion of the Hamiltonian Eq. (\[reducedsingleparticleH\]) with lower energy branch as a function of the reduced magnetic fields $\beta$ for fixed $\alpha=10$ and different values of $\theta=0$ (a), and $\frac{1}{4}\pi$ (b). The inset in (b) shows that the ground states crossing for certain values of $\beta$ at $\theta=\pi/4\neq0$, while there is no crossing in (a) at $\theta=0$. \[spectrum\]](fig1b.eps "fig:"){width="0.8\linewidth"}]{}
The physics of Eq. \[eq:ham\_1\] can be illustrated by considering the single-particle properties first. After introducing the characteristic length scale of the confining trap $l_T=\sqrt{\hbar/m\omega}$, the dimensionless Hamiltonian is rewritten as $$\begin{aligned}
\frac{H_0}{\hbar\omega}=\int d \vec{\rho}^2 && \hspace{-.45cm} \hat{\phi}(\vec{\rho})^{\dag} \Big \{ -\frac{\vec{\nabla}^2}{2} + \beta \rho \left ( \cos\theta \hat{r} +\sin\theta \hat{\varphi} \right ) \cdot \vec{\sigma} \nn \\
&+& \alpha \vec{k} \cdot \vec{\sigma} + \frac{1}{2}\rho^2 \Big \}\hat{\phi}(\vec{\rho}), \label{reducedsingleparticleH}\end{aligned}$$ where $\alpha=\lambda/(m\omega l_T)$ and $\beta=\Lambda l_T/(\hbar \omega)$ are the dimensionless SOC and magnetic field strengths, respectively; the normalized condensates wave-function is defined as $$\phi(\vec{\rho})=\frac{l_T}{\sqrt{N}}\Psi(\vec{r}=l_T\vec{\rho})$$ with $N$ the total number of atoms;
Since the total angular momentum $\hbar j_z=\hbar l_z+\frac{\hbar}{2}\sigma_z$ is conserved for this typical Hamiltonian, we can use it to label the single-particle states. If the magnetic field along the radial direction, i.e., $\theta=l \pi$, the Hamiltonian also supports a generalized parity symmetry described by $i\sigma_y P_x$, namely =0, \[eq:symm\] with $P_x$ the reflection operation about the $y$-axis satisfying $P_x: (x,y)\rightarrow (-x,y)$. Therefore for given eigenstates $\phi_m=[f(\rho) e^{i m \varphi} ,
g(\rho) e^{i (m+1) \varphi}]^T$ with $j_z=(m+1/2)$, the above symmetry indicates that these two states $\{\phi_m, (i\sigma_y P_x)\phi_m\}$ are degenerate for $H_0(\theta=l \pi)$. This symmetry is broken when $\theta\neq l \pi$.
Due to the coupling between the real space magnetic field and momentum space SO coupling, the single particle ground states exhibit interesting properties at large values of $\alpha$ and $\beta$. In momentum space, the low energy state moves to a circle with the radius determined by $\alpha$. The momentum space single-particle eigenstates break into two bands $\psi^{\pm}(\vec k)$ with the corresponding eigenvalues $E^{\pm}_{\vec{k}}/(\hbar \omega)=\frac{1}{2}
(|\vec{k}|^2\pm 2 \alpha |\vec{k}|)$ and eigenstates $\frac{1}{\sqrt 2}[1,\pm e^{i\theta_k}]^T$, respectively. For the lower band which we focus on, the spin orientation is $\langle\vec{\sigma}\rangle=(-\cos\theta_{\vec{k}}, -\sin\theta_{\vec{k}})$, which is anti-parallel to $\vec{k}$. On the other hand, in the real space, for a large value of $\beta$, the potential energy in real space is minimized around the circle with the the radius $r/l_T=\beta$ with a spatial dependent spin polarization. Therefore around this space circle, the local wavevector at a position $\vec r$ is aligned along the direction of the local magnetic field to minimize the energy. The projection of the local wavevector along the tangent direction of the ring gives rise to the circulation, and thus the ground state carries large angular momentum $m$ which is estimated as m2/(2/)=. Therefore, by varying the angle $\theta$, a series of ground states are obtained with their angular momentum ranging from $0$ to $\alpha\beta \gg 1$. This is very different from the usual method to generate giant vortex, where fast rotating the trap is needed [@fetter2009].
[![The density and phase profiles of the single-particle ground states for fixed $\alpha=6$, $\beta=1$, and different $\theta=\frac{1}{40} \pi$ (a), $\frac{2}{5}\pi$ (b). From left to right: the density and phase profiles for spin-up and spin-down components, respectively. \[densityphaseSP\] ](fig2.eps "fig:"){width="\linewidth"}]{}
For $\beta \gg 1$, the low energy wavefunctions mainly distribute around the circle $\rho=\beta$. As shown in Appendix \[sect:dispersion\], the approximated wavefunctions for the lowest band ($n=1$) is written as $$\begin{aligned}
\phi_{n=1,j_z}(\rho,\varphi) &\simeq& \frac{1}{2\pi^{\frac{3}{4}}\rho^{\frac{1}{2}}}
e^{-\frac{(\rho-\beta)^2}{2}}e^{i\rho\alpha \cos\theta } \nonumber \\
&\times&
\left [ \begin{array}{c} e^{i [m\varphi-\frac{\theta}{2}]} \\
-e^{i [(m+1)\varphi+\frac{\theta}{2}]} \end{array} \right ],\end{aligned}$$ where $\varphi$ is the azimuthal angle. The corresponding energy dispersion is approximated as $$\begin{aligned}
E_{n,j_z}\approx n+\frac{1-\alpha^2-\beta^2}{2}
+ \frac{(j_z-\alpha\beta\sin\theta)^2}{2(\alpha^2\cos^2\theta+\beta^2)}.
\label{eq:dispersion}\end{aligned}$$ For given values of $\alpha$ and $\beta$, $E_{n,j_z}$ is minimized at $j_z \simeq \alpha\beta\sin\theta$, which is consistent with the above discussion. In the case of $\theta= l \pi$, two states with $m=l$ and $-(l+1)$ are degenerated due to the symmetry defined in Eq. \[eq:symm\]. Interestingly, Eq. \[eq:dispersion\] also indicates that for integer $\alpha\beta\sin\theta=l$, an approximate degeneracy occurs for $m=l$ and $l-1$.
Fig. \[spectrum\] shows the single-particle dispersion of different angular momentum eigenstates along with the radius $\beta$ for different values of $\theta$. For $\theta=0$, the dispersion with different $j_z$ never cross each other Fig. \[spectrum\] (1a). The values of $j_z$ for the ground state are always $j_z=\frac{1}{2}$ or $-\frac{1}{2}$ due to the symmetry Eq. \[eq:symm\]. When $\theta=\pi/4\neq0$, the spectra cross at certain parameter values, and the ground-state can be degenerate even without additional symmetries as shown in Fig. \[spectrum\] (1b), which is consistent with above discussions. For $\beta \gg 1$, the probability density of the ground state single particle wavefunction mainly distributes around a ring with $\rho=\beta$. Interestingly, the phase distribution exhibits the typical Archimedean spirals with the equal-phase line satisfying $\rho \sim m \varphi$ (or $\rho\sim(m+1) \varphi$) (see Fig. \[densityphaseSP\] for details).
[![The profiles of the condensate wavefunctions of the spin-up component for $\alpha=11$, $\beta=6$, and $\theta=\frac{\pi}{2}$. The interaction parameters are $g=15$ ($a$), 35 ($b$), 75 ($c$), and 100 ($d$), respectively. We note that (c) and (d) exhibit similar profiles but with different $q$. From top to bottom: the density and phase profiles in real space, and the momentum distributions which mainly are located around the circle $|k|=\alpha$. \[profilepiover2\]](fig3.eps "fig:"){width="1.0\linewidth"}]{}
Phase transitions induced by interaction {#sect:interaction}
========================================
In this section, we consider the interaction effect which will couple single-particle eigenstates with different values of $j_z$. It is interesting to consider the possible vortex configurations in various parameter regimes, which has been widely considered in the case of the fast rotating BECs.
If the dimensionless interaction parameter $g=g_{2D}N/(\hbar\omega l_T^2)$ is small, it is expected that the ground state still remains in a giant-vortex state, which is similar to the non-interacting case. The envelope of the variational wave-function is approximated as $$\begin{aligned}
\phi_{j_z}(\rho,\varphi) \sim \frac{1}{2\pi^{\frac{3}{4}}\sqrt{\sigma\rho}}
e^{-\frac{(\rho-\beta)^2}{2\sigma^2}}e^{i\rho\alpha \cos\theta }
\left [ \begin{array}{c} e^{i [m\varphi-\frac{\theta}{2}]}
\\ -e^{i [(m+1)\varphi+\frac{\theta}{2}]} \end{array} \right ] \nn\end{aligned}$$ with $\sigma$ the radial width of the condensates. Around a thin ring inside the cloud with the radius $\rho$, in order to maintain the overall phase factor $e^{i m \varphi}$, the magnitude of the local momentum along the azimuth direction is determined by $k_{\varphi}=m/\rho$. Depending on the width $\sigma$ of the cloud, the linewidth of $k_{\varphi}$ is proportional to $\delta k_{\varphi}=m\sigma/\beta^2$. In momentum space, this leads to the expansion of the distribution around the ring with $|k|=\alpha$. The increasing of the kinetic energy mainly comes from the term $\hat{E}_{\varphi}=(j_z/\rho-\alpha\sin\theta)^2/2$, which is estimated as $\langle \hat{E}_{\varphi} \rangle_{j_z}$. Details derivation of various energy contributions can be found in Appendix B.
[![Ground state profiles of the condensates for $\alpha=11$, $\beta=6$, $\theta=\pi/3$ with different interaction $g=85$(a), $105$(b), $125$(c), and $145$(d) respectively. From top to bottom: density and phase profiles of the spin-up component, momentum distribution in the lower band along the circle $|k|=\alpha$. The orientation of the ellipse-shaped vortices is determined by $\theta$. See text for details. []{data-label="profilepiover3"}](fig4.eps "fig:"){width="0.98\linewidth"}]{}
Increasing the interaction strength $g$ expands the cloud and leads to larger width $\sigma$ and $\delta k_{\varphi}$, which makes the above variational state energetically unfavorable. In order to minimize the total energy, the condensates tend to involve additional vortices such that the local momentum mainly distributes around the circle $|k|=\alpha$ with smaller $\delta k_{\varphi}$ . Fig. (\[profilepiover2\]) and (\[profilepiover3\]) show the typical ground-state configurations for selected parameters. The phase accumulations around the inwards and outwards boundaries of the cloud are $2\pi m_+$ and $2\pi m_-$ respectively. Therefore, there are $q=m_+-m_-$ vortices involved and distributed symmetrically inside the condensates. Between two nearest vortices, the local wavefunction can be approximately determined as a plane-wave state. Therefore, their corresponding distribution in momentum space is also composed of $q$ peaks located symmetrically around the circle $|k|=\alpha$.
As further increase of the interaction strength, the condensates break into more pieces by involving more vortices. The number of the vortices is qualitatively determined by the competition of the azimuthal kinetic energy and the kinetic energy introduced by the vortices. Specifically, if $q$ vortices locate in the middle of the cloud around the circle $\rho_0 \simeq \beta$, then for the inwards part of the condensates with $\rho < \rho_0$, the mean value of the angular momentum can be approximated as $j_{z,-}\approx j_z-q/2$, while for the regime with $\rho > \rho_0$, we have $j_{z,+}\approx j_z+q/2$. The corresponding kinetic energy along the azimuthal direction is modified as $$\begin{aligned}
\langle \hat{E}_{\varphi} \rangle&=&\langle \hat{E}_{\varphi} \rangle_{j_z} + \frac{q}{2\beta^2}(\frac{q}{4}-\frac{\sigma \alpha \sin\theta}{\sqrt{\pi}}).
\label{eq:energy1}\end{aligned}$$ This indicates that, to make the vortex-lattice state favorable, we must have $(\frac{q}{4}-\frac{\sigma \alpha \sin\theta}{\sqrt{\pi}})<0$. In the limit case with $\theta=0$, this condition is always violated. Therefore, the ground state remains to be an eigenstate of $j_z$ with $j_z=\pm \frac{1}{2}$ even for large interaction strength.
![ Enlarged density and phase profiles around single vortex. The two figures (a) and (b) are the corresponding parts adapted from Fig. (3c) and (4d) respectively.[]{data-label="singlevortex"}](fig5.eps){width="0.78\linewidth"}
We note the vortices display an ellipse-like shape with two main axis, as shown in Fig. \[singlevortex\]. The phase profile is twisted, and the constant phase front exhibits a dislocation around vortex cores. Along the direction of local wavevector $\vec{k}$, the vortex density profile is determined by the length scale $2\pi l_T/\alpha$. While perpendicular to the direction of local $\vec{k}$, the vortex profile is dominated by the healing length $\xi$ due to interaction. Therefore, the vortex density distribution is determined by two different length scales in mutually orthogonal directions, which results in ellipse-shaped vortices. Changing the interaction strength and SO coupling alerts the ratio of the two length scales, thus changes the eccentricity of the ellipses. Additionally, changing the angle $\theta$ also changes the direction of local magnetic fields, and thus modifies the orientation of the vortices, as shown in Fig. \[profilepiover2\] and Fig. \[profilepiover3\].
On the other hand, the introduction of vortices lead to the increase of kinetic energy due to the modification of the density profile. This can be estimated as $0.19q/(\beta\sigma \alpha \xi)$, where $\xi=1/\sqrt{2 g n_0}$ is the dimensionless healing length with $n_0$ the bulk density of the clouds. The total energy changing due to the presence of the vortices can be written as $$\begin{aligned}
\Delta E=\frac{q}{2\beta^2}(\frac{q}{4}-\frac{\sigma \alpha \sin\theta}{\sqrt{\pi}}) + \frac{0.19q}{\beta\sigma \alpha \xi}.
\label{eq:energy2}\end{aligned}$$
[![ (a) Critical angle $\theta_c$ as a function of SO coupling strength $\alpha$ for fixed values of $\beta=4$ and $g=800$. (b) $\theta_c$ decreases as the increase of interaction parameter $g$ for fixed $\alpha=10$, $6$, and $\beta=4$. \[criticaltheta\] ](fig6a.eps "fig:"){width="0.8\columnwidth"} ![ (a) Critical angle $\theta_c$ as a function of SO coupling strength $\alpha$ for fixed values of $\beta=4$ and $g=800$. (b) $\theta_c$ decreases as the increase of interaction parameter $g$ for fixed $\alpha=10$, $6$, and $\beta=4$. \[criticaltheta\] ](fig6b.eps "fig:"){width="0.8\columnwidth"}]{}
Several interesting features can be extracted from Eq. \[eq:energy2\]. For fixed parameters $g$, $\alpha$, and $\beta$, there always exists a critical $\theta_c$ such that $\Delta E =0$ is satisfied. When $\theta<\theta_c$, then $\Delta E>0$, which indicates that a giant-vortex ground state is always favored. As increasing $\alpha$, $\theta_c$ satisfying $\Delta E =0$ becomes smaller. At $\theta>\theta_c$, the ground state exhibits a lattice-type structure along the ring with a giant vortex core. The values of $q$ is determined by minimizing $\Delta E$ with respect to $g$, $\alpha$, and $\beta$, respectively. Fig. (\[criticaltheta\]) shows $\theta_c$ as a function of SO strength $\alpha$ at which the transition from a giant-vortex state to a vortex-lattice state occurs. When $\alpha$ is small, a giant-vortex state is favored for all values of $\theta$. As $\alpha$ increases, $\theta_c$ drops quickly initially and decreases much slower when $\alpha$ becomes large as shown in Fig. (\[criticaltheta\]) (a). In Fig. (\[criticaltheta\]) (b), it shows that as increasing the interaction $g$, it is becomes easier to drive the system into the vortex lattice state.
[![Phase diagram in the $\alpha-g$ plane for $\beta=6$ with different $\theta=\pi/3$ ((a)) and $\pi/2$ ((b)). The number $\underline{M}$ means that the condensates support a vortex-lattice-type ground state with $M$ momentum peaks along the circle $|k|=\alpha$. The regime with shadow in (b) indicates the ground state shows multi-layer structure as increasing the interaction strength. Other phases are defined as follows: GV(giant-vortex state),IM (intermediate regime). \[phasediagram\] ](fig7a.eps "fig:"){width="0.8\columnwidth"} ![Phase diagram in the $\alpha-g$ plane for $\beta=6$ with different $\theta=\pi/3$ ((a)) and $\pi/2$ ((b)). The number $\underline{M}$ means that the condensates support a vortex-lattice-type ground state with $M$ momentum peaks along the circle $|k|=\alpha$. The regime with shadow in (b) indicates the ground state shows multi-layer structure as increasing the interaction strength. Other phases are defined as follows: GV(giant-vortex state),IM (intermediate regime). \[phasediagram\] ](fig7b.eps "fig:"){width="0.8\columnwidth"}]{}
Fig. (\[phasediagram\]) shows the phase diagram in the $\alpha-g$ plane for a fixed $\beta=6$ for different values of $\theta$. For a fixed $\alpha$ and at small values of $g$, the system remains to be a giant-vortex state until $g$ reaches its critical value $g_c$. When $g>g_c$, the system enters into an intermediate regime in which vortices start to enter into the condensates from boundaries. The momentum distribution also breaks into several disconnected segments. More single quantum vortices are generated in the condensates as further increasing the interaction strength. The vortices distribute symmetrically along the ring and separate the condensates into pieces. Between two neighboring vortices, the condensates are approximated by local plane-wave states. The momentum distribution composes of multi-peaks symmetrically located around the circle $|k|=\alpha$. Increasing $g$ also increases the number of the single quantum $M$ inside the condensates, hence increases the number of peaks in momentum space. For a smaller value $\theta=\pi/3$, the critical $g_c$ is increased, which means that stronger interactions are needed to drive the system into the vortex-lattice states. Interestingly, the intermediate regime is also greatly enlarged. This is consistent with the limit case $\theta=0$, where the system remains to be a giant-vortex state even in the case of large interaction strength.
[![Density and momentum distributions about the ground states of the condensates for $\theta=\pi/3$ with $g=400$((a)), $1000$((b)), and $\theta=\pi/2$ with $g=400$((c)), $1000$((d)). Other parameters are the same with figure (\[profilepiover3\]). We note that since the two spin components share almost the same profiles, only the densities of the spin-up component are shown for simplicity. \[multiplelayer\] ](fig8.eps "fig:"){width="0.98\columnwidth"}]{}
More ellipse-shaped vortices are formed as further increasing the interaction strength, which are self-organized into a multiple layered ring structure, as shown in Fig.(\[multiplelayer\]). Around each ring, vortices distributed symmetrically. The number of the vortices between different layers can be not equal due to their different radius. Therefore the distribution in momentum space becomes asymmetric, and exhibits complex multi-peak structures around the circle $|k|=\alpha$.
Experimental consideration {#sect:experiment}
==========================
The Hamiltonian Eq. \[eq:ham\_1\] considered above can be dynamically generated on behalf of a series of gradient magnetic pulses [@anderson2013; @xu2013]. Starting with the typical single-particle Hamiltonian $H_s=\frac{p^2}{2m}+\frac{1}{2}m\omega^2 r^2$, in the first time step, we employ a pair of magnetic pulses $U_1$ and $U_1^\dagger$, defined as as U\_1=e\^[i (x \_x+y \_y)/]{}, at time $t=2n\tau$, $(2n+1)\tau$ respectively. Secondly, a typical effective gradient coupling, , is applied during the whole time duration $[(2n+1)\tau, 2(n+1)\tau]$. Combining these two time steps, an effective dynamical evolution $U=e^{-i H_0 \tau}$, which implements the desired dynamics. In practice, the gradient magnetic pulse in the first cycle can be simulated with quadrupole fields as $\vec{B}=(x,y,-2z)$. When the condensates is strongly confined in the $xy$ plane, the influence of the nonzero gradient along $z$-axis can be neglected. The effective gradient coupling in the second cycle can be implemented with the help of atom-laser coupling. For instance, a standard two-set Raman beams with blue-detuning [@liu2014] can realize an effective coupling , where the wavevectors $\vec{k}_1$ and $\vec{k}_2$ in the $xy$ plane can be chosen as $\vec{k}_1=k(\cos\theta, -\sin\theta)$ and $
\vec{k}_2=k(\sin\theta, \cos\theta)$. When $2\pi/k$ is much larger than the trap length $l_T$, the required effective coupling is approximately obtained. Finally, the phases discussed in the context can be detected by monitoring their corresponding density and momentum distributions using the setup of time of flight.
Concluding remarks {#sect:conclusions}
==================
To summarize, we have discussed the ground state phase diagram of SO coupled BECs subject to gradient magnetic fields. Theoretical and numerical analyses indicate that the system supports various interesting vortex physics, including the single-particle giant-vortex states with tunable vorticity, multiple layered vortex-lattice-ring states, and the ellipse-shaped vortex profiles. Therefore, the combination of SO coupling and the gradient magnetic fields provides a powerful method to engineer various vortex states without rotating the trap. We hope our work will stimulate further research of searching for various novel states in SO coupled bosons subject to effective gradient magnetic fields.
Acknowledgement
===============
X.F. Z., Z.W. Z., and G.C. G. acknowledge the support by NSFC (Grant Nos. 11004186, 11474266,11174270), National Basic Research Program of China (Grants No. 2011CB921204 and No. 2011CBA00200). C. W. is supported by the NSF DMR-1410375 and AFOSR FA9550-14-1-0168, and also acknowledges the support from the National Natural Science Foundation of China (11328403).
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Single particle eigenstates for large $\beta$ {#sect:dispersion}
=============================================
We start with the dimensionless Hamiltonian $$\begin{aligned}
\frac{H_0}{\hbar\omega}=\int d \vec{\rho}^2 \hat{\phi}(\vec{\rho})^{\dag} \Big \{ -\frac{\vec{\nabla}^2}{2} + \beta \rho \left ( \cos\theta \hat{r} +\sin\theta \hat{\varphi} \right ) \cdot \vec{\sigma} + \alpha \vec{k} \cdot \vec{\sigma} + \frac{1}{2}\rho^2 \Big \}\hat{\phi}(\vec{\rho}).\end{aligned}$$ Since the total angular momentum is conserved, the single particle eigenstates can be written as $\phi_m=[f(\rho) e^{i m \varphi} ,
g(\rho) e^{i (m+1) \varphi}]^T$ with $j_z=(m+1/2)$. Substitute this wavefunction into their corresponding Schrödinger equations, we obtain $$\begin{aligned}
\left\{\frac{\hat{p}^2_{\rho}}{2} + \frac{j_z^2}{2\rho^2} +\frac{(\rho\sigma_x-\beta)^2}{2} -\alpha [(\hat{p}_{\rho} \cos\theta + \frac{j_z}{\rho}\sin\theta ) \sigma_x \right. \nn \\
&& \hspace{-8.5cm} \left. + (\frac{j_z}{\rho}\cos\theta-\hat{p}_{\rho} \sin\theta ) \sigma_y ] + \frac{j_z\sigma_z}{2 \rho^2} \right \} \left[ \begin{array}{c} \tilde{f}(\rho) \\ \tilde{g}(\rho) \end{array} \right ]= E \left[ \begin{array}{c} \tilde{f}(\rho) \\ \tilde{g}(\rho) \end{array} \right ], \nn\end{aligned}$$ where $\tilde{f}(\rho)=f(\rho) e^{i\theta/2}$, $\tilde{g}(\rho)=g(\rho)e^{-i\theta/2}$, $\hat{p}_{\rho}=-i(\frac{\partial}{\partial \rho}+\frac{1}{2\rho})$ is the momentum operator along the radical direction. For large $\beta \gg 1$, these functions mainly distribute around the circle $\rho=\beta$ in the plane, so we consider the superposition $F^{\pm}(\rho)=\frac{1}{2}[\tilde{f}(\rho)\pm \tilde{g}(\rho)])$, which satisfies the following approximated equations as $$\begin{aligned}
\left(\frac{\hat{p}^2_{\rho}}{2} \mp \alpha \cos\theta \hat{p}_{\rho} + \frac{j_z^2}{2\rho^2} \mp \alpha \sin\theta \frac{j_z}{\rho} +\frac{\rho^2}{2} \mp \beta \rho\right ) F^{\pm}(\rho) \nn \\
\pm i \alpha \left( \hat{p}_{\rho} \sin\theta - \frac{j_z}{\rho} \cos\theta \right) F^{\mp}(\rho) = E_{j_z} F^{\pm}(\rho). \nn\end{aligned}$$ The above equation indicates that to minimize the kinetic energy, we need $\langle \vec{p}_{\rho} \rangle \simeq \alpha\cos\theta$. Around $\rho=\beta$, we have the approximated solutions as $F^{\pm}(\rho) \sim H_n(\rho\pm\beta)e^{-(\rho\pm\beta)^2/2}e^{\pm i\alpha \cos\theta \rho}$ with $H_n(r)$ the usual $n$-th Hermite polynomial. Therefore, $F^+$ is negligible since we always have $\rho>0$. The solution now can be written as $\tilde{f}(\rho)\simeq\tilde{g}(\rho)\propto H_n(\rho-\beta)e^{-(\rho-\beta)^2/2}e^{i\alpha \cos\theta \rho}$. So we obtain the approximated wavefunctions for the lowest band ($n=1$) as $$\begin{aligned}
\phi_{n=1,j_z} \simeq \frac{1}{(2\pi)^{\frac{3}{4}}\rho^{\frac{1}{2}}}e^{-\frac{(\rho-\beta)^2}{2}}e^{i\rho\alpha \cos\theta }\left [ \begin{array}{c} e^{i [m\varphi-\frac{\theta}{2}]} \\ -e^{i [(m+1)\varphi+\frac{\theta}{2}]} \end{array} \right ].\end{aligned}$$ The dispersion is estimated as $$\begin{aligned}
E_{n,j_z}= n+\frac{1-\alpha^2-\beta^2}{2} + \frac{(j_z-\alpha\beta\sin\theta)^2}{2(\alpha^2\cos^2\theta+\beta^2)},\end{aligned}$$ which is minimized when $j_z \simeq \alpha\beta\sin\theta$, so for the kinetic term along the tangential direction $\hat{E}_{\varphi}=(\frac{j_z}{\rho}-\alpha\sin\theta)^2/2$.
Energy estimation of vortix lattice states around the ring
==========================================================
For weak interaction, the condensates expands along the radial direction as the parameter $g$ is increased. When $g$ is large enough, to lower the kinetic energy, the system tends to involve vortices located around a ring inside the condensates, which separate the wavefunction into two parts. Inside the vortex-ring, the wavefunction for the spin-up component is approximated as a giant vortex with the phase factor $e^{i 2\pi m_-\phi}$, while outside the ring, the mean angular momentum carried by single particle is approximated as $m_+\hbar$. The difference $q=m_+-m_-$ represents the vortex number inside the condensates. Therefore the variational ground-state can be approximated as follows $$\begin{aligned}
\phi(\rho,\phi)= \left \{
\begin{array}{c}
\phi_{j_z-q/2}(\rho,\phi) \mbox{ when } \rho \in (0,\beta), \\
\phi_{j_z+q/2}(\rho,\phi) \mbox{ when } \rho \in (\beta,\infty).
\end{array}
\right.\end{aligned}$$ We also assumes that around the circle $\rho=\beta$, vortices are involved and self-organized to compensate the phase mismatch so that the the whole wavefunction is well-defined.
The increase of the kinetic energy mainly comes from the term $\hat{E}_{\varphi}=(\frac{j_z}{\rho}-\alpha\sin\theta)^2/2$. For ground state without lattices, the energy is estimated and denoted as $$\begin{aligned}
\langle \hat{E}_{\varphi} \rangle_{j_z} = \int^{\infty}_0 d\rho \int^{2\pi}_0 d\varphi \rho \phi^{\dag}_{j_z} \frac{(j_z/\rho-\alpha\sin\theta)^2}{2} \phi_{j_z},\end{aligned}$$ where we use $\langle \rangle_{j_z}$ to denote the mean values over the trivial variational function $\phi_{j_z}$ Similarly, for GV states and large $\beta \gg 1$, the corresponding energy increasing is estimated as $$\begin{aligned}
\langle \hat{E}_{\varphi} \rangle &=& \int^{\infty}_{\beta} d\rho \int^{2\pi}_0 d\varphi \rho \phi^{\dag}_{j_z+\frac{q}{2}} \frac{[(j_z+\frac{q}{2})/\rho-\alpha\sin\theta]^2}{2} \phi_{j_z+\frac{q}{2}} + \int^{\beta}_0 d\rho \int^{2\pi}_0 d\varphi \rho \phi^{\dag}_{j_z-\frac{q}{2}} \frac{[(j_z-\frac{q}{2})/\rho-\alpha\sin\theta]^2}{2} \phi_{j_z-\frac{q}{2}} \nn \\
&\approx& \langle \hat{E}_{\varphi} \rangle_{j_z} +\langle \frac{q^2}{8\rho^2} \rangle_{j_z} + \int^{\infty}_{0} d\rho \int^{2\pi}_0 d\varphi \rho \phi^{\dag}_{j_z} \frac{q}{2\rho}(\frac{j_z}{\rho}-\alpha\sin\theta) \phi_{j_z} \nn \\
&\approx & \langle \hat{E}_{\varphi} \rangle_{j_z} + \frac{q}{2\beta^2}(\frac{q}{4}-\frac{\sigma \alpha \sin\theta}{\sqrt{\pi}}),\end{aligned}$$ which gives the Eq. (\[eq:energy1\]) shown in the context.
Around each vortex core, both the density and phase are twisted as shown in Fig. \[singlevortex\]. To take into account the energy contribution of these vortices, we choose the following approximated local wavefunction as (see Fig.(8a) for $\theta=\pi/2$) $$\psi \simeq \phi_0 \left[\frac{e^{-ik_+x}}{e^{-\sqrt{2}y/\xi}+1}+\frac{e^{-i(k_-x+\pi)}}{e^{\sqrt{2}y/\xi}+1}\right] \left [ \begin{array}{c} 1 \\ 1 \end{array} \right ],$$ where for simplicity we have set the origin to the vortex core, $\phi_0$ is the bulk wavefunction away from the vortex cores and is estimated as $|\phi_0|=\sqrt{n_0}=[2\pi^{\frac{3}{4}}\sqrt{\sigma\rho}]^{-1}$, $k_{\pm}$ represent the corresponding local wavevectors of the condensates inwards and outwards the vortex ring. $\xi=(2gn_0)^{-1/2}$ is usually called as the healing length as can be seen by considering the above wavefunction along the line $x=0$ $$\frac{1}{e^{-\sqrt{2}y/\xi}+1} - \frac{1}{e^{\sqrt{2}y/\xi}+1} = \tanh[y/(\sqrt{2}\xi)],$$ which is consistent with the estimation shown in [@book]. We note that the approximated wavefuntion is only valid when $k_-x \in [-\pi,\pi]$ and $y\in[-2\xi,2\xi]$. We also requires $k_+ \simeq 2 k_-\simeq\alpha$ such that a vortex is formed around the origin as indicated by numerics. Since energy gain along the tangential direction has been taking into account in $\hat{E}_{\varphi}$, in this typical case, the changing of kinetic energy along the direction perpendicular to the local wavevector $\vec{k}$ can then be approximated by $$\int_{k_-x=-\pi}^{k_-x=\pi} dx \int_{-2\xi}^{2\xi} dy \psi^{\dag} \Big[-\frac{\nabla_y^2}{2} \Big]\psi = \frac{\sqrt{2}\tanh(\sqrt{2})^3}{3\sqrt{\pi}\beta\sigma\alpha\xi} \approx \frac{0.19}{\beta\sigma \alpha \xi}.$$ We note that the above analysis also applies for $\theta \neq 0$. Taking into account all vortices inside the two-components condensates, we obtain the total energy introduced due to fluctuation of the vortex density profile as $0.19q/(\beta\sigma \alpha \xi)$.
|
---
abstract: 'We have investigated dynamical heterogeneity in a highly supercooled liquid using molecular-dynamics simulations in three dimensions. Dynamical heterogeneity can be characterized by three quantities: correlation length $\xi_4$, intensity $\chi_4$, and lifetime $\tau_{\text{hetero}}$. We evaluated all three quantities consistently from a single order parameter. In a previous study (H. Mizuno and R. Yamamoto, Phys. Rev. E [**82**]{}, 030501(R) (2010)), we examined the lifetime $\tau_{\text{hetero}}(t)$ in two time intervals $t=\tau_\alpha$ and $\tau_{\text{ngp}}$, where $\tau_\alpha$ is the $\alpha$-relaxation time and $\tau_{\text{ngp}}$ is the time at which the non-Gaussian parameter of the Van Hove self-correlation function is maximized. In the present study, in addition to the lifetime $\tau_{\text{hetero}}(t)$, we evaluated the correlation length $\xi_4(t)$ and the intensity $\chi_4(t)$ from the same order parameter used for the lifetime $\tau_{\text{hetero}}(t)$. We found that as the temperature decreases, the lifetime $\tau_{\text{hetero}}(t)$ grows dramatically, whereas the correlation length $\xi_4(t)$ and the intensity $\chi_4(t)$ increase slowly compared to $\tau_{\text{hetero}}(t)$ or plateaus. Furthermore, we investigated the lifetime $\tau_{\text{hetero}}(t)$ in more detail. We examined the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$ and found that as the time interval $t$ increases, $\tau_{\text{hetero}}(t)$ monotonically becomes longer and plateaus at the relaxation time of the two-point density correlation function. At the large time intervals for which $\tau_{\text{hetero}}(t)$ plateaus, the heterogeneous dynamics migrate in space with a diffusion mechanism, such as the particle density.'
author:
- Hideyuki Mizuno
- Ryoichi Yamamoto
bibliography:
- 'paper.bib'
title: 'Dynamical heterogeneity in a highly supercooled liquid: Consistent calculations of correlation length, intensity, and lifetime'
---
INTRODUCTION
============
As liquids are cooled toward the glass transition temperature $T_g$, a drastic slowing occurs for dynamical properties such as the structural relaxation time, the diffusion constant, and the viscosity, while only small changes are detected in static properties [@ediger_1996; @Debenedetti_2001]. Despite the extremely widespread use of glass in industry, the formation process and dynamic properties of this material are still poorly understood. The goal of theoretical investigations of the glass transition is to understand the universal mechanism that gives rise to the drastic slowing of dynamical properties. Numerous studies have attempted to explain the fundamental mechanisms of the slowing of the dynamics observed in fragile glass (i.e., the sharp increase in viscosity near the glass transition). However, the physical mechanisms behind this slowing have not been successfully identified.
Recently, “dynamical heterogeneities” in glass-forming liquids have attracted much attention. The dynamics of glass-forming liquids are not only drastically slow but also become progressively more heterogeneous upon approaching the glass transition. Dynamical heterogeneities have been detected and visualized through simulations of soft-sphere systems [@muranaka_1994; @hurley_1995; @yamamoto1_1998; @yamamoto_1998; @perera_1999; @cooper_2004], hard-sphere systems [@doliwa_2002], and Lennard-Jones (LJ) systems [@donati_1998], and through experiments performed on colloidal dispersions using particle-tracking techniques [@Marcus1999; @kegel_2000; @weeks_2000]. Insight into the mechanisms of dynamical heterogeneities will lead to a better understanding of the slowing of the dynamics near the glass transition.
The properties of dynamical heterogeneity can be characterized by the following three quantities: the correlation length, the intensity, and the lifetime. In a system displaying dynamical heterogeneity, the particles can be divided into “slow” and “fast” sub-sets. The slow and fast particles form cooperative correlated regions, and these slow and fast regions migrate in space over time. These three quantities (the correlation length, the intensity, and the lifetime) can describe the static and dynamic properties of the slow and fast regions. The intensity measures the average variance of the slow and fast regions, and the correlation length characterizes the spatial extent of the slow and fast regions. The lifetime represents the time scale at which the slow and fast regions migrate in the space.
The correlation length, intensity, and lifetime can be investigated using the correlation functions of the particle dynamics. In fact, we can evaluate the correlation length $\xi_4$ and the intensity $\chi_4$ by calculating the four-point correlation functions that correspond to the static structure factors of the particle dynamics. Several simulations [@yamamoto1_1998; @perera_1999; @glotzer_2000; @doliwa2000; @lacevic_2003; @berthier_2004; @toninelli_2005; @chandler_2006; @stein_2008; @karmakar_2010; @karmakar2_2010; @flenner_2010; @flenner_2011], experiments [@ediger_2000; @berthier_2005; @Ferrier2007; @narumi_2011], and mode-coupling theories [@biroli_2004; @biroli_2006; @szamel_2010] have estimated $\xi_4$ and $\chi_4$ using four-point correlation functions and have revealed that $\xi_4$ and $\chi_4$ increase with decreasing temperature (or an increase in the volume fraction in the case of hard-sphere systems). Furthermore, we can quantify the lifetime $\tau_{\text{hetero}}$ using the multiple-time extensions of the four-point correlation functions (i.e., the multi-time correlation functions) that correspond to the time correlation functions of the particle dynamics. Recent simulations have quantified $\tau_{\text{hetero}}$ using multi-time correlation functions [@yamamoto_1998; @flenner_2004; @leonard_2005; @kim_2009; @kim_2010]. Various experiments, including photobleaching techniques and nuclear magnetic resonance, have also measured $\tau_{\text{hetero}}$ [@ediger_2000; @richert_2002; @wang_1999; @wang_2000; @rohr_1991; @bohmer_1991; @russell_2000]. It was reported that $\tau_{\text{hetero}}$ increases dramatically with decreasing temperature or an increase in the volume fraction and can exceed the $\alpha$-relaxation time near the glass transition.
As mentioned above, there have been many studies on the correlation length, the intensity, and the lifetime of dynamical heterogeneity near the glass transition. However, knowledge of and measurements relating to the lifetime are still limited. Moreover, individual studies have been restricted to only the correlation length and the intensity or only the lifetime, and the relationship between the length and time scales of dynamical heterogeneity remains controversial despite its importance [@kim_2009; @kim_2010].
The aim of the present study is to examine all the three quantities (the correlation length, the intensity, and the lifetime of dynamical heterogeneity) consistently. We performed molecular-dynamics (MD) simulations and investigated dynamical heterogeneity using the correlation functions of the particle dynamics. In our previous study [@mizuno_2010], we evaluated the lifetime $\tau_{\text{hetero}}(t)$ in two different time intervals: the $\alpha$-relaxation time $\tau_\alpha$ and the time $\tau_{\text{ngp}}$ at which the non-Gaussian parameter of the Van Hove self-correlation function is maximized. In the present study, in addition to the lifetime $\tau_{\text{hetero}}(t)$, we quantified the correlation length $\xi_4(t)$ and the intensity $\chi_4(t)$ from the same order parameter used when calculating $\tau_{\text{hetero}}(t)$. Furthermore, we examined the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$ to understand the lifetime and the dynamic properties of dynamical heterogeneity in more detail.
The paper is organized as follows. In Sec. \[pdynamics\], we explain the correlation functions of the particle dynamics. We show that dynamical heterogeneity can be systematically examined using the correlation functions of the particle dynamics. In Sec. \[model\], we briefly review our MD simulation and present some results from conventional density correlation functions. In Secs. \[result1\] and \[result2\], the results for dynamical heterogeneity are presented. In Sec. \[result1\], we first show three quantities, the correlation length, the intensity, and the lifetime, which are consistently calculated from a single order parameter. In Sec. \[result2\], we next present the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$. In Sec. \[conclusion\], we summarize our results.
CORRELATION FUNCTIONS OF PARTICLE DYNAMICS {#pdynamics}
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As we mentioned, dynamical heterogeneity can be characterized by three quantities: the correlation length, the intensity, and the lifetime. In this section, we introduce the correlation functions of the particle dynamics and demonstrate that these three quantities can be systematically evaluated in terms of the correlation functions of the particle dynamics.
The conventional two-point correlation function $F({k},t)$ represents the correlation of local fluctuations $\delta n(\vec{k},t)$ with some order parameter, such as particle density. The expression $\delta n(\vec{k},t)$ is the Fourier component of the fluctuations at time $t$, and $F({k},t) = \langle \delta n(\vec{k},t) \delta n(-\vec{k},0) \rangle$, where $k=\left| \vec{k} \right|$. When $t=0$, $S({k}) \equiv F(k,t=0)$ is the spatial correlation of $\delta n(\vec{k},0)$ (i.e., the static structure factor), and we can examine the static structure of the order parameter by the wavenumber dependence of $S({k})$. When $t>0$, $F(k,t)$ describes the particle dynamics in the time interval $[0,t]$, averaged over the initial time and space. As the time interval $t$ increases, $F({k},t)$ decays in the stretched exponential form, $$\frac{F({k},t)}{F({k},0)} \sim \exp \left( -\left( \frac{t}{\tau({k})}\right)^\beta \right),
\label{F2}$$ where $\tau({k})$ is the relaxation time of the two-point correlation function that represents the characteristic time scale of the average particle dynamics.
![(Color) Schematic illustration of the time configuration of the correlation functions of the particle dynamics: (a) the spatial correlation function of the particle dynamics in the time interval $[0,t]$ and (b) the time correlation function of the particle dynamics between the two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$. []{data-label="ts"}](fig1.eps)
To examine the structure and motion of spatially heterogeneous dynamics, we must calculate the correlation of the local fluctuations, $\delta Q_{{k}}(\vec{q},t_0,t)$, in the particle dynamics. The expression $\delta Q_{{k}}(\vec{q},t_0,t)$ is the Fourier component $\vec{q}$ of the fluctuations in the particle dynamics that are associated with a microscopic wavenumber ${k}$ in the time interval $[t_0,t_0+t]$. $F({k},t)$ is equal to $Q_{{k}}(\vec{q},t_0,t)$ averaged over the initial time $t_0$ and space, i.e., $F({k},t) \sim \langle Q_{{k}}(\vec{q},t_0,t) \rangle$. The correlation function defined by $$\begin{aligned}
S_{4,{k}}({q},t)=
\langle \delta Q_{{k}}(\vec{q},0,t) \delta Q_{{k}}(-\vec{q},0,t) \rangle,
\end{aligned} \label{K4}$$ represents the spatial correlation of the particle dynamics in the time interval $[0,t]$. The time configuration of $S_{4,{k}}({q},t)$ is schematically illustrated in Fig. \[ts\](a). $S_{4,{k}}({q},t)$ is the four-point correlation function. We can examine the structure of the particle dynamics by the wavenumber dependence of $S_{4,{k}}({q},t)$. At small wavenumbers of $q$, $S_{4,k}(q,t)$ can be approximated by the simple Ornstein-Zernike (OZ) form [@yamamoto1_1998; @lacevic_2003], $$S_{4,k}(q,t) = \frac{\chi_{4,k}(t)}{1 + q^2 \xi_{4,k}(t)^2}, \label{OZ}$$ where $\xi_{4,k}(t)$ is the correlation length of the heterogeneous dynamics and $\chi_{4,k}(t)$ is the intensity of the heterogeneous dynamics, which is the long-wavelength limit of $S_{4,k}(q,t)$, i.e., $\chi_{4,k}(t) = \lim_{q \rightarrow 0} S_{4,k}(q,t)$. Note that $\chi_{4,k}(t)$ is termed the four-point dynamical susceptibility. According to previous studies [@karmakar_2010; @karmakar2_2010; @flenner_2010; @flenner_2011], to quantify $\xi_{4,k}(t)$ and $\chi_{4,k}(t)$ accurately, we need to use a large system or determine $\chi_{4,k}(t)$ through other means (not OZ fitting). In their work [@flenner_2010; @flenner_2011], E. Flenner et al. claimed that accurate determinations of $\xi_{4,k}(t)$ and $\chi_{4,k}(t)$ can be made by fitting $S_{4,k}(q,t)$ to the OZ form in the range of $q \xi_{4,k}(t) < 1.5$. They used a large system composed of $8 \times 10^4$ particles to obtain accurate fits for the OZ form. In the present study, we used a large system with $10^5$ particles to fit $S_{4,k}(q,t)$ to the OZ form accurately and obtain the values of $\xi_{4,k}(t)$ and $\chi_{4,k}(t)$.
Furthermore, the time correlation function defined by $$\begin{aligned}
F_{4,{k}}({q},t_s,t)=
\langle \delta Q_{{k}}(\vec{q},t_s+t,t) \delta Q_{{k}}(-\vec{q},0,t) \rangle,
\end{aligned} \label{F4}$$ represents the correlation of the particle dynamics between the two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$. The value $t_s$ is the time separation between the two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$. The time configuration of $F_{4,{k}}({q},t_s,t)$ is schematically illustrated in Fig. \[ts\](b). $F_{4,{k}}({q},t_s,t)$ is the multiple-time extension of the four-point correlation function [@kim_2009; @kim_2010]. As the time separation $t_s$ increases, $F_{4,{k}}({q},t_s,t)$ with fixed $t$ decays in the stretched exponential form, $$\begin{aligned}
\frac{F_{4,{k}}(q,t_s,t)}{F_{4,{k}}({q},0,t)}
\sim \exp\left( - \left( \frac{t_s}{\tau_{4,{k}}({q},t)} \right)^c \right),
\end{aligned} \label{F42}$$ where $\tau_{4,{k}}({q},t)$ is the relaxation time of the correlation of the particle dynamics. We determined the lifetime $\tau_{\text{hetero}}(t)$ of the heterogeneous dynamics as $\tau_{4,{k}}({q},t)$ at $q=0.38$. We need longer trajectories of the simulations to quantify $\tau_{\text{hetero}}(t)$ than to quantify $\xi_{4,k}(t)$ and $\chi_{4,k}(t)$. In the present study, we used a smaller system with $10^4$ particles to calculate $\tau_{\text{hetero}}(t)$. As explained above, we can systematically evaluate the correlation length, the intensity, and the lifetime of dynamical heterogeneity by calculating the correlation functions of the particle dynamics.
SIMULATION MODEL AND RESULTS FROM THE DENSITY CORRELATION FUNCTIONS {#model}
===================================================================
Simulation model
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We performed MD simulations in three dimensions on binary mixtures of two different atomic species, 1 and 2, with a cube of constant volume $V$ as the basic cell, surrounded by periodic boundary image cells. The particles interacted via their soft-sphere potentials, $v_{a b}(r)= \epsilon (\sigma_{a b}/r)^{12}$; where r is the distance between two particles, $\sigma_{a b} = (\sigma_a + \sigma_b)/2$, and $a,b \in 1,2$. The interaction was truncated at $r=3 \sigma_{a b}$. The mass ratio was $m_2/m_1=2$, and the diameter ratio was $\sigma_2/\sigma_1=1.2$. This diameter ratio avoided system crystallization and ensured that an amorphous supercooled state formed at low temperatures [@miyagawa_1991]. As mentioned in Sec. \[pdynamics\], we used two systems: a small system with $N_1 = N_2 =5 \times 10^3$ ($N=N_1+N_2=10^4$) particles and a large system with $N_1 = N_2 =5 \times 10^4$ ($N=N_1+N_2=10^5$) particles. The large system was used to quantify the correlation length $\xi_{4,k}(t)$ and the intensity $\chi_{4,k}(t)$, and the small system was used to quantify the lifetime $\tau_{\text{hetero}}(t)$. In the present paper, the following dimensionless units are used: length, $\sigma_1$; temperature, $\epsilon/k_B$; and time, $\tau_0 = (m_1 \sigma_1^2/\epsilon)^{1/2}$. The particle density was fixed at the high value of $\rho = N/V = 0.8$. The system lengths were $L=V^{1/3} = 23.2$ and $50.0$ for the small and large systems, respectively. Simulations were performed at $T=0.772,\ 0.473,\ 0.352,\ 0.306,\ 0.289,\ 0.267$, and $0.253$. Note that the freezing point of the corresponding one-component model is approximately $T=0.772\ (\Gamma_{\text{eff}}=1.15)$ [@miyagawa_1991]. Here, $\Gamma_{\text{eff}}$ is the effective density, which is a single parameter characterizing this model. At $T=0.253\ (\Gamma_{\text{eff}}=1.52)$, the system is in a highly supercooled state. We used the leapfrog algorithm with time steps of $0.005$ when integrating the Newtonian equation of motion. At each temperature, the system was carefully equilibrated under the canonical condition so that no appreciable aging effect was detected for various quantities, including the pressure and the density correlation function. Once equilibrium was established, data were taken under the microcanonical condition. The length of the data collection runs was at least 100 times the $\alpha$-relaxation time, $\tau_\alpha$, for the small system and 10 times $\tau_\alpha$ for the large system. Information regarding this model, such as the static structure factor, the intermediate scattering function, and the mean square displacement, can be found in previous works [@yamamoto1_1998; @kim_2000].
Single-particle and collective-particle diffusive motion {#sincol}
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Before showing the results for dynamical heterogeneity in supercooled liquids, we present some results from an investigation of the average particle dynamics using conventional density correlation functions. Let us consider the density correlation functions $F_{sa}(k,t)$ and $F_a(k,t)\ (a \in 1,2)$, defined by $$\begin{aligned}
& F_{sa}(k,t) = \left< \frac{1}{N_a} \sum_{j=1}^{N_a} \delta n_{aj}(\vec{k},t) \delta n_{aj}(-\vec{k},0) \right>, \\
& F_a(k,t) = \langle \delta n_a(\vec{k},t) \delta n_a(-\vec{k},0) \rangle, \label {dc}
\end{aligned}$$ where $\delta n_{aj}(\vec{k},t) = \exp[ -i \vec{k} \cdot \vec{r}_{aj}(t)]$ is the Fourier component $\vec{k}$ of the tagged particle density fluctuations of particle species $a$, and $\delta n_a(\vec{k}, t) = \sum^{N_a}_{j=1} \exp[ -i \vec{k} \cdot \vec{r}_{aj}(t)]$ is the Fourier component $\vec{k}$ of the density fluctuations of particle species $a$. The terms $F_{sa}(k,t)$ and $F_a(k,t)$ describe the single-particle and collective-particle motion, respectively [@simpleliquid]. We calculated $F_{sa}(k,t)$ and $F_a(k,t)$ for a wide range of wavenumbers, $k=0.35 - 40$. As seen in Eq. (\[F2\]), $F_{sa}(k,t)$ and $F_a(k,t)$ decay in the stretched exponential form, $$\begin{aligned}
& \frac{F_{sa}(k,t)}{F_{sa}(k,0)} = \exp \left( -\left( \frac{t}{\tau_{sa}(k)} \right)^{\beta_s} \right),\\
& \frac{F_a(k,t)}{F_a(k,0)} = \exp \left( -\left( \frac{t}{\tau_{ca}(k)} \right)^{\beta_c} \right),
\end{aligned}$$ where $\tau_{sa}(k)$ and $\tau_{ca}(k)$ are the wavenumber-dependent relaxation times of $F_{sa}(k,t)$ and $F_a(k,t)$, respectively.
In Fig. \[tauk\], $\tau_{sa}(k)$ and $\tau_{ca}(k)$ are plotted for particle species 1 and 2 as functions of the wavenumber $k$. In Fig. \[tauk\](a), $\tau_{sa}(k)$ approaches $\tau_{sa}(k) = D_{sa}^{-1} k^{-2}$ at small wavenumbers $k$, where $D_{sa}$ is the diffusion constant of the single-particle motion of particle species $a$. The term $D_{sa}$ is calculated by $D_{sa} = \lim_{t \rightarrow \infty} \langle [\Delta\vec{r}_a(t)]^2 \rangle/6t$; where $\langle [\Delta\vec{r}_a(t)]^2 \rangle$ is the mean square displacement of particle species $a$; $\langle [\Delta\vec{r}_a(t)]^2 \rangle = \langle \sum_{j=1}^{N_a} [\Delta\vec{r}_{aj}(t)]^2/N_a \rangle$; $\Delta\vec{r}_{aj}(t) = \vec{r}_{aj}(t)-\vec{r}_{aj}(0)$. The diffusion constant of particle species 1 is larger than that of particle species 2 at every temperature. However, $\tau_{ca}(k)$ also approaches $\tau_{ca}(k) \sim k^{-2}$ at small $k$ in Fig. \[tauk\](b). This behavior indicates that the collective-particle motion is also diffusive at large length scales, and we can define the diffusion constant of the collective-particle motion $D_a$ as $$D_a=\lim_{k \to 0} \tau_{ca}^{-1}(k) k^{-2}.$$ We can see that $D_1$ and $D_2$ are almost identical between $D_{s1}$ and $D_{s2}$ in Fig. \[tauk\](b).
![(Color) The wavenumber dependence of (a) $\tau_{sa}(k)$ and (b) $\tau_{ca}(k)$ for particle species $a=1$ and $2$. Temperatures are $0.772 - 0.253$ from the lowest curve to the highest. The black line is $\tau_{sa}(k) = D_{sa}^{-1} k^{-2}$, where $D_{sa}$ is the diffusion constant of the single-particle motion calculated from the mean square displacement. []{data-label="tauk"}](fig2.eps)
![ The wavenumber dependence of (a) $S_{\rho\rho}(k)$ and (b) $\tau_{\rho\rho}(k)$. The temperatures are $0.772 - 0.253$ from the lowest curve to the highest. []{data-label="rhorho"}](fig3.eps)
Correlation between the static structure factor and the relaxation time
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Next, we consider a density variable defined by $$\rho_{\text{eff}}(\vec{k},t) = \sigma_1^3 n_1(\vec{k},t) + \sigma_2^3 n_2(\vec{k},t),$$ that represents the degree of particle packing and is the effective one-component density of our binary mixture [@yamamoto1_1998]. We calculated the static structure factor $S_{\rho\rho}(k)$ and the time correlation function $F_{\rho\rho}(k,t)$, defined by $$\begin{aligned}
& S_{\rho\rho}(k) = \frac{1}{N} \langle \delta \rho_{\text{eff}}(\vec{k},0) \delta \rho_{\text{eff}}(-\vec{k},0) \rangle ,\\
& F_{\rho\rho}(k,t) = \langle \delta \rho_{\text{eff}}(\vec{k},t) \delta \rho_{\text{eff}}(-\vec{k},0) \rangle.
\end{aligned}$$ The terms $S_{\rho\rho}(k)$ and $F_{\rho\rho}(k,t)$ represent the static structure and the average particle dynamics of the effective one-component fluids, respectively. Note that $F_{\rho\rho}(k,t)$ decays in the stretched exponential form with the relaxation time $\tau_{\rho\rho}(k)$, as in Eq. (\[F2\]).
Figure \[rhorho\] shows the wavenumber dependence of $S_{\rho\rho}(k)$ in (a) and $\tau_{\rho\rho}(k)$ in (b). We can see that $S_{\rho\rho}(k)$ and $\tau_{\rho\rho}(k)$ are maximized and minimized at almost the same wavenumbers $k$. For example, both $S_{\rho\rho}(k)$ and $\tau_{\rho\rho}(k)$ have first-peak values around $k=2\pi$. Thus, there is a correlation between $S_{\rho\rho}(k)$ and $\tau_{\rho\rho}(k)$, i.e., between the static structure and the particle dynamics. A similar correlation was found in the Lennard-Jones (LJ) model [@degennes1], water [@degennes2], and the polymer model [@degennes3]. According to Ref. [@degennes3], $\tau_{\rho\rho}(k)$ is modulated by $S_{\rho\rho}(k)$, and this modulation can be understood as a consequence of “de Gennes narrowing” [@degennes0].
![ Time intervals $\tau_\alpha$ and $\tau_{\text{ngp}}$ versus the inverse temperature $1/T$. We use these two time intervals to define the local dynamics. []{data-label="taua"}](fig4.eps)
RESULTS I: CORRELATION LENGTH, INTENSITY, AND LIFETIME {#result1}
======================================================
In this section, we present the results of three quantities: the correlation length, the intensity, and the lifetime of dynamical heterogeneity. We consistently evaluated these three quantities using a single order parameter representing the particle dynamics and its correlation functions. In our previous study [@mizuno_2010], we evaluated the lifetime $\tau_{\text{hetero}}(t)$. For this study, in addition to the lifetime $\tau_{\text{hetero}}(t)$, we quantified the correlation length $\xi_4(t)$ and the intensity $\chi_4(t)$ from the same order parameter used when calculating $\tau_{\text{hetero}}(t)$. As in our previous study [@mizuno_2010], we used two time intervals, $\tau_\alpha$ and $\tau_{\text{ngp}}$, to define the local dynamics. The value $\tau_\alpha$ is the $\alpha$-relaxation time defined by $F_{s1}(k_m,\tau_{\alpha}) = e^{-1}$, where $F_{s1}(k,t)$ is the self-part of the density time correlation function for particle species 1 as defined in Eq. (\[dc\]) and $k_m = 2\pi$ is the first-peak wavenumber of the static structure factor. The value $\tau_{\text{ngp}}$ is the time at which the non-Gaussian parameter $\alpha_2(t)$ [@rahman_1964] of the Van Hove self-correlation function defined as $\alpha_2(t) = 3\langle [\Delta\vec{r}_1(t)]^4 \rangle/5\langle [\Delta\vec{r}_1(t)]^2 \rangle^2-1$ is maximized. In Fig. \[taua\], we show $\tau_\alpha$ and $\tau_{\text{ngp}}$ as functions of the inverse temperature $1/T$. At $T=0.306$, $\tau_\alpha \simeq \tau_{\text{ngp}}$, and $\tau_\alpha$ grows exponentially larger than $\tau_{\text{ngp}}$ with decreasing temperature at $T<0.306$. This trend agrees with other simulation results for LJ systems [@kob_1995_1; @kob_1995_2].
The structure of the heterogeneous dynamics
-------------------------------------------
We first examined the structure of the heterogeneous dynamics. We calculated the displacement of each particle of species 1 in the time interval $[t_0, t_0 + t]$; $\Delta \vec{r}_{1j}(t_0,t) =\vec{r}_{1j}(t_0+t)-\vec{r}_{1j}(t_0) \ (j=1,2,...,N_1)$, and the particle mobility $a_{1j}^2(t_0,t)$ of each particle was defined as $$a_{1j}^2(t_0,t) = \frac {[\Delta \vec{r}_{1j}(t_0,t)]^2}{\langle [\Delta \vec{r}_{1j}(t_0,t)]^2 \rangle}.$$ In Fig. \[hetero1\], we show the spatial distribution of the particle mobility at $T=0.253$, which is the lowest temperature in our simulations. In the figure, the particles are drawn as spheres with radii $a_{1j}^2(t_0,t)$ located at $$\vec{R}_{1j}(t_0,t) = \frac{1}{2}[\vec{r}_{1j}(t_0) + \vec{r}_{1j}(t_0+t)].$$ Notice that $a_{1j}^2(t_0,t)\ge 1$ ($a_{1j}^2(t_0,t)<1$) means that the particle $j$ moves more (less) than the mean value of the single-particle displacement, i.e., particle $j$ is mobile (immobile). In Fig. \[hetero1\], the red (blue) spheres represent $a_j^2(t_0,t)\ge 1$ ($a_j^2(t_0,t)<1$). In the figure, the two heterogeneity structures in $\tau_\alpha$ and $\tau_{\text{ngp}}$ are both significant but differ considerably. There are many more red spheres in \[hetero1\](a) than in \[hetero1\](b), but there are much larger red spheres in \[hetero1\](b) than in \[hetero1\](a). This pattern means that many mobile particles contribute to the heterogeneity in the time interval $\tau_\alpha$, but in the time interval $\tau_\text{ngp}$, relatively few particles are mobile and contribute to the heterogeneity.
![(Color) The distribution of the particle mobility $a_{1j}^2(t_0,t)$ for particle species 1. The time intervals are (a) $[t_0,t_0+\tau_\alpha] (t=\tau_\alpha)$ and (b) $[t_0,t_0+\tau_\text{ngp}] (t=\tau_\text{ngp})$. The temperature is $0.253$. The radii of the spheres are $a_{1j}^2(t_0,t)$, and the centers are at $\vec{R}_{1j}(t_0,t)$. The red and blue spheres represent $a_{1j}^2(t_0,t) \ge 1$ (mobile particles) and $a_{1j}^2(t_0,t)<1$ (immobile particles), respectively. []{data-label="hetero1"}](fig5.eps)
![ The wavenumber dependence of $S_{4}(q,t)$ for particle species 1 at $T=0.772-0.253$. The time interval $t$ is $\tau_\alpha$ in (a) and $\tau_{\text{ngp}}$ in (b). Note that $S_{4}(q,t)$ was calculated with a larger system, $N = 10^5$. []{data-label="sdq"}](fig6.eps)
We also calculated the spatial correlation function of the particle dynamics expressed in Eq. (\[K4\]). We considered the local fluctuations in the particle mobility of particle species 1 defined by $$\delta \hat{{\mathcal{D}}}_1(\vec{r},t_0,t) = \sum_{j=1}^{N_1} \delta a_{1j}^2(t_0,t) \delta( \vec{r} - \vec{R}_{1j}(t_0,t) ),$$ or the Fourier component $\vec{q}$ of $\delta \hat{{\mathcal{D}}}_1(\vec{r},t_0,t)$, $$\delta {\mathcal{D}}_1(\vec{q},t_0,t) = \sum_{j=1}^{N_1} \delta a_{1j}^2(t_0,t) \exp[-i\vec{q} \cdot \vec{R}_{1j}(t_0,t) ], \label{dop}$$ where $\delta a_{1j}^2(t_0,t) = a_{1j}^2(t_0,t) - 1$ is the fluctuation of the particle mobility of the particle $j$. The order parameter $\delta {\mathcal{D}}_1(\vec{q},t_0,t)$ represents the local fluctuations in the particle dynamics in the time interval $[t_0,t_0+t]$. Note that the same order parameter as that of $\delta {\mathcal{D}}_1(\vec{q},t_0,t)$ is used to evaluate the lifetime in our previous study [@mizuno_2010]. We used $\delta {\mathcal{D}}_1(\vec{q},t_0,t)$ as $\delta Q_{{k}}(\vec{q},t_0,t)$ in Eq. (\[K4\]), and the correlation function defined by $$S_{4}(q,t) = \frac{1}{N_1} \langle \delta {\mathcal{D}}_1(\vec{q},0,t) \delta {\mathcal{D}}_1(-\vec{q},0,t) \rangle, \label{sd4eq}$$ corresponds to $S_{4,{k}}({q},t)$. The term $S_{4}(q,t)$ represents the spatial correlation of the particle dynamics in the time interval $[0,t]$. The time configuration of $S_{4}(q,t)$ is schematically illustrated in Fig. \[ts\](a). We were able to examine the structure of the heterogeneous dynamics using the wavenumber dependence of $S_{4}(q,t)$.
Figure \[sdq\] shows the wavenumber dependence of $S_{4}(q,t)$ of particle species 1 for $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. We calculated $S_{4}(q,t)$ using a larger system with $N = 10^5$ to quantify the correlation length and the intensity accurately. At small wavenumbers of $q$ (long-distance scales), the correlations in $\tau_\alpha$ and $\tau_{\text{ngp}}$ both become large with decreasing temperature in a similar manner. However, at large wavenumbers of $q$ (short-distance scales), $S_{4}(q,\tau_{\text{ngp}})$ grows larger than $S_{4}(q,{\tau_\alpha})$ at low temperatures, which reflects that more highly mobile particles exist in the time interval $\tau_\text{ngp}$, as can be seen in the visualization shown in Fig. \[hetero1\].
![ (a) The correlation length $\xi_4(t)$ versus $\tau_\alpha$. (b) The intensity $\chi_{4}(t)$ versus the correlation length $\xi_4(t)$. The time intervals are $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. The straight lines are power law fits: $\xi_4(\tau_\alpha) \sim \tau_\alpha^{0.1 \pm 0.01}$ and $\chi_4(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{3.2 \pm 0.1}$. The dashed curve is a fit to $\xi_4(\tau_\alpha) \sim (\ln {\tau_\alpha})^{0.77 \pm 0.12}$. []{data-label="xi"}](fig7.eps)
![ The correlation length $\xi_4(t)$ and the intensity $\chi_{4}(t)$ calculated by using $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$, i.e., the distribution of the logarithm of the square particle displacements. (a) $\xi_4(t)$ versus $\tau_\alpha$. (b) $\chi_{4}(t)$ versus $\xi_4(t)$. The time intervals are $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. The straight lines are power law fits: $\xi_4(\tau_\alpha) \sim \tau_\alpha^{0.1 \pm 0.01}$ and $\chi_4(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{3.5 \pm 0.2}$. The dashed curve is a fit to $\xi_4(\tau_\alpha) \sim (\ln {\tau_\alpha})^{0.91 \pm 0.12}$. []{data-label="xilog"}](fig8.eps)
The correlation length, the intensity, and the lifetime of the heterogeneous dynamics
-------------------------------------------------------------------------------------
We next quantified the correlation length and the intensity of the heterogeneous dynamics in $\tau_\alpha$ and $\tau_{\text{ngp}}$. As mentioned for Eq. (\[OZ\]), at small wavenumbers of $q$, $S_{4}(q,t)$ can be approximated by the simple OZ form, $$S_{4}(q,t) = \frac{\chi_{4}(t)}{1 + q^2 \xi_{4}(t)^2}, \label{OZd}$$ where $\xi_{4}(t)$ is the correlation length and $\chi_{4}(t)$ is the intensity. The values $\xi_{4}(t)$ and $\chi_{4}(t)$ correspond to $\xi_{4,k}(t)$ and $\chi_{4,k}(t)$ in Eq. (\[OZ\]), respectively. To obtain accurate values of $\xi_{4}(t)$ and $\chi_{4}(t)$, $S_{4}(q,t)$ was carefully fitted to the OZ form in the range of $q \xi_{4}(t) < 1.5$ [@flenner_2010; @flenner_2011].
Figure \[xi\] shows $\xi_{4}(t)$ versus $\tau_\alpha$ in \[xi\](a) and $\chi_{4}(t)$ versus $\xi_{4}(t)$ in \[xi\](b) for $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. For the time interval $t=\tau_\alpha$, we examined the scaling relationships between $\tau_\alpha$, $\xi_{4}(\tau_\alpha)$, and $\chi_{4}(\tau_\alpha)$. As in Fig. \[xi\](a), we obtained a power law fit $\xi_{4}(\tau_\alpha) \sim \tau_\alpha^{1/z}$ with $1/z=0.1$. The scaling exponent $0.1$ is very small. We also found that a fit to $\xi_4(\tau_\alpha) \sim (\ln {\tau_\alpha})^{1/\theta}$ with $1/\theta = 0.77$ provides a better description of the data over a larger range of $\tau_\alpha$, so there is a slower-than-power law increase of $\xi_4(\tau_\alpha)$ with $\tau_\alpha$. Therefore, $\xi_{4}(\tau_\alpha)$ increases much more slowly compared to $\tau_\alpha$ as the temperature decreases. This result is consistent with the most recent results [@flenner_2010; @flenner_2011]. Furthermore, we obtained the scaling relationship $\chi_{4}(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{2-\eta}$ with $2-\eta=3.2$ in Fig. \[xi\](b). Our scaling exponent $3.2$ agrees well with Ref. [@flenner_2010].
On the other hand, when the time interval $t$ is $\tau_{\text{ngp}}$, $\xi_4(\tau_{\text{ngp}})$ reaches a value near $\tau_\alpha \simeq 250$ and then plateaus as $\tau_\alpha$ increases in Fig. \[xi\](a), while $\chi_4(\tau_{\text{ngp}})$ monotonically gets large in Fig. \[xi\](b). Here, we remark that even in the range where $\xi_4(\tau_{\text{ngp}})$ plateaus, $\chi_4(\tau_{\text{ngp}})$ continues to increase with $\tau_\alpha$. This behavior of $\xi_4(\tau_{\text{ngp}})$ and $\chi_4(\tau_{\text{ngp}})$ may be due to the choice of the order parameter $\delta \hat{{\mathcal{D}}}_{1}(\vec{r},t_0,t)$. $\delta \hat{{\mathcal{D}}}_{1}(\vec{r},t_0,t)$ represents the distribution of the square particle displacements, so the mobile particles with large displacements contribute to the structure factor $S_{4}(q,t)$ much more than the immobile particles. At $t=\tau_{\text{ngp}}$ and at low temperatures, our simulation data show that only a small number of particles tend to have very large displacements compared to the other mostly immobile particles, which can be also seen in Fig. \[hetero1\]. In this situation, the intensity $\chi_4(\tau_{\text{ngp}})$ can increase from the contribution of particles with very large displacements, whereas the correlation length $\xi_4(\tau_{\text{ngp}})$ cannot increase because the number of mobile particles involved in the correlated regions decreases.
So, we used an another order parameter $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$ defined as $$\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t) = \sum_{j=1}^{N_1} \ln a_{1j}^2(t_0,t) \delta( \vec{r} - \vec{R}_{1j}(t_0,t) ).$$ The order parameter $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$ represents the distribution of the logarithm of the square particle displacements. We quantified $\xi_4(t)$ and $\chi_4(t)$ in terms of the structure factor of $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$. Notice that in this case, the immobile particles with small displacements contribute to the structure factor to the same extent as the mobile particles because of the logarithm operation. Figure \[xilog\] shows $\xi_4(t)$ and $\chi_4(t)$ calculated by $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$. At the time interval $t=\tau_\alpha$, the scaling relationships between $\tau_\alpha$, $\xi_4(\tau_\alpha)$ and $\chi_4(\tau_\alpha)$ are almost same as those examined by $\delta \hat{{\mathcal{D}}}_{1}(\vec{r},t_0,t)$. On the other hand, at the time interval $t=\tau_{\text{ngp}}$, both $\xi_4(\tau_{\text{ngp}})$ and $\chi_4(\tau_{\text{ngp}})$ reach values near $\tau_\alpha \simeq 250$ and then plateau as $\tau_\alpha$ increases, i.e., both $\xi_4(\tau_{\text{ngp}})$ and $\chi_4(\tau_{\text{ngp}})$ show the plateau of the heterogeneity. This result indicates that the behavior that $\chi_4(\tau_{\text{ngp}})$ continues to increase in Fig. \[xi\](a) is due to the choice of the order parameter $\delta \hat{{\mathcal{D}}}_{1}(\vec{r},t_0,t)$. For the confirmation purpose, we also calculated the lifetime $\tau_{\text{hetero}}(t)$ by using $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$ and checked that the lifetimes of both $\delta \hat{{\mathcal{D}}}_{1}(\vec{r},t_0,t)$ and $\delta \hat{{\mathcal{D}}}_{\log 1}(\vec{r},t_0,t)$ have almost same values and behave in the same manner with decreasing temperature.
In our previous study, we already determined the scaling relationships between the lifetime $\tau_{\text{hetero}}(t)$ and $\tau_\alpha$, i.e., $\tau_{\text{hetero}}(\tau_\alpha) \sim \tau_\alpha^{1.08}$ and $\tau_{\text{hetero}}(\tau_{\text{ngp}}) \sim \tau_\alpha^{0.91}$. At the time interval $t=\tau_\alpha$, we can determine the scaling relationship between the correlation length and the lifetime: $$\tau_{\text{hetero}}(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{10.8},$$ or $$\tau_{\text{hetero}}(\tau_\alpha) \sim \exp(k \xi_4(\tau_\alpha)^{1.3}).$$ The scaling exponent $10.8$ of the power law scaling is very large, and there is an exponential growth of $\tau_\text{hetero}(\tau_\alpha)$ with $\xi_4(\tau_\alpha)$. Furthermore, at the time interval $t=\tau_\text{ngp}$, although the correlation length $\xi_4(\tau_{\text{ngp}})$ plateaus, the lifetime $\tau_{\text{hetero}}(\tau_{\text{ngp}})$ continues to increase dramatically with decreasing temperature. Thus, we can conclude that the lifetime $\tau_{\text{hetero}}$ gets large dramatically with decreasing temperature, whereas the correlation length $\xi_4$ and the intensity $\chi_4$ increase slowly compared to $\tau_{\text{hetero}}$ or plateau, i.e., the time scale of dynamical heterogeneity grows faster than the length scale and the intensity of dynamical heterogeneity.
RESULTS II: TIME-INTERVAL DEPENDENCE OF LIFETIME {#result2}
================================================
In this section, we present the results for the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$. To examine the lifetime in more detail, we evaluated the lifetime $\tau_{\text{hetero}}(t)$ for various time intervals $t$ and determine how the lifetime $\tau_{\text{hetero}}(t)$ depends on the time interval.
![(Color) The time-interval dependence of the distribution of the particle mobility $a_{1j}^2(t_0,t)$ for particle species 1. The time intervals are (a) $t=0.001\tau_\alpha$, (b) $t=0.1\tau_\alpha$, (c) $t=\tau_\alpha$ and (d) $t=100\tau_\alpha$. The temperature is $0.267$. The radii of the spheres are $a_{1j}^2(t_0,t)$, and the centers are at $\vec{R}_{1j}(t_0,t)$. See also the caption for Fig. \[hetero1\]. []{data-label="hetero2"}](fig9.eps)
![The wavenumber dependence of $S_4(q,t)$ at $T=0.306$. The time intervals are (a) $0.05\tau_\alpha$, $0.1\tau_\alpha$, $0.5\tau_\alpha$, and $\tau_\alpha$ from the lowest curve to the highest, and (b) $\tau_\alpha$, $5\tau_\alpha$, $10\tau_\alpha$, $50\tau_\alpha$, $100\tau_\alpha$, and $300\tau_\alpha$ from the highest curve to the lowest. $S_4(q,t)$ is maximized at $t=\tau_\alpha$. The dashed curve represents the static structure factor $S_{11}(q)$. []{data-label="sd4q"}](fig10.eps)
The structure of the heterogeneous dynamics {#strct}
-------------------------------------------
First, we examined the time-interval dependence of the structure of the heterogeneous dynamics. The spatial distribution of the particle mobility $a_{1j}^2(t_0,t)$ for particle species 1 is shown in Fig. \[hetero2\], in which the time interval increases from $t=0.001 \tau_\alpha$ to $100\tau_\alpha$. The temperature is $0.267$. We can recognize that the heterogeneity is much weaker at the short time interval $t = 0.001 \tau_\alpha$ in \[hetero2\](a). As the time interval $t$ increases, the heterogeneity increases and is maximized between $t = 0.1 \tau_\alpha$ in \[hetero2\](b) and $t=\tau_\alpha$ in \[hetero2\](c). As the time interval $t$ increases further, the heterogeneity decreases and is weakened at $t = 100 \tau_\alpha$ in \[hetero2\](d).
We also calculated the spatial correlation function $S_{4}(q,t)$ defined in Eq. (\[sd4eq\]). Figure \[sd4q\] shows the wavenumber dependence of $S_{4}(q,t)$ for various time intervals $t$. We can see that as the time interval $t$ gets large, $S_{4}(q,t)$ increases, is maximized, and then decreases in the region of $q$. This behavior agrees with the visualization of the heterogeneity structure shown in Fig. \[hetero2\]. It is also seen that at large time intervals, $S_{4}(q,t)$ maintains a constant value, independent of the wavenumber $q$. This result indicates that at large time intervals, particle mobilities are uniformly distributed throughout space, i.e., the structure of the particle dynamics is spatially homogeneous. Note that the static structure factor $S_{11}(q)$ also becomes constant at small $q$ (long-distance scale).
The motion of the heterogeneous dynamics {#mhd}
----------------------------------------
Next, the motion of heterogeneous dynamics was investigated in detail. In our previous study [@mizuno_2010], we investigated the motion of the heterogeneous dynamics at the time intervals $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. Our results suggested that the heterogeneous dynamics might migrate in space with a diffusion-like mechanism. In the present study, we examined the motion of the heterogeneous dynamics at various time intervals $t$. As in our previous study [@mizuno_2010], we calculated the time correlation function of the particle dynamics expressed in Eq. (\[F4\]). We used $\delta {\mathcal{D}}_1(\vec{q},t_0,t)$ as $\delta Q_{{k}}(\vec{q},t_0,t)$ in Eq. (\[F4\]), and the correlation function $S_{{\mathcal{D}}}(q,t_s,t)$, $$S_{{\mathcal{D}}}(q,t_s,t) = \langle \delta {\mathcal{D}}_1(\vec{q},t_s+t,t) \delta {\mathcal{D}}_1(-\vec{q},0,t) \rangle,$$ corresponds to $F_{4,{k}}({q},t_s,t)$ and represents the correlation of the particle dynamics between two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$. The time configuration of $S_{{\mathcal{D}}}(q,t_s,t)$ is schematically illustrated in Fig. \[ts\](b). As mentioned for Eq. (\[F42\]), when the time separation $t_s$ increases, $S_{{\mathcal{D}}}(q,t_s,t)$ with fixed $t$ decays in the stretched exponential form, $$\begin{aligned}
\frac{S_{{\mathcal{D}}}(q,t_s,t)}{S_{{\mathcal{D}}}(q,0,t)}
\sim \exp\left( - \left( \frac{t_s}{\tau_{h}(q,t)} \right)^c \right),
\end{aligned}$$ where $\tau_{h}(q,t)$ is the wavenumber-dependent relaxation time of $S_{{\mathcal{D}}}(q,t_s,t)$ and represents the time scale at which the heterogeneous dynamics move in the space. The expression $\tau_{h}(q,t)$ corresponds to $\tau_{4,{k}}({q},t)$ in Eq. (\[F42\]). We were able to examine the motion of the heterogeneous dynamics using the wavenumber dependence of $\tau_{h}(q,t)$.
![The wavenumber dependence of $\tau_h(q,t)$ at $T=0.306$. The time intervals are $0.05\tau_\alpha$, $0.1\tau_\alpha$, $0.5\tau_\alpha$, $\tau_\alpha$, $5\tau_\alpha$, $10\tau_\alpha$, $30\tau_\alpha$, $50\tau_\alpha$, $70\tau_\alpha$, $100\tau_\alpha$, and $300\tau_\alpha$ from the lowest curve to the highest. The dashed curve is the relaxation time $\tau_{c1}(q)$ of the two-point density correlation function. []{data-label="tauhtsq306"}](fig11.eps)
In Fig. \[tauhtsq306\], we show the wavenumber dependence of $\tau_h(q,t)$ for various time intervals $t$. As the time interval $t$ increases, at small wavenumbers $q$, $\tau_h(q,t)$ increases monotonically and approaches the relaxation time $\tau_{c1}(q)$ of the two-point density correlation function shown in Fig. \[tauk\]. This result indicates that the heterogeneous dynamics at large time intervals behave like the particle density, the motion of which is diffusive at a long distance scale. Notice that $\tau_h(q,t)$ at large $t$ and small $q$ is proportional to $q^{-2}$, which indicates this diffusion mechanism. We also checked that $S_{{\mathcal{D}}}(q,t_s,t)$ decays in the exponential form at large $t$ and small $q$, as does the two-point density correlation function.
![The wavenumber dependence of $\tau_{hs}(q,t)$ at $T=0.306$. The time intervals are $0.05\tau_\alpha$, $0.1\tau_\alpha$, $0.5\tau_\alpha$, $\tau_\alpha$, $5\tau_\alpha$, $10\tau_\alpha$, $30\tau_\alpha$, $50\tau_\alpha$, $70\tau_\alpha$, $100\tau_\alpha$, and $300\tau_\alpha$ from the lowest curve to the highest. The dashed curve is the relaxation time $\tau_{s1}(q)$ of the self-part of the two-point density correlation function. []{data-label="tauhtssq306"}](fig12.eps)
Furthermore, we calculated the self-part of $S_{{\mathcal{D}}}(q,t_s,t)$, defined by: $$S_{{\mathcal{D}}s}(q,t_s,t) = \left< \frac{1}{N_1} \sum_{j=1}^{N_1} \delta {\mathcal{D}}_{1j}(\vec{q},t_s+t,t)\delta {\mathcal{D}}_{1j}(\vec{q},0,t) \right>,$$ where $$\delta {\mathcal{D}}_{1j}(\vec{q},t_0,t)= \delta a_{1j}^2(t_0,t) \exp[-i\vec{q} \cdot \vec{R}_{1j}(t_0,t) ].$$ The correlation function $S_{{\mathcal{D}}s}(q,t_s,t)$ represents the correlation of the individual particle dynamics between two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$. Like $S_{{\mathcal{D}}}(q,t_s,t)$, as the time separation $t_s$ increases, $S_{{\mathcal{D}}s}(q,t_s,t)$ decays in the stretched exponential form with the relaxation time $\tau_{hs}(q,t)$. The value of $\tau_{hs}(q,t)$ is the time scale at which the individual heterogeneous dynamics move in space. Figure \[tauhtssq306\] shows the wavenumber dependence of $\tau_{hs}(q,t)$ for various time intervals. As the time interval $t$ increases, $\tau_{hs}(q,t)$ increases monotonically at small $q$ and approaches the relaxation time $\tau_{s1}(q)$ of the self-part of the two-point density correlation function shown in Fig. \[tauk\]. Like $\tau_{h}(q,t)$, $\tau_{hs}(q,t)$ is also proportional to $q^{-2}$ at large $t$ and small $q$. Thus, both the collective-particle behavior and the single-particle behavior of the heterogeneous dynamics at large time intervals are diffusive, like the motion of particle density at long-distance scales.
That $\tau_h(q,t)$ and $\tau_{hs}(q,t)$ approach the relaxation times $\tau_{c1}(q)$ and $\tau_{s1}(q)$ of the two-point density correlation functions can possibly be interpreted as follows. The correlation function $S_{{\mathcal{D}}}(q,t_s,t)$ can be written as $$\begin{aligned}
& S_{{\mathcal{D}}}(q,t_s,t) = \bigg< \sum_{j=1}^{N_1} \sum_{k=1}^{N_1} \delta a_{1j}^2(t_s+t,t) \delta a_{1k}^2(0,t) \times \\
& \qquad \quad \quad \exp[-i\vec{q} \cdot (\vec{R}_{1j}(t_s+t,t)-\vec{R}_{1k}(0,t)) ] \bigg>.
\end{aligned}$$ From this expression, there can be two types of relaxation of $S_{{\mathcal{D}}}(q,t_s,t)$ with increasing time of separation $t_s$. One is the relaxation due to fluctuations in the particle mobility, i.e., due to the term “$\delta a_{1j}^2(t_s+t,t) \delta a_{1k}^2(0,t)$”. The other is the relaxation due to particle motion, i.e., due to the term “$\exp[-i\vec{q} \cdot (\vec{R}_{1j}(t_s+t,t)-\vec{R}_{1k}(0,t)) ]$”, which is the same as the relaxation of the density correlation functions. So, the total relaxation time $\tau_{h}(q,t)$ of $S_{{\mathcal{D}}}(q,t_s,t)$ is determined by the two times $\tau_{\delta a}(t)$ and $\tau_{c1}(q)$, where $\tau_{\delta a}(t)$ is the time scale at which particle mobility fluctuates, and $\tau_{c1}(q)$ is the time scale of particle motion and is the relaxation time of the density correlation function. Here, for simplicity, we assume that $S_{{\mathcal{D}}}(q,t_s,t)$ relaxes with these two times $\tau_{\delta a}(t)$ and $\tau_{c1}(q)$ in the form $$\begin{aligned}
& \frac{S_{{\mathcal{D}}}(q,t_s,t)}{S_{{\mathcal{D}}}(q,0,t)} \sim \exp\left( - \left( \frac{t_s}{\tau_{\delta a}(t)} \right)^c \right) \times \exp\left( - \left( \frac{t_s}{\tau_{c1}(q)} \right)^c \right).
\label{2times}
\end{aligned}$$ The relaxation time $\tau_h(q,t)$ is then mainly determined by the smaller time scale of $\tau_{\delta a}(t)$ and $\tau_{c1}(q)$. When the time interval $t$ is small, particle mobilities fluctuate faster than the particles move, which means that $\tau_{\delta a}(t)$ is smaller than $\tau_{c1}(q)$. In this case, $\tau_h(q,t)$ is determined by the time $\tau_{\delta a}(t)$. As the time interval $t$ increases, then the fluctuations in particle mobilities becomes slower, and the lifetime $\tau_h(q,t)$ increases accordingly. As the time interval $t$ increases further, particle mobilities fluctuate more slowly than the particles move, which means that $\tau_{\delta a}(t)$ becomes larger than $\tau_{c1}(q)$. In this case, $\tau_h(q,t)$ is determined by the time $\tau_{c1}(q)$ instead of $\tau_{\delta a}(t)$. Thus, at large time intervals, because particle mobilities fluctuate very slowly, the relaxation time $\tau_h(q,t)$ can be determined by the time scale of particle motions, which is the relaxation time of the density correlation function.
![The time-interval dependence of $\tau_{\text{hetero}}(t)$. Temperatures are 0.473 in (a), 0.306 in (b), 0.267 in (c), and 0.253 in (d). The time interval $t$ and the lifetime $\tau_{\text{hetero}}(t)$ are normalized by $\tau_\alpha$. The dotted line indicates the value of $\tau_{c1}(q)$ at $q=0.38$, where $\tau_{c1}(q)$ is the relaxation time of the two-point density correlation function. []{data-label="tauhts"}](fig13.eps)
The lifetime of the heterogeneous dynamics
------------------------------------------
Finally, we determined the lifetime $\tau_{\text{hetero}}(t)$ of the heterogeneous dynamics as $\tau_{h}(q,t)$ at $q=0.38$. We show the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$ for various temperatures in Fig. \[tauhts\]. As the time interval $t$ increases, $\tau_{\text{hetero}}(t)$ increases monotonically, and at large time intervals, $\tau_{\text{hetero}}(t)$ approaches and is limited to the relaxation time $\tau_{c1}(q=0.38)$ of the two-point density correlation function. At large time intervals for which $\tau_{\text{hetero}}(t)$ plateaus, the heterogeneous dynamics migrate in space with a diffusion mechanism like that of particle density, as we showed in Figs. (\[tauhtsq306\]) and (\[tauhtssq306\]).
SUMMARY {#conclusion}
=======
In this study, we have investigated three quantities that characterize dynamical heterogeneity: the correlation length $\xi_4(t)$, the intensity $\chi_4(t)$, and the lifetime $\tau_{\text{hetero}}(t)$. The intensity $\chi_4(t)$ measures the average variance of the slow and fast regions, and the correlation length $\xi_4(t)$ characterizes the spatial extent of the slow and fast regions. The lifetime $\tau_{\text{hetero}}(t)$ represents the time scale at which the slow and fast regions migrate in space. We evaluated all three quantities using a single order parameter representing the particle dynamics and its correlation functions. To define the particle dynamics, we used two time intervals, $t=\tau_\alpha$ and $\tau_{\text{ngp}}$. We found that at low temperatures, the lifetime $\tau_{\text{hetero}}(t)$ increases dramatically with decreasing temperature. In contrast, the correlation length $\xi_4(t)$ and the intensity $\chi_4(t)$ increase slowly compared to the lifetime or plateau. At the time interval $t=\tau_\alpha$, we obtained the scaling relationships between $\xi_4(t)$, $\chi_4(t)$, and $\tau_{\text{hetero}}(\tau_\alpha)$: $\chi_4(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{3.2}$ and $\tau_{\text{hetero}}(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{10.8}$ or $\tau_{\text{hetero}}(\tau_\alpha) \sim \exp(k \xi_4(\tau_\alpha)^{1.3})$. The scaling exponent $10.8$ of $\tau_{\text{hetero}}(\tau_\alpha) \sim \xi_4(\tau_\alpha)^{10.8}$ is very large, and there is an exponential growth of $\tau_\text{hetero}(\tau_\alpha)$ with $\xi_4(\tau_\alpha)$. Furthermore, at the time interval $t=\tau_\text{ngp}$, although the correlation length $\xi_4(\tau_{\text{ngp}})$ plateaus as the temperature decreases, the lifetime $\tau_{\text{hetero}}(\tau_{\text{ngp}})$ continues to increase dramatically. Thus, we can conclude that the lifetime $\tau_{\text{hetero}}$ gets large dramatically with decreasing temperature, whereas the correlation length $\xi_4$ and the intensity $\chi_4$ increase slowly compared to $\tau_{\text{hetero}}$ or plateau, i.e., the time scale of dynamical heterogeneity grows faster than the length scale and the intensity of dynamical heterogeneity.
Furthermore, we investigated the time-interval dependence of the lifetime $\tau_{\text{hetero}}(t)$. As the time interval $t$ increases, $\tau_{\text{hetero}}(t)$ increases monotonically. At large time intervals, the lifetime $\tau_{\text{hetero}}(t)$ approaches and is limited to the relaxation time of the two-point density correlation function. At those large time intervals, the wavenumber-dependent lifetimes $\tau_h(q,t)$ at small wavenumbers $q$ (long-distance scales) almost coincide with the relaxation time $\tau_{c1}(q)$ of the two-point density correlation function and are proportional to $q^{-2}$. Therefore, the heterogeneous dynamics migrate in space with a diffusion mechanism like that of particle density. Note that at large time intervals, particle mobilities $a_{1j}^2(t_0,t)$ are uniformly distributed in space, and the heterogeneity structure is much weaker.
We wish to acknowledge Dr. K. Kim for useful comments and discussions.
|
---
abstract: 'Bayesian learning is built on an assumption that the model space contains a true reflection of the data generating mechanism. This assumption is problematic, particularly in complex data environments. Here we present a Bayesian nonparametric approach to learning that makes use of statistical models, but does not assume that the model is true. Our approach has provably better properties than using a parametric model and admits a trivially parallelizable Monte Carlo sampling scheme that affords massive scalability on modern computer architectures. The model-based aspect of learning is particularly attractive for regularizing nonparametric inference when the sample size is small, and also for correcting approximate approaches such as variational Bayes (VB). We demonstrate the approach on a number of examples including VB classifiers and Bayesian random forests.'
author:
- 'S. P. Lyddon, S. G. Walker, C. C. Holmes'
bibliography:
- 'library.bib'
title: Nonparametric learning from Bayesian models with randomized objective functions
---
Introduction {#sec:intro}
============
Bayesian updating provides a principled and coherent approach to inference for probabilistic models [@robert2007bayesian], but is predicated on the model class being true. That is, given a generative model $F_\theta(x)$ parametrized by a finite-dimensional parameter $\theta$, then for some parameter value $\theta_0 \in \Theta$ it is that $x \sim F_{\theta_0}(x)$. In reality all models are false. If the data is simple and small, and the models are sufficiently rich, then the consequences of model misspecification may not be severe. However, data is increasingly being captured at scale, both in terms of the number of observations as well as the diversity of data modalities. This poses a risk in conditioning on an assumption that the model is true.
In this paper we discuss a scalable approach to Bayesian nonparametric learning (NPL) from models without the assumption that $x \sim F_{\theta_0}(x)$. To do this we use a nonparametric prior for $F_0$ that is centred on a model but does not assume the model to be true. A concentration parameter, $c$, in the nonparametric prior quantifies trust in the baseline model and this is subsequently reflected in the relative posterior influence given to the model-based inference for parameter $\theta$, compared to a purely empirical approach. In particular, $c \rightarrow \infty$ recovers the standard model-based Bayesian update while $c \rightarrow 0 $ leads to a Bayesian bootstrap estimator for the object of interest.
Our method provides both a means of regularizing a nonparametric update and correcting misspecified or approximate posterior inferences. This may be useful in a number of situations, including:
- Nonparametric regularization: where we wish to use a Bayesian NP approach but we would like to include a regularization term, centered on a conventional model, that induces stability and parametric structure on the problem.
- Model misspecification: where we have used a parametric Bayesian model and we are concerned that the model may be misspecified.
- Approximate posteriors: where for expediency we have used an approximate posterior, such as in Variational Bayes (VB), and we wish to account for the approximation.
- Direct updating of the prior: where the parameters have meaning but little trust is placed in the validity of the generative likelihood, such as for classification trees that define piecewise constant predictions regardless of the smoothness of the actual decision boundaries.
- Information sharing under privacy considerations: where we might be willing to share pseudo-data generated from a posterior model, but not the actual training data or model.
- Direct updating from utility-functions: where the sole purpose of the modelling task is to perform some action or take a decision under a well-specified utility function.
Our work builds upon previous ideas including [@Newton1994] who introduced the weighted likelihood bootstrap (WLB) as a way of generating approximate samples from the posterior of a well-specified Bayesian model. [@Lyddon2018] highlighted that the WLB in fact provides an exact representation of uncertainty for the model parameters that minimize the Kullback-Leibler (KL) divergence, $d_{\text{KL}}(F_0,F_\theta)$, between the unknown data-generating distribution and the model likelihood $f_\theta(x)$, and hence is well motivated regardless of model validity. These approaches however do not allow for the inclusion of prior knowledge and do not provide a Bayesian update as we do here.
A major underlying theme behind our paper, and indeed an open field for future research, is the idea of obtaining targeted posterior samples via the maximization of a suitably randomized objective function. The WLB randomizes the log-likelihood function, effectively providing samples which are randomized maximum likelihood estimates, whereas we randomize a more general objective function under a NP posterior. The randomization takes into account knowledge associated with the parametric model.
Foundations of Nonparametric Learning {#sec:foundations}
=====================================
We begin with the simplest scenario, namely \[S1\], concerning a possibly misspecified model before moving on to more complicated situations. All of what follows can be considered from a viewpoint of NP regularization and hence we don’t explicitly deal with \[S0\] from herein.
Bayesian updating of misspecified models
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Suppose we have a parametric statistical model, ${\mathcal{F}}_\Theta = \{ f_\theta(\cdot); \ \theta \in \Theta ) \}$, where for each $\theta \in \Theta \subseteq {\mathbb{R}}^p$, $f_\theta : {\mathcal{X}}\rightarrow {\mathbb{R}}$ is a probability density. The conventional approach to Bayesian learning involves updating a prior distribution to a posterior through Bayes’ theorem. This approach is well studied and well understood [@Bernardo2006], but formally assumes that the model space captures the truth. We will derive a posterior update under weaker assumptions.
Suppose that ${\mathcal{F}}_\Theta$ has been selected for the purpose of a prediction, or a decision, or some other modelling task. Consider the thought experiment where the modeller somehow gains access to Nature’s true sampling distribution for the data, $F_0(x)$, which does not necessarily belong to ${\mathcal{F}}_\Theta$. How should they then update their model?
With access to $F_0$ the modeller can simply request an infinite training set, ${x}_{1:\infty} \stackrel{iid}{\sim} F_0$, and then update to the posterior ${\pi}(\theta | {x}_{1:\infty})$. Under an infinite sample size all uncertainty is removed and for regular models the posterior concentrates at a point mass at $\theta_0$, the parameter value maximizing the expected log-likelihood, assuming that the prior has support there.
$$\begin{aligned}
\label{eq:inf_sample}
\theta_0 = \operatorname*{arg\,max}_{\theta\in\Theta} \lim_{n\to\infty} n^{-1} \sum_{i=1}^{n} \log f_\theta(x_i) = \operatorname*{arg\,max}_{\theta\in\Theta} \int_{{\mathcal{X}}} \log f_\theta(x)\,d F_0.\end{aligned}$$
It is straightforward to see that ${\theta}_0$ minimizes the KL divergence from the true data-generating mechanism to a density in ${\mathcal{F}}_\Theta$ $$\label{eq:theta_0}
\theta_0 = \operatorname*{arg\,max}_{\theta \in \Theta} \int_{\mathcal{X}}\log f_\theta(x) dF_0(x) = \operatorname*{arg\,min}_{\theta\in\Theta} \int_{\mathcal{X}}\log \frac{f_0(x)}{f_\theta(x)} dF_0(x).$$ This is true regardless of whether $F_0$ is in the model space of ${\mathcal{F}}_\Theta$ and is well-motivated as the target of statistical model fitting [@Akaike1981a; @burnham2003model; @Walker2013; @Bissiri2016].
Uncertainty in the value of $\theta_0$ flows directly from uncertainty in $F_0$. In reality, of course $F_0$ is unknown, but being “Bayesian” we can place a prior on it, $\pi(F)$, for $F \in {\mathcal{F}}$, that should reflect our honest uncertainty about $F_0$. Typically the prior should have broad support unless we have special knowledge to hand, which is a problem with a parametric modelling approach that only supports a family of distribution functions indexed by a finite-dimensional parameter. The Bayesian NP literature provides a range of priors for this sole purpose, see for example [@hjort2010bayesian]. Once a prior for $F$ is chosen, the correct way to propagate uncertainty about $\theta$ comes naturally from the posterior distribution for the law ${\cal L}[\theta(F) | x_{1:n}]$, via ${\cal L}[F | x_{1:n}]$, where $\theta(F) = \operatorname*{arg\,max}_{\theta\in\Theta} \int \log f_\theta(x) dF(x)$. The posterior on the parameter of interest is then captured in the marginal by treating $F$ as a latent probability measure, $$\label{eq:f_marg}
\tilde{\pi}(\theta | x_{1:n} ) = \int_{\mathcal{F}}\pi(\theta, dF | x_{1:n} ) = \int_{\mathcal{F}}\pi(\theta | F) \pi(dF | x_{1:n} )
$$ where $\pi(\theta | F)$ assigns probability $1$ to $\theta=\theta(F)$. We use $\tilde{\pi}$ to denote the NP update to distinguish it from the conventional Bayesian posterior, noting that in general the nonparametric posterior will be different to the standard Bayesian update, $\tilde{\pi}(\theta | x_{1:n}) \not\equiv \pi(\theta | x_{1:n})$, as they are conditioning on different prior states of knowledge. In particular, as stated above, $\pi(\theta | x_{1:n})$ assumes that $F_0 \in {\mathcal{F}}_\Theta$.
An NP prior using a MDP
-----------------------
For our purposes, the mixture of Dirichlet processes (MDP) [@Antoniak1974] is a convenient vehicle for specifying prior beliefs $\pi(F)$ centered on parametric models[^1]. The MDP prior can be written as $$\label{eq:MDP}
[F \mid \theta] \sim \text{DP}(c, f_\theta(\cdot) ) ; \qquad \theta \sim \pi(\theta).$$ This is a mixture of standard Dirichlet processes with mixing distribution or hyper-prior $\pi(\theta)$, and concentration parameter $c$. We write this as $F \sim \text{MDP}(\pi(\theta), c, f_\theta(\cdot) )$.
The MDP provides a practical, simple posterior update. From the conjugacy property of the DP applied to (\[eq:MDP\]), we have the conditional posterior update given data $x_{1:n}$, as $$\label{eq:MDP_posterior_condl}
[F \mid \theta, x_{1:n}] \sim \text{DP}\left( c+n, \ \frac{c}{c+n}f_\theta(\cdot) + \frac{1}{c+n} \sum_{i=1}^n \delta_{x_i}(\cdot) \right)$$ where $\delta_x$ denotes the Dirac measure at $x$. We see the representation of $c$ as an effective sample size governing the trust in $f_{\theta}(x)$. Thus the marginal posterior distribution for ${\cal L}[F | x_{1:n}]$ can be written as $$\label{eq:mdp-marg}
\pi(dF \mid x_{1:n} ) = \int_\Theta \pi(dF \mid \theta, x_{1:n} ) \, \pi(\theta \mid x_{1:n} ) \, d\theta,$$ i.e. $$\label{eq:MDP_posterior_full}
[F | x_{1:n}] \sim \text{MDP}\left( \pi(\theta | x_{1:n}), \ c+n, \ \frac{c}{c+n} f_\theta(\cdot) +\frac{1}{c+n}\sum\limits_{i=1}^n \delta_{x_i}(\cdot) \right).$$ The mixing distribution $\pi(\theta | x_{1:n})$ coincides with the parametric Bayesian posterior [@Antoniak1974], $\pi(\theta | x_{1:n}) \propto\prod_{i=1}^n~f_{\theta}(x_i)~\pi(\theta)$, assuming there are no ties in the data, although as noted above it does not follow that the NP marginal $\tilde{\pi}(\theta | x_{1:n})$ is equivalent to the parametric Bayesian posterior $\pi(\theta | x_{1:n}) $.
We can see from the form of the conditional MDP (\[eq:MDP\_posterior\_condl\]) that the sampling distribution of the centering model, $f_\theta(x)$, regularizes the influence of the data $\sum_{i=1}^n \delta_{x_i}(\cdot)$. The resulting NP posterior (\[eq:mdp-marg\]) combines the information from the posterior distribution of the centering model $\pi(\theta | x_{1:n})$ with the information in the empirical distribution of the data. This leads to a simple and highly parallelizable Monte Carlo sampling scheme as shown below.
Monte Carlo conditional maximization
------------------------------------
The marginal (\[eq:f\_marg\]) facilitates a Monte Carlo estimator for functionals of interest, $G = \int_\Theta g(\theta) \tilde{\pi}(\theta | x_{1:n} ) d \theta$, by sampling $\pi( \theta, dF | x_{1:n})$ jointly from the posterior, $$\begin{aligned}
\label{eq:FF_marg}
\int_\Theta g(\theta) \tilde{\pi}(\theta | x_{1:n} ) d \theta & \approx & \frac{1}{B} \sum_{i=1}^{B} g(\theta^{(i)}) \nonumber \\
\theta^{(i)} ~ = ~ \theta(F^{(i)}) & = & \operatorname*{arg\,max}_{\theta \in \Theta} \int_{\mathcal{X}}\log f_{\theta}(x) dF^{(i)}(x) \label{eq:FF_marg1} \\
F^{(i)} & \sim & \pi(dF | x_{1:n}). \label{eq:FF_marg2}\end{aligned}$$ This involves an independent Monte Carlo draw (\[eq:FF\_marg2\]) from the MDP marginal followed by a conditional maximization of an objective function (\[eq:FF\_marg1\]) to obtain $\theta(F)$. This Monte Carlo conditional maximization (MCCM) sampler is highly amenable to fast implementation on distributed computer architectures. The resulting estimator has $1/B$ order of variance independent of the dimension of $\theta$ and, unlike MCMC, there are no issues of convergence nor need for a user-defined burn-in phase.
However, we do need to sample from the standard Bayesian posterior directly. This makes the approach particularly attractive to fast, tractable, approximations for $\pi(\theta | x_{1:n})$, such as a variational Bayes (VB). The NP update corrects for the approximation in a computationally efficient manner, leading to an exact posterior with optimal properties as shown below.
A more general construction {#sec:general}
---------------------------
So far we have assumed, hypothetically, that:
(i) the modeller is interested in learning about $\theta_0=\operatorname*{arg\,max}_\theta\int\log f_\theta(x)dF_0(x)$, rather than $\alpha_0 = \operatorname*{arg\,max}_\alpha \int u(x, \alpha) dF_0(x)$ more generally, for a utility function $u(x, \alpha)$.
(ii) the mixing distribution $\pi(\theta | x_{1:n})$ of the MDP posterior is constructed from the same centering model that defines the target parameter, $\theta_0 = \operatorname*{arg\,max}_\theta \int \log f_\theta(x) dF_0(x)$.
Both of these assumptions can be relaxed. For the latter case, it is valid to center the MDP on a tractable baseline posterior $\pi(\gamma | x_{1:n})$ and still learn about $\theta_0$ in (\[eq:theta\_0\]) through the marginal $\tilde{\pi}(\theta | x_{1:n} )$ as in (\[eq:f\_marg\]) obtained via $\theta(F)$ and $$\sim \text{MDP}\left( \ \pi(\gamma | x_{1:n}), \ c+n, \ \frac{c}{c+n} f_\gamma(\cdot) + \frac{1}{c+n} \sum_i\delta_{x_i} \right).$$ For the former, we can use the mapping $\alpha(F) = \operatorname*{arg\,max}_\alpha \int u(x, \alpha) dF(x)$ to derive the NPL posterior $\tilde{\pi}(\alpha | x_{1:n} ) = \int \pi(\alpha | F) \pi(dF | x_{1:n} )$ on actions or parameters maximizing some expected utility under a model-centered MDP posterior.
This highlights a major theme of the paper in the idea of obtaining posterior samples, $\pi(\alpha | x_{1:n})$, via maximization of a suitably randomized objective function. In generality the target is $\alpha_0 = \operatorname*{arg\,max}_{\alpha} \int u(x, \alpha) dF_0(x)$, obtained by maximizing an objective function, and the randomization arises from the uncertainty in $F_0$ through $\pi(F | x_{1:n})$ that takes into account the information, and any misspecification, associated with the parametric centering model.
A Posterior Bootstrap Algorithm {#Subsec:posterior bootstrap}
-------------------------------
We will use the general construction of Section \[sec:general\] to describe a sampling algorithm. We assume we have access to samples from the posterior mixing distribution, $\pi(\gamma | x_{1:n})$, in the MDP. In the case of model misspecification, \[S1\], if the data contains no ties, this is simply the parametric Bayesian posterior under $\{f_{\gamma}(x), \pi(\gamma)\}$, for which there is a large literature of computational methods available for sampling - see for example [@robert2005]. If there are ties then we refer the reader to [@Antoniak1974] or note that we can simply break ties by adding a new pseudo-variable, such as $x^* \sim N(0, \epsilon^2)$ for small $\epsilon$.
The sampling algorithm, found in Algorithm \[algo:mdp-training\], can be thought of as a mixture of Bayesian bootstraps. After a sample is drawn from the mixing posterior, $\pi(\gamma | x_{1:n})$, a posterior pseudo-sample is generated, $x_{1:T} \stackrel{iid}{\sim} f_{\gamma^{(i)}}(x)$, and added to the dataset, which is then randomly weighted. The parameter under this implicit distribution function is then computed as the solution of an optimization problem.
Note for the special case of correcting model misspecification \[S1\], we have $\gamma \equiv \theta$, $f_\gamma(\cdot) \equiv f_\theta(\cdot)$, $\pi(\gamma | x_{1:n}) \equiv \pi(\theta | x_{1:n})$, $\alpha \equiv \theta$, $u(x, \alpha) \equiv \log f_\theta(x)$, so that the posterior sample is given by $$\tilde{\theta}^{(i)} = \operatorname*{arg\,max}_\theta \left\{ \sum\limits_{j=1}^n w_j^{(i)} \log f_\theta(x_j) + \sum\limits_{j=1}^T w_{n+j}^{(i)} \log f_\theta(x_{n+j}^{(i)}) \right\}.$$ where $w^{(i)} \sim \text{Dirichlet}(\cdot)$ following Algorithm \[algo:mdp-training\] and $x_{(n+1):(n+T)}^{(i)}$ are $T$ synthetic data observations drawn from the parametric sampling distribution under $\theta^{(i)}$ which itself is drawn from $\pi(\theta | x_{1:n})$. We leave the concentration parameter $c$ to be set subjectively by the practitioner, representing faith in the parametric model. Some further guidance to the setting of $c$ can be found in Appendix \[app:setting\_c\].
Adaptive Nonparametric Learning: aNPL
-------------------------------------
Instead of the Dirichlet distribution approximation to the Dirichlet process, a stick-breaking procedure [@Sethuraman1991] with a termination threshold set as a proportion, $\epsilon \in (0,1)$, of the expected model weight, $\frac{c}{c+n}$, has a number of desirable properties. This procedure entails following the usual DP stick-breaking construction for the model component of the MDP posterior, by repeatedly drawing $\text{Beta}(1,c)$-distributed stick breaks, but terminates when the unaccounted for probability measure $\prod_j (1 - v_j)$, multiplied by the average mass assigned to the model, $\frac{c}{c+n}$, drops below some threshold $\epsilon$ set by the user. This adaptive nonparametric learning (aNPL) algorithm is written out in full for the model misspecification setting \[S1\] in Appendix \[app:posterior\_bootstrap\_misspec\].
One advantage of this approach is that a number of theoretical results then hold, as for large enough $n$, under this adaptive scheme the parametric model is in effect ‘switched off’, and in effect the MDP with $c=0$ is used to generate posterior samples when $n$ passes some user-controlled value. This is an interesting notion in itself. For small samples, we prefer the regularization that our model provides, though as $n$ grows the average probability mass assigned to the model decays like $(c+n)^{-1}$, as seen in (\[eq:MDP\_posterior\_condl\]). In the adaptive version, we agree a hard threshold at which point we discard the model entirely and allow the data to speak for itself. We set this point at a level such that we are a priori comfortable that there is enough information in the sample alone with which to quantify uncertainty in our parameter of interest. For example, $\epsilon = 10^{-4}$ and $c=1$ only utilises the centring model for $n < 10,000$. Further, we could use this idea to set $c$, as the quantity is determined if a tolerance level, $\epsilon$, and a threshold $n$ over which the parametric model would be discarded, are provided by the practitioner.
Properties {#sec:properties}
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NPL has a number of desirable properties.
#### Honesty about correctness of model.
Uncertainty in the data-generating mechanism is quantified via a NP update that takes into account the model likelihood, prior, and concentration parameter $c$. Uncertainty about model parameters flows from uncertainty surrounding the data-generating mechanism, $F_0(x)$.
#### Incorporation of prior information.
The prior for $\theta$ is naturally incorporated as a mixing distribution for the MDP. This is in contrast to a number of Bayesian methods with similar computational properties but that do not admit a prior [@Newton1994; @Chamberlain2003].
#### Parallelized bootstrap computation.
As shown in Section \[Subsec:posterior bootstrap\], NPL is trivially parallelizable through a Bayesian bootstrap and can be coupled with misspecified or approximate models to deliver highly scalable and exact inference.
#### Consistency.
Under mild regularity, all posterior mass concentrates in any neighbourhood of $\theta_0$ as defined in (\[eq:theta\_0\]), as the number of observations tends to infinity. This follows from an analogous property of the DP (see, for example [@hjort2010bayesian]).
#### Standard Bayesian inference is recovered as $\boldsymbol{c \to \infty}$.
This follows from the property of the DP that it converges to the prior degenerate at the base probability distribution in the limit of $c\rightarrow\infty$.
#### Non-informative learning with $\boldsymbol{c =0}$.
If no prior or model is available, setting $c=0$ recovers the WLB. This has an exact interpretation as an objective NP update [@Lyddon2018], where the asymptotic properties of the misspecified WLB were studied. A result of relevance to us is presented in the following theorem.
> \[theorem1\] Let $\tilde{\theta}_n$ be a WLB sample of a parameter defined in (\[eq:theta\_0\]) or via aNPL, given $n$ observations $x_{1:n}$, and let $P_{c=0 \,}$ be its probability measure. Under regularity conditions, for any Borel set $A \subset \mathbb{R}^p$, as $n \rightarrow \infty$ we have $$P_{c=0} \left\{ n^{1/2} \left( \tilde{\theta}_n - \hat{\theta}_n \right) \in A \mid x_{1:n} \right\} \ \rightarrow \ P( z \in A),$$ a.s. $x_{1:\infty}$, where $z \sim N_d\{ \, 0, \, J( \theta_0)^{-1} I(\theta_0) J(\theta_0)^{-1} \, \}$, with $$\begin{aligned}
> I(\theta) &= \int \nabla \log f_\theta(x) \nabla \log f_\theta( x)^T \,dF_0(x),\\
> J(\theta) &= - \int \nabla^2 \log f_\theta( x) \,dF_0(x),\end{aligned}$$ where $\nabla$ is the gradient operator with respect to $\theta$, and $\hat{\theta}_n$ is the MLE.
>
> See [@Lyddon2018], Theorem 1.
In fact the theorem presented in [@Lyddon2018] is for a general utility function, consistent with point (i) of Section \[sec:general\]. The covariance matrix $J( \theta_0)^{-1} I(\theta_0) J(\theta_0)^{-1}$ is a well-known quantity in the robust statistics literature, sometimes called the sandwich covariance matrix [@Cox1961; @Huber1967; @White1982a]. It’s the asymptotic covariance matrix for a maximum likelihood estimator under potential misspecification; if well specified this coincides with the Bayesian posterior, but in general does not. [@Muller2013] showed that replacing a Bayesian posterior covariance matrix with the sandwich covariance matrix can provide a reduction in frequentist risk, asymptotically, for some decision problems. We will see next that for large samples the misspecified Bayesian posterior distribution is predictively suboptimal as well.
#### A superior asymptotic uncertainty quantification to Bayesian updating.
A natural way to compare posterior distributions is by measuring their predictive risk, defined as the expected KL divergence of the posterior predictive to $F_0$. We consider only the situation where there is an absence of strong prior information, following [@Shimodaira2000; @Fushiki2005].
We say that predictive $\pi_1$ asymptotically dominates $\pi_2$ up to $o(n^{-k})$ if for all distributions $q$ there exists a non-negative and possibly positive real-valued functional $K(q(\cdot))$ such that: $${\mathbb{E}}_{x_{1:n}\sim q} \,d_{\text{KL}}(q(\cdot), \pi_2(\,\cdot \mid x_{1:n} ) ) - {\mathbb{E}}_{x_{1:n} \sim q}\, d_{\text{KL}}(q(\cdot), \pi_1(\,\cdot \mid x_{1:n}) ) \,=\, K(q(\cdot)) + o(n^{-k}).$$
We have the following theorem about the asymptotic properties of the MDP with $c=0$, which covers aNPL, as the model component of aNPL is ignored for large enough $n$.
> The MDP posterior predictive with $c=0$ asymptotically dominates the standard Bayesian posterior predictive up to $o(n^{-1})$.
>
> In [@Fushiki2005] the bootstrap predictive is shown to asymptotically dominate the standard Bayesian predictive up to $o(n^{-1})$. In Theorem 1 of [@Fushiki2010], the predictive of the MDP with $c=0$ (equivalently called the Bayesian bootstrap) and the bootstrap predictive are shown to be equal up to $o_p(n^{-3/2})$. A Taylor expansion argument can show that the predictive risk of the MDP has the same asymptotic expansion up to $o(n^{-1})$ as that of the bootstrap. Thus Theorem 2 of [@Fushiki2005] can be proven with the MDP predictive in place of the bootstrap predictive. Thus the predictive of the MDP with $c=0$ must also dominate the standard Bayesian predictive.
Illustrations {#sec:illustrations}
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Exponential family, \[S1\]
--------------------------
Suppose the centering model is an exponential family with parameter $\theta$ and sufficient statistic $s(x)$, $$\label{eq:exp_family}
{\mathcal{F}}_\Theta = \left\{ f_\theta(x) = g(x) \exp \left\{ \theta^T s(x) - K(\theta) \right\}; \ \theta \in \Theta \right\}.$$
Under assumed regularity, by differentiating under the integral sign of (\[eq:theta\_0\]) we find that our parameter of interest must satisfy ${\mathbb{E}}_{F_0} s(x) = \nabla_\theta K(\theta_0)$. For a particular $F$ drawn from the posterior bootstrap, the expected sufficient statistic is $$\nabla_\theta K(\tilde{\theta}^{(i)}) = \lim_{T\to \infty} \left\{ \sum\limits_{j=1}^n w_j^{(i)} s(x_j) + \sum\limits_{j=n+1}^{n+T} w_j^{(i)} s(x_j^{(i)}) \right\}.$$ with $\tilde{\theta}$ the posterior parameter value, and weights $w_{1:(n+T)}$ arising from the Dirichlet distribution as set out in Algorithm \[algo:mdp-training\], and $x_j^{(i)} \sim f_\theta(\cdot)$, with $\theta$ drawn from the parametric posterior. This provides a simple geometric interpretation of our method, as (random) convex combinations of (randomly-weighted) empirical sufficient statistics and model sufficient statistics from the parametric posterior. The distribution of the random weights is governed by $c$ and $n$ only. Our method generates stochastic maps from misspecified posterior samples to corrected NP posterior samples, by incorporating empirical information in the data over and above that captured by the model.
Updating approximate posteriors \[S2\]: Variational Bayes uncertainty correction
--------------------------------------------------------------------------------
Variational approximations to Bayesian posteriors are a popular tool for obtaining fast, scalable, but approximate Bayesian posterior distributions [@bishop2006pattern; @Blei2017]. The approximate nature of the variational update can be accounted for using our approach. Figure \[fig:bishop\_gaussian\] shows a mean-field normal approximation $q$ to a correlated normal posterior $p$, an example similar to one from [@bishop2006pattern], Section 10.1. We generated 100 observations from a bivariate normal distribution, centered at $(\frac{1}{2},\frac{1}{2})$, with dimension-wise variances both equal to $1$ and correlation equal to $0.9$, and independent normal priors on each dimension, both centered at $0$ with variance $100$. Each posterior plotted is based on $10,000$ posterior samples.
By applying the posterior bootstrap with a VB posterior (VB-NPL) in place of the Bayes posterior, we recover the correct covariance structure for decreasing prior concentration $c$, and the effect of the VB centering model is clear. If instead of $d_{\text{KL}}(q,p)$ we use $d_{\text{KL}}(p,q)$ as the objective, as it is for expectation propagation, the model posterior uncertainty may be overestimated, but is still corrected for using the posterior bootstrap.
![Posterior 95% probability contour for a bivariate Gaussian, comparing VB-NPL with $c\in\{1,10^2,10^3,10^4\}$ (red, orange, green, blue respectively) to the known Bayes posterior (grey dashed line) and a VB approximation (black dashed line).[]{data-label="fig:bishop_gaussian"}](Bishop_95_17jun18.pdf)
We demonstrate this in practice through a VB logistic regression model fit to the Statlog German Credit dataset, containing 1000 observations and 25 covariates (including intercept), from the UCI ML repository [@Dua:2017], preprocessing via [@Fernandez-Delgado2014]. An independent normal prior with variance $100$ was assigned to each covariate, and 1000 posterior samples were generated for each method. We obtain a mean-field VB sample using automatic differentiation variational inference (ADVI) in Stan [@Kucukelbir2015a]. In Fig. \[fig:beta\_21\_22\] we show that the NP update effectively corrects the VB approximation for small values of $c$. Of course, local variational methods can provide more accurate posterior approximations to Bayesian logistic posteriors [@Jaakkola1997], though these too are approximations, that NP updating can correct.
When generating pseudo-samples for use in the posterior bootstrap, both features and classes are required. Given a set of features and a parameter draw from the mixing distribution, classes are generated according to the probability specified by the logistic distribution. For the features, in this example (and the example in Section \[subsec:BRF\]) we repeatedly re-use the features of the dataset. In some applications additional features, independent but identically distributed as those in the dataset, may be available. In this case, these could instead be the basis of our pseudo-dataset.
#### Comparison with Bayesian logistic regression.
The conventional Bayesian logistic regression assumes the true log-odds of each class is linear in the predictors, and would use MCMC for inference [@Polson2013]. The MCMC samples, shown as points in Fig. \[fig:beta\_21\_22\], show a good match to the NPL update but MCMC requires a user defined burn-in and convergence checking. The runtime to generate 1 million samples by MCMC (discarding an equivalent burn-in), was 33 minutes, compared to 21 seconds with NPL, using an m5.24xlarge AWS instance with 96 vCPUs; a speed-up of 95 times. Additionally NPL has provably better predictive properties, as detailed in Section \[sec:properties\].
![Posterior contour plot for $\beta_{22}$ vs $\beta_{21}$, for VB-NPL (green) and VB (blue), for three different values of the concentration parameter $c$. Scatter plot is a sample from a Bayesian logistic posterior (red) via Polya-Gamma scheme.[]{data-label="fig:beta_21_22"}](vb_logit_scatter_sb.pdf)
Directly updating the prior \[S3\]: Bayesian Random Forests, using synthetic generated data \[S4\] {#subsec:BRF}
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Random forests (RF) [@Breiman2001] is an ensemble learning method that is widely used and has demonstrated excellent general performance [@Fernandez-Delgado2014]. We construct a Bayesian RF (BRF), via NP learning with decision trees, under a prior mixing distribution (scenario \[S3\] in Section \[sec:intro\]). This enables the incorporation of prior information, via a prior prediction function, in a principled way that is not available to RF. The step-like generative likelihood function arising from the tree partition structure does not reflect our beliefs about the true sampling distribution; the trees are just a convenient compression of the data. Because of this we simply update the prior in the MDP by specifying $\pi(\gamma | x_{1:n}) = \pi(\gamma)$. Details of our implementation of BRF can be found in Appendix \[app:brf\].
#### Information sharing with pseudo-observations \[S4\].
We emulate the situation whereby two analysts, each with a separate dataset, wish to share information, but a privacy restriction prevents direct accessing of both datasets. To test our method, we split our dataset equally into an external training, internal training and test dataset. We compared an RF trained on the internal data to two different BRFs. The first is a non-informative BRF, obtained by setting $c=0$, and trained on the internal dataset only. The second is an informative BRF (so $c>0$), trained on an augmented dataset containing the internal dataset and pseudo-samples containing information from the external dataset, suitably weighted according to Algorithm \[algo:mdp-training\]. These pseudo samples are comprised of the external features, along with class labels that are predictions from a non-informative BRF that has been trained on the external dataset. We assume that the posterior and predictive adequately preserves privacy. As a benchmark, we compared these methods to an RF trained on the combined internal and external datasets, a setup that incorporates all of the data but breaches the privacy constraints. See Figure \[fig:diagram-brf\] for a pictorial description of the four methods. We compared the test accuracy, over 100 repetitions, for a number of values of $c$, generating 10,000 prior pseudo-samples for the BRF method with $c>0$, with all forests containing 100 trees.
![Depiction of the four methods tested in our BRF example.[]{data-label="fig:diagram-brf"}](brf-flow-30may.pdf)
Boxplots comparing our methods for 13 datasets from the UCI ML repository [@Dua:2017] can be found in Fig. \[fig:boxplots\]. For small $c$, we find that BRF and RF have similar performance, but as $c$ increases, more weight is given to the externally-trained component and we find that BRF outperforms RF. The best performance of our BRF tends to occur when $c$ is set equal to the number of samples in the external training dataset, in line with our intuition of the role of $c$ as an effective sample size. A figure demonstrating this can be found in Appendix \[app:brf\]. Overall, the informative BRF provides a performance boost over RF; in fact the BRF accuracy is close to that obtained by the RF trained on both the internal and external datasets but without the need to share the data or the model.
![Boxplot of classification accuracy minus that of RF, for 13 UCI datasets and our 3 other methods.[]{data-label="fig:boxplots"}](box_all_10000_10June.pdf)
Direct updating of utility functions \[S5\]: population median
--------------------------------------------------------------
We demonstrate inference for a population median, under a misspecified Gaussian model, with parameter of interest $\alpha_0 = \operatorname*{arg\,min}_\alpha \int \vert \alpha-x \vert dF_0(x)$, and an MDP prior centered at a ${\mathcal{N}}(\theta,1)$ with prior $\pi(\theta) = {\mathcal{N}}(0,10^2)$. We use the posterior bootstrap to generate posterior samples that incorporate the prior model information with that from the data. Figure \[fig:posterior\_median\] presents histograms of posterior medians given a sample of $20$ observations from a skew-normal distribution with mean $0$, variance $1$ and median approximately $-0.2$. The left-most histogram is sharply peaked at the sample median but does not have support outside of $(x_{\min}, x_{\max})$. As $c$ grows smoothness from the parametric model is introduced to a point where the normal location parameter is used.
![Posterior histogram for median (left to right) $c=0,20,80,1000$. Right-most: posterior expected loss as a function of observation $x$. Dotted line shows the loss to the sample median.[]{data-label="fig:posterior_median"}](posterior_median2.pdf)
Discussion {#sec:discussion}
==========
We have introduced a new approach for scalable Bayesian nonparametric learning (NPL) for parametric models that both facilitates prior regularization via a baseline model, and corrects for model misspecification by incorporating an empirical component that has greater influence as the number of observations grows. A concentration parameter $c$ encodes subjective beliefs on the validity of the model; $c=\infty$ recovers Bayesian updating under the baseline model, and $c=0$ ignores the model entirely. Under regularity conditions, asymptotically, our method closely matches parametric Bayesian updating if the posited model were indeed true, and will provide a superior predictive if the model is misspecified. Additionally, our construction admits a trivially parallelizable sampler without the drawbacks of common MCMC-based samplers.
Our approach can be seen to blur the boundaries between Bayesian and frequentist inference. Conventionally, the Bayesian approach conditions on data and treats the unknown parameter of interest as if it was a random variable with some prior on a known model class. Whereas, the frequentist approach treats the object of inference as a fixed but unknown constant and characterizes uncertainty through the finite sample variability of an estimator targetting this value. Here we randomize an objective function (an estimator) according to a Bayesian nonparametric prior on the sampling distribution leading to a quantification of subjective beliefs on the value that would be returned by the estimator under an infinite sample size. In Appendix \[app:models\] we provide a review of the foundations of statistical models in this context.
At the heart of our approach is the notion of Bayesian updating via randomized objective functions through the posterior bootstrap. The posterior bootstrap acts on an augmented dataset, comprised of data and posterior predictive synthetic-samples, under which randomized maximum likelihood estimators provide a well-motivated quantification of uncertainty without assuming much about the data-generating mechanism. The precursor to this is the weighted likelihood bootstrap, which utilized a simpler form of randomization that ignored prior information. Our approach provides scope for quantifying uncertainty for more general machine learning models by randomizing their objective function suitably.
Posterior Bootstrap for model misspecification {#app:posterior_bootstrap_misspec}
==============================================
We present our posterior bootstrap for the particular case of correction of posterior samples under model misspecification. We use the stick-breaking construction here to sample from the model component, which is applicable in the general setting of Section \[sec:general\] of the paper as well.
Setting the MDP concentration parameter {#app:setting_c}
=======================================
The MDP concentration parameter $c$ represents the subjective faith we have in the model: setting $c=0$ means we discard the baseline model completely; whereas for $c=\infty$ we obtain the Bayesian posterior, equivalent to knowing that the model is true.
One way that the concentration parameter may be set in practice would be to calibrate a priori uncertainty via the variance of the population mean. It is straightforward to show that under an $\text{MDP}(\pi(\theta), c, f_\theta(\cdot) )$, the variance in the mean functional is given by $$\text{var}\, \mu(\theta) \ + \ \frac{1}{1+c}\, {\mathbb{E}}\,\sigma^2(\theta)$$ where $\mu(\theta)$ is the mean of $x | \theta$ under the mode, and $\sigma^2(\theta)$ is the variance of $x | \theta$, and the variance and expectation in the expression above are with respect to the prior $\pi$. Focus then is on the elicitation of the true variance in this functional, inclusive of model uncertainty; once we have this we can solve for $c$.
Bayesian Random Forests - further details {#app:brf}
=========================================
We implemented our BRF by rewriting the fitting routine for the Scikit-learn RandomForestClassifier class [@scikit-learn]. Instead of applying bootstrap aggregation as RF does, for each tree we construct an augmented dataset, containing the internal training data and external pseudo-data. We then fit the decision tree to the augmented dataset, suitably weighted.
If the concentration parameter $c$ is set to zero then no prior data is necessary; we just need to generate sample weights $w_{1:n} \sim \text{Dirichlet}(1,\ldots,1)$ and fit the weighted tree. When $c>0$ we use a BRF with $c=0$ to generate the external pseudo-data, by fitting it to the external data. External features and class labels generated by predictions from this prior-trained BRF are used as the prior pseudo-data to construct the augmented dataset. If $T$ pseudo-observations are generated, $x_{(n+1):(n+T)}$ are generated and augmented to our training data $x_{1:n}$, we generate a vector of $\text{Dirichlet}(1,\ldots,1,c/T,\ldots,c/T)$ weights, relating to observations $(x_1,\ldots,x_n,x_{n+1},\ldots,x_{n+T})$, with which to fit our tree. Stick-breaking could be used similarly.
When training RF decision trees, terminal node splits are made until the leaves only contain training data from a single class. This is computationally problematic for our BRF method, as after augmenting our internal training data with a large pseudo-dataset of samples, trees may need to grow ever deeper until leaf purity is attained. To avert this issue, we threshold the proportion of sample weight required to be present across samples at a node before a split can take place. This can be done via the `min_weight_fraction_leaf` argument of the Scikit-learn RF class. In our testing we set a weight proportion threshold of $0.5(n+c)^{-1}$.
Internal, external and test data were obtained by equal sized class-stratified splits. Each forest contained 100 trees, and 10,000 external pseudo-samples were generated for the BRFs with $c>0$. Predictions were made in our BRF as usual by majority vote across the forest of trees.
A plot of the average classification accuracy as a function of $c$ is given for the Bank dataset in Figure \[fig:bank\]. As discussed in Section \[subsec:BRF\], for small $c$ the external data is given little weight and our method performs similar to an RF trained only on the internal data. As $c$ grows, the accuracy improves, peaking around where $c$ is equal to the number of external samples. This peak performance is roughly the same as that attained by an RF trained on both the external and internal data.
![Mean classification accuracy vs $c$ for RF (black solid), RF-all (black dashed), and BRF (red) for varying $c$. over 100 repetitions, for the Bank dataset. Error bars represent one standard error from the mean. The BRF with $c=0$ has a mean accuracy of 0.895 and a standard error of 0.0004.[]{data-label="fig:bank"}](acc_c_bank_100_10000.pdf)
Note that when setting $c=0$ our method is equivalent to Bayesian bootstrapping random decision trees. [@Taddy2015] uses the Bayesian bootstrap as an underlying model of the data-generating mechanism, viewing the randomly weighted trees generated as a posterior sample over a functional of the unknown data-generating mechanism, similar to our construction. Previously, a number of attempts have been made in the literature to construct Bayesian models for decision trees [@Chipman1998; @Denison1998] but the associated MCMC sampling routines tend to mix poorly. Our method, whilst remaining honest that our trees are poor generative models, is very similar to RF in nature and performance, however has the additional benefit of enabling the user to incorporate prior information via a prediction function, in a principled manner.
The nature of statistical models {#app:models}
================================
To understand the scope and impact of model misspecification as covered in this paper it is interesting to consider the motivation for statistical modelling and the goal of the statistical method. Fisher considered the latter in the seminal paper [@Fisher1922].
> “the object of statistical methods is the reduction of data. A quantity of data, which usually by its mere bulk is incapable of entering the mind, is to be replaced by relatively few quantities which shall adequately represent the whole, or which, in other words, shall contain as much as possible, ideally the whole, of the relevant information contained in the original data.”
Although conditional on a model being ‘true’ it can be possible for a reduction to preserve the information contained in the data, in practical reality this will lead to an information loss. [@cox1979theoretical] explain that modelling aids the interpretation of the phenomena underlying the data, helping to separate out the meaningful features from the accidental. [@box1992bayesian] points out that a Bayesian approach to statistical inference allows the scientist to focus more on modelling aspects, as the other elements of the analysis such as inference and decision follow automatically once the model and prior is specified. These perspectives have particular relevance in the modern era of large, complex datasets, and hence models. Information loss can be manifest, and features of interest to the practitioner may be relatively simple, in comparison to the data-generating mechanism. The Bayesian approach in this instance can be particularly demanding as a global model of the scientific phenomenon is required.
Traditionally, model misspecification was mainly dealt with as part of the model-building process. As an example, [@Box1976] describes the scientific method as an iteration between an inductive step, where a tentative theory (model) is put forward, followed by a deductive step, where deductions conditional on this theory may lead to acceptance or rejection of the theory. From this perspective, if model misspecification exists, this would be picked up in the deductive step, and picked up in a subsequent inductive step. This idea is quite widely accepted; posterior predictive checks are a common means or performing the model checking [@Gelman2012].
One problem with this approach is that is says little about what to do if this iterative process does not converge, for reasons of some type of resource constraint. This had led to some more recent research in the area of model misspecification, where the consequences of a model being wrong are considered. One idea is to change the model in some way, to make it more robust to misspecification [@Royall2003; @Grunwald2012; @Holmes2017]. Another idea is to work directly with beliefs for specific parameters of interest. [@Bissiri2016] developed a decision-theoretic approach around this idea, showing that a belief update for such functionals should exist, and further providing the functional form of posterior distributions for decision functionals that maximise an expected utility. All this is done under the assumption that the true data-generating mechanism is unknown. Applications of such an approach can be found in [@Syring2015a], and [@Lyddon2018], where the calibration of a data-to-prior scale is performed by information-matching to a Bayesian bootstrap. In related work [@Watson2016a] explored the behaviour of such decision functionals in KL-neighbourhoods of the approximating model.
We have developed a methodology that deals with the problem of model misspecificatino directly and transparently, by using the information from a Bayesian model to center beliefs about the underlying data-generating mechanism. As well as being generally applicable to models and more targeted parameters, our idea extends to providing an uncertainty correction for approximate models such as VB.
[^1]: The MDP should not to be confused with the Dirichlet process mixture model (DPMM) [@Lo1984].
|
---
abstract: 'Let $F \in \mathbf{Z}[{\boldsymbol}{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of $F$ with $|{\boldsymbol}{x}| {\leqslant}X$, weighted by $\lambda(x_1)$.'
address: |
School of Mathematics\
University of Bristol\
Bristol\
BS8 1TW\
UK
author:
- V Vinay Kumaraswamy
title: Averages of Hecke eigenvalues over thin sequences
---
Introduction
============
The study of averages of arithmetic functions along thin sequences is a central topic in analytic number theory. For instance, the sum $\textstyle\sum_{n {\leqslant}X}a(p(n))$, where $p(n)=n^2+bn+c$ is an integer polynomial, and $a(n)$ are Fourier coefficients of automorphic forms, has been widely studied. For this sum, Hooley [@H] established an asymptotic formula with a power-saving error term when $a(n) = \tau(n)$, the divisor function, and $p(n)$ is irreducible. The case when $a(n)$ are Fourier coefficients of cusp forms was first settled by Blomer [@B], and later refined by Templier and Tsimerman [@TT]. However, the analogous sum over the primes, i.e. the sum $\textstyle\sum_{n {\leqslant}X}\Lambda(p(n))$, where $\Lambda(n)$ is the von Mangoldt function, is much harder to estimate, and this is a long standing open problem.
Mean values of arithmetic functions over polynomials of higher degree are poorly understood; obtaining an asymptotic formula for the sum $\textstyle\sum_{n {\leqslant}X} \tau(n^3+2)$ would represent a significant breakthrough in the subject. However, in the case of polynomials in more than variable, several results have been established. Among the most striking results in this regime are by Friedlander and Iwaniec [@FI], on the existence of infinitely many primes of the form $x^2+y^4$, and by Heath-Brown [@HB01], on primes of the form $x^3+2y^3$.
Analogously, for the divisor function, sums of the form $\textstyle\sum_{m,n
{\leqslant}X}\tau(|B(m,n)|)$, where $B(u,v)$ is an integral binary form of degree $3$ or $4$, have been investigated by several authors. For irreducible binary cubic forms, Greaves [@G2] gave an asymptotic formula for the aforementioned sum, and the sum over irreducible quartic forms was handled by Daniel [@D]. The case when $B(m,n)$ is not irreducible has also been considered; for example, such sums have been of much interest in problems relating to Manin’s conjecture for del Pezzo surfaces. See [@Br], where cubic forms are considered, and [@BB1], [@BB2], [@BT] and [@HB02] that treat the case of quartic forms.
Continuing in the same vein as the aforementioned results is the following theorem, which follows from our main theorem. In principle, our result corresponds to the case when a cubic form $B(m,n)$ splits over $\mathbf{Q}$ as the product of a linear and a quadratic form.
\[cor1\] Let $\lambda(n)$ be normalised Fourier coefficients of a holomorphic Hecke cusp form $f$ of full level and weight $k$, and let $r(n)$ be the number of representations of an integer as a sum of two squares. Let $A$ and $B$ be non-zero integers. Then there exists $\delta > \frac{1}{60}$, independent of $A$ and $B$, such that $$\sum_{m,n {\leqslant}X}r(Am^2+Bn^2)\lambda(m) \ll_{f,A,B} X^{2-\delta}.$$
Although we have stated this with the $r$-function, our methods could potentially be adapted to deal with the divisor function. It is worth emphasising that existing results on divisor sums over binary cubic and quartic forms have largely relied on arguments involving the geometry of numbers, and one cannot expect to be able to establish Theorem \[cor1\] by relying solely on these methods. Instead, we will draw from techniques in the theory of automorphic forms. Next, we move to our main theorem.
Let $F \in \mathbf{Z}[{\boldsymbol}{x}]$ be an integral quadratic form in four variables and let $w \in C_0^{\infty}(\mathbf{R}^4)$ be a smooth function with support in $[1/2,2]^4$. Let $$N_F(X) = \sum_{F({\boldsymbol}{x})=0}w\left(\frac{{\boldsymbol}{x}}{X}\right),$$ count integral solutions to $F=0$ of height less than $X$. As $X \to \infty$, Heath-Brown [@HB Theorems 6,7] established an asymptotic formula for $N_F(X)$ with a power-saving error term (see also recent work of Getz [@G] where this is refined and a second order main term is given). Now, given an arithmetic function $a(n):\mathbf{N} \to \mathbf{C}$, it is natural to ask if we can count solutions to $F=0$ in which one of the variables is weighted by $a(n)$. More precisely, let $$\label{eq:nwfa} N(a;X) = N_F(a;X) = \sum_{F({\boldsymbol}{x})=0}w\left(\frac{{\boldsymbol}{x}}{X}\right)a(x_1),$$ where $F$ and $w$ are as above. For instance, if $a = \Lambda$ then $N(a;X)$ counts weighted solutions to $F=0$ where one of the co-ordinates is prime. The inhomogeneous case (i.e. counting solutions to $F({\boldsymbol}{x})=N$ for non-zero $N$), however, has been well-studied. Tsang and Zhao [@TZ] showed that every sufficiently large integer $N \equiv 4 \pmod{24}$ can be written in the form $p_1^2 + P_2^2 + P_3^2 + P_4^2$, where $p_1$ is a prime, and each $P_i$ has at most $5$ prime factors.
In this note, we investigate the case where the $a(n)$ are Fourier coefficients of a holomorphic cusp form, and not all the variables are weighted. Suppose that a holomorphic cusp form $f(z)$ has Fourier expansion $$f(z) = \sum_{n=1}^{\infty} \lambda(n)n^{\frac{k-1}{2}}e(nz),$$ and then set $a(n) = \lambda(n)$. Our main result is
\[mainthm\] Let $F \in \mathbf{Z}[{\boldsymbol}{x}]$ be a non-singular diagonal quadratic form in $4$ variables, and let $w$ be a smooth function with compact support in $[1/2,2]^4$. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke cusp form $f$ of full level and weight $k$. Then for all ${\varepsilon}> 0$ we have $$N(\lambda;X) \ll_{{\varepsilon},f,F,w} X^{2-\frac{1}{6}+{\varepsilon}}.$$
From Heath-Brown’s work on estimating $N_F(X)$ and Deligne’s bound for $\lambda(n)$, we obtain the ‘trivial’ bound $N(\lambda;X) \ll_{{\varepsilon}} X^{2+{\varepsilon}}$. Consequently, Theorem \[mainthm\] detects cancellation for $\lambda(n)$ along thin sequences. Moreover, although we shan’t pursue it here, our method of proof allows us to handle slightly more general quadratic forms of the shape $A_1x_1^2+Q(x_2,x_3,x_4)$. It would also be very interesting to obtain an analogue of Theorem \[mainthm\] with two arithmetic weights.
Our proof of Theorem \[mainthm\] uses the $\delta$-method, which in its current form has its genesis in the work of Duke, Friedlander and Iwaniec [@DFI] on the subconvexity problem for $GL_2$ $L$-functions. In our present work, it is more convenient to use a variant of this method developed by Heath-Brown [@HB].
One could also consider estimating $N(\lambda;X)$ by parametrising solutions to $F=0$. To illustrate this, let $F = x_1x_2-x_3^2-x_4^2$. Solutions to $F=0$ in $\mathbf{P}^3$ can be parametrised as $[y_2^2+y_3^2:y_1^2:y_1y_2:y_1y_3]$, with $[y_1:y_2:y_3]\in \mathbf{P}^2$. Thus studying $N(\lambda;X)$ reduces to studying sums of the form $$\begin{split}
\sum_{g {\leqslant}X}\hspace{0.2cm} \sum_{\substack{(y_1,y_2^2+y_3^2)=g \\ y_2^2+y_3^2 {\leqslant}gX \\ y_1^2 {\leqslant}gX \\ (y_1,y_2,y_3)=1}}\lambda(y_2^2+y_3^2).
\end{split}$$ The innermost sum can potentially be analysed by the methods developed in [@TT], although the additional GCD condition makes it a challenging prospect.
We end our introduction by highlighting the key ideas in the proof of Theorem \[mainthm\]. As is typical when applying the $\delta$-method, an application of Poisson summation in the unweighted variables leads us to study sums that are essentially of the form $$\label{eq:intro}
X\sum_{\substack{{\boldsymbol}{c}' \in \mathbf{Z}^3 \\ |{\boldsymbol}{c}'| \ll 1}}\sum_{q \ll X}q^{-\frac{3}{2}}\sum_{n \ll X} \lambda(n)T(A_1n^2,F^{-1}(0,{\boldsymbol}{c}');q)I_q(n,{\boldsymbol}{c}').$$ Here $I_q(n,{\boldsymbol}{c}')$ is an exponential integral, $F^{-1}$ is the quadratic form dual to $F$, and $T(m,n;q)$ is a certain one-dimensional exponential sum of modulus $q$ which, on average, admits square-root cancellation (for fixed $m$, say). The derivatives $\frac{\partial^j}{\partial n^j}I_q(n,{\boldsymbol}{c}')$ depend polynomially on $X/q$, and determining how to control them is one of the main challenges we shall face.
Using Deligne’s bound for $\lambda(n)$ and the bound $I_q(n,{\boldsymbol}{c}') \ll 1$, we see that the sum in is $O(X^{2+{\varepsilon}}).$ This will be our starting point, and our objective is to make some saving in the $n$-sum. In this endeavour, three not unrelated strategies present themselves: exploiting cancellation from sums of Hecke eigenvalues, Mellin inversion, and the Voronoi summation formula. We shall make use of all three methods to successfully analyse the $n$-sum.
If $F^{-1}(0,{\boldsymbol}{c}') = 0$ then $T(A_1n^2,0;q)$ is essentially a Gauss sum. For fixed $q$, we shall see that this sum vanishes unless $n$ satisfies certain congruence properties modulo divisors of $q$. Moreover, $T(A_1n^2,0;q)$ is $O(q^{1/2})$ on average, leaving us to get cancellation for sums of the form $\sum_{n \equiv 0 \pmod{d}}\chi(n)\lambda(n)I_q(n,{\boldsymbol}{c}'),$ for $\chi$ a Dirichlet character with conductor $e$, and $[d,e] \mid q.$
On account of the classical bound $\sum_{n {\leqslant}X}e_q(\alpha n)\lambda(n) \ll X^{1/2}\log X$ (which is uniform in $\alpha \in \mathbf{R}$), it is natural to try and estimate the $n$-sum by partial summation. However it appears difficult to derive good bounds for $\partial I_q(n,{\boldsymbol}{c}')/ \partial n $ unless $q$ is large. Instead, we are able to control the Mellin transform of $I_q(n,{\boldsymbol}{c}')$ by means of a stationary phase argument, and this is one of the main novelties of this paper. The subsequent application of Mellin inversion to estimate the $n$-sum naturally leads to requiring a subconvexity estimate for twists of $L(s,f)$ by Dirichlet characters, and this allows us to save a small power of $X$ in the $n$-sum.
On the other hand, if $F^{-1}(0,{\boldsymbol}{c}')$ does not vanish, Voronoi’s formula works well when $q$ is a small power of $X$. Indeed, if $w$ has support in $[X,2X]$ and its derivatives satisfy the bound $w^{(j)}(x) \ll_j x^{-j}$, Voronoi’s identity transforms the sum $\textstyle\sum \lambda(n)e_q(an)w(n)$ to a ‘short’ sum of length about $q^2/X$, when $(a,q)=1$. However, in our current regime, the derivatives of $I_q(n,{\boldsymbol}{c}')$ are too large for small $q$, and we must balance these opposing forces to make a saving in the $n$-sum. When $q$ is large, partial summation becomes a viable option, and we are able to demonstrate cancellation in the $n$-sum.
We end by remarking that the methods used in this note appear to extend to cover the case when $f$ is not holomorphic. In this case, we have the bound $$\lambda(n) \ll_{{\varepsilon},f} n^{\frac{7}{64}+{\varepsilon}}$$ due to Kim and Sarnak [@KS], but this does not affect the analysis significantly. With more effort, one could also establish a similar result for forms with arbitrary level and central character.
Finally, if $f$ is not a cusp form, we will have to account for the appearance of a main term, but the analysis of the error terms will remain unchanged. Although we omit the details, the proof of Theorem \[mainthm\] can be suitably modified to give an asymptotic formula for $N(a;X)$ when $a(n) = \tau(n)$ or $r(n)$.
I would like to thank my supervisor, Tim Browning, for suggesting this problem to me, and for his detailed comments on earlier drafts of the paper, which have significantly improved its exposition. I would also like to thank E. Mehmet K[i]{}ral for a helpful conversation, and Efthymios Sofos and Raphael Steiner for their comments on an earlier version of the paper. Part of this work was done while I was a Program Associate in the Analytic Number Theory Program at the Mathematical Sciences Research Institute, Berkeley, USA, during Spring Semester 2017, which was supported by the American National Science Foundation under grant no. DMS-1440140.
Notation {#notation .unnumbered}
--------
We write $4$-tuples ${\boldsymbol}{c} = (c_1,\ldots,c_4)$ as $\mathbf{c} = (c_1,{\boldsymbol}{c'})$, where ${\boldsymbol}{c'} = (c_2,c_3,c_4)$ is a 3-tuple. Let $S(m,n;q) = \textstyle\sum_{x \pmod{q}}^{*}e_q(mx+n\overline{x})$ denote the standard Kloosterman sum, and let $c_q(m) = S(m,0;q)$ be Ramanujan’s sum. For an integer $n$, $v_p(n)$ will denote its valuation at a prime $p$. If $F$ is a non-singular quadratic form, we denote by $F^{-1}$ the form dual to $F$; by $\Delta$ we denote the discriminant of $F$. We use the notation ${\boldsymbol}{1}_{S}$ to denote the indicator function of a set $S$. All implicit constants will be allowed to depend on the quadratic form $F$, the cusp form $f$ and the weight function $w$. Any further dependence will be indicated by an appropriate subscript.
Preliminaries
=============
Summation formulae
------------------
The following lemma is a standard application of Poisson summation.
\[poisson1\] Let $w(x)$ be a smooth function with compact support. Then $$\sum_{m \equiv b {\hspace{-2.9mm}\pmod{q}}} w(m) = \frac{1}{q} \sum_{m \in \mathbf{Z}} \widehat{w}\left(\frac{m}{q}\right)e_q(bm),$$ and $\widehat{w}$ denotes the Fourier transform of $w$.
Next, we state a form of the Voronoi summation formula. For a proof, we refer the reader to [@FGKM Proposition 2.1].
\[progr\] Let $g(x)$ be a smooth function with compact support, and $\lambda(m)$ be the normalised Fourier coefficients of a cusp form of weight $k$ and full level. We then have $$\sum_{m \equiv b {\hspace{-2.9mm}\pmod{q}}} \lambda(m)g(m) = \frac{1}{q} \sum_{d \mid q} \sum_{m=1}^{\infty}\lambda(m)S(b,m;d)\check{g}_d(m),$$ where $$\label{eq:hankeltransform}
\check{g}_d(m) = \frac{2\pi i^{k}}{d} \int_{0}^{\infty} g(x) J_{k-1}\left(\frac{4\pi}{d}\sqrt{xm}\right) \, dx,$$ is a Hankel-type transform of $g$.
\[trunc\] Let $g \in C^{\infty}_{0}(\mathbf{R})$ be a smooth function with support in $[1/2,2]$ and let $\|g\|_{N,1}$ be its Sobolev norm of order $N$. Then for any $l {\geqslant}0$ we have $$\int_{0}^{\infty} g(x)J_{k-1}\left(t\sqrt{x}\right) \, dx \ll_l \min\left\{\|g\|_{\infty},\|g\|_{1,l}t^{-(l+1/2)}\right\}.$$
Denote the left hand side above by $I(t)$. Although this is a standard argument, we present a proof from [@FGKM Proposition 2.3]. Set $\alpha = t^{-2}$. Making the change of variables $x \to \alpha y^2$ we see that $$I(t) = 2\alpha \int_{0}^{\infty} g(\alpha y^2) y J_{k-1}(y) \, dy.$$ Using the fact (see (35) in [@FGKM]) that $$\frac{d}{dx}x^kJ_k(x) = x^kJ_{k-1}(x),$$ and by repeated integration by parts we have $$I(t) = 2\alpha \int_{0}^{\infty} \left\{\sum_{0{\leqslant}v {\leqslant}l} \xi_{v,l} (\alpha y^2)^v g^{(v)}(\alpha y^2)\right\} \frac{J_{k-1+l}(y)}{y^{l-1}} \, dy,$$ for some constants $\xi_{v,l}$. Since $J_{k-1+l}(y) \ll (1+y)^{-1/2}$ and $y \asymp \alpha^{-\frac{1}{2}}$, we see that $$I(t) \ll_l \|g\|_{1,l}t^{-(l+1/2)}.$$ This completes the proof.
Some facts about $L$-functions
------------------------------
In this section, we collect some standard facts about $L$-functions; [@IK Chapter 5] is a useful reference. Let $f$ be a Hecke eigenform of weight $k$ and full level with normalised Fourier coefficients $\lambda(n)$ as before. Let $\chi$ be a primitive Dirichlet character with conductor $D$. For $\sigma > 1$ let $$\label{eq:lsfotimeschi}
L(s,f \otimes \chi) = \sum_{n=1}^{\infty} \frac{\chi(n)\lambda(n)}{n^s}.$$ Then $L(s,f \otimes \chi)$ has analytic continuation to the entire complex plane, satisfies a functional equation, and has an Euler product $$\label{eq:eulerproduct}
L(s, f\otimes \chi) = \prod_{p} \left(1-\frac{\chi(p)\lambda(p)}{p^s} + \frac{\chi^2(p)}{p^{2s}}\right)^{-1}$$ for $\sigma > 1$. Applying the Phragmén-Lindelöf principle in the region $\frac{1}{2} {\leqslant}\sigma {\leqslant}1$ to $L(s,f\otimes \chi^{})$, we get that $$\label{eq:phraglind}
L(s,f\otimes \chi) \ll_{{\varepsilon},f} (D(1+|t|))^{1-\sigma+{\varepsilon}},$$ for any ${\varepsilon}> 0$. When $\sigma = \frac{1}{2}$, we can improve on . We record the following subconvexity bounds for $L(s,f\otimes \chi)$. It follows from [@BMN] that there exists $A > 0$ such that for all ${\varepsilon}> 0$ we have $$\label{eq:weyl}
L(s,f\otimes \chi) \ll_{{\varepsilon},f} D^A(1+|t|)^{\frac{1}{3}+{\varepsilon}}.$$ Although they are not used here, ‘hybrid’ subconvexity bounds for $L(s,f \otimes \chi)$ are also known, thanks to the work of Blomer and Harcos [@BH], and Munshi [@M]: there exists $\delta >0$ such that for all ${\varepsilon}>0$ we have $$L(s,f \otimes \chi) \ll_{{\varepsilon},f} (D(1+|t|))^{\frac{1}{2}-\delta+{\varepsilon}}.$$
Setting up the $\delta$-method
==============================
Let $$\delta(n) = \begin{cases} 1 &\mbox{$n = 0$, } \\
0 &\mbox{otherwise.}\end{cases}$$ By [@HB Theorem 1] there exists a function $h:\mathbf{R}^{+} \times \mathbf{R} \to \mathbf{R}$ such that for any $Q {\geqslant}1$, $$\delta(n) = c_Q Q^{-2} \sum_{q=1}^{\infty} \sideset{}{^{*}}\sum_{a {\hspace{-2.9mm}\pmod{q}}} e_q(an) h\left(\frac{q}{Q},\frac{n}{Q^2}\right),$$ where $c_Q = 1 + O_A(Q^{-A})$. The function $h(x,y)$ vanishes unless $x {\leqslant}\min(1,2|y|)$, its derivatives satisfy the bound $$\label{eq:derh}\frac{\partial^{{a+b}}}{\partial x^a y^b}h(x,y) \ll_N x^{-1-a-b}\left(x^N+\min\left(1,\frac{x}{|y|}\right)^N\right),$$ and $h(x,y)$ resembles the $\delta$ distribution in the following sense: $$\int_{\mathbf{R}} h(x,y)f(y) \, dy = f(0) + O_{f,N}(x^N).$$ Using the $\delta$-symbol to detect the equation $F({\boldsymbol}{x}) = 0$ we see that $$\label{eq:1}
N(\lambda,X) = c_QQ^{-2}\sum_{q=1}^{\infty}\sideset{}{^{*}}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{x} \in \mathbf{Z}^4}\lambda(x_1)e_q(aF({\boldsymbol}{x}))w\left(\frac{{\boldsymbol}{x}}{X}\right)h\left(\frac{q}{Q},\frac{F({\boldsymbol}{x})}{Q^2}\right).$$ We will take $Q=X$ in our application of the $\delta$-method, since $F({\boldsymbol}{x})$ is typically of size $X^2$ when ${\boldsymbol}{x}$ is of size $X$. For the rest of this note, we fix the quadratic form to be $$F({\boldsymbol}{x}) = A_1x_1^2 + \ldots + A_4x_4^2,$$ for non-zero integers $A_1, \ldots, A_4$.
Applying the Poisson summation formula
--------------------------------------
Letting $Q=X$ and breaking up the sum in into residue classes modulo $q$ we get $$\begin{split}
N(\lambda;X) = c_QQ^{-2}\sum_{q=1}^{\infty}\,&\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b})) \\ &\times \sum_{{\boldsymbol}{x} \equiv {\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}\lambda(x_1)w\left(\frac{{\boldsymbol}{x}}{X}\right)h\left(\frac{q}{Q},\frac{F({\boldsymbol}{x})}{Q^2}\right).
\end{split}$$ Applying Lemma \[poisson1\] in the $x_2,x_3$ and $x_4$ variables we get that $$\label{eq:prevoronoi}
\begin{split}
N(\lambda;X) = c_QX\sum_{q=1}^{\infty}q^{-3}&\sum_{{\boldsymbol}{c}' \in \mathbf{Z}^3}\hspace{0.2cm}\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}') \\
&\times\sum_{c_1 \equiv b_1 {\hspace{-2.9mm}\pmod{q}}}\lambda(c_1)I_q({\boldsymbol}{c}),
\end{split}$$ where if $r=q/X$, $$I_q({\boldsymbol}{c}) = \int_{\mathbf{R}^3} w(c_1/X,{\boldsymbol}{z})h(r,F(c_1/X,{\boldsymbol}{z}))e_{r}(-{\boldsymbol}{c}'.{\boldsymbol}{z})\, d{\boldsymbol}{z}.$$ By properties of the $h$-function we see that $q \ll X$, or equivalently, $r \ll 1$.
Set ${\boldsymbol}{u}' = r^{-1}{\boldsymbol}{c}'$, $$F_q(b_1,s) = \sum_{n \equiv b_1 {\hspace{-2.9mm}\pmod{q}}}\frac{\lambda(n)}{n^s},$$ and $$\label{eq:iqcs}
I_q({\boldsymbol}{c}',s) = \int_{\mathbf{R}^+\times \mathbf{R}^3} w({\boldsymbol}{x})h(r,F({\boldsymbol}{x}))e(-{\boldsymbol}{u}'.{\boldsymbol}{x}')x_1^{s-1}\, d{\boldsymbol}{x}.$$ For $\frac{1}{2} {\leqslant}\sigma {\leqslant}2$, integrating by parts we see that $$\label{eq:iqhparts}
\begin{split}
I_q({\boldsymbol}{c}',s) &\ll_N |s|^{-N} \left|\int \frac{\partial^N}{\partial x_1^N}\left\{w({\boldsymbol}{x})h(r,F({\boldsymbol}{x}))\right\}x_1^{s+N-1}e(-{\boldsymbol}{u}'.{\boldsymbol}{x}')d{\boldsymbol}{x}\right| \\
&\ll_N r^{-1-N}|s|^{-N},
\end{split}$$ by . For $\sigma > 1$, we have $F_q(b_1,s) \ll 1$, as the Dirichlet series converges absolutely in this region. By the Mellin inversion theorem, we therefore have $$\begin{split}
N(\lambda;X) = c_QX\sum_{q \ll X}q^{-3}&\sum_{{\boldsymbol}{c}' \in \mathbf{Z}^3}\hspace{0.2cm}\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}') \\ &\times\frac{1}{2\pi i}\int_{(\sigma)}X^s F_q(b_1,s) I_q({\boldsymbol}{c}',s)\, ds,
\end{split}$$ whenever $\sigma > 1$.
We end this section by recording an alternate expression for $N(\lambda;X)$. Applying Lemma \[progr\] to the $c_1$ variable in we see that $$\label{eq:postvoronoi}
N(\lambda;X) = c_QX^{2}\sum_{q\ll X}q^{-4}\sum_{\substack{{\boldsymbol}{c} \in \mathbf{Z}^4\\c_1 {\geqslant}1}}\lambda(c_1) \sum_{d \mid q}S_{d,q}({\boldsymbol}{c})I_{d,q}({\boldsymbol}{c}),$$ where $$\label{eq:s2}
\begin{split}
S_{d,q}({\boldsymbol}{c}) &= \sideset{}{^{*}}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}} e_q(aF({\boldsymbol}{b})+{\boldsymbol}{b'}.{\boldsymbol}{c}')S(b_1,c_1;d)
\end{split}$$ and $$\label{eq:i2}
\begin{split}
I_{d,q}({\boldsymbol}{c}) &= \frac{2\pi i^{k}}{d} \int_{\mathbf{R}^+\times\mathbf{R}^3} w\left({\boldsymbol}{x}\right) h\left(r,F({\boldsymbol}{x})\right) J_{k-1}\left(\frac{4\pi}{d}\sqrt{c_1Xx_1}\right) e\left(-{\boldsymbol}{u}'.{\boldsymbol}{x'}\right) \, d{\boldsymbol}{x}.
\end{split}$$
Integral estimates
==================
Preliminaries
-------------
Let $$\label{eq:testtest}
w_0(x) = \begin{cases} \exp(-(1-x^2)^{-1}), &\mbox{$|x| < 1$} \\
0 &\mbox{$|x| {\geqslant}1$},
\end{cases}$$ be a smooth function with compact support and let $$\gamma(x) = w_0\left(\frac{x}{100\max_{i=1,2,3,4}|A_i|}\right).$$ Then $\gamma(F({\boldsymbol}{x})) \gg 1$ whenever ${\boldsymbol}{x} \in \operatorname{supp}(w)$. Recall that $r=q/X$ and let $$\label{eq:g}g(r,y) = h(r,y)\gamma(y).$$ Then $g$ has compact support, and by [@HB Lemma 17] we have the following bound for its Fourier transform, $$\label{eq:prt}
\begin{split}
p_{r}(t) &= p(t) = \int_{\mathbf{R}} g(r,y)e(-ty) \, dy \ll_j (r|t|)^{-j}.
\end{split}$$
The above bound shows that $p(t)$ has polynomial decay unless $|t| \ll r^{-1-o(1)}.$
We also record a certain dissection argument due to Heath-Brown [@HB Lemma 2]. Let $w_0$ be as in , and let $c_0 = \textstyle\int_{\mathbf{R}} w_0(x) \,dx.$ For ${\boldsymbol}{u},{\boldsymbol}{v} \in \mathbf{R}^3$ define $$w_{\delta}(x_1, {\boldsymbol}{u},{\boldsymbol}{v}) = c_0^{-3}w_0^{(3)}({\boldsymbol}{u})w(x_1,\delta {\boldsymbol}{u} + {\boldsymbol}{v}),$$ where $$w_0^{(3)}({\boldsymbol}{u}) = \prod_{i=1}^3 w_0(u_i).$$ Then $$\label{eq:dissection}
\int_{\mathbf{R}^3} w_{\delta}\left(x_1,\frac{{\boldsymbol}{x}'-{\boldsymbol}{y}'}{\delta},{\boldsymbol}{y}'\right) \, d{\boldsymbol}{y}' = \delta^3 w({\boldsymbol}{x}).$$
Estimates for $I_q({\boldsymbol}{c})$
-------------------------------------
Recall that $$I_q({\boldsymbol}{c}) = \int_{\mathbf{R}^3} w(c_1/X,{\boldsymbol}{z})h(r,F(c_1/X,{\boldsymbol}{z}))e(-{\boldsymbol}{u}'.{\boldsymbol}{z})\, d{\boldsymbol}{z}.$$ We have the following estimates.
\[iqctrivial\] $I_q({\boldsymbol}{c}) \ll 1.$
This follows from [@HB Lemma 15].
\[lemmatruncatec’\] Let $N{\geqslant}0$ and suppose that ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$. Then $$I_q({\boldsymbol}{c}) \ll_{N} \frac{X}{q}|{\boldsymbol}{c}'|^{-N}.$$
This follows from [@HB Lemma 19].
As a consequence of Lemma \[lemmatruncatec’\], we find that $I_q({\boldsymbol}{c}) \ll_A X^{-A}$ if $|{\boldsymbol}{c}'| \gg X^{{\varepsilon}}$. It remains to examine the behaviour of $I_q({\boldsymbol}{c})$ when $|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$.
\[lemmastationaryphase\] Let ${\varepsilon}> 0$. Suppose that $1 {\leqslant}|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$. Then, for $j = 0,1$ we have $$\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) \ll_{{\varepsilon}} (r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}\left(r^{-j}\left(\frac{c_1}{X^2}\right)^{j} + \frac{j}{X}\right)|{\boldsymbol}{u}'|^{-\frac{1}{2}}.$$
Since $r \ll 1$, we have $|{\boldsymbol}{u}'| \gg 1$ under the hypotheses of the lemma. By we have $$I_q({\boldsymbol}{c}) = \int_{\mathbf{R}} p(t) \int _{\mathbf{R}^3}\widetilde{w}(c_1/X,{\boldsymbol}{z})e(tF(c_1/X,{\boldsymbol}{z})-{\boldsymbol}{u}'.{\boldsymbol}{z})\, d{\boldsymbol}{z} \, dt,$$ where $$\widetilde{w}(c_1/X,{\boldsymbol}{z}) = \frac{w(c_1/X,{\boldsymbol}{z})}{\gamma(F(c_1/X,{\boldsymbol}{z}))}.$$ For $j \in \left\{0,1\right\}$, $$\begin{split}
\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) &= \left(\frac{4\pi iA_1c_1}{X^2}\right)^j\int t^j p(t) \int \widetilde{w}(c_1/X,{\boldsymbol}{z})e(tF(c_1/X,{\boldsymbol}{z})-{\boldsymbol}{u}'.{\boldsymbol}{z}) \, d{\boldsymbol}{z} \, dt \\
&\quad \quad + \frac{j}{X}\int p(t)\int \frac{\partial}{\partial c_1}\widetilde{w}(c_1/X,{\boldsymbol}{z})e(tF(c_1/X,{\boldsymbol}{z})-{\boldsymbol}{u}'.{\boldsymbol}{z})\,d{\boldsymbol}{z} \, dt.
\end{split}$$ Denote the integrals over ${\boldsymbol}{z}$ by $I_1(t)$ and $I_2(t)$ respectively. Using [@HBP Lemma 3.1] we have the following bounds for $I_k(t)$: $$I_k(t) \ll \min\left(1,|t|^{-\frac{3}{2}}\right),$$ and if $|{\boldsymbol}{u}'| \gg |t|$ then $$I_k(t) \ll_{N} |{\boldsymbol}{u}'|^{-N},$$ for $k = 1,2.$
By , we have the bounds, $$\int_{|t| \ll |{\boldsymbol}{u}'|} |t|^j|p(t)| \ll |{\boldsymbol}{u}'|^{1+j},$$ and $$\int_{|t| \gg |{\boldsymbol}{u}'|} |p(t)||t|^{j-\frac{3}{2}} \,dt \ll_j r^{-j}|{\boldsymbol}{u}'|^{-\frac{1}{2}}.$$ As a result, $$\label{eq:iqpartial1}
\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) \ll_N \left(\frac{c_1}{X^2}\right)^j\left(|{\boldsymbol}{u}'|^{1+j-N}+r^{-j}|{\boldsymbol}{u}'|^{-\frac{1}{2}}\right) + \frac{j}{X}\left(|{\boldsymbol}{u}'|^{1-N} + |{\boldsymbol}{u}'|^{-\frac{1}{2}}\right).$$
We will see that this is satisfactory for the lemma unless $|{\boldsymbol}{c}'| \ll 1,$ essentially. If this is the case, we proceed as follows. By [@HB Lemma 15] we have $$\label{eq:iqcj}
\begin{split}
\frac{\partial }{\partial c_1}I_q({\boldsymbol}{c}) &= \left(\frac{2A_1c_1}{X^2}\right)\int_{\mathbf{R}^3}w(c_1/X,{\boldsymbol}{z})\frac{\partial h(r,F(c_1/X,{\boldsymbol}{z}))}{\partial c_1}e(-{\boldsymbol}{u}'.{\boldsymbol}{z}') \, d{\boldsymbol}{z} \, + \\
&\quad \quad \frac{1}{X} \int \frac{\partial w(c_1/X,{\boldsymbol}{z})}{\partial c_1}h(r,F(c_1/X,{\boldsymbol}{z})e(-{\boldsymbol}{u}'.{\boldsymbol}{z})\, d{\boldsymbol}{z} \\
&\ll \left(\frac{2A_1c_1}{X^2}\right) \int r^{-1}\left\{1+\min \left(1,\frac{r^2}{F(c_1/X,{\boldsymbol}{z})^2}\right) \right\}\, d{\boldsymbol}{z}\, + \\
&\quad \quad \frac{1}{X} \int \left\{1+\min \left(1,\frac{r^2}{F(c_1/X,{\boldsymbol}{z})^2}\right) \right\}\, d{\boldsymbol}{z} \\
&\ll r^{-1}\left(\frac{c_1}{X^2}\right)+ \frac{1}{X},
\end{split}$$ by and by the observation that the measure of the set of ${\boldsymbol}{z}$ for which $|F(\frac{c_1}{X},{\boldsymbol}{z})| \ll \nu$ is $O(\nu)$. Consequently, $$\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) \ll r^{-j}\left(\frac{c_1}{X^2}\right)^{j} + \frac{j}{X},$$ for $j=0,1$.
We are now in place to finish the proof of the lemma. Suppose first that $|{\boldsymbol}{u}'| \ll r^{-2{\varepsilon}/3}$, then $|{\boldsymbol}{u}'|^{\frac{1}{2}-{\varepsilon}} \ll r^{-{\varepsilon}}$. In this case, $$\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) \ll r^{-j}\left(\frac{2c_1}{X^2}\right)^{j} + \frac{j}{X} \ll (r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}\left(r^{-j}\left(\frac{2c_1}{X^2}\right)^{j} + \frac{j}{X}\right)|{\boldsymbol}{u}'|^{-\frac{1}{2}}.$$ Suppose next that $|{\boldsymbol}{u}'| \gg r^{-\frac{2{\varepsilon}}{3}}$ then choosing $N$ large enough in we get $$\frac{\partial^j}{\partial c_1^j}I_q({\boldsymbol}{c}) \ll \left(r^{-j}\left(\frac{c_1}{X^2}\right)^{j} + \frac{j}{X}\right)|{\boldsymbol}{u}'|^{-\frac{1}{2}}.$$ This completes the proof of the lemma.
The reader should compare the preceding result to [@HB Lemma 22].
Estimates for $I_q({\boldsymbol}{c}',s)$
----------------------------------------
Recall from that $$I_q({\boldsymbol}{c}',s) = \int_{\mathbf{R}^+\times \mathbf{R}^3} w({\boldsymbol}{x})h(r,F({\boldsymbol}{x}))e(-{\boldsymbol}{u}'.{\boldsymbol}{x}')x_1^{s-1}\, d{\boldsymbol}{x}.$$ The following ‘trivial’ bound follows from [@HB Lemma 15]. Our task for the rest of the section will be to improve on this bound.
\[iqcstrivialbound\] Let $\frac{1}{2} {\leqslant}\sigma {\leqslant}2$. We have $I_q({\boldsymbol}{c}',s) \ll 1.$
By Fourier inversion we write $$\label{eq:star}
I_q({\boldsymbol}{c}',s) = \int_{\mathbf{R}} p(\alpha) \int_{\mathbf{R}^+\times \mathbf{R^3}} \widetilde{w}({\boldsymbol}{x})e(\alpha F({\boldsymbol}{x}) - {\boldsymbol}{u}'.{\boldsymbol}{x}')x_1^{s-1} \, d{\boldsymbol}{x} \, d\alpha,$$ where $$\label{eq:widew}
\widetilde{w}({\boldsymbol}{x}) = \frac{w({\boldsymbol}{x})}{\gamma(F({\boldsymbol}{x}))}.$$
### Case 1: ${\boldsymbol}{c}' = {\boldsymbol}{0}$
We have the following
\[lemmaczero\] Suppose that ${\boldsymbol}{c}' = {\boldsymbol}{0}$ and $\frac{1}{2} {\leqslant}\sigma {\leqslant}2$. We have $$I_q({\boldsymbol}{0},s) = I_q(s) \ll_A \min\left\{1,|s|^{-A}\right\}.$$
We write $$\begin{split}
I_q(s) &= \int_{\mathbf{R}} p(u) \int_{\mathbf{R}^{+}\times \mathbf{R}^3} \widetilde{w}({\boldsymbol}{x})e(\Psi(u,{\boldsymbol}{x}))\, d{\boldsymbol}{x} \, du,
\end{split}$$ with $\Psi(u,{\boldsymbol}{x}) = uF({\boldsymbol}{x})+\frac{(s-1) \log x_1}{2\pi i}.$ Since $F$ is diagonal and $w$ is supported in the box $[1/2,2]^4$, we see that $$\nabla F({\boldsymbol}{x}) \gg 1$$ in the support of $w$; as a result we find that $\nabla \Psi \gg |u|$. Furthermore, if $|u| \ll |s|$ we see that $\nabla \Psi \gg |s|$. Therefore, we have by [@HB Lemma 10] and that for $A {\geqslant}0$ $$\begin{split}
I_q(s) &\ll_A |s|^{-A}\int_{|u| \ll |s|}|p(u)| \, du + \int_{|u| \gg |s|}|u|^{-A}|p(u)| \, du \\
&\ll_A |s|^{-A},
\end{split}$$ since for all $j=j_1+\ldots + j_4 {\geqslant}2$ we have $$\left|\frac{\partial^{j}\Psi(u,{\boldsymbol}{x})}{\partial^{j_1} x_1\ldots \partial^{j_4}x_4}\right| \ll_j |s|.$$ This completes the proof.
### Case 2: ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$
Next we will give estimates for $I_q({\boldsymbol}{c}',s)$ in the spirit of Lemmas \[lemmatruncatec’\] and \[lemmastationaryphase\]. Let $$\Psi({\boldsymbol}{x}') = \alpha F(0,{\boldsymbol}{x}') - {\boldsymbol}{u}'.{\boldsymbol}{x}'.$$ By we have $$I_q({\boldsymbol}{c}',s) = \int_{\mathbf{R}} p(\alpha) \int_{\mathbf{R}^+} x_1^{s-1}e(\alpha A_1x_1^2)\int_{\mathbf{R}^3} \widetilde{w}({\boldsymbol}{x})e(\Psi({\boldsymbol}{x}')) \, d{\boldsymbol}{x}' \, dx_1 \, d\alpha.$$ If $|\alpha| \ll |{\boldsymbol}{u}'|$, then $\nabla \Psi({\boldsymbol}{x}') \gg |{\boldsymbol}{u}'|,$ and as a result, the integral over ${\boldsymbol}{x}'$ is $O(|{\boldsymbol}{u}'|^{-N})$ by [@HB Lemma 10]. Therefore, since the integral over ${\boldsymbol}{x}'$ is trivially $O(1)$, we have the bound $$I_q({\boldsymbol}{c}',s) \ll_N |{\boldsymbol}{u}'|^{-N}\int_{|\alpha|\ll |{\boldsymbol}{u}'|}|p(\alpha)| \, d\alpha + \int_{|\alpha| \gg |{\boldsymbol}{u}'|}|p(\alpha)| \, d\alpha.$$ Therefore, by we have established the following result.
\[lemmaperronlarge\] Suppose that ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$ and $\frac{1}{2} {\leqslant}\sigma {\leqslant}2$. Then $$I_q({\boldsymbol}{c}', s) \ll \min\left\{1, r^{-1}|{\boldsymbol}{c}'|^{-N}\right\}.$$
Observe that Lemma \[lemmaperronlarge\] implies that $I_q({\boldsymbol}{c}',s) \ll_A X^{-A}$ unless $|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$.
### An estimate for $I_q({\boldsymbol}{c}',s)$ when $1 {\leqslant}|{\boldsymbol}{c}'|\ll X^{{\varepsilon}}$
Let $s= \sigma + it$. In the range $1 {\leqslant}|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$ we proceed as follows. Integrating by parts we get $$\label{eq:postparts}
\begin{split}
I_q({\boldsymbol}{c}',s) &= \frac{1}{s}\int_{\mathbf{R}} p(\alpha) \int_{\mathbf{R}^+\times\mathbf{R}^3} \frac{\partial}{\partial x_1}\widetilde{w}({\boldsymbol}{x})x_1^{\sigma} e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1 + \Psi({\boldsymbol}{x}')) \, d{\boldsymbol}{x} \, d\alpha \, + \\
& \frac{4\pi i A_1}{s}\int_{\mathbf{R}} \alpha p(\alpha) \int_{\mathbf{R}^+\times\mathbf{R}^3} \widetilde{w}({\boldsymbol}{x})x_1^{\sigma+1} e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1 + \Psi({\boldsymbol}{x}')) \, d{\boldsymbol}{x} \, d\alpha \\
&= \frac{1}{s}I_1 + \frac{4\pi i A_1}{s}I_2,
\end{split}$$ say. Observe that $$I_1 = \int \frac{\partial w({\boldsymbol}{x})}{\partial x_1}h(r,F({\boldsymbol}{x}))x_1^s\, d{\boldsymbol}{x}.$$ Lemma \[iqcstrivialbound\] applied to the test function $\partial w({\boldsymbol}{x})/\partial x_1$ shows that $$I_1 \ll 1.$$ Similarly, observe that $$I_2 = \frac{1}{2\pi i A_1} \int w({\boldsymbol}{x})\frac{\partial}{\partial x_1}h(r,F({\boldsymbol}{x}))x_1^{s+1} \, d{\boldsymbol}{x}.$$ Arguing as in we see that $$\label{eq:i2bound}
I_2 \ll r^{-1}.$$ Our goal will be to remove the factor $r^{-1}$ in the bound for $I_2$. Indeed, we will show that $I_2 \ll X^{{\varepsilon}}$. Our approach is modeled on the proof of [@HB Lemma 22].
Applying to $\widetilde{w}({\boldsymbol}{x})x_1^{\sigma+1}$ we get that $$\begin{split}
I_2 & =\delta^{-3} \int \alpha p(\alpha) \times \\
&\quad \quad \int \int w_{\delta}\left(x_1,\frac{{\boldsymbol}{x}'-{\boldsymbol}{y}}{\delta},{\boldsymbol}{y}\right)e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1 + \Psi({\boldsymbol}{x}'))\, d{\boldsymbol}{x} \, d{\boldsymbol}{y} \, d\alpha.
\end{split}$$ Let ${\boldsymbol}{x}' = {\boldsymbol}{y} + \delta {\boldsymbol}{z}$. By virtue of $w$ being compactly supported, we see that $|{\boldsymbol}{y}| \ll 1,$ and we arrive at the inequality $$\begin{split}
I_2 &{\leqslant}\int \int |\alpha p(\alpha)| \times \\ &\quad \quad \left|\int w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1 +\Psi({\boldsymbol}{y}+\delta {\boldsymbol}{z})) \, dx_1 \, d{\boldsymbol}{z}\right| \, d\alpha \, d{\boldsymbol}{y}.
\end{split}$$ with $w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z}) = w_{\delta}\left(x_1,{\boldsymbol}{z},{\boldsymbol}{y}\right).$ Observe that the partial derivatives $$\frac{\partial^{j_1+\ldots + j_4}}{\partial x_1^{j_1}\ldots \partial x_4^{j_4}}w_{{\boldsymbol}{y}}({\boldsymbol}{x}) \ll 1,$$ for all $j_i {\geqslant}0$ (the implicit constant depends polynomially on $\sigma$, the coefficients of $F$, and the support of $w$). Henceforth, we will take $\delta = |{\boldsymbol}{u}'|^{-\frac{1}{2}}$.
Let $R {\geqslant}1$ be a parameter to be chosen later. Suppose first that $R^3 {\leqslant}|{\boldsymbol}{u}'| {\leqslant}r^{-1}R$. Let ${\boldsymbol}{y} = (y_2,y_3,y_4)$. As in the proof of [@HB Lemma 22], we say that a pair $({\boldsymbol}{y},\alpha)$ is ‘good’, if $$|\nabla \Psi({\boldsymbol}{0})| = |{\boldsymbol}{u}'|^{-\frac{1}{2}}|2\alpha (A_2y_2,A_3y_3,A_4y_4)- {\boldsymbol}{u}'| {\geqslant}R\max \left\{|\alpha|/|{\boldsymbol}{u}'|, 1\right\},$$ and that $({\boldsymbol}{y},\alpha)$ is ‘bad’ otherwise. If $({\boldsymbol}{y}, \alpha)$ is ‘good’ then [@HB Lemma 10] shows that $$\int \left|\int w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})e(\Psi({\boldsymbol}{y}+\delta {\boldsymbol}{z})) \, d{\boldsymbol}{z}\right|\, dx_1 \ll_N R^{-N}.$$ For the ‘bad’ pairs we will bound the $x_1$ integral using a stationary phase argument and to bound the ${\boldsymbol}{z}$ integral trivially.
Suppose that $({\boldsymbol}{y},\alpha)$ is bad. Since $|{\boldsymbol}{u}'|^{-\frac{1}{2}} \ll R^{-\frac{3}{2}},$ observe that $|{\boldsymbol}{y}| \gg_{w,F} 1$ if $R$ is large enough in terms of the coefficients of $F.$ Moreover, $|{\boldsymbol}{y}| \ll 1$, trivially, and as a result, we see that $|{\boldsymbol}{u}'| \ll |\alpha| \ll |{\boldsymbol}{u}'|.$ Therefore, if $({\boldsymbol}{y},\alpha)$ is bad, we get $$\label{eq:badest}
|2\alpha(A_2y_2,A_3y_3,A_4y_4) - {\boldsymbol}{u}'| \ll |{\boldsymbol}{u}'|^{\frac{1}{2}} \ll R|{\boldsymbol}{u}'|^{\frac{1}{2}}.$$ We have$$\begin{split}
I_2 &\ll_N r^{-2}R^{-N} + \\ &\quad \int |\alpha p(\alpha)| \int \left| \int w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1 +\Psi({\boldsymbol}{x}')) \, dx_1 \, d{\boldsymbol}{z}\right| \, d{\boldsymbol}{y}\, \, d\alpha,
\end{split}$$ where the integral is over the bad pairs $({\boldsymbol}{y},\alpha)$. Integrating trivially over ${\boldsymbol}{z}$ we are left with $$\begin{split}
\int w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})e(\alpha A_1x_1^2 + &\tfrac{t}{2\pi}\log x_1 +\Psi({\boldsymbol}{x}')) \, dx_1 \, d{\boldsymbol}{z} \\ &{\leqslant}\int \left|\int w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})e(\alpha A_1x_1^2 + \tfrac{t}{2\pi}\log x_1) \, dx_1 \right| \, d{\boldsymbol}{z}.
\end{split}$$ To evaluate the integral over $x_1$ we need the following result.
\[lemmasecond\] Let $w$ have compact support in $[1/2,2]$, and suppose that $A \neq 0$. Then for all $N {\geqslant}0$ we have $$\label{eq:secondder}
\int w(x)e(Ax^2 + B\log x) \, dx \ll_N \max\left\{|A|^{-\frac{1}{2}},|A|^{-N}\right\}.$$
Applying the lemma to the test function $w_{{\boldsymbol}{y}}(x_1,{\boldsymbol}{z})$ with $A = A_1 \alpha$ and $B=t/2\pi$, and integrating trivially over ${\boldsymbol}{z}$, we see that $$\begin{split}
I_2 &\ll_N r^{-2}R^{-N} + \int \int_{|\alpha| \asymp |{\boldsymbol}{u}'|} |p(\alpha)| |\alpha|^{\frac{1}{2}} \, d\alpha \, d{\boldsymbol}{y},
\end{split}$$ where the integral over ${\boldsymbol}{y}$ is over those ${\boldsymbol}{y}$ such that $({\boldsymbol}{y},\alpha)$ is ‘bad’. By , the measure of the set of ${\boldsymbol}{y}$ such that $({\boldsymbol}{y},\alpha)$ is ‘bad’ is $O(|{\boldsymbol}{u}'|^{-\frac{3}{2}}R^3).$ Putting everything together, we have shown that if $r^{-1}R {\geqslant}|{\boldsymbol}{u}'| {\geqslant}R^3$ then $$\label{eq:i2fine}I_2 \ll r^{-2}R^{-N} + R^3.$$
All that remains now is to choose $R$. Suppose first that $|{\boldsymbol}{u}'| \ll r^{-{\varepsilon}/2}$. Then, $|{\boldsymbol}{u}'|^{1-{\varepsilon}} \ll r^{-{\varepsilon}}.$ In this case, we make use of the trivial bound to get, $$I_2 \ll r^{-1} \ll (r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}r^{-1}|{\boldsymbol}{u}'|^{-1} \ll_{{\varepsilon}} (r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}|{\boldsymbol}{c}'|^{-1}.$$
If, on the other hand, $|{\boldsymbol}{u}'| \gg r^{-{\varepsilon}/2}$, choose $R = (r^{-1}|{\boldsymbol}{u}|')^{{\varepsilon}/12}$. By taking $N$ sufficiently large, we get from that $$I_2 \ll_N R^3 +r^{-2}R^{-N} \ll_{{\varepsilon}} (r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}|{\boldsymbol}{c}'|^{-1}.$$ As a result, we have shown
\[csmall\] Let ${\varepsilon}> 0$. Suppose that ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$ and $\sigma > 0$. Then $$I_q({\boldsymbol}{c}',s) \ll_{{\varepsilon}} \frac{1}{|s|}X^{{\varepsilon}}.$$
We bring matters to a close by giving a proof of Lemma \[lemmasecond\].
Let $\Psi(x) = A x^2 + B\log x$. Suppose that $|B| {\geqslant}8 |A|$. We then have, $|\Psi'(x)| {\geqslant}B$ in the support of $w$. And $|\Psi''(x)| = |2A - \frac{B}{2x^2}| \ll |B|$, $|\Psi^{(n)}(x)| = \left|\frac{B}{n!x^n}\right| \ll_n |B|$ for all $n {\geqslant}3$. By [@HB Lemma 10] we get $$\int w(x)e(\Psi(x)) \, dx \ll |B|^{-N} \ll |A|^{-N}.$$ Suppose next that $|B| {\leqslant}\frac{1}{8} |A|$. In this case, $|\Psi'(x)| {\geqslant}|A|$ in the support of $w$, and we also see that $|\Psi^{(n)}(x)| \ll_n |A|$ for all $n {\geqslant}2$. Consequently, $$\int w(x)e(\Psi(x)) \, dx \ll |A|^{-N}$$ in this case as well.
Suppose finally that $\frac{1}{8}|A| {\leqslant}|B| {\leqslant}8|A|$. If $AB$ is positive, then it is easy to see that $|\Psi'(x)| \gg |A|$, and as a result $\textstyle\int w(x)e(\Psi(x)) \, dx \ll |A|^{-N}$. Finally, suppose that $AB$ is negative. Then in this case $|\Psi''(x)| = |2A - \frac{B}{2x^2}| \gg |A|$. We need the following refinement of the second derivative test due to Tao [@T Lemma 2.5]
Let $w(x)$ have support in $[1/2,2]$ and let $\Psi(x)$ be a smooth function such that $|\Psi''(x)| {\geqslant}c$, for some $c > 0$. Then $$\int w(x)e(\Psi(x)) \, dx \ll c^{-\frac{1}{2}}.$$
The proof can be found in Tao’s lecture notes, and we include it here for the sake of completeness. Since $$\int w(x)e(\Psi(x))\, dx = - \int_{1/2}^2 w'(x) \int_{1/2}^x e(\Psi(y)) \, dy \, dx,$$ it is sufficient to prove the lemma in the unweighted case, $$\int_{1/2}^2 e(\Psi(x)) \, dx.$$ Let $\tau$ be a parameter to be chosen in due course. Observe that by our assumption on $\Psi''$, $|\Psi'(x)| {\geqslant}\tau$ except for an interval of length $O(\tau/c)$. Furthermore, on the remaining portion of the interval $[1/2,2]$, $\Psi'(x)$ is monotonic. As a result, $$\int_{1/2}^2 e(\Psi(x)) \, dx \ll \frac{1}{\tau} + \tau/c,$$ by the first derivative test. Choosing $\tau$ appropriately completes the proof.
Applying the lemma with $c = |A|$ we get our desired estimate for .
Estimates for $I_{d,q}({\boldsymbol}{c})$
-----------------------------------------
Let ${\boldsymbol}{u}' = r^{-1}{\boldsymbol}{c}'$. It follows from that $$I_{d,q}({\boldsymbol}{c}) = \frac{2\pi i^k}{d}\int_{\mathbf{R}^{+} \times \mathbf{R}^{3}} w\left({\boldsymbol}{x}\right) h\left(r,F({\boldsymbol}{x})\right)J_{k-1}\left(\frac{4\pi}{d}\sqrt{c_1x_1X}\right)e\left(-{\boldsymbol}{u'}.{\boldsymbol}{x'}\right) \, d{\boldsymbol}{x}.$$ Furthermore, $$I_{d,q}({\boldsymbol}{c}) = \frac{2\pi i^k}{d}\int_{\mathbf{R}^{+} \times \mathbf{R}^{3}} \widetilde{w}\left({\boldsymbol}{x}\right) g(r,F({\boldsymbol}{x}))J_{k-1}\left(\frac{4\pi}{d}\sqrt{c_1x_1X}\right)e\left(-{\boldsymbol}{u'}.{\boldsymbol}{x'}\right) \, d{\boldsymbol}{x},$$ where $\widetilde{w}({\boldsymbol}{x})$ is as in , and $g(x,y)$ is as in .
### Estimates for large ${\boldsymbol}{c}'$
We begin by applying Lemma \[trunc\] with $$\psi(x_1) = \psi_{{\boldsymbol}{x}'}(x_1) = \tilde{w}({\boldsymbol}{x})g(r,F({\boldsymbol}{x}))e(-{\boldsymbol}{u}'.{\boldsymbol}{x}'),$$ treated as a function in the variable $x_1$, and $t = 4\pi(c_1X)^{\frac{1}{2}}/d$ we have, $$I_{d,q}({\boldsymbol}{c}) \ll_N \frac{1}{d}\left(4\pi\frac{\sqrt{c_1X}}{d}\right)^{1/2}\|\psi\|_{N,1}\left(4\pi\frac{\sqrt{c_1X}}{d}\right)^{-N}.$$ Since $\frac{\partial^n}{\partial y^n}h(r,y) \ll r^{-1-n}$, we have the bound $\|\psi\|_{N,1} \ll_N r^{-1-N}$, which gives us $$I_{d,q}({\boldsymbol}{c}) \ll_{N} (dr)^{-1}\left(4\pi\frac{\sqrt{c_1X}}{d}\right)^{1/2} \left(\frac{X}{q}\frac{d}{\sqrt{c_1X}}\right)^{N}.$$
Hence we have the following
\[truncatem1\] For all $N {\geqslant}0$ we have $$I_{d,q}({\boldsymbol}{c}) \ll_{N} (dr)^{-1}\left(\frac{d^{\frac{1}{2}}}{(c_1X)^{\frac{1}{4}}}\right)\min\left\{\left(\frac{X}{q}\frac{d}{\sqrt{c_1X}}\right)^{N-1}, |{\boldsymbol}{c}'|^{-N}\right\}$$ As a result, $I_{d,q}({\boldsymbol}{c}) \ll_A X^{-A}$ whenever $c_1 \gg X^{1+{\varepsilon}}/(q/d)^2$, or $|{\boldsymbol}{c}'| \gg X^{{\varepsilon}}$.
The first bound follows from the preceding discussion. The second follows from [@HB Lemma 19], and the bound $J_{k-1}(x) \ll (1+x)^{-1/2}$ and taking $N > 2A/{\varepsilon}$.
### Analysis of $I_{d,q}({\boldsymbol}{c})$ using stationary phase
By Lemma \[truncatem1\] we may suppose that $1 {\leqslant}c_1 \ll X^{1+{\varepsilon}}/(q/d)^2$, and that $1 {\leqslant}|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$. In this range, we have the following estimate for $I_{d,q}({\boldsymbol}{c}).$
\[stphasehb\] Suppose that ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$. Let ${\varepsilon}>0$. Then $$I_{d,q}({\boldsymbol}{c}) \ll_{{\varepsilon}} \left(1+\frac{\sqrt{c_1X}}{d}\right)^{-\frac{1}{2}}\left(\frac{q}{|{\boldsymbol}{c}'|dX}\right)X^{{\varepsilon}}.$$
Since this is a straightforward adaptation of [@HB Lemma 22], and the fact that $J_{k-1}(z) \ll (1+z)^{-\frac{1}{2}},$ we will only point out the necessary changes to the argument. We adopt the notation from [*loc. cit.*]{}, and run Heath-Brown’s argument for the weight function $J_{k-1}\left(4\pi\frac{\sqrt{c_1x_1X}}{d}\right)w({\boldsymbol}{x}),$ but the dissection argument, (8.2) [*loc. cit.*]{} is applied only in the ${\boldsymbol}{x}'$ variable, in the spirit of the proof of Lemma \[csmall\] above. Moreover, the notion of ‘good’ and ‘bad’ pairs of vectors $({\boldsymbol}{y},t)$ is independent of the $x_1$ variable, and we differentiate by parts only in the ${\boldsymbol}{x}'$ variable, and run the stationary phase argument in the ${\boldsymbol}{x}'$ variable. Finally, we remark that we have the uniform bound, $$J_{k-1}\left(4\pi\frac{\sqrt{c_1x_1X}}{d}\right)w({\boldsymbol}{x}) \ll_{w} \left(1+\frac{\sqrt{c_1X}}{d}\right)^{-\frac{1}{2}}$$ in the support of $w$.
Exponential sums
================
We begin by establishing certain multiplicativity results for the exponential sums that we will encounter in the proof of Theorem \[mainthm\]. Let $S_{d,q}({\boldsymbol}{c})$ be the exponential sum in . We have
\[multiplicativitylemma\] Suppose that $d = u_1u_2$ and $q = v_1v_2$ with $(u_1v_1,u_2v_2) = 1$. Then the following holds, $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= S_{u_1,v_1}(\overline{u_2}^2c_1,\overline{v_2}{\boldsymbol}{c}')S_{u_2,v_2}(\overline{u_1}^2c_1,\overline{v_1}{\boldsymbol}{c}') \\
&= S_{u_1,v_1}(v_2\overline{u_2^2}c_1,{\boldsymbol}{c}')S_{u_2,v_2}(v_1\overline{u_1^2}c_1,{\boldsymbol}{c}').
\end{split}$$
Let $a=v_2a_1+v_1a_2$ where $a_i$ run modulo $v_i$. Let ${\boldsymbol}{b} = v_2\overline{v_2}{\boldsymbol}{s} + v_1\overline{v_1}{\boldsymbol}{t}$ where $s$ (respectively $t$) runs modulo $v_1$ (respectively $v_2$). Then $$\label{eq:basicmult}
\begin{split}
e_{q}(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}') &= e_{v_1}(a_1F({\boldsymbol}{s})+{\boldsymbol}{s}'.\overline{v_2}{\boldsymbol}{c}')e_{v_2}(a_2F({\boldsymbol}{t})+{\boldsymbol}{t}'.\overline{v_1}{\boldsymbol}{c}').
\end{split}$$ Also, $$\begin{split}
S(b_1,c_1;u_1u_2) &= S(\overline{u_2}s_1,\overline{u_2}c_1;u_1)S(\overline{u_1}t_1,\overline{u_1}c_1;u_2) \\
&= S(s_1,\overline{u_2}^2c_1;u_1)S(t_1,\overline{u_1}^2c_1;u_2).
\end{split}$$ This gives us the first multiplicativity statement. For the second, replace ${\boldsymbol}{s}$ (resp. ${\boldsymbol}{t}$) by $v_2{\boldsymbol}{s}$ (resp. $v_1{\boldsymbol}{t}$).
Let $$\label{eq:sq(n)}
A_q({\boldsymbol}{c}) = \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b}' {\hspace{-2.9mm}\pmod{q}}}e_q(aF(c_1,{\boldsymbol}{b}')+{\boldsymbol}{b}'.{\boldsymbol}{c}').$$
\[multforaq\] Let $q=v_1v_2$ with $(v_1,v_2)=1$. We have $$\begin{split}
A_q({\boldsymbol}{c}) &= A_{v_1}(c_1,\overline{v_2}{\boldsymbol}{c}')A_{v_2}(c_1,\overline{v_1}{\boldsymbol}{c}')\\
&=A_{v_1}(\overline{v_2}^2c_1,{\boldsymbol}{c}')A_{v_2}(\overline{v_1}^2c_1,{\boldsymbol}{c}').
\end{split}$$
The proof is similar to the proof above: let $d=1$ and write $a$ and ${\boldsymbol}{b}'$ as in Lemma \[multiplicativitylemma\]. We have, $$e_q(aF(c_1,{\boldsymbol}{b}')+{\boldsymbol}{b}'.{\boldsymbol}{c}') = e_{v_1}(a_1F(n,{\boldsymbol}{x}')+{\boldsymbol}{x}'.\overline{v_2}{\boldsymbol}{c}')e_{v_2}(a_2F(n,{\boldsymbol}{y}')+{\boldsymbol}{y}'.\overline{v_1}{\boldsymbol}{c}').$$ Therefore, $$A_q({\boldsymbol}{c}) = A_{v_1}(c_1,\overline{v_2}{\boldsymbol}{c}')A_{v_2}(c_1,\overline{v_1}{\boldsymbol}{c}').$$ The lemma follows by replacing ${\boldsymbol}{x}'$ by $v_2{\boldsymbol}{x}'$, ${\boldsymbol}{y}'$ by $v_1{\boldsymbol}{y}'$, and by replacing $a_1$ by $\overline{v_2}^2a_1$ and $a_2$ by $\overline{v_1}^2a_2.$
Evaluation of $S_q(n)$
----------------------
Set $$\label{eq:definesqn}
S_q(n) = A_q(n,{\boldsymbol}{0}).$$ Lemma \[multforaq\] shows that $S_{q_1q_2}(n) = S_{q_1}(n)S_{q_2}(n)$ whenever $(q_1,q_2)=1$. Therefore, it suffices to evaluate $S_q(n)$ at prime powers $q=p^k$. In doing so, we will encounter the following exponential sums, and it will be useful to have their evaluation at hand. For $p > 2$ define $$\label{eq:spknpm}
\begin{split}
\mathcal{S}^{-}(p^k,n) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^k}}}e_{p^k}(an^2)\left(\frac{a}{p}\right) \text{, and} \\
\mathcal{S}^{+}({p^k},n) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^k}}}e_{p^k}(an^2) = c_{p^k}(n^2).
\end{split}$$
\[spknpmeval\] Suppose that $p$ is an odd prime. Then we have $$\begin{split}
\mathcal{S}^{-}({p^k},n) &= \begin{cases} 0 &\text{ $k$ is even,}\\
\epsilon_p p^{k-\frac{1}{2}}{\boldsymbol}{1}_{v_p(n^2) = k-1} &\text{ otherwise.} \end{cases} \\
\mathcal{S}^{+}(p^k,n) &= \begin{cases} {\varphi}(p^k){\boldsymbol}{1}_{p^{k/2} \mid n} &\text{if $k$ is even}, \\
p^k\left({\boldsymbol}{1}_{p^{\frac{k+1}{2}} \mid n} - \frac{{\boldsymbol}{1}_{p^{\frac{k-1}{2}} \mid n}}{p}\right) &\text{ if $k$ is odd.}\end{cases}
\end{split}$$
For $\mathcal{S}^{-}$, we write $a = u + pv$ and $$\begin{split}
\mathcal{S}^{-}(p^k,n) &= \sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{p}}}\left(\frac{u}{p}\right)e_{p^k}(un^2)\sum_{v {\hspace{-2.9mm}\pmod{p^{k-1}}}}e_{p^{k-1}}(vn^2) \\
&= p^{k-1}{\boldsymbol}{1}_{p^{k-1}\mid n^2}\sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{p}}}\left(\frac{u}{p}\right)e_{p}(un^2/p^{k-1}) \\
&= \epsilon_p p^{k-\frac{1}{2}}{\boldsymbol}{1}_{p^{k-1}\Vert n^2}\left(\frac{n^2/p^{k-1}}{p}\right).
\end{split}$$ By definition, $\mathcal{S}^{+}(p^k,n)$ is Ramanujan’s sum, and its evaluation is well-known.
Next, we recall some basic facts about Gauss sums. Let $$\delta_n=\begin{cases} 0, & \mbox{ if }n\equiv 0\bmod{2}, \\ 1 & \mbox{ if } n\equiv 1\bmod{2}, \end{cases}
\quad
\epsilon_n=\begin{cases} 1, & \mbox{ if }n\equiv 1\bmod{4}, \\ i, & \mbox{ if } n\equiv 3\bmod{4}.
\end{cases}$$ The following result is recorded in [@BB Lemma 3], but it goes back to Gauss.
\[gaussevlemma\] Define $$\mathcal{G}(s,t;q) = \sum_{b {\hspace{-2.9mm}\pmod{q}}} e_q(sb^2+tb).$$ Suppose that $(s,q)=1$. Then $$\mathcal{G}(s,t;q)=
\begin{cases}
\epsilon_q \sqrt{q}\left(\frac{s}{q}\right) e\left(-\frac{\overline{4s}t^2}{q}\right) & \text{
if $q$ is odd,}\\
2 \delta_t \epsilon_v\sqrt{v} \left(\frac{2s}{v}\right) e\left(-\frac{\overline{8s}t^{2}}{v}\right) &
\text{ if $q=2v$, with $v$ odd,}\\
(1+i) \epsilon_s^{-1} (1-\delta_t) \sqrt{q}\left(\frac{q}{s}\right) e\left(-\frac{\overline{s}t^{2}}{4q}\right)
& \text{
if $4\mid q$.}
\end{cases}$$ If $(s,q) \neq 1$, $\mathcal{G}(s,t;q) = 0$ unless $(s,q) \mid t$, in which case we have $$\mathcal{G}(s,t;q) = (s,q)\mathcal{G}\left(\frac{s}{(s,q)},\frac{t}{(s,q)};\frac{q}{(s,q)}\right).$$
For the remainder of the section we will assume that coefficients $A_1,\ldots,A_4$ satisfy the following condition.
\[a0\]\[Condition $A_0$\] Let $l_1,l_2,l_3$ and $l_4$ be non-zero integers, and let $q = \prod_{p^{k_p} \Vert q}$. We say that the tuple $(q;l_1,l_2,l_3,l_4)$ satisfies Condition $A_0$ if for each odd prime $p \mid q$, we have $k_p {\geqslant}\max\left\{v_p(l_1), v_p(l_2), v_p(l_3), v_p(l_4)\right\}$, if $p \nmid \Delta$, and if $k_p {\geqslant}\max\left\{2+v_p(l_1), v_p(l_2), v_p(l_3), v_p(l_4)\right\}$ for $p \mid \Delta$. If $p=2$, we require that $k_2 {\geqslant}3+\max_{1 {\leqslant}i {\leqslant}4}v_2(l_i).$
For a prime $p$ and $1 {\leqslant}i {\leqslant}4$, let $a_i(p) = v_p(A_i)$. Define the following product of Jacobi symbols, $$\label{eq:prodjacobi}
J(p^k) = J_{\Delta}(p^k) = \prod_{i=2}^4 \left(\frac{\overline{A_1}^{-1}\overline{A_i}}{p^{k-a_i}}\right),$$ if $p \neq 2$, and $$\label{eq:prodjacobi2}
J(2^k) = \prod_{2 {\leqslant}i {\leqslant}4}\left(\frac{2^{k-a_i}}{\overline{A_1}^{-1}\overline{A_i}}\right),$$ where $\overline{A_i} = A_i/(p^{a_i},A_i).$ $J(q)$ is then defined multiplicatively for arbitrary $q$.
### Evaluation of $S_q(n)$ for odd $q$
To state our result on the evaluation of $S_q(n)$ for odd $q$ we define the following invariant. Let $$\label{eq:epsilonforodd}
\epsilon(p^k) = \epsilon_{\Delta}(p^k) = \begin{cases} 1 &\text{ if each $k-a_i {\hspace{-2.9mm}\pmod{2}}$ has the same parity for $ 2 {\leqslant}i {\leqslant}4$} \\
\epsilon_p^2 &\text{ otherwise},
\end{cases}$$ and extend the definition of $\epsilon(q)$ to odd $q$ by multiplicativity.
\[evalsqnqodd\] Let $q=p^k$, and $p \neq 2$ and suppose that $(q,A_1,\ldots,A_4)$ satisfies Condition $A_0$. If $k-a_2-a_3-a_4$ is even, $$S_q(n) = q^{\frac{3}{2}}J(p^k)p^{a_1}c_{p^{k-a_1}}(n^2)\epsilon(p^k)\prod_{i=2}^4p^{\frac{a_i}{2}}.$$ If $k-a_2-a_3-a_4$ is odd, then $S_q(n)$ vanishes unless $k-a_1$ is odd, in which case, $$S_q(n) = {\boldsymbol}{1}_{v_p(n^2)=k-a_1-1}p^{-\frac{1}{2}}q^{\frac{5}{2}}J(p^k)\epsilon(p^k)\prod_{i=2}^4p^{\frac{a_i}{2}}.$$
We have $$\begin{split}
S_q(n) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^k}}}e_q(aA_1n^2)\prod_{i=2}^4\sum_{b_i {\hspace{-2.9mm}\pmod{q}}}e_q(aA_ib_i^2).
\end{split}$$ Applying Lemma \[gaussevlemma\] to each of the sums over $b_i$ we get that $$\begin{split}
S_q(n) &= p^{\frac{3k}{2}}\prod_{i=2}^4 {\left(\frac{\overline{A_i}}{p^{k-a_i}}\right)}p^{\frac{a_i}{2}}\epsilon_{p^{k-a_i}} {\sideset{}{^*}\sum}_{a {\hspace{-2.9mm}\pmod{p^k}}}e_{p^k}(aA_1n^2)\prod_{i=2}^4 {\left(\frac{a}{p^{k-a_i}}\right)}\\
&= p^{\frac{3k}{2}}p^{a_1}\mathcal{S}^{\pm}(p^{k-a_1},n)J(p^k)\prod_{i=2}^4\epsilon_{p^{k-a_i}}p^{\frac{a_i}{2}},
\end{split}$$ depending on the parity of $k-a_2-a_3-a_4 \pmod{2}$. If $k-a_2-a_3-a_4$ is even, then by Lemma \[spknpmeval\] $$\label{eq:pmiddeltanot2even}
S_q(n) = q^{\frac{3}{2}}J(p^k)p^{a_1}c_{p^{k-a_1}}(n^2)\prod_{i=2}^4p^{\frac{a_i}{2}}\epsilon_{p^{k-a_i}}.$$ If $k-a_2-a_3-a_4$ is odd, we have $$\begin{split}
S_q(n) &= q^{\frac{3}{2}}p^{a_1}J(p^k)\epsilon_p\prod_{i=2}^4\epsilon_{p^{k-a_i}}p^{\frac{a_i}{2}}\mathcal{S}^{-}(p^{k-a_1},n) \\
&= {\boldsymbol}{1}_{v_p(n^2)={k-a_1-1}}J(p^k)p^{-\frac{1}{2}}q^{\frac{5}{2}}\epsilon_p\prod_{i=2}^4p^{\frac{a_i}{2}}\epsilon_{p^{k-a_i}}.
\end{split}$$ This completes the proof of the lemma.
### Evaluation of $S_{p^k}(n)$ for $p=2$
Let $q=2^k$. By Lemma \[gaussevlemma\], $$\label{eq:s2kn1}
\begin{split}
S_{q}(n) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{2^k}}}e_{2^k}(aA_1n^2)\prod_{i=2}^42^{a_i}\sum_{b_i {\hspace{-2.9mm}\pmod{2^{k-a_i}}}}e_{2^{k-a_i}}\left(a\overline{A_i}b_i^2\right) \\
&= (1+i)^3 2^{\frac{3k+a_2+a_3+a_4}{2}}\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{2^k}}}e_{2^k}(aA_1n^2)\prod_{2 {\leqslant}i {\leqslant}4}\epsilon^{-1}_{a\overline{A_i}}\left(\frac{2^{k-a_i}}{a\overline{A_i}}\right) \\
&= J(2^k) 2^{a_1+\frac{3k+a_2+a_3+a_4}{2}}\sum_{v {\hspace{-2.9mm}\pmod{2^{k-2-a_1}}}}e_{2^{k-2}}(vn^2)\times \\
&\quad \quad \quad \sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{4}}}e_{2^{k-2-a_1}}(un^2)\prod_{2 {\leqslant}i {\leqslant}4}(1+i)\epsilon^{-1}_{u\overline{A_1}^{-1}\overline{A_i}}\left(\frac{2^{k-a_i}}{u+4v}\right).
\end{split}$$
Suppose first that $k-a_2-a_3-a_4$ is even, in which case $\prod_{2 {\leqslant}i {\leqslant}4}\left(\frac{2^{k-a_i}}{.}\right)=1$. In this case, the sum over $v$ vanishes unless $n^2 \equiv 0 \pmod{2^{k-2-a_1}}$. Let $T$ denote the sum over $u$. We have $$\begin{split}
T &= (1+i)^3\sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{4}}}e_{4}(un^2/2^{k-2-a_1})\prod_{2 {\leqslant}i {\leqslant}4}\epsilon^{-1}_{u\overline{A_1}^{-1}\overline{A_i}}.
\end{split}$$ Define the invariant $$\gamma(\Delta) = \prod_{2 {\leqslant}i {\leqslant}4} \epsilon^{-1}_{\overline{A_1}^{-1}\overline{A_i}},$$ which is a fourth root of unity that depends on the coefficients of $F$. A simple calculation reveals that $$\prod_{2{\leqslant}i {\leqslant}4} \epsilon^{-1}_{\overline{A_1}^{-1}\overline{A_i}}\prod_{2 {\leqslant}i {\leqslant}4} \epsilon^{-1}_{3\overline{A_1}^{-1}\overline{A_i}} = i.$$ Let $$\label{eq:delta+}
\delta^{+}(n,2^k) = \frac{(1+i)^3}{4}\begin{cases} \left(\gamma(\Delta) + \frac{i}{\gamma(\Delta)}\right) &\text{ if $\frac{n^2}{2^{k-2-a_1}} \equiv 0 {\hspace{-2.9mm}\pmod{4}}$} \\
i \left(\gamma(\Delta) - \frac{i}{\gamma(\Delta)}\right) &\text{ if $\frac{n^2}{2^{k-2-a_1}} \equiv 1 {\hspace{-2.9mm}\pmod{4}}$} \\
- \left(\gamma(\Delta) + \frac{i}{\gamma(\Delta)}\right) &\text{ if $\frac{n^2}{2^{k-2-a_1}} \equiv 2 {\hspace{-2.9mm}\pmod{4}}$}.
\end{cases}$$ Then $T = 4\delta^{+}(n,2^k).$ Substituting back into we get $$S_q(n) = {\boldsymbol}{1}_{2^{k-2-a_1} \mid n^2}\delta^{+}(n,k)J(2^k)2^{\frac{5k+a_2+a_3+a_4}{2}} .$$ Suppose finally that $k-a_2-a_3-a_4$ is odd. Proceeding as in , but writing $a = u + 8v$, we get that $$\begin{split}
S_q(n) &= 2^{\frac{3k+a_2+a_3+a_4}{2}}J(2^k)\sum_{v {\hspace{-2.9mm}\pmod{2^{k-3}}}} e_{2^{k-3-a_1}}(vn^2)\times \\
&\quad \quad \quad \sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{8}}}e_{8}(un^2/2^{k-3-a_1})\left(\frac{2}{u}\right)\prod_{2 {\leqslant}i {\leqslant}4}(1+i)\epsilon^{-1}_{u\overline{A_1}^{-1}\overline{A_i}}
\end{split}$$ The sum over $v$ vanishes unless $2^{k-3-a_1} \mid n^2$ - in which case, up to a factor of $(1+i)^3$, the inner sum is $$e_8(n^2/2^{k-3-a_1})\gamma(\Delta) - \frac{ie_8(3n^2/2^{k-3-a_1})}{\gamma(\Delta)} - e_8(5n^2/2^{k-3-a_1})\gamma(\Delta) + \frac{ie_8(7n^2/2^{k-3-a_1})}{\gamma(\Delta)},$$ which vanishes unless $2^{k-3-a_1} \Vert n^2$. Therefore, this forces $n^2/2^{k-3-a_1} \equiv 1 \pmod{8}$. Let $$\label{eq:delta-}
\delta^{-}(n,2^k) = \frac{-1}{\sqrt{2}}\left(\gamma(\Delta)+\frac{1}{\gamma(\Delta)}\right).$$ Then we see that the $u$-sum evaluates to $8\delta^-(n,2^k)$. As a result, $$S_q(n) = {\boldsymbol}{1}_{v_2(n^2)=k-3-a_1}J(2^k)\delta^-(n,2^k)2^{\frac{5k+a_2+a_3+a_4}{2}}.$$
\[sqn2k\] Let $q=2^k$, and notation as above. Suppose that $(q;A_1,\ldots,A_4)$ satisfies Condition $A_0$. Then $$S_q(n) = 2^{\frac{5k+a_2+a_3+a_4}{2}}J(2^k)\begin{cases} \delta^{+}(n,2^k) {\boldsymbol}{1}_{2^{k-2-a_1} \mid n^2} &\text{ if $k-a_2-a_3-a_4$ is even}\\
\delta^{-}(n,2^k) {\boldsymbol}{1}_{v_2(n^2)=k-3-a_1} &\text{ if $k-a_2-a_3-a_4$ is odd.}\end{cases}$$
### A description of $S_q(n)$ for general $q$ {#sectionsqn}
Having evaluated $S_q(n)$ at prime-powers, we will now record a qualitative description of $S_q(n)$ for general $q$. Write $q= q_{odd}q_{even}$ where $(q_{odd},q_{even})=1$ and $q_{even} \mid 2^{\infty}$. Write $q_{odd}=q_1q_2$, where $q_1$ is composed entirely of primes $p \mid q$ such that $k_p-\textstyle\sum_{i=2}^4a_i(p)$ is odd, and $q_2$ is composed entirely of primes such that $k_p-\textstyle\sum_{i=2}^4a_i(p)$ is even. By Lemma \[evalsqnqodd\] we see that $S_{q_1}(n)$ vanishes $k_p - a_1(p)$ is odd, for each $p \mid q_1$. To this end define $$\label{eq:iota}
\iota(q_1) = \begin{cases} 1 &\text{ $k_p - a_1(p)$ is odd for each $p \mid q_1$} \\
0 &\text{otherwise.}
\end{cases}$$ Let $s(q)$ denote the squarefree kernel of an integer $q$. By Lemma \[multforaq\] we have $$S_q(n) = S_{q_{even}}(n)S_{q_1}(n)S_{q_2}(n).$$ For an integer $q$ define $${\varrho}(q) = \prod_{\substack{p^k \Vert q \\ \text{$k$ is odd}}} p^{\frac{k+1}{2}} \prod_{\substack{p^k \Vert q \\ \text{$k$ is even}}}p^{\frac{k}{2}}$$ and let $$\widetilde{q_1} = {\varrho}\left(\frac{q_1}{s(q_1)(q_1,A_1)}\right).$$ Invoking Lemma \[evalsqnqodd\], we see that $S_{q_1}(n)$ vanishes unless $n = \widetilde{q_1}m$ with $(m,s(q_1))=1$. Let $\chi^0_{s(q_1)}$ be the principal character modulo $s(q_1)$. We have, $$\label{eq:sq1n}
S_{q_1}(n) = \iota(q_1)\epsilon(q_1)J(q_1)\prod_{i=2}^{4}(q_1,A_i)^{\frac{1}{2}}{\boldsymbol}{1}_{\widetilde{q_1} \mid n}\chi^0_{s(q_1)}(n/\widetilde{q_1})\frac{q_1^{\frac{5}{2}}}{s(q_1)^{\frac{1}{2}}}.$$ Similarly, $$\label{eq:sq2n}
S_{q_2}(n) = q_2^{\frac{3}{2}} \epsilon(q_2)J(q_2) \prod_{i=2}^{4}(q_2,A_i)^{\frac{1}{2}}(q_2,A_1)c_{q_2/(q_2,A_1)}(n^2).$$ To give an explicit description of $c_{q_2/(q_2,A_1)}(n^2)$, decompose $q_2 = q_3q_4$, with $$\begin{split}
q_3 &= \prod_{\substack{p^k \Vert q_2 \\ k-a_1(p) \text{ is odd }}}p^k \\
q_4 &= \prod_{\substack{p^k \Vert q_2 \\ k-a_1(p) \text{ is even }}}p^k.
\end{split}$$ Then it follows from the definitions of $q_3$ and $q_4$, and Lemma \[spknpmeval\] that $$\label{eq:cr4}
\begin{split}
c_{q_4/(q_4,A_1)}(n^2) &= {\varphi}\left(\frac{q_4}{(q_4,A_1)}\right){\boldsymbol}{1}_{\sqrt{q_4/(q_4,A_1)}\mid n} \\
&= \frac{q_4}{(q_4,A_1)} \prod_{p \mid q_4/(q_4,A_1)}\left(1-\frac{1}{p}\right){\boldsymbol}{1}_{{\varrho}(q_4/(q_4,A_1))\mid n}
\end{split}$$ and $$\begin{split}
c_{q_3/(q_3,A_1)}(n^2) &= \prod_{p^k \Vert q_3/(q_3,A_1)} p^k \left({\boldsymbol}{1}_{p^k \mid n^2} - \frac{{\boldsymbol}{1}_{p^{k-1}\mid n^2}}{p}\right) \\
&= \frac{q_3}{(q_3,A_1)}\sum_{d \mid \frac{q_3}{(q_3,A_1)}}\frac{\mu(d)}{d}{\boldsymbol}{1}_{\frac{q_3/(q_3,A_1)}{d} \mid n^2} \\
&= \frac{q_3}{(q_3,A_1)}\sum_{d \mid \frac{q_3}{(q_3,A_1)}}\frac{\mu(d)}{d}{\boldsymbol}{1}_{{\varrho}\left(\frac{q_3/(q_3,A_1)}{d}\right) \mid n}.
\end{split}$$ Substituting back into , we obtain $$\label{eq:sq2explicit}
S_{q_2}(n) = \epsilon(q_2)J(q_2)\prod_{i=2}^4(q_2,A_i)^{\frac{1}{2}}\frac{{\varphi}(q_4/(q_4,A_1))}{q_4/(q_4,A_1)}q_2^{\frac{5}{2}}\sum_{d \mid \frac{q_3}{(q_3,A_1)}}\frac{\mu(d)}{d}{\boldsymbol}{1}_{{\varrho}\left(\frac{q_2/(q_2,A_1)}{d}\right) \mid n}.$$ Combining with Lemma \[sqn2k\] to evaluate $S_{q_{even}}(n)$, we obtain the following result.
\[sqnproposition\] Suppose that $(q;A_1,\ldots,A_4)$ satisfies condition $A_0$. Then there exist integers $s_q(F)$, $\theta \mid q$ and $\kappa \mid q$ such that $S_q(n) = {\boldsymbol}{1}_{\substack{\theta \mid n}}{\boldsymbol}{1}_{(n/\theta,\kappa)=1}s_q(F)$. Moreover, $s_q(F)$ is independent of $n$ and it satisfies the bound $|s_q(F)| \ll_{\Delta} q^{\frac{5}{2}}$. In addition, $s_q(F)$ is multiplicative in $q$, and if $q$ is square-free then $s_q(F) \ll_{\Delta} q^2$. We also have $\theta \gg_{\Delta} {\varrho}\left(\frac{q}{s(q)(q,A_1)}\right)$ and $\kappa \ll_{\Delta} s(q)$.
The existence of $\theta$ and $\kappa$, and the lower bound for $\theta$ follow from , and Lemma \[sqn2k\]. It is clear that $S_q(n) \ll q^{\frac{5}{2}}$, and this gives our bound for $|s_q(F)|$. The multiplicativity of $s_q(F)$ follows from the multiplicativity of $S_q(n)$.
Finally, suppose that $q$ is square-free. Since $(p,A_1,\ldots,A_4)$ satisfies Condition $A_0$ for each $p \mid q$, we see that $q_2=1$. As a result, $|s_q(F)| \ll q^2$, by . This completes the proof of the proposition.
Exponential sums in the case where $F^{-1}(0,{\boldsymbol}{c}')=0$ and ${\boldsymbol}{c}' \neq {\boldsymbol}{0}$ {#aq0}
----------------------------------------------------------------------------------------------------------------
Having evaluated $S_q(n) = A_q(n,{\boldsymbol}{0})$ explicitly, we will now relate it to the more general sum $A_q(n,{\boldsymbol}{c}')$ with $F^{-1}(0,{\boldsymbol}{c}')=0.$
\[replacec’0\] Let ${\boldsymbol}{c}' \neq {\boldsymbol}{0} \in \mathbf{Z}^3$ and let $n \in \mathbf{N}$. Let $p$ be a prime and $q = p^k$. For $2 {\leqslant}i {\leqslant}4$ suppose that $c_i \equiv 0 \pmod{p^{v_p(A_i)}}$, and that $F^{-1}(0,{\boldsymbol}{c}') = 0$. Let $A_q(n,{\boldsymbol}{c}')$ be as in . Suppose that $(q,A_2,A_3,A_4)$ satisfies condition $A_0$ (Definition \[a0\]). Then $$A_q(n,{\boldsymbol}{c}') = A_q(n,{\boldsymbol}{0}) = S_q(n).$$
Let $a_i = v_p(A_i)$, as before. Then the sum over ${\boldsymbol}{b}'$ in is $$\label{eq:ul}
\prod_{i=2}^4 p^{a_i} \sum_{b_i {\hspace{-2.9mm}\pmod{p^{k-a_i}}}}e_{p^{k-a_i}}(aA_i/p^{a_i}b_i^2 + b_i c_i/p^{a_i}).$$ To ease notation, let $A_i' = A_i/p^{a_i}$ and let $c_i' = c_i/p^{a_i}$. If $p \neq 2$, by Lemma \[gaussevlemma\] the above expression evaluates to $$\label{eq:ul2}
\prod_{i=2}^4 p^{\frac{k+a_i}{2}}\epsilon_{p^{k-a_i}}\left(\frac{aA_i'}{p^{k-a_i}}\right)e_{p^{k-a_i}}(-\overline{4a A_i'} c_i'^2),$$ since by hypothesis $(A_i',p)=1$. Make a change of variables $\overline{a} \to -4A_2'A_3'A_4'b$, and observe that for $i=2,3,4$ we have $e_{p^{k-a_i}}(-\overline{4aA_i'}c_i'^2) = e_{p^{k-a_i}}(b
\prod_{\substack{2 {\leqslant}j {\leqslant}4 \\ i \neq j}}A_j'c_i'^2)$. Consequently, the expression in is $$\prod_{i=2}^4 p^{\frac{k+a_i}{2}}\epsilon_{p^{k-a_i}}\left(\frac{-4b\prod_{\substack{2 {\leqslant}j {\leqslant}4 \\ i \neq j}}A_j'}{p^{k-a_i}}\right)e_{p^k}(p^{a_2}A_3'A_4'c_2'^2+p^{a_3}A_2'A_4'c_3'^2+p^{a_4}A_2'A_3'c_4'^2).$$ However, since $F^{-1}(0,{\boldsymbol}{c}') = A_3A_4c_2^2+\ldots+A_2A_3c_4^2 = 0$, we have $p^{a_2}A_3'A_4'c_2'^2 + p^{a_3}A_2'A_4'c_3'^2 + p^{a_4}A_2'A_3'c_4'^2 = 0$. Consequently, the exponential factor above is $=1,$ and we see that $A_q(n,{\boldsymbol}{c}')$ is independent of ${\boldsymbol}{c}'$ and this completes the proof for odd $p$. A similar argument works when $p=2.$
Auxillary estimates
-------------------
Recall the sum $S_{d,q}({\boldsymbol}{c})$ from . We begin by recording a version of [@HB Lemma 28].
\[lemma0\] Let $q=p^t$, $d=p^{\delta}$ with $t {\geqslant}2$ and $\delta {\leqslant}t$. Suppose that $p \nmid 2\Delta$ and $F^{-1}(0,{\boldsymbol}{c}') \neq 0$. Then $S_{d,q}({\boldsymbol}{c})$ vanishes unless $p \mid F^{-1}(0,{\boldsymbol}{c}')$.
In the expression $$S_{d,q}({\boldsymbol}{c}) = \sideset{}{^*}\sum_{z {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{z})\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^t}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{p^t}}}e_{p^t}(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}'+p^{t-\delta}b_1z)$$ set $a = u + pv$ to see that $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= \sideset{}{^*}\sum_{z {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{z})\sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{p}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_{p^t}(uF({\boldsymbol}{b})+{\boldsymbol}{b}.(p^{t-\delta}z,{\boldsymbol}{c}'))\times \\ &\quad \quad \sum_{v {\hspace{-2.9mm}\pmod{p^{t-1}}}}e_{p^{t-1}}(vF({\boldsymbol}{b})) \\
&= p^{t-1}\sideset{}{^*}\sum_{z {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{z})\sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{p}}}\sum_{\substack{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{p^t}} \\ F({\boldsymbol}{b}) \equiv 0 {\hspace{-2.9mm}\pmod{p^{t-1}}}}}e_{p^t}(uF({\boldsymbol}{b})+{\boldsymbol}{b}.(p^{t-\delta}z,{\boldsymbol}{c}')).
\end{split}$$ Writing ${\boldsymbol}{b} = {\boldsymbol}{x} + p^{t-1}{\boldsymbol}{y}$, we get that $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= p^{t-1}\sideset{}{^*}\sum_{z {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{z})\times \\ &\quad\quad \sideset{}{^*}\sum_{u {\hspace{-2.9mm}\pmod{p}}}\sum_{\substack{{\boldsymbol}{x} {\hspace{-2.9mm}\pmod{p^{t-1}}} \\ F({\boldsymbol}{x}) \equiv 0 {\hspace{-2.9mm}\pmod{p^{t-1}}}}}e_{p^t}(uF({\boldsymbol}{x}) + p^{t-\delta}x_1z +{\boldsymbol}{x}'.{\boldsymbol}{c}')\times \\ &\quad \quad \quad \sum_{{\boldsymbol}{y} {\hspace{-2.9mm}\pmod{p}}}e_p({\boldsymbol}{y}.(u\nabla F({\boldsymbol}{x}) + (p^{2t-\delta-1}z,{\boldsymbol}{c}')).
\end{split}$$ As $2t {\geqslant}2+\delta$, the sum over ${\boldsymbol}{y}$ vanishes unless $\nabla F({\boldsymbol}{x}) \equiv -\overline{u}(0,{\boldsymbol}{c}') \pmod{p}$. Since $p \nmid 2\Delta$, this is the same as the condition ${\boldsymbol}{x} \equiv -\overline{2u}M^{-1}(0,{\boldsymbol}{c}') \pmod{p}$, where $M$ is the matrix corresponding to the quadratic form $F$. Observe that this forces $F({\boldsymbol}{x}) \equiv \overline{4u^2}F^{-1}(0,{\boldsymbol}{c}') \pmod{p}$. Consequently, the sum over ${\boldsymbol}{x}$ vanishes unless $F^{-1}(0,{\boldsymbol}{c}') \equiv 0 \pmod{p}$, and the lemma follows.
Let $$\label{eq:tqr}
T_q(r) = \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b}) + b_1r + {\boldsymbol}{b}'.{\boldsymbol}{c}').$$ and $$\label{eq:tq}
T_q = \sum_{r {\hspace{-2.9mm}\pmod{q}}}|T_q(r)|.$$ In the proof of Theorem \[mainthm\], we will need good control on the average order of $T_q$. We have
\[c’0\] Suppose that $F^{-1}(0,{\boldsymbol}{c}') \neq 0$ and $|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$. Then $$\sum_{q {\leqslant}X}T_q \ll_{{\varepsilon}} X^{4+{\varepsilon}}.$$
Observe that $T_q$ is multiplicative in $q$. Write $q = uv$ where $u$ is square-free and $v$ is square-full. Let $N = 2|\Delta||F^{-1}(0,{\boldsymbol}{c}')|.$ Further factorise $v = v_1v_2$, with the property that $(v_1,N)=1$ and $p \mid N$ for any prime $p$ that divides $v_2$. Thus we are led to estimating $T_u$, $T_{v_1}$ and $T_{v_2}$ individually.
If $p \nmid 2\Delta$, it follows from [@HB Lemma 26] that $$T_p = p^2 \sum_{r_p {\hspace{-2.9mm}\pmod{p}}}|c_p(F^{-1}(r_p,{\boldsymbol}{c}'))| {\leqslant}3p^3.$$ Furthermore, if $p \mid 2\Delta$, observe that $T_p \ll_F 1.$ Hence we have $$T_u \ll u^3 3^{\omega(u)}.$$ By [@HB Lemma 25] we see that $$T_{v_2} \ll v_2^4.$$ To deal with $T_{v_1}$ we make the following claim.
Suppose that $p \nmid 2\Delta$ and that $F^{-1}(0,{\boldsymbol}{c}') \neq 0$. Let $r \pmod{p^t}$ and $p \mid r$. Then we claim that $T_{p^t}(r) = 0$ unless $p \mid F^{-1}(0,{\boldsymbol}{c}')$.
To see this, we argue as in the proof of Lemma \[lemma0\] to see that $$\begin{split}
T_{p^t}(r) &= p^{t+3}{\sideset{}{^*}\sum}_{u {\hspace{-2.9mm}\pmod{p}}}\sum_{{\boldsymbol}{x} {\hspace{-2.9mm}\pmod{p^{t-1}}}}e_{p^t}(uF({\boldsymbol}{x}) + rx_1 + {\boldsymbol}{x}'.{\boldsymbol}{c}'),
\end{split}$$ where the ${\boldsymbol}{x}$-sum is also subject to the conditions $F({\boldsymbol}{x}) \equiv 0 \pmod{p^{t-1}}$ and $2M{\boldsymbol}{x} \equiv -\overline{u}(r,{\boldsymbol}{c}') \pmod{p}$, and $M$ is the matrix associated to the quadratic form $F$. It is then easy to see that if $p \mid r$ then $p \mid x_1$, and this in turn implies that $p \mid F^{-1}(0,{\boldsymbol}{c}'),$ as claimed.
It now follows that $$T_{v_1} = {\sideset{}{^*}\sum}_{r {\hspace{-2.9mm}\pmod{v_1}}}|T_{v_1}(r)|.$$ By applying Lemma \[gaussevlemma\] to each term in $T_{v_1}$ we get $$\begin{split}
T_{v_1} &= v_1^2 {\sideset{}{^*}\sum}_{r {\hspace{-2.9mm}\pmod{v_1}}}|c_{v_1}(\overline{A_1}r^2+F^{-1}(0,{\boldsymbol}{c}'))| \\
&{\leqslant}v_1^2 {\sideset{}{^*}\sum}_{r {\hspace{-2.9mm}\pmod{v_1}}}(v_1,r^2+F^{-1}(0,{\boldsymbol}{c}')) \ll_{{\varepsilon}} X^{{\varepsilon}}v_1^3,
\end{split}$$ since $|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$. As a result, $$\begin{split}
\sum_{q {\leqslant}X}T_q &\ll X^{{\varepsilon}}\sum_{\substack{v_2 {\leqslant}X \\ p \mid v_2 \implies p \mid N}}v_2^4\sum_{uv_1 {\leqslant}X/v_2}(uv_1)^3 \ll_{{\varepsilon}} X^{4+{\varepsilon}}
\end{split}$$ since $$\sum_{\substack{v {\leqslant}X \\ p \mid v \implies p \mid N}} 1 \ll_{{\varepsilon}} (NX)^{{\varepsilon}}.$$ This completes the proof of the lemma.
Notice that we do not need the condition $F^{-1}(0,{\boldsymbol}{c}') \neq 0$ to estimate the sum over the square-free part. However, we have used this fact to restrict the number of terms in the $v$-sum. Without this observation, Lemma \[c’0\] would only hold with the weaker upper bound $O(X^{\frac{9}{2}+{\varepsilon}}).$
Next, we analyse the sum $S_{d,q}({\boldsymbol}{c})$. Observe that Lemma \[multiplicativitylemma\] shows that it suffices to consider the case where $q=p^k$ is a prime power. Lemma \[lemma0\] shows that for $(p,2\Delta)=1$, if $p^2 \mid q$ then $S_{d,q}({\boldsymbol}{c})$ vanishes unless $p \mid |F^{-1}(0,{\boldsymbol}{c}')|.$ If $d =1$, then $S_{1,q}({\boldsymbol}{c}) = S_{1,q}(0,{\boldsymbol}{c}')$.
Let $d = p^{\delta}$ and $q = p^{\kappa}$, with $\delta {\leqslant}\kappa$. Recall that $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^{\kappa}}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{p^{\kappa}}}}e_{p^{\kappa}}(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}')S(b_1,c_1;p^{\delta}).
\end{split}$$ Suppose first that $p \neq 2$. For $1 {\leqslant}i {\leqslant}4$, let $p^{a_i} = (A_i,p^{\kappa}).$ By Lemma \[gaussevlemma\] we see that $S_{d,q}({\boldsymbol}{c})$ vanishes unless $c_i \equiv 0 \pmod{p^{a_i}}$ and in this case, $$\label{eq:sdqb'}
\begin{split}
S_{d,q}({\boldsymbol}{c}) &= p^{\frac{3k+a_2+a_3+a_4}{2}}\prod_{i=2}^4\epsilon_{p^{\kappa-a_i}}\left(\frac{A_i/p^{a_i}}{p^{\kappa-a_i}}\right)\times \\ &\quad \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^{\kappa}}}}\prod_{i=2}^4\left(\frac{a}{p^{\kappa - a_i}}\right)e_{p^{\kappa - a_i}}(-\overline{4aA_i/p^{a_i}}(c_i/p^{a_i})^2) \\
&\quad \quad \sum_{b_1 {\hspace{-2.9mm}\pmod{p^{\kappa}}}}e_{p^{\kappa}}(aA_1b_1^2)S(b_1,c_1;p^{\delta}).
\end{split}$$
If $(p,\Delta)=1$, and $d = q = p$, we have $a_i=0$, and notice that the sum over $b_1$ in is $$\begin{split}
&= \sideset{}{^*}\sum_{x {\hspace{-2.9mm}\pmod{p}}}e_p(c_1\overline{x})\sum_{b_1 {\hspace{-2.9mm}\pmod{p}}}e_p(aA_1b_1^2+b_1x) \\
&= \epsilon_p p^{\frac{1}{2}}\left(\frac{aA_1}{p}\right)\sideset{}{^*}\sum_{x {\hspace{-2.9mm}\pmod{p}}}e_p(c_1 \overline{x} - \overline{4aA_1}x^2).
\end{split}$$ Consequently, $$\begin{split}
S_{p,p}({\boldsymbol}{c}) &= p^2\left(\frac{\Delta}{p}\right) \sideset{}{^*}\sum_{a,x {\hspace{-2.9mm}\pmod{p}}}e_p(c_1\overline{x} - \overline{4a}F^{-1}(x,{\boldsymbol}{c}')) \\
&= p^2\left(\frac{\Delta}{p}\right)\left\{{\varphi}(p)\sideset{}{^*}\sum_{\substack{x {\hspace{-2.9mm}\pmod{p}}\\ F^{-1}(x,{\boldsymbol}{c}') \equiv 0 {\hspace{-2.9mm}\pmod{p}}}}e_p(c_1\overline{x}) - \sideset{}{^*}\sum_{\substack{x {\hspace{-2.9mm}\pmod{p}} \\ F^{-1}(x,{\boldsymbol}{c}') \not\equiv 0 {\hspace{-2.9mm}\pmod{p}}}}e_p(c_1\overline{x})\right\}.
\end{split}$$ Hence $|S_{p,p}({\boldsymbol}{c})| {\leqslant}3p^3.$
We can also handle these sums in greater generality. Suppose that $p \neq 2$. The sum over $b_1$ in vanishes (by Lemma \[gaussevlemma\]) unless $\delta {\leqslant}\kappa - a_1$, and in this case, by Lemma \[gaussevlemma\], the sum is $$\label{eq:sdqb'b1}
\begin{split}
&={\sideset{}{^*}\sum}_{x {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{x})\sum_{b_1 {\hspace{-2.9mm}\pmod{p^{\kappa}}}}e_{p^{\kappa}}(aA_1b_1^2+p^{\kappa-\delta}b_1x) \\
&= \epsilon_{p^{k-a_1}}p^{\frac{k+a_1}{2}}\left(\frac{aA_1/p^{a_1}}{p^{\kappa-a_1}}\right)\sideset{}{^*}\sum_{x {\hspace{-2.9mm}\pmod{p^{\delta}}}}e_{p^{\delta}}(c_1\overline{x}-\overline{4aA_1/p^{k-a_1}}p^{\kappa-\delta-a_1}x^2).
\end{split}$$ If $\delta {\leqslant}\frac{\kappa-a_1}{2}$, we see that $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= p^{2k+\frac{a_1+a_2+a_3+a_4}{2}}\prod_{i=1}^4\epsilon_{p^{k-a_i}}\left(\frac{A_i/p^{a_i}}{p^{k-a_i}}\right)c_{p^\delta}(c_1)\times \\ &\quad \quad \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{p^\kappa}}}\left(\frac{A_1/p^{a_1}}{p^{\kappa-a_1}}\right)\prod_{i=2}^4\left(\frac{A_i/p^{a_i}}{p^{\kappa-a_i}}\right)e_{p^{\kappa - a_i}}(-\overline{4aA_i/p^{a_i}}(c_i/p^{a_i})^2) \\
&\ll_{\Delta} p^{3k+\delta} \ll q^{7/2}.
\end{split}$$ If $\delta > \frac{\kappa-a_1}{2},$ clearing denominators in , we get $$\begin{split}
S_{d,q}({\boldsymbol}{c}) &= p^{3\kappa-\delta-a_1+\frac{a_1+a_2+a_3+a_4}{2}}\prod_{i=1}^4\epsilon_{p^{k-a_i}}\left(\frac{A_i/p^{a_i}}{p^{k-a_i}}\right)\times \\
&\quad \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\left(\frac{A_1/p^{a_1}}{p^{\kappa-a_1}}\right)\prod_{i=2}^4\left(\frac{A_i/p^{a_i}}{p^{\kappa-a_i}}\right)e_{p^{\kappa - a_i}}(-\overline{4aA_i/p^{a_i}}(c_i/p^{a_i})^2)\times \\
&\quad \quad \sideset{}{^*}\sum_{x {\hspace{-2.9mm}\pmod{p^{2\delta-\kappa+a_1}}}}e_{p^{2\delta-\kappa+a_1}}(c_1\overline{x}-\overline{4aA_1/p^{k-a_1}}x^2) \\
&\ll_{\Delta} q^{\frac{7}{2}},
\end{split}$$ by [@BV Lemma 3.1] applied to the sum over $x$. A similar analysis holds when $p = 2$, except we have the slightly worse bound (see [@BV Lemma 3.2]) $$S_{d,q}({\boldsymbol}{c}) \ll 2^{\frac{15\kappa}{4}}$$ in this case. Therefore, we have shown the following
\[lemmasdqorder\] Suppose that $d \mid q = p^{\kappa}$. If $\kappa {\geqslant}2$, $F^{-1}(0,{\boldsymbol}{c}') \neq 0$, and $p \nmid 2\Delta F^{-1}(0,{\boldsymbol}{c}')$, then $S_{d,q}({\boldsymbol}{c}) = 0$. If $\kappa {\geqslant}2$, $p = 2$ we have $$S_{d,q}({\boldsymbol}{c}) \ll_{\Delta} q^{\frac{7}{2}+\frac{1}{4}},$$ and if $p \neq 2$ and $\kappa {\geqslant}2$, we have $$S_{d,q}({\boldsymbol}{c}) \ll_{\Delta} q^{\frac{7}{2}}.$$ Finally, if $(p,2\Delta)=1$ and $q = p$. Then $S_{1,p}({\boldsymbol}{c}) \ll p^2(p,F^{-1}(0,{\boldsymbol}{c}'))$, and $|S_{p,p}({\boldsymbol}{c})| {\leqslant}3p^{3}.$ If $p \mid 2\Delta$, then, $S_{1,p}({\boldsymbol}{c}) \ll_{\Delta} 1$ and $S_{p,p}({\boldsymbol}{c}) \ll_{\Delta} 1.$
Proof of Theorem \[mainthm\]
=============================
It follows from and Lemma \[lemmatruncatec’\] that for any ${\varepsilon}> 0$, $$\begin{split}
N(\lambda;X) = c_QX\sum_{q \ll X}q^{-3}&\sum_{|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}}\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}') \\
&\times \sum_{c_1 \equiv b_1 {\hspace{-2.9mm}\pmod{q}}}\lambda(c_1)I_q({\boldsymbol}{c}) + O(1).
\end{split}$$ Our task now is to show that the right hand side is $o(X^2)$. The analysis of the exponential sum is predicated on the vanishing or non-vanishing of $F^{-1}(0,{\boldsymbol}{c}')$. Define the sets $$\begin{split}
\mathcal{C}_0 &= \left\{{\boldsymbol}{c}' \in \mathbf{Z}^3, |{\boldsymbol}{c}'| \ll X^{{\varepsilon}} : F^{-1}(0,{\boldsymbol}{c}') = 0 \right\}, \\
\mathcal{C}_1 &= \left\{{\boldsymbol}{c}' \in \mathbf{Z}^3, |{\boldsymbol}{c}'| \ll X^{{\varepsilon}} : F^{-1}(0,{\boldsymbol}{c}') \neq 0 \right\}.
\end{split}$$ For $i = 0,1$ let $N^{(i)}(\lambda;X)$ denote the contribution from ${\boldsymbol}{c}' \in \mathcal{C}_i.$ We will show that there exists a $\delta > 0$ such that $N^{(i)}(\lambda; X) \ll X^{2-\delta}$. We start with $N^{(0)}(\lambda;X)$.
Contribution from $N^{(0)}(\lambda;X)$
--------------------------------------
Let $|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$ such that $F^{-1}(0,{\boldsymbol}{c}')=0$. Recall the sum $A_q({\boldsymbol}{c})$ from . Set $$\label{eq:n0flc'}
N^{(0)}(\lambda,{\boldsymbol}{c}';X) = \sum_{q \ll X}q^{-3}\sum_{c_1=1}^{\infty}\lambda(c_1)A_q({\boldsymbol}{c})I_q({\boldsymbol}{c}).$$ Then $N^{(0)}(\lambda;X) = c_QX\textstyle\sum_{{\boldsymbol}{c}' \in \mathcal{C}_0}N^{(0)}(\lambda,{\boldsymbol}{c}';X).$
We begin by writing $q = rs$, a product of coprime integers, as follows. Recalling Condition $A_0$ (Definition \[a0\]), let $$r= \prod_{\substack{p^k \Vert q \\ (p^k;A_1,\ldots,A_4) \text{ satisfies } \\ \text{ Condition } A_0}} p^k$$ be the greatest divisor of $q$ that satisfies Condition $A_0$. By Lemma \[multforaq\] we have $A_q({\boldsymbol}{c}) = A_r(\overline{s}^2c_1,{\boldsymbol}{c}')A_s(\overline{r}^2c_1,{\boldsymbol}{c}')$. Lemma \[gaussevlemma\] shows that $A_r({\boldsymbol}{c})$ vanishes unless $c_i \equiv 0 \pmod{p^{v_p(A_i)}}$, for $2 {\leqslant}i {\leqslant}4$, so without loss of generality, we may assume that ${\boldsymbol}{c}'$ satisfies this condition. Furthermore, Lemma \[replacec’0\] applies to the sum $A_r(\overline{s}^2c_1,{\boldsymbol}{c}')$, by construction of $r$, and observe that $s \ll |\Delta| \ll 1$. As a result we have $$\label{eq:splitintocongruence}
\begin{split}
\sum_{c_1=1}^{\infty}\lambda(c_1)A_q({\boldsymbol}{c})I_q({\boldsymbol}{c}) &= \sum_{\sigma {\hspace{-2.9mm}\pmod{s}}}A_s(\overline{r}^2\sigma,{\boldsymbol}{c}')\sum_{c_1 \equiv \sigma {\hspace{-2.9mm}\pmod{s}}}\lambda(c_1)A_r(\overline{s}^2c_1,{\boldsymbol}{0})I_q({\boldsymbol}{c}) \\
&= \sum_{\sigma {\hspace{-2.9mm}\pmod{s}}}A_s(\overline{r}^2\sigma,{\boldsymbol}{c}')\sum_{c_1 \equiv \sigma {\hspace{-2.9mm}\pmod{s}}}\lambda(c_1)A_r(c_1,{\boldsymbol}{0})I_q({\boldsymbol}{c}) \\
&= \sum_{\sigma {\hspace{-2.9mm}\pmod{s}}}A_s(\overline{r}^2\sigma,{\boldsymbol}{c}')\Sigma_r(\sigma,s),
\end{split}$$ say. We apply Proposition \[sqnproposition\] to $A_r(c_1,{\boldsymbol}{0}) = S_r(c_1)$; let $\theta$ and $\kappa$ be as in the statement of the proposition. Then $\theta, \kappa \mid r$, and we have $$\begin{split}
\Sigma_r(\sigma,s) &= s_r(F)\sum_{\substack{\theta c_1 \equiv \sigma {\hspace{-2.9mm}\pmod{s}} \\ (c_1,\kappa)=1}}\lambda(\theta c_1)I_q(\theta c_1,{\boldsymbol}{c}') \\
&= s_r(F)\sum_{\substack{c_1 \equiv \overline{\theta}\sigma {\hspace{-2.9mm}\pmod{s}}\\ (c_1,\kappa)=1}} \lambda(\theta c_1)I_q(\theta c_1,{\boldsymbol}{c}').
\end{split}$$ Clearing denominators, and using multiplicative characters to cut out the congruence condition $c_1 \equiv \overline{\theta}\sigma \pmod{s}$, we see that $$\label{eq:complicatedsum1}
\Sigma_r(\sigma,s) = \frac{s_r(F)}{{\varphi}(\hat{s})}\sum_{\chi {\hspace{-2.9mm}\pmod{\hat{s}}}}\overline{\chi}(\overline{\theta}\widetilde{\sigma})\sum_{(c_1,\kappa)=1}\chi(c_1)\lambda((\sigma,s)\theta c_1)I_q((\sigma,s)\theta c_1,{\boldsymbol}{c}'),$$ where $\hat{\sigma} = \sigma/(\sigma,s)$ and $\widetilde{\sigma} = \sigma/(\sigma,s)$. To analyse the inner sum, we need the following
\[newprop\] Let $\chi$ be a Dirichlet character modulo $D$, and let $\theta,\kappa$ be positive integers. Then there exists $A > 0$, such that for all ${\varepsilon}>0$ we have $$\sum_{(n,\kappa)=1} \chi(n)\lambda(\theta n)I_q(\theta n, {\boldsymbol}{c}') \ll_{{\varepsilon}} (\theta \kappa)^{{\varepsilon}}D^A\frac{X^{5/6+{\varepsilon}}}{\theta^{\frac{1}{2}}q^{\frac{1}{3}}}.$$
Let $S(\chi,\theta)$ be the sum in question. Define the Dirichlet series $$F_{\chi,\theta}(s) = \sum_{(n,\kappa)=1}^{\infty}\frac{\chi(n)\lambda(\theta n)}{n^s}.$$ Since $f$ is a newform, $$\lambda(mn) = \sum_{d \mid (m,n)}\mu(d)\lambda(m/d)\lambda(n/d).$$ As a result, for $\sigma >1$ $$\label{eq:factorisation}
\begin{split}
F_{\chi,\theta}(s) &= \sum_{\substack{\beta \mid \theta \\ (\beta,\kappa=1)}}\frac{\mu(\beta)\chi(\beta)\lambda(\theta/\beta)}{\beta^s}\sum_{(n,\kappa)=1} \frac{\chi(n)\lambda(n)}{n^s} \\
&= P(\chi,\theta,\kappa)L(s,f\otimes \chi),
\end{split}$$ where $$P(\chi,\theta,\kappa) = \prod_{\substack{p^l \Vert \theta \\ (p,\kappa)=1}} \left(\lambda(p^l) - \frac{\chi(p)\lambda(p^{l-1})}{p^s}\right)\prod_{p \mid \kappa}\left(1-\frac{\lambda(p)\chi(p)}{p^s} + \frac{\chi^2(p)}{p^{2s}}\right).$$ Recall from that $L(s,f\otimes \chi)$ has an Euler product and if $\chi^*$ is the primitive character, of conductor $D^*$ say, that induces $\chi$, observe that $$L(s, f\otimes \chi) = \prod_{p \mid D/D^*}\left(1-\frac{\lambda(p)\chi^{*}(p)}{p^s} + \frac{\chi^*(p)^2}{p^{2s}}\right)L(s, f\otimes \chi^*).$$ Applying to $L(s,f\otimes \chi^{*})$ for $\frac{1}{2} {\leqslant}\sigma {\leqslant}1$ we get that $$\label{eq:phlind}F_{\chi,\theta}(s) \ll_{{\varepsilon}} (b\kappa)^{{\varepsilon}}(D{^*}^2(1+|t|))^{1-\sigma+{\varepsilon}},$$ and by we get that $$\label{eq:weylforf}
F_{\chi,\theta}(s) \ll_{{\varepsilon}} (\theta\kappa)^{{\varepsilon}}(D{^*}^A(1+|t|)^{\frac{1}{3}+{\varepsilon}},$$ when $\sigma = \frac{1}{2}.$ Recall the integral $I_q({\boldsymbol}{c}',s)$ from . By the Mellin inversion theorem, we have for any $\sigma > 1$ that $$S(\chi,\theta) = \frac{1}{2\pi i}\int_{(\sigma)} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds.$$ Next, we move the line of integration to $\sigma = \frac{1}{2}$ and use . To this end, fix ${\varepsilon}> 0$ and set $T=r^{-1}X^{{\varepsilon}}$. By we have $$\begin{split}
\int_{(\sigma)} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds &= \int_{\sigma-iT}^{\sigma+iT} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds \, + \\
&\quad O\left(X^{\sigma}r^{-N-1}\int_{|t| {\geqslant}T}|t|^{-N} \, dt\right).
\end{split}$$ The error term is $$O\left(X^{\sigma}r^{-2}X^{(1-N){\varepsilon}}\right).$$ Choosing $N$ large enough we get that $$\begin{split}
\int_{(\sigma)} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds &= \int_{\sigma-iT}^{\sigma+iT} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds \, + O_N(X^{-N}).
\end{split}$$ By and the horizontal integrals are bounded as follows, $$\begin{split}
\int_{\sigma\pm iT}^{\frac{1}{2} \pm iT} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q(s) \, ds
&\ll X^{\sigma-(N-2){\varepsilon}}r^{-\frac{1}{2}}.
\end{split}$$ Once again, choosing $N$ large enough, we get that $$\int_{(\sigma)} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds = \int_{\frac{1}{2}-iT}^{\frac{1}{2}+iT} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds \, + O_N(X^{-N}).$$ By Lemmas \[lemmaczero\] and \[csmall\], and we have $$\begin{split}
\int_{\frac{1}{2}-iT}^{\frac{1}{2}+iT} \left(\frac{X}{\theta}\right)^s F_{\chi,\theta}(s)I_q({\boldsymbol}{c}',s) \, ds
&\ll_{{\varepsilon}} (\theta\kappa)^{{\varepsilon}}D^A\left(\frac{X}{\theta}\right)^{\frac{1}{2}}T^{\frac{1}{3}+{\varepsilon}}\int |I_q({\boldsymbol}{c}',s)| \, ds \\
&\ll_{{\varepsilon}} (\theta\kappa)^{{\varepsilon}}D^A\left(\frac{X}{\theta}\right)^{\frac{1}{2}}T^{\frac{1}{3}+{\varepsilon}} \ll_{{\varepsilon}} (\theta\kappa)^{{\varepsilon}}D^A\frac{X^{\frac{5}{6}+{\varepsilon}}}{\theta^{\frac{1}{2}}q^{\frac{1}{3}}}.
\end{split}$$ This completes the proof of the proposition.
Applying Propositions \[newprop\] and \[sqnproposition\] to the inner sum in we get that $$\Sigma_r(\sigma,s) \ll_{{\varepsilon}} |s_r(F)|X^{\frac{5}{6}+{\varepsilon}},$$ since $\operatorname{cond}(\chi) \ll s \ll_\Delta 1$. As a result, we have get $$\begin{split}
N^{(0)}(\lambda,{\boldsymbol}{c}';X) &\ll \sum_{q \ll X} \frac{|s_r(F)|X^{\frac{5}{6}+{\varepsilon}}}{q^{3}},
\end{split}$$ where $r \mid q$ is the largest divisor of $q$ that such that $(r;A_1,\ldots,A_4)$ satisfies Condition $A_0$. Let $q = uv$ where $u$ is square-free and $v$ is square-full. Since $s_q(F)$ is multiplicative, we have by Proposition \[sqnproposition\] that $$\begin{split}
N^{(0)}(\lambda,{\boldsymbol}{c}';X) &\ll X^{\frac{5}{6}+{\varepsilon}}\sum_{v \ll X} v^{-\frac{1}{2}}\sum_{u \ll X/v}\frac{1}{u} \\
&\ll_{{\varepsilon}} X^{\frac{5}{6}+{\varepsilon}},
\end{split}$$ since the number of square-full $v {\leqslant}X$ is $O(X^{\frac{1}{2}}).$ Summing over ${\boldsymbol}{c}' \in \mathcal{C}_0$ we obtain the bound $$\label{eq:n0lambdax}
N^{(0)}(\lambda;X) \ll_{{\varepsilon}} X^{2-\frac{1}{6}+{\varepsilon}}.$$
Contribution from $N^{(1)}(\lambda;X)$
--------------------------------------
Next we examine $N^{(1)}(\lambda;X)$. For ${\boldsymbol}{c}' \in \mathcal{C}_1$ define $$\begin{split}
N_q^{(1)}(\lambda,{\boldsymbol}{c}';X) &= \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}e_q(aF({\boldsymbol}{b})+{\boldsymbol}{b}'.{\boldsymbol}{c}')\\ &\quad \quad \times \sum_{c_1 \equiv b_1 {\hspace{-2.9mm}\pmod{q}}}\lambda(c_1)I_q({\boldsymbol}{c}).
\end{split}$$ Then $$N^{(1)}(\lambda;X) = c_QX\sum_{{\boldsymbol}{c}' \in \mathcal{C}_1}\sum_{q\ll X}q^{-3}N_q^{(1)}(F,\lambda,{\boldsymbol}{c}').$$ By and Lemma \[truncatem1\] we see that $$\label{eq:nq1}
N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)= \frac{X}{q}\sum_{d \mid q}\sum_{c_1 \ll X^{1+{\varepsilon}}/(q/d)^2}\lambda(c_1) S_{d,q}({\boldsymbol}{c})I_{d,q}({\boldsymbol}{c}) + O_N(X^{-N}).$$ Write $q=uv$ where $(u,2\Delta)=1$ is square-free, and $v$ is square-full and is composed of primes dividing $N=2|\Delta F^{-1}(0,{\boldsymbol}{c}')|$. Further, decompose $v = v_0v_1$ where $v_0$ is the $2^{\infty}$-part of $v$. By Lemma \[lemmasdqorder\] we have $$S_{d,q}({\boldsymbol}{c}) \ll (u,F^{-1}(0,{\boldsymbol}{c}'))u^{3+{\varepsilon}}v_0^{\frac{15}{4}}v_1^{\frac{7}{2}}.$$ Applying Lemma \[stphasehb\] to estimate $I_{d,q}({\boldsymbol}{c}),$ obtain the bound $$\begin{split}
\sum_{c_1 \ll X^{1+{\varepsilon}}/(q/d)^2} |\lambda(c_1)I_{d,q}({\boldsymbol}{c})| &\ll X^{{\varepsilon}}\left(\frac{q}{d^{\frac{1}{2}}X^{\frac{5}{4}}}\right)\sum_{c_1 \ll X^{1+{\varepsilon}}/(q/d)^2}c_1^{-\frac{1}{4}} \\
&\ll X^{{\varepsilon}}\left(\frac{q}{d^{\frac{1}{2}}X^{\frac{5}{4}}}\right)\left(1+\frac{X^{\frac{3}{4}}}{(q/d)^{\frac{3}{2}}}\right) \\
&\ll \frac{qX^{{\varepsilon}}}{X^{\frac{5}{4}}} + \frac{q^{\frac{1}{2}}X^{{\varepsilon}}}{X^{\frac{1}{2}}}.
\end{split}$$ Inserting our bound for $S_{d,q}({\boldsymbol}{c})$ into we have shown
\[pr1\] Suppose that $1 {\leqslant}|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$. With notation as above, we have $$N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)\ll_{{\varepsilon}} u^3v_0^{\frac{15}{4}}v_1^{\frac{7}{2}}\left(\frac{1}{X^{\frac{1}{4}}} + \frac{X^{\frac{1}{2}}}{q^{\frac{1}{2}}}\right)X^{{\varepsilon}}.$$
We can also estimate the sum over $c_1$ in using partial summation: employing additive characters to detect the congruence condition $c_1 \equiv b_1 \pmod{q}$ we have $$\begin{split}
N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)&= \frac{1}{q}\sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{\substack{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}} \\ r {\hspace{-2.9mm}\pmod{q}}}}e_q(aF({\boldsymbol}{b})+b_1r+{\boldsymbol}{b}'.{\boldsymbol}{c}')\sum_{c_1=1}^{\infty}\lambda(c_1)e_q(-rn)I_q({\boldsymbol}{c}) \\
&{\leqslant}\frac{1}{q}\sum_{r {\hspace{-2.9mm}\pmod{q}}}\lvert \sideset{}{^*}\sum_{a {\hspace{-2.9mm}\pmod{q}}}\sum_{\substack{{\boldsymbol}{b} {\hspace{-2.9mm}\pmod{q}}}}e_q(aF({\boldsymbol}{b})+b_1r+{\boldsymbol}{b}'.{\boldsymbol}{c}')\rvert \\
&\quad \quad \quad \quad \times \lvert\sum_{c_1=1}^{\infty}\lambda(c_1)e_q(-rc_1)I_q({\boldsymbol}{c})\rvert.
\end{split}$$ By Lemma \[lemmastationaryphase\] we have $$\begin{split}
\sum_{c_1=1}^{\infty}\lambda(c_1)e_q(-rn)I_q({\boldsymbol}{c})&= - \int \sum_{c_1 {\leqslant}x} \lambda(c_1)e_q(-rc_1) \frac{\partial}{\partial x}I_q(x,{\boldsymbol}{c}')\, dx \\
&\ll \frac{(r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}|{\boldsymbol}{u}'|^{-\frac{1}{2}}}{qX}\int_{1}^X x^{\frac{3}{2}}\log x \, dx \, + \\
&\quad \quad \frac{(r^{-1}|{\boldsymbol}{u}'|)^{{\varepsilon}}|{\boldsymbol}{u}'|^{-\frac{1}{2}}}{X}\int_{1}^X x^{\frac{1}{2}}\log x \, dx \\
&\ll_{{\varepsilon}} \frac{X^{1+{\varepsilon}}}{q^{\frac{1}{2}}} + q^{\frac{1}{2}}X^{{\varepsilon}},
\end{split}$$ using the bound $$\sum_{n {\leqslant}z} \lambda(n)e(\alpha n) \ll_f z^{\frac{1}{2}}\log z,$$ which is uniform in $\alpha$. Recall $T_q$ from . We have shown
\[pr2\] Suppose that $1 {\leqslant}|{\boldsymbol}{c}'| \ll X^{{\varepsilon}}$ and $F^{-1}(0,{\boldsymbol}{c}') \neq 0$. Then, $$N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)\ll_{{\varepsilon}} \left(\frac{X^{1+{\varepsilon}}}{q^{\frac{3}{2}}} +\frac{X^{{\varepsilon}}}{q^{\frac{1}{2}}}\right)T_q.$$
With Propositions \[pr1\] and \[pr2\] in place, we can complete our analysis of $N^{(1)}(\lambda;X)$. Let $1 {\leqslant}Y \ll X$ be a parameter to be chosen later. Then $$N^{(1)}(F,\lambda,{\boldsymbol}{c}') = \sum_{q {\leqslant}Y}q^{-3} N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)+ \sum_{q > Y}q^{-3}N_q^{(1)}(\lambda,{\boldsymbol}{c}';X).$$ Using Proposition \[pr1\] to estimate the sum up to $Y$, we get $$\begin{split}
\sum_{q {\leqslant}Y}q^{-3} N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)\ll X^{\frac{1}{2}+{\varepsilon}}\sum_{v \ll Y}v^{-\frac{7}{2}}v_0^{\frac{15}{4}}v_1^{\frac{7}{2}}\sum_{u \ll Y/v}u^{-\frac{1}{2}} \ll (XY)^{\frac{1}{2}+{\varepsilon}},
\end{split}$$ since $$\sum_{\substack{v \ll Y \\ p \mid v \implies p \mid N}} 1 \ll_{{\varepsilon}} (NY)^{{\varepsilon}}.$$ Applying Proposition \[pr2\] to the second sum, we get by Lemma \[c’0\] that $$\sum_{q > Y}q^{-3}N_q^{(1)}(\lambda, {\boldsymbol}{c}';X)\ll_{{\varepsilon}} X^{1+{\varepsilon}}Y^{-\frac{1}{2}} + X^{\frac{1}{2}+{\varepsilon}}.$$ The optimal choice for $Y$ is $Y = X^{\frac{1}{2}},$ and this gives us $$\label{eq:n1}
N^{(1)}(\lambda;X) \ll_{{\varepsilon}} X^{1+\frac{3}{4}+{\varepsilon}}.$$ Combined with this completes the proof of Theorem \[mainthm\].
Deduction of Theorem \[cor1\] from Theorem \[mainthm\]
========================================================
This follows by way of a standard argument in moving from estimates for sums with a smooth cut-off to sums with a sharp cut-off. For the sake of brevity, we provide a brief outline of the proof. Let $1 {\leqslant}P {\leqslant}X$ be a parameter that we will choose later, and let $\alpha(x)$ be a non-negative, smooth function with support in $[1,X+X/P]$ such that $\alpha(x) =1$ on $[X/P,X]$ with derivatives satisfying $\alpha^{(j)}(x) \ll_j P^j/x^j$ for all $j {\geqslant}0.$ Let ${\varepsilon}>0$. Observe that $$\sum_{m,n {\leqslant}X}r(Am^2+Bn^2)\lambda(m) = \sum_{m,n}\alpha(m)\alpha(n)\lambda(m)r(Am^2+Bn^2) + O_{{\varepsilon}}(X^{2+{\varepsilon}}/P).$$ Applying a smooth partition of unity, it suffices to consider the sum $$\sum_{m,n}W(m/X)W(n/X)r(Am^2+Bn^2)\lambda(m)$$ with $\operatorname{supp}(W) \in [X/2,X]$, satisfying $W^{(j)}(x) \ll_j P^j/X^j.$ Notice that $$\begin{split}
\sum_{m,n}W(m/X)W(n/X)&r(Am^2+Bn^2)\lambda(m) = \\ &\sum_{\substack{u,v \in \mathbf{Z} \\ Am^2+Bn^2-u^2-v^2=0}}W(m/X)W(n/X)\lambda(m).
\end{split}$$ Applying a smooth partition of unity for the $u,v$ variables, we are left with the sum $$\sum_{F({\boldsymbol}{x})=0}w(X^{-1}{\boldsymbol}{x})\lambda(x_1),$$ where $w$ is a smooth function with support in $[1/2,2]$ and with Sobolev norm $\|w\|_{N,1} \ll_N P^N,$ and $F({\boldsymbol}{x}) = Ax_1^2+Bx_2^2-x_3^2-x_4^2.$ It can be shown that the error term in Theorem \[mainthm\] depends polynomially on $\|w\|_{1,1}$ (see, for instance, [@K Proposition 2.1]), and as a result, we obtain $$\sum_{F({\boldsymbol}{x})=0}w(X^{-1}{\boldsymbol}{x})\lambda(x_1) \ll_{{\varepsilon}} P^8X^{2-\frac{1}{6}+{\varepsilon}}.$$ The optimal choice for $P$ is $P=X^{\frac{1}{54}-{\varepsilon}},$ and the theorem follows.
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abstract: 'The magnetic moment per Co$^{4+}$ of P2($\gamma$)-type Na$_x$CoO$_2$ in Curie-Weiss metal regime is revisited and examined under a newly proposed modification of the octahedral crystal electric field (CEF). The proposed model explains the origin of the existence of the intermediate state S=1 for Co$^{3+}$ through an exciton-like elementary excitation of the narrow gap between the split $e_g$ and $t_{2g}$ groups. The CoO$_2$ layer is proposed to be constructed from the tilted edge-sharing square-planar CoO$_2$ chains with inter-chain coupling. The square-planar CEF of CoO$_2$ requires covalent bond formation between Co and the four in-plane neighboring oxygens, while the oxygens sitting in the neighboring chains can be viewed as apical oxygens of low effective charge for a CoO$_6$ pseudo-octahedron. The reason why angle-resolved photoemission spectroscopy (ARPES) failed to observe the local-density approximation (LDA) predicted $e_g''$ hole pockets at $k_z$=0 and the reversed order of the $t_{2g}$ splittings between LDA and CEF calculations can also be resolved using the proposed model.'
author:
- 'G. J. Shu$^1$'
- 'F. C. Chou$^{1,2}$'
title: |
Revisiting the crystal electric field of P2-$Na_xCoO_2$\
for the intermediate spin state of Co$^{3+}$
---
 The crystal structure of Na$_x$CoO$_2$ is presented with the CoO$_2$ and Na layers in the space group $P6_3/mmc$. The conventional electronic configuration of Co$^{3+}$ (3d$^6$) in an octahedral crystal field splits into $e_g-t_{2g}$ groups, and the additional splitting of $t_{2g}$ ($e_g'-a_{1g}$) is due to trigonal distortion. In contrary to the spectrum of lower $a_{1g}$ predicted from the CEF calculation, the $a_{1g}$ is higher based on LDA calculations.[@Singh2000; @Landron2008]](fig-NaxCoO2){width="3.5in"}
Layered P2-type (or $\gamma$-type) Na$_x$CoO$_2$ is an important class of material for both fundamental and applied aspects, e.g., it is a good candidate thermoelectric material with a high thermoelectric figure-of-merit (ZT) at high temperature;[@Terasaki1997] has a great similarity to the important battery cathode electrode material Li$_x$CoO$_2$;[@Hertz2008], has superconductivity, as observed in Na$_{1/3}$CoO$_2$ after hydration;[@Takada2003] and has an intriguing Curie-Weiss metal behavior as a strongly correlated electron material.[@Chou2004] The crystal structure of Na$_x$CoO$_2$ can be viewed as a Na layer sandwiched between CoO$_2$ triangular lattice layers as illustrated in Fig. \[fig-NaxCoO2\]. A 2D triangular lattice of CoO$_2$ formed by the closely packed CoO$_6$ octahedra is commonly observed in many cobalt oxide compounds in contrast to the high T$_c$ cuprate materials, which contain a 2D square lattice of CuO$_2$. The CoO$_2$ layer has been the basic building block of many layered cobalt oxide compounds, including Na$_x$CoO$_2$, Pb$_2$Sr$_2$Co$_2$O$_y$, and Co$_3$Co$_4$O$_9$, of various types of staging and the unavoidable incommensurability is due to layer mismatch.[@Sootsman2009] The physical properties of materials with a hole-doped CoO$_2$ layer have often been interpreted from the mixed valence of Co$^{3+}$ and Co$^{4+}$. However, the early d-orbital energy levels for Co under the octahedral crystal field assumption (see Fig. \[fig-NaxCoO2\]) cannot be used to reasonably interpret the Curie-Weiss metal behavior, especially concerning the puzzling questions of why the theoretically predicted $e_g'$ hole pockets have not been observed experimentally, and why the $t_{2g}$ splitting ($a_{1g}-e_g'$) is reversed between predictions from LDA and CEF calculations.[@Zhou2005; @Landron2008]
The existence of an intermediate spin (S=1) for Co$^{3+}$ has been proposed from the ellipsometry experimental results for Na$_{0.82}$CoO$_2$.[@Bernhard2004] Daghofer *et al.* proposed that Hund’s rule coupling across a narrow gap between the split $e_g$ and $t_{2g}$ groups could permit the existence of S=1 within the proposed spin-orbit-polaron model, i.e., the intermediate spin (IS) state of S=1 for Co$^{3+}$ may exist in addition to the original low spin (LS) state of S=0.[@Daghofer2006] However, in view of the large gap in $\Delta$ of $\sim$1 eV between $e_g-t_{2g}$,[@Johannes2005] it is difficult to construct one octahedral CEF that has an actual severe distortion. That is, the oxygens must be removed from the original apical positions, which is structurally or chemically nearly impossible due to the CoO$_2$ layer formed by the closely packed edge-sharing octahedra. In addition, the energy levels of $t_{2g}$ splitting predicted from the *ab initio* calculations are strangely reversed relative to the CEF prediction.[@Landron2008] Landron *et al.* have carefully examined this issue but failed to uncover connections with the metal-ligand hybridization, the long-range crystalline field, the screening effects, and the orbital relaxation; the tentative conclusion still vaguely points to the mixing of $e_g$ and $t_{2g}$.[@Landron2008]
 Average spin susceptibilities of Na$_x$CoO$_2$ (x=0.67-0.82) in a 1 Tesla applied field. The inset shows normalized $1/(\chi$-$\chi_{\circ}$) vs. (T-${\Theta}$) to illustrate the x-dependence of the Curie constant by the different slopes corresponding to 1/C. ](fig-chiT){width="3.5in"}
x 0.67 0.71 0.75 0.82
------------------------------------- ----------- ----------- ----------- -----------
fitting range 60-300K 65-300K 60-300K 125-300K
$\chi_\circ$ (cm$^3$/mole) 0.0004(5) 0.0003(7) 0.0003(6) 0.0001(5)
C (cm$^3$-K/mole) 0.0675(5) 0.1082(1) 0.1311(3) 0.1423(2)
$\Theta$(K) -103.5(7) -70.2(5) -106.1(7) -102.5(3)
$\mu_{eff}$ ($\mu_B$ per Co) 0.7350 0.9305 1.0243 1.0671
$\mu_{eff}$ ($\mu_B$ per Co$^{4+}$) 1.279 1.728 2.049 2.515
$\alpha$ -0.084\* 0.00068 0.049 0.091
: \[table-mueff\] Curie-Weiss law, $\chi(T)=\chi_\circ + C/(T-\Theta)$, fitting for polycrystalline samples of $Na_xCoO_2$ in an applied field H= 1 Tesla.
\*This value is unphysical, which implies that the itinerant contribution is no longer negligible.
We have revisited this issue experimentally through the analysis of the effective magnetic moment for a series of Na$_x$CoO$_2$ samples in the Curie-Weiss metal regime. Single crystal samples of Na$_x$CoO$_2$ of x$\sim$0.8 were grown using the optical floating-zone method in an oxygen atmosphere. The Na contents of the single crystal samples were fine-tuned to x=0.67, 0.71, 0.75, and 0.82 using an electrochemical intercalation technique and verified with electron microprobe analysis (EPMA), within an error of $\pm$0.01, as described previously.[@Shu2007] Average spin susceptibility ($\chi$(T)=M/H) data were acquired from a pulverized powder sample under an applied magnetic field of 1 Tesla in the temperature range of 5-300K, as shown in Fig. \[fig-chiT\]. The spin susceptibilities can be fitted satisfactorily with the Curie-Weiss law, $\chi(T)=\chi_\circ + C/(T-\Theta)$, in the paramagnetic regime and are summarized in Table \[table-mueff\]. Assuming the amount of Co$^{4+}$ ($(t_{2g})^5$) with S=1/2 can be determined directly from the Na content x without oxygen vacancies, it is expected that the $\mu_{eff}$ values per Co$^{4+}$ should be 1.732 $\mu_B$ with g=2. However, it is clear that except for the sample of x=0.71, the $\mu_{eff}$ per Co$^{4+}$ values are progressively and significantly higher than 1.732 $\mu_B$ for x $>$ 0.71, as indicated in Table \[table-mueff\]. Similar results have also been reported consistently in all early published works,[@Chou2004; @Wang2003; @Luo2004; @Rhyee2008] including for Li$_x$CoO$_2$, which has a similar CoO$_2$ 2D triangular lattice.[@Hertz2008] While a Na content higher than $\sim$0.71 is expected to contain a lower level of Co$^{4+}$ and less doped itinerant holes, the increasing Curie constant remains unclear.
{width="4.5in"}
While the CoO$_2$ layer of Na$_x$CoO$_2$ has been viewed as closely packed edge-sharing octahedra that forms a 2D triangular lattice, the distortion from the flattened CoO$_2$ layer along the rhombohedral (111) axis requires the distortion of octahedral CEF, whereupon the $t_{2g}$ degeneracy is lifted. The octahedral distortion has been reflected on the subtle difference of Co-O bond lengths and O-Co-O bond angles in CoO$_6$ through synchrotron X-ray structure refinement before.[@Huang2009] In the crystal field theory for elongated octahedral CEF, as illustrated in Fig. \[fig-CEF\], $e_g$ splits into $b_{1g}$-$a_{1g}$ and $t_{2g}$ splits into $b_{2g}$-$e_g'$, where“a/b", “e", and “t" denote single, double, and triple degeneracy in the convention of group theory, respectively.[@Moore] The two groups of the $e_g$ doublet and $t_{2g}$ triplet have a large gap of a $\Delta_{oct}$ of $\sim$1 eV, and the elongation of the apical Co-O distance lifts the degeneracy of both $e_g$ and $t_{2g}$ by lowering the z-related levels within each group. The extreme case of such a distortion can be observed in the square-planar CEF that is derived by removing apical oxygens to infinity, which lowers the z-related levels significantly, as demonstrated in Fig. \[fig-CEF\](d). However, since the first published LDA band calculation for Na$_x$CoO$_2$, the $t_{2g}$ splitting has been commonly mislabeled as $a_{1g}$-$e_g'$ in the physics community (see Fig. \[fig-NaxCoO2\]),[@Singh2000] and $a_{1g}=(d_{xy}+d_{yz}+d_{zx})/\sqrt{3}$ and $e_g'=[(d_{zx}-d_{yz})/\sqrt{2}, (2d_{xy}-d_{yz}-d_{zx})/\sqrt{6}]$ after an axis transformation of the principal z-axis from the apical oxygen direction to the (111) direction in rhombohedral symmetry.[@Zhou2005; @Landron2008] Note that the relative positions of the ligand does not change after axis transformation, and an identical $t_{2g}$ splitting, as a result of octahedral CEF distortion, can be described for any selection of an axes system. The most common distortion in the rhombohedral symmetry description is the thickness change of the CoO$_2$ layer, which can be roughly represented by a small elongation from the original perfect CoO$_6$ octahedron.
 The CoO$_2$ plane can also be viewed as edge-sharing square-planar chains with inter-chain coupling instead of edge-sharing octahedra. The pseudo-octahedron of CoO$_6$ and Co-O-Co at 90$^\circ$ are shown in the insets.](fig-chain){width="3.5in"}
If the CoO$_2$ layer is viewed as closely packed edge-sharing octahedra, it is impossible to create one distorted octahedral CEF by adjusting the apical Co-O bond lengths alone, i.e., to create a distortion severe enough for both the $e_g$ and $t_{2g}$ groups to split wide and leave one small gap between them. Instead of assuming that the CoO$_2$ layer which is composed of edge-sharing octahedra, we propose an alternative view; the CoO$_2$ layer could be composed of edge-sharing square-planar CoO$_2$ chains with nontrivial inter-chain coupling, as illustrated in Fig. \[fig-chain\]. There are two main reasons to make such an assumption; the first is because the earlier view of octahedral CEF with trigonal distortion failed to provide a reasonable argument to allow the formation of a narrow gap between the $e_g$ and $t_{2g}$ groups, and the second is that each oxygen with two electrons missing from its $2p$-orbital can form, at most, two covalent bonds with the neighboring Co atoms in a 90$^\circ$ coordination, as illustrated in the inset of Fig. \[fig-chain\]. Chemically, it is impossible for oxygen to form three covalent bonds with the neighboring three Co atoms, as implied in the earlier octahedral picture.
The modified view of CEF is effectively in between an elongated octahedral CEF and a square-planar CEF. It is possible that the Co-O only has four covalent $\sigma$-bonds within each square-planar CoO$_2$ unit, and the apical oxygens should not be viewed as individual ligands but as weak effective charges dressed by the neighboring edge-sharing chains, as illustrated in Fig. \[fig-chain\]. The CoO$_2$ plane originally assumed to be in edge-sharing octahedra is now viewed as edge-sharing square-planar chains with inter-chain coupling, i.e., all of the oxygens in each square-planar edge-sharing chain also serve as the effective apical oxygen for the neighboring square-planar CoO$_2$ unit to form a pseudo-octahedron. The actual crystal field can now be understood as a unique octahedral CEF with apical oxygens of a much less effective charge. In fact similar energy splitting can also be generated through the addtional magnetic couplings of $J_{diag}$ and $J'$ per polaron unit defined in the calculations by Daghofer *et al.*[@Daghofer2006] Starting from the square-planar CEF, the effective charge of a distorted octahedral CEF may be formed when the $a_{1g}$ (d$_{z^2}$) level is raised slightly above the $b_{2g}$ (d$_{xy}$) level. A clear level inversion between $a_{1g}$ and $b_{2g}$ between the square-planar and elongated octahedral CEF occurs once the effective charge of the apical oxygen is properly tuned through the inter-chain coupling .
 The low spin (LS) state and intermediate spin (IS) state for Co$^{3+}$ (3d$^6$) derived from the proposed CEF. Thermal or strain energy at ambient temperature would allow activation between $a_{1g}$-$b_{2g}$ of the narrow gap $\delta$ on the order of $\sim$10 meV.](fig-spin){width="3.5in"}
Following our proposed model, which leads to the possible existence of a narrow gap between the $e_g$ and $t_{2g}$ groups, i.e., the small gap ($\delta$) between $a_{1g}$-$b_{2g}$ shown in Fig. \[fig-CEF\](c), we may re-examine the spins of the 3d electrons in the Co ions. It is commonly accepted that the LS states of Co$^{3+}$ ($3d^6$) and Co$^{4+}$ ($3d^5$) are S=0 and S=1/2, respectively. In the newly constructed CEF with a narrow gap between $a_{1g}$-$b_{2g}$ of $\sim$10 meV,[@Zhou2005] the thermal or stain energy at ambient temperature would be sufficient to activate electrons from the filled $b_{2g}$ to the empty $a_{1g}$, as illustrated in Fig. \[fig-spin\]. In fact, Hund’s rule coupling is an alternative description for the existence of IS state when the pairing energy is able to overcome the small CEF gap.[@Daghofer2006] For the IS state (S=1) of Co$^{3+}$ generated through the activation process, the quantity and positions of S=1 must fluctuate in a Boltzmann distribution of population and at random positions. It is reasonable to assume that the Curie constant should change as a function of temperature, i.e., increased S=1 should be observed at higher temperatures because of increased S=1 activation across the narrow $a_{1g}-b_{2g}$ gap for Co$^{3+}$.
 Temperature dependence of Curie constants for x=0.82 and 0.71, where Curie-Weiss law fittings were performed using the selected temperature sections of $\pm$50K centered between 100-250K. The inset shows Arrhenius plot for the number of activated S=1 state of Co$^{3+}$ ($\alpha$(T)) with activation energy ($\delta$) between $a_{1g}-b_{2g}$ fitted to be $\sim$17 meV. ](fig-CT){width="3.5in"}
The Curie-Weiss law for the localized spins is an approximation of Brillouin function under the condition of $\mu H/k_B T \ll1$, i.e., the Curie constant should be temperature independent at low field and high temperature regime. As a double check to the validity of the applied Curie-Weiss law fitting, the fitted Curie constant for x=0.71 does show a temperature independent value at high temperature above $\sim$150K (see Fig. \[fig-CT\]), and only deviates at low temperature when the $\mu H/k_B T\ll1$ condition is violated. On the other hand, we find that the Curie constant has a subtle temperature dependence when the Curie-Weiss law is fitted using different temperature sections, especially for x=0.82 as shown in Fig. \[fig-CT\]. The temperature dependence of C(T) (C=$N\mu_{eff}^2/3k_B$) could be coming from the $N(T)$ through added number of localized spins, or from the $\mu_{eff}(T)$ through added couplings that enhance the size of the magnetic moment of the localized spins effectively. T. Moriya has shown that the spin fluctuation of FM coupled itinerant electrons in the strongly correlated system can also be described with a Curie’s law defined in parallel to that of the localized spins.[@MoriyaBook] While we cannot rule out that the observed Curie constant has contribution from the correlated itinerant electrons completely, especially from the RKKY type coupling of itinerant electrons to the localized spins of S=1/2 for Co$^{4+}$,[@Balicas2008] the impact of these couplings should be diminishing at higher temperature. However, the increasing C(T) as a function of temperature for x=0.82 (see Fig. \[fig-CT\]) does not support, or at least implies that the scenario of coupling enhanced effective moment cannot be the dominant factor. Quite the contrary, the increasing C(T) suggests that more localized spins are added to the system at higher temperature, which agrees with a picture that more spins of S=1 are generated from the thermally activated IS state of Co$^{3+}$.
The confusing excess magnetic moment per Co$^{4+}$ beyond S=1/2 presented in Table \[table-mueff\] can now be analyzed quantitatively using the proposed model. Using $Na_{0.82}CoO_2$ as an example, the Curie constant (C=N$\mu_{eff}^2$/3$k_B$) has three possible contributions from the LS state of Co$^{4+}$ with S=1/2 ($\mu_{Co^{4+}}^{LS}$=1.732$\mu_B$), the LS state of Co$^{3+}$ with S=0 ($\mu_{Co^{3+}}^{LS}$=0), and the IS state of Co$^{3+}$ with S=1($\mu_{Co^{3+}}^{IS}$=2.828$\mu_B$). Under the constraint of total N=N$_{Co^{4+}}^{LS}$+N$_{Co^{3+}}^{LS}$+N$_{Co^{3+}}^{IS}$, the $\mu_{eff}$ per Co ion can be analyzed with a modified Curie-Weiss law ($\chi = \chi_\circ + \frac{C}{T-\Theta}$) of $$\begin{aligned}
C= \frac{N_{Co^{4+}}^{LS}(\mu_{Co^{4+}}^{LS})^2+N_{Co^{3+}}^{LS}(\mu_{Co^{3+}}^{LS})^2+N_{Co^{3+}}^{IS}(\mu_{Co^{3+}}^{IS})^2}{3k_B}.
\label{eq:one}\end{aligned}$$
We may estimate the fraction ($\alpha$) of Co$^{3+}$ at the activated IS state (S=1) at any instance from $$\begin{aligned}
\mu_{eff}^2 = (1-0.82)\times1.732^2+0.82[(1-\alpha)\times0^2+\alpha\times2.828^2].
\label{eq:one}\end{aligned}$$
The fractions of Co$^{3+}$ at S=1, $\alpha$, is estimated to be nearly 9$\%$ under 1 Tesla in the temperature range of 125-300K (see Table \[table-mueff\]), which quantitatively agrees with a description that at most one Co$^{3+}$ in a Na di-vacancy formed $\sqrt{13}$a superlattice (1-$\frac{11}{13}$$\sim$9$\%$) is activated, i.e., for each di-vacancy formed supercell with 2 Co$^{4+}$ at the corners and 11 Co$^{3+}$ in the middle, only one of the Co$^{3+}$ (random in time and position) is activated from S=0 to S=1. In the meantime, there is almost no IS state activation for x=0.71, as indicated by the $\alpha$$\approx$0 value observed in Table \[table-mueff\], which is reasonable because x=0.71 has a smaller $\sqrt{12}$a superlattice size formed by mixed Na tri- and quadri-vacancies of larger Co$^{4+}$ clusters.[@Chou2004; @Shu2009] We must note that the regular magnetic field strength does not provide enough Zeeman energy to keep the electron stabilized at the IS state of S=1; however, the narrow gap activation process would keep a fixed amount of Co$^{3+}$ ions at the IS state at random positions. In fact, such a narrow gap activation process is equivalent to an elementary excitation, similar to that of the quasiparticle exciton used as a method of energy transport in condensed matter without actual net charge transport.
The temperature dependence of Curie constant C(T) for x=0.82 (Fig. \[fig-CT\]) implies that the number of localized spins is raised at higher temperature, which is in accord with the proposed model of thermally activated IS state (S=1) for Co$^{3+}$. Following the same quantitative analysis that extracts the fraction of Co$^{3+}$ at S=1, $\alpha$, the $a_{1g}-b_{2g}$ gap (i.e., $\delta$ defined in Fig. \[fig-spin\]) can be estimated from the fitting of activation energy of $\alpha$(T)$\sim$e$^{(-\delta/k_B T)}$ in Arrhenius law, as shown in the inset of Fig. \[fig-CT\] with log scale. The activation energy for Co$^{3+}$ activated from the ground state of S=0 to the excited state of S=1 between the narrow gap of $a_{1g}-b_{2g}$ is fitted to be $\delta$$\sim$17 meV, which provides a reasonable order experimentally and is in agreement with that estimated from the crystal field calculation.[@Zhou2005; @Daghofer2006]
The current model can also resolve the puzzling contradiction concerning why LDA calculations predicted $e_g'$ hole pockets along $\Gamma - K$ at $k_z$=0 have never been observed in ARPES experiments.[@Singh2000; @Yang2007] A strong electronic correlation has been proposed to be responsible for the missing $e_g'$ hole pockets, i.e., the $e_g'$ band could be pushed below the Fermi level.[@Zhou2005] Based on our proposed CEF model, the absence of $e_g'$ hole pockets is not surprising at all because the $e_g'$ band should be filled in the valence band further below both $a_{1g}$ and $b_{2g}$, which is consistent with the description of a large inter-orbital repulsion U$^\prime$ proposed by Zhou *et al.*[@Zhou2005] The inconsistent $a_{1g}$-$e_g'$ level splitting predicted from *ab initio* calculations and the octahedral CEF with trigonal distortion originated from the mislabeling of $a_{1g}$ as shown in Fig. \[fig-CEF\]. The actual CEF should be viewed as unexpectedly weak ligands in the apical positions of a pseudo-octahedral CEF, and the best starting point of a CEF description is the square-planar CEF followed by an increasing inter-chain coupling between the edge-sharing square-planar chains. The actual narrow gap is between the $a_{1g}$ (which belongs to the $e_g$ group) and $b_{2g}$ (which belongs to the $t_{2g}$ group) levels, as illustrated in Fig. \[fig-CEF\](c), following the standard crystal field labeling by symmetry.
Acknowledgment {#acknowledgment .unnumbered}
==============
We are grateful to Patrick A. Lee for many helpful discussions. FCC acknowledges support from NSC-Taiwan under project number NSC 101-2119-M-002-007. GJS acknowledges support from NSC-Taiwan under project number NSC 100-2112-M-002-001-MY3.
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abstract: |
Bike-sharing systems are a means of smart transportation in urban environments with the benefit of a positive impact on urban mobility. In this paper we are interested in studying and modeling the behavior of features that permit the end user to access, with her/his web browser, the status of the Bike-Sharing system. In particular, we address features able to make a prediction on the system state. We propose to use a machine learning approach to analyze usage patterns and learn computational models of such features from logs of system usage.
On the one hand, machine learning methodologies provide a powerful and general means to implement a wide choice of predictive features. On the other hand, trained machine learning models are provided with a measure of predictive performance that can be used as a metric to assess the cost-performance trade-off of the feature. This provides a principled way to assess the runtime behavior of different components before putting them into operation.
author:
- Davide Bacciu
- Stefania Gnesi
- Laura Semini
bibliography:
- 'bibliography.bib'
title: Using a Machine Learning Approach to Implement and Evaluate Product Line Features
---
Introduction
============
Product line engineering provides a way to manage variability during the entire design process [@van2001notion] and is an important means for identifying variability needs early on. In this context, a feature represents a *‘logical unit of behavior that is specified by a set of functional and quality requirements’* [@Soltani2012]. A *Feature Model* is a compact representation of the commonalities and variabilities of the system, expressed as mandatory and optional features. Variability is achieved through the selection of the features that will be present in the final product.
In *attributed feature models*, quantitative, non-functional characteristics of features are captured by attributes that are assigned to each feature. The use of attributed feature models is specifically useful for the decision-making process [@Soltani2012], as each stakeholder can make decisions taking into consideration both the features and the characteristics of the final product. A number of techniques allow the configuration of feature models based on both functional and non-functional requirements [@Soltani2012; @benavides2013automated; @Kacper2013].
In this paper, we want to explore the possibility of applying a machine learning (ML) approach to implement features and, at the same time, evaluate them to derive meaningful values to fill the attributed feature model. We concentrate on predictive features that are able to analyse the current state and some historical data, and provide some information to the user. More in general, the purpose of the analysis is to evaluate the features and their possible combinations to help a stakeholder in deciding which product of a line to deploy, making the best possible compromise between cost and usefulness. The stakeholder may be a client that wants to buy a product, or, like in this case, a supplier that wants to know, before deployment, if a feature prediction accuracy will be worth the cost. Finally, also when the cost is not an issue, it is interesting to have an assessment of the features before putting them in operation, to be sure they are accurate.
We focus, to explain our ideas, on Bike-sharing systems (BSS), which are a sustainable means of smart transportation with a positive impact on urban mobility. The quantitative analysis of bike-sharing systems, seen as collective adaptive systems (CAS) is a case study of the European project QUANTICOL (http://www.quanticol.eu). CAS consist of a large number of spatially distributed entities, which may be competing for shared resources even when collaborating to reach common goals. The importance of the CAS in the context of urban mobility and in achieving societal goals means that it is necessary to carry out comprehensive analysis of their design and to investigate all aspects of their behavior before they are put into operation.
In previous work [@TerBeek2014; @TerBeekQC2014; @TerBeek2013], this case study was presented and defined a discrete feature model, specifying several kinds of nonfunctional quantitative properties and behavioral characteristics. In particular, in [@TerBeek2013] we established a chain of tools, each used to model a different aspect of the system, from feature modeling to product derivation and from quantitative evaluation of the attributes of products to model checking value-passing modal specifications.
This paper puts forward the idea of using Machine Learning (ML) methodologies to learn computational models of the features from BSS usage data. Throughout these methodologies, it is possible to learn the (unknown) relationship between the feature and its inputs by exploiting historical data representing examples of such input-output map. A trained model can then be used to provide predictions on future values of the feature in response to new input information, i.e. providing an implementation of the feature component. The advantage of such an approach is twofold: on the one hand, such methodologies provide a powerful and general means to realize a wide choice of predictive features for which there exist sufficient (and significative) historical data. On the other hand, trained ML models are provided with a measure of predictive performance that can be used as a metric to assess the cost-performance trade-off of the feature. This provides a principled way to assess the runtime behavior of different components before putting then into operation
Case Study
==========
Many cities are currently adopting fully automated public bike-sharing systems (BSS) as a green urban mode of transportation. The concept is simple: a user arrives at a station, takes a bike, uses it for a while and returns it to another station. A BSS can be conveniently considered and designed as a product line.
The automation of these systems permits to monitor the stations, to control if borrowed bikes are returned, to let the user pay for the usage, etc. In particular, a basic service of the system keeps track of all bikes and maintains a complete picture of which bikes are docked to each station and which ones are currently hired. For hired bikes, the system keeps track of the user name and departure station and time.
A user is interested in knowing the status of some station. In the case of the station being empty, the system may make a prediction and infer if there is a bike that will be returned to that station soon.
We can describe the feature model of the [*Status*]{} subsystem taking into account the above condition and hence it comprises a mandatory feature (for the basic service) and two optional ones, that can represent two different ways to predict the arrival of a bike, as shown in Figure \[fig:FeatMod\].
![Status subsystem: the Feature Model[]{data-label="fig:FeatMod"}](FeatureModel.JPG){width="50.00000%"}
Feature [*AllBikesNow*]{}, mandatory in each product of the line, keeps updated the current status of the service and tells how many bikes are parked in each station. This is used by the Bike-Sharing system administrators to know if some stations are empty or full and bikes need to be redistributed. The final user can access the status using their web browser before going to the station.
The [*LocationPreview*]{} feature predicts if a bike is going to arrive at a given station, and estimates the needed time. It makes use of a GPS to locate the bike: knowing the departure station and the path so far, the feature can predict the probability it will arrive at a station of interest in the next few minutes, and calculate the expected arrival time. Learning models are trained using historical traces of the BSS system usage.
Feature [*UserProfile*]{} also offers the same kind of prediction, but uses different data: the log, for each user, of all the uses of the Bike-Sharing system. For each use of the system, the log contains: departure time and station; arrival time and station. Analysing these data, [*UserProfile*]{} can predict if one of the bikes currently in use will arrive at the station of interest. Again, the feature returns a probability and the expected arrival time.
In the next sections we provide a brief overview of ML and we discuss how this methodologies can be exploited to realize the [*LocationPreview*]{} and [*UserProfile*]{} features of our BSS use case. Further, we summarize the main ideas underlying performance assessment in ML models.
An Introduction to Machine Learning
===================================
Machine learning provides computational models and methodologies to realize data-driven adaptive approaches to data analysis, pattern discovery and recognition, as well as to the predictive modeling of input-output data relationships. The term data-driven refers to the fact that ML approaches rely on (numerical) information encoded in the data, which is typically vectorial (i.e. multivariate data in a vector space) but can also be of relational type (i.e. compound information with a graph-based representation where edges encode relationships between the atomic information pieces) [@srl].
ML is an active and wide research field comprising several paradigms, e.g. neural-inspired, probabilistic, kernel-based approaches, and addressing a variety of computational learning task types. For the purpose of product line feature modeling and evaluation, we focus on ML models and algorithms targeted at solving [*supervised learning tasks*]{}. Supervised learning refers to a specific class of ML problems that comprise learning of an (unknown) map $M: \mathcal{X} \rightarrow \mathcal{Y}$ between input information $x \in \mathcal{X}$ (e.g. a vector of attributes) and an output prediction $y \in \mathcal{Y}$ (in general, a vector of different dimensionality with respect to the input). Such an unknown map is learned from couples $\mathcal{D} = \{(x_1,y_1),\dots,(x_N,y_N)\}$ of input-output data, referred to as [*training examples*]{}, following a numerical routine targeted at the optimization of an error/performance function $E(\mathcal{D})$ which measures the quality of the predictions generated by the ML model.
ML models are characterized by two operational phases. The first is the [*training*]{} (or [*learning*]{}) phase, where ground-truth teaching information (encoded in training samples) is used to adapt the parameters regulating the response of the model so that its error (performance) $E(\mathcal{D})$ is reduced (increased, respectively). The [*testing*]{} (or [*prediction*]{}) phase, instead, supplies a trained model with novel input information (typically unseen at training time) to generate run-time predictions (i.e. to compute the learned map on novel data). The two phases are not always disjoint: [*incremental learning*]{} approaches exist that allow to continuously adapt the parameters of a ML model while this keeps providing its predictions in response to new input data.
In general, the final quality of the ML model predictions is influenced, on the one hand, by the quality of the training data, which should represent a sufficient and significative sample of the relationship to be modeled, and, on the other hand, by the adequacy of the learning model for the specific computational learning task. In this sense, different tasks, associated with different features to be modeled, may require to use learning models with different capabilities: in the following section, we analyze the nature of the tasks associated with BSS features prediction and we discuss which ML approaches are best suited to address them.
Machine Learning for BSS Features
=================================
Supervised learning approaches can be used to address modeling of product line features in our BSS scenario using logs of previous bike usage as training samples for the ML model. Here, we focus on the realization of the [*UserProfile*]{} and [*LocationPreview*]{} features. These two features are paradigmatic of two classes of learning tasks which require learning models of different nature and capabilities, i.e. static models for vectorial data and dynamic models for sequential data.
User Profile
------------
The [*UserProfile*]{} feature requires to predict the destination station and arrival time of a bike given information on its pickup details and having knowledge of the BSS usage of the person who has picked-up the bike. A ML approach to realize such feature requires to train a different ML model for each user, using its personal usage logs as training data. In other words, a training dataset contains vector couples $(x_n,y_n)$ where the input attributes $x_n$ are a numerical encoding of the departure time and station, while the output $y_n$ encodes the associated time with destination and arrival station. At run-time, the feature prediction will be obtained by selecting the trained learning model associated with the user who has picked-up the bike and supplying it with the details of pickup time and station. Training of the learning models can also be performed at run-time: for instance, when a new customer subscribes to the service, the system starts collecting his/her usage information; as soon as sufficient usage data is collected, it is used to train a new learning model specific for the usage patterns of the customer. The same approach can be used to maintain the knowledge encoded in an existing customer model up-to-date: new examples are added to the log as the customer uses the system and a re-training of the learning model is performed as soon as a sufficient amount of new data is collected.
The prediction of the arrival station is an instance of a [*classification problem*]{} whose objective is to assign an input pattern to one of $K$ different and finite classes (bike stations, in our case study). The prediction of the time to destination, on the other hand, is an example of a [*regression task*]{}, where we are required to predict a generic (possibly continuous valued) output in response to the input pattern. These two problems can be more effectively solved by resorting to two separate and specialized learning models. The prediction of a probability estimate of class membership in place of a hard class assignment can be easily achieved by using a one-of-$K$ encoding of the classifier output coupled with a [*soft-max*]{} between the $K$ outputs. The one-of-$K$ encoding represents the fact that the $n$-th sample belongs to class $k$ (out of $K$) by a $K$ dimensional output vector $y_k$ having zeros on all components except for its $k$-th element $y_n(k)$ which is set to one. As a result, a trained learning model provided with an input $x$ at testing phase will produce a $K$-dimensional prediction $\tilde{y}$: the corresponding soft-max output $\bar{y}$ will again be a $K$-dimensional vector whose $k$-th element is $$\bar{y}(k) = \frac{\tilde{y}(k)}{\sum_{l = 1}^K \tilde{y}(l)}.$$
Data involved in the [*UserProfile*]{} feature is of static type, that is each training sample is a pair of identically and independently distributed vectors. The majority of the learning models in the literature have been designed to deal with such static vectorial data. For the purpose of implementing the [*UserProfile*]{} feature it is worth mentioning [*Support Vector Machines*]{} (SVMs), a family of supervised learning models which construct separating hyperplanes between the training vectors and exploit them to perform classification and regression [@cristSVM]. SVMs build on the concept of a linear separator (i.e. the separating hyperplane) and extend it to deal with non-linear problem by exploiting the so-called [*kernel trick*]{}, that is an implicit map of the input vector into an high-dimensional feature space by means of a non-linear map induced by a kernel function. SVM are highly effective classifiers and regressors for a wide-class of learning problems and several stable implementations are freely available [@svmlight; @libsvm; @svmTorch]. SVM training can be computationally demanding due to hyperparameters search, which can be a limiting factor for their use in run-time training. Further, it is difficult to interpret the result of a trained SVM. When interpretation of the results is an issue, probabilistic learning models found wide application: Naive Bayes and logistic regression are popular approaches [@nb], although based on strong probabilistic assumptions which can be relaxed by resorting to more general Bayesian Networks [@Pearl:2000].
Location Preview
----------------
The [*LocationPreview*]{} feature predicts the same output as the [*UserProfile*]{} feature using different input data, that are GPS trajectories corresponding to journeys performed by the BSS users. Trajectory data encodes a form of dynamical information of different nature with respect to the static vectorial data in [*UserProfile*]{}, requiring a radically different ML approach. A GPS trajectory is a form of sequential data, a type of structured information where the observation at a given point of the sequence is dependent on the context provided by the preceding or succeeding elements of the sequence. Such contextual information plays a role also in the learning task where, for instance, the decision on which will be the arrival station corresponding to a GPS trajectory cannot be taken based on the observation of a single element of the sequence, but should rather take into account the context provided by the full sequence or by a part of it. This requires learning models that can take into consideration such contextual information when computing their predictions, that are ML models for sequential/timeseries data. A straightforward approach to the problem is to use models for static data (such as those seen for the [*UserProfile*]{} feature) feeding them with a fixed-size chunk of the input sequence. This window of observations can be slid across the full length of the sequence, providing a prediction for each sequence element that can take into consideration the surrounding elements up to the window length. The key issue of such an approach is how to determine the correct size of the window for each learning problem. To address this issue, learning models have been proposed that are capable of maintaining a memory of the history of the input signals and to use it to compute their predictions. [*Recurrent Neural Networks*]{} (RNNs) [@rnn] are ML models that have been proposed specifically to deal with the dynamics of sequential information. They extended the original artificial neural networks paradigm with feedback connections that introduce a dynamic memory of the neuron activation which can be used to encode short to long term dependencies among the elements of the sequence, depending on the specific network architecture. In this context, the use of Reservoir Computing (RC) [@rc] has gained increasing interest as a modeling method for RNN, due to its ability in conjugating computational efficiency with the RNN capability of dealing with learning in temporal sequence domains. The underlying idea of the RC approach is to use a layer of sparsely connected recurrent neuron whose connections are initialized and left untrained; adaptation of the neural weights is restricted to the layer of output neurons. This allows to considerably reduce the computational complexity of training, which is a key issue if this is performed at run-time. RC models appear well suited for the implementation of the [*LocationPreview*]{} feature: in particular, they have already shown considerable efficacy in closely related learning tasks, such as the prediction of the destination room of trajectories of users walking in indoor environments [@esn].
Discussion
----------
The work reported in [@miningLDA] applies ML to BSS data but it focuses mainly on mining usage models of BSS with the aim of identifying template behaviors which can be used as demand profiles for system management. In contrast, we propose to use ML as a modeling tool to build system features that can be used at run-time by the user application. At the same time, as explained in the next section, we propose to use the learning model error functions as part of the pre-deployment analysis to assess the cost-performance trade-off of the features to be included in the final BSS deployment. Finally, we also take into account the dynamic nature of trajectory data by using appropriate ML models, such as the RC approach, instead of adapting static models to perform spatio-temporal data analysis [@giot2014predicting]
Performance evaluation of ML models
===================================
A key aspect of ML models is the assessment of their predictive performance. Good ML practice envisages a three-step process to build effective predictive learning models and reliably assess their performance.
1. Training, which consists in adapting the parameters of the learning models using training data and numerical routines that optimize the model performance function (error).
2. Model selection, which consists in estimating performance achieved by different learning models, including different hyper-parameter settings (i.e. model-tuning parameters set by the developer), in order to select the best model (with respect to the performance function)
3. Final assessment, which consists in evaluating the performance of the selected model on new data, providing a measure of the generalization performance of the ultimately chosen model.
Clearly, the latter step can be interpreted as a robust estimation of the performance of the feature implemented by the learning model when deployed in the run-time system. As such, it can be used as part of the product line to straightforwardly assess the efficiency-cost trade-off of the features implemented by ML models. Note that such an evaluation step can, in principle, exploit data logged by an existing system deployed by another client and different from the one being developed. Clearly, such estimated performance will provide an indication which will resemble the actual deployment only if the usage data available is coherent with the expected usage of the system under development. For instance, it has to be expected that trajectory data for cities with considerably different BSS scales and topologies will not provide an adequate ground for comparison.
The three steps above can be implemented throughout a cross-validation scheme. The popular $K$-fold cross-validation would partition the available usage data into $K$ equally sized subsets, using $K-1$ groups for the first step while using the hold-out subset to assess the model performance (second step). This procedure is repeated for each of the $K$ possible choices of the held out group and the performance is then averaged over such $K$ choices. In the simplest scheme, this latter performance is used as final assessment of the model (third step). However, when key model selection choices are required in the second step, these are taken on the $K$-fold averaged performance, while the final assessment is computed on a completely external test set of hold-out data never used in the $K$-fold process. The actual form of the performance measure depends on the nature of the learning task, but it typically evaluates the discrepancy between the output predicted by the learning models and the desired (ground-truth) output. The Mean Absolute Error (MAE) is a popular choice to estimate the performance in regression task as the absolute value of the difference between the model output and the expected target output, averaged over the number of samples under consideration. For classification tasks, performance is often assessed as class accuracy $$acc_i = \frac{TP_i + TN_i}{N_i}$$ where $N_i$ is the number of samples in the $i$-th class, while $TP_i$ and $TN_i$ are the number of true positive and true negative classifications predicted by the model for the $i$-th class.
Conclusions
===========
We have discussed how a machine learning approach can be used to both implement and evaluate predictive Product Line Features. We addressed the case study of the European project QUANTICOL, concerning the quantitative analysis of bike-sharing systems (BSS). The features required by the case study are paradigmatic of two classes of learning tasks which require learning models of different nature and capabilities, i.e. static models for vectorial data and dynamic models for sequential data.
Such models are trained on historical usage data to realize a deployable implementation of the feature. In addition to that, a trained computational learning model is characterized by a measure of predictive performance that can be used to assess the cost-performance trade-off of the feature before putting it into operation.
We are currently training and validating a learning model for the [*UserProfile*]{} feature using real-world usage data comprising more than 280.000 entries on the form
$$\langle UserID, \; leave \; station, \; leave \; date \; and \; time, \; return \; station, \; return \; date \; and \; time \rangle$$
covering all hires in Pisa across two years. As concerns the [*LocationPreview*]{} feature, since bikes in Pisa are not equipped with GPS, we will look for data from different towns, which may in any case provide a measure of predictive performance and let the stakeholders assess whether it is worth buying GPS trackers. A different solution is to use data coming from a simulation.
A general question is indeed associated with the concept of continuous learning, that is deciding when to activate model training and how to keep the feature up to date with respect to the availability of new usage data. Such choices can have an impact on the prediction accuracy as well as on the stability of the learning model. For this specific aspect, we will initially rely on expert knowledge, but we will also explore possible automated decision processes.
To assess the cost-performance trade-off of the features, we plan to use Clafer, a general-purpose modeling language designed to represent domains, meta-models, components and variability models, like Feature models. Clafer has already been applied for modeling and optimization of product lines [@Kacper2013] [@Antkiewicz2013] [@Murashkin2013].
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has been partly supported by the EU FP7-ICT FET-Proactive project QUANTICOL (600708) and by the Italian MIUR project CINA (PRIN 2010LHT4KM).
|
---
abstract: 'We performed magnetic and ferroelectric measurements, associated with Landau theory and symmetry analysis, in order to clarify the situation of the $\rm YMnO_3$ system, a classical example of type I multiferroics. We found that the only magnetic group compatible with all experimental data (neutrons scattering, magnetization, polarization, dielectric constant, second harmonic generation) is the $P6''_3$ group. In this group a small ferromagnetic component along [**c**]{} is induced by the Dzyaloshinskii-Moriya interaction, and observed here in SQUID magnetization measurements. We found that the ferromagnetic and antiferromagnetic components can only be switched simultaneously, while the magnetic orders are functions of the polarization square and therefore insensitive to its sign.'
author:
- Kiran Singh
- 'Marie-Bernadette Lepetit$^*$'
- 'Charles Simon$^*$'
- Natalia Bellido
- Stéphane Pailhès
- Julien Varignon
- Albin De Muer
title: 'Analysis of the multiferroicity in the hexagonal manganite $\rm YMnO_3$'
---
Introduction
============
Hexagonal $\rm YMnO_3$ presents ferroelectricity and antiferromagnetism [@Bertaut_struct; @Smolens_FE_AFM] and can be considered as the prototype of “type I” ferroelectric antiferromagnetic materials in which the details of the magnetoelectric coupling can be studied.
Despite numerous investigations since the pioneer work of Yakel [ *et al.*]{} in 1963 [@Bertaut_struct], the exact crystalline and magnetic structures are still under debate. The temperature of the ferroelectric (FE) transition is for example not completely clear. Located by some authors at 920K [@Tc=920K], recent X rays measurements proposed 1258K [@Gibbs_FE]. These discrepancies are not fully understood and are possibly due to some changes in the oxygen deficiency when the sample is heated. Despite these discrepancies, we can try to summarize the knowledge of this ferroelectric transition as follows. (i) A transition corresponding to a unit-cell tripling and a change in space group from centrosymmetric $P6_3/m mc$ (\#194) to polar $P6_3cm$ (\#185) is observed in this temperature range. In this respect $\rm YMnO_3$ is a typical example of an improper ferroelectric [@improp; @Moise], opening the field to the new concept of hybrid improper ferroelectricity [@hybridFE]. (ii) Indeed, the symmetric group $P6_3/mmc$ reduces to $P6_3cm$ by a rotation of the $\rm MnO_5$ polyhedra. A displacement of the yttrium atoms with respect to the manganese atoms along the c axis of the structure induces a [**c**]{} axis polarization [@Katsu_neutrons; @VAken_FE]. (iii) Furthermore, a possible intermediate phase with the space group $P6_3/mcm$ can be derived from group theory [@Abraham_FE], however it was not observed in the recent measurements [@Gibbs_FE], neither confirmed by symmetry-mode analysis [@Moise]. The authors rather observe some evidence for an iso-symmetric phase transition at about 920 K, which involves a sharp decrease in the estimated polarization. This transition correlates with several previous reports of anomalies in physical properties in this temperature region [@Choi], but is not really understood.
At $T_N=74\rm\,K$, $\rm YMnO_3$ undergoes a paramagnetic (PM) to antiferromagnetic (AFM) transition. The magnetism arises from $\rm
Mn^{3+}$ ions, in $3d^4$ configuration, with spins equal to 2 (high spin). Neutrons diffraction measurements [@Bertaut_magn1; @Lee; @Munoz_neutrons; @Tapan] showed that the structure is antiferromagnetic with moments in the [ **ab**]{}-plane. Following Bertaut [*et al.*]{} [@Bertaut_magn2], Muñoz [*et al.*]{} [@Munoz_neutrons], proposed for the symmetry of the antiferromagnetic order the $\Gamma_1$ (totally symmetric) irreducible representation of the $P6_3cm$ group ; this order corresponds to the$V_1$ order pictured in figure \[fig:VW\]. More recently, a spin polarized analysis showed that the group is rather $P6_3$ or $P6_3'$ [@Tapan]. Finally, in a second harmonic optical generation work, Fröhlich [*et al.*]{} rather concluded to a very different order associated with the $P6_3’cm’$ magnetic group [@Frohlich_SHG; @Fiebig_SHG] ; this order corresponds to the $W_2$ order pictured in figure \[fig:VW\]. Let us note that, while Bertaut [*et al.*]{} and Muñoz [*et al.*]{} performed a full symmetry analysis, checking all possible irreducible representations for the magnetic ordering, Brown and Chatterji, as well as Fiebig [ *et al.*]{} only considered the $\Gamma_1$ representation of the tested symmetry groups. One should however remember that the magnetic order is the spin part of the system wave-function and as such can belong to any of the irreducible representation of the magnetic symmetry group. On another hand, the polarization behaves as the density matrix and thus can only belong to the totally symmetric $\Gamma_1$ representation in groups with only one-dimensional irreducible representations.
[**$V_1$**]{}\
[**$V_2$**]{}\
[**$V_3$**]{}
Associated with the AFM order, several authors reported a ferromagnetic (FM) component associated with a spin canting along the [**c**]{} direction. First suggested [@Bertaut_magn1], and observed by Bertaut [*et al.*]{} [@Bertaut_magn2], this FM component was later observed in the isotypic compound $\rm ScMnO_3$ by Xu [*et al.*]{} [@Xu_FM_Sc] as well as Bieringer and Greedan [@Bieringer_FM_Sc]. Attributed to $\rm Mn_3O_4$ impurities by Fiebig [*et al*]{} [@Fiebig_SHG], a FM component disappearing at $T_N$ was later observed in neutrons scattering by one of us [@Pailhes_FM]. The controversy about the existence of such a component is thus still opened. One could argue that the weakness of the proposed canting removes most of the interest of its existence, however as we will see in the present paper the existence of a FM component has many consequences on the symmetry group of the magnetic structure as well as the interpretation of the $\rm YMnO_3$ properties.
Let us finally quote the existence of a giant magneto-elastic coupling observed by powder neutron diffraction at the magnetic transition [@Park08; @Patnaik]. Very large atomic displacements (up to 0.1Å ) are induced by the magnetic ordering without any identified change of the symmetry group. The influence of such displacements on the polarization or dielectric constant in the magnetic phase was however never reported on single crystal (such measurements exist in thin films) while this information is crucial for the assertion of the assumed magneto-electric coupling seen by domain imaging using second harmonic generation measurements [@Fiebig_domaines].
The present paper aims at building a coherent description for the magnetic structure of the $\rm YMnO_3$ compound, which will account for all the experimental observations and resolve their apparent contradictions.
Can we get some further insight from the experiments?
=====================================================
Experimental details
--------------------
All the measurements reported in the present work were performed on the same single crystal, grown long time ago in Groningen by G. Nénert, from the group of T. Palstra. The sample size for dielectric measurements is $\rm a=1.1\,mm$, $\rm b=1.5\,mm$ and $\rm
c=0.3\,mm$. Magnetic measurements were performed with a QD MPMS-5 SQUID magnetometer. Dielectric and polarization measurements were respectively performed in a QD PPMS-14 with Agilent 4284A LCR meter and Keithley 6517A. Magnetic fields above 14T (and up to 25T) were achieved in the LNCMI Grenoble. The experimental setup for the dielectric constant measurements was the same as in Caen, while the LNCMI setup was used for the magnetization. Antiferromagnetic neutron diffraction peaks were measured on 4F triple axis spectrometer in Laboratoire Léon Brillouin in Saclay on the same single crystal.
The antiferromagnetic transition
--------------------------------
We performed neutron scattering experiments on a neutron triple axis spectrometer and checked the crystal orientation and crystalline quality. The 100 magnetic peak is associated with the antiferromagnetic order parameter. On fig. \[fig:neutrons\], the temperature dependence of its amplitude is reported, showing the magnetic transition at $T_N=74\rm\,K$. On the same figure, we reported the [**ab**]{} component of the dielectric constant, $\varepsilon$, which presents an anomaly at $T_N$. Let us note that the [**c**]{} component of $\varepsilon$ does not present any anomaly at this temperature (not shown). The strong similarity, below $T_N$, between the temperature dependence of the antiferromagnetic order parameter, and the non linear part of $\varepsilon$, suggests that they are closely related, and thus infers the existence of a magneto-electric coupling. One should emphasize the fact that the anomaly of the dielectric constant is not a divergence as expected in the case of a linear magneto-electric coupling. This proof of a non-linear magneto-electric coupling is of utter importance as we will see in the next section.
The polarization and the dielectric constant
--------------------------------------------
This magneto-electric coupling can also be asserted from the polarization and dielectric constant measurements in the magnetic phase.
We performed polarization measurements along the [**c**]{}-axis (the only one allowed by symmetry). A strong reduction of the polarization amplitude is observed below $T_N$ (see fig. \[fig:pol\]) : $2 \rm \,
\mu C/cm^2$ at 30K, to be compared with the $5.5 \rm \,\mu C/cm^2$ measured at room temperature [@Kim_pol].
These polarization values are compatible with the estimated ones, obtained both as $\vec P=\sum_iq_i\vec r_i$ and from our first principle calculations. We computed the polarization using density functional theory and a Berry phases approach at the atomic structures given in reference at 10K and 300K. The calculations were performed with the B1PW hybrid functionals that was specifically designed for the treatment of ferroelectric oxides [@B1PW]. At 300K we found a polarization of $5\rm\,\mu
C/cm^2$. in full agreement with experimental values. At 10K, the polarization is strongly reduced to $1\rm\,\mu C/cm^2$ to be compared with the experimental result of $2 \rm \, \mu C/cm^2$ at 30K.
In addition, we measured the polarization versus the magnetic field. Since this effect is expected to be very small, we used a procedure consisting in ramping many times the magnetic field from $-14\rm\,T$ to $+14\rm\,T$ and extracting the periodic signal from the raw data. One can see on $P(H)$ taken in the magnetic phase (fig. \[fig:polH\]) an anomaly that can be associated with a meta-magnetic transition.
This anomaly can also be followed on the dielectric constant, $\varepsilon$, as a function of applied field and temperature. The meta-magnetic transition phase diagram, characteristic of an antiferromagnetic compound under magnetic field, can so be built (see fig. \[fig:eps\_H\]).
[(a)]{}\
\
[(b)]{}\
The searched of the meta-magnetic transition is a classical method to observe the antiferromagnetism. Indeed, in usual systems, the magnetization (or magnetic susceptibility) versus magnetic field presents an anomaly at the AFM/FM transition under applied field. In $\rm YMnO_3$ our ability to see this transition on electric degrees of freedom (polarization and dielectric constant) clearly proves the existence of a coupling between the polarization and the magnetic order parameter, as first proposed by Huang [*et al.*]{} [@Huang].
The ferromagnetic component
---------------------------
As mentioned in the introduction, one of us (S. Pailhès) observed in a non-polarized neutrons scattering experiment, a Bragg peak that was associated with a ferromagnetic component [@Pailhes_FM]. Indeed, this Bragg peak, at $\vec q=(2,-1,1)$, can neither be associated with the antiferromagnetic order within the ([**a**]{},[**b**]{}) plane, nor with the nuclear extinction rules, since for the $P6_3cm$ symmetry group imposes $2l=0$. In addition it disappears at $T_N$, as expected from a canted AFM order. One objection can however be made against this interpretation. The existence of two $\rm MnO_3$ layers per unit cell (respectively at $z=0$ and $z=1/2$) , forbids to rule out the possibility of an antiferromagnetic coupling between the [**c**]{} components of the canted magnetic moments of each layer ($W_3$ versus $V_3$ order of figure \[fig:VW\]).
We thus performed precise magnetic measurements on a SQUID magnetometer at low magnetic field, and we did observe a small FM component (see fig. \[fig:FM\]). The sample was cooled down from 100K (still above $T_N$) to 10K either under an applied magnetic field along the [**c**]{} axis of the crystal (Field Cooled =FC) or without any field (Zero Field Cooled = ZFC). After cooling, the magnetization was always measured ramping the temperature up under the applied field. This procedure, assuming that the applied field is too small to reverse the magnetization, clearly evidenced the existence of a ferromagnetic component along the [**c**]{} axis (see fig. \[fig:FM\]). The applied magnetic field is 0.05T.
Can we build a theoretical description compatible with the above experimental facts?
====================================================================================
Let us summarize the facts we learned from experiments.
- $\rm YMnO_3$ exhibits a magneto-electric coupling between the AFM and the FE orders.
- This magneto-electric coupling is non linear. The immediate consequence of this is that the AFM order parameter cannot be in the same irreducible representation than the FE order parameter, that is the polarization. The latter being in the totally symmetric irreducible representation : $\Gamma_1$, the AFM order cannot belong to the $\Gamma_1$ irreducible representation of the magnetic group. Assuming that the magnetic order found by Bertaut [@Bertaut_magn1; @Bertaut_magn2] and Muñoz [@Munoz_neutrons] is correct, it means that the magnetic group is not $P6_3cm$ as assumed by these authors. See $V_1$ of fig. \[fig:VW\] for a picture of this order.
- There is a weak FM component along the [**c**]{} axis.
- Even if essentially quenched by the crystal field splitting of the Mn $3d$ orbitals, the spin-orbit coupling and thus the Dzyaloshinskii–Moriya (DM) interaction always exists provided it is symmetry allowed. This is the case with the AFM magnetic order $V_1$ found in neutrons scattering, since the spins vorticity is non nil. The DM interaction should thus induces a FM component (even if small) along the [**c**]{} direction.
- Finally the AFM and FM order parameters are not linearly coupled. Indeed, they present different behaviors around the transition (see figure 7 of ref. or figures \[fig:neutrons\] and \[fig:FM\] of the present paper).
According to the above analysis the $P6_3cm$ group cannot be the system magnetic group. Can we find a magnetic subgroup of the crystallographic group $P6_3cm$ compatible with all the above experimental requirements? The following symmetry group analysis tells us that only one magnetic group is compatible with (i) the $V_1$ AFM order, (ii) the fact that this order is not in the $\Gamma_1$ irreducible representation and (iii) the existence of a FM component along the [**c**]{} direction. This group is the $P6_3'$ magnetic group. Indeed, we first examined the magnetic groups associated with the $P6_3cm$ crystallographic group, that is
$P6_3cm$ :
: discarded since the AFM order $V_1$ belongs to $\Gamma_1$ and the FM component $V_3$ is not allowed (does not belong to the same representation as $V_1$);
$P6_3'c'm$ :
: discarded since $V_1$ and $V_3$ do not belong to the same representation (FM component not allowed);
$P6_3'cm'$ :
: discarded since the FM component is not allowed;
$P6_3c'm'$ :
: discarded since the FM component is not allowed.
Since none of them is compatible with the experimental requirements we looked further in their subgroups and thus abandoned the mirror planes.
The $\bf P6_3$ group
: was discarded since $V_1$ belongs to $\Gamma_1$, which is incompatible with the absence of a linear magneto-electric coupling.
Finally the $\bf P6_3'$ group
: is the only group compatible with all the requirements.
Let us remember that the $P6_3'$ magnetic group was strongly suggested by Brown and Chatterji [@Tapan] from the polarimetric study of neutron diffraction. In fact, they were the first to suggest that the mirror planes are incompatible with the $\rm
YMnO_3$ magnetic group.
Let us now see whether we can account for all the experimental results in a Landau analysis. We established that the magnetic transition should be a transition between the paramagnetic (PM) phase belonging to the $P6_3cm$ group, and the antiferromagnetic (AFM) phase belonging to the $P6_3'$ group. In the $P6_3'$ group the $\Gamma_4$ irreducible representation, to whom both the AFM ($V_1$) and the FM ($V_3$) order parameters belong, is three times represented, namely by the $V_1$, $V_2$ and $V_3$. The Landau theory must thus involve all three magnetic order parameters in addition to the change in the ferroelectric polarization. $V_1$ and $V_2$ are easily represented by the toroidal ($\vec A$) and divergence ($B$) components of the in-plane spins component, while $V_3$ is the out of plane component associated with the magnetization ($\vec M$). For each unit cell one can thus define $$\begin{aligned}
\vec A &=& \frac{1}{6r} \sum_i \vec r_i \wedge \vec S_i =
\frac{1}{6r} \sum_i \vec r_i \wedge \vec S_{ab,i} \\
B &=& \frac{1}{6r} \sum_i \vec r_i \cdot \vec S_i =
\frac{1}{6r} \sum_i \vec r_i \cdot \vec S_{ab,i} \\
\vec M &=& \frac{1}{6} \sum_i \vec S_i = \frac{1}{6} \sum_i \vec S_{c,i} \end{aligned}$$ where the summations over $i$ run over the six Mn atoms of the unit cell ; the $\vec r_i$ refer to the in plane components of the Mn atoms position vectors (note that $\sum_i \vec r_i = \vec 0$ and $\forall i\,
|\vec r_i|=r$) ; the $\vec S_i$ are the Mn atomic spins ($\vec S_i =
\vec S_{ab,i} + \vec S_{c,i}$ where $\vec S_{ab,i}$ is the in-plane component of the Mn spins and $\vec S_{c,i}$ is the [**c**]{} axis component).
$\vec A$ and $\vec M$ are vectors along the [**c**]{} direction while $B$ is a scalar. Let use write $A = S_{ab}\cos{\varphi}$ and $B=S_{ab}\sin{\varphi}$ and point out that the intensity of the 100 AFM magnetic peak (fig. \[fig:neutrons\]) is proportional to ${S_{ab}}^2$ whatever the angle $\varphi$. In the paramagnetic state, i.e. for $T > T_N$, $S_{ab} = 0$, but the polarization $P$ is not zero. This is one of the important issue of this compound. $\vec P$ is not a driving order parameter for the magnetic transition ; however, since its value presents a singularity at $T_N$, it is a secondary order parameter. Its contribution should thus be taken into account in the Landau free energy and can only contain even powers of $P$, as imposed by the higher temperature paraelectric to ferroelectric transition. The free energy can thus be expressed up to the power 4 of the order parameters $$\begin{aligned}
F &=& \overbrace{\alpha_2(T-T_N) (A^2+B^2) + \alpha_4(A^2+B^2)^2}^{\text{AFM energy}} \\
&& \overbrace{-\beta_2(P^2-{P_0}^2) + \beta_4(P^4-{P_0}^4)}^{\text{change in the FE energy}} \\
&& \overbrace{ + \gamma_2 M^2 + \gamma_4 M^4}^{\text{FM energy}} \\
&& \overbrace{ + c_4 (A^2+B^2)(P^2-{P_0}^2)}^{\text{AFM/FE coupling}} \\
&& \overbrace{ + d_4 M^2(P^2-{P_0}^2)}^{\text{FM/FE coupling}} \\
&& \overbrace{ + e_4 (A^2+B^2)M^2}^{\text{AFM/FM coupling}} \\
&& \overbrace{ + z_4 (P^2-{P_0}^2) \vec A \cdot \vec M}^{\text{Dzyaloshinskii–Moriya interaction}} \end{aligned}$$ where $\alpha_2$, $\alpha_4$, $\beta_2$, $\beta_4$, $\gamma_2$, $c_4$, $d_4$, $e_4$, $z_4$ are the temperature independent Landau expansion coefficients. If one notes $t=T_N-T$, and $\vec{\delta
P}=\vec P-\vec P_0$ the gradient of the free energy writes as $$\begin{aligned}
\frac{\partial F}{\partial S_{ab}} &=&
S_{ab}\left[ -2\alpha_2\,t + 4\alpha_4\,{S_{ab}}^2
+ 2c_4 \delta\!P\,(2P_0+\delta\!P) \right. \\ && \left.
+ 2e_4 M^2 \right]
+ z_4 \, \cos{\varphi}\,M\, \delta\!P\,(2P_0+\delta\!P) =0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial\delta\!P} &=&
(P_0+\delta\!P) \left[ -2\beta_2
+ 4\beta_4\,({P_0}^2 + 2{P_0}\delta\!P +{\delta\!P}^2)
\right. \\ && \left.
+ 2c_4 \, {S_{ab}}^2
+ 2d_4 \, M^2 \quad
+ 2z_4 \, S_{ab}\,M\,\cos{\varphi}\right] = 0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial \varphi} &=&
z_4 \, S_{ab}\,M\,\sin{\varphi}\, \delta\!P\,(2P_0+\delta\!P) = 0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial M} &=&
M\,\left[ 2\gamma_2 + 4\gamma_4 M^2
+ 2d_4 \,\delta\!P\,(2P_0+\delta\!P)
+ 2e_4 {S_{ab}}^2 \right] \\ && +
z_4 \, S_{ab} \, \cos{\varphi}\, \delta\!P\,(2P_0+\delta\!P) = 0\end{aligned}$$
From the experimental results we know that $M \ll {S_{ab}}^2$ and $\delta\!P$. We thus expect that if $S_{ab}\propto t^\nu$, $\delta\!P\propto t^\mu$ and $M\propto t^\eta$, we will have in the vicinity of the transition $\eta > \mu$ and $\eta > 2\nu$. In an order by order expansion of the free energy gradient as a function of $t$, one can thus suppose either that $\eta > \nu+1$ ($M\ll S_{ab}t$) or that $\eta \sim \nu+1$ ($M\sim S_{ab}t$). It is easy to show that the first hypothesis leads to a contradiction. Let us thus assume that $M\sim S_{ab}t$. One gets at the zeroth order in $t$ $$\begin{aligned}
\frac{\partial F}{\partial\delta\!P} =
P_0 \left[ -2\beta_2 + 4\beta_4\,{P_0}^2 \right] = 0
&\quad \Leftrightarrow \quad& P_0^2 = \frac{\beta_2}{2\beta_4}\end{aligned}$$ and at the following order $$\begin{aligned}
&& \left\{
\begin{array}[c]{lcl}
\frac{\partial F}{\partial S_{ab}} &: \quad &
S_{ab}\left[ -2\alpha_2\,t + 4\alpha_4\,{S_{ab}}^2 \
+ 4c_4P_0 \delta\!P \right] =0 \\[1ex]
\frac{\partial F}{\partial\delta\!P} &: \quad &
8\beta_4\,{P_0}^2\delta\!P + 2c_4 P_0\, {S_{ab}}^2 = 0
\end{array} \right. \\[2ex]
%%
& \Leftrightarrow &
\left\{
\begin{array}[c]{lcl}
{S_{ab}}^2 &=& \frac{\alpha_2\,\beta_2}{2\alpha_4\,\beta_2 - {c_4}^2{P_0}^2}\,t \\[1ex]
\delta\!P &=& - \frac{\alpha_2\,c_4P_0}{4\alpha_4\,\beta_2 - 2{c_4}^2{P_0}^2}\,t
\end{array} \right. \\\end{aligned}$$ $$\begin{aligned}
\frac{\partial F}{\partial \varphi} &: \quad &
2z_4 P_0\, S_{ab}\, M\,\sin{\varphi}\, \delta\!P = 0
\quad \Leftrightarrow \quad
\sin{\varphi}=0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial M} &: \quad &
2\gamma_2 M + 2z_4P_0 \, S_{ab} \, \cos{\varphi}\, \delta\!P = 0
\\ &\Leftrightarrow &
M =
\frac{{\alpha_2}^{3/2}\,{\beta_2}^{1/2} c_4z_4{P_0}^2}
{2\gamma_2\,(2\alpha_4\,\beta_2 - {c_4}^2{P_0}^2)^{3/2}}
\,\cos{\varphi}\,t^{3/2} \end{aligned}$$ We thus retrieve the $V_1$ order for the AFM spins arrangement ($\sin{\varphi}=0$) ; the decrease in the polarization amplitude under the Néel transition ($\delta P<0$), the fact that the FM order parameter is much weaker than both the AFM one and the change in the polarization ($\nu=1/2$, $\mu=1$, $\eta=3/2$), and finally the fact the FM and AFM order parameters are not linearly related at $T_N$. The polarization and the square of the AFM order parameter are predicted to vary linearly in $t$ at the magnetic transition, as a classical second order phase transition. In fact, as it is for most magnetic phase transitions, higher order terms in the free energy make the temperature dependence over a large scale of temperature different from the mean field prediction. Here for example, the best fit for ${S_{ab}}^2$ is a power law in $t^{1/3}$ (not shown in fig. \[fig:neutrons\]).
Coming back to the anomaly of the dielectric constant at $T_N$ and using the second derivative of $F$ with respect to $P$, one gets in the first order in $t$ $$\begin{aligned}
\frac{1}{\chi_e} &=& \frac{\partial^2 F}{\partial{\delta\!P}^2} =
4\beta_2 - 6 \, c_4\, S^2 + 2c_4 \, S^2
= 4 (\beta_2 - c_4\, S^2)
\\[2ex]
\text{and}\quad \varepsilon &=& 1+\chi_e\\
&=& 1 + \frac{1}{4\beta_2} + \frac{c_4}{4{\beta_2}^2}\, S^2 \end{aligned}$$ Comparing the above expression with the experimental data of fig. \[fig:neutrons\], the Landau analysis correctly predicts the critical shape of $\varepsilon$ versus the AFM order parameter ${S_{ab}}^2$.
As a first conclusion one can state that the above Landau analysis seems in perfect agreement with all the experimental data. The most important consequence of it is that one cannot switch the direction of any of the magnetic orders — clockwise vs counter clockwise rotation of the antiferromagnetic order (sign of $A$) or direction of the magnetization (sign of $M$) — by switching $P$. Indeed, one has $P =
P_0 \left(1- \frac{c_4}{4\beta_4} \, A^2\right)$, and $ M =
-\frac{z_4}{\gamma_2}\, A \, P_0\delta\!P$ thus a change in the sign of $P$ will leave the sign of both $A$ and $M$ unchanged. On the contrary, $A$ and $M$ are switched simultaneously.
Are there other options?
========================
If one supposes that the weak FM component is artefactual, then there are three different groups compatible with the $V_1$ AFM order and the absence of a linear magneto-electric coupling, that is : $P6_3'c'm$, $P6_3'cm'$ and $P6_3c'm'$. In such a case however it is difficult to explain why the $\vec D_{ij}.(\vec S_i\wedge \vec S_j)$ Dzyaloshinskii-Moriya interaction does not yield a FM component along the $\bf c$ direction. Giving up the FM component thus means giving up the $V_1$ ordering for the AFM order.
Is there another AFM order compatible with the neutrons scattering experiments? Following Bertaut [@Bertaut_magn3] and Muñoz [@Munoz_neutrons], there is indeed another AFM order possibly compatible with the neutrons diffraction data, even if with a significantly worse agreement factor than $V_1$ ($R_\text{MAG}=10.8\%$ instead of $7.6\%$ [@Munoz_neutrons]). This order is pictured as $W_2$ in fig. \[fig:VW\]. It is compatible with a non-linear magneto-electric coupling in the $P6_3cm$, $P6_3'c'm$, $P6_3c'm'$ and $P6_3$ magnetic groups. In the $P6_3cm$, $P6_3'c'm$ and $P6_3c'm'$ groups it is associated in its irreducible representation with the $W_3$ order, while in the $P6_3$ magnetic groups both the $W_1$ and $W_3$ orders belong to the representation of $W_2$. At this point let us note that the $W_3$ AFM order is compatible with the $(2,-1,1)$ peak observed by Pailhès [*et al*]{} [@Pailhes_FM] in neutrons scattering.
What Landau’s theory tells us ?
--------------------------------
In the $P6_3cm$, $P6_3'c'm$ and $P6_3c'm'$ groups the Landau analysis yields $$\begin{aligned}
F &=& \alpha_2(T-T_N) B'^2 + \alpha_4B'^4 \quad + \quad
\gamma_2 M'^2 + \gamma_4 M'^4\\
&& -\beta_2(P^2-{P_0}^2) + \beta_4(P^4-{P_0}^4) \\
&& + c_4 B'^2(P^2-{P_0}^2)
+ d_4 M'^2(P^2-{P_0}^2)
+ e_4 B'^2M'^2\end{aligned}$$ where $B'$ and $M'$ are the order parameters respectively associated with $W_2$ and $W_3$. $$\begin{aligned}
B' &=& \frac{1}{6r} \sum_i (-1)^i \vec r_i \cdot \vec S_i =
\frac{1}{6r} \sum_i (-1)^i \vec r_i \cdot \vec S_i^{a,b} \\
\vec M' &=& \frac{1}{6} \sum_i (-1)^i \vec S_i =
\frac{1}{6} \sum_i (-1)^i \vec S_{c,i} \end{aligned}$$ $$\begin{aligned}
\frac{\partial F}{\partial S_{ab}} &=&
S_{ab}\left[ -2\alpha_2\,t + 4\alpha_4\,{S_{ab}}^2
+ 2c_4 \delta\!P\,(2P_0+\delta\!P) \right. \\ && \left.
+ 2e_4 M'^2 \right]
=0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial\delta\!P} &=&
(P_0+\delta\!P) \left[ -2\beta_2
+ 4\beta_4\,({P_0}^2 + 2{P_0}\delta\!P +{\delta\!P}^2)
\right. \\ && \left.
+ 2c_4 \, {S_{ab}}^2
+ 2d_4 \, M'^2 \right] = 0 \\[2ex]
%%%%%%%%%%
\frac{\partial F}{\partial M'} &=&
M'\,\left[ 2\gamma_2 + 4\gamma_4 M'^2
+ 2d_4 \,\delta\!P\,(2P_0+\delta\!P) \right. \\ && \left.
+ 2e_4 {S_{ab}}^2 \right]
= 0\end{aligned}$$ These equations give $$\begin{aligned}
P_0^2 &=& \frac{\beta_2}{2\beta_4} \\ %\quad
%\text{at the }0^\text{th}\text{ order} \\
{S_{ab}}^2 = \frac{\alpha_2\,\beta_2}{2\alpha_4\,\beta_2 - {c_4}^2{P_0}^2}\,t
&\text{and}&
\delta\!P = -\frac{\alpha_2\,c_4P_0}{4\alpha_4\,\beta_2-2{c_4}^2{P_0}^2}\,t
\\[1ex]
\text{and finally} \qquad M'&=& 0\end{aligned}$$
One sees that these results are equivalent to the previous derivation as far as $S_{ab}$ and $P$ are concerned. However, if $M'=0$ is coherent with the neutrons scattering results of ref. [@Munoz_neutrons], it is not compatible with the existence of the $(2,-1,1)$ peak observed by Pailhès [*et al*]{} [@Pailhes_FM]. Indeed, in this representation the $(2,-1,1)$ peak measures the intensity of the order parameter $M'$.
What about the second harmonic generation experiments?
------------------------------------------------------
The second harmonic spectra are due to $d$–$d$ electronic transitions within the $\rm Mn^{3+}$ ions (see figure \[fig:shg\]). The non linear succeptibility is dominated by the term starting from the atomic ground state ($S=2$) and can be written as $$\begin{aligned}
\epsilon_0 \; \chi_{\alpha\beta\gamma} &=&
\sum_{m,k}\frac{ \langle 0|\hat P_\alpha|m\rangle
\langle m|\hat P_\beta|k\rangle
\langle k|\hat P_\gamma|0\rangle }
{( \hbar\omega_{mi}-2 \hbar\omega)( \hbar\omega_{ki}- \hbar\omega)} \end{aligned}$$ where $|0\rangle$ is the $\rm Mn^{3+}$ atomic ground state ; $|k\rangle$ and $|m\rangle$ span the $d$–$d$ $\rm Mn^{3+}$ excited states, $\hbar\omega_{ki}$ and $\hbar\omega_{mi}$ being their excitation energies ; $\hat P_\alpha$ are the dipolar moment operators along the $\alpha$ direction.
We thus evaluated both the in-plane, $\epsilon_0 \; \chi_{xxx}$ and $\epsilon_0 \; \chi_{yyy}$, and the out-of-plane components, $\epsilon_0 \; \chi_{zxx},\,\epsilon_0 \; \chi_{zyy}$, for the different magnetic groups and orders discussed in this paper. The detailed calculations can be found in Appendix.
For the magnetic groups associated with the $P6_3cm$ space group we found [$$\begin{aligned}
\label{eq:chi_P63cm}
\epsilon_0 \, \chi_{xxx}(z\!=\!0) &=&
\epsilon_0 \, \chi_{xxx}^0(z\!=\!0)
\\ && \hspace{-15ex}
+ \left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}\right]
\langle S_x\rangle_{{\rm Mn}_1}
\nonumber \\ && \hspace{-15ex}
+ \left[
\frac{B}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{B'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}\right]
\langle S_x\rangle_{{\rm Mn}_1}^2
\nonumber \\[2ex]
% ---
\epsilon_0 \, \chi_{yyy}(z\!=\!0) &=& 0
\nonumber \\[2ex]
% ---
\epsilon_0 \, \chi_{zxx}(z\!=\!0) &=&
\epsilon_0 \, \chi_{zxx}^0(z\!=\!0) %\\ && \hspace{-15ex}
+ \frac{C}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}
\nonumber \\ && \hspace{-18ex}
+ \frac{D}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}^2
%\\ && \hspace{-15ex}
+ \frac{E}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}^3 \nonumber
\end{aligned}$$ ]{} and similarly for the $z\!=\!1/2$ layer ($\rm Mn_1$ being replaced by $\rm Mn_4$). $\chi_{xxx}^0(z\!=\!0)$ is the spin independent (FE) tensor, $\varepsilon_{i}$ the energy of the iron orbital $\varphi_i\simeq 3d_i$ and $\langle S_\alpha\rangle_{{\rm Mn}_i}$ the $\alpha$ component of the spin of the $\rm Mn_i$ atom. $x,y,z$ are orthogonal axes, $x$ being along the crystallographic $a$ direction and $k$ along the crystallographic $c$ direction.
From these results one can derive the following conclusions.
- Within the symmetry rules associated with a $P6_3cm$ crystal group the second harmonic signal can only be sensitive to magnetic orders in which $\langle S_x\rangle_{{\rm Mn}_1}\ne 0$ and/or $\langle S_x\rangle_{{\rm Mn}_4}\ne 0$.
- The experimental data [@Frohlich_SHG] that sees a magnetic contribution to the in-plane component of $\chi$ are thus incompatible with the $V_1$ magnetic order as previously shown by Iizuka-Sakano [@Ii_SHG] and coherently with our previous analysis.
- We showed that the only possible magnetic order compatible with a $P6_3cm$ crystal group is $W_2$ in which $\langle
S_x\rangle_{{\rm Mn}_1} = \langle S_x\rangle_{{\rm Mn}_4}\ne
0$. According to equations \[eq:chi\_P63cm\] this order predicts a magnetic contribution to the in-plane component of $\chi$, but also to the out-of-plane one $\chi_{zxx}=\chi_{zyy}$. While the first one is in agreement with the experimental findings, no magnetic signal was found in the out-of-plane SHG signal.
- The $P6_3cm$ crystal group and associated magnetic groups are thus not only incompatible with the existence of a FM component and the $(2,-1,1)$ peak observed by neutrons scattering but also with the SHG experimental data.
Let us thus go back to the $P6_3'$ magnetic group and remember that, up to now, this group was found compatible with all experimental facts. The calculation yields the following form for the $\chi$ components (only the contributions associated with the $V_1$ and $V_3$ magnetic orders compatible with the $P6_3'$ magnetic group are retained) [$$\begin{aligned}
\epsilon_0 \, \chi_{xxx}(z=0)&=& \epsilon_0 \, \chi_{xxx}^0(z=0)
\\ && \hspace{-18ex}
+ {\bf i }\left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right]
\langle S_y\rangle_{\rm Mn_1} \langle S_z\rangle_{\rm Mn_1}
\\ && \hspace{-18ex}
+ \left[
\frac{B}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{B'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right] \langle S_y\rangle_{\rm Mn_1}^2
\\[2ex]
% ---- yyy
\epsilon_0 \, \chi_{yyy}(z=0)&=& \epsilon_0 \, \chi_{yyy}^0(z=0)
\\ && \hspace{-16ex}
+ \left[
\frac{C}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{C'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right]
\langle S_y\rangle_{\rm Mn_1}
\\ && \hspace{-17ex}
+ {\bf i}\, \left[ \left(
\frac{D}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{D'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right) \langle S_y\rangle_{\rm Mn_1}^2
\right. \\ && \left. \hspace{-14.6ex} +
\left(
\frac{E}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{E'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right)
\right]
\langle S_z\rangle_{\rm Mn_1}
\\[2ex]
% --- zxx
\epsilon_0 \chi_{zxx}(z=0)&=& \epsilon_0 \chi_{zyy}(z=0)
\;=\;
\epsilon_0 \chi_{zxx}^0
\\ && \hspace{-14ex}
\;+\; \left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right] \langle S_y\rangle_{\rm Mn_1}^2\end{aligned}$$ ]{} Let us remember that the $V_1$ and $V_3$ orders cannot be reversed independently ($\langle S_y\rangle_{\rm Mn_1}\langle S_z\rangle_{\rm
Mn_1} >0$ whatever the magnetic domain), and that $\langle
S_y\rangle_{\rm Mn_1}=-\langle S_y\rangle_{\rm Mn_4}$, $\langle
S_z\rangle_{\rm Mn_1}=\langle S_z\rangle_{\rm Mn_4}$. One thus sees immediately that $\chi_{xxx}$ and $\chi_{zxx}$ depend only on $\langle
S_y\rangle_{\rm Mn_1}^2$ and should thus be insensitive to the magnetic domains. On the contrary, $\chi_{yyy}$ depend on $\langle
S_z\rangle_{\rm Mn_1}$ and should thus exhibit a sensitivity to the magnetic domains at two different frequencies ; namely $2\hbar\omega=\varepsilon_{z^2}-\varepsilon_{xy}$ and $2\hbar\omega=\varepsilon_{z^2}-\varepsilon_{x^2-y^2}$, differing by $\Delta E'$. Those results are in full agreement with the experimental data reported on reference .
Conclusion
==========
In the present paper we showed from joined experimental evidences and theoretical analysis that the AFM transition in $\rm YMnO_3$ is associated with three order parameters, namely the AFM one (primary order parameter), the extra-component of the polarization along [ **c**]{} and the ferromagnetic component along the [**c**]{} axis induced by the Dzyaloshinskii-Moriya interaction (secondary order parameters). Moreover the analysis of the magnetic transition showed the absence of linear coupling between them and thus a hierarchy. Taking into account the different experimental observations (magnetic and transport macroscopic measurements, neutrons scattering data, optical second harmonic responses), as well as the presence of the DM coupling, it appears that the $P6_3'$ magnetic group is the only possible one. In the past, many publications tried to address this question with different conclusions, but all of them present unsolved questions or problems we tried to address in the present work. For example, the importance of a ferromagnetic component was underlined by Bertaut, but corresponds in his samples to a parasitic phase ; some authors have discarded magnetic groups, assuming that the magnetic order should belong to the $\Gamma_1$ irreducible representation of the symmetry group, and so forgetting that despite being by far the most frequent, this is not the only possibility and any of the group representation is valid for the wave function. In fact, the absence of a divergence in the dielectric constant at the phase transition implies that the magneto-electric coupling is not linear, and thus that the polarization and the AFM order cannot belong to the same irreducible representation. The polarization being of $\Gamma_1$ symmetry, the magnetic order cannot belong to the totally symmetric representation $\Gamma_1$. This, in addition to the presence of the small ferromagnetic component, implies that the only possible group is here $P6_3'$. In this group, a change in the sign of the polarization, P, will let both the weak magnetization, M, and the AFM order parameter, A, unchanged. On the contrary, A and M will be switched simultaneously. For possible applications, this type of multiferroic cannot be used to switch the magnetization with an electric field, but rather to switch antiferromagnetism with an intense magnetic field, providing memories which are only little sensitive to magnetic fields.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank G. Nénert and TM. Palstra for providing them with the sample, the IDRIS and CRIHAN French computer centers for providing them with computer time.
Appendix {#appendix .unnumbered}
========
General considerations {#general-considerations .unnumbered}
----------------------
In the following appendix the SHG equations are expressed in term of an orthogonal $x,y,z$ set of axes. The $x$ axis is along the $a$ direction, that is associated with one of the O–Mn bonds in the $z=0$ layer (O is the $(0,0,z\simeq0)$ in-plane oxygen), the $y$ axis its in-plane orthogonal and the $z$ axis is along the $c$ direction.
The three-fold rotation axis is present in any of the groups proposed in this paper. We can thus use it in order to express the $\chi_{\alpha\beta\gamma}$ tensor for the $z=0$ layers as a function of its value for the $\rm Mn_1$ ion (see fig. \[fig:VW\] for the ions labeling), and for the $z=1/2$ layer as a function of its value for the $\rm Mn_4$ ion. One gets easily $$\begin{aligned}
\chi_{xxx}(z=0) &=&
\frac{3}{4}\left(\chi_{xxx}({\rm Mn}_1)- \sum\chi_{xyy}({\rm Mn}_1)\right)\\
\chi_{zxx}(z=0) &=&
\frac{3}{2}\left(\chi_{xxz}({\rm Mn}_1)+ \chi_{yyz}({\rm Mn}_1)\right)\\
\chi_{xzz}(z=0) &=& 0 \end{aligned}$$ and similarly for $z=1/2$ with $\rm Mn_4$ or for $\chi_{yyy},
\chi_{zyy}, \chi_{yzz}$. The summation $\sum\chi_{xyy}$ must be intended as a sum over all similar terms, that is $\sum\chi_{xyy}=
\chi_{xyy} + \chi_{yxy} + \chi_{yyx}$.
Starting from the high temperature phase, we will proceed in perturbation (up to the first order in the wave functions, second order in energy) to include the different symmetry breaking at the FE and AFM transitions, as well as the spin-orbit interaction. In the $P6_3/mmc$ high temperature group, the Mn ions are located on sites of $D_{3h}$ symmetry and one gets the following $3d$ zeroth order orbitals (associated with a nil non linear succeptibility tensor) $$\begin{aligned}
\varphi_{z^2} &=& d_{z^2} \\
%
\varphi_{xz} &=& d_{xz} \\
%
\varphi_{yz} &=& d_{yz} \\
%
\varphi_{x^2-y^2} &=& c d_{x^2-y^2} + c' p_x \\
%
\varphi_{xy} &=& c d_{xy} + c'p_y\end{aligned}$$ At this point let us notice that the $d_{x^2-y^2}$ and $p_x$ (as well as the $d_{xy}$ and $p_y$) Mn orbitals belong to the same irreducible representation and are thus hybridized through the metal-ligand interactions.
The magnetic groups associated with the $P6_3cm$ crystal group {#the-magnetic-groups-associated-with-the-p6_3cm-crystal-group .unnumbered}
--------------------------------------------------------------
Going through the FE transition toward the $P6_3cm$ group, the Mn ions goes from a $D_{3h}$ site to a $C_s$ symmetry site, thus the degeneracies between $\varphi_{xz}\,/\,\varphi_{yz}$ and the $\varphi_{x^2-y^2}\,/\,\varphi_{xy}$ orbitals are lifted by respectively $\delta E$ and $\delta E'$. At the first order of perturbation in this symmetry breaking and in the spin orbit coupling, one gets the following orbitals $$\begin{aligned}
\label{eq:orbcm}
\varphi_{z^2} & =& d_{z^2} + \varsigma p_z
+ \mu d_{xz} + \nu (c d_{x^2-y^2} + c' p_x)
\nonumber \\ && %\quad
+ \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2}
\left[\langle S_x\rangle d_{xz} + \langle S_y\rangle d_{yz}\right]
\nonumber \\[2ex]
%
\varphi_{xz} & =& d_{xz} + \lambda (c d_{x^2-y^2} + c' p_x)
- \mu d_{z^2} + \varsigma' p_z \nonumber \\ && %\quad
- \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2} \langle S_x\rangle d_{z^2}
+ i\frac{\aleph}{\delta E}\langle S_z\rangle d_{yz}
\nonumber \\ && %\quad
+ \frac{c\aleph}{\Delta\varepsilon_1}
\left[\langle S_x\rangle (cd_{x^2-y^2}+c' p_x)
+ \langle S_y\rangle (cd_{xy}+ c'p_y)\right]\nonumber \\[2ex]
%
\varphi_{yz} & =& d_{yz} + \lambda (c d_{xy} + c' p_y)
\nonumber \\ && %\quad
- \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2}\langle S_y\rangle d_{z^2}
+ i\frac{\aleph}{\delta E}\langle S_z\rangle d_{xz}
\nonumber \\ && %\quad
+ \frac{c\aleph}{\Delta\varepsilon_1}
\left[ -\langle S_y\rangle (cd_{x^2-y^2}+c' p_x)
+ \langle S_x\rangle (cd_{xy}+ c'p_y)\right] \nonumber \\[2ex]
%
\varphi_{x^2-y^2} &=& c d_{x^2-y^2} + c' p_x - \lambda d_{xz} - \nu d_{z^2}
+ \varsigma'' p_z
\nonumber \\ && %\quad
- \frac{c \aleph}{\Delta\varepsilon_1}
\left[\langle S_x\rangle d_{xz} - \langle S_y\rangle d_{yz}\right]
\nonumber \\ && %\quad
+ i \frac{c^2 2 \aleph}{\delta E'}
\langle S_z\rangle (cd_{xy}+ c'p_y) \nonumber \\[2ex]
%
\varphi_{xy} &=& c d_{xy} + c'p_y - \lambda d_{yz}\nonumber \\ && %\quad
- \frac{c\aleph}{\Delta\varepsilon_1}
\left[\langle S_y\rangle d_{xz} + \langle S_x\rangle d_{yz}\right]
\nonumber \\ && %\quad
+ i\frac{ c^2 2 \aleph}{\delta E'}\langle S_z\rangle (cd_{x^2-y^2}+c' p_x)\end{aligned}$$ where $\aleph$ is the spin-orbit coupling constant, and $\langle S_j
\rangle$ the average values of the spin operators associated with ground state spin order. $\Delta\varepsilon_1$ is the excitation energy from the degenerated $\varphi_{xz}$ or $\varphi_{yz}$ orbitals toward the $\varphi_{z^2}$ one, $\Delta\varepsilon_2$ is the excitation energy from the degenerated $\varphi_{x^2-y^2}$, $\varphi_{xy}$ orbitals toward the $\varphi_{xz} $ or $\varphi_{yz}$ ones. $\lambda, \mu, \nu, \varsigma,
\varsigma', \varsigma''$ are the first order mixing coefficients associated with the $P6_3/mmc \rightarrow P6_3cm$ symmetry breaking.
For any of the magnetic groups associated with the $P6_3cm$ spatial group, the non linear succeptibility tensor will involve the following transitions (authorized light polarization is shown on top of the arrows while the orbitals irreps are given in parentheses)
$$\left\{ \begin{array}[c]{cl}
\varphi_{xz} (A) \overset{xz}{\longrightarrow} \varphi_{z^2} (A) &
\left\{\begin{array}[c]{c@{\quad}l}
\varphi_{x^2-y^2} (A) \overset{x,z}{\longrightarrow} \varphi_{xz} (A)&
\varphi_{z^2} (A) \overset{x,z}{\longrightarrow} \varphi_{x^2-y^2} (A)\\
\varphi_{xy} (A') \overset{y}{\longrightarrow} \varphi_{xz} (A)&
\varphi_{z^2} (A) \overset{y}{\longrightarrow} \varphi_{xy} (A')\\
\end{array} \right. \\[3ex]
%
\varphi_{yz} (A') \overset{y}{\longrightarrow} \varphi_{z^2} (A) &
\left\{\begin{array}[c]{c@{\quad}l}
\varphi_{x^2-y^2} (A) \overset{y}{\longrightarrow} \varphi_{yz} (A')&
\varphi_{z^2} (A) \overset{x,z}{\longrightarrow} \varphi_{x^2-y^2} (A)\\
\varphi_{xy} (A') \overset{x,z}{\longrightarrow} \varphi_{yz} (A')&
\varphi_{z^2} (A) \overset{y}{\longrightarrow} \varphi_{xy} (A')\\
\end{array} \right.
\end{array} \right\}$$
Using the above diagram and the orbitals given in equations \[eq:orbcm\] one can show that [$$\begin{aligned}
\label{eq:chi_P63cm} \epsilon_0 \,
\chi_{xxx}(z\!=\!0) &=& \epsilon_0 \, \chi_{xxx}^0(z\!=\!0) \\ &&
\hspace{-15ex} + \left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega} +
\frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}\right]
\langle S_x\rangle_{{\rm Mn}_1} \nonumber \\ && \hspace{-15ex} +
\left[
\frac{B}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega} +
\frac{B'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}\right]
\langle S_x\rangle_{{\rm Mn}_1}^2
\nonumber \\[2ex]
% ---
\epsilon_0 \, \chi_{yyy}(z\!=\!0) &=& 0
\nonumber \\[2ex]
% ---
\epsilon_0 \, \chi_{zxx}(z\!=\!0) &=&
\epsilon_0 \, \chi_{zxx}^0(z\!=\!0) %\\ && \hspace{-15ex}
+ \frac{C}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}
\nonumber \\ && \hspace{-18ex}
+ \frac{D}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}^2
%\\ && \hspace{-15ex}
+ \frac{E}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}^3 \nonumber
\end{aligned}$$ ]{} and similarly for the $z\!=\!1/2$ layer. $\chi_{xxx}^0(z\!=\!0)$ is the spin independent (FE) tensor and $\varepsilon_{i}$ the energy of orbital $\varphi_i$.
For the $W_2$ magnetic order one has $\langle S_x\rangle_{{\rm
Mn}_1}=\langle S_x\rangle_{{\rm Mn}_4}$ thus if $\epsilon_0 \,
\chi_{xxx}^0$ and $\chi_{zxx}$ include all the magnetic domain independent terms [$$\begin{aligned}
\epsilon_0 \, \chi_{xxx}
&=& \epsilon_0 \, \chi_{xxx}^0 \\ && \hspace{-15ex} + 2 \left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega} +
\frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}\right]
\langle S_x\rangle_{{\rm Mn}_1}
\nonumber \\[2ex]
% ---
\epsilon_0 \, \chi_{zxx}^0 &=&
2\frac{C}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}
\nonumber \\ && %\hspace{-18ex}
+ 2 \frac{E}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
\langle S_x\rangle_{{\rm Mn}_1}^3 \nonumber
\end{aligned}$$ ]{} It results that in this scheme both the in-plane and out-of-plane signal should be sensitive to the magnetic domains.
On the contrary, the $V_1$ magnetic order should not display any SHG signal since $\langle S_x\rangle_{{\rm Mn}_1}=\langle
S_x\rangle_{{\rm Mn}_4}=0$.
The $P6'_3$ magnetic group {#the-p6_3-magnetic-group .unnumbered}
--------------------------
Let us now look at the $P6_3'$ magnetic group. The associated crystal group is $P6_3$ in which the Mn ions are on a $P_1$ symmetry site. In this group the Fe $3d$ orbitals can be expressed as [$$\begin{aligned}
\label{eq:orb63}
\varphi_{z^2} &=& d_{z^2} + \varsigma p_z
+ \mu d_{xz} + \nu (c d_{x^2-y^2} + c' p_x)
\nonumber \\ && %\quad
+ \mu' d_{yz}
+ \nu' [c d_{xy} + c'p_y]
\nonumber \\ && %\quad
+ \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2}
\left[\langle S_x\rangle d_{xz} + \langle S_y\rangle d_{yz}\right]
\nonumber \\[2ex]
%
\varphi_{xz} &=& d_{xz} + \lambda (c d_{x^2-y^2} + c' p_x)
- \mu d_{z^2} + \varsigma' p_z \nonumber \\ && %\quad
+ \tau d_{yz} + \upsilon [c d_{xy} + c'p_y] \nonumber \\ && %\quad
- \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2} \langle S_x\rangle d_{z^2}
+ i\frac{\aleph}{\delta E}\langle S_z\rangle d_{yz}
\nonumber \\ && %\quad
+ \frac{c\aleph}{\Delta\varepsilon_1}
\left[\langle S_x\rangle (cd_{x^2-y^2}+c' p_x)
+ \langle S_y\rangle (cd_{xy}+ c'p_y)\right]
\nonumber \\[2ex]
%
\varphi_{yz} &=& d_{yz} + \lambda (c d_{xy} + c' p_y)
\nonumber \\ && %\quad
- \mu' d_{z^2} + \varsigma''' p_z h - \tau d_{xz}
+ \upsilon'[ c d_{x^2-y^2} + c' p_x]\nonumber \\ && %\quad
- \frac{\sqrt{3}\aleph}{\Delta\varepsilon_2}\langle S_y\rangle d_{z^2}
+ i\frac{\aleph}{\delta E}\langle S_z\rangle d_{xz}
\nonumber \\ && %\quad
+ \frac{c\aleph}{\Delta\varepsilon_1}
\left[ -\langle S_y\rangle (cd_{x^2-y^2}+c' p_x)
+ \langle S_x\rangle (cd_{xy}+ c'p_y)\right] \nonumber \\[2ex]
%
\varphi_{x^2-y^2} &=& [c d_{x^2-y^2} + c' p_x] - \lambda d_{xz}
- \nu d_{z^2} + \varsigma'' p_z
\nonumber \\ && %\quad
- \upsilon' d_{yz} + \tau'[c d_{xy} + c'p_y] \nonumber \\ && %\quad
- \frac{c \aleph}{\Delta\varepsilon_1}
\left[\langle S_x\rangle d_{xz} - \langle S_y\rangle d_{yz}\right]
\nonumber \\ && %\quad
+ i \frac{c^2 2 \aleph}{\delta E'}
\langle S_z\rangle (cd_{xy}+ c'p_y) \nonumber \\[2ex]
%
\varphi_{xy} &=& [c d_{xy} + c'p_y] - \lambda d_{yz}
\nonumber \\ && %\quad
- \nu'd_{z^2} + \varsigma'''' p_z - \tau'[c d_{x^2-y^2} + c' p_x]
- \upsilon d_{xz} \nonumber \\ && %\quad
- \frac{c\aleph}{\Delta\varepsilon_1}
\left[\langle S_y\rangle d_{xz} + \langle S_x\rangle d_{yz}\right]
\nonumber \\ && %\quad
+ i\frac{ c^2 2 \aleph}{\delta E'}\langle S_z\rangle (cd_{x^2-y^2}+c' p_x)\end{aligned}$$ ]{} where $\mu',\nu', \tau, \tau', \upsilon, \upsilon',\varsigma''',
\varsigma''''$ are the first order perturbation coefficients associated with the $P6_3cm \rightarrow P6_3$ symmetry breaking. The non linear succeptibility tensor will thus involve the following transitions
$$\left\{ \begin{array}[c]{clcl}
\varphi_{xz} &\overset{x,y,z}{\longrightarrow} &\varphi_{z^2} &
\left\{\begin{array}[c]{clc@{\qquad}clc}
\varphi_{x^2-y^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{xz} &
\varphi_{z^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{x^2-y^2} \\
\varphi_{xy} &\overset{x,y,z}{\longrightarrow}& \varphi_{xz} &
\varphi_{z^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{xy}\\
\end{array} \right.\\[5ex]
%
\varphi_{yz} &\overset{x,y,z}{\longrightarrow}& \varphi_{z^2} &
\left\{\begin{array}[c]{clc@{\qquad}clc}
\varphi_{x^2-y^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{yz} &
\varphi_{z^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{x^2-y^2} \\
\varphi_{xy} &\overset{x,y,z}{\longrightarrow}& \varphi_{yz} &
\varphi_{z^2} &\overset{x,y,z}{\longrightarrow}& \varphi_{xy} \\
\end{array} \right.
\end{array} \right\}$$
As it is expected that the $P6_3cm \rightarrow P6_3$ punctual symmetry breaking is very weak (not observed in X-ray diffraction up to now), in the calculation of the second harmonic succeptibility tensor we will thus neglect the terms in $\mu',\nu', \tau, \tau', \upsilon,
\upsilon',\varsigma''', \varsigma''''$. Using the above diagram and the orbitals given in equations \[eq:orb63\] one can show that the SHG tensor has the following form (only the contributions associated with the $V_1$ and $V_3$ magnetic orders compatible with the $P6'_3$ magnetic group are retained) [ $$\begin{aligned}
\epsilon_0 \, \chi_{xxx}(z=0)&=& \epsilon_0 \, \chi_{xxx}^0(z=0)
\\ && \hspace{-18ex}
+ {\bf i }\left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right]
\langle S_y\rangle_{\rm Mn_1} \langle S_z\rangle_{\rm Mn_1}
\\ && \hspace{-18ex}
+ \left[
\frac{B}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{B'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right] \langle S_y\rangle_{\rm Mn_1}^2
\\[2ex]
% ---- yyy
\epsilon_0 \, \chi_{yyy}(z=0)&=& \epsilon_0 \, \chi_{yyy}^0(z=0)
\\ && \hspace{-16ex}
+ \left[
\frac{C}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{C'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right]
\langle S_y\rangle_{\rm Mn_1}
\\ && \hspace{-17ex}
+ {\bf i}\, \left[ \left(
\frac{D}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{D'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right) \langle S_y\rangle_{\rm Mn_1}^2
\right. \\ && \left. \hspace{-14.6ex} +
\left(
\frac{E}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{E'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right)
\right]
\langle S_z\rangle_{\rm Mn_1}
\\[2ex]
% --- zxx
\epsilon_0 \chi_{zxx}(z=0)&=& \epsilon_0 \chi_{zyy}(z=0)
\;=\;
\epsilon_0 \chi_{zxx}^0
\\ && \hspace{-14ex}
\;+\; \left[
\frac{A}{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{A'}{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right] \langle S_y\rangle_{\rm Mn_1}^2\end{aligned}$$ ]{} Using $\langle S_y\rangle_{\rm Mn_1}\langle S_z\rangle_{\rm
Mn_1} >0$ whatever the magnetic domain, $\langle
S_y\rangle_{\rm Mn_1}=-\langle S_y\rangle_{\rm Mn_4}$ and $\langle
S_z\rangle_{\rm Mn_1}=\langle S_z\rangle_{\rm Mn_4}$, one gets [ $$\begin{aligned}
\epsilon_0 \, \chi_{xxx}&=& \epsilon_0 \, \chi_{xxx}^0
\\[2ex]
% ---- yyy
\epsilon_0 \, \chi_{yyy}&=& \epsilon_0 \, \chi_{yyy}^0
\\ && \hspace{-10ex}
+ 2{\bf i}\, \left[
\frac{D \langle S_y\rangle_{\rm Mn_1}^2+E }
{\varepsilon_{z^2}-\varepsilon_{x^2-y^2}-2\hbar\omega}
+ \frac{D'\langle S_y\rangle_{\rm Mn_1}^2+E'}
{\varepsilon_{z^2}-\varepsilon_{xy}-2\hbar\omega}
\right]
\langle S_z\rangle_{\rm Mn_1}
\\[2ex]
% --- zxx
\epsilon_0 \chi_{zxx}&=& \epsilon_0 \chi_{zyy}
\;=\;
\epsilon_0 \chi_{zxx}^0 \end{aligned}$$ ]{} One sees immediately that $\chi_{xxx}$ and $\chi_{zxx}$ should be insensitive to the magnetic order, while $\chi_{yyy}$ should exhibit a sensitivity to the magnetic domains at two different frequencies.
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---
abstract: '[The ionic atmospheres around nucleic acids play important roles in biological function. Large-scale explicit solvent simulations coupled to experimental assays such as anomalous small-angle X-ray scattering (ASAXS) can provide important insights into the structure and energetics of such atmospheres but are time- and resource-intensive. In this paper, we use classical density functional theory (cDFT) to explore the balance between ion-DNA, ion-water, and ion-ion interactions in ionic atmospheres of RbCl, SrCl$_2$, and CoHexCl$_3$ (cobalt hexammine chloride) around a B-form DNA molecule. The accuracy of the cDFT calculations was assessed by comparison between simulated and experimental ASAXS curves, demonstrating that an accurate model should take into account ion-ion correlation and ion hydration forces, DNA topology, and the discrete distribution of charges on [the DNA backbone]{} . As expected, these calculations revealed significant differences between monovalent, divalent, and trivalent cation distributions around DNA. About half of the DNA-bound Rb$^+$ ions penetrate into the minor groove of the DNA and half adsorb on the DNA [backbone]{} . The fraction of cations in the minor groove decreases for the larger Sr$^{2+}$ ions and becomes zero for CoHex$^{3+}$ ions, which all adsorb on the DNA [backbone]{} . The distribution of CoHex$^{3+}$ ions is mainly determined by Coulomb and steric interactions, while ion-correlation forces play a central role in the monovalent Rb$^+$ distribution and a combination of ion-correlation and hydration forces affect the Sr$^{2+}$ distribution around DNA. [This does not imply that correlations in CoHex solutions are weaker or stronger than for other ions. Steric inaccessibility of the grooves to large CoHex ions leads to their binding at the DNA surface. In this binding mode, first-order electrostatic interactions (Coulomb) dominate the overall binding energy as evidenced by low sensitivity of ionic distribution to the presence or absence of second-order electrostatic correlation interactions.]{}]{} [Please address correspondence to Nathan Baker (nathan.baker@pnnl.gov).]{}'
author:
- 'Maria L. Sushko[^1], Dennis G. Thomas[^2], Suzette A. Pabit[^3], Lois Pollack[^4], Alexey V. Onufriev[^5], Nathan A. Baker[^6]'
bibliography:
- 'cdft.bib'
title: 'The role of correlation and solvation in ion interactions with B-DNA'
---
Introduction {#introduction .unnumbered}
============
Interactions with ions stabilize nucleic acid secondary and tertiary structure, have a major impact on DNA packing in cells, and strongly influence protein and drug binding [@Rau:1984; @Rau:1984a; @Knobler:2009; @Xiang:2009; @Draper:2008; @Draper:2005; @Woodson:2005; @Misra:1994; @Misra:1994a]. A fraction of counterions bind to specific sites on nucleic acids and can be detected in crystallographic structures [@Frederiksen:2009], while other counterions form a dynamic ion atmosphere around DNA, diffusing along the molecule and exchanging with ions in bulk solution [@Freisinger:2007]. Mean-field approaches such as Manning counterion condensation [@Manning:1978] and Poisson-Boltzmann (PB) [@Lamm:2003; @Baker:2004; @Fixman:1979; @Anderson:1990] theory have been used to obtain insight into ion distributions around biomolecules and ion-mediated interactions between macro-ions [and have been compared with experimental data with some success [@Chu:2007; @Giambasu:2014; @Pabit:2009; @Bai:2007].]{} While successful in describing some properties of nucleic acids in electrolyte solutions (e.g., RNA p$K_a$ shifts [@Tang:2007], monovalent ion concentration linkages to ligand-DNA binding [@Shkel:2012; @Misra:1994; @Misra:1994a; @Misra:1998], and low valency ion distributions around DNA), these mean-field methods often fail when the ion charge concentration increases. For example, PB models cannot capture the displacement of Na$^+$ by Mg$^{2+}$ around DNA in mixed solutions [@Chu:2007] or ion-mediated DNA-DNA attractive interactions [@Kornyshev:1999]. By imposing the constraint that a fraction of the counterions are bound (condensed) to polyelectrolyte and part form ionic atmosphere in the mean-field counterion condensation theory, it has been possible to reproduce attraction between like-charged polyelectrolytes in the presence of monovalent counterions in the intermediate range of separations [@Kornyshev:1999; @Ray:1994; @Manning:2011; @Perico:2011; @Pietronave:2008]. Manning suggests that the origin of this effect lies in the increase in entropy due to the increase in the effective volume available for condensed counterions as two DNA molecules approach [@Manning:2011]. Such condensation implies penetration of ions through the DNA hydration layer and their partial desolvation to form direct bonds with DNA [@Long:2006; @Allahyarov:2003; @Giambasu:2014]. Describing this process requires atomistic or coarse-grained representation of the macro-ion, which captures both the discreteness of charge distribution on the DNA [backbone]{} and DNA topology, as well as a model for ion desolvation. Such characteristics are not currently present in the PB equation or other popular models of biomolecular electrostatics.
These failures suggest that, to reliably describe ion distribution around nucleic acids, the theoretical model must be refined to include more detailed interactions and incorporate higher-order non-mean-field interactions such as fluctuations. Such extensions of PB approach have been developed for simple geometries (e.g., plates, rods, spheres, etc.) to include second-order terms representing the interactions between fluctuations in ionic densities [@GonzalezAmezcua:2001; @Kjellander:1988; @Blum:1975; @Henderson:1978; @Hoye:1978; @Jiang:2001; @Jiang:2001a]. These extended models and molecular simulations [@Guldbrand:1986; @Rouzina:1996; @Rouzina:1996a; @Arenzon:1999; @Levin:1999; @Shklovskii:1999; @Shklovskii:1999a; @Netz:2001; @Naji:2004] as well as experimental data [@Danilowicz:2009; @Lin:1978; @Drifford:1985; @Zero:1984; @Matsuoka:1991; @Sedlak:1993; @Sedlak:1994; @Sedlak:1996; @Sedlak:1997; @Sedlak:1992; @Sedlak:1992a; @Bockstaller:2001] predict attraction between like-charged objects in the presence of multivalent electrolytes.
In this study, we establish a minimal model based on classical density functional theory (cDFT) to systematically study the influence of the discrete DNA molecular charge representation, ion-ion correlations, and ion-solvent interactions on the distribution of monovalent and multivalent ions around highly charged macromolecules. We show that this model is able to accurately reproduce the results of anomalous small-angle X-ray scattering (ASAXS) experiments [@Pabit:2010; @Andresen:2008; @Pabit:2009; @Pabit:2009a] for B-DNA in RbCl, SrCl$_2$ and CoHexCl$_3$ solutions. As expected, ion-ion correlations play a significant role in the accurate prediction of ASAXS curves. However, our results also demonstrate the importance of ion solvation in cation-DNA interactions and show that for doubly-charged cations these interactions can be as important as ion-ion correlations in modeling ion distributions around DNA.
Methods {#methods .unnumbered}
=======
DNA models {#dna-models .unnumbered}
----------
We used two coarse-grained models for the DNA macro-ion in the cDFT simulations: an infinitely long cylinder with a uniform line charge density along its $z$-axis (charge distribution -1 e per 0.17 nm and the 2 nm cylinder diameter) and a model with a discrete charge distribution (Fig. \[fig:dna-models\]). [Na$^+$ counterions present at 0.78 M concentration in all DNA calculations.]{} The discrete charge distribution of the second model is described by three particle types: two helical arrays of charged spheres that represent the phosphate groups (charge -1 e, diameter 0.42 nm), two helical arrays of neutral spheres (diameter 0.42 nm) that represent the sugar/base groups, and an array of overlapping neutral spheres (diameter 0.78 nm) defining the DNA axis [@Allahyarov:2005]. The positions of these spheres were chosen to mimic B-form DNA using a cylindrical coordinate system $\left( r^s_j, \phi^s_j, z^s_j \right)$ for DNA [backbone]{} $s$ and [base pair]{} $j$. The phosphate spheres have coordinates $r^s_j = 0.89$ nm, $\phi_j^s = \phi^s_0 + 36 j$ degrees, and $z_j^s = z_0^s + 0.34 j$ nm; the sugar/base spheres have coordinates $r_i^s = 0.59$ nm, $\phi_j^s = \phi^s_0 + 36 j$ degrees, and $z_j^s = z_0^s + 0.34 j$ nm; and the axis spheres have coordinates $r = 0$ nm, $\phi = 0$ degrees, and $z_j = 0.5 + 0.34 j$ nm. There are [10 base pairs]{} $(j = 0, \ldots, 9)$ per turn of B-DNA; the angular cylindrical coordinates for [backbone]{} start at $\phi^{(1)}_0 = 0$ and $\phi^{(2)}_0 = 154$ degrees, respectively.
(a) ![Macroion models used in classical DFT simulations: (a) Model of a cylinder with uniform axial charge density; (b) discrete charge model.[]{data-label="fig:dna-models"}](images/cylinder "fig:"){width="0.6in"} (b) ![Macroion models used in classical DFT simulations: (a) Model of a cylinder with uniform axial charge density; (b) discrete charge model.[]{data-label="fig:dna-models"}](images/DNA_3D_fig "fig:"){width="1in"}
Computational models {#computational-models .unnumbered}
--------------------
A variety of computational models were used with the DNA models described above to assess the influence of different energetic contributions on DNA-ion interactions. These models are summarized in Table \[tab:models\] and described in detail in the following sections.
Model Ion-ion elec. correl. Ion-ion steric correl. Ion-solvent interact. Water struct. change Rb$^+$ exp. agreement Sr$^{2+}$ exp. agreement CoHex$^{3+}$ exp. agreement
------------------------ ----------------------- ------------------------ ----------------------- ---------------------- ----------------------- --------------------------- -----------------------------
NLPB no no no no no\* no\* yes
cDFT, no correlation no yes no yes no\* no\* yes
cDFT, no ion solvation yes yes no yes yes no\* yes
cDFT, full model yes yes yes yes yes yes yes
: A summary of the different computational models used in this paper to assess the influence of different energetic contributions (ion-ion electrostatic correlations, ion-ion steric correlations, ion-solvent interactions, and water structural changes) on DNA-ion distributions and compare the resulting distribution functions with experimental ASAXS data. The rows provide model descriptions while the columns indicate which physical phenomena are included by the models. [Additional quantitative data on model results are included in Table \[tab:results\].]{}[]{data-label="tab:models"}
\
\*Agreement with experiment can be obtained by fitting ion radii.
### Classical density functional theory (cDFT) {#classical-density-functional-theory-cdft .unnumbered}
Classical DFT (cDFT) has been previously used to determine the equilibrium distributions of multicomponent salt solutions surrounding DNA [@Wu:2007; @Cao:2005; @Ovanesyan2014]. In our cDFT models, the aqueous salt solution was modeled as a dielectric medium with $\epsilon = 78.5$, charged spherical particles representing ions, and neutral spherical particles representing water molecules. The concentration of spherical “water molecules" was 55.5 M, chosen to model experimental water density. The solutions considered in this work were aqueous NaCl, RbCl, SrCl$_2$ and CoHexCl$_3$ electrolytes in Na$^+$ buffer. These electrolytes were chosen based on the availability of the experimental data for these systems [@Pabit:2010; @Andresen:2008]. We used experimental crystalline ionic diameters for mobile ions: $\sigma_{{\text{Na}}} = 0.204$ nm, $\sigma_{{\text{CoHex}}} = 1.166$ nm, $\sigma_{{\text{Sr}}} = 0.252$ nm, $\sigma_{{\text{Rb}}}$ = 0.322 nm, $\sigma_{\text{Cl}} = 0.362$ nm, and $\sigma_{\text{water}} = 0.275$ nm [@Marcus:1988]. The ion charges were $q_{{\text{Na}}} = +1$, $q_{{\text{CoHex}}} = +3$, $q_{{\text{Sr}}} = +2$, $q_{{\text{Rb}}} = +1$, $q_{\text{Cl}}$ = -1, and $q_{\text{water}} = 0$. Parameterization of the cDFT model was performed against experimental data for the concentration dependence of mean activity coefficients in bulk electrolyte solutions (see Supporting Information). All calculations were performed at 298 K temperature.
To determine the equilibrium water and ion distributions via cDFT, the total Helmholtz free energy functional is minimized with respect to the densities of all the species in the presence of rigid DNA models. For this optimization, it is convenient to partition the total free energy of the system into so-called ideal (${\mathcal{F}}^{\text{id}}$) and excess components (${\mathcal{F}}^{\text{ex}}$) [@Wu:2007]. The ideal free energy corresponds to the non-interacting system and is determined by the configurational entropy contributions from water and small ions, $${\mathcal{F}}^{{\text{id}}} =
k T \sum_i^N \int_\Omega \left( \rho_i({{\bf{r}}}) \log \rho_i({{\bf{r}}}) - \rho_i({{\bf{r}}}) \right) d {{\bf{r}}}$$ where $k$ is Boltzmann’s constant, $T$ is the temperature, $\rho_i : \Omega \mapsto [0,1]$ is the density profile of ion species $i$, $N$ is the number of ion species, ${{\bf{r}}} \in \Omega$ is the ion coordinate, and $\Omega \in \mathbb{R}^3$ is the calculation domain. The excess free energy is generally not known exactly but can be approximated by $${\mathcal{F}}^{{\text{ex}}} \approx {\mathcal{F}}^{{\text{ex}}}_{\text{hs}} + {\mathcal{F}}^{{\text{ex}}}_{\text{corr}} + {\mathcal{F}}^{{\text{ex}}}_{\text{C}} + {\mathcal{F}}^{{\text{ex}}}_{\text{solv}}$$ where ${\mathcal{F}}^{{\text{ex}}}_{\text{hs}}$ is the hard-sphere repulsion term, ${\mathcal{F}}^{{\text{ex}}}_{\text{corr}}$ is the [ion-ion electrostatic correlation]{} term, ${\mathcal{F}}^{{\text{ex}}}_{\text{C}}$ is the direct Coulomb term, and ${\mathcal{F}}^{{\text{ex}}}_{\text{solv}}$ is the [ion-solvent interaction]{} term included in some cDFT calculations (as described below).
The [ion-ion steric correlation]{} term excess free energy describes ion and water many-body interactions in condensed phase due to density fluctuations and can be approximated by Fundamental Measure Theory [@Yu:2002] as $${\mathcal{F}}^{{\text{ex}}}_{\text{hs}} \approx \int_\Omega \Phi^{\text{hs}}\left[ n_w \left( {{\bf{r}}} \right) \right] d {{\bf{r}}}$$ where the functional $\Phi^{\text{hs}}$ has the form [@Da:2014] $$\begin{gathered}
\Phi_{\text{hs}}(\boldsymbol{r}) = -n_0\ln(1-n_3)+\frac{n_1n_2}{1-n_3} +
\left[\frac{1}{36\pi n_3^2}\ln(1-n_3) + \frac{1}{36\pi n_3(1-n_3)^2} \right] n_2^3 \\
- \frac{\boldsymbol{n}_1\cdot\boldsymbol{n}_2}{1-n_3}
- \left[ \frac{1}{12\pi n_3^2}\ln(1-n_3)+\frac{1}{12\pi n_3(1-n_3)^2} \right] n_2(\boldsymbol{n}_2\cdot\boldsymbol{n}_2).
\label{hard-sphere-weighted-density-modified}\end{gathered}$$ where $n_\alpha$ and $\boldsymbol{n}_\beta$ are the scalar and vector weighted averages of the density distribution functions $\rho_i(\boldsymbol{r})$ and are defined by: $$\begin{aligned}
n_\alpha (\boldsymbol{r}) & = \sum_i \int_\Omega \rho_i(\boldsymbol{r}')\omega_i^{(\alpha)}(\boldsymbol{r}'-\boldsymbol{r})\,{\rm d}\boldsymbol{r}',\;\; \text{~for~} \alpha = 0, 1, 2, 3 \\
\boldsymbol{n}_\beta(\boldsymbol{r}) & = \sum_i \int_\Omega \rho_i(\boldsymbol{r}')\boldsymbol{\omega}_i^{(\beta)}(\boldsymbol{r}'-\boldsymbol{r})\,{\rm d}\boldsymbol{r}',\;\; \text{~for~} \beta = 1, 2.\end{aligned}$$ In the limit of a bulk hard-sphere fluid in the absence of external fields, vector densities $\boldsymbol{n}_1$ and $\boldsymbol{n}_2$ vanish. In the same limit, the four scalar weighted densities reduce to the sum of bulk densities for all species ($n_0$) and the 1D ($n_1$), 2D ($n_2$), and 3D ($n_3$) packing fractions. The “weight functions" $\omega_i^{(\alpha)}$ and $\boldsymbol{\omega}_i^{(\beta)}$, characterizing the geometry of particles ([ion-ion steric correlations]{} with radius $R_i$ for ion species $i$), are given by [@Da:2014] $$\begin{aligned}
\omega_i^{(3)}(\boldsymbol{r})&=\theta (|\boldsymbol{r}|-R_i)\\
\omega_i^{(2)}(\boldsymbol{r})&=|\nabla\theta(|\boldsymbol{r}|-R_i)|=\delta(|\boldsymbol{r}|-R_i)
\label{w2}\\
\boldsymbol{\omega}_i^{(2)}(\boldsymbol{r}) & =\nabla\theta(|\boldsymbol{r}|-R_i)=\frac{\boldsymbol{r}}{r}\delta(|\boldsymbol{r}|-R_i)
\label{wv2}\\
\omega_i^{(0)}(\boldsymbol{r})&=\omega_i^{(2)}(\boldsymbol{r})/(4\pi R_i^2)
\label{w0}\\
\omega_i^{(1)}(\boldsymbol{r})&=\omega_i^{(2)}(\boldsymbol{r})/(4\pi R_i)
\label{w1}\\
\boldsymbol{\omega}_i^{(1)}(\boldsymbol{r}) & = \boldsymbol{\omega}_i^{(2)}(\boldsymbol{r})/(4\pi R_i).
\label{wv1}\end{aligned}$$ In the preceding formula, $\theta$ is the Heaviside step function, with $\theta(x)=0$ for $x>0$ and $\theta(x) = 1$ for $x\leq 0$, and $\delta$ denotes the Dirac delta function.
The [ion-ion electrostatic interaction]{} term (${\mathcal{F}}^{{\text{ex}}}_{\text{corr}}$) can be derived using the Mean Spherical Approximation [@Blum:1975; @Hoye:1978] $$\begin{gathered}
{\mathcal{F}}^{{\text{ex}}}_{\text{corr}} = {\mathcal{F}}^{{\text{ex}}}_{\text{corr}}\left[\left\{ \rho_i^{{\text{bulk}}} \right\} \right]
- kT \int_\Omega \sum_i^N c^{(1)}_i \left( \rho_i({{\bf{r}}}) - \rho_i^{\text{bulk}}\right) d {{\bf{r}}} \\
- \frac{kT}{2} \int_\Omega \int_\Omega \sum_{i,j}^N c^{(2)}_{ij}
\left( \rho_i({{\bf{r}}}) - \rho_i^{{\text{bulk}}} \right)
\left( \rho_j({{\bf{r}}}') - \rho_j^{{\text{bulk}}} \right)
d {{\bf{r}}} d {{\bf{r}}}'\end{gathered}$$ where $\rho_i^{{\text{bulk}}}$ is the bulk concentration of ion species $i$ and the first term describes ion correlation free energy in bulk electrolyte solution in the absence of DNA. The first-order direct correlation functions are defined as $$c_i^{(1)} = -\frac{\mu_i}{kT},$$ where $\mu_i$ is the chemical potential of ion species $i$. The second-order direct correlation functions are defined as $$c_{ij}^{(2)}\left( {{\bf{r}}} - {{\bf{r}}}' \right) =
\begin{cases}
-\frac{q_i q_j}{kT \epsilon} \left( \frac{2B}{\sigma_{ij}} - \left( \frac{B}{\sigma_{ij}} \right)^2 \left| {{\bf{r}}} - {{\bf{r}}}' \right| - \frac{1}{\left| {{\bf{r}}} - {{\bf{r}}}' \right|} \right) & \left| {{\bf{r}}} - {{\bf{r}}}' \right| \leq \sigma_{ij} \\
0 & \left| {{\bf{r}}} - {{\bf{r}}}' \right| > \sigma_{ij},
\end{cases}$$ where $q_i$ is the charge of ion species $i$, $\epsilon$ is the dielectric constant of the solvent, $\sigma_{ij} = \left( \sigma_i + \sigma_j \right)/2$ is the hard-sphere contact distance between ions of diameters $\sigma_i$ and $\sigma_j$, $B$ is given by $$B = \frac{1}{\xi} \left( \xi + 1 - \sqrt{1 + 2 \xi} \right),$$ $\xi = \kappa \sigma_{ij}$, $\kappa$ is the inverse Debye length $\kappa^2 = l_B \sum_i q_i^2 \rho_i^{{\text{bulk}}}$, $l_B = \frac{e^2}{kT\epsilon}$ is the Bjerrum length, and $e$ is the unit charge. The direct Coulomb free energy term can be calculated exactly $${\mathcal{F}}^{{\text{ex}}}_{\text{C}} = \frac{kT l_B}{2} \int_\Omega \int_\Omega \sum_{i,j}^N \frac{q_i q_j}{\left| {{\bf{r}}} - {{\bf{r}}}' \right|} \rho_i({{\bf{r}}}) \rho_j({{\bf{r}}}') d {{\bf{r}}} d {{\bf{r}}}'.$$ Finally, the [ion-solvent interaction]{} term ${\mathcal{F}}^{{\text{ex}}}_{\text{solv}}$ models ion-water interactions with a square well potential $$V({{\bf{r}}} - {{\bf{r}}}') = \begin{cases}
\infty & \left| {{\bf{r}}} - {{\bf{r}}}' \right| < \sigma_{ij} \\
-\varepsilon & \sigma_{ij} \leq \left| {{\bf{r}}} - {{\bf{r}}}' \right| \leq \sigma_{ij} + h \\
0 & \sigma_{ij} + h < \left| {{\bf{r}}} - {{\bf{r}}}' \right|,
\end{cases}$$ where $\varepsilon$ is the well depth, and $h$ is the well width. For the current study, $h = 0.2 \varsigma$ is the well width for interactions between ions and water and $\varsigma$ is the sum of radii of interacting particles [@Cao:2005]. The following well depths were calculated using SPC/E water using the parameters from Horinek et al: $\varepsilon_{{\text{Sr}}} = 0.01038$ eV, $\varepsilon_{{\text{Cl}}} = 0.0053894$ eV, $\varepsilon_{{\text{Rb}}} = \epsilon_{{\text{CoHex}}} = 0.0021$ eV [@Horinek:2009]. Simulations of concentration dependence of ion activity coefficients in RbCl and CoHex${\text{Cl}}_3$ solutions demonstrated that adding attractive ion-water interactions does not affect the ion chemical potential (see Supporting Information).
Minimization of the excess free energy functional ${\mathcal{F}}^{{\text{ex}}}$ with respect to the water and ion densities gives $$\rho_i({{\bf{r}}}) = \exp \left( \frac{\mu_i}{kT} - \frac{1}{kT} \frac{\delta {\mathcal{F}}^{{\text{ex}}}}{\delta \rho_i({{\bf{r}}})} \right).$$ We solve Poisson’s equation $$-\nabla \cdot \epsilon({{\bf{r}}}) \nabla \varphi({{\bf{r}}}) = \sum_i q_i \rho_i({{\bf{r}}})
\label{eqn:poisson}$$ for the electrostatic potential ($\varphi({{\bf{r}}})$) where $\epsilon({{\bf{r}}})$ is the dielectric coefficient. For an infinitely long uniformly charged cylinder in electroneutral conditions, the potential $$\varphi(r) = \frac{4 \pi}{\epsilon} \int_r^\infty t \log \left( \frac{r}{t} \right) \sum_i^N q_i \rho_i(t) dt.$$ Using this potential for the cylinder model and a numerical solution to Poisson’s equation (Eq. \[eqn:poisson\]) for the 3D DNA model, the expression for the densities is $$\begin{aligned}
\rho_i({{\bf{r}}}) &=& \exp \left( \frac{\mu_i}{kT} - \frac{q_i \varphi({{\bf{r}}})}{kT}- \frac{1}{kT} \frac{\delta \left( {\mathcal{F}}^{{\text{ex}}}_{\text{hs}} + {\mathcal{F}}^{{\text{ex}}}_{\text{corr}} + {\mathcal{F}}^{{\text{ex}}}_{\text{solv}} \right)}{\delta \rho_i({{\bf{r}}})} \right).
\label{eqn:new-dens}\end{aligned}$$ The resulting system of Eqs. \[eqn:poisson\] and \[eqn:new-dens\] was solved iteratively to self-consistency using the numerical procedure described [in detail]{} by Meng [@Da:2014]. In particular, equilibrium ion density distributions were obtained using a relaxed Gummel iterative procedure for 3D systems and Picard iterations in 1D. Convergence was considered to be achieved when the maximum difference between the input and the output density profiles between iterations was smaller than $10^{-6}$. [The solution of Eqs. \[eqn:poisson\] and \[eqn:new-dens\] encompasses the equilibrium distribution of the densities of all ion species, corresponding to the minimum of the total free energy; the corresponding free energies for each contribution; and the chemical potentials. “Panoramic” density distributions representing angular distributions of ions on DNA backbone and in minor grooves were calculated along the corresponding helical shells. For each, angle the ion densities were averaged within the shells over $r$ and $z$. For ions on the DNA backbone and in the minor grooves, the radial positions of the shells was defined as $1 < r < (1 + \sigma)$ nm and $0.5 < r < 1$ nm, respectively.]{}
Three main features distinguish our approach from previous cDFT models [@Goel:2008; @Goel:2011; @Patra:1999]. First, our model includes a full representation of the coarse-grained DNA topology and a discrete distribution of charges. Second, we use Pauling diameters for ions and van der Waals diameters for water molecules as opposed to previous restricted models where all species have the same diameter. Finally, our model includes water-ion attractive interactions.
Anomalous small-angle X-ray scattering curve calculations {#anomalous-small-angle-x-ray-scattering-curve-calculations .unnumbered}
---------------------------------------------------------
ASAXS profiles were calculated using the ion density distributions $\rho_i(r)$ around DNA. In the 3D model, ion densities were averaged in cylindrical coordinates over the cylinder azimuthal angle $\phi$ and length $z$ for each radial distance $r$ from the DNA axis. The excess form factor for ion species $\alpha$ was calculated as $$F_{\text{ion}, \alpha}(Q) = a_\alpha \int \rho_\alpha(r) e^{-\imath Q r} dr,$$ where $a_\alpha$ is a constant related to the average electron density of ion species $\alpha$ and $Q$ is the scattering vector. In the current study, we only consider the excess form factor due to cation species; the chloride anion has no ASAXS response. Furthermore, we only consider a single cation species at a time so that $F_{\text{ion}}(Q) = F_{\text{ion}, \alpha}(Q)$.
The excess form factor of DNA ($F_{\text{DNA}}(Q)$) was calculated using AquaSAXS [@Poitevin:2011]. from the form factor of DNA in vacuo ($F_{\text{DNA}}^{\text{vac}}(Q)$), the form factor of the volume of water excluded by DNA ($F_{\text{DNA}}^{\text{excl}}(Q)$), and the form factor of hydration shell of the DNA ($F_{\text{hsh}}(Q)$): $$F_{\text{DNA}}(Q) = F_{\text{DNA}}^{\text{vac}}(Q) - \rho_w F_{\text{DNA}}^{\text{excl}}(Q) + \rho_w F_{\text{hsh}}(Q),$$ where $\rho_w$ is the bulk density of water. The form factor of the hydration shell is calculated using water density maps, $\rho_{\text{hsh}}(r)$, obtained via AquaSol [@Koehl:2010], which employs the Poisson-Boltzmann formalism with water treated as an assembly of self-oriented dipoles: $$F_{\text{hsh}}(Q) = b \int \left( \frac{\rho_{\text{hsh}}(r)}{\rho_w} - 1 \right) e^{-\imath Q r} dr$$ where $b$ is a scale factor to adjust the hydration shell contribution (usually $b=1.0$) and integration is performed over the region where solvent density deviates from the bulk by a factor larger (in magnitude) than $\pm 10^{-4}$.
The ASAXS intensity is then calculated from these quantities as $$I(Q) = 2\left( f'_{\text{ion}}(E_{1})-f'_{\text{ion}}(E_{2}) \right) \left(f_{\text{DNA}}N_{\text{ion}}F_{\text{DNA}}(Q)F_{\text{ion}}(Q)
+
f_{\text{ion0}}N^{2}_{\text{ion}}F_{\text{ion}}(Q)^{2} \right) + \left(f'^{2}_{\text{ion}}(E_{1}) - f'^{2}_{\text{ion}}(E_{2})\right) N^{2}_{\text{ion}}F^{2}_{\text{ion}}(Q)$$ where $f'_{\text{ion}}(E_{i})$ is the energy-dependent real part of ion anomalous scattering factor,$E_1$ is the energy far from the X-ray absorption edge of the ion, $E_2$ is the energy near the edge where ion scattering is suppressed by absorption, $f_{\text{ion0}}$ is the energy independent solvent-corrected scattering factor, $f_{\text{DNA}}$ is the effective number of electrons from DNA and $N_{\text{ion}}$ is the number of excess ions around DNA [@Pabit:2010] (see Supporting Information for more details). Since experimental data are available in arbitrary units, theoretical intensities were uniformly scaled with a common scaling factor, chosen to match the experimental and calculated intensities, obtained using 3D cDFT-full model, at low $Q$.
Results {#results .unnumbered}
=======
[Tables \[tab:models\] and \[tab:results\] provide qualitative and quantitative results on the performance of the models. These results are described in greater detail below.]{}
Model Rb$^+$ Sr$^{2+}$ CoHex$^{3+}$
------------------------ ------------ ------------ --------------
NLPB 23.0 13.6 5.6
cDFT, no correlation 23.4 13.7 5.6
cDFT, no ion solvation 34.9 25.1 5.7
cDFT, full model 34.6 16.6 5.7
Experiment 34 $\pm$ 3 19 $\pm$ 2 No data
: [Numbers of condensed ions for the models described in Table \[tab:models\] and experimental results [@Pabit:2010] (where available).]{}[]{data-label="tab:results"}
Comparison between DNA Model systems {#comparison-between-dna-model-systems .unnumbered}
------------------------------------
The uniformly charged cylinder model (Fig. \[fig:dna-models\] left) represents a one-dimensional case for which ionic distribution is only a function of the radial distance from the cylinder axis. [[This 1D model produces monotonically decreasing with the distance from the cylinder surface density distributions of monovalent and multivalent counterions (Fig. \[fig:cdft-cylinder\]). Competitive cation condensation in mixed 5 mM CoHexCl$_3$ + 20 mM NaCl solutions results in preferential CoHex$^{3+}$ condensation on the cylinder surface: sodium ions are not found in the immediate vicinity of the DNA (Fig. \[fig:cdft-cylinder\]b). This competition is in qualitative agreement with experimental observations of a negligible effect of Na$^+$ on CoHex$^{3+}$ binding when NaCl concentration is below 40 mM [@Braunlin1987].]{}]{}
(a) ![Ionic distributions around a uniformly charged cylinder. [The distance $x/d$ is the distance ($x$) from the cylinder surface scaled by the ion radius ($d$).]{} (a) Solutions of 100 mM NaCl, 100 mM RbCl, and 10 mM SrCl$_2$ in a 1 mM NaCl buffer; concentration profiles are shown for Na$^+$ (solid line), Rb$^+$ (dashed line), and Sr$^{2+}$ (dot-dashed line). (b) Solutions of 5 mM CoHexCl$_3$ in a 20 mM NaCl buffer; concentration profiles are shown for Na$^+$ (solid line), CoHex$^{3+}$ (dashed line), and Cl$^-$ (dot-dashed line).[]{data-label="fig:cdft-cylinder"}](images/1d_rb_sr "fig:"){width="0.45\linewidth"} (b) ![Ionic distributions around a uniformly charged cylinder. [The distance $x/d$ is the distance ($x$) from the cylinder surface scaled by the ion radius ($d$).]{} (a) Solutions of 100 mM NaCl, 100 mM RbCl, and 10 mM SrCl$_2$ in a 1 mM NaCl buffer; concentration profiles are shown for Na$^+$ (solid line), Rb$^+$ (dashed line), and Sr$^{2+}$ (dot-dashed line). (b) Solutions of 5 mM CoHexCl$_3$ in a 20 mM NaCl buffer; concentration profiles are shown for Na$^+$ (solid line), CoHex$^{3+}$ (dashed line), and Cl$^-$ (dot-dashed line).[]{data-label="fig:cdft-cylinder"}](images/1d_cohex "fig:"){width="0.45\linewidth"}
[For monovalent ions, the 1D cDFT calculations predict 91.5% and 77.5% DNA charge neutralization by Na$^+$ and Rb$^+$, respectively. Such differences in monovalent cation condensation on DNA were not observed experimentally [@Andresen:2004], demonstrating a fundamental deficiency of a uniformly charged cylinder model for simulating ionic atmosphere around DNA. For divalent ions, the 1D cDFT calculations predict charge inversion at the DNA surface in SrCl$_2$ solution. Note that charge inversion in the presence of multivalent salts has also been observed in cDFT and MC simulations for a cylinder DNA model [@Goel:2008; @Goel:2011]. However, we do not see this effect in our more detailed 3D DNA geometry simulations (see below). Finally, for trivalent ions, 90% DNA charge neutralization is found within 5 CoHex radii from the cylinder surface or within the region where excess CoHex concentration is present (Fig. \[fig:cdft-cylinder\]b).]{}
We also performed 3D cDFT calculations of the same electrolyte solutions surrounding the helical discrete charge model (Fig. \[fig:dna-models\] right). Fig. \[fig:cdft-Rb-helical\] shows cDFT results for the monovalent ion Rb$^+$.
(a) ![[[Cation distributions around DNA calculated using (a) cDFT with the full model, (b) cDFT with no ion-correlation interactions (cDFT-nc, dotted line), and (b,c) NLPB. In panels (a-c), panoramic Rb$^+$ density distributions are shown on the DNA backbone (black lines) and in the minor groove (red lines) as defined in the manuscript text. The inset in panel (b) shows a zoom-in into a low-density region. Panoramic views of cation distributions around DNA in 100 mM NaCl are shown in blue in panels (a) and (c) for comparison. The radial Rb$^+$ density distribution calculated from the full cDFT model is shown in panel (d).]{}]{}[]{data-label="fig:cdft-Rb-helical"}](images/den_panoramic_rb_na_wi "fig:"){width="0.45\linewidth"} (b) ![[[Cation distributions around DNA calculated using (a) cDFT with the full model, (b) cDFT with no ion-correlation interactions (cDFT-nc, dotted line), and (b,c) NLPB. In panels (a-c), panoramic Rb$^+$ density distributions are shown on the DNA backbone (black lines) and in the minor groove (red lines) as defined in the manuscript text. The inset in panel (b) shows a zoom-in into a low-density region. Panoramic views of cation distributions around DNA in 100 mM NaCl are shown in blue in panels (a) and (c) for comparison. The radial Rb$^+$ density distribution calculated from the full cDFT model is shown in panel (d).]{}]{}[]{data-label="fig:cdft-Rb-helical"}](images/den_panoramic_rb_na_ni "fig:"){width="0.45\linewidth"} (c) ![[[Cation distributions around DNA calculated using (a) cDFT with the full model, (b) cDFT with no ion-correlation interactions (cDFT-nc, dotted line), and (b,c) NLPB. In panels (a-c), panoramic Rb$^+$ density distributions are shown on the DNA backbone (black lines) and in the minor groove (red lines) as defined in the manuscript text. The inset in panel (b) shows a zoom-in into a low-density region. Panoramic views of cation distributions around DNA in 100 mM NaCl are shown in blue in panels (a) and (c) for comparison. The radial Rb$^+$ density distribution calculated from the full cDFT model is shown in panel (d).]{}]{}[]{data-label="fig:cdft-Rb-helical"}](images/den_panoramic_rbcl_ni_zoom "fig:"){width="0.45\linewidth"} (d) ![[[Cation distributions around DNA calculated using (a) cDFT with the full model, (b) cDFT with no ion-correlation interactions (cDFT-nc, dotted line), and (b,c) NLPB. In panels (a-c), panoramic Rb$^+$ density distributions are shown on the DNA backbone (black lines) and in the minor groove (red lines) as defined in the manuscript text. The inset in panel (b) shows a zoom-in into a low-density region. Panoramic views of cation distributions around DNA in 100 mM NaCl are shown in blue in panels (a) and (c) for comparison. The radial Rb$^+$ density distribution calculated from the full cDFT model is shown in panel (d).]{}]{}[]{data-label="fig:cdft-Rb-helical"}](images/rb_rdf "fig:"){width="0.45\linewidth"}
As shown in the panoramic density profiles, cDFT predicts a two-peak radial density distribution of Rb$^+$ [(Fig. \[fig:cdft-Rb-helical\])]{}: first peak at around 0.6 nm is due to cation penetration into DNA minor grooves and the second peak at 1.2 nm to Rb$^+$ condensation on DNA [backbones]{}. These data are in good quantitative agreement with molecular dynamics results obtained using TIP3P water model [@Giambasu:2014]. Fig. \[fig:cdft-Sr-helical\] shows cDFT results for the divalent ion Sr$^{2+}$.
(a) ![[[Panoramic density distributions of Sr$^{2+}$ ions on (a) the DNA [backbone]{} and in (b) the DNA minor grooves obtained 10 mM SrCl$_2$. Backbone and minor groove definitions are provided in the text. Results are shown for the full cDFT model (solid red line), cDFT with no corrlations (cDFT-nc, dotted line), and the NLPB model (solid black line). Panel (c) shows radial densities of Sr$^{2+}$ ions around a DNA molecule calculated using the full cDFT model (solid line) and the cDFT model without ion-water attractive interactions (dotted line).]{}]{}[]{data-label="fig:cdft-Sr-helical"}](images/sr_panor_3curve "fig:"){width="0.45\linewidth"} (b) ![[[Panoramic density distributions of Sr$^{2+}$ ions on (a) the DNA [backbone]{} and in (b) the DNA minor grooves obtained 10 mM SrCl$_2$. Backbone and minor groove definitions are provided in the text. Results are shown for the full cDFT model (solid red line), cDFT with no corrlations (cDFT-nc, dotted line), and the NLPB model (solid black line). Panel (c) shows radial densities of Sr$^{2+}$ ions around a DNA molecule calculated using the full cDFT model (solid line) and the cDFT model without ion-water attractive interactions (dotted line).]{}]{}[]{data-label="fig:cdft-Sr-helical"}](images/sr_panor_mg "fig:"){width="0.45\linewidth"} (c) ![[[Panoramic density distributions of Sr$^{2+}$ ions on (a) the DNA [backbone]{} and in (b) the DNA minor grooves obtained 10 mM SrCl$_2$. Backbone and minor groove definitions are provided in the text. Results are shown for the full cDFT model (solid red line), cDFT with no corrlations (cDFT-nc, dotted line), and the NLPB model (solid black line). Panel (c) shows radial densities of Sr$^{2+}$ ions around a DNA molecule calculated using the full cDFT model (solid line) and the cDFT model without ion-water attractive interactions (dotted line).]{}]{}[]{data-label="fig:cdft-Sr-helical"}](images/sr_rdf "fig:"){width="0.45\linewidth"}
In the case of Sr$^{2+}$, the effect of [ion-solvent interactions]{} can be clearly seen in the density distribution of Sr$^{2+}$ with respect to the DNA axis [(Fig. \[fig:cdft-Sr-helical\])]{}. While both cDFT models - with and without [ion-solvent interactions]{} - produce two-peak Sr$^{2+}$ density distributions at the same positions with respect to the DNA axis, the density distributions are qualitatively different. In particular, the model without [ion-solvent interactions]{} predicts much higher Sr$^{2+}$ concentration in the DNA grooves than on [the backbone]{} , while the trend is reversed in the model with [ion-solvent interactions]{}. The 3D cDFT results for trivalent CoHexCl$_3$ solutions are shown in Fig. \[fig:cdft-CoHex-helical\]b and are very similar to those obtained from the 1D cDFT model.
(a) ![(a) Panoramic density distributions of CoHex$^{3+}$ ions on DNA [backbone]{} obtained using [[the full cDFT (solid red line) model, the cDFT model with no correlations (cDFT-nc, dotted line), and the NLPB model (solid black line) for 0.5 mM CoHexCl$_3$. The NLPB and cDFT-nc curves have been shifted by 0.5 mM for clarity. (b) Radial [density]{} distributions of CoHex$^{3+}$ ions around the DNA molecule. Note: as seen in panel (b), there is zero CoHex density$^{3+}$ in the minor groove, so the corresponding panoramic density is not shown.]{}]{}[]{data-label="fig:cdft-CoHex-helical"}](images/cohex_panor_3curve "fig:"){width="0.45\linewidth"} (b) ![(a) Panoramic density distributions of CoHex$^{3+}$ ions on DNA [backbone]{} obtained using [[the full cDFT (solid red line) model, the cDFT model with no correlations (cDFT-nc, dotted line), and the NLPB model (solid black line) for 0.5 mM CoHexCl$_3$. The NLPB and cDFT-nc curves have been shifted by 0.5 mM for clarity. (b) Radial [density]{} distributions of CoHex$^{3+}$ ions around the DNA molecule. Note: as seen in panel (b), there is zero CoHex density$^{3+}$ in the minor groove, so the corresponding panoramic density is not shown.]{}]{}[]{data-label="fig:cdft-CoHex-helical"}](images/cohex_rdf "fig:"){width="0.45\linewidth"}
Finally, we used the results of our cDFT and NLPB calculations to determine ASAXS profiles as described in the *Methods* section. The results of these calculations for Rb$^+$ and Sr$^{2+}$, together with experimental data, are shown in Fig. \[fig:rb-sr-asaxs\].
(a) ![Simulated and experimental ASAXS profiles for 25 bp DNA in (a) 100 mM RbCl and (b) 10 mM SrCl$_2$ solutions. [[Experimental data [@Andresen:2008; @Pabit:2010] are shown as black dots. This figure shows simulation results using the full 3D cDFT model (blue lines), the 3D cDFT model without ion-water interactions (red lines), the full 1D cDFT model (blue dots), and the NLPB model (green lines and dots).]{}]{}[]{data-label="fig:rb-sr-asaxs"}](images/rb_asaxs_cdft_exp_nlpb "fig:"){width="0.45\linewidth"} (b) ![Simulated and experimental ASAXS profiles for 25 bp DNA in (a) 100 mM RbCl and (b) 10 mM SrCl$_2$ solutions. [[Experimental data [@Andresen:2008; @Pabit:2010] are shown as black dots. This figure shows simulation results using the full 3D cDFT model (blue lines), the 3D cDFT model without ion-water interactions (red lines), the full 1D cDFT model (blue dots), and the NLPB model (green lines and dots).]{}]{}[]{data-label="fig:rb-sr-asaxs"}](images/sr_asaxs_exp_cdft_hydr "fig:"){width="0.45\linewidth"}
Similar results for CoHexCl$_3$ are shown in Fig. \[fig:cohex-asaxs\].
![[[Simulated and experimental ASAXS profiles for 25 bp DNA in 0.5 mM CoHexCl$_3$ DNA solutions. Experimental data (unpublished) are shown with a thick black line and those from Andresen et al [@Andresen:2008] with a blue line. 1D cDFT results are shown with a thin black line and 1D NLPB results with a red line. The 3D cDFT and NLPB data coincide and are shown with a green line.]{}]{}[]{data-label="fig:cohex-asaxs"}](images/cohex_asaxs_1d_3d_exp){width="0.45\linewidth"}
Discussion {#discussion .unnumbered}
==========
Comparison to Manning condensation {#comparison-to-manning-condensation .unnumbered}
----------------------------------
The cDFT calculations of ionic distributions for the uniformly charged cylinder model (Fig. \[fig:cdft-cylinder\]) reproduce the Manning condensation limits [@Manning:1978] with approximately 1M concentrations of singly-charged cations at the cylinder surface. A complete 3D cDFT model also reproduces the Manning condensation limit for monovalent cations: the [concentration of condensed Rb$^+$]{} on [the backbone]{} and in minor groove is about 1 M (Fig. \[fig:cdft-Rb-helical\]). [The 1 M limit for monovalent ion concentrations at DNA surface corresponds to 76% compensation of native B-form DNA charge by condensed counterions. Manning’s theory predicts that the concentration of condensed counterions is independent of bulk salt concentration in the range of 0.0001-–0.1 M and increases slightly for higher ionic strengths of monovalent electrolyte solution (80% charge compensation for 0.5 M and 83% for 1 M solutions).]{} Additionally, the 3D cDFT model predicts that the multivalent ions form much denser layers at the DNA surface than the monovalent cations (Figures \[fig:cdft-Rb-helical\], \[fig:cdft-Sr-helical\], and \[fig:cdft-CoHex-helical\]), consistent with Manning theory. The good correlation between our non-mean-field cDFT model (with full ion-ion correlations included) and the mean-field Manning theory is somewhat surprising, particularly given the significant differences observed in the total condensed ion densities between cDFT and the mean-field NLPB approaches. However, Manning theory indirectly accounts for interactions beyond first-order electrostatics through partitioning the total ion density into condensed ions and the surrounding ionic atmosphere. This accounts for the success of Manning theory in predicting the [condensed concentrations]{} of 1:1 electrolyte counterions on DNA as observed in experiments [@Manning:1978] and recent MD simulations [@Giambasu:2014].
Ion interaction with DNA grooves {#ion-interaction-with-dna-grooves .unnumbered}
--------------------------------
By definition, the cylinder model does not allow ion penetration inside DNA and therefore yields well-known monotonically decreasing counterion distributions shown in Fig. \[fig:cdft-cylinder\]. Thus, the model is not adequate for describing the interaction between DNA and small weakly solvated Na$^+$ and Rb$^+$ ions, which are known to penetrate into the minor grooves of DNA [@Giambasu:2014; @Robbins:2014]. However, our more detailed helical charge model allows ion penetration. Simulations of RbCl solutions using this model showed that about half of the condensed Rb$^+$ ions are bound to the minor groove of the DNA molecule (Fig. \[fig:cdft-Rb-helical\]). The distributions of cation densities on the DNA [backbone]{} and in the minor groove are highly structured: they exhibit a periodicity correlated with the periodic spacing of phosphate groups on [the DNA backbone]{} . In contrast, cation distributions in the major groove are mostly featureless (see Fig. S1 in Supporting Information), in agreement with previous simulations and experimental data [@Allahyarov:2003]. Penetration of some cations into the grooves lowers the effective charge density on the DNA, limiting cation condensation on [the backbone]{} .
Increasing cation valency correlates with a stronger preference of cation binding to phosphate groups on the DNA [backbone]{} (Figures \[fig:cdft-Rb-helical\], \[fig:cdft-Sr-helical\], and \[fig:cdft-CoHex-helical\]). A similar preference for CoHex$^{3+}$ binding to phosphates of B-DNA was also observed in MD simulations [@Tolokh:2014] and is determined by the strong electrostatic attraction of the trivalent cations to phosphate groups, CoHex$^{3+}$-CoHex$^{3+}$ repulsion, and steric inaccessibility of B-DNA minor groove to the large CoHex$^{3+}$ ions. As shown in Figures \[fig:cdft-Sr-helical\] and \[fig:cdft-CoHex-helical\], both Sr$^{2+}$ and CoHex$^{3+}$ ions preferentially bind to every fourth phosphate on the strand. Further away from the DNA axis, the Sr$^{2+}$ density variations along the angular cylindrical coordinate have the same period as the period of the angular phosphate distribution [(Fig. \[fig:cdft-Sr-helical\])]{}. The period of the density variations for CoHex$^{3+}$ [(Fig. \[fig:cdft-CoHex-helical\])]{} is two times larger than for Sr$^{2+}$; i.e., some Sr$^{2+}$ions can penetrate into the minor groove, while CoHex$^{3+}$ ions bind exclusively to phosphate groups on the [backbone]{} .
Influence of correlation on ion distributions {#influence-of-correlation-on-ion-distributions .unnumbered}
---------------------------------------------
To investigate the influence of ion correlation forces on the distribution of ions around DNA, we used a cDFT model without ion-correlation interactions (cDFT-nc) as well as the nonlinear Poisson-Boltzmann (NLPB) model, which also lacks correlation (Figures \[fig:cdft-Rb-helical\], \[fig:cdft-Sr-helical\], and \[fig:cdft-CoHex-helical\]). Both models without correlations yield qualitatively different ion distributions than the 3D cDFT calculations which include correlations. In the presence of correlations, sterically allowed ions accumulate in the minor groove; in the absence of correlations, ions accumulate near phosphate groups on the exterior of the DNA strand. The largest qualitative difference between NLPB and cDFT ion distributions was observed for the Rb$^+$ density distribution. In NLPB, Rb$^+$ ions decorate the phosphate groups, driven by Coulombic interactions; the panoramic distribution of Rb$^+$ ions condensed on [the backbone]{} in NLPB model has a larger peak at 45$^\circ$ and a smaller one at 135$^\circ$ (Figures \[fig:cdft-Rb-helical\], \[fig:cdft-Sr-helical\], and \[fig:cdft-CoHex-helical\]). In contrast, ion-ion correlations reduce the effective electrostatic repulsion between cations promoting their penetration into the grooves.Due to stronger Coulomb interactions between multiply-charged cations the effect of correlations is weaker for Sr$^{2+}$ and CoHex$^{3+}$ resulting in the decrease in the fraction of counterions in the grooves with ion radius and charge (Fig. \[fig:cdft-Sr-helical\] and Fig. \[fig:cdft-CoHex-helical\]). For CoHex$^{3+}$ the concentration of counterions in the grooves becomes insignificant.As a result, NLPB and cDFT predict qualitatively similar panoramic density distributions on DNA [the DNA backbone]{} for Sr$^{2+}$ and CoHex$^{3+}$ (Figures \[fig:cdft-Sr-helical\] and \[fig:cdft-CoHex-helical\]).
[As illustrated in the figures and Tables \[tab:models\] and \[tab:results\], the]{} models without correlations (NLPB and cDFT-nc) are very similar to each other, indicating the major influence of correlation on even low charge-density (monovalent) ion behavior. This result contrasts the conclusion that correlations are insignificant in monovalent electrolytes from early theories of ion correlations [@Grochowski.2008]. However, these theories considered electrolytes at uniformly charged surfaces ignoring the influence of the discreteness of charge distribution on fluctuations in ionic atmosphere. Not surprisingly, these models do not capture the experimentally observed attraction between like-charged polyelectrolytes in low concentration monovalent electrolytes [@Sedlak:1996; @Manning:2011]. Recent molecular dynamic simulations also point to the importance of non-mean-field interactions between biomolecules and monovalent electrolytes manifested in a more structured ionic atmosphere than that predicted by NLPB [@Giambasu:2014; @Robbins:2014]. The small difference between the cDFT-nc and NLPB models (in the height of the double peak around 45$^\circ$) for Rb$^+$ is due to the solvent excluded-volume effects included in the cDFT-nc model and absent from NLPB theory (Fig. \[fig:cdft-Rb-helical\]).
Ion correlations also influence ion-specific details in density distributions for counterions of the same valency. 3D cDFT results demonstrate that smaller Na$^+$ ions tend to accumulate on [the DNA backbone]{} and minor groove while Rb$^+$ ions are more evenly distributed along the DNA helix (see Fig. \[fig:cdft-Rb-helical\]a). In the cDFT-nc and NLPB models with no correlation, the differences between Na$^+$ and Rb$^+$ distributions are significantly smaller (see Fig. \[fig:cdft-Rb-helical\]b), suggesting that ion-correlation interactions are responsible for this effect.
Comparison with ASAXS experiments {#comparison-with-asaxs-experiments .unnumbered}
---------------------------------
ASAXS profiles calculated using the 3D cDFT model show very good agreement with experimental data [@Andresen:2008] for RbCl solutions (Fig. \[fig:rb-sr-asaxs\]). The shapes of the scattering curves are very similar in the 1D cDFT and 3D cDFT models, with similar average numbers of condensed counterions: the 1D and 3D cDFT calculations predict 34.9 and 34.6 condensed Rb$^{+}$ ions, respectively. Both predictions are within error of the experimental measurement of 34 $\pm$ 3 ions [@Pabit:2010]. However, the distribution of Rb$^+$ ions is different in these models: all condensed cations decorate the cylinder surface (by definition) in the 1D model while half of the condensed cations are on [the DNA backbone]{} and the other half are in minor grooves in the 3D models. In contrast, the 3D NLPB model shows a significant deviation of the simulated scattering curve from the experimental one (Fig. \[fig:rb-sr-asaxs\]). As discussed in the previous section, penetration of some cations into DNA grooves reduces the negative electrostatic potential acting on cations in solution. In the absence of any interactions beyond Coulomb forces, this penetration leads to lower concentrations of cations on the DNA surface and lower [concentrations of condensed counterions]{}. Previous NLPB simulations demonstrated that adjusting the ionic radius of Rb$^+$ to its hydrated radius and prohibiting ion penetration into the DNA hydration shell can lead to closer agreement between 3D NLPB results and experiment [@Andresen:2004] – but at the price of an incorrect ion distribution around DNA.
Comparison of calculated and experimental SrCl$_2$ data highlight the importance of [ion-solvent interactions]{} on ion distributions around DNA [(Fig. \[fig:rb-sr-asaxs\])]{}. Sr$^{2+}$ ions have a significant hydration energy; approximately 3 times higher than that of monovalent alkali metal ions. These strong cation-water interactions lower the entropy of water molecules around cations, but introduce a higher enthalpy cost for partial ion desolvation [@Lightstone:2001]. Simulations with the solvent approximated as dielectric continuum do not account for such desolvation, limiting interactions in the system to first- and second-order electrostatic interactions: Coulomb and ion-correlation forces. To understand the importance of these desolvation contributions, we used two variants of the cDFT model: one with attractive cation-water interactions and another without. As shown in Fig. \[fig:rb-sr-asaxs\], ASAXS curves calculated using cDFT without desolvation contributions [via ion-solvent interactions]{} deviate significantly from the experimental data and the ASAXS curves calculated using the complete cDFT model. On the other hand, the experimental ASAXS curves agree with those calculated from the complete cDFT model. The importance of desolvation is also emphasized by the fact that inclusion of such interactions is essential for reproducing the chemical potentials of divalent cations but is not required for weakly hydrated alkali metal ions or CoHex$^{3+}$ (see Supporting Information). In summary, [ion-solvent interactions]{} are important for accurately modeling ion-DNA interactions: desolvation reduces the excess chemical potential of cations and anions, lowering the effective concentration of electrolyte and weakening ion-ligand interactions. Surprisingly, an NLPB model [that]{} includes neither [ion-solvent interactions]{} nor [ion-ion correlations]{} reproduces the experimental ASAXS curves for Sr$^{2+}$ (Fig. \[fig:rb-sr-asaxs\]), although some differences are obvious in the more detailed radial [density]{} functions (Fig. \[fig:cdft-Sr-helical\]). This agreement is serendipitous and is due to cancellation of errors from the lack of ion-correlation, that favors ion accumulation in the grooves, and [ion solvation]{}, that limits ion concentration in the grooves.
Unexpectedly, the trivalent CoHexCl$_3$ solution is the simplest ion to model around DNA; CoHex$^{3+}$ can be reliably described by first-order electrostatics (i.e., direct Coulomb interactions). CoHex$^{3+}$ ions decorate DNA [backbone]{} phosphates and do not penetrate inside B-DNA grooves [(Fig. \[fig:cdft-CoHex-helical\])]{}. All models explored in this paper show reasonable agreement between the calculated ASAXS profiles and the experimental data (Fig. \[fig:cohex-asaxs\]), and is consistent with all-atom MD simulations in explicit solvent [@Tolokh:2014]. For these triply-charged ions, the good agreement between cDFT and NLPB is caused by the dominance of first-order electrostatics in ion-DNA interactions over higher-order ion-ion correlations. Moreover, the large CoHex$^{3+}$ ionic diameter creates a steric barrier for ion penetration inside the grooves, rendering the 1D cylinder models adequate for calculating the average number of condensed CoHex$^{3+}$ ions. Finally, because the diameter of CoHex$^{3+}$ is large, the field at its surface is comparable to Na$^+$ and the effects of [ion-solvent interactions]{} are lower than for the smaller divalent Sr$^{2+}$ ions. It follows from the current study that the models required to describe CoHex$^{3+}$ around a single B-DNA [backbone]{} are relatively simple. However, we expect that ion correlation forces will dominate DNA-DNA interactions between multiple [backbones]{} due to the higher local phosphate charge density.
Conclusions {#conclusions .unnumbered}
===========
We have studied the details of ionic atmospheres around DNA molecule for 1:1, 2:1 and 3:1 electrolytes using a combination of cDFT and NLPB methods. Our calculations demonstrated that ion-ion correlation interactions induce counterion penetration into the DNA grooves, unless sterically prohibited by large ion radii. In particular, ion binding in the grooves – compared to binding on [the backbone]{} – has a profound effect on ion-induced nucleic acid condensation as demonstrated in our previous work [@Tolokh:2014]. [Ion-solvent]{} interactions have an opposite effect: when the enthalpy cost of desolvation is high (e.g., for Sr$^{2+}$ ions), ion-water interactions limit ion penetration into the DNA grooves. Partial compensation of these two opposing effects explains the success of NLPB in reproducing the average number of condensed cations and the shape of the ASAXS curves of the ion-counting experiments. In contrast, cDFT model without ion-desolvation interactions was found to systematically overestimate ion concentration in DNA grooves. Through the comparison of several cDFT models and experimental data, we demonstrated that a minimum model to describe ion-polyelectrolyte interactions should include long-range correlations arising from density and charge density fluctuations in electrolyte solution as well as short-range ion (de)solvation forces. The latter interactions are often ignored in reduced models of electrolyte solutions limiting their applicability to the classes of weakly solvated ions. Ion hydration forces are particularly pronounced in solutions of multiply-charged ions and give significant contribution to ion activity and, therefore, to ion-polyelectrolyte interactions. Our results highlight important aspects of the properties of electrolyte solutions influencing ionic atmosphere around biomolecules that may significantly impact DNA condensation and biomolecules-ligand interactions. [One caveat of the current work is its neglect of DNA sequence-specific effects which have been shown to influence ion binding in some cases [@Halle1998; @Shui1998; @McFail-Isom1999; @Denisov2000; @Pasi2015]. The goal of our initial research was to understand the general characteristics of DNA-ion interactions that drive the behavior of different ionic species around DNA. In the future, we plan to extend the DNA model to include such sequence-dependent structural variations.]{}
Author contributions {#author-contributions .unnumbered}
====================
MLS, DGT, and SAP performed the research, analyzed the data, and helped write the manuscript. LP, AVO, and NAB designed the research, analyzed the data, and assisted in writing the manuscript.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by NIH Grant R01 GM099450. NAB and MLS acknowledge fruitful discussions of hydration effects with Chris Mundy and Greg Schenter. The cDFT calculations were performed using PNNL Institutional Computing resources at Pacific Northwest National Laboratory. PNNL is a multiprogram national laboratory operated for DOE by Battelle under Contract DE-AC05-76RL01830.
SUPPLEMENTARY MATERIAL {#supplementary-material .unnumbered}
======================
An on-line supplement to this article can be found by visiting BJ Online at `http://www.biophysj.org`.
[^1]: Physical Sciences Division, Pacific Northwest National Laboratory, Richland, WA 99352.
[^2]: Biological Sciences Division, Pacific Northwest National Laboratory, Richland, WA 99352
[^3]: School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853-3501.
[^4]: School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853-3501.
[^5]: Department of Computer Science and Department of Physics, Virginia Tech, Blacksburg, VA 24061.
[^6]: Computational and Statistical Analytics Division, Pacific Northwest National Laboratory, Richland, WA 99352; Division of Applied Mathematics, Brown University.
|
---
abstract: 'We extend a field theoretic approach for the investigation of the electronic charge-current density response of crystalline systems to arbitrary applied electromagnetic fields. The approach leads to the introduction of microscopic polarization and magnetization fields, as well as free charge and current densities, the dynamics of which are described by a lattice gauge theory. The spatial averages of such quantities constitute the fields of macroscopic electrodynamics. We implement this formalism to study the orbital electronic response of a class of insulators to applied uniform dc electric and magnetic fields at zero temperature. To first-order in the applied fields, the free charge and current densities vanish; thus the response of the system is characterized by the first-order modifications to the microscopic polarization and magnetization fields. Associated with the dipole moment of the microscopic polarization (magnetization) field is a macroscopic polarization (magnetization), for which we extract various response tensors. We focus on the orbital magnetoelectric polarizability (OMP) tensor, and find the accepted expression as derived from the “modern theory of polarization and magnetization.” Since our results are based on the spatial averages of microscopic fields, we can identify the distinct contributions to the OMP tensor from the perspective of this microscopic theory, and we establish the general framework in which extensions to finite frequency can be made.'
author:
- 'Perry T. Mahon'
- 'J. E. Sipe'
bibliography:
- 'dcResponse\_Updated.bib'
title: 'Magnetoelectric Polarizability: A Microscopic Perspective'
---
Introduction
============
Interest in describing the response of insulators to external electromagnetic fields dates back to the earliest studies of electricity and magnetism. In pioneering work near the start of the twentieth century, Lorentz [@Lorentz] based his definition of the macroscopic polarization and magnetization fields on a physical picture of molecules with electric and magnetic moments [@Jackson], and from that perspective addressed the response of the macroscopic quantities to applied fields.
Near the end of the twentieth century a new approach, called the “modern theory of polarization and magnetization,” was introduced [@Resta1994; @Resta2005; @Resta2006; @Niu2007]. Largely focused on the static response of crystalline materials to uniform fields, the microscopic underpinning was now electronic Bloch eigenfunctions [@RestaX], or alternately the spatially localized Wannier functions that could be constructed from them. But a macroscopic perspective was taken to define the polarization and magnetization. For example, if one imagined a slow variation in material parameters leading to a macroscopic current density $\boldsymbol{J}$, the polarization (or at least its change) could be defined through $\boldsymbol{J}=d\boldsymbol{P}/dt$ [@KingSmith1993]. Thus, instead of a microscopic picture of the underlying position and motion of charges *leading* to the definition of macroscopic quantities, as it did for Lorentz, links to a microscopic picture *follow* from the definitions. In the “modern theory,” the macroscopic polarization was found to be related to the dipole moment of a Wannier function and its associated nucleus [@Resta1994]. The ambiguity of which nucleus to associate with a given Wannier function – the “closest,” or one some number of lattice spacings away? – leads to a “quantum of ambiguity” in the macroscopic polarization itself. Such ambiguities are inherent to the “modern theory,” and can generally be related to the behavior and description of charges and currents at the surface of a finite sample [@VanderbiltBook].
We have recently argued [@Mahon2019] that it is useful to expand upon the approach of Lorentz by introducing *microscopic* polarization and magnetization fields in bulk crystals, and defining the corresponding macroscopic fields as their spatial averages; in general, microscopic “free” charges and currents, and their spatial averages, are also introduced, and the resulting description takes the form of a generalized lattice gauge theory. The strategy employed is an extension of that used to introduce microscopic polarization and magnetization fields for atoms and molecules [@Healybook; @PZW], which itself is an extension of Lorentz’ characterization of molecules by a series of multipole moments. Such microscopic polarization and magnetization fields allow for the visualization of electronic dynamics in the sense that perturbative modifications to these microscopic fields arising due to the applied electromagnetic field can be found and exhibited if one has the Wannier functions in hand; existing schemes, which are primarily ab initio based [@WF1; @WF2], can be used to construct such Wannier functions. In the usual “long-wavelength limit” of optics, where the applied electric field is varying in time but its variation in space is neglected, we recover the standard results for crystalline solids. In other instances where comparison with the “modern theory” is possible, such as the response of the polarization to a static or uniform applied electric field, or the response of systems expected to exhibit the quantum anomalous Hall effect, we also find agreement [@Mahon2019].
The approach we implement has the advantage that it can be employed to describe the effects of generally spatially varying, time-dependent applied electromagnetic fields [^1]. Exploring its characterization of this generalized optical response, in the linear and nonlinear regimes, is our main program. There is, however, an interesting overlap of our program with that of the “modern theory,” and that is in calculating the static response of the polarization to an applied, uniform magnetic field, and the static response of the magnetization to an applied, uniform electric field. This phenomenon is termed the *magnetoelectric effect*, and for a class of insulators [^2] in the “frozen-ion” approximation, where spin contributions are also neglected, both of these responses are characterized by the orbital magnetoelectric polarizability (OMP) tensor [@Qi2008; @Vanderbilt2009; @Malashevich2010; @Essin2010; @Swiecicki2014; @VanderbiltBook], $$\begin{aligned}
\alpha^{il}=\left.\frac{\partial P^i}{\partial B^l}\right|_{\substack{\boldsymbol{E}=0\\\boldsymbol{B}=0}}=\left.\frac{\partial M^l}{\partial E^i}\right|_{\substack{\boldsymbol{E}=0\\\boldsymbol{B}=0}}, \label{OMP}\end{aligned}$$ to first-order in the applied fields. Unlike its generalization to finite frequency, the OMP tensor is non-vanishing only when both spatial-inversion and time-reversal symmetry are broken in the unperturbed system [^3], and it is composed of two distinct terms: the Chern-Simons contribution and the cross-gap contribution. The former is isotropic and entirely a property of the subspace of originally occupied states, while the latter involves both occupied and excited states of the unperturbed system. The Chern-Simons contribution has generated particular interest in the literature because of its topological features; there is a discrete ambiguity in its value, which can be used to identify $\mathbb{Z}_{2}$ topological insulators [@VanderbiltBook; @Kane2010; @Tokura2019]. As well, while the analytic structure of the cross-gap contribution is of the form one would expect to find from the usual treatment of linear response using a Kubo formalism, the Chern-Simons contribution takes a rather unexpected form.
The expression for the OMP tensor found via the “modern theory” is well established [@Essin2010; @Malashevich2010]. In this paper we present a calculation of the OMP tensor within our framework of identifying microscopic polarization and magnetization fields. It is a special case of our general approach, in which by using a set of orthogonal functions that are well-localized spatially one can associate a portion of a total quantity with the point about which each of these functions is localized; a total quantity can be decomposed into “site” contributions. With this, our goal is to formulate the relation of these “site” quantities to the applied electric and magnetic fields evaluated at that site. The OMP tensor is extracted upon taking these electric and magnetic fields to be uniform in space and independent of time. Our results are in complete agreement with those of the “modern theory,” as would be expected, but in the process we achieve some insight into the microscopic origin of the distinct contributions to the response. In particular, we can compare our results for the OMP tensor with what would be expected in a “molecular crystal limit,” a model in which on each lattice site there is a molecule with orbitals that share no common support with the orbitals of molecules on other lattice sites. And with the development of the formalism presented here we position ourselves to extend this approach to describe material response at finite frequency.
After some preliminary discussion to begin Sec. \[Sect:1\], we extend the formalism [@Mahon2019] where necessary in order to calculate the response of the site quantities to arbitrary applied electromagnetic fields. This is a very general development, and only in the later sections do we restrict ourselves to the limit of uniform and static applied electric and magnetic fields. The calculations are made in Sec. \[Sect:2\]; in Sec. \[Sect:3\] we show that the accepted expression for the OMP tensor is reproduced. We also calculate the OMP tensor in the molecular crystal limit by two approaches. The first is a direct molecular physics calculation, and the second is by taking the appropriate limit of our general expressions; they agree, as they should. We also discuss the nature of both the Chern-Simons and cross-gap contributions from the perspective of this microscopic theory. In Sec. \[Sect:Conclusion\] we conclude.
Perturbative modifications to the single-particle density matrix {#Sect:1}
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The electronic response of a crystalline insulator is a consequence of the evolution of the fermionic electron field operator, $\widehat{\psi}(\boldsymbol{x},t)$. We assume that, in the Heisenberg picture, the dynamics of this object is governed by $$\begin{aligned}
i\hbar\frac{\partial\widehat{\psi}(\boldsymbol{x},t)}{\partial t}=\Big[H_{0}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}_{\text{mc}}(\boldsymbol{x},t)\big)+e\phi(\boldsymbol{x},t)\Big]\widehat{\psi}(\boldsymbol{x},t), \label{fieldEvo}\end{aligned}$$ where $e=-|e|$ is the electron charge, $$\begin{aligned}
\boldsymbol{\mathfrak{p}}_{\text{mc}}(\boldsymbol{x},t)=\boldsymbol{\mathfrak{p}}(\boldsymbol{x})-\frac{e}{c}\boldsymbol{A}(\boldsymbol{x},t),\end{aligned}$$ and $H_0\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)$ is the differential operator that governs the dynamics of the electron field in the unperturbed infinite crystal. The presence of an applied classical electromagnetic field, characterized by its vector and scalar potentials, $\boldsymbol{A}(\boldsymbol{x},t)$ and $\phi(\boldsymbol{x},t)$, has been included through the usual minimal coupling prescription. In writing (\[fieldEvo\]) the independent particle approximation is made, neglecting any interactions other than between the electron field and the applied electric and magnetic fields, which are taken to be the macroscopic Maxwell fields. These external fields are assumed to be applied at times greater than an initial time at which the system is taken to be in its unperturbed ground state, and the expectation values of pairs of (Heisenberg) field operators $\widehat{\psi}(\boldsymbol{x},t)$ and their adjoints in the unperturbed ground state are used to construct the minimal coupling Green functions [@Mahon2019].
Implementing the frozen-ion approximation, we take $$\begin{aligned}
H_0\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)=\frac{\big(\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)^{2}}{2m}+V(\boldsymbol{x}),\label{H}\end{aligned}$$ where $V(\boldsymbol{x})$ is the spatially periodic lattice potential that characterizes the crystal structure and satisfies $V(\boldsymbol{x})=V(\boldsymbol{x}+\boldsymbol{R})$ for all Bravais lattice vectors $\boldsymbol{R}$, and $$\begin{aligned}
\boldsymbol{\mathfrak{p}}(\boldsymbol{x})=\frac{\hbar}{i}\boldsymbol{\nabla}-\frac{e}{c}\boldsymbol{A}_{\text{static}}(\boldsymbol{x}).
\label{physicalP}\end{aligned}$$ In (\[physicalP\]) we have allowed for the presence of an “internal,” static, cell-periodic magnetic field described by the vector potential $\boldsymbol{A}_{\text{static}}(\boldsymbol{x})$, where $\boldsymbol{A}_{\text{static}}(\boldsymbol{x})=\boldsymbol{A}_{\text{static}}(\boldsymbol{x}+\boldsymbol{R})$. The inclusion of such an “internal” field respects the discrete translational symmetry of the crystal, but generically leads to a Hamiltonian (\[H\]) with broken time-reversal symmetry, which will be important in what follows. In future publications we plan to include both the spin-orbit and Coulomb interactions that we neglect here.
In periodic systems, a set of exponentially localized Wannier functions (ELWFs), $\{W_{\alpha\boldsymbol{R}}(\boldsymbol{x})\equiv\braket{\boldsymbol{x}}{\alpha\boldsymbol{R}}\}$, can generally be constructed [@Brouder; @Marzari2012; @Vanderbilt2012; @Panati; @Troyer2016] via $$\begin{aligned}
\ket{\alpha\boldsymbol{R}}=\sqrt{\frac{\Omega_{uc}}{(2\pi)^3}}\int_{\text{BZ}}d\boldsymbol{k}e^{-i\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}}\sum_{n}U_{n\alpha}(\boldsymbol{k})\ket{\psi_{n\boldsymbol{k}}}, \label{WF}\end{aligned}$$ where $\Omega_{uc}$ is the unit cell volume, and $$\begin{aligned}
\psi_{n\boldsymbol{k}}(\boldsymbol{x})\equiv\braket{\boldsymbol{x}}{\psi_{n\boldsymbol{k}}}=\frac{1}{\sqrt{(2\pi)^3}}e^{i\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{x}}u_{n\boldsymbol{k}}(\boldsymbol{x})\end{aligned}$$ are eigenfunctions of (\[H\]) that are normalized over the infinite crystal such that $\braket{\psi_{m\boldsymbol{k}'}}{\psi_{n\boldsymbol{k}}}=\delta_{nm}\delta(\boldsymbol{k}-\boldsymbol{k}')$. A periodic gauge choice is made such that the Bloch eigenvectors $\ket{\psi_{n\boldsymbol{k}}}$ and the unitary matrix $U(\boldsymbol{k})$ are periodic over the first Brillouin zone [^4]. Associated with each Bloch eigenfunction $\psi_{n\boldsymbol{k}}(\boldsymbol{x})$ is an energy $E_{n\boldsymbol{k}}$ and a cell-periodic function $u_{n\boldsymbol{k}}(\boldsymbol{x})\equiv\braket{\boldsymbol{x}}{{n\boldsymbol{k}}}$ satisfying the orthogonality relation $\left({m\boldsymbol{k}}|{n\boldsymbol{k}}\right)=\delta_{nm}$; we adopt the notation $$\begin{aligned}
(g|h)\equiv\frac{1}{\Omega_{uc}}\int_{\Omega_{uc}}g^{*}(\boldsymbol{x})h(\boldsymbol{x})d\boldsymbol{x},\end{aligned}$$ for functions $g(\boldsymbol{x})\equiv\braket{\boldsymbol{x}}{g}$ and $h(\boldsymbol{x})\equiv\braket{\boldsymbol{x}}{h}$ that are periodic over a unit cell, where the integration is over any unit cell. Also, we restrict our study to three-dimensional systems [^5]. Here and below $n$ is a band index, $\alpha$ is a type index, and $\hbar\boldsymbol{k}$ denotes a crystal-momentum within the first Brillouin zone.
In this paper we initiate our considerations with the zero temperature ground state of an insulator, and consider the class of insulators for which the set of occupied (unoccupied) energy eigenfunctions map to a set of occupied (unoccupied) ELWFs. This class includes ordinary insulators [^6] and $\mathbb{Z}_2$ topological insulators [@Vanderbilt2012], but excludes, for example, Chern insulators [@Troyer2016]. Therefore, $U_{n\alpha}(\boldsymbol{k})\neq0$ only if both $n$ and $\alpha$ label occupied or unoccupied band and orbital types. We associate a filling factor $f_n$ with band $n$ that is either $0$ or $1$, and a filling factor $f_{\alpha}$ with orbital type $\alpha$ that is also either $0$ or $1$; the latter can be inferred directly from the occupancy of the bands used in the construction of a particular ELWF.
The ELWFs (\[WF\]) can generally be expressed as $$\begin{aligned}
\braket{\boldsymbol{x}}{\alpha\boldsymbol{R}}=\sqrt{\Omega_{uc}}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^{3}}e^{i\boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{x}-\boldsymbol{R})}\braket{\boldsymbol{x}}{\alpha\boldsymbol{k}},\end{aligned}$$ where another set of cell-periodic functions, $\{\braket{\boldsymbol{x}}{\alpha\boldsymbol{k}}\}$, have been introduced and are formed via $$\begin{aligned}
\ket{\alpha\boldsymbol{k}}=\sum_{n}U_{n\alpha}(\boldsymbol{k})\ket{n\boldsymbol{k}}.\end{aligned}$$ Here the $\{\braket{\boldsymbol{x}}{\alpha\boldsymbol{k}}\}$ are in general *not* the cell-periodic parts of energy eigenfunctions. The sets of vectors $\{\ket{\alpha\boldsymbol{k}}\}$ and $\{\ket{n\boldsymbol{k}}\}$ are related by a (in general) multiband gauge transformation characterized by $U(\boldsymbol{k})$ [@VanderbiltBook]. Although we use Roman and Greek subscripts in the notation $U_{n\alpha}(\boldsymbol{k})$, as we will primarily be considering a transformation from the cell-periodic part of energy eigenfunctions to cell-periodic functions that are not associated with energy eigenfunctions, this need not necessarily hold. Indeed, the simplest type of gauge transformation, although of course it would not generally lead to $\ket{\alpha\boldsymbol{k}}$ associated with ELWFs, is one that involves each band individually, and is of the form $U_{n\alpha}(\boldsymbol{k})=\delta_{n\alpha}e^{-i\lambda_{n}(\boldsymbol{k})}$, where, for a given $n$ and $\boldsymbol{k}$, $\lambda_{n}(\boldsymbol{k})\in\mathbb{R}$. Rather than a special limit of the general multiband transformation, this could be considered as simply a new choice of Bloch eigenstates; under Bloch’s theorem, energy eigenstates are uniquely defined only within a $\boldsymbol{k}$-dependent phase [@Marzari2012], even at $\boldsymbol{k}$ points where there is no degeneracy. Here, however, it is considered as one type of $U_{n\alpha}(\boldsymbol{k})$, associated with a gauge transformation of the U$(1)$ type. That is, we consider the vectors $\{\ket{n\boldsymbol{k}}\}$ fixed at the start, and use the term “gauge dependent” for quantities that depend generally on the $U_{n\alpha}(\boldsymbol{k})$ and their derivatives, including $U_{n\alpha}(\boldsymbol{k})$ of the U$(1)$ type.
The ELWFs are an important element of our approach, because we use them to introduce “site” quantities, and define the macroscopic polarization and magnetization in terms of their moments, as we discuss in detail below. Of course, not all gauge transformations of an initial set $\left\{\ket{n\boldsymbol{k}}\right\}$ will lead to ELWFs via (\[WF\]), as indicated by the example given above. Nonetheless, the response tensors, such as $\alpha^{il}$, must be such that the resulting charge and current densities in the bulk are not only invariant with respect to the choices of $U_{n\alpha}(\boldsymbol{k})$ that lead to ELWFs, but in fact to *all* choices of $U_{n\alpha}(\boldsymbol{k})$; that is, the charge and current densities in the bulk must be gauge invariant. This is plausible because it would be possible – although we would argue much less convenient and less interesting physically – to calculate those charge and current densities directly from the minimal coupling Hamiltonian without ever introducing Wannier functions. And we shall see that this gauge invariance does indeed hold.
A useful identity [@Marzari2012] is $$\begin{aligned}
\int W^*_{\beta\boldsymbol{R}}(\boldsymbol{x})x^aW_{\alpha\boldsymbol{0}}(\boldsymbol{x})d\boldsymbol{x}=\frac{\Omega_{uc}}{(2\pi)^{3}}\int_{\text{BZ}}d\boldsymbol{k} e^{i\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}}\tilde{\xi}^a_{\beta\alpha}(\boldsymbol{k}),
\label{firstMoment}\end{aligned}$$ where $$\begin{aligned}
\tilde{\xi}^a_{\beta\alpha}(\boldsymbol{k})\equiv i\left({\beta\boldsymbol{k}}|\partial_a{\alpha\boldsymbol{k}}\right)
\label{connectionWannier}\end{aligned}$$ is the non-Abelian Berry connection associated with the set $\{\ket{\alpha\boldsymbol{k}}\}$; here and below, superscript indices indicate Cartesian components [^7], repeated Cartesian components are summed over, and we adopt the shorthand $\partial_a\equiv\partial/\partial k^a$. The object (\[connectionWannier\]) is related to the non-Abelian Berry connection associated with the set $\{\ket{n\boldsymbol{k}}\}$, $$\begin{aligned}
\xi^a_{mn}(\boldsymbol{k})\equiv i\left({m\boldsymbol{k}}|\partial_a{n\boldsymbol{k}}\right),
\label{eq:Bloch_connection}\end{aligned}$$ via $$\begin{aligned}
\sum_{\alpha\beta}U_{m\beta}(\boldsymbol{k})\tilde{\xi}^a_{\beta\alpha}(\boldsymbol{k})U^{\dagger}_{\alpha n}(\boldsymbol{k})=\xi^a_{mn}(\boldsymbol{k})+\mathcal{W}^a_{mn}(\boldsymbol{k}). \label{connection}\end{aligned}$$ Here we have defined the Hermitian matrix $\mathcal{W}^a$ [@VanderbiltBook], populated by elements $$\begin{aligned}
\mathcal{W}^a_{mn}(\boldsymbol{k})\equiv i\sum\limits_{\alpha}\big(\partial_aU_{m\alpha}(\boldsymbol{k})\big)U^\dagger_{\alpha n}(\boldsymbol{k}), \label{W}\end{aligned}$$ which, for the class of insulators we consider, is non-zero only if $f_m=f_n$. Under the aforementioned periodic gauge choice, all objects appearing in (\[connection\]) are periodic over the first Brillouin zone. In what follows, the $\boldsymbol{k}$ dependence of the preceding objects is usually kept implicit.
A consequence of the Hamiltonian (\[H\]) and the resulting dynamics (\[fieldEvo\]) of the electron field operator is that the differential operators associated with the spatial components of the conserved current take the form $$\begin{aligned}
J^a_{\text{mc}}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x});t\big)=J^a\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}_{\text{mc}}(\boldsymbol{x},t)\big)=\frac{e}{m}\mathfrak{p}^a_{\text{mc}}(\boldsymbol{x},t),\end{aligned}$$ in the usual fashion [^8], where $J^a\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)$ are the analogous differential operators arising for the unperturbed system. Another useful identity is $$\begin{aligned}
\int\psi^*_{n'\boldsymbol{k}'}(\boldsymbol{x})\mathfrak{p}^a(\boldsymbol{x})\psi_{n\boldsymbol{k}}(\boldsymbol{x})d\boldsymbol{x}=\mathfrak{p}^a_{n'n}(\boldsymbol{k})\delta({\boldsymbol{k}-\boldsymbol{k}'}), \label{eq:full_integral}\end{aligned}$$ where the matrix elements are $$\begin{aligned}
&\mathfrak{p}^a_{n'n}(\boldsymbol{k})=\delta_{n'n}\frac{m}{\hbar}\partial_a E_{n\boldsymbol{k}}+\frac{im}{\hbar}\big(E_{n'\boldsymbol{k}}-E_{n\boldsymbol{k}}\big)\xi^a_{n'n}(\boldsymbol{k}). \label{pMatrixElements}\end{aligned}$$ This can be shown by breaking the integral in (\[eq:full\_integral\]) into the sum of integrals over unit cells; the sum over lattice vectors yields the Dirac delta function in (\[eq:full\_integral\]), and the expression for (\[pMatrixElements\]) follows from the form of the integral over the unit cell and the use of (\[eq:Bloch\_connection\]). Indeed, a more general form of (\[eq:full\_integral\],\[pMatrixElements\]) can be derived involving the matrix elements of $\mathfrak{p}^{a}(\boldsymbol{x})$ in the basis of the cell-periodic functions $\braket{\boldsymbol{x}}{\alpha\boldsymbol{k}} $ using the same strategy [^9].
In previous work [@Mahon2019] we considered the calculation of the expectation values of the electronic charge and current density operators for a crystalline insulator perturbed by an electromagnetic field. Noting that the lesser, equal time single-particle Green function can be used to find such quantities, we employed a set of spatially localized, “adjusted Wannier functions” as a basis in which to expand the electron field operator in an effort to associate portions of the full electronic Green function with individual lattice sites. Upon identifying such “site” Green functions, and thereby identifying “site” charge and current densities, we defined microscopic polarization and magnetization fields associated with each lattice site using the same functions that are used in atomic and molecular physics to relate the microscopic polarization and magnetization fields of atoms and molecules to the microscopic charge and current densities [^10]; we call these functions “relators.” The full polarization and magnetization fields are given by summing the respective site contributions. We then insisted that these microscopic polarization and magnetization fields, together with the electronic charge and current density expectation values, satisfy the expressions arising in classical macroscopic electrodynamics relating such quantities. This led to the identification of microscopic “free” electronic charge and current densities, which take predictable forms. At zero temperature the first-order modifications to both the free charge and current densities due to an applied electromagnetic fields vanish for the class of insulators considered here, even for applied fields in the x-ray regime [@Mahon2019]. As a consequence, the first-order perturbative modifications to the expectation values of the electronic charge and current density operators resulting from an applied electromagnetic field can be found directly from the first-order modifications to the microscopic polarization and magnetization fields.
A quantity central to the calculation of both the microscopic polarization and magnetization fields is the single-particle density matrix, $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)$. Thus, a starting point in describing the effect of an applied electromagnetic field to a crystalline insulator is identifying how the applied fields affect this object. The single-particle density matrix evolves according to [@Mahon2019] $$\begin{aligned}
& i\hbar\frac{\partial\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)}{\partial t}=\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t)\eta_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t),\label{EDM-eom}\end{aligned}$$ where $$\begin{aligned}
\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t)&=\delta_{\nu\beta}\delta_{\boldsymbol{R}_{2}\boldsymbol{R}'}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_1,\boldsymbol{R}';t)}\bar{H}_{\alpha\boldsymbol{R};\mu\boldsymbol{R}_{1}}(t)\nonumber\\
& -\delta_{\mu\alpha}\delta_{\boldsymbol{R}_{1}\boldsymbol{R}}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_2,\boldsymbol{R}';t)}\bar{H}_{\nu\boldsymbol{R}_{2};\beta\boldsymbol{R}'}(t)\nonumber \\
& -e\Omega^0_{\boldsymbol{R}'}(\boldsymbol{R},t)\delta_{\nu\beta}\delta_{\mu\alpha}\delta_{\boldsymbol{R}_{2}\boldsymbol{R}'}\delta_{\boldsymbol{R}_{1}\boldsymbol{R}}.\label{scriptF}\end{aligned}$$ The definitions of $\Delta(\boldsymbol{R}_1,\boldsymbol{R}_2,\ldots,\boldsymbol{R}_N;t)$, $\Omega^0_{\boldsymbol{R}'}(\boldsymbol{R},t)$, and $\bar{H}_{\alpha\boldsymbol{R};\mu\boldsymbol{R}_{1}}(t)$ are as given earlier [@Mahon2019], and are provided in the Appendices. The first of these quantities is related to the magnetic flux through the surface generated by connecting the points $(\boldsymbol{R}_1,\boldsymbol{R}_2,\ldots,\boldsymbol{R}_N)$ with straight lines, when the usual choice of straight-line paths for the relators is adopted. The second is related to the electric field along the path connecting points $(\boldsymbol{R}',\boldsymbol{R})$. The third quantity can be understood as a generalized “hopping” matrix element. Each of terms appearing in (\[scriptF\]) is gauge invariant in the electromagnetic sense, and consequently so is $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)$.
In Appendix \[AppendixA\] we show that $$\begin{aligned}
&e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\bar{H}_{\alpha\boldsymbol{R};\mu\boldsymbol{R}''}(t)=e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\bar{H}_{\alpha\boldsymbol{R};\mu\boldsymbol{R}''}(\boldsymbol{R}_{a},t)-e\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\delta_{\alpha\mu}\delta_{\boldsymbol{R}\boldsymbol{R}''}, \label{RaRefSite}\end{aligned}$$ and $$\begin{aligned}
&e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\bar{H}_{\mu\boldsymbol{R}'';\beta\boldsymbol{R}'}(t)=e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}_{b},\boldsymbol{R}';t)}\bar{H}_{\mu\boldsymbol{R}'';\beta\boldsymbol{R}'}(\boldsymbol{R}_{b},t)-e\Omega_{\boldsymbol{R}'}^{0}(\boldsymbol{R}_{b},t)\delta_{\mu\beta}\delta_{\boldsymbol{R}'\boldsymbol{R}''}, \label{RbRefSite}\end{aligned}$$ for *any* lattice sites $\boldsymbol{R}_{a}$ and $\boldsymbol{R}_{b}$. We have defined $$\begin{aligned}
&\bar{H}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(\boldsymbol{R}_{a},t)\equiv\nonumber\\
&\quad\int\chi_{\mu\boldsymbol{R}_{1}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R}_{1},\boldsymbol{x},\boldsymbol{R}_{a};t)}\left(\mathcal{H}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t)+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}_{1};t)}{\partial t}+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}_{2};t)}{\partial t}\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}_{2};t)}\chi_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x},t)d\boldsymbol{x}\nonumber \\
&\quad-\frac{i\hbar}{2}\int e^{i\Delta(\boldsymbol{R}_{1},\boldsymbol{x},\boldsymbol{R}_{a};t)}\left(\chi_{\mu\boldsymbol{R}_{1}}^{*}(\boldsymbol{x},t)\frac{\partial\chi_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x},t)}{\partial t}-\frac{\partial\chi_{\mu\boldsymbol{R}_{1}}^{*}(\boldsymbol{x},t)}{\partial t}\chi_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x},t)\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}_{2};t)}d\boldsymbol{x},\label{barHelements}
\end{aligned}$$
and $$\begin{aligned}
\mathcal{H}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t)=H_{0}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t)\big)-e\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t), \label{Hcal}\end{aligned}$$ where $$\begin{aligned}
\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t)=\boldsymbol{\mathfrak{p}}(\boldsymbol{x})-\frac{e}{c}\boldsymbol{\Omega}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t), \label{frakP}\end{aligned}$$ as before [@Mahon2019]. Here $\boldsymbol{\Omega}_{\boldsymbol{R}_a}(\boldsymbol{x},t)$ is related to the magnetic field along the path connecting points $(\boldsymbol{R}_a,\boldsymbol{x})$, and is defined in Appendix \[AppendixB\]. The functions in the set $\{\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)\}$ are generally not orthogonal, but they are related to the mutually orthogonal “adjusted Wannier functions” introduced earlier [@Mahon2019]. In the limit of a weak applied magnetic field, a perturbative expansion can be constructed for each $\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)$ [@Mahon2019], and the lowest order terms are $$\begin{aligned}
\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)&=W_{\alpha\boldsymbol{R}}(\boldsymbol{x})-\frac{i}{2}\sum_{\beta\boldsymbol{R}'}W_{\beta\boldsymbol{R}'}(\boldsymbol{x})\nonumber\\
&\times\left[\int W_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{y})\Delta(\boldsymbol{R}',\boldsymbol{y},\boldsymbol{R};t)W_{\alpha\boldsymbol{R}}(\boldsymbol{y})d\boldsymbol{y}\right]+\ldots\end{aligned}$$ Choosing $\boldsymbol{R}_{a}=\boldsymbol{R}_{b}$, one can then re-express (\[scriptF\]) as $$\begin{aligned}
\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t)&=\delta_{\nu\beta}\delta_{\boldsymbol{R}_{2}\boldsymbol{R}'}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}_{1},\boldsymbol{R}';t)}\bar{H}_{\alpha\boldsymbol{R};\mu\boldsymbol{R}_{1}}(\boldsymbol{R}_{a},t)\nonumber \\
&-\delta_{\mu\alpha}\delta_{\boldsymbol{R}_{1}\boldsymbol{R}}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{2},\boldsymbol{R}_{a},\boldsymbol{R}';t)}\bar{H}_{\nu\boldsymbol{R}_{2};\beta\boldsymbol{R}'}(\boldsymbol{R}_{a},t)\nonumber \\
&-\hbar\frac{\partial\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}';t)}{\partial t}\delta_{\nu\beta}\delta_{\mu\alpha}\delta_{\boldsymbol{R}_{2}\boldsymbol{R}'}\delta_{\boldsymbol{R}_{1}\boldsymbol{R}}.\end{aligned}$$
To prepare for later perturbative analysis, we expand all quantities in powers of the applied electromagnetic field such that $$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)&=\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(0)}+\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(t)+\ldots,\\
\bar{H}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(\boldsymbol{R}_{a},t)&=H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(0)}+\bar{H}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},t)+\ldots,\end{aligned}$$ etc., where the superscript $(0)$ denotes the contribution to the total quantity that is independent of the applied fields, the superscript $(1)$ denotes the contribution that is linear in the electric and magnetic fields, and so on. Using (\[EDM-eom\]), and equating terms appearing with the same powers of the applied fields, the zeroth-order term $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(0)}$ is found to satisfy the same equation of motion as the unperturbed single-particle density matrix, and so $$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(0)}=f_{\alpha}\delta_{\alpha\beta}\delta_{\boldsymbol{R}\boldsymbol{R}'}, \label{EDMunpert}\end{aligned}$$ while from (\[barHelements\]) it is found that $$\begin{aligned}
{H}^{(0)}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}=\int W^*_{\mu\boldsymbol{R}_{1}}(\boldsymbol{x})H_0\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)W_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x})d\boldsymbol{x}.
\label{unpertHelements}\end{aligned}$$ Next, implementing the usual Fourier series analysis via $$\begin{aligned}
g(t)=\sum_{\omega}g(\omega)e^{-i\omega t}, \label{Fouier}\end{aligned}$$ the first-order modification to the single-particle density matrix due to an applied electromagnetic field can be identified, via (\[EDM-eom\]), as
$$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(\omega)&= -\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}d\boldsymbol{k}'\frac{\braket{\alpha\boldsymbol{R}}{\psi_{i\boldsymbol{k}}}\braket{\psi_{i\boldsymbol{k}}}{\mu\boldsymbol{R}_{1}} H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},\omega)\braket{\nu\boldsymbol{R}_{2}}{\psi_{j\boldsymbol{k}'}}\braket{\psi_{j\boldsymbol{k}'}}{\beta\boldsymbol{R}'}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}-\hbar(\omega+i0^{+})}\nonumber\\
&+\frac{i}{2}f_{\beta\alpha}\int W_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x})\Big(\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};\omega)+\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}_{a};\omega)\Big)W_{\beta\boldsymbol{R}'}(\boldsymbol{x})d\boldsymbol{x} \label{EDM1}\end{aligned}$$
(Appendix \[AppendixD\]), where $f_{ji}\equiv f_j-f_i$ and $f_{\beta\alpha}\equiv f_{\beta}-f_{\alpha}$; recall that for the class of insulators of interest here there are well defined filling factors associated with the orbital type indices, but in general this is not so. In identifying (\[EDM1\]), we have also introduced $$\begin{aligned}
& H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},\omega)\equiv\int W_{\mu\boldsymbol{R}_{1}}^{*}(\boldsymbol{x})\mathcal{H}_{\boldsymbol{R}_{a}}^{(1)}(\boldsymbol{x},\omega)W_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x})d\boldsymbol{x}. \label{Hcorr}\end{aligned}$$
Often in optics one is interested in applied fields that vary little over electron correlation lengths which, for the class of insulators we consider, are on the order of the lattice constant. In Appendix \[AppendixB\] we show that if this approximation is made, and if $|\boldsymbol{y}-\boldsymbol{x}|$ and $|\boldsymbol{z}-\boldsymbol{x}|$ are on the order of lattice constants, then $$\begin{aligned}
\boldsymbol{\Omega}_{\boldsymbol{y}}(\boldsymbol{x},\omega)&\simeq\frac{1}{2}\boldsymbol{B}(\boldsymbol{y},\omega)\cross(\boldsymbol{x}-\boldsymbol{y}),\label{omegaVec}\\
\Omega_{\boldsymbol{y}}^{0}(\boldsymbol{x},\omega)&\simeq\left(\boldsymbol{x}-\boldsymbol{y}\right)\boldsymbol{\cdot}\boldsymbol{E}(\boldsymbol{y},\omega)\nonumber\\
&+\frac{1}{2}(x^{a}-y^{a})(x^{b}-y^{b})F^{ab}(\boldsymbol{y},\omega),
\label{omega0}\\
\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};\omega)&\simeq-\frac{e}{2\hbar c}\boldsymbol{B}(\boldsymbol{y},\omega)\boldsymbol{\cdot}\big[\left(\boldsymbol{x}-\boldsymbol{y}\right)\cross(\boldsymbol{z}-\boldsymbol{y})\big], \label{delta}\end{aligned}$$ where $\boldsymbol{E}(\boldsymbol{y},t)$ is the applied electric field, $\boldsymbol{B}(\boldsymbol{y},t)$ is the applied magnetic field, and $$\begin{aligned}
F^{ab}(\boldsymbol{y},t)\equiv\frac{1}{2}\frac{\partial E^{a}(\boldsymbol{y},t)}{\partial y^{b}}+\frac{1}{2}\frac{\partial E^{b}(\boldsymbol{y},t)}{\partial y^{a}}. \label{eq:Fdef}\end{aligned}$$ Furthermore, under this approximation, and implementing (\[H\]) and (\[Fouier\]), the first-order modification to (\[Hcal\]) is found to be $$\begin{aligned}
\mathcal{H}^{(1)}_{\boldsymbol{R}_{a}}(\boldsymbol{x},\omega)=-\frac{e}{mc}\Omega^a_{\boldsymbol{R}_a}(\boldsymbol{x},\omega)\mathfrak{p}^a(\boldsymbol{x})-e\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},\omega).\label{Hcal1}\end{aligned}$$ In the above we have used straight-line paths in the relators, and in what follows, this choice is always made. Eq. (\[EDM1\])-(\[Hcal1\]) make clear the motivation for introducing the arbitrary lattice site $\boldsymbol{R}_a$; it serves as a reference site for the applied electromagnetic field. This will prove useful when considering the response of a quantity associated with site $\boldsymbol{R}$ to a spatially varying electromagnetic field; choosing $\boldsymbol{R}_a=\boldsymbol{R}$, the modification to that site quantity is related to the applied field strength at that site.
However, in this paper we restrict ourselves to uniform applied electric and magnetic fields in the dc limit. Thus $F^{ab}(\boldsymbol{x},t)$ vanishes and $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(\omega)\neq0$ only if $\omega=0$. We keep only the non-vanishing, first-order perturbative modifications arising from the applied electric and magnetic fields, which we denote by $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(E)}$ and $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(B)}$, respectively, such that $$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(\omega=0)=\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(E)}+\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(B)}. \label{EDMlinear}\end{aligned}$$
Implementing (\[EDM1\]), the first-order perturbative modification to the single-particle density matrix due to a uniform applied dc electric field is found to be $$\begin{aligned}
&\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(E)}=eE^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\frac{e^{i\boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{R}-\boldsymbol{R}')}U^\dagger_{\alpha i}\xi^l_{ij}U_{j\beta}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}} \label{EDMe}\end{aligned}$$ (Appendix \[AppendixD\]), where $\boldsymbol{E}\equiv\boldsymbol{E}(\boldsymbol{R}_a,\omega=0)$, for any $\boldsymbol{R}_a$, is now the uniform dc electric field. The result (\[EDMe\]) is consistent with previous work [@Mahon2019], where this expression was derived via a different method. Notably (\[EDMe\]) is written as a single Brillouin zone integral, unlike (\[EDM1\]). This feature is expected upon comparison to the usual perturbative treatment, in which, when spatial variation of the electric field is neglected, a $\boldsymbol{k}$-conserving interaction term arises that results in perturbative modifications being given by single Brillouin zone integrals [@SipeBook]. While here we assume the applied electric field is uniform, more generally this reduction to a single $\boldsymbol{k}$ integral holds as a consequence of the approximation that the applied electric field is to vary little over electron correlation lengths, allowing a Taylor series expansion about each lattice site. In the limit of uniform applied fields of interest here, the $\boldsymbol{R}_{a}$ dependence of (\[EDMe\]) vanishes.
There are two distinct contributions to the corresponding first-order perturbative modification arising from a uniform applied dc magnetic field, $$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(B)}&=\frac{e}{4\hbar c}\epsilon^{lab}B^l\frac{\Omega_{uc}}{(2\pi)^{3}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}e^{i\boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{R}-\boldsymbol{R}')}U^\dagger_{\alpha i}{\mathscr{B}}^{ab}_{ij}(\boldsymbol{k})U_{j\beta}\nonumber\\
&+\frac{e}{4\hbar c}\epsilon^{lab}B^l\frac{\Omega_{uc}}{(2\pi)^{3}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}e^{i\boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{R}-\boldsymbol{R}')}\left\{\big(\partial_a U^\dagger_{\alpha i}\big)U_{j \beta}-U^\dagger_{\alpha i}\big(\partial_a U_{j\beta}\big)\right\}\xi^b_{ij}, \label{EDMb2}\end{aligned}$$ where we have defined $$\begin{aligned}
{\mathscr{B}}^{ab}_{ij}(\boldsymbol{k})&\equiv i\sum_{n}\left\{\frac{E_{n\boldsymbol{k}}-E_{j\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{in}\xi^b_{nj}+\frac{E_{n\boldsymbol{k}}-E_{i\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{in}\xi^b_{nj}
\right\}-2\frac{\partial_a(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^b_{ij},\end{aligned}$$ and where $\epsilon^{lab}$ is the Levi-Civita symbol; $\boldsymbol{B}\equiv\boldsymbol{B}(\boldsymbol{R}_{a},\omega=0)$, for any $\boldsymbol{R}_{a}$, is the uniform dc magnetic field. We mention that the two terms appearing in (\[EDMb2\]) are not simply the individual contributions of the terms of (\[EDM1\]). Moreover, in the first-order perturbative modifications of the quantities considered below, the first term of (\[EDMb2\]) gives rise to gauge invariant contributions, while the final term gives rise to gauge dependent contributions [^11].
Electronic response to applied uniform dc $\boldsymbol{E}$ and $\boldsymbol{B}$ fields {#Sect:2}
======================================================================================
We now use the perturbative modifications to the single-particle density matrix (\[EDMlinear\]) to calculate the first-order response of the electric and magnetic dipole moments – found from the microscopic polarization and magnetization fields, respectively – to applied uniform dc electric and magnetic fields. In this section we restrict earlier [@Mahon2019], more general expressions to this limit. Thus a function previously written in terms of frequency components $g(\omega)$ (\[Fouier\]), will simply be given by the single non-vanishing component $g\equiv g(\omega=0)$. Furthermore, as previously mentioned, to first-order in the applied fields, the free charge and current densities vanish at zero temperature for the class of insulators we consider [@Mahon2019], as would be expected physically, and so those quantities do not appear here.
Summary of formalism
--------------------
The microscopic polarization and magnetization fields can be decomposed into site contributions [@Mahon2019], such that $$\begin{aligned}
\boldsymbol{p}(\boldsymbol{x})&=\sum_{\boldsymbol{R}}\boldsymbol{p}_{\boldsymbol{R}}(\boldsymbol{x}), \nonumber\\
\boldsymbol{m}(\boldsymbol{x})&=\sum_{\boldsymbol{R}}\boldsymbol{m}_{\boldsymbol{R}}(\boldsymbol{x}), \end{aligned}$$ where the sums range over all Bravais lattice vectors $\boldsymbol{R}$. In the limit of applied uniform dc electric and magnetic fields treated perturbatively, the discrete translational symmetry that was present in the unperturbed Hamiltonian (\[H\]) is now lost (cf. (\[fieldEvo\])), but the polarization and magnetization fields associated with each lattice site are still physically equivalent; by this we mean $$\begin{aligned}
\boldsymbol{p}_{\boldsymbol{R}+\boldsymbol{R}'}(\boldsymbol{x})&=\boldsymbol{p}_{\boldsymbol{R}}(\boldsymbol{x}-\boldsymbol{R}'), \\
\boldsymbol{m}_{\boldsymbol{R}+\boldsymbol{R}'}(\boldsymbol{x})&=\boldsymbol{m}_{\boldsymbol{R}}(\boldsymbol{x}-\boldsymbol{R}'),\end{aligned}$$ for any $\boldsymbol{R}$ and $\boldsymbol{R}'$. This is a result of the fact that it is the electric and magnetic fields, *not* the vector and scalar potentials, that enter in the expressions that follow. Thus, in its perturbative response, the system retains its periodic nature in the limit of applied uniform dc fields.
Each site polarization field, $\boldsymbol{p}_{\boldsymbol{R}}(\boldsymbol{x})$, is related to the electronic charge density associated with that site, $\rho_{\boldsymbol{R}}(\boldsymbol{x})$, via the “relator” ${s}^i(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})$ (see Appendix \[AppendixB\]), as shown previously [@Mahon2019]. It is given by $$\begin{aligned}
{p}^i_{\boldsymbol{R}}(\boldsymbol{x})& \equiv\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int{s}^i(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})\rho_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber \\
&\quad\qquad\qquad\times\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'},\label{siteP}\end{aligned}$$ where above and below the sums range over all lattice vectors and orbital types, and $$\begin{aligned}
\rho_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x},\boldsymbol{R})&=\frac{e}{2}\big(\delta_{\boldsymbol{R}\boldsymbol{R}'}+\delta_{\boldsymbol{R}\boldsymbol{R}''}\big)e^{i\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}'')}\nonumber \\
&\qquad\times\chi_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{x})\chi_{\alpha\boldsymbol{R}''}(\boldsymbol{x}). \label{rhoElements}\end{aligned}$$ Meanwhile, there are two contributions to each site magnetization field $$\begin{aligned}
\boldsymbol{m}_{\boldsymbol{R}}(\boldsymbol{x})=\bar{\boldsymbol{m}}_{\boldsymbol{R}}(\boldsymbol{x})+\tilde{\boldsymbol{m}}_{\boldsymbol{R}}(\boldsymbol{x}).\label{siteM}\end{aligned}$$ The first of these, $\bar{\boldsymbol{m}}_{\boldsymbol{R}}(\boldsymbol{x})$, corresponds to the “local” or “atomic-like” contribution to each site magnetization field, and is related to the electronic current density associated with that site, $\boldsymbol{j}_{\boldsymbol{R}}(\boldsymbol{x})$, via the “relator” $\alpha^{ib}(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})$ (see Appendix \[AppendixB\]) [^12]. It is given by $$\begin{aligned}
\bar{m}_{\boldsymbol{R}}^{i}(\boldsymbol{x})&\equiv\frac{1}{c}\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int\alpha^{ib}(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})j_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{b}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber\\
&\quad\qquad\qquad\times\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'},\label{siteMbar}\end{aligned}$$ where $$\begin{aligned}
&\boldsymbol{j}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x},\boldsymbol{R})=\nonumber\\ &\frac{1}{4}\delta_{\boldsymbol{R}\boldsymbol{R}''}e^{i\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}'')}\chi_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{x})\Big[\boldsymbol{J}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R})\big)\chi_{\alpha\boldsymbol{R}''}(\boldsymbol{x})\Big]\nonumber\\
& +\frac{1}{4}\delta_{\boldsymbol{R}\boldsymbol{R}'}\Big[\boldsymbol{J}^{*}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R})\big)\chi_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{x})\Big]e^{i\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}'')}\chi_{\alpha\boldsymbol{R}''}(\boldsymbol{x})\nonumber \\
& +\frac{1}{4}\delta_{\boldsymbol{R}\boldsymbol{R}''}\left[\boldsymbol{J}^{*}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R})\big)\Big(e^{i\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}'')}\chi_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{x})\Big)\right]\chi_{\alpha\boldsymbol{R}''}(\boldsymbol{x})\nonumber \\
& +\frac{1}{4}\delta_{\boldsymbol{R}\boldsymbol{R}'}\chi_{\beta\boldsymbol{R}'}^{*}(\boldsymbol{x})\left[\boldsymbol{J}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R})\big)\Big(e^{i\Delta(\boldsymbol{R}',\boldsymbol{x},\boldsymbol{R}'')}\chi_{\alpha\boldsymbol{R}''}(\boldsymbol{x})\Big)\right].\label{jElements} \end{aligned}$$ It is not obvious that there is another contribution to each site magnetization field, as each atomic-like contribution is found from a site electronic current density, and these collectively compose the total current. However, in extended systems where the charge-current density associated with each site need *not* be conserved, there is in general an additional term, $\tilde{\boldsymbol{m}}_{\boldsymbol{R}}(\boldsymbol{x})$ [@Mahon2019]. Adopting the terminology of the “modern theory,” it corresponds to the “itinerant” contribution to each site magnetization field, and is given by $$\begin{aligned}
\tilde{m}_{\boldsymbol{R}}^{i}(\boldsymbol{x})&\equiv\frac{1}{c}\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int\alpha^{ib}(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})\tilde{j}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{b}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber\\
&\quad\qquad\qquad\times\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}. \label{siteMtilde}\end{aligned}$$ Here $$\begin{aligned}
& \boldsymbol{\tilde{j}}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x},\boldsymbol{R})=\frac{1}{2}\big(\delta_{\boldsymbol{R}\boldsymbol{R}'}+\delta_{\boldsymbol{R}\boldsymbol{R}''}\big)\boldsymbol{\tilde{j}}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x}), \label{jTildeSite}\end{aligned}$$ with $$\begin{aligned}
{\tilde{j}}^b_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x}) & =-\sum_{\boldsymbol{R}_3}\int{s}^b(\boldsymbol{x};\boldsymbol{z},\boldsymbol{R}_3)\Gamma_{\boldsymbol{R}_3}^{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}(\boldsymbol{z})d\boldsymbol{z}\nonumber \\
& -\frac{1}{2}\sum_{\boldsymbol{R}_1\boldsymbol{R}_2}{s}^b(\boldsymbol{x};\boldsymbol{R}_2,\boldsymbol{R}_1)\varsigma_{\boldsymbol{R}_2\boldsymbol{R}_1}^{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'},\label{jTilde}\end{aligned}$$ where $$\begin{aligned}
\varsigma_{\boldsymbol{R}_2\boldsymbol{R}_1}^{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'} &=\frac{e}{i\hbar}\Big(\delta_{\boldsymbol{R}'\boldsymbol{R}_2}\delta_{\boldsymbol{R}''\boldsymbol{R}_1}\bar{H}_{\beta\boldsymbol{R}_2;\alpha\boldsymbol{R}_1} \nonumber\\
& \quad\qquad-\delta_{\boldsymbol{R}''\boldsymbol{R}_2}\delta_{\boldsymbol{R}'\boldsymbol{R}_1}\bar{H}_{\beta\boldsymbol{R}_1;\alpha\boldsymbol{R}_2}\Big),\end{aligned}$$ and $$\begin{aligned}
\Gamma_{\boldsymbol{R}_3}^{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}(\boldsymbol{x})&=\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{j}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{x},\boldsymbol{R}_3)\nonumber\\
&+\frac{1}{i\hbar}\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\rho_{\nu\boldsymbol{R}_{2};\mu\boldsymbol{R}_{1}}(\boldsymbol{x},\boldsymbol{R}_3)\mathfrak{F}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}. \label{Gamma}\end{aligned}$$
Typically it is the response of the electric and magnetic dipole moments to external perturbations that one is interested in studying; these correspond to the spatial integrals of the microscopic polarization and magnetization fields [@Mahon2019] respectively, $$\begin{aligned}
\mu^i_{\boldsymbol{R}}\equiv\int p^i_{\boldsymbol{R}}(\boldsymbol{x})d\boldsymbol{x} \text{\space ,\space} \nu^i_{\boldsymbol{R}}\equiv\int m^i_{\boldsymbol{R}}(\boldsymbol{x})d\boldsymbol{x}.\end{aligned}$$ From (\[siteP\]) we find $$\begin{aligned}
\mu^i_{\boldsymbol{R}}&=\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int\big(y^i-R^i\big)\rho_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}, \label{mu}\end{aligned}$$ and $\boldsymbol{\nu}_{\boldsymbol{R}}=\bar{\boldsymbol{\nu}}_{\boldsymbol{R}}+\tilde{\boldsymbol{\nu}}_{\boldsymbol{R}}$, where, from (\[siteMbar\]), $$\begin{aligned}
\bar{\nu}^i_{\boldsymbol{R}}&=\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\frac{\epsilon^{iab}}{2c}\int\big(y^a-R^a\big)j_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{b}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber\\
&\quad\qquad\qquad\times\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'} \label{nuBar}\end{aligned}$$ is the atomic-like contribution to the magnetic dipole moment and, from (\[siteMtilde\]), $$\begin{aligned}
\tilde{\nu}^i_{\boldsymbol{R}}&=\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\frac{\epsilon^{iab}}{2c}\int\big(y^a-R^a\big)\tilde{j}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{b}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber\\
&\quad\qquad\qquad\times\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'} \label{nuTilde}\end{aligned}$$ is the itinerant contribution to the magnetic dipole moment. The macroscopic polarization and magnetization can be found from their respective dipole moments introduced above, and are taken to be $$\begin{aligned}
\boldsymbol{P}=\frac{\boldsymbol{\mu_R}}{\Omega_{uc}} \text{\space,\space} \boldsymbol{M}=\frac{\boldsymbol{\nu_R}}{\Omega_{uc}}. \label{PM}\end{aligned}$$ In the limit of applied uniform dc fields, both the electric and magnetic dipole moments are independent of $\boldsymbol{R}$, and as a consequence, the macroscopic polarization and magnetization are uniform.
Unperturbed expressions
-----------------------
We begin by confirming that our microscopic treatment yields the standard expressions for the unperturbed ground state macroscopic polarization and magnetization, more usually constructed from macroscopic arguments [@Resta2010]. While (\[mu\],\[nuBar\],\[nuTilde\]) have been defined to include only valence and conduction electron contributions, the contributions from ion cores can be identified as well (see Mahon *et al.* [@Mahon2019]). However, we focus only on the former contributions here.
Expanding (\[mu\]) in powers of the applied electromagnetic field, we find the zeroth-order term to be $$\begin{aligned}
\mu^{i(0)}_{\boldsymbol{R}}&=\sum\limits_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int \big(y^i-R^i\big)\rho^{(0)}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y} \right]\eta^{(0)}_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'} \nonumber\\
&=e\sum_{\alpha}f_{\alpha}\int W^*_{\alpha\boldsymbol{0}}(\boldsymbol{x})x^iW_{\alpha\boldsymbol{0}}(\boldsymbol{x})d\boldsymbol{x} \nonumber\\
&=e\frac{\Omega_{uc}}{(2\pi)^3}\sum_{j}f_{j}\int_{\text{BZ}}d\boldsymbol{k}\left(\xi^i_{jj}+\mathcal{W}^i_{jj}\right), \label{unpertP}\end{aligned}$$ where we have used (\[firstMoment\])-(\[connection\]). This corresponds to the unperturbed ground state electric dipole moment, and upon implementing (\[PM\]), the usual expression [@Resta1994] for $\boldsymbol{P}^{(0)}$ is reproduced. It is well known that the unperturbed macroscopic polarization is unique modulo a “quantum of ambiguity.” This ambiguity originates from the gauge dependence of (\[unpertP\]), and it has been shown that the gauge dependent term of (\[unpertP\]) contributes only to this “quantum” [@Resta1994]. Importantly, it is only the diagonal elements of the $\mathcal{W}^i$ matrix that appear in (\[unpertP\]), and as a result, even a U$(1)$ gauge transformation can give rise to this “quantum of ambiguity.” This is discussed further in Sec. \[Sect:3\].
Turning to the magnetic dipole moment, we expand (\[nuBar\]) and (\[nuTilde\]) in powers of the applied electromagnetic field. The zeroth-order terms are found to be $$\begin{aligned}
\bar{\nu}^{i(0)}_{\boldsymbol{R}}&=\sum_{\alpha\boldsymbol{R}'}f_{\alpha}\left[\frac{\epsilon^{iab}}{2c}\int\big(y^a-R^a\big)j^{b(0)}_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right] \nonumber \\
&=\frac{e}{2mc}\epsilon^{iab}\sum_{\alpha}f_{\alpha}\int W^*_{\alpha\boldsymbol{0}}(\boldsymbol{x})x^a\mathfrak{p}^b(\boldsymbol{x})W_{\alpha\boldsymbol{0}}(\boldsymbol{x})d\boldsymbol{x}, \label{nuBarUnpert}\end{aligned}$$ and $$\begin{aligned}
&\tilde{\nu}^{i(0)}_{\boldsymbol{R}}\nonumber\\
&=\sum_{\alpha\boldsymbol{R}'}f_{\alpha}\left[\frac{\epsilon^{iab}}{2c}\int\big(y^a-R^a\big)\tilde{j}^{b(0)}_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right] \nonumber \\
&=\frac{e}{2\hbar c}\epsilon^{iab}\sum_{\alpha\gamma\boldsymbol{R}_1}f_{\alpha}R_1^a\text{Im}\Big[H^{(0)}_{\alpha\boldsymbol{0};\gamma\boldsymbol{R}_1}\int W^*_{\gamma\boldsymbol{R}_1}(\boldsymbol{x})x^bW_{\alpha\boldsymbol{0}}(\boldsymbol{x})d\boldsymbol{x}\Big], \label{nuTildeUnpert}\end{aligned}$$ which, together, form the unperturbed ground state magnetic dipole moment, $\boldsymbol{\nu}^{(0)}_{\boldsymbol{R}}=\bar{\boldsymbol{\nu}}^{(0)}_{\boldsymbol{R}}+\tilde{\boldsymbol{\nu}}^{(0)}_{\boldsymbol{R}}$. Separately (\[nuBarUnpert\]) and (\[nuTildeUnpert\]) are “multiband gauge dependent.” By this we mean that only if there are intersections amongst occupied bands is it gauge dependent; in the isolated valence band limit both (\[nuBarUnpert\]) and (\[nuTildeUnpert\]) become gauge invariant. Nonetheless, even if the occupied bands are not isolated the sum of (\[nuBarUnpert\],\[nuTildeUnpert\]) is generally a gauge invariant quantity and thus there is no ambiguity in the value of the unperturbed macroscopic magnetization. Implementing (\[PM\]), the usual expression [@Resta2005; @Resta2006] for $\boldsymbol{M}^{(0)}$ is reproduced.
First-order perturbative modifications {#Sect:3c}
--------------------------------------
We now turn to the first-order modifications to $\boldsymbol{\mu}_{\boldsymbol{R}}$ and $\boldsymbol{\nu}_{\boldsymbol{R}}$ and thus, through (\[PM\]), to $\boldsymbol{P}$ and $\boldsymbol{M}$ due to an applied electromagnetic field. Generally, the “site quantities” we consider are of the form $$\begin{aligned}
\Lambda^i_{\boldsymbol{R}}& \equiv\sum_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\Lambda^i_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{R})\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}, \label{siteQuantity}\end{aligned}$$ where $\eta_{\alpha\boldsymbol{R''};\beta\boldsymbol{R'}}$ is the single-particle density matrix, and where $\Lambda_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{i}(\boldsymbol{R})$, which we call a “site quantity matrix element,” is given by $$\begin{aligned}
\Lambda^i_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{R})=\frac{1}{2}\big(\delta_{\boldsymbol{RR}'}+\delta_{\boldsymbol{RR}''}\big)\Lambda^i_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}; \label{siteElements}\end{aligned}$$ see (\[rhoElements\],\[mu\]) for $\boldsymbol{\mu}_{\boldsymbol{R}}$, and (\[jElements\],\[jTildeSite\],\[nuBar\],\[nuTilde\]) for $\boldsymbol{\nu}_{\boldsymbol{R}}$.
### Dynamical and compositional modifications
The first-order modification to (\[siteQuantity\]) due to an applied electromagnetic field thus has two types of contributions, $$\begin{aligned}
\Lambda_{\boldsymbol{R}}^{i(1)}=\Lambda_{\boldsymbol{R}}^{i(1;\text{I})}+\Lambda_{\boldsymbol{R}}^{i(1;\text{II})},\label{siteModification}\end{aligned}$$ the first arising from the combination of the unperturbed expression for $\Lambda_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{i}(\boldsymbol{R})$ and the first-order modification to $\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}$ due to the applied fields, $$\begin{aligned}
\Lambda_{\boldsymbol{R}}^{i(1;\text{I})}\equiv\frac{1}{2}\sum_{\alpha\beta\boldsymbol{R}'}\left(\Lambda_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{i(0)}\eta_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}}^{(1)}+\Lambda_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}}^{i(0)}\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}\right),\label{lambdaDynamical}\end{aligned}$$ and the second from the combination of the first-order modification to $\Lambda_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}^{i}(\boldsymbol{R})$ due to the applied fields and the unperturbed expression for $\eta_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}$, (\[EDMunpert\]), $$\begin{aligned}
\Lambda_{\boldsymbol{R}}^{i(1;\text{II})}\equiv\sum_{\alpha\boldsymbol{R}'}f_{\alpha}\Lambda_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}^{i(1)}(\boldsymbol{R})=\sum_{\alpha}f_{\alpha}\Lambda_{\alpha\boldsymbol{R};\alpha\boldsymbol{R}}^{i(1)}.\label{lambdaSite}\end{aligned}$$
We refer to the first contribution (\[lambdaDynamical\]) as “dynamical” because it arises from modifications to the single-particle density matrix, which characterizes the electron hopping amplitude between various lattice sites and orbital types, due to the applied fields. Notably in the response of the site quantity associated with $\boldsymbol{R}$ this lattice site always appears as at least one lattice site in the single-particle density matrix components. This is expected physically; the site quantity associated with $\boldsymbol{R}$ is affected by electrons moving between different orbital types at that lattice site, and by electrons moving from the region nearest $\boldsymbol{R}$ to regions nearest other $\boldsymbol{R}'$. Contributions to (\[lambdaDynamical\]) arising from $\eta_{\beta\boldsymbol{R'};\alpha\boldsymbol{R}}$ and $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R'}}$, with $\boldsymbol{R'}\neq\boldsymbol{R}$, are a consequence of the extended nature of the system, in which electrons are not confined to regions of space; such contributions vanish in the limit that the crystalline solid is considered simply as a periodic array of “isolated molecules,” which we call the “molecular crystal limit” (see Sec. \[Sect:limits\]). Conversely, contributions to (\[lambdaDynamical\]) arising from $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}}$ take the form of single-site modifications.
The second contribution (\[lambdaSite\]) to the first-order modification of a site quantity associated with $\boldsymbol{R}$ depends on the first-order modification to the site quantity matrix element associated only with lattice site $\boldsymbol{R}$, $\Lambda_{\alpha\boldsymbol{R};\alpha\boldsymbol{R}}^{i(1)}$, and with orbital types $\alpha$ that are originally occupied. It is not associated with any change in the single-particle density matrix, but rather with the dependence of the associated site quantity matrix elements on the applied fields themselves; thus we call it a “compositional” modification. It leads to a dependence of the moments $\boldsymbol{\mu}_{\boldsymbol{R}}$ and $\boldsymbol{\nu}_{\boldsymbol{R}}$ on those fields, even though in this contribution (\[lambdaSite\]) the populations remain as they were before the fields were applied. There is a familiar analog to this in the response of an atom to an applied magnetic field. Considering a single electron, the initial operator for the magnetic dipole moment $\boldsymbol{\nu}_{\text{atom}}=(e/2mc)\boldsymbol{\mathfrak{X}}\cross\boldsymbol{\mathfrak{P}},$ where here $\boldsymbol{\mathfrak{X}}$ and $\boldsymbol{\mathfrak{P}}$ are the position and momentum operators of the electron, becomes $\boldsymbol{\nu}_{\text{atom}}\rightarrow(e/2mc)\boldsymbol{\mathfrak{X}}\cross(\boldsymbol{\mathfrak{P}}-e\boldsymbol{A}(\boldsymbol{\mathfrak{X}})/c)$ when a magnetic field is applied. For a uniform applied magnetic field we can take $\boldsymbol{A}(\boldsymbol{\mathfrak{X}})=(\boldsymbol{B}\cross\boldsymbol{\mathfrak{X}})/2$, giving $$\begin{aligned}
\boldsymbol{\nu}_{\text{atom}}=\frac{e}{2mc}\boldsymbol{\mathfrak{X}}\cross\boldsymbol{\mathfrak{P}}-\frac{e^{2}}{4mc^{2}}\Big(\mathfrak{X}^{2}\boldsymbol{B}-(\boldsymbol{\mathfrak{X}}\boldsymbol{\cdot}\boldsymbol{B})\boldsymbol{\mathfrak{X}}\Big). \label{eq:diamagnetic}\end{aligned}$$ The second term gives a contribution when the expectation value is taken, even in the ground state (say a $1s$ orbital), and gives the diamagnetic response of the atom. The contributions from the compositional modification to (\[siteModification\]) are of this form. Nonetheless, in the extended systems we consider it is important to note that during the perturbative analysis many lattice sites and all orbital types may be involved; for instance, observe that (\[jTildeSite\])-(\[Gamma\]) would be used in constructing (\[nuTilde\]).
We also distinguish between the first-order modifications arising from the applied electric field and the applied magnetic field, such that $$\begin{aligned}
\Lambda_{\boldsymbol{R}}^{i(1)}=\Lambda_{\boldsymbol{R}}^{i(E)}+\Lambda_{\boldsymbol{R}}^{i(B)},\end{aligned}$$ where each of these modifications is composed of a dynamical and a compositional term, $$\begin{aligned}
\Lambda_{\boldsymbol{R}}^{i(E)}&=\Lambda_{\boldsymbol{R}}^{i(E;\text{I})}+\Lambda_{\boldsymbol{R}}^{i(E;\text{II})}, \\ \Lambda_{\boldsymbol{R}}^{i(B)}&=\Lambda_{\boldsymbol{R}}^{i(B;\text{I})}+\Lambda_{\boldsymbol{R}}^{i(B;\text{II})}.\end{aligned}$$
### Induced polarization
#### Response to electric field.
We begin by considering modifications due to an applied electric field. From (\[rhoElements\]), and the fact that $\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x})$ only depends on the applied magnetic field and $\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R})=0$, it is clear that $$\begin{aligned}
\rho^{(E)}_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}(\boldsymbol{y},\boldsymbol{R})=0, \label{rhoE}\end{aligned}$$ and so there is no compositional modification to $\boldsymbol{\mu}^{(E)}_{\boldsymbol{R}}$. The first-order modification is entirely dynamical, as described above, and given by $$\begin{aligned}
\mu_{\boldsymbol{R}}^{i(E)}&=\sum\limits_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int \big(y^i-R^i\big)\rho^{(0)}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y} \right]\eta^{(E)}_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'}\nonumber\\
&=e^2E^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\frac{\xi^l_{ij}\xi^i_{ji}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}.\label{muE}\end{aligned}$$ Implementing (\[PM\]), we find the usual result from perturbation theory [@Aversa]. This modification is gauge invariant, in that the final line of (\[muE\]) is independent of the unitary transformation $U_{n\alpha}(\boldsymbol{k})$ [^13].
#### Response to magnetic field.
The first-order modification due to an applied magnetic field has non-vanishing dynamical and compositional modifications. Considering first the compositional modification, we find $$\begin{aligned}
\mu_{\boldsymbol{R}}^{i(B;\text{II})}
&=\sum\limits_{\alpha\boldsymbol{R}'}f_\alpha\left[\int \big(y^i-R^i\big)\rho^{(B)}_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y}\right]\nonumber\\
&=\frac{e^2}{2\hbar c}\epsilon^{lab}B^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{\alpha\gamma}f_\alpha\int_{\text{BZ}}d\boldsymbol{k} \text{Re}\big[\tilde{\xi}^i_{\alpha\gamma}\partial_b \tilde{\xi}^a_{\gamma\alpha}\big].
\label{muBb}\end{aligned}$$ We later simplify this term; notably it is gauge dependent. Recalling the magnetic field modifications to the single-particle density matrix (\[EDMb2\]), we find $$\begin{aligned}
&\mu_{\boldsymbol{R}}^{i(B;\text{I})}\nonumber\\
&=\sum\limits_{\alpha\beta\boldsymbol{R}'\boldsymbol{R}''}\left[\int \big(y^i-R^i\big)\rho^{(0)}_{\beta\boldsymbol{R}';\alpha\boldsymbol{R}''}(\boldsymbol{y},\boldsymbol{R})d\boldsymbol{y} \right]\eta^{(B)}_{\alpha\boldsymbol{R}'';\beta\boldsymbol{R}'} \nonumber\\
&=\frac{e^2}{2\hbar c}\epsilon^{lab}B^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\Bigg\{\frac{\partial_b(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{ij}\xi^i_{ji}\nonumber\\
&\quad\qquad\qquad+\sum_{n}\frac{E_{n\boldsymbol{k}}-E_{j\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^a_{in}\xi^b_{nj}\xi^i_{ji}\big]\Bigg\} \nonumber\\
&+\frac{e^2}{2\hbar c}\epsilon^{lab}B^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ijn}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\text{Re}\big[i\xi^i_{jn}\mathcal{W}^a_{ni}\xi^b_{ij}\big],\label{muBa}\end{aligned}$$ which is also gauge dependent. The two sets of terms appearing in the final equality of (\[muBa\]) originate individually from the two terms of (\[EDMb2\]). In going from the first to the final equality we have implemented (\[firstMoment\],\[connection\]), and used $\mathcal{W}^i_{mn}\neq 0$ only if $f_m= f_n$, which holds for the class of insulators considered here. Similar arguments are used in the following subsection when finding (\[nuBarE\],\[nuTildeEa\]). In Sec. \[Sect:3\] we explicitly combine (\[muBb\]) and (\[muBa\]), and show that the usual OMP tensor is reproduced [@Essin2010].
### Induced magnetization
In this work, we only consider modifications to the site magnetic dipole moment arising from an applied electric field. We defer to a later study the response of the magnetic dipole moment to a magnetic field, as considered in this framework.
#### Response of atomic-like contribution to electric field.
This involves the first-order modification to (\[nuBar\]) due to an applied electric field. There is no compositional modification, as from (\[jElements\]) it is clear that $$\begin{aligned}
j^{b(E)}_{\alpha\boldsymbol{R}';\alpha\boldsymbol{R}'}(\boldsymbol{y},\boldsymbol{R})=0,\end{aligned}$$ following the argument leading to (\[rhoE\]), and so this modification is entirely dynamical. Using (\[EDMe\]) we find $$\begin{aligned}
&\bar{\nu}_{\boldsymbol{R}}^{i(E)}= \nonumber \\
&\quad\frac{e^2}{4\hbar c}\epsilon^{iab}E^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\Bigg\{\frac{\partial_b(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{ji}\xi^l_{ij}\nonumber\\
&\quad\qquad\qquad+2\sum_{n}\frac{E_{n\boldsymbol{k}}-E_{i\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^a_{jn}\xi^b_{ni}\xi^l_{ij}\big]\Bigg\} \nonumber\\
&\quad+\frac{e^2}{2\hbar c}\epsilon^{iab}E^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ijn}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\Bigg\{\text{Re}\big[i\xi^l_{ij}\mathcal{W}^b_{jn}\xi^a_{ni}\big]\nonumber\\
&\quad\qquad\qquad+\frac{E_{n\boldsymbol{k}}-E_{j\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^l_{ij}\mathcal{W}^a_{jn}\xi^b_{ni}\big]\Bigg\}. \label{nuBarE}\end{aligned}$$
#### Response of itinerant contribution to electric field.
This involves the first-order modification to (\[nuTilde\]) due to an applied electric field, and has non-vanishing modifications of both compositional and dynamical origin. The compositional modification is $$\begin{aligned}
&\tilde{\nu}_{\boldsymbol{R}}^{i(E;\text{II})}=\frac{e^2}{2\hbar c}\epsilon^{iab}E^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{\alpha\gamma}f_\alpha\int_{\text{BZ}}d\boldsymbol{k} \text{Re}\big[\tilde{\xi}^l_{\alpha\gamma}\partial_b \tilde{\xi}^a_{\gamma\alpha}\big],\label{nuTildeEb}\end{aligned}$$ and the dynamical modification is $$\begin{aligned}
&\tilde{\nu}_{\boldsymbol{R}}^{i(E;\text{I})}=\nonumber\\
&\quad\frac{e^2}{4\hbar c}\epsilon^{iab}E^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\Bigg\{\frac{\partial_b(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{ji}\xi^l_{ij}\nonumber\\
&\quad\qquad\qquad-2\sum_{n}\frac{E_{n\boldsymbol{k}}-E_{j\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^l_{ij}\mathcal{W}^a_{jn}\xi^b_{ni}\big]\Bigg\}. \label{nuTildeEa}\end{aligned}$$
Both (\[nuBarE\]) and (\[nuTildeEb\]) are gauge dependent in general, while (\[nuTildeEa\]) is multiband gauge dependent. Very generally, there is a simplification that occurs when (\[nuBarE\],\[nuTildeEb\],\[nuTildeEa\]) are summed to form the total magnetic dipole moment: the term appearing in the final line of (\[nuTildeEa\]) cancels with the term appearing in the final line of (\[nuBarE\]), and as a result the gauge dependent terms appearing in the total $\boldsymbol{\nu}_{\boldsymbol{R}}^{(E)}$ do not explicitly depend on the energies $E_{n\boldsymbol{k}}$.
Microscopic origin of contributions to the OMP tensor {#Sect:3}
=====================================================
Constructing the OMP tensor
---------------------------
The OMP tensor, which characterizes the first-order response of the macroscopic polarization to an external uniform dc magnetic field, is defined through $$\begin{aligned}
P^{i(B)}=\alpha^{il}B^l,\end{aligned}$$ and, from (\[PM\]), (\[muBb\]), and (\[muBa\]), is found to be $$\begin{aligned}
\alpha^{il}&=\frac{e^2}{2\hbar c}\epsilon^{lab}\sum_{ij}f_{ji}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\Bigg\{\frac{\partial_b(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\xi^a_{ji}\xi^i_{ij}\nonumber\\
&\quad\qquad\qquad+\sum_{n}\frac{E_{n\boldsymbol{k}}-E_{i\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^a_{jn}\xi^b_{ni}\xi^i_{ij}\big]\Bigg\} \nonumber\\
&+\frac{e^2}{2\hbar c}\epsilon^{lab}\sum_{ijn}f_{ji}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\text{Re}\big[i\xi^i_{ij}\mathcal{W}^b_{jn}\xi^a_{ni}\big]\nonumber\\
&+\frac{e^2}{2\hbar c}\epsilon^{lab}\sum_{\alpha\gamma}f_\alpha \int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\text{Re}\big[\tilde{\xi}^i_{\alpha\gamma}\partial_b \tilde{\xi}^a_{\gamma\alpha}\big],\label{alpha}\end{aligned}$$ after some manipulation of the band indices. If we now define an analogous tensor characterizing the first-order response of the macroscopic magnetization to an applied uniform dc electric field, we find that, upon combining (\[nuBarE\]), (\[nuTildeEb\]), and (\[nuTildeEa\]), this response is described by the same $\alpha$ tensor introduced above, but with the order of the indices switched, such that $$\begin{aligned}
M^{i(E)}=\alpha^{li}E^l.\end{aligned}$$ This is the usual result in the $\hbar\omega\ll E_{\text{gap}}$ limit, which is effectively the condition we have initially assumed.
In the previous section we made a concerted effort to identify dynamical and compositional contributions to the various first-order modifications; distinguishing between these proves useful here. Because the $\alpha^{il}$ tensor characterizes the linear response of both $P^i$ to $B^l$, and $M^l$ to $E^i$, we can focus on the terminology associated with only the magnetization. In what follows, we illustrate how the three types of non-vanishing terms – the dynamical atomic-like, the compositional itinerant, and the dynamical itinerant modifications – combine to give an OMP tensor having the usual form $$\begin{aligned}
\alpha^{il}={\alpha}_{\text{G}}^{il}+\alpha_{\text{CS}}^{il}, \label{alphaDecomp}\end{aligned}$$ where ${\alpha}_{\text{G}}^{il}$ contains only cross-gap contributions to the response and $\alpha_{\text{CS}}^{il}$, the Chern-Simons contribution, is a property of the originally occupied subspace. We find that the cross-gap response, ${\alpha}_{\text{G}}^{il}$, originates from a combination of the dynamical atomic-like (\[nuBarE\]) and dynamical itinerant (\[nuTildeEa\]) terms, whereas $\alpha_{\text{CS}}^{il}$ has contributions from the dynamical atomic-like term as well, but also from the compositional itinerant term (\[nuTildeEb\]).
In order to proceed, we first re-express (\[alpha\]) in terms the cell-periodic functions $\braket{\boldsymbol{x}}{n\boldsymbol{k}}$, and re-write the sums to be over occupied states (the set $\{\ket{v\boldsymbol{k}}\}$) and unoccupied states (the set $\{\ket{c\boldsymbol{k}}\}$). Taking the term in (\[alpha\]) involving the ratio of energy differences to start, which can be traced back to (\[nuBarE\]), and adopting the shorthand $\ket{n}\equiv\ket{{n\boldsymbol{k}}}$, we find
$$\begin{aligned}
&\epsilon^{lab}\sum_{ijn}f_{ji}\frac{E_{n\boldsymbol{k}}-E_{i\boldsymbol{k}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}}\text{Re}\big[i\xi^a_{jn}\xi^b_{ni}\xi^i_{ij}\big]= \nonumber\\
&\quad 2\epsilon^{lab}\Bigg\llbracket-\sum_{cvv'}\frac{E_{v\boldsymbol{k}}-E_{v'\boldsymbol{k}}}{E_{v\boldsymbol{k}}-E_{c\boldsymbol{k}}}\text{Re}\big[\left(\partial_bv|v'\right)\left(\partial_av'|c\right)\left(c|\partial_iv\right)\big]+\sum_{cc'v}\frac{E_{c\boldsymbol{k}}-E_{c'\boldsymbol{k}}}{E_{v\boldsymbol{k}}-E_{c\boldsymbol{k}}}\text{Re}\big[\left(\partial_bv|c'\right)\left(c'|\partial_ac\right)\left(c|\partial_iv\right)\big]\Bigg\rrbracket\nonumber\\
&\quad+\epsilon^{lab}\Bigg\llbracket2\sum_{cvv'}\text{Re}\big[\left(\partial_iv|c\right)\left(c|\partial_av'\right)\left(v'|\partial_bv\right)\big]+\sum_{cv}\text{Re}\big[\left(\partial_iv|c\right)\left(\partial_ac|\partial_bv\right)\big]\Bigg\rrbracket.
\label{eq:work1}\end{aligned}$$
The first set of $\llbracket...\rrbracket$’s are identified as cross-gap contributions and will be included in ${\alpha}_{\text{G}}^{il}$. The second set of $\llbracket...\rrbracket$’s, together with the penultimate and final terms of (\[alpha\]), form $\alpha_{\text{CS}}^{il}$; the penultimate term of (\[alpha\]) originates from the gauge dependent term arising in the dynamical atomic-like modification (\[nuBarE\]) that does not explicitly depend on energy, and the final term of (\[alpha\]) originates from the compositional itinerant modification (\[nuTildeEb\]). We find $$\begin{aligned}
\alpha^{il}_{\text{CS}}&=\frac{e^2}{2\hbar c}\epsilon^{lab}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\Bigg\llbracket2\sum_{cvv'}\text{Re}\big[\left(\partial_iv|c\right)\left(c|\partial_av'\right)\left(v'|\partial_bv\right)\big]+\sum_{cv}\text{Re}\big[\left(\partial_iv|c\right)\left(\partial_ac|\partial_bv\right)\big]\Bigg\rrbracket\nonumber\\
&+\frac{e^2}{2\hbar c}\epsilon^{lab}\sum_{ijn}f_{ji}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\text{Re}\big[i\xi^i_{ij}\mathcal{W}^b_{jn}\xi^a_{ni}\big]+\frac{e^2}{2\hbar c}\epsilon^{lab}\sum_{\alpha\gamma}f_\alpha \int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\text{Re}\big[\tilde{\xi}^i_{\alpha\gamma}\partial_b \tilde{\xi}^a_{\gamma\alpha}\big] \nonumber\\
&=-\delta^{il}\frac{e^2}{2\hbar c}\epsilon^{abc}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\Bigg\{\left(\sum_{vv'}\xi^a_{vv'}\partial_b\xi^c_{v'v}-\frac{2i}{3}\sum_{vv'v_1}\xi^a_{vv'}\xi^b_{v'v_1}\xi^c_{v_1v}\right)+\sum_{vv'}(\partial_b\mathcal{W}^a_{vv'})\mathcal{W}^c_{v'v}-\frac{2i}{3}\sum_{vv'v_1}\mathcal{W}^a_{vv'}\mathcal{W}^b_{v'v_1}\mathcal{W}^c_{v_1v}\Bigg\}
\label{alphaCS}\end{aligned}$$ (Appendix \[AppendixCS\]), which is the usual Chern-Simons contribution to the OMP tensor [@Qi2008; @Vanderbilt2009; @Essin2010; @Malashevich2010], and is multiband gauge dependent in the sense introduced after Eq. (\[nuTildeUnpert\]). Due to this gauge dependence, this contribution to the OMP tensor is multivalued, but, like $\boldsymbol{P}^{(0)}$, has been shown to be unique modulo a quantum of indeterminacy [@VanderbiltBook]. Furthermore, this contribution is isotropic, and the corresponding quantity vanishes in systems of spatial dimension less than three.
The remaining terms compose ${\alpha}^{il}_{\text{G}}$, in accordance with (\[alphaDecomp\]). These terms originate from the dynamical atomic-like modification (\[nuBarE\]), and the dynamical itinerant modification (\[nuTildeEa\]). We find $$\begin{aligned}
{\alpha}^{il}_{\text{G}}&=\frac{e^2}{\hbar c}\epsilon^{lab}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^3}\Bigg\{-\sum_{cv}\frac{\partial_b( E_{c\boldsymbol{k}}+E_{v\boldsymbol{k}})}{E_{v\boldsymbol{k}}-E_{c\boldsymbol{k}}}\text{Re}\big[\left(\partial_av|c\right)\left(c|\partial_iv\right)\big]-\sum_{cvv'}\frac{E_{v\boldsymbol{k}}-E_{v'\boldsymbol{k}}}{E_{v\boldsymbol{k}}-E_{c\boldsymbol{k}}}\text{Re}\big[\left(\partial_bv|v'\right)\left(\partial_av'|c\right)\left(c|\partial_iv\right)\big]\nonumber\\
&\qquad\qquad\qquad\qquad\qquad+\sum_{cc'v}\frac{E_{c\boldsymbol{k}}-E_{c'\boldsymbol{k}}}{E_{v\boldsymbol{k}}-E_{c\boldsymbol{k}}}\text{Re}\big[\left(\partial_bv|c'\right)\left(c'|\partial_ac\right)\left(c|\partial_iv\right)\big]\Bigg\},
\label{alphaG}\end{aligned}$$
which is in agreement with the usual expression for the cross-gap contribution [@Essin2010; @Malashevich2010]. Importantly this contribution is found to be gauge invariant; this is a result of the fact that the multiband gauge dependent terms appearing in (\[nuBarE\]) and (\[nuTildeEa\]) cancelled one another.
It has been pointed out that the net gauge dependence of the OMP tensor, through the Chern-Simons contribution, has no effect on the charge-current density in the bulk [@Vanderbilt2009; @Essin2010]. We present a slightly different formulation of that argument here, but starting with the same assumption used in earlier arguments: While the derivation we have presented here, as well as that in the approach of the “modern theory,” holds strictly only for uniform dc applied fields, we can expect that for electric and magnetic fields varying very slowly in both space and time the same response tensors $\alpha_{\text{CS}}^{il}$ and $\alpha_{\text{G}}^{il}$ can be used to good approximation [^14]. For such slowly varying fields, the first-order response to the applied fields in the bulk of a medium can be expressed as $$\begin{aligned}
\boldsymbol{P}^{(1)}(\boldsymbol{x},t)&=\boldsymbol{P}'^{(1)}(\boldsymbol{x},t)+\alpha_{\text{CS}}\boldsymbol{B}(\boldsymbol{x},t),\nonumber\\
\boldsymbol{M}^{(1)}(\boldsymbol{x},t)&=\boldsymbol{M}'^{(1)}(\boldsymbol{x},t)+\alpha_{\text{CS}}\boldsymbol{E}(\boldsymbol{x},t), \label{eq:CSrelations} \end{aligned}$$ where we have taken $\alpha_{\text{CS}}^{il}=\delta^{il}\alpha_{\text{CS}}$, and at the level of linear response we have $$\begin{aligned}
P'^{i(1)}(\boldsymbol{x},t)&=\chi_{E}^{il}E^{l}(\boldsymbol{x},t)+\alpha_{\text{G}}^{il}B^{l}(\boldsymbol{x},t),\\
M'^{i(1)}(\boldsymbol{x},t)&=\chi_{B}^{il}B^{l}(\boldsymbol{x},t)+\alpha_{\text{G}}^{li}E^{l}(\boldsymbol{x},t),\end{aligned}$$ where $\chi_{E}^{il}$ and $\chi_{B}^{il}$ are the usual linear electric and magnetic susceptibilities. Inside the bulk material we then immediately find that the macroscopic charge and current densities that result, $$\begin{aligned}
\rho^{(1)}(\boldsymbol{x},t)&=-\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{P}^{(1)}(\boldsymbol{x},t),\nonumber\\
\boldsymbol{J}^{(1)}(\boldsymbol{x},t)&=\frac{\partial\boldsymbol{P}^{(1)}(\boldsymbol{x},t)}{\partial t}+c\boldsymbol{\nabla}\cross\boldsymbol{M}^{(1)}(\boldsymbol{x},t), \label{eq:Jrho_PM}\end{aligned}$$ can also be written as $$\begin{aligned}
\rho^{(1)}(\boldsymbol{x},t)&=-\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{P}'^{(1)}(\boldsymbol{x},t),\nonumber\\
\boldsymbol{J}^{(1)}(\boldsymbol{x},t)&=\frac{\partial\boldsymbol{P}'^{(1)}(\boldsymbol{x},t)}{\partial t}+c\boldsymbol{\nabla}\cross\boldsymbol{M}'^{(1)}(\boldsymbol{x},t). \label{eq:Jrho_PMtilde}\end{aligned}$$ That is, the contributions from the Chern-Simons response coefficient, $\alpha_{\text{CS}}$, completely cancel each other. In deriving (\[eq:Jrho\_PMtilde\]) from (\[eq:Jrho\_PM\]) we have used Faraday’s law and Gauss’ law for magnetism, $$\begin{gathered}
c\boldsymbol{\nabla}\cross\boldsymbol{E}(\boldsymbol{x},t)+\frac{\partial\boldsymbol{B}(\boldsymbol{x},t)}{\partial t}=0, \nonumber\\
\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B}(\boldsymbol{x},t)=0, \label{eq:FG} \end{gathered}$$ which of course must be assumed to hold for $\boldsymbol{E}(\boldsymbol{x},t)$ and $\boldsymbol{B}(\boldsymbol{x},t)$, no matter how slowly they are varying in space and time. Interestingly, the relation between $\{ \boldsymbol{P}^{(1)}(\boldsymbol{x},t),\boldsymbol{M}^{(1)}(\boldsymbol{x},t)\} $ and $\{ \boldsymbol{P}'^{(1)}(\boldsymbol{x},t),\boldsymbol{M}'^{(1)}(\boldsymbol{x},t)\} $ can be understood as another kind of “gauge dependence.” Very generally, such sets of fields lead to the same charge-current densities when $$\begin{aligned}
\boldsymbol{P}'^{(1)}(\boldsymbol{x},t)&=\boldsymbol{P}^{(1)}(\boldsymbol{x},t)+\boldsymbol{\nabla}\cross\boldsymbol{a}(\boldsymbol{x},t),\nonumber\\
\boldsymbol{M}'^{(1)}(\boldsymbol{x},t)&=\boldsymbol{M}^{(1)}(\boldsymbol{x},t)-\frac{1}{c}\frac{\partial\boldsymbol{a}(\boldsymbol{x},t)}{\partial t}+\boldsymbol{\nabla}b(\boldsymbol{x},t),%\label{eq:gauge_transformation} \end{aligned}$$ for a general vector field $\boldsymbol{a}(\boldsymbol{x},t)$ and a general scalar field $b(\boldsymbol{x},t);$ here the sets of fields $\{\boldsymbol{P}^{(1)}(\boldsymbol{x},t),\boldsymbol{M}^{(1)}(\boldsymbol{x},t)\}$ and $\{\boldsymbol{P}'^{(1)}(\boldsymbol{x},t),\boldsymbol{M}'^{(1)}(\boldsymbol{x},t)\}$ of (\[eq:CSrelations\]) are related by $$\begin{aligned}
b(\boldsymbol{x},t)&=0,\nonumber\\ \boldsymbol{a}(\boldsymbol{x},t)&=c\alpha_{\text{CS}}\int_{-\infty}^{t}\boldsymbol{E}(\boldsymbol{x},t')dt', %\label{eq:GT} \end{aligned}$$ which can be easily confirmed using (\[eq:FG\]), under the condition that at $t=-\infty$ the system is unperturbed.
In the past, the origin of the ambiguity arising in the OMP tensor has, like that in the unperturbed macroscopic polarization, been understood in the context of finite sized systems [@VanderbiltBook; @Lee2012; @Souza2017]. At a surface where $\alpha^{il}$ cannot be treated as uniform the argument in the above paragraph breaks down, and the surface current that will arise shows an ambiguity that reflects the well-known gauge dependence of $\alpha_{\text{CS}}$ [@Qi2008; @Vanderbilt2009]. However, it seems there should be an equivalent bulk interpretation. That certainly holds for the unperturbed macroscopic polarization; in a calculation where the energy eigenstates are chosen and fixed at the start, a “quantum of ambiguity” arises from the gauge dependent term of (\[unpertP\]). This term has been shown [@Resta1994] to depend only on the phase of the determinant of $U(\boldsymbol{k})$, and as such, is not qualitatively different whether the bands are isolated or not. Indeed, this ambiguity can be understood at the level of a U$(1)$ gauge transformation, which in the simplest of cases takes the form $U_{n\alpha}(\boldsymbol{k})=\delta_{n\alpha}e^{-i\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}}$; this corresponds to changing the site with which each Wannier function is associated and, in turn, this changes $\boldsymbol{P}^{(0)}$ by a discrete amount, proportional to $\boldsymbol{R}$. However, such an interpretation cannot be used to understand the ambiguity associated with the OMP tensor; the gauge dependent term of (\[alphaCS\]) vanishes in the limit of isolated bands, and thus must interpreted on the more general grounds of a multiband gauge transformation. Yet, at some level these ambiguities appear to be related, since the terms giving rise to them are constructed from the same object, the $\mathcal{W}^a$ matrix; (\[unpertP\]) contains only diagonal matrix elements, while (\[alphaCS\]) contains only off-diagonal matrix elements. Perhaps it is from this perspective that a bulk interpretation of the discrete ambiguity associated (\[alphaCS\]) can be formulated.
Limiting cases {#Sect:limits}
--------------
In this section we explore the magnetoelectric response in the limit of isolated molecules. First we construct the response tensor for a single molecule, and then use that to construct the response tensor of a crystal in the “molecular crystal limit,” which we take to be a model where there is a molecule at each lattice site with orbitals that have no common support with orbitals of molecules associated with other lattice sites. Finally, we show that the molecular crystal limit so obtained is in agreement with the appropriate limit of our general expressions (\[alphaCS\],\[alphaG\]).
### A single molecule
As pointed out earlier (Appendix D of Ref. [@Mahon2019]), the response of a molecule to an applied electromagnetic field can be treated via the same approach we have used here for a crystal, by constructing microscopic polarization and magnetization fields from the electronic Green function, with the expectation values of the electric and magnetic dipole moments following from the single-particle density matrix (\[mu\],\[nuBar\],\[nuTilde\]). However, for a molecule (or atom) it is also possible to follow a more common strategy in molecular physics [@Healybook; @PZW], where microscopic polarization and magnetization *operators* are introduced, leading to operators associated with the electric and magnetic dipole moments. We present that approach here (Appendix C of Mahon *et al.* [@Mahon2019]) to better make the connection between this calculation and molecular physics.
We take the initially unperturbed system to be described by (\[H\],\[physicalP\]), where we include an $\boldsymbol{A}_{\text{static}}(\boldsymbol{x})$ – which of course need not be periodic, since we are considering a localized system – to guarantee the breaking of time-reversal symmetry, and consider a $V(\boldsymbol{x})$ that vanishes as $|\boldsymbol{x}|\rightarrow\infty$ and that does not satisfy inversion symmetry about any point. The latter condition could arise in a molecule because of a non-centrosymmetric configuration of the nuclei, or even in an atomic system because of an imposed dc electric field that is considered part of the unperturbed Hamiltonian. We consider a “special point” $\boldsymbol{R}=\boldsymbol{0}$, which for a molecule could be taken to be, say, the center of mass of the ion cores and for an atom could be taken as the position of the ion core. In the frozen-ion approximation the contribution of the ions to the multipole moments of a molecule will not affect the perturbative response calculation we make, so we neglect them. We introduce a “special point” electron field operator (see Appendix C of Mahon *et al.* [@Mahon2019]) $\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)$, $$\begin{aligned}
& \widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)=e^{-i\Phi(\boldsymbol{x},\boldsymbol{0};t)}\widehat{\psi}(\boldsymbol{x},t),\end{aligned}$$ where $$\begin{aligned}
& \Phi(\boldsymbol{x},\boldsymbol{0};t)\equiv\frac{e}{\hbar c}\int s^{a}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{0})A^{a}(\boldsymbol{w},t)d\boldsymbol{w},\end{aligned}$$ and $\boldsymbol{A}(\boldsymbol{x},t)$ is again the vector potential describing the applied electromagnetic field. Then the Hamiltonian operator governing the evolution of $\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)$ is $$\begin{aligned}
&\mathsf{\widehat{H}}_{\text{sp}}(t)=\int\widehat{\psi}_{\text{sp}}^{\dagger}(\boldsymbol{x},t)\Big(H_{0}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{0};t)\big)\nonumber\\
&\qquad\qquad\qquad\qquad\qquad-e\Omega_{\boldsymbol{0}}^{0}(\boldsymbol{x},t)\Big)\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)d\boldsymbol{x},
\label{eq:Hsp}\end{aligned}$$ where $\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{0};t)$ is given by (\[frakP\]) and $\Omega_{\boldsymbol{0}}^{0}(\boldsymbol{x},t)$ by (\[eq:Omega\_0\]); we have $$\begin{aligned}
i\hbar\frac{\partial\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)}{\partial t}=\left[\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t),\mathsf{\widehat{H}}_{\text{sp}}(t)\right].\end{aligned}$$ We begin the evolution at a time $t_{0}$ before the electromagnetic field is applied, taking as the (Heisenberg) ket the ground state $\ket{G}$; at this time we have $\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t_{0})=\widehat{\psi}(\boldsymbol{x},t_{0})\equiv\widehat{\psi}(\boldsymbol{x})$. The dynamics can equivalently be described in a Schrödinger picture where the field operator is fixed at $\widehat{\psi}(\boldsymbol{x}$) and the ket evolves from $\ket{G}$ at $t_{0}$ according to a Hamiltonian operator $\mathsf{\widehat{H}}(t)$ having the form of (\[eq:Hsp\]), but with $\widehat{\psi}_{\text{sp}}(\boldsymbol{x},t)$ replaced by $\widehat{\psi}(\boldsymbol{x})$. Using the approximate expressions (\[omegaVec\],\[omega0\]) for $\boldsymbol{\Omega_{0}}(\boldsymbol{x},t)$ and $\Omega_{\boldsymbol{0}}^{0}(\boldsymbol{x},t)$, and neglecting the variation of the electric and magnetic fields over the atom or molecule, we can write $\widehat{\mathsf{H}}(t)$ as $$\begin{aligned}
& \widehat{\mathsf{H}}(t)=\widehat{\mathsf{H}}^{0}-\widehat{\boldsymbol{\mu}}\boldsymbol{\cdot}\boldsymbol{E}(t)-\widehat{\boldsymbol{\nu}}_{\text{P}}\boldsymbol{\cdot}\boldsymbol{B}(t)-\frac{1}{2}\widehat{\boldsymbol{\nu}}_{\text{D}}(t)\boldsymbol{\cdot}\boldsymbol{B}(t),\label{eq:Hatom_use}\end{aligned}$$ where $\boldsymbol{E}(t)\equiv\boldsymbol{E}(\boldsymbol{0},t)$ and $\boldsymbol{B}(t)\equiv\boldsymbol{B}(\boldsymbol{0},t)$, $$\begin{aligned}
& \widehat{\mathsf{H}}^{0}=\int\widehat{\psi}^{\dagger}(\boldsymbol{x})H_{0}\big(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\big)\widehat{\psi}(\boldsymbol{x})d\boldsymbol{x},\end{aligned}$$ and the operator for the electric dipole moment $\widehat{\boldsymbol{\mu}}$, and operators for the paramagnetic ($\widehat{\boldsymbol{\nu}}_{\text{P}}$) and diamagnetic ($\widehat{\boldsymbol{\nu}}_{\text{D}}(t))$ contributions to the magnetic dipole moment operator $\widehat{\boldsymbol{\nu}}(t)=\widehat{\boldsymbol{\nu}}_{\text{P}}+\widehat{\boldsymbol{\nu}}_{\text{D}}(t)$ are given by $$\begin{aligned}
\widehat{\boldsymbol{\mu}}&=e\int\widehat{\psi}^{\dagger}(\boldsymbol{x})\boldsymbol{x}\widehat{\psi}(\boldsymbol{x})d\boldsymbol{x},\\
\widehat{\boldsymbol{\nu}}_{\text{P}}&=\frac{e}{2mc}\int\widehat{\psi}^{\dagger}(\boldsymbol{x})\Big(\boldsymbol{x}\cross\boldsymbol{\mathfrak{p}}(\boldsymbol{x})\Big)\widehat{\psi}(\boldsymbol{x})d\boldsymbol{x},\\
\widehat{\boldsymbol{\nu}}_{\text{D}}(t)&=-\frac{e^{2}}{4mc^{2}}\int\widehat{\psi}^{\dagger}(\boldsymbol{x})\Big((\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{x})\boldsymbol{B}(t)\\
&\qquad\qquad\qquad\qquad\qquad-\big(\boldsymbol{x}\boldsymbol{\cdot}\boldsymbol{B}(t)\big)\boldsymbol{x}\Big)\widehat{\psi}(\boldsymbol{x})d\boldsymbol{x}.\end{aligned}$$ Note that the last expression is the field theoretic analogue of (\[eq:diamagnetic\]). Taking $\widehat{\mathsf{H}}^{0}\ket{G}=E_{G}\ket{G}$, we then calculate the response, to first-order, to applied static fields $\boldsymbol{E}$ and $\boldsymbol{B}$. The diamagnetic term in (\[eq:Hatom\_use\]) will make no contribution to the response to first order, and if we denote the excited states by $\ket{H}$ (with energies $E_{H}$), then by standard perturbation theory the response of the expectation value of the electric dipole moment operator to the magnetic field, $\expval{\widehat{\boldsymbol{\mu}}}_{\text{atom}}^{(B)}$, and the response of the expectation value of the magnetic dipole moment operator to the electric field, $\expval{\widehat{\boldsymbol{\nu}}} _{\text{atom}}^{(E)}$, are given by $$\begin{aligned}
& \expval{\widehat{\mu}^{i}} _{\text{atom}}^{(B)}=\check{\alpha}^{il}B^{l},\nonumber\\
& \expval{\widehat{\nu}^{i}} _{\text{atom}}^{(E)}=\check{\alpha}^{li}E^{l},
\label{eq:atom_moments} \end{aligned}$$ where $$\begin{aligned}
& \check{\alpha}^{il}=2\text{Re}\sum_{H}\frac{\bra{G}\widehat{\mu}^{i}\ket{H}\bra{H}\widehat{\nu}_{\text{P}}^{l}\ket{G}}{E_{H}-E_{G}}.\label{eq:alpha_first}\end{aligned}$$ We expand the field operator $\widehat{\psi}(\boldsymbol{x})$ in terms of orbitals $W_{v}(\boldsymbol{x})$ that are occupied in the ground state and orbitals $W_{c}(\boldsymbol{x})$ that are not, $$\begin{aligned}
& \widehat{\psi}(\boldsymbol{x})=\sum_{v}\widehat{c}_{v}W_{v}(\boldsymbol{x})+\sum_{c}\widehat{c}_{c}W_{c}(\boldsymbol{x}),\label{eq:orbital_expand}\end{aligned}$$ where $\widehat{c}_{v}$ and $\widehat{c}_{c}$ are fermionic annihilation operators generating single-particle eigenfunctions of $H_{0}(\boldsymbol{x},\mathfrak{p}(\boldsymbol{x}))$ with energies $E_{v}$ and $E_{c}$ respectively, with $$\begin{aligned}
& \widehat{c}_{c}\ket{G}=0 \text{\space,\space} \widehat{c}_{v}^{\dagger}\ket{G}=0.\end{aligned}$$ Then the expression (\[eq:alpha\_first\]) becomes $$\begin{aligned}
& \check{\alpha}^{il}=2\text{Re}\sum_{c,v}\frac{\left(\mu^{i}\right)_{vc}\left(\nu_{\text{P}}^{l}\right)_{cv}}{E_{c}-E_{v}},\label{eq:alpha_atom2}\end{aligned}$$ where $$\begin{aligned}
& \left(\mu^{i}\right)_{cv}=ex_{cv}^{i},\end{aligned}$$ with $$\begin{aligned}
& x_{cv}^{i}\equiv\int W_{c}^{*}(\boldsymbol{x})x^iW_{v}(\boldsymbol{x})d\boldsymbol{x},\label{eq:xcv}\end{aligned}$$ and $$\begin{aligned}
\left(\nu_{\text{P}}^{l}\right)_{cv}&=\frac{e}{2mc}\epsilon^{lab}\int W_{c}^{*}(\boldsymbol{x})x^{a}\mathfrak{p}^{b}(\boldsymbol{x})W_{v}(\boldsymbol{x})d\boldsymbol{x}\\
& =\frac{e}{2mc}\epsilon^{lab}\sum_{n}x_{cn}^{a}\int W_{n}^{*}(\boldsymbol{x})\mathfrak{p}^{b}(\boldsymbol{x})W_{v}(\boldsymbol{x})d\boldsymbol{x} \\
&=\frac{ie}{2\hbar c}\epsilon^{lab}\sum_{n}(E_{n}-E_{v})x_{cn}^{a}x_{nv}^{b}.\end{aligned}$$ Here we have inserted a complete set of states $\{\ket{n}\}$, with single-particle energies $\{E_{n}\}$, into the first expression, where the label $n$ ranges over all $v$ and $c$, and in going to the third line we have used the commutation relation of $\boldsymbol{x}$ and $H_{0}(\boldsymbol{x},\mathfrak{p}(\boldsymbol{x}))$ to write the matrix element of $\mathfrak{p}^{b}(\boldsymbol{x})$ in terms of $x_{nv}^{b}$ in the usual way. From (\[eq:alpha\_atom2\]) we then have $$\begin{aligned}
& \check{\alpha}^{il}=\frac{e^{2}}{\hbar c}\epsilon^{lab}\sum_{vcn}\frac{E_{n}-E_{v}}{E_{c}-E_{v}}\text{Re}\left[ix_{vc}^{i}x_{cn}^{a}x_{nv}^{b}\right].\label{eq:alphabar_work}\end{aligned}$$ As in a solid, this vanishes unless both time-reversal symmetry and inversion symmetry are broken: For if there is time-reversal symmetry the orbitals $W_{n}(\boldsymbol{x})$ can be chosen to be real and the quantity inside the brackets is purely imaginary, while if there is inversion symmetry the matrix elements $x_{vc}^{i}$ themselves vanish.
Splitting the sum over $n$ in (\[eq:alphabar\_work\]) into a sum over filled states $v'$ and a sum over empty states $c'$ we can write $$\begin{aligned}
& \check{\alpha}^{il}=\check{\alpha}_{\text{G}}^{il}+\check{\alpha}_{\text{CS}}^{il},\label{eq:alphabar_decomp}\end{aligned}$$ where $$\begin{aligned}
\check{\alpha}_{\text{G}}^{il}&=\frac{e^{2}}{\hbar c}\epsilon^{lab}\sum_{vcc'}\frac{E_{c'}-E_{c}}{E_{c}-E_{v}}\text{Re}\left[ix_{vc}^{i}x_{cc'}^{a}x_{c'v}^{b}\right]\nonumber\\
& +\frac{e^{2}}{\hbar c}\epsilon^{lab}\sum_{vv'c}\frac{E_{v'}-E_{v}}{E_{c}-E_{v}}\text{Re}\left[ix_{vc}^{i}x_{cv'}^{a}x_{v'v}^{b}\right]\label{eq:alphabar_CG} \end{aligned}$$ and $$\begin{aligned}
\check{\alpha}_{\text{CS}}^{il}=\frac{e^{2}}{\hbar c}\epsilon^{lab}\text{Re}\left[i\sum_{vcc'}x_{vc}^{i}x_{cc'}^{a}x_{c'v}^{b}\right].\end{aligned}$$ In the second of these expressions we sequentially put $$\begin{aligned}
\sum_{c'}=\sum_{n}-\sum_{v'} \text{\space , \space} \sum_{c}=\sum_{n'}-\sum_{v''},\end{aligned}$$ where both $n$ and $n'$ range over all states; noting that the sums over $n$ and $n'$ give no net contribution, we have $$\begin{aligned}
& \check{\alpha}_{\text{CS}}^{il}=\frac{e^{2}}{\hbar c}\text{Re}\left[i\epsilon^{lab}\sum_{vv'v''}x_{vv''}^{i}x_{v''v'}^{a}x_{v'v}^{b}\right],\end{aligned}$$ where the quantity in brackets is real. In three dimensions at least two of $i,l,a,b$ must be identical; if $i\neq l$ the expression is found to vanish, and in general we can write $$\begin{aligned}
& \check{\alpha}_{\text{CS}}^{il}=\delta^{il}\frac{ie^{2}}{3\hbar c}\epsilon^{cab}\sum_{vv'v''}x_{vv''}^{c}x_{v''v'}^{a}x_{v'v}^{b}.\label{eq:alphabar_CS}\end{aligned}$$
In the presence of uniform applied fields $\boldsymbol{E}$ and $\boldsymbol{B}$, and in the frozen-ion approximation, each magnetoelectric response tensor component $\check{\alpha}^{il}$ of a molecule can then be written (\[eq:alphabar\_decomp\]) as the sum of a Chern-Simons-like term (\[eq:alphabar\_CS\]) and a cross-gap-like term (\[eq:alphabar\_CG\]). As in a solid, the Chern-Simons-like contribution depends only on the occupied orbitals, and describes an isotropic response, regardless of how complicated might be the structure of the molecule. Note also that in a situation where all relevant initially occupied (unoccupied) orbitals are degenerate, $E_{c}=E_{c'}$ ($E_{v}=E_{v'}),$ the cross-gap-like term vanishes, in line with the vanishing of the cross-gap term in a solid when all relevant initially occupied (unoccupied) bands are degenerate [@Essin2010].
### The molecular crystal limit
We can now construct the “molecular crystal limit,” in which we consider a periodic array of molecules where the orbitals associated with a molecule at a given lattice site share no common support with those of molecules associated with other lattice sites; again, we take the external electric and magnetic fields to which the molecules respond to be the macroscopic Maxwell fields. Since the first-order modifications to the moments associated with each molecule are given by (\[eq:atom\_moments\]), using the expressions (\[PM\]) for $\boldsymbol{P}$ and $\boldsymbol{M}$, together with the defining equation (\[OMP\]) for the orbital magnetoelectric polarizability tensor, from (\[eq:alphabar\_decomp\],\[eq:alphabar\_CG\],\[eq:alphabar\_CS\]) we have $$\begin{aligned}
& \alpha^{il}=\alpha_{\text{G}}^{il}+\alpha_{\text{CS}}^{il},\label{eq:alphadecomp_MC}\end{aligned}$$ where simply $$\begin{aligned}
& \alpha_{\text{G}}^{il}=\frac{\check{\alpha}^{il}_{\text{G}}}{\Omega_{uc}}\text{\space , \space} \alpha_{\text{CS}}^{il}=\frac{\check{\alpha}^{il}_{\text{CS}}}{\Omega_{uc}},\label{eq:alphaCS_MC}\end{aligned}$$ with $\check{\alpha}^{il}_{\text{G}}$ and $\check{\alpha}^{il}_{\text{CS}}$ given by (\[eq:alphabar\_CG\],\[eq:alphabar\_CS\]) respectively.
Alternately, rather than building up the molecular crystal limit by assembling a collection of molecules, we can imagine starting with a full bandstructure calculation and taking the limit where the Wannier functions associated with each lattice site have no common support with the Wannier functions associated with a different lattice site. We also take the ELWFs to be eigenfunctions of the unperturbed Hamiltonian, (\[H\]), and so $E_{n\boldsymbol{k}}\rightarrow E_{n}$. These conditions lead to simplifications in the general expressions (\[alphaCS\],\[alphaG\]), and when they are employed the result should reproduce the molecular crystal limit (\[eq:alphadecomp\_MC\],\[eq:alphaCS\_MC\]). A first simplification is that, since the bands are flat, $\partial_{a}(E_{i\boldsymbol{k}}+E_{j\boldsymbol{k}})\rightarrow0$. Further, taking the orbitals introduced in (\[eq:orbital\_expand\]) to be the ELWFs $W_{v\boldsymbol{0}}(\boldsymbol{x})$ and $W_{c\boldsymbol{0}}(\boldsymbol{x})$, the flat bands can be identified by taking $U_{n\alpha}(\boldsymbol{k})=\delta_{n\alpha}$ in (\[WF\]) relating $\ket{\alpha\boldsymbol{R}}$ with $\ket{\psi_{n\boldsymbol{k}}}$. Hence the Hermitian matrices $\mathcal{W}^{a}$ (\[W\]) vanish, and we need not distinguish between the connections (\[connectionWannier\],\[eq:Bloch\_connection\]); the lack of common support for orbitals associated with different lattice sites, together with (\[firstMoment\]), also implies that $\xi_{cv}^{a}$, etc., are independent of $\boldsymbol{k}$, and from (\[firstMoment\]) and (\[connectionWannier\],\[eq:Bloch\_connection\]) we take $$\begin{aligned}
& \xi_{cv}^{a}(\boldsymbol{k})=i\left(c\boldsymbol{k}|\partial_{a}v\boldsymbol{k}\right)\rightarrow x_{cv}^{a},\end{aligned}$$ where we have used (\[eq:xcv\]) and $$\begin{aligned}
& \Omega_{uc}\int_{\text{BZ}}\frac{d\boldsymbol{k}}{(2\pi)^{3}}=1.\end{aligned}$$ Applying these molecular crystal conditions to the general expressions (\[alphaCS\],\[alphaG\]), we indeed recover (\[eq:alphaCS\_MC\]), as expected. Note that in this limit there is no gauge dependence in $\alpha_{\text{CS}}^{il}$, since the bands are isolated, and as well both $\alpha_{\text{CS}}^{il}$ and $\alpha_{\text{G}}^{il}$ arise solely from dynamical contributions. For while a calculation of the magnetic susceptibility *would* involve a compositional contribution due to the diamagnetic response, even in the molecular crystal limit (see the discussion around (\[eq:diamagnetic\])), the contributions to the magnetoelectric response in that limit are purely dynamical; were the calculation for a single molecule here done in terms of the Green function strategy used for a crystal, the response would result from changes in $\eta_{\alpha\boldsymbol{0};\beta\boldsymbol{0}}.$ Of course, in the molecular crystal limit all responses are what we have called atomic-like rather than itinerant.
Microscopic origin of $\alpha_{\text{G}}^{il}$ and $\alpha_{\text{CS}}^{il}$
----------------------------------------------------------------------------
While the qualitative features of the two contributions to the OMP tensor have been discussed earlier [@Essin2010], the microscopic nature of the approach implemented here can provide further insight into the character of the cross-gap and Chern-Simons contributions.
Since both of these contributions are non-vanishing in the molecular crystal limit, neither can simply be understood as entirely a consequence of the delocalized nature of Bloch electrons. For the Chern-Simons contribution, this is in agreement with earlier work [@Essin2010] where a particular model for a molecule at a lattice site was constructed that exhibits a Chern-Simons-like response; our expression (\[eq:alphabar\_CS\]) for the Chern-Simons-like response of an arbitrary molecule generalizes that result. However, neither contribution can be understood as a purely “localized molecule-like contribution” either, because the full expressions for $\alpha_{\text{CS}}^{il}$ (\[alphaCS\]) and $\alpha_{\text{G}}^{il}$ (\[alphaG\]) contain terms that vanish in the molecular crystal limit; this is again in agreement with earlier arguments [@Essin2010].
When moving from the molecular crystal limit of the Chern-Simons and cross-gap tensors, where the only contributions are atomic-like, to the full crystal expressions, both acquire itinerant contributions. In addition, while the cross-gap tensor is purely dynamical in nature, both in the molecular crystal limit and more generally, the Chern-Simons tensor acquires a compositional modification when moving from the molecular crystal limit to the general expression for a crystal. This suggests that perhaps it is through this compositional modification that a bulk interpretation of the discrete ambiguity associated with the Chern-Simons tensor can be constructed. We plan to explore this conjecture in a future publication.
Conclusion {#Sect:Conclusion}
==========
We have implemented a previously developed [@Mahon2019] microscopic theory of polarization and magnetization to study the zero temperature, orbital electronic response of a class of insulators to applied uniform dc electric and magnetic fields. To first-order in the applied fields, the free charge and current densities vanish [@Mahon2019]; thus the perturbative modifications to both the electronic charge and current density expectation values due to the applied fields can be found directly from the modifications to the microscopic polarization and magnetization fields. Associated with the dipole moment of the microscopic polarization (magnetization) field is a macroscopic polarization (magnetization), for which various response tensors can be extracted.
A quantity central in any calculation implementing this microscopic theory is the single-particle density matrix. We began by re-expressing the equation governing its dynamics to include an arbitrary lattice site, $\boldsymbol{R}_a$, which is to be used as a reference site for the applied electromagnetic field when calculating modifications to site quantities. This strategy will be useful in future work, where we plan to take into account the spatial variations of the electric and magnetic fields. However, in the calculation reported here we have restricted ourselves to the limit of uniform applied dc electric and magnetic fields. We began by reproducing the usual electric susceptibility [@Aversa], and then found the orbital magnetoelectric polarizability tensor. Generally, the OMP tensor is written as a combination of the isotropic, Chern-Simons contribution and the cross-gap contribution; this microscopic theory reproduces the usual result [@Qi2008; @Vanderbilt2009; @Essin2010; @Malashevich2010].
In the course of the perturbative analysis it became evident that there are generally two distinct types of modifications that contribute to response tensors. The first type arose from modifications to the single-particle density matrix due to the applied fields, and were termed “dynamical.” The other type arose from modifications to the diagonal elements of site quantity specific matrices, and were termed “compositional.” The electric susceptibility was found to arise from only a dynamical modification. In our analysis of the magnetoelectric response, we found three terms with distinct microscopic origins combine to form the OMP tensor: a dynamical modification to the atomic-like contribution, and dynamical and compositional modifications to the itinerant contribution. We found the Chern-Simons contribution to arise from a combination of parts of the atomic-like dynamical modification, and the itinerant compositional modification. The cross-gap contribution was found to arise from the remainder of the atomic-like dynamical modification, and the itinerant dynamical modification. We have also compared our expressions with those that arise in the molecular crystal limit, a model in which a periodic array of isolated molecules is considered.
As is well known, the first-order macroscopic bulk charge and current densities that result from applied uniform dc electric and magnetic fields are gauge invariant, but at a surface where the bulk arguments underlying this result are not valid, gauge-dependent currents emerge. The ambiguity associated with such surface currents can be studied in detail via the microscopic theory that underpins the approach taken here; we plan to consider this in future work. We also intend to extend the calculations presented in this paper to take into account the frequency dependence of the response tensors in general. This will lead to a description of frequency-dependent magnetoelectric effects, such as optical activity, for which the breaking of both spatial-inversion symmetry and time-reversal symmetry in the underlying material system is not required. A microscopic understanding of the mechanisms giving rise to such effects is now accessible via this formalism.
Acknowledgments
===============
We thank Ivo Souza, Rodrigo A. Muniz, Sylvia Swiecicki, and Julen Ibanez-Azpiroz for insightful discussions. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). P.T.M. acknowledges a PGS-D scholarship from NSERC.
Appendices
==========
Introducing arbitrary lattice sites {#AppendixA}
===================================
Here we work out an expression for $$\begin{aligned}
& e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\bar{H}_{\alpha\boldsymbol{R};\eta\boldsymbol{R}''}(t)=\mathscr{A}+\mathscr{B}+\mathscr{C}+\mathscr{D} \label{eq:workout_Hbar}\end{aligned}$$ (see Eq. (35) of [@Mahon2019] for the definition of $\bar{H}_{\alpha\boldsymbol{R};\eta\boldsymbol{R}''}(t)$), where $$\begin{aligned}
& \mathscr{A}=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}'';t))\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x},\\
& \mathscr{B}=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R};t))\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)\right)^{*}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}d\boldsymbol{x},\\
& \mathscr{C}=-\frac{e}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\left(\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{x},t)+\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{x},t)\right)\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x},\\
& \mathscr{D}=-\frac{i\hbar}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}\left(\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\frac{\partial\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)}{\partial t}-\frac{\partial\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)}{\partial t}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)\right)d\boldsymbol{x}.\end{aligned}$$ Looking at the first of these terms, we note that $$\begin{aligned}
& \boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}'';t)=e^{-i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)},\end{aligned}$$ so $$\begin{aligned}
& \mathscr{A}=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}e^{-i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}.\end{aligned}$$ Now $$\begin{aligned}
\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)+\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)-\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t) &=\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)\\
&=\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)+\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t),\end{aligned}$$ and so $$\begin{aligned}
& \mathscr{A}=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}.\end{aligned}$$ Similarly, since $$\begin{aligned}
& \boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R};t)=e^{-i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)},\end{aligned}$$ we have $$\begin{aligned}
\mathscr{B}&=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(e^{-i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)\right)^{*}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}d\boldsymbol{x}\\
& =\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)\right)^{*}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}d\boldsymbol{x}.\end{aligned}$$ Now $$\begin{aligned}
\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)+\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)+\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)
&=\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'',\boldsymbol{R}';t)\\
&=\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)+\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t),\end{aligned}$$ so $$\begin{aligned}
\mathscr{B}&=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)\right)^{*}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}d\boldsymbol{x}\\
&=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(H_{0}^{*}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))\left(e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\right)\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}.\end{aligned}$$ And then $$\begin{aligned}
\mathscr{A}+\mathscr{B}&=\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}\\
& +\frac{1}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\left(H_{0}^{*}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))\left(e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\right)\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}\\
& =e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}H_{0}(\boldsymbol{x},\boldsymbol{\mathfrak{p}}(\boldsymbol{x},\boldsymbol{R}_{a};t))e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}.\end{aligned}$$ The last form is not “explicitly Hermitian,” but it will be convenient. Next, since $$\begin{aligned}
& \Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)+\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)+\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)=\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)+\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t),\label{eq:delta_identity} \end{aligned}$$ we can write $$\begin{aligned}
\mathscr{A}+\mathscr{B}+\mathscr{C}&=
e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}\mathcal{H}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}\\
&+e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\int e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\left(e\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)-\frac{e}{2}\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{x},t)-\frac{e}{2}\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{x},t)\right)\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x},\end{aligned}$$ where we have used (\[Hcal\]). Now $$\begin{aligned}
e\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)-\frac{e}{2}\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{x},t)-\frac{e}{2}\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{x},t)
& =\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)-\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{x},t)\right)+\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)-\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{x},t)\right)\\
& =\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)+\Omega_{\boldsymbol{x}}^{0}(\boldsymbol{R}'',t)\right)+\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)+\Omega_{\boldsymbol{x}}^{0}(\boldsymbol{R},t)\right)\\
& =\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)+\Omega_{\boldsymbol{x}}^{0}(\boldsymbol{R}'',t)+\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{R}_{a},t)\right)\\
& +\frac{e}{2}\left(\Omega_{\boldsymbol{R}_{a}}^{0}(\boldsymbol{x},t)+\Omega_{\boldsymbol{x}}^{0}(\boldsymbol{R},t)+\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\right)-\frac{e}{2}\left(\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{R}_{a},t)+\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\right)\\
& =\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}{\partial t}+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}{\partial t}-\frac{e}{2}\left(\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{R}_{a},t)+\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\right),\end{aligned}$$ and recalling (\[eq:delta\_identity\]) we have $$\begin{aligned}
\mathscr{A}+\mathscr{B}+\mathscr{C}&=
e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\\
& \times\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}\left(\mathcal{H}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t)+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}{\partial t}+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}{\partial t}\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}\\
& -\frac{e}{2}e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\left(\Omega_{\boldsymbol{R}''}^{0}(\boldsymbol{R}_{a},t)+\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\right)\int e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}.\end{aligned}$$ As the function $\chi_{\alpha\boldsymbol{R}}(\boldsymbol{x},t)$ satisfies the modified orthogonality relation [@Mahon2019] $$\begin{aligned}
\int e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}=\delta_{\alpha\eta}\delta_{\boldsymbol{R}\boldsymbol{R}''},\end{aligned}$$ we find $$\begin{aligned}
\mathscr{A}+\mathscr{B}+\mathscr{C}&=e^{i\Delta(\boldsymbol{R},\boldsymbol{R}_{a},\boldsymbol{R}'',\boldsymbol{R}';t)}\\
& \times\int\chi_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x},t)e^{i\Delta(\boldsymbol{R},\boldsymbol{x},\boldsymbol{R}_{a};t)}\left(\mathcal{H}_{\boldsymbol{R}_{a}}(\boldsymbol{x},t)+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}{\partial t}+\frac{\hbar}{2}\frac{\partial\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};t)}{\partial t}\right)e^{i\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}'';t)}\chi_{\eta\boldsymbol{R}''}(\boldsymbol{x},t)d\boldsymbol{x}\\
& -e\Omega_{\boldsymbol{R}}^{0}(\boldsymbol{R}_{a},t)\delta_{\alpha\eta}\delta_{\boldsymbol{R}\boldsymbol{R}''},\end{aligned}$$ and so from (\[eq:workout\_Hbar\]) we have (\[RaRefSite\]), where we have defined $\bar{H}_{\alpha\boldsymbol{R};\eta\boldsymbol{R}''}(\boldsymbol{R}_{a},t)$ as in (\[barHelements\]). We also require $$\begin{aligned}
e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}';t)}\bar{H}_{\eta\boldsymbol{R}'';\beta\boldsymbol{R}'}(t)&=e^{i\Delta(\boldsymbol{R}'',\boldsymbol{R}',\boldsymbol{R};t)}\bar{H}_{\eta\boldsymbol{R}'';\beta\boldsymbol{R}'}(t)\\
& =e^{i\Delta(\boldsymbol{R}'',\boldsymbol{R}_{a},\boldsymbol{R}',\boldsymbol{R};t)}\bar{H}_{\eta\boldsymbol{R}'';\beta\boldsymbol{R}'}(\boldsymbol{R}_{a},t)-e\Omega_{\boldsymbol{R}'}^{0}(\boldsymbol{R}_{a},t)\delta_{\eta\beta}\delta_{\boldsymbol{R}'\boldsymbol{R}''}\\
& =e^{i\Delta(\boldsymbol{R},\boldsymbol{R}'',\boldsymbol{R}_{a},\boldsymbol{R}';t)}\bar{H}_{\eta\boldsymbol{R}'';\beta\boldsymbol{R}'}(\boldsymbol{R}_{a},t)-e\Omega_{\boldsymbol{R}'}^{0}(\boldsymbol{R}_{a},t)\delta_{\eta\beta}\delta_{\boldsymbol{R}'\boldsymbol{R}''},\end{aligned}$$ where we have used (\[barHelements\]). Since the lattice site $\boldsymbol{R}_{a}$ is arbitrary, we can as well write (\[RbRefSite\]) for any lattice site $\boldsymbol{R}_{b}$.
\
Modifications to the single-particle density matrix {#AppendixD}
===================================================
Beginning with the equation of motion for the single-particle density matrix $$\begin{aligned}
i\hbar\frac{\partial\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)}{\partial t}=\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t)\eta_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t),
\label{EDMeom}\end{aligned}$$ then expanding all quantities in powers of the applied fields, and then matching powers, at zeroth-order we find $$\begin{aligned}
&i\hbar\frac{\partial\eta^{(0)}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)}{\partial t}=\\
&\quad\sum_{\lambda\boldsymbol{R}''}\Big({H}^{(0)}_{\alpha\boldsymbol{R};\lambda\boldsymbol{R}''}\eta^{(0)}_{\lambda\boldsymbol{R}'';\beta\boldsymbol{R}'}(t)-\eta^{(0)}_{\alpha\boldsymbol{R};\lambda\boldsymbol{R}''}{H}^{(0)}_{\lambda\boldsymbol{R}'';\beta\boldsymbol{R}'}\Big),\nonumber \end{aligned}$$ where ${H}^{(0)}_{\alpha\boldsymbol{R};\lambda\boldsymbol{R}''}$ is given by (\[unpertHelements\]). Thus $\eta^{(0)}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)$ evolves as the unperturbed single-particle density matrix, and consequently $$\begin{aligned}
\eta^{(0)}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)=f_\alpha\delta_{\alpha\beta}\delta_{\boldsymbol{RR}'},\end{aligned}$$ as expected.
From (\[EDMeom\]) it is found that the first-order modification to the single-particle density matrix due to an applied electromagnetic field evolves according to $$\begin{aligned}
&i\hbar\frac{\partial\eta^{(1)}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}(t)}{\partial t}=\nonumber\\
&\quad\sum_{\lambda\boldsymbol{R}''}\Big({H}^{(0)}_{\alpha\boldsymbol{R};\lambda\boldsymbol{R}''}\eta^{(1)}_{\lambda\boldsymbol{R}'';\beta\boldsymbol{R}'}(t)-\eta^{(1)}_{\alpha\boldsymbol{R};\lambda\boldsymbol{R}''}{H}^{(0)}_{\lambda\boldsymbol{R}'';\beta\boldsymbol{R}'}\Big)\nonumber\\
&\quad+\sum_{\mu\boldsymbol{R}_{1}}f_{\mu}\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\mu\boldsymbol{R}_{1}}(t)^{(1)}, \label{EDMeom1}\end{aligned}$$ the final term of which is found to be $$\begin{aligned}
\sum_{\mu\boldsymbol{R}_{1}}f_{\mu}\mathfrak{F}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{\mu\boldsymbol{R}_{1};\mu\boldsymbol{R}_{1}}(t)^{(1)}=f_{\beta\alpha}\bar{H}_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(\boldsymbol{R}_{a},t).\end{aligned}$$ It is useful to define the intermediate quantity $$\begin{aligned}
\eta_{i\boldsymbol{k};j\boldsymbol{k}'}(t)=\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\braket{\psi_{i\boldsymbol{k}}}{\mu\boldsymbol{R}_{1}} \eta_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}(t)\braket{ \nu\boldsymbol{R}_{2}}{\psi_{j\boldsymbol{k}'}},\label{transf}\end{aligned}$$ for which, from (\[EDMeom1\]), we find $$\begin{aligned}
& i\hbar\frac{\partial\eta_{i\boldsymbol{k};j\boldsymbol{k}'}^{(1)}(t)}{\partial t}=\big(E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}\big)\eta_{i\boldsymbol{k};j\boldsymbol{k}'}^{(1)}(t)\\
&\quad+f_{ji}\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\braket{\psi_{i\boldsymbol{k}}}{\mu\boldsymbol{R}_{1}}\bar{H}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},t)\braket{\nu\boldsymbol{R}_{2}}{\psi_{j\boldsymbol{k}'}}.\end{aligned}$$ Then, implementing the usual Fourier analysis via (\[Fouier\]), we find $$\begin{aligned}
& \eta_{i\boldsymbol{k};j\boldsymbol{k}'}^{(1)}(\omega)=\\
&\quad-f_{ji}\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\frac{\braket{\psi_{i\boldsymbol{k}}}{\mu\boldsymbol{R}_{1}}\bar{H}_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},\omega)\braket{\nu\boldsymbol{R}_{2}}{\psi_{j\boldsymbol{k}'}}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}-\hbar(\omega+i0^{+})},\end{aligned}$$ where $0^{+}$ entering in the denominator describes the turning on of the applied electromagnetic field at $t>-\infty$. Finally, using (\[barHelements\]) and the inverse of (\[transf\]), we find
$$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(1)}(\omega)&= -\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}d\boldsymbol{k}'\frac{\braket{\alpha\boldsymbol{R}}{\psi_{i\boldsymbol{k}}}\braket{\psi_{i\boldsymbol{k}}}{\mu\boldsymbol{R}_{1}}H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},\omega)\braket{\nu\boldsymbol{R}_{2}}{\psi_{j\boldsymbol{k}'}}\braket{\psi_{j\boldsymbol{k}'}}{\beta\boldsymbol{R}'}}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}-\hbar(\omega+i0^{+})}\nonumber\\
&-\frac{i}{2}f_{\beta\alpha}\int W_{\alpha\boldsymbol{R}}^{*}(\boldsymbol{x})\Big(\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R};\omega)+\Delta(\boldsymbol{R}_{a},\boldsymbol{x},\boldsymbol{R}';\omega)\Big)W_{\beta\boldsymbol{R}'}(\boldsymbol{x})d\boldsymbol{x}, \label{EDMpt}
\end{aligned}$$
where $H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(1)}(\boldsymbol{R}_{a},\omega)$ is given by (\[Hcorr\]). We now apply this result to the case of applied uniform dc fields and find the first-order modification to the single-particle density matrix due to an applied electric field, Eq. (\[EDMe\]). The second term of (\[EDMpt\]) vanishes trivially, and for the first we make use of (\[Hcorr\],\[Hcal1\],\[omega0\]) to find $$\begin{aligned}
H_{\mu\boldsymbol{R}_{1};\nu\boldsymbol{R}_{2}}^{(E)}(\boldsymbol{R}_{a},\omega)=-eE^l\int W^*_{\mu\boldsymbol{R}_{1}}(\boldsymbol{x})(x^l-R^l_a)W_{\nu\boldsymbol{R}_{2}}(\boldsymbol{x})d\boldsymbol{x},\end{aligned}$$ where $\boldsymbol{E}\equiv\boldsymbol{E}(\boldsymbol{R}_a,\omega=0)$ for any $\boldsymbol{R}_a$, in this limit. Then, recalling (\[WF\]), we find $$\begin{aligned}
\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(E)}&=eE^l\frac{\Omega^2_{uc}}{(2\pi)^{6}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}d\boldsymbol{k}'\frac{e^{i(\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}-\boldsymbol{k}'\boldsymbol{\cdot}\boldsymbol{R}')}U^{\dagger}_{\alpha i}(\boldsymbol{k})U_{j\beta}(\boldsymbol{k}')}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}}\\
&\quad\qquad\qquad\qquad\qquad\times\sum_{\mu\nu\boldsymbol{R}_{1}\boldsymbol{R}_{2}}e^{-i\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}_1}e^{i\boldsymbol{k}'\boldsymbol{\cdot}\boldsymbol{R}_2}U_{i\mu}(\boldsymbol{k})\left(\int W^*_{\mu\boldsymbol{R}_{1}-\boldsymbol{R}_2}(\boldsymbol{x})x^lW_{\nu\boldsymbol{0}}(\boldsymbol{x})d\boldsymbol{x}\right)U^\dagger_{\nu j}(\boldsymbol{k}')\nonumber\\
&=eE^l\frac{\Omega^3_{uc}}{(2\pi)^{9}}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}d\boldsymbol{k}'d\boldsymbol{k}_1\frac{e^{i(\boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{R}-\boldsymbol{k}'\boldsymbol{\cdot}\boldsymbol{R}')}U^{\dagger}_{\alpha i}(\boldsymbol{k})U_{j\beta}(\boldsymbol{k}')}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}'}}\\
&\quad\qquad\qquad\qquad\qquad\times\sum_{\mu\nu}U_{i\mu}(\boldsymbol{k})\tilde{\xi}^l_{\mu\nu}(\boldsymbol{k}_1)U^\dagger_{\nu j}(\boldsymbol{k}')\sum_{\boldsymbol{R}_{1}}e^{-i(\boldsymbol{k}-\boldsymbol{k_1})\boldsymbol{\cdot}\boldsymbol{R}_1}\sum_{\boldsymbol{R}_{2}}e^{i(\boldsymbol{k}'-\boldsymbol{k}_1)\boldsymbol{\cdot}\boldsymbol{R}_2}\nonumber\\
&=eE^l\frac{\Omega_{uc}}{(2\pi)^3}\sum_{ij}f_{ji}\int_{\text{BZ}}d\boldsymbol{k}\frac{e^{i\boldsymbol{k}\boldsymbol{\cdot}(\boldsymbol{R}-\boldsymbol{R}')}U^\dagger_{\alpha i}(\boldsymbol{k})\xi^l_{ij}(\boldsymbol{k})U_{j\beta}(\boldsymbol{k})}{E_{i\boldsymbol{k}}-E_{j\boldsymbol{k}}},\end{aligned}$$
where we have used the identity $$\begin{aligned}
\frac{\Omega_{uc}}{(2\pi)^{3}}\sum_{\boldsymbol{R}}e^{i(\boldsymbol{k}-\boldsymbol{k}')\boldsymbol{\cdot}\boldsymbol{R}}=\delta(\boldsymbol{k}-\boldsymbol{k}'),\end{aligned}$$ as well as (\[firstMoment\]) and (\[connection\]), in going to the final expression. Notice that the approximation of an applied electric field that varies little on the scale of the lattice constant gives rise to a simplified form of (\[Hcorr\]), which in turn allows $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(E)}$ to be written as a single Brillouin zone integral. To derive the expression for $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(B)}$ a similar procedure is followed; however, one must also use (\[pMatrixElements\]). In this case, the approximation that the magnetic field varies little on the scale of the lattice constant allows $\eta_{\alpha\boldsymbol{R};\beta\boldsymbol{R}'}^{(B)}$ to be written as a single Brillouin zone integral.
Nearly uniform electromagnetic fields {#AppendixB}
=====================================
We now work out some of the general expressions for our relators and quantities dependent on them in the limit of nearly uniform applied fields. By this we mean that we keep the electric field and its first derivatives at the “expansion point,” but only the magnetic field at that point. We use a straight-line path; see [@Mahon2019], where we find $$\begin{aligned}
s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y}) & =\int_{0}^{1}(x^{i}-y^{i})\delta(\boldsymbol{w}-\boldsymbol{y}-u(\boldsymbol{x}-\boldsymbol{y}))du,\nonumber \\
\alpha^{jk}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y}) & =\epsilon^{jmk}\int_{0}^{1}(x^{m}-y^{m})\delta(\boldsymbol{w}-\boldsymbol{y}-u(\boldsymbol{x}-\boldsymbol{y}))udu,\label{eq:straight-line_use}\end{aligned}$$ for our relators, and we will consider the quantities $$\begin{aligned}
& \Omega_{\boldsymbol{y}}^{j}(\boldsymbol{x},t)=\int\alpha^{lj}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})B^{l}(\boldsymbol{w},t)d\boldsymbol{w},\\
& \Omega_{\boldsymbol{y}}^{0}(\boldsymbol{x},t)=\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})E^{i}(\boldsymbol{w},t)d\boldsymbol{w}, \label{eq:Omega_0} \end{aligned}$$ and $$\begin{aligned}
& \frac{\hbar c}{e}\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};t)= \nonumber\\
& \int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})A^{i}(\boldsymbol{w},t)d\boldsymbol{w}+\int s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})A^{i}(\boldsymbol{w},t)d\boldsymbol{w}\nonumber\\
& +\int s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})A^{i}(\boldsymbol{w},t)d\boldsymbol{w}.\end{aligned}$$ The “expansion point” here is $\boldsymbol{y}$. We first consider $$\begin{aligned}
& \Omega_{\boldsymbol{y}}^{j}(\boldsymbol{x},t)\simeq B^{l}(\boldsymbol{y},t)\int\alpha^{lj}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})d\boldsymbol{w}. \end{aligned}$$ Now $$\begin{aligned}
\int\alpha^{lj}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})d\boldsymbol{w} & =\epsilon^{lmj}\int_{0}^{1}(x^{m}-y^{m})udu\\
& =\frac{1}{2}\epsilon^{lmj}(x^{m}-y^{m}),\end{aligned}$$ so $$\begin{aligned}
\boldsymbol{\Omega}_{\boldsymbol{y}}(\boldsymbol{x},t)\simeq\frac{1}{2}\boldsymbol{B}(\boldsymbol{y},t)\cross(\boldsymbol{x}-\boldsymbol{y}).\end{aligned}$$ Next, we consider $$\begin{aligned}
\Omega_{\boldsymbol{y}}^{0}(\boldsymbol{x},t)& \simeq E^{i}(\boldsymbol{y},t)\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})d\boldsymbol{w}\nonumber \\
& +\frac{\partial E^{i}(\boldsymbol{y},t)}{\partial y^{k}}\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})(w^{k}-y^{k})d\boldsymbol{w}.\end{aligned}$$ The terms we need are $$\begin{aligned}
\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})d\boldsymbol{w} & =\int_{0}^{1}(x^{i}-y^{i})du=x^{i}-y^{i},\label{eq:s_zero_moment}\end{aligned}$$ and $$\begin{aligned}
\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{y})(w^{k}-y^{k})d\boldsymbol{w}&=\int_{0}^{1}(x^{i}-y^{i})(x^{k}-y^{k})udu\nonumber \\
& =\frac{1}{2}(x^{i}-y^{i})(x^{k}-y^{k}) \label{eq:s_one_moment}\end{aligned}$$ so we have $$\begin{aligned}
& \Omega_{\boldsymbol{y}}^{0}(\boldsymbol{x},t)\\
& \simeq(x^{i}-y^{i})E^{i}(\boldsymbol{y},t)+\frac{1}{2}(x^{i}-y^{i})(x^{k}-y^{k})\frac{\partial E^{i}(\boldsymbol{y},t)}{\partial y^{k}}\\
& =(x^{i}-y^{i})E^{i}(\boldsymbol{y},t)\\
& +\frac{1}{2}(x^{i}-y^{i})(x^{k}-y^{k})\left(\frac{1}{2}\frac{\partial E^{i}(\boldsymbol{y},t)}{\partial y^{k}}+\frac{1}{2}\frac{\partial E^{k}(\boldsymbol{y},t)}{\partial y^{i}}\right)\\
& =\left(\boldsymbol{x}-\boldsymbol{y}\right)\boldsymbol{\cdot}\boldsymbol{E}(\boldsymbol{y},t)+\frac{1}{2}(x^{i}-y^{i})(x^{k}-y^{k})F^{ik}(\boldsymbol{y},t),\end{aligned}$$ where we have used (\[eq:Fdef\]). Finally, we look at $\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};t)$. This can be done “by hand” when the magnetic field is uniform, but in what follows we work it out formally. $$\begin{aligned}
&\frac{\hbar c}{e}\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};t)\\
& \simeq A^{i}(\boldsymbol{y},t)\int\left(s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})+s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})+s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})\right)d\boldsymbol{w}\\
& +\frac{\partial A^{i}(\boldsymbol{y},t)}{\partial y^{j}}\int(w^{j}-y^{j})\Big(s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})\\
& \qquad\qquad\qquad\qquad+s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})+s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})\Big)d\boldsymbol{w}+\ldots\end{aligned}$$ From (\[eq:s\_zero\_moment\]) we have $$\begin{aligned}
& \int\left(s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})+s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})+s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})\right)d\boldsymbol{w}=0,\end{aligned}$$ while $$\begin{aligned}
& \int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})(w^{j}-y^{j})d\boldsymbol{w}\\
& =\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})(z^{j}-y^{j})d\boldsymbol{w}+\int s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})(w^{j}-z^{j})d\boldsymbol{w}\\
& =(x^{i}-z^{i})(z^{j}-y^{j})+\frac{1}{2}(x^{i}-z^{i})(x^{j}-z^{j})\\
& =((x^{i}-y^{i})-(z^{i}-y^{i}))(z^{j}-y^{j})\\
& +\frac{1}{2}((x^{i}-y^{i})-(z^{i}-y^{i}))((x^{j}-y^{j})-(z^{j}-y^{j}))\\
& =\left(\frac{1}{2}(x^{i}-y^{i})(z^{j}-y^{j})-\frac{1}{2}(z^{i}-y^{i})(x^{j}-y^{j})\right)\\
& +\frac{1}{2}(x^{i}-y^{i})(x^{j}-y^{j})-\frac{1}{2}(z^{i}-y^{i})(z^{j}-y^{j}),\end{aligned}$$ where in the third line we have used (\[eq:s\_zero\_moment\],\[eq:s\_one\_moment\]); similarly $$\begin{aligned}
& \int s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})(w^{j}-y^{j})d\boldsymbol{w}=-\frac{1}{2}(x^{i}-y^{i})(x^{j}-y^{j}),\end{aligned}$$ and finally $$\begin{aligned}
& \int s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})(w^{j}-y^{j})d\boldsymbol{w}=\frac{1}{2}(z^{i}-y^{i})(z^{j}-y^{j}).\end{aligned}$$ So, in all $$\begin{aligned}
& \int(w^{j}-R^{j})\left(s^{i}(\boldsymbol{w};\boldsymbol{x},\boldsymbol{z})+s^{i}(\boldsymbol{w};\boldsymbol{y},\boldsymbol{x})+s^{i}(\boldsymbol{w};\boldsymbol{z},\boldsymbol{y})\right)d\boldsymbol{w}\\
& =\frac{1}{2}(x^{i}-y^{i})(z^{j}-y^{j})-\frac{1}{2}(z^{i}-y^{i})(x^{j}-y^{j}),\end{aligned}$$ and $$\begin{aligned}
& \frac{\hbar c}{e}\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};t)\\
& \simeq\frac{1}{2}\frac{\partial A^{i}(\boldsymbol{y},t)}{\partial y^{j}}\left((x^{i}-y^{i})(z^{j}-y^{j})-(z^{i}-y^{i})(x^{j}-y^{j})\right)\\
& =\frac{1}{2}\left(\frac{\partial A^{i}(\boldsymbol{y},t)}{\partial y^{j}}-\frac{\partial A^{j}(\boldsymbol{y},t)}{\partial y^{i}}\right)(x^{i}-y^{i})(z^{j}-y^{j}),\end{aligned}$$ and since $$\begin{aligned}
& \frac{\partial A^{j}(\boldsymbol{y},t)}{\partial y^{i}}-\frac{\partial A^{i}(\boldsymbol{y},t)}{\partial y^{j}}=\epsilon^{kij}B^{k}(\boldsymbol{y},t),\end{aligned}$$ we have $$\begin{aligned}
\frac{\hbar c}{e}\Delta(\boldsymbol{x},\boldsymbol{z},\boldsymbol{y};t) & \simeq-\frac{1}{2}\boldsymbol{B}(\boldsymbol{y},t)\boldsymbol{\cdot}\big[\left(\boldsymbol{x}-\boldsymbol{y}\right)\cross(\boldsymbol{z}-\boldsymbol{y})\big].\end{aligned}$$ We collect all of these approximate expressions in (\[omega0\],\[omegaVec\],\[delta\]). The expansions of $\Omega_{\boldsymbol{y}}^{j}(\boldsymbol{x},t)$ and $\Omega_{\boldsymbol{y}}^{0}(\boldsymbol{x},t)$ derived here can also be derived using a formal expansion of the relators (\[eq:straight-line\_use\]) about $u=0$.
Constructing the Chern-Simons contribution to the OMP tensor {#AppendixCS}
============================================================
Here we outline the steps in going from the first to the second equality of Eq. (\[alphaCS\]). The final term of the first line of (\[alphaCS\]) can be re-expressed, using (\[connection\]), as proportional to the Brillouin zone integral of
$$\begin{aligned}
&\epsilon^{lab}\sum_{\alpha\gamma}f_\alpha\tilde{\xi}^i_{\alpha\gamma}\partial_b \tilde{\xi}^a_{\gamma\alpha}\\
&=\epsilon^{lab}\sum_{\alpha\gamma}\sum_{nmps}f_n(\xi^i_{ps}+\mathcal{W}^i_{ps})U_{s\gamma}
\partial_b\big(U^\dagger_{\gamma m}(\xi^a_{mn}+\mathcal{W}^a_{mn})U_{n\alpha}\big)U^\dagger_{\alpha p} \\
&=\epsilon^{lab}\sum_{nm}f_n(\xi^i_{nm}+\mathcal{W}^i_{nm})\partial_b(\xi^a_{mn}+\mathcal{W}^a_{mn})+\epsilon^{lab}\sum_{nmps}f_n(\xi^i_{ps}+\mathcal{W}^i_{ps})(\xi^a_{mn}+\mathcal{W}^a_{mn})\big(i\mathcal{W}^b_{sm}\delta_{np}-i\delta_{sm}\mathcal{W}^b_{np}\big) \\
&=\epsilon^{lab}\sum_{nms}f_n(\xi^i_{nm}+\mathcal{W}^i_{nm})(i\xi^b_{ms}\xi^a_{sn}-i\mathcal{W}^b_{ms}\mathcal{W}^a_{sn})\\
&+i\epsilon^{lab}\sum_{nms}f_n\Big((\xi^i_{ns}+\mathcal{W}^i_{ns})\mathcal{W}^b_{sm}(\xi^a_{mn}+\mathcal{W}^a_{mn})-(\xi^i_{sm}+\mathcal{W}^i_{sm})(\xi^a_{mn}+\mathcal{W}^a_{mn})\mathcal{W}^b_{ns}\Big),\end{aligned}$$
where we have used the identities $\epsilon^{iab}\partial_b\xi^a_{mn}=i\epsilon^{iab}\sum_{s}\xi^b_{ms}\xi^a_{sn}$ and $\epsilon^{iab}\partial_b\mathcal{W}^a_{mn}=-i\epsilon^{iab}\sum_{s}\mathcal{W}^b_{ms}\mathcal{W}^a_{sn}$; in the following we will also often use $\mathcal{W}^a_{mn}\neq0$ only if $f_m=f_n$. We now consider the contributions to (\[alphaCS\]) at each order in $\mathcal{W}$. We first consider the terms quadratic in $\mathcal{W}$; their contribution to (\[alphaCS\]) is proportional to the Brillouin zone integral of $$\begin{aligned}
&\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(-\xi^i_{nm}\mathcal{W}^b_{ms}\mathcal{W}^a_{sn}+\mathcal{W}^i_{ns}\mathcal{W}^b_{sm}\xi^a_{mn}+\xi^i_{ns}\mathcal{W}^b_{sm}\mathcal{W}^a_{mn}-\mathcal{W}^i_{sm}\xi^a_{mn}\mathcal{W}^b_{ns}-\xi^i_{sm}\mathcal{W}^a_{mn}\mathcal{W}^b_{ns}\Big) \\
&=\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(\mathcal{\xi}^a_{mn}\mathcal{W}^i_{ns}\mathcal{W}^b_{sm}+\xi^b_{mn}\mathcal{W}^a_{ns}\mathcal{W}^i_{sm}+\xi^i_{sm}\mathcal{W}^b_{mn}\mathcal{W}^a_{ns}\Big) \\
&=\delta^{il}\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(\mathcal{\xi}^a_{mn}\mathcal{W}^l_{ns}\mathcal{W}^b_{sm}+\xi^b_{mn}\mathcal{W}^a_{ns}\mathcal{W}^l_{sm}+\xi^l_{sm}\mathcal{W}^b_{mn}\mathcal{W}^a_{ns}\Big) \\
&=\delta^{il}\epsilon^{cab}\text{Re}\sum_{nms}if_n\mathcal{\xi}^a_{nm}\mathcal{W}^c_{ms}\mathcal{W}^b_{sn},\end{aligned}$$ where in going from the second to third line we have used the fact that, in three-dimensions, at least two of $i,l,a,b$ must be identical; if $i\neq l$ the expression is found to vanish. This sort of argument is often used in what follows. The contribution to (\[alphaCS\]) that is linear in $\mathcal{W}$ (notice the penultimate term of (\[alphaCS\]) also contributes here) is proportional to the Brillouin zone integral of $$\begin{aligned}
&\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(\mathcal{W}^i_{nm}\xi^b_{ms}\xi^a_{sn}+\big(\xi^i_{ns}\mathcal{W}^b_{sm}\xi^a_{mn}-\xi^i_{sm}\xi^a_{mn}\mathcal{W}^b_{ns}\big)\Big)-\epsilon^{lab}\sum_{ijn}f_{ji}\text{Re}\big[i\xi^i_{ji}\mathcal{W}^b_{in}\xi^a_{nj}\big] \\
&=\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(\mathcal{W}^i_{nm}\xi^b_{ms}\xi^a_{sn}-\xi^i_{sm}\xi^a_{mn}\mathcal{W}^b_{ns}\Big)+\epsilon^{lab}\sum_{ijn}f_{i}\text{Re}\big[i\xi^i_{ji}\mathcal{W}^b_{in}\xi^a_{nj}\big] \\
&=\delta^{il}\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(\xi^b_{ms}\xi^a_{sn}\mathcal{W}^l_{nm}+\xi^l_{sm}\xi^b_{mn}\mathcal{W}^a_{ns}+\xi^a_{sm}\xi^l_{mn}\mathcal{W}^b_{ns}\Big) \\
&=\delta^{il}\epsilon^{cab}\text{Re}\sum_{nms}if_n\xi^b_{nm}\xi^a_{ms}\mathcal{W}^c_{sn}.\end{aligned}$$ Now the combined contribution of the linear and quadratic in $\mathcal{W}$ terms is proportional to $$\begin{aligned}
&\delta^{il}\epsilon^{abc}\sum_{nms}f_n\int_{\text{BZ}}d\boldsymbol{k}\text{Re}\Big[i\xi^b_{nm}\xi^a_{ms}\mathcal{W}^c_{sn}+i\mathcal{\xi}^a_{nm}\mathcal{W}^c_{ms}\mathcal{W}^b_{sn}\Big] \\
&=\delta^{il}\epsilon^{abc}\sum_{ns}f_n\int_{\text{BZ}}d\boldsymbol{k}\text{Re}\Big[\partial_b\xi^a_{ns}\mathcal{W}^c_{sn}+i\sum_{m}\mathcal{\xi}^a_{nm}\mathcal{W}^c_{ms}\mathcal{W}^b_{sn}\Big] \\
&=\delta^{il}\epsilon^{abc}\sum_{ns}f_n\int_{\text{BZ}}d\boldsymbol{k}\text{Re}\Big[-\xi^a_{ns}\partial_b\mathcal{W}^c_{sn}+i\sum_{m}\mathcal{\xi}^a_{nm}\mathcal{W}^c_{ms}\mathcal{W}^b_{sn}\Big] \\
&=0,\end{aligned}$$ where we have used an integration by parts on the initially linear in $\mathcal{W}$ term. This “miraculous cancellation” is also presented in Appendix C of Vanderbilt [@VanderbiltBook]. Thus, the cubic in $\mathcal{W}$ contribution is the only gauge dependent term that has not vanished, or been cancelled. Its contribution to (\[alphaCS\]) is proportional to the Brillouin zone integral of $$\begin{aligned}
&\epsilon^{lab}\text{Re}\sum_{nms}if_n\Big(-\mathcal{W}^i_{nm}\mathcal{W}^b_{ms}\mathcal{W}^a_{sn}+\mathcal{W}^i_{ns}\mathcal{W}^b_{mn}\mathcal{W}^a_{sm}-\mathcal{W}^i_{sm}\mathcal{W}^a_{mn}\mathcal{W}^b_{ns}\Big) \nonumber\\
&=-\epsilon^{lab}\text{Re}\sum_{vv'v_1}i\mathcal{W}^i_{vv'}\mathcal{W}^a_{v'v_1}\mathcal{W}^b_{v_1v} \nonumber\\
&=-\delta^{il}\frac{\epsilon^{cab}}{3}\text{Re}\sum_{vv'v_1}i\mathcal{W}^c_{vv'}\mathcal{W}^a_{v'v_1}\mathcal{W}^b_{v_1v} \nonumber\\
&=-\delta^{il}\epsilon^{abc}\text{Re}\left[\sum_{vv_1}\big(\partial_b\mathcal{W}^a_{vv_1}\big)\mathcal{W}^c_{v_1v}-\frac{2i}{3}\sum_{vv_1v'}\mathcal{W}^a_{vv'}\mathcal{W}^b_{v'v_1}\mathcal{W}^c_{v_1v}\right].
\label{CSgauge}\end{aligned}$$ The contribution of this term to (\[alphaCS\]) is proportional to the Brillouin zone integral of the well-known term arising from the gauge-transformation of the Chern-Simons 3-form (see, e.g., Eq. C.19 of Vanderbilt [@VanderbiltBook]). The contribution independent of $\mathcal{W}$ is proportional (notice the first term of (\[alphaCS\]) contributes here) to the Brillouin zone integral of $$\begin{aligned}
&\epsilon^{lab}\text{Re}\sum_{nms}if_n\xi^i_{nm}\xi^b_{ms}\xi^a_{sn}+\epsilon^{lab}\Bigg\llbracket 2\sum_{cvv'}\text{Re}\big[\left(\partial_iv|c\right)\left(c|\partial_av'\right)\left(v'|\partial_bv\right)\big]+\sum_{cv}\text{Re}\big[\left(\partial_iv|c\right)\left(\partial_ac|\partial_bv\right)\big]\Bigg\rrbracket \nonumber\\
&=\epsilon^{lab}\text{Re}\sum_{vm}\big[\left(\partial_iv|m\right)\left(\partial_bm|\partial_av\right)-\left(\partial_iv|c\right)\left(\partial_bc|\partial_av\right)\big]+2\epsilon^{lab}\sum_{cvv'}\text{Re}\big[\left(\partial_iv|c\right)\left(c|\partial_av'\right)\left(v'|\partial_bv\right)\big] \nonumber\\
&=\epsilon^{lab}\text{Re}\sum_{vv'}\big[\left(\partial_iv|v'\right)\left(\partial_bv'|\partial_av\right)\big]+2\epsilon^{lab}\sum_{cvv'}\text{Re}\big[\left(\partial_iv|c\right)\left(c|\partial_av'\right)\left(v'|\partial_bv\right)\big],
\label{CS}\end{aligned}$$ which is equivalent to Eqs. (A11a)+(A11b) of Essin *et al.* [@Essin2010]. Then, in all, (\[alphaCS\]) results from the combination of (\[CSgauge\])+(\[CS\]).
[^1]: While in past work [@Mahon2019] and in this paper we treat the applied electromagnetic field classically, quantum mechanical effects can, in principle, be taken into account.
[^2]: This includes both ordinary and $\mathbb{Z}_2$ topological insulators. We discuss this further below.
[^3]: More precisely, the OMP tensor vanishes modulo a discrete ambiguity when time-reversal or inversion symmetry are present in the unperturbed system.
[^4]: More precisely, $\ket{\psi_{n\boldsymbol{k}+\boldsymbol{G}}}=\ket{\psi_{n\boldsymbol{k}}}$ and $U(\boldsymbol{k}+\boldsymbol{G})=U(\boldsymbol{k})$ for any reciprocal lattice vector $\boldsymbol{G}$. For more details, see, e.g., Vanderbilt [@VanderbiltBook].
[^5]: The derived expressions can later be applied to lower dimensional systems by confining the Bloch and Wannier functions to the appropriate subspace of $\mathbb{R}^3$. However, in systems with spatial dimension less than three, the Chern-Simons contribution vanishes. Thus, three-dimensional systems are of primary interest here.
[^6]: By “ordinary insulator” we mean crystalline insulators supporting sets of bands, all with a given occupation, for which there exists no topological obstruction to choosing a smooth gauge that can respect some underlying symmetry of the system. For instance, there exists no obstruction to choosing a time-reversal or inversion symmetric gauge for a system with the same discrete symmetry.
[^7]: As the metric for spatial components is identity, covariant and contravariant objects transform trivially into one another.
[^8]: See, e.g., Peskin and Schroeder [@Peskin].
[^9]: Rodrigo A. Muniz, J. L. Cheng, and J. E. Sipe, in preparation
[^10]: For a review and references to original work see Ref. [@PZW].
[^11]: As previously discussed, in this paper we move the gauge freedom of the energy eigenstates, and thus the gauge dependence of the connections $\xi^a_{nm}$, into the $U_{n\alpha}$ matrices.
[^12]: The relator $\alpha^{ib}(\boldsymbol{x};\boldsymbol{y},\boldsymbol{R})$ is not to be confused with the OMP tensor $\alpha^{il}$ introduced in Eq. (\[OMP\]).
[^13]: As previously discussed, in this paper we move the gauge freedom of the energy eigenstates, and thus the gauge dependence of the connections $\xi^a_{nm}$, into the $U_{n\alpha}$ matrices.
[^14]: See, e.g., Zhong *et al.* [@Souza2016].
|
---
author:
- 'M. Bruno, F. Gulminelli[^1] , F. Cannata, M. D’Agostino, F. Gramegna, G. Vannini'
date: 'Received: date / Revised version: date'
title: 'Bimodal pattern in the fragmentation of Au quasi-projectiles '
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
The existence of different phases for infinite nuclear matter is predicted by theoretical calculations since the early 80’s [@history]. Then, the possibility of observing a nuclear liquid-gas phase transition in the laboratory has been deduced from several experimental observations associated to the multi-fragmentation of finite nuclei. These observations indicate the occurrence of a state change in finite nuclei, which is interpreted to be the finite system counterpart of a phase transition [@chomaz].
Many of these signals are qualitative, and therefore cannot give information on the detailed trajectory (in terms of pressure, volume, temperature, isospin) followed by the system from one phase to the other. Other signals give in principle quantitative information, but can be distorted. Indeed the products of the reactions are detected asymptotically and not at the production time, and therefore they need to be corrected for secondary decay [@decay]. These corrections are, at least partially, model dependent and induce systematic errors which are difficult to estimate quantitatively.
To overcome these difficulties, it is important to perform a systematic study of different phase transition signals. The best would be to exploit new generation 4$\pi$ apparatuses, in order to be able to investigate several signals at the same time, with the same experimental data samples, and with a complete or quasi-complete detection [@praga]. Waiting for these new apparatuses, some of the signals indicating a phase transition have been obtained with measurements performed by the Multics [@mcs], together with the Miniball [@mini], multi-detectors. In the last few years we have investigated in detail the properties of quasi-projectiles detected in Au + Au reactions at 35 A.MeV, with a fixed source charge, and at different excitation energies. The following signals have been obtained:
1. the average size of the heaviest fragment (tentatively associated to the liquid part) decreases for increasing excitation energy of the nuclear system [@dag] with a power law distribution of exponent $\beta\approx 0.31$;
2. temperature measurements result compatible [@mil; @reliability] with a “plateau” in the caloric curve [@caloric];
3. critical exponents have been extracted [@dag], close to the values expected within the liquid-gas universality class;
4. the size distribution presents a scaling $\grave{a}$ la Fisher [@critical];
5. interaction energy fluctuations, corrected for side-feeding, were shown to overcome the statistical expectation in the canonical ensemble, corresponding to a negative branch of the microcanonical heat capacity for a system in thermodynamical equilibrium [@negative].
Some of these signals are coherent with the findings of other experimental collaborations with different data sets [@ma; @natowitz; @eos; @isis]. In particular, the last two signals have been confirmed in central reaction measurements performed with the Multics [@dag2; @bruno] and with the Indra [@reliability; @leneindre] apparatuses. Some of these behaviors were also observed in other finite physical systems undergoing a transformation interpreted as a first order phase transition, namely in the melting of atomic clusters [@negclu; @turchi] and in the fragmentation of hydrogen clusters [@neg3].
Recently [@PRE], a new topological observable has been proposed to recognize first order phase transitions. When a finite system undergoes such a transition, the most probable value of the order parameter changes discontinuously, while the associated distribution is bimodal close to the transition point, i.e. it shows two separate peaks, corresponding to the two different phases [@binder; @dasgupta]. In the case of transitions with a finite latent heat, this behavior is in agreement with the Yang-Lee theorem for the distribution of zeroes of the canonical partition sum in the complex temperature plane [@zeroes], and equivalent to the presence of a curvature anomaly in the microcanonical entropy $S(E)$ [@intruder].
Since many different correlated observables can serve as order parameters in a finite system, the task is to choose an order parameter which can be accessible from the experimental side [@WCI]. This is the case for observables related to the measured charges. The INDRA collaboration [@indra] has proposed as order parameter the variable Z$_{sym}=\frac {Z_1-Z_2} {Z_1+Z_2}$, where Z$_1$ and Z$_2$ are the charge of the largest and the second largest fragments detected, in each event, in the decay of an excited source. An indication of a bimodal distribution was obtained for this quantity. Signals of bimodality in different observables have been obtained in experiments with different projectile-target combinations, and in different energy ranges [@ma; @WCI; @bell].
In ref. [@WCI] it has been pointed out that the variable Z$_{sym}$ can present a spurious bimodality in small three-dimensional percolation lattices close to the percolation threshold. This behavior is due to finite size, and makes a bimodality in Z$_{sym}$ an ambiguous signature of the transition. On the other side, the size $A_1$ or charge $Z_1$ of the largest fragment have distributions which for any lattice size are consistent with the critical percolation behavior [@campi]. These observables were then suggested as more apt to discriminate between a first order phase transition, a critical phenomenon, and a smooth cross-over.
In this paper we investigate if these different signals are present in our data. We also discuss the relation of the bimodality signal with other phase transitions indicators obtained for our data.
The experiment {#sec:1}
==============
The measurements and the analysis have been extensively described elsewhere [@dag]. Here we recall that the measurements were performed at the K1200-NSCL Cyclotron of the Michigan State University. The Multics [@mcs] and Miniball [@mini] arrays were coupled to measure light charged particles and fragments with a geometrical acceptance of the order of 87% of 4$\pi$. The events have been recorded if at least two different modules have been fired. Similarly to Ref. [@indra], the selection of the quasi-projectile (QP) source have been made by a shape analysis, keeping all the fragments with $Z\geq3$. The fragments have been considered as belonging to the QP, if forward emitted in the ellipsoid reference frame. The complete source has been obtained by doubling the forward emitted light particles in the backward direction, in order to minimize the contamination of light particles emitted by a possible mid-velocity source. At the end of this procedure, only events with total charge within 10% of the Au charge have been considered for the analysis, in order to study the decay of a well detected constant size source, in a wide range of excitation energies.
In order to visualize the source characteristics in the selected events, the fragment (Z$\geq 3$) charge density distribution [@lecolley] is shown in Fig. \[vpar\] as a function of the fragment velocity in the QP reference frame. The ensemble averaged charge density $\langle\rho_Z(v_{par})\rangle$ is defined as
$$\langle\rho_Z(v_{par})\rangle = \Big\langle {\sum Z(v_{par}) \over
\sum Z} \Big\rangle$$
where $\rho_Z(v_{par})$ is the event-by-event distribution in the velocity $v_{par}$ for the collected charge fraction. This observable represents the distribution of the collected charge bound in fragments along the direction of the QP velocity. In Fig. \[vpar\] the continuous line (full points) represents the charge density for fragments accepted for the QP, the dashed line (open squares) for fragments of the QT. This latter is consistent with the filtered simulation of a QT source symmetric to the QP (dotted line - open circles). QP and QT can be easily recognized, showing that the imposed conditions are effective to select events where the contamination of a mid-velocity source is negligible (for more details see Refs. [@dag; @reliability]).
The characteristics of these events have been examined by analyzing the isotropy of the fragment angular distribution in the quasi-projectile reference frame [@negative; @dag2], and by comparing the data to predictions of a statistical multifragmentation model [@SMM]. The general conclusion is that an important degree of equilibration appears to be reached by the excited quasi-projectile sources in the whole range of excitation energies. For more details, see Refs. [@dag; @reliability; @negative; @dag2].
In Refs. [@dag; @reliability] a further condition to characterize the QP was added, i.e. that the velocity of the heaviest fragment is larger than 75% of the projectile velocity. This condition indeed has been replaced with the “completeness” of the event [@WCI; @indra] which reflects on the limitations on the parallel momentum, since it results less correlated to the variables we want to study. This does not affect the distribution shown in Fig. \[vpar\], and all the signals of phase transition do not change significantly. To quantify this statement, we present in Fig. \[cneg\] the power-law behavior of the average charge of the largest fragment, normalized to the charge of the source, as a function of the excitation energy, and the normalized partial-energy fluctuations, leading to the estimate of a negative branch for the microcanonical heat capacity [@negative]. The power law in the Z$_1$ distribution and the partial energy fluctuations are very scarcely affected by the condition on the velocity of largest fragment, apart from some very small variations at the higher energies [@reliability; @negative].
These observations mean that the detection apparatus appears especially effective in the complete detection of purely binary collisions. The same is not true for all reaction mechanisms, which may need different detection systems to be addressed. Only a fraction of well detected peripheral collisions can be interpreted as the independent statistical decay of two isotropic sources [@colin]. For instance within the INDRA apparatus it has been pointed out that for 80 A.MeV Au+Au collisions, these events represent about the 20% of the total number of complete events [@leneindre2], and depend on the selection criteria adopted [@bonnet]. In our case the statistical events represent about the 30% of the measured events as reported in Ref. [@bo2000], and as can be inferred from Fig. 2b) of Ref. [@dag]. The different in the percentage of statistical events could also be due to the different trigger conditions of two and four modules fired, used in our and Indra measurements, respectively.
Signals of bimodality {#sec:2}
=====================
In the liquid-gas phase transition, the largest fragment detected in each event is a natural order parameter, because of its correlation with the particle density in the grancanonical ensemble [@PRE; @big]. The variable proposed in Ref. [@indra; @rivet] in turn is trivially correlated to the largest cluster size, and in addition brings further information on the global fragmentation pattern. This means that it should be possible to observe a bimodal distribution for the charge of the largest fragment or the asymmetry, if one considers a system close to the transition temperature.
The global distributions of Z$_1$ and Z$_{sym}$ are shown in Fig. \[whole\], for all QP events selected as explained in section 2. Because of the impact parameter geometry, this distribution is clearly dominated by peripheral collisions at low deposited energy, leading to a heavy Z$_1\approx 75$ residue with a large asymmetry Z$_{sym}\approx 0.9$.
We can however also notice the presence of a large bump, corresponding to much lighter residues and much more symmetric fragmentation patterns. For this bump to be interpreted as an indication of bimodality, we should show that:
- the two different decay patterns can be obtained in the de-excitation of the same source,
- they correspond to the same temperature.
Let us first concentrate on the source definition. As we have already stressed in the last section, we are considering only events with a detected charge in the forward QP hemisphere close to the original $Au$ charge. This guarantees a good detection, but does not constrain the reaction mechanism or the number of sources, since the system is symmetric. Fig. \[vpar\] shows that the selected events are consistent with a purely binary kinematics, meaning that the bump at low charge shown by Fig.\[whole\] cannot be ascribed to a reduced size of the excited source. However Fig. \[vpar\] is obtained with the whole set of events, which is largely dominated by peripheral collisions. We may then wonder whether a (small) contamination of central collisions, leading to an important stopping in the center of mass, may be responsible of a decrease of the QP source size in the dissipative reactions corresponding to the low Z$_1$ bump.
In Fig. \[twobody\] we plot the velocity in the laboratory frame of the QP source as a function of the excitation energy, with cuts of Z$_1>$ 50 (full points) and Z$_1 <$ 50 (open points). The source velocity expected for a two body (QP-QT) kinematics, obtained via energy and momentum conservation in the hypothesis of an equal sharing of the excitation energy by the two collision partners, is shown by the dashed line [@dag]. We can see that for all calorimetrically reconstructed excitation energies, and for both Z$_1$ cuts, the observed behavior is compatible with purely binary collisions. This comparison shows that also the lighter Z$_1$’s, corresponding to constant size QP remnants, do not come from central reactions.
This discussion implies that we can safely consider the data as characteristic of the de-excitation of constant size source in a wide range of excitation energies.
Let us now come to the central question of data sorting. The global distributions of Fig. \[whole\] reflect the excitation energy deposit imposed by the dynamics of the entrance channel, and cannot be considered as belonging to a single statistical ensemble. If a sorting cannot be avoided, it is also clear that the shape of the distributions will depend on the sorting choice.
The two de-excitation modes visible in Fig. \[whole\] are associated to very different excitation energies. If they represent two different phases, this means that the associated phase transition should have a non zero latent heat, as it is the case for regular liquid-gas. Therefore, the sorting variable should not impose a too strong constraint on the deposited energy, such that the two phases can be accessed in the same bin. In particular, in the liquid-gas phase transition, Z$_1$ is known to be bimodal in the canonical ensemble which allows huge energy fluctuations, while no bimodality is observed in the microcanonical ensemble with fixed energy [@PRE].
To search for a possible bimodal behavior, we should then in principle sort the data in temperature bins, i.e. in canonical ensembles. This is not possible experimentally, but we can choose a sorting variable as close as possible to a canonical temperature. Moreover, as suggested by previous papers [@indra; @rivet], the sorting observable should better not be auto-correlated with fragments and light particles emitted by the QP source. To fulfill these requirements, as in [@indra], the transverse energy Et$_{12}$ = $\sum_Z E_Z sin^2 (\theta_Z)$ of the light particles ($Z\leq2$) emitted by the quasi-target source has been chosen, which is only loosely correlated to the QP observables. The QT had been much larger than the QP, this sorting could be considered as a canonical one. This is not our case, but it has still the advantage to allow for relatively large energy fluctuations, as it is needed to explore two phases that could be separated by a non zero latent heat.
It has to be noted that efficiency limitations for fragments in the backward direction do not allow to have a total reconstruction of the quasi-target source (see Fig. \[vpar\]). Therefore the selection on QT characteristics can only be performed on light particles; in particular the transverse energy Et$_{12}$ of these particles is detected with good efficiency. The distribution of Et$_{12}$ for the selected events is shown in Fig. \[etra\].
This sorting can be assimilated to an impact parameter sorting. The width of the transverse energy bins, equally spaced, was chosen of 40 MeV. Only the first six bins have sufficient statistics to be considered for the subsequent analysis.
The excitation energy constraint implied by this sorting is explored in Fig. \[estar\], which shows the distribution of the calorimetric $\varepsilon$\* in the six transverse energy bins. We can notice from this figure that the variables Et$_{12}$ and $\varepsilon$\* are loosely correlated, and a relatively wide distribution of $\varepsilon$\* is obtained in most of the bins of transverse energy. It is well known [@reliability] that the calorimetric measurement is not perfect, and the incomplete detection creates a spurious width in the energy distribution. Since this spurious width never exceed 1 MeV per nucleon, it is clear from Fig.\[estar\] that the sorting in Et$_{12}$ bins cannot be considered as a microcanonical selection, where no bimodality would be expected.
The charge of the heaviest fragment Z$_1$ is represented as a function of the asymmetry Z$_{sym}$ of the two heaviest in Fig. \[bim\]. We can see that the maximal probability does not monotonically change with the centrality selection. The most probable fragmentation pattern, characterized by a residue exhausting most of the available charge and an important asymmetry between the two largest fragments, abruptly changes between the fourth and the fifth bin with the apparition of a second peak. This peak represents multifragmentation events, with the largest fragment comparable in size to the other emitted clusters. In the sixth bin this second peak tends to become more prominent, even if the situation would be more clear with higher statistics. Such a discontinuous behavior agrees with the expectations from a phase transition, and with the findings of the INDRA collaboration on peripheral Xe + Sn and Au + Au collisions [@indra].
Projecting the plots of Fig. \[bim\] onto the two axes, we can note that the best indication of bimodality appears on the Z$_1$ variable (see Fig. \[projbig\]), whereas the plot of the asymmetry Z$_{sym}$ does not show a clear bimodal behavior (see Fig. \[projasym\]). The largest fragment size distribution, peaked around $Z_1\approx$ 70 up to the fourth bin, shows a maximum around $Z_1\approx 30$ in the sixth bin, passing through a configuration (fifth bin) where a minimum in the probability appears to be associated to the intermediate patterns, even if the statistics should definitely be improved. This strongly suggests a first order phase transition [@PRE].
The indication of bimodality is less clear in the Z$_{sym}$ variable, which shows a wide distribution in the last bins without any pronounced minimum ( see Fig. \[projasym\]). However the sudden change in the most probable fragmentation pattern, shown in the bidimensional correlations of Fig. \[bim\], is clearly seen in both variables. The most probable value of Z$_1$ and Z$_{sym}$ is shown in Fig \[jump\] as a function of the transverse energy. Both variables show a sudden decrease from the evaporation dominated pattern, to the multifragmentation dominated one. This behavior is again in agreement with the findings of ref. [@indra] and consistent with the expectation from a phase transition. In principle a first order phase transition should be associated to a discontinuous jump [@PRE], while a continuous power law behavior would characterize a second order phase transition. This however would be true only if the sorting variable could be assimilated to a thermodynamical temperature. In the microcanonical ensemble, even a first order transition is associated to a continuous behavior of the order parameter. As a consequence, the power law behavior of the average size of the largest cluster as a function of excitation energy (see Fig. \[cneg\]), can be observed both in the case of a critical behavior and in the coexistence zone of a first order phase transition [@big; @richert]. In the experimental case, the arbitrariness of the sorting and the absence of a physical external bath does not allow to draw definite conclusions.
A better understanding on the nature and order of the observed phase change can be achieved form Fig. \[phase\], which shows the distribution of the largest cluster charge and excitation energy in the Et$_{12}$ region (fourth + fifth bins) where the sudden change in the fragmentation pattern is observed. As we have already mentioned, a first order phase transition should imply a non zero latent heat, meaning that the two “phases” observed at the same “temperature” should be associated to different excitation energies.
We can see in Fig. \[phase\] that indeed the two decay modes observed in the $Z_1$ distribution correspond to different values of the calorimetric excitation energy: the cut $Z_1$=50 that roughly identifies the two modes (see Fig. \[whole\]) succeeds in splitting the energy distributions into two separate components (right part of Fig. \[phase\]), even if the distance of the centroids is too small to produce a clear bimodality in the excitation energy distribution. The lower (higher) $Z_1$ component roughly corresponds to a deposited energy higher (lower) than $3.5$ A.MeV (left part of Fig. \[phase\]). This behavior is consistent with the expectations from a first order phase transition smoothed by finite size effects.
If we interpret the two $Z_1$ distributions as two coexisting phases, it would be tempting to estimate the latent heat of the transition from the energy distance between the two peaks. The “liquid” peak points to an excitation energy $E_1^*\approx 2$ A.MeV which nicely agrees with the global $Z_1$-$E^*$ correlation shown in the left part of Fig. \[cneg\], and with the indication of the fluctuation measurement shown in the right part of the same figure. The “vapor” contribution peaks at $E_2^*\approx 5$ A.MeV, a lower value respect to the location of the second divergence in the fluctuation analysis. This discrepancy may be due to the intrinsic limitations of the transverse energy sorting, that does not allow sufficient energy fluctuations respect to a physical heat bath; it may also point to an incomplete exploration of the high energy phase space in our data sample, that cuts the distributions on the high energy side.
To summarize, the results of Fig.\[phase\] tend to suggest that the observed sudden change from evaporation to multifragmentation can be associated to first order phase transition. Higher statistics samples obtained with collisions at higher beam energy could allow to be conclusive about the compatibility between fluctuations and bimodality [@bonnet]. In addition a detailed study of the convexity properties of the distributions is needed [@gul; @bonnet2].
Conclusions {#sec:3}
===========
In this paper we have presented a new analysis of the 35 A.MeV quasi-projectile Au+Au data collected with the Multics-Miniball apparatus. The distributions of the largest cluster charge and of the charge asymmetry between the two largest clusters detected in each event have been studied. These data allow to analyze, with a limited statistics, the de-excitation of a constant size quasi-projectile source within a large range of dissipated energy. A clear transition from an evaporative to a multifragmentation pattern has been observed. The shape of the distributions have been studied to search for a possible bimodal behavior, which would allow to interpret this transition in the de-excitation mode as the finite system counterpart of a first order phase transition. The asymmetry distribution does not present a clear structure, while the largest fragment charge appears bimodal.
The same data have shown several different signals that coherently point to a first order liquid-gas-like phase transition. We recall here the determination of thermodynamically consistent critical exponents, both in a moment analysis [@dag] and in an analysis “à la Fisher” [@critical], and the fluctuation peak in the partial energy distribution, with an absolute value of fluctuations consistent with the existence of a negative branch for the microcanonical heat capacity [@reliability; @negative].
Concerning bimodality, the search of a convexity in the distribution of the largest fragment emitted in each collisional event appears a very direct, and therefore interesting signature of a first order phase transition. The intrinsic weakness associated to this signal, namely the arbitrariness in the choice of the sorting variable and of the sorting bin width, will be the object of future investigations [@gul]. To confirm the significance of these findings, it will be important to verify that the dynamics of the entrance channel and the different bias associated to different detectors do not influence the final results. To this aim, the bimodality distributions for systems produced with different entrance channels and detected with different apparatuses should be compared in details. Moreover, the stability of the signals as a function of the different ways of sorting the events should be checked.
In particular, the qualitative trend of the data appears consistent with the recent findings of the INDRA collaboration [@indra] at higher incident energies, but the behavior of the asymmetry variable Z$_{sym}$ is not equivalent in the two data sets and deserves a further analysis.
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[^1]: *Member of the Institut Universitaire de France*
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abstract: |
We demonstrate successful “dry” refrigeration of quantum fluids down to $T=0.16$mK by using copper nuclear demagnetization stage that is pre-cooled by a pulse-tube-based dilution refrigerator. This type of refrigeration delivers a flexible and simple sub-mK solution to a variety of needs including experiments with superfluid $^3$He. Our central design principle was to eliminate relative vibrations between the high-field magnet and the nuclear refrigeration stage, which resulted in the minimum heat leak of $Q=4.4$nW obtained in field of 35mT.
For thermometry, we employed a quartz tuning fork immersed into liquid $^3$He. We show that the fork oscillator can be considered as self-calibrating in superfluid $^3$He at the crossover point from hydrodynamic into ballistic quasiparticle regime.
address:
- 'O.V.Lounasmaa Laboratory, Aalto University, 00076 AALTO, Finland'
- 'O.V.Lounasmaa Laboratory, Aalto University, 00076 AALTO, Finland'
- 'BlueFors Cryogenics Ltd, Arinatie 10, 00370 Helsinki, Finland'
- 'O.V.Lounasmaa Laboratory, Aalto University, 00076 AALTO, Finland'
- 'O.V.Lounasmaa Laboratory, Aalto University, 00076 AALTO, Finland'
author:
- 'I.Todoshchenko'
- 'J.-P.Kaikkonen'
- 'R.Blaauwgeers'
- 'P.J.Hakonen'
- 'A.Savin'
title: 'Dry demagnetization cryostat for sub-millikelvin helium experiments: refrigeration and thermometry'
---
Introduction
============
The currently ongoing conversion of sub-Kelvin refrigerators to cryogen-free platforms [@Uhlig02] facilitates new applications (microscopy, imaging, medicine, space applications, security, astronomy, etc) as such systems can be run practically anywhere. Nowadays, dry dilution refrigerators equipped with large superconducting magnets and base temperatures below 10mK are offered by many suppliers as standard products. Many demanding quantum experiments require a further reduction of temperature.
To meet the above demand, the next challenge is to extend the operation of dry systems to sub-millikelvin temperatures. Nuclear demagnetization cooling [@Lounasmaa] combined with a commercially available dry dilution refrigerator as a precooling systems is one of the options to reach the microKelvin regime. Recently, successful operation of such a cryogen-free experimental platform down to 600$\mu$K was demonstrated by Batey [*et al.*]{} [@Saunders2013] using a PrNi$_5$ nuclear stage. However, this temperature is the practical limit for PrNi$_5$ because of its large intrinsic field, in contrast to copper which enables much lower temperatures. Additionally, copper is more available, easier to handle, and it provides better thermal conductivity when compared with PrNi$_5$. As a consequence, most of the modern sub-mK nuclear demagnetization refrigerators employ Cu for the nuclear cooling stage [@Berglund72; @Bradley84; @Huiki86; @Borovik-Romanov87; @Gloss88; @Xu92]. However, Cu stage is more sensitive to external heat loads because of the very demanding precooling conditions to fully polarize the nuclear spins in copper. Furthermore, due to the high electrical conductivity, a good-quality Cu nuclear stage is extremely prone to eddy current heating. All these reasons make nuclear cooling with copper much more difficult to implement on a dry system, but, if successful, also much more rewarding.
Nuclear demagnetization cryostat is quite complicated and laborious machine which includes typically a large liquid helium bath, a dilution unit, and a large-bore 8-9 Tesla solenoid. The refrigerator requires a large amount of work to construct and to keep operational [@Lounasmaa], and furthermore, its operation needs daily attention. Consequently, for a long time, the investigation of the superfluid $^3$He was a prerogative of big laboratories. This situation will change once a pulse-tube-based dry dilution refrigerator is combined efficiently with a nuclear cooling stage. Such a combination will allow lengthy sub-mK experiments with minimal attention by the operator of the refrigerator.
The aim of this work was to make the first demonstration of superfluid $^3$He refrigeration on a “dry” nuclear demagnetization cryostat all the way down to 0.2mK. Our central design principle was to eliminate relative vibrations between the high-field magnet and the nuclear refrigeration stage. This principle was found to work quite well and a heat leak of $Q=4.4$nW was obtained at 35mT. The heat leak was found to scale as $B^2$ upto $\sim100$mT, which is a clear sign of vibrational heating due to eddy currents. For thermometry, we employed a quartz tuning fork immersed into liquid $^3$He, which indicated 0.16 mK for the lowest temperature. Furthermore, we show that such fork oscillator can be considered as self-calibrating in superfluid $^3$He at the crossover point from hydrodynamic into ballistic quasiparticle regime.
The paper is organized as follows. In the next section the technical details of the cryostat are given. In Section 3 we discuss the thermometry with the tuning fork. The performance of the cryostat is described in Section 4 where we describe the cooling cycle and give the values for the lowest temperatures achieved, the heat leaks in different fields, thermal conductivity of the heat switch, and other details. In Conclusions we summarize the results of our experiments and their analysis.
The cryogen-free demagnetization refrigerator
=============================================
Our cryostat is based on the commercially available BF-LD400 dry dilution cryostat from BlueFors Cryogenics [@BlueFors]. It has a two-stage pulse tube refrigerator with a base temperature of 3K for the second stage. For condensation of $^3$He-$^4$He mixture, the system employs a 2bar compressor, which can be switched off during continuous circulation. The dilution unit cools down to 7mK and provides 550$\mu$W of cooling power at 100mK. The cryostat has a set of radiation shields thermally anchored at 60K, 3K, 0.7K ($^3$He evaporator) and at the mixing chamber temperature. The cryostat is equipped with a 9T magnet from American Magnetics which is thermally anchored to the 2nd stage of the pulse tube cooler. In our BF-LD400 the still pumping line and the pulse tube mounting were fitted with the damper systems provided by Bluefors Ltd so that the pulse tube and the turbo pump were mechanically well decoupled from the top flange of the cryostat.
The nuclear stage of the so-called “Helsinki design” [@Pekola84; @Berglund89] was made of a single cylindrical copper piece in which a set of slits were machined in order to reduce eddy currents which appear when the enclosed magnetic flux is varying, see Fig.\[fig:spacer\]. The heat switch which connects the stage and the mixing chamber consists of seven bended aluminium foils (50mm$\times$10mm$\times$0.5mm each) diffusion welded to two copper rods, out of which one is bolted to the nuclear stage and the other one to the mixing chamber flange. Before welding, all parts of the heat switch were annealed: copper at 900$^\circ$C in $2\cdot10^{-3}$mbar of air for two weeks and aluminium at 550$^\circ$C in better than $10^{-5}$mbar vacuum for 1 hour. A small superconducting solenoid, surrounded by a cylindrical niobium shield, is mounted on the heat switch. A field of 20mT is employed to drive the aluminium from the superconducting state to the normal state.
The nuclear stage is first pre-cooled in a field of $B=8$T; during this process, the heat switch is in the normal state and thermally connects the nuclear stage to the mixing chamber via electronic thermal conductivity. When the stage approaches the mixing chamber temperature, the heat switch is turned to superconducting state where most of the electrons are bound into Cooper pairs and cannot conduct heat, so that only phonon conductivity is left which is very small at 10 - 20 mK. The magnetic field is then slowly decreased and, in ideal adiabatic conditions, the temperature of copper nuclei is lowered proportionally to $B$ [@Lounasmaa]. In reality, some amount of entropy is lost during the demagnetization process due to heat leaks which are to be reduced well below 1$\mu$W for successful cooling.
![\[fig:spacer\] a) Photograph of the copper nuclear stage with a total length of 438mm. b) Cross-sectional view of the nucelear stage and radiation shields with spacers: 1 – nuclear stage, 2 – mixing chamber radiation shield, 3 – slotted ring, 4 – 0.7K radiation shield, 5 – thread, 6 – M4 nylon bolt, 7 – bore of the 9T magnet, 8 – plastic spring. ](stage_spacer.eps){width="0.95\linewidth"}
Owing to gradients in the polarizing field, eddy currents will be generated in constant field if the nuclear stage vibrates with respect to the magnet. In the case of a “dry” dilution refrigerator, the level of mechanical vibrations is significantly higher compared with traditional liquid-He-based refrigerators. Despite the fact that the pulse tube cooler was partly decoupled mechanically from the rest of our cryostat by flexible links, the vibrational heat load to the nuclear stage due to eddy currents was found to be a critical factor for the cooling power of the stage. Initially, there were no spacers between the radiation shields, the magnet, and the nuclear stage, so that the long tail of the nuclear stage vibrated with respect to the magnet, and the eddy currents were producing a heat load of few $\mu$W to the stage in an 8T field. Addition of spacers, however, reduced the eddy current heating by an order of magnitude.
The employed spacer system is illustrated in Fig.\[fig:spacer\]b. We have developed an easy way to make spacers between the shields, the magnet and the copper stage for fixing them together without significant changes to the shields. The spacers are made of slotted brass rings with four threaded holes for nylon bolts. The rings are placed between two shields and between the 20-mK shield and the stage. The outer shield has four holes for bolts which tightly press the inner shield, while the ring abuts the outer one, see Fig.\[fig:spacer\]b. The outer shield has plastic springs to go tightly inside the bore of the magnet. All parts of the stage-shields-magnet assembly are thus strongly mechanically fixed together so that they vibrate as a single piece without significant movements with respect to one another.
The experimental cell for liquid $^3$He is embedded inside the top part of the nuclear stage in a form of a cylinder, and silver sinter with an effective area of about 20m$^2$ is baked on the cell walls. In this first experiment, only an oscillating fork was installed inside the cell. Approximately 0.6mole of $^3$He of 200ppm $^4$He purity was condensed to the cell. This resulted in a partially filled chamber with a free liquid-vapor interface.
Thermometry
===========
The thermometry of helium sample in the sub-millikelvin range is quite a difficult task. The thermal boundary resistance, Kapitza resistance, between (dielectric) helium and the copper refrigerant increases as $1/T^3$. This means that even tiny heat leak to helium sample will saturate temperature of helium at the level determined by the heat leak and surface area of thermal contact, while the copper is much colder. Because of this thermal decoupling of helium from the environment it is absolutely necessary to measure the temperature of liquid $^3$He directly. There are three practical ways to determine liquid $^3$He temperature by measuring a) nuclear magnetic susceptibility of helium with NMR b) melting pressure with a sensitive [*in situ*]{} transducer, and c) density of quasiparticles with a mechanical oscillator. Unfortunately, the temperature dependence of susceptibility of superfluid $^3$He saturates below 0.5$T_c$, and thus NMR thermometry is not useful at the lowest temperatures. Moreover, the sensitivity of the melting curve thermometer also rapidly decreases at low temperatures. In contrast to these two thermometers, the sensitivity of a mechanical oscillator rapidly increases with lowering temperature [@Lancaster_Nature83]. In the ballistic regime below $\sim0.3\,T_c$, where the mean free path of quasiparticles becomes larger than the size of the oscillator, the fluid-induced damping decreases exponentially, $\Delta f\propto\exp{(-\Delta/T)}$ where $\Delta f$ is the width of the resonance of the oscillator and $\Delta$ is the superfluid energy gap. This behavior has been predicted theoretically by Gu$\rm\acute{e}$nault [*et al.*]{} [@Lancaster_JLTP86] and demonstrated experimentally by Todoshchenko [*et al.*]{} [@Todoshchenko02] using a vibrating wire in combination with melting curve thermometry.
![\[fig:ballistic\_mc\] Preliminary experiment on the self-calibrated fork thermometry in superfluid $^3$He at the melting pressure: a) Crossover from the hydrodynamic to the ballistic regime of oscillating fork at $\ln{(\Delta f/{\rm Hz})}\approx4.5$. The lowest temperature is 0.12$T_c$, according to Eq.(\[eq:scaling\]). b) Logarithm of the width of the resonance as a function of inverse reduced temperature measured using melting curve thermometer. The fit shows slope -2, in agreement with Eq.(\[eq:scaling\]) where $\Delta=2.0\,T_c$ [@Todoshchenko02]. The lowest temperature is 0.14$T_c$. ](ballistic_mc.eps){width="0.95\linewidth"}
Mechanical oscillator thermometer is, however, a secondary thermometer and as such it needs to be calibrated against some other thermometer. Even in the ballistic regime, it needs calibration at least at one temperature in order to scale the width of the resonance with respect to it. Fortunately, the oscillator can be used to calibrate itself. The idea is to use the central frequency of the resonance as an independent single-point (“fixed point") thermometer. In the hydrodynamic regime there is an additional mass attached effectively to the oscillator due to the viscous motion of quasiparticle excitations around it. Upon cooling, the mean free path of quasiparticles increases rapidly, and less and less quasiparticles around the oscillator feel its motion and the effective mass of the oscillator decreases. The central frequency of the resonance is inversely proportional to the square root of the effective mass, and it saturates once the mean free path becomes longer than the size of the oscillator. Hence, the point of the crossover from the hydrodynamic regime into the ballistic regime is manifested by a saturation of the dependence of central frequency $f_0$ as a function of the width of the resonance $\Delta f$ (see Fig.\[fig:ballistic\_mc\]a).
The onset of the ballistic regime is quite exactly determined because the dependence of the mean free path on temperature is very steep [@Ono82]. The calibration procedure is thus to measure the width $\Delta f_b$ at which central frequency $f_0$ saturates and attribute this width to certain temperature $T_b$ depending on the size of the oscillator. Then the temperatures below $T_b$ can be calculated according to an exponential scaling law
$$\frac{\Delta f}{\Delta f_b}=\exp{[\frac{\Delta}{T_b}-\frac{\Delta}{T}]},
\label{eq:scaling}$$
where $\Delta f_b$ refers to the width at the onset temperature $T_b$. The superfluid energy gap $\Delta$ is temperature-independent below 0.5$T_c$ [@Muhlschlegel59] and varies with pressure from 1.8$T_c$ at zero bar to 2.0$T_c$ at the melting pressure [@Todoshchenko02]. This simple exponential law together with the evident single-point calibration at the onset of the ballistic regime makes the mechanical oscillator as a very sensitive “primary” thermometer.
The self-calibration property has proven to work reliably at the melting pressure according to independent simultaneous temperature measurements using melting curve thermometry; this experiment was done on a regular nuclear demagnetization cryostat for $^3$He [@Manninen14]. The melting curve thermometer was a capacitive Straty-Adams strain pressure gauge [@Straty-Adams] with $\sim5\,\mu$bar accuracy. The resonator in these cross-check experiments was a commercially available quartz tuning fork with 0.6mm wide tines, which had exactly the same dimentions as in the present work. According to the calculations by Ono [*et al.*]{} [@Ono82], the onset of the ballistic regime for this size is at $T_b=0.25\,T_c$. The crossover from hydrodynamic to ballistic behavior is viewed best by plotting the central frequency $f_0$ versus logarithm of the resonance width $\Delta f$ as shown in Fig.\[fig:ballistic\_mc\]a. The onset of the ballistic regime, which we determine as the maximum of the second derivative of the function $f_0=f_0(\ln{(\Delta f/{\rm Hz})})$, occurs at $\ln{(\Delta f_b/{\rm Hz})}=4.5$. The lowest measured width corresponds to $\ln{(\Delta f_{min}/{\rm Hz})}=-3.9$ which, according to Eq.(\[eq:scaling\]), yields $T_{min}=\Delta/(4.5+3.9+\Delta/T_b)$. By substituting $\Delta=2.0\,T_c$ for the superfluid energy gap at high pressure and $T_b=0.25\,T_c$ for the onset we find $T_{min}=0.12\,T_c$. The apparent arbitrariness in the determination of the onset location, [*i. e.*]{} $T_b$, does not affect significantly the accuracy of the method due to sharpness of the crossover. Indeed, if we take 4 or 5 instead of 4.5 for the $\ln{(\Delta f_b/{\rm Hz})}$, it will result in a very small change of the minimum temperature, ranging from 0.125$T_c$ to 0.118$T_c$. An independent measurement of $\ln{(\Delta f/{\rm Hz})}$ [*vs*]{} $T$ using melting curve thermometry, Fig.\[fig:ballistic\_mc\]b, yields 0.14$T_c$ for the minimum temperature. The agreement with the result $T_{min}=0.12\,T_c$ based on the self-calibration method is quite good if one considers the relatively large heat capacity of solid $^3$He which keeps the solid-liquid interface at a temperature slightly higher than that of the liquid.
![\[fig:heatleak\] a) Temperature of liquid $^3$He inside the nuclear stage. Magnetization towards 8T started at time $t=0$ and was completed at $t=2.4$h. At $t=74$h the heat switch was turned off and the magnetic field sweep from 8T down to 35mT was started. Superfluid transition was reached at time $t=90.5$h in a field of 250mT. The curve is the fit of the precool process using Eq.(\[eq:precool\]) with 1.4$\mu\Omega$ for the resistance of the heat switch and with a heat leak of $P=250$nW, see the text for detailes. b) The heat leak to the nuclear stage as a function of the final demagnetization field $B_f$. Open symbols – empty cell, closed symbols – cell is partially filled with liquid $^3$He. The straight line corresponds to the expected, eddy-current-induced $B^2$ dependence at low fields.](precool_heatleak.eps){width="0.95\linewidth"}
In addition to the fork resonances, the temperature of the demagnetization stage was also recorded by a noise thermometer manufactured by Physical-Technical Institute (Physikalisch-Technische Bundesanstalt) in Berlin using the electronics from Magnicon Ltd. [@Beyer2013] The sensitive element of the noise thermometer is a copper block, the thermal noise currents of which are recorded by a SQUID that is picking up the magnetic flux induced by the noise currents. The noise thermometer was mechanically attached to the nuclear demagnetization stage with 5mm long threaded stud at the end of the copper block. We found that this system worked well on the dry dilution refrigerator despite the magnetic noise from the pulse tube heat exchangers [@Eshraghi2009]. It was found that temperatures down to 0.4mK could be recorded on the nuclear stage using the noise thermometer. The saturation of the noise thermometer at temperatures below 0.4mK is due to the heat leak of $\sim200$pW from the SQUID bias.
Performance of the refrigerator
===============================
Fig.\[fig:heatleak\]a depicts a full operation cycle of our demagnetization cryostat. The cooldown starts by running the dilution unit at a high circulation rate of $\sim800$$\mu$mol$/$s, which cools the nuclear stage below 20mK through the heat switch which is kept in the conducting (normal) state by its small magnet. Prior to the demagnetization cycles, the experimental chamber inside the nuclear stage, was filled by 0.6 moles of $^3$He at low pressure to immerse the tuning fork fully inside liquid. In the normal fluid state above $T_c=0.93$mK [@Greywall86], the width of the fork resonance $\Delta f$ is inversely proportional to temperature [@Eltsov2007]. The calibration of the fork for measurements in the normal state of $^3$He was made at $T_c$ during the demagnetization process.
![\[fig:ballistic\] a) Logarithm of the width of the resonance in liquid $^3$He at zero pressure. Demagnetization was started at time $t=0$ and was finished at $t=17.5$h. On cooling, $T_c$ was passed at $t\simeq16.5$h after which the width $\Delta f$ rapidly decreased by two orders of magnitude. At $t = 18$h, warm-up began and $T_c$ was passed again at $t=24$h. The temperature axis on the right is obtained from the ballistic exponent below 0.3mK and using the width data by Eltsov [*et al.*]{} [@Eltsov2007] at 0.3mK $<T<$ 0.9mK scaled to the size of our fork. Temperature axis refers only to superfluid state from 16.5 to 24 hours. b) Crossover from the hydrodynamic to the ballistic regime for the motion of the oscillating fork (from left to right). The saturation of the central frequency $f_0$ occurs at $\ln{(\Delta f/{\rm Hz})} \simeq 4.5$. The lowest temperature is 0.19$T_c$ according to the self-calibration method. ](ballistic.eps){width="0.95\linewidth"}
In Fig.\[fig:heatleak\]a, the field in the main magnet was ramped up from 0 to 8T in 2.5hours. Precooling of the nuclear stage at 8T was continued for about three days; this time is mostly determined by the thermal conductance of the present heat switch and the heat capacity of the nuclear stage. Our copper nuclear stage has effectively 3.5kg in the high field region, which means that its nuclear heat capacity is $C\simeq 1.7\cdot10^{-4}B^2/T^2$JK$/$T$^2=A/T^2$ [@Lounasmaa], where field-dependent constant $A$ equals 0.012JK in 8T field. The heat balance can be written as
$$Pdt-\frac{A}{T^2}dT-\frac{T^2-T_0^2}{\gamma}dt=0$$
where $P$ is the heat leak to the nuclear stage, $\gamma$ is the thermal resistance of the heat switch, and $T_0=14$mK is the ultimate temperature of the dilution unit under 8T conditions. After integration, we find
$$\label{eq:precool}
t-t_1=\frac{\gamma A}{\Theta^2}~[~\frac{1}{T_1}-\frac{1}{T}+
\frac{1}{2\Theta}\ln{\frac{(T_1-\Theta)}{(T_1+\Theta)}\frac{(T+\Theta)}{(T-\Theta)}}~],$$
where $\Theta=\sqrt{P\gamma+T_0^2}$ is a characteristic temperature of the demagnetization refrigerator, which accounts for the temperature of the dilution unit, for the heat leak to the nuclear stage, and for the quality of the heat switch.
Eq.(\[eq:precool\]) fits well the experimentally measured temperature during precooling as seen in Fig.\[fig:heatleak\]a. From the fit we find the heat leak to the nuclear stage $P=250$nW and the thermal resistance of the switch $\gamma=120$K$^2/$W. If the temperature difference $\delta T=T-T_0$ over the heat switch is relatively small, then the heat flux through the switch can be written as $\dot{Q}=(2T/\gamma)\delta T$. It is more convenient to express the thermal resistance in units of electrical (temperature-independent) resistance $R=\gamma L/2=1.4\,\mu\Omega$, where $L=2.4\cdot10^{-8}$W$\Omega/$K$^2$ is the Lorenz number. After the precool, at time $t = 75$h, the heat switch is turned to superconducting state and the field is slowly swept down to 50mT.
We applied fork thermometry with self-calibration (see Sect.3) to measure the lowest temperature achieved in superfluid $^3$He on our refrigerator. Similar fork as in the experiment at the melting pressure was employed meaning that the onset temperature $T_b$ for the ballistic regime must again be 0.25$T_c$. Fig.\[fig:ballistic\]a displays the logarithm of the fork resonance width $\Delta f$ as a function of time. Near $T_c$, $\Delta f$ becomes of the order of the central frequency $f_0$ and the resonance is basically lost until the liquid enters the superfluid state at $t = 16.5$h. With growing superfluid fraction, the width decreases rapidly over two orders of magnitude and, finally at $\Delta f\sim300$Hz, the fork is in the regime where Eq.(\[eq:scaling\]) becomes applicable. The liquid still cools a bit further down to $\Delta f=40$Hz which corresponds to $T=0.19$$T_c=0.17$mK according to Eq.(\[eq:scaling\]). While warming, $T_c$ is passed again at $t=24.0$h. From the warmup rate we can calculate the heat leak to the nuclear stage to be about 25nW in a final demagnetization field $B_f$ of 50mT. During the next demagnetization cycle to a field of 35mT, we have measured even a smaller fork resonance width of 15Hz corresponding to $T=0.16$mK. Note that the sensitivity of the fork thermometer is so high at lowest temperatures that the decrease of the width by 2.5 times means only 5% difference in temperature.
During these demagnetization experiments, we also followed the nuclear stage temperature by noise thermometry. Unfortunately, the reading from the noise thermometer was limited to $T > 0.40$mK due to overheating of the copper block by the SQUID bias current. At $T > 1$mK, the noise thermometer worked well and was instrumental in our heat leak measurements at high $T$.
At low values of the final demagnetization fields $B_f$, the heat leak to the nuclear stage was found to depend linearly on $B_f^2$ as illustrated in Fig.\[fig:heatleak\]b. This means that the warmup rate does not actually depend on the field as both the heat capacity of the stage and the heat leak are proportional to $B_f^2$ (provided that $B_f \gg B_{int}=0.3$mT). At fields higher than 50-100mT, however, the field dependence of the heat leak becomes gradually slower and the warmup rate decreases correspondingly. Indeed, instead of a typical 7 hours below $T_c$, we stayed in our best cool downs at $B_f = 100$mT in the superfluid state for 14 hours. This time span, which can be extended to $\sim 20-25$ hours by further demagnetization from 100 mT, can be regarded as fully adequate for basic experiments in superfluid $^3$He. The minimum heat leak to the nuclear stage of 4.4nW was measured in 35mT field with empty cell. Typically, the heat leaks with the cell filled with $^3$He were somewhat higher compared to empty cell because of the viscous heating of $^3$He due to vibrations.
Conclusions
===========
We have demonstrated cooling of $^3$He well below the superfluid transition temperature $T=0.93$ mK at saturated vapor pressure using a helium-free “dry” demagnetization refrigerator. Our work is the first to cool helium-3 down to 0.16mK on a LHe-bath-free refrigerator, and thus we open an opportunity for deep sub-mK investigations on a commercially available pulse tube refrigerator straightforwardly equipped with a copper adiabatic nuclear demagnetization stage. In our work, we have introduced a new simple method to measure the temperature of superfluid $^3$He via the properties of mechanical fork oscillators without the need for other thermometers for calibration. The method has proven to work accurately at the melting pressure when compared against an independent melting curve thermometer.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Juha Tuoriniemi, Matti Manninen, and Juho Rysti who took part in the thermometry experiments at the melting pressure. The work was supported by the Academy of Finland (contracts no.135908 and 250280, LTQ CoE and FIRI2010) and the EU 7th Framework Programme (FP7/2007–2013, grant Microkelvin). This research project made use of the Aalto University Cryohall infrastructure.
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|
---
abstract: 'This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. After formulating a Lopatinskiǐ condition, elliptic estimates are established near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provide a rigorous framework for the mode analysis of aAdS spacetimes for the full range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils with a spectral parameter in the boundary condition is shown, which has applications to linear stability of certain aAdS spacetimes.'
address: 'Department of Mathematics, Evans Hall, University of California, Berkeley, CA 94720, USA'
author:
- Oran Gannot
bibliography:
- 'adsbib.bib'
title: 'Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes'
---
The study of linear fields on asymptotically anti-de Sitter (aAdS) spaces has stimulated new interest in boundary value problems for a class of singular elliptic equations, wherein the operator $D_x^2 + (\nu^2 -1/4)x^{-2}$ acts on one of the variables [@enciso:2013; @holzegel:2012:jhde; @holzegel:2015; @holzegel:2013:cmp; @vasy:2012:apde; @warnick:2013:cmp; @warnick:2015:cmp]. To formulate this class of operators more precisely, consider a product manifold $[0,\varepsilon) \times \partial X$, where $\partial X$ is compact. The model for what we call a Bessel operator has the form $$P(x,y,D_x,D_y) = D_x^2 + (\nu^2 - 1/4)x^{-2} + A(x,y,D_y),$$ where $(x,y) \in (0,\varepsilon)\times \partial X$ and $A$ is a family of second order differential operators on $\partial X$ depending smoothly on $x\in [0,\varepsilon)$. The parameter $\nu$ is required to be real and strictly positive. In the study of linear waves on aAdS spacetimes, $\nu$ is related to the mass of a scalar field — see Section \[sect:motivation\] for more details. The condition $\nu > 0$ corresponds to the Breitenlohner–Freedman bound [@breitenlohner:1982:plb; @breitenlohner:1982:ap].
Boundary data for this problem are formally defined by the following weighted restrictions: $$\gamma_- u = x^{\nu-1/2}u|_{\partial X}, \quad \gamma_+ u = x^{1-2\nu}\partial_x (x^{\nu - 1/2}u)|_{\partial X}.$$ Some care is needed to give precise meaning to these restrictions — see Section \[subsect:traces\], along with an earlier discussion in Warnick [@warnick:2013:cmp]. The boundary operators in this paper are of the form $T = T^- \gamma_- + T^+ \gamma_+ $, where $T^-, \, T^+$ are differential operators on $\partial X$ of order at most one and zero respectively. This paper is concerned with solvability of the boundary value problem $$\label{eq:introBVP}
\begin{cases}
P u = f \text{ on $X$} \\
T u = g \text{ on $\partial X$}
\end{cases}$$ when $0 < \nu <1$, and the simpler equation $$\label{eq:introeqn}
Pu = f \text{ on $X$}$$ when $\nu \geq 1$. No boundary conditions are imposed when $\nu \geq 1$, since in that case $u$ must satisfy a Dirichlet boundary condition. The difference between the cases $0 < \nu < 1$ and $\nu \geq 1$ is explained in more detail in the introduction to Section \[sect:ellipticBVP\].
Ellipticity of the Bessel operator $P$ is defined in Section \[subsect:ellipticity\]. As in the study of smooth boundary value problems, there is also a notion of ellipticity for , given by a natural Lopatinskiǐ condition on the pair $(P,T)$. This condition is introduced in Section \[subsect:lopatinskii\]. Elliptic estimates are proved in Theorem \[theo:bvptheo\] of Section \[subsect:variable\]. When the operators $P,\, T$ depend polynomially on a spectral parameter $\lambda$, there is a notion of parameter-ellipticity for both $P$ and the boundary value problem . Theorem \[theo:bvptheosemiclassical\] of Section \[subsect:semiclassicalvariable\] provides elliptic estimates in terms of parameter-dependent norms, which are uniform as $|\lambda|\rightarrow \infty$ in the cone of ellipticity.
For the global problem, consider a compact manifold ${
\overline{X}
}$ where $[0,\varepsilon) \times \partial X$ is identified with a collar neighborhood of $\partial X$. Suppose that the restriction of $P$ to this collar is a Bessel operator — see Section \[sect:prelim\] for details. As in the case of smooth boundary value problems, estimates for $P$ near $\partial X$ may often be combined with estimates in the interior $X$ to establish the Fredholm property (including some cases where $P$ fails to be everywhere elliptic on $X$). In Section \[sect:fredholm\], a sufficient condition of this type is discussed. Furthermore, in the presence of a spectral parameter $\lambda$, unique solvability is established for $\lambda$ in the cone of ellipticity provided $|\lambda|$ is sufficiently large.
Section \[sect:motivation\] recalls the notion of an aAdS metric, which is the primary motivation for this paper. It is shown that the Fourier transformed Klein–Gordon operator is indeed a Bessel operator whose order $\nu$ depends on the Klein–Gordon mass. One goal is to study the Klein–Gordon equation by Fourier synthesis once its spectral family is understood, corresponding to the study of normal or quasinormal modes. This paper provides a rigorous framework for future work in that direction.
For stationary aAdS spacetimes with compact time slices and an everywhere timelike Killing field, Section \[sect:completeness\] describes a class of boundary conditions which yield a complete set of normal modes associated to a discrete set of eigenvalues. Of particular interest are time-periodic boundary conditions which depend on $\partial_t$ (hence depend on the spectral parameter after a Fourier transform). This is important for the study of modes with transparent or dissipative boundary conditions, along with superradiance phenomena [@avis:1978:prd; @holzegel:2015; @winstanley:2001].
In a forthcoming paper [@gannot:2014:kerr], discreteness of quasinormal frequencies is also established for massive fields on Kerr–AdS black holes with arbitrary rotation speed. These frequencies replace eigenvalues in scattering problems [@cardoso:2014:jhep; @dias:2012:cqg:b; @gannot:2014:kerr; @horowitz:2000:prd; @konoplya:2011; @warnick:2015:cmp]. When $0 < \nu < 1$, arbitrary boundary conditions satisfying the Lopatinskiǐ condition may be imposed on the field (although of course one does not have any completeness statement).
The approach of this paper is inspired by the texts of Roitberg [@roitberg:1996] and Kozlov–Mazya–Rossman [@kozlov:1997]. This approach is particularly suited to the singular nature of Bessel operators, and allows for the study of boundary value problems in low regularity spaces as needed in applications to general relativity — see Section \[sect:fredholm\]. All the methods are classical, using only homogeneity properties of differential operators. The key is exploiting the theory of “twisted” derivatives as first emphasized in [@warnick:2013:cmp]. This is based on the classical observation that the one-dimensional Bessel operator $D_x^2 + (\nu^2 - 1/4)x^{-2}$ admits a factorization as the product of a first order order operator and its adjoint; this first order operator is then treated as an elementary derivative.
Using a variational approach, Holzegel [@holzegel:2012:jhde], Warnick [@warnick:2013:cmp; @warnick:2015:cmp], and Holzegel–Warnick [@holzegel:2012wt] derive some of the same (or similar) elliptic estimates. However, only the “classical” self-adjoint boundary conditions are handled when $0 < \nu < 1$; these are the Dirichlet ($T = \gamma_-$) and Robin boundary conditions ($T = \gamma_+ + \beta \gamma_- $ with real-valued $\beta$). The approach taken here accounts for a larger class of non-self-adjoint boundary conditions, which is optimal among differential boundary conditions.
The results of this paper should also be compared to earlier works of Vasy [@vasy:2012:apde], Holzegel [@holzegel:2012:jhde] on aAdS spaces, where a more restrictive measure is used to define the space of square integrable functions. In those works, the square integrability condition is equivalent to the generalized Dirichlet boundary condition. This limits the range of applications, since different boundary conditions are employed throughout the physics literature on aAdS spaces [@avis:1978:prd; @berkooz:2002:jhep; @breitenlohner:1982:plb; @breitenlohner:1982:ap; @dias:2013:jhep; @ishibashi:2004:cqg; @witten:2001]
There is also a general microlocal approach to degenerate boundary value problems developed by Mazzeo–Melrose [@mazzeo:1987], Mazzeo [@mazzeo:1991], Mazzeo–Vertman [@mazzeo:2013], among many others. The work of Vasy [@vasy:2012:apde] on aAdS spaces makes use of this technology. In particular, the elliptic theory in [@mazzeo:2013] could likely reproduce the results of this paper, and is also applicable to much more general classes of elliptic operators. On the other hand, the approach developed here is directly motivated by the physics literature. For instance, the Sobolev spaces used in this paper were originally defined in [@warnick:2013:cmp] to give precise meaning to the energy renormalization implicit in the work of Breitenlohner–Freedman [@breitenlohner:1982:plb]. There is also a simplicity advantage in using physical space methods, rather than a more sophisticated microlocal approach.
Bessel operators of the precise kind studied here arise in numerous contexts outside of general relativity with negative cosmological constant, both mathematical and physical. See for instance the monograph of Kipriyanov [@kipriyanov:1997] and the substantial literature surrounding “generalized axially symmetric potentials” [@erdelyi:1956; @gilbert:1962; @huber:1954; @weinstein:1953; @wittstein:2014].
Preliminaries {#sect:prelim}
=============
Conventions for differential operators
--------------------------------------
If $P$ is a smooth differential operator on a manifold $Y$, then in local coordinates, $$\label{eq:difflocalcoords}
P = \sum_{|\alpha| \leq m} a_{\alpha}(y) D_y^\alpha.$$ In that case the order of $P$ is said to no greater than $m$. The smallest $m$ for which there exists a nonzero coefficient $a_{\alpha}, \, |\alpha| = m$ in some coordinate representation is the order of $P$, written $\mathrm{ord}(P)$. If the order of $P$ is no greater than $m$, the symbol $\sigma_m(P)$ of $P$ with respect to $m$ is the polynomial function on $T^*Y$ given in local coordinates by $$\sigma_m(P)(y,\eta) = \sum_{|\alpha|=m} a_\alpha(y) \eta^\alpha, \quad (y,\eta) \in T^*Y.$$ Thus if $m$ is strictly larger than the order of $P$, then $\sigma_m(P) = 0$. The space of differential operators of order no greater than $m$ is denoted $\mathrm{Diff}^m(Y)$. If $P$ has order no greater than $m$ with $m < 0$, then $P = 0$; conversely if $P = 0$, then $P$ can assigned any negative order. This convention will be useful throughout Section \[sect:ellipticBVP\].
The class of parameter-dependent differential operators on a manifold $Y$ is defined as follows: $P \in \mathrm{Diff}_{(\lambda)}^m(Y)$ if in local coordinates, $$P(y,D_y;\lambda) = \sum_{j + |\alpha| \leq m} a_{\alpha,j}(x) \lambda^j D_y^\alpha.$$ The parameter-dependent order $\mathrm{ord}^{(\lambda)}(P)$ is defined by assigning to $\lambda^j$ the same weight as a derivative of order $j$. Thus the parameter-dependent principal symbol of $P$ is given by $$\sigma^{(\lambda)}_{m}(A) = \sum_{j + |\alpha| = m} a_{\alpha,j}(t)\lambda^j \eta^\alpha ,\quad (y,\eta,\lambda) \in T^*Y \times \mathbb{C}.$$ If $Y$ is a compact manifold without boundary, the parameter -dependent Sobolev norms on $Y$ are defined by $$\VERT u \VERT^2_{H^s(Y)} = |\lambda|^{2s} \| u \|^2_{H^0(Y)} + \| u \|^2_{H^s(Y)},$$ and a parameter-dependent operator $P \in \mathrm{Diff}^m_{(\lambda)}(Y)$ is bounded $P: H^s(Y) \rightarrow H^{s-m}(Y)$ uniformly with respect to $|\lambda|$ in these norms.
Manifolds with boundary {#subsect:manifoldwithboundary}
-----------------------
Let ${
\overline{X}
} = X \cup \partial X$ denote an $n$-dimensional manifold with compact boundary $\partial X$ and interior $X$. A boundary defining function for $\partial X$ is a function $x \in C^\infty({
\overline{X}
})$ satisfying $$x^{-1}(0) = \partial X, \quad x > 0 \text{ on } X, \quad {dx}|_{\partial X} \neq 0.$$ Given $x$, there exists an open subset ${
\overline{W}
} \supseteq \partial X$, a number $\varepsilon > 0$, and a diffeomorphism $\phi : [0,\varepsilon) \times \partial X \rightarrow {
\overline{W}
}$ such that $x \circ \phi$ agrees with the projection $[0,\varepsilon) \times \partial X$. A collar neighborhood of this type can be constructed as follows: choose a Riemannian metric $g$ on ${
\overline{X}
}$ and consider the unit normal vector field $N = \nabla_g x/|\nabla_g x|_{g}$. This is well defined on $\{0 \leq x < \varepsilon\}$ provided $\varepsilon > 0$ is chosen sufficiently small to ensure $dx \neq 0$. Then let $$\phi(s,y) = \exp(sN)(y)$$ where $\exp(sN)$ is the flow of $N$ for $s \in [0,\varepsilon)$ and $y \in \partial X$. A collar diffeomorphism of this type is said to be compatible with $x$. Unless otherwise specified, a manifold with boundary ${
\overline{X}
}$ will always refer to ${
\overline{X}
}$ equipped with a distinguished boundary defining function $x$ and a choice of compatible collar diffeomorphism $\phi$ (note that $\phi$ also depends on a metric $g$ used to define the vector field $N$).
Bessel operators {#subsect:bessel}
----------------
Given $\nu \in \mathbb{R}$, formally define the differential operator $\partial_\nu$ by the formula $${{\partial}_\nu}= \partial_x + (\nu-1/2){x^{-1}} = x^{1/2-\nu}\partial_x x^{\nu-1/2}.$$ Furthermore, let ${{\partial}_\nu}^* = -x^{\nu-1/2}\partial_x x^{1/2-\nu}$, which is the formal adjoint of ${{\partial}_\nu}$ with respect to Lebesgue measure on ${\mathbb{R}_+}$. Similarly, let ${D_\nu}= -i {{\partial}_\nu}$ and ${D_\nu}^* = i{{\partial}_\nu}^*$. Note that $$|{D_\nu}|^2 := {D_\nu}^*{D_\nu}= D_x^2 + (\nu^2 - 1/4)x^{-2}$$ is the one-dimensional Bessel operator in Schrödinger form.
Let ${
\overline{X}
}$ denote a manifold with compact boundary. A differential operator $P \in \mathrm{Diff}^2(X)$ is called a Bessel operator of order $\nu > 0$ if there exist $$A = A(x,y,D_y) \in \mathrm{Diff}^2(\partial X), \quad B= B(x,y,D_y) \in \mathrm{Diff}^1(\partial X),$$ depending smoothly on $x \in [0,\varepsilon)$, such that $B(0,y,D_y) = 0$ and $$\label{eq:besseloperator}
\phi^* P = |{D_\nu}|^2 + B(x,y,D_y){D_\nu}+ A(x,y,D_y)$$ The set of such operators is denoted by $\mathrm{Bess}_\nu(X)$.
The requirement that $|{D_\nu}|^2$ appears with unit coefficient is not at all essential. If the coefficient is a positive function smooth up to $x = 0$, then the quotient of $P$ by this coefficient is a Bessel operator as above, and this normalization does not affect any of the arguments in Sections \[sect:ellipticBVP\], \[sect:fredholm\].
The class of Bessel operators depends strongly on the pair $(x,\phi)$, where $x$ is a boundary defining function and $\phi$ is a collar diffeomorphism compatible with $x$ in the sense of Section \[subsect:manifoldwithboundary\]. To make this explicit, write $\mathrm{Bess}_\nu(X;x,\phi)$.
\[lem:equivalentbessel\] Let $x, \rho$ be two boundary defining functions with associated collar diffeomorphisms $\phi_x,\phi_\rho$. Given $p \in {
\overline{W}
}_x$, write $$\phi_x^{-1}(p) = (x(p),y(p)) \in [0,\varepsilon_x)\times \partial X.$$ Let $\Phi = \phi_x^{-1}\circ \phi_\rho$, so that $\Phi(t,z) = ((x\circ \phi_\rho)(t,z), (y\circ \phi_\rho)(t,z))$ for each $(t,z) \in [0,\varepsilon)\times \partial X$. If $$\label{eq:evendiffeo}
\partial_t^{2}(x\circ \phi_\rho)(0,\cdot) = 0, \quad \partial_t (y\circ\phi_\rho)(0,\cdot) = 0$$ and $P \in \mathrm{Bess}_\nu(X;\rho,\phi_\rho)$, then there exists a positive function $f \in C^\infty({
\overline{W}
}_x\cap{
\overline{W}
}_\rho)$ such that $ fP \in \mathrm{Bess}(X;x,\phi_x)$.
The condition gives $$(x\circ \phi_\rho)(t,z) = t \partial_t (x\circ \phi_\rho) (0,z) + \mathcal{O}(t^3), \quad (y\circ\phi_\rho)(t,z) = (y\circ \phi_\rho)(0,z) + \mathcal{O}(t^2).$$ The result can be established from the composition rule for derivatives in local coordinates.
\[lem:xrho\] Suppose that $x$ is a boundary defining function, and that $\phi_x$ is given by the flow of $\nabla_g x/|\nabla_g x|_g$ with respect to a fixed metric $g$ defined near $\partial X$. Let $\rho = e^w x$ and $h=e^{2w}g$, where $w$ is smooth near $\partial X$ and $\partial_x w(0) =0$. If $\phi_\rho$ is given by the flow of $\nabla_{h} \rho/ |\nabla_{h}\rho|_{h}$, then $\Phi := \phi_x^{-1} \circ \phi_\rho$ satisfies .
First write $$x(\phi_\rho(t,z)) = e^{-w(\phi_\rho(t,z))} \rho(\phi_\rho(t,z)) =e^{-w(\phi_\rho(t,z))}t,$$ which gives $$\partial_t^2 (x\circ\phi_\rho)(t,z) = e^{-w(\phi_\rho(t,z))}\left( -2\partial_t (w\circ \phi_\rho)+ t \partial_t (w\circ\phi_\rho)^2 + t \partial_t^2 (w \circ\phi_\rho) \right)(t,z).$$ This quantity vanishes at $t=0$ provided $\partial_t (w\circ\phi_\rho)(0,z)=0$, as assumed in the lemma. Next, write $$\nabla_h \rho = x e^w \nabla_h w + e^w \nabla_h x = e^{-w}(x \nabla_g w + \nabla_g x).$$ Thus $\nabla_h \rho = e^{-w}\nabla_g x$ and $$\frac{\nabla_h \rho}{|\nabla_h \rho|_h} = e^w \frac{\nabla_g x}{|\nabla_g x|}$$ when restricted to $\partial X$. Therefore $\partial_t (y\circ\phi_\rho)(0,z)$ is proportional to $(\nabla_h \rho)_zy$ for any $z \in \partial X$, since $\phi_\rho(0,z) = z \in \partial X$. Further, $(\nabla_h \rho)_z y$ is proportional to $(\nabla_g x)_z y$, which is proportional to $\partial_t (y \circ \phi_x)(0,z)$. But $(y\circ \phi_x)(t,z) = z$ for all $t$, so the derivative of this constant function vanishes, finishing the proof.
In Lemma \[lem:xrho\] the relation $h = e^{2w}g$ can be replaced by $h = e^{v}g$ for any smooth $v$, but in practice $h$ will be related to $g$ as stated.
A (smooth, positive) density $\mu$ on ${
\overline{X}
}$ is said to be of product type near $\partial X$ if $$\phi^*\mu = |dx| \otimes \mu_{\partial X}$$ for a fixed density $\mu_{\partial X}$ on $\partial X$.
It is always possible to choose a density of product type near $\partial X$. This is useful in light of the next lemma. If ${
\overline{X}
}$ is compact, then $L^2(X)$ may be defined as the space of square integrable functions with respect to any smooth density $\mu$ on ${
\overline{X}
}$, in particular one of product type near $\partial X$.
\[lem:closedunderadjoint\] Suppose that $\mu$ is of product type near $\partial X$. If $P \in \mathrm{Bess}_\nu(X)$, then $P^* \in \mathrm{Bess}_\nu(X)$, where $P^*$ is the formal adjoint of $P$ with respect to $\mu$.
The pullback of $P^*$ to $(0,\varepsilon)\times \partial X$ is given by $$|{D_\nu}|^2 + {D_\nu}^* B^* + A^*,$$ where $A^*,B^*$ are the formal adjoints of $A, B$ with respect to $\mu_{\partial X}$.
On the other hand, ${D_\nu}^* B^* = B^* {D_\nu}^* + [D_x,B^*]$. Furthermore, since $B = xB_1$ for a first order operator $B_1$ on $\partial X$ depending smoothly on $x\in[0,\varepsilon)\times \partial X$, it follows that $$B^*{D_\nu}^* = xB_1^*{D_\nu}^* = B_1^*(xD_x - i(1/2-\nu)) = B^*{D_\nu}- i(1/2-\nu)B_1^*,$$ which completes the proof since the multiple of $B_1^*$ as well as $[D_x,B^*]$ may be absorbed into $A^*$.
For the local theory, it is convenient to work on ${\mathbb{T}^n_+}= {\mathbb{R}_+}\times {\mathbb{T}^{n-1}}$, where ${\mathbb{T}^{n-1}}= \left( \mathbb{R}/2\pi \mathbb{Z} \right)^{n-1}$. The set of Bessel operators on ${\mathbb{T}^n_+}$ is defined with respect to the canonical product decomposition on ${\mathbb{T}^n_+}$. Thus $P \in \mathrm{Bess}_\nu({\mathbb{T}^n_+})$ if $$P(x,y,{D_\nu},D_y) = |{D_\nu}|^2 + \sum_{|\beta|\leq 1} b_\beta(x,y) D_y^\beta {D_\nu}+ \sum_{|\alpha| \leq 2} a_{\alpha}(x,y)D_y^\alpha$$ for $b_\beta \in xC^\infty({
\overline{{\mathbb{T}^n_+}}
})$ and $a_{\alpha} \in C^\infty({
\overline{{\mathbb{T}^n_+}}
})$. When working on ${\mathbb{T}^n_+}$, the functions $b_\beta, \, a_{\alpha}$ are referred to as the *coefficients* of $P$
Fix a coordinate chart $Y \subseteq \partial X$ and a diffeomorphism $\theta : Y \rightarrow V$, where $V$ is an open subset of $\mathbb{T}^{n-1}$. Setting $$U = \phi([0,\varepsilon) \times Y),$$ the map $\psi : U \rightarrow [0,\varepsilon) \times V$ given by $\psi = (1\times \theta)\circ \phi^{-1}$ defines a boundary coordinate chart on ${
\overline{X}
}$. Given $P \in \mathrm{Bess}_\nu(X)$, there clearly exists $P_U \in \mathrm{Bess}_\nu({\mathbb{T}^n_+})$ such that $$Pu = P_U(u \circ \psi)$$ for each $u \in {C}_c^\infty(U^\circ)$. Furthermore, it is always possible to arrange it so that the coefficients of $P_U$ (in the sense of the previous paragraph) are constant outside a compact subset of ${
\overline{{\mathbb{T}^n_+}}
}$.
Ellipticity and the boundary symbol {#subsect:ellipticity}
-----------------------------------
Given $P \in \mathrm{Bess}_\nu(X)$ which near $\partial X$ has the form $$P = |{D_\nu}|^2 + B{D_\nu}+ A,$$ let $A_0(y,D_y) = A(0,y,D_y)$. Ellipticity of $P$ at a point $p \in \partial X$ is defined via the function $$\label{eq:besselsymbol}
\xi^2 + \sigma_2(A_0)(p,\eta),$$ which is a homogeneous polynomial of degree two in $(\xi,\eta) \in \mathbb{R} \times T_p^* \partial X$.
\[def:elliptic\] The Bessel operator $P \in \mathrm{Bess}_\nu(X)$ is said to be (properly) elliptic at $p \in \partial X$ if for each $\eta \in T_p^*\partial X \setminus 0$ the polynomial $$\label{eq:properlyellipticpolynomial}
\xi \mapsto \xi^2 + \sigma_2(A_0)(p,\eta)$$ has no real roots.
Ellipticity at $p \in \partial X$ is equivalent to the statement that the homogeneous polynomial $\xi^2 + \sigma_2(A_0)(p,\eta)$ is elliptic in $(\xi,\eta)$. Thus, ellipticity implies the existence of nonreal roots $\pm \xi(p,\eta)$, where ${\operatorname{Im}}\xi(p,\eta) < 0$ by convention.
For each $(y,\eta) \in T^* \partial X \setminus 0$, the symbol $\sigma_2(A_0)(y,\eta)$ determines a family of one dimensional Bessel operators given by $$\label{eq:boundarysymbol}
\widehat{P}_{(y,\eta)} = |{D_\nu}|^2 + \sigma_2(A_0)(y,\eta).$$ The operator $\widehat{P}_{(y,\eta)}$ is called the boundary symbol operator of $P$. Let $\mathcal{M}_+(y,\eta)$ denote the space of solutions to the equation $$\widehat{P}_{(y,\eta)}u = 0$$ which are bounded as $x\rightarrow \infty$. Ellipticity at $p\in \partial X$ implies that $\dim \mathcal{M}_+(p,\eta) = 1$ for each $\eta \in T_p^* \partial X \setminus 0$. Indeed, the space of solutions to $\widehat{P}_{(p,\eta)}u = 0$ is spanned by the modified Bessel functions $$\left\lbrace x^{1/2}K_\nu(i\xi(p,\eta)x),\, x^{1/2}I_\nu(i\xi(p,\eta)x)\right\rbrace.$$ Since ${\operatorname{Re}}i\xi(p,\eta) > 0$, it follows that $$x^{1/2}K_\nu(i\xi(p,\eta)x) = \mathcal{O}\left(e^{-x/C}\right), \quad x \rightarrow \infty,$$ while the second solution grows exponentially [@olver:2014 Chapter 7.8]. Thus only the first solution can possible lie in $\mathcal{M}_+(p,\eta)$.
Parameter-dependent Bessel operators {#subsect:semiclassicalbessel}
------------------------------------
\[def:semiclassicalbesseloperator\] Let ${
\overline{X}
}$ denote a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. A differential operator $P(\lambda) \in \mathrm{Diff}^2_{(\lambda)}(X)$ is called a parameter-dependent Bessel operator of order $\nu > 0$ if there exist $$A(\lambda) = A(x,y,D_y;\lambda) \in \mathrm{Diff}^2_{(\lambda)}(\partial X), \quad B(\lambda) = B(x,y,D_y;\lambda)\in \mathrm{Diff}^1_{(\lambda)}(\partial X)$$ depending smoothly on $x \in [0,\varepsilon_x)$, such that $B(0,y,D_y;\lambda) = 0$ and $$\phi^*_x P(\lambda) = |{D_\nu}|^2 + B(x,y,D_y;\lambda){D_\nu}+ A(x,y,D_y;\lambda)$$ The set of such operators is denoted by $\mathrm{Bess}^{(\lambda)}_\nu(X)$.
Ellipticity with parameter is defined by replacing the standard principal symbol of $A$ with its parameter-dependent version. Begin by fixing an angular sector $\Lambda \subseteq \mathbb{C}$.
\[def:semiclassicalelliptic\] A parameter-dependent Bessel operator $P(\lambda)$ is said to be (properly) parameter-elliptic with respect to $\Lambda$ at $p \in \partial X$ if for each $(\eta,\lambda) \in T_p^*\partial X \times \Lambda \setminus 0$, the polynomial $$\label{eq:besselsymbolsemiclassical}
\xi \mapsto \xi^2 + \sigma_2^{(\lambda)}(A_0)(p,\eta;\lambda)$$ has no real roots.
Similarly, for $(y,\eta,\lambda) \in T^* \partial X \times \Lambda \setminus 0$, define $$\widehat{P}_{(y,\eta;\lambda)} = |{D_\nu}|^2 + \sigma^{(\lambda)}_2 (A_0)(y,\eta;\lambda),$$ and then let $\mathcal{M}_+(y,\eta;\lambda)$ denote the space of solutions to $\widehat{P}_{(y,\eta;\lambda)}u=0$ which are bounded as $x\rightarrow \infty$. As before, this space is one-dimensional.
Motivation: asymptotically anti-de Sitter manifolds {#sect:motivation}
===================================================
This section recalls the notion of an asymptotically anti-de Sitter (aAdS) metric. Then, a convenient expression for the Klein–Gordon equation is given with respect to a certain product decomposition near the conformal boundary. By means of a Fourier transform, the initial boundary value problem for the Klein–Gordon equation is reduced to the study of the boundary value problem for a stationary partial differential equation depending polynomially on the spectral parameter. The corresponding operator is a Bessel operator whose order $\nu$ depends on the Klein–Gordon parameter; the condition $\nu > 0$ translates into the well known Breitenlohner–Freedman bound.
aAdS metrics
------------
Let ${
\overline{X}
}$ denote an $n$-dimensional compact manifold with boundary as in Section \[subsect:manifoldwithboundary\], and set ${
\overline{M}
} = \mathbb{R} \times {
\overline{X}
}$. Here $t \in \mathbb{R}$ will denote a global time coordinate.
There is a one-to-one correspondence between boundary defining functions $\rho$ on ${
\overline{M}
}$ satisfying $\partial_t\rho = 0$ and boundary defining functions $x$ for ${
\overline{X}
}$: given $\rho$ with $\partial_t\rho = 0$, define $x(\cdot) = \rho(t_0,\cdot)$ for any $t_0 \in \mathbb{R}$; conversely, given $x$, define $\rho(t,\cdot) = x(\cdot)$.
A boundary defining function $\rho$ on $M$ satisfying $\partial_t \rho = 0$ is said to be *stationary*.
A smooth Lorentzian metric $g$ on ${
\overline{M}
}$ is said to asymptotically simple if there exists a boundary defining function $\rho \in C^\infty({
\overline{M}
})$ such that
1. the Lorentzian metric $\bar{g} = \rho^2 g$ admits a smooth extension to ${
\overline{M}
}$,
2. the restriction $\bar{g}|_{T\partial M}$ is again Lorentzian.
The map $g \mapsto \bar{g}|_{\partial M}$ depends on $g$ and a choice of boundary defining function. However, the conformal class $[\bar{g}|_{T \partial M}]$ depends only on $g$, since any two boundary defining functions differ by a positive multiple. Also note that if $g$ is asymptotically simple, then $d\rho$ is spacelike for $\bar{g}|_{\partial M}$.
An asymptotically simple manifold $({
\overline{M}
},g)$ is said to be aAdS if there exists a boundary defining function $\rho$ such that $|d\rho|_{\bar g}^2 = -1$ on $\partial M$.
The aAdS property does not depend on the choice of boundary defining function. In addition to being aAdS, suppose that $g$ is *stationary* in the sense that $\partial_t$ is a Killing vector field for $g$. For the remainder of this section, all metrics are assumed to be stationary. This allows for a reduction to compact time slices.
\[lem:stationarygamma\] Suppose $\gamma \in [\bar{g}|_{T\partial M}]$ is stationary. Then there exists a stationary boundary defining function $x$ such that $x^2 g|_{T\partial M} = \gamma$.
Choose any boundary defining function $\tilde{\rho}$ such that $\tilde{\rho}^2 g|_{T\partial M} = \gamma$, and define $x$ on ${
\overline{X}
}$ by $$x(p) = \tilde{\rho}(0,p), \quad p \in {
\overline{X}
}.$$ Since $\gamma$ is stationary, it follows that $x^2 g|_{T\partial M} = \gamma$.
Suppose $g$ and $\gamma \in [\bar{g}|_{T\partial M}]$ are stationary, with $x$ as in Lemma \[lem:stationarygamma\]. Let $s\mapsto \phi(s,\cdot)$ denote the flow on ${
\overline{M}
}$ corresponding to $\nabla_{x^2 g} x$; note this vector field commutes with $\partial_t$.
\[lem:productdecomp\] There exists $\varepsilon > 0$ and an open neighborhood ${
\overline{\mathcal{C}}
}$ of $\partial M$ in ${
\overline{M}
}$ such that $$\phi : [0,\varepsilon)_s \times \partial M \rightarrow {
\overline{\mathcal{C}}
}$$ is a diffeomorphism.
Since $\partial_t$ and the infinitesimal generator $\nabla_{x^2 g}x$ commute, it follows that $\phi(s,\cdot)$ commutes with the flow of $\partial_t$, namely the time translations. Since $\partial X$ is compact, there exists $\varepsilon > 0$ and an open neighborhood $U$ of $\{t = 0\} \times \partial X$ in $\partial M$ such that $\phi$ is a diffeomorphism from $[0,\varepsilon) \times U$ onto its range. It now suffices to commute $\phi$ with time translations $(t_0,p) \mapsto (t_0 + t,p)$ to construct a diffeomorphism valid for all times.
\[lem:geodesicbdf\] Suppose that $({
\overline{M}
}, g)$ is an aAdS spacetime. If $g$ and $\gamma_0 \in [\bar{g}|_{T\partial M}]$ are stationary, then there exists a unique stationary boundary defining function $x$ such that:
1. $x^2 g|_{T\partial M} = \gamma_0$,
2. $|dx|^2_{x^2 g} = -1$ in a collar neighborhood ${
\overline{\mathcal{C}}
}$ of $\partial M$ as in Lemma \[lem:productdecomp\].
The proof of [@graham:1991 Lemma 5.2] goes through essentially unchanged. Let $\rho$ be any stationary boundary defining function such that $\rho^2 g|_{T\partial M} = \gamma_0$. Write $\bar{g} = \rho^2 g$ and $$x = e^w \rho.$$ Then the equation $|dx|^2_{x^2 g} = -1$ is equivalent to $$\label{eq:eqforx}
-1 = | d(e^w \rho) |^2_{e^{2w}\bar g} = |d\rho|^2_{\bar g} + 2\rho \left< dw, d\rho \right>_{\bar g} + \rho^2 |dw|^2_{\bar g}.$$ This is the same as $$2(\nabla_{\bar{g}}\rho)w + \rho |dw|^2_{\bar g} = -\rho^{-1}(1 + |d\rho|^2_{\bar g}),$$ which is a noncharacteristic first order equation for $w$ since $d\rho$ is spacelike. Since $\rho$ and $\bar{g}$ are stationary, this equation can be solved uniquely for all times with the initial condition $w|_{\partial M} = 0$.
If $x$ satisfies the condition described in Lemma \[lem:geodesicbdf\], then $x$ is said to be a special boundary defining function. Note the integral curves of $\nabla_{x^2 g} x$ are geodesics near $\partial M$. Thus the Gauss lemma implies that $$\label{eq:modelmetric}
\phi^*_x(g) = \frac{-ds^2 + \gamma(s)}{s^2}$$ on $[0,\varepsilon) \times \partial M$, where $\phi_x$ is the diffeomorphism considered in Lemma \[lem:productdecomp\]. Here $s\mapsto \gamma(s)$ is a smooth family of stationary Lorentzian metrics on $\partial M$, such that $\gamma(0) = \gamma_0$. Furthermore, $\phi^*_xx = s$, so by an abuse of notation it is convenient to write $g = x^{-2}(-dx^2 + \gamma(x))$ on the collar neighborhood ${
\overline{\mathcal{C}}
}$.
\[defi:modulox3\] The metric $g$ as in Lemma \[lem:geodesicbdf\] is said to even modulo $\mathcal{O}(x^3)$ (in the sense of Guillarmou [@guillarmou:2005]) if there exists a two-tensor $\gamma_1$ on $\partial M$ such that $$\gamma(s) = \gamma_0 + s^2 \gamma_1 + \mathcal{O}(s^3).$$
As in [@guillarmou:2005 Proposition 2.1], this evenness property is instrinsic to the conformal class $[\bar{g}|_{T\partial M}]$ in the sense that it does not depend on the particular representative. The fundamental class of aAdS metrics which are even modulo $\mathcal{O}(x^3)$ are the Einstein aAdS metrics, see [@anderson2004structure Section 2] for example. This includes all of the physically motivated aAdS spaces. The Einstein condition also enforces additional conditions on the expansion of $\gamma(s)$ which are not exploited here — in the asymptotically hyperbolic setting, see Mazzeo–Pacard [@mazzeo2011constant Section 2].
The following criterion is frequently easy to verify for aAdS spacetimes.
Suppose that $\rho$ is stationary. If $\rho^2 g$ has an expansion $$\phi_\rho^*(\rho^2 g)(s,y,ds,dy) = -ds^2 + \gamma_0(y,dy) + \gamma_1(y,ds,dy)s + \mathcal{O}(s^2)$$ with $\gamma_1 = 0$, then $g$ is even modulo $\mathcal{O}(x^3)$.
Let $x$ denote the unique stationary boundary defining function associated to the representative $\rho^2 g$. Write $x = e^{w} \rho$ for some $w$. Then implies that after dividing by $\rho$, $$2\partial_\rho w = \mathcal{O}(\rho),$$ so $\partial_\rho w = 0$ at the boundary. The result can then be deduced from Lemma \[lem:xrho\].
The Klein–Gordon equation {#subsect:kleingordon}
-------------------------
Fix a stationary aAdS spacetime $({
\overline{M}
}, g)$, equipped with a geodesic boundary defining function $x$ as in Lemma \[lem:geodesicbdf\]. Furthermore, suppose that $g$ is even modulo $\mathcal{O}(x^3)$ in the sense of Definition \[defi:modulox3\]. In light of the product decomposition $\eqref{eq:modelmetric}$ it follows that near $\partial M$, $$\Box_g = x^2 D_x^2 + i(n - 1 + e(x))xD_x + x^2\Box_{\gamma(x)},$$ where $x \mapsto e(x)$ is a smooth family of functions on $\partial X$ such that $$e(x) = x^2 e_0 + \mathcal{O}(x^3)$$ for some $e_0 \in C^\infty(\partial X)$. Indeed, $e(x) = (1/2) x\partial_x \log(\det \gamma(x))$, and $\det \gamma(x) = \det \gamma_0 + \mathcal{O}(x^2)$. Given $\nu > 0$, let $$P_g = x^{-(n+1)/2}( \Box_g + \nu^2 - n^2/4) x^{(n-1)/2},$$ which corresponds to conjugating the Klein–Gordon operator with mass $\nu^2 - n^2/4$ by $x^{(n-1)/2}$ and then multiplying by $x^2$. Explicitly, $$P_g = D_x^2 +(\nu^2 - 1/4)x^{-2} + ix^{-1}e(x)D_x + \Box_{\gamma(x)}.$$ In particular $P_g = \Box_{\bar{g}} + \nu^2 - 1/4$, where $\bar{g} = x^2 g$ is smooth up to $\partial M$. Using that $g$ is stationary, it is possible to define the spectral family $$\label{eq:stationarykleingordon}
P(\lambda) = e^{i\lambda t} P_g e^{-i\lambda t}$$ which is naturally a differential operator on ${
\overline{X}
}$ for each $\lambda \in \mathbb{C}$.
\[lem:stationarykleingordon\] Let $({
\overline{M}
}, g)$ denote a stationary aAdS spacetime. Suppose $x$ is a geodesic boundary defining function, and $g$ is even modulo $\mathcal{O}(x^3)$. Define the rescaled and conjugated stationary Klein–Gordon operator $P(\lambda)$ on $X$ by . Then $P(
\lambda)$ is a parameter-dependent Bessel operator on $X$ of order $\nu$ with respect to $x$.
As indicated above, $$P(\lambda) = D_x^2 + ix^{-1}e(x)D_x + (\nu^2-1/4)x^{-2} + e^{i\lambda t}{\Box}_{\gamma(x)} e^{-i\lambda t}$$ for $x \in [0,\varepsilon)$. Since $e(x) = x^2 e_0 + \mathcal{O}(x^3)$, it follows that $x^{-1}e(x) \in xC^\infty({
\overline{\mathcal{C}}
})$. Furthermore, $x^{-1}e(x)D_x$ can be replaced by $x^{-1}e(x){D_\nu}$ since the two differ by a smooth bounded function. Finally, $e^{i\lambda t}{\Box}_{\gamma(x)}e^{-i\lambda t}$ may be considered as a family of parameter-dependent second order operators on $\partial X$, depending smoothly on $x\in [0,\varepsilon)$.
\[lem:adsiselliptic\] Suppose that $\partial_t$ is timelike for $\gamma_0$ in the sense that $\gamma_0(\partial_t,\partial_t) > 0$. Then $P(\lambda)$ is elliptic on $\partial X$ in the sense of Section \[subsect:ellipticity\].
Consider the metric $g_0 = -dx^2 + \gamma_0$ on $[0,\infty) \times \partial X$. If $\eta \cdot dy$ is the canonical one-form on $\partial X$, then the function is formally obtained by evaluating $$-g_0^{-1}(\xi\, dx + \eta\cdot dy - \lambda\, dt,\xi \, dx + \eta\cdot dy - \lambda\, dt)$$ at $\lambda = 0$. But if $\partial^\flat_t$ denotes the covector dual to $\partial_t$ with respect to $g_0$, then $$\lambda = -g_0^{-1}(\partial_t^\flat,\xi\,dx + \eta\cdot dy -\lambda \, dt)$$ If this latter quantity vanishes, then the covector $\xi \, dx + \eta \cdot dy$ is orthogonal to the timelike covector $\partial_t^\flat$. Thus $(\xi,\eta) \mapsto -g_0^{-1}(\xi \, dx + \eta \cdot dy, \xi \, dx + \eta \cdot dy)$ is positive definite for each $y \in \partial X$.
\[lem:adsisparameterelliptic\] Suppose that $dt$ is timelike for $\gamma_0$ in the sense that $\gamma_0^{-1}(dt,dt) > 0$. Then $P(\lambda)$ is parameter-elliptic on $\partial X$ in the sense of Section \[subsect:semiclassicalbessel\] with respect to any angular sector $\Lambda\subseteq \mathbb{C}$ disjoint from $\mathbb{R}\setminus 0$.
In the notation of Lemma \[lem:adsiselliptic\] above, for $\lambda \in \mathbb{C}$ the function $\eqref{eq:besselsymbolsemiclassical}$ is formally obtained by evaluating $$\begin{aligned}
\label{eq:symbol}
-&g_0^{-1}(\xi\, dx + \eta\cdot dy -{\operatorname{Re}}\lambda\, dt, \xi\, dx + \eta\cdot dy - {\operatorname{Re}}\lambda \,dt) + ({\operatorname{Im}}\lambda)^2 \gamma_0^{-1}(dt,dt) \notag \\
+ &2i\, ({\operatorname{Im}}\lambda) \,g_0^{-1}(\xi\, dx + \eta\cdot dy - {\operatorname{Re}}\lambda \, dt, dt).\end{aligned}$$ If the imaginary part of this quantity vanished but ${\operatorname{Im}}\lambda \neq 0$, then $\xi \, dx + \eta \cdot dy - {\operatorname{Re}}\lambda \,dt$ would be orthogonal to $dt$. Thus $g_0^{-1}$ would negative definite on vectors of this form, which shows that the real part does not vanish.
Function spaces and mapping properties {#sect:functionmapping}
======================================
The purpose of this section is to define Sobolev-type spaces ${\mathcal{H}}^{s}$ based on the elementary derivatives ${{\partial}_\nu}$ and $|{D_\nu}|^2$, both on ${\mathbb{T}^n_+}$ and on a manifold with boundary. Finally, it is shown that Bessel operators act continuously between these spaces.
The exposition is closest to that of [@warnick:2013:cmp], where these “twisted” Sobolev spaces were first introduced in the context of aAdS geometry. The relationship between ${\mathcal{H}}^1$ and certain weighted Sobolev spaces was exploited both in [@warnick:2013:cmp] and also in the closely related study of asymptotically hyperbolic spaces in [@gonzalez:2010]. Similar spaces are also defined in [@katrahov:1981], and in [@katrakhov:2011:de; @zemanian] as related to the Hankel transform.
Throughout this section the spaces $L^2({\mathbb{T}^n_+})$ and $L^2({\mathbb{T}^{n-1}})$ are defined with respect to ordinary Lebesgue measure, and $H^m({\mathbb{T}^{n-1}})$ denotes the standard Sobolev space of order $m$ on ${\mathbb{T}^{n-1}}$. The notation ${\mathcal{H}}^0({\mathbb{T}^n_+}) := L^2({\mathbb{T}^n_+})$ is also frequently used.
The weighted space $H^1_\mu({\mathbb{T}^n_+})$ {#subsect:weighted}
----------------------------------------------
Given $\mu \in \mathbb{R}$, let $$H_\mu^1({\mathbb{T}^n_+}) = \{ u \in {\mathscr{D}}'({\mathbb{T}^n_+}): x^{\frac{\mu}{2}} \partial^\alpha u \in L^2({\mathbb{T}^n_+}) \text{ for } |\alpha| \leq 1 \},$$ which is a Hilbert space under the norm $$\| u \|^2_{H^1_\mu({\mathbb{T}^n_+})} = \sum_{|\alpha| \leq 1} \| x^{\frac{\mu}{2}} \partial^\alpha u \|^2_{L^2({\mathbb{T}^n_+})}.$$ Furthermore, let $\mathring{H}_\mu^1({\mathbb{T}^n_+})$ denote the closure of $C_c^\infty({\mathbb{T}^n_+})$ in $H_\mu^1$. These spaces are well studied, see Lions [@lions:1961], Grisvard [@grisvard:1963] for example.
\[lem:weightedensity\] The following hold for $\mu \in \mathbb{R}$.
1. If $|\mu| < 1$, then $C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is dense in $H^1_\mu({\mathbb{T}^n_+})$.
2. If $|\mu| \geq 1$, then $H^1_\mu({\mathbb{T}^n_+})= \mathring{H}^1_\mu({\mathbb{T}^n_+})$.
Proofs of these facts may be found in [@grisvard:1963; @lions:1961].
Given a Hilbert space $E$, let $H_\mu^1({\mathbb{R}_+};E)$ denote the Hilbert space of $E$-valued distributions $u \in {\mathscr{D}}'({\mathbb{R}_+};E)$ such that $$x^{\frac{\mu}{2}} u \in L^2({\mathbb{R}_+};E), \quad x^{\frac{\mu}{2}} u' \in L^2({\mathbb{R}_+};E),$$ equipped with obvious norm. The Sobolev embedding theorem in this setting, [@grisvard:1963 Proposition 1.1’], says that $H^1_\mu({\mathbb{R}_+};E) \hookrightarrow C^0({
\overline{{\mathbb{R}_+}}
};E)$ for $\mu < 1$, thus the map $u \mapsto u(0)$ is continuous $H^1_\mu({\mathbb{R}_+};E) \rightarrow E$ . Since $H^1_\mu({\mathbb{T}^n_+}) \subseteq H^1_\mu({\mathbb{R}_+};L^2({\mathbb{T}^{n-1}}))$, it follows that any $u \in H^1_\mu({\mathbb{T}^n_+})$ admits a trace $$\label{eq:L2trace}
u \mapsto u|_{{\mathbb{T}^{n-1}}} \in L^2({\mathbb{T}^{n-1}}).$$ Furthermore, the kernel of $u \mapsto u|_{{\mathbb{T}^{n-1}}}$ is $\mathring{H}^1_\mu({\mathbb{T}^n_+})$ — see [@grisvard:1963 Proposition 1.2]. The next lemma improves upon the regularity of this restriction.
\[lem:trace\] Suppose that $\mu <1$. Then the restriction $$u \mapsto u|_{{\mathbb{T}^{n-1}}}, \quad u \in C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$$ extends uniquely to continuous map $\gamma: H_\mu^1({\mathbb{T}^n_+}) \rightarrow H^{(1-\mu)/2}({\mathbb{T}^{n-1}})$, and furthermore $\gamma$ admits a continuous right inverse.
By the Sobolev embedding, any $\varphi \in C_c^\infty({
\overline{{\mathbb{R}_+}}
}) \subseteq H^1_\mu({\mathbb{R}_+})$ admits an estimate of the form $$| \varphi(0) |^2 \leq C \int_{{\mathbb{R}_+}} x^{\mu} \left( | \varphi |^2 + | \varphi' |^2 \right) dx.$$ Apply this inequality to the function $\varphi(sx)$, and then choose $s$ (depending on $\varphi$) satisfying $$\int_{\mathbb{R}} x^{\mu} |\varphi|^2 \, dx = s^2 \int_{\mathbb{R}} x^{\mu} |\varphi'|^2 \, dx.$$ This yields the estimate $$\label{eq:interpinequality}
| \varphi(0) |^2 \leq 2 C \left( \int_{{\mathbb{R}_+}} x^{\mu} | \varphi|^2 dx \right)^{(1-\mu)/2}\left( \int_{{\mathbb{R}_+}} x^{\mu} | \varphi' |^2 dx \right)^{(1+\mu)/2}.$$ Now consider $u \in H^1_\mu({\mathbb{T}^n_+})$ and let $\hat{u}(q)$ denote its Fourier coefficients, where $q \in \mathbb{Z}^{n-1}$. It suffices to apply the inequality to $\hat{u}(q)$, which lies in $H^1_\mu({\mathbb{R}_+})$ for each $q \in \mathbb{Z}^{n-1}$. Multiplying by $\left< q \right>^{1-\mu}$ and summing over all $q$, it follows that $$\| \gamma u \|_{H^{(1-\mu)/2}} \leq C \| u \|_{H^1_\mu({\mathbb{R}_+})}.$$ The unique continuation of $\gamma$ follows from the density of $C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ in $H^1_\mu({\mathbb{T}^n_+})$. That $\gamma$ admits a right inverse is also straightforward, see Lemma \[lem:tracelift\] for a closely related result.
The trace $u \mapsto \gamma u$ defined in Lemma \[lem:trace\] agrees with the restriction given by since they both agree on the dense set $C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$.
The space ${\mathcal{H}}^1({\mathbb{T}^n_+})$ {#subsect:sob1}
---------------------------------------------
Given $\nu \in \mathbb{R}$, define $${\mathcal{H}}^1({\mathbb{T}^n_+}) = \{ u \in {\mathscr{D}}'({\mathbb{T}^n_+}): {{\partial}_\nu}^j \partial_y^\alpha u \in L^2({\mathbb{T}^n_+}) \text{ for $j + |\alpha| \leq 1$} \},$$ where ${{\partial}_\nu}^j \partial_y^\alpha u$ is taken in the sense of distributions on ${\mathbb{T}^n_+}$. Then ${\mathcal{H}}^1({\mathbb{T}^n_+})$ is a Hilbert space when equipped with the norm $$\| u \|^2_{{\mathcal{H}}^1({\mathbb{T}^n_+})} = \sum_{j + |\alpha| \leq 1} \| {{\partial}_\nu}^j\partial_y^\alpha u \|^2_{L^2({\mathbb{T}^n_+})},$$ The space ${\mathcal{H}}^1_*({\mathbb{T}^n_+})$ is defined analogously by replacing ${{\partial}_\nu}$ with its formal adjoint ${{\partial}_\nu}^*$. Let $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$ denote the closure of $C_c^\infty({\mathbb{T}^n_+})$ in ${\mathcal{H}}^1({\mathbb{T}^n_+})$, and similarly for $\mathring{{\mathcal{H}}}^1_*({\mathbb{T}^n_+})$.
\[lem:usualsobolev\] If $\nu \neq 0$, then $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+}) = \mathring{H}^1({\mathbb{T}^n_+})$ with an equivalence of norms.
Let $u \in C_c^\infty({\mathbb{T}^n_+})$. Note that $$0 \leq \left< {{\partial}_\nu}u, {{\partial}_\nu}u \right>_{{\mathbb{T}^n_+}} = \| \partial_x u \|^2_{L^2({\mathbb{T}^n_+})} + (\nu^2 -1/4)\| x^{-1} u \|^2_{L^2({\mathbb{T}^n_+})},$$ so Hardy’s inequality in one dimension follows by plugging in $\nu =0$. From this and for any $\nu \in \mathbb{R}$, $$\| \partial_\nu u \|_{L^2({\mathbb{T}^n_+})} \leq \| \partial_x u \|_{L^2({\mathbb{T}^n_+})} + |\nu - 1/2| \| x^{-1} u \|_{L^2({\mathbb{T}^n_+})} \leq C_\nu \| \partial_x u \|_{L^2({\mathbb{T}^n_+})}.$$ Conversely, If $|\nu| \geq 1/2$, then $\nu^2 - 1/4 \geq 0$, so $$\| \partial_x u \|_{L^2({\mathbb{T}^n_\sharp})} \leq \| {{\partial}_\nu}u \|_{L^2({\mathbb{T}^n_\sharp})}.$$ If $0 < |\nu| < 1/2$, then $\nu^2 -1/4 < 0$, so $$(4\nu^2 -1)\| \partial_x u \|^2_{L^2({\mathbb{T}^n_+})} \leq (\nu^2 -1/4)\| x^{-1} u \|_{L^2({\mathbb{T}^n_+})}.$$ This also gives $4\nu^2 \| \partial_x u \|^2_{L^2({\mathbb{T}^n_+})} \leq \| {{\partial}_\nu}u \|^2_{L^2({\mathbb{T}^n_+})}$. In either case, adding the $L^2({\mathbb{T}^n_+})$ norms of $\partial_y^\alpha u$ for $|\alpha|\leq 1 $ shows that $$C_\nu^{-1}\| u \|_{H^1({\mathbb{T}^n_+})} \leq \| u \|_{{\mathcal{H}}^1({\mathbb{T}^n_+})} \leq C_\nu \| u \|_{H^1({\mathbb{T}^n_+})}$$ for a constant $C_\nu> 0$ depending on $\nu \neq 0$. The result follows from the density of $C_c^\infty({\mathbb{T}^n_+})$ in both spaces.
The basic observation concerning ${\mathcal{H}}^1({\mathbb{T}^n_+})$ is that the map ${\mathscr{D}}'({\mathbb{T}^n_+}) \rightarrow {\mathscr{D}}'({\mathbb{T}^n_+})$ given by $u \mapsto x^{\nu-1/2}u$ restricts to an isometric isomorphism $${\mathcal{H}}^1({\mathbb{T}^n_+}) \rightarrow H^1_{1-2\nu}({\mathbb{T}^n_+}).$$ It follows from Lemma \[lem:weightedensity\] that $x^{1/2-\nu}C^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is dense in ${\mathcal{H}}^1({\mathbb{T}^n_+})$ if $0 < \nu < 1$, and ${\mathcal{H}}^1({\mathbb{T}^n_+}) = \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$ if $\nu \geq 1$. Using Lemma \[lem:trace\], it is also possible to define weighted traces of ${\mathcal{H}}^1({\mathbb{T}^n_+})$ functions, as will be explained in Section \[subsect:traces\].
The space ${\mathcal{H}}^2({\mathbb{T}^n_+})$
---------------------------------------------
Given $\nu > 0$, define $${\mathcal{H}}^2({\mathbb{T}^n_+}) = \{ u \in {\mathcal{H}}^1({\mathbb{T}^n_+}): {{\partial}_\nu}u \in \mathcal{H}^1_*({\mathbb{T}^n_+}), \text{ and } \partial_y^\alpha u \in {\mathcal{H}}^1({\mathbb{T}^n_+}) \text{ for } |\alpha| \leq 1 \}.$$ Then ${\mathcal{H}}^2({\mathbb{T}^n_+})$ becomes a Hilbert space when equipped with the norm $$\label{eq:H2nor}
\| u \|^2_{{\mathcal{H}}^2({\mathbb{T}^n_+})} = \| {{\partial}_\nu}^*{{\partial}_\nu}u \|^2_{L^2({\mathbb{T}^n_+})} + \sum_{|\alpha| \leq 1} \| \partial_y^\alpha u \|^2_{{\mathcal{H}}^1({\mathbb{T}^n_+})}.$$ Although $x^{1/2-\nu}C_c^\infty({\mathbb{T}^n_+})$ is dense in ${\mathcal{H}}^1({\mathbb{T}^n_+})$ when $0 < \nu < 1$, this is not the case for ${\mathcal{H}}^2({\mathbb{T}^n_+})$. In fact, $x^{1/2-\nu}C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is not contained in ${\mathcal{H}}^2({\mathbb{T}^n_+})$ unless $\nu = 1/2$. An appropriate dense space of smooth functions is defined in Section \[subsect:traces\].
Weighted traces {#subsect:traces}
---------------
It follows from Lemma \[lem:trace\] that the weighted restriction $$u \mapsto x^{\nu- 1/2}u|_{{\mathbb{T}^{n-1}}}, \quad u \in x^{1/2-\nu}C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$$ extends uniquely to a continuous map $\gamma_- : {\mathcal{H}}^1({\mathbb{T}^n_+}) \rightarrow H^\nu({\mathbb{T}^{n-1}})$, and furthermore $\gamma_-$ admits a continuous right inverse. This is true for all $\nu > 0$. Similarly, there exists a weighted restricted $$\gamma_-^* : \mathcal{H}^1_*({\mathbb{T}^n_+}) \rightarrow H^{1-\nu}({\mathbb{T}^{n-1}}),$$ initially defined for $u \in x^{\nu-1/2}C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ by $\gamma_-^* u = x^{1/2 -\nu}u|_{{\mathbb{T}^{n-1}}}$. However, note that $\gamma_-^*$ is now defined for $\nu < 1$ — indeed, ${\mathcal{H}}^1_*({\mathbb{T}^n_+})$ is isomorphic to $H^1_{2\nu -1}({\mathbb{T}^n_+})$, and the trace on $H^1_{2\nu -1}({\mathbb{T}^n_+})$ is only defined for $2\nu -1 < 1$. Since $u \in {\mathcal{H}}^2({\mathbb{T}^n_+})$ implies ${{\partial}_\nu}u \in {\mathcal{H}}^1_*({\mathbb{T}^n_+})$, there exists a second trace $$\gamma_+ : \mathcal{H}^{2}({\mathbb{T}^n_+}) \rightarrow H^{1-\nu}({\mathbb{T}^{n-1}})$$ given by the composition $\gamma_+ = \gamma_-^* \circ {{\partial}_\nu}$. The trace $\gamma_+$ therefore only exists for $0 < \nu < 1$, while $\gamma_-$ is well defined for $\nu > 0$.
\[defi:cnu\] Given $\nu > 0$, let ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ denote the following spaces of functions.
1. If $0 < \nu < 1$, then ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ consists of $u \in C^\infty({\mathbb{T}^n_+})$ of the form $$\label{eq:Cnu}
u(x,y) = x^{1/2-\nu}u_-(x^2,y) + x^{1/2+\nu}u_+(x^2,y)$$ for some $u_\pm \in C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$.
2. If $\nu \geq 1$, then ${\mathcal{F}_\nu({\mathbb{T}^n_+})}= C_c^\infty({\mathbb{T}^n_+})$.
Note that ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is contained in ${\mathcal{H}}^s({\mathbb{T}^n_+})$ for each $s=0,1,2$. If $0 < \nu < 1$, then ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is not typically contained in $x^{1/2-\nu}C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ (unless $\nu = 1/2$); on the other hand, traces of $u\in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ are still easily computed from the definitions.
\[lem:directtrace\] Suppose that $0 < \nu < 1$. If $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ satisfies , then $$\label{eq:tracevalues}
\gamma_- u = u_-(0,\cdot), \quad \gamma_+ u = 2\nu u_+(0,\cdot)$$
If $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ satisfies , then $x^{\nu - 1/2}u(x,y) = u_-(x^2,y) + x^{2\nu}u_+(x^2,y)$. Since this function is continuous on ${\mathbb{R}_+}$ with values in $C^\infty({\mathbb{T}^{n-1}})$, it follows that $\gamma_- u = u_-(0,\cdot)$, see the remark after Lemma \[lem:trace\]. A similar argument shows that $\gamma_+ u = 2\nu u_+(0,\cdot)$.
\[lem:traceproperty\] Suppose that $\nu > 0$ and $s = 0,1,2$. Then ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is dense in ${\mathcal{H}}^s({\mathbb{T}^n_+})$.
A proof is provided in Appendix \[appendix1\].
Suppose that $0 < \nu < 1$. Then there exist unique continuous maps $$\gamma_\mp : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow H^{s-1\pm\nu}({\mathbb{T}^{n-1}})$$ such that if $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ satisfies , then $\gamma_- u = u_-(0,\cdot)$ and $\gamma_+ u = 2\nu u_+(0,\cdot)$. Here $\gamma_-$ is defined for $s=1,2$, while $\gamma_+$ is only defined for $s=2$.
Combining Lemma \[lem:traceproperty\] with Lemma \[lem:directtrace\] shows that the map $$u \mapsto u_-(0,\cdot), \quad u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$$ admits a unique extension ${\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow H^{\nu}({\mathbb{T}^{n-1}})$ for $s = 1,2$. The additional regularity $\gamma_- u \in H^{1+\nu}({\mathbb{T}^{n-1}})$ for $u \in {\mathcal{H}}^2({\mathbb{T}^n_+})$ follows from the equality $\gamma_\pm \partial_y^\alpha u = \partial_y^\alpha \gamma_\pm u$ for $u\in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and each multiindex $\alpha$. Similarly, the map $$u \mapsto 2\nu u_+(0,\cdot), \quad u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$$ admits a unique extension ${\mathcal{H}}^2({\mathbb{T}^n_+}) \rightarrow H^{1-\nu}({\mathbb{T}^{n-1}})$.
As noted above, $\gamma_-$ can be defined for all $\nu > 0$ (rather than just $0 < \nu < 1$) but this fact is only ever used in Lemma \[app:nugeq1\] of the Appendix.
Dual spaces
-----------
Throughout, ${\mathcal{H}}^0({\mathbb{T}^n_+}) = L^2({\mathbb{T}^n_+})$ is identified with its own antidual ${\mathcal{H}}^0({\mathbb{T}^n_+})'$ via the Riesz representation. Given $s = 1,2$, let $${\mathcal{H}}^{-s}({\mathbb{T}^n_+}) = {\mathcal{H}}^{s}({\mathbb{T}^n_+})'$$ denote the corresponding antiduals. Since the inclusion $\iota : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \hookrightarrow {\mathcal{H}}^0({\mathbb{T}^n_+})$ is dense, ${\mathcal{H}}^0({\mathbb{T}^n_+})$ is identified with a dense subspace of ${\mathcal{H}}^{-s}({\mathbb{T}^n_+})$ via the map $\iota^* : {\mathcal{H}}^{0}({\mathbb{T}^n_+}) \hookrightarrow {\mathcal{H}}^{-s}({\mathbb{T}^n_+})$. Thus if $s \geq 0$ and $u, v \in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$, then the image $\iota^* u$ in ${\mathcal{H}}^{-s}({\mathbb{T}^n_+})$ acts on $v$ via the ${\mathcal{H}}^0({\mathbb{T}^n_+})$ pairing $$\iota^*u(v) = \langle u, v\rangle_{{\mathbb{T}^n_+}}.$$ Because ${\mathcal{H}}^{s}({\mathbb{T}^n_+})$ is dense in ${\mathcal{H}}^{-s}({\mathbb{T}^n_+})$, there is no ambiguity in using the notation $$\langle f, v \rangle_{{\mathbb{T}^n_+}}:= f(v), \quad f \in {\mathcal{H}}^{-s}({\mathbb{T}^n_+}),\;v \in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$$ in general.
A Fourier characterization {#subsect:fourier}
--------------------------
Given $s = 0,1,2$, any $u \in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$ has well defined Fourier coefficients $$\hat{u}(q) = (2\pi)^{-(n-1)/2} \int_{[-\pi,\pi]^{n-1}} e^{-i\left<q,y\right>} u(\cdot,y) \, dy, \quad q \in \mathbb{Z}^{n-1}.$$ It is easily seen $\hat{u}(q) \in {\mathcal{H}}^{s}({\mathbb{R}_+})$ for each fixed $q \in \mathbb{Z}^{n-1}$.
This may be extended uniquely by duality: given $f \in \mathcal{H}^{-s}({\mathbb{T}^n_+})$, let $\hat{f}(q) \in {\mathcal{H}}^{-s}({\mathbb{R}_+})$ denote the functional $$\label{eq:dualfourier}
\langle \hat{f}(q) , v \rangle_{{\mathbb{T}^n_+}} = (2\pi)^{-(n-1)/2} \langle f, e^{i\left<q,y\right>} v \rangle_{{\mathbb{T}^n_+}},$$ where $v \in {\mathcal{H}}^s({\mathbb{R}_+})$. Given $\tau > 0$ and $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$, let $$\label{eq:dilationdef}
(S_\tau u)(x,y) = u(\tau x,y)$$ denote the action of dilation in the normal variable. This clearly extends to a bounded map $S_{\tau} : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s}({\mathbb{T}^n_+})$ for $s = 0,1,2$. Furthermore, $S_{\tau}$ may be extended uniquely to ${\mathcal{H}}^{-s}({\mathbb{T}^n_+})$ by duality: given $f \in \mathcal{H}^{-s}({\mathbb{T}^n_+})$, define $$\left< S_\tau f ,v \right>_{{\mathbb{T}^n_+}} = \tau^{-1} \left<f , S_{\tau^{-1}} v \right>_{{\mathbb{T}^n_+}}$$ for $v \in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$.
\[lem:fouriernorm\] Given $s=0,\pm 1,\pm 2$, $$\| u \|^2_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} = \sum_{q\in \mathbb{Z}^{n-1}} \left<q\right>^{2s-1} \| S_{\left<q\right>^{-1}} \hat{u}(q) \|^2_{{\mathcal{H}}^{s}({\mathbb{R}_+})}.$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$.
When $s\geq 0$ this follows immediately from Parseval and Fubini’s theorems. When $s < 0$, the proof is exactly the same as in [@kozlov:1997 Lemma 2.3.1].
The space ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$ {#subsect:tildespace}
-----------------------------------------------------------
If ${
\underline{t}
} = (t_1,\ldots, t_k)$, define $$H^{{
\underline{t}
}}({\mathbb{T}^{n-1}}) := \prod_{j = 1}^k H^{t_k}({\mathbb{T}^{n-1}}).$$ Keeping this notation in mind, let ${
\underline{\nu}
} = (1 - \nu, 1+\nu)$ and then set $$\label{eq:gamma}
{
\underline{\gamma}
} = {
\begin{pmatrix}\ifx\relax\relax\else\\\fi\gamma_-\\#3\end{pmatrix}
}.$$ Following [@kozlov:1997; @roitberg:1996], define the following spaces for $0 < \nu < 1$. Given $s = 0,\,\pm 1,\,\pm 2$, let ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$ denote the set of all $$(u, \phi_-, \phi_+ ) \in {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \times H^{s - {
\underline{\nu}
}}({\mathbb{T}^{n-1}})$$ such that
1. $\phi_- = \gamma_- u$ and $\phi_+ = \gamma_+ u$ if $s =2$,
2. $\phi_- = \gamma_- u$ and $\phi_+$ is arbitrary if $s = 1$,
3. $\phi_\pm$ are arbitrary if $s \leq 0$.
A typical element of ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$ will be denoted $(u, \underline{\phi})$, where $\underline{\phi} = (\phi_-, \phi_+)$. The norm of $(u,{
\underline{\phi}
})$ is given by $$\| (u,{
\underline{\phi}
}) \|^2_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})} = \| u \|^2_{{\mathcal{H}}^s({\mathbb{T}^n_+})} + \| {
\underline{\phi}
} \|^2_{H^{s-{
\underline{\nu}
}}({\mathbb{T}^{n-1}})}.$$ If $s = 2$, then $u \mapsto (u, {
\underline{\gamma}
})$ provides an isomorphism $${\mathcal{H}}^{2}({\mathbb{T}^n_+}) \rightarrow {\widetilde{\mathcal{H}}}^{2}({\mathbb{T}^n_+}).$$ On the other hand, if $s \leq 1$, then the two spaces ${\mathcal{H}}^{s}({\mathbb{T}^n_+}),\, {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$ are fundamentally different.
\[lem:tildedensity\] Suppose that $0 < \nu < 1$. Then for each $s=0,\pm 1, \pm 2$, the set $$\{ (u, {
\underline{\gamma}
}u) : u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}\}$$ is dense in ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$.
It suffices to prove this for $s\geq 0$, since ${\widetilde{\mathcal{H}}}^0({\mathbb{T}^n_+})$ is dense in ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$ if $s < 0$. Assuming that $s \geq 0$ and $(u,{
\underline{\phi}
})\in {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$, choose $u_n \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ such that $u_n \rightarrow u$ in ${\mathcal{H}}^s({\mathbb{T}^n_+})$. If $\chi \in C_c^\infty({
\overline{{\mathbb{R}_+}}
})$ satisfies $\chi = 1$ near $x=0$, define $$u_{n,\varepsilon} = u_n - \left( x^{1/2-\nu}(\gamma_- u_n - \phi_-) + (2\nu)^{-1} x^{1/2+\nu}(\gamma_+ u_n - \phi_+) \right)\chi(\varepsilon^{-1}x).$$ Clearly $u_{n,\varepsilon} \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and $\gamma_\pm u_{n,\varepsilon} = \phi_\pm$. Furthermore, since $s\geq 0$, it is particularly easy to check that $u_{n,\varepsilon} \rightarrow u_n$ in ${\mathcal{H}}^{s}({\mathbb{T}^n_+})$ for $n$ fixed and $\varepsilon \rightarrow 0$. Thus it is possible find a sequence $\varepsilon_n \rightarrow 0$ such that $u_{n,\varepsilon_n} \rightarrow u$ in ${\mathcal{H}}^s({\mathbb{T}^n_+})$ as $n \rightarrow \infty$, and hence $$(u_{n,\varepsilon_n},{
\underline{\gamma}
} u_{n,\varepsilon_n}) \rightarrow (u,{
\underline{\phi}
})$$ in ${\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$.
Recall from Section \[subsect:fourier\] the dilation $S_\tau$, defined by . Note that $$(\gamma_- \circ S_\tau) u = \tau^{1/2-\nu} \gamma_- u, \quad (\gamma_+ \circ S_\tau) u = \tau^{1/2+\nu} \gamma_+ u$$ for each $\tau > 0$ and $u\in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. It follows that $S_\tau$ may be extended uniquely to ${\widetilde{\mathcal{H}}}^{s,k}({\mathbb{T}^n_+})$ by defining $$S_\tau (u,{
\underline{\phi}
}) = (S_\tau u , \tau^{1/2-\nu} \phi_-, \tau^{1/2+\nu} \phi_+ ).$$ It follows from Lemma \[lem:fouriernorm\] and the usual Fourier characterization of $H^m({\mathbb{T}^{n-1}})$ that $$\| (u,{
\underline{\phi}
}) \|^2_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})} = \sum_{q\in \mathbb{Z}^{n-1}} \left<q\right>^{2s-1} \| S_{\left<q\right>^{-1}} (\hat{u}(q),{
\underline{\hat{\phi}}
}(q)) \|^2_{{\mathcal{H}}^{s}({\mathbb{R}_+})}$$ for each $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$.
Parameter-dependent norms {#subsect:semiclassicalnorms}
-------------------------
When considering the action of parameter-dependent Bessel operators, one must consider modified norms on the spaces defined so far. Given $s = 0,1,2$ and $u \in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$, let $$\VERT u \VERT^2_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} = \sum_{j=0}^s |\lambda|^{2(s-j)} \| u \|^2_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})}.$$ Furthermore, if $f \in {\mathcal{H}}^{-s}({\mathbb{T}^n_+})$, let $$\VERT f \VERT_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})} = \sup \{ |\left< f, u \right>_{{\mathbb{T}^n_+}}|: \VERT v \VERT_{{\mathcal{H}}^s({\mathbb{T}^n_+})} = 1\}.$$ Recall the standard parameter-dependent norms $\VERT v \VERT_{{\mathcal{H}}^m({\mathbb{T}^{n-1}})}$ on $H^m({\mathbb{T}^{n-1}})$ as in Section \[subsect:semiclassicalbessel\]. Given $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$, set $$\VERT (u,{
\underline{\phi}
}) \VERT^2_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})} = \VERT u \VERT^2_{{\mathcal{H}}^s({\mathbb{T}^n_+})} + \VERT {
\underline{\phi}
} \VERT^2_{H^{s-{
\underline{\nu}
}}({\mathbb{T}^{n-1}})}.$$ These parameter-dependent norms have the property that there exists $C>0$ independent of $\lambda$ such that $$\VERT u \VERT_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})} \leq C |\lambda|^{-1} \VERT u \VERT_{{\mathcal{H}}^s({\mathbb{T}^n_+})}, \quad \VERT (u,{
\underline{\phi}
}) \VERT_{{\widetilde{\mathcal{H}}}^{s-1}({\mathbb{T}^n_+})} \leq C |\lambda|^{-1} \VERT (u,{
\underline{\phi}
}) \VERT_{{\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})}$$ for $u \in {\mathcal{H}}^s({\mathbb{T}^n_+})$ and $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$, respectively.
Mapping properties {#subsect:mappingpropertiesonRNP}
------------------
In this section, mapping properties of Bessel operators on ${\mathbb{T}^n_+}$ are examined. The analogue of Green’s formulas is established, which allows the extension of $P$ to spaces with low regularity. Recall from Section \[subsect:bessel\] that $P \in \mathrm{Bess}_\nu({\mathbb{T}^n_+})$ means that $$\label{eq:besselmappingrnp}
P = |{D_\nu}|^2 + B(x,y,D_y){D_\nu}+ A(x,y,D_y),$$ where $B \in \mathrm{Diff}^1({\mathbb{T}^{n-1}}), A \in \mathrm{Diff}^2({\mathbb{T}^{n-1}})$ depend smoothly on $x \in {
\overline{{\mathbb{R}_+}}
}$ and $B(0,y,D_y)= 0$. Throughout this section, assume that the coefficients of $A,B$ are constant outside a compact subset of ${
\overline{{\mathbb{T}^n_+}}
}$. The boundedness of each term in $P$ will be exmined individually.
Before proceeding, it is necessary to consider certain multipliers of ${\mathcal{H}}^s({\mathbb{T}^n_+})$ when $s\geq 0$. The commutation relations $$\label{eq:commutatorformulas}
[ {{\partial}_\nu}, \varphi ] = \partial_x \varphi = [ {{\partial}_\nu}^*, \varphi ], \quad [|{D_\nu}|^2, \varphi] = -\partial_x^2 \varphi - 2(\partial_x \varphi)\partial_x.$$ will be used throughout the following lemma.
\[lem:leibniz\] Suppose that $\varphi \in C^\infty(\overline{{\mathbb{T}^n_+}})$ is bounded along with all of its derivatives, and consider multiplication by $\varphi$ as a continuous map ${\mathscr{D}}'({\mathbb{T}^n_+}) \rightarrow {\mathscr{D}}'({\mathbb{T}^n_+})$.
1. For $s=0,1$, multiplication by $\varphi$ restricts to a continuous map $${\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s}({\mathbb{T}^n_+}).$$
2. If $\partial_x \varphi|_{{\mathbb{T}^{n-1}}}= 0$, then multiplication by $\varphi$ restricts to a continuous map $${\mathcal{H}}^{2}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{2}({\mathbb{T}^n_+}).$$
In either of these two cases, $$\label{eq:multiplierestimate}
\| \varphi u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq \| \varphi \|_{C^0({
\overline{{\mathbb{T}^n_+}}
})} \| u \|_{\mathcal{H}^{s}} + C_s\| u \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})},$$ where $C_s \geq 0 $ depends on the first $s$ derivatives of $\varphi$, and $C_0 = 0$.
The continuity statement is obvious for $s=0$. For $s = 1$ it follows from the first commutator formula . When $s=2$, the additional condition $\partial_x \phi |_{{\mathbb{T}^{n-1}}} = 0$ is needed to ensure that $$u \mapsto (\partial_x \varphi)\partial_x u$$ is bounded ${\mathcal{H}}^{1}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+})$: the vanishing of $\partial_x \phi$ implies $(\partial_x \varphi)\partial_x = (\partial_x \varphi){{\partial}_\nu}$ modulo multiplication by a smooth function, which acts continuously by the first part. The estimate follows as well from .
Lemma \[lem:leibniz\] result may also be extended to ${\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$ by defining $$\varphi(u,{
\underline{\phi}
}) := (\varphi u, \varphi|_{{\mathbb{T}^{n-1}}}{
\underline{\phi}
}),$$ and using that standard Sobolev spaces on ${\mathbb{T}^{n-1}}$ are closed under multiplication by smooth functions.
The hypotheses of Lemma \[lem:leibniz\] can not be improved when $s=2$. In other words ${\mathcal{H}}^2({\mathbb{T}^n_+})$ is not closed under multiplication by arbitrary $C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ functions. On the other hand, if $\varphi \in C_c^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is constant in a neighborhood of ${\mathbb{T}^{n-1}}$, then ${\mathcal{H}}^s({\mathbb{T}^n_+})$ is closed under multiplication by $\varphi$ for each $s=0,1,2$.
Now consider the term $|{D_\nu}|^2$, which is clearly bounded $$|{D_\nu}|^2 : {\mathcal{H}}^2({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+}).$$ The distinction between $0< \nu < 1$ and $\nu \geq 1$ plays an important role when extending this action. Suppose that $0 < \nu < 1$, and let $J$ denote the usual symplectic matrix, $$J = \left( \begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array} \right).$$ Then the following formulae are valid for each $u,v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$: $$\begin{aligned}
\left< |{D_\nu}|^2 u,v \right>_{{\mathbb{T}^n_+}} &= \left< u, |{D_\nu}|^2 v \right>_{{\mathbb{T}^n_+}} + \left< {
\underline{\gamma}
}u,J{
\underline{\gamma}
}v \right>_{{\mathbb{T}^{n-1}}}, \label{eq:TD1} \\
\left< |{D_\nu}|^2 u,v \right>_{{\mathbb{T}^n_+}} &= \left< {D_\nu}u,{D_\nu}v \right>_{{\mathbb{T}^n_+}} + \left< \gamma_+ u, \gamma_- v \right>_{{\mathbb{T}^{n-1}}} \label{eq:TD2}\end{aligned}$$ Since ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is dense, is valid for $v \in {\mathcal{H}}^2({\mathbb{T}^n_+})$, and is valid for $v \in {\mathcal{H}}^1({\mathbb{T}^n_+})$.
\[lem:TDgreen1\] Let $0 < \nu < 1$ and $s=0,1,2$. Then there exists $C>0$ such that $$\| |{D_\nu}|^2u\|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \| (u,{
\underline{\gamma}
}u) \|_{{\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})}.$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$.
For $s=2$ this follows since the norms $\| u \|_{{\mathcal{H}}^2({\mathbb{T}^n_+})}$ and $\| (u,{
\underline{\gamma}
}u)\|_{{\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+})}$ are equivalent for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. The case $s=1$ follows from , and the case $s=0$ follows from .
As a consequence of Lemma \[lem:TDgreen1\], the map $(u,{
\underline{\gamma}
} u) \mapsto |{D_\nu}|^2 u, \, u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ admits a unique extension as a bounded operator $$|{D_\nu}|^2 : {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$$ for $s=0,1,2$ and $0 < \nu < 1$. The situation is simpler when $\nu \geq 1$, since in that case ${\mathcal{F}_\nu({\mathbb{T}^n_+})}= C_c^\infty({\mathbb{T}^n_+})$ is dense in ${\mathcal{H}}^s({\mathbb{T}^n_+})$. The analogues of , are given by $$\begin{aligned}
\left< |{D_\nu}|^2 u,v \right>_{{\mathbb{T}^n_+}} &= \left< u, |{D_\nu}|^2 v \right>_{{\mathbb{T}^n_+}}, \label{eq:TD3} \\
\left< |{D_\nu}|^2 u, v \right>_{{\mathbb{T}^n_+}} &= \left< {D_\nu}u, {D_\nu}v \right>_{{\mathbb{T}^n_+}} \label{eq:TD4},\end{aligned}$$ valid for each $u, v\in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. The analogue of Lemma \[lem:TDgreen1\] is the following.
\[lem:TDgreen2\] Let $\nu \geq 1$ and $s=0,1,2$. Then there exists $C>0$ such that $$\| |{D_\nu}|^2 u \|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \| u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})}$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$.
From Lemma \[lem:TDgreen2\], it follows that the map $u \mapsto |{D_\nu}|^2u, u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ admits a unique continuous extension as a bounded operator $|{D_\nu}|^2 : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$ for $s=0,1,2$ and $\nu \geq 1$.
Next consider a typical term in $B{D_\nu}$. Such a term may be written as $b(x,y) D_y^\beta {D_\nu}$, where $b \in x C^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is constant for $x$ large and $|\beta| \leq 1$. The following result holds for all $\nu >0$, since there are no boundary terms when integrating by parts.
\[lem:Bbounded\] Suppose that $b \in x C^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is constant for $x$ large and $|\beta| \leq 1$. Then $$b D_y^\beta {D_\nu}: {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-|\beta|-1}({\mathbb{T}^n_+})$$ is bounded for each $s =0,1,2$. Furthermore, there exists $c >0$ depending on $s,\beta$ and $C \geq 0$ depending on $b,s,\beta,r$ such that $$\label{eq:Bbounded}
\| b D_y^{\beta} {D_\nu}u \|_{{\mathcal{H}}^{s-|\beta|-1}({\mathbb{T}^n_+})} \leq c r\|b\|_{C^1({
\overline{{\mathbb{T}^n_+}}
})}\| u \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} + C\| u \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})}.$$ for each $u\in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$ such that $\operatorname{supp}u \subseteq \{ 0 \leq x \leq r \}$.
The boundedness result is clear for $s=2$. For $s=0,1$, it follows from the same considerations as in Lemma \[lem:closedunderadjoint\]: define $B = bD_y^\beta$, and note that $B = B_1 x = xB_1$ where $B_1$ is smooth up to $x=0$. Thus $$B{D_\nu}= {D_\nu}^* B + i(1-2\nu)B_1 + [B,D_x]$$ So for each $u,v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$, $$\left< B{D_\nu}u, v \right>_{{\mathbb{T}^{n-1}}} = \left< {D_\nu}u, B^* v \right> = \left< u, B^* {D_\nu}v - i(1-2\nu)B_1^* v + [B,D_x]^* v \right>_{{\mathbb{T}^n_+}}.$$ The first equality implies boundedness for $s=1$, while the second implies boundedness for $s=2$.
Similarly, clearly holds for $s=2$. To prove the other cases, begin by writing $b = xb_1$, where $b_1$ is smooth up to $x=0$. Also define $q = [D_y^\beta,b]$, and $q = xq_1$, so the functions $q,q_1$ are smooth up to $x=0$ (and vanish if $|\beta|=0$).
$(1)$ If $s=1$, then for $u, v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$, $$\left< bD_y^\beta D_\nu u, v \right>_{{\mathbb{T}^n_+}} = \left< b {D_\nu}u, D_y^\beta v \right>_{{\mathbb{T}^n_+}} - \left< u, {D_\nu}\,{
\overline{q}
}\,v - i(1-2\nu){
\overline{q_1}
}v \right>_{{\mathbb{T}^n_+}}.$$ Thus $$\| bD_y^\beta u \|_{{\mathcal{H}}^{s-|\beta|-1}({\mathbb{T}^n_+})} \leq \| b D_\nu u \|_{{\mathcal{H}}^0({\mathbb{T}^n_+})} + C \| u \|_{{\mathcal{H}}^0({\mathbb{T}^n_+})},$$ whence the result follows.
$(2)$ Similarly for $s=0$, if $u, v \in \mathcal{F}_\nu$, then $$\left< bD_y^\beta {D_\nu}u, v \right>_{{\mathbb{T}^n_+}} = \left< b u, {D_\nu}^* D_y^\beta v \right>_{{\mathbb{T}^n_+}} - \left< u, {D_\nu}^* \, {
\overline{q}
}\,v + [D_x,b]^* D_y^\beta v \right>_{{\mathbb{T}^n_+}}.$$ To handle the first term write $$\left< bu, {D_\nu}^* D_y^\beta v \right> = \left< b_1 u, x {D_\nu}^* D_y^\beta v \right> = \left< bu, ({D_\nu}- i(1-2\nu))D_y^\beta v \right>,$$ which gives exactly $$|\left< bu , {D_\nu}^* D_y^\beta v \right>_{{\mathbb{T}^n_+}}| \leq \| bu\|_{{\mathcal{H}}^0({\mathbb{T}^n_+})}\| v \|_{{\mathcal{H}}^{1+|\beta|}({\mathbb{T}^n_+})},$$ as desired. Now consider the second term, which in absolute value is bounded by $$|\left< u, {D_\nu}^* \, {
\overline{q}
}\,v + {
\overline{q}
}\, D_y^\beta v \right>_{{\mathbb{T}^n_+}}| \leq \| u \|_{{\mathcal{H}}^{-1}({\mathbb{T}^n_+})}\left( \| {D_\nu}^*\,{
\overline{q}
}\,v \|_{{\mathcal{H}}^{1}({\mathbb{T}^n_+})} + \| {
\overline{q}
}\, D_y^\beta v \|_{{\mathcal{H}}^{1}({\mathbb{T}^n_+})} \right).$$ The second term in parentheses on the right hand side is bounded by a constant times $\| v \|_{{\mathcal{H}}^{1+|\beta|}({\mathbb{T}^n_+})}$ according to Lemma \[lem:leibniz\]. For the first term on the right hand side, the only part of the ${\mathcal{H}}^{1}({\mathbb{T}^n_+})$ norm which can’t be handled as above is the summand $\| {D_\nu}{D_\nu}^* \,{
\overline{q}
}\, v\|_{L^2({\mathbb{T}^n_+})}$. For this, $${D_\nu}{D_\nu}^* \, x{
\overline{q_1}
} = {D_\nu}\, (x {D_\nu}- i(2-2\nu)) \,{
\overline{q_1}
} = x|{D_\nu}|^2 {
\overline{q_1}
} - i(3-2\nu){D_\nu}{
\overline{q_1}
}.$$ Using , $$x|{D_\nu}|^2 {
\overline{q_1}
} = x{
\overline{q_1}
}|{D_\nu}|^2 - 2x(\partial_x\,{
\overline{q_1}
}){{\partial}_\nu}- x(\partial_x^2 \,{
\overline{q_1}
}) - (1-2\nu)\partial_x({
\overline{q_1}
}),$$ which is bounded ${\mathcal{H}}^2({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+})$. This shows that $$\| {D_\nu}^* \,{
\overline{q}
}\, v \|_{{\mathcal{H}}^{1}({\mathbb{T}^n_+})} \leq C\| v\|_{{\mathcal{H}}^{1+|\beta|}({\mathbb{T}^n_+})},$$ which completes the proof.
Lemma \[lem:Bbounded\] implies that $bD_y^\beta {D_\nu}$ is also bounded ${\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-|\beta|-1}({\mathbb{T}^n_+})$ since the projection ${\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^s({\mathbb{T}^n_+})$ onto the first factor is continuous.
Finally, a typical term in the operator $A$ can be written as $a(x,y)D_y^\alpha$, where $|\alpha| \leq 2$ and $a \in C^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is constant outside a compact subset of ${
\overline{{\mathbb{T}^n_+}}
}$.
\[lem:Abounded\] Suppose that $a \in C^\infty({
\overline{{\mathbb{T}^n_+}}
})$ is constant for $x$ large.
1. If $s=0,1$ and $|\alpha| \leq 2$, then $a D_y^\alpha : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-|\alpha|}({\mathbb{T}^n_+})$ is bounded.
2. If $s=0,1,2$ and $0 < |\alpha| \leq 2$, then $a D_y^\alpha : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-|\alpha|}({\mathbb{T}^n_+})$ is bounded
Furthermore, suppose that $a(0,p) = 0$ for $p\in{\mathbb{T}^{n-1}}$. Then there exists $c>0$ depending on $s,\alpha$ and $C \geq 0$ depending on $a,s,\alpha,r$ such that in each of the above cases, $$\label{eq:Abounded}
\| a D_y^{\alpha} u \|_{{\mathcal{H}}^{s-|\alpha|}({\mathbb{T}^n_+})} \leq cr\|a\|_{C^1({
\overline{{\mathbb{T}^n_+}}
})}\| u \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} + C \| u \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})}.$$ for each $u\in {\mathcal{H}}^{s}({\mathbb{T}^n_+})$ such that $\operatorname{supp}u \subseteq \{ (x,y) \in {
\overline{{\mathbb{T}^n_+}}
}: |x| + |y-p| < r \}$.
$(1)$ First suppose that $s=0,1$. The boundedness result is clear if $s=1$ and $|\alpha| \leq 1$ or $s=0$ and $|\alpha| =0$. Otherwise, suppose that $s=1$ and $|\alpha| = 2$. Write $aD_y^\alpha = \sum_{|\gamma| = 1} D_y^\gamma A_{\gamma}$ for smooth tangential operators $A_{\gamma}(y,D_y)$ of order at most one. Then for each $u,v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and $|\gamma| = 1$, $$|\left< D_y^\gamma A_\gamma u, v \right>_{{\mathbb{T}^n_+}}| = | \left< A_{\gamma}u, D_y^\gamma v \right>_{{\mathbb{T}^n_+}} | \leq C\| u \|_{{\mathcal{H}}^1({\mathbb{T}^n_+})} \| v \|_{{\mathcal{H}}^1({\mathbb{T}^n_+})}.$$ On the other hand, suppose that $s=0$. Then $$|\left< aD_y^\alpha u, v \right>_{{\mathbb{T}^n_+}} | = | \left< u, D_y^\alpha av \right>_{{\mathbb{T}^n_+}}| \leq C\| u \|_{{\mathcal{H}}^0({\mathbb{T}^n_+})} \| v \|_{{\mathcal{H}}^{|\alpha|}({\mathbb{T}^n_+})}$$ for $1 \leq |\alpha| \leq 2$.
$(2)$ The only case not handled above is $s=2$, in which case it follows from Lemma \[lem:leibniz\] that $aD_y^\alpha$ is bounded ${\mathcal{H}}^2({\mathbb{T}^n_+})\rightarrow {\mathcal{H}}^{s-|\alpha|}({\mathbb{T}^n_+})$ provided $|\alpha| \neq 0$.
$(3)$ This follows from the same arguments as in $(1)$ and $(2)$.
To summarize the above discussion, write $A = \sum_{|\alpha| \leq 2} a_{\alpha} D_y^\alpha$ (non uniquely) in the form $$A = \sum_{|\alpha| \leq 1} D_y^\alpha A_\alpha$$ for some $A_\alpha \in \mathrm{Diff}^1({\mathbb{T}^{n-1}})$ which depends smoothly on $x \in {\mathbb{R}_+}$. Recall that $P^*$ is also a Bessel operator, accordig Lemma \[lem:closedunderadjoint\]. Then there are the two Green’s formulas $$\begin{aligned}
&\langle Pu , v \rangle_{{\mathbb{T}^n_+}} = \langle u, P^* v \rangle_{{\mathbb{T}^n_+}} + \langle {
\underline{\gamma}
} u, J{
\underline{\gamma}
}v \rangle_{{\mathbb{T}^{n-1}}}, \label{eq:green2} \\
&\langle Pu , v \rangle_{{\mathbb{T}^n_+}} = \left< {D_\nu}u, {D_\nu}v \right>_{{\mathbb{T}^n_+}} + \left< {D_\nu}u, B^*v \right>_{{\mathbb{T}^n_+}} + \sum_{|\alpha| \leq 1} \left< A_\alpha u, D_y^\alpha v \right>_{{\mathbb{T}^n_+}} + \left< \gamma_+ u, \gamma_- v \right>_{{\mathbb{T}^{n-1}}} \label{eq:green1},\end{aligned}$$ valid for each $u, v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$.
\[lem:besselextension1\] Let $0 < \nu < 1$ and $s=0,1,2$. Then there exists $C>0$ depending on $s$ such that $$\| Pu \|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \| (u,{
\underline{\gamma}
}u) \|_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})}$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. Thus the map $(u,{
\underline{\gamma}
}u) \mapsto Pu, \, u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ admits a unique extension as a bounded operator $$P : {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$$ for $s=0,1,2$ and $0 < \nu < 1$. When $s=0,1$, this extension is determined by , .
This is a direct consequence of Lemmas \[lem:TDgreen1\], \[lem:Bbounded\], \[lem:Abounded\].
The situation is simpler when $\nu \geq 1$: the analogues of , are given by $$\begin{aligned}
&\langle Pu , v \rangle_{{\mathbb{T}^n_+}} = \langle u, P^* v \rangle_{{\mathbb{T}^n_+}} \label{eq:green4} \\
&\langle Pu , v \rangle_{{\mathbb{T}^n_+}} = \left< {D_\nu}u, {D_\nu}v \right>_{{\mathbb{T}^n_+}} + \left< {D_\nu}, B^*v \right>_{{\mathbb{T}^n_+}} + \sum_{|\alpha| \leq 1} \left< A_\alpha u, D_y^\alpha v \right>_{{\mathbb{T}^n_+}}\label{eq:green3},\end{aligned}$$ valid for each $u, v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. As before, is in fact valid for $v \in {\mathcal{H}}^2({\mathbb{T}^n_+})$, while $\eqref{eq:green3}$ is valid for $v \in {\mathcal{H}}^1({\mathbb{T}^n_+})$.
\[lem:besselextension2\] Let $\nu \geq 1$ and $s = 0,1,2$. Then there exists $C>0$ such that $$\| Pu \|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \| u \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})}.$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. the map $u \mapsto Pu, \, u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ admits a unique extension as a bounded operator $$P : {\mathcal{H}}^{s}({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$$ for $s=0,1,2$ and $\nu \geq 1$. When $s=0,1$ this extension is determined by , . The action of $P$ on ${\mathcal{H}}^s({\mathbb{T}^n_+})$ is simply the restriction of $P : {\mathscr{D}}'({\mathbb{T}^n_+}) \rightarrow {\mathscr{D}}'({\mathbb{T}^n_+})$ to ${\mathcal{H}}^s({\mathbb{T}^n_+})$.
This is a direct consequence of Lemmas \[lem:TDgreen2\], \[lem:Abounded\], \[lem:Bbounded\].
Suppose that $0< \nu < 1$. If $s=0,1$, then an element $f \in {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$ is not uniquely determined by a distribution in $\mathscr{D}'({\mathbb{T}^n_+})$. On the other hand, $f$ may certainly be restricted to a functional on $\mathring{{\mathcal{H}}}^{s}({\mathbb{T}^n_+})$, which is determined uniquely by a distribution since $C_c^\infty({\mathbb{T}^n_+})$ is dense in this space by definition. Given $s=0,1,2$ and $u \in {\mathcal{H}}^s({\mathbb{T}^n_+}), \,f \in {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$, the equation $Pu = f$ can be interpreted in this weak sense, namely $$\left< u, P^*v \right>_{X} = \left<f, v \right>_X$$ for all $v \in C_c^\infty({\mathbb{T}^n_+}) \subseteq \mathring{{\mathcal{H}}}^{2-s}({\mathbb{T}^n_+})$. For $s=2$ this is just the statement that $Pu = f$ in distributions. Furthermore, if $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$ and $P(u,{
\underline{\phi}
}) = f$, then $Pu = f$ weakly.
Now suppose that $P \in \mathrm{Bess}_{\nu}^{(\lambda)}({\mathbb{T}^n_+})$ is a parameter-dependent Bessel operator. Recalling the definition of the parameter-dependent norms as in Section \[subsect:semiclassicalnorms\], it is straightforward to show that the following hold.
1. If $0 < \nu < 1$ and $s=0,1,2$, then there exists $C>0$ such that $$\VERT P(\lambda)(u,{
\underline{\phi}
}) \VERT_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \VERT (u,{
\underline{\phi}
}) \VERT_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+})}$$ for each $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$.
2. If $\nu \geq 1$ and $s=0,1,2$, then there exists $C>0$ such that $$\VERT P(\lambda)u \VERT_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} \leq C \VERT u \VERT_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})}$$ for each $u \in {\mathcal{H}}^s({\mathbb{T}^n_+})$.
There are also straightforward extensions of Lemmas \[lem:TDgreen1\], \[lem:TDgreen2\], \[lem:Bbounded\], \[lem:Abounded\] for parameter-dependent norms.
Function spaces on a manifold {#subsect:spaceonmanifold}
-----------------------------
Consider a compact manifold with boundary ${
\overline{X}
}$, equipped with a distinguished boundary defining function $x$ and collar diffeomorphism $\phi$ as in Section \[subsect:manifoldwithboundary\].
\[defi:cnuonmanifold\] Given $\nu > 0$, let $\mathcal{F}_\nu(X)$ denote the following spaces of functions.
1. If $0 < \nu < 1$, then $\mathcal{F}_\nu(X)$ consists of $u \in C^\infty(X)$ such that $$\label{eq:cnuonmanifold}
(u \circ \phi)(x,y) = x^{1/2-\nu}u_-(x^2,y) + x^{1/2+\nu}u_+(x^2,y)$$ for some $u_\pm \in C^\infty([0,\sqrt{\varepsilon}) \times \partial X)$.
2. If $\nu \geq 1$, then $\mathcal{F}_\nu = C_c^\infty(X)$.
Fix a finite open cover ${
\overline{X}
} = \bigcup_i U_i$ by coordinate charts $(U_i, \psi_i)$, such that either $$U_i \cap \partial X = \emptyset, \quad \psi_i : U_i \rightarrow \psi_i(U_i) \subseteq \mathbb{T}^n,$$ or if $U_i \cap \partial X \neq \emptyset$ then $$U_i = \phi([0,\varepsilon) \times Y_i), \quad \psi_i = (1 \times \theta_i)\circ \phi^{-1}$$ for a coordinate chart $(Y_i, \theta_i)$ on $\partial X$. This of course implies that $\partial X = \bigcup_i Y_i$, where the union is taken over all $i$ such that $U_i \cap \partial X \neq \emptyset$. Now take a partition of unity of the form $$\sum_{i} \chi_i^2 = 1, \quad \chi_i \in C_c^\infty(U_i),$$ with the additional property that if $U_i \cap \partial X \neq \emptyset$, then $\chi_i$ has the form $$\chi_i = (\alpha \beta_i) \circ \phi^{-1},$$ for functions $\alpha \in C_c^\infty([0,\varepsilon)), \, \beta_i \in C_c(Y_i)$, where $\alpha = 1$ near $x=0$. Note that if $u \in \mathcal{F}_\nu(X)$ then $\chi_i u$ may be identified with an element of ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ via the coordinate map $\psi_i$. Keeping this in mind, define $$\| u \|_{i,{\mathcal{H}}^{s}(X)} := \| (\chi_i u) \circ \psi_i^{-1} \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})},$$ for $s=0,\pm 1,\pm 2$ and $u \in \mathcal{F}_\nu(X)$.
\[defi:normonmanifold\] Given $s=0,\pm 1,\pm 2$, let $$\| u \|^2_{{\mathcal{H}}^s(X)} = \sum_i \| u \|^2_{i,{\mathcal{H}}^{s}(X)}.$$ Then define $${\mathcal{H}}^s(X) = \text{closure of } \mathcal{F}_\nu(X) \text{ in the ${\mathcal{H}}^s(X)$ norm}.$$
To prove that ${\mathcal{H}}^s(X)$ is independent of the choice of covering $U_i$ and partition of unity $\chi_i$, the following two results are needed
\[lem:invariance1\] Let $Y,Y'$ be open subsets of ${\mathbb{T}^{n-1}}$, and suppose that $\Phi : Y \rightarrow Y'$ is a diffeomorphism between them. Suppose that $K\subseteq {
\overline{{\mathbb{R}_+}}
}\times Y$ is compact. Then for each $s=0,\pm1,\pm2$ there exists $C>0$ such that $$C^{-1} \| u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq \| u \circ (1\times \Phi) \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq C\| u\|_{{\mathcal{H}}^s({\mathbb{T}^n_+})}$$ for each $u \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ with $\operatorname{supp}u \subseteq K' := (1\times \Phi)(K)$,
For $s =0,1,2$ this follows immediately from the change of variables formula in the tangential direction. The cases $s=-1,-2$ follow by duality: choose $\alpha \in C_c^\infty({
\overline{{\mathbb{R}_+}}
})$ and $\beta \in C_c^\infty(Y')$ such that $\chi := \alpha\beta = 1$ on $K'$. Then for each $v \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$, $$\frac{|\left< u, v \right>_{{\mathbb{T}^n_+}}|}{\| v \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})}} \leq C_1^{-1} \frac{|\left< u, \chi v \right>_{{\mathbb{T}^n_+}}|}{\| \chi v \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})}},$$ where $\| \chi v \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})} \leq C_1 \| v\|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})}$. Thus $$\| u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq C_1^{-1}\sup_{\| w \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})} =1} |\left< u, w \right>_{{\mathbb{T}^n_+}}|,$$ where the supremum is taken over all $w \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ such that $\operatorname{supp}w \subseteq K'$. Then $$\left< u, w \right>_{{\mathbb{T}^n_+}} = \left< u\circ(1\times \Phi), w_1 \right>_{{\mathbb{T}^n_+}},$$ where $w_1(x,y) = J(y)(w\circ(1\times\Phi))(x,y)$ and $J$ is the Jacobian determinant of $1\times \Phi$. Since $J$ depends only on $y$, $$\| w_1 \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})} \leq C_2\| w \|_{{\mathcal{H}}^{-s}({\mathbb{T}^n_+})},$$ which shows that $\| u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq C \| u \circ (1\times \Phi) \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})}$ for some $C>0$. The same argument is now applied with $\Phi$ replaced by $\Phi^{-1}$ to conclude the reverse inequality.
More generally, the space ${\mathcal{H}}^s({\mathbb{T}^n_+})$ are invariant under diffeomorphisms $\Phi$ satisfying the conditions in Lemma \[lem:equivalentbessel\]
Before stating the next result, recall that for $M$ a (non-compact) manifold without boundary, the standard Sobolev spaces $H^s_{\mathrm{loc}}(M),\, H^s_\mathrm{comp}( M)$ are defined as follows: Fix a locally finite open cover $M = \bigcup_j X_j$ by coordinate charts $(X_j, \psi_j)$ where $\psi_j: X_j \rightarrow \mathbb{R}^{n}$, and a subordinate partition of unity $\sum_j \chi_j^2 = 1$ where $\chi_j \in C_c^\infty(X_j)$. Then $H^s_{\mathrm{comp}}(M)$ is defined as all compactly supported distributions $u\in \mathcal{E}'(M)$ such that the norm $$\label{eq:standardsobolev}
\| u \|^2_{H^s(M)} := \sum_j \| (\chi_j u)\circ \psi_j \|^2_{H^s(\mathbb{R}^{n-1})}.$$ is finite. The local spaces $H^2_\mathrm{loc}(M)$ then consist of distributions $u \in \mathcal{D}'(M)$ such that $\chi u \in H^s_{\mathrm{comp}}(M)$ for each $\chi \in C_c^\infty(M)$. Let $H^s_K(M)$ denote the space of all $u \in H^s_{\mathrm{comp}}(M)$ whose supports are contained in a fixed closed set $K\subset M$. If $M$ is compact, then $H^s_K(M)$ is complete under the norm \[eq:standardsobolev\]. Furthermore, these spaces do not depend on any of the choices used to define them.
\[lem:invariance2\] Let $K$ be a compact subset of ${\mathbb{T}^n_+}$. Then for each $s=0,\pm1,\pm2$ there exists $C>0$ such that $$C^{-1} \| u \|_{H^s({\mathbb{T}^n_+})} \leq \| u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq C\| u \|_{H^s({\mathbb{T}^n_+})}$$ for each $u \in C_c^\infty({\mathbb{T}^n_+})$ such that $\operatorname{supp}u \subseteq K$.
This is again straightforward for $s=0,1,2$, while the cases $s=-1,-2$ follow by duality.
The combination of Lemmas \[lem:invariance1\], \[lem:invariance2\] show that the spaces ${\mathcal{H}}^s(X)$ do not depend on any of the choices used to define them.
Fix a density on ${
\overline{X}
}$ of product type near $\partial X$. Let $\left< \cdot, \cdot \right>_X$ denotes the inner product on $L^2(X;\mu)$. For each $s=0,\pm 1,\pm2$ $$| \left< u , v \right>_X | \leq C \| u \|_{{\mathcal{H}}^{s}(X)} \| v \|_{{\mathcal{H}}^{-s}(X)},$$ where $u,v \in \mathcal{F}_\nu$. Furthermore, $\left< \cdot, \cdot \right>_X$ extends to a nondegenerate pairing ${\mathcal{H}}^s(X) \times {\mathcal{H}}^{-s}(X) \rightarrow \mathbb{C}$.
This can be reduced to the case on ${\mathbb{T}^n_+}$ via the coordinate charts $(U_i,\psi_i)$ and partition of unity $\chi_i$ used to define ${\mathcal{H}}^s(X)$ (it is here that choosing a quadratic partition of unity is particularly convenient)
Thus ${\mathcal{H}}^{-s}(X)$ is naturally identified with the antidual of ${\mathcal{H}}^s(X)$ via the inner product induced by $\mu$ on ${\mathcal{H}}^0(X)$. When $0 < \nu < 1$, it is also possible to show that the maps $$u \mapsto u_-(0,\cdot), \quad u \mapsto 2\nu u_+(0,\cdot)$$ for $u \in \mathcal{F}_\nu(X)$ satisfying admit continuous extensions $\gamma_\mp$ such that $$\gamma_\mp : {\mathcal{H}}^s(X) \rightarrow H^{s-1\pm\nu}(\partial X).$$ It is understood that $\gamma_-$ exists for $s=1,2$, while $\gamma_+$ exists for $s=2$. The spaces ${\widetilde{\mathcal{H}}}^{s}(X)$ are then defined exactly as in Section \[subsect:tildespace\].
Let $P \in \mathrm{Bess}_\nu(X)$. Then the following hold.
1. If $0 < \nu <1$ and $s=0,1,2$, then there exists $C>0$ such that $$\| Pu \|_{{\mathcal{H}}^{s-2}(X)} \leq C \| (u,{
\underline{\gamma}
}u) \|_{{\widetilde{\mathcal{H}}}^{s}(X)}.$$ for each $u \in \mathcal{F}_\nu(X)$.
2. If $\nu \geq 1$ and $s=0,1,2$, then there exists $C>0$ such that $$\| Pu \|_{{\mathcal{H}}^{s-2}(X)} \leq C \| u \|_{{\mathcal{H}}^{s}(X)}.$$ for each $u \in \mathcal{F}_\nu(X)$.
As in Section \[subsect:mappingpropertiesonRNP\], it follows that $(u,{
\underline{\gamma}
}u) \mapsto Pu$ admits a unique extension to ${\widetilde{\mathcal{H}}}^{s}(X)$ for $0 < \nu <1$, and $u \mapsto Pu$ has a unique continuous extension to ${\mathcal{H}}^s(X)$ for $\nu \geq 1$.
The traces $\gamma_\pm$ can be formulated in terms of the boundary defining function $x$. This is clear for $\gamma_-$, which is just the restriction of $x^{\nu-1/2}u$ to the boundary. If $\partial_x$ is the unique vector field on ${
\overline{X}
}$ which pulls back via $\phi_x$ to the coordinate vector field $\partial_x$, then $\gamma_+$ is the restriction of $x^{1-2\nu}\partial_x(x^{\nu-1/2}u)$ to the boundary.
Suppose that $x$ and $\rho$ are boundary defining functions satisfying the conditions in Lemma \[lem:xrho\]. Then the traces $\gamma_\pm$ defined with respect to $x$ agree with those defined with respect to $\rho$.
This can be checked in local coordinates using Lemmas \[lem:equivalentbessel\], \[lem:xrho\]. However, note that the spaces $\mathcal{F}_\nu(X)$ defined with respect to $x$ and $\rho$ do not agree.
The parameter-dependent norms on ${\mathcal{H}}^s(X)$ are defined by replacing $\| \cdot \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})}$ with $\VERT \cdot \VERT_{{\mathcal{H}}^s({\mathbb{T}^n_+})}$ in Definition \[defi:normonmanifold\], and similarly for ${\widetilde{\mathcal{H}}}^s(X)$. Then $P$ is uniformly bounded in $\lambda$ with respect to these norms.
The following compactness result is established in the appendix. A different proof can be found in [@holzegel:2012wt Section 6]. It is used in Section \[sect:fredholm\] to prove the Fredholm property for certain boundary value problems.
\[lem:compact\] [[@holzegel:2012wt Section 6]]{} Let $\nu >0$ and $\mu$ be a density of product type near $\partial X$.
1. The inclusion ${\mathcal{H}}^1(X) \hookrightarrow {\mathcal{H}}^0(X)$ is compact.
2. The injection ${\mathcal{H}}^0(X) \hookrightarrow {\mathcal{H}}^{-1}(X)$ induced by the $L^2(X;\mu)$ inner product is compact.
3. If $0< \nu < 1$, then ${\widetilde{\mathcal{H}}}^{1}(X)\hookrightarrow {\widetilde{\mathcal{H}}}^0({\mathbb{T}^n_+})$ and the injection ${\widetilde{\mathcal{H}}}^0(X)\hookrightarrow {\widetilde{\mathcal{H}}}^1(X)'$ induced by the $L^2(X;\mu), L^2(\partial X; \mu_{\partial X})$ inner products are compact.
$(1)$ For a proof, see the appendix. The other cases $(2), (3)$ follow by duality.
Finally, it is important to consider the action of standard pseudodifferential operators whose Schwartz kernels are compactly supported in $X \times X$. The class of such pseudodifferential operators of order $m$ is denoted by $\Psi^{m}_{\mathrm{comp}}(X)$. If $Q \in \Psi^m_\mathrm{comp}(X)$, then there exists a compact subset $K \subseteq X$ such that $\operatorname{supp}Qu \subseteq K$ for each $u \in C^\infty(X)$.
\[lem:pseudodifferentialbounded\] Suppose that $Q \in \Psi^2_\mathrm{comp}(X)$. Then there exists a compact subset $K \subseteq X$ such that for each $s = 0,\pm 1, \pm 2$, the map $$u \mapsto Qu, \quad u \in \mathcal{F}_\nu$$ extends uniquely to a bounded map $$Q : \begin{cases} {\widetilde{\mathcal{H}}}^{s}(X) \rightarrow H^{s-2}_K(X) &\text{ if } 0< \nu < 1,\\
{\mathcal{H}}^{s}(X) \rightarrow H^{s-2}_K(X) &\text{ if } \nu \geq 1. \end{cases}$$
It suffices to prove this in local coordinates, where the result follows from Lemma \[lem:invariance2\].
It is also necessary to consider the class of (compactly supported) parameter-dependent pseudodifferential operators, denoted here by $\Psi^{m,(\lambda)}_\mathrm{comp}(X)$ — see [@shubin:2001 Chapter II.9] for this class of operators, or [@zworski:2012 Chapter 4], [@dimassi:1999 Chapter 7] for an equivalent semiclassical description. In that case, if $Q \in \Psi^{2,(\lambda)}_\mathrm{comp}(X)$, then the boundedness result of Lemma \[lem:pseudodifferentialbounded\] holds uniformly for the appropriate parameter-dependent norms.
Graph norms {#subsect:graphnorm}
-----------
Throughout this section, assume that $0 < \nu < 1$. Following [@roitberg:1996 Chapter 6.1], an alternative characterization of the spaces ${\widetilde{\mathcal{H}}}^s(X)$ is given. Given $s=0,1,2$ and a Bessel operator $P$, define the norm $$\| u \|_{{\mathcal{H}}^s_P(X)} = \| u \|_{{\mathcal{H}}^s(X)} + \| Pu \|_{{\mathcal{H}}^{s-2}(X)}$$ for $u \in \mathcal{F}_\nu(X)$.
\[lem:graphnorm\] Give $s=0,1,2$ there exists $C>0$ such that $$C^{-1} \| u \|_{{\mathcal{H}}^s_P(X)} \leq \| (u, {
\underline{\gamma}
}u) \|_{{\widetilde{\mathcal{H}}}^s(X)} \leq C^{-1} \| u \|_{{\mathcal{H}}^s_P(X)}$$ for each $u \in \mathcal{F}_\nu(X)$.
The first inequality above holds according to Lemma \[lem:besselextension1\]. For the converse, first recall that the trace map ${
\underline{\gamma}
} : {\widetilde{\mathcal{H}}}^{s}(X) \rightarrow H^{s-{
\underline{\nu}
}}(\partial X)$ has a continuous right inverse $\mathcal{K}$, which furthermore maps $C^\infty(\partial X) \times C^\infty(\partial X) \rightarrow \mathcal{F}_\nu(X)$ — see Lemma \[lem:tracelift\] in the Appendix. Fix $u \in \mathcal{F}_\nu(X)$ and define the linear form $\ell$ on $C^\infty(\partial X)\times C^\infty(\partial X)$ by $$\ell({
\underline{\psi}
}) = \left< Pu, v \right>_{X} - \left< u, P^*v \right>_X,$$ where $v \in \mathcal{F}_\nu(X)$ is any element satisfying ${
\underline{\gamma}
}v = {
\underline{\psi}
}$. The form $\ell$ is well defined, since by Green’s formula it is independent of the choice of $v$; in particular, it is possible to take $v = \mathcal{K}({
\underline{\psi}
})$. Here the duality is induced by a fixed density of product type near $\partial X$. Then $$|\ell({
\underline{\psi}
})| \leq C_1 (\| Pu \|_{{\mathcal{H}}^{s-2}(X)} + \|u \|_{{\mathcal{H}}^s(X)}) \| \mathcal{K}({
\underline{\psi}
}) \|_{{\widetilde{\mathcal{H}}}^{2-s}(X)} \leq C_2 \| u \|_{{\mathcal{H}}^s_P(X)} \| {
\underline{\psi}
} \|_{H^{2-s-{
\underline{\nu}
}}(\partial X)}.$$ By Hahn–Banach and the Riesz theorem, there exists a unique ${
\underline{\phi}
} \in H^{s-2+{
\underline{\nu}
}}(\partial X)$ such that for each $u, v \in \mathcal{F}_\nu(X)$, $$\left< Pu, v \right>_X - \left< u, P^*v \right>_X = \left< {
\underline{\phi}
}, {
\underline{\gamma}
}v \right>_{\partial X}, \quad \| {
\underline{\phi}
} \|_{H^{s-2+{
\underline{\nu}
}}(\partial X)} \leq C_2\|u \|_{{\mathcal{H}}^s_P(X)}.$$ The constant $C_2$ in the latter inequality is independent of $u \in \mathcal{F}_\nu(X)$. On the other hand, Green’s formula implies that $\ell({
\underline{\psi}
}) = -\left< J{
\underline{\gamma}
}u, {
\underline{\psi}
} \right>_{\partial X}$ for all ${
\underline{\psi}
}$, so in particular ${
\underline{\phi}
} = -J{
\underline{\gamma}
}u$. Since $$\| {
\underline{\gamma}
} u \|_{H^{s-{
\underline{\nu}
}}(\partial X)} = \| {
\underline{\phi}
} \|_{H^{s-2+{
\underline{\nu}
}}(\partial X)},$$ it follows that $\| (u, {
\underline{\gamma}
}u) \|_{{\widetilde{\mathcal{H}}}^s(X)} \leq C_2 \| u \|_{{\mathcal{H}}^s_P(X)}$.
Let ${\mathcal{H}}^s_P(X)$ denote the closure of $\mathcal{F}_\nu(X)$ in the norm $\| \cdot \|_{{\mathcal{H}}^s_P(X)}$. Since $(u,{
\underline{\gamma }
}u),\, u \in \mathcal{F}_\nu(X)$ is dense in ${\widetilde{\mathcal{H}}}^s(X)$, it follows from Lemma \[lem:graphnorm\] that ${\mathcal{H}}^s_P(X)$ is naturally isomorphic to ${\widetilde{\mathcal{H}}}^s(X)$ via the closure of the map $u \mapsto (u,{
\underline{\gamma}
}u)$. Moreover, any element of ${\mathcal{H}}^s_P(X)$ can be identified with a unique pair $(u,f)$, where $u \in {\mathcal{H}}^s(X), \, f \in {\mathcal{H}}^{s-2}(X)$, and $Pu = f$ in the weak sense (described at the end of Section \[subsect:mappingpropertiesonRNP\]).
Elliptic boundary value problems {#sect:ellipticBVP}
================================
This section concerns boundary value problems for Bessel operators on a compact manifold with boundary ${
\overline{X}
}$ as in Section \[subsect:manifoldwithboundary\]. When $0 < \nu < 1$, these are of them form $$\label{eq:BVP}
\begin{cases} Pu = f & \text{ on } X\\
Tu = g & \text{ on } \partial X.
\end{cases}$$ Here $P \in \mathrm{Bess}_\nu(X)$ is Bessel operator which is elliptic in the sense of Section \[subsect:ellipticity\] on $\partial X$, and $$T = T^+ \gamma_+ + T^- \gamma_-$$ for some differential operators $T^\pm$ on the boundary, to be specified in the next section. The boundary operator $T$ is only relevant when $0 < \nu < 1$. When $\nu \geq 1$, one considers the simpler equation $$Pu = f \text{ on } X.$$
To highlight the difference between the cases $0 < \nu < 1$ and $\nu \geq 1$, fix $p \in \partial X$ and consider the model equation on ${\mathbb{R}_+}$ determined by the boundary symbol operator, $$\label{eq:ellipticitymodel}
\widehat{P}_{(p,\eta)}u = f.$$ referring to Section \[subsect:ellipticity\] for notation. Suppose that $P$ is elliptic at $p\in \partial X$. Any two solutions to the equation differ by an element of the kernel of $\widehat{P}_{(p,\eta)}$. If $u \in \ker \widehat{P}_{(p,\eta)}$ satisfies $u \in L^2((1,\infty))$, then necessarily $u \in \mathcal{M}_+(p,\eta)$. On the other hand, if $\nu$ is not an integer, then $$\label{eq:besselorigin}
K_\nu(s) = \frac{\pi}{2}\frac{I_{-\nu}(s) - I_{\nu}(s)}{\sin(\nu\pi)},$$ where $I_\nu$ is the modified Bessel function of the first kind [@olver:2014 Chapter 7.8] (if $\nu$ is an integer, equality holds in the sense of limits). In particular, if $0 < \nu < 1$, then $I_{\pm \nu}(s) = \mathcal{O}(s^{\pm \nu})$. Consequently $\ker \widehat{P}_{(p,\eta)} \cap L^2({\mathbb{R}_+}) = \mathcal{M}_+(p,\eta)$, and hence $\widehat{P}_{(p,\eta)}$ cannot be an isomorphism between any $L^2$ based spaces: in general, must be augmented by boundary conditions so that the $L^2$ kernel is trivial. Of course, all of these observations are classical when $\nu = \frac{1}{2}$ (boundary value problems in the smooth setting).
This is in contrast to the situation when $\nu \geq 1$. In that case, $\sqrt{x}K_\nu(i\xi(p,\eta) x)$ is not square integrable near the origin, and so the $L^2$ kernel of $\widehat{P}_{(p,\eta)}$ is always trivial. Hence specifying $f$ on the right hand side of (in an appropriate function space) will uniquely determine a solution $u$. Thus in the case $\nu \geq 1$, it is not necessary to impose any boundary conditions apart from the square integrability requirement.
In the self-adjoint setting, the heuristic above is the limit point/limit circle criterion of Weyl on self-adjoint extensions of symmetric ordinary differential operators with regular singular points — see [@zettl:2010] for an exhaustive modern treatment, and [@bachelot:2011:jmpa; @ishibashi:2004:cqg] for discussions in the context of AdS cosmology.
Boundary conditions {#subsect:BC}
-------------------
This section is only relevant in the case $0 < \nu < 1$. Choose differential operators $$T^- \in \mathrm{Diff}^1(\partial X), \quad T^+ \in \mathrm{Diff}^0(\partial X),$$ noting that $T_+$ is just multiplication by a smooth function on $\partial X$. Then set $T = T^\pm \gamma_\pm$, where Einstein summation is implied. A natural question is how to define the “leading order” term in $T$. Suppose that $\mu \in \{1-\nu,2-\nu,1+\nu\}$ and $$\label{eq:orderlessthanm}
\mathrm{ord}(T^-) - \nu \leq \mu - 1 , \quad \mathrm{ord}(T^+) + \nu \leq \mu - 1.$$ Then $T$ is said to have $\nu$-order less than or equal to $\mu$, written as $\mathrm{ord}_\nu(T) \leq \mu$. Note that if $\mathrm{ord}_\nu(T) \leq \mu$, then $B: \mathcal{H}^{2}(X) \rightarrow H^{2 - \mu}(\partial X)$ is continuous.
Suppose that $\mu \in \{1-\nu,2-\nu,1+\nu\}$ and $\mathrm{ord}_\nu(T) \leq \mu$. Define the family of operators $$\widehat{T}_{(y,\eta)} = \sigma_{\lceil \mu-1+\nu \rceil}(T^-) \gamma_- + \sigma_{\lceil\mu-1-\nu\rceil}(T^+)\gamma_+,$$ indexed by $(y,\eta) \in T^* \partial X$. Thus each $(y,\eta) \in T^*\partial X$ gives rise to a one-dimensional boundary operator $\widehat{T}_{(y,\eta)}$.
The boundary value problem {#subsect:theBVP}
--------------------------
Although boundary value problems of the form $\eqref{eq:BVP}$ are ultimately of interest, for duality purposes it is convenient to consider a more general type of problem. Fix $J \in \mathbb{N}$, and choose
- $\mu_k \in \{1-\nu,2-\nu,1+\nu\}$ for $k \in \{1,\ldots, J+1\}$,
- numbers $\tau_j \in \mathbb{R}$ for $j \in \{1,\ldots J\}$, not necessarily integers.
Let $T = (T_1,\ldots, T_{J+1})^\top$ denote a $(J+1) \times 1$ matrix of boundary operators, such that $\mathrm{ord}_\nu(T_k) \leq \mu_k$. Furthermore, for each $k \in \{1,\ldots,J+1\}$ and $j \in \{1,\ldots,J\}$, suppose $C_{k,j} \in \mathrm{Diff}^*(\partial X)$ is a differential operator on $\partial X$ such that $$\mathrm{ord}(C_{k,j}) \leq \tau_j + \mu_k.$$ Let $C$ denote the $(J+1) \times J$ matrix with entries $C_{k,j}$. Given these prerequisites, consider the modified boundary value problem $$\label{eq:modifiedBVP}
\begin{cases}
P u = f & \text{ on $X$} \\
T u + C\underline{u} = \underline{g} & \text{ on $\partial X$},
\end{cases}$$ where $\underline{u} = (u_1,\ldots u_J), \, \underline{g} = (g_1,\ldots, g_{J+1})$ are collections of functions on $\partial X$. In order to associate an operator to this problem, note that $Tu$ may be written in the form $$Tu = G {
\underline{\gamma}
} u,$$ where $G$ is the $(J+1) \times 2$ matrix $$G = \left( \begin{array}{cc}
T^{-}_1 & T^+_1 \\
\vdots & \vdots \\
T^-_{J+1} & T^+_{J+1}
\end{array} \right).$$ Throughout, it is always understood that $G$ is associated with $T$ in this way. Finally, set ${
\underline{\mu}
} = (\mu_1,\ldots,\mu_{J+1})$ and ${
\underline{\tau}
} = (\tau_1, \ldots, \tau_{J})$. Then let $\mathscr{P}$ denote the map $$\mathscr{P}(u, \underline{\phi}, \underline{u}) = (P(u,\underline{\phi}), G\underline{\phi} + C \underline{u}).$$ This is also written as $\mathscr{P} = \{P,T,C\}$.
\[lem:BVPcontinuous\] The map $\mathscr{P} = \{P,T,C\}$ is bounded $$\mathscr{P} :
{\widetilde{\mathcal{H}}}^{s}(X) \times H^{s+\underline{\tau}}(\partial X) \rightarrow {\mathcal{H}}^{s-2}(X) \times H^{s-\underline{\mu}}(\partial X)$$ for each $s=0,1,2$.
The mapping properties follows from the results of Section \[subsect:spaceonmanifold\].
The adjoint boundary value problem {#subsect:adjointproblem}
----------------------------------
Fix a density $\mu$ which is of product type near $\partial X$. Let $P^*$ denote the formal $L^2(X;\mu)$ adjoint of $P$; then $P^*$ is also a Bessel operator in light of Lemma \[lem:closedunderadjoint\]. Let $C^*, \, G^*$ denote the formal $L^2(\partial X;\mu_{\partial X})$ adjoints of $C, \, G$. Define the problem $$\label{eq:modifiedBVPadjoint}
\begin{cases}
P^* v = f & \text{ on $X$}, \\
J {
\underline{\gamma}
} v + G^* \underline{v} = \underline{g} & \text{ on $\partial X$}, \\
C^* \underline{v} = \underline{h} & \text{ on $\partial X$},
\end{cases}$$ where $\underline{v} = (v_1,\ldots,v_{J+1}), \, (\underline{g},\underline{h}) = (g_1,g_2,h_1,\ldots h_J)$ are functions on $\partial X$.
Although Green’s formula was previously only established for the formal adjoint of a Bessel operator on ${\mathbb{T}^n_+}$, it is clear that also holds here when the appropriate $\mu$ and $\mu_{\partial X}$ inner products are substituted on $X$ and $\partial X$: $$\left<Pu, v \right>_X + \left< Tu + C{
\underline{u}
}, {
\underline{v}
} \right>_{(\partial X)^{J+1}} = \left< u, P^* v \right>_X + \left< {
\underline{\gamma}
}u, G^* {
\underline{v}
} + J{
\underline{\gamma}
} v \right>_{(\partial X)^2} + \left<{
\underline{u}
}, C^* {
\underline{v}
}\right>_{(\partial X)^J}.$$ In light of this, the problem is said to be the formal adjoint of . Also notice that has the same form as . The corresponding operator is denoted by $\mathscr{P}^*$.
The Lopatinskiǐ condition {#subsect:lopatinskii}
-------------------------
The standard Lopatinskiǐ condition for smooth elliptic boundary value problems (see [@lions:1968; @roitberg:1996]) has a natural generalization to the situation here. Begin by choosing $c_{k,j} \in \mathbb{Z}$ (not necessarily nonnegative) such that $$\mathrm{ord}(C_{k,j}) \leq c_{k,j} \leq \tau_j + \mu_k,$$ and then define the matrix $\widehat{C}_{(y,\eta)}$ with entries $$(\widehat{C}_{(y,\eta)})_{k,j} = \sigma_{c_{k,j}}(C_{k,j})(y,\eta).$$ Thus $(y,\eta) \mapsto \widehat{C}_{(y,\eta)}$ is a function on $T^*\partial X$ with values in matrices over $\mathbb{C}$. Furthermore, define $\widehat{G}_{(y,\eta)}$ by the equality $$\widehat{G}_{(y,\eta)}{
\underline{\gamma}
}u = \widehat{T}_{(y,\eta)}u.$$
The matrix $\widehat{C}_{(y,\eta)}$ depends strongly on the choice of $c_{k,j}$, and is not necessarily obtained by calculating the principal symbol of $C_{k,j}$ entry-wise with respect to the order of $C_{k,j}$. The numbers $c_{k,j}$ in general will depend on the choice of $\tau_j, \mu_k$ as well. Similarly, $\widehat{G}_{(y,\eta)}$ is in general different from the principal symbol of $G$ calculated entry-wise with respect to the order of each entry.
\[defi:lopatinskii\] Suppose $P$ is elliptic on $\partial X$. The boundary operators $(T,C)$ are said to satisfy the Lopatinskiǐ condition with respect to $P$ if for each fixed $p \in \partial M$ and $\eta \in T^*_p\partial X \setminus 0$, the only element $(u,\underline{u}) \in \mathcal{M}_+(p,\eta) \times \mathbb{C}^J$ satisfying $$\widehat{T}_{(p,\eta)}u + \widehat{C}_{(p,\eta)}\underline{u} = 0 \\$$ is the trivial solution $(u,\underline{u}) = 0$. The boundary value problem , or equivalently the operator $\mathscr{P} = \{P,T,C\}$, is said to be elliptic on $\partial X$ if $P$ is elliptic on $\partial X$ in the sense of Definition \[def:elliptic\] and $(T,C)$ satisfy the Lopatinskiǐ condition on $\partial X$ with respect to $P$.
As in Section \[subsect:ellipticity\], the one-dimensional space $\mathcal{M}_+(y,\eta)$ is spanned by the function $$\label{eq:u_+}
u(y,\eta;x) =\frac{2^{1-\nu} \Gamma(1-\nu) \sin(\pi\nu) }{\pi} (i\xi(y,\eta))^{\nu} \sqrt{x}K_\nu(i\xi(y,\eta) x).$$ The argument is chosen so that all of the quantities are positive when $\xi$ lies on the negative imaginary axis. This choice of normalization for $u$ is motivated by the following:
\[lem:scatteringmatrix\] Let $0 < \nu < 1$. Then $$\gamma_- \left(u(y,\eta)\right) = 1, \quad \gamma_+ \left(u(y,\eta)\right) = -2\nu \frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}\left( \frac{i\xi(y,\eta)}{2} \right)^{2\nu}.$$
This follows from the asymptotic behavior $$K_\nu(i\xi x) \sim \frac{\pi}{2\sin(\pi\nu)} \left(\frac{1}{\Gamma(1-\nu)} \left(\frac{i\xi x}{2}\right)^{-\nu} - \frac{1}{\Gamma(1+\nu)} \left(\frac{i\xi x}{2}\right)^{\nu} \right)$$ as $x \rightarrow 0^+$.
It is clear from Lemma \[lem:scatteringmatrix\] that the Dirichlet condition $T = \gamma_-$ and Neumann condition $T = \gamma_+$ satisfy the Lopatinski[ǐ]{} condition with respect to any elliptic Bessel operator. The same is therefore true for the Robin condition $T = \gamma_+ + T^- \gamma_-$, where $T^- \in C^\infty(\partial X)$.
Consider a boundary condition $T = \gamma_+ + T^- \gamma_-$, where $T^-$ is a nonzero vector field on $\partial X$.
1. If $1/2 < \nu < 1$, then $\widehat{T}_{(y,\eta)} = \gamma_+$ for arbitrary $T^-$. Thus $T$ satisfies the Lopatinski[ǐ]{} conditions with respect to any elliptic Bessel operator.
2. If $ \nu = 1/2$, then $T$ is a classical oblique boundary condition. The Lopatinski[ǐ]{} condition is satisfied if $T^-$ is a real vector field for example, but can otherwise fail.
3. If $0 < \nu < 1/2$, then $$\widehat{T}_{(y,\eta)} = \sigma_1(T^-)(y,\eta) \gamma_-.$$ Since $\sigma_1(T^-)(y,\eta)$ is linear in $\eta$, it must have a nontrivial zero at each $y \in \partial X$ provided the dimension of the underlying manifold $X$ is at least four (or three if $T^-$ is real). In that case the Lopatinski[ǐ]{} condition necessarily fails at every point on the boundary.
Consider the operator $\Delta_\nu = |{D_\nu}|^2 + D_y^2 + D_z^2$ acting on $(0,1) \times \mathbb{T}^2$, where $(y,z)$ are standard coordinates on $\mathbb{T}^2 = (R/2\pi\mathbb{Z})^2$. Clearly $\Delta_\nu$ is an elliptic Bessel operator. Consider the boundary value problem $$\begin{cases} \Delta_\nu u = f, \\ T_0 u = g, \quad T_1 u = 0, \end{cases}$$ where $T_1 u = u|_{x=1}$, and $T_0 = (\partial_y - \partial_z)\gamma_-$. This is not a Fredholm problem, since there is an infinite dimensional kernel: for each $n \geq 0$, consider the function $$u_n(x,y,z) =\left(\sqrt{x}K_\nu(nx) - \frac{K_\nu(n)}{K_\nu(-n)}\sqrt{x}K_\nu(-nx) \right)e^{in(y+z)}.$$ The family of $u_n$ is linearly independent and each $u_n$ solves the boundary value problem. If $0 < \nu < 1/2$, then $T_0' = \gamma_+ + T_0$ is a compact perturbation of the original problem; thus the problem with $T_0'$ replacing $T_0$ is not Fredholm either. If $1/2 \leq \nu < 1$ then the problem with $T_0'$ satisfies the Lopatinski[ǐ]{} condition, so is indeed Fredholm.
Before proceeding with the next lemma, suppose that $P \in \mathrm{Bess}_\nu(X)$ and $\mu$ is a density of product type near $\partial X$. If $P$ is elliptic at $p \in \partial X$, then so is $P^*$, since the function is simply replaced by its complex conjugate.
\[lem:adjointiselliptic\] Suppose that $\mathscr{P} = \{P,T,C\}$ is elliptic. If $\mu$ is a density of product type near $\partial X$ and $\mathscr{P}^*$ is the corresponding adjoint boundary value problem, then $\mathscr{P}^*$ is also elliptic.
Since ellipticity only depends on various “principal symbols”, it is easy to see that $$\widehat{P^*}_{(y,\eta)} = \widehat{P}^*_{(y,\eta)},$$ where the latter adjoint is calculated with respect to the standard $L^2({\mathbb{R}_+})$ inner product. Similarly $$\widehat{T^*}_{(y,\eta)} = \widehat{T}^*_{(y,\eta)}, \quad \widehat{C^*}_{(y,\eta)} = \widehat{C}^*_{(y,\eta)},$$ where the latter adjoints are taken in the sense of matrices over $\mathbb{C}$.
Suppressing the dependence on $(y,\eta)$, Green’s formula implies that $$\begin{gathered}
\langle \widehat{P}u, v \rangle_{{\mathbb{R}_+}} + \langle \widehat{T} u + \widehat{C} {
\underline{u}
}, {
\underline{v}
} \rangle_{\mathbb{C}^{J+1}} = \langle u,\widehat{P}^* v \rangle_{X} + \langle {
\underline{\gamma}
}u, \widehat{G}^* {
\underline{v}
} + J{
\underline{\gamma}
} v \rangle_{\mathbb{C}^2} + \langle {
\underline{u}
}, \widehat{C}^* {
\underline{v}
} \rangle_{\mathbb{C}^J}.\end{gathered}$$ The goal is to prove that if the right hand side vanishes, then $(v,\underline{v}) = 0$. The proof relies on Lemma \[lem:1daprioris=2\] below (whose proof is of course independent of the present lemma). As in the proof of Lemma \[lem:1daprioris=2\], the Lopatinskiǐ condition implies that $$(u,\underline{u}) \mapsto \widehat{T}u + \widehat{C}{
\underline{u}
}$$ is an isomorphism between the spaces $\mathcal{M}_+ \times \mathbb{C}^J \rightarrow \mathbb{C}^{J+1}$. So choose $(u,\underline{u}) \in \mathcal{M}_+ \times \mathbb{C}^J$ such that $$\widehat{T}u + \widehat{C}{
\underline{u}
} = {
\underline{v}
}.$$ Since $\widehat{A}u = 0$, it follows from Green’s formula that $\underline{v} = 0$. On the other hand, from Lemma \[lem:1daprioris=2\] it is always possible to solve the inhomogeneous equation $$\begin{cases} \widehat{A} u = v, \\
\widehat{T} u + \widehat{C} {
\underline{u}
} = 0,
\end{cases}$$ whence Green’s formula implies that $v = 0$ as well.
The Dirichlet Laplacian {#subsect:dirichletlaplacian}
-----------------------
The results of this section are applied in one dimension to Section \[subsect:ccestimate\], and in higher dimensions to Appendix \[app:nugeq1\]. Define the Bessel operator $\Delta_\nu \in \mathrm{Bess}_\nu({\mathbb{T}^n_+})$ by $$\Delta_\nu = |{D_\nu}|^2 + \Delta_{{\mathbb{T}^{n-1}}},$$ where $\Delta_{{\mathbb{T}^{n-1}}} = \sum_{i=1}^{n-1} D_{y^{i}}^2$ is the positive Laplacian on ${\mathbb{T}^{n-1}}$. Consider the continuous, nonnegative Hermitian form $$\label{eq:sesquilinearform}
\ell(u,v) := \langle {D_\nu}u, {D_\nu}v\rangle_{{\mathbb{T}^n_+}} + \sum_{i=1}^{n-1} \left<D_{y^i} u, D_{y^i}u \right>_{{\mathbb{T}^n_+}}$$ on $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$. Associated to this form is the unbounded self-adjoint operator $L$ on $L^2({\mathbb{T}^n_+})$ with domain $$D(L) = \{ u \in \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+}) : v \mapsto \ell(u,v) \text{ is continuous on $L^2({\mathbb{T}^n_+})$} \}.$$ Standard manipulations show that $$\label{eq:D(L)}
D(L) = \mathring{{\mathcal{H}}}^{1}({\mathbb{T}^n_+}) \cap \{ u \in L^2({\mathbb{T}^n_+}): \Delta_\nu u \in L^2({\mathbb{T}^n_+}) \},$$ and $Lu = \Delta_\nu u$ in the sense of distributions for each $u \in D(L)$. The domain $D(L)$ is equipped with the graph norm.
In one dimension it is obvious tht $D(L) = {\mathcal{H}}^2({\mathbb{R}_+})\cap \mathring{{\mathcal{H}}}^1({\mathbb{R}_+})$, with an equivalence of norms via the open mapping theorem. This is also true in higher dimensions, but is not immediate from the definition
The next lemma follows from the Lax–Milgram theorem.
\[lem:laxmilgram\] Let $\nu >0$. For each $a \in \mathbb{C} \setminus (-\infty,0]$ the inverse $(L + a)^{-1}$ exists, and maps $$(L + a)^{-1}: \begin{cases} \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})' \rightarrow \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+}),\\
L^2({\mathbb{T}^n_+}) \rightarrow D(L). \end{cases}$$
Since $a \notin \ (-\infty,0]$ the form $\ell_a(u,v) = \ell(u,v) + a\left<u,v\right>_{{\mathbb{T}^n_+}}$ is coercive on $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$, so $\ell_a(u,v)$ defines an inner product on $\mathring{{\mathcal{H}}}^1$ equivalent to the usual one. The Lax–Milgram theorem guarantees that for each $f \in \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})'$ there exists a unique $u \in \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$ such that $\ell_a(u,v) = \left<f,v\right>$, and the mapping $u \mapsto f$ is continuous $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})' \rightarrow \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$.
Furthermore, the unbounded operator associated to $\ell_a$ is clearly $L + a$ (acting in the distributional sense) so $L + a : D(L) \rightarrow L^2({\mathbb{T}^n_+})$ is bijective. Since this map is continuous when $D(L)$ is equipped with the graph norm, it is an isomorphism by the open mapping theorem.
Elliptic Bessel operators on ${\mathbb{R}_+}$ {#subsect:ellipticbesselonRP}
---------------------------------------------
In this section, fix an operator $P$ on ${\mathbb{R}_+}$ of the form $$\label{eq:1dbessel}
P = |{D_\nu}|^2 + a, \quad a \in \mathbb{C}.$$ Thus $\xi \mapsto \xi^2 + a$ has no real roots precisely when $a \notin (-\infty,0]$. In that case, $P$ is said to be *regular*. This is distinguished from ellipticity of $P$ since the principal symbol of multiplication by $a$ as a second order operator is zero (in other words, the boundary symbol operator is $|{D_\nu}|^2$ and not $|{D_\nu}|^2 + a$). Furthermore, if $0 < \nu < 1$, fix boundary conditions $(T,C)$. Thus $T$ is just a column vector of $J$ boundary operators $T_k = T_{k}^\pm \gamma_\pm$ with $T^\pm_k \in \mathbb{C}$, and $C$ is a $(J+1) \times J$ matrix with $\mathbb{C}$-valued entries.
Regularity of the operator $\mathscr{P} = \{P,T,C\}$ is defined as just the Lopatinski[ǐ]{} condition: let $\mathcal{M}_+$ denote the space of bounded solutions to the equation $Pu = 0$. Then $\mathscr{P}$ is regular if the only element $(u, {
\underline{u}
}) \in \mathcal{M}_+ \times \mathbb{C}^J$ satisfying $Tu + C{
\underline{u}
}=0$ is the trivial solution.
\[prop:1dapriori\] Suppose that $P$ given by is regular, and that $\mathscr{P} = \{P,T,C\}$ is regular if $0 < \nu < 1$.
1. If $0 < \nu < 1$, then $\mathscr{P}$ is an isomorphism $${\widetilde{\mathcal{H}}}^{s}({\mathbb{R}_+}) \times \mathbb{C}^J \rightarrow {\mathcal{H}}^{s-2}({\mathbb{R}_+}) \times \mathbb{C}^{1+J}$$ for each $s = 0,1,2$. The operator norm of $\mathscr{P}^{-1}$ depends continuously on $a$ and the coefficients of $G$ and $C$
2. If $\nu \geq 1$, then $P$ is an isomorphism $${\mathcal{H}}^s({\mathbb{R}_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{R}_+})$$ for each $s=0,1,2$. The operator norm of $P^{-1}$ depends continuously on $a$.
The proof of this proposition is split up across several Lemmas.
\[lem:1daprioris=2\] Proposition \[prop:1dapriori\] holds when $0 < \nu < 1$ and $s=2$.
Since ${\widetilde{\mathcal{H}}}^{2}({\mathbb{R}_+})$ is isomorphic to ${\mathcal{H}}^{2}({\mathbb{R}_+})$ via the map $v \mapsto (v,{
\underline{\gamma}
}v)$, it is sufficient to prove the lemma with ${\mathcal{H}}^{2}({\mathbb{R}_+})$ replacing ${\widetilde{\mathcal{H}}}^{2}({\mathbb{R}_+})$. By the regularity condition, $P$ is injective. Indeed any solution in ${\mathcal{H}}^{2}({\mathbb{R}_+})$ to the equation $P u = 0$ must lie in $\mathcal{M}_+$, and the Lopatinskiǐ condition implies that such a solution is unique. It remains to show surjectivity.
Fix $(f,g) \in {\mathcal{H}}^0({\mathbb{R}_+}) \times \mathbb{C}^{J+1}$. From Lemma \[lem:laxmilgram\], it follows that that the equation $$P u = f$$ has a solution $u_1 \in \mathcal{H}^2({\mathbb{R}_+})\cap \mathring{{\mathcal{H}}}^1({\mathbb{R}_+})$. It then suffices to let $(u_2, {
\underline{u}
}) \in \mathcal{M}_+ \times \mathbb{C}^J$ solve $$\begin{cases}
P u_2 = 0,\\
T u_2 + C{
\underline{u}
} = g - T u_1, \\
\end{cases}$$ This is possible since $$(u, \underline{u} ) \mapsto T u + C \underline{u}$$ as a map between the finite dimensional vector spaces $\mathcal{M}_+ \times \mathbb{C}^J \rightarrow \mathbb{C}^{J+1}$ is injective, hence an isomorphism. Setting $u = u_1 + u_2$ shows that $\mathscr{P}(u,{
\underline{u}
}) = (f,g)$. It is also easy to see that the operator norm of $\mathscr{P}$ depends continuously on $a$ and the coefficients of $G$ and $C$, which implies the same for the operator norm of $\mathscr{P}^{-1}$ via the resolvent identity.
\[lem:1daprioris=0\] Proposition \[prop:1dapriori\] holds when $0 < \nu < 1$ and $s=0$.
Since the formal adjoint operator $\mathscr{P}^*$ is also regular according to Lemma \[lem:adjointiselliptic\], the map $${\mathcal{H}}^{2}({\mathbb{R}_+}) \times \mathbb{C}^{1+J} \rightarrow {\mathcal{H}}^{0}({\mathbb{R}_+}) \times \mathbb{C}^2 \times \mathbb{C}^{J}$$ given by $$(v,\underline{v}) \mapsto (P^* v, \, J{
\underline{\gamma}
} v + G^*\underline{v}, \, C^*\underline{v})$$ is an isomorphism according to Lemma \[lem:1daprioris=2\]. But in that case, a direct calculation shows that $\mathscr{P}^*$ agrees with the Hilbert space adjoint $\mathscr{P}'$ of $$\mathscr{P} : {\widetilde{\mathcal{H}}}^{0}({\mathbb{R}_+}) \times \mathbb{C}^J \rightarrow \mathcal{H}^{-2}({\mathbb{R}_+}) \times \mathbb{C}^{1+J}.$$ Since $\mathscr{P}'$ is an isomorphism, $\mathscr{P}$ is an isomorphism on the stated spaces as well.
To prove Proposition \[prop:1dapriori\] for $s=1$, the following regularity result is needed.
\[lem:1dregularity\] Let $0 < \nu < 1$. Suppose that $(u,\underline{\phi}) \in {\widetilde{\mathcal{H}}}^0({\mathbb{R}_+})$ satisfies $P(u,\underline{\phi}) \in {\mathcal{H}}^{-1}({\mathbb{R}_+})$. Then $(u,\underline{\phi}) \in {\widetilde{\mathcal{H}}}^{1}({\mathbb{R}_+})$.
Let $f \in \mathring{{\mathcal{H}}}^1({\mathbb{R}_+})'$ denote the restriction of the functional $P(u,\underline{\phi})$ to $\mathring{{\mathcal{H}}}^1({\mathbb{R}_+})$. This implies that $f = Pu$ in the sense of distributions. By Lemma \[lem:laxmilgram\], there exists a unique $\tilde u \in \mathring{{\mathcal{H}}}^1({\mathbb{R}_+})$ such that $P \tilde u = f$ in the distributional sense. Thus in sense of distributions on ${\mathbb{R}_+}$, $$P(u - \tilde u) = 0.$$ Since $u$ and $\tilde{u}$ are square integrable, it follows that $u - \tilde{u} \in \mathcal{M}_+$. Thus it is certainly true that $$u = (u - \tilde{u}) + \tilde{u} \in {\mathcal{H}}^1({\mathbb{R}_+}).$$ It remains remains to prove that $\phi_- = \gamma_- u$. A priori $(u,\underline{\phi}) \in {\widetilde{\mathcal{H}}}^0({\mathbb{R}_+})$, so for each $v\in {\mathcal{H}}^2({\mathbb{R}_+})$, $$\langle f , v \rangle_{{\mathbb{R}_+}} = \langle u , P^* v \rangle_{{\mathbb{R}_+}} - \phi_+ (\gamma_- v) + \phi_- (\gamma_+ v).$$ Using that $u \in {\mathcal{H}}^1({\mathbb{R}_+})$, this may be rewritten as $$\langle f , v \rangle_{{\mathbb{R}_+}} - \langle {D_\nu}u, {D_\nu}v \rangle_{{\mathbb{R}_+}} - a \langle u , v \rangle_{{\mathbb{R}_+}} + \phi_+(\gamma_-v) = (\phi_- - \gamma_- u)\gamma_+ v$$ for each $v\in {\mathcal{H}}^2({\mathbb{R}_+})$. But the left hand side extends to a continuous functional on ${\mathcal{H}}^1({\mathbb{R}_+})$, which is not true of the right hand side unless $\phi_- = \gamma_- u$, thus completing the proof.
\[lem:1daprioris=1\] Proposition \[prop:1dapriori\] holds when $0 < \nu < 1$ and $s=1$.
The regularity result of Lemma \[lem:1dregularity\] combined with Lemma \[lem:1daprioris=0\] shows that $\mathscr{P}$ defines a continuous bijection, hence an isomorphism $${\widetilde{\mathcal{H}}}^1({\mathbb{R}_+}) \times \mathbb{C}^J \rightarrow {\mathcal{H}}^{-1}({\mathbb{R}_+}) \times \mathbb{C}^{J+1}$$ as stated.
\[lem:1dapriorinugeq1\] Proposition \[prop:1dapriori\] holds when $\nu \geq 1$.
If $\nu \geq 1$, then ${\mathcal{H}}^{s}({\mathbb{R}_+}) = \mathring{{\mathcal{H}}}^s({\mathbb{R}_+})$ for $s\geq 0$. Thus it suffices to apply Lemma \[lem:laxmilgram\] directly when $s=1,2$. The case $s=0$ is handled by duality, similar to \[lem:1daprioris=0\].
The combination of Lemmas \[lem:1daprioris=2\], \[lem:1daprioris=1\], \[lem:1daprioris=0\], \[lem:1dapriorinugeq1\] establishes Proposition \[prop:1dapriori\]
Elliptic Bessel operators on ${\mathbb{T}^n_+}$ with constant coefficients {#subsect:ccestimate}
--------------------------------------------------------------------------
Throughout this section, $P$ denotes a constant coefficient Bessel operator on ${\mathbb{T}^n_+}$, $$\label{eq:ccbessel}
P({D_\nu},D_y) = |{D_\nu}|^2 + A(D_y)$$ If $0 < \nu < 1$, then $P$ is also augmented by boundary conditions $(T,C)$ with constant coefficients: thus each boundary operator is of the form $T_k(D) = T_k^\pm(D_y) \gamma_\pm$, and each entry of $C(D_y)$ has constant coefficients.
The principal part of $P$ is the operator $$P^\circ({D_\nu},D_y) = |{D_\nu}|^2 + A^\circ(D_y),$$ where $A^\circ(D_y)$ is the standard principal part of $A$. The principal parts of $(T(D_y),C(D_y))$ are defined to be the unique boundary operators $(T^\circ(D_y),C^\circ(D_y))$ satisfying $$T^\circ(\eta) = \widehat{T}_\eta, \quad C^\circ(\eta) = \widehat{C}_\eta$$ for each $\eta \in \mathbb{R}^{n-1}$. Finally, define $\mathscr{P}^\circ({D_\nu},D_y) = \{ P^\circ({D_\nu},D_y),T^\circ(D_y),C^\circ(D_y)\}$. Ellipticity of either $P$ or $\mathscr{P}$ depends only on these principal parts.
\[lem:higherdimapriori\] Assume that $P$ and $\mathscr{P}$ are elliptic. Furthermore, assume that the one dimensional operators $P({D_\nu},q)$ (if $\nu \geq 1$) and $\mathscr{P}({D_\nu},q)$ (if $0< \nu < 1$) are regular for each $q\in\mathbb{Z}^{n-1}$.
1. If $0 < \nu < 1$, then $$\mathscr{P}: {\mathcal{H}}^2({\mathbb{T}^n_+}) \times H^{2+{
\underline{\tau}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+}) \times {\mathcal{H}}^{2-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})$$ is an isomorphism.
2. If $\nu \geq 1$, then $$P: {\mathcal{H}}^2({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+})$$ is an isomorphism.
$(1)$ Let $0 < \nu <1$. By ellipticity, $$\mathscr{P}^\circ({D_\nu},\left<q\right>^{-1}q): {\mathcal{H}}^2({\mathbb{R}_+}) \times \mathbb{C}^J \rightarrow \mathbb{C}^{1+J}$$ is an isomorphism for each $q \in \mathbb{Z}^{n-1}$, according to Proposition \[prop:1dapriori\]. Since $\left<q\right>^{-1}q$ ranges over a compact subset of $\mathbb{R}^{n-1}$, the operator norm of $\mathscr{P}^\circ({D_\nu},\left<q\right>^{-1}q)^{-1}$ is bounded uniformly with respect to $q \in \mathbb{Z}^{n-1}\setminus 0$. On the other hand, the homogeneity of $P^\circ$ implies $$\tau^{-2}S_{-\tau}P^\circ({D_\nu},D_y)S_\tau = P^\circ({D_\nu},\tau^{-1}D_y), \quad \tau^{-{
\underline{\mu}
}+1/2} T^\circ(D_y)S_\tau = T^\circ(\tau^{-1} D_y).$$ Using $\tau = \left<q\right>$, this implies that the operator norm corresponding to the problem $$\begin{cases}
\left<q\right>^{-2}S_{\left<q\right>^{-1}}{P}({D_\nu},q)S_{\left<q\right>}v = \phi, \\
\left<q\right>^{-{\mu}_k + 1/2}T(q) S_{\left<q\right>}v + \sum_{i=1}^J \left<q\right>^{-\tau_j - \mu_k}C_{k,j}(q){
\underline{v}
} = {
\underline{\psi}
}
\end{cases}$$ tends to that of $\mathscr{P}^\circ({D_\nu},\left<q\right>^{-1} D_y)$ as $|q| \rightarrow \infty$. Thus the former problem is invertible for $q \in \mathbb{Z}^{n-1}$ with operator norm uniformly bounded in $q$. Apply this invertibility result to the functions $$v = S_{\left<q\right>^{-1}} \hat{u}(q), \quad \underline{v} = (\left<q\right>^{\tau_1 + 1/2} \hat{{
\underline{u}
}}_1(q), \ldots, \left<q\right>^{\tau_J + 1/2} \hat{{
\underline{u}
}}_J(q)).$$ This implies that $$\begin{gathered}
\| S_{\left<q\right>^{-1}} \hat{u}(q) \|^2_{{\mathcal{H}}^2({\mathbb{T}^n_+})} + \left< q \right>^{1+2{
\underline{\tau}
}} \| \hat{{
\underline{u}
}}(q) \|^2_{\mathbb{C}^J} \\ \leq C \left( \left< q \right>^{-4} \| S_{\left<q\right>^{-1}} P({D_\nu},q)\hat{u}(q) \|^2 + \left<q\right>^{1-2{
\underline{\mu}
}} \|T(q)\hat{u}(q) + C_{k,j}(q) \hat{u}(q) \|^2_{\mathbb{C}^{J+1}} \right). \label{eq:fourierestimate}
\end{gathered}$$ From it follows that $\mathscr{P}$ is injective. Now multiply this equation by $\left<q\right>^{2s-1} = \left<q\right>^{3}$ and sum over $q\in \mathbb{Z}^{n-1}$. Then Lemma \[lem:fouriernorm\] shows that the Fourier series for $(u,{
\underline{u}
})$ converges in ${\mathcal{H}}^2({\mathbb{T}^n_+})\times H^{2+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})$. Combined with the fact that $\mathscr{P}({D_\nu},q)$ is invertible for each $q\in\mathbb{Z}^{n-1}$, this shows that $\mathscr{P}$ is surjective.
$(2)$ The proof when $\nu \geq 1$ follows as above, disregarding the boundary operators.
Assume that $P$ and $\mathscr{P}$ are elliptic. Furthermore, assume that the one dimensional operators $P({D_\nu},q)$ (if $\nu \geq 1$) and $\mathscr{P}({D_\nu},q)$ (if $0< \nu < 1$) are regular for each $q\in\mathbb{Z}^{n-1}$.
1. If $0 < \nu < 1$, then $$\mathscr{P}: {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+}) \times {\mathcal{H}}^{s-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})$$ is an isomorphism for $s=0,1,2$.
2. If $\nu \geq 1$, then $$P: {\mathcal{H}}^s({\mathbb{T}^n_+}) \rightarrow {\mathcal{H}}^{s-2}({\mathbb{T}^n_+})$$ is an isomorphism for $s=0,1,2$.
\[cor:higherdimapriori\]
$(1)$ It remains to handle the cases $s=0,1$. First consider $s=0$. As in the proof of Lemma \[lem:1daprioris=0\], the formal adjoint $$\mathscr{P}^*: {\mathcal{H}}^2({\mathbb{T}^n_+}) \times H^{{
\underline{\mu}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+}) \times H^{{
\underline{\nu}
}-2}({\mathbb{T}^{n-1}}) \times H^{-{
\underline{\tau}
}}({\mathbb{T}^{n-1}})$$ agrees with the adjoint of $$\mathscr{P}: {\widetilde{\mathcal{H}}}^0({\mathbb{T}^n_+}) \times H^{{
\underline{\tau}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^{-2}({\mathbb{T}^n_+}) \times H^{-{
\underline{\mu}
}}({\mathbb{T}^{n-1}}).$$ Now $\mathscr{P}^*$ satisfies the same hypotheses as $\mathscr{P}$ in regards to the application of Lemma \[lem:higherdimapriori\], so is an isomorphism. This implies that $\mathscr{P}'$ is an isomorphism, hence so is $\mathscr{P}$ on the stated spaces.
The case $s=1$ follows from combined with Lemma \[lem:1daprioris=1\]: indeed, multiplying the analogue of by $\left<q\right>^{2s-1} = \left<q\right>$ and using the invertibility result from Lemma \[lem:1daprioris=1\] shows that $\mathscr{P}$ is surjective on ${\widetilde{\mathcal{H}}}^1({\mathbb{T}^n_+})\times H^{1+{
\underline{\tau}
}}({\mathbb{T}^n_+})$ (as well as injective by the $s=0$ case).
$(2)$ As usual, when $\nu \geq 1$ the proof follows by dropping the boundary terms.
If $P({D_\nu},D_y)$ is elliptic, then $P^\circ({D_\nu},D_y + \tfrac{1}{2})$ satisfies the hypotheses of Lemma \[lem:higherdimapriori\]. Similarly, if $0< \nu < 1$ and $\mathscr{P}({D_\nu},D_y)$ is elliptic, then $\mathscr{P}^\circ({D_\nu},D_y+\tfrac{1}{2})$ also satisfies the hypotheses of Lemma \[lem:higherdimapriori\].
Elliptic Bessel operators on ${\mathbb{T}^n_+}$ with variable coefficients
--------------------------------------------------------------------------
In this section, let $P$ be a Bessel operator on ${\mathbb{T}^n_+}$ of the form $$P(x,y,{D_\nu},D_y) = |{D_\nu}|^2 + B(x,y,D_y){D_\nu}+ A(x,y,D_y),$$ where the coefficients of $A, B$ are constant outside a compact subset of ${
\overline{{\mathbb{T}^n_+}}
}$. If $0 < \nu < 1$, then $P$ is also augmented by boundary conditions $(T(y,D_y),C(y,D_y))$. Introduce the notation $$\begin{gathered}
P^{(0)}({D_\nu},D_y) := P^\circ(0,0,{D_\nu},D_y+\tfrac{1}{2}),\\ T^{(0)}(D_y) = T^\circ(0,D_y+\tfrac{1}{2}), \quad C^{(0)}(D_y) = C^\circ(0,D_y+ \tfrac{1}{2}).\end{gathered}$$ According to Lemma \[lem:higherdimapriori\], if $P$ and $\mathscr{P}$ are elliptic, then $P^{(0)}$ (if $\nu \geq 1$) and $\mathscr{P}^{(0)}$ (if $0 < \nu < 1$) are isomorphisms on the appropriate spaces.
Given $\rho > 0$, define the Fourier multiplier $K_\rho = {1}_{|x| \geq \rho}(D_y)$. This operator acts both on Sobolev spaces $H^m({\mathbb{T}^{n-1}})$, as well as on ${\mathcal{H}}^s({\mathbb{T}^n_+})$ (or ${\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$) via the results of Section \[subsect:fourier\]. If $m > m'$ then clearly $$\label{eq:SrhoH}
\| K_\rho \phi \|_{H^{m'}({\mathbb{T}^{n-1}})} \leq \left<\rho\right>^{m-m'} \| \phi \|_{H^m({\mathbb{T}^{n-1}})}$$ for $\phi \in H^m({\mathbb{T}^{n-1}})$. Similarly, $$\label{eq:SrhoSob}
\| K_\rho u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq \left<\rho \right>^{-|\alpha|} \| D_y^\alpha u \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})} \leq \left<\rho \right>^{-|\alpha|} \| u \|_{{\mathcal{H}}^{s+|\alpha|}({\mathbb{T}^n_+})}$$ for $u \in {\mathcal{H}}^s({\mathbb{T}^n_+})$, provided $s+|\alpha| \leq 2$. A similar statement holds for $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+})$.
\[lem:localizedvariableRNP\] Assume that $P$ and $\mathscr{P}$ are elliptic. Then there exists $\delta >0$ such that the following hold.
1. Let $0 < \nu < 1$ and $s=0,1,2$. Suppose that $(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})$ satisfies $$\begin{gathered}
\operatorname{supp}u \subseteq \{(x,y)\in{
\overline{{\mathbb{T}^n_+}}
}: |x| + |y| < \varepsilon \}, \\ \operatorname{supp}{
\underline{\phi}
} \subseteq \{y \in {\mathbb{T}^{n-1}}: |y| < \delta \}, \quad \operatorname{supp}{
\underline{u}
} \subseteq \{ y \in {\mathbb{T}^{n-1}}: |y| < \delta\}.
\end{gathered}$$ Then $$\begin{aligned}
\label{eq:variablecoefficientestimateRNP}
\| (u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})} &\leq C( \| \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \|_{{
{\mathcal{H}}^{s - 2}({\mathbb{T}^n_+}) \times H^{s-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})
}} \notag \\ &+ \| (u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s-1}(X) \times H^{s-1+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})} ),
\end{aligned}$$ where $C>0$ does not depend on $(u,{
\underline{\phi}
},{
\underline{u}
})$. In addition, if $s=0,1$ and $$\mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\mathcal{H}}^{s-1}({\mathbb{T}^n_+}) \times {\mathcal{H}}^{s-{
\underline{\mu}
}+1}({\mathbb{T}^{n-1}}),$$ then $(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^{s+1}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}+1}({\mathbb{T}^n_+})$.
2. Let $\nu \geq 1$ and $s=0,1,2$. Suppose that $u \in {\mathcal{H}}^2({\mathbb{T}^n_+})$ satisfies $$\operatorname{supp}u \subseteq \{(x,y)\in{
\overline{{\mathbb{T}^n_+}}
}: |x| + |y| < \delta \}$$ Then $$\| u \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} \leq C ( \| Pu \|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} + \| u \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})}),$$ where $C>0$ does not depend on $u$. In addition, if $s=0,1$ and $Pu \in {\mathcal{H}}^{s-1}({\mathbb{T}^n_+})$, then $u \in {\mathcal{H}}^{s+1}({\mathbb{T}^n_+})$.
$(1)$ For concreteness, assume that $s=1$ and $\mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\mathcal{H}}^0({\mathbb{T}^n_+}) \times H^{2-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})$. If $(f,{
\underline{g}
}) = \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
})$, consider the identity $$\begin{gathered}
\label{eq:perturbidentity}
\mathscr{P}^{(0)}(u,{
\underline{\phi}
},{
\underline{u}
}) + (\mathscr{P}-\mathscr{P}^{(0)})(K_\rho (u,{
\underline{\phi}
}) ,K_\rho {
\underline{u}
}) \\ = (f,{
\underline{g}
}) - (\mathscr{P}-\mathscr{P}^{(0)})((1-K_\rho)(u,{
\underline{\phi}
}), (1-K_\rho){
\underline{u}
}).
\end{gathered}$$ Noting that the term $(P-P^{(0)})(u,{
\underline{\phi}
})$ depends only on $u$ (and not on ${
\underline{\phi}
})$, it follows from Lemmas \[lem:Bbounded\], \[lem:Abounded\] and that $$\begin{aligned}
\|(P-P^{(0)})K_\rho u\|_{{\mathcal{H}}^0({\mathbb{T}^n_+})} &\leq C_1 \delta \| u\|_{{\mathcal{H}}^2({\mathbb{T}^n_+})} + C_2 \| K_\rho u \|_{{\mathcal{H}}^1({\mathbb{T}^n_+})}\\ &\leq (C_1\delta + C_2 \left< \rho \right>^{-1}) \| (u,{
\underline{\phi}
}) \|_{{\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+})},
\end{aligned}$$ for positive constants $C_1, C_2$. By standard interpolation inequalities on $H^m({\mathbb{T}^{n-1}})$, $$\begin{aligned}
\| (T_k - T^{(0)})_k K_\rho {
\underline{\phi}
} \| &\leq C_3\delta \| {
\underline{\phi}
} \|_{H^{2-{
\underline{\nu}
}}({\mathbb{T}^{n-1}})} + C_4 \| K_\rho {
\underline{\phi}
} \|_{H^{1-{
\underline{\nu}
}}({\mathbb{T}^{n-1}})} \\ &\leq (C_3 \delta + C_4 \left<\rho \right>^{-1}) C_5 \| (u,{
\underline{\phi}
}) \|_{{\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+})}.
\end{aligned}$$ For this, one should consider the cases $0< \nu < 1/2, \, \nu =1/2$, and $1/2 < \nu < 1$ separately, but they all yield the same type of the estimate. Similarly, $$\| (C - C^{(0)})K_\rho {
\underline{u}
} \|_{H^{2-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})} \leq ( C_6 \delta + C_7\left< \rho \right>^{-1}) \| {
\underline{u}
} \|_{H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})}.$$ These inequalities imply that the operator norm of $$(u,{
\underline{\phi}
},{
\underline{u}
}) \mapsto (\mathscr{P} - \mathscr{P}^{(0)})(K_\rho u,K_\rho {
\underline{u}
})$$ can be made arbitrarily small by choosing $\delta >0$ small and $\rho > 0$ large. Since $\mathscr{P}^{(0)}$ is invertible with domain ${\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+}) \times H^{2+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})$, it follows that the operator on the left hand side of is invertible for $\delta$ small and $\rho$ large.
On the other hand, the map $$(u,{
\underline{\phi}
},{
\underline{u}
}) \mapsto (\mathscr{P}-\mathscr{P}^{(0)})((1-K_\rho)(u,{
\underline{\phi}
}), (1-K_\rho){
\underline{u}
})$$ is bounded ${\widetilde{\mathcal{H}}}^1({\mathbb{T}^n_+}) \times H^{1+{
\underline{\tau}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^0({\mathbb{T}^n_+}) \times H^{2-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})$. In particular, $(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+}) \times H^{2+{
\underline{\tau}
}({\mathbb{T}^{n-1}})}$, and the estimate holds. Of course this also implies that holds for arbitrary $(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^2({\mathbb{T}^n_+})\times H^{2+\tau}({\mathbb{T}^{n-1}})$ as well. The exact same argument establishes the regularity result for $s=0$, as well for $s=0,1$.
$(2)$ As usual, the case $\nu \geq 1$ can be handled by a simpler argument not involving the boundary operators.
Lemma \[lem:localizedvariableRNP\] can be semi-globalized via a partition of unity argument.
\[cor:localizedvariableRNP\] Assume that $P$ and $\mathscr{P}$ are elliptic at $\partial X$. There exists $\delta > 0$ such that if $\varphi, \chi \in C_c^\infty([0,\delta))$ satisfy $\varphi = 1$ near $x=0$ and $\chi = 1$ near $\operatorname{supp}\varphi$, then the following hold.
1. Let $0 < \nu < 1$ and $s=0,1,2$. Then $$\begin{aligned}
\label{eq:variablecoefficientestimateRNPsemiglobal}
\| \varphi(u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})} &\leq C( \| \varphi \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \|_{{
{\mathcal{H}}^{s - 2}({\mathbb{T}^n_+}) \times H^{s-{
\underline{\mu}
}}({\mathbb{T}^{n-1}})
}} \notag \\ &+ \|\chi (u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s-1}({\mathbb{T}^n_+}) \times H^{s-1+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})} ),
\end{aligned}$$ for each $(u,{
\underline{\phi}
},{
\underline{u}
})\in {\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \times {\mathcal{H}}^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})$. In addition, if $s=0,1$ and $$\varphi \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\mathcal{H}}^{s-1}({\mathbb{T}^n_+}) \times {\mathcal{H}}^{s-{
\underline{\mu}
}+1}({\mathbb{T}^{n-1}}),$$ then $\varphi (u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^{s+1}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}+1}({\mathbb{T}^{n-1}})$.
2. Let $\nu \geq 1$ and $s=0,1,2$. Then $$\label{eq:nugeq1RNPsemiglobal}
\| \varphi u \|_{{\mathcal{H}}^{s}({\mathbb{T}^n_+})} \leq C ( \| \varphi Pu \|_{{\mathcal{H}}^{s-2}({\mathbb{T}^n_+})} + \| \chi u \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+})}),$$ for each $u \in {\mathcal{H}}^s({\mathbb{T}^n_+})$. In addition, if $s=0,1$ and $\varphi Pu \in {\mathcal{H}}^{s-1}({\mathbb{T}^n_+})$, then $\varphi u \in {\mathcal{H}}^{s+1}({\mathbb{T}^n_+})$.
\[Sketch of proof for $0< \nu <1$\] By compactness of ${\mathbb{T}^{n-1}}$ it is possible to choose $\delta$ and a finite cover ${\mathbb{T}^{n-1}}= \bigcup_i U_i$ such that Lemma \[lem:localizedvariableRNP\] is valid for $(u,{
\underline{\phi}
},{
\underline{u}
})$ supported in $[0,\delta) \times U_i$. Fix a partition of unity $\beta_i$ subordinate to $U_i$, and choose $\gamma_i$ supported in $U_i$ that $\gamma_i = 1$ on $\operatorname{supp}\beta_i$. For $\varphi, \chi$ as in the statement of the corollary, $$\begin{gathered}
\| \varphi(u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s}({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})} \leq C_1\| \varphi \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})\times H^{s-{
\underline{\mu}
}({\mathbb{T}^{n-1}})}}\\
+ \sum_i \| [\mathscr{P},\beta_i \varphi] \gamma_i \chi (u,{
\underline{\phi}
},{
\underline{\mu}
}) \|_{{\mathcal{H}}^s({\mathbb{T}^n_+})\times H^{s-{
\underline{\mu}
}({\mathbb{T}^{n-1}})}} \\
+ C_2 \| \varphi (u,{
\underline{\phi}
},{
\underline{u}
}) \|_{{\mathcal{H}}^{s-1}({\mathbb{T}^n_+}) \times H^{s-1+{
\underline{\tau}
}}({\mathbb{T}^{n-1}})}.
\end{gathered}$$ Writing $\varphi_i \beta_i \varphi$, the commutator $[\mathscr{P},\beta_i\varphi]$ is given by $$(u,{
\underline{\phi}
},{
\underline{u}
}) \mapsto (P(\varphi_i u, \beta_i {
\underline{\phi}
}) - \varphi_i P(u,{
\underline{\phi}
}), [G,\beta_i] {
\underline{\phi}
} + [C,\beta_i]{
\underline{u}
} ).$$ It is then straightforward to check that this operator has the requisite mapping properties. The regularity statement is established in the same way.
As usual, the norms of the lower order terms on the right hand sides of , can be taken in less regular Sobolev spaces by iterating Corollary \[cor:localizedvariableRNP\]. Similarly, the regularity result can also be iterated.
Elliptic Bessel operators on a compact manifold with boundary {#subsect:variable}
-------------------------------------------------------------
The main theorem in this section establishes elliptic estimates and elliptic regularity for elliptic Bessel operators on a compact manifold with boundary ${
\overline{X}
}$.
\[theo:bvptheo\] Let ${
\overline{X}
}$ be a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. Assume that $P \in \mathrm{Bess}_\nu(X)$ is elliptic at $\partial X$ in the sense of Section \[subsect:ellipticity\]. If $0 < \nu < 1$, then assume $P$ is augmented by boundary conditions $(T,C)$ such that $\mathscr{P} = \{P,T,C\}$ is elliptic at $\partial X$. There exists $0 < \delta < \varepsilon$ such that if $\varphi, \chi \in C_c^{\infty}(\{ 0 \leq x < \delta\})$ satisfy $\varphi = 1$ near $\partial X$ and $\chi = 1$ near $\operatorname{supp}\varphi$, then the following hold.
1. Let $0 < \nu < 1$ and $s=0,1,2$. Then $$\begin{aligned}
\| \varphi(u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s}(X) \times H^{s+{
\underline{\tau}
}}(\partial X)} &\leq C( \| \varphi \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \|_{{
{\mathcal{H}}^{s - 2}(X) \times H^{s-{
\underline{\mu}
}}(\partial X)
}} \notag \\ &+ \|\chi (u, {
\underline{\phi}
}, {
\underline{u}
}) \|_{{\widetilde{\mathcal{H}}}^{s-1}(X) \times H^{s-1+{
\underline{\tau}
}}(\partial X)} ),
\end{aligned}$$ for each $(u,{
\underline{\phi}
},{
\underline{u}
})\in {\widetilde{\mathcal{H}}}^{s}(X) \times {\mathcal{H}}^{s+{
\underline{\tau}
}}(\partial X)$. In addition, if $s=0,1$ and $$\varphi \mathscr{P}(u,{
\underline{\phi}
},{
\underline{u}
}) \in {\mathcal{H}}^{s-1}(X) \times {\mathcal{H}}^{s-{
\underline{\mu}
}+1}(\partial X),$$ then $\varphi (u,{
\underline{\phi}
},{
\underline{u}
}) \in {\widetilde{\mathcal{H}}}^{s+1}(X) \times H^{s+{
\underline{\tau}
}+1}(\partial X)$.
2. Let $\nu \geq 1$ and $s=0,1,2$. Then $$\| \varphi u \|_{{\mathcal{H}}^{s}(X)} \leq C ( \| \varphi Pu \|_{{\mathcal{H}}^{s-2}(X)} + \| \chi u \|_{{\mathcal{H}}^{s-1}(X)}),$$ for each $u \in {\mathcal{H}}^s(X)$. In addition, if $s=0,1$ and $\varphi Pu \in {\mathcal{H}}^{s-1}(X)$, then $\varphi u \in {\mathcal{H}}^{s+1}(X)$.
The global problem may be reduced to a local problem on ${\mathbb{T}^n_+}$ via coordinate charts and a partition of unity. Indeed, cover the collar neighborhood ${
\overline{W}
}$ by finitely many coordinate charts $(U_i,\psi_i)$ of the form $$U_i = \psi([0,\varepsilon) \times Y_i), \quad \psi_i = (1 \times \theta_i)\circ \phi^{-1},$$ where $(Y_i, \theta_i)$ is a coordinate chart on $\partial X$. Choose a partition of unity $\beta_i$ subordinate to $Y_i$ and a function $\alpha \in C_c^\infty(0\leq x < \varepsilon)$ such that $\alpha =1$ near $\{ 0 \leq x < \delta\}$. Then set $\chi_i = \alpha\beta_i$ and apply the results of Corollary \[cor:localizedvariableRNP\] to each element $$((\chi_i u)\circ \psi_i^{-1}, (\beta_i {
\underline{\phi}
}) \circ \theta_i^{-1}, (\beta_i {
\underline{u}
}) \circ \theta_i^{-1}) \in {\widetilde{\mathcal{H}}}^s({\mathbb{T}^n_+}) \times H^{s+{
\underline{\tau}
}}({\mathbb{T}^{n-1}}).$$ Again there will be various commutator terms which give only lower order contributions, as in Corollary \[cor:localizedvariableRNP\].
As in the remark following Corollary \[cor:localizedvariableRNP\], the error terms in Theorem \[theo:bvptheo\] can taken in weaker Sobolev spaces by iteration.
Recall the definition of ${\mathcal{H}}^s_P(X)$ in Section \[subsect:graphnorm\]. Theorem \[theo:bvptheo\] can be used to show that ${\mathcal{H}}^s_P(X)$ (or equivalently ${\widetilde{\mathcal{H}}}^s(X)$) may be identified with the space of all pairs $(u,f) \in {\mathcal{H}}^s(X)\times{\mathcal{H}}^{s-2}(X)$ such that $Pu =f $ in the weak sense, see also [@roitberg:1996 Chapter 6.1].
\[lem:graphnorm2\] Let $0 < \nu < 1$, and suppose that $P$ is elliptic at $\partial X$. Then for $s=0,1,2$, $${\mathcal{H}}^s_P(X) = \{ (u,f) \in {\mathcal{H}}^{s}(X) \times {\mathcal{H}}^{s-2}(X): \, Pu =f \text{ weakly} \},$$ where the space on the right hand side is equipped with the ${\mathcal{H}}^s_P(X)$ norm.
As in the remark following Lemma \[lem:graphnorm\], ${\mathcal{H}}^s_P(X)$ is contained in the space on the right hand side. For the converse, suppose that $u \in {\mathcal{H}}^s(X)$ and $f = Pu \in {\mathcal{H}}^{s-2}(X)$ weakly. Similar to the proof of Lemma \[lem:graphnorm\], consider the functional $$\ell({
\underline{\psi}
}) = \left< u, P^*v \right>_X - \left< f, v \right>_X, \quad {
\underline{\psi}
} \in C^\infty(\partial X)\times C^\infty(\partial X)$$ where $v$ is any member of $\mathcal{F}_\nu(X)$ such that ${
\underline{\gamma}
} v = {
\underline{\psi}
}$. Since $Pu = f$ weakly, $\ell$ is well defined (namely it does not depend on the choice of $v$). In particular, one may take $v = \mathcal{K}({
\underline{\psi}
})$, where $\mathcal{K}$ is a bounded right inverse mapping $C^\infty(\partial X) \times C^\infty(\partial X) \rightarrow \mathcal{F}_\nu(X)$. Thus $$\ell({
\underline{\psi}
}) \leq C_1 \| u \|_{{\mathcal{H}}^s_P(X)} (\| \mathcal{K}({
\underline{\psi}
}) \|_{{\mathcal{H}}^{2-s}(X)} + \| {
\underline{\psi}
} \|_{H^{2-s-{
\underline{\nu}
}}(\partial X)}) \leq C_2 \| u \|_{{\mathcal{H}}^s_P(X)} \| {
\underline{\psi}
}\|_{H^{2-s-{
\underline{\nu}
}}(\partial X)}$$ By Hahn–Banach and the Riesz theorem, there exists a unique ${
\underline{\phi}
} \in H^{s-{
\underline{\nu}
}}(\partial X)$ such that $$\left< u, P^*v \right>_X - \left< f, v \right>_X = \left< J{
\underline{\phi}
}, {
\underline{\gamma}
}v \right>_{\partial X}.$$ for each $v \in \mathcal{F}_\nu(X)$. Consider the pair $(u, {
\underline{\phi}
})$; a priori this is an element of ${\widetilde{\mathcal{H}}}^0(X)$. On the other hand, for each $v \in \mathcal{F}_\nu(X)$, $$\begin{aligned}
\left< P(u,{
\underline{\phi}
}),v \right>_X &= \left< u, P^*v \right>_X + \left< {
\underline{\phi}
}, J{
\underline{\gamma}
}v \right>_{\partial X} \\
&= \left<f ,v\right>_X,
\end{aligned}$$ so $P(u,{
\underline{\phi}
}) = f$. Since $f \in {\mathcal{H}}^{s-2}(X)$ and $\phi_- \in H^{s-1+\nu}(\partial X)$, Theorem \[theo:bvptheo\] implies that $(u,{
\underline{\psi}
}) \in {\widetilde{\mathcal{H}}}^s(X)$ since the the boundary value problem $\{P, \gamma_-\}$ is elliptic at $\partial X$. According to Lemma \[lem:graphnorm\], this means that the pair $(u,f)$ can be identified with an element of ${\mathcal{H}}^s_P(X)$.
Suppose that $P$ is elliptic at $\partial X$ and let $s = 0,1$. If $u \in {\mathcal{H}}^s(X)$ and $Pu \in {\mathcal{H}}^0(X)$ in distributions, then there is a canonical $f \in {\mathcal{H}}^{s-2}(X)$ such that $Pu = f$ weakly, namely the element $Pu \in {\mathcal{H}}^0(X)\hookrightarrow {\mathcal{H}}^{s-2}(X)$ itself. According to Lemma \[lem:graphnorm2\], to this choice of $f$ there is a uniquely associated $\phi \in {\mathcal{H}}^{s-{
\underline{\nu}
}}(\partial X)$ such that $P(u,{
\underline{\phi}
}) = Pu$ and the norm $\| (u,{
\underline{\phi}
}) \|_{{\widetilde{\mathcal{H}}}^s(X)}$ is equivalent to $\| u \|_{{\mathcal{H}}^s(X)} + \| Pu \|_{{\mathcal{H}}^{2-s}(X)}$. Adding $\| Pu \|_{{\mathcal{H}}^0(X)}$ to both of these norms shows that the spaces $$\{u \in {\mathcal{H}}^s(X): Pu \in {\mathcal{H}}^0(X) \} \text{ and } \{ (u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^s(X): P(u,{
\underline{\phi}
}) \in {\mathcal{H}}^0(X) \}$$ coincide, with an equivalence between the natural graph norms. This will be exploited in Section \[subsect:globalass\].
Parameter-elliptic boundary value problems {#subsect:semiclassicalvariable}
------------------------------------------
This section concerns elliptic estimates for parameter-dependent Bessel operators. The exposition is deliberately brief, since most of the definitions and facts in this section are straightforward adaptations from the non-parameter-dependent setting. In particular the main theorem of this section, Theorem \[theo:bvptheosemiclassical\], is stated without proof. The interested reader is referred to [@roitberg:1996 Chap. 9] for an indication of how the proofs should be modified in the parameter-dependent setting.
Fix a compact manifold with boundary ${
\overline{X}
}$ with the usual data of a boundary defining function and collar diffeomorphism. Let $P(\lambda) \in \mathrm{Bess}_\nu^{(\lambda)}(X)$ be a parameter-dependent Bessel operator; if $0 < \nu < 1$, then $P(\lambda)$ is augmented by boundary conditions as in Section \[subsect:theBVP\]. The boundary conditions themselves may depend on the spectral parameter $\lambda$, namely one considers $(T(\lambda),C(\lambda))$ where $$T_k(\lambda) = T^\pm(\lambda) \gamma_\pm, \quad T_-(\lambda) \in \mathrm{Diff}^1_{(\lambda)}(\partial X), \, T^+(\lambda) \in \mathrm{Diff}^0(\partial X).$$ and $C_{k,j} \in \mathrm{Diff}^*_{(\lambda)}(\partial X)$. It is necessary to formulate a parameter-dependent Lopatinskiǐ condition for $(T(\lambda),C(\lambda))$. Suppose that $\mu \in \{1-\nu,2-\nu,1+\nu\}$ and $\mathrm{ord}^{(\lambda)}_\nu(T(\lambda)) \leq \mu$. Here the order of $T$ with respect to $\nu$ is defined in the parameter-dependent sense, namely factors of $\lambda$ are given the same weight as a derivative tangent to $\partial X$. Define the family of operators $$\widehat{T}_{(y,\eta;\lambda)} = \sigma^{(\lambda)}_{\lceil \mu-1+\nu \rceil}(T^-(\lambda)) \gamma_- + \sigma^{(\lambda)}_{\lceil\mu-1-\nu\rceil}(T^+(\lambda))\gamma_+,$$ indexed by $(y,\eta,\lambda) \in T^* \partial X \times \mathbb{C}$. Thus each $(y,\eta,\lambda) \in T^*\partial X \times \mathbb{C}$ gives rise to a one-dimensional boundary operator $\widehat{T}_{(y,\eta;\lambda)}$.
Next, choose $c_{k,j} \in \mathbb{Z}$ such that $$\mathrm{ord}^{(\lambda)}(C_{k,j}(\lambda)) \leq c_{k,j} \leq \tau_j + \mu_k,$$ and then define the matrix $\widehat{C}_{(y,\eta)}$ with entries $$(\widehat{C}_{(y,\eta;\lambda)})_{k,j} = \sigma_{c_{k,j}}^{(\lambda)}(C_{k,j}(\lambda))(y,\eta;\lambda).$$ Again the order of $C_{k,j}(\lambda)$ is taken in the parameter-dependent sense.
Suppose that $P(\lambda)$ is parameter-elliptic on $\partial X$ with respect to an angular sector $\Lambda$. The boundary operators $(T(\lambda),C(\lambda))$ are said to satisfy the parameter-dependent Lopatinskiǐ condition with respect to $P$ and $\Lambda$ if for each $p \in \partial X$ and $(\eta,\lambda) \in T^*_p\partial X \times \Lambda \setminus 0$, the only element $(u,\underline{u}) \in \mathcal{M}_+(p,\eta,\lambda) \times \mathbb{C}^J$ satisfying $$\widehat{T}_{(p,\eta,\lambda)}u + \widehat{C}_{(p,\eta,\lambda)}\underline{u} = 0 \\$$ is the trivial solution $(u,\underline{u}) = 0$. The operator $\mathscr{P}(\lambda) = \{P(\lambda),T(\lambda),C(\lambda)\}$, is said to be parameter elliptic if $P(\lambda)$ is parameter-elliptic and $(T(\lambda),C(\lambda))$ satisfy the parameter-dependent Lopatinskiǐ condition on $\partial X$ with respect to $P(\lambda)$ and $\Lambda$.
If $P(\lambda)$ is parameter-elliptic in an angular sector $\Lambda$ and $T$ is a $\lambda$-independent boundary condition satisfying the Lopatinski[ǐ]{} condition, then $\{P(\lambda),T\}$ is parameter-elliptic with respect to $\Lambda$.
If $1/2 < \nu < 1$, then any boundary condition of the form $T = \gamma_+ + T^-(\lambda)\gamma_-$, where $T^-(\lambda) \in \mathrm{Diff}^1_{(\lambda)}(\partial X)$, satisfies the parameter-dependent Lopatinski[ǐ]{} condition with respect to any angular sector.
In the notation of Theorem \[theo:bvptheo\], the main theorem of this section is the following. As remarked in the introduction to this section, it is provided without proof.
\[theo:bvptheosemiclassical\] Let ${
\overline{X}
}$ be a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. Assume that $P(\lambda) \in \mathrm{Bess}_\nu^{(\lambda)}(X)$ is parameter-elliptic at $\partial X$ with respect to an angular sector $\Lambda$ in the sense of Section \[subsect:semiclassicalbessel\]. If $0 < \nu < 1$, then assume $P(\lambda)$ is augmented by parameter-dependent boundary conditions $(T(\lambda),C(\lambda))$ such that $\mathscr{P}(\lambda) = \{P(\lambda),T(\lambda),C(\lambda)\}$ is elliptic with respect to $\Lambda$. There exists $0 < \delta < \varepsilon$ such that if $\varphi, \chi \in C_c^{\infty}(\{ 0 \leq x < \delta\})$ satisfy $\varphi = 1$ near $\partial X$ and $\chi = 1$ near $\operatorname{supp}\varphi$, then the following hold.
1. Let $0 < \nu < 1$ and $s=0,1,2$. Then $$\begin{aligned}
\VERT \varphi(u, {
\underline{\phi}
}, {
\underline{u}
}) \VERT_{{\widetilde{\mathcal{H}}}^{s}(X) \times H^{s+{
\underline{\tau}
}}(\partial X)} &\leq C( \VERT \varphi \mathscr{P}(\lambda)(u,{
\underline{\phi}
},{
\underline{u}
}) \VERT_{{
{\mathcal{H}}^{s - 2}(X) \times H^{s-{
\underline{\mu}
}}(\partial X)
}} \notag \\ &+ \VERT \chi (u, {
\underline{\phi}
}, {
\underline{u}
}) \VERT_{{\widetilde{\mathcal{H}}}^{s-1}(X) \times H^{s-1+{
\underline{\tau}
}}(\partial X)} ),
\end{aligned}$$ for each $(u,{
\underline{\phi}
},{
\underline{u}
})\in {\widetilde{\mathcal{H}}}^{s}(X) \times {\mathcal{H}}^{s+{
\underline{\tau}
}}(\partial X)$ and $\lambda \in \Lambda$.
2. Let $\nu \geq 1$ and $s=0,1,2$. Then $$\VERT \varphi u \VERT_{{\mathcal{H}}^{s}(X)} \leq C ( \VERT \varphi P(\lambda)u \VERT_{{\mathcal{H}}^{s-2}(X)} + \VERT \chi u \VERT_{{\mathcal{H}}^{s-1}(X)}),$$ for each $u \in {\mathcal{H}}^s(X)$ and $\lambda \in \Lambda$.
Conormal regularity {#subsect:conormal}
-------------------
So far only regularity at the ${\mathcal{H}}^2$ level has been discussed. Higher order regularity is defined in terms of a scale of conormal Sobolev spaces relative to ${\mathcal{H}}^s$. Let ${
\overline{X}
}$ be a compact manifold with boundary with a fixed boundary defining function $x$ and collar neighborhood ${
\overline{W}
}$. Then let ${
\overline{X}
}_\mathrm{even}$ denote the manifold ${
\overline{X}
}$ equipped with a new smooth structure: on the collar ${
\overline{W}
} \simeq [0,\varepsilon)_x\times \partial X$, functions are smooth if in the normal direction they depend on $x^2$ (rather than just $x$).
Define the Lie algebra $\mathcal{V}_b({
\overline{X}
}_\mathrm{even})$ of smooth vector fields on ${
\overline{X}
}_\mathrm{even}$ which are tangent to $\partial X$. In local coordinates $x,y_1,\ldots,y_{n-1}$ on the collar, elements of $\mathcal{V}_b({
\overline{X}
}_\mathrm{even})$ are $C^\infty({
\overline{X}
}_\mathrm{even})$ linear combinations of $x\partial_x$ and $\partial_{y_i}$.
\[lem:conormalcommutator\] If $P \in \mathrm{Bess}_\nu(X)$ and $V \in \mathcal{V}_b({
\overline{X}
}_\mathrm{even})$ satisfies $Vx = x + \mathcal{O}(x^3)$, then $[P,V] \in \mathrm{Bess}_\nu(X)$.
The hypothesis implies that in local coordinates, $$V (x,y)= a(x^2,y)x\partial_x + b^i(x^2,y) \partial_{y_i}$$ where $a(0,y) = 1$. Note that $$[ |{{\partial}_\nu}|^2, x\partial_x ] = -2 |{{\partial}_\nu}|^2, \quad [{{\partial}_\nu},x\partial_x ] = -{{\partial}_\nu}.$$ Also from , if $a \in C^\infty({
\overline{X}
}_\mathrm{even})$, then $$[|{{\partial}_\nu}|^2,a] = \hat{a}x\partial_\nu + \tilde{a}$$ for $\hat{a},\tilde{a} \in C^\infty({
\overline{X}
}_\mathrm{even})$, as well as $[x\partial_\nu,a] \in C^\infty({
\overline{X}
}_\mathrm{even})$. The result follows immediately from these observations.
Given $k \in \mathbb{N}$ and $s = 0,1,2$, the space ${\mathcal{H}}^{s,k}(X)$ is defined as $${\mathcal{H}}^{s,k}(X) = \{ u \in {\mathcal{H}}^s(X): V_1 \cdots V_k u \in {\mathcal{H}}^s(X) \text{ for any } V_1,\ldots,V_k \in \mathcal{V}_b({
\overline{X}
}_\mathrm{even}) \}.$$ Fixing a finite generating set $\mathscr{V}$ for $\mathcal{V}_b({
\overline{X}
}_\mathrm{even})$, this space can be given the topology of a Hilbert space by inductively defining the norms $$\| u \|^2_{{\mathcal{H}}^{s,k}(X)} = \sum_{V\in \mathscr{V}} \| Vu \|^2_{{\mathcal{H}}^{s,k-1}(X)}.$$ A different choice of generating set yields an equivalent norm. Note that over any compact $K\subseteq X$, there is an equivalence between functions in ${\mathcal{H}}^{s,k}(X)$ and $H^{s+k}(X)$ which are supported on $K$. In addition, all of the density results which hold for ${\mathcal{H}}^{s}(X)$ also hold for ${\mathcal{H}}^{s,k}(X)$.
If $s=0,1$, then in fact $${\mathcal{H}}^{s,k}(X) = \{ u \in {\mathcal{H}}^s(X): V_1 \cdots V_k u \in {\mathcal{H}}^s(X) \text{ for any } V_1,\ldots,V_k \in \mathcal{V}_b({
\overline{X}
}) \}.$$ Thus only ${\mathcal{H}}^{2,k}(X)$ necessitates the introduction of a new smooth structure on ${
\overline{X}
}$.
Let $P \in \mathrm{Bess}_\nu(X)$ and $k \in \mathbb{N}$.
1. If $\nu >0$, then $P: {\mathcal{H}}^{2,k}(X)\rightarrow {\mathcal{H}}^{0,k}(X)$ is bounded.
2. If $0 < \nu < 1$ and $T$ is a boundary operator such that $\mathrm{ord}_\nu(T) \leq \mu$, then $T : {\mathcal{H}}^{2,k}(X) \rightarrow H^{k+2-\mu}(\partial X)$ is bounded.
$(1)$ Any $V \in \mathcal{V}_b({
\overline{X}
}_\mathrm{even})$ can be written as $V = a V_1$, where $a \in C^\infty({
\overline{X}
}_\mathrm{even})$ and $V_1 x = x+ \mathcal{O}(x^3)$. The result can now be deduced from Lemma \[lem:conormalcommutator\].
$(2)$ Given a vector field $Z$ on $\partial X$, there exists $V \in \mathcal{V}_b({
\overline{X}
}_\mathrm{even})$ such that $V|_{\partial X} = Z$. Then $Z(\gamma_\pm u) = \gamma_\pm(Vu)$ for each $u\in {\mathcal{H}}^{2,k}(X)$; this is certainly true on $\mathcal{F}_\nu(X)$ and extends by density.
Fix a generating set $\mathscr{V}= \{V_0,V_1,\ldots,V_N\}$ for $\mathcal{V}_b({
\overline{W}
}_\mathrm{even})$ as follows: set $V_0 = x\partial_x$, and then choose a collection of vector fields $V_1,\ldots,V_N$ on $\partial X$ which span $T\partial X$. Then $V_0,\ldots ,V_N$ may be considered as vector fields on $[0,\varepsilon)_x\times\partial X$, hence on ${
\overline{W}
}$. Note that the flow of $V_0$ is given by $\exp(hV_0)(x,y) = (e^h x, y)$, where $(x,y) \in [0, \varepsilon)_x\times \partial X$. Given $V \in \mathscr{V}$, let $$\varrho^h_V u = (u \circ \exp(hV) - u)/h$$ denote the associated difference quotient.
Suppose that $u \in {\mathcal{H}}^{2,k}(X)$ is supported in $\{ 0 \leq x < \delta\}$, where $0 <\delta < \varepsilon$. Observe that there exists $h_0 > 0$ depending on $\delta$ such that $\varrho^h_{V_0}u$ is well defined for $0 < h_0 <h$; the difference quotients corresponding to $V_1,\ldots,V_N$ are defined for all $h$. The first step is to calculate the commutator of $P$ with $\varrho_{V_0}^h$; this is illustrated for $[|{D_\nu}|^2,\varrho_{V_0}^h]$. First note that $$[\partial_x, \varrho_{V_0}^h]u = h^{-1}(e^h -1)(\partial_x u)\circ \exp(hV_0).$$ A short calculation gives $$[|{D_\nu}|^2, \varrho_{V_0}^h]u = h^{-1}(1-e^{2h})(|{D_\nu}|^2u)\circ\exp(hV_0),$$ which shows that $$\| [|{D_\nu}|^2,\varrho_{V_0}^h] u \|_{{\mathcal{H}}^{0,k}(X)} \leq C \| u \|_{{\mathcal{H}}^{2,k}(X)}$$ for $0 < h_0 < h$, where $C>0$ does not depend on $u$ or $h$. Continuing this calculation shows that $\| [P, \varrho_V^h]u \|_{{\mathcal{H}}^{0,k}(X)} \leq C \| u \|_{{\mathcal{H}}^{2,k}(X)}$ for any $V \in \mathscr{V}$. As for the boundary operators, one has $$\gamma_- (u\circ \exp(hV_0)) = \gamma_- u,\quad \gamma_+ (u\circ \exp(hV_0)) = e^{(1/2+\nu)s}\gamma_+ u,$$ so $\gamma_- \circ \varrho_{V_0}^h = 0$ and $\gamma_+ \circ \varrho_{V_0}^h = (e^{(1/2+\nu)s}-1)\gamma_+u$. Similarly, $$\| [T,\varrho_{V_i}^h] u \|_{H^{k+2-\mu}(\partial X)} \leq C \| u \|_{{\mathcal{H}}^{2,k}(X)}$$ for $i = 1,\ldots, N$, uniformly in $h$.
\[theo:conormal\] Let ${
\overline{X}
}$ be a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. Assume that $P \in \mathrm{Bess}_\nu(X)$ is elliptic at $\partial X$ in the sense of Section \[subsect:ellipticity\]. If $0 < \nu < 1$, then assume $P$ is augmented by a boundary condition $T$ such that $\mathscr{P} = \{P,T\}$ is elliptic at $\partial X$. There exists $0 < \delta < \varepsilon$ such that if $\varphi, \chi \in C_c^{\infty}(\{ 0 \leq x < \delta\})$ satisfy $\varphi = 1$ near $\partial X$ and $\chi = 1$ near $\operatorname{supp}\varphi$, then the following hold.
1. Let $0 < \nu < 1$. If $\chi u \in {\mathcal{H}}^{2}(X)$ and $\chi Pu \in {\mathcal{H}}^{0,k}(X), \, Tu \in H^{k+2-\mu}(\partial X)$ for some $k \in \mathbb{N}$, then $\varphi u \in {\mathcal{H}}^{2,k}(X)$. Furthermore, $$\| \varphi u \|_{{\mathcal{H}}^{2,k}(X)} \leq C\left(\| \chi \mathscr{P}u \|_{{\mathcal{H}}^{0,k}\times H^{k+2-\mu}(\partial X)} + \| \chi u \|_{{\mathcal{H}}^{0}(X)} \right),$$ where $C>0$ does not depend on $u$.
2. Let $\nu \geq 1$. If $\chi u \in {\mathcal{H}}^{2}(X)$ and $\chi Pu \in {\mathcal{H}}^{0,k}(X)$ for some $k \in \mathbb{N}$, then $\varphi u \in {\mathcal{H}}^{2,k}(X)$. Furthermore, $$\| \varphi u \|_{{\mathcal{H}}^{2,k}(X)} \leq C\left(\| \chi Pu \|_{{\mathcal{H}}^{0,k}} + \| \chi u \|_{{\mathcal{H}}^{0}(X)} \right),$$ where $C>0$ does not depend on $u$.
The proof is by induction; the case $k=0$ is Theorem \[theo:bvptheo\]. Suppose that the result holds for $k \in \mathbb{N}$; combined with the calculations preceeding the theorem, this gives that $\varrho_V^h \varphi u \in {\mathcal{H}}^{2,k}(X), V \in \mathscr{V}$ is well defined and uniformly bounded for $h$ sufficiently small. Standard functional analysis (extracting a weakly convergent subsequence, etc.) proves that $V \varphi u \in {\mathcal{H}}^{2,k}(X)$ for every $V \in \mathscr{V}$, with a corresponding estimate. This allows one to conclude the result for $k+1$.
Asymptotic expansions
---------------------
Using Mellin transform techniques, it is straightforward to give asymptotic expansions for solutions of certain Bessel equations. This section is a special case of far more general expansions; see [@mazzeo:1991 Section 7] for example. The approach taken here is essentially the same as [@vasy2010wave Lemma 4.13]. The space $\dot{C}^\infty(X)$ refers to smooth functions on $X$ which vanish to infinite order at $\partial X$.
\[[[@vasy2010wave Lemma 4.13]]{}\] \[lem:frobenius\] Suppose that $P \in \mathrm{Bess}_\nu(X)$ for $\nu > 0$, and $g_\pm \in C^\infty(\partial X)$. Then there exist $v_\pm$ such that $P(x^{1/2+\nu}v_+ + x^{1/2-\nu}v_-) \in \dot{C}^\infty(X)$ and $v_\pm|_{\partial X} = g_\pm$ with the following properties.
1. If $2\nu \notin \{3,5,7,\ldots\}$, then $v_\pm \in C^\infty({
\overline{X}
})$. In addition $v_\pm - g_\pm \in x^2 C^\infty({
\overline{X}
})$.
2. If $2\nu \in \{3,5,7,\ldots\}$, then $v_+ \in C^\infty({
\overline{X}
})$ and $$v_- \in C^\infty({
\overline{X}
}) + x^{2\nu}(\log x) C^\infty({
\overline{X}
}),$$ where $a_j \in C^\infty(\partial X)$.
Suppose that $P \in \mathrm{Bess}_\nu(X)$ is elliptic at $\partial X$. Write $P$ in the form $$P = |{D_\nu}|^2 - E$$ near $\partial X$, where $E \in \mathrm{Diff}^2_b({
\overline{X}
})$ and $\mathrm{Diff}^m_b({
\overline{X}
})$ are the operators of order at most $m$ generated by vector fields in $\mathcal{V}_b({
\overline{X}
})$. If $0 < \nu < 1$, then also fix a boundary condition $T$ such that $\{P,T\}$ is elliptic at $\partial X$. The equation $Pu =f $ can be expressed as $$\label{eq:bequation}
|{D_\nu}|^2u = x^2(Eu + f)$$ Formally, the Mellin transform of the left hand side of is $$(s+1/2-\nu)(s+1/2+\nu)Mu(s,\cdot), \quad s\in \mathbb{C}.$$ Now suppose that $u \in {\mathcal{H}}^0(X)$ and $f \in \dot{C}^\infty(X) $. Also suppose that $Tu \in C^\infty(\partial X)$ when $0 < \nu < 1$. If $u$ is supported sufficiently close to $\partial X$, then $u \in {\mathcal{H}}^{2,k}(X)$ by Theorem \[theo:conormal\] for any $k \in \mathbb{Z}$. In that case, the left hand side of is square integrable with respect to the measure $d({\operatorname{Im}}s)$ along the line $\{ {\operatorname{Re}}s = 1/2\}$. Furthermore, since $u \in {\mathcal{H}}^{2,k}(X) \subseteq {\mathcal{H}}^{0,k}(X)$ and $E \in \mathrm{Diff}^2_b({
\overline{X}
})$, the right hand side of is an element of $x^2 {\mathcal{H}}^0(X)$. Define ${\mathcal{H}}^{s,\infty}(X) = \cap_{k\geq 0} {\mathcal{H}}^{s,k}(X)$.
Suppose that $P$ and $\{P,T\}$ are elliptic at $\partial X$. If $u \in {\mathcal{H}}^0(X)$ and $$Pu \in \dot{C}^\infty(X),\quad Tu \in C^\infty(\partial X),$$ then the following hold.
1. Let $0 < \nu < 1$. Then there exist $u_\pm \in C^\infty({
\overline{X}
})$ such that $$u = x^{1/2+\nu}u_+ + x^{1/2-\nu}u_-.$$ In addition $u_\pm - g_\pm \in x^2 C^\infty({
\overline{X}
})$, where $g_- = \gamma_- u$ and $2\nu g_+ = \gamma_+ u$.
2. Let $\nu \geq 1$. If $2\nu \notin \{ 3,5,7,\ldots\}$, then there exists $u_\pm \in C^\infty({
\overline{X}
})$ such that $$u = x^{1/2+\nu}u_+ + x^{1/2-\nu}u_-,$$ where $u_- \in x^k C^\infty({
\overline{X}
})$ for some $k \in \mathbb{N}$ satisfying $k > \nu - 1$. If $2\nu \in \{3,5,7,\ldots\}$, then the same statement holds but with $u_- \in x^k C^\infty({
\overline{X}
}) + x^{2\nu}(\log x) C^\infty({
\overline{X}
})$.
For a more details, again see [@vasy2010wave Lemma 4.13]. By cutting off $u$ (which does not affect the condition $Pu \in \dot{C}^\infty(X)$), it may be assumed that $u$ is supported near $\partial X$, hence a function on $(0,\varepsilon) \times \partial X$. Let $$h = Eu + f \in {\mathcal{H}}^0((0,\varepsilon)\times\partial X).$$ Write $l_\nu(s) = (s+1/2+\nu)(s+1/2-\nu)$ and then take the Mellin transform to yield $$Mu(s,\cdot) = l_\nu(s)^{-1} M h(s+2,\cdot).$$ First suppose that $2\nu$ is not an integer, so the roots of $l_\nu(s)$ are simple. Since $M h(s+2,\cdot)$ is holomorphic for ${\operatorname{Re}}s > 1/2$, this provides a meromorphic extension of $Mu(s,\cdot)$ from $\{ {\operatorname{Re}}s > 1/2\}$ to $\{{\operatorname{Re}}s > -3/2\}$ with simple poles at the roots of $l_\nu(s)$ in the strip $\{ -3/2 < {\operatorname{Re}}s < 1/2\}$. Since $u \in {\mathcal{H}}^{0,\infty}(X)$ the residues are smooth functions on $\partial X$.
Now take the inverse Mellin transform by deforming the contour to any line $\{ {\operatorname{Re}}s = -3/2 + \varepsilon \}$. Note that $Mu(s,\cdot)$ has two poles in $\{-3/2 < {\operatorname{Re}}s < 1/2\}$ if $0 < \nu < 1$, and no poles in this region if $\nu \geq 1$. In the former case, $$u = x^{1/2+\nu}g_+ + x^{1/2-\nu}g_- + u_1$$ for $g_\pm \in C^\infty(\partial X)$ and $u_1 \in x^2{\mathcal{H}}^{0,\infty}(X) \cap {\mathcal{H}}^{2,\infty}(X)$, while in the latter case $u \in x^2 {\mathcal{H}}^{0,\infty}(X) \cap {\mathcal{H}}^{2,\infty}(X)$. In the first case, choose $v_{\pm}$ as in Lemma \[lem:frobenius\], where $v_\pm = g_\pm + x^2C^\infty({
\overline{X}
})$. Thus $$u - x^{1/2-\nu}v_+ - x^{1/2+\nu}v_- \in x^2 {\mathcal{H}}^{0,\infty}(X) \cap {\mathcal{H}}^{2,\infty}(X)$$ and $P(u - x^{1/2+\nu}v_+ - x^{1/2-\nu}v_-) \in \dot{C}^\infty(X)$. Applying the same argument gives the next terms in the expansion, which come from the poles at $s = -5/2 \pm \nu$. This may be continued indefinitely, but note that after this second step there may appear powers of the form $x^{r+1/2\pm\nu}$ (for $r > 2)$ rather than just $x^{2r+1/2\pm\nu}$ (unless additional evenness assumptions are made on $E$). A similar argument applies in the second case, where $Mu(s,\cdot)$ is first continued further to left until an indicial root is crossed. When $2\nu$ is an integer, one picks up a logarithmic factor when taking the inverse Mellin transform, corresponding to a pole of multiplicity two.
The Fredholm alternative and unique solvability {#sect:fredholm}
===============================================
Global assumptions {#subsect:globalass}
------------------
Let ${
\overline{X}
}$ denote a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. Consider a pseudodifferential operator $P \in \Psi^2(X)$ of the form $P = P_1 + P_2$, where $$P_1 \in \mathrm{Bess}_\nu(X), \quad P_2 \in \Psi_{\mathrm{comp}}^2(X)$$ Assume that $P_1$ is elliptic at $\partial X$ in the sense of Section \[subsect:ellipticity\]. Furthermore, if $0 < \nu < 1$, fix a scalar boundary condition $T$ with $\mathrm{ord}_\nu(T) \leq \mu$; this is just for simplicity, whereas matrix boundary conditions arise in the adjoint problem. Assume that $\mathscr{P} = \{P,T\}$ is elliptic at $\partial X$ as well. Since $P_2$ has a compactly supported Schwartz kernel, this is just a statement about the operator $\{ P_1, T\}$.
Without any assumptions on the behavior of $P$ away from $\partial X$, there is no reason to expect that $P$ or $\mathscr{P}$ are Fredholm. This section outlines some additional global assumptions which guarantee a Fredholm problem. The simplest of these assumptions is that $P$ is everywhere elliptic (in the standard sense) on $X$, but in view of applications to general relativity, this is overly restrictive. Indeed, operators which arise in the study of quasinormal modes on black holes spacetimes have the property that their ellipticity degenerates at the event horizon. Moreover, rotating Kerr–AdS black holes contain an ergoregion, so that the corresponding operator is not everywhere elliptic even in the black hole exterior.
The global assumptions on $P$ presented next are motivated by recent work of Vasy [@vasy:2013], which applies to the setting of rotating black holes. More generally, these assumptions are typical for situations where coercive estimates are proved via propagation results.
Given $\nu > 0$, define the space $$\mathcal{Y} =
\begin{cases}
\{ u \in {\mathcal{H}}^1(X): Pu \in {\mathcal{H}}^0(X), \, Tu \in H^{2-\mu}(\partial X) \}& \text{ if $0< \nu < 1$},\\
\{ u \in {\mathcal{H}}^1(X): Pu \in {\mathcal{H}}^0(X) \} & \text{ if $\nu \geq 1$}
\end{cases}$$ where $Pu$ is taken as a distribution on $X$. That $Tu$ is well defined follows from Lemma \[lem:graphnorm2\]. Equip $\mathcal{Y}$ with the norm $$\| u \|_{\mathcal{Y}} =
\begin{cases}
\| u \|_{{\mathcal{H}}^1(X)} + \| Pu \|_{{\mathcal{H}}^0(X)} + \| Tu \|_{H^{2-\mu}(\partial X)} & \text{ if $0 < \nu < 1$} \\
\| u \|_{{\mathcal{H}}^1(X)} + \| Pu \|_{{\mathcal{H}}^0(X)} & \text{ if $\nu \geq 1$}.
\end{cases}$$ According to the discussion following Lemma \[lem:graphnorm2\], the space $\mathcal{Y}$ is equivalent to $$\label{eq:alternativespace}
\{(u,{
\underline{\phi}
})\in{\widetilde{\mathcal{H}}}^1(X): \mathscr{P}(u,{
\underline{\phi}
}) \in {\mathcal{H}}^0(X) \times H^{2-\mu}(\partial X)\}$$ for $0 < \nu <1$ when the latter space is equipped with the norm $\| (u,{
\underline{\phi}
}) \|_{{\widetilde{\mathcal{H}}}^1(X)} + \|\mathscr{P}(u,{
\underline{\phi}
}) \|_{{\mathcal{H}}^0(X)\times H^{2-\mu}(\partial X)}$.
\[lem:Yk\] The space $\mathcal{Y}$ has the following properties.
1. $\mathcal{Y}$ is complete.
2. $\mathcal{F}_\nu(X)$ is dense in $\mathcal{Y}$
3. If $0 < \nu < 1$, then $\mathscr{P}:\mathcal{Y} \rightarrow {\mathcal{H}}^0(X) \times H^{2-\mu}(\partial X)$ is bounded.
4. If $\nu \geq 1$, then $P : \mathcal{Y} \rightarrow {\mathcal{H}}^{0}(X)$ is bounded.
5. If $\zeta \in C_c^\infty(X)$ and $K = \operatorname{supp}\zeta$, then for each $m < 1$ the map $\mathcal{Y} \rightarrow H^m_K(X)$ given by $u \mapsto \zeta u$ is compact.
$(1)$ For $0 < \nu < 1$, use the alternative description of $\mathcal{Y}$: suppose that $(u_n,{
\underline{\phi_n}
}) \in \mathcal{Y}$ is a Cauchy sequence. This implies that there exists $$(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^{1}(X), \quad (w,{
\underline{w}
}) \in {\mathcal{H}}^{0}(X) \times H^{2-\mu}(\partial X)$$ such that $$(u_n,{
\underline{\phi_n}
}) \rightarrow (u,{
\underline{\phi}
}) \text{ in } {\widetilde{\mathcal{H}}}^{1}(X), \quad \mathscr{P}(u_n,{
\underline{u_n}
}) \rightarrow (w,{
\underline{w}
}) \text{ in } {\mathcal{H}}^{0}(X) \times H^{2-\mu}(\partial X).$$ Certainly $(u_n,{
\underline{\phi_n}
}) \rightarrow (u,{
\underline{\phi}
})$ in ${\widetilde{\mathcal{H}}}^1(X)$, and then by continuity $\mathscr{P}(u_n,{
\underline{\phi_n}
}) \rightarrow \mathscr{P}(u,{
\underline{\phi}
})$ in ${\mathcal{H}}^{-1}(X) \times H^{1-\mu}(\partial X)$. This implies that $\mathscr{P}(u,{
\underline{\phi}
}) = (w,{
\underline{w}
})$ since the natural map $${\mathcal{H}}^{0}(X) \times H^{2-\mu}(\partial X) \hookrightarrow {\mathcal{H}}^{-1}(X) \times H^{1-\mu}(\partial X),$$ is injective. Thus $\mathcal{Y}$ is complete. A simpler proof works when $\nu \geq 1$.
$(2)$ Again assume that $0 < \nu < 1$. Fix a cutoff $\chi$ such that $\chi = 1$ in a neighborhood of $\partial X$. If $\chi$ is supported sufficiently close to $\partial X$, then $\chi u\in {\mathcal{H}}^2(X)$ by Theorem \[theo:bvptheo\]. Thus there is certainly a sequence $u_n \in \mathcal{F}_\nu(X)$ such that $u_n \rightarrow \chi u$ in ${\mathcal{H}}^1(X)$ and $Pu_n \rightarrow P\chi u$ in ${\mathcal{H}}^0(X)$, along with $Tu_n \rightarrow Tu$. If $\varphi = 1$ near $\partial X$ and $\chi =1$ near $\operatorname{supp}\varphi$, then also $\varphi u_n \rightarrow \varphi u$ and $\varphi Pu_n \rightarrow \varphi Pu$. This also implies that $P(\varphi u_n) \rightarrow P(\varphi u)$ since $$[P,\varphi]u_n \rightarrow [P,\varphi]u = [P,\varphi] u$$ in ${\mathcal{H}}^0(X)$ by continuity. On the other hand, the same reasoning above combined with the mollification argument in [@vasy:2013 Section 2.6] shows the existence of a sequence $v_n \in C^\infty(X)$ such that $(1-\varphi)v_n \rightarrow (1-\varphi)u$ in ${\mathcal{H}}^1(X)$ and $P((1-\varphi)v_n) \rightarrow P((1-\varphi)u)$ in ${\mathcal{H}}^0(X)$. It then suffices to take the sequence $\varphi u_n + (1-\varphi)v_n \in \mathcal{F}_\nu(X)$ which converges to $u$ in $\mathcal{Y}$.
$(3)$ The boundedness of $u \mapsto Pu$ as a map $\mathcal{Y} \rightarrow {\mathcal{H}}^0(X)$ holds by construction of $\mathcal{Y}$. As in the previous part, $\chi u \in {\mathcal{H}}^2(X)$ so $T: \mathcal{Y} \rightarrow H^{2-\mu}(\partial X)$ is also bounded. This establishes the boundedness of $\mathscr{P} = \{P,T\}$.
$(4)$ See $(3)$ above.
$(5)$ The map $u \rightarrow \zeta u$ is bounded ${\mathcal{H}}^{1}(X) \rightarrow H_K^{1}(X)$, which embeds compactly in $H_K^m(X)$.
Typically, one constructs a partition of unity of the form $$1 = \varphi + \sum_{i = 1}^N A_i + R,$$ where $A_i \in \Psi^0_{\mathrm{comp}}(X),\,R \in \Psi^{-\infty}_\mathrm{comp}(X)$ are pseudodifferential operators with compactly supported Schwartz kernels, and $\varphi \in C^\infty({
\overline{X}
})$ satisfies $\varphi = 1$ near $\partial X$. Under various hypotheses on $P$ (now considered as an element of $\Psi^2(X)$), it is often the case that there exists $B_i,X_i \in \Psi_\mathrm{comp}^0(X)$ such that $$\label{eq:vasyestimate}
\| A_i u \|_{H^1(X)} \leq C \| B_i Pu \|_{H^0(X)} + \| X_i u \|_{H^m(X)}, \quad m < 1$$ for each $u \in C^\infty(X)$. Since the operators $B_i,\,X_i$ and $R$ have compactly supported Schwartz kernels, it is possible to combine with the results of Theorem \[theo:bvptheo\] to conclude the following type of a priori estimate: if $0 < \nu < 1$, then $$\label{eq:AP0} \tag{AP0}
\| u \|_{{\mathcal{H}}^1(X)} \leq C \left( \| \mathscr{P}u \|_{{\mathcal{H}}^0(X)\times H^{2-\mu}(X)} + \| u \|_{{\mathcal{H}}^0(X)} + \| \chi u \|_{H^m(X)} \right)$$ for each $u \in \mathcal{F}_\nu(X)$, while if $\nu \geq 1$ then $$\label{eq:AP1} \tag{AP1}
\| u \|_{{\mathcal{H}}^1(X)} \leq C \left( \| Pu \|_{{\mathcal{H}}^0(X)} + \| u \|_{{\mathcal{H}}^0(X)} + \| \chi u \|_{H^m(X)}\right)$$ for each $u \in \mathcal{F}_\nu(X)$. Since $\mathcal{F}_\nu(X)$ is dense in $\mathcal{Y}$, the estimate implies $$\| u \|_{\mathcal{Y}} \leq C \left( \| \mathscr{P}u \|_{{\mathcal{H}}^0(X)\times H^{2-\mu}(X)} + \| u \|_{{\mathcal{H}}^0(X)} + \| \chi u \|_{H^m(X)} \right)$$ for each $u \in \mathcal{Y}$, and similarly for . It is standard that , imply $\mathscr{P}:\mathcal{Y} \rightarrow {\mathcal{H}}^0(X) \times H^{2-\mu}(\partial X)$ and $P : \mathcal{Y} \rightarrow {\mathcal{H}}^0(X)$ have finite dimensional kernels provided $m < 1$ — see Lemma \[lem:fdkernel\].
Suppose that $0 < \nu <1$. In order to prove that $\mathscr{P}$ has finite dimensional cokernel, it is necessary to introduce spaces associated with the formal adjoint $\mathscr{P}^*$ and Hilbert space adjoint $\mathscr{P}'$. Fix a density $\mu$ on ${
\overline{X}
}$ of product type near $\partial X$. A priori, $\mathscr{P}^*$ is bounded $${\widetilde{\mathcal{H}}}^{0}(X) \times H^{\mu-2}(\partial X) \rightarrow {\mathcal{H}}^{-2}(X) \times H^{{
\underline{\nu}
}-2}(\partial X).$$ Recall that if $(f,{
\underline{g}
}) = \mathscr{P}^*(v,{
\underline{\psi}
},{
\underline{v}
})$, then $$\label{eq:adjointvsformal1}
\left< u, f \right>_{X} + \left< {
\underline{w}
}, {
\underline{g}
} \right>_{\partial X} = \left< Pu, v \right>_X + \left< {
\underline{w}
} - {
\underline{\gamma}
}u, J{
\underline{\psi}
} \right>_{\partial X} + \left< G{
\underline{w}
}, {
\underline{v}
} \right>_{\partial X},$$ where the dualities on $X$ and $\partial X$ are induced by $\mu$ and $\mu_{\partial X}$. Now define the space $$\widetilde{\mathcal{X}} = \{ (v,{
\underline{\psi}
},{
\underline{v}
}) \in {\widetilde{\mathcal{H}}}^{0}(X) \times H^{\mu -2}(\partial X): \mathscr{P}^*(v,{
\underline{\psi}
},{
\underline{v}
}) \in {\mathcal{H}}^{-1}(X) \times H^{{
\underline{\nu}
}-1}(\partial X) \}.$$ The corresponding space for $\nu \geq 1$ is defined to be $$\mathcal{Z} = \{ u \in {\mathcal{H}}^0(X): Pu \in {\mathcal{H}}^{-1}(X) \}.$$ The spaces $\widetilde{\mathcal{X}}$ and $\mathcal{Z}$ have properties similar to those in Lemma \[lem:Yk\]. In particular, the set of all $(v,{
\underline{\gamma}
}v,{
\underline{v}
})$ such that $v \in \mathcal{F}_\nu(X)$ and ${
\underline{v}
} \in C^\infty(\partial X)$ is dense in $\widetilde{\mathcal{X}}$. Similarly, $\mathcal{F}_\nu(X)$ is dense in $\mathcal{Z}$ for $\nu \geq 1$.
The analogue of , is formulated next for the adjoint problems. First suppose that $0 < \nu < 1$. The a priori estimate is $$\begin{aligned}
\| (v,{
\underline{\gamma}
}v,{
\underline{v}
}) \|_{{\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)} &\leq C ( \| \mathscr{P}^*(v,{
\underline{v}
}) \|_{{\mathcal{H}}^{-1}(X) \times H^{{
\underline{\nu}
}-1}(\partial X)} \notag \\ &+ \| (v, {
\underline{v}
}) \|_{{\widetilde{\mathcal{H}}}^{-1}(X) \times H^{\mu - 3}(\partial X)} + \| \chi v \|_{H^{m}(X)} ) \tag{AP0*} \label{eq:AP0*}\end{aligned}$$ for each $(v,{
\underline{v}
}) \in \mathcal{F}_\nu(X) \times C^\infty(\partial X)$. By density this implies the same estimate for $(v,{
\underline{\psi}
},{
\underline{v}
}) \in \widetilde{\mathcal{X}}$. When $\nu \geq 1$ the estimate is $$\tag{AP1*} \label{eq:AP1*}
\| v \|_{{\mathcal{H}}^0(X)} \leq C( \| P^*v \|_{{\mathcal{H}}^{-1}(X)} + \| v \|_{{\mathcal{H}}^{-1}(X)} + \| \chi v \|_{H^m(X)}$$ for each $v \in \mathcal{F}_\nu(X)$. For , to be useful, one should require $m < 0$.
As with the direct problem, it is frequently possible to combine the local estimates of Theorem \[theo:bvptheo\] with interior estimates via a pseudodifferential partition of unity to show that the adjoint estimates , hold.
When $0 < \nu <1$, the formally adjoint operator $\mathscr{P}^*$ should be compared with the Hilbert space adjoint $$\mathscr{P}' : {\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X) \rightarrow {\widetilde{\mathcal{H}}}^2(X)',$$ defined by $$\left< (u,{
\underline{\phi}
}), \mathscr{P}'(v,{
\underline{v}
}) \right>_{X} = \left< Pu, v \right>_X + \left< Tu, {
\underline{v}
} \right>_{\partial X}.$$ Recall that the inclusion of ${\widetilde{\mathcal{H}}}^2(X) \hookrightarrow {\widetilde{\mathcal{H}}}^1(X)$ is dense. Consequently ${\widetilde{\mathcal{H}}}^{1}(X)'$ may be identified with a dense subspace of $ {\widetilde{\mathcal{H}}}^{2}(X)'$, where this identification is induced by the $\mu$–inner product. In order to describe ${\widetilde{\mathcal{H}}}^1(X)'$, note that that there is an isomorphism $$\Phi: {\widetilde{\mathcal{H}}}^1(X) \rightarrow {\mathcal{H}}^1(X) \times H^{-\nu}(\partial X)$$ given by $\Phi(u,{
\underline{\phi}
}) = (u,\phi_+)$; the inverse of $\Phi$ is $\Phi^{-1}(u,\phi_+) = (u,\gamma_-u,\phi_+)$. Thus for each $\alpha \in {\widetilde{\mathcal{H}}}^1(X)'$ there exist unique $f \in {\mathcal{H}}^{-1}(X), \, g_+ \in H^{\nu}(\partial X)$ such that $$\alpha(u,{
\underline{\phi}
}) = \left<f, u \right>_X + \left< g_+, \phi_+ \right>_{\partial X}.$$ Furthermore, note that if $g_- \in H^{-\nu}(\partial X)$, then the functional given by $u \mapsto \left< g_-, \gamma_- u \right>_{\partial X}$ is an element of ${\mathcal{H}}^1(X)'$. Thus it may be represented in the form $u \mapsto \left<f_-, u \right>_X$ for a unique $f_- \in {\mathcal{H}}^{-1}(X)$. The next lemma summarizes this discussion.
\[lem:alpharep\] Each $\alpha \in {\widetilde{\mathcal{H}}}^{1}(X)'$ admits a representation $$\label{eq:alpharep}
\alpha(u,{
\underline{\phi}
}) = \left< f,u \right>_X + \left< {
\underline{g}
}, {
\underline{\phi}
} \right>_{\partial X},$$ where $f \in {\mathcal{H}}^{-1}(X)$ and ${
\underline{g}
} \in H^{{
\underline{\nu}
}-1}(\partial X)$. Furthermore, $\| \alpha \|_{{\widetilde{\mathcal{H}}}^{1}(X)'}$ is equivalent to the norm $$\inf \{ \| f \|_{{\mathcal{H}}^{-1}(X)} + \|{
\underline{g}
} \|_{H^{{
\underline{\nu}
}-1}(\partial X)} \},$$ where the infimum is taken over all $f,{
\underline{g}
}$ such that holds.
Now define $\widetilde{\mathcal{Z}} = \{ (v,{
\underline{v}
}) \in {\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X): \mathscr{P}'(v,{
\underline{v}
}) \in {\widetilde{\mathcal{H}}}^1(X)' \}$.
\[lem:propertyBadjoint\] Suppose that holds. Then $$\begin{aligned}
\| (v,{
\underline{v}
}) \|_{{\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)} &\leq C ( \| \mathscr{P}'(v,{
\underline{v}
}) \|_{{\widetilde{\mathcal{H}}}^1(X)'} \notag \\ &+ \| (v,{
\underline{v}
}) \|_{{\mathcal{H}}^{-1}(X) \times H^{\mu - 3}(\partial X)} + \| \chi v \|_{H^{m}(X)} ) \tag{AP0'} \label{eq:fdcokernelapriori2}\end{aligned}$$ for each $(v,{
\underline{v}
}) \in \widetilde{\mathcal{Z}}$.
Since $\mathscr{P}'(v,{
\underline{v}
}) \in {\widetilde{\mathcal{H}}}^1(X)'$, there exists $f \in {\mathcal{H}}^{-1}(X)$ and ${
\underline{g}
} \in H^{{
\underline{\nu}
}-1}(\partial X)$ such that the action of $\mathscr{P}'(v,{
\underline{v}
})$ on $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^1(X)$ is given by $$\label{eq:alpharep2}
(u, {
\underline{\phi}
}) \mapsto \left< f , u \right>_X + \left< g, {
\underline{\phi}
} \right>_{\partial X}.$$ Now let ${
\underline{\psi}
} = JG^*{
\underline{v}
} - J{
\underline{g}
}$, so that $J{
\underline{\psi}
} + G^* {
\underline{v}
} = {
\underline{g}
}$. Furthermore, note that ${
\underline{\psi}
} \in H^{-{
\underline{\nu}
}}(\partial X)$, so $(v,{
\underline{\psi}
})$ may be considered as an element of ${\widetilde{\mathcal{H}}}^0(X)$. Referring back to , it follows that $\mathscr{P}^*(u,{
\underline{\psi}
},{
\underline{v}
}) = (f,g)$. This shows that $(u,{
\underline{\psi}
},{
\underline{v}
}) \in \widetilde{\mathcal{X}}$, so $$\begin{aligned}
\| (v,{
\underline{v}
}) \|_{{\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)} &\leq C ( \| f \|_{{\mathcal{H}}^{-1}(X)} + \| g \|_{H^{{
\underline{\nu}
}-1}(\partial X)} \\ &+ \| (v, {
\underline{v}
}) \|_{{\mathcal{H}}^{-1}(X) \times H^{\mu - 3}(\partial X)} + \| \chi v \|_{H^{m}(X)} )\end{aligned}$$ by . In the last line, this used the fact that $$\| {
\underline{\psi}
} \|_{ H^{-1- {
\underline{\nu}
}}(\partial X) } \leq C (\| {
\underline{v}
} \|_{H^{\mu - 3}} + \| g \|_{H^{{
\underline{\nu}
}-1}(\partial X)} ).$$ It now suffices to take the infimum over all $f,{
\underline{g}
}$ satisfying , and then appeal to Lemma \[lem:alpharep\].
The Fredholm property {#subsect:thefredholmproperty}
---------------------
In this section, the Fredholm property is established whenever , , , hold. A complete proof is given for the more complicated case $0 < \nu < 1$.
\[lem:fdkernel\] Let $0 < \nu < 1$.
1. If holds with $m < 1$, then the operator $$\mathscr{P} : \mathcal{Y} \rightarrow {\mathcal{H}}^{0}(X) \times H^{2-\mu}(\partial X)$$ has a finite dimensional kernel.
2. If holds with $m < 0$, then the operator $$\mathscr{P}': {\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X) \rightarrow {\mathcal{H}}^{-2}(X) \times H^{{
\underline{\nu}
} - 2}(\partial X)$$ has a finite dimensional kernel
$(1)$ This is immediate from the compactness of the inclusion $\mathcal{Y} \hookrightarrow {\mathcal{H}}^0(X)$ and the multiplication operator $\chi : \mathcal{Y} \rightarrow H^m_{\operatorname{supp}\chi}(X)$, combined with .
$(2)$ Clearly the kernel of $\mathscr{P}'$ restricted to ${\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X)$ is equal to the kernel of $\mathscr{P}'$ restricted to $\widetilde{\mathcal{Z}}$. The result follows from the same type of compactness considerations as in $(1)$, using .
In light of Lemma \[lem:fdkernel\], let $\mathcal{K}$ denote the finite dimensional kernel of $\mathscr{P}'|_{\widetilde{\mathcal{Z}}}$.
\[lem:fdcokerneladjoint\] Let $0 < \nu < 1$, and assume that holds. Suppose that $$(h, {
\underline{k}
}) \in {\mathcal{H}}^{0}(X) \times H^{2 -\mu}(\partial X)$$ lies in the annihilator of $\mathcal{K}$ via the duality between ${\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X)$ and ${\mathcal{H}}^{0}(X) \times H^{2 - \mu}(\partial X)$. Then there exists $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^1(X)$ such that $\mathscr{P}(u,{
\underline{\phi}
}) = (h,{
\underline{k}
})$.
This fact is more or less standard, but a complete proof is included for the readers convenience.
$(1)$ Fix a (closed) subspace $V$ of ${\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X)$ which is complementary to the finite-dimensional space $\mathcal{K} \subset \widetilde{\mathcal{Z}}$. Then there exists $C' > 0$ such that $$\label{eq:adjointglobalestimate2}
\| (v,{
\underline{v}
}) \|_{{\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)} \leq C' \| \mathscr{P}'(v,{
\underline{v}
}) \|_{{\widetilde{\mathcal{H}}}^{1}(X)'}$$ for each $(v,{
\underline{v}
}) \in V \cap \widetilde{\mathcal{Z}}$. If this were not true, there would exist a sequence $(v_n,{
\underline{v_n}
}) \in V \cap \widetilde{\mathcal{Z}}$ such that $$\| (v_n,{
\underline{v_n}
}) \|_{{\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)} = 1, \quad \| \mathscr{P}' (v_n,{
\underline{v_n}
}) \|_{{\widetilde{\mathcal{H}}}^{1}(X)'} \rightarrow 0.$$ By weak compactness of the ${\mathcal{H}}^{0}(X) \times H^{\mu-2}(\partial X)$-unit ball, it may be assumed that $(v_n,{
\underline{v_n}
})$ is weakly convergent. Since $V$ is closed, it follows that $(v_n,{
\underline{v_n}
}) \rightarrow (v,{
\underline{v}
})$ weakly for some $(v,{
\underline{v}
}) \in V$. Thus $$\mathscr{P}'(v_n,{
\underline{v_n}
}) \rightarrow \mathscr{P}'(v,{
\underline{v}
})$$ weakly in ${\widetilde{\mathcal{H}}}^{2}(X)'$. This means $\mathscr{P}'(v,{
\underline{v}
})= 0$ since $\mathscr{P}'(v_n,{
\underline{v_n}
}) \rightarrow 0$ (in norm) in ${\mathcal{H}}^{1}(X)'$. Thus $(v,{
\underline{v}
}) \in V \cap \mathcal{K}$, which implies $(v,{
\underline{v}
}) = 0$, since $V$ complements the kernel.
Now by compactness there exists a subsequence $(v_{n_j},{
\underline{v_{n_j}}
})$, such that $\chi v_{n_j}$ is convergent in $H^{m}_{\operatorname{supp}\chi}(X)$ and $(v_{n_j},{
\underline{v_{n_j}}
})$ is convergent in ${\mathcal{H}}^{-1}(X) \times H^{\mu -3}(\partial X)$. Then implies that $(v_{n_j},{
\underline{v_{n_j}}
})$ is Cauchy, hence convergent in ${\mathcal{H}}^{0}(X) \times H^{\mu -2}(\partial X)$. This limit must be $(v,{
\underline{v}
})$, but that contradicts $(v,{
\underline{v}
}) = 0$ since $(v_n,{
\underline{v_n}
})$ has unit norm in ${\mathcal{H}}^0(X) \times H^{\mu-2}(\partial X)$. This completes the proof of \[eq:adjointglobalestimate2\].
$(2)$ Now suppose that $(h, k) \in {\mathcal{H}}^{0}(X) \times H^{2 -\mu}(\partial X)$ is in the annihilator of $\mathcal{K}$. Define the antilinear functional $\ell$ on the range of $\mathscr{P}'|_{\widetilde{\mathcal{Z}}}$ by the formula $$\mathscr{P}'(v,{
\underline{v}
}) \mapsto \langle h, v \rangle_X + \langle k, {
\underline{v}
} \rangle_{\partial X},$$ where $(v,{
\underline{v}
}) \in \widetilde{\mathcal{Z}}$. This is well defined, since if $\mathscr{P}'(v,{
\underline{v}
}) = 0$ and $(v,{
\underline{v}
}) \in \widetilde{\mathcal{Z}}$, then $(v,{
\underline{v}
}) \in \mathcal{K}$, hence the right hand side vanishes.
For each $(v,{
\underline{v}
}) \in V \cap \widetilde{\mathcal{Z}}$, one has by $$\langle h, v \rangle_X + \langle k,{
\underline{v}
} \rangle_{\partial X} \leq C\left ( \| (h,{
\underline{k}
}) \|_{{\mathcal{H}}^{0}(X)\times H^{2-\mu}(\partial X)} \right) \left(\| \mathscr{P}^* (v,{
\underline{v}
}) \|_{{\widetilde{\mathcal{H}}}^1(X)'}\right).$$ Since this is invariant under adding elements of $\mathcal{K}$, it is in fact true for $(v,{
\underline{v}
}) \in \widetilde{\mathcal{Z}}$. Thus $\ell$ is bounded on the range of $\mathscr{P}'|_{\widetilde{\mathcal{Z}}}$.
Now extend $\ell$ to an antilinear functional on ${\widetilde{\mathcal{H}}}^{1}(X)'$ by the Hahn–Banach theorem. Then there exists a unique $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^{1}(X)$ such that $\ell(\alpha) = \alpha(u,{
\underline{\phi}
})$, and furthermore $$\ell(\mathscr{P}'(v,{
\underline{v}
})) = \left< h, v \right>_X + \left< {
\underline{k}
}, {
\underline{v}
} \right>_{\partial X}$$ whenever $(v,{
\underline{v}
}) \in \widetilde{\mathcal{Z}}$.
The claim is that $\mathscr{P}(u,{
\underline{\phi}
}) = (h,{
\underline{k}
})$. To see this, approximate $(u,{
\underline{\phi}
})$ in ${\widetilde{\mathcal{H}}}^1(X)$ by a sequence $(u_n, {
\underline{\phi_n}
}) \in {\widetilde{\mathcal{H}}}^2(X)$. Certainly $\mathscr{P}(u_n,{
\underline{\phi_n}
}) \rightarrow \mathscr{P}(u,{
\underline{\phi}
})$ in ${\mathcal{H}}^{-1}(X) \times H^{1-\mu}(\partial X)$. Furthermore, the pairing between $(u_n,{
\underline{\phi_n}
})$ and $\mathscr{P}'(v,{
\underline{v}
})$ is given by $$\left< Pu_n, v \right>_{\partial X} + \left< Bu_n, {
\underline{v}
} \right>_{\partial X}.$$ for each $(v,{
\underline{v}
}) \in {\mathcal{H}}^1(X) \times H^{\mu-1}(\partial X)$. Thus for $(v,{
\underline{v}
}) \in \mathcal{F}_\nu \times C^\infty(\partial X) \subseteq {\mathcal{H}}^1(X) \times H^{\mu-1}(\partial X)$, this converges to $\left< Pu, v \right>_X + \left< Tu, {
\underline{v}
} \right>_{\partial X}$. But on the other hand it converges to $\left< h, v \right>_X + \left< {
\underline{k}
}, {
\underline{v}
} \right>_{\partial X}$ since $\mathcal{F}_\nu(X) \times C^\infty(\partial X) \subseteq \widetilde{\mathcal{Z}}$ as well. Thus $\mathscr{P}(u,{
\underline{\phi}
}) = (h,{
\underline{k}
})$, since $\mathcal{F}_\nu(X) \times C^\infty(\partial X)$ is dense in ${\mathcal{H}}^1(X) \times H^{\mu-1}(\partial X)$ .
\[theo:fredholm\] Let $\nu > 0$ and $P$ as in Section \[subsect:globalass\] be elliptic at $\partial X$. If $0 < \nu < 1$, then let $T$ denote a scalar boundary operator satisfying $\mathrm{ord}_\nu(T) \leq \mu$, such that $\mathscr{P} = \{P,T\}$ is elliptic at $\partial X$.
1. Suppose that $0 < \nu < 1$. If $\mathscr{P}$ satisfies with $m< 1$ and with $m < 0$, then $$\mathscr{P}: \mathcal{Y} \rightarrow {\mathcal{H}}^0(X) \times H^{2- \mu}(\partial X)$$ is Fredholm.
2. Suppose that $\nu \geq 1$. If $P$ satisfies with $m < 1$ and with $m<0$, then $$P: \mathcal{Y} \rightarrow {\mathcal{H}}^0(X)$$ is Fredholm.
$(1)$ Lemma \[lem:fdkernel\] shows the kernel is finite dimensional. On the other hand, Lemma \[lem:fdcokerneladjoint\] shows that the equation $\mathscr{P}(u,{
\underline{\phi}
}) = (h,{
\underline{k}
}) $ has a solution $(u,{
\underline{\phi}
}) \in {\widetilde{\mathcal{H}}}^1(X)$ for $(h,{
\underline{k}
})$ in a space of finite codimension in ${\mathcal{H}}^0(X)\times H^{2-\mu}(X)$; clearly this $(u,{
\underline{\phi}
})$ can be identified with a unique element of $\mathcal{Y}$, namely $u$.
$(2)$ When $\nu \geq 1$, there is a natural analogue of Lemma \[lem:fdcokerneladjoint\]. Since the arguments are simpler when there is no boundary operator, the proofs are omitted.
Unique solvability
------------------
In this section, again let ${
\overline{X}
}$ denote a compact manifold with boundary as in Section \[subsect:manifoldwithboundary\]. This time, consider a pseudodifferential operator $P(\lambda) \in \Psi^{2,(\lambda)}(X)$ of the form $P(\lambda) = P_1(\lambda) + P_2(\lambda)$, where $$P_1(\lambda) \in \mathrm{Bess}^{(\lambda)}_\nu(X), \quad P_2(\lambda) \in \Psi_{\mathrm{comp}}^{2,(\lambda)}(X).$$ Assume that $P_1(\lambda)$ is parameter-elliptic at $\partial X$ with respect to an angular sector $\Lambda$ in the sense of Section \[subsect:semiclassicalbessel\]. If $0 < \nu < 1$, fix a scalar boundary condition $T(\lambda)$ with $\mathrm{ord}^{(\lambda)}_\nu(T(\lambda)) \leq \mu$, and assume that $\mathscr{P}(\lambda) = \{P(\lambda),T(\lambda)\}$ is parameter-elliptic at $\partial X$ with respect to $\Lambda$. It is also assumed that the ‘principal parts’ of $P(\lambda), T(\lambda)$ do not depend on $\lambda$, so the spaces $\mathcal{Y}$ are independent of $\lambda$.
The parameter-dependent versions of , , , are obtained by replacing the norms $\| \cdot \|$ with their uniform counterparts $\VERT \cdot \VERT$, and insisting that the estimates hold for all $\lambda \in \Lambda$.
\[theo:Lambdafredholm\] Let $\nu > 0$ and $P(\lambda), \mathscr{P}(\lambda), \Lambda$ be as above. Suppose that the parameter-dependent versions of , , , hold.
1. Let $0 < \nu < 1$. There exists $R> 0$ such that $$\mathscr{P}(\lambda): \mathcal{Y} \rightarrow {\mathcal{H}}^0(X) \times H^{2- \mu}(\partial X)$$ is an isomorphism for $\lambda \in \Lambda$ satisfying $|\lambda| > R$.
2. Let $\nu \geq 1$. Then there exists $R> 0$ such that $$P(\lambda): \mathcal{Y} \rightarrow {\mathcal{H}}^0(X)$$ is an isomorphism for $\lambda \in \Lambda$ satisfying $|\lambda| > R$.
The parameter-dependent versions of , show that $\mathscr{P}(\lambda)$ and $\mathscr{P}(\lambda)'$ respectively are injective on the appropriate spaces (for $\lambda \in \Lambda$ with $|\lambda|$ sufficiently large). This implies that $\mathscr{P}(\lambda)$ is an isomorphism for $|\lambda|$ sufficiently large. Similar remarks hold for $P$ when $\nu \geq 1$.
Completeness of generalized eigenfunctions {#sect:completeness}
==========================================
In this section, sufficient conditions are given which guarantee an elliptic parameter-dependent Bessel operator has a complete set of generalized eigenvectors. Completeness of eigenvectors for non-self adjoint boundary value problems has a long history, going back to classic works of Keldysh [@keldyvs1951characteristic], Browder [@browder1953eigenfunctions], Schechter [@schechter1959remarks], Agmon [@agmon1962eigenfunctions], among many others. The results of this section apply to large classes Bessel operator pencils with a spectral parameter in the boundary condition, and two-fold completeness is established (which is stronger than just completeness).
One application of this section is to describe a class of boundary conditions for which linearized scalar perturbations of global anti-de Sitter space have complete sets of normal modes. Recent numerical and perturbative studies have hinted at a relationship between the linear spectra of such perturbations and possible nonlinear instability mechanisms [@balasubramanian2014holographic; @bizon:2014:grg; @bizon:2011:prl; @bizon2015resonant; @buchel2015conserved; @dias:2012:cqg; @dias:2012:cqg:b]. These normal modes have been studied by separation of variable techniques, but there has not appeared a general criterion guaranteeing completeness of normal modes (nor even the discreteness of normal frquencies) for general, possibly time-periodic, boundary conditions. The results of this section also apply to more general stationary aAdS spacetimes with compact time slices, where $\partial_t$ is Killing but the spacetime is not necessarily static.
Two-fold completeness
---------------------
The main reference for this section is [@markus2012introduction Chapter II] Let ${
\overline{X}
}$ be a manifold with boundary, and let $P(\lambda) \in \mathrm{Bess}^{(\lambda)}_\nu(X)$ be a parametet-dependent Bessel operator such that $P(\lambda)$ is parameter-elliptic at $\partial X$ in the sense of Section \[subsect:ellipticity\], and $P(\lambda)$ is parameter-elliptic on $X$ in the usual sense. If $0< \nu < 1$, let $T(\lambda)$ be a scalar parameter-dependent boundary operator such that $\mathscr{P}(\lambda) = \{P(\lambda),T(\lambda)\}$ is parameter-elliptic at $\partial X$.
Parameter-ellipticity of $P(\lambda)$ implies a decomposition $$P(\lambda) = P_2 + \lambda P_1 + \lambda^2 P_0,$$ where $P_0 \in C^\infty({
\overline{X}
})$ does not vanish. Dividing by $P_0$, it may be assumed that $P(\lambda)$ is of the form $P(\lambda) = P_2 + \lambda P_1+ \lambda^2$. The boundary operator $T(\lambda)$ is written as $T(\lambda) = T_1 + \lambda T_0$.
If $0 < \nu < 1$, a complex number $\lambda_0 \in \mathbb{C}$ is said to be an eigenvalue of $\mathscr{P}(\lambda)$ if there exists $u_0 \in {\mathcal{H}}^2(X)$ such $\mathscr{P}(\lambda_0)u_0 = 0$. Corresponding to an eigenvalue $\lambda_0$, a sequence $(u_0,\ldots,u_k)$ with $u_0 \neq 0$ is said to be a chain of generalized eigenvectors if $$\begin{gathered}
P(\lambda_0)u_p + \frac{1}{1!}\partial_\lambda P(\lambda_0) u_{p-1} + \frac{1}{2!} \partial_\lambda P(\lambda_0) u_{p-2} = 0, \\
T(\lambda_0)u_p + \frac{1}{1!}T(\lambda_0)u_{p-1} = 0\end{gathered}$$ for $p = 0,\ldots, k$. Thus $(u_0,\ldots,u_k)$ is a chain of generalized eigenvectors with eigenvalue $\lambda_0$ if and only if the function $$u(t) = e^{\lambda_0 t} \sum_{j=0}^k \frac{t^k}{k!} u_{k-j}$$ solves the (time-dependent) equation $\mathscr{P}(\partial_t)u(t) = 0$. Such a solution $u(t)$ is called elementary. To each elementary solution is associated the Cauchy data $(u(0),\partial_t u(0))$. The set of generalized eigenvectors (for all possible eigenvalues) is said to be two-fold complete in a Hilbert space $H$ continuously embedded in ${\mathcal{H}}^0(X)\times {\mathcal{H}}^0(X)$ if the span of all Cauchy data $(u(0),\partial_t u(0))$ corresponding to elementary solutions (for all eigenvalues) is dense in $H$. The same definition holds if $\nu \geq 1$, this time replacing $\mathscr{P}(\lambda)$ with $P(\lambda)$.
A general criterion concerning two-fold completeness is given by [@yakubov1993completeness Theorem 3.4]; that theorem is a refinement of the standard reference [@dunford1963linear Corollary XI.9.31].
\[prop:completeness\] Let $P(\lambda), T(\lambda)$ be defined as above. Fix rays $\Gamma_1,\ldots, \Gamma_s$ through the origin of the complex plane such the angle between any two adjacent rays is less than or equal to $\pi/n$, where $\dim X = n$.
1. Let $0 < \nu < 1$. If $\mathscr{P}(\lambda)$ is elliptic with respect to $\Gamma_1,\ldots, \Gamma_s$, then the eigenvalues of $\mathscr{P}(\lambda)$ are discrete and the set of generalized eigenvectors is two-fold complete in the space $\{(v_1,v_2)\in {\mathcal{H}}^2(X)\times {\mathcal{H}}^1(X): T_0 v_2 + T_1 v_1 = 0 \}$.
2. Let $\nu \geq 1$. If $P(\lambda)$ is elliptic with respect to $\Gamma_1,\ldots, \Gamma_s$, then the eigenvalues of $P(\lambda)$ are discrete and the set of generalized eigenvectors is two-fold complete in the space ${\mathcal{H}}^2(X)\times {\mathcal{H}}^1(X)$.
$(1)$ To apply [@yakubov1993completeness Theorem 3.4], it must be verified that the singular values of the embeddings $J_k : {\mathcal{H}}^{k}(X)\hookrightarrow {\mathcal{H}}^{k-1}(X)$ satisfy $s_j(J_k) \leq Cj^{-1/n}$ for $k = 1,2$, and that the space $\{(v_1,v_2)\in {\mathcal{H}}^2(X)\times {\mathcal{H}}^1(X): T_0 v_2 + T_1 v_1 = 0 \}$ is dense in ${\mathcal{H}}^1(X)\times {\mathcal{H}}^0(X)$.
The claim about the singular values follows from Lemma \[lem:singularvaluesmanifold\]. To verify the density claim, first fix a sequence $\lambda_n\in\mathbb{C}$ such that $|\lambda_n|$ tends to infinity along one of the rays of ellipticity, say $\Gamma_1$.
Given $(u_1,u_2) \in {\mathcal{H}}^1(X)\times {\mathcal{H}}^0(X)$, take a sequence $(v_1^n,v_2^n) \in {\mathcal{H}}^2(X)\times {\mathcal{H}}^1(X)$ such that $(v_1^n,v_2^n) \rightarrow (u_1,u_2)$ in ${\mathcal{H}}^1(X)\times {\mathcal{H}}^0(X)$ as $n \rightarrow \infty$. According to Theorem \[theo:Lambdafredholm\], the operator $$\mathscr{P}(\lambda_n)^{-1} : {\mathcal{H}}^0(X) \times H^{2-\mu}(X) \rightarrow {\mathcal{H}}^2(X)$$ exists for $n$ sufficiently large, where $\mu = \mathrm{ord}_\nu^{(\lambda)}(T(\lambda))$. Note that $T_0 v_2^n + T_1 v_1^n \in H^{2-\mu}(\partial X)$. Let $$w_1^n = \mathscr{P}(\lambda_n)^{-1}(0,-T_0 v_2^n - T_1 v_1^n),$$ so $w_1^n$ lies in ${\mathcal{H}}^2(X)$, and set $w_2^n = \lambda w_1^n$. Then $$(v_1^n + w_1^n, v_2^n + w_2^n) \in \{(v_1,v_2)\in {\mathcal{H}}^2(X)\times {\mathcal{H}}^1(X): T_0 v_2 + T_1 v_1 = 0 \}.$$ Furthermore, according to Theorems \[theo:bvptheosemiclassical\], \[theo:Lambdafredholm\] the solution $w_1^n$ satisfies $$|\lambda|^{2-s} \| w_1^n \|_{{\mathcal{H}}^s(X)} \leq C \|T_0 v_2^n + T_1 v_1^n \|_{H^{2-\nu}(\partial X)}$$ for $s = 0,1$. The right hand side is uniformly bounded in $H^{2-\mu}(X)$ as $n\rightarrow \infty$, so $(w_1^n,w_2^n) \rightarrow 0$ in ${\mathcal{H}}^1(X)\times {\mathcal{H}}^0(X)$. This shows that $(v_1^n + w_1^n, v_2^n + w_2^n) \rightarrow (u_1,u_2)$, establishing the density.
$(2)$ For $\nu \geq 1$ the singular value estimates remain the same, and the density result is trivial.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper is based on work supported by NSF grants DMS-1201417 and DMS-1500852. I would like to thank Maciej Zworski for his encouragement on this project.
Proof of lemma \[lem:traceproperty\] {#appendix1}
====================================
The proof of Lemma \[lem:traceproperty\] is broken up into several stages. Recall in this section that $\gamma_\pm$ are defined as in the beginning of Section \[subsect:traces\] without any mention of the space ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$.
\[lem:absolutelycontinuoustrace\] Let $\nu > 0$.
1. If $u \in {\mathcal{H}}^{1}({\mathbb{T}^n_+})$ and $\gamma_- u = 0$, then for a.e. $y \in {\mathbb{T}^{n-1}}$, $$u(x,y) = x^{1/2-\nu}\int_0^x t^{\nu-1/2} {{\partial}_\nu}u(t,y) \,dt.$$
2. Suppose in addition that $0 < \nu < 1$. If $u \in {\mathcal{H}}^{2}({\mathbb{T}^n_+})$, and ${
\underline{\gamma}
}u = 0$, then for a.e. $y \in {\mathbb{T}^{n-1}}$, $$u(x,y) = x^{1/2-\nu}\int_0^x t^{-2\nu + 1} \int_0^t s^{1/2-\nu} {{\partial}_\nu}^*{{\partial}_\nu}u(s,y)\, ds\, dt.$$
These two facts follow from the Sobolev embedding for weighted spaces, as in Section \[subsect:weighted\]. In the first case, for a.e. $y\in {\mathbb{T}^{n-1}}$ the function $x \mapsto x^{\nu-1/2}u(x,y)$ is absolutely continuous on ${
\overline{{\mathbb{R}_+}}
}$, and $\gamma_- u = 0$ implies that $x^{\nu - 1/2}u(x,y) \rightarrow 0$ as $x\rightarrow 0$ for a.e. $y \in {\mathbb{T}^{n-1}}$. The the result follows from the fundamental theorem of calculus. A similar argument applies in the second case, in which the functions $x \mapsto x^{\nu-1/2}u(x,y), \, x \mapsto x^{1/2-\nu}{{\partial}_\nu}u(x,y)$ are absolutely continuous on ${
\overline{{\mathbb{R}_+}}
}$ for a.e. $y\in {\mathbb{T}^{n-1}}$, and vanish at $x=0$.
\[lem:mathring\] Let $0 < \nu < 1$. Then $\mathring{{\mathcal{H}}}^{1}({\mathbb{T}^n_+})= \ker \gamma_-$, and $\mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+}) = \ker {
\underline{\gamma}
}$.
The first equality comes from [@grisvard:1963 Proposition 1.2]. It remains to show the second equality.
$(1)$ First show that if $u \in {\mathcal{H}}^{2}({\mathbb{T}^n_+})$ and ${
\underline{\gamma}
}u = 0$, then $u \in \mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+})$. Begin by assuming that $u$ has compact support in ${
\overline{{\mathbb{T}^n_+}}
}$; this is possible, since if $\chi \in C_c^\infty({
\overline{{\mathbb{R}_+}}
})$ satisfies $\chi = 1$ near $x=0$, then it is easy to see that $u$ is approximated in ${\mathcal{H}}^2({\mathbb{T}^n_+})$ by the functions $\chi(x/n)u$ as $n \rightarrow \infty$.
Next, fix $\chi \in C_c^\infty({\mathbb{R}_+})$ satisfying
$0 \leq \chi \leq 1$,
$\chi(x) = 0$ for $0 \leq x \leq 1$,
$\chi(x) = 1$ for $x\geq 2$
. Then let $\chi_n(x) = \chi(nx)$ and consider the sequence $u_n = \chi_n u$. Then $u_n$ has compact support in ${\mathbb{T}^n_+}$, which implies that $u_n \in H^2({\mathbb{T}^n_+})$ since the ${\mathcal{H}}^2({\mathbb{T}^n_+})$ and $H^2({\mathbb{T}^n_+})$ norms are comparable on compact subsets of ${\mathbb{T}^n_+}$. But the compact support also implies that $u_n \in \mathring{H}^2({\mathbb{T}^n_+})$ by the well known characterization of $\mathring{H}^2({\mathbb{T}^n_+})$. This implies $u_n$ can be approximated by compactly supported functions in the $H^2({\mathbb{T}^n_+})$ norm, all of whose supports are contained in a fixed compact subset of ${\mathbb{T}^n_+}$. Again by the comparability of norms, this implies $u_n \in \mathring{{\mathcal{H}}}^2({\mathbb{T}^n_+})$.
It now suffices to prove that $u_n \rightarrow u$ in ${\mathcal{H}}^{2}({\mathbb{T}^n_+})$, since $\mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+})$ is closed. This is easily deduced from Lemma \[lem:absolutelycontinuoustrace\], imitating the proof of [@evans:2010 Chapter 5.5, Theorem 2] for instance.
$(2)$ The inclusion $\mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+}) \subseteq \ker {
\underline{\gamma}
}$ is clear, since ${
\underline{\gamma}
} = 0$ for each $u \in C_c^\infty({\mathbb{T}^n_+})$, and hence ${
\underline{\gamma}
} = 0$ for each $u \in \mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+})$ by density and continuity.
\[lem:tracelift\] There exists a map $$\mathcal{K}: C^\infty({\mathbb{T}^{n-1}}) \times C^\infty({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{F}_\nu({\mathbb{T}^n_+})}$$ such that ${
\underline{\gamma}
} \circ \mathcal{K} = 1$ on $C^\infty({\mathbb{T}^{n-1}})\times C^\infty({\mathbb{T}^{n-1}})$ and $\mathcal{K}$ extends by continuity $$\mathcal{K} : H^{s-{
\underline{\nu}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^{s}({\mathbb{T}^n_+})$$ for each $s=0,\pm 1,\pm 2$. In particular, if $s=2$ then $\mathcal{K}$ is a right inverse for ${
\underline{\gamma}
}$
Let $\varphi \in C^\infty_c(\overline{{\mathbb{R}_+}})$ be such that $\varphi = 1$ near $x =0$, and set $$v_-(x) = x^{1/2-\nu} \varphi(x^2), \quad v_+(x) = (2\nu)^{-1} x^{1/2+\nu} \varphi(x^2),$$ so $v_\pm \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$. Given $(f_-,f_+) \in C^\infty({\mathbb{T}^{n-1}}) \times C^\infty({\mathbb{T}^{n-1}})$, define $u_\pm(x,y)$ by its Fourier coefficients, $$\hat{u}_\pm(x,q) = \left<q \right>^{-(1/2\pm \nu)} \hat{f}_\pm(q) v_\pm(\left<q\right>x).$$ Then $u_\pm\in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and $\gamma_\pm (u_- + u_+) = f_\pm$ in the sense of Lemma \[lem:directtrace\]. Appealing to Section \[subsect:fourier\] shows that the map defined by $$\mathcal{K}(f_-,f_+) := u_- + u_+$$ extends by continuity to a map $\mathcal{K}:H^{s-{
\underline{\nu}
}}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^s({\mathbb{T}^n_+})$. If $s = 2$, then ${
\underline{\gamma}
}$ is bounded on ${\mathcal{H}}^2({\mathbb{T}^n_+})$, and ${
\underline{\gamma}
}\circ \mathcal{K}$ is the identity on $H^{s-{
\underline{\nu}
}}({\mathbb{T}^{n-1}})$ by Lemma \[lem:directtrace\] and continuity.
Suppose that $0< \nu <1$. Then ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is dense in ${\mathcal{H}}^{2}({\mathbb{T}^n_+})$ for each $s = 0,1,2$.
This is clear when $s=0$. The proof is given here in the case $s= 2$; the case $s=1$ is simpler, and can be handled similarly. Suppose that $u \in {\mathcal{H}}^{2}({\mathbb{T}^n_+})$, and let $\tilde{u} = \mathcal{K}({
\underline{\gamma}
}u)$. Then ${
\underline{\gamma}
}(u - \tilde u) = 0$, so $u - \tilde u \in \mathring{{\mathcal{H}}}^{2}({\mathbb{T}^n_+})$ by Lemma \[lem:mathring\]. It follows that there exists a sequence $u_j \in C_c^\infty({\mathbb{T}^n_+})$ such that $u_j \rightarrow u-\tilde{u}$ in ${\mathcal{H}}^{2}({\mathbb{T}^n_+})$.
On the other hand, approximate ${
\underline{\gamma}
}u$ by a sequence $\underline{v}_j \in C^\infty({\mathbb{T}^{n-1}}) \times C^\infty({\mathbb{T}^{n-1}})$, and hence $\tilde{u}_j = \mathcal{K}\underline{v}_j$ satisfy $\tilde{u}_j \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and $\tilde{u}_j \rightarrow \tilde u$ in ${\mathcal{H}}^{2}({\mathbb{T}^n_+})$. Therefore, $u_j + \tilde{u}_j \in {\mathcal{F}_\nu({\mathbb{T}^n_+})}$ and $u_j + \tilde{u}_j \rightarrow u$, which shows that ${\mathcal{F}_\nu({\mathbb{T}^n_+})}$ is dense in ${\mathcal{H}}^{2,k}({\mathbb{T}^n_+})$.
\[app:nugeq1\] Suppose that $ \nu \geq 1$. Then $C_c^\infty({\mathbb{T}^n_+})$ is dense in ${\mathcal{H}}^{s}({\mathbb{T}^n_+})$ for each $s = 0,1,2$.
This result clearly holds for $s=0$, and by the results of Section \[subsect:sob1\], it also holds for $s=1$. For $s=2$, Lemma \[lem:laxmilgram\] implies that $\Delta_\nu + 1$ is an isomorphism from $D(L)$ onto $L^2({\mathbb{T}^n_+})$. The first step is write down an explicit formula for the inverse $(\Delta_\nu + 1)^{-1}$ acting on $L^2({\mathbb{T}^n_+})$.
Introduce the Zemanian space $\mathcal{Z}_\nu({\mathbb{R}_+})$ [@zemanian Chapter 5] by $$\mathcal{Z}_\nu({\mathbb{R}_+}) = \{ v(x) = x^{1/2+\nu}v_+(x^2): v_+(x) \in \mathcal{S}(\mathbb{R}) \},$$ where $\mathcal{S}(\mathbb{R})$ is the space of Schwartz functions on $\mathbb{R}$. Note that $\mathcal{Z}_\nu({\mathbb{R}_+})$ is contained in ${\mathcal{H}}^s({\mathbb{R}_+})$ for each $s=0,1,2$. Given $v \in \mathcal{Z}_\nu({\mathbb{R}_+})$, define the Hankel transform $$\left(\mathscr{H}_\nu v\right)(\xi) = \int_{{\mathbb{R}_+}} (\xi x)^{1/2} J_\nu(\xi x) v(x) dx.$$ Referring to [@zemanian Chapter 5], it is well known that
$\mathscr{H}_\nu$ is an automorphism of $\mathcal{Z}_\nu({\mathbb{R}_+})$,
$\mathscr{H}_\nu^2 = I$,
$\mathscr{H}_\nu$ is isometric with respect to the $L^2({\mathbb{R}_+})$ norm.
If $f = v \otimes w$ where $v \in \mathcal{Z}_\nu({\mathbb{R}_+})$ and $w\in C^\infty({\mathbb{T}^{n-1}})$, then $$f = (2\pi)^{-(n-1)/2} \sum_{q\in\mathbb{Z}^{n-1}}\int_{{\mathbb{R}_+}} e^{i\left<q,y\right>} (\xi x)^{1/2} J_\nu(\xi x) (\mathscr{H}_\nu v)(\xi)\hat{w}(q) \,d\xi,$$ and $$\| f \|^2_{L^2({\mathbb{T}^n_+})} = (2\pi)^{-(n-1)/2}\sum_{q\in\mathbb{Z}^{n-1}}\int_{{\mathbb{R}_+}} |(\mathscr{H}_\nu v)(\xi) \hat{w}(q)|^2 d\xi.$$ Given $f = v \otimes w$ as above, let $$\label{eq:u}
u = (2\pi)^{-(n-1)/2} \sum_{q\in\mathbb{Z}^{n-1}}\int_{{\mathbb{R}_+}} (1+ |q|^2 + |\xi|^2)^{-1} e^{i\left<q,y\right>} (\xi x)^{1/2} J_\nu(\xi x) (\mathscr{H}_\nu v)(\xi)\hat{w}(q) \,d\xi.$$ Then $u$ has an expansion $$\label{eq:zemanian}
u(x,y) = x^{1/2+\nu}u_+(x^2,y),$$ where $u_+(x,y) \in \mathcal{S}(\mathbb{R} \times {\mathbb{T}^{n-1}})$ is rapidly decaying in the $x$ variable [@zemanian Chaper 5]. Denote this space by $\mathcal{Z}_\nu({\mathbb{T}^n_+})$, which is contained in ${\mathcal{H}}^2({\mathbb{T}^n_+})\cap\mathring{{\mathcal{H}}}^1_0({\mathbb{T}^n_+})$. Note that there is a continuous inclusion $${\mathcal{H}}^2({\mathbb{T}^n_+})\cap\mathring{{\mathcal{H}}}^1_0({\mathbb{T}^n_+}) \hookrightarrow D(L).$$ Since $(\Delta_\nu + 1)u = f$, it follows that $u$ is the unique solution in $D(L)$ to the equation $(\Delta_\nu + 1)u = f$. Furthermore, $$\| u \|_{{\mathcal{H}}^2({\mathbb{T}^n_+})} \leq C \| f \|_{L^2({\mathbb{T}^n_+})},$$ where $C>0$ does not depend on $u$ or $f$. Lemma \[lem:laxmilgram\] and the open mapping theorem imply that $D(L) = {\mathcal{H}}^2({\mathbb{T}^n_+}) \cap \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$ with an equivalence of norms; since functions $v\otimes w$ as above are dense in $L^2({\mathbb{T}^n_+})$, the space $\mathcal{Z}_\nu({\mathbb{T}^n_+})$ is dense in ${\mathcal{H}}^2({\mathbb{T}^n_+})\cap\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$. Finally, if $\nu \geq 1$ then ${\mathcal{H}}^2({\mathbb{T}^n_+}) = {\mathcal{H}}^2({\mathbb{T}^n_+}) \cap \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_+})$ by the density result for $s=1$.
Given $u \in \mathcal{Z}_\nu({\mathbb{T}^n_+})$, there exists a sequence $v_n \in C_c^\infty({\mathbb{T}^n_+})$ such that $v_n \rightarrow u$ weakly in ${\mathcal{H}}^2({\mathbb{T}^n_+})$. To see this, fix $\chi \in C_c^\infty({\mathbb{R}_+})$ satisfying
$0 \leq \chi \leq 1$,
$\chi(x) = 0$ for $0 \leq x \leq 1$,
$\chi(x) = 1$ for $x\geq 2$
, and then let $\chi_n(x) = \chi(nx)$. The claim is that $\chi_n u \rightarrow u$ weakly in ${\mathcal{H}}^2({\mathbb{T}^n_+})$ after passing to a subsequence if necessary. Take for example $$\begin{aligned}
|{D_\nu}|^2(\chi_n(x) - 1)u(x,y) &= (\chi_n(x)-1)|{D_\nu}|^2 u(x) \\ &+ 2 n \chi_n'(x) \left( (1/2+\nu) x^{-1/2+\nu} u_+(x^2,y) + 2x^{3/2+\nu}\partial_x u_+(x^2,y) \right) \\ &+ n^2 \chi_n''(x) u(x,y).
\end{aligned}$$ The first term tends to zero in $L^2({\mathbb{T}^n_+})$ norm. The $L^2({\mathbb{T}^n_+})$ norm squared of the third term is bounded by a constant times $$n^4 \int_{{\mathbb{T}^{n-1}}} \int_{1/n}^{2/n} x^{1+2\nu} |u_+(x^2,y)|^2 \, dx\, dy \leq C n^4 \int_{1/n}^{2/n} x^{1+2\nu} = \mathcal{O}(n^{2\nu -2}).$$ So $n^2 \chi''_n(x)u(x,y)$ is bounded in $L^2({\mathbb{T}^n_+})$ for $\nu \geq 1$ (and converges to zero if $\nu > 1$). The second term is similarly bounded in $L^2({\mathbb{T}^n_+})$. It is also clear the second and third terms converge to zero in $\mathscr{D}'({\mathbb{R}_+})$. Extracting a weakly convergent subsequence, this implies that $|{D_\nu}|^2(\chi_n -1)u(x,y)$ tends to zero weakly along a subsequence. Repeating this argument for the other terms whose $L^2({\mathbb{T}^n_+})$ norms define the ${\mathcal{H}}^2({\mathbb{T}^n_+})$ norm (as in ), it follows that $\chi_n u \rightarrow u$ weakly in ${\mathcal{H}}^2({\mathbb{T}^n_+})$. Furthermore, by truncating $\chi_n u$ at successively larger values of $x > 0$, one may find a sequence of $C_c^\infty({\mathbb{T}^n_+})$ functions $v_n$ such that $v_n \rightarrow u$ weakly in ${\mathcal{H}}^2({\mathbb{T}^n_+})$.
Now suppose that $u \in {\mathcal{H}}^2({\mathbb{T}^n_+})$ satisfies $$\left< u, v \right>_{{\mathcal{H}}^2({\mathbb{T}^n_+})} = 0$$ for all $v \in C_c^\infty({\mathbb{T}^n_+})$. Choose a sequence $u_m \in \mathcal{Z}_\nu({\mathbb{T}^n_+})$ such that $u_m \rightarrow u$ in ${\mathcal{H}}^2({\mathbb{T}^n_+})$. But if $v_{m,n}$ is an associated sequence of $C_c^\infty({\mathbb{T}^n_+})$ functions converging weakly to $u_m$ (as constructed above), then $$\left< u, u_m \right>_{{\mathcal{H}}^2({\mathbb{T}^n_+})} = \lim_{n \rightarrow \infty} \left< u, v_{m,n} \right>_{{\mathcal{H}}^2({\mathbb{T}^n_+})} = 0.$$ Passing to the limit $m\rightarrow \infty$ gives $u=0$, so $C_c^\infty({\mathbb{T}^n_+})$ is dense.
Compactness and embeddings of Schatten class
============================================
Let ${\mathbb{T}^n_\sharp}= {\mathbb{T}^{n-1}}\times (0,1)$. The spaces ${\mathcal{H}}^s({\mathbb{T}^n_\sharp}), \, \mathring{{\mathcal{H}}}^s({\mathbb{T}^n_\sharp})$ are defined as before. The goal of this section is to study Schatten class properties of the embeddings of these spaces into $L^2(\mathbb{T}^n_\sharp)$ for $s=1,2$. First, compactness properties are examined — this is done differently in [@holzegel:2012wt Section 6], but the approach taken here immediately yields the Schatten property.
The first observation is that the embedding $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})$ is compact for $\nu > 0$, since $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) = \mathring{H}^1({\mathbb{T}^n_\sharp})$ according to Lemma \[lem:usualsobolev\]. To prove compactness of the embedding ${\mathcal{H}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})$ requires slightly more work.
Recall the following facts: first, $$\sqrt{x}K_\nu(x) \sim (\pi/2)^{1/2}e^{-x}, \quad \sqrt{x}I_\nu(x) \sim (1/2\pi)^{1/2} e^x,$$ valid for real $x \rightarrow \infty$. Furthermore, $$|\sqrt{x}K_\nu(x)| \leq C\left(\frac{1+x^\nu}{1+x^{1/2}}\right) x^{1/2-\nu}e^{-x}$$ for all $x > 0$. Combined with the equation satisfied by $\sqrt{x}K_\nu(x)$, this gives $$\label{eq:Knugrowth1}
\int_0^1 |\sqrt{x} K_\nu(\tau x)|^2 \, dx \leq C\tau^{-2}, \quad \int_0^1 |(|{D_\nu}|^2 \sqrt{x} K_\nu(\tau x))|^2 \, dx \leq C\tau^2$$ for $\tau > 0$. Combining with Lemma \[lem:scatteringmatrix\], and integration by parts shows that $$\label{eq:Knugrowth2}
\int_0^1 |({{\partial}_\nu}\sqrt{x}K_\nu(\tau x))|^2 \, dx \leq C.$$ By Laplace’s method, integrals of $\sqrt{x}I_\nu(\tau x)$ may be evaluated as well. In particular, $$\begin{gathered}
\int_0^1 |\sqrt{x}I_\nu(\tau x)|^2 \sim \frac{1}{4\pi} \tau^{-2} e^{2\tau}, \quad \int_0^1 |({{\partial}_\nu}\sqrt{x}I_\nu(\tau x))|^2 \sim \frac{1}{4\pi} e^{2\tau}, \label{eq:Inugrowth1} \\ \quad \int_0^1 |(|{D_\nu}|^2\sqrt{x}I_\nu(\tau x))|^2 \sim \frac{1}{4\pi} \tau^{2} e^{2\tau}. \label{eq:Inugrowth2}\end{gathered}$$ as $\tau \rightarrow \infty$. The following construction of a Poisson operator is a refinement of Lemma \[lem:tracelift\].
\[lem:poisson\] Let $0 < \nu < 1$. There exists a map $\mathcal{K}_0 : C^\infty({\mathbb{T}^{n-1}}) \rightarrow \mathcal{F}_\nu({\mathbb{T}^n_\sharp})$ such that $$(\gamma_-\circ \mathcal{K}_0)\phi = \phi, \quad (\mathcal{K}_0\phi) (1,\cdot) = 0, \quad (\Delta_\nu + 1)\mathcal{K}_0\phi = 0$$ for each $\phi \in C^\infty({\mathbb{T}^{n-1}})$. The map $\mathcal{K}_0$ extends by continuity $$\mathcal{K}_0 : H^{s-1+\nu}({\mathbb{T}^{n-1}}) \rightarrow {\mathcal{H}}^s({\mathbb{T}^n_\sharp})$$ for each $s=0,\pm1,\pm2$. Similarly, for each $\nu > 0$ there exists a map $\mathcal{K}_1: C^\infty({\mathbb{T}^{n-1}}) \rightarrow \mathcal{F}_\nu({\mathbb{T}^n_\sharp})$ such that $$(\gamma_- \circ \mathcal{K}_1)\phi = 0, \quad (\mathcal{K}_1 \phi)(1,\cdot) = \phi, \quad (\Delta_\nu+1)\mathcal{K}_1\phi = 0,$$ and $\mathcal{K}_1$ extends by continuity $$\mathcal{K}_1: H^{s-1/2}({\mathbb{T}^{n-1}})\rightarrow {\mathcal{H}}^s({\mathbb{T}^n_\sharp})$$ for $s=0,\pm1,\pm2$.
Only $\mathcal{K}_1$ is constructed in detail; the construction of $\mathcal{K}_0$ is similar but somewhat simpler. For each $q \in \mathbb{Z}^{n-1}$, consider the function $$v(x,q) = \Gamma(1-\nu) \left(\frac{\left<q\right>}{2}\right)^\nu \frac{\sqrt{x}K_\nu(\left<q\right>x)I_\nu(\left<q\right>) - \sqrt{x}I_\nu(\left<q\right>x)K_\nu(\left<q\right>)}{ \frac{\pi}{2\sin\pi\nu}I_\nu(\left<q\right>) -K_\nu(\left<q\right>) }.$$ Note that $v(q) \in \mathcal{F}_\nu({\mathbb{T}^n_\sharp})$ by asymptotics of Bessel functions. Furthermore, $$(|{D_\nu}|^2 + \left<q \right>^2)v(x,q) = 0,$$ and $v(1,q) = 0, \gamma_-v(q) = 1$.
Given $f \in C^\infty({\mathbb{T}^{n-1}})$, let $$\hat{u}(x,q) = \hat{f}(q)v(x,q).$$ According to , , , , the map sending $f$ to the function $u$ with Fourier coefficients $\hat{u}(q)$ is bounded $H^{s-1+\nu}({\mathbb{T}^{n-1}})\rightarrow {\mathcal{H}}^s({\mathbb{T}^n_+})$ for $s=0,\pm1,\pm2$.
Recall that ${\mathcal{H}}^1({\mathbb{T}^n_\sharp})$ is a Hilbert space, equipped with the scalar product $$\left< u, v \right>_{{\mathcal{H}}^1({\mathbb{T}^n_\sharp})} =
\left< {D_\nu}u, {D_\nu}v \right>_{{\mathbb{T}^n_\sharp}} + \sum_{i=1}^{n-1} \left< D_{y_i} u, D_{y_i} v \right>_{{\mathbb{T}^n_\sharp}} + \left< u, v \right>_{{\mathbb{T}^n_\sharp}}.$$ Suppose that $0 < \nu < 1$. Lemma \[lem:poisson\] implies that $\left< \mathcal{K}_0\phi_0 + \mathcal{K}_1\phi_1 ,v\right>_{{\mathcal{H}}^1({\mathbb{T}^n_\sharp})}=0$ for all $\phi_0,\phi_1 \in C^\infty({\mathbb{T}^{n-1}})$ and $v \in C_c^\infty({\mathbb{T}^n_\sharp})$. By continuity this holds true for $\phi_0 \in H^\nu({\mathbb{T}^{n-1}}),\,\phi_1 \in H^{1/2}({\mathbb{T}^{n-1}})$ and $v \in \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$. Consider the orthogonal decomposition $${\mathcal{H}}^1({\mathbb{T}^n_\sharp}) = \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \oplus \mathcal{X},$$ where $\mathcal{X}$ is the orthogonal complement of $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$ in ${\mathcal{H}}^1({\mathbb{T}^n_\sharp})$. Let $\gamma_1$ denote the restriction to $\{ x =1 \}$. Lemma \[lem:poisson\] shows that the map $$\Gamma: u \mapsto (\gamma_-u,\gamma_1 u)$$ is an isomorphism from $\mathcal{X}$ onto $H^{\nu}({\mathbb{T}^{n-1}}) \times H^{1/2}({\mathbb{T}^{n-1}})$. The inverse of $\Gamma$ is given by $\mathcal{K}_0 + \mathcal{K}_1: (\phi_0,\phi_1)\mapsto \mathcal{K}_0\phi_0 + \mathcal{K}_1 \phi_1$.
A similar discussion applies if $\nu \geq 1$. In that case, $\gamma_1 : \mathcal{X} \rightarrow H^{1/2}({\mathbb{T}^{n-1}})$ is an isomorphism, with inverse $\mathcal{K}_1$.
\[lem:compactembedding\] If $\nu > 0$, then the embeddings $${\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \hookrightarrow {\mathcal{H}}^1({\mathbb{T}^n_\sharp}), \quad {\mathcal{H}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})$$ are compact.
$(1)$ First suppose that $0< \nu < 1$. Write the orthogonal decomposition ${\mathcal{H}}^1({\mathbb{T}^n_\sharp}) = \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \oplus \mathcal{X}$. The inclusion of $\mathcal{X}$ into $L^2({\mathbb{T}^n_\sharp})$ can be factored as $$\label{eq:compactfactor}
\mathcal{X} \xrightarrow{\Gamma} H^\nu({\mathbb{T}^{n-1}})\times H^{1/2}({\mathbb{T}^{n-1}}) \hookrightarrow H^{-1+\nu}({\mathbb{T}^{n-1}}) \times H^{-1/2}({\mathbb{T}^{n-1}}) \xrightarrow{\mathcal{K}_0+\mathcal{K}_1} L^2({\mathbb{T}^n_\sharp}),$$ noting that $\mathcal{K}_0 + \mathcal{K}_1: H^{-1+\nu}({\mathbb{T}^{n-1}})\times H^{-1/2}({\mathbb{T}^{n-1}}) \rightarrow L^2({\mathbb{T}^n_\sharp})$ is an extension by continuity of the same map acting $H^\nu({\mathbb{T}^{n-1}})\times H^{1/2}({\mathbb{T}^{n-1}}) \rightarrow \mathcal{X}$. This is compact since the inclusion of $H^\nu({\mathbb{T}^{n-1}}) \times H^{1/2}({\mathbb{T}^{n-1}})$ into $H^{-1+\nu}({\mathbb{T}^{n-1}}) \times H^{-1/2}({\mathbb{T}^{n-1}})$ is compact.
The space ${\mathcal{H}}^2({\mathbb{T}^n_\sharp})$ may be identified with a closed subspace $H$ of ${\mathcal{H}}^1({\mathbb{T}^n_\sharp})^{n+1} \times {\mathcal{H}}^1_{*}({\mathbb{T}^n_\sharp})$ via the mapping $$u \mapsto (u,\partial_{y_1}u,\ldots, \partial_{y_{n-1}}u, {{\partial}_\nu}u).$$ With this in mind, the embedding ${\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \hookrightarrow {\mathcal{H}}^1({\mathbb{T}^n_\sharp})$ is identified with the embedding $$\label{eq:H2compact}
{\mathcal{H}}^1({\mathbb{T}^n_\sharp})^{n+1} \times {\mathcal{H}}^1_{*}({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})^{n+2},$$ restricted to $H$. But the inclusion ${\mathcal{H}}^1_*({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})$ is compact by the first part as well (since $0 < 1-\nu < 1$), so is compact.
$(2)$ Now suppose that $\nu \geq 1$. The same argument as in the first part shows that ${\mathcal{H}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})$ is compact. Next, by the same reductions as above, it suffices to consider the inclusion ${\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \cap \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$. As in Section \[subsect:dirichletlaplacian\], let $L$ denote the self-adjoint operator with distributional action $\Delta_\nu$ and form domain $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$. Now the embedding of $D((L+1)^{1/2}) = \mathring{{\mathcal{H}}}^1({\mathbb{T}^{n-1}})$ into $L^2({\mathbb{T}^n_\sharp})$ is compact, hence so is each embedding $D((L+1)^{N}) \hookrightarrow D((L+1)^n)$ for $N > n$. But as in Lemma \[app:nugeq1\], the domain $D(L) = D(L+1)$ (with the graph norm) is equivalent to ${\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \cap \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$ (with its usual norm).
Let $L$ denote the self-adjoint operator with distributional action given by $\Delta_\nu$ and form domain $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$ (see Section \[subsect:dirichletlaplacian\]). Lemma \[lem:compactembedding\] and Lax–Milgram guarantee that this operator has discrete spectrum. The eigenvalues and eigenvectors are well known. The eigenvalues are $|q|^2 + j_{\nu,n}^2+1$, where $q \in \mathbb{Z}^{n-1}$ and $j_{\nu,n}$ is the $n$’th positive root of the Bessel function $J_\nu$. The corresponding eigenfunction is $$\sqrt{x}J_\nu(j_{\nu,n}x)\otimes e^{i\left<q,y\right>}.$$ The zeros $j_{\nu,n}$ satisfy the asymptotic formula $$j_{\nu,n} = \left( n + \tfrac{1}{2}\nu - \tfrac{1}{4}\right)\pi + \mathcal{O}(n^{-1})$$ as $n \rightarrow \infty$. The eigenvalues of the compact operator $(L+1)^{-1/2}$ are therefore $(1+|q|^2 +j_{\nu,n}^2)^{-1/2}$, and if they are listed in descending order $\lambda_1 > \ldots > \lambda_j > \ldots > 0$ (with multiplicity) then $$\lambda_j \leq Cj^{-1/n}.$$ for some $C>0$.
\[lem:schattenlocal\] If $\nu > 0$, then the singular values of the embedding $$J_1: {\mathcal{H}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp}), \quad J_2 : {\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \hookrightarrow {\mathcal{H}}^1({\mathbb{T}^n_\sharp})$$ satisfy $s_j(J_i) < C j^{-1/n}$.
$(1)$ First suppose that $0 < \nu < 1$. Let $\Pi$ denote the orthogonal projection onto $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$, so $J_1 = J_1\Pi + J_1(1-\Pi)$. Since $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$ is the form domain of $L$, the operator $(L+1)^{-1/2}$ is an isomorphism acting $L^2({\mathbb{T}^n_\sharp}) \rightarrow \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$. Write $$J_1 \Pi= J _1 (L + 1)^{-1/2} (L+1)^{1/2} \Pi.$$ The composition $J_1 (L + 1)^{-1/2}$ is self-adjoint and positive definite on $L^2({\mathbb{T}^n_\sharp})$, so its singular values are the $\lambda_j$ which satisy $\lambda_j \leq Cj^{-1/n}$. Furthermore, $(L + 1)^{1/2}\Pi$ is bounded $\mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \rightarrow L^2({\mathbb{T}^n_\sharp})$, so the inequality $s_j(AB) \leq s_j(A)\|B\|$ shows that $s_j(J_1\Pi) \leq Cj^{-1/n}$. On the other hand, the term $J_1(1-\Pi)$ factors through the map in . It is well known that the inclusion $$H^{\nu}({\mathbb{T}^{n-1}}) \times H^{1/2}({\mathbb{T}^{n-1}}) \hookrightarrow H^{\nu-1}({\mathbb{T}^{n-1}}) \times H^{-1/2}({\mathbb{T}^{n-1}})$$ has singular values bounded by $Cj^{-1/(n-1)}$. The inequality $s_{i+j-1}(A+B) \leq s_{i}(A) + s_{j}(B)$ applied to the sum $J_1 = J_1\Pi + J_1(1-\Pi)$ shows that $s_j(J_1) \leq Cj^{-1/n}$. The same method of proof applies when $\nu \geq 1$
$(2)$ Now consider $J_2$. In the case $0 < \nu < 1$, the space ${\mathcal{H}}^2({\mathbb{T}^n_\sharp})$ is identified with the closed subspace $H \subseteq {\mathcal{H}}^1({\mathbb{T}^n_\sharp})^{n+1}\times {\mathcal{H}}^1_*({\mathbb{T}^n_\sharp})$ as in the proof of Lemma \[lem:compactembedding\]. Since the singular values of the embedding ${\mathcal{H}}^1({\mathbb{T}^n_\sharp})^{n+1}\times {\mathcal{H}}^1_*({\mathbb{T}^n_\sharp}) \hookrightarrow L^2({\mathbb{T}^n_\sharp})^{n+2}$ are bounded by $Cj^{-1/n}$, the same is true of the embedding ${\mathcal{H}}^2({\mathbb{T}^n_\sharp})\hookrightarrow {\mathcal{H}}^1({\mathbb{T}^n_\sharp})$. For $\nu \geq 1$ it would also suffice to bound the singular values of $J_2': {\mathcal{H}}^2({\mathbb{T}^n_\sharp}) \cap \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp}) \hookrightarrow \mathring{{\mathcal{H}}}^1({\mathbb{T}^n_\sharp})$. But $J_2' = J_2'(L+1)^{-1/2}(L+1)^{1/2}$, and the singular values of $J_2'(L+1)^{-1/2}$ are again the $\lambda_j \leq Cj^{-1/n}$.
Lemma \[lem:schattenlocal\] easily extends to the case of a compact manifold with boundary.
\[lem:singularvaluesmanifold\] Let ${
\overline{X}
}$ be a compact manifold with boundary. If $\nu > 0$, then the embeddings $$J_1: {\mathcal{H}}^1(X) \hookrightarrow {\mathcal{H}}^0(X), \quad J_2: {\mathcal{H}}^2(X) \hookrightarrow {\mathcal{H}}^1(X)$$ are compact, and the singular values of $J_i$ satisfy $s_j(J_i) \leq Cj^{-1/n}$.
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abstract: |
We present an approach for the efficient calculation of vibrational Raman intensities in periodic systems within density functional theory. The Raman intensities are computed from the second order derivative of the electronic density matrix with respect to a uniform electric field. In contrast to previous approaches, the computational effort required by our method for the evaluation of the intensities is negligible compared to that required for the calculation of vibrational frequencies. As a first application, we study the signature of 3- and 4-membered rings in the the Raman spectra of several polymorphs of SiO$_2$, including a zeolite having 102 atoms per unit cell. \
author:
- Michele Lazzeri and Francesco Mauri
title: 'First principles calculation of vibrational Raman spectra in large systems: signature of small rings in crystalline SiO$_2$'
---
Vibrational Raman spectroscopy [@bruesch] is one of the most widely used optical techniques in materials science. It is a standard method for quality control in production lines. It is very effective in determining the occurrence of new phases or structural changes at extreme conditions (high pressure and temperature), where it is often preferred to the more difficult and less readily available x-ray diffraction experiments based on synchrotron sources [@hemley87]. Moreover, it can be used in the absence of long-range structural order as for liquid or amorphous materials [@sharma81; @hosono01; @pasquarello98]. The theoretical determination of Raman spectra is highly desirable, since it can be used to associate Raman lines to specific microscopic structures.
Density functional theory (DFT) [@DFT] can be used to determine with high accuracy both frequencies and intensities of Raman spectra. Vibrational frequencies can be efficiently determined using [*first order response*]{} [@DFPT; @gonze95]. Within this approach Raman intensities (RI) calculation is also possible, but requires a computational time significantly larger and is not practical for large systems. Thus, while many examples of frequency calculations have been reported so far [@DFPT], RI were predicted from first-principles in a very limited number of cases involving systems with a small number of atoms [@baroni86; @umari01; @putrino02]. In this Letter we show that it is possible to obtain RI in extended solids with a computational cost negligible with respect to that required for the frequency determination. The efficiency of our approach will lead [*ab-initio*]{} calculations to become a routine instrument for the interpretation of experimental Raman data. Our method is based on [*second order response*]{} to DFT. In particular, we compute the second order derivative of the electronic density matrix with respect to a uniform electric field, using pseudopotentials and periodic boundary conditions. As a first application we calculate Raman spectra of several SiO$_2$ polymorphs, including a zeolite having 102 atoms per unit cell [@zeolite].
In a Raman spectrum the peak positions are fixed by the frequencies $\omega_\nu$ of the optical phonons with null wavevector. In non-resonant Stokes Raman spectra of harmonic solids, the peak intensities $I^\nu$ can be computed within the Placzek approximation [@bruesch] as: $$\label{eq1}
I^\nu\propto |{\bf e}_i\cdot\tensor{\bf A}^{\nu}\cdot{\bf e}_s|^2
\frac{1}{\omega_{\nu}}(n_\nu+1),$$ where ${\bf e}_i$ (${\bf e}_s$) is the polarization of the incident (scattered) radiation, $n_\nu=(\exp(\hbar\omega_\nu/k_{\rm B}T)-1)^{-1}$, $T$ is the temperature, and $$\label{eq2}
A^\nu_{lm}=\sum_{k\gamma}
\frac{\partial^3{\cal E}^{\rm el}}{\partial E_l \partial E_m
\partial u_{k\gamma}}
\frac{w^\nu_{k\gamma}}{\sqrt{M_{\gamma}}}.$$ Here ${\cal E}^{\rm el}$ is the electronic energy of the system, $E_l$ is the $l^{th}$ Cartesian component of a uniform electric field, $u_{k\gamma}$ is the displacement of the $\gamma^{th}$ atom in the $k^{th}$ direction, $M_{\gamma}$ is the atomic mass, and $w^\nu_{k\gamma}$ is the orthonormal vibrational eigenmode $\nu$.
Linear response [@DFPT; @gonze95] can be used to determine $\omega_\nu$, ${\bf w}^\nu$, and also the dielectric tensor $\tensor{\bm{\epsilon}}^\infty$ defined as $\epsilon^\infty_{lm}=\delta_{lm}-(4\pi/{\Omega})
\partial^2{\cal E}^{\rm el}/({\partial E_l \partial E_m})$, where $\Omega$ is the cell volume. RI have been computed [@baroni86; @umari01] through Eq. (\[eq1\]), obtaining $\tensor{\bf A}^\nu$ by finite-differences derivation of $\tensor{\bm{\epsilon}}^\infty$ with respect $u_{k\gamma}$. This approach requires 36$N^{\rm at}$ linear response calculations, where $N^{\rm at}$ is the number of atoms. Thus, the scaling of the RI calculation is the same as that of the frequency calculation with a much larger prefactor. This has limited the applications of this approach to small systems. RI have also been computed from the the dynamical autocorrelation functions of $\tensor{\bm{\epsilon}}^\infty$ in a molecular dynamics (MD) run [@putrino02]. This approach also copes with liquids or anharmonic solids, but is very demanding, requiring the calculation of $\tensor{\bm{\epsilon}}^\infty$ at each MD step.
Alternatively, RI can be obtained knowing the second order derivative of the DFT density matrix $\rho=\sum_v |\psi_v\rangle\langle\psi_v|$, being $|\psi_v\rangle$ the normalized occupied Kohn-Sham (KS) eigenstates [@DFT]. In fact, according to the well known Hellmann-Feynman theorem $$\frac{\partial {\cal E}^{\rm el}}{\partial u_{k\gamma}}= 2\,
{\rm Tr}\left\{\rho\frac{\partial v^{\rm ext}}{\partial u_{k\gamma}}\right\},$$ where ${\rm Tr}\{O\}$ is the trace of the operator $O$, and $v^{\rm ext}$ is the external ionic potential (the KS self-consistent potential is $V^{\rm KS}=V^{\rm Hxc}+v^{\rm ext}$, where $V^{\rm Hxc}$ is the sum of the Hartree and the exchange-correlation potential). Thus $$\label{eq6}
\frac{\partial^3{\cal E}^{\rm el}}{\partial E_l \partial E_m\partial u_{k\gamma}}= 2\,
{\rm Tr}\left\{\left(\frac{\partial^2 \rho}{\partial E_l \partial E_m}
\right)\frac{\partial v^{\rm ext}}{\partial u_{k\gamma}}\right\}.$$ The $\partial^2\rho/(\partial E_l \partial E_m)$ calculation requires six second-order calculations, instead of the 36$N^{\rm at}$ first-order calculations needed for the finite-differentiation [@umari01]. Because of this better size-scaling, the $\tensor{\bf A}^\nu$ calculation through Eq. (\[eq6\]) is much more efficient and the time for RI calculation is negligible compared to that for $\omega_\nu$ in large systems.
The approach based on Eq. (\[eq6\]) has already been used in isolated molecules [@frisch86] but never in extended systems. Indeed, in solids the calculation of $\partial^2\!\rho/(\partial E_l \partial E_m)$ is not trivial because the position operator, required by the electric field perturbation, is ill-defined in periodic boundary conditions. Because of this, although a formalism to calculate derivatives of $\rho$ at any order was proposed by Gonze already in 1995 [@gonze95], only very recently Nunes and Gonze [@nunes01] were able to include perturbations due to macroscopic electric fields. To do that, they use the polarization-Berry phase formalism [@kingsmith93], arguing that this concept remains valid in the presence of finite electric fields. This approach has been applied so far to a one dimensional non-self-consistent model [@nunes01]. In the following we give an expression for the second derivative of $\rho$, that does not require the Berry phase formalism to cope with uniform electric fields, and we use it to compute $\tensor{\bf A}^\nu$ in real systems with a DFT self-consistent Hamiltonian.
The derivative of $\rho$ with respect to two generic perturbation parameters $\lambda$ and $\mu$ is: $$\begin{aligned}
\label{sec3}
\frac{\partial^2\rho}{\partial\lambda\partial\mu}&=&
\sum_v
\left(
|P\eta_v^{(\lambda,\mu)}\rangle
\langle\psi_v|+
|P\frac{\partial\psi_v}{\partial\lambda}\rangle
\langle \frac{\partial\psi_v}{\partial\mu}P|+
\right. \nonumber \\
& &
\left.
-\sum_{v'} |\psi_{v'}\rangle
\langle\frac{\partial\psi_{v'}}{\partial\lambda}P|P
\frac{\partial\psi_v}{\partial\mu}\rangle
\langle\psi_v|\right) + cc,\end{aligned}$$ where $P=({\bf 1} -\rho)$ is the projector on the empty state subspace, the sums over $v$ and $v'$ run over the occupied states, and $|\eta_v^{(\lambda,\mu)}\rangle$ are the second derivatives of the occupied KS-orbitals in the parallel-transport gauge [@gonze95]. According to our derivation: $$\begin{aligned}
\label{eq30}
|P\frac{\partial\psi_v}{\partial\lambda}\rangle&=&
\tilde G_v \left[ \frac{\partial V^{\rm KS}}{\partial \lambda},\rho
\right] |\psi_v\rangle, \\
\label{sec1}
|P\eta_v^{(\lambda,\mu)}\rangle &=&
\tilde G_v \left\{~
\frac{\partial^2V^{\rm KS}}{\partial\lambda\partial\mu}
~+~
\left[ \frac{\partial V^{\rm KS}}{\partial\lambda},
\frac{\partial\rho}{\partial\mu} \right]
\right.~+ \nonumber \\
&&
\left.
+~\left[ \frac{\partial V^{\rm KS}}{\partial\mu},
\frac{\partial\rho}{\partial\lambda} \right]
~\right\}|\psi_v\rangle .\end{aligned}$$ Here, $$\tilde G_v = \sum_c \frac{|\psi_c\rangle\langle\psi_c|}{\epsilon_v-\epsilon_c}$$ is the Green function operator projected on the empty states $|\psi_c\rangle$ [@note1], $[A,B]=AB-BA$, and the first derivative of the density matrix is: $$\label{eq31}
\frac{\partial \rho}{\partial \mu} = \sum_v
|P\frac{\partial\psi_v}{\partial\mu}\rangle\langle\psi_v|+cc.$$ Since $\partial V^{\rm KS}/\partial \lambda$ and $\partial^2 V^{\rm KS}/(\partial \lambda \partial \mu)$ depend on $\partial \rho/\partial \lambda$, $\partial \rho/\partial \mu$, and $\partial^2\! \rho/(\partial \lambda \partial \mu)$, Eqs. (\[sec3\]-\[eq31\]), should be solved self-consistently.
The advantage of the present formulation, compared to that of Ref. [@gonze95], lies in the introduction of the commutators of Eqs. (\[eq30\],\[sec1\]). Thanks to the commutators, all the quantities needed with our formalism are well defined in an extended insulator, even if the perturbation $\mu$ or $\lambda$ are the component $E_l$ of a a uniform electric field, i.e. if $\partial V^{\rm KS}/\partial \lambda =
-e r_l + \partial V^{\rm Hxc}/\partial{E_l}$ [@nota2], being $r_l$ the $l^{th}$ Cartesian component of the position operator ${\bf r}$, and $e$ the electron charge. In particular, in an insulator, the commutators $[{\bf r},\rho]$ and $[{\bf r}, \partial\rho/\partial \mu]$ in Eqs. (\[eq30\],\[sec1\]) are well defined, bounded operators, since the density matrix is localized ($\langle {\bf r}'' |\rho| {\bf r}' \rangle$ goes to zero exponentially for $|{\bf r}''-{\bf r}'|\rightarrow \infty$).
Finally, in a periodic system, the right-hand side of Eq. (\[sec1\]) can be easily computed in terms of the $|u^{\bf k}_i\rangle$, that are the periodic parts of the Bloch-wavefunctions $|\psi^{\bf k}_i\rangle$ with reciprocal-lattice vector ${\bf k}$, using the substitutions: $$\begin{aligned}
\langle\psi^{\bf k}_c|\left[r_l,\rho\right]|
\psi^{\bf k}_v\rangle= i
\sum_{v'} \langle u^{\bf k}_c|
\frac{\partial |u^{\bf k}_{v'}\rangle
\langle u^{\bf k}_{v'}|}{\partial k_l} |u^{\bf k}_v\rangle \\
\langle\psi^{\bf k}_{c}|\left[r_l,\frac{\partial \rho}{\partial E_m}\right]|
\psi^{\bf k}_{v}\rangle= i\sum_{v'}
\langle u^{\bf k}_{c}|
\frac{\partial |P_{{\bf k}}\frac{\partial u^{\bf k}_{v'}}{\partial E_m}\rangle
\langle u^{\bf k}_{v'}|}{\partial k_l} |u^{\bf k}_{v}\rangle
\label{trics}\end{aligned}$$ where $l$ and $m$ are Cartesian indexes, $c$ is an empty band index, $v$ and $v'$ are occupied band indexes, and $P_{{\bf k}}$ is the projector on the empty subspace of the point ${\bf k}$. In our implementation, the derivative with respect to $k_l$ in the right-hand side of Eq. (\[trics\]) is computed numerically by finite-differences, using an expression independent from the arbitrary wavefunction-phase, as in Refs. [@dalcorso94; @nunes01].
![Vibrational Raman spectra of various SiO$_2$ polymorph powders. Measurements are from Refs. [@coesite]. Theoretical frequencies are rescaled by +5%, and the spectra are convoluted with a uniform Gaussian broadening having 4.0 cm$^{-1}$ width.[]{data-label="fig1"}](Fig1.ps){width="85mm"}
$N$ 2 10 28 60 110 182
------------------------------------ ------- ------ ------ ------ ------ ------
$\gamma_{\rm SOR}$ 8.54 5.30 5.32 5.39 5.40 5.40
$\gamma_{\rm FD}$ 18.99 7.09 5.69 5.45 5.41 5.40
$\gamma_{\rm FD}$ Ref. [@baroni86] 7.10
: Raman activity in Si computed with our approach ($\gamma_{\rm SOR}$), and by finite differences ($\gamma_{\rm FD}$). $N$ is the number of inequivalent k-points. []{data-label="tab1"}
We test our approach on Si in the diamond phase, where the Raman activity is determined by $\gamma = a \partial\epsilon^{\infty}_{11}/\partial u$ [@baroni86], where $a= 10.20~a.u.$ is the lattice spacing and $u$ the displacement of one atom along the $(1,1,1)$ direction [@technicalities]. We compute $\gamma$ for various grids of k-points, using both our second order response method ($\gamma_{\rm SOR}$) and by finite differentiation with respect to the atomic displacement ($\gamma_{\rm FD}$), Tab. \[tab1\]. At convergence the two approaches are completely equivalent.
As a second application, we consider tetrahedral SiO$_2$. In this class of materials, that includes the all-silica zeolites, the quartz, cristobalite, tridymite and coesite polymorphs of SiO$_2$, and vitreous silica (v-SiO$_2$), each Si atom is tetrahedrally coordinated to four O atoms and each O atom is bonded to two Si atoms. The properties of these systems can be effectively described in terms of the $n$-membered rings ($n$-MRs) of tetrahedra contained in their structure [@sharma81; @hosono01; @pasquarello98]. E.g., a clear correlation between the presence of 3- and 4-MRs and the degradation of optical v-SiO$_2$ fibers under UV radiation has been observed [@hosono01]. In the v-SiO$_2$ Raman spectra the two sharp peaks at 490 cm$^{-1}$ ($D_1$ line) and 606 cm$^{-1}$ ($D_2$ line), have been attributed to the breathing mode (BM) of the O atoms towards the ring center of 4-MRs and 3-MRs, respectively [@sharma81]. This attribution has been confirmed by DFT vibrational frequency calculations [@pasquarello98]. The attribution would be further supported by experimental measurements on well characterized crystalline polymorphs containing 3- and 4-MRs. However, the strong Raman peak at 520 cm$^{-1}$ in coesite, a phase that contains 4-MRs, is shifted by 30 cm$^{-1}$ with respect to the $D_1$ line in v-SiO$_2$, and no Raman measurements has been published on the H-ZSM-18 zeolite, that is the only known SiO$_2$ crystalline polymorph with 3-MRs [@zeolite]. Interestingly this zeolite contains 4-MRs as well.
To clarify this topic, we compute the Raman spectra of $\alpha$-quartz, coesite, $\alpha$-cristobalite, and H-ZSM-18 [@technicalities; @technicalities2]. In Fig. \[fig1\], we compare our results with the available experimental spectra [@coesite]. The vibrational frequencies are systematically underestimated by 5% by our calculation. To simplify the comparison with the experiments, in Fig. \[fig1\] and \[fig2\], the theoretical frequencies are multiplied by a scaling factor of 1.05. The ability of the method in reproducing quantitatively all the measured features is evident.
In order to associate Raman peaks of Fig. \[fig1\] to the small-ring BMs, we project the vibrational eigenmode ${\bf w}^\nu$ on the subspace generated by the BMs of a given kind of rings, $\cal R$, and on the corresponding complementary subspace, $\bf \bar{\cal R}$. We use the two resulting projected vectors to decompose $\tensor {\bf A}^\nu$ so that $\tensor {\bf A}^\nu=\tensor {\bf A}^\nu_{\cal R}+\tensor {\bf A}^\nu_{\bf \bar{\cal R}}$. Since $I^\nu$ is quadratic in $\tensor{\bf A}^\nu$, see Eq. (\[eq1\]), $I^\nu=
I^\nu_{\cal R} + I^\nu_{\bf \bar{\cal R}}+ I^\nu_{\rm overlap}$, where $I^\nu_{\rm overlap}$ is the term bilinear in $\tensor {\bf A}^\nu_{\cal R}$ and $\tensor {\bf A}^\nu_{\bf \bar{\cal R}}$. A Raman peak can be associated to a ring BM (i.e. the Raman activity is solely due to the BM ) if, and only if, $I^\nu_{\bf {\cal R}}\gg |I^\nu_{\rm overlap}|$.
The structure of H-ZSM-18 [@zeolite] contains two equivalent 3-MRs and two kinds of 4-MRs which we will call 4-MRs$_0$, and 4-MRs$_1$ [@rings]. In Fig. \[fig2\], we show the projected Raman spectra of the zeolite and the coesite. In the H-ZSM-18 spectrum, the peaks at 485, and 615 cm$^{-1}$ are very well described by the BM of 4-MRs$_0$ and 3-MRs, respectively. A direct analysis of the vibrational eigenmodes shows that both BMs are decoupled from other modes. The frequencies of the two peaks are very close to those of the measured $D_1$ and $D_2$ lines in v-SiO$_2$ (490, and 606 cm$^{-1}$), thus confirming that these lines are due to rings BMs [@sharma81; @pasquarello98]. However, the presence of small-MRs in a structure does not guarantee, in general, the occurrence of completely decoupled BMs. This is the case of the 4-MRs in coesite and the 4-MRs$_1$ in the zeolite, whose BMs exhibit a large $|I^\nu_{\rm overlap}|$, see Fig. \[fig2\]. These overlaps imply the existence of a coupling with other modes, that, in turn, explains the 30 cm$^{-1}$ difference between the 4-MRs frequency of coesite and that of the $D_1$ line of v-SiO$_2$. A comparable frequency shift from the $D_1$ line is observed, with opposite sign, for the 4-MRs$_1$ BMs in the zeolite.
![Raman intensities projected on the breathing modes of various rings labeled 3-MRs, and 4-MRs$_x$ (see the text). For clarity, the overlap intensity ($I^\nu_{\rm overlap}$ in the text) is shifted vertically.[]{data-label="fig2"}](Fig2.ps){width="85mm"}
In conclusion, with the aim of building an instrument for the routine interpretation of Raman spectra, we developed a method for the efficient calculation of Raman intensity. We computed the Raman spectra of SiO$_2$ polymorphs containing up to 102 atoms. We found that: i) not all the small-membered-rings have decoupled breathing modes, ii) the H-ZSM-18 zeolite provides decoupled breathing mode of 4- and 3-membered rings, whose frequencies nicely coincide with the $D_1$ and $D_2$ lines of vitreous silica. An experimental determination of the Raman spectra of this zeolite can thus provide an experimental calibration for the determination of the density of decoupled small membered rings in vitreous silica.
Calculations were performed at IDRIS supercomputing center. Our approach was implemented in the PWSCF code [@pwscf].
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The operator $\tilde G_v$ can be efficiently applied to any vector by solving a linear system, following Refs. [@DFPT; @gonze95].
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|
---
abstract: |
We study the structure of 01-matrices avoiding a pattern $P$ as an interval minor. We focus on critical $P$-avoiders, i.e., on the $P$-avoiding matrices in which changing a 0-entry to a 1-entry always creates a copy of $P$ as an interval minor.
Let $Q$ be the $3\times 3$ permutation matrix corresponding to the permutation $231$. As our main result, we show that for every pattern $P$ that has no rotated copy of $Q$ as interval minor, there is a constant $c_P$ such that any row and any column in any critical $P$-avoiding matrix can be partitioned into at most $c_P$ intervals, each consisting entirely of 0-entries or entirely of 1-entries. In contrast, for any pattern $P$ that contains a rotated copy of $Q$, we construct critical $P$-avoiding matrices of arbitrary size $n\times n$ having a row with $\Omega(n)$ alternating intervals of 0-entries and 1-entries.
author:
- 'Vít Jelínek [^1]'
- 'Stanislav Ku[č]{}era [^2]'
bibliography:
- 'matrices.bib'
title: 'On the structure of matrices avoiding interval-minor patterns[^3]'
---
Introduction
============
A *binary matrix* is a matrix with entries equal to 0 or 1. All matrices considered in this paper are binary. The study of extremal problems of binary matrices has been initiated by the papers of Bienstock and Győri [@bienstock] and of Füredi [@furedi]. Since these early works, most of the research in this area has focused on the concept of forbidden submatrices: a matrix $M$ is said to contain a pattern $P$ as a submatrix if we can transform $M$ into $P$ by deleting some rows and columns, and by changing 1-entries into 0-entries. This notion of submatrix is a matrix analogue of the notion of subgraph in graph theory.
The main problem in the study of pattern-avoiding matrices is to determine the extremal function ${\mathrm{ex}}(n;P)$, defined as the largest number of 1-entries in an $n\times n$ binary matrix avoiding the pattern $P$ as submatrix. This is an analogue of the classical Turán-type problem of finding a largest number of edges in an $n$-vertex graph avoiding a given subgraph. Despite the analogy, the function ${\mathrm{ex}}(n;P)$ may exhibit an asymptotic behaviour not encountered in Turán theory. For instance, for the pattern[^4] $P={\left({\begin{smallmatrix}\bullet&&\bullet&\\&\bullet&&\bullet\end{smallmatrix}}
\right)}$ Füredi and Hajnal [@furedi_hajnal] proved that ${\mathrm{ex}}(n;P)=\Theta(n\alpha(n))$, where $\alpha(n)$ is the inverse of the Ackermann function.
The asymptotic behaviour of ${\mathrm{ex}}(n;P)$ for general $P$ is still not well understood. Füredi and Hajnal [@furedi_hajnal] posed the problem of characterising the *linear* patterns, i.e., the patterns $P$ satisfying ${\mathrm{ex}}(n;P)=O(n)$. Marcus and Tardos [@marcus] proved that ${\mathrm{ex}}(n;P)=O(n)$ whenever $P$ is a *permutation matrix*, i.e., $P$ has exactly one 1-entry in each row and each column. This result, combined with previous work of Klazar [@klazar_fh], has confirmed the long-standing Stanley–Wilf conjecture. However, the problem of characterising linear patterns is still open despite a number of further partial results [@crowdmath; @fulek; @keszegh; @tardos; @geneson; @pettie_degrees].
Fox [@fox] has introduced a different notion of containment among binary matrices, based on the concept of interval minors. Informally, a matrix $M$ contains a pattern $P$ as an interval minor if we can transform $M$ into $P$ by contracting adjacent rows or columns and changing 1-entries into 0-entries; see Section \[sec-prelims\] for the precise definition. In this paper, we mostly deal with containment and avoidance of interval minors rather than submatrices. Therefore, the phrases *$M$ avoids $P$* or *$M$ contains $P$* always refer to avoidance or containment of interval minors, and the term *$P$-avoider* always refers to a matrix that avoids $P$ as interval minor.
In analogy with ${\mathrm{ex}}(n;P)$, it is natural to consider the corresponding extremal function ${\mathrm{ex}_{\preccurlyeq}}(n;P)$ as the largest number of 1-entries in an $n\times
n$ matrix that avoids $P$ as an interval minor. If $M$ contains $P$ as a submatrix, it also contains it as an interval minor, and therefore ${\mathrm{ex}_{\preccurlyeq}}(n;P)\le{\mathrm{ex}}(n;P)$. Moreover, it can be easily seen that for a permutation matrix $P$ the two notions of containment are equivalent, and hence ${\mathrm{ex}_{\preccurlyeq}}(n;P)={\mathrm{ex}}(n;P)$.
Fox [@fox] used interval minors as a key tool in his construction of permutation patterns with exponential Stanley–Wilf limits. In view of the results of Cibulka [@cibulka09], this is equivalent to constructing a permutation matrix $P$ for which the limit of the ratio ${\mathrm{ex}}(n;P)/n$ (which is equal to ${\mathrm{ex}_{\preccurlyeq}}(n;P)/n$) is exponential in the size of $P$.
Even before the work of Fox, interval minors have been implicitly used by Guillemot and Marx [@gm], who proved that a permutation matrix $M$ which avoids as interval minor a fixed complete square pattern (i.e., a square pattern with all entries equal to 1) admits a type of recursive decomposition of bounded complexity. This result can be viewed as an analogue of the grid theorem from graph theory [@rs], which states that graphs avoiding a large square grid as a minor have bounded tree-width. Guillemot and Marx used their result on forbidden interval minors to design a linear-time algorithm for testing the containment of a fixed pattern in a permutation.
Subsequent research into interval-minor avoidance has focused on avoiders of a complete matrix. In particular, Mohar et al. [@mohar] obtained exact values for the extremal function for matrices simultaneously avoiding a complete pattern of size $2\times \ell$ and its transpose, and they obtained bounds for patterns of size $3\times \ell$. Their results were further generalised by Mao et al. [@mao] to a multidimensional setting.
While the functions ${\mathrm{ex}}(n;P)$ exhibit diverse forms of asymptotic behaviour, the function ${\mathrm{ex}_{\preccurlyeq}}(n;P)$ is linear for every nontrivial pattern $P$. This is a consequence of the Marcus–Tardos theorem and the fact that any binary matrix is an interval minor of a permutation matrix; see Fox [@fox]. Therefore, in the interval-minor avoidance setting, it is not as natural to classify patterns by the growth of ${\mathrm{ex}_{\preccurlyeq}}(n;P)$ alone as in the submatrix avoidance setting.
In our paper, we instead classify the patterns $P$ based on the structure of the $P$-avoiders. We introduce the notion of *line complexity* of a binary matrix $M$, as the largest number of maximal runs of consecutive 0-entries in a single row or a single column of $M$. We focus on the *critical* $P$-avoiders, which are the matrices that avoid $P$ as interval minor, but lose this property when any 0-entry is changed into a 1-entry.
Our main result is a sharp dichotomy for line complexity of critical $P$-avoiders. Let $Q_1,\dotsc,Q_4$ be defined as follows: $$Q_1={\left({\begin{smallmatrix} &\bullet& \\\bullet& & \\ & &\bullet\end{smallmatrix}}
\right)},\ Q_2={\left({\begin{smallmatrix} &\bullet& \\ &
&\bullet\\\bullet& & \end{smallmatrix}}
\right)},\ Q_3={\left({\begin{smallmatrix}\bullet& & \\ & &\bullet\\ &\bullet&
\end{smallmatrix}}
\right)}\text{ and }
Q_4={\left({\begin{smallmatrix} & &\bullet\\\bullet& & \\ &\bullet& \end{smallmatrix}}
\right)}.$$ We show that if a pattern $P$ avoids the four patterns $Q_i$ as interval minors (or equivalently, as submatrices), then the line-complexity of every critical $P$-avoider is bounded by a constant $c_P$ depending only on $P$. On the other hand, if $P$ contains at least one of the $Q_i$, then there are critical $P$-avoiders of size $n\times n$ with line complexity $\Omega(n)$, for any $n$.
After properly introducing our terminology and proving several simple basic facts in Section \[sec-prelims\], we devote Section \[sec-bounded\] to the statement and proof of our main result. In Section \[sec-further\], we discuss the possibility of extending our approach to general minor-closed matrix classes, and present several open problems.
Preliminaries {#sec-prelims}
=============
#### Basic notation.
For integers $m$ and $n$, we let $[m,n]$ denote the set $\{m,m+1,\dotsc,n\}$. We will also use the notation $[m,n)$ for the set $[m,n-1]$, $(m,n]$ for the set $[m+1,n]$, and $[n]$ for $[1,n]$. We will avoid using $(m,n)$ for $[m+1,n-1]$, however; instead, we will use the notation $(m,n)$ to denote ordered pairs of integers.
We write ${{\{0,1\}}^{m\times n}}$ for the set of binary matrices with $m$ rows and $n$ columns. We will always assume that rows of matrices are numbered top-to-bottom starting with 1, that is, the first row is the topmost.
For a matrix $M\in{{\{0,1\}}^{m\times n}}$, we let $M(i,j)$ denote the value of the entry in row $i$ and column $j$ of $M$. We say that the pair $(i,j)$ is a *1-entry* of $M$ if $M(i,j)=1$, otherwise it is a *0-entry*. The set of 1-entries of a matrix $M\in{{\{0,1\}}^{m\times n}}$ is called the *support* of $M$, denoted by $\operatorname{supp}(M)$; formally, $\operatorname{supp}(M)=\{(i,j)\in[m]\times[n];\;M(i,j)=1\}$.
We say that a binary matrix $M'$ *dominates* a binary matrix $M$, if the two matrices have the same number of rows and the same number of columns, and moreover, $\operatorname{supp}(M)\subseteq\operatorname{supp}(M')$. In other words, $M$ can be obtained from $M'$ by changing some 1-entries into 0-entries.
For a matrix $M\in{{\{0,1\}}^{m\times n}}$ and for a set of row-indices $R\subseteq[m]$ and column-indices $C\subseteq[n]$, we let $M[R\times C]$ denote the submatrix of $M$ induced by the rows in $R$ and columns in $C$. More formally, if $R=\{r_1<r_2<\dotsb<r_k\}$ and $C=\{c_1<c_2<\dotsb<c_\ell\}$, then $M[R\times
C]$ is a matrix $P\in{{\{0,1\}}^{k\times \ell}}$ such that $P(i,j)=M(r_i,c_j)$ for every $(i,j)\in[k]\times[\ell]$.
A *line* in a matrix $M$ is either a row or a column of $M$. We view a line as a special case of a submatrix. For instance, the $i$-th row of a matrix $M\in{{\{0,1\}}^{m\times n}}$ is the submatrix $M[\{i\}\times[n]]$. A *horizontal interval* is a submatrix formed by consecutive entries belonging to a single row, i.e., a submatrix of the form $M[\{i\}\times[j_1,j_2]\}$ where $i$ is a row index and $j_1,j_2$ are column indices. Vertical intervals are defined analogously.
We say that a submatrix of $M$ is *empty* if it does not contain any 1-entries.
For a matrix $M\in{{\{0,1\}}^{m\times n}}$ and an entry $e\in[m]\times[n]$, we let $M\Delta e$ denote the matrix obtained from $M$ by changing the value of the entry $e$ from 0 to 1 or from 1 to 0.
#### Interval minors.
A *row contraction* in a matrix $M\in{{\{0,1\}}^{m\times n}}$ is an operation that replaces a pair of adjacent rows $r$ and $r+1$ by a single row, so that the new row contains a 1-entry in a column $j$ if and only if at least one of the two original rows contained a 1-entry in column $j$. Formally, the row contraction transforms $M$ into a matrix $M'\in{\{0,1\}}^{(m-1)\times n}$ whose entries are defined by $$M'(i,j)=\begin{cases}
M(i,j)\text{ if }i<r,\\
\max\{M(r,j),M(r+1,j)\}\text{ if }i=r,\\
M(i+1,j)\text{ if }i>r.
\end{cases}$$ A column contraction is defined analogously.
We say that a matrix $P\in{{\{0,1\}}^{k\times \ell}}$ is an *interval minor* of a matrix $M\in{{\{0,1\}}^{m\times n}}$, denoted $P{\preccurlyeq}M$, if we can transform $M$ by a sequence of row contractions and column contractions to a matrix $P'\in{{\{0,1\}}^{k\times \ell}}$ that dominates $P$. When $P$ is an interval minor of $M$, we also say that $M$ *contains* $P$, otherwise we say that $M$ *avoids* $P$, or $M$ is $P$-avoiding.
![A pattern $P$ and a matrix $M$ that contains $P$. The thick lines indicate a partition of $M$ containing $P$, and the shaded 1-entries form an image of $P$.[]{data-label="fig-contain"}](contain)
There are several alternative ways to define interval minors. One possible approach uses the concept of matrix partition. For $P\in{{\{0,1\}}^{k\times \ell}}$ and $M\in{{\{0,1\}}^{m\times n}}$, a *partition of $M$ containing $P$* is the sequence of row indices $r_0,r_1,\dotsc,r_k$ and column indices $c_0,c_1,\dotsc,c_\ell$ with $0\le r_0<r_1<\dotsb<r_k\le m$ and $0\le c_0<c_1<\dotsb<c_\ell\le n$, such that for every 1-entry $(i,j)$ of $P$, the submatrix $M[(r_{i-1},r_i]\times(c_{j-1},c_j]]$ has at least one 1-entry. See Figure \[fig-contain\].
An *embedding* of a matrix $P\in{{\{0,1\}}^{k\times \ell}}$ into a matrix $M\in{{\{0,1\}}^{m\times n}}$ is a function $\phi\colon [k]\times[\ell]\to[m]\times[n]$ with the following properties:
- If $e=(i,j)$ is a 1-entry of $P$, then $\phi(e)$ is a 1-entry of $M$.
- Let $e_1=(i_1,j_1)$ and $e_2=(i_2,j_2)$ be two entries of $P$, and suppose that $\phi(e_1)=(i^*_1,j^*_1)$ and $\phi(e_2)=(i^*_2,j^*_2)$. If $i_1<i_2$ then $i^*_1<i^*_2$, and if $j_1<j_2$ then $j^*_1<j^*_2$.
Notice that in an embedding $\phi$ of $P$ into $M$, two entries of $P$ belonging to the same row may be mapped to different rows of $M$, and similarly for columns.
In practice, it is often inconvenient and unnecessary to specify completely an embedding of $P$ into $M$. In particular, it is usually unnecessary to specify the image of all the 0-entries in $P$. This motivates the notion of partial embedding, which we now formalise. Consider again binary matrices $P\in{{\{0,1\}}^{k\times \ell}}$ and $M\in{{\{0,1\}}^{m\times n}}$. Let $S$ be a nonempty subset of $[k]\times[\ell]$. We say that a function $\psi\colon S\to[m]\times[n]$ is a *partial embedding* of $P$ into $M$ if the following holds:
- If $e=(i,j)$ is a 1-entry of $P$, then $e$ is in $S$ and $\psi(e)$ is a 1-entry of $M$.
- An entry $e=(i,j)\in S$ is mapped by $\psi$ to an entry $\psi(e)=(i^*,j^*)$ of $M$ satisfying the following inequalities: $i\le i^*$, $j\le j^*$, $k-i\le m-i^*$ and $\ell-j\le n-j^*$. Informally, the entry $\psi(e)$ is at least as far from the top, left, bottom and right edge of the corresponding matrix as the entry $e$.
- Let $e_1=(i_1,j_1)$ and $e_2=(i_2,j_2)$ be two entries in $S$, with $\psi(e_1)=(i^*_1,j^*_1)$ and $\psi(e_2)=(i^*_2,j^*_2)$. If $i_1< i_2$ then $i_2-i_1\le
i^*_2-i^*_1$, and if $j_1<j_2$ then $j_2-j_1\le j^*_2-j^*_1$.
For a partial embedding $\psi$ of a pattern $P$ into a matrix $M$, the *image of $P$* (with respect to $\psi$) is the set of entries $\{\psi(e);\; e\in\operatorname{supp}(P)\}$ in the matrix $M$. Note that all the entries in the image of $P$ are 1-entries.
\[lem-minor\] For matrices $P\in{{\{0,1\}}^{k\times \ell}}$ and $M\in{{\{0,1\}}^{m\times n}}$ the following properties are equivalent.
- $P$ is an interval minor of $M$.
- $M$ has a partition containing $P$.
- $P$ has an embedding into $M$.
- $P$ has a partial embedding into $M$.
We will prove the implications $2\implies 1\implies 3\implies 4\implies 2$.
To see that 2 implies 1, suppose $M$ has a partition containing $P$, determined by row indices $r_0,r_1,\dotsc,r_k$ and column indices $c_0,c_1,\dotsc,c_\ell$, where we may assume that $r_0=c_0=0$, $r_k=m$ and $c_\ell=n$. We may then contract the rows from each interval of the form $(r_{i-1},r_i]$ into a single row, and contract the columns from each interval $(c_{i-1},c_i]$ to a single column, to obtain a matrix $P'\in{{\{0,1\}}^{k\times \ell}}$ that dominates $P$.
To see that 1 implies 3, suppose that $P$ is an interval minor of $M$. This means that there is a sequence of matrices $M_0, M_1, M_2,\dotsc, M_s$ with $s=(m-k)+(n-\ell)$, where $M_0\in{{\{0,1\}}^{k\times \ell}}$ is a matrix that dominates $P$, and for each $i\in[s]$, the matrix $M_{i-1}$ can be obtained from $M_i$ by contracting a pair of adjacent rows or columns. We can then easily observe that for every $i=0,1,\dotsc,s$ there is an embedding $\phi_i$ of $P$ into $M_i$. Indeed, reasoning by induction, the embedding $\phi_0$ is the identity map, and for a given $i\in[s]$, if there is an embedding $\phi_{i-1}$ of $P$ into $M_{i-1}$, then an embedding $\phi_i$ can be obtained by an obvious modification of $\phi_{i-1}$.
Clearly, 3 implies 4, since every embedding is also a partial embedding.
To show that 4 implies 2, assume that $\psi\colon S\to[m]\times[n]$ is a partial embedding of $P$ into $M$. We will define a sequence of row indices $0\le r_0<r_1<\dotsb<r_k\le m$ with these two properties:
- For each entry $e\in S$ that belongs to row $i$ of $P$, the entry $\psi(e)$ belongs to a row $i^*$ of $M$ for some $i^*\in(r_{i-1},r_i]$.
- If $S$ contains at least one entry from row $i$ in $P$, then $S$ contains an entry $e$ in row $i$ such that $\psi(e)$ is in row $r_i$ of $M$.
We define the numbers $r_i$ inductively, starting with $r_0=0$. Suppose that $r_0,\dotsc,r_{i-1}$ have been defined, for some $i\ge 1$. If $S$ contains no entry from row $i$ of $P$, define $r_i=r_{i-1}+1$. On the other hand, if $S$ contains an entry from row $i$, we let $r_i$ be the largest row index of $M$ such that $\psi$ maps an entry from row $i$ of $P$ to an entry in row $r_i$ of $M$. Notice that any entry $e\in S$ that does not belong to the first $i$ rows of $P$ must be mapped by $\psi$ to an entry strictly below row $r_i$ of $M$, otherwise $\psi$ would not satisfy the properties of a partial embedding.
In an analogous way, we also define a sequence of column indices $0\le
c_0<c_1<\dotsb<c_{\ell}\le n$. These sequences will satisfy that for every $e=(i,j)\in S$ we have $\psi(e)\in (r_{i-1},r_i]\times(c_{j-1},c_j]$. Since $\psi$ is a partial embedding, $S$ contains all the 1-entries of $P$, and $\psi$ maps these 1-entries to 1-entries of $M$. In particular, the sequences $(r_i)_{i=0}^k$ and $(c_j)_{j=0}^\ell$ form a partition of $M$ containing $P$.
#### Minor-closed classes.
For a matrix $P$, we let ${Av_{{\preccurlyeq}}\left(P\right)}$ denote the set of all binary matrices that do not contain $P$ as an interval minor. We call the matrices in ${Av_{{\preccurlyeq}}\left(P\right)}$ the *avoiders* of $P$, or *$P$-avoiders*.
More generally, if ${\mathcal{F}}$ is a set of matrices, we let ${Av_{{\preccurlyeq}}\left({\mathcal{F}}\right)}$ denote the set of binary matrices that avoid all elements of ${\mathcal{F}}$ as interval minors.
We call a set ${\mathcal{C}}$ of binary matrices a *minor-closed class* (or just *class*, for short) if for every matrix $M\in{\mathcal{C}}$, all the interval minors of $M$ are in ${\mathcal{C}}$ as well. Clearly, ${Av_{{\preccurlyeq}}\left({\mathcal{F}}\right)}$ is a class, and for every class ${\mathcal{C}}$ there is a (possibly infinite) set ${\mathcal{F}}$ such that ${\mathcal{C}}={Av_{{\preccurlyeq}}\left({\mathcal{F}}\right)}$. A *principal class* is a class of matrices determined by a single forbidden pattern, i.e., a class of the form ${Av_{{\preccurlyeq}}\left(P\right)}$ for a matrix $P$.
For a class ${\mathcal{C}}$ of matrices, we say that a matrix $M\in{\mathcal{C}}$ is *critical for ${\mathcal{C}}$* if the change of any 0-entry of $M$ to a 1-entry creates a matrix that does not belong to ${\mathcal{C}}$. In other words, $M\in{\mathcal{C}}$ is critical for ${\mathcal{C}}$ if it is not dominated by any other matrix in ${\mathcal{C}}$. For a pattern $P$, we let ${Av_{crit}\left(P\right)}$ be the set of critical matrices for ${Av_{{\preccurlyeq}}\left(P\right)}$, and similarly for a set of patterns ${{\mathcal{F}}}$, ${Av_{crit}\left({\mathcal{F}}\right)}$ is the set of all critical matrices for ${Av_{{\preccurlyeq}}\left({\mathcal{F}}\right)}$.
Simple examples of $P$-avoiders {#ssec-example}
-------------------------------
We conclude this section by presenting several examples of avoiders of certain simple patterns. These examples will play a role in Section \[sec-bounded\], in the proof of our main result. We begin with a very simple example, which we present without proof.
\[obs-Rk\] Let $R_k$ be the matrix with 1 row and $k$ columns, whose every entry is a 1-entry (see Figure \[fig-rkdk\]). A matrix $M\in{{\{0,1\}}^{m\times n}}$ avoids $R_k$ if and only if $M$ has at most $k-1$ nonempty columns. Consequently, $M$ is a critical $R_k$-avoider if and only if $\operatorname{supp}(M)$ is a union of $\min\{k-1,n\}$ columns.
![The patterns $R_k$, $D_k$ and ${\overline{D}}_k$.[]{data-label="fig-rkdk"}](rkdk)
Next, we will consider the diagonal patterns $D_k\in{\{0,1\}}^{k\times k}$, defined by $\operatorname{supp}(D_k)=\{(i,i);\; i\in[k]\}$, and their mirror image ${\overline{D}}_k\in
{\{0,1\}}^{k\times k}$, defined by $\operatorname{supp}({\overline{D}}_k)=\{(i,k-i+1);\; i\in[k]\}$ (see again Figure \[fig-rkdk\]). To describe the avoiders of these patterns, we first introduce some terminology.
Let $e$ and $e'$ be two entries of a matrix $M$. An *increasing walk* from $e$ to $e'$ in $M$ is a set of entries $W=\{e_i=(r_i,c_i);\; i=0,\dotsc,t\}$ such that $e_0=e$, $e_t=e'$, and for every $i\in[t]$ we have either $r_i=r_{i-1}$ and $c_i=c_{i-1}+1$ (that is, $e_i$ is to the right of $e_{i-1}$), or $r_i=r_{i-1}-1$ and $c_i=c_{i-1}$ (that is, $e_i$ is above $e_{i-1}$). A *decreasing walk* is defined analogously, except now $e_i$ is either to the right or below $e_{i-1}$.
We say a matrix $M$ is an *increasing matrix* if $\operatorname{supp}(M)$ is a subset of an increasing walk. A *decreasing matrix* is defined analogously. See Figure \[fig-walk\].
![An increasing matrix (left) and a decreasing matrix (right). The shaded entries form an increasing and a decreasing walk in the respective matrices.[]{data-label="fig-walk"}](walk){width="90.00000%"}
\[pro-diag\] A matrix $M\in{{\{0,1\}}^{m\times n}}$ avoids the pattern $D_k$ if and only if $M$ contains $k-1$ increasing walks $W_1,\dotsc,W_{k-1}$ from $(m,1)$ to $(1,n)$ such that $$\operatorname{supp}(M)\subseteq W_1\cup W_2\cup\dotsb\cup W_{k-1}.$$
Clearly, if $M$ contains $D_k$, then $M$ has $k$ 1-entries no two of which can belong to a single increasing walk, and therefore $\operatorname{supp}(M)$ cannot be covered by $k-1$ increasing walks.
Suppose now that $M$ avoids $D_k$. Consider a partial order $\triangleleft$ on the set $\operatorname{supp}(M)$, defined as $(i,j)\triangleleft(i',j')\iff i<i'$ and $j<j'$. Since $M$ avoids $D_k$, this order has no chain of length $k$. By the classical Dilworth theorem [@dilworth], $\operatorname{supp}(M)$ is a union of $k-1$ antichains of $\triangleleft$. We may easily observe that each antichain of $\triangleleft$ is contained in an increasing walk.
Proposition \[pro-diag\] shows, in particular, that a matrix $M$ avoids the pattern $D_2={\left({\begin{smallmatrix}\bullet& \\ &\bullet\end{smallmatrix}}
\right)}$ if and only if $M$ is an increasing matrix. By symmetry, $M$ avoids ${\overline{D}}_2$ if and only if it is a decreasing matrix.
Another direct consequence of the proposition is the following corollary, describing the structure of critical $D_k$-avoiders.
\[cor-diag\] A critical $D_k$-avoiding matrix $M$ contains $k-1$ increasing walks $W_1,\dotsc,W_{k-1}$ from $(m,1)$ to $(1,n)$ such that $\operatorname{supp}(M)= W_1\cup W_2\cup\dotsb\cup W_{k-1}$.
Note that Corollary \[cor-diag\] only gives a necessary condition for a matrix to be a critical $D_k$-avoider, therefore it is not a characterisation of critical $D_k$-avoiders. With only a little bit of extra effort, we could state and prove such a characterisation, but we omit doing so, as we do not need it for our purposes.
A simple but useful observation is that adding an empty row or column to the boundary of a pattern affects the $P$-avoiders in a predictable way. We state it here without proof.
\[obs-empty\] Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern, and let $P'\in{\{0,1\}}^{k\times(\ell+1)}$ be the pattern obtained by appending an empty column to $P$; in other words, we have $P'[[k]\times[\ell]]=P$, and the last column of $P'$ is empty. Then a matrix $M\in{{\{0,1\}}^{m\times n}}$ avoids $P'$ if and only if the matrix obtained by removing the last column from $M$ avoids $P$. Consequently, $M$ is in ${Av_{crit}\left(P'\right)}$ if and only if all the entries in the last column of $M$ are 1-entries, and the preceding columns form a matrix from ${Av_{crit}\left(P\right)}$. Analogous properties hold for a pattern $P''$ obtained by prepending an empty column in front of all the columns of $P$, and also for rows instead of columns.
Line complexity {#sec-bounded}
===============
In the previous section, we have seen several examples of matrices avoiding a fixed pattern as interval minor. At a glance, it is clear that these matrices are highly structured. We would now like to make the notion of ‘highly structured matrices’ rigorous, and generalize it to other forbidden patterns.
We will focus on the local structure of matrices, i.e., the structure observed by looking at a single row or column. For a forbidden pattern $P$ with at least two rows and two columns, it is not hard to see that any binary vector can appear as a row or column of a $P$-avoiding matrix.
However, the situation changes when we restrict our attention to critical $P$-avoiders. In the examples of critical $P$-avoiders we saw in Subsection \[ssec-example\], the 1-entries in each row or column were clustered into a bounded number of intervals. In particular, for these patterns $P$, there are only at most polynomially many vectors of a given length that may appear as rows or columns of a critical $P$-avoider.
In this section, we study this phenomenon in detail. We show that it generalizes to many other forbidden patterns $P$, but not all of them. As our main result, we will present a complete characterisation of the patterns $P$ exhibiting this phenomenon.
Let us begin by formalising our main concepts.
A *horizontal 0-run* in a matrix $M$ is a maximal sequence of consecutive 0-entries in a single row. More formally, a horizontal interval $M[\{r\}\times[c_1,c_2]]$ is a *horizontal 0-run* if all its entries are 0-entries, $c_1=1$ or $M(r,c_1-1)=1$, and $c_2=n$ or $M(r,c_2+1)=1$. Symmetrically, a vertical interval is a *vertical 0-run* if it is a maximal vertical interval that only contains 0-entries. In the same manner, we define a (horizontal or vertical) *1-run* to be a maximal interval of consecutive 1-entries in a single line of $M$.
Note that each line in a matrix $M$ can be uniquely decomposed into an alternating sequence of 0-runs and 1-runs.
Let $M$ be a binary matrix. The *complexity* of a line of $M$ is the number of 0-runs contained in this line. The *row-complexity* of $M$ is the maximum complexity of a row of $M$, i.e., the least number $k$ such that each row has complexity at most $k$. Similarly, the *column-complexity* of $M$ is the maximum complexity of a column of $M$.
For a class of matrices ${\mathcal{C}}$, we define its *row-complexity*, denoted ${r\left({\mathcal{C}}\right)}$, as the supremum of the row-complexities of the critical matrices in ${\mathcal{C}}$. We say that ${\mathcal{C}}$ is *row-bounded* if ${r\left({\mathcal{C}}\right)}$ is finite, and *row-unbounded* otherwise. Symmetrically, we define the *column-complexity* ${c\left({\mathcal{C}}\right)}$ of ${\mathcal{C}}$ and the property of being *column-bounded* and *column-unbounded*. We say that a class ${\mathcal{C}}$ is *bounded* if it is both row-bounded and column-bounded; otherwise, it is *unbounded*.
We stress that when defining the row-complexity and column-complexity of a class of matrices, we only take into account the matrices that are critical for the class.
We are now ready to state our main result.
\[thm-main\] Let $P$ be a pattern. The class ${Av_{{\preccurlyeq}}\left(P\right)}$ is row-bounded if and only if $P$ does not contain any of $Q_1, Q_2, Q_3, Q_4$ as an interval minor, where $$Q_1={\left({\begin{smallmatrix} &\bullet& \\\bullet& & \\ & &\bullet\end{smallmatrix}}
\right)},\ Q_2={\left({\begin{smallmatrix} &\bullet& \\ &
&\bullet\\\bullet& & \end{smallmatrix}}
\right)},\ Q_3={\left({\begin{smallmatrix}\bullet& & \\ & &\bullet\\ &\bullet&
\end{smallmatrix}}
\right)}\text{ and }
Q_4={\left({\begin{smallmatrix} & &\bullet\\\bullet& & \\ &\bullet& \end{smallmatrix}}
\right)}.$$
Before we prove Theorem \[thm-main\], we point out two of its direct consequences.
\[cor-first\] For a pattern $P$, these statements are equivalent:
- ${Av_{{\preccurlyeq}}\left(P\right)}$ is row-bounded.
- ${Av_{{\preccurlyeq}}\left(P\right)}$ is column-bounded.
- ${Av_{{\preccurlyeq}}\left(P\right)}$ is bounded.
\[cor-second\] Let ${{\mathcal{C}}}={Av_{{\preccurlyeq}}\left(P\right)}$ and ${{\mathcal{C}}}'={Av_{{\preccurlyeq}}\left(P'\right)}$ be principal classes, and suppose that ${{\mathcal{C}}}\subseteq{{\mathcal{C}}}'$ (or equivalently, $P{\preccurlyeq}P'$). If ${{\mathcal{C}}}'$ is bounded, then ${{\mathcal{C}}}$ is bounded as well.
Although each of these two corollaries is stating a seemingly basic property of the boundedness notion, we are not able to prove either of them without first proving Theorem \[thm-main\]. We also remark that neither of the two corollaries can be generalized to non-principal classes of matrices, as we will see in Section \[sec-further\].
Let us say that a pattern $P$ is *row-bounding* if ${Av_{{\preccurlyeq}}\left(P\right)}$ is row-bounded, otherwise $P$ is *non-row-bounding*. Similarly, $P$ is *bounding* if ${Av_{{\preccurlyeq}}\left(P\right)}$ is bounded and *non-bounding* otherwise.
Let ${{\mathcal{Q}}}$ be the set of patterns $\{Q_1,Q_2,Q_3,Q_4\}$. Theorem \[thm-main\] states that a pattern $P$ is row-bounding if and only if $P$ is in ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$. To prove this, we will proceed in several steps. We first show, in Subsection \[ssec-nonbound\], that if $P$ contains a pattern from ${{\mathcal{Q}}}$, then $P$ is not row-bounding. This is the easier part of the proof, though by no means trivial. Next, in Subsection \[ssec-bound\], we show that every pattern in ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ is row-bounding. This part is more technical, and requires a characterisation the structure of the patterns in ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$.
Non-row-bounding patterns {#ssec-nonbound}
-------------------------
Our goal in this subsection is to show that any pattern $P$ that contains one of the matrices from ${{\mathcal{Q}}}$ is not row-bounding. Let us therefore fix such a pattern $P$. Without loss of generality, we may assume that $Q_1{\preccurlyeq}P$.
\[thm-unbound\] For every matrix $P$ such that $Q_1{\preccurlyeq}P$, the class ${Av_{{\preccurlyeq}}\left(P\right)}$ is row-unbounded.
Refer to Figure \[fig-manyints\]. Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern containing $Q_1$ as an interval minor. In particular, there are row indices $1\le r_1<r_2<r_3\le k$ and column indices $1\le c_1<c_2<c_3\le \ell$ such that $P(r_1,c_2)=P(r_2,c_1)=P(r_3,c_3)=1$.
For an arbitrary integer $p$, we will show how to construct a matrix in ${Av_{crit}\left(P\right)}$ of row-complexity at least $p$. We first describe a matrix $M\in{{\{0,1\}}^{m\times n}}$ with $m=r_1+p(r_3-r_1)+(k-r_3)$ and $n=(c_1-1)+p(c_3-c_1+1)+(\ell-c_3)$.
In the matrix $M$, the leftmost $c_1-1$ columns, the rightmost $\ell-c_3$ columns, the topmost $r_1-1$ rows and the bottommost $k-r_3$ rows have all entries equal to 1. We call these entries the *frame* of $M$.
In the $r_1$-th row of $M$, there are $p$ 0-entries appearing in columns $c_2+i(c_3-c_1+1)$ for $i=0,\dotsc,p-1$, and the remaining entries in row $r_1$ are 1-entries.
The remaining entries of $M$, that is, the entries in rows $r_1+1,\dotsc,m-(k-r_3)$ and columns $c_1,\dotsc,n-(\ell-c_3)$, form a submatrix with $p(r_3-r_1)$ rows and $p(c_3-c_1+1)$ columns. We partition these entries into rectangular blocks, each block with $r_3-r_1$ rows and $c_3-c_1+1$ columns. For $i,j\in\{0,\dotsc,p-1\}$, let $B_{i,j}$ be such a block, with top-left corner in row $r_1+1+i(r_3-r_1)$ and column $c_1+j(c_3-c_1+1)$. The entries in $B_{i,j}$ are all equal to 1 if $i+j=p-1$, otherwise they are all equal to 0.
We claim that the matrix $M$ avoids $P$. To see this, assume there is an embedding $\phi$ of $P$ into $M$, and consider where $\phi$ maps the three 1-entries $e_1=(r_1,c_2)$, $e_2=(r_2,c_1)$, and $e_3=(r_3,c_3)$. Note that none of these three entries can be mapped into the frame of $M$, and moreover, neither $e_2$ nor $e_3$ can be mapped to the $r_1$-th row of $M$. In particular, $\phi(e_3)$ is inside a block $B_{i,j}$ for some $i+j=p-1$. Since $\phi(e_2)$ is to the top-left of $\phi(e_3)$, it must belong to the same block $B_{i,j}$. It follows that $\phi(e_2)$ is in the leftmost column of $B_{i,j}$, which is the column $c_1+j(c_3-c_1+1)$, and $\phi(e_3)$ in its rightmost column, i.e., the column $c_3+j(c_3-c_1+1)$. Therefore, $\phi(e_1)$ is in column $c_2+j(c_3-c_1+1)$; however, all the entries in this column where $\phi$ could map $e_1$ are 0-entries. Therefore $M$ is in ${Av_{{\preccurlyeq}}\left(P\right)}$.
The matrix $M$ is not necessarily a critical $P$-avoider. However, we can transform it into a critical $P$-avoider by greedily changing 0-entries to 1-entries as long as the resulting matrix stays in ${Av_{{\preccurlyeq}}\left(P\right)}$. By this process, we obtain a matrix $M'\in{Av_{crit}\left(P\right)}$ that dominates $M$. We claim that the $r_1$-th row of $M'$ is the same as the $r_1$-th row of $M$. This is because changing any 0-entry in the $r_1$-th row of $M$ to a 1-entry produces a matrix containing the complete pattern $1^{k\times\ell}$ as a submatrix, and in particular also containing $P$ as a minor.
We conclude that the matrix $M'\in{Av_{crit}\left(P\right)}$ has row-complexity at least $p$, showing that ${Av_{{\preccurlyeq}}\left(P\right)}$ is indeed row-unbounded.
![Illustration of the proof of Theorem \[thm-unbound\]. Left: a pattern $P$ with a shaded image of $Q_1$. Right: a $P$-avoider with a shaded row of complexity $p=4$.[]{data-label="fig-manyints"}](manyints.pdf){width="\textwidth"}
Row-bounding patterns {#ssec-bound}
---------------------
We now prove the second implication of Theorem \[thm-main\], that is, we show that any pattern $P$ avoiding the four patterns in ${{\mathcal{Q}}}$ is row-bounding (and therefore, by symmetry, also column-bounding). We first prove a result describing the structure of the patterns $P\in{Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$.
We say that a matrix $M$ can be *covered by $k$ lines* if there is a set of lines $\ell_1,\dots,\ell_k$ such that each 1-entry of $M$ belongs to some $\ell_i$. The following fact is a version of the classical K onig–Egerváry theorem. We present it here without proof; a proof can be found, e.g., in Kung [@kung].
\[fac-eger\] A matrix $M$ cannot be covered by $k$ lines if and only if $M$ contains a set of $k+1$ 1-entries, no two of which are in the same row or column.
\[pro:boundedints\] If a pattern $P$ belongs to ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$, then
1. $P$ avoids the pattern $D_2={\left({\begin{smallmatrix}\bullet&\\&\bullet\end{smallmatrix}}
\right)}$, or
2. $P$ avoids the pattern ${\overline{D}}_2={\left({\begin{smallmatrix}&\bullet\\ \bullet&\end{smallmatrix}}
\right)}$, or
3. $P$ can be covered by three lines.
Assume $P$ cannot be covered by three lines. By Fact \[fac-eger\], $P$ contains four 1-entries $e_1=(r_1,c_1)$, $e_2=(r_2,c_2)$, $e_3=(r_3,c_3)$ and $e_4=(r_4,c_4)$, no two of which are in the same row or column. We may assume that $r_1<r_2<r_3<r_4$. Moreover, since $P$ does not contain any pattern from ${{\mathcal{Q}}}$, we see that any three entries among $e_1, e_2, e_3, e_4$ must form an image of $D_3$ or of ${\overline{D}}_3$. Consequently, the four entries $e_i$ form an image of $D_4$ or of ${\overline{D}}_4$, i.e., we must have either $c_1<c_2<c_3<c_4$ or $c_1>c_2>c_3>c_4$. Suppose that $c_1<c_2<c_3<c_4$ holds, the other case being symmetric.
![Illustration of the proof of Proposition \[pro:boundedints\].[]{data-label="fig-boundedints"}](boundedints)
We will now show that $P$ avoids the pattern ${\overline{D}}_2$. Note first that the submatrix $P[[r_3]\times[c_3]]$ avoids ${\overline{D}}_2$, since an image of ${\overline{D}}_2$ there would form an image of $Q_1$ with $e_4$. Therefore, by Proposition \[pro-diag\], all the 1-entries in $P[[r_3]\times[c_3]]$ belong to a single decreasing walk from $(1,1)$ to $e_3$. Symmetrically, all 1-entries in the submatrix $P[[r_2,k]\times[c_2,\ell]]$ belong to a decreasing walk from $e_2$ to $(k,\ell)$.
Moreover, there can be no 1-entry in $P[(r_3,k]\times [1,c_2)]$ or in $P[[1,r_2)\times(c_3,\ell]]$, since such a 1-entry would form a forbidden pattern with $e_2$ and $e_3$. We conclude that all the 1-entries of $P$ belong to a single decreasing walk and therefore $P$ avoids ${\overline{D}}_2$.
We note that Proposition \[pro:boundedints\] is not an equivalent characterisation of patterns from ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$, since a matrix covered by three lines may contain a pattern from ${{\mathcal{Q}}}$. Later, in Lemma \[lem-2types\], we will give a more precise description of the avoiders of ${{\mathcal{Q}}}$ that cannot be covered by two lines.
#### Relative row-boundedness.
Before we prove that each pattern $P$ in the set ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ is row-bounding, we need some technical preparation. First of all, we shall need a more refined notion of row-boundedness, which considers individual 1-entries of the pattern $P$ separately.
Let $P$ be a pattern, let $e$ be a 1-entry of $P$, let $M$ be a $P$-avoiding matrix, and let $f$ be a 0-entry of $M$. Recall that $M\Delta f$ is the matrix obtained from $M$ by changing the entry $f$ from 0 to 1. We say that the entry $f$ of $M$ is *critical for $e$ (with respect to $P$)* if there is an embedding of $P$ into $M\Delta f$ that maps $e$ to $f$. Moreover, if $z$ is a 0-run in $M$, we say that $z$ is *critical for $e$* if at least one 0-entry in $z$ is critical for $e$.
Note that a $P$-avoiding matrix is critical for ${Av_{{\preccurlyeq}}\left(P\right)}$ if and only if each 0-entry of $M$ is critical for at least one 1-entry of $P$.
Let $e$ be a 1-entry of a pattern $P$. Let $M$ be a matrix avoiding $P$. The *complexity of a row $r$ of $M$ relative to $e$* is the number of 0-runs in row $r$ that are critical for $e$. The *row-complexity of $M$ relative to $e$* is the maximum complexity of a row of $M$ relative to $e$, and the *row-complexity of ${Av_{{\preccurlyeq}}\left(P\right)}$ relative to $e$*, denoted ${r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}$, is the supremum of the row-complexities of the matrices in ${Av_{{\preccurlyeq}}\left(P\right)}$ relative to $e$. When ${r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}$ is finite, we say that ${Av_{{\preccurlyeq}}\left(P\right)}$ is *row-bounded relative to $e$* and *$e$ is row-bounding*, otherwise ${Av_{{\preccurlyeq}}\left(P\right)}$ is *row-unbounded relative to $e$*.
Notice that in the definition of ${r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}$, we are taking supremum over all the matrices in ${Av_{{\preccurlyeq}}\left(P\right)}$, not just the critical ones. This makes the definition more convenient to work with, but it does not make any substantial difference. In fact, for a pattern $P$ with a row-bounding 1-entry $e$, the row-complexity relative to $e$ in ${Av_{{\preccurlyeq}}\left(P\right)}$ is maximized by a critical $P$-avoider. To see this, suppose that $M$ is a $P$-avoiding matrix, $M^+$ is any critical $P$-avoiding matrix that dominates $M$, and $f$ is a 0-entry of $M$ that is critical for $e$; then $f$ is necessarily also a 0-entry in $M^+$, and is still critical for $e$ in $M^+$. Therefore, the row-complexity of $M^+$ relative to $e$ is at least as large as the row-complexity of $M$ relative to $e$.
Observe that the following inequalities hold for any pattern $P$: $$\max_{e\in\operatorname{supp}(P)} {r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)} \le {r\left({Av_{{\preccurlyeq}}\left(P\right)}\right)} \le \sum_{e\in\operatorname{supp}(P)}
{r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}.$$ In particular, a pattern $P$ is row-bounding if and only if each 1-entry of $P$ is row-bounding.
\[lem-columns\] Let $P$ be a pattern, and let $M$ be a $P$-avoiding matrix. Let $z$ be a horizontal 0-run of $M$, and let $f\in z$ be a 0-entry in this 0-run. Assume that there is an embedding $\phi$ of $P$ into $M\Delta f$. Then $P$ has a 1-entry $e$ mapped by $\phi$ to $f$, and moreover, every entry of $P$ in the same column as $e$ is mapped by $\phi$ to a column containing an entry from $z$.
Clearly, $\phi$ must map a 1-entry of $P$ to the entry $f$, otherwise $\phi$ would also be an embedding of $P$ into $M$ and $M$ would not be $P$-avoiding.
Suppose now that $z=\{r\}\times[c_1,c_2]$ for a row $r$ and columns $c_1\le
c_2$. Let $e'$ be an entry of $P$ in the same column as $e$. Suppose that $\phi$ maps $e'$ to an entry in column $c$, with $c\not\in[c_1,c_2]$. Assume that $c<c_1$, the case $c>c_2$ being analogous. Then we may modify $\phi$ to map $e$ to the 1-entry $(r,c_1-1)$ instead of $f$, obtaining an embedding of $P$ into $M$, which is a contradiction.
#### Criteria for relative row-boundedness.
Let us first point out a trivial but useful fact: if $\overline P\in{{\{0,1\}}^{k\times \ell}}$ is a pattern obtained from a pattern $P$ by reversing the order of rows (i.e., turning $P$ upside down) then a 1-entry $e=(i,j)$ of $P$ is row-bounding if and only if the corresponding 1-entry $\overline e=(k-i+1,j)$ of $\overline P$ is row-bounding. Analogous properties hold for reversing the order of columns or 180-degree rotation. Similarly, operations that map rows to columns, such as transposition or 90-degree rotation, will map row-bounding 1-entries to column-bounding ones and vice versa.
We will now state several general criteria for row-boundedness of 1-entries, which we will later use to show that any ${{\mathcal{Q}}}$-avoiding pattern is row-bounding.
\[lem-leftmost\] If $P\in{{\{0,1\}}^{k\times \ell}}$ is a pattern with a row $r\in[k]$ and a column $c\in[\ell]$ such that $\operatorname{supp}(P)\subseteq (\{r\}\times[\ell]) \cup (
[m]\times[c,\ell])$, then every 1-entry of $P$ in the interval $\{r\}\times[c]$ is row-bounding (see Figure \[fig-leftmost\]).
![Illustration of Lemma \[lem-leftmost\] and Lemma \[lem-H\]. The shaded areas are the possible locations of 1-entries. The 1-entries in the cells marked by crosses are row-bounding.[]{data-label="fig-leftmost"}](leftmost "fig:")![Illustration of Lemma \[lem-leftmost\] and Lemma \[lem-H\]. The shaded areas are the possible locations of 1-entries. The 1-entries in the cells marked by crosses are row-bounding.[]{data-label="fig-leftmost"}](lemHtyp1 "fig:")
![Illustration of Lemma \[lem-leftmost\] and Lemma \[lem-H\]. The shaded areas are the possible locations of 1-entries. The 1-entries in the cells marked by crosses are row-bounding.[]{data-label="fig-leftmost"}](lemHtyp2 "fig:")![Illustration of Lemma \[lem-leftmost\] and Lemma \[lem-H\]. The shaded areas are the possible locations of 1-entries. The 1-entries in the cells marked by crosses are row-bounding.[]{data-label="fig-leftmost"}](lemHtyp3 "fig:")
Let $e=(r,j)$ be a 1-entry of $P$ with $j\le c$. Let $M\in{{\{0,1\}}^{m\times n}}$ be a $P$-avoider, let $f=(r',c')$ be a 0-entry of $M$ critical for $e$, and let $z$ be the horizontal 0-run containing $f$.
We claim that in the row $r'$ of $M$, there are fewer than $j$ 1-entries to the left of $f$. Suppose this is not the case, i.e., row $r'$ contains $j$ distinct 1-entries $f'_1, f'_2,\dotsc,f'_j$, numbered left to right, all of them to the left of $f$.
Let $\phi$ be an embedding of $P$ into $M\Delta f$ which maps $e$ to $f$. Recall from Lemma \[lem-columns\] that all the entries in column $j$ of $P$ are mapped to columns intersecting $z$. In particular, all the entries from column $j$ are mapped to the right of $f'_j$.
We define a partial embedding $\psi$ of $P$ into $M$, as follows. Firstly, $\psi$ maps the entries $(r,1), (r,2),\dotsc,(r,j)$ of $P$ to the 1-entries $f'_1, f'_2,\dotsc,f'_j$ of $M$. Next, $\psi$ maps each 1-entry of $P$ that is not among $(r,1), (r,2),\dotsc,(r,j)$ to the same entry as $\phi$. We easily see that $\psi$ is a partial embedding of $P$ into $M$, a contradiction.
Therefore, there are fewer than $j$ 1-entries in row $r'$ to the right of $f$, and hence row $r$ has at most $j$ 0-runs critical for $e$. Consequently, ${r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}\le j$ and $e$ is row-bounding.
The assumptions of Lemma \[lem-leftmost\] are satisfied when $c$ is the leftmost nonempty column of a pattern $P$ and $r$ is an arbitrary row. We state this important special case as a separate corollary.
\[cor-leftmost\] Any 1-entry in the leftmost nonempty column of a pattern $P$ is row-bounding.
\[lem-H\] Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern with a row $r$, and two distinct columns $c_1<c_2$, such that all the 1-entries of $P$ in row $r$ belong to the interval $\{r\}\times[c_1,c_2]$. Moreover, if $c$ is a column index with $c_1<c<c_2$, then $P$ has no 1-entry in column $c$ except possibly for the entry $(r,c)$. Suppose furthermore that $P$ satisfies one of the following three conditions (see Figure \[fig-leftmost\]):
- All the 1-entries of $P$ above row $r$ are in a single row $r_1<r$, and all the 1-entries below row $r$ are in a single row $r_2>r$.
- All the 1-entries of $P$ above row $r$ are in a single row $r_1<r$, and all the 1-entries below row $r$ are in the submatrix $P[(r,k]\times[c_2,\ell]]$.
- All the 1-entries of $P$ above row $r$ are in the submatrix $P[[1,r)\times[c_1]]$, and all the 1-entries below row $r$ are in the submatrix $P[(r,k]\times[c_2,\ell]]$.
Then every 1-entry in the interval $\{r\}\times[c_1,c_2]$ is row-bounding.
Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern satisfying the assumptions, and let $d=c_2-c_1+1$. We will show that for each 1-entry $e\in \{r_2\}\times[c_1,c_2]$ of $P$ and every $P$-avoiding matrix $M\in{{\{0,1\}}^{m\times n}}$, there are at most $d$ 0-runs critical for $e$ in each row of $M$.
For contradiction, assume that $M$ has a row $r'$ with at least $d+1$ 0-runs critical for $e$. Let $f$ and $f'$ be the leftmost and the rightmost 0-entries critical for $e$ in row $r'$. By assumption, $M$ has at least $d$ 1-entries in row $r'$ between $f$ and $f'$. Let $f_1,f_2,\dotsc,f_d$ be $d$ such 1-entries, numbered left to right.
Let $\phi$ be an embedding of $P$ into $M\Delta f$ which maps $e$ to $f$, and let $\phi'$ be an embedding of $P$ into $M\Delta f'$ which maps $e$ to $f'$. Let us describe a partial embedding $\psi$ of $P$ into $M$. Firstly, $\psi$ maps the entries $(r,c_1),\allowbreak(r,c_1+1),\allowbreak\dotsc,(r,c_2)$ to the entries $f_1,f_2,\dotsc,f_d$ in row $r'$ of $M$. Next, $\psi$ maps each 1-entry in $M[[m]\times[c_1]]$ except $(r,c_1)$ to the same entry as $\phi$, and $\psi$ maps the 1-entries in $M[[m]\times[c_2,n]]$ except $(r,c_2)$ to the same entry as $\phi'$. We easily check that this makes $\psi$ a partial embedding of $P$ into $M$: note that from Lemma \[lem-columns\], it follows that $\phi$ maps all the entries in column $c_1$ of $P$ to entries strictly to the left of $f_1$, and $\phi'$ maps entries in column $c_2$ to entries strictly to the right of $f_d$.
This is impossible, since $M$ is $P$-avoiding. Therefore, every row of a $P$-avoiding matrix has at most $d$ 0-runs critical for $e$, and $e$ is row-bounding.
\[lem-I\] Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern with two rows $r_1\le r_2$ and a column $c$, such that for every $r\in[r_1,r_2]$, $P$ has no 1-entry in row $r$ except possibly for the entry $(r,c)$. Suppose moreover, that $P$ satisfies one of the following conditions (see Figure \[fig-I\]):
- All the 1-entries of $P$ above row $r_1$ are in column $c$ or in the row $r_1-1$, and all the 1-entries below row $r_2$ are in column $c$ or in the row $r_2+1$.
- All the 1-entries of $P$ above row $r_1$ are in column $c$ or in the row $r_1-1$, and all the 1-entries below row $r_2$ are in the submatrix $P[(r_2,k]\times [c,\ell]]$.
- All the 1-entries of $P$ above row $r_1$ are in the submatrix $P[[1,r_1)\times[c]]$, and all the 1-entries below row $r_2$ are in the submatrix $P[(r_2,k]\times [c,\ell]]$.
Then every 1-entry in the interval $[r_1,r_2]\times\{c\}$ is row-bounding.
![Illustration of Lemma \[lem-I\]. The shaded areas correspond to possible locations of 1-entries. The 1-entries in cells marked by crosses are row-bounding.[]{data-label="fig-I"}](lemI)
Let $P$ be a pattern satisfying the assumptions of the lemma, and let $e =(r,c)$ be its 1-entry, with $r\in[r_1,r_2]$. Let $M$ be a $P$-avoider. We claim that every row of $M$ has at most one 0-run critical for $e$. For contradiction, suppose that row $i$ of $M$ has two 0-runs $z_L$ and $z_R$ critical for $e$, where $z_L$ is to the left of $z_R$. Let $f_L\in
z_L$ and $f_R\in z_R$ be two 0-entries critical for $e$ in the two 0-runs.
Let $\phi_L$ be an embedding of $P$ into $M\Delta f_L$ with $\phi_L(e)=f_L$, and $\phi_R$ be an embedding mapping $P$ into $M\Delta f_R$ with $\phi_R(e)=f_R$. We will describe a partial embedding $\psi$ of $P$ into $M$.
Since $f_L$ and $f_R$ are in distinct 0-runs, $M$ has a 1-entry $f$ that lies in row $i$ between $f_L$ and $f_R$. We put $\psi(e)=f$. For any other 1-entry $e'\in\operatorname{supp}(P)\setminus\{e\}$, we will define $\psi(e')$ to be equal to either $\phi_L(e')$ or $\phi_R(e')$, by the following rules.
For a 1-entry $e'$ which is strictly to the left of column $c$, we let $\psi(e')=\phi_L(e')$ and for a 1-entry $e'$ strictly to the right of column $c$, we let $\psi(e)=\phi_R(e')$.
It remains to deal with the 1-entries in column $c$. For a 1-entry $e'$ in $[r_1,r)\times\{c\}$, we choose $\psi(e')$ to be the lower of the two entries $\phi_L(e')$ and $\phi_R(e')$, i.e., we choose the entry that has larger row-index. If $\phi_L(e')$ and $\phi_R(e')$ are in the same row, we choose $\psi(e')$ arbitrarily from the two options.
For a 1-entry $e'$ in $[1,r_1)\times\{c\}$, we distinguish two possibilities. If $P$ is of Type 1 or Type 2, that is, all 1-entries above row $r_1$ are in column $c$ or row $r_1-1$, we choose $\psi(e')$ to be the higher of the two entries $\phi_L(e')$ and $\phi_R(e')$. If, on the other hand, $P$ is of Type 3, so all 1-entries above row $r_1$ are in columns $1,\dotsc,c$, we put $\psi(e')=\phi_L(e')$.
We proceed symmetrically for 1-entries below row $r$. For a 1-entry $e'\in
(r,r_2]\times\{c\}$, we choose $\psi(e')$ to be the higher of the two entries $\phi_L(e')$ and $\phi_R(e')$, breaking ties arbitrarily. For a 1-entry $e'\in
(r_2,k]\times\{c\}$, if $P$ is of Type 1, we let $\psi(e')$ be the lower of $\phi_L(e')$ and $\phi_R(e')$, and if $P$ is of Type 2 or 3, we put $\psi(e')=\phi_R(e')$.
Note that we may deduce from Lemma \[lem-columns\] that $\phi_L$ maps all the entries in column $c$ of $P$ to entries strictly to the left of $f$, and $\phi_R$ maps entries from column $c$ to entries strictly to the right of $f$. We may then easily verify that the mapping $\psi$ is a partial embedding of $P$ into $M$. This contradiction shows that the entry $e=(r,c)$ is row-bounding.
\[lem-I2\] Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern with two rows $r_1<r_2$ and two columns $c_1<c_2$ of one of the following two types (see Figure \[fig-I2\]):
- $\operatorname{supp}(P)\subseteq ([r_1,r_2]\times\{c_1\})\cup\big(
\{r_1,r_2\}\times ([c_1]\cup\{c_2\})\big)$.
- $\operatorname{supp}(P)\subseteq ([r_1,r_2]\times\{c_1\})\cup\big(
\{r_2\}\times
([c_1]\cup\{c_2\})\big)\cup([r_1]\times\{c_2\})$.
If $e=(r_1,c_1)$ is a 1-entry of $P$, then it is row-bounding.
![Illustration of the proof of Lemma \[lem-I2\]. As before, the shaded areas correspond to possible locations of 1-entries, and the 1-entry $e$, marked by a cross, is row-bounding.[]{data-label="fig-I2"}](lemI2){width="90.00000%"}
Suppose that $P\in{{\{0,1\}}^{k\times \ell}}$ satisfies the assumptions of the lemma, and that the entry $e=(r_1,c_1)$ is a 1-entry. Let $e'$ be the entry $(r_2,c_2)$ of $P$. Notice that if $e'$ is a 0-entry, we can deduce that $e$ is row-bounding by Lemma \[lem-leftmost\] (for Type 1) or by Lemma \[lem-H\] (for Type 2). Assume therefore that $e'$ is a 1-entry of $P$.
Let $M\in{{\{0,1\}}^{m\times n}}$ be a $P$-avoider. We will show that every row of $M$ has at most $\ell(\ell+1)$ 0-runs critical for $e$. Suppose that a row $r'$ of $M$ has more than $\ell(\ell+1)$ 0-runs critical for $e$. Among these 0-runs, we select a subsequence $z_1,z_2,\dotsc,z_{\ell+1}$ numbered left to right, with the property that for each $i\in[\ell]$, $M$ has at least $\ell$ 1-entries in row $r'$ between $z_i$ and $z_{i+1}$, and $M$ also has at least $\ell$ 1-entries in row $r'$ to the right of $z_{\ell+1}$.
For each $i\in[\ell+1]$, let $f_i$ be a 0-entry in $z_i$ critical for $e$, and let $\phi_i$ be an embedding of $P$ into $M\Delta f_i$ that maps $e$ to $f_i$. For $i\in[\ell]$, let $w_i$ be the interval of entries that lie between $z_i$ and $z_{i+1}$ in row $r'$ of $M$, and let $w_{\ell+1}$ be the interval of entries in row $r'$ to the right of $z_{\ell+1}$. Recall that each $w_i$ contains at least $\ell$ 1-entries. Let $g_i$ be the leftmost entry in $w_i$, which is necessarily a 1-entry, because $z_i$ is a maximal interval of 0-entries. Finally, let $h_i=(p_i,q_i)$ be the 1-entry $\phi_i(e')$ (recall that $e'=(r_2,c_2)$ is a 1-entry of $P$).
![Illustration of the proof of Lemma \[lem-I2\]: the structure of a $P$-avoiding matrix with many intervals critical for $e$ in row $r'$.[]{data-label="figI2pf"}](lemI2pf){width="\textwidth"}
Let us define a partial embedding $\psi$ of $P$ into $M$. We let $\psi$ map the entry $(r_1,c_2)$ to the 1-entry $g_{\ell+1}$, and if $P$ is of Type 2, then for every 1-entry $e''$ in the interval $[1,r_1)\times\{c_2\}$, we define $\psi(e'')=\phi_{\ell+1}(e'')$. Note that all the entries we mapped so far are to the right of $f_{\ell+1}$.
To define $\psi$ for the remaining 1-entries of $P$, we will distinguish several situations, depending on the positions of the entries $h_i=(p_i,q_i)$.
If, for some $i\in[\ell]$, the entry $h_i$ is to the right of the rightmost column of $w_i$, we put $\psi(e)=g_i$, and for every 1-entry $e''$ of $P$ for which $\psi$ has not yet been defined, we put $\psi(e'')=\phi_i(e'')$. To see that the mapping $\psi$ is a partial embedding of $P$ into $M$, it is enough to observe that all the 1-entries in column $c_2$ of $P$ are mapped by $\psi$ to entries strictly to the right of $w_{i}$, while by Lemma \[lem-columns\], all the 1-entries in column $c_1$ are mapped to the columns intersecting the interval $z_i$, except for the entry $e$, which is mapped to $g_i$. There are therefore at least $\ell-1$ columns which separate the image of any entry from column $c_1$ from the image of any entry from column $c_2$. With this in mind, it is easy to check that $\psi$ is indeed a partial embedding.
Suppose that the situation from the previous paragraph does not occur, that is, for every $i\in[\ell]$, the entry $h_i$ is not to the right of the rightmost column intersecting $w_i$. Since $h_i$ must by construction be to the right of the column containing $f_i$, we know that the column $q_i$ containing $h_i$ intersects either $z_i$ or $w_i$. In particular, we have $q_1<q_2<\dotsb<q_{\ell+1}$.
Assume now, that for some $i\in[\ell]$, the inequality $p_i\le p_{i+1}$ holds. We now complete the mapping $\psi$ as follows: we put $\psi(e)=g_i$, $\psi(e')=h_{i+1}$, and for all the 1-entries $e''$ of $P$ not yet mapped (i.e., the 1-entries in columns $1,\dotsc,c_1$ except $e$), we put $\psi(e'')=\phi_i(e'')$. The mapping $\phi$ is again a partial embedding of $P$ into $M$.
It remains to deal with the situation when we have $p_1>p_2>\dotsb>p_\ell>p_{\ell+1}$, which means that the 1-entries $h_1,h_2,\dotsc,h_{\ell+1}$ form an image of the diagonal pattern ${\overline{D}}_{\ell+1}$. We complete the mapping $\psi$ as follows: a 1-entry of the form $(r_1,j)$ for $j\le c_1$ is mapped to the entry $g_j$, a 1-entry of the form $(r_2,j)$ for any $j\in[\ell]$ is mapped to $h_j$, and any 1-entry $e''\in[r_1+1,r_2)\times\{c_1\}$ is mapped to $\phi_\ell(e'')$. Note that for $j<c_1$, the mapping $\psi$ maps the 1-entries in column $j$ to 1-entries in columns intersecting $z_j\cup w_j$, and for $j=c_1$, the 1-entries in column $j$ get mapped to columns intersecting $z_j\cup w_j\cup z_\ell$.
In all cases, we found a partial embedding $\psi$ of $P$ into $M$, which is a contradiction. Therefore, each row of $M$ has at most $\ell(\ell+1)$ 0-runs critical for $e$, and $e$ is row-bounding.
#### Row-boundedness of specific patterns.
We now have enough technical tools to establish that any pattern $P$ from ${Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ is row-bounding. Recall from Proposition \[pro:boundedints\] that any $P\in{Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ avoids $D_2$ or ${\overline{D}}_2$ or can be covered by three lines.
We will first look at patterns that can be covered by fewer than three lines, and show that they are all row-bounding.
\[lem-1row2col\] A pattern $P$ that has at most two nonempty columns or at most one nonempty row is row-bounding.
It follows from Lemma \[lem-leftmost\] and trivial symmetries that every 1-entry of $P$ is row-bounding, hence $P$ is row-bounding.
\[lem-tworows2\] If $P\in{{\{0,1\}}^{k\times \ell}}$ is a pattern with two nonempty rows, then $P$ is row-bounding.
We will show that for every 1-entry $e$ of $P$, we have ${r\left({Av_{{\preccurlyeq}}\left(P\right)},e\right)}\le
\ell^2$.
In view of Observation \[obs-empty\], we may assume that only the first row and the last row of $P$ are nonempty. Let $e$ be a 1-entry of $P$, and suppose without loss of generality that $e$ is in the first row, i.e., $e=(1,c)$ for some $c$.
Given a matrix $M\in{Av_{{\preccurlyeq}}\left(P\right)}$, consider an arbitrary row $r$ of $M$. For contradiction, suppose that the row $r$ has $\ell^2+1$ distinct 0-runs $z_1,\dots,z_{\ell^2+1}$ critical for $e$, numbered left to right. Let $c_i$ denote the leftmost column intersecting $z_i$, and for $i\le \ell^2$, let $X_i$ denote the set of column indices $[c_i, c_{i+1})$. Observe that for every $i\le\ell^2$, $M$ has at least one 1-entry in the interval $\{r\}\times X_i$.
![The matrix $M$ considered in the proof of Lemma \[lem-tworows2\].[]{data-label="fig-tworows"}](tworows){width="\textwidth"}
Let $B_i$ be the submatrix $M[[r+k-1,m]\times X_i]$ of $M$ (see Figure \[fig-tworows\]). Note that if there are at least $\ell$ distinct values of $i$ for which $B_i$ contains at least one 1-entry, then the matrix $M$ contains the pattern $P$.
Suppose therefore that $B_i$ is empty for each $i$ up to at most $\ell-1$ exceptions. In particular, there is an index $j\in[\ell^2]$ such that the $\ell$ consecutive submatrices $B_j, B_{j+1}, \dotsc, B_{j+\ell-1}$ are all empty.
Recall that $e=(1,c)$ is a 1-entry of $P$, and that all the 1-entries of $P$ are in rows 1 and $k$. Let $c'$ be a column index such that $e'=(k,c')$ is a 1-entry of $P$, and $|c-c'|$ is as small as possible. Suppose without loss of generality that $c\le c'$ and let $d:=c'-c$.
Let $f$ be a 0-entry in $z_j$ critical for $e$, and let $\phi$ be an embedding of $P$ into $M\Delta f$ that maps $e$ to $f$. Note that by Lemma \[lem-columns\], $\phi$ maps the entries in column $c$ of $P$ to entries in columns intersecting $z_j$, and in particular, the entry $(k,c)$ is mapped inside $B_j$. Since $B_j$ is empty, $(k,c)$ is a 0-entry and in particular, $c'$ is greater than $c$.
It follows that the 1-entry $e'=(k,c')$ is mapped strictly to the right of the column containing $f$, and since $B_j,\dotsc,B_{j+\ell-1}$ are all empty, $e'$ must be mapped to the right of the columns in the set $X_{j+\ell-1}$.
We now define a partial embedding $\psi$ of $P$ into $M$ as follows: the $d+1$ entries in $P[\{1\}\times[c,c']]$ get mapped into $M[\{r\}\times (X_j\cup
X_{j+1}\cup\dotsb\cup X_{j+d})]$ by $\psi$ (recall that $\{r\}\times X_i$ contains at least one 1-entry for each $i$). The remaining 1-entries of $P$ are mapped by $\psi$ in the same way as by $\phi$. Then $\psi$ is a partial embedding of $P$ into $M$, a contradiction.
\[lem-rowcol\] A pattern $P$ that can be covered by one row and one column is row-bounding.
Suppose that $P\in{{\{0,1\}}^{k\times \ell}}$ is covered by row $r$ and column $c$. By Lemma \[lem-leftmost\], all the 1-entries in $P[\{r\}\times[c]]$ are row-bounding, and by symmetry, the 1-entries in $P[\{r\}\times[c,\ell]]$ are row-bounding as well. By Lemma \[lem-I\], the 1-entries in $P[[1,r)\times\{c\}]$ and $P[(r,k]\times\{c\}]$ are also row-bounding.
Lemmas \[lem-1row2col\], \[lem-tworows2\] and \[lem-rowcol\] imply that any pattern that can be covered by two lines is row-bounding. We now proceed with the remaining cases of Proposition \[pro:boundedints\].
\[lem-walkpat\] A pattern $P\in{{\{0,1\}}^{k\times \ell}}$ that avoids $D_2$ or ${\overline{D}}_2$ is row-bounding.
Suppose that $P$ avoids ${\overline{D}}_2$, the other case being symmetric. From Proposition \[pro-diag\], we know that $P$ is a decreasing pattern. Every 1-entry of $P$ is row-bounding either by Lemma \[lem-H\] (Type 3), or by Lemma \[lem-I\] (Type 3), and therefore $P$ is row-bounding.
What follows is the last and the most difficult case of our analysis, which deals with patterns that are not increasing or decreasing and cannot be covered by two lines.
\[lem-2types\] Let $P\in{Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ be a pattern that contains both $D_2$ and ${\overline{D}}_2$, and that cannot be covered by two lines. Then $P$ can be transformed by a rotation or a reflection to a pattern $P_0$ of one of these two types (see Figure \[fig-2types\]).
- $P_0$ has three rows $r<r'<r''$ and two columns $c<c'$ with $$\operatorname{supp}(P_0)\subseteq \big(\{r'\}\times [c,c'] \big)\cup\{(r,c), (r'',c),
(r,c'),(r'',c')\}.$$
- $P_0$ has two rows $r<r'$ and two columns $c<c'$ with $$\operatorname{supp}(P_0)\subseteq \big( \{r\}\times[c,c']\big)\cup \big(
\{r'\}\times[c]\big)\cup \big([r]\times \{c'\}\big)\cup\{(r',c')\}.$$
![The two types of ${{\mathcal{Q}}}$-avoiders considered in Lemma \[lem-2types\]. The shaded areas are the possible positions of 1-entries.[]{data-label="fig-2types"}](2types){width="90.00000%"}
Let $P\in{{\{0,1\}}^{k\times \ell}}$ be a pattern satisfying the assumptions of the lemma. Since $P$ cannot be covered by two lines, by Fact \[fac-eger\], $P$ contains three 1-entries $e_1=(r_1,c_1)$, $e_2=(r_2,c_2)$ and $e_3=(r_3,c_3)$, with $r_1<r_2<r_3$, and such that the columns $c_1,c_2,c_3$ are all distinct. Since $P$ avoids the patterns from ${{\mathcal{Q}}}$, we must have either $c_1<c_2<c_3$ or $c_1>c_2>c_3$. Without loss of generality, assume $c_1<c_2<c_3$.
By Proposition \[pro:boundedints\], $P$ can be covered by three lines. Suppose first that the three lines that cover $P$ are the rows $r_1$, $r_2$ and $r_3$. Suppose moreover, that the three 1-entries were chosen in such a way that $c_1$ is as large as possible, while $c_2$ and $c_3$ are as small as possible; see Figure \[fig-2typespf\] (left). In particular, row $r_1$ of $P$ has no 1-entry in any of the columns $[c_1+1,c_2)$, otherwise we could choose a larger value of $c_1$. Similarly, row $r_2$ has no 1-entry in columns $[c_1+1,c_2)$ and row $r_3$ has no 1-entry in columns $[c_2+1,c_3)$.
Moreover, since $P$ avoids the four patterns from the set ${{\mathcal{Q}}}$, row $r_1$ has no 1-entry in columns $[c_2+1,c_3)$ or $(c_3,\ell]$, row $r_2$ has no 1-entry in columns $[1,c_1)$ or $(c_3,\ell]$, and row $r_3$ has no 1-entry in columns $[1,c_1)$ or $[c_1+1,c_2)$.
Therefore, apart from the three 1-entries $e_i$, a 1-entry of $P$ can appear in one of the three intervals $\alpha=\{r_1\}\times[1,c_1)$, $\beta=\{r_2\}\times(c_2,c_3]$ and $\gamma=\{r_3\}\times(c_3,\ell]$, or be equal to one of the five entries $a=(r_2,c_1)$, $b=(r_3,c_1)$, $c=(r_1,c_2)$, $d=(r_3,c_2)$ or $e=(r_1,c_3)$; see Figure \[fig-2typespf\] (left). Note that $a$ and $c$ cannot be simultaneously equal to 1, otherwise they would form a forbidden pattern with $e_3$, and similarly, if $\beta$ contains a 1-entry then $d=0$, if $\alpha$ contains a 1-entry then $b=0$, and if $\gamma$ contains a 1-entry then $e=0$.
Since $P$ contains a copy of ${\overline{D}}_2$, at least one of $b$ and $e$ must be a 1-entry. Let us go through the cases that may occur.
#### Case I: $b=1$.
If $b=1$ then $\alpha$ is empty. We have two subcases:
- Then $c=0$ and $d=0$. If $\gamma$ is empty, then $P$ is a Type 1 matrix, with $c=c_1$, $c'=c_3$, and $(r,r',r'')=(r_1,r_2,r_3)$. If $\gamma$ is nonempty, then $e=0$, and $P$ is a mirror image of a Type 2 matrix, with $(r,r')=(r_2,r_3)$ and $(c,c')=(c_3,c_1)$.
- If $\gamma$ is nonempty, then $e=0$ and since at most one of $a$ and $c$ is nonempty, rotating $P$ counterclockwise by 90 degrees yields a Type 2 matrix. If $\gamma$ is empty, then either $a=0$ and $P$ is the transpose of a Type 1 matrix, or $a=1$, and therefore $c=0$, and at least one of $d$ and $e$ is a 0-entry, resulting in a Type 1 matrix or a rotated Type 2 matrix.
#### Case II: $b=0$.
If $b=0$, then $e=1$, otherwise $P$ would avoid ${\overline{D}}_2$. Consequently, $\gamma$ is empty. If $\beta$ were empty as well, then $P$ would be symmetric to a matrix from case I by a 180-degree rotation. We may therefore assume that $\beta$ is nonempty, and hence $d=0$. At most one of $a$ and $c$ can be a 1-entry, and in either case we get an upside-down copy of a Type 2 matrix.
![${{\mathcal{Q}}}$-avoiders covered by rows $r_1$, $r_2$ and $r_3$ (left), by rows $r_1$, $r_2$ and column $c_3$ (center), and by rows $r_1$, $r_3$ and column $c_2$ (right). The shaded entries are potential 1-entries, the dots represent the three 1-entries $e_1$, $e_2$ and $e_3$.[]{data-label="fig-2typespf"}](2typespf){width="\textwidth"}
This completes the analysis of matrices that can be covered by 3 rows. Suppose now that $P$ can be covered by two rows and one column. As each of the three entries $e_1$, $e_2$ and $e_3$ must be covered by a distinct line, there are three possibilities: either $P$ is covered by rows $r_1$ and $r_2$ and column $c_3$; or $P$ is covered by rows $r_1$ and $r_3$ and column $c_2$; or $P$ is covered by rows $r_2$ and $r_3$ and column $c_1$. The last possibility is symmetric to the first one, so we only consider the first two.
Suppose $P$ is covered by rows $r_1$ and $r_2$ and column $c_3$. Choose $c_1$ and $c_2$ to be as large as possible, and $r_3$ to be as small as possible. Together with the absence of patterns from ${{\mathcal{Q}}}$, this means that apart from the 1-entries $e_1$, $e_2$ and $e_3$, all the remaining 1-entries must be inside the intervals $\alpha$, $\beta$ and $\gamma$ or at the positions $a$, $b$ or $c$ depicted in Figure \[fig-2typespf\] (center). Moreover, if $a=1$ then $\beta$ is empty. Therefore, $P$ is an upside-down copy of a matrix of Type 2, with the role of column $c$ played by $c_1$ if $a=0$ or by $c_2$ if $a=1$.
Let us now suppose that $P$ is covered by rows $r_1$ and $r_3$ and column $c_2$. See Figure \[fig-2typespf\] (right). Suppose $c_1$ is largest possible and $c_3$ smallest possible. We make no assumptions about $r_2$, to keep the configuration symmetric. All the 1-entries are in the intervals $\alpha$, $\beta$, $\gamma$ and $\delta$ or at the positions $a$ and $b$ depicted in the figure. Since $P$ contains ${\overline{D}}_2$, at least one of $a$ and $b$ is a 1-entry. Suppose without loss of generality that $a=1$. Then $\alpha$ is empty. If $\delta$ is nonempty, then $b=0$, and $P$ is a Type 2 matrix rotated 90 degrees clockwise. Otherwise $\delta$ is empty and $P$ is a rotated Type 1 matrix.
The cases when $P$ can be covered by three columns, or by two columns and a row, are symmetric to the cases handled so far by a 90-degree rotation.
We now have all the ingredients to complete the proof of our main result.
![Illustration of the proof of Theorem \[thm-rbound\]. The symbols indicate the criteria used to prove row-boundedness of the 1-entries in the two types of patterns of Lemma \[lem-2types\].[]{data-label="fig-rowbound"}](rowbound)
\[thm-rbound\] Every pattern $P\in{Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$ is row-bounding.
Choose a $P\in{Av_{{\preccurlyeq}}\left({{\mathcal{Q}}}\right)}$. By Proposition \[pro:boundedints\], either $P$ can be covered by three lines, or it avoids $D_2$, or it avoids ${\overline{D}}_2$. If $P$ avoids one of the two patterns of size 2, then it is row-bounding by Lemma \[lem-walkpat\]. If it can be covered by two lines, it is row-bounding by Lemmas \[lem-1row2col\], \[lem-tworows2\] and \[lem-rowcol\]. Finally, if $P$ contains both $D_2$ and ${\overline{D}}_2$ and cannot be covered by two lines, Lemma \[lem-2types\] shows that, up to symmetry, $P$ corresponds to a matrix of Type 1 or Type 2. We therefore need to argue that the matrices of these two types, as well as their transposes, are row-bounding. See Figure \[fig-rowbound\].
If $P$ is of Type 1, its 1-entries in column $c$ or in column $c'$ are row-bounding by Corollary \[cor-leftmost\], and those in row $r'$ are row-bounding by Lemma \[lem-H\].
If $P$ is the transpose of a Type 1 matrix, then its 1-entries in columns $r$ and $r''$ are row-bounding by Corollary \[cor-leftmost\], and those in column $r'$ by Lemmas \[lem-I\] and \[lem-I2\].
If $P$ is of Type 2, the 1-entries in row $r'$ and in column $c'$ are row-bounding by Lemma \[lem-leftmost\] and Corollary \[cor-leftmost\], and those in row $r$ are row-bounding by Lemma \[lem-H\].
Finally, if $P$ is the transpose of a Type 2 matrix, the 1-entries in column $r'$ and in row $c'$ are row-bounding by Lemma \[lem-leftmost\] and Corollary \[cor-leftmost\], and the remaining 1-entries are covered by Lemmas \[lem-I\] and \[lem-I2\].
Theorems \[thm-unbound\] and \[thm-rbound\] together imply Theorem \[thm-main\].
Further directions and open problems {#sec-further}
====================================
#### Boundedness of non-principal classes.
So far, we only considered principal classes of matrices, i.e., classes determined by a single forbidden pattern. It is natural to ask to what extent our results generalize to arbitrary minor-closed classes of matrices, or at least to classes determined by a finite number of forbidden patterns.
All our row-boundedness results for principal classes are based on the study of row-bounding 1-entries in a pattern $P$. This approach extends straightforwardly to the setting of multiple forbidden patterns. In particular, for a set ${{\mathcal{F}}}$ of patterns, a pattern $P\in{{\mathcal{F}}}$ and a 1-entry $e$ of $P$, we say that $e$ is *row-bounding in ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$* if each row of a matrix $M\in{Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ has only a bounded number of 0-runs critical for $e$ with respect to $P$. Note that if ${{\mathcal{F}}}$ is finite, then ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded if and only if each 1-entry of each pattern $P\in{{\mathcal{F}}}$ is row-bounding in ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$.
Note also that, by definition, if $e$ is a 1-entry of $P$ that is row-bounding in ${Av_{{\preccurlyeq}}\left(P\right)}$, then for every set of patterns ${{\mathcal{F}}}$ that contains $P$, the entry $e$ is also row-bounding in ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$, since ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is a subclass of ${Av_{{\preccurlyeq}}\left(P\right)}$. Therefore, all the criteria for row-bounding entries that we derived in Subsection \[ssec-bound\] are applicable to non-principal classes as well.
We have seen in Corollary \[cor-first\] that a principal class is row-bounded if and only if it is column-bounded. Our next example shows that this property does not generalize to non-principal classes.
\[pro-counter\] For the set of patterns ${{\mathcal{F}}}=\{D_4,P\}$ with $$P={\left({\begin{smallmatrix} &\bullet& \\\bullet& & \\ &\bullet& \\ &
&\bullet\end{smallmatrix}}
\right)} \text{ and } D_4={\left({\begin{smallmatrix}\bullet& & & \\ &\bullet& & \\ & &\bullet& \\
& &&\bullet\end{smallmatrix}}
\right)},$$ the class ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded but not column-bounded.
To prove that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is not column-bounded, we apply the transpose of the construction of Theorem \[thm-unbound\], and observe that the constructed matrix avoids $D_4$ (see Figure \[fig-colunbound\] (left)).
To prove that the class ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded, observe first that all the 1-entries in $D_4$ are row-bounding by Lemma \[lem-H\], the leftmost and the rightmost 1-entry of $P$ are row-bounding by Corollary \[cor-leftmost\], and the 1-entry $(3,2)$ of $P$ is row-bounding by Lemma \[lem-I\]. It thus remains to show that the entry $e=(1,2)$ of $P$ is row-bounding in ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$.
![Left: illustration that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ has unbounded column-complexity relative to the entry $e=(2,1)$ of $P$. Right: illustration of the proof that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded.[]{data-label="fig-colunbound"}](colunbound "fig:"){height="25.00000%"} ![Left: illustration that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ has unbounded column-complexity relative to the entry $e=(2,1)$ of $P$. Right: illustration of the proof that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded.[]{data-label="fig-colunbound"}](colunbound2 "fig:"){height="25.00000%"}
We will show that each matrix $M\in{Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ has at most two 0-runs critical for $e$ in any given row. Refer to Figure \[fig-colunbound\] (right). For contradiction, suppose that there are three 0-runs $z_1< z_2< z_3$ in a row $r$ of $M$. Let $g_1$ be a 1-entry of $M$ that lies in row $r$ between $z_1$ and $z_2$, let $g_2$ be a 1-entry of $M$ in row $r$ between $z_2$ and $z_3$, and let $f$ be a 0-entry in $z_3$ critical for the entry $e$. Let $\phi$ be an embedding of $P$ into $M\Delta f$ with $\phi(e)=f$.
Consider the three 1-entries $h_1=\phi(2,1)$, $h_2=\phi(3,2)$ and $h_3=\phi(4,3)$. If $h_1$ is in a column strictly to the right of $g_1$, then $g_1$ forms an image of $D_4$ with the three $h_i$s, a contradiction. If, on the other hand, $h_1$ is not to the right of $g_1$, then $h_1$ is strictly to the left of $g_2$, and $g_2$ forms an image of $P$ with the three $h_i$s (recall that $h_3$ is to the right of $f=\phi(1,2)$, and therefore also to the right of $g_2$). This shows that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is row-bounded.
Recall from Corollary \[cor-second\], that any principal subclass of a bounded principal class is again bounded. The example of Proposition \[pro-counter\] shows that this result does not generalize to non-principal classes: indeed, the class ${Av_{{\preccurlyeq}}\left(D_4\right)}$ is bounded by Theorem \[thm-main\] (or by Corollary \[cor-diag\]), while its subclass ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is not bounded.
On the positive side, it is not hard to show that row-boundedness (and therefore also boundedness) is closed under union and intersection of classes.
\[pro-boundunion\] If ${{\mathcal{C}}}_1$ and ${{\mathcal{C}}}_2$ are row-bounded classes of matrices, then the classes ${{\mathcal{C}}}_1\cup{{\mathcal{C}}}_2$ and ${{\mathcal{C}}}_1\cap{{\mathcal{C}}}_2$ are row-bounded as well.
Let $K_i$ be the row-complexity of the class ${{\mathcal{C}}}_i$, for $i\in\{1,2\}$. Since every matrix that is critical for ${{\mathcal{C}}}_1\cup{{\mathcal{C}}}_2$ is also critical for ${{\mathcal{C}}}_1$ or for ${{\mathcal{C}}}_2$, we observe that ${{\mathcal{C}}}_1\cup{{\mathcal{C}}}_2$ has row-complexity at most $\max\{K_1,K_2\}$. In particular, ${{\mathcal{C}}}_1\cup{{\mathcal{C}}}_2$ is row-bounded.
Let us argue that ${{\mathcal{C}}}_1\cap{{\mathcal{C}}}_2$ is row-bounded as well. We claim that ${{\mathcal{C}}}_1\cap{{\mathcal{C}}}_2$ has row-complexity at most $K:=K_1+K_2$. For contradiction, suppose that there is a matrix $M$ critical for ${{\mathcal{C}}}_1\cap{{\mathcal{C}}}_2$ with row-complexity at least $K+1$.
Let $r$ be a row of $M$ with maximum complexity, let $z_1, z_2,\dotsc,z_{K+1}$ be a sequence of 0-runs in this row, and let $f_i$ be a 0-entry in $z_i$. By criticality of $M$, we know that for each $i\in[K+1]$, the matrix $M\Delta f_i$ does not belong to ${{\mathcal{C}}}_1$ or does not belong to ${{\mathcal{C}}}_2$.
In particular, there are either at least $K_1+1$ values of $i$ for which $M\Delta f_i$ is not in ${{\mathcal{C}}}_1$, or at least $K_2+1$ values of $i$ for which $M\Delta f_i$ is not in ${{\mathcal{C}}}_2$. Suppose without loss of generality that the former situation occurs. Let $M^+$ be a critical matrix for the class ${{\mathcal{C}}}_1$ that dominates the matrix $M$. If $f_i$ is a 0-entry of $M$ such $M\Delta f_i$ is not in ${{\mathcal{C}}}_1$, then $f_i$ is also a 0-entry of $M^+$. It follows that $M^+$ has at least $K_1+1$ 0-runs in row $r$, which is impossible, since $K_1$ is the row-complexity of ${{\mathcal{C}}}_1$.
In contrast with Proposition \[pro-boundunion\], an intersection of two unbounded classes is not necessarily unbounded, as we will now show. Consider the two patterns $Q_1={\left({\begin{smallmatrix} &\bullet&
\\\bullet& & \\ & &\bullet\end{smallmatrix}}
\right)}$ and $Q_2={\left({\begin{smallmatrix} &\bullet& \\ &
&\bullet\\\bullet& & \end{smallmatrix}}
\right)}$, and recall from Theorem \[thm-main\] that both ${Av_{{\preccurlyeq}}\left(Q_1\right)}$ and ${Av_{{\preccurlyeq}}\left(Q_2\right)}$ are unbounded classes.
\[pro-unbinter\] The class ${Av_{{\preccurlyeq}}\left(\{Q_1,Q_2\}\right)}={Av_{{\preccurlyeq}}\left(Q_1\right)}\cap{Av_{{\preccurlyeq}}\left(Q_2\right)}$ is bounded.
Let us first show that every 1-entry of the two patterns $Q_1$ and $Q_2$ is row-bounding for ${{\mathcal{C}}}:={Av_{{\preccurlyeq}}\left(\{Q_1,Q_2\}\right)}$. For a 1-entry that belongs to the first or the last column of either pattern, this follows from Corollary \[cor-leftmost\].
Consider the 1-entry $e=(1,2)$ of the pattern $Q_1$. We claim that each row in a matrix $M\in{{\mathcal{C}}}$ has at most two 0-runs critical for $e$. Suppose that a matrix $M\in{{\mathcal{C}}}$ has a row $r$ with three 0-runs $z_1<z_2<z_3$ critical for $e$. Let $f_i$ be a 0-entry in $z_i$ critical for $e$, and let $g_i$ be a 1-entry in row $r$ between $z_i$ and $z_{i+1}$, for $i\in\{1,2\}$.
For $i\in\{1,2,3\}$, let $\phi_i$ be an embedding of $Q_1$ into $M\Delta f_i$ that maps $e$ to $f_i$. Consider the three 1-entries $h_1=\phi_1(2,1)$, $h_2=\phi_1(3,3)$, and $h_3=\phi_3(3,3)$. Let $p_i$ and $q_i$ be the row and the column containing $h_i$.
![Illustration of the row-boundedness (left) and column-boundedness (right) of ${Av_{{\preccurlyeq}}\left(\{Q_1,Q_2\}\right)}$.[]{data-label="fig-unbinter"}](unbinter){width="90.00000%"}
Note that $g_1$ cannot be to the left of column $q_2$, since then $h_1$, $g_1$ and $h_2$ would form an image of $Q_1$. It follows that $g_1$ is in the column $q_2$ or to the right of it, and consequently, we have $q_2<q_3$. Moreover, if $p_3>p_1$, then $h_1$, $g_2$ and $h_3$ form an image of $Q_1$, so $p_3$ is no larger than $p_1$ and hence $p_3<p_2$. But then $h_2$, $g_2$ and $h_3$ form an image of $Q_2$, a contradiction.
By symmetry, the 1-entry $(1,2)$ of $Q_2$ is row-bounding as well, and therefore ${{\mathcal{C}}}$ is row-bounded.
Let us now argue that ${{\mathcal{C}}}$ is column-bounded. It is enough to show that the 1-entry $e'=(2,1)$ of $Q_1$ is column-bounding for ${{\mathcal{C}}}$, the rest follows from symmetry and from Corollary \[cor-leftmost\]. Suppose that a matrix $M\in{{\mathcal{C}}}$ has a column $c$ with three 0-runs critical for $e'$. In particular, column $c$ contains a 0-entry $f'$ critical for $e'$ such that below $f'$, there are at least two 1-entries $g'_1$ and $g'_2$ in column $c$ of $M$. Suppose that $g'_1$ is above $g'_2$.
Let $\phi$ be an embedding of $Q_1$ into $M\Delta f'$ with $\phi(e')=f'$. Define $h'_1=\phi(1,2)$ and $h'_2=\phi(3,3)$. Let $r'$ be the row containing $h'_2$. If $g'_1$ is above row $r'$, then $g'_1$, $h'_1$ and $h'_2$ form a copy of $Q_1$, and if $g'_1$ is not above row $r'$, then $g'_2$ is below row $r'$ and $g'_2$, $h'_1$ and $h'_2$ form a copy of $Q_2$, a contradiction.
#### Open problems.
A natural question arising from our results is to extend the dichotomy of Theorem \[thm-main\] to non-principal classes of matrices.
For which sets ${{\mathcal{F}}}$ of patterns is the class ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ row-bounded? Can we characterize such sets ${{\mathcal{F}}}$, at least when ${{\mathcal{F}}}$ is finite?
The notion of complexity we used in this paper is quite crude, in the sense that it only takes into account single lines of the corresponding matrix. It is reasonable to expect that matrices from a class of unbounded complexity possess nontrivial properties that could be revealed by a more refined approach.
Is there a refinement of our complexity notion that would provide nontrivial insight into the structure of critical matrices in unbounded classes?
Throughout the paper, we focused on distinguishing bounded classes from unbounded ones. We made no attempts to obtain tight estimates for the actual value of the complexity of a bounded class. This might be a line of research worth pursuing.
What is the highest possible value of ${r\left({Av_{{\preccurlyeq}}\left(P\right)}\right)}$, over all row-bounding patterns $P$ of a given size $k\times \ell$? For which pattern is this maximum attained?
By Observation \[obs-empty\], if $P^+$ is a pattern obtained by adding an empty row or column to the boundary of a pattern $P$, then ${Av_{{\preccurlyeq}}\left(P\right)}$ has the same complexity as ${Av_{{\preccurlyeq}}\left(P^+\right)}$, and the avoiders of $P^+$ can be easily described in terms of the avoiders of $P$.
It is, however, more challenging to deal with a pattern $P^+$ obtained by inserting an empty line into the interior of a pattern $P$. Theorem \[thm-main\] implies that $P$ is bounding if and only if $P^+$ is bounding, but we are not aware of any direct proof of this.
Let $P^+$ be a pattern obtained from a pattern $P$ by inserting a new empty row or column to an arbitrary position inside $P$. Can we bound ${r\left({Av_{{\preccurlyeq}}\left(P^+\right)}\right)}$ in terms of ${r\left({Av_{{\preccurlyeq}}\left(P\right)}\right)}$? Can we describe the avoiders of $P^+$ in terms of the avoiders of $P$? If ${{\mathcal{F}}}$ is a set of patterns and ${{\mathcal{F}}}^+$ a set of patterns obtained by inserting empty rows and columns to the patterns in ${{\mathcal{F}}}$, is it true that ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}^+\right)}$ is bounded if and only if ${Av_{{\preccurlyeq}}\left({{\mathcal{F}}}\right)}$ is?
[^1]: Computer Science Institute, Charles University, Faculty of Mathematics and Physics, Malostranské náměstí 25, Prague, Czech Republic, `jelinek@iuuk.mff.cuni.cz`
[^2]: Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, United Kingdom `s.kucera@lse.ac.uk`
[^3]: Supported by project 16-01602Y of the Czech Science Foundation (GAČR), and by the Neuron Foundation for the Support of Science.
[^4]: We use the convention of representing 1-entries in binary matrices by dots and 0-entries by blanks.
|
---
abstract: 'We study the superconducting current of a Josephson junction (JJ) coupled to an external nanomagnet driven by a time dependent magnetic field both without and in the presence of an external AC drive. We provide an analytic, albeit perturbative, solution for the Landau-Lifshitz (LL) equations governing the coupled JJ-nanomagnet system in the presence of a magnetic field with arbitrary time-dependence oriented along the easy axis of the nanomagnet’s magnetization and in the limit of weak dimensionless coupling $\epsilon_0$ between the JJ and the nanomagnet. We show the existence of Shapiro-like steps in the I-V characteristics of the JJ subjected to a voltage bias for a constant or periodically varying magnetic field and explore the effect of rotation of the magnetic field and the presence of an external AC drive on these steps. We support our analytic results with exact numerical solution of the LL equations. We also extend our results to dissipative nanomagnets by providing a perturbative solution to the Landau-Lifshitz-Gilbert (LLG) equations for weak dissipation. We study the fate of magnetization-induced Shapiro steps in the presence of dissipation both from our analytical results and via numerical solution of the coupled LLG equations. We discuss experiments which can test our theory.'
author:
- 'Roopayan Ghosh$^{(1)}$, Moitri Maiti$^{(2)}$, Yury M. Shukrinov,$^{(2,3)}$ and K. Sengupta$^{(1)}$'
title: Magnetization induced dynamics of a Josephson junction coupled to a nanomagnet
---
Introduction {#intro}
============
The physics of Josephson junctions (JJs) has been the subject of intense theoretical and experimental endeavor for decades [@likharev1]. The interest in the physics of such JJs has received renewed attention in recent years in the context of Majorana modes in unconventional superconductors [@kit1; @kwon1; @jay1; @gil1]. Indeed, it has been theoretically predicted [@kit1; @kwon1]and experimentally observed [@rokhin1] that such junctions may serve as a test bed for detection of Majorana end modes in unconventional superconductors. It has been shown that the presence of such end modes lead to fractional Josephson effect [@kit1; @kwon1] and results in the absence of odd Shapiro steps [@shapiro1] when such junctions are subjected to an external AC drive [@evenlit; @kiril1; @commentsub].
Recently molecular nanomagnets have been studied as potential candidates for qubit realization owing to their long magnetization relaxation times at low temperatures [@time1]. Such a realization is expected to play a central role in several aspects of quantum information processing [@loss1] and spintronics using molecular magnets [@ref1; @ref2]. These systems have potential for high-density information storage and are also excellent examples of finite-size spin systems which are promising test-beds for addressing several phenomena in quantum dynamics viz. quantum-tunneling of the magnetization [@tunnel], quantum information [@qi], entanglement [@ent1] to name a few. The study of the spin dynamics of the nanomagnets is a crucial aspect of all such studies. One way to probe such dynamics is to investigate the spin response in bulk magnets using inelastic neutron scattering and subsequent finite-size extrapolation to obtain the inelastic neutron scattering spectra for a single molecule [@spin1]. Other, more direct, methods include determination of the real-space dynamical two-spin correlations in high-quality crystals of nanomagnets [@spin2] and transport measurements through nanomagnets [@spin3]. Another probe of such magnetization dynamics, which we shall focus on in the present study, involves interaction of the nanomagnet with a JJ; the modulation of the Josephson current and the Shapiro steps in the current-voltage (I-V) characteristics is known to bear signature of the nature of the spin precession of the nanomagnet [@spin4].
The physics of such a JJ-nanomagnet system has therefore received significant attention both theoretically and experimentally. It has been theoretically studied in Refs. , where the effects of spin-flip of the nanomagnet on the Josephson current was charted out. More recently, several theoretical studies have been carried out on a variety of aspects of such systems including effect of superconducting correlations on the spin dynamics of the nanomagnet [@nuss1], the influence of spin-orbit coupling of a single spin on the Josephson current [@nazarov1], and the effect of deposition of single magnetic molecules on superconducting quantum interference devices(SQUIDs) made out of such junctions [@delan1]. Another interesting phenomenon which has been widely studied in this context is magnetization switching [@cai1; @buzhdin1; @yury1] which constitutes magnetization reversal of the nanomagnet by a externally driven JJ. In addition, the possibility of a Josephson current to induce exchange interaction between ferromagnetic layers has been studied in Ref.. Furthermore, the dynamics of both JJs with misaligned ferromagnetic layers and those coupled to single or multiple ferromagnetic layers have also been studied numerically [@linder1]. It has also been shown that the presence of a single classical spin of a molecular magnet precessing under the action of a constant magnetic field coupled to a JJ may lead to generation of finite spin current whose polarization axis rotates with same frequency as the classical spin [@holm1]. Such theoretical studies were complemented by experimental work on these systems [@wernsdorfer1]. More recently, magnetization reversal of a single spin using a JJ subjected to a static field and a weak linearly polarized microwave radiation has been demonstrated in Ref. . The possibility of the presence of Shapiro-like steps in the I-V characteristics of such coupled JJ-nanomagnet for constant applied magnetic field has also been pointed out in Ref. . However, to the best of our knowledge, most of these studies do not provide any analytic treatment of the coupled JJ-nanomagnet system even at a classical level where they are known to be governed by the Landau-Lifshitz-Gilbert (LLG) equations [@gil1]. Moreover, the current-voltage (I-V) characteristics of a JJ in the presence of a nanomagnet with time-dependent magnetic fields and in the presence of external AC drive has not been studied systematically so far.
In this work we study a JJ coupled to a nanomagnet with a fixed easy-axis anisotropy direction (chosen to be $\hat y$) in the presence of an arbitrary time dependent external magnetic field along $\hat y$. For nanomagnets with weak anisotropy, we find an analytic perturbative solution to the coupled Landau-Lifshitz (LL) equations in the limit of weak coupling between the nanomagnet and the JJ. Using this solution, we show that a finite DC component of the supercurrent, leading to Shapiro-like steps in the I-V characteristics of a voltage-biased JJ, can occur, in the absence of any external radiation, for either a constant or a periodically time varying magnetic field. Our theoretical analysis provides exact analytic results for the position of such steps. We study the stability of these steps against change in the direction of the applied magnetic field and increase of the dimensionless coupling strength $\epsilon_0$ between the JJ and the nanomagnet. We also provide a detailed analysis of the fate of this phenomenon in the presence of an external AC drive and demonstrate that the presence of such a drive leads to several new fractions (ratio between the applied DC voltage and the drive frequency) at which the supercurrent develops a finite DC component leading to Shapiro steps in the I-V characteristics. We support our analytical results with numerical study of the systems which allows exact, albeit numerical, investigation of the dynamics of the coupled JJ and nanomagnet system. We also extend our study to systems with dissipation via perturbative analytic and exact numerical solution of the coupled LLG equations and study the behavior of the steps with increasing dissipation. Finally, we discuss experiments which may test our theory, discuss the significance of our results in context of junctions of unconventional superconductors hosting Majorana end modes, and point out the distinction between voltage and current biased junctions in the present context. We note that the analytic solution of the LLG equations that we present here as well as the presence of additional, more robust, Shapiro steps for periodically time-varying magnetic fields have not been reported so far; thus our work may serve as an ingredient for development of new detectors for magnetization of a nanomagnet in coupled JJ-nanomagnet systems.
![A schematic representation of the JJ-nanomagnet system showing the position of the nanomagnet (shown schematically by the filled oval and an arrow representing the direction of its instantaneous magnetization) and the JJ (orthogonal to the direction of the magnetic field). The inset shows the angles $\theta$ and $\phi$ used to specify the direction of the nanomagnet’s magnetization $\vec m = \vec M/|\vec M|$.[]{data-label="fig1"}](fig1.pdf)
The plan of the rest of this work is as follows. In Sec.\[ansol\], we provide the analytic solution of the LL equations governing the coupled nanomagnet and JJ. This is followed by an analogous solution for LLG equations describing the coupled JJ-nanomagnet system in the presence of dissipation in Sec.\[gilres\]. Next, in Sec. \[numsol\], we back the analytical results with exact numerics and discuss details of Shapiro-step like features in the I-V of the JJ for constant or periodic magnetic field. Finally, we chart out our main results, discuss experiments which can test our theory, and conclude in Sec. \[dissec\].
Formalism and analytical solution {#ansol}
=================================
In this section, we obtain analytic solution to the LL equations for the weakly coupled JJ-nanomagnet system. We shall sketch the general derivation of our result for arbitrary time-dependent magnetic field in Sec. \[genres\] and then apply these results to demonstrate the existence of Shapiro-like steps for constant and periodic magnetic fields in Sec. \[pecresults\]. The extension of these results for dissipative magnets will be charted out in Sec.\[gilres\].
Perturbative solution {#genres}
---------------------
The coupled JJ-nanomagnet system is schematically shown in Fig.\[fig1\]. In what follows we consider a JJ along $\hat x$ and choose the easy axis of the nanomagnet along $\hat y$; the radius vector $\vec r$ between the nanomagnet and the JJ thus lies in the $x-y$ plane as shown in Fig. \[fig1\]. The energy functional governing the JJ and the nanomagnet is given by [@cai1] $$\begin{aligned}
E &=& E_1 + E_2 \nonumber\\
E_1 &=& -K M_y^2 - M_y B(t), \quad E_2 = -E_J \cos \gamma,
\label{en1}\end{aligned}$$ where $K>0$ denotes the magnetization anisotropy constant, $\vec
B(t)\parallel {\hat y}$ is the external magnetic field which can have arbitrary time dependence, and $E_J$ is the Josephson energy of the junction. The phase difference $\gamma$ across the junction is given by
$$\begin{aligned}
\gamma(t) &=& \gamma_0(t) + \gamma_1(t), \nonumber\\
\gamma_0(t) &=& \gamma_{00}+ \frac{2e}{\hbar}
\int^t V_0(t') dt' = \gamma_{00} + \omega_0' \int^t g(t') dt' \nonumber\\
\gamma_1(t) &=& - \frac{2 \pi}{\Phi_0} \int {\vec dl} \cdot {\vec
A(\vec r)}, \label{phasedef}\end{aligned}$$
where $\gamma_{00}$ is the intrinsic DC phase of the JJ, $\gamma_0$ is the phase generated by the external voltage, $V_0(t)=V_0 g(t)$ is the applied voltage across it, $\omega_0'=2eV_0/\hbar$ is the Josephson frequency of the junction, $g(t)$ is a dimensionless function specifying the time dependence of the applied voltage, $\Phi_0= h c/2e$ is the flux quantum, $h= 2 \pi \hbar$ with $\hbar$ being the Planck constant, $e$ is the charge of an electron, and $c$ is the speed of light. The vector potential ${\vec A(\vec r)}$ is given by $$\begin{aligned}
{\vec A(\vec r,t)} &=& \mu_0 (\vec M(t) \times \vec r)/(4 \pi |\vec
r|^3). \label{vecpot}\end{aligned}$$ Note that in our chosen geometry, as shown in Fig. \[fig1\], $\vec dl \parallel {\hat x}$ and $\vec r$ lies in the $x-y$ plane, so that $$\begin{aligned}
\gamma_1(t) &=& -k_0 M_z(t)/|\vec M|, \nonumber\\
k_0 &=& \mu_0 |\vec M| l/(2 \Phi_0 a \sqrt{l^2+a^2}), \label{mpot}\end{aligned}$$ where the geometrical factor $k_0$ can be tuned by tuning the distance $a$ between the JJ with the nanomagnet (Fig. \[fig1\]). Moreover, in this geometry, the orbital effect of the magnetic field do not affect the phase of the JJ since $ \vec dl \cdot \vec A_B
\sim \vec dl \cdot (\vec B \times \vec r) =0$. In this geometry, the LL equations for the nanomagnet reads $$\begin{aligned}
\frac{d \vec M}{dt} &=& \gamma_g (\vec M \times \vec B_{\rm eff})
\label{beffeq} \\
\vec B_{\rm eff} &=& -\frac{\delta E}{\delta \vec M} = B(M_y) \hat y
+ \frac{E_J k_0}{|\vec M|} \sin(\gamma_0(t)+\gamma_1(t))\hat z
\nonumber\end{aligned}$$ where $B(M_y)= K M_y + B(t)$ and $\gamma_g$ is the gyromagnetic ratio [@comment1]. These LL equations are to be solved along with the constraint of constant $|\vec M|$; in what follows we shall set $|\vec M|=M_0$. We note that Eq. \[beffeq\] do not include dissipation which shall be treated in Sec. \[gilres\]. Thus the solutions obtained in this section can be treated as limiting case of very weakly dissipating nanomagnets. We also note that our analysis do not take into account the change in $I_s$ arising from the spin-flip scattering induced by the coupling of the JJ with the nanomagnet [@balat1; @holm1]. This can be justified by the fact that in our geometry, the nanomagnet does not reside atop the junction and thus we expect the spin-flip scattering matrix elements to be small. Further, even with a significant contribution from spin-flip scattering, such effects become important when the Larmor frequency of the magnetization $\omega_L \ge \Delta_0/\hbar$ [@holm1] which is not the regime that we focus on. This issue is discussed further in Sec. \[dissec\].
Eq. \[beffeq\] represents a set of non-linear equations which, in most cases, need to be solved numerically. Here we identify a limit in which these equations admits an analytic, albeit perturbative, solution for arbitrary $B(t)$. To this end we define the following dimensionless quantities $$\begin{aligned}
\vec m &=& \vec M/M_0 = (\sin \theta \cos \phi, \cos \theta, \sin
\theta \sin \phi) \nonumber\\
\omega_B (t) &=& B(M_y)/B_1, \quad \epsilon_0 = k_0 E_J/(B_1 M_0)
\nonumber\\
B(t) &=& B_1 f(t), \quad \tau= \gamma_g B_1 t, \quad
\omega_0=\omega'_0/(\gamma_g B_1) \label{paramet1}\end{aligned}$$ where $f(t)$ is a dimensionless function specifying the time dependence of the magnetic field, $\omega_0$ is the dimensionless Josephson frequency (scaled with the frequency associated with the magnetic field $B_1$), and $B_1$ is the amplitude of the external magnetic field. In what follows we shall seek perturbative solution for $\vec m$ in the weak coupling and weak anisotropy limit (for which $\epsilon_0, \, K M_0/B_1 \ll 1$ and $k_0 \le 1$) to first order in $\epsilon_0$ and $K$. In terms of the scaled variables, the LL equations (Eq. \[beffeq\]) can be written in terms of $\theta$ and $\phi$ as $$\begin{aligned}
\frac{d \phi}{d\tau} &=& \omega_B(\tau) - \epsilon_0 \cot \theta
\sin
\phi \sin(\gamma_0(\tau) - k_0 \sin\theta \sin\phi) \nonumber\\
\frac{d \theta}{d \tau} &=& \epsilon_0 \cos\phi \sin(\gamma_0(\tau)
- k_0 \sin\theta \sin\phi). \label{tpeq1}\end{aligned}$$ with the initial condition $ \phi(\tau=0)=0$ and $\theta(\tau=0)=
\theta_0$. We note that the choice of this initial condition for $\theta$ and $\phi$ amounts to choosing the initial magnetization of the nanomagnet in the $x-y$ plane: $\vec M= (M_1, M_2,0)$ where $\cos \theta_0 = M_2/M_0$, and $M_1^2 +M_2^2= M_0^2$. We choose $\theta_0$ such that $\cot \theta_0 <1$ and the perturbative solutions that we present remains valid as long as $\epsilon_0
\cot(\theta) \ll 1$. We have checked that this limit is satisfied in all our numerical simulations described in Sec. \[numsol\].
The perturbative solutions of Eq. \[tpeq1\] can be obtained by writing $$\begin{aligned}
\theta(\tau) &=& \delta \theta(\tau), \quad \phi(\tau)= z(\tau) +
\delta
\phi(\tau) \nonumber\\
z(\tau) &=& K \cos(\theta_0) M_0 \tau/B_1 + \int_0^{\tau} d\tau
f(\tau) \label{pertsol}\end{aligned}$$ where $\delta \theta(\tau)$ and $\delta \phi(\tau)$ satisfies, to first order in $\epsilon_0$ and $K$ \[[*i.e.*]{}, neglecting terms ${\rm O}(\epsilon_0^2)$, ${\rm O}(K\epsilon_0)$ and ${\rm O}(k_0
\epsilon_0)$\], $$\begin{aligned}
\frac{d \delta \phi}{d\tau} &=& -\epsilon_0 \cot (\theta_0) \sin
(z(\tau)) \nonumber\\
&& \times \sin(\gamma_0(\tau) - k_0 \sin(\theta_0) \sin(z(\tau)))
\label{tpeq2} \\
\frac{d \delta \theta}{d \tau} &=& \epsilon_0 \cos(z(\tau))
\sin(\gamma_0(\tau) - k_0 \sin(\theta_0) \sin(z(\tau))). \nonumber\end{aligned}$$ The solution of Eq. \[tpeq2\] is straightforward and can be written as $$\begin{aligned}
\delta \theta(\tau) &=& \epsilon_0 \int_0^{\tau} dt'
\left[\cos(z(t'))
\sin[\gamma_0(t') \right. \nonumber\\
&& \left. - k_0 \sin(\theta_0) \sin(z(t'))] \right]\nonumber\\
\delta \phi(\tau) &=& -\epsilon_0 \cot \theta_0 \int_0^{\tau} d t'
\left[
\sin(z(t')) \sin [\gamma_0(t') \right. \nonumber\\
&& \left.- k_0 \sin(\theta_0) \sin(z(t'))]\right] \label{solgen}\end{aligned}$$
Eqs. \[pertsol\], \[tpeq2\], and \[solgen\] constitute the central result of this work. These equations describe the dynamics of a nanomagnet in the presence of weak coupling with a JJ. We note that in obtaining these results, we have neglected the normal state resistance of the JJ which can be safely done for tunnel junctions or for weak links with large resistance and small capacitance [@cai1]. We also note that the domain of validity of these solutions require $\delta \theta(\tau), \delta \phi(\tau) \le
z(\tau)$ at all times; we shall discuss this domain in the context of specific drives in Sec. \[pecresults\]. We now use these solutions to study the behavior of the supercurrent of the JJ given by $$\begin{aligned}
I_s = I_c \sin \left[ \gamma_0(\tau) - k_0 \sin(\phi(\tau))
\sin(\theta(\tau)) \right] \label{cricur}\end{aligned}$$ for several possible magnetic field profiles. Here $I_c$ is the critical current of the JJ. Although Eq. \[cricur\] yields $I_s$ for any magnetic field profile, in what follows we shall concentrate on constant and periodically varying magnetic fields since they allow for Shapiro-step like features in the I-V characteristics of a voltage biased JJ.
Before ending this subsection, we note that the solutions for $\vec
M$ is stable against small fluctuations of the direction of the applied magnetic field. To see this, we write the external magnetic field $\vec B$ is applied in an arbitrary direction in the $x-y$ plane: $\vec B= B_1 f(t) ( \sin(\alpha_0), \cos(\alpha_0),0)$ with $K \alpha_0 \ll K$. Next, we move to a rotated coordinate frame for which the magnetization $\vec m'$ is related to $\vec m$ by $$\begin{aligned}
\left( \begin{array} {c} m'_x \\ m'_y \\ m'_z \end{array} \right)
&=& \left( \begin{array} {ccc} \cos \alpha_0 & -\sin \alpha_0 & 0 \\
\sin \alpha_0 & \cos \alpha_0 & 0 \\ 0 & 0 & 1 \end{array} \right)
\left( \begin{array} {c} m_x \\ m_y \\ m_z \end{array} \right)\end{aligned}$$ We proceed by using the parametrization $ \vec m' = (\sin \theta'
\cos \phi', \cos \theta', \sin \theta' \sin \phi')$. In this representation, the initial values of $\vec m'$ are given by $$\begin{aligned}
m'_x &=& \sin(\theta_0-\alpha_0), \,\, m'_y=
\cos(\theta_0-\alpha_0), \, m'_z=0\end{aligned}$$ where $\theta_0$ and $\phi_0=0$ depicts the initial condition for $\vec m$. Next, repeating the same algebraic steps as outlined earlier in the section, one finds that the equations governing $\theta'$ and $\phi'$ are given by $$\begin{aligned}
\frac{d \theta^{\prime}}{d\tau} &=& \epsilon_0 \cos(\phi^{\prime})
\sin(\gamma_0(\tau)-k \sin(\theta^{\prime}) \sin(\phi^{\prime})) \\
\frac{d \phi^{\prime}}{d \tau} &=& \omega'_B(\tau)- \epsilon_0
\cot(\theta^{\prime}) \sin(\phi^{\prime}) \nonumber\\
&& \times \sin(\gamma_0 (\tau)-k_0 \sin (\theta^{\prime})
\sin(\phi^{\prime}) \nonumber\\
\omega'_B(\tau) &=& K (\cos(\alpha_0) \cos(\theta')+ \sin(\alpha_0)
\sin(\theta') \sin(\phi'))/B_1 \nonumber\\
&& + f(\tau) \simeq \omega_B(\tau) + {\rm O}(K\alpha_0)
\label{roteq}\end{aligned}$$ Note that the analytic solution to Eq. \[roteq\] can only be obtained when terms ${\rm O}(K \alpha_0)$ can be neglected. In this case, the perturbative solution to Eq. \[roteq\] can be obtained in the same way as done before in this section. The final result is $$\begin{aligned}
\theta'(\tau) &=& \delta \theta'(\tau), \quad \phi'(\tau)= z(\tau) +
\delta
\phi'(\tau) \nonumber\\
\delta \theta(\tau) &=& \epsilon_0 \int_0^{\tau} dt'
\left[\cos(z(t'))
\sin[\gamma_0(t') \right. \nonumber\\
&& \left. - k_0 \sin(\theta_0 -\alpha_0) \sin(z(t'))] \right]\nonumber\\
\delta \phi(\tau) &=& -\epsilon_0 \cot \theta_0 \int_0^{\tau} dt'
\left[
\sin(z(t')) \sin [\gamma_0(t') \right. \nonumber\\
&& \left.- k_0 \sin(\theta_0 -\alpha_0) \sin(z(t'))]\right]
\label{solgen2}\end{aligned}$$ The behavior of these solutions shall be checked against exact numerics in Sec. \[numsol\].
Constant and Periodically varying magnetic fields {#pecresults}
-------------------------------------------------
In this section, we apply our perturbative results on constant and periodically time-varying magnetic fields for which the I-V characteristics of the JJ may have Shapiro-like steps. While this effect has been discussed, using a somewhat different geometry, in Ref. for constant magnetic field, we demonstrate its presence for periodic magnetic fields.
[*Constant magnetic field*]{}: This case was studied in Ref.. For an external constant voltage, $g(t)=1$ and one has $\gamma_0 = \omega_0 \tau +\gamma_{00}$, where $\gamma_{00}$ is the intrinsic phase difference across the JJ at $t=0$. Further, in this case, $f(t)=1$, and $z(\tau)= \omega_c \tau$ where $\omega_c= 1+ K M_2/B_1$. Thus the supercurrent to the leading order and for $\epsilon_0, K \ll 1$, is given by $$\begin{aligned}
I_s &\simeq& I_c \sin(\omega_0 \tau + \gamma_{00} -k_0 \sin(\theta_0) \sin(\omega_c \tau)) \nonumber\\
&=& I_c \sum_n J_n [k_0 \sin (\theta_0)]
\sin[(\omega_0-n\omega_c)\tau + \gamma_{00}] \label{solzero}\end{aligned}$$ which indicates the presence of a finite DC component of $I_s$ leading to Shapiro steps in the I-V characteristics of the JJ-nanomagnet system at $$\begin{aligned}
\omega_0=n^0 \omega_c. \label{condcons1}\end{aligned}$$ To study the stability of these steps we consider the solution to ${\rm O}(\epsilon_0)$. For constant magnetic field, the ${\rm
O}(\epsilon_0)$ correction to $\vec M$ can be obtained from Eq.\[solgen\] which, after some straightforward algebra, yields for $z(\tau)=\omega_c \tau$ $$\begin{aligned}
\delta \theta(\tau) &=& - \epsilon_0 \sum_{n} J_n(k_0
\sin(\theta_0)) \nonumber \\
&& \times \sum_{s=\pm1} \frac{\cos[(\omega_0
-(n-s)\omega_c)\tau+\gamma_{00}]-\cos\gamma_{00}}{\omega_0
-(n-s)\omega_c} \nonumber\\
\delta \phi(\tau) &=& \epsilon_0 \cot(\theta_0) \sum_{n} J_n(k_0
\sin(\theta_0))
\label{pertt1}\\
&& \times \sum_{s=\pm1} s \frac{\sin[(\omega_0
-(n-s)\omega_c)\tau+\gamma_{00}]-\sin \gamma_{00}}{\omega_0
-(n-s)\omega_c} \nonumber\end{aligned}$$ We note that for $n=n_{\pm} = n^0 \mp 1$, both $\delta \theta$ and $\delta \phi$ grows linearly in time. These terms turn out to be the most important corrections to the zeroth order solution near $\omega_0=n^0 \omega_c$ and leads to the destabilization of the steps as $\epsilon_0$ increases. We also note that such terms restrict validity of the perturbative expansion up to a finite time $T_p$ so that $\epsilon_0 \cot(\theta_0) J_{n_{\pm}}(k
\sin(\theta_0)) T_p \sim 1$; we shall discuss this in more details while comparing our perturbative results with exact numerics in Sec. \[numsol\]. The supercurrent to first order in $\epsilon_0$ and $K$ is thus given by $$\begin{aligned}
I_s &\simeq& I_c \sin(\omega_0 \tau - k_0 \sin (\theta_0+ \delta
\theta(\tau))
\sin(\omega_c \tau+\delta \phi(\tau))) \label{firstorder1} \nonumber\\\end{aligned}$$ The behavior of the DC component of $I_s$ in the presence of these corrections is charted out in Sec. \[numsol\].
[*Periodic Magnetic fields*]{}: In this case, we choose a periodic magnetic field so that $f(\tau)= \cos(\omega_1 \tau)$, where $\omega_1$ is the external drive frequency measured in units of $\gamma_g B_1$ [@commentmag]. For this choice, one has $ z(\tau)
= \omega_2 \tau + \sin(\omega_1 \tau)/\omega_1$, where $\omega_2=
\gamma_g K M_2/B_1$. Thus the zeroth order solution for the supercurrent $I_s^{\rm periodic}$ reads $$\begin{aligned}
I_s ^{\rm periodic} &\simeq& I_c \sin[ \omega_0 \tau + \gamma_{00} -
k_0 \sin(\theta_0) \nonumber\\
&& \times \sin ( \omega_2 \tau
+ \sin(\omega_1 \tau)/\omega_1) ] \nonumber\\
&=& I_c \sum_{n_1, n_2} J_{n_1}(k_0 \sin(\theta_0))
J_{n_2}(n_1/\omega_1) \nonumber\\
&& \times \sin[\gamma_{00} +(\omega_0 -n_2 \omega_1 - n_1 \omega_2)
\tau].\end{aligned}$$ We note that this solution admits a finite DC component of $I_s^{\rm
periodic}$ and hence Shapiro-like steps for $(n_1,n_2)=(n_1^0,
n_2^0)$ for which $$\begin{aligned}
\omega_0 - n_2^0 \omega_1 - n_1^0 \omega_2=0. \label{condper1}\end{aligned}$$ The amplitude of these peaks depend on product of two Bessel functions unlike the ones found for constant magnetic field [@cai1]; moreover, the condition for their occurrence depends on two distinct integers which allows the peaks to occur in the absence of any DC voltage across the junction. The condition for occurrence of such peaks are given by $\omega_0=0$ and $\omega_2= n_2^0
\omega_1/n_1^0$; they provide examples of Shapiro steps without any voltage bias across a JJ and have no analog in their constant magnetic field counterparts.
The first order corrections to these solutions can be obtained in a manner analogous to one used for constant magnetic field. The final result is given by $$\begin{aligned}
I_s^{\rm periodic} &\simeq& I_c \sin \big[\omega_0 \tau - k_0
\sin(\theta_0 +\delta \theta_p(\tau)) \nonumber\\
&& \times \sin(\omega_2 \tau+ \sin(\omega_1 \tau)/\omega_1
+ \delta \phi_p(\tau))\big] \label{firstorder2} \nonumber\\\end{aligned}$$ where $\delta \theta_p$ and $\delta \phi_p$ are given by
$$\begin{aligned}
\delta \theta_p &=&
-\frac{\epsilon_0}{2}[\sum_{n_1,n_2,n_3}J_{n_1}(\frac{1}{\omega_1})J_{n_2}(k_0
\sin(\theta_0))J_{n_3}(\frac{n_2}{\omega_1})
\sum_{s=\pm1}\frac{\cos[\gamma_{00} +(\omega_0 -(n_3+s
n_1)\omega_1-(n_2+s)\omega_2)\tau]-\cos(\gamma_{00})}{\omega_0-(n_3+s
n_1)\omega_1-(n_2+s)\omega_2}] \label{percorr} \\
\delta \phi_p &=& \frac{\epsilon_0}{2} \cot(\theta_0)
[\sum_{n_1,n_2,n_3}J_{n_1}(\frac{1}{\omega_1})J_{n_2}(k_0
\sin(\theta_0))J_{n_3}(\frac{n_2}{\omega_1})\sum_{s=\pm1}
s \frac{\sin[\gamma_{00}+(\omega_0-(n_3+s
n_1)\omega_1-(n_2+s)\omega_2)\tau]-\sin(\gamma_{00})}{\omega_0-(n_3+s
n_1)\omega_1-(n_2+s)\omega_2}] \nonumber\end{aligned}$$
We note that the main contribution to the zeroth order results again comes from terms linear in time which occurs for $$\begin{aligned}
\omega_0-(n_3^s +s n_1^s)\omega_1-(n_2^s+s)\omega_2 &=& 0
\label{condpertur}\end{aligned}$$ for $s=\pm 1$. The perturbation theory thus remain valid for $\tau
\le T_p^{'}$ so that $$\begin{aligned}
\epsilon_0 T'_p J_{n_1^s}(\frac{1}{\omega_1})J_{n_2^s}(k_0
\sin(\theta_0))J_{n_3^s}(\frac{n_2^0}{\omega_1}) \le 1.
\label{pervalcond1}\end{aligned}$$ The behavior of the DC component of $I_s^{\rm periodic}$ as a function of $\epsilon_0$, as obtained from Eq. \[firstorder2\] is discussed and compared to exact numerics in Sec. \[numsol\].
[*External AC drive*]{}: Next, we consider the behavior of $I_s$ in the presence of both an external magnetic field and an AC field of amplitude $A$ and frequency $\omega_A$ so that $\gamma_0 (t)=
\gamma_{00}+ \omega_0 \tau + A \sin(\omega_A \tau)/\omega_A$. First we consider a constant magnetic field for which $f(\tau)=1$. In this case, using Eqs. \[pertsol\] and \[cricur\], one obtains, to zeroth order in $\epsilon_0$ $$\begin{aligned}
I_s &\simeq& I_c \sin \Big[\omega_0 \tau + A \sin(\omega_A
\tau)/\omega_A \nonumber\\
&&- k_0 \sin(\theta_0) \sin \omega_c \tau +\gamma_{00} \Big]
\nonumber\\
&\simeq& I_c \sum_{n_1,n_2} J_{n_1}(A/\omega_A)
J_{n_2}(k_0\sin(\theta_0))
\nonumber\\
&& \times \sin(\gamma_{00} + (\omega_0 + n_1 \omega_A - n_2\omega_c)
\tau)\end{aligned}$$ A finite DC component of $I_s$ leading to Shapiro like steps thus appear in the I-V characteristics for integers $n_1^0,n_2^0$ which satisfies $$\begin{aligned}
\omega_0 + n_1^0 \omega_A - n_2^0 \omega_c=0. \label{condcons2}\end{aligned}$$ The condition of occurrence of these peaks mimics those for periodic magnetic field in the absence of external AC drive and the peak amplitude depends on the product of two Bessel functions. We note that the resultant Shapiro steps may occur for low AC drive frequencies and thus could, in principle, be amenable to easier experimental realization.
Next, we consider a periodically varying magnetic field in the presence of external radiation. In this case, one has, $f(\tau)=\omega_2+\cos(\omega_1 \tau)$. Using Eq. \[pertsol\], one has $z(\tau)=\omega_2 \tau + \sin(\omega_1 \tau)/\omega_1$ which leads to (Eq. \[cricur\]) $$\begin{aligned}
I_s &\simeq& I_c \sin \Big[\omega_0\tau + A \sin(\omega_A
\tau)/\omega_A - k_0 \sin(\theta_0) \times \nonumber\\
&& \sin (\omega_2 \tau+ \sin(\omega_1 \tau))/\omega_1+ \gamma_{00} \Big] \nonumber\\
&\simeq& I_c \sum_{n_1,n_2, n_3 } J_{n_1}(A/\omega_A)
J_{n_2}(k_0\sin(\theta_0)) J_{n_3}(n_2/\omega_2) \nonumber\ \\
&& \times \sin(\gamma_{00} + (\omega_0 + n_1 \omega_A - n_2 \omega_2
- n_3 \omega_1) \tau)\end{aligned}$$ Thus the presence of the steps now occurs for a set of integers $(n_1^0,n_2^0,n_3^0)$ which satisfies $$\begin{aligned}
\omega_0 + n_1^0 \omega_A - n_2^0 \omega_2 -n_3^0 \omega_1=0.
\label{condper2}\end{aligned}$$ The perturbative $O(\epsilon_0)$ corrections to the above solutions can be carried out in similar manner to that outlined above.
Dissipative nanomagnets {#gilres}
-----------------------
In this section, we include the dissipative (Gilbert) term in the LLG equations to model dissipative nanomagnets and seek a solution to these equations in the limit weak dissipation and weak coupling between the JJ and the nanomagnets. The resultant LLG equations are given by $$\begin{aligned}
\frac {d \vec M}{dt} &=& \gamma_g (\vec M \times \vec B_{\rm
eff})-\frac{\eta \gamma_g}{M_0} \vec M \times d \vec M/dt
\label{llg1}\end{aligned}$$ where $\eta$ is a dimensionless constant specifying the strength of the dissipative term. Following the same parametrization as in Eqs.\[paramet1\], we find that one can express the LLG equations, in terms of $\theta$ and $\phi$, as $$\begin{aligned}
\frac{d \phi}{d\tau} &=& \omega_B(\tau) - \epsilon_0 \cot \theta
\sin
\phi \sin(\gamma_0(\tau) - k_0 \sin(\theta) \sin(\phi)) \nonumber\\
\frac{d \theta}{d \tau} &=& \epsilon_0 \cos(\phi)
\sin(\gamma_0(\tau) -
k_0 \sin(\theta) \sin(\phi)) \nonumber\\
&& - \eta \omega_B(\tau) \sin(\theta). \label{tpgil1}\end{aligned}$$ where we have neglected terms $O(\epsilon_0 \eta)$ and ${\rm
O}(\eta^2)$. We note that the effect of the dissipative term manifests itself in $\theta$ but not in $\phi$; this fact can be understood as a consequence of the fact that to leading order $\vec
M \times (\vec M \times \vec B_{\rm eff})$ lies along $\hat y$ and hence only effects the dynamics of $M_y$ which depend only on $\theta$. For small $\epsilon_0$ and $\eta$, Eq. \[tpgil1\] therefore admits a perturbative solution $$\begin{aligned}
\phi(\tau) &=& z(\tau) + \delta \phi(\tau), \quad \theta(\tau)=
\delta \theta_d(\tau)
\nonumber\\
\delta \theta_d (\tau) &=& 2 \arctan[\tan(\theta_0/2)e^{-\eta
z(\tau)}] +\delta \theta(\tau). \label{thetadsol}\end{aligned}$$ where $z(\tau)$, $\delta \theta(\tau)$, and $\delta \phi(\tau)$ are given by Eq. \[pertsol\] and we have neglected terms ${\rm
O}(\epsilon_0 \eta)$. The supercurrent, in the presence of the dissipative term is given by $$\begin{aligned}
I_s &=& I_c \sin [\gamma_0(\tau)- k_0 \sin(\theta_d(\tau))
\sin(\phi(\tau))] \label{supcurrent}\end{aligned}$$ The fate of the DC component of $I_s$ leading to Shapiro-like steps in the presence of the dissipative term shall be checked numerically in Sec. \[numsol\] for both periodic and constant applied magnetic fields.
Numerical results {#numsol}
=================
In this section, we analyze the coupled JJ-nanomagnet system both without and in the presence of dissipation and compare these results, wherever applicable, to the theoretical results obtained in Sec. \[genres\]. In what follows, we focus on cases of constant or periodically varying magnetic fields since Shapiro-step like features are expected to appear in the I-V characteristics of the JJ only for these protocols. The LLG equations for magnetization solved numerically to generate the data for the plots are given by $$\begin{aligned}
\frac{d m_x}{d \tau}&=&[-\beta_1 (1+\eta ^2 m_x^2)-\beta_2 \eta
(m_z+ \eta m_x m_y) \nonumber\\
&& + \beta_3 \eta (m_y- \eta m_x
m_z)]/(1+\eta ^2) \nonumber \\
\frac{d m_y}{d \tau}&=&[-\beta_2 (1+\eta ^2 m_y^2)-\beta_3 \eta
(m_x+ \eta m_z m_y) \nonumber\\
&& + \beta_1 \eta (m_z- \eta m_x m_y)]/(1+\eta ^2)
\nonumber\\
\frac{d m_z}{d \tau}&=&[-\beta_3 (1+\eta ^2 m_z^2)-\beta_1 \eta
(m_y+ \eta m_z m_x) \nonumber\\
&& + \beta_2 \eta (m_x- \eta m_z m_y)]/(1+\eta ^2)
\label{cartnum}\end{aligned}$$ where $$\begin{aligned}
\beta_1 &=&-(f(\tau) \cos \alpha_0 +K' m_y) m_z \nonumber\\
&& +\epsilon_0 m_y \sin(\gamma_0(\tau)-k_0 m_z) \label{betadef} \\
\beta_2&=& -\epsilon_0 m_x \sin(\gamma_0(\tau)-k_0 m_z) +
\sin(\alpha_0) f(\tau) m_z \nonumber\\ \ \beta_3&=& (f(\tau)
\cos(\alpha_0) +K' m_y) m_x -\sin(\alpha_0) f(\tau) m_y \nonumber\end{aligned}$$ In these equations $f(\tau)=1$ for constant and $f(\tau)=\cos(\omega_1 \tau)$ for the periodically varying magnetic fields, $\alpha_0=0$ indicates an applied magnetic field along $\hat
y$, we have set $\theta_0=\pi/3$ and $\gamma_{00}=\pi/2$ for all simulations, and $K'= KM_0/B_1$. Note that Eq. \[cartnum\] reduces to the usual LL equations for $\eta=0$. The supercurrent is then computed using the values of $m_{z}$ obtained from Eq.\[cartnum\]: $I_s = I_c \sin(\gamma_0(\tau)- k_0 m_z)$.
![(a) Comparison between theoretical (red dots) and numerical (black solid line) values of $I_s(\tau)/I_c$ as a function of time at late times $\tau \ge 2\times 10^4$ for a constant magnetic field $\omega_B=0.5$ along $\hat y$. Other parameters are $\epsilon_0=10^{-4}$, $\eta=0$, $k_0=0.05$, $K=0.0001$ and $\omega_0=0.5$. (b) Plot of time $T'$ after which the theoretical and analytic results for $I_s(t)$ deviates by more that $1\%$ at the peak position as a function of $\epsilon_0$.[]{data-label="fig2"}](fig2a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![(a) Comparison between theoretical (red dots) and numerical (black solid line) values of $I_s(\tau)/I_c$ as a function of time at late times $\tau \ge 2\times 10^4$ for a constant magnetic field $\omega_B=0.5$ along $\hat y$. Other parameters are $\epsilon_0=10^{-4}$, $\eta=0$, $k_0=0.05$, $K=0.0001$ and $\omega_0=0.5$. (b) Plot of time $T'$ after which the theoretical and analytic results for $I_s(t)$ deviates by more that $1\%$ at the peak position as a function of $\epsilon_0$.[]{data-label="fig2"}](fig2b.pdf "fig:"){width="4.2cm" height="3.2cm"}
To compare the theoretical results with exact numerics, we first compare the values $I_s(\tau)/I_c$ computed theoretically (Eq.\[firstorder1\]) with exact numerical result. For comparing the two results, we have fixed the external voltage $\omega_0= \omega_B$ which leads to a Shapiro step in the I-V characteristics of the JJ with $n^0=1$. As discussed in Sec. \[pecresults\], one expects one of the perturbative terms to grow linearly in time in this case; the presence of this linear term is expected to invalidate the perturbative theoretical results for $\tau >T' \sim
\epsilon_0^{-1}$. In Fig. \[fig2\](a), we show the comparison between theoretical and numerical values of $I_s(\tau)/I_c$ at $\tau>2\times 10^4$ for $\epsilon_0= 10^{-4}$; we find that the numerical and analytical values differ by less than $5\%$ even at late times ($t \simeq 2 T'$). In Fig. \[fig2\](b), we plot $T'$, which is the minimum time at which the deviation between theoretical and numerical values of $I_s(\tau)/I_c$ reaches $1\%$ near the peak position, as a function of $\epsilon_0$; the result shows the expected decrease of $T' \sim 1/\epsilon_0$ as $\epsilon_0$ increases. In Fig. \[fig3\](a), we carry out a similar comparison for dissipative nanomagnets with $\eta=0.0001$; we find that $T'$ decreases with $\epsilon_0$ in a qualitatively similar manner to the non-dissipative case. However, we note that the value of $T'$ with finite $\eta$ (Fig. \[fig3\](a)) is larger than its $\eta=0$ counterpart (Fig. \[fig2\](b)); this feature is a consequence of opposite signs of the correction terms due to $\epsilon_0$ (Eq.\[pertt1\]) and $\eta$ (Eq. \[thetadsol\]). For small $\eta$ and $\epsilon_0$, these corrections tend to mutually cancel leading to a better stability of the zeroth order result which results in higher value of $T'$. Finally, in Fig. \[fig3\](b), we plot $T'_p$ for a periodically varying magnetic field with $\omega_1=1.2$. As expected from Eq. \[pervalcond1\], $T'_p \sim
100 \epsilon_0 ^{-1} \gg \epsilon_0^{-1}$ which implies much better stability for the Shapiro-like steps for periodic magnetic field compared to their constant field counterparts.
Next, we study the presence of a finite DC component of $I_s$ in the case of a constant applied magnetic field along $\hat y$ ($f(\tau)=1$ and $\alpha_0=0$) in the absence of dissipation ($\eta=0$) and external AC voltage ($\gamma_0(\tau)= \omega_0
\tau$). The results of our study is shown in Fig. \[fig4\] where we plot $I_s^{\rm DC}/I_c$, with $I_s^{\rm DC}$ given by $$\begin{aligned}
I_s^{\rm DC} = \frac{1}{T_{\rm max}} \int_0^{T_{\rm max}} I_s(t')
dt' = I_s(\omega=0),\end{aligned}$$
![(a) Plot of $T'$ as a function of $\epsilon_0$ for $\eta=0.0001$. (b) Plot of $T'$ as a function of $\epsilon_0$ for periodic protocol with $\omega_B=0$. $\eta=0$, and $\omega_1=1.2$. All other parameters are same as in Fig. \[fig2\].[]{data-label="fig3"}](fig3a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![(a) Plot of $T'$ as a function of $\epsilon_0$ for $\eta=0.0001$. (b) Plot of $T'$ as a function of $\epsilon_0$ for periodic protocol with $\omega_B=0$. $\eta=0$, and $\omega_1=1.2$. All other parameters are same as in Fig. \[fig2\].[]{data-label="fig3"}](fig3b.pdf "fig:"){width="4.2cm" height="3.2cm"}
as a function of $\omega_0$ for a fixed constant $\omega_B$. Here $T_{\rm max} = 40,000$ represents the maximum time up to which we average $I_s(\tau)$. Note that $I_s(\tau)$ is chosen so that increasing it any further does not lead to a change in the peak height for $\epsilon_0=0$. As shown in Fig. \[fig4\](a), (b) and (c), we find that for $\epsilon_0 \ll 1$, $I_s^{\rm DC}$ shows sharp peaks at $\omega_0 = \omega_B, 2 \omega_B$ corresponding to $n^0=1,2$ in Eq. \[condcons1\]; the position of this peaks match exactly with our theoretical results. However, the peak heights turn out to be smaller than that predicted by theory and they rapidly decrease with increasing $\epsilon_0$. This mismatch between theoretical and numerical results is a consequence of the linearly growing perturbative terms $\sim \epsilon_0$ in expression for $\delta \theta(\tau)$ and $\delta \phi(\tau)$ (Eq. \[pertt1\]) which invalidate the theoretical result for $T' \sim
\epsilon_0^{-1}$. Thus for constant magnetic field and moderate $\epsilon_0 >0.01$, the step-like feature predicted in Eq.\[condcons1\] disappears. In Fig. \[fig4\](d), we study the behavior of the peak with variation of $\alpha_0$. We find that the height of the peak increases with $\alpha_0$ for small $\alpha_0$ in accordance with the theoretical prediction of Sec. \[genres\]. For larger $\alpha_0>\alpha_0^{\rm max}$, the peak height starts to decrease and the peak height becomes almost half of its maximum for $\alpha_0= \pi/2$ when $\vec B
\parallel \hat x$.
![ Plot $I_s(\omega=0)/I_c \equiv I_s^{\rm DC}/I_c$ as a function of the Josephson frequency $\omega_0= 2e V_0/(\hbar
\gamma_gB_1)$ for a constant magnetic field $\omega_B \simeq 1$ with $K=0.0001$, $k_0=0.1$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n^0=1$ and $n^0=2$ as predicted by theoretical analysis. (d) Plot of the peak height for the $n^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=10 K =0.0001$ and $k_0=0.1$. The red dots correspond to results from perturbative theoretical analysis near $\alpha_0=0$. []{data-label="fig4"}](fig4a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![ Plot $I_s(\omega=0)/I_c \equiv I_s^{\rm DC}/I_c$ as a function of the Josephson frequency $\omega_0= 2e V_0/(\hbar
\gamma_gB_1)$ for a constant magnetic field $\omega_B \simeq 1$ with $K=0.0001$, $k_0=0.1$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n^0=1$ and $n^0=2$ as predicted by theoretical analysis. (d) Plot of the peak height for the $n^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=10 K =0.0001$ and $k_0=0.1$. The red dots correspond to results from perturbative theoretical analysis near $\alpha_0=0$. []{data-label="fig4"}](fig4b.pdf "fig:"){width="4.2cm" height="3.2cm"} ![ Plot $I_s(\omega=0)/I_c \equiv I_s^{\rm DC}/I_c$ as a function of the Josephson frequency $\omega_0= 2e V_0/(\hbar
\gamma_gB_1)$ for a constant magnetic field $\omega_B \simeq 1$ with $K=0.0001$, $k_0=0.1$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n^0=1$ and $n^0=2$ as predicted by theoretical analysis. (d) Plot of the peak height for the $n^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=10 K =0.0001$ and $k_0=0.1$. The red dots correspond to results from perturbative theoretical analysis near $\alpha_0=0$. []{data-label="fig4"}](fig4c.pdf "fig:"){width="4.2cm" height="3.2cm"} ![ Plot $I_s(\omega=0)/I_c \equiv I_s^{\rm DC}/I_c$ as a function of the Josephson frequency $\omega_0= 2e V_0/(\hbar
\gamma_gB_1)$ for a constant magnetic field $\omega_B \simeq 1$ with $K=0.0001$, $k_0=0.1$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n^0=1$ and $n^0=2$ as predicted by theoretical analysis. (d) Plot of the peak height for the $n^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=10 K =0.0001$ and $k_0=0.1$. The red dots correspond to results from perturbative theoretical analysis near $\alpha_0=0$. []{data-label="fig4"}](fig4d.pdf "fig:"){width="4.2cm" height="3.2cm"}
![Plot $I_s^{\rm DC}/I_c$ for a periodically varying magnetic field $B= B_1 \sin \omega_1 t$ as a function of $\omega_1$ with $K=0.0001$, $k_0=0.1$, $\omega_0=1.2$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n_2^0=1,2,3,4$ (from right to left) as predicted by theoretical analysis. (d) Plot of the peak height for the $n_2^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=K =0.0001$ and $k_0=0.1$.[]{data-label="fig5"}](fig5a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ for a periodically varying magnetic field $B= B_1 \sin \omega_1 t$ as a function of $\omega_1$ with $K=0.0001$, $k_0=0.1$, $\omega_0=1.2$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n_2^0=1,2,3,4$ (from right to left) as predicted by theoretical analysis. (d) Plot of the peak height for the $n_2^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=K =0.0001$ and $k_0=0.1$.[]{data-label="fig5"}](fig5b.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ for a periodically varying magnetic field $B= B_1 \sin \omega_1 t$ as a function of $\omega_1$ with $K=0.0001$, $k_0=0.1$, $\omega_0=1.2$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n_2^0=1,2,3,4$ (from right to left) as predicted by theoretical analysis. (d) Plot of the peak height for the $n_2^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=K =0.0001$ and $k_0=0.1$.[]{data-label="fig5"}](fig5c.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ for a periodically varying magnetic field $B= B_1 \sin \omega_1 t$ as a function of $\omega_1$ with $K=0.0001$, $k_0=0.1$, $\omega_0=1.2$ and (a)$\epsilon_0=0.0001$ (b)$\epsilon_0=0.001$ and (c) $\epsilon_0=0.01$. The position of the peaks corresponds to $n_2^0=1,2,3,4$ (from right to left) as predicted by theoretical analysis. (d) Plot of the peak height for the $n_2^0=1$ peak as a function of the angle $\alpha_0$ made by $\vec B$ with $\hat y$ for $\epsilon_0=K =0.0001$ and $k_0=0.1$.[]{data-label="fig5"}](fig5d.pdf "fig:"){width="4.2cm" height="3.2cm"}
Next, we study the characteristics of the peaks in $I^{\rm DC}_s$ for periodically varying magnetic field for which $f(\tau)=
\cos(\omega_1 \tau)$. In Figs. \[fig5\](a), (b), and (c), we plot $I^{\rm DC}/I_c$ a function of $\omega_1$ for a fixed $\omega_0=
1.2$, $\alpha_0=\omega_2=\eta=0$, and for several values of $\epsilon_0$. We find that the position of the peaks corresponds to integer values of $n_2^0$ (as indicated in the caption of Fig.\[fig5\]) in complete accordance with Eq. \[condper1\] with $\omega_2=0$. Moreover, in contrast to the constant magnetic field case, the peaks of $I^{\rm DC}_s$ are much more stable against increasing $\epsilon_0$. This features of the peaks can be understood as follows. For periodic magnetic field with $\omega_2=0$, the zeroth order solution is given by $z(\tau) =
\sin(\omega_1 \tau)/\omega_1$; thus the perturbative terms $\delta
\theta(\tau)$ and $\delta \phi(\tau)$ (Eq. \[percorr\]) involve product of Bessel functions. This renders the effective perturbative parameter to be $\epsilon_0^{\rm eff} \simeq \epsilon_0
J_{n_1^s}(\frac{1}{\omega_1})J_{n_2^s}(k_0
\sin(\theta_0))J_{n_3^s}(\frac{n_2^0}{\omega_1})$ (Eqs,\[condpertur\] and \[pervalcond1\]). Consequently, the effect of the perturbative correction to the weak coupling solution is drastically reduced in this case leading to a better stability of peak height with increasing $\epsilon_0$. Thus periodic magnetic fields are expected to lead to enhanced stability of Shapiro steps compared to their constant field counterparts. Finally in Fig.\[fig5\] (d), we show the variation of the peak height of $I^{\rm
DC}_s$ as a
![(a) Plot of $I_s^{\rm DC}/I_c$ in the presence of an AC field $\omega(\tau)=\omega_0 + A \sin(\omega_A \tau)/\omega_A$ with $A=0.1$, $\omega_A=0.2$, $\epsilon_0=K=0.0001$, $k_0=0.1$ as a function of $\omega_0$ for constant magnetic field $\omega_B=1$. (b) Similar plot as a function of $\omega_1$ for periodic magnetic field with $\omega_0=1.2$. All the peak positions conform to the theoretical prediction in Sec. \[pecresults\].[]{data-label="fig6"}](fig6a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![(a) Plot of $I_s^{\rm DC}/I_c$ in the presence of an AC field $\omega(\tau)=\omega_0 + A \sin(\omega_A \tau)/\omega_A$ with $A=0.1$, $\omega_A=0.2$, $\epsilon_0=K=0.0001$, $k_0=0.1$ as a function of $\omega_0$ for constant magnetic field $\omega_B=1$. (b) Similar plot as a function of $\omega_1$ for periodic magnetic field with $\omega_0=1.2$. All the peak positions conform to the theoretical prediction in Sec. \[pecresults\].[]{data-label="fig6"}](fig6b.pdf "fig:"){width="4.2cm" height="3.2cm"}
function of $\alpha_0$. We again find similar non-monotonic behavior of the peak height as a function of $\alpha_0$; the reason for this is similar to that already discussed in the context of constant magnetic field case. However, in the present case, the correction terms are much smaller and the peak height is accurately predicted by the zeroth order perturbative results: $ I^{\rm DC}_s/I_c \sim
2J_{n_1^0}(k_0\sin(\theta_0 -\alpha_0)) J_{n_2^0}(n_1^0/\omega_1)$. This is most easily checked by noting that the peak height vanishes for $\alpha_0= \theta_0= \pi/3$ for which $ J_{n_1^0}(0) =
\delta_{n_1^0 0}$ leading to vanishing of the peak for $n_2^0=1$.
Next, we study the behavior of the system in the presence of an applied AC field of amplitude $A$ and frequency $\omega_A$. In the presence of such a field $\omega (\tau)= \omega_0 + A \sin(\omega_A
\tau)/\omega_A$. In Fig. \[fig6\](a), we show the behavior of the peaks of $I^{\rm DC}_s$ as a function of $\omega_0$ for a fixed $\omega_A=0.2$ and $A = 0.1$ in the presence of a constant magnetic field. The peaks in $I^{\rm DC}_s$ occur at $\omega_0=0.4,0.6,0.8,1$ (from left to right); each of these correspond to two sets of $(n_1^0,n_2^0)= (3,1)\, {\rm and}\, (-2,0),\, (2,1)\,{\rm and}\,
(-3,0),\, (1,1)\, {\rm and}\, (-4,0)$ and $(0,1)\,{\rm and}\,
(-5,0)$ respectively as predicted in Eq. \[condcons2\]. In Fig.\[fig6\](b), we investigate the behavior for $I^{\rm DC}_s$ for a periodically varying magnetic field as a function of $\omega_1$ for $\omega_0=1.2$ and for same amplitude and frequency of the AC field. We find several peaks in $I^{\rm DC}_s$; each of these peaks corresponds to a fixed set of integers $(n_1^0, n_3^0)$ (Eq.\[condper2\] with $\omega_2=0$) as shown in Table \[table1\].
[?[0.25mm]{} c | c |c ?[0.25mm]{} c| c| c ?[0.25mm]{}]{} $\omega_1$ & $n_1^0$ & $n_3^0$ &$\omega_1$ & $n_1^0$ & $n_3^0$\
1.2 & 0 & 1 & 0.3 & 0 & 4\
1 & -1 & 1 &0.28 & 1 & 5\
0.8 & 2 & 2 &0.24 & 0 & 5\
0.6 & 0 & 2 &0.2 & -3 & 3\
0.4 & -2 & 2 &0.2 & 0 & 6\
0.4 & 0 & 3 &0.08 & 0 & 15\
. (b) Variation of the peak height (for $n^0=1$) as a function of $\log \epsilon_0$ and $\log \eta$ showing the presence of a line in the $\epsilon_0-\eta$ plane for which the peak height is maximal.[]{data-label="fig7"}](fig7a.pdf "fig:"){width="4.2cm" height="3.2cm"} . (b) Variation of the peak height (for $n^0=1$) as a function of $\log \epsilon_0$ and $\log \eta$ showing the presence of a line in the $\epsilon_0-\eta$ plane for which the peak height is maximal.[]{data-label="fig7"}](fig7b.pdf "fig:"){width="4.2cm" height="3.cm"}
.(b) Variation of the peak height (for $n_0=1$) as a function of $\log \epsilon_0$ and $\log \eta$ showing the region of maximal peak height.[]{data-label="fig8"}](fig8a.pdf "fig:"){width="4.2cm" height="3.2cm"} .(b) Variation of the peak height (for $n_0=1$) as a function of $\log \epsilon_0$ and $\log \eta$ showing the region of maximal peak height.[]{data-label="fig8"}](fig8b.pdf "fig:"){width="4.2cm" height="3.cm"}
![(a)Plot of $I_s^{\rm DC}/I_c$ for a constant magnetic field $\omega_0=1.0$ as a function of $eta$ with $\epsilon_0=0.0004$.(b) Similar plot for periodic magnetic field with $\omega_1=1.2$. All other parameters are same in Fig. \[fig4\].[]{data-label="fig9"}](fig9a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![(a)Plot of $I_s^{\rm DC}/I_c$ for a constant magnetic field $\omega_0=1.0$ as a function of $eta$ with $\epsilon_0=0.0004$.(b) Similar plot for periodic magnetic field with $\omega_1=1.2$. All other parameters are same in Fig. \[fig4\].[]{data-label="fig9"}](fig9b.pdf "fig:"){width="4.2cm" height="3.cm"}
Next, we study the effect of dissipation on these peaks by plotting $I^{\rm DC}_s$ as a function of $\omega_0$ in Fig. \[fig7\](a) (for constant magnetic field) and as a function of $\omega_1$ in Fig. \[fig8\](a) (periodic magnetic field) for $\eta=0.0001$. As seen in both cases, the position of the peaks remain same as that for $\eta=0$ in accordance with the analysis of Sec. \[gilres\]. The variation of the peak height as a function of $\log \epsilon_0$ and $\log \eta$ is shown in Figs. \[fig7\](b) for a constant magnetic field. We find that the maximal peak-height occur along a line in the $\epsilon_0-\eta$ plane. This can be seen more clearly by plotting $I^{\rm DC}_s/I_c$ as a function of $\eta$ for a fixed $\epsilon_0$ as shown in Fig. \[fig9\](a); the figure displays a clear peak in $I_s^{\rm DC}$ at $\epsilon_0 \simeq
\eta$. This can be understood from Eq. \[thetadsol\] and \[pertt1\] as follows. For the constant magnetic field, $z(\tau)=
\omega_c \tau$; consequently for small $\eta$, the correction to the zeroth order solution from the dissipative term varies linearly with $\eta$ (Eq. \[thetadsol\]) $$\begin{aligned}
\delta \theta_d (\tau) \simeq \delta \theta(\tau) + \theta_0 +
\sin(2\theta_0) \eta \omega \tau + ... \label{constmagdis}\end{aligned}$$ where the ellipsis indicate higher order term in $\eta$. This correction has opposite sign to the $\tau$-linear correction terms (terms corresponding to $n=n^0 \mp 1$ in Eq. \[pertt1\]) arising due to a finite $\epsilon_0$ in $\delta \theta(\tau)$. The corrections from $\eta$ and $\epsilon_0$ with opposite signs cancel along some specific line $\epsilon_0-\eta$ plane leading to enhanced better stability of the zeroth order solution and hence enhanced peak height. We note that the angle of this line depends on details of the relative magnitude of the correction terms. Thus we find that the presence of dissipation in a nanomagnet may lead to enhancement of the Shapiro-like steps for constant magnetic fields.
In contrast, as shown in \[fig8\](b), the peak height is almost independent of $\eta$ for small $\eta$ for periodically varying magnetic field. This can also be clearly seen from Fig.\[fig9\](b) where $I_s^{\rm DC}$ is shown to be indepedent of $\eta$ for small $\eta$ at fixed $\epsilon_0$. For such fields, $z(\tau)= \omega_2 \tau + \sin(\omega_1 \tau)/\omega_1$, where $\omega_2 = \gamma_g K M_2/B_1 \ll \omega_1$ for our choice of parameters. In this case, one can write, for small $\eta$ $$\begin{aligned}
\delta \theta_d (\tau) \simeq \delta \theta(\tau) + \theta_0 +
\sin(2\theta_0) \eta \sin(\omega_1 \tau)/\omega_1+ ...
\label{constmagdis}\end{aligned}$$ where the ellipsis indicate higher order term in $\eta$. Thus the correction term is bounded and provides an oscillatory contribution to $\theta(\tau)$. For small $\eta$, it is insignificant compared to the correction term from $\epsilon_0$ and hence the peak height stays almost independent of $\eta$. Thus we find that the role of dissipation is minimal for small $\eta$ in case of periodically varying magnetic fields. The oscillatory variation of the peak height as a function of $\epsilon_0$ for a fixed $\eta$ can be traced to its dependence on product of three Bessel functions as can be seen from Eq. \[percorr\].
Finally, we briefly study the effect of increasing $T_{\rm max}$ in our numerical study. The relevance of this lies in the fact that for any finite $\epsilon_0$ and $\eta$, our analytical results hold till $\tau \sim T'$ (constant magnetic field) or $\tau \sim T'_p$ (periodic magnetic field) while the DC signal receives contribution from all $T$. Thus it is necessary to ensure that these deviations do not lead to qualitatively different results for the DC response. To this end, we plot the height of the peak value of $I_{s}^{DC}$ as a function of $1/T_{\rm max}$ in Fig. \[fig10\]. We find from Fig. \[fig10\](a) that for constant magnetic field, the peak height indeed extrapolates to zero indicating that the Shapiro steps will be destabilized due to perturbative corrections if $I_s$ is averaged over very long time. However, we note from Fig.\[fig10\](c), $I_s^{\rm DC}$ could retain a non-zero value in the presence of a finite dissipation parameter $\eta$. This could be understood since the effect of damping, as shown in Fig.\[fig8\], negates that of $\epsilon_0$ on the peak value of $I_s^{\rm DC}$. Furthermore, from Figs. \[fig10\](b) and (d), we note that for the periodic magnetic fields the extrapolated value of $I_s^{\rm DC}$ is a finite which is lead to finite Shapiro steps in the I-V characteristics of theses JJs. Thus we expect that the Shapiro-step like features in the I-V characteristics of the JJ to be much more stable for periodically varying magnetic fields.
![Plot $I_s^{\rm DC}/I_c$ as a function of $1/T_{\rm max}$ for (a) constant magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (b) periodically varying magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (c) constant magnetic field with $\epsilon_0=\eta=0.001$ and (d) Periodically varying magnetic field with $\epsilon_0=\eta=0.001$. All other parameters are same as in Figs. \[fig4\] and \[fig5\].[]{data-label="fig10"}](fig10a.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ as a function of $1/T_{\rm max}$ for (a) constant magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (b) periodically varying magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (c) constant magnetic field with $\epsilon_0=\eta=0.001$ and (d) Periodically varying magnetic field with $\epsilon_0=\eta=0.001$. All other parameters are same as in Figs. \[fig4\] and \[fig5\].[]{data-label="fig10"}](fig10b.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ as a function of $1/T_{\rm max}$ for (a) constant magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (b) periodically varying magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (c) constant magnetic field with $\epsilon_0=\eta=0.001$ and (d) Periodically varying magnetic field with $\epsilon_0=\eta=0.001$. All other parameters are same as in Figs. \[fig4\] and \[fig5\].[]{data-label="fig10"}](fig10c.pdf "fig:"){width="4.2cm" height="3.2cm"} ![Plot $I_s^{\rm DC}/I_c$ as a function of $1/T_{\rm max}$ for (a) constant magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (b) periodically varying magnetic field with $\epsilon_0=0.001$ and $\eta=0$, (c) constant magnetic field with $\epsilon_0=\eta=0.001$ and (d) Periodically varying magnetic field with $\epsilon_0=\eta=0.001$. All other parameters are same as in Figs. \[fig4\] and \[fig5\].[]{data-label="fig10"}](fig10d.pdf "fig:"){width="4.2cm" height="3.2cm"}
Discussion {#dissec}
==========
In this work, we have studied a coupled JJ-nanomagnet system and analyzed the behavior of the supercurrent in the JJ in the presence of a finite coupling to the nanomagnet. We have provided a perturbative analytical solution to both the LL and the LLG equations governing the magnetization dynamics of the nanomagnet for arbitrary time-dependent magnetic field applied along the easy axis of the nanomagnet in the presence of weak coupling to the JJ and for weak dissipation. We have estimated the regime of validity of our perturbative results. We note here that whereas we have mainly focussed on the dynamics of the critical current of the junction in the presence of the nanomagnet in this work, the dynamics of the nanomagnet itself may also have interesting features [@buzhdin1; @yury1]; we leave this issue for future work.
Using these results, we have studied the behavior of the supercurrent of the JJ for constant and periodically varying magnetic fields. The reason for choice of such magnetic fields are that they are the only ones which lead to a fixed DC component of $I_s(t)$ which in turn leads to Shapiro step-like features in the I-V characteristics of a voltage-biased JJ. We note that while such features are known for constant magnetic field from earlier works [@cai1; @buzhdin1], the presence of such peaks in $I_s^{\rm DC}$ has not been theoretically reported for periodically time-varying magnetic fields. Moreover, we show, both from our analytical results and by performing exact numerics which supports these results, that the peaks in $I_{s}^{\rm DC}$ for periodically varying magnetic field are much more robust against increase of both $\epsilon_0$ and $\eta$ compared to their constant field counterparts; we therefore expect such peaks to be more experimentally accessible. We have also studied the behavior of such JJ-nanomagnet systems in the presence of external AC voltage. The presence of such a voltage leads to more peaks in $I_{s}^{\rm DC}$ whose positions are accurately predicted by our theoretical analysis. We note that our analysis, which is carried out at zero temperature, is expected to be valid at low temperature where $k_B T \ll \Delta_0, g \mu_B B$ so that the the presence of thermal noise can be neglected. However, we point out that the effect of such noise term in our formalism can be addressed by adding a (white) noise term in the Gilbert equations following standard procedure [@noise1].
Our analysis could be easily extended to unconventional superconductors hosting Majorana end states. For these junctions, the current-voltage relation is $4 \pi$ periodic and given by $I_s =
I_c \sin (\gamma(t)/2)$ [@kit1; @kwon1; @kiril1]. An analysis using this I-V relation immediately reveals that the Shapiro steps will be present for $\omega_0= 2n^0 \omega_c$ for constant magnetic field (Eq. \[condcons1\]) and $\omega_0 = 2(n_2^0 \omega_1+n_1^0
\omega_2)$ for periodic magnetic fields (Eq. \[condper1\]). The additional factor of $2$ is a consequence of $4\pi$ periodicity mentioned above. Thus coupling such JJs with Majorana end modes to nanomagnets in the presence of a magnetic field may lead to new experimental signatures of such end modes.
The experimental verification our work would involve preparing a voltage biased JJ-nanomagnet system with sufficiently small values of $\epsilon_0$. The current in such a junction, assuming a resistive junction, is given by $$\begin{aligned}
I(t) = I_c \sin(\phi(t)-k_0 m_z) + V_0/R + \frac{\hbar k_0}{2 e R}
dm_z/dt \label{vbias}\end{aligned}$$ where $V_0$ is the bias voltage, $R$ is the resistance of the junction and $\phi(t) = 2 e V_0 t/\hbar = \omega_0 t$. Thus the DC component of the current will show additional spikes when Eqs.\[condcons1\] (constant magnetic field) or \[condper1\] (periodic magnetic field) is satisfied. We note that it is essential to have a voltage bias to observe these steps. This can be seen from the fact that for current-biased junctions, the phase $\phi(t)$ is not locked to a fixed value of $\omega_0 t$ but has to be obtained from the solution of $$\begin{aligned}
I = \frac{\hbar}{2eR} \partial_t \phi(t) + I_c \sin[\phi(t) - k_0
m_z] - \frac{\hbar k_0}{2 e R} dm_z/dt, \label{cbias}\end{aligned}$$ where $I$ is the bias current. Thus $\phi$ becomes a function of $m_z$; we have checked both using perturbative analytic methods [@kiril1] and exact numerics that in this case no steps exist. This situation is to be contrasted to the case of standard Shapiro steps induced via external radiation of amplitude $A$ where steps can be shown to exist for both current and voltage biases [@likharev1; @kiril1]. Thus the present system would require a voltage-biased junction for observation of Shapiro steps.
For experimental realization of such a system, this we envisage a 2D thin film superconducting junction in the $x-y$ plane coupled to the nanomagnet as shown in Fig. \[fig1\]. We note that value of Josephson energy and superconducting gap in a typical niobium film are $E_J \sim 40$K and $\Delta_0 \sim 3$ meV respectively. Thus for a typical magnetic field $\simeq 100$ Gauss, for which $\gamma_g
B_1= 0.28$GHz, one could estimate an $\epsilon_0 \simeq 0.0005$ for $k_0 \simeq 0.1$. The Larmor frequency $\omega_L$ associated with such magnetic field would be order of GHz while the spin-flip processes responsible for any change in the Josephson current in niobium junctions would be valid only if $\omega_L \ge
\Delta_0/\hbar \sim 4.5$THz conforming the regime for our perturbative analysis to hold. Moreover the critical current in such junctions is $I_c = 2 e E_J / \hbar \simeq 1.5 \mu$A. Thus the peaks in DC would correspond to $I_s^{\rm DC} \simeq 10$nA which is well within detection capability of standard experiments. Typically such peaks would require a voltage bias of $0.1 \mu$V which is also well within present experimental capability. We also note here that such experiments should also be possible with 1D junctions which have been prepared experimentally in recent times using nanowires with spin-orbit coupling [@rokhin1].
To conclude, we have provided a perturbative analytic results for supercurrent of a coupled JJ-nanomagnet system in the limit of weak coupling between them and in the presence of a time dependent field applied to the system. Using this analytic result and exact numerical solution of the LL and the LLG equations, we predict existence of peaks in $I_s^{\rm DC}$ for both constant and periodic magnetic fields which are expected to provide Shapiro-like steps in the I-V characteristics of a voltage-biased JJ without the presence of external AC drive. We have analyzed the effect of finite dissipation of the nanomagnet and the presence of external AC drive on these peaks and discussed experiments which can test our theory.
[*Acknowledgement:*]{} KS acknowledges DST/RFBR grant INT/RUS/RFBR/P-249. RG acknowledges SPM fellowship from CSIR for support. The reported study was partially funded by RFBR according to the research projects 15-29-01217 and 16-52-45011, India.
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---
abstract: 'We compute the one-loop corrections to $Z$ decay properties from dimension-6 operators in the Standard Model Effective Field Theory (SMEFT) that contribute also to anomalous 3-gauge boson couplings and examine the relative sensitivity of the two processes to the anomalous couplings. The size of the contributions is of order a few percent, of the same size as Standard Model electroweak corrections. This is part of a program of computing electroweak quantities to one-loop in the SMEFT: these calculations are needed for a future global fit to limit the coefficients of the dimension-six Wilson coefficients consistently at one loop.'
author:
- Sally Dawson
- Ahmed Ismail
bibliography:
- 'zpaper.bib'
title: SMEFT Corrections to $Z$ Boson Decays
---
Introduction {#sec:intro}
============
The development of the precision electroweak program at the LHC is a major task for the coming decade. At present, the interactions of the Higgs boson and the electroweak gauge bosons appear to have approximately Standard Model (SM) like interactions and there is no sign of new massive particles. These points together imply that deviations from the SM can be analyzed in an effective field theory framework[@Giudice:2007fh; @Brivio:2017vri].
In the Standard Model Effective Field Theory (SMEFT), deviations from the SM are parameterized in terms of a tower of higher dimension operators, $O_k^d$, $$\mathcal{L}=\mathcal{L}_{SM}+\Sigma_d\Sigma_k{C_k^d O_k^d\over \Lambda^{d-4}}\, ,$$ where the operators, $O_k^d$, contain only SM fields and are invariant under $SU(3)\times SU(2)\times U(1)$. The complete set of dimension-$6$ operators was first compiled in Refs. [@Buchmuller:1985jz; @Grzadkowski:2010es] and the Feynman rules in this basis (“Warsaw Basis”) are conveniently given in Ref. [@Dedes:2017zog]. The new physics is completely contained in the coefficient functions, $C_k^d$. The scale of the assumed UV complete theory is $\Lambda$, and we assume $\Lambda \gg v=246$ GeV. For a weakly coupled theory, the corrections to SM predictions are dominated by the dimension-$6$ contributions.
Predictions for Higgs production and decay, along with $VV$ ($W^\pm,Z,\gamma$) interactions are well known at tree level in the SMEFT[@Giudice:2007fh; @Brivio:2017vri; @Falkowski:2015fla; @Contino:2014aaa]. Including also contributions to the oblique parameters, limits on the allowed sizes of the SMEFT coefficients can be extracted in a global fit to Higgs signal rates and gauge boson pair production[@Butter:2016cvz; @Ellis:2018gqa; @Berthier:2016tkq; @Falkowski:2017pss]. A precision Higgs and electroweak physics program, however, requires SMEFT calculations beyond the leading order if matching between the experimental results and theory is to be eventually done at the few percent level.
The program of calculating SMEFT quantities beyond leading order is in its infancy. One-loop calculations exist for $H\rightarrow \gamma
\gamma$[@Hartmann:2015aia; @Hartmann:2015oia; @Dedes:2018seb], $H\rightarrow b {\overline b}$[@Gauld:2016kuu; @Gauld:2015lmb] and the unphysical $H\rightarrow ZZ$ and $H\rightarrow W^+W^-$ processes[@Dawson:2018pyl; @Dawson:2018liq]. The one-loop Yukawa, $y_t$, and $\lambda={M_H^2\over 2 v^2}$ contributions to $Z$ decays are also known[@Hartmann:2016pil]. In addition to effects in the electroweak sector, one-loop contributions from top-quark operators can significantly affect Higgs production rates at the LHC[@Degrande:2012gr; @Vryonidou:2018eyv].
In this paper, we compute the $1$-loop corrections to the partial $Z$ decay widths due to the dimension-$6$ operators that contribute to $pp\rightarrow W^+W^-$ and compare the sensitivity of the two processes. These operators are particularly interesting because for transverse gauge boson production they contribute to different helicity amplitudes [@Azatov:2016sqh; @Baglio:2017bfe], such that their interference with the SM does not grow with energy unless decays or higher order corrections are considered [@Azatov:2017kzw; @Panico:2017frx]. Along with anomalous 3-gauge boson couplings, we include in our calculation the shifts in the $Z$ decay widths due to anomalous fermion couplings, which have important contributions not only to the $Z$ widths[@Dawson:1994fa], but also to gauge boson pair production[@Baglio:2017bfe; @Zhang:2016zsp; @Alves:2018nof]. Low energy data places strong limits on deviations from the SM and information from $Z$ decays is particularly interesting due to the precision of the LEP measurements. Consistent fits to the LEP data require the inclusion of the complete set of SMEFT operators, along with the one-loop predictions. Our calculation is a step in this direction, and is related to previous studies of the loop effects of gauge boson self-couplings on precision electroweak observables [@Hagiwara:1992eh; @Hagiwara:1993ck; @Alam:1997nk; @Mebane:2013cra; @Mebane:2013zga].
In Section \[sec:smeft1\], we review the basics of the one-loop SMEFT calculation and in Section \[sec:zdec\] the calculation of $Z\rightarrow f {\overline f}$ in the SMEFT is summarized, with analytic formulae presented in a series of appendices. Numerical results are given in Section \[sec:zdec\].
SMEFT at one-loop {#sec:smeft1}
=================
In this work we consider modifications of the $Z f {\overline f}$ and $W^+W^-V$ ($V=Z,\gamma$) vertices. We consider only operators that contribute to both $q {\overline q}\rightarrow W^+W^-$ [@Baglio:2017bfe; @Alves:2018nof] and to $Z\rightarrow f {\overline f}$.
The fermion vertices can be parameterized as, $$\begin{aligned}
\mathcal{L}_f&=&g_ZZ_\mu\biggl[
\biggl(g_L^{Zf}+\delta g_{L}^{Zf}\biggr)
{\overline f}_L\gamma_\mu f_L\
+\biggl[g_R^{Zf}+\delta g_{R}^{Zf}\biggr]
{\overline f}_R\gamma_\mu f_R +(f\rightarrow f^\prime)\biggr]\nonumber \\
&&+{g\over \sqrt{2}}\biggl\{W_\mu\biggl[(1+\delta g_{L}^W){\overline f}_L\gamma_\mu f_L^\prime
+\delta g_R^W
{\overline f}_R\gamma_\mu f_R^\prime\biggr] +h.c.\biggr\}\, ,
\label{eq:dgdef}
\end{aligned}$$ where $g_Z\equiv e/(c_W^{}s_W^{})= g/c_W$ and $f$ ($f^\prime$) denotes up-type (down-type) quarks. The SM fermion couplings are: $$\begin{aligned}
g_R^{Zf}&=&-s_W^2 Q_f\quad{\rm and}\quad g_L^{Zf}=T_3^f -s_W^2 Q_f,\end{aligned}$$ where $T_3^f=\pm \displaystyle \frac{1}{2}$ and $Q_f$ are the weak isospin and electric charge of the fermions, respectively.
Assuming CP conservation, the most general Lorentz invariant $3-$gauge boson couplings can be written as [@Gaemers:1978hg; @Hagiwara:1986vm] $$\begin{aligned}
L_{V}=
-ig_{WWV}\left(g_1^V\left(W^+_{\mu\nu}W^{-\mu}V^\nu-W_{\mu\nu}^-W^{+\mu}V^\nu\right)+\kappa^VW^+_\mu
W^-_\nu V^{\mu\nu}+\frac{\lambda^V}{M^2_W}W^+_{\rho\mu}{W^{-\mu}}_\nu V^{\nu\rho}\right),
\label{eq:lagdef}\end{aligned}$$ where $g_{WW\gamma}=e$ and $g_{WWZ}=g c_W$. For the $3-$gauge boson couplings we define $g_1^V = 1+\delta g_1^V$, $\kappa_{}^V=
1+\delta\kappa_{}^V$, and in the SM, $\delta g_1^V = \delta\kappa_{}^V
= \lambda_{}^V = 0$. Because of gauge invariance we always have $\delta g_1^\gamma = 0$. We assume $SU(2)$ invariance, which implies that the coefficients are related by, $$\begin{aligned}
\delta g_L^W&=&\delta g_L^{Zf}-\delta g_L^{Zf'},
\nonumber \\
\delta g_1^Z&=& \delta \kappa_{}^Z+{s_W^2\over c_W^2}\delta \kappa_{}^\gamma,
\nonumber \\
\lambda_{}^\gamma &=& \lambda_{}^Z,
\label{eq:su2rel}\end{aligned}$$ leaving three independent effective couplings.
We work in the Warsaw basis [@Buchmuller:1985jz; @Grzadkowski:2010es] and the dimension-$6$ operators contributing to the 3-gauge boson vertices are, $$\begin{aligned}
\mathcal{O}_{W}&=& \epsilon^{abc} W_\mu^{a\nu}W_\nu^{b\rho}W_\rho^{c\mu}\nonumber \\
\mathcal{O}_{HWB}&=& \Phi^\dagger\sigma^a\Phi W^a_{\mu\nu}B^{\mu\nu}\nonumber \\
\label{eq:ops}\end{aligned}$$ where $D_\mu \Phi=(\partial_\mu -i\,\frac{g}{2}\sigma^a
W^a_\mu-i\frac{g'}{2}B_\mu)\Phi$, $W^a_{\mu\nu}=\partial_\mu W^a_\nu
-\partial_\nu W^a_\mu+g\varepsilon^{abc}W^b_\mu W^c_\nu$, and $\Phi$ is the Higgs doublet field with a vacuum expectation value $\langle\Phi\rangle =
(0,v/\sqrt{2})^{\rm T}$. Two other operators involving the Higgs and gauge bosons make important contributions to the effective $Zf\bar{f}$ vertices, $$\begin{aligned}
\mathcal{O}_{HW}&=& (\Phi^\dagger\Phi)W^a_{\mu\nu}W^{a\mu\nu}\nonumber \\
\mathcal{O}_{HB}&=&(\Phi^\dagger\Phi)B_{\mu\nu}B^{\mu\nu}\, .
\label{ops:ex}\end{aligned}$$ $\mathcal{O}_{HW}$ and $\mathcal{O}_{HB}$ contribute to the 1-loop renormalization of the input parameters, as discussed in the next section.
We take as our input parameters $M_W, M_Z$ and $G_{\mu}$. All other parameters are defined in terms of the input parameters. The Lagrangian of interest to us is: $$\begin{aligned}
\mathcal{L}&=&-\frac14 W_{\mu\nu}^a W_{\mu\nu}^a-\frac14 B_{\mu\nu}B_{\mu\nu}
+\frac1{\Lambda^2}\left(C_{HW} \mathcal{O}_{HW} +C_{H B} \mathcal{O}_{HB} +C_{HWB} \mathcal{O}_{HWB}
+C_{W} \mathcal{O}_{W}
\right).\end{aligned}$$ We define “barred” fields, ${\overline W}_\mu\equiv (1-C_{H W} v^2/\Lambda^2)W_\mu$ and ${\overline B}_\mu\equiv (1-C_{H B}v^2/\Lambda^2)B_\mu$ and “barred” gauge couplings, ${\overline g}\equiv (1+C_{H W} v^2/\Lambda^2)g$ and ${\overline g^\prime}\equiv (1+C_{H B}v^2/\Lambda^2)g^\prime$ so that ${\overline W}_\mu {\overline g}= W_\mu g$ and ${\overline B}_\mu {\overline g^\prime}= B_\mu g^\prime $. The “barred” fields have their kinetic terms properly normalized and the covariant derivatives have the canonical form. The masses of the W and Z fields (poles of the propagators) are, in terms of the “barred” couplings [@Dedes:2017zog; @Alonso:2013hga], $$\begin{aligned}
M_W^2&=&\frac{{\overline g}^2 v^2}4,\nonumber\\
M_Z^2&=&\frac{({\overline g^\prime}^2+{\overline g}^2) v^2}4+\frac{v^4}{\Lambda^2}\left(\frac18 ({\overline g^\prime}^2+{\overline g}^2) C_{H D}
+\frac12 {\overline g^\prime}{\overline g}C_{H WB} \right).
\label{eq:wdef}\end{aligned}$$ The extra terms in the definition of the $Z$ mass are due to the rotation, $(W_\mu^3,B_\mu)\to(Z_\mu,A_\mu)$, that is proportional to $C_{HWB}$[^1]. We can define $\cos\theta_W\equiv c_W$ in terms of $M_W$ and $M_Z$, $$\begin{aligned}
c_W^2&\equiv& {M_W^2\over M_Z^2}\nonumber \\
&=&{{\overline g}_2^2\over
({\overline g}_1^2+{\overline g}_2^2) }\biggl(1+{\delta c_W^2\over c_W^2}\biggr)
\, ,
\end{aligned}$$ and ${\delta s_W^2\over s_W^2}=-{\delta c_W^2\over c_W^2}$. Comparing with Eq. \[eq:wdef\], $${\delta s_W^2} =-{s_{W} c_{W}\over c_{W}^2-s_{W}^2}{v^2\over\Lambda^2}C_{HWB}\, .
\label{eq:sdef}$$ In Eq. \[eq:sdef\] we can use, $c_{W}={M_W\over M_Z}$ to ${\cal {O}}({v^2\over \Lambda^2})$.
We find the following mappings between the SMEFT coefficients, $C_{HWB}$ and $C_W$, and the effective couplings, $$\begin{aligned}
\delta g_1^Z &=& -{\delta s_W^2\over c_W^2}\nonumber\\
\delta \kappa^Z&=& -2\delta s_W^2\nonumber \\
\delta \kappa_{}^\gamma &=&-{c_W^2-s_W^2\over s_W^2}\delta s_W^2 \nonumber \\
\lambda_{}^V&=& \frac{v}{\Lambda^2} 3 M_W^{} C_{W}\nonumber\\
\delta g_L^W &=& \delta s_W^2\nonumber\\
\delta g_R^W&=&0\, \nonumber\\
\delta
g^{Zf}_{L,R}&=&Q_f \delta s_W^2 ,\nonumber\\
\label{eq: mapcoef}\end{aligned}$$ The shifts including the SMEFT operators that we have omitted can be found in Refs. [@Berthier:2015oma; @Zhang:2016zsp].
Results {#sec:zdec}
=======
At tree level, the decay amplitude for $Z \to f(p) \bar{f}(p')$ in the SMEFT is, (including only those terms that contribute also to 3- gauge boson vertices), $$\mathcal{M}_0 =2M_{Z0}\sqrt{\sqrt{2}G_{\mu 0}}\biggl\{T_3^f-Q_f\biggl(1-{M_{W0}^2\over M_{Z0}^2}\biggr)
+Q_f {M_{W0}\over M_{Z0}}\sqrt{1-{M_{W0}^2\over M_{Z0}^2}}{v^2\over\Lambda^2}
C_{HWB} \biggr\} \bar{u}(p) \slashed{\epsilon}^*(p + p') v(p')\, ,$$ where the subscript $"0"$ indicates the unrenormalized tree level value, and in the $C_{HWB}$ term we can take $v^2={1\over \sqrt{2} G_\mu}$.
At one loop, there are contributions from corrections to the input parameters and fields to $\mathcal{M}_0$, $Z-\gamma$ mixing, and the one-particle irreducible loop corrections to the decay, $\mathcal{M}_1$. The virtual decay amplitude is, $$\begin{split}
\mathcal{M}_{\mathrm{1-loop}} &= \left( 1 + \delta C_{HWB} \frac{\partial}{\partial C_{HWB}} + \delta G_\mu \frac{\partial}{\partial G_\mu} + \delta M_Z^2 \frac{\partial}{\partial M_Z^2} + \delta M_W^2 \frac{\partial}{\partial M_W^2} + \frac{1}{2} \delta Z_Z + \delta Z_f \right) \mathcal{M}_0 \\&\quad - \mathcal{M}_\gamma \frac{\Pi_{\gamma Z}(M_Z^2)}{M_Z^2} + \mathcal{M}_1\, .
\label{eq:zdecayamp}
\end{split}$$ Here $\mathcal{M}_\gamma$ is the amplitude for $\gamma \to f(p) \bar{f}(p')$, which is $$\mathcal{M}_\gamma ={2M_W }\sqrt{\sqrt{2}G_\mu}Q_f\biggl\{\sqrt{1-{M_W^2\over M_Z^2}}-{M_W\over M_Z}{v^2\over\Lambda^2}C_{HWB}\biggr\} \bar{u}(p) \slashed{\epsilon}^*(p + p') v(p')\, ,$$ in the SMEFT. In Eq. \[eq:zdecayamp\], $Z_f$ and $Z_Z$ are the wavefunction renormalizations of the external $Z$ boson and the fermions. We use on-shell renormalization for all quantities, except for the Wilson coefficients which are renormalized using ${\overline{MS}}$ subtraction. In general, the coefficients are renormalized as[@Alonso:2013hga; @Grojean:2013kd], $$C_i(\mu)=C_{0,i}
-{1\over 32\pi^2 \hat{\epsilon}}\gamma_{ij}C_j,$$ where $\mu$ is the renormalization scale, $\gamma_{ij}$ is the one-loop anomalous dimension and $\hat{\epsilon}^{-1}\equiv\epsilon^{-1}-\gamma_E+\log(4\pi)$ is related to the regulator $\epsilon$ for integrals evaluated in $d=4-2\epsilon$ dimensions.
The renormalization of $G_\mu$ in the SMEFT, including both logarithms and constant contributions, can be found in the appendix of Ref. [@Dawson:2018pyl]. The shifts in the SM input parameters as well as the external field wave function renormalizations follow from the 2-point functions in Appendix \[sec:appa\].
We calculate the contributions to Eq. \[eq:zdecayamp\] to $\mathcal{O}(\frac{1}{\Lambda^2})$, neglecting higher order terms whose impact would be expected to be comparable to that of dimension-$8$ operators. The 1PI loop amplitude $\mathcal{M}_1$ is given in Appendix \[sec:appb\]. We use FeynArts [@Hahn:2000kx] and FeynCalc [@Mertig:1990an; @Shtabovenko:2016sxi] to calculate loop amplitudes with the SMEFT package for FeynRules [@Alloul:2013bka; @Christensen:2009jx]. Explicit analytic expressions for the loop integrals have been computed using the FeynHelpers interface [@Shtabovenko:2016whf] between FeynCalc and Package-X [@Patel:2015tea]. As a check of our calculation, we demonstrate that the UV poles in Eq. \[eq:zdecayamp\] cancel completely in Appendix A.
There are IR divergences arising from loops with massless photons, appearing in the fermion wave function renormalization and $\mathcal{M}_1$. We regulate these divergences with a photon mass, $M_Z \beta$. We find, $$\begin{aligned}
Re\biggl(
Z_f\mathcal{M}_0+\mathcal{M}_1\biggr)&= \sqrt{G_\mu}
{M_Z M_W^2\over {\sqrt[4]{2} \pi^2}}
Q_f^2\log \beta
(3+\log \beta)\sqrt{1-{M_W^2\over M_Z^2}}
\biggl\{
-G_{\mu} \sqrt{1-{M_W^2\over M_Z^2}} \left[ T_3^f-Q_f\biggl(1-{M_W^2\over M_Z^2} \biggr) \right] \\
&+\frac{C_{HWB}}{\sqrt{2} \Lambda^2} {M_W\over M_Z}\biggl[
2T_3^f-3Q_f\biggl(1-{M_W^2\over M_Z^2}\biggr)\biggr]\biggr\}\, . {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:virtg}\end{aligned}$$
The above divergences give $\beta$-dependent terms in the decay width, which are in turn canceled by real photon emission that contributes to both soft and collinear singularities. The SMEFT calculation of $Z \to f \bar{f} \gamma$ proceeds analogously to the well-known SM result [@Ellis:1991qj] with additional terms proportional to $C_{HWB}$. The result is $$\begin{aligned}
\left| \mathcal{M}(Z \to f \bar{f} \gamma) \right|^2 &=& {4G_\mu^2M_W^2M_Z^2 \over \pi^2}Q_f^2
\biggl\{
\biggl(T_3^f-Q_f s_W^2\biggr)\biggl[
-\log (\beta ) (3+ \log \beta )
+\frac{3
E_0^2}{2 M_Z^2} \nonumber \\&&
- \log \left(\theta_0\right)\biggl(
\frac{3 E_0}{M_Z}+\log \left(\frac{M_Z}{2
E_0}-1\biggr)-\frac{3}{4}\right)-\frac{E_0}{M_Z}+
\frac{5 \pi ^2}{12}-\frac{87}{16}\biggr]\cdot
\biggl[-s_W^2\biggl (T_3^f-Q_f s_W^2\biggr)
\nonumber\\ &&+{2c_Ws_W v ^2\over\Lambda^2}C_{HWB}(T_3^f-2Q_f s_W^2)\biggr]\biggr\}\, ,
\label{eq:photreal}\end{aligned}$$ where $s_W^2=1-{M_W^2\over M_Z^2}, c_W={M_W\over M_Z}$ in Eq. \[eq:photreal\] and $\theta_0$ and $E_0$ are the angular and energy cutoff for observing the photon, and depend on the detector sensitivities[@Kniehl:1991xe; @Schwartz:2013pla].
After summing Eq. \[eq:virtg\] and the contributions from virtual and real photon emission, taking into account the fermion wave function renormalization, there is no $\beta$ dependence, verifying the cancellation of the IR divergences. In our numerical results below, we take $\theta_0 = 1^{\degree}$ and $E_0 = 1~\mathrm{GeV}$.
Effective $Z$ Vertices
----------------------
From Eq. \[eq:zdecayamp\], we obtain the contribution to the $Z \to f \bar{f}$ decay width from $C_{HWB}, C_{HW}, C_{HB}$ and $C_W$, still working to $\mathcal{O}(\frac{1}{\Lambda^2})$. We write our result in terms of effective fermion couplings as $$\Gamma(Z \to f_i \bar{f}_i) = \frac{G_{\mu} M_Z^3}{6 \sqrt{2} \pi} N_c (g_i^f)^2$$ where $i = L,R$ indicates the fermion helicity and we neglect fermion masses. For a fermion with charge $Q_f$ and weak isospin $T_3^f$, the effective coupling is $$\begin{aligned}
g^f & =& (g^f)_{\mathrm{SM}} \bigg\{1
+ \biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2{1\over D(Q_f,T_3^f) } \biggl[ -0.23 C_{HWB} Q_f^4+1.5 T_3^f C_{HWB} Q_f^3 \nonumber \\ &&+\left(-1.9
C_{HWB} (T_3^{f})^2+0.15 C_{HB}+0.15 C_{HW}+11.0
C_{HWB}+0.19 C_{W}\right) Q_f^2 \\ &&+ T_3^f (-0.67
C_{HB}-0.69 C_{HW}-49.0 C_{HWB}-0.85 C_{W})
Q_f+(T_3^{f})^2 (0.0084 C_{HB}+0.029 C_{HW}-0.23
C_{HWB}+0.032 C_{W}) \nonumber
\biggr]\biggr\}\, ,\end{aligned}$$ where $$D(Q_f,T_3^f)={Q_f^4 - 8.7 T_3^f Q_f^3 + \left( 17 (T_3^{f})^2 - 76 \right) Q_f^2 + 660 T_3^f Q_f - 1400 (T_3^{f})^2 }\, .$$ The relatively large size of the $C_{HWB}$ coefficients is due to the fact that they contribute at tree level. For our numerical results we use, $$\begin{aligned}
G_\mu&=&1.1663787(6)\times 10^{-5}~\mathrm{GeV}^{-2}\nonumber \\
M_Z&=&91.1876\pm .0021~\mathrm{GeV}\nonumber \\
M_W&=&80.385\pm .015~\mathrm{GeV}\nonumber\\
M_H&=&125.09\pm 0.21\pm 0.11 ~\mathrm{GeV}\nonumber\\
M_t&=&173.1\pm0.6~\mathrm{GeV} \, .
\label{eq:input}\end{aligned}$$
In particular, the SM fermion vertex couplings are $$\begin{split}
g_L^\nu &= (g_L^\nu)_{\mathrm{SM}} \biggl[ 1 + \delta g_L^{Z\nu}
+ \biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{ -6.0 \cdot 10^{-6} C_{HB}-2.1 \cdot 10^{-5} C_{HW} + 1.6 \cdot 10^{-4} C_{HWB} -2.3 \cdot 10^{-5} C_{W} \biggr\}\biggr] \\
g_L^e &= (g_L^e)_{\mathrm{SM}} \biggl[ 1 + \delta g_L^{Ze}
+\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{ 0.0019 C_{HB}+0.0019 C_{HW} + 0.043 C_{HWB} +0.0023 C_{W} \biggr\}\biggr] \\
g_R^e &= (g_R^e)_{\mathrm{SM}} \biggl[ 1 + \delta g_R^{Ze}
+\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{-0.0020 C_{HB}-0.0020 C_{HW} - 0.033 C_{HWB} -0.0025 C_{W} \biggr\}\biggr] \\
g_L^u &= (g_L^u)_{\mathrm{SM}} \biggl\{ 1 + \delta g_L^{Zu} +
\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{9.3 \cdot 10^{-4} C_{HB}+ 9.3 \cdot 10^{-4} C_{HW} + 0.021 C_{HWB} +0.0011 C_{W} \biggr\}\biggr] \\
g_R^u &= (g_R^u)_{\mathrm{SM}} \biggl[ 1 + \delta g_R^{Zu}
+\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{ -0.0020 C_{HB}-0.0020 C_{HW} - 0.034 C_{HWB} -0.0025 C_{W} \biggr\}\biggr] \\
g_L^d &= (g_L^d)_{\mathrm{SM}} \biggl[1 + \delta g_L^{Zd} +
\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{3.7 \cdot 10^{-4} C_{HB}+ 3.6 \cdot 10^{-4} C_{HW} + 0.0080 C_{HWB} + 4.5 \cdot 10^{-4} C_{W} \biggr\}\biggr] \\
g_R^d &= (g_R^d)_{\mathrm{SM}} \biggl[ 1 + \delta g_R^{Zd}+
\biggl({1~\mathrm{TeV}\over \Lambda}\biggr)^2
\biggl\{ -0.0020 C_{HB}-0.0020 C_{HW} - 0.034 C_{HWB} -0.0025 C_{W} \biggr\}\biggr]\, .
\label{eq:couplings}
\end{split}$$ For $b_L$, the coefficient in front of $C_W$ is $2.7 \cdot 10^{-4}$ rather than $4.5 \cdot 10^{-4}$ because of top mass effects. The tree level contributions of $C_{HWB}$ are contained in the $\delta g_{L,R}^{Zf}$ contributions as given in Eq. \[eq: mapcoef\].
![Limits on the coefficients of the operators $\mathcal{O}_W$ and $\mathcal{O}_{HWB}$ which contribute to $Z$ decay at one loop and tree level, respectively. All other operators are set to zero, and the region between the solid brown lines is allowed by $Z$ pole measurements given our calculation. For comparison, the region between the dashed blue lines is allowed by the same LEP precision data considering only the impact of the operator $\mathcal{O}_{HWB}$ that modifies the $S$ parameter. The region between the magenta dotted lines is allowed by measurements of $VV$ production at the LHC, to which $\mathcal{O}_W$ contributes at tree level. The region in the plane that is allowed by all measurements is shown in red.[]{data-label="fig:opplots"}](zdecay_eft_fit.pdf){width="49.00000%"}
![Limits on the coefficients of the operators $\mathcal{O}_{HB}$ and $\mathcal{O}_{HW}$ which contribute to $Z$ decay at one loop. The solid brown lines are as in Fig. \[fig:opplots\]. (The allowed region is between the lines). For comparison, the region allowed by the more constraining measurement of the $H \to \gamma \gamma$ signal strength is shown in red between the blue dashed lines.[]{data-label="fig:ggpplots"}](zdecay_eft_fit2.pdf){width="49.00000%"}
[|c|c|c||C|C|C|C|C|C|C|]{} & & &\
&&& $g_L^\nu$ & $g_L^\ell$ & $g_L^b$ & $g_L^c$ & $g_R^\ell$ & $g_R^b$ & $g_R^c$\
$g_L^\nu$ & $0.50199 \pm 0.00020$ & $0.50075 \pm 0.00077$ & 1.00 &&&&&&\
$g_L^\ell$ & $-0.26919 \pm 0.00020$ & $-0.26939 \pm 0.00022$ & -0.32 & 1.00 &&&&&\
$g_L^b$ & $-0.42114 \pm 0.00045$ & $-0.4182 \pm 0.0015$ & 0.05 & -0.27 & 1.00 &&&&\
$g_L^c$ & $0.34674 \pm 0.00017$ & $0.3453 \pm 0.0036$ & -0.02 & 0.04 & -0.09 & 1.00 &&&\
$g_R^\ell$ & $0.23208 \pm 0.00018$ & $0.23186 \pm 0.00023$ & 0.25 & 0.34 & -0.37 & 0.07 & 1.00 &&\
$g_R^b$ & $0.077420 \pm 0.000052$ & $0.0962 \pm 0.0063$ & 0.00 & -0.33 & 0.88 & -0.14 & -0.35 & 1.00 &\
$g_R^c$ & $-0.15470 \pm 0.00011$ & $-0.1580 \pm 0.0051$ & 0.00 & 0.08 & -0.17 & 0.30 & 0.08 & -0.13 & 1.00\
These effective couplings are bounded by LEP measurements at the $Z$ pole. We proceed to take the limits of [@ALEPH:2005ab] on the $Z$-fermion couplings to constrain the SMEFT operators. We minimize a $\chi^2$ function constructed using the LEP measurements of the quantities $(g_L^\nu, g_L^e, g_R^e, g_L^u, g_R^u, g_L^b, g_R^b)$ and their correlations shown in Table \[tab:lep\], including the uncertainties on the input parameters. While $Z$ pole measurements constrain all of the operators in Eq. \[eq:couplings\], we focus on the implications of our calculation for the operators $\mathcal{O}_W$, $\mathcal{O}_{HB}$ and $\mathcal{O}_{HW}$, which do not contribute to $Z$ decay at tree level. We seek to minimize the quantity $$(\chi^2)_{\mathrm{LEP}} = \left(\vec{g}_{\mathrm{SMEFT}} - \vec{g}_{\mathrm{exp}}\right)^\mathrm{T} V^{-1} \left(\vec{g}_{\mathrm{SMEFT}} - \vec{g}_{\mathrm{exp}} \right)\, ,$$ where $\vec{g} = \left( g_L^\nu, g_L^\ell, g_L^b, g_L^c, g_R^\ell, g_R^b, g_R^c \right)$ and $V$ is the covariance matrix constructed from the errors and correlations above. We use Eq. \[eq:couplings\] together with the SM predictions of Table \[tab:lep\] to calculate $\vec{g}_{\mathrm{SMEFT}}$. Since we set light fermion masses to zero in our SMEFT analysis, the effective couplings for the down (up) quark apply equally to the $b$ ($c$) quark, with the exception of the $b_L$ for which top quark corrections apply as specified below Eq. \[eq:couplings\].
In Fig. \[fig:opplots\], we show the resulting 90% CL limits in 2-dimensional planes of the coefficients of these operators along with that of $\mathcal{O}_{HWB}$, which affects electroweak couplings at tree level. The coefficients of all other operators are set to zero. We compare our results to processes in which the SMEFT operators contribute at tree level. The limit of [@Alves:2018nof], set using LHC Run I data [@Khachatryan:2015sga; @Aad:2016wpd; @Aad:2016ett; @Khachatryan:2016poo], is converted in our notation to $$-0.17 < C_W \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2 < 0.18\, .$$ For $\mathcal{O}_W$, we use the limits of [@Alves:2018nof] obtained by using 8 TeV LHC gauge boson pair production in leptonic final states [@Khachatryan:2015sga; @Aad:2016wpd; @Aad:2016ett; @Khachatryan:2016poo].
For $\mathcal{O}_{HB}$ and $\mathcal{O}_{HW}$, we use limits [@Dawson:2018liq] from the calculation of $H \to \gamma \gamma$ [@Hartmann:2015aia; @Dedes:2018seb; @Dawson:2018liq] in the SMEFT, as compared to measurements of $H \to \gamma \gamma$ at Run 1 and 2 of the LHC [@Khachatryan:2016vau; @Aaboud:2018xdt; @Sirunyan:2018ouh]. The SMEFT calculation of $H \to \gamma \gamma$ [@Dawson:2018liq] gives, $$\begin{aligned}
\mu_{\gamma\gamma}&=&{\Gamma(H\rightarrow \gamma\gamma)\over \Gamma(H\rightarrow \gamma\gamma)\mid_{SM}}\nonumber \\
&=&1-40.15{C_{HB}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2-13.08{C_{HW}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2
+22.40{C_{HWB}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2\, .\end{aligned}$$ Then, using the average $\mu_{\gamma\gamma} = 1.09 \pm 0.10$ of current LHC Higgs measurements [@Khachatryan:2016vau; @Aaboud:2018xdt; @Sirunyan:2018ouh], we find, $$\begin{aligned}
-0.003 < \mid {C_{HB}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2+.33{C_{HW}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2\
-.55{C_{HWB}} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2\mid < 0.007\, ,\end{aligned}$$ or taking only one non-zero coupling at a time with the conservative bound $|\mu_{\gamma\gamma} - 1| < 0.29$, $$\begin{split}
\left|C_{HB} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2\right| &< 0.007 \\
\left|C_{HW} \left( \frac{1~\mathrm{TeV}}{\Lambda} \right)^2\right| &< 0.02\, .
\end{split}$$ $\mathcal{O}_{HWB}$ corresponds to the oblique parameter $S$ [@Peskin:1990zt; @Peskin:1991sw], whose limit we take from the Gfitter collaboration[@Haller:2018nnx] of $S = 0.04 \pm 0.11$ to set the $2 \sigma$ bound, $$-0.004 < C_{HWB} < 0.006 \, .$$
The existing bounds in Fig. \[fig:ggpplots\] are stronger than those that we obtain directly from $Z$ pole measurements. Nevertheless, they provide complementary information, and in particular in Fig. \[fig:ggpplots\], the interplay between the limits on $C_W$ and $C_{HWB}$ demonstrates the power of electroweak precision measurements to constrain couplings that only contribute at loop level. In the case of the operators $C_{HB}$ and $C_{HW}$ which directly affect $H \to \gamma \gamma$, Higgs precision is already significantly more effective than $Z$ pole measurements in setting limits, due to the loop suppression of these operators’ contributions to $Z$ decay.
Conclusions {#sec:concl}
===========
Precision measurements of electroweak physics will eventually necessitate higher order calculations of BSM contributions. The SMEFT framework takes a general approach to potential new UV physics by parametrizing its effects in terms of higher dimension operators involving the SM fields. In this work, we have furthered the applicability of the SMEFT to probe new physics by considering the one loop corrections to $Z$ decay from operators which contribute to gauge boson production.
While the contributions of the operators $\mathcal{O}_W$, $\mathcal{O}_{HB}$ and $\mathcal{O}_{HW}$ are small relative to those of the operators that modify the $Z$ coupling to fermions at tree level, the relative size of all of the SMEFT operators is fixed by the new physics. In particular, integrating out a heavy SM singlet scalar could naturally give these operators without changing the leading $Z$ couplings to the fermions [@deBlas:2017xtg]. In such a scenario, it would be essential to have the higher order contributions of the BSM physics to all possible processes. In this regard our calculation provides a useful prediction, relating the effects of new physics in $Z$ decay to those in other electroweak processes provided the states responsible for deviations from the SM are heavy enough to be integrated out.
A full calculation of $Z$ decay at one loop in the SMEFT would provide even more complete information about the influence of higher dimensional operators on $Z$ physics. With this as well as other higher order calculations of electroweak processes, in the future a global fit at NLO in the SMEFT could be performed to bound the sizes of all possible dimension-$6$ SMEFT operators.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Ayres Freitas and Pier Paolo Giardino for useful discussions. SD is supported by the U.S. Department of Energy under Grant Contract DE-SC0012704. AI is supported by the U.S. Department of Energy under Grant Contract DE-SC0015634 and by PITT PACC.
UV poles {#sec:poles}
========
The cancellation of UV poles follows from the individual contributions: Numerically with $\Lambda = 1~\mathrm{TeV}$, the pieces are as follows. $$\begin{aligned}
\frac{\partial M_0}{\partial C_{HWB}} \delta C_{HWB}
&= \frac{1}{\epsilon} \Big\{ Q_f \left(\left(2.7\times 10^{-5}\right) C_{HB}+\left(2.7\times
10^{-5}\right) C_{HW}+\left(4.6\times 10^{-4}\right)
C_{HWB}+\left(2.6\times 10^{-5}\right) C_{W}\right) \Big\} + \mathcal{O}(\epsilon^0) \\
\frac{\partial M_0}{\partial G_{\mu}} \delta G_{\mu} &= \frac{1}{\epsilon} \Big\{ \left(2.1\times
10^{-5}\right) \left(Q_f (C_{HWB}+8.8)-40
T_3^f\right) (\xi -5.5) \Big\} + \mathcal{O}(\epsilon^0) \\
\frac{\partial M_0}{\partial M_Z^2} \delta M_Z^2 &= \frac{1}{\epsilon} \Big\{ Q_f
\big(\left(-1.4\times 10^{-4}\right) \xi C_{HW}-\left(3.2\times
10^{-4}\right) C_{HW}-\left(6.7\times 10^{-4}\right)
C_{HWB}+\left(7.7\times 10^{-4}\right) C_{W} \\&\quad +C_{HB}
\left(\left(-3.9\times 10^{-5}\right) \xi -9.1\times
10^{-5}\right)-\left(1.4\times 10^{-6}\right) C_{HWB} \xi
-\left(1.5\times 10^{-3}\right) \xi +1.2\times 10^{-2}\big) \\&\quad +T_3^f
\big(\left(7.7\times 10^{-5}\right) \xi C_{HW}+\left(1.8\times
10^{-4}\right) C_{HW}+\left(5.3\times 10^{-5}\right)
C_{HWB}-\left(4.3\times 10^{-4}\right) C_{W} \\&\quad +C_{HB}
\left(\left(2.2\times 10^{-5}\right) \xi +5.1\times
10^{-5}\right)+\left(4.1\times 10^{-5}\right) C_{HWB} \xi
+\left(8.2\times 10^{-4}\right) \xi -6.5\times 10^{-3}\big) \Big\} + \mathcal{O}(\epsilon^0) \\
\frac{\partial M_0}{\partial M_W^2} \delta M_W^2 &= \frac{Q_f}{\epsilon} \Big\{
\left(-9.2\times 10^{-4}\right) C_{W}+C_{HWB}
\left(5.3\times 10^{-4}-\left(5.2\times 10^{-5}\right) \xi
\right)+C_{HW} \left(\left(1.5\times 10^{-4}\right) \xi +3.0\times
10^{-4}\right) \\&\quad +\left(1.3\times 10^{-3}\right) \xi -1.2\times
10^{-2} \Big\} + \mathcal{O}(\epsilon^0) \\
-M_{\gamma }\frac{\Pi_{\gamma Z}\left(M_Z^2\right)}{M_Z^2} &= \frac{Q_f}{\epsilon} \Big\{
\left(3.7\times 10^{-5}\right) \xi C_{HWB}-\left(2.3\times
10^{-4}\right) C_{HWB}+\left(1.9\times 10^{-4}\right)
C_{W}+C_{HW} \big(\left(-3.5\times 10^{-5}\right) \xi
\\&\quad -5.2\times 10^{-5}\big)+C_{HB} \left(\left(3.5\times
10^{-5}\right) \xi +5.2\times 10^{-5}\right)-\left(1.7\times
10^{-4}\right) \xi +1.3\times 10^{-3} \Big\} + \mathcal{O}(\epsilon^0) \\
\frac{Z_Z}{2}M_0 &= \frac{1}{\epsilon} \Big\{ T_3^f \big(\left(-7.7\times 10^{-5}\right) \xi
C_{HW}-\left(1.8\times 10^{-4}\right) C_{HW}-\left(1.3\times
10^{-4}\right) C_{HWB}+\left(3.2\times 10^{-4}\right)
C_{W} \\&\quad +C_{HB} \left(\left(-2.2\times 10^{-5}\right) \xi
-5.1\times 10^{-5}\right)-\left(6.7\times 10^{-5}\right) C_{HWB}
\xi -\left(7.8\times 10^{-4}\right) \xi -3.1\times 10^{-4}\big) \\&\quad +Q_f
\big(\left(1.7\times 10^{-5}\right) \xi C_{HW}+\left(4.0\times
10^{-5}\right) C_{HW}+\left(2.1\times 10^{-5}\right)
C_{HWB}-\left(7.0\times 10^{-5}\right) C_{W} \\&\quad +C_{HB}
\left(\left(4.9\times 10^{-6}\right) \xi +1.1\times
10^{-5}\right)-\left(4.7\times 10^{-6}\right) C_{HWB} \xi
+\left(1.7\times 10^{-4}\right) \xi +6.9\times 10^{-5}\big) \Big\} + \mathcal{O}(\epsilon^0) \\
Z_f M_0 &= \frac{\xi}{\epsilon} \Big\{ Q_f^3 \left(1.3\times 10^{-4}-\left(4.3\times 10^{-5}\right)
C_{HWB}\right) +Q_f^2 T_3^f \left(\left(1.9\times 10^{-4}\right)
C_{HWB}-8.3\times 10^{-4}\right) \\&\quad +Q_f T_3^{f2} \left(2.6\times
10^{-3}-\left(3.0\times 10^{-4}\right) C_{HWB}\right)
-\left(1.6\times 10^{-3}\right) T_3^f \Big\} + \mathcal{O}(\epsilon^0) \\
M_1 &= \frac{1}{\epsilon} \Big\{ Q_f^3 \left(\left(4.3\times 10^{-5}\right) C_{HWB}-1.3\times
10^{-4}\right) \xi +Q_f T_3^{f2} \big(\left(3.0\times 10^{-4}\right)
C_{HWB}-2.6\times 10^{-3}\big) \xi \\&\quad +T_3^f
\big(\left(8.3\times 10^{-4}\right) \xi Q_f^2+\left(1.2\times
10^{-4}\right) C_{W}+\left(2.4\times 10^{-3}\right) \xi
+C_{HWB} \big(\left(-1.9\times 10^{-4}\right) \xi
Q_f^2 \\&\quad +\left(2.5\times 10^{-5}\right) \xi +7.6\times
10^{-5}\big)+2.3\times 10^{-3}\big) \Big\} + \mathcal{O}(\epsilon^0)\end{aligned}$$
The sum vanishes for any given fermion.
$2$-point functions {#sec:appa}
===================
In this appendix, we show the two-point functions in $R_\xi$ gauge due to the SMEFT operators that also contribute to gauge boson pair production. Previous results for the gauge boson two-point functions in other operator bases appear in [@Alam:1997nk; @Chen:2013kfa].
In $D=4-2\epsilon$ dimensions, the two-point function for a massless fermion with weak isospin $T_3^f$ and charge $Q_f$ is $$\begin{aligned}
\Sigma(p) &= \frac{1}{8 \pi ^2} \Bigg( \frac{2 \sqrt{2} A_0(M_W^2) G_{\mu} T_3^{f2} \left((D-2) M_W^2-p^2\right)}{p^2}+2 \sqrt{2} A_0(M_W^2 \xi) G_{\mu} T_3^{f2} \\&\quad +\frac{1}{M_Z^4 p^2} \bigg(A_0(M_Z^2)
\left((D-2) M_Z^2-p^2\right) \left(M_W^2 Q_f+M_Z^2 (T_3^f-Q_f)\right) \Big(\sqrt{2} G_{\mu} \left(M_W^2 Q_f+M_Z^2
(T_3^f-Q_f)\right) \\&\quad +2 M_W Q_f C_{HWB} \sqrt{M_Z^2-M_W^2}\Big)\bigg) \\&\quad -\frac{A_0(M_Z^2 \beta) (D-2) M_W^2 Q_f^2
\left(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z)+2 M_W C_{HWB} \sqrt{M_Z^2-M_W^2}\right)}{M_Z^2
p^2} \\&\quad +\frac{A_0(M_Z^2 \xi) \left(M_W^2 Q_f+M_Z^2 (T_3^f-Q_f)\right) \left(\sqrt{2} G_{\mu} \left(M_W^2 Q_f+M_Z^2 (T_3^f-Q_f)\right)+2
M_W Q_f C_{HWB} \sqrt{M_Z^2-M_W^2}\right)}{M_Z^4} \\&\quad -\frac{B_0(0, M_Z^2 \beta, M_Z^2 \beta) M_W^2 (\xi -1) Q_f^2 \left(\sqrt{2}
G_{\mu} (M_W-M_Z) (M_W+M_Z)+2 M_W C_{HWB} \sqrt{M_Z^2-M_W^2}\right)}{M_Z^2} \\&\quad +\frac{2 \sqrt{2}
B_0(p^2, 0, M_W^2) G_{\mu} T_3^{f2} \left(p^2-M_W^2\right) \left((D-2) M_W^2+p^2\right)}{p^2}-2 \sqrt{2} B_0(p^2, 0, M_W^2 \xi) G_{\mu} T_3^{f2}
\left(p^2-M_W^2 \xi \right) \\&\quad -\frac{1}{M_Z^4 p^2}\bigg(B_0(p^2, 0, M_Z^2) (M_Z-p) (M_Z+p) \left((D-2) M_Z^2+p^2\right) \left(M_W^2 Q_f+M_Z^2
(T_3^f-Q_f)\right) \Big(\sqrt{2} G_{\mu} \big(M_W^2 Q_f \\&\quad +M_Z^2 (T_3^f-Q_f)\big)+2 M_W Q_f C_{HWB}
\sqrt{M_Z^2-M_W^2}\Big)\bigg) \\&\quad +\frac{1}{M_Z^2
p^2}\bigg(B_0(p^2, 0, M_Z^2 \beta) M_W^2 Q_f^2 \left(\beta (D-2) M_Z^2-p^2 (D+\xi -3)\right)
\Big(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z) \\&\quad +2 M_W C_{HWB} \sqrt{M_Z^2-M_W^2}\Big)\bigg) \\&\quad +\frac{1}{M_Z^4} \bigg(B_0(p^2, 0, M_Z^2 \xi) \left(M_Z^2 \xi -p^2\right) \left(M_W^2 Q_f+M_Z^2 (T_3^f-Q_f)\right) \Big(\sqrt{2} G_{\mu} \left(M_W^2
Q_f+M_Z^2 (T_3^f-Q_f)\right) \\&\quad +2 M_W Q_f C_{HWB} \sqrt{M_Z^2-M_W^2}\Big)\bigg) +\frac{1}{M_Z^2} \bigg(C_0(0, p^2, p^2, M_Z^2 \beta, M_Z^2 \beta, 0)
M_W^2 (\xi -1) Q_f^2 \left(p^2-\beta M_Z^2\right) \\&\quad \times \left(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z)+2 M_W
C_{HWB} \sqrt{M_Z^2-M_W^2}\right)\bigg) \Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ We have regulated IR divergences with a photon of mass $M_Z \beta$, and use standard FeynCalc notation [@Mertig:1990an] for the Passarino-Veltman functions.
This leads to the wave function renormalization $$\begin{aligned}
Z_f &= \frac{1}{8 \pi ^2 \epsilon } \Bigg(\frac{1}{M_Z^2} \bigg(Q_f^2 \left(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z)+2 M_W C_{HWB} \sqrt{M_Z^2-M_W^2}\right) \big(2
\xi \epsilon \big(M_W^2 (\log (\beta )-\log (\xi )) \\&\quad +M_Z^2 (\log (M_Z^2 / \mu^2)+\log (\pi \xi ))\big)+2 M_W^2 (\xi -1)
\epsilon +M_Z^2 ((2 (\gamma -1) \xi +3) \epsilon -2 \xi )\big)\bigg) \\&\quad -2 Q_f T_3^f \left(\sqrt{2} G_{\mu} (M_W-M_Z)
(M_W+M_Z)+ M_W C_{HWB} \sqrt{M_Z^2-M_W^2}\right) (2 \xi \epsilon (2 \log (M_Z^2 / \mu^2)+\log (\pi \xi ))-2 \xi \\&\quad +(2 (\gamma -1) \xi +3) \epsilon )-\sqrt{2} G_{\mu} T_3^{f2} \big(\left(2 M_W^2+M_Z^2\right) (2 \xi \epsilon (\log (\pi
\xi ))+2 \xi ((\gamma -1) \epsilon -1)+3 \epsilon )+4 M_W^2 \xi \epsilon \log (M_W^2 / \mu^2) \\&\quad +2 M_Z^2 \xi \epsilon \log
(M_Z^2 / \mu^2)\big)\Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ For the $b_L$, there are corrections proportional to the top mass, leading to an additional wave function renormalization which in Feynman gauge is $$Z_{b_L} = Z_f \left( Q_f = -\frac{1}{3}, T_3^f = -\frac{1}{2} \right) + \frac{G_{\mu} M_T^2 \left( 2 \epsilon \log \left(\frac{\mu^2}{M_W^2}\right) - 2 \gamma \epsilon + \epsilon - 2 \epsilon \log \pi + 2 \right)}{16 \sqrt{2} \pi^2 \epsilon}$$
The transverse $W$ two-point function is $$\begin{aligned}
\Pi_{WW}^T(p^2) &= \frac{1}{16 p^2 \pi ^2} \Bigg(-\frac{18 \sqrt{2} B_0(p^2, 0, 0) (D-2) G_{\mu} M_W^2
p^4}{D-1} -\frac{1}{(D-1) M_Z^2} \bigg(C_0(0, p^2, p^2, 0, 0, M_W^2 \xi) \left(M_W^4-p^4\right)
\\&\quad \times \left(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z)+2
M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\right)
(\xi -1) \left(p^2-M_W^2 \xi \right)^2 \bigg) \\&\quad -\frac{6
\sqrt{2} A_0(M_T^2) G_{\mu} M_W^2 \left(M_T^2-(D-2)
p^2\right)}{D-1}+\frac{6 \sqrt{2} B_0(p^2, 0, M_T^2) G_{\mu} M_W^2
(M_T-p) (M_T+p) \left(M_T^2+(D-2)
p^2\right)}{D-1} \\&\quad -\frac{1}{D-1} \bigg(B_0(p^2, M_H^2, M_W^2) M_W^2 \Big(8 (D-1)
\left(-M_H^2+M_W^2+p^2\right) C_{HW} p^2 \\&\quad +\sqrt{2}
G_{\mu} \left(p^4-2 \left(M_H^2+(3-2 D) M_W^2\right)
p^2+\left(M_H^2-M_W^2\right)^2\right)\Big) \bigg) +\frac{1}{D-1} \bigg(
A_0(M_H^2) \Big(\sqrt{2} G_{\mu} M_W^2
\big(M_H^2-M_W^2 \\&\quad -(D-2) p^2\big)-2 (D-1) p^2
\left(4 M_W^2+p^2\right)
C_{HW}\Big)\bigg)-\frac{1}{(D-1) M_Z^4} \bigg(B_0(p^2, M_W^2, M_Z^2) M_W
(M_W+M_Z-p) (M_W+ \\&\quad M_Z+p)
\left((M_W-M_Z)^2-p^2\right) \Big(36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} M_W^2 M_Z^2 C_{W}
p^2+\sqrt{2} G_{\mu} M_W \big(M_W^4 \\&\quad +2 (2 D-3)
\left(M_Z^2+p^2\right) M_W^2+M_Z^4+p^4+2 (2 D-3)
M_Z^2 p^2\big)+2 \sqrt{M_Z^2-M_W^2}
\big(M_W^4+(2 D-3) \left(M_Z^2+2 p^2\right)
M_W^2 \\&\quad +p^4+(2 D-3) M_Z^2 p^2\big)
C_{HWB}\Big)\bigg)+\frac{1}{(D-1)
M_Z^2} \bigg(C_0(0, p^2, p^2, 0, 0, M_W^2)
\left(M_W^2-p^2\right)^2 \big(M_W^4 \\&\quad +2 (2 D-3) p^2
M_W^2+p^4\big) \left(\sqrt{2} G_{\mu}
\left(M_Z^2-M_W^2\right)-2 M_W
\sqrt{M_Z^2-M_W^2} C_{HWB}\right) (\xi -1) \bigg) \\&\quad +\frac{1}{(D-1)
M_Z^4} \bigg(B_0(p^2, M_W^2 \xi, M_Z^2 \xi) M_W
\left(M_W^4-p^4\right) \left(\sqrt{2} G_{\mu} M_W+2
\sqrt{M_Z^2-M_W^2} C_{HWB}\right) \\&\quad \times
\left(p^4-2 \left(M_W^2+M_Z^2\right) \xi
p^2+\left(M_W^2-M_Z^2\right)^2 \xi ^2\right)\bigg)+\frac{1}{(D-1)
M_Z^4} \bigg( B_0(p^2, M_W^2, M_Z^2 \xi) M_W
\left(M_W^2-p^2\right)^2 \\&\quad \times \left(\sqrt{2} G_{\mu} M_W+2
\sqrt{M_Z^2-M_W^2} C_{HWB}\right)
\left(M_W^4+\left((4 D-6) p^2-2 M_Z^2 \xi \right)
M_W^2+\left(p^2-M_Z^2 \xi \right)^2\right) \bigg) \\&\quad -\frac{1}{D-1} \bigg( A_0(M_W^2 \xi) \Big(2 \left(2 (D-1)
C_{HW} p^4+M_W \sqrt{M_Z^2-M_W^2} C_{HWB}
p^2-M_W^3 \sqrt{M_Z^2-M_W^2}
C_{HWB}\right) \\&\quad +\sqrt{2} G_{\mu} \left((\xi -1) M_W^4-2 p^2
(D+\xi -3) M_W^2+p^4 (4 D+\xi -7)\right)\Big) \bigg) -\frac{1}{(1-D) M_Z^2} \bigg(2
B_0(p^2, 0, M_W^2) \\&\quad \times \Big(18 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
M_W (M_W-M_Z) (M_W+M_Z) (M_W-p)^2 p^2
C_{W} (M_W+p)^2 \\&\quad +\sqrt{2} G_{\mu} (M_W-M_Z)
(M_W+M_Z) \left(M_W^2+p^2\right) \left((2 D+\xi -4)
M_W^4-2 p^2 (4 D+\xi -6) M_W^2+p^4 (2 D+\xi
-4)\right) \\&\quad + M_W \sqrt{M_Z^2-M_W^2}
\left(M_W^2+p^2\right) C_{HWB} \left((2 D+2 \xi -5)
M_W^4-2 p^2 (6 D+2 \xi -9) M_W^2+p^4 (2 D+2 \xi
-5)\right)\Big)\bigg) \\&\quad -\frac{1}{(D-1)
M_Z^2} \bigg(2 B_0(p^2, 0, M_W^2 \xi)
(M_W-p) (M_W+p) \bigg(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \Big(\xi M_W^4-p^2
\big(\xi ^2+2 \xi \\&\quad +2 D-3\big) M_W^2-p^4 (2 D+\xi
-4)\Big)- M_W \sqrt{M_Z^2-M_W^2}
C_{HWB} \big((\xi -2) \xi M_W^4+2 p^2 \left(\xi ^2+D (\xi
+2)-3\right) M_W^2 \\&\quad +p^4 (2 D+2 \xi -5)\big)\bigg)\bigg)+\frac{1}{(D-1) M_Z^4} \bigg(A_0(M_Z^2 \xi) \Big(\sqrt{2} G_{\mu} \big((\xi
+1) M_W^6+\left(4 (D-2) p^2-2 M_Z^2 \xi \right)
M_W^4 \\&\quad +\left(2 M_Z^2 p^2 \xi -p^4 (4 D+\xi -7)\right)
M_W^2-(D-1) M_Z^4 p^2\big) M_W^2+2
\Big(-(D-1) M_Z^4 C_{HW} p^4 \\&\quad +2 M_W^3 M_Z^2
\sqrt{M_Z^2-M_W^2} \left(p^2-M_W^2\right) C_{HWB}
\xi +M_W^3 \sqrt{M_Z^2-M_W^2} (M_W-p)
(M_W+p) C_{HWB} \big((\xi +1) M_W^2 \\&\quad +p^2 (4 D+\xi
-7)\big)\Big)\Big)\bigg)+\frac{1}{D-1} \bigg(A_0(M_W^2) \Big(36
\sqrt[4]{2} (D-2) \sqrt{G_{\mu}} M_W C_{W} p^4 \\&\quad +2
M_W \left((3-2 D) \sqrt{M_Z^2-M_W^2}
C_{HWB} M_W^2+4 (D-1) p^2 C_{HW} M_W+(3-2 D)
\sqrt{M_Z^2-M_W^2} p^2 C_{HWB}\right) \\&\quad +\sqrt{2}
G_{\mu} \left((\xi +1) M_W^4-M_H^2
M_W^2-\left(M_Z^2+2 p^2 (2 (D-3) D+\xi +3)\right)
M_W^2+p^4 (4 D+\xi -7)\right)\Big)\bigg) \\&\quad -\frac{1}{(D-1) M_Z^4} \bigg(A_0(M_Z^2)
M_W \Big(36 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
M_W^2 M_Z^2 \left(M_W^2-M_Z^2-p^2\right)
C_{W} p^2+\sqrt{2} G_{\mu} M_W \big((\xi +1) M_W^6 \\&\quad +2
\left(2 (D-2) \left(M_Z^2+p^2\right)-M_Z^2 \xi \right)
M_W^4+\left((9-4 D) M_Z^4+2 p^2 \left(2 (D-2)^2+\xi \right)
M_Z^2-p^4 (4 D+\xi -7)\right) M_W^2 \\&\quad -M_Z^4
\left(M_Z^2+p^2\right)\big)+2
\sqrt{M_Z^2-M_W^2} C_{HWB} \big((\xi +1)
M_W^6+\left((2 D-\xi -5) M_Z^2+4 (D-2) p^2\right)
M_W^4 \\&\quad +\left(-2 (D-2) M_Z^4+p^2 (2 D (2 D-7)+\xi +13)
M_Z^2-p^4 (4 D+\xi -7)\right) M_W^2-2 (D-1) M_Z^4
p^2\big)\Big)\bigg) \\&\quad +\frac{1}{(D-1) M_Z^4} \bigg( B_0(p^2, M_Z^2, M_W^2 \xi) M_W
(M_W-p) (M_W+p) \Big(-\sqrt{2} G_{\mu} M_W
\left(M_W^2-2 M_Z^2+p^2\right) \big(M_Z^4+ \\&\quad \left((4
D-6) p^2-2 M_W^2 \xi \right) M_Z^2+\left(p^2-M_W^2 \xi
\right)^2\big)-2 \sqrt{M_Z^2-M_W^2}
C_{HWB} \big(\xi ^2 M_W^6-\xi \left(M_Z^2 (\xi
+2)-p^2 (\xi -2)\right) M_W^4 \\&\quad +\left((2 \xi +1) M_Z^4+2 p^2
(-\xi +D (\xi +2)-3) M_Z^2+p^4 (1-2 \xi )\right)
M_W^2-M_Z^6+p^6+(2 D-5) M_Z^2 p^4 \\&\quad +3 (3-2 D) M_Z^4
p^2\big)\Big) \bigg) \Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ which yields the mass shift $$\begin{aligned}
\delta M_W^2 &= \Pi_{WW}^T(M_W^2) \\
&= \frac{1}{16 \pi ^2} \Bigg(\frac{A_0(M_H^2) \left(\sqrt{2} G_{\mu} \left(M_H^2-(D-1)
M_W^2\right)-10 (D-1) M_W^2
C_{HW}\right)}{D-1} \\&\quad -\frac{6 \sqrt{2} A_0(M_T^2) G_{\mu}
\left(M_T^2-(D-2) M_W^2\right)}{D-1}-\frac{1}{D-1} \bigg(A_0(M_W^2)
\bigg(-36 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}} M_W^3
C_{W} \\&\quad +\sqrt{2} G_{\mu} \left(4 (D-3) (D-1)
M_W^2+M_H^2+M_Z^2\right)+4 M_W
\left((2 D-3) C_{HWB} \sqrt{M_Z^2-M_W^2}-2 (D-1)
M_W C_{HW}\right)\bigg)\bigg) \\&\quad -2 A_0(M_W^2 \xi) M_W^2
\left(\sqrt{2} G_{\mu}+2
C_{HW}\right)+\frac{1}{(D-1) M_Z^2} \bigg(A_0(M_Z^2) \Big(36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} M_W^3 M_Z^2 C_{W} \\&\quad +\sqrt{2}
G_{\mu} \left(-4 (D-2) (D-1) M_W^4+4 (D-2) M_W^2
M_Z^2+M_Z^4\right)-4 M_W C_{HWB}
\sqrt{M_Z^2-M_W^2} \big(2 (D-2) (D-1) M_W^2 \\&\quad +(3-2 D)
M_Z^2\big)\Big) \bigg) -A_0(M_Z^2 \xi)
M_W^2 \left(\sqrt{2} G_{\mu}+2
C_{HW}\right)-\frac{18 \sqrt{2} B_0(M_W^2, 0, 0) (D-2) G_{\mu}
M_W^4}{D-1} \\&\quad +\frac{6 \sqrt{2} B_0(M_W^2, 0, M_T^2) G_{\mu}
(M_T-M_W) (M_T+M_W) \left((D-2)
M_W^2+M_T^2\right)}{D-1}+\frac{1}{M_Z^2} \bigg(16 B_0(M_W^2, 0, M_W^2) M_W^4
\\&\quad \times \left(\sqrt{2} G_{\mu} \left(M_Z^2-M_W^2\right)-2
M_W C_{HWB}
\sqrt{M_Z^2-M_W^2}\right) \bigg) -\frac{1}{D-1} \bigg(B_0(M_W^2, M_H^2, M_W^2)
\Big(\sqrt{2} G_{\mu} \big(4 (D-1) M_W^4 \\&\quad +M_H^4-4
M_H^2 M_W^2\big)-8 (D-1) M_W^2
C_{HW} \left(M_H^2-2
M_W^2\right)\Big)\bigg)+\frac{1}{(D-1) M_Z^2} \bigg(B_0(M_W^2, M_W^2, M_Z^2) \left(4
M_W^2-M_Z^2\right) \\&\quad \times \Big(36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} M_W^3 M_Z^2 C_{W}+\sqrt{2}
G_{\mu} \left(4 (D-1) M_W^4+4 (2 D-3) M_W^2
M_Z^2+M_Z^4\right) \\&\quad +4 M_W C_{HWB}
\sqrt{M_Z^2-M_W^2} \left(2 (D-1) M_W^2+(2 D-3)
M_Z^2\right)\Big)\bigg)\Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$
The transverse $Z$ two-point function is $$\begin{aligned}
\Pi_{ZZ}^T(p^2) &= \frac{1}{48
\pi ^2} \Bigg(-\frac{1}{(D-1)
M_Z^2} \bigg(B_0(p^2, 0, 0) (D-2) \Big(\sqrt{2} G_{\mu} \left(160
M_W^4-200 M_Z^2 M_W^2+103 M_Z^4\right) \\&\quad +40
M_W \left(8 M_W^2-5 M_Z^2\right)
\sqrt{M_Z^2-M_W^2} C_{HWB}\Big) p^2\bigg)+\frac{1}{(D-1)
M_Z^2} \bigg(2 A_0(M_T^2) (D-2) \Big(\sqrt{2} G_{\mu} \big(32
M_W^4 \\&\quad -40 M_Z^2 M_W^2+17 M_Z^4\big)+8
M_W \left(8 M_W^2-5 M_Z^2\right)
\sqrt{M_Z^2-M_W^2} C_{HWB}\Big)\bigg) -\frac{1}{(1-D)
M_Z^2} \bigg(B_0(p^2, M_T^2, M_T^2) \\&\quad \times \Big(\sqrt{2} G_{\mu} \left(2
M_T^2 \left(-64 M_W^4+80 M_Z^2 M_W^2+(9 D-43)
M_Z^4\right)-(D-2) \left(32 M_W^4-40 M_Z^2
M_W^2+17 M_Z^4\right) p^2\right) \\&\quad -8 M_W
\left(8 M_W^2-5 M_Z^2\right) \sqrt{M_Z^2-M_W^2}
\left(4 M_T^2+(D-2) p^2\right) C_{HWB}\Big)\bigg)-\frac{1}{M_Z^2} \bigg(3 A_0(M_Z^2 \xi) \Big(\sqrt{2} G_{\mu}
M_Z^4+2 p^2 \\&\quad \times \Big((C_{HW}-C_{HB})
M_W^2+\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)\Big)\bigg)+\frac{1}{(D-1)
M_Z^2} \bigg(3
B_0(p^2, M_W^2, M_W^2) \\&\quad \times \left(4 M_W^2-p^2\right) \Big(36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} p^2 C_{W} M_W^3+4
\sqrt{M_Z^2-M_W^2} \left(2 (D-1) M_W^2+(2 D-3)
p^2\right) C_{HWB} M_W \\&\quad +\sqrt{2} G_{\mu} \left(4 (D-1)
M_W^4+4 (2 D-3) p^2 M_W^2+p^4\right)\Big)\bigg)+\frac{3 \sqrt{2} B_0(p^2, M_W^2 \xi, M_W^2 \xi) G_{\mu}
\left(M_Z^4-p^4\right) \left(p^2-4 M_W^2 \xi \right)}{(D-1)
M_Z^2} \\&\quad +\frac{1}{(D-1) p^2} \bigg(3 A_0(M_Z^2) \Big(\sqrt{2} G_{\mu}
\left(-M_H^2+M_Z^2+p^2\right) M_Z^2+8 (D-1)
p^2 \Big((C_{HW}-C_{HB})
M_W^2 \\&\quad +\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)\Big)\bigg)-\frac{1}{(D-1) p^2} \bigg(3
B_0(p^2, M_H^2, M_Z^2) \Big(\sqrt{2} G_{\mu} \big(p^4 \\&\quad -2 \left(M_H^2+(3-2
D) M_Z^2\right) p^2+\left(M_H^2-M_Z^2\right)^2\big)
M_Z^2+8 (D-1) p^2
\left(-M_H^2+M_Z^2+p^2\right)
\Big((C_{HW}-C_{HB})
M_W^2 \\&\quad +\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)\Big)\bigg)+\frac{1}{(D-1) M_Z^2
p^2} \bigg(3
A_0(M_H^2) \Big(\sqrt{2} G_{\mu} M_Z^4
\left(M_H^2-M_Z^2-(D-2) p^2\right) \\&\quad -2 (D-1) p^2
\left(4 M_Z^2+p^2\right) \left((C_{HW}-C_{HB})
M_W^2+\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\right)\Big)\bigg) \\&\quad +\frac{1}{(D-1) M_Z^2
p^2} \bigg(6 B_0(p^2, M_W^2, M_W^2 \xi) \Big(\sqrt{2} G_{\mu}
\left(-M_Z^4+p^4+2 M_W^2 (M_Z-p) (M_Z+p)\right)
\big((\xi -1)^2 M_W^4 \\&\quad +2 p^2 (2 D-\xi -3) M_W^2+p^4\big)+2
M_W \sqrt{M_Z^2-M_W^2} (M_Z-p)
(M_Z+p) C_{HWB} \big((\xi -1)^2 M_W^4 \\&\quad +2 (D-2) p^2
(\xi +1) M_W^2+(3-2 D) p^4\big)\Big)\bigg)-\frac{1}{(D-1) M_Z^2 p^2} \bigg(6 A_0(M_W^2 \xi) \Big(2 \Big((D-1) M_Z^2
C_{HB} p^4 \\&\quad -(D-1) M_W^2 (C_{HB}-C_{HW})
p^4+M_W \sqrt{M_Z^2-M_W^2} \left(M_Z^2+(D-2)
p^2\right) C_{HWB} p^2 \\&\quad +M_W^3 \sqrt{M_Z^2-M_W^2}
(M_Z-p) (M_Z+p) C_{HWB} (\xi -1)\Big)+\sqrt{2}
G_{\mu} \big(2 (M_Z-p) (M_Z+p) (\xi -1)
M_W^4 \\&\quad -(M_Z-p) (M_Z+p) \left((\xi -1) M_Z^2+p^2
(4 D+\xi -7)\right) M_W^2+(D-1) M_Z^4
p^2\big)\Big)\bigg) \\&\quad +\frac{1}{(D-1) M_Z^2 p^2} \bigg(6 A_0(M_W^2) \Big(36
\sqrt[4]{2} (D-2) \sqrt{G_{\mu}} M_W^3 C_{W}
p^4+\sqrt{2} G_{\mu} \big(2 \big(M_Z^2 (\xi -1) \\&\quad -p^2 (2 (D-3)
D+\xi +3)\big) M_W^4+\left(-(\xi -1) M_Z^4-2 p^2
M_Z^2+p^4 (4 D+\xi -7)\right) M_W^2+M_Z^4 p^2\big) {\addtocounter{equation}{1}\tag{\theequation}}\\&\quad +2
M_W \sqrt{M_Z^2-M_W^2} C_{HWB}
\left(\left(M_Z^2 (\xi -1)-p^2 (4 (D-3) D+\xi +7)\right)
M_W^2+(2 D-3) p^2
\left(M_Z^2+p^2\right)\right)\Big)\bigg)\Bigg)\end{aligned}$$ which yields the mass shift $$\begin{aligned}
\delta M_Z^2 &= \Pi_{ZZ}^T(M_Z^2) \\
&= -\frac{1}{48
\pi ^2} \Bigg(-\frac{1}{D-1} \bigg(3 A_0(M_H^2) \Big(\sqrt{2} G_{\mu} \left(M_H^2-(D-1)
M_Z^2\right)-10 (D-1) \Big(M_W C_{HWB}
\sqrt{M_Z^2-M_W^2} \\&\quad +M_W^2
(C_{HW}-C_{HB})+M_Z^2
C_{HB}\Big)\Big) \bigg)-\frac{1}{(D-1) M_Z^2} \bigg(2 A_0(M_T^2) (D-2) \Big(\sqrt{2}
G_{\mu} \left(32 M_W^4-40 M_W^2 M_Z^2+17
M_Z^4\right) \\&\quad +8 M_W C_{HWB} \left(8
M_W^2-5 M_Z^2\right)
\sqrt{M_Z^2-M_W^2}\Big)\bigg)-\frac{1}{(D-1)
M_Z^2} \bigg(6
A_0(M_W^2) \Big(36 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
M_W^3 M_Z^2 C_{W} \\&\quad +\sqrt{2} G_{\mu} \left(-4 (D-2)
(D-1) M_W^4+4 (D-2) M_W^2 M_Z^2+M_Z^4\right)-4
M_W C_{HWB} \sqrt{M_Z^2-M_W^2}
\big(2 (D-2) (D-1) M_W^2 \\&\quad +(3-2 D) M_Z^2\big)\Big)\bigg)+6 A_0(M_W^2 \xi) \left(\sqrt{2} G_{\mu} M_Z^2+2
\left(M_W C_{HWB}
\sqrt{M_Z^2-M_W^2}+M_W^2
(C_{HW}-C_{HB})+M_Z^2
C_{HB}\right)\right) \\&\quad +\frac{3 A_0(M_Z^2) \left(\sqrt{2} G_{\mu}
\left(M_H^2-2 M_Z^2\right)-8 (D-1)
\left(M_W C_{HWB} \sqrt{M_Z^2-M_W^2}+M_W^2
(C_{HW}-C_{HB})+M_Z^2
C_{HB}\right)\right)}{D-1} \\&\quad +3 A_0(M_Z^2 \xi) \left(\sqrt{2}
G_{\mu} M_Z^2+2 \left(M_W C_{HWB}
\sqrt{M_Z^2-M_W^2}+M_W^2
(C_{HW}-C_{HB})+M_Z^2
C_{HB}\right)\right) \\&\quad +\frac{B_0(M_Z^2, 0, 0) (D-2) \left(\sqrt{2}
G_{\mu} \left(160 M_W^4-200 M_W^2 M_Z^2+103
M_Z^4\right)+40 M_W C_{HWB} \left(8
M_W^2-5 M_Z^2\right)
\sqrt{M_Z^2-M_W^2}\right)}{D-1} \\&\quad +\frac{1}{D-1} \bigg(3 B_0(M_Z^2, M_H^2, M_Z^2)
\Big(\sqrt{2} G_{\mu} \left(4 (D-1) M_Z^4+M_H^4-4
M_H^2 M_Z^2\right)-8 (D-1) \left(M_H^2-2
M_Z^2\right) \\&\quad \times \left(M_W C_{HWB}
\sqrt{M_Z^2-M_W^2}+M_W^2
(C_{HW}-C_{HB})+M_Z^2
C_{HB}\right)\Big)\bigg)+\frac{1}{(D-1) M_Z^2} \bigg(B_0(M_Z^2, M_T^2, M_T^2) \\&\quad \times \Big(\sqrt{2}
G_{\mu} \left(2 M_T^2 \left((43-9 D) M_Z^4+64
M_W^4-80 M_W^2 M_Z^2\right)+(D-2) M_Z^2 \left(32
M_W^4-40 M_W^2 M_Z^2+17 M_Z^4\right)\right) \\&\quad +8
M_W C_{HWB} \left(8 M_W^2-5
M_Z^2\right) \sqrt{M_Z^2-M_W^2} \left((D-2)
M_Z^2+4 M_T^2\right)\Big)\bigg)+\frac{1}{(D-1) M_Z^2} \bigg(3
B_0(M_Z^2, M_W^2, M_W^2) \\&\quad \times \left(M_Z^2-4 M_W^2\right) \Big(36 \sqrt[4]{2}
(D-2) \sqrt{G_{\mu}} M_W^3 M_Z^2
C_{W}+\sqrt{2} G_{\mu} \left(4 (D-1) M_W^4+4 (2 D-3)
M_W^2 M_Z^2+M_Z^4\right) \\&\quad +4 M_W
C_{HWB} \sqrt{M_Z^2-M_W^2} \left(2 (D-1)
M_W^2+(2 D-3) M_Z^2\right)\Big)\bigg) \Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ and the wave function renormalization $$\begin{aligned}
\delta Z_Z &= -\frac{\partial \Pi_{ZZ}^T(p^2)}{p^2}\bigg|_{p^2 = M_Z^2} \\
&= \frac{1}{48 \pi ^2} \Bigg( \frac{3 \sqrt{2} A_0(M_Z^2) G_{\mu}
\left(M_Z^2-M_H^2\right)}{(D-1) M_Z^2}+\frac{6
A_0(M_Z^2 \xi) \left((C_{HW}-C_{HB})
M_W^2+\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2
C_{HB}\right)}{M_Z^2} \\&\quad +\frac{B_0(M_Z^2, M_T^2, M_T^2) (D-2)
\left(\sqrt{2} G_{\mu} \left(32 M_W^4-40 M_Z^2
M_W^2+17 M_Z^4\right)+8 M_W \left(8
M_W^2-5 M_Z^2\right) \sqrt{M_Z^2-M_W^2}
C_{HWB}\right)}{(D-1) M_Z^2} \\&\quad +\frac{(D-2) B'_0(M_Z^2, 0, 0)
\left(\sqrt{2} G_{\mu} \left(160 M_W^4-200 M_Z^2
M_W^2+103 M_Z^4\right)+40 M_W \left(8
M_W^2-5 M_Z^2\right) \sqrt{M_Z^2-M_W^2}
C_{HWB}\right)}{D-1} \\&\quad +\frac{B_0(M_Z^2, 0, 0) (D-2) \left(\sqrt{2}
G_{\mu} \left(160 M_W^4-200 M_Z^2 M_W^2+103
M_Z^4\right)+40 M_W \left(8 M_W^2-5
M_Z^2\right) \sqrt{M_Z^2-M_W^2}
C_{HWB}\right)}{(D-1) M_Z^2} \\&\quad +\frac{1}{(D-1)
M_Z^2} \bigg(B'_0(M_Z^2, M_T^2, M_T^2)
\Big(\sqrt{2} G_{\mu} \Big(2 \left(64 M_W^4-80 M_Z^2
M_W^2+(43-9 D) M_Z^4\right) M_T^2+(D-2) M_Z^2
\big(32 M_W^4 \\&\quad -40 M_Z^2 M_W^2+17
M_Z^4\big)\Big)+8 M_W \left(8 M_W^2-5
M_Z^2\right) \sqrt{M_Z^2-M_W^2} \left(4
M_T^2+(D-2) M_Z^2\right) C_{HWB}\Big)\bigg) \\&\quad +\frac{1}{(D-1)
M_Z^2} \bigg(3 A_0(M_H^2) \Big(\sqrt{2} G_{\mu}
(M_H-M_Z) (M_H+M_Z)+2 (D-1)
\Big((C_{HW}-C_{HB})
M_W^2 \\&\quad +\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)\Big) \bigg) +\frac{1}{(D-1)
M_Z^2} \bigg(3 B_0(M_Z^2, M_H^2, M_Z^2) \Big(8 (D-1) M_Z^2
\Big((C_{HW}-C_{HB})
M_W^2 \\&\quad +\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)-\sqrt{2} G_{\mu} M_H^2
\left(M_H^2-2 M_Z^2\right)\Big)\bigg) +\frac{1}{D-1} \bigg(3 B'_0(M_Z^2, M_H^2, M_Z^2) \\&\quad \times \Big(\sqrt{2} G_{\mu}
\left(M_H^4-4 M_Z^2 M_H^2+4 (D-1) M_Z^4\right)-8
(D-1) \left(M_H^2-2 M_Z^2\right)
\Big((C_{HW}-C_{HB})
M_W^2 \\&\quad +\sqrt{M_Z^2-M_W^2} C_{HWB}
M_W+M_Z^2 C_{HB}\Big)\Big)\bigg)+\frac{1}{(D-1) M_Z^2} \bigg(3
B'_0(M_Z^2, M_W^2, M_W^2) \left(M_Z^2-4 M_W^2\right) \\&\quad \times \Big(36 \sqrt[4]{2}
(D-2) \sqrt{G_{\mu}} M_Z^2 C_{W} M_W^3+4
\sqrt{M_Z^2-M_W^2} \left(2 (D-1) M_W^2+(2
D-3) M_Z^2\right) C_{HWB} M_W \\&\quad +\sqrt{2} G_{\mu}
\left(4 (D-1) M_W^4+4 (2 D-3) M_Z^2
M_W^2+M_Z^4\right)\Big)\bigg)+\frac{1}{(D-1) M_Z^2} \bigg(3
B_0(M_Z^2, M_W^2, M_W^2) \\&\quad \times \Big(-72 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
\left(2 M_W^2-M_Z^2\right) C_{W} M_W^3-8
\sqrt{M_Z^2-M_W^2} \left((3 D-5) M_W^2+(3-2
D) M_Z^2\right) C_{HWB} M_W \\&\quad +\sqrt{2} G_{\mu}
\left((44-28 D) M_W^4+16 (D-2) M_Z^2 M_W^2+3
M_Z^4\right)\Big)\bigg) +\frac{6 \sqrt{2}
B_0(M_Z^2, M_W^2 \xi, M_W^2 \xi) G_{\mu} \left(M_Z^2-4 M_W^2 \xi
\right)}{D-1} \\&\quad +\frac{1}{(D-1) M_Z^4} \bigg(12 B_0(M_Z^2, M_W^2, M_W^2 \xi) \Big(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \big((\xi -1)^2
M_W^4+2 M_Z^2 (2 D-\xi -3)
M_W^2 \\&\quad +M_Z^4\big)+ M_W
\sqrt{M_Z^2-M_W^2} C_{HWB} \left((\xi -1)^2
M_W^4+2 (D-2) M_Z^2 (\xi +1) M_W^2+(3-2 D)
M_Z^4\right)\Big)\bigg) \\&\quad +\frac{1}{(D-1)
M_Z^4} \bigg(12 A_0(M_W^2)
M_W \Big(-18 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
M_W^2 C_{W} M_Z^2+
\sqrt{M_Z^2-M_W^2} C_{HWB} \\&\quad \times \left((\xi -1)
M_W^2+(3-2 D) M_Z^2\right)+\sqrt{2} G_{\mu} M_W
\left(M_W^2 (\xi -1)-M_Z^2 (2 D+\xi -4)\right)\Big)\bigg) \\&\quad + \frac{1}{(D-1) M_Z^4} \bigg(12 A_0(M_W^2 \xi) \Big(\sqrt{2} G_{\mu}
\left(M_Z^2 (2 D+\xi -4)-M_W^2 (\xi -1)\right)
M_W^2+ \Big((D-1) C_{HB} M_Z^4 {\addtocounter{equation}{1}\tag{\theequation}}\\&\quad -(D-1)
M_W^2 (C_{HB}-C_{HW}) M_Z^2+(D-2) M_W
\sqrt{M_Z^2-M_W^2} C_{HWB} M_Z^2-M_W^3
\sqrt{M_Z^2-M_W^2} C_{HWB} (\xi
-1)\Big)\Big)\bigg) \Bigg)\end{aligned}$$
The $\gamma - Z$ two-point function is $$\begin{aligned}
\Pi_{\gamma Z}^T(p^2) &= \frac{1}{48 \pi ^2} \Bigg(\frac{3 \sqrt{2} B_0(p^2, M_W^2 \xi, M_W^2 \xi) G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(4 M_W^2 \xi -p^2\right)
p^4}{(D-1) M_W M_Z^2} \\&\quad +\frac{3 A_0(M_Z^2 \xi)
\left(2 C_{HWB} M_W^2+2 \sqrt{M_Z^2-M_W^2}
(C_{HB}-C_{HW}) M_W-M_Z^2 C_{HWB}\right)
p^2}{M_Z^2} \\&\quad +\frac{20 B_0(p^2, 0, 0) (D-2) M_W \left(\sqrt{2}
G_{\mu} \sqrt{M_Z^2-M_W^2} \left(5 M_Z^2-8
M_W^2\right)+ M_W \left(16 M_W^2-13
M_Z^2\right) C_{HWB}\right) p^2}{(D-1) M_Z^2} \\&\quad +6
B_0(p^2, M_H^2, M_Z^2) \left(-M_H^2+M_Z^2+p^2\right)
\left(2 C_{HWB} M_W^2+2 \sqrt{M_Z^2-M_W^2}
(C_{HB}-C_{HW}) M_W-M_Z^2 C_{HWB}\right) \\&\quad +6
A_0(M_Z^2) \left(-2 C_{HWB} M_W^2+2
\sqrt{M_Z^2-M_W^2} (C_{HW}-C_{HB})
M_W+M_Z^2 C_{HWB}\right) \\&\quad -\frac{3 A_0(M_H^2)
\left(2 M_Z^2+p^2\right) \left(-2 C_{HWB} M_W^2+2
\sqrt{M_Z^2-M_W^2} (C_{HW}-C_{HB})
M_W+M_Z^2 C_{HWB}\right)}{M_Z^2} \\&\quad +\frac{1}{(D-1)
M_Z^2} \bigg(4
B_0(p^2, M_T^2, M_T^2) M_W \left(4 M_T^2+(D-2) p^2\right)
\Big(\sqrt{2} G_{\mu} \sqrt{M_Z^2-M_W^2} \left(5
M_Z^2-8 M_W^2\right) \\&\quad + M_W \left(16
M_W^2-13 M_Z^2\right) C_{HWB}\Big)\bigg) +\frac{1}{(D-1) M_Z^2} \bigg(8 A_0(M_T^2) (D-2) M_W \Big(\sqrt{2} G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(8 M_W^2-5
M_Z^2\right) \\&\quad + M_W \left(13 M_Z^2-16
M_W^2\right) C_{HWB}\Big)\bigg) -\frac{1}{(D-1) M_W
M_Z^2} \bigg(3
B_0(p^2, M_W^2, M_W^2) \left(4 M_W^2-p^2\right) \\&\quad \times \Big(-36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} \sqrt{M_Z^2-M_W^2} p^2 C_{W}
M_W^3+2 \left(2 M_W^2-M_Z^2\right) \left(2
(D-1) M_W^2+(2 D-3) p^2\right) C_{HWB} M_W \\&\quad -\sqrt{2}
G_{\mu} \sqrt{M_Z^2-M_W^2} \left(4 (D-1) M_W^4+4 (2
D-3) p^2 M_W^2+p^4\right)\Big)\bigg)+\frac{1}{(D-1) M_W M_Z^2 p^2} \bigg(6 B_0(p^2, M_W^2, M_W^2 \xi) \\&\quad \times \Big(\sqrt{2} G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(p^4+M_W^2 \left(M_Z^2-2
p^2\right)\right) \left((\xi -1)^2 M_W^4+2 p^2 (2 D-\xi -3)
M_W^2+p^4\right) \\&\quad - M_W \left(\left(M_Z^2-2
p^2\right) M_W^2+M_Z^2 p^2\right) C_{HWB} \left((\xi
-1)^2 M_W^4+2 (D-2) p^2 (\xi +1) M_W^2+(3-2 D)
p^4\right)\Big)\bigg) \\&\quad -\frac{1}{(D-1) M_Z^2 p^2} \bigg(6
A_0(M_W^2 \xi) \Big( \Big(-\left(M_Z^2-2 p^2\right)
C_{HWB} (\xi -1) M_W^4-p^2 C_{HWB} \left(\xi
M_Z^2+2 (D-2) p^2\right) M_W^2 \\&\quad -2 (D-1)
\sqrt{M_Z^2-M_W^2} p^4 (C_{HB}-C_{HW})
M_W+(D-2) M_Z^2 p^4 C_{HWB}\Big) \\&\quad +\sqrt{2} G_{\mu}
M_W \sqrt{M_Z^2-M_W^2} \left(\left(M_Z^2-2
p^2\right) (\xi -1) M_W^2+p^2 \left((3-2 D) M_Z^2+p^2 (4
D+\xi -7)\right)\right)\Big)\bigg) \\&\quad +\frac{1}{(D-1) M_Z^2 p^2} \bigg(6
A_0(M_W^2) \Big(36 \sqrt[4]{2} (D-2) \sqrt{G_{\mu}}
M_W^2 \sqrt{M_Z^2-M_W^2} C_{W} p^4+\sqrt{2}
G_{\mu} M_W \sqrt{M_Z^2-M_W^2} \\&\quad \times \left((4 D+\xi -7)
p^4-M_Z^2 p^2+M_W^2 \left(M_Z^2 (\xi -1)-2 p^2 (2
(D-3) D+\xi +3)\right)\right)+ C_{HWB} \big(\big(2 p^2
(4 (D-3) D+\xi +7) \\&\quad -M_Z^2 (\xi -1)\big) M_W^4-p^2 \left((2
D (2 D-5)+\xi +4) M_Z^2+2 (2 D-3) p^2\right) M_W^2+(2 D-3)
M_Z^2 p^4\big)\Big)\bigg)\Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ which yields the on-shell mixing $$\begin{aligned}
\Pi_{\gamma Z}(M_Z^2) &= \frac{1}{48 \pi ^2} \Bigg(-\frac{3 \sqrt{2} B_0(M_Z^2, M_W^2 \xi, M_W^2 \xi) G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(M_Z^2-4 M_W^2 \xi
\right) M_Z^2}{(D-1) M_W}+9 A_0(M_H^2) \\&\quad \times \left(2
C_{HWB} M_W^2+2 \sqrt{M_Z^2-M_W^2}
(C_{HB}-C_{HW}) M_W-M_Z^2 C_{HWB}\right)+3
A_0(M_Z^2 \xi) \Big(2 C_{HWB} M_W^2 \\&\quad +2
\sqrt{M_Z^2-M_W^2} (C_{HB}-C_{HW})
M_W-M_Z^2 C_{HWB}\Big)+6 A_0(M_Z^2)
\Big(-2 C_{HWB} M_W^2 \\&\quad +2 \sqrt{M_Z^2-M_W^2}
(C_{HW}-C_{HB}) M_W+M_Z^2 C_{HWB}\Big)+6
B_0(M_Z^2, M_H^2, M_Z^2) \left(M_H^2-2 M_Z^2\right) \Big(-2
C_{HWB} M_W^2 \\&\quad +2 \sqrt{M_Z^2-M_W^2}
(C_{HW}-C_{HB}) M_W+M_Z^2
C_{HWB}\Big)+\frac{1}{D-1} \bigg(20 B_0(M_Z^2, 0, 0) (D-2) M_W \\&\quad \times
\left(\sqrt{2} G_{\mu} \sqrt{M_Z^2-M_W^2} \left(5
M_Z^2-8 M_W^2\right)+ M_W \left(16
M_W^2-13 M_Z^2\right) C_{HWB}\right)\bigg) \\&\quad +\frac{1}{(D-1)
M_Z^2} \bigg(4
B_0(M_Z^2, M_T^2, M_T^2) M_W \left(4 M_T^2+(D-2) M_Z^2\right)
\Big(\sqrt{2} G_{\mu} \sqrt{M_Z^2-M_W^2} \left(5
M_Z^2-8 M_W^2\right) \\&\quad + M_W \left(16
M_W^2-13 M_Z^2\right) C_{HWB}\Big)\bigg) +\frac{1}{(D-1) M_Z^2} \bigg(8 A_0(M_T^2) (D-2) M_W \Big(\sqrt{2} G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(8 M_W^2-5
M_Z^2\right) \\&\quad + M_W \left(13 M_Z^2-16
M_W^2\right) C_{HWB}\Big)\bigg)+\frac{1}{(D-1) M_W
M_Z^2} \bigg(3
B_0(M_Z^2, M_W^2, M_W^2) \left(4 M_W^2-M_Z^2\right) \\&\quad \times \Big(36 \sqrt[4]{2}
(D-2) \sqrt{G_{\mu}} M_Z^2
\sqrt{M_Z^2-M_W^2} C_{W} M_W^3-2
\left(2 M_W^2-M_Z^2\right) \left(2 (D-1) M_W^2+(2 D-3)
M_Z^2\right) C_{HWB} M_W \\&\quad +\sqrt{2} G_{\mu}
\sqrt{M_Z^2-M_W^2} \left(4 (D-1) M_W^4+4 (2 D-3)
M_Z^2 M_W^2+M_Z^4\right)\Big)\bigg) \\&\quad +\frac{1}{(D-1)
M_W M_Z^2} \bigg( 6 B_0(M_Z^2, M_W^2, M_W^2 \xi) (M_W-M_Z)
(M_W+M_Z) \Big( M_W C_{HWB}
\big((\xi -1)^2 M_W^4 \\&\quad +2 (D-2) M_Z^2 (\xi +1)
M_W^2+(3-2 D) M_Z^4\big)-\sqrt{2} G_{\mu}
\sqrt{M_Z^2-M_W^2} \big((\xi -1)^2 M_W^4+2
M_Z^2 (2 D-\xi -3) M_W^2 \\&\quad +M_Z^4\big)\Big) \bigg) -\frac{1}{(D-1)
M_Z^2} \bigg(6 A_0(M_W^2) \Big(-36 \sqrt[4]{2} (D-2)
\sqrt{G_{\mu}} M_W^2 \sqrt{M_Z^2-M_W^2}
C_{W} M_Z^2 \\&\quad +\sqrt{2} G_{\mu} M_W
\sqrt{M_Z^2-M_W^2} \left(M_W^2 (4 (D-3) D+\xi
+7)-M_Z^2 (4 D+\xi -8)\right)- C_{HWB} \big((8
(D-3) D+\xi +15) M_W^4 \\&\quad -M_Z^2 \left(4 D^2-6 D+\xi -2\right)
M_W^2+(2 D-3) M_Z^4\big)\Big)\bigg) +\frac{1}{(D-1)
M_Z^2} \bigg(6 A_0(M_W^2 \xi) {\addtocounter{equation}{1}\tag{\theequation}}\\&\quad \times \Big(\sqrt{2} G_{\mu} M_W
\sqrt{M_Z^2-M_W^2} \left(M_W^2 (\xi -1)-M_Z^2 (2
D+\xi -4)\right)+ \Big((C_{HWB}-C_{HWB} \xi )
M_W^4 \\&\quad +M_Z^2 C_{HWB} (2 D+\xi -4) M_W^2+2 (D-1)
M_Z^2 \sqrt{M_Z^2-M_W^2} (C_{HB}-C_{HW})
M_W-(D-2) M_Z^4 C_{HWB}\Big)\Big)\bigg)\Bigg)\end{aligned}$$
Vertex functions {#sec:appb}
================
The one loop amplitude for $Z(p + p') \to f(p) \bar{f}(p')$, the decay of a $Z$ boson to a pair of massless fermions with weak isospin $T_3^f$ and charge $Q_f$, is $$\mathcal{M}_1 = V \bar{u}(p) \slashed{\epsilon}^*(p + p') v(p')$$ where the vertex function is $$\begin{aligned}
V &= \frac{\sqrt{G_{\mu}}}{4 \cdot 2^{3/4} M_Z^5 \pi ^2} \Bigg( \frac{\sqrt{2} B_0(M_Z^2, M_W^2 \xi, M_W^2 \xi)
G_{\mu} T_3^f \left(M_Z^2-4 M_W^2 \xi \right)
M_Z^6}{D-1}+2 \sqrt{2} C_0(M_Z^2, 0, 0, 0, 0, M_Z^2) (D-8) \big(Q_f
M_W^2 \\&\quad +M_Z^2 (T_3^f-Q_f)\big)^2 \left(2 G_{\mu} \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right)+3 \sqrt{2}
M_W \sqrt{M_Z^2-M_W^2} Q_f C_{HWB}\right)
M_Z^4 \\&\quad +4 C_0(M_Z^2, 0, 0, 0, 0, M_Z^2 \beta) M_W^2 Q_f^2 \Big(\sqrt{2}
G_{\mu} (M_W-M_Z) (M_W+M_Z) \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad + M_W
\sqrt{M_Z^2-M_W^2} \left(3 Q_f M_W^2+M_Z^2 (2
T_3^f-3 Q_f)\right) C_{HWB}\Big) (\beta (-D+2 \beta +8)+2)
M_Z^4+8 \sqrt{2} B_0(0, 0, M_Z^2) \big(Q_f M_W^2 \\&\quad +M_Z^2
(T_3^f-Q_f)\big)^2 \left(2 G_{\mu} \left(Q_f M_W^2+M_Z^2
(T_3^f-Q_f)\right)+3 \sqrt{2} M_W
\sqrt{M_Z^2-M_W^2} Q_f C_{HWB}\right) M_Z^2 \\&\quad -4
\sqrt{2} C_{00}(0, M_Z^2, 0, M_Z^2, 0, 0) (D-2) \left(Q_f M_W^2+M_Z^2
(T_3^f-Q_f)\right)^2 \Big(2 G_{\mu} \left(Q_f M_W^2+M_Z^2
(T_3^f-Q_f)\right) \\&\quad +3 \sqrt{2} M_W
\sqrt{M_Z^2-M_W^2} Q_f C_{HWB}\Big) M_Z^2+8
C_0(M_Z^2, 0, 0, 0, 0, M_W^2) \left(2 M_W^6-(D-8) M_Z^2 M_W^4+2
M_Z^4 M_W^2\right) T_3^{f2} \\&\quad \times \left(\sqrt{2} G_{\mu}
\left(M_Z^2 (Q_f-T_3^f)-M_W^2 (Q_f-2
T_3^f)\right)- M_W \sqrt{M_Z^2-M_W^2}
(Q_f-2 T_3^f) C_{HWB}\right) M_Z^2 \\&\quad + \frac{1}{D-2} \bigg(4 C_0(M_Z^2, 0, 0, M_W^2, M_W^2, 0)
M_W^5 T_3^f \Big(3 \sqrt[4]{2} (3 D-8) \sqrt{G_{\mu}}
M_W^2 C_{W} M_Z^2+2 \sqrt{2} (D-2) G_{\mu}
M_W \left(M_W^2+2 M_Z^2\right) \\&\quad +2 (D-2)
\sqrt{M_Z^2-M_W^2} \left(M_W^2+M_Z^2\right)
C_{HWB}\Big) M_Z^2\bigg)+8 B_0(0, 0, M_W^2) M_W^2
T_3^f \Big(\frac{6 \sqrt[4]{2} (D-3) \sqrt{G_{\mu}}
M_Z^2 C_{W} M_W^3}{D-2} \\&\quad +
\sqrt{M_Z^2-M_W^2} \left((4 (Q_f-2 T_3^f) T_3^f+2)
M_W^2+M_Z^2 (4 (Q_f-2 T_3^f) T_3^f+1)\right)
C_{HWB} M_W+2 \sqrt{2} G_{\mu}
\left(M_W^2+M_Z^2\right) \\&\quad \times \left((2 (Q_f-2 T_3^f)
T_3^f+1) M_W^2+2 M_Z^2 T_3^f
(T_3^f-Q_f)\right)\Big) M_Z^2+\frac{1}{(D-2)
(D-1)} \bigg(B_0(M_Z^2, M_W^2, M_W^2) T_3^f
\\&\quad \times \Big(-6 \sqrt[4]{2} (D-1) (3 D-8) \sqrt{G_{\mu}}
M_Z^2 \left(2 M_W^2-M_Z^2\right) C_{W}
M_W^3-2 (D-2) \sqrt{M_Z^2-M_W^2} \big(4
(D-1) M_W^4 \\&\quad +2 (D-3) M_Z^2 M_W^2+(3-2 D)
M_Z^4\big) C_{HWB} M_W-\sqrt{2} (D-2) G_{\mu}
\left(2 M_W^2-M_Z^2\right) \big(4 (D-1) M_W^4+4 (2
D-3) M_Z^2 M_W^2 \\&\quad +M_Z^4\big)\Big) M_Z^2\bigg)-16 B_0(0, 0, M_Z^2 \beta) M_W^2 Q_f^2 \Big(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad + M_W
\sqrt{M_Z^2-M_W^2} \left(3 Q_f M_W^2+M_Z^2 (2
T_3^f-3 Q_f)\right) C_{HWB}\Big) (\beta +1) M_Z^2-4
B_0(0, M_Z^2 \beta, M_Z^2 \beta) M_W^2 Q_f^2 \\&\quad \times \Big(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad + M_W
\sqrt{M_Z^2-M_W^2} \left(3 Q_f M_W^2+M_Z^2 (2
T_3^f-3 Q_f)\right) C_{HWB}\Big) (\xi -1) M_Z^2 \\&\quad +\frac{1}{D-1} \bigg(2
B_0(M_Z^2, M_W^2, M_W^2 \xi) T_3^f \Big(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \big((\xi -1)^2
M_W^4 +2 M_Z^2 (2 D-\xi -3)
M_W^2+M_Z^4\big) \\&\quad + M_W
\sqrt{M_Z^2-M_W^2} C_{HWB} \big((\xi -1)^2
M_W^4+2 (D-2) M_Z^2 (\xi +1) M_W^2 \\&\quad +(3-2 D)
M_Z^4\big)\Big) M_Z^2\bigg)+\frac{1}{D-1} \bigg(2 A_0(M_W^2 \xi)
T_3^f \Big( M_W \sqrt{M_Z^2-M_W^2}
C_{HWB} \big(M_Z^2 \big(-8 (D-1) T_3^{f2}+4 (D-1) Q_f
T_3^f \\&\quad +2 D-3\big)-M_W^2 (\xi -1)\big)+\sqrt{2} G_{\mu}
\big(-(\xi -1) M_W^4+M_Z^2 \left(-4 Q_f T_3^f+4 \left(-2
(D-1) T_3^{f2}+D Q_f T_3^f+D\right)+\xi -6\right) M_W^2 \\&\quad -4
(D-1) M_Z^4 (Q_f-T_3^f) T_3^f\big)\Big)
M_Z^2\bigg)+2 A_0(M_W^2) T_3^f \bigg(4 \left(-2
M_W^2-M_Z^2\right) T_3^f \Big(\sqrt{2} G_{\mu}
\left((Q_f-2 T_3^f) M_W^2+M_Z^2
(T_3^f-Q_f)\right) \\&\quad + M_W \sqrt{M_Z^2-M_W^2}
(Q_f-2 T_3^f) C_{HWB}\Big)+\frac{M_W}{(D-2) (D-1)} \bigg(\Big(-6 \sqrt[4]{2}
(D-4) (D-1) \sqrt{G_{\mu}} M_W^2 C_{W}
M_Z^2 \\&\quad -(D-2) \sqrt{M_Z^2-M_W^2} C_{HWB}
\left((4 D-\xi -3) M_W^2+(2 D-3) M_Z^2\right)-\sqrt{2} (D-2)
G_{\mu} M_W \big((4 D-\xi -3) M_W^2 \\&\quad +M_Z^2 (4 D+\xi
-6)\big)\Big)\bigg)\bigg) M_Z^2-2 \sqrt{2} A_0(M_Z^2)
\left(Q_f M_W^2+M_Z^2 (T_3^f-Q_f)\right)^2 \Big(2 G_{\mu}
\left(Q_f M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad +3 \sqrt{2}
M_W \sqrt{M_Z^2-M_W^2} Q_f
C_{HWB}\Big)+2 \sqrt{2} A_0(M_Z^2 \xi) \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right)^2 \Big(2 G_{\mu} \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad +3 \sqrt{2}
M_W \sqrt{M_Z^2-M_W^2} Q_f C_{HWB}\Big)+8
A_0(M_Z^2 \beta) M_W^2 Q_f^2 \Big(\sqrt{2} G_{\mu}
(M_W-M_Z) (M_W+M_Z) \left(Q_f
M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad + M_W
\sqrt{M_Z^2-M_W^2} \left(3 Q_f M_W^2+M_Z^2 (2
T_3^f-3 Q_f)\right) C_{HWB}\Big)+2 B_0(M_Z^2, 0, 0) \bigg(2
M_W^2 M_Z^2 \\&\quad \times \Big(\sqrt{2} G_{\mu} (M_W-M_Z)
(M_W+M_Z) \left(Q_f M_W^2+M_Z^2
(T_3^f-Q_f)\right) \\&\quad + M_W \sqrt{M_Z^2-M_W^2}
\left(3 Q_f M_W^2+M_Z^2 (2 T_3^f-3 Q_f)\right)
C_{HWB}\Big) (-D+2 \beta +7) Q_f^2 \\&\quad +\sqrt{2} (D-6)
\left((T_3^f-Q_f) M_Z^3+M_W^2 Q_f M_Z\right)^2 \Big(2
G_{\mu} \left(Q_f M_W^2+M_Z^2 (T_3^f-Q_f)\right) +3 \sqrt{2}
M_W \sqrt{M_Z^2-M_W^2} Q_f
C_{HWB}\Big) \\&\quad +4 M_W^2 M_Z^2 \left((D-7) M_Z^2-2
M_W^2\right) T_3^{f2} \Big(\sqrt{2} G_{\mu} \left((Q_f-2
T_3^f) M_W^2+M_Z^2 (T_3^f-Q_f)\right) \\&\quad +
M_W \sqrt{M_Z^2-M_W^2} (Q_f-2 T_3^f)
C_{HWB}\Big)\bigg)\Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$ For the $b_L$, there are also top mass effects, and the vertex function in Feynman gauge is $$\begin{aligned}
V &= \frac{\sqrt{G_{\mu}}}{216 \cdot 2^{3/4} M_Z^3 \pi ^2} \Bigg(-108 M_Z^2 \bigg(2 \sqrt{M_Z^2-M_W^2} C_{HWB}
C_0\left(M_Z^2,0,0,M_W^2,M_W^2,M_T^2\right) M_W^2 \\&\quad +2 \sqrt{2} G_{\mu}
\big(C_0\left(M_Z^2,0,0,M_W^2,M_W^2,M_T^2\right) M_W^2+B_0\left(0,M_T^2,M_W^2\right)-M_Z^2
C_1\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right) \\&\quad +(D-2) C_{00}\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right)\big)
M_W+2 \sqrt{M_Z^2-M_W^2} C_{HWB} B_0\left(0,M_T^2,M_W^2\right) \\&\quad +2 D \sqrt{M_Z^2-M_W^2}
C_{HWB} C_{00}\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right)-4 \sqrt{M_Z^2-M_W^2} C_{HWB}
C_{00}\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right) \\&\quad +3 \sqrt[4]{2} \sqrt{G_{\mu}} M_Z^2 C_{W}
\big(C_0\left(M_Z^2,0,0,M_W^2,M_W^2,M_T^2\right) M_W^2+B_0\left(0,M_T^2,M_W^2\right) \\&\quad +\left(M_Z^2-2
M_W^2\right) C_1\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right)+(D-4)
C_{00}\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right)\big)\bigg) M_W^3 \\&\quad +18 M_Z^2 \bigg(4 \sqrt{2} (D-6) G_{\mu}
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W^4+4 D \sqrt{M_Z^2-M_W^2} C_{HWB}
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W^3 \\&\quad -24 \sqrt{M_Z^2-M_W^2} C_{HWB}
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W^3 \\&\quad -\sqrt{2} G_{\mu} \left(\left(4 (D-6) M_T^2+(D+2) M_Z^2\right)
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right)+8 (D-2)
C_{00}\left(0,M_Z^2,0,M_W^2,M_T^2,M_T^2\right)\right) M_W^2 \\&\quad -4 D M_T^2 \sqrt{M_Z^2-M_W^2}
C_{HWB} C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W \\&\quad +24 M_T^2 \sqrt{M_Z^2-M_W^2}
C_{HWB} C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W \\&\quad -8 M_Z^2 \sqrt{M_Z^2-M_W^2}
C_{HWB} C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) M_W \\&\quad -8 D \sqrt{M_Z^2-M_W^2} C_{HWB}
C_{00}\left(0,M_Z^2,0,M_W^2,M_T^2,M_T^2\right) M_W+16 \sqrt{M_Z^2-M_W^2} C_{HWB}
C_{00}\left(0,M_Z^2,0,M_W^2,M_T^2,M_T^2\right) M_W \\&\quad +4 \left(\sqrt{2} G_{\mu} \left(4 M_W^2-M_Z^2\right)+4
M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\right) B_0\left(0,M_T^2,M_W^2\right)+(D-6) \bigg(\sqrt{2} G_{\mu} \left(4
M_W^2-M_Z^2\right) \\&\quad +4 M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\bigg)
B_0\left(M_Z^2,M_T^2,M_T^2\right)+2 \sqrt{2} G_{\mu} M_Z^2 \big(\left(2 (D-3) M_T^2+M_Z^2\right)
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right) \\&\quad +(D-2)
C_{00}\left(0,M_Z^2,0,M_W^2,M_T^2,M_T^2\right)\big)\bigg) M_W^2+4 \bigg(6 \sqrt{M_Z^2-M_W^2}
C_{HWB} M_W^3 \\&\quad +\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z) \left(2 M_W^2+M_Z^2\right)\bigg) \big(((D-6) \beta -2)
C_0\left(M_Z^2,0,0,0,0,M_Z^2 \beta \right) M_Z^2+4 B_0\left(0,0,M_Z^2 \beta \right) \\&\quad +(D-6)
B_0\left(M_Z^2,0,0\right)-2 (D-2) C_{00}\left(0,M_Z^2,0,M_Z^2 \beta ,0,0\right)\big) M_W^2+108 M_T^2 M_Z^2
\Big(2 \sqrt{2} G_{\mu} M_W (M_W-M_Z) (M_W+M_Z) \\&\quad + \left(2 M_W^2-M_Z^2\right) \sqrt{M_Z^2-M_W^2}
C_{HWB}\Big) C_0\left(M_Z^2,0,0,M_W^2,M_W^2,M_T^2\right) M_W+108 M_T^2 M_Z^2 \Big(\sqrt{2} G_{\mu}
\left(M_Z^2-2 M_W^2\right) \\&\quad -2 M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\Big)
C_{00}\left(0,M_Z^2,0,M_T^2,M_W^2,M_W^2\right)+18 M_T^2 M_Z^2 \bigg(4 \Big(\sqrt{2} G_{\mu} (M_W-M_Z)
(M_W+M_Z) \\&\quad + M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\Big)
B_0\left(M_Z^2,M_T^2,M_T^2\right)+\bigg(\sqrt{2} G_{\mu} \left(\left(M_Z^2-4 M_W^2\right) M_T^2+4 M_W^2
(M_W-M_Z) (M_W+M_Z)\right) \\&\quad +4 M_W \left(M_W^2-M_T^2\right) \sqrt{M_Z^2-M_W^2} C_{HWB}\bigg)
C_0\left(M_Z^2,0,0,M_T^2,M_T^2,M_W^2\right)-8 \bigg(\sqrt{2} G_{\mu} (M_W-M_Z) (M_W+M_Z) \\&\quad +
M_W \sqrt{M_Z^2-M_W^2} C_{HWB}\bigg) C_{00}\left(0,M_Z^2,0,M_W^2,M_T^2,M_T^2\right)\bigg)-\left(2
M_W^2+M_Z^2\right)^2 \bigg(\sqrt{2} G_{\mu} \left(2 M_W^2+M_Z^2\right) \\&\quad +6 M_W \sqrt{M_Z^2-M_W^2}
C_{HWB}\bigg) \bigg((D-8) C_0\left(M_Z^2,0,0,0,0,M_Z^2\right) M_Z^2+4 B_0\left(0,0,M_Z^2\right)+(D-6)
B_0\left(M_Z^2,0,0\right) \\&\quad -2 (D-2) C_{00}\left(0,M_Z^2,0,M_Z^2,0,0\right)\bigg) \Bigg) {\addtocounter{equation}{1}\tag{\theequation}}\end{aligned}$$
[^1]: We will neglect the contribution to $M_Z$ that is proportional to $C_{H D}$.
|
---
abstract: 'Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to large sampling variance, the number of non-zeros is uneven among rows of the projection matrix which is generated in one trial, and more data information may be lost after dimension reduction. To break this bottleneck, based on random sampling without replacement in statistics, this paper builds a stable sparse subspace embedded matrix (S-SSE), in which non-zeros are uniformly distributed. It is proved that the S-SSE is stabler than the existing matrix, and it can maintain Euclidean distance between points well after dimension reduction. Our empirical studies corroborate our theoretical findings and demonstrate that our approach can indeed achieve satisfactory performance.'
address:
- 'College of Physical Education (Main Campus), Zhengzhou University, 100 Science Avenue, Zhengzhou, China'
- 'School of Mathematics and Statistics, Xidian University, 266 Xinglong Section, Xifeng Road, Xi’an, China'
author:
- Li Chen
- Shuisheng Zhou
- Jiajun Ma
bibliography:
- 'dim\_reduce.bib'
title: Stable Sparse Subspace Embedding for Dimensionality Reduction
---
dimensionality reduction ,feature projection ,random projection ,sparse ,stable
Introduction
============
Dimensionality reduction, which projects original features into a lower dimensional space, has been a prevalent technique in dealing with high dimensional datasets, because it is able to remove redundant features, reduce memory usage, avoid the curse of dimensionality and improve efficiency of machine learning algorithm. As a preprocessing step, dimensionality reduction has been applied to a variety of problems including $k$-means clustering [@Boutsidis2010; @Sinha2018icml; @Cai2017knowledge], support vector machines classification [@shi2012margin; @zhang2013recovering; @kumar2008randomized; @paul2014random], $k$-nearest neighbors classification [@deegalla2006reducing], least squares regression, and low rank approximation [@Kenneth2017]. However, how to design efficient and effective dimensionality reduction algorithm is a serious challenge problem.
The goal of dimensionality reduction is to approximate a large matrix $X$ with a much smaller sketch $\hat X$ such that the solution to a given problem on $\hat X$ is a good approximation on $X$. Some works obtain $\hat X$ by low-rank approximation (also known as singular value decomposition (SVD) or principal component analysis(PCA)[@Michael2015]). Given a dataset $X\in\mathbb{R}^{m \times n}$, consisting of $m$ data points each having $n$ features, SVD requires $O(mn\min\{m,n\})$ time to reduce data dimensionality from $n$ to $d$ ($d\ll n$), which is prohibitively large even for moderate size datasets. By imposing sparse regularization, some sparse PCA based methods are proposed for dimension reduction, see [@Zou2006spca][@Shen2008spca][@leng2009aspca]. These low-rank approximation methods can preserve data information well, but they are all based on minimization optimization problems, so it is very hard to solve them and the computation is time consuming. To overcome this obstacle, we study random projection (RP) techniques in this article.
RP multiplies $X$ by the transpose of a random matrix $R\in\mathbb{R}^{d \times n}$, i.e. $\hat{X}=XR^\top \in \mathbb{R}^{m \times d}$, where $d$ is independent of $m$ and $n$, to satisfy $\|R\mathbf{x}\|_2\approx \|\mathbf{x}\|_2$ simultaneously for all samples $\mathbf{x}\in \mathbb{R}^n$ in $X$. It has been applied in various fields, such as image data [@Farhad2017], text documents [@bingham2001random], face recognition [@goel2005face], privacy preserving distributed data mining [@liu2006random], etc. Compared to SVD-based dimensionality reduction approaches, RP reduces the running time to at most $O(mnd)$. The critical factor affecting the efficiency and effectiveness of RP is the random matrix $R$. A good $R$ is able to make the process of dimensionality reduction efficient, and can well preserve the Euclidean distances between pairwise points after dimensionality reduction. There are a number of literatures on designing $R$. In [@arriaga1999algorithmic], the entries of $R$ (denoted by $R_{ij}$) obey standard normal distribution having mean $0$ and variance $1$, i.e. $R_{ij}\sim N(0,1)$. Achlioptas [@Achlioptas2001] demonstrates that $R_{ij}$ can also have values $+1$ or $-1$ with probability $1/2$, which we denote as $U(1,-1)$. It is proved that $R_{ij}$ in this method have mean $0$ and variance $1$, and the distribution of $R_{ij}$ is symmetric about the origin with $\mathbb{E}(R_{ij}^2) = 1$. This property is sufficient to prove that $(1+\epsilon)$-approximate holds after dimensionality reduction [@Achlioptas2001; @arriaga1999algorithmic]. Comparing to $R_{ij}\sim N(0,1)$, the advantage of $R_{ij}\sim U(1,-1)$ is that the computation of the projection only contains summations and subtractions, but no multiplications, hence the computation is simple. However, because random matrices $R$ are both dense in these two methods, the computational complexity of multiplication $XR^\top$ are both $O(nnz(X)d)$, where $nnz(X)$ denotes the number of nonzero entries in $X$, and $nnz(X)=mn$ when $X$ is dense. This complexity is lower than SVD-based dimensionality reduction approaches as $d\ll n <m$ normally, but it is still high.
0.15in
Method Type Dimensions \#nonzeros per column Time for $XR^\top$
------------------------- --------- ------------------------------------------- -------------------------------------- --------------------------------------------
[@dasgupta1999learning] Density $O(\frac{\log m}{\epsilon^{2}})$ $O(\frac{\log m}{\epsilon^{2}})$ $O(\frac{nnz(X) \log m}{\epsilon^{2}})$
[@Achlioptas2001] Density $O(\frac{\log m}{\epsilon^{2}})$ $O(\frac{\log m}{\epsilon^{2}})$ $O(\frac{nnz(X)\log m}{\epsilon^{2}})$
[@Achlioptas2001] Sparse $O(\frac{\log m}{\epsilon^{2}})$ $O(\frac{\log m}{3\epsilon^{2}})$ $O(\frac{nnz(X)\log m}{3\epsilon^{2}})$
[@Daniel2014] Sparse $O(\frac{\log (1/\delta)}{\epsilon^{2}})$ $O(\frac{\log(1/\delta)}{\epsilon})$ $O(\frac{nnz(X)\log(1/\delta)}{\epsilon})$
This Paper Sparse $O(\frac{\log (1/\delta)}{\epsilon^{2}})$ 1 $O(nnz(X)$
\[tab:diff\_methods\] -0.1in
To further reduce the complexity of RP, researchers turn their attention to sparse matrices. The complexity of the multiplication $XR^\top$ is $O(nnz(X)\varrho)$ when $R$ is a sparse matrix, where $\varrho<d$ is the number of nonzero entries in per row. The smaller $\varrho$ is, the less computational cost of RP is. In [@Achlioptas2001] and [@Li2006], $R_{ij}\in \{+\sqrt{\kappa}, -\sqrt{\kappa}\}$ with probability $1/2\kappa$, otherwise $0$, where $\kappa\geq3$ such as $\kappa=\sqrt{n}$ or $\kappa=n/\log{n}$. In each row of this matrix, $d/\kappa$ entries are non-zeros, where $d=O(\epsilon^{-2}\log m)$. In [@Daniel2014], $R_{ij}=\eta_{ij}\sigma_{ij}/\sqrt{\varrho}$, where $\sigma_{ij}$ are independent and uniform in $\{-1,+1\}$, $\eta_{ij}$ are indicator random variables for $R_{ij}\neq 0$. Each column of this matrix exactly has $\varrho\geq 2(2\epsilon-\epsilon^2)^{-1}log(1/\delta)>1$ nonzero entries, where $0< \epsilon, \delta <1/2$. These methods are all able to get $(1+\epsilon)$-approximation of Euclidean distance between points. However, $\varrho$ are all larger than 1, thus $R$ are not sufficiently sparse. Recently, Clarkson et al. [@Kenneth2017] and Liu et al. [@liu2017sparse] constructed a very sparse embedded (SE) matrix $R$ with $R_{ij}\in\{+1,-1,0\}$. In $R$, each column only contains one nonzero entry. The computational complexity of the multiplication $XR^\top$ is only $O(nnz(X))$, which is the lowest as far as we know. Table \[tab:diff\_methods\] summarizes the properties of the above mentioned methods. There is one defect in the existing RP matrices that the positions of nonzero entries in each column of $R$ are random. Although the row labels of non-zero entries in each column are obtained by uniform sampling with replacement from $\{1,\ldots,d\}$, such sampling manner leads to a large variance, therefore the number of non-zeros is uneven among rows of the RP matrix that is generated in one trial, which may cause more data information loss after dimension reduction and leads to bad Euclidean distance preservation between points. Moreover, the large variance also causes the generated RP matrices instability, and further leads to the performance of dimension reduction unstable.
To improve stability of the sparse RP matrices as well as reduce variance of the number of nonzero entries among rows in matrix, we use the ideas of randomly sampling without replacement in statistics. To the best of our knowledge, this is the first attempt to improve the stability of RP matrices, and our method is simple and effective. The main contributions are summarized as follows.
- The stable sparse subspace embedded matrix is constructed for dimension reduction. In this construction, the idea of uniform sampling without replacement is adopted to obtain the position of nonzero entries in the matrix. In the constructed matrix, each row contains $\lfloor\frac{n}{d}\rfloor$ or $\lfloor\frac{n}{d}\rfloor+1$ nonzero entries, and each column contains only one nonzero.
- We prove that our matrix is stabler than SE matrix [@liu2017sparse].
- It is proved that embedding the original data into dimension $d=O(\epsilon^{-2}\log (1/\delta))$ is sufficient to preserve all the pairwise Euclidean distances up to $1\pm\epsilon$.
- Experimental results verify our theoretical analysis, and illustrate that our algorithm outperforms other compared dimension reduction methods.
The rest of this paper is organized as follows. Section 2 gives notations used in this paper and introduces theoretical basis of random projections. Section 3 describes sparse embedding method. We propose our stable sparse subspace embedding in section 4 and present its analysis in section 5. Experimental results are presented in section 6. Finally, we summarize the whole article and point out a few questions in section 7.
Preliminaries
=============
Notations and linear algebra
----------------------------
$X\in \mathbb{R}^{m\times n}$ is the dataset with $m$ samples and $n$ features. We denote $d$ as the number of reduced features. All logarithms are base-$2$ by $\log$. For a positive integer $n$, we use $[n]$ to denote the set $[1,\ldots,n]$. $\lceil\cdot\rceil$ denotes the smallest integer greater than a number, and $\lfloor\cdot\rfloor$ denotes the largest integer less than a number. $\mathbb{P}(\cdot)$ is the probability of an event. A vector $\mathbf{x}$ is assumed to be a row vector, and $\mathbf{x}^\top$ denotes its transpose. For a vector $\mathbf{x}\in \mathbb{R}^n$, $\|\mathbf{x}\|_2=\sqrt{\sum_{i=1}^n x_i^2}$. For a matrix $R\in \mathbb{R}^{d\times n}$, $\|R\|_F=\sqrt{\sum_{i,j} R_{ij}^2}$ and $\|R\|_2=\sup_{\|\mathbf{x}\|_2=1}\|R\mathbf{x}\|_2=\sqrt{\lambda_{max}(R^\top R)}$, i.e. the square root of the largest eigenvalue of $R^\top R$. $R_{i \cdot }$ denotes all the entries of the $i$-th row in $R$.
Theoretical basis of random projections
---------------------------------------
RP is a computationally efficient and sufficiently accuracy method as respect to preserving Euclidean distance after dimension reduction. The theoretical basis of RP arises from the following lemma:
\[lem:JL\] **(Johnson-Lindenstrauss Lemma [@Johnson1984; @Daniel2014])** For any real numbers $0<\epsilon, \delta<1/2$, there exists an absolute constant $C>0$, such that for any integer $d=C\epsilon^{-2}\log(1/\delta)$, there exists a probability distribution $\mathcal{D}$ on $d\times n$ real matrices such that for any fixed $\mathbf{x}\in \mathbb{R}^n$, $$\mathbb{P}_{R\sim \mathcal{D}}((1-\epsilon)\|\mathbf{x}\|_2\leq \|R\mathbf{x}\|_2\leq(1+\epsilon)\|\mathbf{x}\|_2)>1-\delta.$$ where $R\sim \mathcal{D}$ indicates that the matrix $R$ is a random matrix with distribution $\mathcal{D}$. $\mathbb{P}$ is the probability of a event.
Using linearity of $R$ and Lemma \[lem:JL\] with $\mathbf{x}=\mathbf{u}-\mathbf{v}$, we get that $R$ satisfies $(1-\epsilon)\|\mathbf{u}-\mathbf{v}\|_2 \leq\|R\mathbf{u}-R\mathbf{v}\|_2 \leq (1+\epsilon)\|\mathbf{u}-\mathbf{v}\|_2$ with probability at least $1-\delta$. Therefore, Johnson-Lindenstrauss lemma illustrates that if points in one space are projected onto a randomly extracted subspace with suitable dimension, then the distance between pairwise points are approximately preserved [@bingham2001random]. In order to satisfy Johnson-Lindenstrauss Lemma, the entries of random projection matrix $R$ should be i.i.d. with zero mean and unit variance [@arriaga1999algorithmic; @Achlioptas2001; @Daniel2014]. For convenience, we define subspace embedded matrix as follows.
\[def:embed\] (Subspace embedded matrix) Given $0< \epsilon, \delta <1$, matrix $R\in \mathbb{R}^{d\times n}$ is a subspace embedded matrix, if for any $\mathbf{x}\in \mathbb{R}^n$, $$\mathbb{P}((1-\epsilon)\|\mathbf{x}\|_2\leq \|R\mathbf{x}\|_2\leq(1+\epsilon)\|\mathbf{x}\|_2)>1-\delta.$$ Moreover, if matrix $R$ is a sparse matrix, then $R$ is a sparse subspace embedded matrix. The probability $\mathbb{P}((1-\epsilon)\|\mathbf{x}\|_2\leq \|R\mathbf{x}\|_2\leq(1+\epsilon)\|\mathbf{x}\|_2)$ is called distance preservation probability.
The Definition \[def:embed\] indicates that matrix $R$ embeds space $\mathbb{R}^n$ into $\mathbb{R}^{d}$ while preserving the distance between points $(1+ \epsilon)$-approximation with the probability larger than $1-\delta$. A good subspace embedded matrix makes the Euclidean distance approximation better, and calculates multiplication $XR^\top$ fast.
Sparse embedding {#sec:SE}
================
The sparse embedding algorithm is listed in Algorithm \[alg:SE\].
\[alg:SE\] Dataset $X\in\mathbb{R}^{m\times n}$. Sparse embedded matrix $R=\Phi Q\in\mathbb{R}^{d\times n}$ and feature extracted matrix $\hat{X}\in \mathbb{R}^{m\times d}$. Build a random map $h$ so that for any $i\in[n]$, $h(i)=j$ for $j\in[d]$ with probability $1/d$, where $0<d<n$. Construct matrix $\Phi\in\{0,1\}^{d\times n}$ with $\Phi_{h(i),i}=1$, and all remaining entries 0. Construct matrix $Q\in \mathbb{R}^{n\times n}$ is a random diagonal matrix whose entries are i.i.d. Rademacher variables. Compute the product $\hat{X}=X(\Phi Q)\top$.
In Algorithm \[alg:SE\], $h$ is a random map so that the row labels of the nonzero entries in $\Phi$ are completely random. This causes that the distribution of nonzero entries is uneven between rows, that is, some rows in $R$ contain more nonzero entries but other rows contain less even none, see Fig.\[fig:matrix\_a\] for an example. In Fig.\[fig:matrix\_a\], the fifth row contains 10 nonzeros. But the eighth row does not contain any nonzeros. For feature extraction $XR^\top$, the fifth row in the SE matrix indicates that ten features of $X$ are linear combined into one feature, which may lead to more information loss. Moreover, the randomness of position of nonzero entry in per column of $R$ results in $R$ instability, because it is equivalent to random sampling from $[d]$ with replacement as the row label of nonzero entry in per column, the variance of which is large.
0.2in
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In the following sections, we build a new sparse subspace embedding matrix and provide theoretical analysis for it in order to overcome the defects of SE.
Stable sparse subspace embedding {#sec:S-SSE}
================================
In this section, we design a new sparse subspace embedded (SSE) matrix: Stable SSE matrix (S-SSE). Algorithm \[alg:1\] gives the construction of S-SSE matrix. In this matrix, each column only has one nonzero entry, which is $+1$ or $-1$ with the same probability. Every row contains almost the same number of non-zeros.
\[alg:1\] Dataset $X\in\mathbb{R}^{m\times n}$. Embedded matrix $R\in\mathbb{R}^{d\times n}$ and feature extracted matrix $\hat{X}$. Set $d=O(\frac{\log (1/\delta)}{\epsilon^2})$. Repeat $[d]$ for $\lceil n/d \rceil$ times and obtain a set $D$. Randomly sample $n$ elements from $D$ without replacement to construct sequence $\mathcal{S}$. Construct matrix ${R}\in\{0,+1,-1\}^{d\times n}$, where ${R}_{\mathcal{S}(i),i}\in\{+1,-1\}$ for $i\in[n]$ with probability $1/2$, and all remaining entries $0$. Compute the multiplication $\hat{X}=XR^\top$.
***Remark 1***. Main difference between S-SSE and SE is the selection of row labels of nonzero entries. SE chooses those by randomly sampling with replacement, whereas our method chooses those by randomly sampling without replacement. The number of nonzero entries in each row of S-SSE matrix is $\lfloor \frac{n}{d} \rfloor$ or $\lceil \frac{n}{d} \rceil$, thus nonzero entries are uniformly distributed among columns of $R$, see Fig.\[fig:matrix\_b\] for an example. Furthermore, because the sampling error of sampling without replacement is smaller than that of sampling with replacement, $R$ constructed by the S-SSE follows a symmetric distribution about zero mean with unit variance better than by the SE, and the S-SSE satisfies the Johnson-Lindenstrauss lemma better [@arriaga1999algorithmic; @Achlioptas2001; @Daniel2014]. This leads to the S-SSE preserving the Euclidean distance better than the SE after dimension reduction, see the experimental results in Figures \[fig:distance\] - \[fig:emsrong-delta\]. Therefore, S-SSE may reduce data information loss after feature extraction comparing to the SE as distance between points is the important data information. Moreover, Section \[sec:stability\] demonstrates that the matrix constructed by our method is stabler than by the SE. ***Remark 2***. The feature extraction is simple by using the S-SSE. It just needs to add or subtract original features in $X$ to form a new feature, i.e. linear combination of features in $X$ corresponding to the column labels of nonzero entries in the row of $R$. The computation complexity of feature extraction is also only $O(nnz(X))$, which is the same as SE method.
Properties of the S-SSE {#sec:properties}
=======================
In this section, we prove two good properties of the S-SSE: stability of matrix and preservation of Euclidean distances.
Stability of matrix {#sec:stability}
-------------------
The following discussion confirms that the S-SSE matrix is stabler than the SE matrix.
The SE and the S-SSE matrices both contain only one nonzero entry in each column. Therefore, the stability of matrices is determined by the change of non-zero entries in rows. We employ the variance of the number of nonzeros in rows to measure the stability of a matrix. Denote the number of nonzeros in rows of the S-SSE matrix as $\mathbf Y$, then the possible values of $\mathbf Y$ are $\lceil \frac{n}{d} \rceil$ or $\lfloor \frac{n}{d} \rfloor$. Denote the number of nonzeros in rows of the SE matrix as $\mathbf Z$. The possible values of $\mathbf Z$ are $0,1,\ldots,n$, because each row of the SE matrix contains $n$ entries, and the position of non-zero entry in each column is selected randomly. Theorem \[thm:stable\] indicates that the expectation of $\mathbf Y$ is the same as that of $\mathbf Z$, while the variance of $\mathbf Y$ is less than that of $\mathbf Z$ when $d\geq 2$.
\[thm:stable\] Denote $\mathbb{E}(\cdot)$ and $Var(\cdot)$ as the expectation and variance of a variable, respectively. The random variables $\mathbf Y$ and $\mathbf Z$ are the number of nonzeros in rows of the S-SSE matrix and the SE matrix, respectively, then $$\mathbb{E}(\mathbf Y)=\mathbb{E}(\mathbf Z), ~~ Var(\mathbf Y)\leq Var(\mathbf Z).$$
Set $n=rd+q$, where $r=\lfloor\frac{n}{d}\rfloor$ and $0\leq q< d$ is an integer, the distribution of $\mathbf Y$ is $$\mathbb{P}(\mathbf{Y}=r)=1-\frac{q}{d}, ~\mathbb{P}(\mathbf{Y}=r+1)=\frac{q}{d}.$$ The expectation of $\mathbf{Y}$ is $$\label{eq:qiwang_Y}
\mathbb{E}(\mathbf{Y})=r(1-\frac{q}{d})+(r+1) \frac{q}{d}=\frac{n}{d}.$$ In addition, because $$\mathbb{E}(\mathbf{Y}^2)=r^2(1-\frac{q}{d})+(r+1)^2 \frac{q}{d},
$$ the variance of $\mathbf{Y}$ is $$\label{eq:fangcha_Y}
Var(\mathbf{Y})=\mathbb{E}(\mathbf{Y}^2)-[\mathbb{E}(\mathbf{Y})]^2=\frac{q}{d}-(\frac{q}{d})^2.$$
In the following, we compute the expectation and variance of $\mathbf{Z}$. Let random event $B$ mean “non-zero is in the $i$-th row" and $\bar{B}$ mean “non-zero is not in the $i$-th row". Because the row label of non-zero entry in each column is randomly chosen, which is equivalent to randomly sampling with replacement from $[d]$, therefore $\mathbb{P}(B)=\frac{1}{d}$ and $\mathbb{P}(\bar{B})=1-\frac{1}{d}$. The random variable $\mathbf{Z}$ is the number of times that $B$ occurs in $n$ Bernoulli trials. Hence $\mathbf{Z}$ obeys the binomial distribution, and the distribution of $\mathbf{Z}$ is $$\mathbb{P}(\mathbf{Z}=k)=C_n^k(\frac{1}{d})^k(1-\frac{1}{d})^{n-k},~k=0,1,\ldots,n.$$ The expectation and variance of $\mathbf{Z}$ are $$\label{eq:qiwang_Z}
\mathbb{E}(\mathbf{Z})=\frac{n}{d},$$ $$\label{eq:fangcha_Z}
Var(\mathbf{Z})=n(\frac{1}{d})(1-\frac{1}{d}).$$
Eqs. and indicate that $\mathbf{E}(\mathbf{Y})=\mathbf{E}(\mathbf{Z})$. Next, we prove $Var(\mathbf{Y})\leq Var(\mathbf{Z})$. If $d=1$, then $Var(\mathbf{Y})= Var(\mathbf{Z})=0$. If $2\leq d\leq n$, then $Var(\mathbf{Z})\geq \frac{n-1}{n}\geq \frac{1}{2}$, while $Var(\mathbf{Y})\leq \frac{1}{4}$, hence $Var(\mathbf{Y})< Var(\mathbf{Z})$. Therefore, $Var(\mathbf{Y})\leq Var(\mathbf{Z})$, where the equality sign holds only when $d=1$.
***Remark 3***. Eq. indicates that the variance of $\mathbf Y$ is related to $q=n ~ \text{mod} ~d$. When $q=0$, then $Var(\mathbf Y)=0$, that is, if $n$ can be divided by $d$ without remainder, then each row of the S-SSE matrix contains the same number of non-zeros. When $q=\frac{d}{2}$, the $Var(\mathbf{Y})$ reaches the maximum $\frac{1}{4}$. In comparison, the $Var(\mathbf{Z})$ is not less than $\frac{1}{4}$. ***Remark 4***. Theorem \[thm:stable\] illustrates that the number of non-zeros in rows of the SE matrix changes greater than that of the S-SSE matrix, which leads to large variety among rows in the SE matrix, and further causes the generated matrices changes greatly. Therefore, the SE matrix is more unstable than S-SSE matrix.
Preservation the Euclidean distances {#sec:preservation}
------------------------------------
In this subsection, we prove that our S-SSE matrix can preserve pairwise Euclidean distance up to $1\pm \epsilon$.
\[lem:Hanson-Wright inequality\] [@Diakonikolas2010; @Daniel2014] Let $B \in \Re^{n\times n}$ be symmetric and $\mathbf{z} \in \{+1, -1\}^n$ be random. Then for all $l \geq 2$, $$\mathbb{E}[|(\mathbf{z}^\top B \mathbf{z}) - tr(B)|^l] \leq C^l \cdot \max\{\sqrt{l}\|B\|_F, l\|B\|_2\}^l$$ where $C>0$ is a universal constant.
\[thm:preserve\] The matrix $R\in \mathbb{R}^{d\times n}$ is constructed by Algorithm \[alg:1\]. Given $0<\epsilon,\delta<\frac{1}{2}$, there exists $d=O(\frac{\log(1/\delta)}{\epsilon^2})$ such that $R$ is a sparse subspace embedding matrix, i.e. for any $\mathbf{x} \in \mathbb{R}^n$, $$\label{eq:thm1}
\mathbb{P}((1-\epsilon)\|\mathbf{x}\|_2\leq \|R\mathbf{x}\|_2\leq(1+\epsilon)\|\mathbf{x}\|_2)>1-\delta.$$
Assume $\mathbf{x}$ is a unit vector, i.e. $\|\mathbf{x}\|_2^2=1$, which can be obtained in data preprocessing step. Therefore, is translated into $$\mathbb{P}(1-\epsilon\leq \|R\mathbf{x}\|_2\leq 1+\epsilon)>1-\delta.$$ It is equal to the following inequation: $$\mathbb{P}(|\|R\mathbf{x}\|_2^2-1|>2\epsilon-\epsilon^2)<\delta.$$ For convenience, we denote $h=\|R\mathbf{x}\|_2^2-1$, then is equal to $$\label{eq:thm2}
\mathbb{P}(|h|>2\epsilon-\epsilon^2)<\delta.$$
We rewrite the entries of matrix $R$ as $R_{ij}=\eta_{ij}\sigma_{ij}$, where $\eta_{ij}$ is an indicator random variable for $R_{ij}\neq 0$, $\sigma_{ij}\in \{+1,-1\}$, then $$h=\|R\mathbf{x}\|_2^2-1=\sum\limits_{t=1}^{d}\sum\limits_{i\neq j\in[n]}\eta_{ti}\eta_{tj}\sigma_{ti}\sigma_{tj}x_ix_j:=\sigma^\top A \sigma,$$ where $A$ is a $dn\times dn$ block diagonal matrix. It can be divided into $d$ blocks with each $n\times n$. For the $t$-th block $A_t$, $$(A_t)_{ij}=
\begin{cases}
\eta_{ti}\eta_{tj}x_ix_j, & i\neq j,\\
0,&i=j.
\end{cases}$$ Then, $$\label{eq:thm3}
\begin{split}
\mathbb{P}(|h|>2\epsilon-\epsilon^2)&=\mathbb{P}(|\sigma^\top A \sigma|>2\epsilon-\epsilon^2)\\
&=\mathbb{P}(|\sigma^\top A \sigma-\mathrm{tr}(A)|>2\epsilon-\epsilon^2)\\
&=\mathbb{P}(|\sigma^\top A \sigma-\mathrm{tr}(A)|^l>(2\epsilon-\epsilon^2)^l)\\
&\leq (2\epsilon-\epsilon^2)^{-l}\mathbb{E}(|\sigma^\top A \sigma-\mathrm{tr}(A)|^l)\\
&\leq (2\epsilon-\epsilon^2)^{-l} C^l \max\{ \sqrt{l}\|A\|_F, l\|A\|_2\}^l
\end{split}$$ where $\mathrm{tr}(A)$ is the trace of the matrix $A$ and $\mathrm{tr}(A)=0$ as $A_{ii}=0$. $l\geq 2$. $C>0$ is some universal constant. The first inequality uses Markov-Bound. The second inequality uses Lemma \[lem:Hanson-Wright inequality\] with $\mathbf{z}=\sigma$ and $B=A$. Next, we compute the bounds of $\|A\|_F$ and $\|A\|_2$.
For any $i\neq j\in [n]$, $\sum_{t=1}^{d}\eta_{ti}\eta_{tj}\leq 1$, which indicates that the number of non-zero entries in the same row is no more than $1$ in two columns. We have $$\label{eq:thm_Fnorm}
\begin{split}
\|A\|_F^2&=\sum\limits_{i\neq j\in [n]}x_i^2x_j^2\sum_{t=1}^{d}\eta_{ti}\eta_{tj}\\
&\leq \sum\limits_{i\neq j\in [n]}x_i^2x_j^2 \leq \|\mathbf{x}\|_2^4 \leq 1.
\end{split}$$
Moreover, we can prove that $$\label{eq:thm_2norm}
\|A\|_2\leq 1.$$
Rewrite $A_t$ as $A_t=\bar{R}_t-\bar{D}_t$, here $(\bar{R}_t)_{ij}=\eta_{ti}\eta_{tj}x_ix_j$, $\bar{D}_t$ is a diagonal matrix with $(\bar{D}_t)_{ii}=\eta_{ti}x_i^2$. Because $\bar{R}_t$ and $\bar{D}_t$ are both positive semidefinite, we have $\|A\|_2\leq \max\{\|\bar{R}_t\|_2, \|\bar{D}_t\|_2\}$. $\|\bar{D}_t\|_2\leq \|\mathbf{x}\|_\infty^2\leq 1$. Denote $v_i=\eta_{ti}x_i$ and $\mathbf{v}\in \mathbb{R}^n$, then $\bar{R}_t=\mathbf{v}\mathbf{v}^\top$ and $\|\bar{R}_t\|^2=\|\mathbf{v}\mathbf{v}^\top\|_2^2\leq \|\mathbf{x}\|_2^2=1$. Therefore, $\|A\|_2\leq 1$. Substitute and into , we obtain $$\label{eq:delta}
\mathbf{P}(|h|>2\epsilon-\epsilon^2)\leq (2\epsilon-\epsilon^2)^{-l}C^ll^l<(\frac{1}{3}\cdot \frac{Cl}{\epsilon^2})^l.$$ Let $C=\frac{C_1}{d}$, where $C_1>0$ is a constant, $l=\log(1/\delta)$. In order to make less than $\delta=(\frac{1}{2})^l$, we need $d>\frac{2}{3}C_1\frac{l}{\epsilon^2}=O(\frac{\log(1/\delta)}{\epsilon^2})$. Therefore, the theorem \[thm:preserve\] is proved.
***Remark 5***. With regard to the SE method, Clarkson et al. proved that $d=O((u/\epsilon)^4\log^2(u/\epsilon))$ can make $\|R\mathbf{x}\|_2=\|\mathbf{x}\|_2$ with probability at least $9/10$ [@Kenneth2017], where $u$ is the rank of $X$. Liu et al. proved that $d=O(\max\{\frac{k+\log(1/\delta)}{\epsilon^2}, \frac{6}{\epsilon^2\delta}\})$ can get the $\epsilon$-approximately optimizing solution of $k$-means clustering [@liu2017sparse]. By comparison, we demonstrate that $d=O(\frac{\log(1/\delta)}{\epsilon^2})$ is sufficient for S-SSE to preserve Euclidean distance up to $(1+ \epsilon)$-approximation, and our proof is simpler.
Experiment
==========
We compare our method S-SSE with several other feature extraction methods to evaluate the performance of the S-SSE. They are listed below:
- **SPCA**: Sparse principal component analysis is proposed by [@Zou2006spca]. SPCA imposes the lasso (elastic net) constraint into the PCA to promote sparse. The matrix deduced by SPCA is a sparse matrix.
- **DE**: The density embedding (DE) method is proposed by [@Achlioptas2001]. In this method, $R$ is dense, $R_{ij}\in \{1,-1\}$ with the same probability.
- **SE**: The sparse embedding (SE) method corresponds to Algorithm \[alg:SE\]. In this method, the position of nonzero entry in each column is randomly chosen.
- **S-SSE**: Stable sparse subspace embedding (S-SSE) corresponds to Algorithm \[alg:1\].
We performed all the experiments on the PC machine with dual Intel core i7-4790 CPUs at 3.60GHz processor and 8 GB of RAM.
Data separability comparison after dimensionality reduction
-----------------------------------------------------------
In order to verify our theoretical analysis in section \[sec:stability\], we performed experiments on a synthetic dataset which consists of four classes. Each class contained 1000 samples with a dimension of 100. Features in four classes were drawn from normal distribution having variance 0.5 and mean 0, 2, 4 and 6, respectively. The dimension was reduced by using the SE and the S-SSE. Figure \[fig:4lei\_data\] shows data distribution when each class containing 100 samples with a dimension of 2.
![Two-dimensional separable dataset with four classes. Each class contains 100 data. Data in four classes were drawn from normal distribution having variance 0.5 and mean 0, 2, 4 and 6, respectively. $x$-axis and $y$-axis are the first and the second feature of the data.[]{data-label="fig:4lei_data"}](4lei_randn){width="50.00000%"}
We adopt separability of dimensionality reduced data to measure the feature extraction performance of the SE and the S-SSE. The separability metric is the ratio of between-class distance and within-class distance, i.e. $$J=\frac{\mathrm{tr}(S_b)}{\mathrm{tr}(S_w)},$$ where $S_w=\sum_{i=1}^c \mathbf{P}_i\frac{1}{N_i}\sum_{j=1}^{N_i} (\mathbf{x}_j^{(i)}-\mathbf{s}_i)(\mathbf{x}_j^{(i)}-\mathbf{s}_i)^\top$ is the within-class dispersion matrix, $S_b=\sum_{i=1}^c \mathbf{P}_i (\mathbf{s}_i-\mathbf{s})(\mathbf{s}_i-\mathbf{s})^\top$ is the between-class dispersion matrix, $c$ is the number of classes, $\mathbf{P}_i$ is the priori probability of the $i$-th class, $N_i$ is the number of samples contained in the $i$-th class, $\mathbf{x}_j^{(i)}$ is the $j$-th sample in the $i$-th class, $\mathbf{s}_i$ is the mean of samples in the $i$-th class, $\mathbf{s}$ is the mean of all samples. The larger the $J$ is, the better the separability is.
In order to obtain unbiased results, we ran programs 1000 times independently for each dimension $d$ and computed mean and variance of $J$. Fig. \[fig:separability\] gives the experimental results. Fig. \[fig:kefen\_mean\] illustrates that values of $J$ are fluctuated around $0.0201$ for different $d$, yet the range of fluctuation is small, about $\pm 0.0003$, which illustrates that the separability of the data is still good after dimensionality reduction by using the SE and the S-SSE. We can also observe that values of $J$ at some $d$ are larger than that at $d=100$, which indicates that feature extraction may improve the separability of the data. With the increasing of $d$, the fluctuation of $J$ decreases, and more and more close to the value of $J$ at $d=100$, which indicates that the separability of dimensionality reduced data becomes stabler as $d$ increases. The fluctuation of $J$ for the S-SSE is smaller than that for the SE, which indicates that the separability of the data dimensionality reduced by using the S-SSE method is stabler than that by using the SE method. Fig. \[fig:kefen\_var\] shows that the variances of $J$ for the SE and the S-SSE both decrease as the dimension increases, which indicates that the larger the reduced dimension is, the stabler the data separability is. For all the $d$, the variances of $J$ for the S-SSE are all smaller than that for the SE, which indicates that the S-SSE is stabler than the SE. Overall, the S-SSE is able to maintain data separability as the SE, but the S-SSE is stabler than the SE, because the random matrix constructed by the S-SSE method is stabler.
Euclidean distance preservation comparison {#sec:exp_preserve}
------------------------------------------
### The variation of relative error $\epsilon$ with $d$
In order to compare the preservation of Euclidean distance for the SE and the S-SSE, we conducted experiments on data with 1000 dimensions to measure the variation of relative error $\epsilon=|\frac{\|xR\|_2}{\|x\|_2}-1|$ with reduced dimension $d$. Entries in the data were randomly chosen from $[0,1]$ or standard normal distribution with mean 0 and variance 1, because real-world datasets are usually normalized to these two distributions before training. The dimension was reduced from 1000 to $d$, where $d$ was set as $20$ to $200$ with interval $20$. For every $d$, experiments were performed 100 times independently and the mean of $\epsilon$ was calculated to obtain unbiased results. Fig. \[fig:distance\] gives the experimental results. It can be shown from Fig. \[fig:distance\] that $\epsilon$ decreases with the increasing of $d$. This is consistent with reality. Moreover, the relative error of the S-SSE is less than that of the SE in most cases. Therefore, the S-SSE can preserve the Euclidean distance better than the SE after dimensionality reduction.
### The variation of distance preservation probability $p$ with $d$ {#sec:1-delta_d}
In order to verify the conclusion of Theorem \[thm:preserve\], and further compare the preservation of Euclidean distance after dimensionality reduction by the SE and the S-SSE, experiments were conducted on one synthetic dataset and two benchmark datasets. We calculate frequency of $\|R\mathbf x\|_2$falling within the interval $[(1-\epsilon)\|\mathbf x\|_2, (1+\epsilon)\|\mathbf x\|_2]$. Experiments were run 10,000 times independently and computed the mean of the frequencies as the distance preservation probability. For convenience, we denote this probability value as $p$, i.e. $p:=\mathbb{P}((1-\epsilon)\|\mathbf{x}\|_2\leq \|R\mathbf{x}\|_2\leq(1+\epsilon)\|\mathbf{x}\|_2)$, which is related to $\epsilon$ and $R$. If $\epsilon$ is fixed at a constant, then the larger $p$ is, the better the Euclidean distance preservation of $R$ is. The synthetic dataset contains 1000 samples with dimension 200, which were uniformly and randomly generated from interval $[0,1]$. The benchmark datasets are DNA and MADELON, whose information is listed in Table \[tab:data\]. To measure the variation of distance preservation probability $p$ with $d$, $\epsilon$ was fixed at $\epsilon=0.1\in (0,0.5)$, and $d$ was set as $20$ to $200$ with interval $20$. Fig. \[fig:d-delta\] gives the experimental results. Fig. \[fig:d-delta\] illustrates that as $d$ increases, $p$ also increases gradually approaching to 1, which indicates that the distance preservation probability increases with the increasing of reduced dimension. With regard to the same $d$, the value of $p$ for the S-SSE is larger than that for the SE, which indicates that the S-SSE method can better preserve Euclidean distance approximation.
### The variation of distance preservation probability $p$ with $\epsilon$
To measure the relationship between distance preservation probability $p$ and relative error $\epsilon$, we fixed $d$ at $80$, $80$ and $100$ for synthetic dataset (the generation method is the same as that in subsection \[sec:1-delta\_d\]), DNA and MADELON, respectively. $\epsilon$ was set as $0.05$ to $0.5$ with interval $0.05$. The experiments were performed 10000 times independently and computed the mean of $p$ as the final results. Fig. \[fig:emsrong-delta\] gives the experimental results. Fig. \[fig:emsrong-delta\] shows that the values of $p$ gradually increase to 1 as $\epsilon$ increases, which indicates that with the enlarging of interval $[(1-\epsilon)\|\mathbf x\|_2, (1+\epsilon)\|\mathbf x\|_2]$, $p$ also increases, which is consistent with reality. The values of $1-p$ for the S-SSE are all smaller than 0.5, which indicates that the condition in Theorem \[thm:preserve\] is reasonable. In addition, given the value of $\epsilon$, $p$ of the S-SSE method is larger than that of the SE method, which indicates that the probability of $\|R\mathbf x\|_2$ falling within the interval $[(1-\epsilon)\|\mathbf x\|_2, (1+\epsilon)\|\mathbf x\|_2]$ after dimension reduction by the S-SSE method is larger than that by the SE method, in other words, S-SSE method can better preserve Euclidean distance approximation.
$k$-means clustering experiments
--------------------------------
Our S-SSE approach can be applied in various Euclidean distance based machine learning algorithms. In these algorithms, $k$-means clustering is one of the most widely used methods, but it is inefficient on dealing with high dimensional datasets. In order to evaluate the performance of the proposed feature extraction method applied in machine learning, this subsection uses the dimensionality reduced data onto the $k$-means clustering and compares the S-SSE against a few other prominent dimensionality reduction methods. For SPCA, we set the number of non-zero entries in each column of principal component directions matrix is 1 to compare the efficient of the SPCA, the SE and our S-SSE. The maximum number of iterations in SPCA is set 3000. We can not get the results of SPCA within three days on GISETTE and SECTOR datasets. Thus, these results are not reported. The datasets can be downloaded from the LIBSVM website [^1]. Table \[tab:data\] lists the information of the datasets, including the number of samples, features and classes.
Datasets \#INSTANCE \#FEATURES \#CLASSES
---------- ------------ ------------ -----------
DNA 3186 180 3
USPS 9298 256 10
MADELON 2000 500 2
MNIST 60000 780 10
GISETTE 7000 5000 2
SECTOR 9619 55197 105
: Information of datasets used in the experiments
\[tab:data\]
In order to compare the effect of feature extraction algorithms the SPCA, the DE, the SE, the SPCA and the S-SSE, we ran standard $k$-means clustering algorithm after dimensionality reduction. We also compare all these algorithms against the standard $k$-means clustering algorithm on the full dimensional datasets. In experiments, Cai’s Litekmeans package [^2] performs very well, hence we employed Cai’s package in our experiments. The results in the figures are the mean of ten runs for each dataset. In each run, $k$-means clustering repeats twenty times, each with a new set of initial centroids, and returns the best one as the clustering output, i.e. in MATLAB, we ran the following command: *litekmeans(X, $k$, ’Replicates’, 20)*.
### Evaluation methodology
To measure the quality of all the methods, we reported the clustering accuracy [@fahad2014survey], e.g. $accuracy=0.9$ implies that $90\%$ of the points are assigned the “correct cluster". We also reported the running time (in seconds) of constructing the matrix $R$ and computing the multiplication $RX^\top$ for all the compared algorithms. All the reported results correspond to the average values of 10 independent runs.
### Results
Experimental results are shown in Figs. \[fig:acc\] - \[fig:time\_XR\]. $x$-axis is compression factor, i.e. the ratio of the number of features after reduction and the number of original features, for instance, $compression~factor=0.3$ indicates that we extract $30\%$ of original features. For SECTOR, the maximum compression factor is set as $0.4$ because its dimension is so extremely high that training it consumes excessive memory.
From Figs. \[fig:acc\] - \[fig:time\_XR\], we can draw the following conclusions:
- Fig. \[fig:acc\] indicates that the S-SSE has superior performance comparing with other RP based methods in terms of accuracy. This verifies our theoretical results. For high-dimension datasets, such as GISETTE, USPS and MNIST, the compression factor can be set as a very small number (0.2, 0.4, 0.3 for these three datases respectively) to obtain satisfactory performance. The SPCA performs well on one dataset DNA, but on the other datasets, its accuracy is lower than other comparison methods. Moreover, with the increasing of extracted dimensions, the accuracy of SPCA may decline because the extracted features may contain noise.
- When compression factor is set as $1$, our algorithm has the same accuracy with standard $k$-means, while the DE and the SE have lower accuracy on some datasets, such as DNA and MADELON. That is because non-zero entries are distributed uniformly in our matrix. When $compression~factor=1$, the S-SSE matrix equals to a identity matrix whose columns are permuted, thus features are unchanged after feature extraction. Whereas, even when $compression~factor=1$, feature extraction by the DE and the SE are still the linear combination of original features rather than the original features themselves, which leads to lower accuracy.
- With regard to running time, the S-SSE and the SE are very similar to each other for constructing $R$ and for computing product $RX^\top$ on all datasets, which means that our method does not increase running time comparing with the SE, while the performance is improved. The DE is the slower method. That is because the DE matrix is not a sparse matrix, generating it and multiplying it with dataset matrix $X$ are time consuming. The SPCA is the slowest method to construct $R$, because SPCA needs to solve a optimization problem to obtain $R$, which is not easy and the computation is extraordinarily time consumption.
Conclusion
==========
High dimensional data has provided a considerable challenge in designing machine learning algorithm. To address this obstacle, researchers apply dimensionality reduction algorithms first instead of directly working with high dimensional data. Random projection is more efficient than low rank based approaches, therefore it attracts a lot of researchers to study. In this study, we design a stable sparse subspace embedding algorithm for dimensionality reduction. It overcomes the disadvantages of the state-of-art sparse embedding methods, such as the instability of matrix, the uneven distribution of nonzeros among columns in matrix. It is proved that the proposed method is stabler than the existing method, and it can preserve $(1+\epsilon)$-approximation after dimensionality reduction. The superior performance of our method are attributed to the uniform distribution of nonzeros in the matrix. The experimental results verify our theoretical analysis and show that compared with other dimensionality reduction methods, the new algorithm is stabler, can better maintain Euclidean distance between points, and can obtain better performance in machine learning algorithm. We conclude this paper with two open questions. Is our stable idea effective for other RP approaches? Does our algorithm perform well on other machine learning algorithms besides $k$-mean clustering?
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the National Natural Science Foundation of China (NNSFC) \[No. 61772020\].
[^1]: <https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/>
[^2]: <http://www.zjucadcg.cn/dengcai/Data/Clustering.html>
|
---
abstract: |
We use the example of playing a 2-player game with entangled quantum objects to investigate the effect of quantum correlation. We find that for simple game scenarios it is *classical* correlation that is the central feature and that these simple quantum games are not sensitive to the quantum part of the correlation. In these games played with quantum objects it is possible to transform a game such as Prisoner’s Dilemma into the game of Chicken. We show that this behaviour, and the associated enhanced equilibrium payoff over playing the game with quantum objects in non-entangled states, is entirely due to the classical part of the correlation.
Generalizing these games to the pure strategy 2-player quantum game where the players have finite strategy sets and a projective joint measurement is made on the output state produced by the players, we show that a given quantum game of this form can always be reproduced by a classical model, such as a communication channel. Where entanglement is a feature of the these 2-player quantum games the matrix of expected outcomes for the players can be reproduced by a classical channel with correlated noise.
author:
- |
*Simon J.D. Phoenix & Faisal Shah Khan*\
Khalifa University, Abu Dhabi, PO Box 127788, UAE
title: The Role of Correlation in Quantum and Classical Games
---
Introduction
============
The field of computer science has been revolutionised by the realisation that computers are physical objects obeying physical laws. Allowing computational devices to access the features of quantum mechanics, and entanglement in particular, has resulted in the potential for quantum devices that can perform certain computations significantly faster than their classical counterparts, lowering the complexity class of the associated problems. It was an innovative and groundbreaking step to ask the same question of classical game theory. Could the introduction of quantum objects and operations to the theory of games also result in a similar revolution allowing resolutions and enhancements not available within a classical treatment?
Since the seminal work of Eisert, Wilkens and Lewenstein (EWL) and Meyer \[1,2\] quantum games have been the subject of much work and controversy. The question of whether a game played with quantum objects can be considered to be quantum mechanical at all has been raised (see, in particular \[3\]) and the necessity of comparing like with like within the context of games has been beautifully formulated by Bleiler \[4\]. Much of the work has focused, naturally, on the use of entangled states (see \[4-21\] for a small selection of the extensive literature). In this paper we focus on 2-player non-cooperative games in which a single projective quantum measurement is performed to generate the measurement results over which the players have preferences. We address the question of whether such games can access the *quantum* nature of any correlation and also question to what extent such games can be considered to be truly quantum mechanical, requiring quantum objects in order to achieve a given result or outcome for the players.
We begin by considering some very simple, and restricted, examples of games played with quantum objects in order to gain an insight into the role of correlation in these systems. It is shown that, for these games, the results depend only on the classical component of the correlation. For these games, although we may begin with preferences over measurement results that are those of one game form, the actual game the players play is a different game altogether. In our first simple example we examine a game played with quantum objects that would appear to be a version of Prisoner’s Dilemma in the first instance. Upon closer inspection, however, it can be seen that the players are *actually* playing the classical game of Chicken[^1] (for an excellent text on game theory see, for example, \[22\]).
We show that the same game transformation can be achieved by a classical game in which the players’ choices are communicated over a noisy channel with a classical correlated noise process. Thus the properties of the quantum game are not dependent upon the quantum part of the correlation. By extending the classical game to the mixed case we see that a classical correlated noise process can lead to similar enhancements of equilibrium payoffs as that claimed for the quantum game with a full quantum strategy set. The role of the classical correlations is further highlighted by consideration of a simple quantum game in which the players communicate their choice by the transmission of their respective particles over some noisy channel such that the quantum interference terms in the density matrix are suppressed. In this scenario we see that an enhanced equilibrium and the transformation of the game is also obtained. These results strongly suggest that the enhancements and the game transformation are due to *classical* correlations in these 2-player games rather than any specifically quantum mechanical feature of the correlation.
We develop a general approach to simple 2-player games played with quantum objects that allows the analysis of a wider class of games than our initial examples. Thus we adopt the perspective that games can be played with quantum objects (we *game* the quantum \[23-25\]) rather than worry about whether such games are proper extensions of some underlying classical game. We show that such games can always be thought of as being *equivalent* to a classical game played with classical coins in the sense that the players analyse the game as if they were playing the equivalent classical game with a potentially different set of preferences to those of the initial quantum game. In this way we can see that a given classical game may sometimes be thought of as a decomposition consisting of a different quantum game with different preferences. Our simple example shows that 2 players can play the game of Chicken by playing a version of Quantum Prisoner’s Dilemma \[23\].
By focusing on the notion of *playable* games, that is there is an implementation of the game with physical objects, we describe the general features of any game whether played with classical or quantum objects. The requirement of playability allows us to develop a model in which the elements necessary for proper comparison of quantum vs classical behaviour are made clear. We believe that this approach, grounded in the physics of the game objects and mechanisms, gives a perspective on quantum games that helps to clarify the issue of just what is quantum mechanical in a quantum game.
Turning Prisoners into Chickens
===============================
Let us consider an attempt to implement the game of Prisoner’s Dilemma (PD) using quantum objects and operations. We shall assume the players (Alice and Bob) are each given a spin-1/2 particle upon which to operate. The players are each allowed only two operations on their own particles; flip or don’t-flip, with respect to the spin-$z$ direction. We shall label these operations as *F* and *I*, respectively. There will be, in general, 4 possible output states that the players can produce, characterized by the choices $\left( I,I\right) ,\left( I,F\right) ,\left( F,I\right) $ and $\left(
F,F\right) $ where the first element refers to the choice of Alice and the second to that of Bob. The output state is subject to a measurement as follows; the spin in the $z$-direction of Alice’s particle is measured and the spin in the $z$-direction of Bob’s particle is measured. The possible measurement results are listed as a tuple $\left( 0,0\right) ,\left(
0,1\right) ,\left( 1,0\right) $ or $\left( 1,1\right) $ where the ‘0’ result indicates spin-down. The measurement results are mapped to an outcome tuple for the players so that $$\begin{aligned}
\left( 0,0\right) & \longrightarrow\left( 3,3\right) \nonumber\\
& \nonumber\\
\left( 0,1\right) & \longrightarrow\left( 0,5\right) \nonumber\\
& \nonumber\\
\left( 1,0\right) & \longrightarrow\left( 5,0\right) \nonumber\\
& \nonumber\\
\left( 1,1\right) & \longrightarrow\left( 1,1\right)\end{aligned}$$ The preferences of the players are thus encapsulated by the assignment of a numerical value as an outcome. If the initial state of the spin-1/2 particles is given, in the measurement basis, by $\left\vert 0\right\rangle _{A}\otimes\left\vert 0\right\rangle _{B}$ which we shall write as $\left\vert
00\right\rangle $ then the payoff matrix becomes
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 3,3\right) $ & $\left( 0,5\right) $\\
> & & \\
> $F$ & $\left( 5,0\right) $ & $\left( 1,1\right) $\end{tabular}
> $$ **Table 1**: *the outcome matrix for an implementation of classical PD using quantum objects*
This is nothing more than the standard description of classical Prisoner’s Dilemma \[22\].
Now let us consider playing the game with a *different* input state of the particles, but keeping everything else the same. The players have the same preferences over the measurement results. We choose an input state that is not mapped onto an eigenstate of the measurement operator by the actions of the players. This means that the results of the measurement will be distributed according to some probability distribution. Let us choose the following initial state $$\left\vert \psi_{0}\right\rangle =\sqrt{\frac{3}{5}}\left\vert 00\right\rangle
+\sqrt{\frac{2}{5}}\left\vert 11\right\rangle$$ The four possible output states are given by $$\begin{aligned}
\left\vert \psi\right\rangle _{II} & =\sqrt{\frac{3}{5}}\left\vert
00\right\rangle +\sqrt{\frac{2}{5}}\left\vert 11\right\rangle \nonumber\\
& \nonumber\\
\left\vert \psi\right\rangle _{IF} & =\sqrt{\frac{3}{5}}\left\vert
01\right\rangle +\sqrt{\frac{2}{5}}\left\vert 10\right\rangle \nonumber\\
& \nonumber\\
\left\vert \psi\right\rangle _{FI} & =\sqrt{\frac{3}{5}}\left\vert
10\right\rangle +\sqrt{\frac{2}{5}}\left\vert 01\right\rangle \nonumber\\
& \nonumber\\
\left\vert \psi\right\rangle _{FF} & =\sqrt{\frac{3}{5}}\left\vert
11\right\rangle +\sqrt{\frac{2}{5}}\left\vert 00\right\rangle\end{aligned}$$ Let us suppose the players choose the operation tuple $\left( I,I\right) $. We can see that if they choose these operations then they will obtain the output tuple $\left( 3,3\right) $ with probability 3/5 and the output tuple $\left( 1,1\right) $ with probability 2/5. Thus, for this choice of operations they will obtain an *expected* outcome tuple of $\left(
11/5,11/5\right) $. Doing a similar calculation for the other possible output states we obtain the matrix for the expected outcome tuples as
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 11/5,11/5\right) $ & $\left( 2,3\right) $\\
> & & \\
> $F$ & $\left( 3,2\right) $ & $\left( 9/5,9/5\right) $\end{tabular}
> \ \ $$ **Table 2**: *the outcome matrix for a game played with quantum objects having preferences over the measurement results in accord with Prisoner’s Dilemma, but in which the input quantum state is entangled according to equation (2)*
The players will use this new matrix to determine their choice of strategy. Their *actual* choice of play is thus determined by this matrix of expected outcomes. However, the matrix of expected outcomes is nothing more than a numerical encapsulation of the preferences of the game of Chicken. The game of Chicken \[22\] can be described by the preferences $$\begin{aligned}
\text{Alice } & \text{: }O_{FI}>O_{II}>O_{IF}>O_{FF}\nonumber\\
& \nonumber\\
\text{Bob } & \text{: }O_{IF}>O_{II}>O_{FI}>O_{FF}$$ where $O_{FI}$, for example, describes the outcome tuple when Alice plays $F$ and Bob plays $I$. In the game of Chicken the usual scenario is to imagine two somewhat irresponsible youths hurtling towards one another in their cars. The winner of the game is the one who doesn’t swerve ($O_{FI}$ in which Alice doesn’t swerve, but Bob does, is Alice’s most preferred outcome). If neither swerve $\left( O_{FF}\right) $ then they crash and this is the least preferred outcome for both players. If both swerve then they are both ‘chickens’, which is more preferable than crashing, but not as preferable as winning. If Alice swerves, but Bob doesn’t, then this is preferable to crashing, but Alice has lost face (she is the chicken and Bob isn’t) and so is not as preferable to her as when they both swerve.
Thus for our quantum game with this entangled state input, despite initially having preferences over the measurement results in accordance with those of Prisoner’s Dilemma, these are transformed into preferences over the expected outcomes that are in accordance with the game of Chicken. The game that the players *actually* play is the classical game of Chicken, despite setting up the game as a quantum version of Prisoner’s Dilemma. The players analyze their choices in terms of this matrix of expected outcomes. At this point it is irrelevant to the players, as far as their game objectives are concerned, whether this matrix has been generated by some complicated quantum process or whether it has been generated by some game in which classical coins are mapped to the outcomes of Chicken. This feature of game transformation has been examined in a general case by consideration of measurement of the Schmidt observables in entangled quantum games \[10,11\].
This simple example, based on the quantum game scenario of Marinatto and Weber (MW) \[21\], illustrates that care must be taken in ascribing quantum behaviour to a game scenario in which the objects used to implement a game are quantum mechanical in nature. Whilst the game scenario of MW raises legitimate concerns about whether it constitutes a proper extension of an underlying classical game \[4\] it is, nevertheless, a perfectly acceptable example of a game played with quantum objects. If we imagine the players are given some black box with dials for their strategy choices, then if they know the payoff function for the possible choices, they will base their final choice upon the analysis of this payoff function. Their transformed preferences over these expected payoffs *define the actual game they are playing*, despite the possibility that their initial preferences over the measurement results were those of another game form. In the example we have discussed we can see that, as far as the players are concerned, the same game can be implemented with either quantum objects and operations or with classical coins. In other words the players cannot tell whether the objects inside their black box are classical or quantum mechanical.
Playable Games and Quantum/Classical Comparisons
================================================
Any game that is actually *playable* must have an implementation in the physical world. The strategy choices represent some manipulation of physical entities be those entities classical coins or quantum particles. For any playable game, quantum or classical, the following elements must be present
- some physical objects prepared in an initial state
- a set of manipulations that can be performed on these objects. The manipulations that are possible are the available strategy choices of the players
- some measurement of the state of these objects after the manipulations of the players have been carried out
- a mapping of the measurement results to some outcomes over which the players have different preferences
In a simple classical game, such as normal form Prisoner’s Dilemma, the initial state and measurement elements are implicit since there is a one-to-one correspondence between the elements, and the strategy choices can be directly related to the measurement results. If we wish to play a game using quantum objects and operations these elements must be made explicit and there is no longer a direct one-to-one correspondence between the strategy choices and the measurement results, in general.
This general description of any playable game, quantum or classical, allows us to make correct comparisons between different game versions. For example, in our simple example of the previous section we assume an entangled input state of the particles. The players can perform local operations on their respective particles, which in this example is restricted to just flip or no-flip. However, the operations performed by the players affect the *entire* quantum state which must be considered to be a single entity. Thus the correct quantum classical comparison to draw is between the quantum game and classical games in which the players have some ability to affect each other’s coins in some way. As we shall see, this can be achieved by the simple expedient of assuming a *classical correlated noise* on the communication of the strategy choices of the players to the device that measures this communication and assigns the payoffs.
Similarly, in comparing classical and quantum games we must, as a minimum condition, give the players strategy sets of the same size. It makes little sense, for example, to compare a classical game in which the players have only 2 strategy choices each with a quantum game in which the players have 4 available strategies each. At best we could describe this as a possible quantum extension of the classical game, but then we must compare the quantum extension with the relevant classical extension of the game in order to draw a direct comparison \[4\]. Thus, if we are to consider entangled quantum games we must compare these to classical games in which correlation features in some way, otherwise the comparison is essentially meaningless. In other words we must compare the relevant extensions of the game in the classical and quantum domains.
Let us consider 2-player games in which the players, Alice and Bob, have the respective available operations $\left\{ \alpha_{1},\alpha_{2},\ldots
,\alpha_{p}\right\} $ and $\left\{ \beta_{1},\beta_{2},\ldots,\beta
_{q}\right\} $. We shall assume that the physical objects to be manipulated are prepared in some initial state $\psi_{in}$. The output is therefore a state of the form $\psi_{out}=\beta_{k}\alpha_{j}\psi_{in}$ where this description is applicable to both quantum and classical game scenarios. This operational perspective also highlights the possibility that certain games can be non-commutative so that the order of play matters \[23\]. There is then some measurement on the output state which yields a set of possible measurement results. These measurement results are the input to the payoff function.
It is customary in entangled quantum games to consider that the players perform independent local operations on their respective particles and to identify a strategy choice with a quantum spin state (or the associated unitary operator that generates this from some specified initial state). This, however, is something of an illusion. If we consider a game of the form considered by EWL \[1\] then the output state is given by$$\psi_{out}=E^{-1}\beta_{k}\alpha_{j}E\psi_{in}=E^{-1}\beta_{k}EE^{-1}\alpha_{j}E\psi_{in}$$ Here we note that $E$ acts on the entire input state and should not be confused with the local operations that are usually assumed in treatments of quantum games. When $E$ is the entanglement operator that produces a maximally entangled state from the ‘ground’ state of two spin-1/2 particles then the strategy sets of the players are equivalent to the sets$$\left\{ \tilde{\alpha}_{1},\tilde{\alpha}_{2},\ldots,\tilde{\alpha}_{p}\right\} ~~\text{and \ }\left\{ \tilde{\beta}_{1},\tilde{\beta}_{2},\ldots,\tilde{\beta}_{q}\right\}$$ where$$\tilde{\alpha}_{j}=E^{-1}\alpha_{j}E~~\text{and \ }\tilde{\beta}_{k}=E^{-1}\beta_{k}E$$ and so the possible manipulations of the players involve directly interacting with the spin of their opponent, even if the sets $\left\{ \alpha_{1},\alpha_{2},\ldots,\alpha_{p}\right\} $ and $\left\{ \beta_{1},\beta
_{2},\ldots,\beta_{q}\right\} $ represent strictly local operations. The identification of a so-called quantum strategy with a spin state of a single particle is, therefore, nothing more than a convenient illusion for entangled quantum games; a game involving entangled states is formally equivalent to a game in which we allow the players entanglement operations as part of their strategy sets.
If we are to make a sensible comparison between quantum and classical games in an attempt to elucidate genuine quantum behaviour we must compare like with like. For games of the EWL type where $E$ is an entanglement operator, therefore, we must compare with the extension of the classical game to include correlation. It is critical, therefore, that the role of correlation is understood in both classical and quantum games. In the following simple game examples we demonstrate that features that may be initially considered to arise from the quantum-mechanical nature of a correlation actually arise from only the classical component of the correlation in an entangled state.
Preservation of Preferences
===========================
The simple example discussed above shows that our original preferences (over the individual measurement results) in a quantum game may be transformed by the measurement process into different preferences over the expected outcomes. It is the probabilistic mapping induced, in general, by the quantum measurement that forces the final analysis of the game in terms of a matrix of expected outcomes, and these expected outcomes can generate tuples that do not correspond to the original ordering of the measurement tuples as expressed by the players’ preferences over the individual measurement results. In our simple example it is the change of input state, whilst keeping all other elements of the physical game unchanged, that ultimately leads to this possibility of the transformation of the preferences. A general input state can be expanded in the basis of the measurement so that we have $\left\vert
\psi_{0}\right\rangle =a\left\vert 00\right\rangle +b\left\vert
01\right\rangle +c\left\vert 10\right\rangle +d\left\vert 11\right\rangle $. It is natural to ask the question as to what are the conditions on the input state, whilst keeping all other elements unchanged, that will strictly preserve the original preferences? Before attempting to give a general perspective on this we consider some special cases.
$\left\vert \psi_{0}\right\rangle =a\left\vert 00\right\rangle
+d\left\vert 11\right\rangle $
--------------------------------------------------------------
Let us generalize the input state given in equation (2) and ask when this preserves the original preferences. We shall not be fully general in this approach because we maintain the specific numerical weightings of PD to express the original preferences, but nevertheless it provides us with an insight into the way the original game can be transformed with different inputs. In fact the transformation (or otherwise) of the preferences depends upon the specific values chosen to represent those preferences. It is possible to choose numerical weightings that respect the preferences such that those preferences are also reflected in the matrix of expected outcomes for any input state, including entangled states. We shall choose, however, the usual initial preferences of PD expressed by the weightings $\left( 5,3,1,0\right)
$ in order to allow some form of comparison.
With this initial entangled input state and the standard PD weightings and noting that $\left\vert a\right\vert ^{2}+\left\vert d\right\vert ^{2}=1$ we find the matrix of expected outcomes
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 1+2\left\vert a\right\vert ^{2},1+2\left\vert a\right\vert
> ^{2}\right) $ & $\left( 5-5\left\vert a\right\vert ^{2},5\left\vert
> a\right\vert ^{2}\right) $\\
> & & \\
> $F$ & $\left( 5\left\vert a\right\vert ^{2},5-5\left\vert a\right\vert
> ^{2}\right) $ & $\left( 3-2\left\vert a\right\vert ^{2},3-2\left\vert
> a\right\vert ^{2}\right) $\end{tabular}
> \ $$ **Table 3**: *the outcome matrix for the simple 2-player game of section 2 with the input state* $\left\vert \psi_{0}\right\rangle =a\left\vert
> 00\right\rangle +d\left\vert 11\right\rangle $
If we write a general outcome matrix in the following way
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $O_{1}$ & $O_{2}$\\
> & & \\
> $F$ & $O_{3}$ & $O_{4}$\end{tabular}
> \ $$ **Table 4**: *general form of the outcome matrix of the pure strategy 2-player game in which players have 2 choices of action each*.
then we can see that the game of Prisoner’s Dilemma occurs when we have the preferences $$\begin{aligned}
P_{A}~~~ & :~~~O_{3}>O_{1}>O_{4}>O_{2}\nonumber\\
& \nonumber\\
P_{B}~~~ & :~~~O_{2}>O_{1}>O_{4}>O_{3}$$ In order to strictly preserve these preferences for our given input state we therefore require that $$5\left\vert a\right\vert ^{2}>1+2\left\vert a\right\vert ^{2}>3-2\left\vert
a\right\vert ^{2}>5-5\left\vert a\right\vert ^{2}$$ which give the conditions under which the preferences of both players are preserved for this input state. We have plotted the expected outcomes for Alice as a function of $p=\left\vert a\right\vert ^{2}$ in Figure \[PCGameFigure1\] below. We can see that there are 3 regions with each region giving a different preference ordering for the outcomes.
The different regions can be determined by consideration of the inequalities in equation (3) and we obtain the regions (where we exclude the boundary points) $$\begin{aligned}
\text{Region I \ \ } & \text{: \ \ }\frac{1}{2}<\left\vert a\right\vert
^{2}<\frac{4}{7}\nonumber\\
& \nonumber\\
\text{Region II \ \ } & \text{: \ \ }\frac{4}{7}<\left\vert a\right\vert
^{2}<\frac{2}{3}\nonumber\\
& \nonumber\\
\text{Region III \ \ } & \text{: \ \ }\frac{2}{3}<\left\vert a\right\vert
^{2}<1\end{aligned}$$ The preferences for the players for these regions are given in the table below
> $$\begin{tabular}
> [c]{c|cccc}
> & & Alice & & Bob\\\hline
> & & & & \\
> $\text{Region I }$ & & $O_{3}>O_{2}>O_{1}>O_{4}$ & & $O_{2}>O_{3}>O_{1}>O_{4}$\\
> & & & & \\
> $\text{Region II}$ & & $O_{3}>O_{1}>O_{2}>O_{4}$ & & $O_{2}>O_{1}>O_{3}>O_{4}$\\
> & & & & \\
> $\text{Region III }$ & & $O_{3}>O_{1}>O_{4}>O_{2}$ & & $O_{2}>O_{1}>O_{4}>O_{3}$\end{tabular}
> \ $$ **Table 5**: *the preferences of the players over the expected outcomes expressed as preference relations for the different entanglement regions where the input state is given by* $\left\vert \psi_{0}\right\rangle
> =a\left\vert 00\right\rangle +d\left\vert 11\right\rangle $
In Region III we can see that the players play Prisoner’s Dilemma, but in Region II they play the game of Chicken.
We can see that for $\left\vert a\right\vert ^{2}>\frac{2}{3}$ (and by symmetry for $\left\vert a\right\vert ^{2}<\frac{1}{3}$ with a switch in the interpretation of cooperate and defect) the players just play Prisoner’s Dilemma, which might suggest that for these values of the parameter $\left\vert a\right\vert ^{2}$ the input state is not ‘quantum’ enough to change the game. We must, however, be careful in making such a claim. Is this changing of the game by inducing a probability distribution over the measurements really a non-classical effect? Let us look at the singlet-type state as input next.
$\left\vert \psi_{0}\right\rangle =b\left\vert 01\right\rangle
+c\left\vert 01\right\rangle $
--------------------------------------------------------------
Once again we maintain the numerical weightings for the original PD game and note that $\left\vert b\right\vert ^{2}+\left\vert c\right\vert ^{2}=1$. With this initial state and the available actions of the players the expected payoff matrix is given by
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 5-5\left\vert b\right\vert ^{2},5\left\vert b\right\vert
> ^{2}\right) $ & $\left( 1+2\left\vert b\right\vert ^{2},1+2\left\vert
> b\right\vert ^{2}\right) $\\
> & & \\
> $F$ & $\left( 3-2\left\vert b\right\vert ^{2},3-2\left\vert b\right\vert
> ^{2}\right) $ & $\left( 5\left\vert b\right\vert ^{2},5-5\left\vert
> b\right\vert ^{2}\right) $\end{tabular}
> \ $$ **Table 6**: *the outcome matrix for the simple 2-player game of section 2 with the input state* $\left\vert \psi_{0}\right\rangle =b\left\vert
> 01\right\rangle +c\left\vert 01\right\rangle $
The expected outcomes for Alice are plotted in Figure \[PCGameFigure2\] below as a function of $p=\left\vert b\right\vert ^{2}$
Of course, these are the same lines as before for the input of section 4.1 but the expected outcomes they represent are different entries in the expected payoff matrix. We note from Figure \[PCGameFigure2\] (and analysis of the conditions for strict preservation of the preferences) that this input state *always* changes the preference relations from the original with none of the new preferences over the expected payoff matrix being equivalent to a game of Prisoner’s Dilemma. The regions where the preferences over the expected outcomes change from one ordering to another are just as before and the preferences for the players for these regions are given in the table below
> $$\begin{tabular}
> [c]{c|cccc}
> & & Alice & & Bob\\\hline
> & & & & \\
> $\text{Region I }$ & & $O_{4}>O_{1}>O_{2}>O_{3}$ & & $O_{1}>O_{4}>O_{2}>O_{3}$\\
> & & & & \\
> $\text{Region II}$ & & $O_{4}>O_{2}>O_{1}>O_{3}$ & & $O_{1}>O_{2}>O_{4}>O_{3}$\\
> & & & & \\
> $\text{Region III }$ & & $O_{4}>O_{2}>O_{3}>O_{1}$ & & $O_{1}>O_{2}>O_{3}>O_{4}$\end{tabular}
> \ $$ **Table 7**: *the preferences of the players over the expected outcomes expressed as preference relations for the different entanglement regions where the input state is given by* $\left\vert \psi_{0}\right\rangle
> =b\left\vert 01\right\rangle +c\left\vert 01\right\rangle $
It would not be surprising in quantum PD with this input state, or indeed with the input state of the previous section with $\frac{1}{2}<\left\vert
a\right\vert ^{2}<\frac{2}{3}$, that the equilibrium payoff might be different to that of standard PD as we are no longer *actually playing* Prisoner’s Dilemma! The entangled states are often taken to be the most ‘non-classical’ states possible. Accordingly, it is always tempting to ascribe any unusual result when working with these states to a ‘quantum’ behaviour. However, as the analysis of Bell’s Theorem shows, pinning down non-classicality is often surprisingly subtle. In Bell’s Theorem for 2 spin-1/2 particles we need to examine correlations between sets of measurements in different spin directions in order to reveal behaviour that can be directly attributed to quantum mechanics in the sense that a ‘classical’ local hidden variable description cannot predict the correct correlations. Determining what is ‘quantum’ in a quantum game is, in our opinion, not a trivial issue.
A Classical Model
-----------------
The feature that the quantum measurement introduces is that the measurement maps the output state of the players onto an eigenstate of the measurement with a probability distribution determined by the amplitudes of the eigenstates in the expansion of the output state in the measurement basis. We can view this as a noise process. The players are trying to communicate a particular choice, but noise on the channel gives rise to an error rate. If the players are aware of the noise and its characteristics then they can build this knowledge into their strategy. This is exactly what we have in the quantum situation. So let’s model the transmission of the players’ choices in a classical game as a communication over a noisy channel in which the players are aware of the noise characteristics and can tailor their choices accordingly. As in the quantum case we will have to deal with a matrix of expected payoffs which could lead to the playing of a different game by transformation of the preferences. Can we achieve this transformation of preferences with such a classical game over a noisy channel?
The simplest case of noise we could consider would be to model the communication as two independent channels with the same error rate $\varepsilon$. A simple calculation shows that such a case preserves the preference relations (or flips them when $\varepsilon>\frac{1}{2}$ but the original PD is recovered with an interchange of the choices cooperate and defect). It is of course formally equivalent to the extension to a mixed game in which the players choose the same probability. A more interesting case occurs when we consider a *correlated* noise such that both channels, or neither, experience an error for a given symbol with a rate $\varepsilon$. Such a correlated noise is, of course, a form of classical noise. We could imagine the players’ signals sent over the same channel and experiencing the same noise, for example. With this kind of noise, if the players send a pair of symbols then either both are correct with probability $1-\varepsilon$ or both are flipped with probability $\varepsilon$.
The expected outcomes for Alice and Bob when they communicate their choices over such a channel are
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 3-2\varepsilon,3-2\varepsilon\right) $ & $\left(
> 5\varepsilon,5-5\varepsilon\right) $\\
> & & \\
> $F$ & $\left( 5-5\varepsilon,5\varepsilon\right) $ & $\left( 1+2\varepsilon
> ,1+2\varepsilon\right) $\end{tabular}
> \ $$ **Table 8**: *the matrix of expected outcomes for classical PD in which the players’ strategy choices are communicated over a channel with correlated noise such that both bits, or neither, are flipped.which is just the same as the expected outcome matrix for the input state* $\left\vert
> \psi_{0}\right\rangle =a\left\vert 00\right\rangle +d\left\vert
> 11\right\rangle $ *considered in section 4.1 for the quantum PD where* $\epsilon=1-\left\vert a\right\vert ^{2}$.
So we can see that a game of PD played over channels with this kind of *correlated* noise will also change the preferences of the players and in Region II the players will be playing Chicken rather than PD. Thus, there is nothing particularly quantum mechanical in nature about the transformation of preferences we obtain for the entangled quantum games considered above.
Mixing and Correlated Noise
---------------------------
Whilst we have not considered the mixed game at all so far, it is instructive to examine the effect of having a classical correlated noise when we extend a (classical) game by mixing. As before, we begin with classical PD but now assume the players will adopt a probabilistic strategy so that Alice chooses to flip (defect) with probability $p$ and Bob chooses to flip with probability $q$. As in the previous section they attempt to communicate their choice over a channel that experiences a correlated noise so that either both bits representing the players’ choice are transmitted error free, or both are flipped. If we assume the error rate is $\varepsilon$ as before then the joint probabilities for obtaining the measured results $I$ and $F$ are as follows: $$\begin{aligned}
P\left( I,I\right) & =\left( 1-p\right) \left( 1-q\right) \left(
1-\varepsilon\right) +pq\varepsilon\nonumber\\
& \nonumber\\
P\left( I,F\right) & =q\left( 1-p\right) \left( 1-\varepsilon\right)
+p\left( 1-q\right) \varepsilon\nonumber\\
& \nonumber\\
P\left( F,I\right) & =p\left( 1-q\right) \left( 1-\varepsilon\right)
+q\left( 1-p\right) \varepsilon\nonumber\\
& \nonumber\\
P\left( F,F\right) & =pq\left( 1-\varepsilon\right) +\left( 1-p\right)
\left( 1-q\right) \varepsilon\end{aligned}$$ The expected outcomes for Alice and Bob now become $$\begin{aligned}
\left\langle O_{A}\left( \varepsilon\right) \right\rangle & =\left[
3+2p-3q-pq\right] \left( 1-\varepsilon\right) +\left[ 1-p+4q-pq\right]
\varepsilon\nonumber\\
& \nonumber\\
\left\langle O_{B}\left( \varepsilon\right) \right\rangle & =\left[
3+2q-3p-pq\right] \left( 1-\varepsilon\right) +\left[ 1-q+4p-pq\right]
\varepsilon\end{aligned}$$ The $\left( 1-\varepsilon\right) $ part of this expected outcome is just the usual expected outcome from the mixed PD without any noise. In this noiseless case the players are forced to the equilibrium position $\left( F,F\right) $ just as the non-mixed game and the expected outcome is $\left( 1,1\right) $. However, the noise term now changes this expected outcome. If $\varepsilon=1$ then the players would play the equilibrium position $\left( I,I\right) $ with an expected outcome of $\left( 1,1\right) $. The actual choice of probability the players make is a function of the error rate $\varepsilon$ and we can see that their best response is given by the choice $p=q=1-\varepsilon
$. This yields the expected outcomes for the players $$\left\langle O_{A}\left( \varepsilon\right) \right\rangle =\left\langle
O_{B}\left( \varepsilon\right) \right\rangle =1+5\varepsilon-5\varepsilon
^{2}$$
This expected outcome is plotted above in Figure \[PCGameFigure3\] and we can see that the maximum value is obtained when $\varepsilon=\frac{1}{2}$ and this gives an expected outcome for the players of $\frac{9}{4}$ which is an improvement on their equilibrium output in the noise-free case (or the all-noise case). This is, of course, nothing more than a uniform distribution of the possible outcomes and the same result is obtained for quantum games in which there is maximal decoherence \[19,20\]. In this case, it is the correlated, but classical, noise that is giving an enhanced equilibrium payoff for the players in all regions except the boundary points (noise-free or all-noise).
Noise in the Quantum Game
-------------------------
Now let us consider the case where the players play their version of Prisoner’s Dilemma with the input state $\left\vert \psi_{0}\right\rangle
=a\left\vert 00\right\rangle +d\left\vert 11\right\rangle $. We shall consider that they attempt to communicate their choice by sending their respective particles over some channel to be measured. Thus we now have a quantum channel. We shall suppose that there is some noise source on this channel. In this case we shall not be too concerned with the details of the noise, but merely suppose that it is sufficient to rapidly suppress the off diagonal coherences in the density matrix. Such a suppression of off-diagonal coherences is, of course, a general feature of open quantum systems \[26-28\]. If we restrict the available operations, as before, to this binary choice of whether to flip or not in the measurement basis, then after this decohering noise process the density matrix description of the state that arrives at the measurement apparatus is given, for each of the possible choices of the players, as $$\begin{aligned}
\rho_{II} & =\left\vert a\right\vert ^{2}\left\vert 00\right\rangle
\left\langle 00\right\vert +\left\vert d\right\vert ^{2}\left\vert
11\right\rangle \left\langle 11\right\vert \nonumber\\
& \nonumber\\
\rho_{IF} & =\left\vert a\right\vert ^{2}\left\vert 01\right\rangle
\left\langle 01\right\vert +\left\vert d\right\vert ^{2}\left\vert
10\right\rangle \left\langle 10\right\vert \nonumber\\
& \nonumber\\
\rho_{II} & =\left\vert a\right\vert ^{2}\left\vert 10\right\rangle
\left\langle 10\right\vert +\left\vert d\right\vert ^{2}\left\vert
01\right\rangle \left\langle 01\right\vert \nonumber\\
& \nonumber\\
\rho_{II} & =\left\vert a\right\vert ^{2}\left\vert 11\right\rangle
\left\langle 11\right\vert +\left\vert d\right\vert ^{2}\left\vert
00\right\rangle \left\langle 00\right\vert\end{aligned}$$ The expected outcomes for the players are given by the expected outcome matrix
> $$\begin{tabular}
> [c]{c|cc}\textit{A}$\backslash$\textit{B} & $I$ & $F$\\\hline
> & & \\
> $I$ & $\left( 1+2\left\vert a\right\vert ^{2},1+2\left\vert a\right\vert
> ^{2}\right) $ & $\left( 5-5\left\vert a\right\vert ^{2},5\left\vert
> a\right\vert ^{2}\right) $\\
> & & \\
> $F$ & $\left( 5\left\vert a\right\vert ^{2},5-5\left\vert a\right\vert
> ^{2}\right) $ & $\left( 3-2\left\vert a\right\vert ^{2},3-2\left\vert
> a\right\vert ^{2}\right) $\end{tabular}
> \ \ \ $$ **Table 9**: *the matrix of expected outcomes for the 2-player game with input state* $\left\vert \psi_{0}\right\rangle =a\left\vert
> 00\right\rangle +d\left\vert 11\right\rangle $ *in which the players particles are sent over a noisy quantum channel that leads to suppression of the off-diagonal coherences.*
Which is precisely the same as that for the game played in the noiseless case considered in section (4.1). In other words, the off-diagonal components in the density matrix in the noiseless case are not contributing to the determination of the expected outcomes. This is only to be expected since any single pair of measurements on the separate systems can, at most, only access half the information contained within the quantum correlation \[29,30\]. If we expressed our entangled state in the Schmidt basis, and made measurements of the Schmidt observables (this is considered within the context of quantum games in \[10,11\]), then we would access precisely half the information contained within the quantum correlation \[29,30\]. In other words, the enhanced equilibrium obtained for quantum games in which a single measurement is made in each the subspaces is a result of a *classical* correlation because the off-diagonal interference terms are not being accessed in such a measurement.
We note that a similar problem has been studied in more generality by Shimamura et. al. \[18\] in which they consider the difference between an entangled state input and its classical counterpart in games of the EWL type (see also Chen et. al. \[19\] who consider a decoherence protocol for Quantum Prisoner’s Dilemma). Both \[18\] and \[19\] are different to the situation we envisage here in which the decoherence occurs during the transmission of the quantum states to the measurement device (or referee). In \[18\] the referee employs the disentangling transformation before measurement and the full space of local unitary operations is allowed by each player. The formal correspondence discussed in section 3 above is no longer applicable because the decoherence destroys the symmetry and the placement of the decohering process in the chain of events becomes significant. Furthermore in both \[18\] and \[19\] the referee performs a disentangling operation before measurement, which amounts to a re-entanglement in the case of a separable mixed state input.
Our purpose here is not to examine the effect of decoherence in general (see for example, \[20\]) but to provide another illustration that, for simple 2-player games of the form considere here, it is correlation, and not quantum correlation, that is the interesting feature. Of course a more general treatment is required to determine the game types and conditions under which quantum correlations do become significant. We shall examine the ramifications of this, and the results of the previous sections, for the interpretation of games such as EWL \[1\] and MW \[21\] elsewhere.
A General Approach
==================
In the preceding sections we examined some very simple, and restricted, quantum games in order to gain some insight into the role of correlation in quantum games. An obvious question is whether the results obtained depend in some way on the nature of the restriction imposed (the choices cooperate or defect being the only operations available to the players).
Consider a game played with quantum mechanical objects and operations that obey the laws of quantum mechanics. The game consists of the following \[23-25\]:
- An input state $\left\vert \psi_{0}\right\rangle $ that is assumed to be known by the players
- Actions available to player *A* described by a finite set of unitary operators $\left\{ \hat{\alpha}_{1},\ldots,\hat{\alpha}_{i},\ldots,\hat{\alpha}_{n}\right\} $
- Actions available to player *B* described by a finite set of unitary operators $\left\{ \hat{\beta}_{1},\ldots,\hat{\beta}_{j},\ldots
,\hat{\beta}_{m}\right\} $
- The actions of the players on the input state produce some output state $\left\vert \psi_{ij}\right\rangle $ that is characterized by the choice of $\hat{\alpha}_{i}$ and $\hat{\beta}_{j}$ by players *A* and *B*, respectively. There are $n\times m$ possible output states from this game for a given input state $\left\vert \psi_{0}\right\rangle $
- A projective measurement $\hat{M}$ on the output state that produces an eigenstate $\left\vert m_{i}\right\rangle $ of the measurement operator where there are $r$ such eigenstates. We assume non-degenerate eigenvalues so that each measurement result can be unambiguously identified with a measurement eigenstate.
- The players each have a different preference relation over the measurement eigenstates. Accordingly, we shall use the terms *preference basis* and *measurement basis* interchangeably. The preference relations therefore induce a preference relation for each player over the set of possible output states $\left\vert \psi_{ij}\right\rangle $
- We shall encapsulate the notion of preference by assigning a numerical value to each measurement eigenstate for each player such that a higher numerical value indicates a greater preference for that player.
- We shall assume that (nominally) each player has some object upon which to act so that the Hilbert space is described by $H=H_{A}\otimes H_{B}$. Note that this does *not*, therefore, imply that the unitary operations available to the players act *only* in their respective subspaces
The players, as noted above, have some preference over the measurement eigenstates so that the output state produced can be expressed in this measurement, or preference, basis as follows$$\left\vert \psi_{ij}\right\rangle =\sum_{i=1}^{r}\left\langle m_{i}\mid
\psi_{ij}\right\rangle ~\left\vert m_{i}\right\rangle$$
Upon measurement, the result $\left\vert m_{l}\right\rangle $ is mapped to a numerical value in accordance with the preference relations as $$\left( \left\vert m_{l}\right\rangle \right) \longrightarrow\left(
\omega_{l}^{A},\omega_{l}^{B}\right)$$ where $\omega_{l}^{A}$ is the outcome for player *A* if the result of the measurement yields the eigenvalue $\left\vert m_{l}\right\rangle $. We can formally combine the measurement and assignment of outcomes into the single Hermitian ‘outcome’ operators $$\begin{aligned}
\hat{\omega}_{A} & ={\displaystyle\sum\limits_{i=1}^{r}}
\omega_{i}^{A}\left\vert m_{i}\right\rangle \left\langle m_{i}\right\vert
\nonumber\\
& \nonumber\\
\hat{\omega}_{B} & ={\displaystyle\sum\limits_{i=1}^{r}}
\omega_{k_{B}}^{B}\left\vert m_{i}\right\rangle \left\langle m_{i}\right\vert\end{aligned}$$ In general, the output state will be a superposition over the preference bases and will not be an eigenstate of the measurement operator (or equivalently the outcome operators). There will therefore be a distribution over the output tuples for any given choice of $\hat{\alpha}_{j}$ and $\hat{\beta}_{k}$. Thus for each choice of $\hat{\alpha}_{j}$ and $\hat{\beta}_{k}$ there will be an *average* outcome tuple expressed as the expected value of the outcome operators $$\left( \left\langle \hat{\omega}_{A}\right\rangle _{jk},\left\langle
\hat{\omega}_{B}\right\rangle _{jk}\right) =\left(
{\displaystyle\sum\limits_{i=1}^{r}}
\omega_{i}^{A}\left\vert \left\langle m_{i}\mid\psi_{jk}\right\rangle
\right\vert ^{2}~,~{\displaystyle\sum\limits_{i=1}^{r}}
\omega_{i}^{B}\left\vert \left\langle m_{i}\mid\psi_{jk}\right\rangle
\right\vert ^{2}\right)$$ Each expected outcome tuple can therefore be thought of as an entry in an $n\times m$ matrix of outcome tuples, just as we would describe any 2-player game. Thus the quantum mechanical game is entirely equivalent, as far as the players are concerned, to a classical game in which the outcomes relating to the choices of the players are described by this matrix. The fact that these outcomes are derived from a quantum measurement and the resultant probabilities is utterly irrelevant. The players play the game *according to the outcomes expressed in this expected payoff matrix*. The game is defined not by their original preferences over the measurement results, but by the induced preferences as expressed by the matrix of expected outcomes. In effect, the quantum mechanical measurement has the potential to change the players’ preferences to those expressed in the expected payoff matrix. So although the players start off with a set of preferences over the results of the measurement they act *as if* they had a new set of preferences, given by the expected outcomes. The players choose their strategies according to this new matrix. *It is this matrix which defines the actual game they are playing*.
The expectation values defined in equation (18) are those considered by Cheon and Tsutsui \[17\], in which they show that each $\left\langle \hat{\omega}_{A}\right\rangle _{jk}$ and $\left\langle \hat{\omega}_{B}\right\rangle
_{jk}$ can be considered to arise from a pseudo-classical part and quantum interference terms. So each element in our matrix of expected payoffs can be thought to arise partly from some quantum interference term in which a different quantum interference term is obtained for each choice of operation (strategy) by Alice and Bob. So whilst each separate entry into the payoff matrix may be thought of in this manner, the entire matrix is just a set of classical probabilities that can, as we argue below, be reproduced by modelling the game classically as a communication channel in which we allow the possibility of classical correlated noise. Once again, we emphasize that it is the matrix of expected payoffs that *defines* the game the players *actually* play, and not whatever complicated physical mechanism we have used to produce this matrix.
It is at this point we must ask what is quantum mechanical about games of this type? It is irrelevant to the players whether the expected payoff matrix that defines the game they are playing has been generated by some quantum process, or whether the entries in the matrix are assigned to measurement of classical coins, just as in any standard classical 2-player game. There is nothing particularly quantum-mechanical about a matrix of tuples and any quantum game of this form can be implemented entirely by classical objects with a given functional mapping of measurement to outcomes. The specific functional decomposition of the game that has generated the final game function is irrelevant to the players; the quantity they analyze is the matrix of expected outcomes. We believe that van Enk and Pike \[3\] were right to be uncomfortable about the ‘quantum’ claim for games of this type.
A matrix of expected outcome tuples is generated in any pure strategy game, quantum or classical, where there is some probability distribution over the measurement results. In the quantum case the measurement induces this distribution, but we can similarly imagine a classical game in which the measurement process is imperfect. The example of the implementation of a classical game as a communication over a noisy channel considered above is just one way of realizing such a distribution of measurement results in the classical case. For a classical game with a distribution over $r$ possible measurement results, $m_{i}$, we have the conditional distribution $P\left(
m_{i}\mid\alpha_{j},\beta_{k}\right) $ which gives the probability of obtaining the measurement result $m_{i}$ given that the players choose the operations $\alpha_{j}$ and $\beta_{k}$. In this case the expected outcomes for Alice are$$\left\langle \omega_{A}\right\rangle _{jk}=\sum_{i=1}^{r}P\left( m_{i}\mid\alpha_{j},\beta_{k}\right) \omega_{i}^{A}$$ with a similar expression for Bob’s outcomes. By modelling the classical game as a communication of a binary number representing the strategy choice of the players we can see that a classical noise process on the channel such that the channel transition probabilities are$$P\left( m_{i}\mid\alpha_{j},\beta_{k}\right) =\left\vert \left\langle
m_{i}\mid\psi_{jk}\right\rangle \right\vert ^{2}$$ will reproduce the results of the quantum game. Thus by assuming a channel with a classical correlated noise we can reproduce the results of a quantum entangled game where we assume a single projective measurement is performed on the resultant output state. The game, classical or quantum, can be thought of as a communication channel where the input symbols are the $n\times m$ strategy choices $\alpha_{j}\beta_{k}$ and the $r$ output symbols are the measurement results $m_{i}$. We are free to model the noise on such a classical channel with any legitimate set of conditional probabilities and these probabilities represent a classical noise (although we may need some peculiar classical noise process to generate the results of a particular quantum game, it is still classical).
The modelling of a pure strategy game as a communication channel in which the input symbol is chosen according to preferences over the output symbols is instructive. For convenience we shall assume the players have strategy sets of equal size where $n=m=2^{\mu}$ so that a classical game can be represented as a channel over which the players each communicate an $\mu$-bit binary string. We can implement this classical game using spin-1/2 particles prepared in the state $\left\vert 00\ldots0\right\rangle $ in some spin basis where the players can perform a flip or a no-flip operation in this basis and the measurement is performed on each particle in this basis. This can be considered to be an expensive quantum implementation of the classical game. The expensive quantum implementation can also be thought of as a quantum communication channel over which qubits are transmitted.
The expensive quantum implementation can now be altered in 3 obvious ways:
- the initial state is prepared as $\left\vert \bar{0}\bar{0}\ldots\bar
{0}\right\rangle $ in some other basis
- the initial state is $\left\vert 00\ldots0\right\rangle $ but the players are given flip and no-flip in some basis aligned at some angle to the basis of the input states
- the initial state is $\left\vert 00\ldots0\right\rangle $ and the players are given flip and no-flip in this basis, but the measurement of the qubits is now performed in some other basis aligned at some angle to the input basis
Of course we can imagine any combination of these things, or consider different bases for each qubit, for example. The point here is that we have changed the quantum implementation so that the output state produced by the players is no longer an eigenstate of the measurement operator and this induces a distribution over the measurement results, which in turn induces preferences over the expected outcomes that can lead to a different game form than that described by the preferences over the individual measurement results. In effect, the measurement induces *noise* on the quantum communication channel.
This changing of the quantum implementation of the game can be thought of as a kind of game extension in which the original game pertains when we adjust the alignments to yield the zero noise case. In order to draw a sensible quantum/classical comparison, therefore, one must compare the noisy quantum channel with a noisy classical channel in the context of the application to the description of a game. It is clear that in the unentangled case the expensive quantum implementation can be modelled as a classical game where $2\mu$ bits replace the $2\mu$ qubits such that the noise characteristics of the classical channel reproduce the measurement-induced noise characteristics of the quantum channel.
If we now extend the quantum implementation to allow entanglement, the above arguments show that, for the situation where a projective measurement is made on the output state in the quantum case, the game can be modelled as a classical communication channel in which we allow the possibility of correlated noise. In both cases the players transmit $2\mu$ bits or qubits over the channel. The single projective measurement is not sensitive enough to distinguish between classical and quantum correlations in these 2-player pure strategy finite games. This is essentially for the same reason that we require more than just a single joint probability distribution to distinguish between hidden variable models and quantum mechanics in tests of Bell’s inequality. In tests of local realism we need to establish the non-existence of the joint distribution $P\left( A,B,C\right) $ that correctly reproduces the marginal distributions $P\left( A,B\right) $ and $P\left( A,C\right) $, for example. Such a distribution only exists if the marginals satisfy the Bell inequality. In a sense, the classical communication over a noisy channel can be thought of as a hidden variable implementation of the quantum game and from this perspective it is not surprising that a 2-player quantum game of the form we have considered does not display non-classical behaviour.
Other authors have considered more general formulations of quantum games. Of particular note is the work of Lee and Johnson \[16\] who show that finite classical games are a strict subset of quantum games, as we would expect. In the context of communcation channels this is expressing the fact that any classical communication channel with a finite input alphabet is a subset of the possible quantum channels. It is how these channels are exploited that determines whether they display quantum or classical characteristics. Lee and Johnson show that a given classical game can have a more efficient implementation using quantum objects in terms of the relative number of bits and qubits, respectively. This is reminiscent of the ability of quantum channels to transmit classical information using fewer qubits \[31\]. Here we are interested in a converse (and more restricted) question ; whether a quantum game can be modelled by a game played with classical objects in the context of the 2-player pure games with finite strategy sets in which a single projective measurement is made on the output. We believe that our analysis offers an insight into where we need to look for game properties that display necessarily quantum-mechanical features.
The more general form of the 2-player game we have considered assumes a finite strategy set for the players, and that a single projective measurement is made on the resultant output state. Furthermore we have assumed a *pure* strategy game. The simple example of the classical mixed game with correlated noise considered above shows that similar considerations may also apply in the more general mixed game case. We consider such situations elsewhere.
Discussion
==========
There is no doubt that the pioneering work \[1,2\] that brought together game theory and quantum mechanics represented a new and original direction in both fields. There has been much work since on various quantum game scenarios, usually focusing on the use of entangled states in games \[1-21\] (we have referenced only a very small selection of the work that has been done). In our previous work \[23-25\] we argued that a game should be seen as something that can actually be played. In other words there is a physical implementation of a game with real objects that obey the laws of physics. With this perspective the necessary elements required to actually play a game can be identified. These elements are; preparation of some initial state, operations by the players on that state, a resulting output state that is subject to a measurement, and a mapping of the results of that measurement to given outcomes. In order to call such a thing a game, rather than just an experiment in physics, we require that the players have some preference over the outcomes, which ultimately determines which operation on the input state they will choose.
In the games we’ve discussed here we have assumed that a single measurement on the output state produced by the players is performed. This measurement remains fixed however many times we play the game. The simple restricted examples examined consider a measurement of spin in the $z$-direction for Alice’s particle and a measurement of spin, also in the $z$-direction, for Bob’s particle. Experimentally, therefore, we are only accessing information about the correlations between these two specific observables. This measurement cannot uncover the full richness of the quantum correlations inherent in an entangled state of 2 particles; it is only accessing *some* of the information about the quantum correlations \[29,30\]. The specific examples of games considered are only allowing us to probe correlations between the spin-$z$ and spin-$z$ measurements, and that can’t give us enough information to decide whether it’s quantum or classical behaviour we’re seeing. The information that can be recovered about the correlations from this kind of fixed measurement is not sufficient to distinguish between the classical and quantum nature of the correlation. The more general analysis for 2-player pure strategy games in which the players can choose from finite strategy sets shows that a single measurement of the output is also insufficient to access the quantum nature of the correlation. In order to do that we need to compare correlations for multi-player games or in 2-player quantum games in which different measurement angles are selected, for example, just as we need to in establishing the experimental violation of Bell’s inequality \[6-9\].
The general analysis of these 2-player pure strategy games shows that the *same* game can be played either with quantum objects in which a single measurement is made, or as a classical communication channel in which the players know the noise characteristics. Where correlation features in the quantum game, when entanglement is introduced, the classical version of the game as a communication channel requires a classical correlated noise. The important point to note is that the single measurement in the quantum case reduces everything to a set of probabilities that can be achieved by an *equivalent* classical communication channel. We do not require quantum objects to play games of this form. In other words, given a set of operations and an associated payoff matrix, there is no way for the players to determine whether they are playing with quantum or classical objects in these games.
The reason we can reproduce these general 2-player pure strategy quantum games as a classical communication channel is that we assume a single (joint) measurement that produces the probability distribution over the results that are then mapped to the expected outcomes via some payoff function. With a single joint measurement the probabilities can always be reproduced by a classical system. Overall, the game is a function that takes some inputs and performs a computation on those inputs. If that computation can be achieved by classical systems then it seems to us that the underlying game is essentially classical even if implemented by quantum objects.
In mathematical terms, then, a 2-player game is nothing more than a function that takes a pair of inputs representing the choices of the players and maps these to outcomes. The physical elements required to play a game are nothing more than a particular functional decomposition of this overall game function. In these terms, therefore, we can see that for the initial example based on the MW protocol \[21\] the function that represents the game of Chicken can be decomposed into functional elements that look like a version of quantum Prisoner’s Dilemma, or it can be functionally decomposed as a game played over a classical communication channel with correlated noise. In general, therefore, if we are to observe genuine quantum behaviour in a game we must consider richer game structures that allow us to probe the quantum regime of the correlation, and effectively perform a quantum computation on the inputs in order to produce the outcomes. Games that allow us to do this are multi-player games \[9,32\] or games in which the final outcomes are determined from comparison of the results of a sequence of games in which different meaurements are made.
**References**
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2. D. Meyer, *Quantum Strategies,* Phys. Rev. Lett., **82**, 1052-1055, (1999)
3. S. J. van Enk and R. Pike, *Classical Rules in Quantum Games*, Phys. Rev. A, **66,** 024306, (2002)
4. S. A. Bleiler, *A Formalism for Quantum Games I - Quantizing Mixtures*, Portland State University, preprint, available at http://arxiv.org/abs/0808.1389, (2008)
5. J. Shimamura, Ş. K. Özdemir, F. Morikoshi and N. Imoto, *Entangled states that cannot reproduce original classical games in their quantum version*, Phys. Lett A, **328**, 20-25, (2004)
6. A. Iqbal and D. Abbott, *Constructing quantum games from a system of Bell’s inequalities*, Phys. Lett. A, **374**/31-32, 3155-3163, (2010)
7. A. Iqbal*, Playing games with EPR-type experiments*, J. Phys. A, **38**/43, 9551-9564, (2005)
8. A. Iqbal and S. Weigert, *Quantum correlation games*, J. Phys. A, **37**, 5873-5885, (2004)
9. C.D. Hill, A.P. Flitney and N.C. Menicucci, *A competitive game whose maximal Nash-equilibrium payoff requires quantum resources for its achievement*, Phys. Lett. A **374,** 3619 (2010)
10. T. Ichikawa, T. Cheon and I. Tsutsui, *Quantum Game Theory Based on the Schmidt Decomposition*, J.Phys.A, **41**, 135303, (2008)
11. T. Ichikawa and I. Tsutsui, *Duality, Phase Structures and Dilemmas in Symmetric Quantum Games*, Ann. Phys., **322**, 531-551, (2007)
12. R. Cleve, P. Hoyer, B. Toner and J. Watrous, *Consequences and Limits of Nonlocal Strategies*, preprint, available at http://arxiv.org/abs/quant-ph/0404076v2
13. R. Renner and S. Wolf, *Towards Characterizing the Non-Locality of Entangled Quantum States*, preprint, available at http://arxiv.org/abs/quant-ph/0211019, (2002)
14. S. E. Landsburg, *Nash Equilibria in Quantum Games*, Proc. Am. Math Soc., **139**, 4423-4434, (2011)
15. G. Dahl and S. Landsburg, *Quantum Strategies in Noncooperative Games*, preprint, available at http://www.landsburg.com/dahlcurrent.pdf
16. C. F. Lee and N. Johnson, *Efficiency and Formalism of Quantum Games*, Phys. Rev. A **67**, 022311 (2003)
17. T. Cheon and I. Tsutsui, *Classical and Quantum Contents of Solvable Game Theory on Hilbert Space*, Physics Letters A, **348** 147-152 (2006)
18. J. Shimamura, S. Özdemir, F. Morikoshi, and N. Imoto, *Quantum and Classical Correlations Between Players in Game Theory*, International Journal of Quantum Information, **2**, 79–89 (2004)
19. L.K. Chen, H. Ang, D. Kianga, L.C. Kwek, and C.F. Loc, *Quantum Prisoner Dilemma under Decoherence*, Physics Letters A, **316**, 317–323 (2003)
20. A. P. Flitney, and D. Abbott, *Quantum Games with Decoherence*, Journal of Physics A, **38**, 449, (2005)
21. L. Marinatto and T. Weber, *A quantum approach to static games of complete information*, Phys. Lett. A, **272**, 291-303, (2000)
22. K. Binmore, *Playing for Real: A Text on Game Theory*, OUP USA, (2007)
23. S. J. D. Phoenix and F. S. Khan, *Playing Games with Quantum Mechanics*, preprint available at http://arxiv.org/abs/1202.4708, (2012)
24. F. S. Khan and S. J. D. Phoenix, *Nash equilibrium in quantum superpositions*, Proceedings of SPIE, **8057,** 80570K-1, (2011)
25. F. S. Khan and S. J. D. Phoenix, *Gaming the Quantum*, Quant. Inf. Comp., **13**, 0231-0244, (2013)
26. W. H. Zurek, *Pointer Basis of Quantum Apparatus: Into what Mixture does the Wave Packet Collapse*?, Phys. Rev D, **24**, 1516–1525, (1981)
27. W. H. Zurek, *Environment-Induced Superselection Rules*, Phys. Rev. D, **26**, 1862–1880, (1982)
28. W. H. Zurek, *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys., **75**, 715, (2003) (available at http://arxiv.org/abs/quant-ph/0105127)
29. S. M. Barnett and S. J. D. Phoenix,* Information Theory, Squeezing and Quantum Correlations*, Phys. Rev. A, **44**, 535, (1991)
30. S. M. Barnett and S. J. D. Phoenix, *Information-Theoretic Limits to Quantum Cryptography*, Phys. Rev. A, **48,** R5, (1993)
31. S.M. Barnett, R. Loudon, D.T. Pegg, and S.J.D. Phoenix, *Communication Using Quantum States*, J. Mod. Opt, **41**, 2351-2373 (1994)
32. A.P. Flitney and D. Abbott, *Multiplayer quantum minority game with decoherence*, Quant. Inf. Comput, **7,** 111, (2007)
[^1]: Of course, ‘changing’ a game by considering an appropriate extension of it is nothing new within game theory. The physics of playable games \[13\] is highlighting this here in a rather dramatic fashion.
|
---
abstract: |
The paper investigates the non-vanishing of $H^1({\mathcal{E}}(n))$, where ${\mathcal{E}}$ is a (normalized) rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$. If $\epsilon$ is the integer defined by the equality $\omega_X = {\mathcal{O}}_X(\epsilon)$, and $\alpha$ is the least integer $t$ such that $H^0({\mathcal{E}}(t)) \ne 0$, then, for a non-stable ${\mathcal{E}}$ ($\alpha \le 0)$ the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also shows that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and also on the second Chern class $c_2$ of ${\mathcal{E}}$. If ${\mathcal{E}}$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $\alpha-2$. The paper considers also the case of a threefold $X$ with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$ but ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$ and gives similar non-vanishing results.\
**Keyword:** rank two vector bundles, smooth threefolds, non-vanishing of 1-cohomology.\
**MSC 2010:** 14J60, 14F05.
author:
- 'E. Ballico'
- 'P. Valabrega'
- 'M. Valenzano'
title: |
Non-vanishing theorems for rank two vector\
bundles on threefolds [^1]
---
Introduction
============
In 1942 G. Gherardelli ([@Ghe]) proved that, if $C$ is a smooth irreducible curve in ${\mathbb{P}}^3$ whose canonical divisors are cut out by the surfaces of some degree $e$ and moreover all linear series cut out by the surfaces in ${\mathbb{P}}^3$ are complete, then $C$ is the complete intersection of two surfaces. Shortly and in the language of modern algebraic geometry: every $e$-subcanonical smooth curve $C$ in ${\mathbb{P}}^3$ such that $h^1({\mathcal{I}}_C(n)) = 0$ for all $n$ is the complete intersection of two surfaces.
Thanks to the Serre correspondence between curves and vector bundles (see [@Hvb], [@H1], [@H2]) the above statement is equivalent to the following one: if ${\mathcal{E}}$ is a rank two vector bundle on ${\mathbb{P}}^3$ such that $h^1({\mathcal{E}}(n)) = 0$ for all $n$, then ${\mathcal{E}}$ splits.
There are many improvements of the above result with a variety of different approaches (see for instance [@CV1], [@CV2], [@Ellia], [@P], [@RV]): it comes out that a rank two vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ is forced to split if $h^1({\mathcal{E}}(n))$ vanishes for just one strategic $n$, and such a value $n$ can be chosen arbitrarily within a suitable interval, whose endpoints depend on the Chern classes and the least number $\alpha$ such that $h^0({\mathcal{E}}(\alpha)) \ne 0$.
When rank two vector bundles on a smooth threefold $X$ of degree $d$ in ${\mathbb{P}}^4$ are concerned, similar results can be obtained, with some interesting difference.
In 1998 Madonna ([@Madonna]) proved that on a smooth threefold $X$ of degree $d$ in ${\mathbb{P}}^4$ there are ACM rank two vector bundles (i.e. whose 1-cohomology vanishes for all twists) that do not split. And this can happen, for a normalized vector bundle ${\mathcal{E}}$ ($c_1\in\{0,-1\})$, only when $ 1-\frac{d+c_1}{2} < \alpha < \frac{d-c_1}{2}$, while an ACM rank two vector bundle on $X$ whose $\alpha$ lies outside of the interval is forced to split.
The following non-vanishing results for a normalized non-split rank two vector bundle on a smooth irreducible thereefold of degree $d$ in ${\mathbb{P}}^4$ are proved in [@Madonna]:
if $\alpha \le 1-\frac{d+c_1}{2}$, then $h^1({\mathcal{E}}(\frac{d-3-c_1}{2}))\ne 0$ if $d+c_1$ is odd, $h^1({\mathcal{E}}(\frac{d-4-c_1}{2}))\ne 0, h^1({\mathcal{E}}(\frac{d-2-c_1}{2}))\ne 0$ if $d+c_1$ is even, while $h^1({\mathcal{E}}(\frac{d-c_1}{2}))\ne 0$ if $d+c_1$ is even and moreover $\alpha \le -\frac{d+c_1}{2}$;
if $\alpha\ge \frac{d-c_1}{2}$, then $h^1({\mathcal{E}}(\frac{d-3-c_1}{2}))\ne 0$ if $d+c_1$ is odd, while $h^1({\mathcal{E}}(\frac{d-4-c_1}{2}))\ne 0$ if $d+c_1$ is even.
In [@Madonna] it is also claimed that the same techniques work to obtain similar non-vanishing results on any smooth threefold $X$ with ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and $h^1({\mathcal{O}}_X(n)) = 0$, for every $n$.
The present paper investigates the non-vanishing of $H^1({\mathcal{E}}(n))$, where ${\mathcal{E}}$ is a rank two vector bundle over any smooth irreducible threefold $X$ of degree $d$ such that ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and $H^1({\mathcal{O}}_X(n)) = 0, \forall n$. Actually we can prove that for such an ${\mathcal{E}}$ there is a wider range of non-vanishing for $h^1({\mathcal{E}}(n))$, so improving the above results.
More precisely, when ${\mathcal{E}}$ is (normalized and) non-stable ($\alpha \le 0$) the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$, where $\epsilon$ is defined by the equality $\omega(X) = {\mathcal{O}}_X(\epsilon)$ (and is $d-5$ if $X \subset {\mathbb{P}}^4$). But we can show that there are other non-vanishing intervals, whose endpoints depend on $\alpha$ and also on the second Chern class $c_2$ of ${\mathcal{E}}$.
If on the contrary ${\mathcal{E}}$ is stable the first cohomology module does not vanish at least between the endpoints $\frac{\epsilon-c_1}{2}$ and $\alpha-2$, but other ranges of non-vanishing can be produced.
We give a few examples obtained by pull-back from vector bundles on ${\mathbb{P}}^3$.
We must remark that most of our non-vanishing results do not exclude the range for $\alpha$ between the endpoints $1-\frac{d+c_1}{2}$ and $\frac{d-c_1}{2}$ (for a general threefold it becomes $-\frac{\epsilon+3+c_1}{2} < \alpha < \frac{\epsilon+5-c_1}{2})$. Actually [@Madonna] produces some examples of nonsplit ACM rank two vector bundles on smooth hypersurfaces in ${\mathbb{P}}^4$, but it can be seen that they do not conflict with our theorems.
As to threefolds with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$, we need to observe that a key point is a good definition of the integer $\alpha$. We are able to prove, by using a boundedness argument, that $\alpha$ exists when ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$ but ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$. In this event the correspondence between rank two vector bundles and two-codimensional subschemes can be proved to hold. In order to obtain non-vanishing results that are similar to the results proved when ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$, we need also use the Kodaira vanishing theorem, which holds in characteristic 0. We can extend the results to characteristic $p > 0$ if we assume a Kodaira-type vanishing condition.
Notation
========
We work over an algebraically closed field ${\mathbf{k}}$ of any characteristic.\
Let $X$ be a non-singular irreducible projective algebraic variety of dimension 3, for short a smooth threefold.\
We fix an ample divisor $H$ on $X$, so we consider the polarized threefold $(X,H)$.\
We denote with ${\mathcal{O}_{X}}(n)$, instead of ${\mathcal{O}_{X}}(nH)$, the invertible sheaf corresponding to the divisor $nH$, for each $n\in{\mathbb{Z}}$.\
For every cycle $Z$ on $X$ of codimension $i$ it is defined its degree with respect to $H$, i.e. $\deg(Z;H) := Z \cdot H^{3-i}$, having identified a codimension 3 cycle on $X$, i.e. a $0$-dimensional cycle, with its degree, which is an integer.\
From now on (with the exception of section 7) we consider a smooth polarized threefold $(X,{\mathcal{O}}_{X}(1))=(X,H)$ that satifies the following conditions:
$\mathbf{(C1)}$
: ${\mathrm{Pic}}(X)\cong{\mathbb{Z}}$ generated by $[H]$,
$\mathbf{(C2)}$
: $H^1(X,{\mathcal{O}_{X}}(n)) = 0$ for every $n\in{\mathbb{Z}}$,
$\mathbf{(C3)}$
: $h^0(O_X(1)) \ne 0$.
By condition $\mathbf{(C1)}$ every divisor on $X$ is linearly equivalent to $aH$ for some integer $a\in{\mathbb{Z}}$, i.e. every invertible sheaf on $X$ is (up to an isomorphism) of type ${\mathcal{O}_{X}}(a)$ for some $a\in{\mathbb{Z}}$, in particular we have for the canonical divisor $K_X \sim \epsilon H$, or equivalently $\omega_{X}\simeq{\mathcal{O}}_{X}(\epsilon)$, for a suitable integer $\epsilon$. Furthermore, by Serre duality condition $\mathbf{(C2)}$ implies that $H^2(X,{\mathcal{O}_{X}}(n)) = 0$ for all $n\in{\mathbb{Z}}$.\
Since by assumption $A^{1}(X)={\mathrm{Pic}}(X)$ is isomorphic to ${\mathbb{Z}}$ through the map $[H]\mapsto 1$, where $[H]=c_{1}({\mathcal{O}}_{X}(1))$, we identify the first Chern class $c_1({\mathcal{F}})$ of a coherent sheaf with a whole number $c_1$, where $c_1({\mathcal{F}}) = c_1 H$.\
The second Chern class $c_2({\mathcal{F}})$ gives the integer $c_2 = c_2({\mathcal{F}})\cdot H$ and we will call this integer the second Chern number or the second Chern class of ${\mathcal{F}}$.\
We set $$d := \deg(X;H) = H^3,$$ so $d$ is the degree of the threefold $X$ with respect to the ample divisor $H$.\
Let $c_1(X)$ and $c_2(X)$ be the first and second Chern classes of $X$, that is of its tangent bundle $TX$ (which is a locally free sheaf of rank 3); then we have $$c_1(X) = [-K_X] = -\epsilon [H],$$ so we identify the first Chern class of $X$ with the integer $-\epsilon$. Moreover we set $$\tau := \deg(c_2(X);H) = c_2(X) \cdot H,$$ i.e. $\tau$ is the degree of the second Chern class of the threefold $X$.\
In the following we will call the triple of integers $(d,\epsilon,\tau)$ the **characteristic numbers** of the polarized threefold $(X,{\mathcal{O}_{X}}(1))$.
We recall the well-known Riemann-Roch formula on the threefold $X$ (see [@Valenzano], proposition 4).
\[gRR\] Let ${\mathcal{F}}$ be a rank $r$ coherent sheaf on $X$ with Chern classes $c_{1}({\mathcal{F}})$, $c_{2}({\mathcal{F}})$ and $c_{3}({\mathcal{F}})$. Then the Euler-Poincaré characteristic of ${\mathcal{F}}$ is $$\begin{aligned}
\chi({\mathcal{F}}) = & \frac{1}{6}\Big(c_{1}({\mathcal{F}})^{3} - 3 c_{1}({\mathcal{F}})\cdot
c_{2}({\mathcal{F}}) + 3 c_{3}({\mathcal{F}})\Big) +
\frac{1}{4}\Big(c_{1}({\mathcal{F}})^{2} - 2 c_{2}({\mathcal{F}})\Big)\cdot
c_{1}(X) + \\
& + \frac{1}{12} c_{1}({\mathcal{F}})\cdot\Big(c_{1}(X)^{2} + c_{2}(X)\Big) +
\frac{r}{24} c_{1}(X)\cdot c_{2}(X)\end{aligned}$$ where $c_{1}(X)$ and $c_{2}(X)$ are the Chern classes of $X$, that is the Chern classes of the tangent bundle $TX$ of $X$.
So applying the Riemann-Roch Theorem to the invertible sheaf ${\mathcal{O}_{X}}(n)$, for each $n\in{\mathbb{Z}}$, we get the Hilbert polynomial of the sheaf ${\mathcal{O}_{X}}(1)$ $$\chi({\mathcal{O}_{X}}(n)) = \frac{d}{6} \left(n - \frac{\epsilon}{2}\right) \left[ \left(n - \frac{\epsilon}{2}\right)^2 + \frac{\tau}{2d} - \frac{\epsilon^2}{4} \right]\!.$$
Let ${\mathcal{E}}$ be a rank 2 vector bundle on the threefold $X$ with Chern classes $c_1({\mathcal{E}})$ and $c_2({\mathcal{E}})$, so with Chern numbers $c_1$ and $c_2$. We assume that ${\mathcal{E}}$ is normalized, i.e. that $c_1 \in\{0,-1\}$. It is defined the integer $\alpha$, the so called first relevant level, such that $h^0({\mathcal{E}}(\alpha)) \ne 0, h^0({\mathcal{E}}(\alpha-1)) = 0$. If $\alpha > 0$, ${\mathcal{E}}$ is called stable, non-stable otherwise.
We set $$\vartheta = \frac{3c_2}{d} - \frac{\tau}{2d} + \frac{\epsilon^2}{4}-\frac{3c_1^2}{4}, \qquad
\zeta_0 = \frac{\epsilon-c_1}{2}, \quad \text{and} \quad w_0 = [\zeta_0]+1,$$ where $[\zeta_0]=$ integer part of $\zeta_0$, so the Hilbert polynomial of ${\mathcal{E}}$ can be written as $$\chi({\mathcal{E}}(n)) = \frac{d}{3}\big(n - \zeta_0\big)\Big[\big(n - \zeta_0\big)^2 - \vartheta\Big].$$ If $\vartheta\ge0$ we set $$\zeta = \zeta_0 + \sqrt{\vartheta}$$ so in this case the Hilbert polynomial of ${\mathcal{E}}$ has the three real roots $\zeta' \le \zeta_0 \le \zeta$ where $\zeta' = \zeta_0 - \sqrt{\vartheta}, \zeta = \zeta_0 + \sqrt{\vartheta}$. We also define $\bar\alpha = [\zeta]+1$.\
The polinomial $\chi({\mathcal{E}}(n))$, as a rational polynomial, has three real roots if and only if $\vartheta\ge0$, and it has only one real root if and only if $\vartheta<0$.\
If ${\mathcal{E}}$ is normalized, we set $$\delta = c_2+d\alpha^2+c_1d\alpha.$$
We have $\delta = 0$ if and only if ${\mathcal{E}}$ splits (see [@VV], Lemma 3.13: the proof works in general).
Unless stated otherwise, we work over the smooth polarized threefold $X$ and *${\mathcal{E}}$ is a normalized non-split rank two vector bundle on $X$*.
About the characteristic numbers $\epsilon$ and $\tau$
======================================================
In this section we want to recall some essentially known properties of the characteristic numbers of the threefold $X$ (see also [@SB] for more general statements). We start with the following remark.
\[OX\] 1. For the fixed ample invertible sheaf ${\mathcal{O}_{X}}(1)$ we have $$h^0({\mathcal{O}_{X}}(n)) \begin{cases} = 0 & \quad\text{for } n < 0 \\
= 1 & \quad \text{for } n = 0 \\
\ne 0 & \quad\text{for } n > 0 \end{cases}$$ and also $h^0({\mathcal{O}_{X}}(m)) - h^0({\mathcal{O}_{X}}(n)) > 0$ for all $n,m\in{\mathbb{Z}}$ with $m > n \ge 0$.\
2. It holds $$\chi({\mathcal{O}_{X}}) = h^0({\mathcal{O}_{X}}) - h^3({\mathcal{O}_{X}}) = 1 - h^0({\mathcal{O}_{X}}(\epsilon)),$$ so we have: $$\chi({\mathcal{O}_{X}}) = 1 \iff \epsilon < 0, \ \ \
\chi({\mathcal{O}_{X}}) = 0 \iff \epsilon = 0, \ \ \
\chi({\mathcal{O}_{X}}) < 0 \iff \epsilon > 0.$$
Let $(X,{\mathcal{O}_{X}}(1))$ be a smooth polarized threefold with [characteristic numbers $(d,\epsilon,\tau)$]{}. Then it holds:
1)
: $\epsilon \ge -4$,
2)
: $\epsilon = -4$ if and only if $X = {\mathbb{P}}^3$, i.e. $(d,\epsilon,\tau)=(1,-4,6)$ and so $\frac{\tau}{2d} - \frac{\epsilon^2}{4} = -1$,
3)
: if $\epsilon = -3$, then $X$ is a hyperquadric in ${\mathbb{P}}^4$, so $(d,\epsilon,\tau)=(2,-3,8)$ and $\frac{\tau}{2d} - \frac{\epsilon^2}{4} = -\frac{1}{4}$,
4)
: $\epsilon\tau$ is a multiple of $24$, in particular if $\epsilon < 0$ then $\epsilon \tau = -24$,
5)
: if $\epsilon\ne 0$, then $\tau > 0$,
6)
: if $\epsilon = 0$, then $\tau > -2d$,
7)
: $\tau$ is always even,
8)
: if $\epsilon$ is even, then $\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -1$,
9)
: if $\epsilon$ is odd, then $\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -\frac{1}{4}$,
10)
: if $\epsilon < 0$, then the only possibilities for $(\epsilon,\tau)$ are the following $$(\epsilon,\tau) \in \{(-4,6), \, (-3,8), \, (-2,12), \, (-1,24) \},$$
For statements **1)**, **2)**, **3)** see [@SB].\
**4)** Observe that $\chi({\mathcal{O}_{X}}) = - \frac{1}{24} \epsilon \tau$ is an integer, and moreover, if $\epsilon < 0$, then $\chi({\mathcal{O}_{X}})=1$.\
**5)** By Remark \[OX\] we have: if $\epsilon > 0$ then $- \frac{1}{24} \epsilon \tau < 0$, while if $\epsilon < 0$ then $- \frac{1}{24} \epsilon \tau > 0$. In both cases we deduce $\tau > 0$.\
**6)** If $\epsilon = 0$, then we have $$\chi({\mathcal{O}_{X}}(n)) = \frac{d}{6} n \left(n^2 + \frac{\tau}{2d} \right),$$ and also $$\chi({\mathcal{O}_{X}}(n)) = h^0({\mathcal{O}_{X}}(n)) > 0 \quad \forall n > 0,$$ therefore we must have $\frac{2d+\tau}{12} > 0$, so $\tau > -2d$.\
**7)** Assume that $\epsilon$ is even, then we have $$d\left(1- \frac{\epsilon}{2}\right)\left(1+ \frac{\epsilon}{2}\right) + \frac{\tau}{2} =
d\left(1-\frac{\epsilon^2}{4}+\frac{\tau}{2d}\right) = 6\, \chi\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}+1\right)\right) \in{\mathbb{Z}}$$ and moreover $d\left(1- \frac{\epsilon}{2}\right)\left(1+ \frac{\epsilon}{2}\right)\in{\mathbb{Z}}$, so $\tau$ must be even.\
If $\epsilon$ is odd, the proof is quite similar.\
**8)** Let $\epsilon$ be even. If it holds $$h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) - h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}-1\right)\right) = \chi\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}+1\right)\right) < 0,$$ then we must have $h^0\left({\mathcal{O}_{X}}\left(\frac{\epsilon}{2}-1 \right)\right) \ne 0$, which implies $$h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) - h^0\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}-1\right)\right) \ge 0,$$ a contradiction. So we must have:$$\chi\left({\mathcal{O}_{X}}\!\left(\frac{\epsilon}{2}+1\right)\right) =
\frac{d}{6}\left(1 + \frac{\tau}{2d} - \frac{\epsilon^2}{4}\right) \ge 0,$$ therefore $$\frac{\tau}{2d} - \frac{\epsilon^2}{4} \ge -1.$$\
**9)** The proof is quite similar to the proof of **8**).\
**10)** If $\epsilon < 0$, then by **1)** we have $\epsilon\in\{-4,-3,-2,-1\}$, and moreover $\epsilon \tau = -24$ by **4)**, so we obtain the thesis.
Non-stable vector bundles ($\alpha \le 0$)
==========================================
**Case $\epsilon \ge 1$.**\
In this subsection we make the following assumptions:
*${\mathcal{E}}$ is a normalized non-split rank two vector bundle with $\alpha \le 0$ and $\epsilon \ge 1$*.\
The case $\epsilon \le 0$ is investigated in the next subsection.
\[nonstable0\] Assume that $\zeta_0 < -\alpha-c_1-1$. Then it holds: $$h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = (n-\zeta_0)\delta$$ for every $n$ such that $\zeta_0 < n \le -\alpha-c_1-1$.
First we assume $c_1 = 0$. It is enough to observe that, from the inequality $n+\alpha \le 0$ and the exact sequence $$0 \to {\mathcal{O}}_X(n-\alpha) \to {\mathcal{E}}(n) \to {\mathcal{I}}(n+\alpha) \to 0$$ we obtain: $h^0({\mathcal{E}}(n)) = h^0({\mathcal{O}}_X(n-\alpha)) = \chi({\mathcal{O}}_X(n-\alpha))+h^0({\mathcal{O}}_X(\epsilon-n+\alpha)) = \chi({\mathcal{O}}_X(n-\alpha))$ since $\epsilon-n+\alpha \le -1$. We also have: $h^0({\mathcal{E}}(\epsilon-n)) = h^0({\mathcal{O}}_X(\epsilon-n-\alpha)) = \chi({\mathcal{O}}_X(\epsilon-n-\alpha))+h^0({\mathcal{O}}_X(n+\alpha)) = \chi({\mathcal{O}}_X(\epsilon-n-\alpha))$.\
Now it is enough to observe that $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = h^0({\mathcal{E}}(n))-h^0({\mathcal{E}}(\epsilon-n)) -\chi({\mathcal{E}}(n)) = \chi({\mathcal{O}}_X(n-\alpha))- \chi({\mathcal{O}}_X(\epsilon-n-\alpha))-\chi({\mathcal{E}}(n))$. If we use the Riemann-Roch formulas for the Euler functions we obtain the required equality.\
Now we assume $c_1 = -1$. We recall that $h^3({\mathcal{E}}(n)) = h^0({\mathcal{E}}(\epsilon-n+1))$. As before we have $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = \chi({\mathcal{O}}_X(n-\alpha))- \chi({\mathcal{O}}_X(\epsilon-n-\alpha+1))-\chi({\mathcal{E}}(n))$ and the computation is very similar.
Observe that the statement of Proposition \[nonstable0\] still holds when $n = \zeta_0$, because the two sides of the equality vanish.
\[nonstable1\] Let us assume that $\zeta_0 < -\alpha-c_1-1$ and let $n$ be such that $\zeta_0 < n \le -\alpha-1-c_1$. Then $h^1({\mathcal{E}}(n)) \ge (n-\zeta_0)\delta$. In particular $h^1({\mathcal{E}}(n)) \ne 0$.
It is enough to observe that $$h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = (n-\zeta_0)\delta$$ and that the right side of the equality is strictly positive for a non-split vector bundle.
Observe that the above theorem describes a non-empty set of integers if and only if $-\alpha -1 -c_ 1 > \zeta_0$; this means $\alpha < -\frac{\epsilon+2+c_1}{2}$, i.e. $\alpha \le -\frac{\epsilon+3+c_1}{2}$. So our assumption on $\alpha$ agrees with the bound of [@Madonna].\
Observe that the inequality on $\alpha$ implies that $\alpha \le -2$ if $\epsilon \ge 1$.
\[nonstable2\] Assume that $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_{1}^2}{4}\ge 0$. Let $n > \zeta_0$ be such that $ \epsilon-\alpha-c_1+1 \le n < \zeta_0+\sqrt{\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_{1}^2}{4}}$ and put $$S(n) =\frac{d}{6}\left(n-\frac{\epsilon-c_1}{2}\right)\!\left[(n-\frac{\epsilon-c_1}{2})^2-6\frac{c_2+d \alpha^2+c_1d\alpha}{d}+\frac{\tau}{2d}-\frac{\epsilon^2}{4}+\frac{3c_1^2}{4}\right]\!.$$ Then $h^1({\mathcal{E}}(n))\ge -S(n) > 0$. In particular $h^1({\mathcal{E}}(n))\ne 0$.
Assume $c_1 = 0$. Under our hypothesis $h^0({\mathcal{E}}(\epsilon-n)) = 0$ and so $-h^1({\mathcal{E}}(n))+h^2({\mathcal{E}}(n)) = \chi({\mathcal{E}}(n)) - h^0({\mathcal{O}}_X(n-\alpha))$. Observe that $\chi({\mathcal{E}}(n)) - h^0({\mathcal{O}}_X(n-\alpha)) -S(n) = \frac{1}{2}nd\alpha(-\epsilon+n+\alpha))+\frac{1}{12}d\alpha(-3\epsilon \alpha+2\alpha^2+\epsilon^2+\frac{\tau}{d}) \le 0$. Therefore we have: $h^1({\mathcal{E}}(n)) \ge h^2({\mathcal{E}}(n))-S(n)$. Hence $h^1({\mathcal{E}}(n))$ may possibly vanish when $$\left(n-\frac{\epsilon}{2}\right)^2-6\frac{c_2+d \alpha^2}{d}+\frac{\tau}{2d}-\frac{\epsilon^2}{4} \ge 0.$$ When $S(n) < 0$, so $-S(n) > 0$, $h^1({\mathcal{E}}(n)) \ge -S(n) > 0$ and in particular it cannot vanish.\
If $c_1 = -1$ the proof is quite similar.
Now we put $\frac{\tau}{2d}-\frac{\epsilon^2}{4} =\lambda$ and consider the following degree $3$ polynomial: $$F(X) = X^3+\left(\lambda-\frac{6\delta}{d}\right)X+\frac{6\alpha \delta}{d}.$$ It is easy to see that, if $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} \le 0$, $F(X)$ is strictly increasing and so it has only one real root $X_0$.
\[nonstable3\] Assume that $\frac{6\delta}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} \le 0$. Let $n$ be such that $ \epsilon-\alpha-c_1+1 \le n < -\alpha+X_0+\zeta_0$, where $X_0$ = unique real root of $F(X)$ . Then $h^1({\mathcal{E}}(n)) \ge -\frac{d}{6}F(n+\alpha-\zeta_0) > -\frac{d}{6}F(X_0) = 0$. In particular $h^1({\mathcal{E}}(n)) \ne 0$.
Assume $c_1 = 0$, the proof being quite similar if $c_1 = -1$.\
It holds (see proposition \[nonstable0\]): $$\begin{aligned}
h^1({\mathcal{E}}(n)) & - h^2({\mathcal{E}}(n)) = \chi({\mathcal{O}}_X(n-\alpha))-\chi({\mathcal{E}}(n)) = \\
& = \left(n-\frac{\epsilon}{2}\right)(c_2+d\alpha^2)+\chi({\mathcal{O}}_X(\epsilon-n-\alpha)) = \\
& = \left(n-\frac{\epsilon}{2}\right)(c_2+d\alpha^2)-\frac{d}{6}\left(n+\alpha-\frac{\epsilon}{2}\right)\left(\left(n+\alpha-\frac{\epsilon}{2}\right)^2+\lambda\right).\end{aligned}$$ If we put: $X = n+\alpha-\frac{\epsilon}{2}$, we obtain: $\frac{d}{6} F(X) = \frac{d}{6}(X^3+(\lambda-\frac{6\delta}{d})X+\frac{6\alpha \delta}{d}) = h^2({\mathcal{E}}(n))-h^1({\mathcal{E}}(n))$. Therefore $h^1({\mathcal{E}}(n)) > -\frac{d}{6}F(n+\alpha-\zeta_0) > -\frac{d}{6}F(X_0) = 0$.
Observe that in Theorems \[nonstable2\] and \[nonstable3\] $\alpha$ can be $0$.
**Case $\epsilon \le 0$.**
In the event that $\epsilon \le -2$, we have $\epsilon-\alpha-c_1+1 \le -\alpha-c_1-1$. Therefore Theorems \[nonstable2\], \[nonstable3\] give something new only beyond $-\alpha-c_1-1.$
First of all we observe that Theorems \[nonstable1\], \[nonstable3\] obviously hold as they are stated also when $\epsilon \le 0$. So we discuss Theorem \[nonstable2\].\
A. $\epsilon \le -2$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(-\epsilon+n+\alpha)+\frac{1}{12}d\alpha(\epsilon^2+\frac{\tau}{d}-3\epsilon \alpha+2\alpha^2) \le 0.$$ The first term of the sum is for sure negative; as for $$\frac{1}{12}d\alpha\left(\epsilon^2+\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(-3\epsilon+2\alpha)$$ we observe that the quantity in brackets has discriminant $$\Delta = \epsilon^2-8\frac{\tau}{d} = 4\left(\frac{\epsilon^2}{4}-\frac{\tau}{2d}+\frac{\tau}{2d}-8\frac{\tau}{d}\right) \le 4(1-15) < 0.$$ Therefore it is positive for all $\alpha \le 0$ and the product is negative.\
B. $\epsilon = -1$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(1+n+\alpha)+\frac{1}{12}d\alpha\left(1+\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(3+2\alpha) \le 0.$$ If $\alpha \le -2$, then it is enough to observe that $\frac{\tau}{d}+3\alpha+2\alpha^2 \ge 0$. If $\alpha = -1$ we have to consider $-\frac{1}{2}n^2d+\frac{1}{12}d\frac{\tau}{d}$ and then we observe that $6n^2+\frac{\tau}{d} > 0$. If $\alpha = 0$ obviously the quantity is $0$.\
C. $\epsilon = 0$.\
In theorem \[nonstable2\] we need to know that $$\frac{1}{2}nd\alpha(n+\alpha)+\frac{1}{12}d\alpha\left(\frac{\tau}{d}\right)+\frac{1}{12}d\alpha^2(2\alpha) \le 0.$$\
It is enough to observe that $2\alpha^2+\frac{\tau}{d} > 0$ by Proposition 3.2, $\mathbf{6)}$ if $\alpha < 0$; otherwise we have a $0$ quantity, and that $n+\alpha \le 0$.
Observe that the case $\alpha = 0$ in Theorem \[nonstable1\] can occur only if $\epsilon \le -c_1-3$.
In theorem \[nonstable2\] we do not use the hypothesis $-\frac{\epsilon+3}{2} \ge \alpha$, but we assume that $6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-1\ge 0$. In theorem \[nonstable3\] we do not use the hypothesis $-\frac{\epsilon+3}{2} \ge \alpha$, but we assume that $6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4} < 0$. Moreover in both theorems there is a range for $n$, the left endpoint being $\epsilon-\alpha-c_1+1$ and the right endpoint being either $\zeta_0+\sqrt{6\frac{c_2+d \alpha^2}{d}-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-1}$ (\[nonstable2\]) or $\zeta_0-\alpha+X_0$ (\[nonstable3\]).\
In [@Madonna] there are examples of ACM nonsplit vector bundles on smooth threefolds in ${\mathbb{P}}^4$, with $-\frac{\epsilon+3+c_1}{2} < \alpha < \frac{\epsilon+5-c_1}{2}$. We want to emphasize that our theorems do not conflict with the examples of [@Madonna]: if $C$ is any curve described in [@Madonna] and lying on a smooth threefold of degree $d$, then our numerical constraints cannot be satisfied (we have checked it directly in many but not all cases).
Let us consider a smooth degree $d$ threefold $X \subset {\mathbb{P}}^4$.\
We have: $$\epsilon = d-5,\ \ \tau = d(10-5d+d^2),\ \ \theta = \frac{3c_2}{d}-\frac{d^2-5+3c_1^2}{4}$$ (see [@Valenzano]). As to the characteristic function of $O_X$ and ${\mathcal{E}}$, it holds: $$\chi({\mathcal{O}}_X(n)) = \frac{d}{6}\left(n-\frac{d-5}{2}\right)\!\left[\left(n-\frac{d-5}{2}\right)^2+\frac{d^2-5}{4}\right]\!,$$ $$\chi({\mathcal{E}}(n)) = \frac{d}{3} \left(n-\frac{d-5-c_1}{2}\right)\!\left[\left(n-\frac{d-5-c_1}{2}\right)^2+\frac{d^2}{4}-\frac{5}{4}+\frac{3c_1^2}{4}-\frac{3c_2}{d}\right]\!.$$ Then it is easy to see that the hypothesis of Theorem \[nonstable2\], i.e. $6\frac{\delta}{d}-\frac{d^2-5+3c_1^2}{4}\ge 0$ is for sure fulfilled if $c_2 \ge 0, \alpha \le -\frac{d-2+c_1}{2}$. In fact we have (for the sake of simplicity when $c_1 = 0)$: $-6\frac{6c_2+d\alpha^2}{d}+\frac{d^2-5}{4} \le \frac{d^2-5}{4}-6\frac{d^2-2d+1}{4} = -\frac{5d^2-12d+11}{4} < 0$.
Condition $\mathbf{(C2)}$ holds for sure if $X$ is a smooth hypersurface of $ {\mathbb{P}}^4$. In general, for a characteristic $0$ base field, only the Kodaira vanishing holds ([@HAL], remark 7.15) and so, unless we work over a threefold $X$ having some stronger vanishing, we need assume, in Theorems \[nonstable1\], \[nonstable2\], \[nonstable3\] that $n-\alpha \notin \{0,...,\epsilon\}$ (which implies, by duality, that also $\epsilon -n+\alpha \notin \{0,...,\epsilon\}$).
Observe that the first assumption ($n-\alpha \notin \{0,...,\epsilon\})$ in the case of Theorem \[nonstable1\] is automatically fulfilled because of the hypothesis $\zeta_0 < -\alpha-c_1-1$, and in Theorems \[nonstable2\] and \[nonstable3\] because of the hypothesis $\epsilon-\alpha-c_1+1 \le n$. In fact $n-\alpha$ is greater than $\epsilon$. But this implies that $\epsilon-n+\alpha < 0$ and so also the second condition is fulfilled, at least when $\epsilon \ge 0$. For the case $\epsilon < 0$ in positive characteristic see [@SB].
Observe that, if $\epsilon < 0$, Kodaira (and so $\mathbf{(C2)}$) holds for every $n$.
For a general discussion, also in characteristic $p > 0$, of this question, see section 7, remark 7.8.
In the above theorems we assume that ${\mathcal{E}}$ is a nonsplit bundle. If ${\mathcal{E}}$ splits, then (see section 2) $\delta = 0$. In Theorem \[nonstable1\] this implies $h^1({\mathcal{E}}(n))-h^2({\mathcal{E}}(n)) = 0$ and so nothing can be said on the non-vanishing.
Let us now consider Theorem \[nonstable2\]. If $\delta = 0$, then we must have: $\zeta_0 < n < \zeta_0+\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}} \le \zeta_0+1$ (the last inequality depending on Proposition 3.2, $\mathbf{8),9)})$. As a consequence $\zeta_0$ cannot be a whole number. Moreover, since we have $2\zeta_0-\alpha+1 \le n < \zeta_0+\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}}$, we obtain that $\zeta_0 < \alpha \le 0$, hence $\epsilon-c_1 \le -1$. If $c_1 = 0$, $\epsilon \in \{-1, -3\}$. If $\epsilon = -3$, then $n$ must satisfy (see Proposition 3.2, $\mathbf{8})$ the following inequalities: $-\frac{3}{2} < n < -1$, which is a contradiction. If $\epsilon = -1$, then, by Proposition 3.2, $\mathbf{8})$ we have $-1+\alpha+1 < -\frac{1}{2}+ \frac{1}{2} = 0$, which implies $\alpha > 0$, a contradiction. If $c_1 = -1$, then $\epsilon \in \{-2, -4\}$. If $\epsilon = -4$, we have $\sqrt{-\frac{\tau}{2d}+\frac{\epsilon^2}{4}-\frac{3c_1^2}{4}} = \frac{1}{2}$, and so we must have: $-\frac{3}{2} < n < -1$, which is impossible. If $\epsilon = -2$, then $\zeta_0 = -\frac{1}{2}$ and so $-2-\alpha+2 < -\frac{1}{2}+\sqrt{1-\frac{3}{4}}$, which implies $-\alpha < 0$ hence $\alpha > 0$, a contradiction with the non-stability of ${\mathcal{E}}$.
Then we consider Theorem \[nonstable3\]. The vanishing of $\delta$ on the one hand implies $\lambda > 0$ and $X_0 = 0$. But on the other hand from our hypothesis on the range of $n$ we see that $\zeta_0 \le -2$, hence $\epsilon = -4, c_1 = 0$. But this contradicts Proposition 3.2, $\mathbf{2)}$.
Stable vector bundles
=====================
In the present section we assume that $\alpha \ge \frac{\epsilon-c_1+5}{2}$, or equivalently that $c_1+2\alpha\ge\epsilon+5$. This means that $\alpha \ge 1$ in any event, so ${\mathcal{E}}$ is stable.
The following lemma holds both in the stable and in the non-stable case.
\[leftvanishing\] If $h^1({\mathcal{E}}(m)) = 0$ for some integer $m \le \alpha-2$, then $h^1({\mathcal{E}}(n)) = 0$ for all $n \le m$.
First of all observe that, by our condition $\mathbf{(C3)}$, from the restriction exact sequence we can obtain in cohomology the exact sequence $$0 \to H^0({\mathcal{E}}(t-1) \to H^0({\mathcal{E}}(t)) \to H^0({\mathcal{E}}_H(t)) \to 0.$$ Then we can follow the proof given in [@VV] for ${\mathbb{P}}^3$ (where condition $\mathbf{(C3)}$ is automatically fulfilled).
\[stable1\] Let ${\mathcal{E}}$ be a rank 2 vector bundle on the threefold $X$ with first relevant level $\alpha$. If $\alpha\ge\frac{\epsilon+5-c_1}{2}$, then $h^1({\mathcal{E}}(n))\ne 0$ for $w_0\le n\le \alpha-2$.
By the hypothesis it holds $w_0 \le \alpha-2$, so we have $h^0({\mathcal{E}}(n))=0$ for all $n\le w_0+1$. Assume $h^1({\mathcal{E}}(w_0))=0$, then by Lemma \[leftvanishing\] it holds $h^1({\mathcal{E}}(n))=0$ for every $n\le w_0$. Therefore we have $$\chi({\mathcal{E}}(w_0)) = h^0({\mathcal{E}}(w_0)) + h^1({\mathcal{E}}(-w_0+\epsilon-c_1)) - h^0({\mathcal{E}}(-w_0+\epsilon-c_1)) = 0.$$ Now observe that the characteristic function has at most three real roots, that are symmetric with respect to $\zeta_0$. Therefore, if $w_0$ is a root, then $w_0 = \zeta_0+\sqrt{\theta}$ and the other roots are $\zeta_0$ and $ \zeta_0-\sqrt{\theta}$. This implies that $\chi({\mathcal{E}}(w_0+1)) > 0$. On the other hand $$\chi({\mathcal{E}}(w_0+1)) = - h^1({\mathcal{E}}(w_0+1)) \le 0,$$ a contradiction. So we must have $h^1({\mathcal{E}}(w_0))\ne0$, then by Lemma \[leftvanishing\] we obtain the thesis.
If ${\mathcal{E}}$ is *ACM*, then $\alpha<\frac{\epsilon+5-c_1}{2}$.
\[stable2\] Let ${\mathcal{E}}$ be a normalized rank 2 vector bundle on the threefold $X$ with $\vartheta\ge0$, then the following hold:
1)
: $h^1({\mathcal{E}}(n))\ne 0$ for $\zeta_0< n < \zeta$.
2)
: $h^1({\mathcal{E}}(n))\ne 0$ for $w_0\le n \le \bar\alpha-2$, and also for $n=\bar\alpha-1$ if $\zeta\notin{\mathbb{Z}}$.
3)
: If $\zeta\in{\mathbb{Z}}$ and $\alpha<\bar\alpha$, then $h^1({\mathcal{E}}(\bar\alpha-1))\ne 0$.
**1)** The Hilbert polynomial of the bundle ${\mathcal{E}}$ is strictly negative for each integer such that $w_0\le n < \zeta$, but for such an integer $n$ we have $h^2({\mathcal{E}}(n))\ge0$ and $h^0({\mathcal{E}}(n))-h^0({\mathcal{E}}(-n+\epsilon-c_1)) \ge 0$ since $n\ge-n+\epsilon-c_1$ for every $n\ge w_0$, therefore we must have $h^1({\mathcal{E}}(n))\ne 0$.\
**2)** It is simply a restatement of 1) in term of $\bar\alpha$, which is, by definition, the integral part of $\zeta+1$.\
**3)** If $\zeta\in{\mathbb{Z}}$, then $\zeta=\bar\alpha-1$, so we have $\chi({\mathcal{E}}(\bar\alpha-1))=\chi({\mathcal{E}}(\zeta))=0$. Moreover $h^0({\mathcal{E}}(\bar\alpha-1))\ne0$ since $\alpha<\bar\alpha$, therefore $h^0({\mathcal{E}}(\bar\alpha-1))-h^3({\mathcal{E}}(\bar\alpha-1)) > 0$, and $h^1({\mathcal{E}}(n)) = 0$ implies $h^1({\mathcal{E}}(m)), \forall m \le n$; hence we must have $h^1({\mathcal{E}}(\bar\alpha-1))\ne 0$ to obtain the vanishing of $\chi({\mathcal{E}}(\bar\alpha-1))$.
If ${\mathcal{E}}$ is *ACM*, then $\vartheta<0$.
Observe that in this section we assume $\alpha \ge \frac{\epsilon-c_1+5}{2}$, in order to have $w_0 \le \alpha-2$ and so to have a non-empty range for $n$ in Theorem \[stable1\].
Observe that in the stable case we need not assume any vanishing of $h^1({\mathcal{O}}_X(n))$.
Observe that split bundles are excluded in this section because they cannot be stable.
Examples
========
We need the following
Let $X \subset {\mathbb{P}}^4$ be a smooth threefold of degree $d$ and let $f$ be the projection onto ${\mathbb{P}}^3$ from a general point of ${\mathbb{P}}^4$ not on $X$, and consider a normalized rank two vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ which gives rise to the pull-back ${\mathcal{F}}= f^*({\mathcal{E}})$. We want to check that $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(-i)$.\
Since $f$ is flat and $\deg (f)=d$, $f_\ast (\mathcal {O}_X)$ is a rank $d$ vector bundle. The projection formula and the cohomology of the hypersurface $X$ shows that $f_\ast (\mathcal {O}_X)$ is ACM. Thus there are integers $a_0\ge \cdots \ge a_{d-1}$ such that $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(a_i)$. Since $h^0(X,\mathcal {O}_X)=1$, the projection formula gives $a_0 = 0$ and $a_i<0$ for all $i>0$. Since $h^0(X,\mathcal {O}_X(1)) = 5 = h^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(1))+h^0(\mathbb {P}^3,\mathcal
{O}_{\mathbb {P}^3})$, the projection formula gives $a_1=-1$ and $a_i \le -2$ for all $i \ge 2$. Fix an integer $t \le d-2$ and assume proved $a_i = -i$ for all $i\le t$ and $a_i < -t$ for all $i>t$. Since $h^0(X,\mathcal {O}_X(t+1))= \binom{t+5}{4}
=
\sum _{i=0}^{t} \binom{t+4-i}{3}$, we get $a_{t+1} = -t-1$ and, if $t+1 \le d-2$, $a_i<-t-1$ for all $i>t+1$. Since $f_\ast (\mathcal {O}_X) \cong \oplus _{i=0}^{d-1} \mathcal {O}_{\mathbb {P}^3}(-i)$, the projection formula gives the following formula for the first cohomology module: $$H^i({\mathcal{F}}(n)) = H^i({\mathcal{E}}(n)) \oplus H^i({\mathcal{E}}(n-1)) \oplus ...\oplus H^i({\mathcal{E}}(n-d+1))$$ all $i$. Observe that, as a consequence of the above equalitiy for $ i = 0$, we obtain that ${\mathcal{F}}$ has the same $\alpha$ as ${\mathcal{E}}$. Moreover the pull-back ${\mathcal{F}}= f^*({\mathcal{E}})$ and ${\mathcal{E}}$ have the same Chern class $c_1$, while $c_2({\mathcal{F}}) = dc_2({\mathcal{E}})$ and therefore $\delta({\mathcal{F}}) = d\delta({\mathcal{E}})$.
**Examples**
**1.** (a stable vector bundle with $c_1 = 0$, $c_2 = 4$ on a quadric hypersurface $X$).\
Choose $d = 2$ and take the pull-back ${\mathcal{F}}$ of the stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.1. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 4$, $\alpha = 1$, $\bar\alpha = 2$, $\zeta_0 = -\frac{3}{2}$, $w_0 = -1$, $\theta = \frac{25}{4}$, $\zeta = -\frac{3}{2}+\sqrt{\frac{25}{4}} = 1 \in {\mathbb{Z}}$. From [@VV], example 4.1, we know that $h^1({\mathcal{E}}) \ne 0$. Since $H^1({\mathcal{F}}(1)) = H^1({\mathcal{E}}(1)) \oplus H^1({\mathcal{E}})$, we have: $ h^1({\mathcal{F}}(1))\ne 0$, one shift higher than it is stated in Theorem \[stable2\], 2.\
**2.** (a non-stable vector bundle with $c_1 = 0$, $c_2 = 45$ on a hypersurface of degree $5$). Choose $d = 5$ and take the pull-back ${\mathcal{F}}$ of the stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.5. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 45$, $\alpha = -3$, $\delta = 90$, $\zeta_0 = 0$. From [@VV], theorem 3.8, we know that $h^1({\mathcal{E}}(12)) \ne 0$. Since $H^1({\mathcal{F}}(16)) = H^1({\mathcal{E}}(16)) \oplus \dots \oplus H^1({\mathcal{E}}(12))$, we have: $ h^1({\mathcal{F}}(16))\ne 0$ (Theorem \[nonstable2\] states that $h^1({\mathcal{F}}(10) \ne 0$).\
**3.** (a stable vector bundle with $c_1 = -1$, $c_2 = 2$ on a quadric hypersurface).\
Let ${\mathcal{E}}$ be the rank two vector bundle corresponding to the union of two skew lines on a smooth quadric hypersurface $Q \subset {\mathbb{P}}^4$. Then its numbers are : $c_1 = -1$, $c_2 = 2$, $\alpha = 1$ and it is known that $h^1({\mathcal{E}}(n)) \ne 0$ if and only if $n = 0$.\
Observe that in this case $\theta = \frac{5}{2} \ge 0, \zeta_0 = -1$, $\bar\alpha = 1$. Therefore theorem \[stable2\] states exactly that $h^1({\mathcal{E}}(0)) \ne 0$, hence this example is sharp.\
**4.** (a non-stable vector bundle with $c_1 = 0$, $c_2 = 8$ on a quadric hypersurface).\
Choose $d = 2$ and take the pull-back ${\mathcal{F}}$ of the non-stable vector bundle ${\mathcal{E}}$ on ${\mathbb{P}}^3$ of [@VV], example 4.10. Then the numbers of ${\mathcal{F}}$ (see Notation) are: $c_1 = 0$, $c_2 = 8$, $\alpha = 0$, $\zeta_0 = -\frac{3}{2}$, $\delta = 8$. We know (see [@VV], example 4.10) that $h^1({\mathcal{E}}(2)) \ne 0, h^1({\mathcal{E}}(3)) = 0$. Since $H^1({\mathcal{F}}(3)) = H^1({\mathcal{E}}(3)) \oplus H^1({\mathcal{E}}(2))$, we have: $ h^1({\mathcal{F}}(3)) \ne 0$, exactly the bound of Theorem \[nonstable2\].
The bounds for a degree $d$ threefold in ${\mathbb{P}}^4$ agree with [@VV], where ${\mathbb{P}}^3$ is considered.
Threefolds with ${\mathrm{Pic}}(X) \ne {\mathbb{Z}}$
====================================================
Let $X$ be a smooth and connected projective threefold defined over an algebraically closed field $\mathbb {K}$. Let ${\mathrm{Num}}(X)$ denote the quotient of ${\mathrm{Pic}}(X)$ by numerical equivalence. Numerical classes are denoted by square brackets $[\,\,]$. We assume ${\mathrm{Num}}(X) \cong \mathbb {Z}$ and take the unique isomorphism $\eta : {\mathrm{Num}}(X)\to \mathbb {Z}$ such that $1$ is the image of a fixed ample line bundle. Notice that $M\in {\mathrm{Pic}}(X)$ is ample if and only if $\eta ([M])>0$.
\[boundedness\] Let $\eta : {\mathrm{Num}}(X) \to \mathbb {Z}$ be as before. Notice that every effective divisor on $X$ is ample and hence its $\eta$ is strictly positive. For any $t\in \mathbb {Z}$ set ${\mathrm{Pic}}_t(X):= \{L\in {\mathrm{Pic}}(X): \eta ([L])=t\}$. Hence ${\mathrm{Pic}}_0(X)$ is the set of all isomorphism classes of numerically trivial line bundles on $X$. The set ${\mathrm{Pic}}_0(X)$ is parametrized by a scheme of finite type ([@La], Proposition 1.4.37). Hence for each $t\in \mathbb {Z}$ the set ${\mathrm{Pic}}_t(X)$ is bounded. Let now ${\mathcal{E}}$ be a rank $2$ vector bundle on $X$. Since ${\mathrm{Pic}}_1(X)$ is bounded there is a minimal integer $t$ such that there is $B\in {\mathrm{Pic}}_t(X)$ and $h^0(E\otimes B) >0$. Call it $\alpha (E)$ or just $\alpha$. By the definition of $\alpha$ there is $B\in {\mathrm{Pic}}_{\alpha}(X)$ such that $h^0(X,{\mathcal{E}}\otimes B) >0$. Hence there is a non-zero map $j: B^\ast \to E$. Since $B^\ast$ is a line bundle and $j\ne 0$, $j$ is injective. The definition of $\alpha$ gives the non-existence of a non-zero effective divisor $D$ such that $j$ factors through an inclusion $B^\ast \to B^\ast (D)$, because $\eta ([D]) >0$. Thus the inclusion $j$ induces an exact sequence $$\label{eqa1}
0 \to B^\ast \to {\mathcal{E}}\to \mathcal {I}_Z\otimes B \otimes \det ({\mathcal{E}}) \to 0$$ in which $Z$ is a closed subscheme of $X$ with pure codimension $2$.\
Observe that $\eta([B]) = \alpha, \eta([B^*]) = -\alpha, \eta([B \otimes det({\mathcal{E}})]) = \alpha +c_1$, hence the exact sequence is quite similar to the usual exact sequence that holds true in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$.
NOTATION:\
We set $\epsilon:= \eta ([\omega _X])$, $\alpha := \alpha ({\mathcal{E}})$ and $c_1:= \eta ([\det ({\mathcal{E}})])$. So we can speak of a normalized vector bundle ${\mathcal{E}}$, with $c_1 \in \{ 0,-1\}$. Moreover we say that ${\mathcal{E}}$ is stable if $\alpha > 0$, nonstable if $\alpha \le 0$. Moreover ª$\zeta_0, \zeta, w_0, \bar \alpha, \theta$ are defined as in section 2.
Fix any $L \in {\mathrm{Pic}}_1(X)$ and set: $d = L^3 = $ degree of $X$.The degree $d$ does not depend on the numerical equivalence class. In fact, if $R$ is numerically equivalent to $0$, then $(L+R)^3 = L^3+R^3+3L^2R+3LR^2 = L^3+0+0+0 = L^3$. Then it is easy to see that the formulas for $\chi({\mathcal{O}}_X(n))$ and $\chi({\mathcal{E}}(n))$ given in section 2 still hold if we consider ${\mathcal{O}}_X \otimes L^{\otimes n}$ and ${\mathcal{E}}\otimes L^{\otimes n}$ (see [@Valenzano]).
\[a1\] (a) Assume the existence of $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$ and $h^0(X,L)>0$. Then for every integer $t>\alpha $ there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M])=t$ and $h^0(X,E\otimes M)>0$.
\(b) Assume $h^0(X,L)>0$ for every $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$. Then $h^0(X,E\otimes M)>0$ for every $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) > \alpha$.
\[a2\] Let ${\mathcal{E}}$ be a normalized rank two vector bundle and assume the existence of a spanned $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$. If char $K > 0$, assume that $\vert R \vert$ induces an embedding of $X$ outside finitely many points. Assume $$\label{eqb1}
2\alpha \le -\epsilon-3-c_1$$ and $h^1(X,{\mathcal{E}}\otimes N)=0$ for every $N\in {\mathrm{Pic}}(X)$ such that $\eta ([N]) \in \{-\alpha -c_1-1,\alpha +2+e\}$. If $h^1(X,B)=0$ for every $B\in {\mathrm{Pic}}(X)$ such that $\eta
([B])=-2\alpha -c_1$, then ${\mathcal{E}}$ splits.
If moreover $h^1(X,M)=0$ for every $M\in {\mathrm{Pic}}(X)$ then it is enough to assume that $h^1(X,{\mathcal{E}}\otimes N)=0$ for every $N\in {\mathrm{Pic}}(X)$ such that $\eta ([N]) = -\alpha -c_1-1$.
By assumption there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) = \alpha$ and $h^0(X,{\mathcal{E}}\otimes M)>0$. Set $A:= M^\ast$. We have seen in remark \[boundedness\] that ${\mathcal{E}}$ fits into an extension of the following type: $$\label{b2}
0 \to A \to {\mathcal{E}}\to \mathcal {I}_C\otimes \det ({\mathcal{E}})\otimes A^\ast \to 0$$ with $C$ a locally complete intersection closed subscheme with pure dimension $1$. Let $H$ be a general element of $\vert R\vert$ and $T$ the intersection of $H$ with another general element of $\vert R\vert$. Observe that $T$, under our assumptions, is generically reduced by Bertini’s theorem-see [@HAL], Theorem II, 8.18 and Remark II, 8.18.1. Since $R$ is spanned, $T$ is a locally complete intersection curve and $C\cap T=\emptyset$. Hence ${\mathcal{E}}\vert T$ is an extension of $\det ({\mathcal{E}})\otimes A^\ast \vert T$ by $A\vert T$. Since T is generically reduced and locally a complete intersection, it is reduced. Hence $h^0(T,M^\ast ) = 0$ for every ample line bundle $M$ on $T$. Since $\omega _T \cong (\omega _X\otimes R^{\otimes 2})\vert T$, we have $\dim (\mbox{Ext}^1(T,\det ({\mathcal{E}})\otimes A^\ast,A)
=h^0(T,\det ({\mathcal{E}})\otimes (A^\ast)^{\otimes 2} \otimes \omega _\otimes R^{\otimes 2}))\vert T)= 0$ (indeed $\eta ([\det ({\mathcal{E}})\otimes (A^\ast )^{\otimes 2}\otimes \omega _X\otimes R^{\otimes 2}])
= 2\alpha +c_1+e+2 <0$). Hence ${\mathcal{E}}\vert T \cong A\vert T\oplus (\det ({\mathcal{E}})\otimes A^\ast )\vert T$. Let $\sigma$ be the non-zero section of $({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast )\vert T$ coming from the projection onto the second factor of the decomposition just given. The vector bundle ${\mathcal{E}}\vert H$ is an extension of $\det ({\mathcal{E}})\otimes A^\ast \vert H$ by $A\vert H$ if and only if $C\cap H =\emptyset$. Since $R$ is ample, $C\cap H = \emptyset$ if and only if $C=\emptyset$. Hence we get simultaneously $C\cap H=\emptyset$ and ${\mathcal{E}}\vert H \cong A\vert H\oplus \det ({\mathcal{E}})\otimes A^\ast \vert H$ if we prove the existence of $\tau \in H^0(H,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast )\vert H)$ such that $\tau \vert T = \sigma$. To get $\tau$ it is sufficient to have $H^1(H,(E\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast )\vert H) =0$. A standard exact sequence shows that $H^1(H,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast )\vert H) =0$ if $h^1(X,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast ) =0$ and $h^2(X,({\mathcal{E}}\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast
\otimes R^\ast) =0$. Since ${\mathcal{E}}^\ast \cong {\mathcal{E}}\otimes \det ({\mathcal{E}})^\ast$, Serre duality gives $h^2(X,(E\otimes (A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast
\otimes R^\ast) =h^1(X,{\mathcal{E}}\otimes A\otimes R^{\otimes 2}\otimes \omega _X)$. Since $\eta ([A\otimes \det ({\mathcal{E}})^\ast \otimes R^\ast ])=-\alpha -c_1-1$ and $\eta ([A\otimes R^{\otimes 2}\otimes \omega _X] )= \alpha +e+2$, we get that $C=\emptyset$. The last sentence follows because $\eta ([A^{\otimes 2}\otimes \det ({\mathcal{E}})^\ast] )= -2\alpha -c_1$.
\[a3\] Instead of the smoothness of $X$ we may assume that $X$ is locally algebraic factorial, i.e. that all local rings $\mathcal {O}_{X,P}$ are factorial. This assumption seems to be essential, because without it a non zero section of $E\otimes M$ with $\eta ([M]) = \alpha (E)$ could vanish on an effective Weil divisor and hence we could not claim the existence of the exact sequence (\[b2\]).
\[a4\] Fix integers $t < z \le \alpha -2$. Assume the existence of $L \in {\mathrm{Pic}}(X)$ such that $\eta ([L])=z$ and $h^1(X,E\otimes L)=0$. If there is $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and $h^0(X,R)>0$, then there exists $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) =t$ and $h^1(X,E\otimes M)=0$. If $h^0(X,R)>0$ for every $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$, then $h^1(X,E\otimes M)=0$ for every $M\in
{\mathrm{Pic}}(X)$ such that $\eta ([M]) =t$.
The proof can follow the lines of Lemma \[leftvanishing\]. In fact consider a line bundle $R$ with $\eta([R]) = 1$ and let $H$ be the zero-locus of a non-zero section of $R$; then we have the following exact sequence: $$0 \to {\mathcal{E}}\otimes L \to {\mathcal{E}}\otimes L \otimes R \to ({\mathcal{E}}\otimes L\otimes R)_H \to 0.$$ Now observe that the vanishing of $h^1(X,{\mathcal{E}}\otimes L)$ implies that $h^0({\mathcal{E}}\otimes L \otimes R)_H = 0$. And now we can argue as in Lemma \[leftvanishing\] (see also [@VV]).
\[a6\] (a) Assume the existence of $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$ and $h^0(X,L)>0$. Then for every integer $t>\alpha $ there is $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M])=t$ and $h^0(X,E\otimes M)>0$.
\(b) Assume $h^0(X,L)>0$ for every $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) =1$. Then $h^0(X,E\otimes M)>0$ for every $M\in {\mathrm{Pic}}(X)$ such that $\eta ([M]) > \alpha$.
In all our results of sections 4 and 5 we use the vanishing of $h^1({\mathcal{O}}_X(n))$ (and by Serre duality of $h^2({\mathcal{O}}_X(n))$), $\forall n$ (or, at least, $\forall n\notin \{0,\cdots,\epsilon\}$), see Remark 4.12.
From now on we need to use similar vanishing conditions and so we introduce the following condition:
$\mathbf{(C4)}$ $h^1(X,L) =0$ for all ${\mathrm{Pic}}(X)$ such that either $\eta ([L]) <0$ or $\eta ([L]) >\epsilon$.\
Observe that $\mathbf{(C4)}$ is always satisfied in characteristic 0 (by the Kodaira vanishing theorem). In positive characteristic it is often satisfied. This is always the case if $X$ is an abelian variety ([@Mumford] p. 150).\
Observe also that, if $\epsilon \le -1$, the Kodaira vanishing and our condition put no restriction on $n$ (see also Remark 4.12).
**Example**. If (\[eqb1\]) holds, then $-2\alpha -c_1 > \epsilon$. Hence we may apply Proposition \[a2\] to $X$. In particular observe that, in the case of an abelian variety with ${\mathrm{Num}}(X) \cong \mathbb {Z}$ or in the case of a Calabi-Yau threefold with ${\mathrm{Num}}(X) \cong \mathbb {Z}$, we have $\epsilon = 0$. Notice that Proposition \[a2\] also applies to any threefold $X$ whose $\omega _X$ is a torsion sheaf.
With the assumption of condition $\mathbf{(C4)}$ the proofs of Theorems \[nonstable1\], \[nonstable2\], \[nonstable3\] can be easily modified in order to obtain the statements below (${\mathcal{E}}$ is normalized, i.e. $\eta ([\det ({\mathcal{E}})]) \in \{-1,0\}$), where, by the sake of simplicity, we assume $\epsilon \ge 0$ (if $\epsilon < 0$, $\mathbf{(C4)}$, which holds by [@SB], implies that all the vanishing of $h^1$ and $h^2$ for all $L\in {\mathrm{Pic}}(X)$ hold).
\[v1\] Assume $\mathbf{(C4)}$, $\alpha \le 0$, the existence of $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and $\zeta _0 < -\alpha -c_1-1$. Fix an integer $n$ such that $\zeta _0 < n \le -\alpha - 1 -c_1$. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge (n-\zeta _0)\delta >0$.
Observe that we should require the following conditions: $n-\alpha \notin \{0,\dots ,\epsilon \}, \epsilon-n+\alpha \notin \{0,\dots ,\epsilon \}$. But they are automatically fulfiled under the assumption that $\zeta_0 < -\alpha-c_1-1$.
\[v2\] Assume $\mathbf{(C4)}$, $\alpha \le 0$, the existence of $R\in {\mathrm{Pic}}(X)$ such that $\eta ([R])=1$ and the same hypotheses of Theorem \[nonstable2\]. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge
-S(n) >0$ ($S(n)$ being defined as in Theorem \[nonstable2\]).
\[v3\] Assumption as in Theorem \[nonstable3\]. Moreover assume $\mathbf{(C4)}$ and $n -\alpha \notin \{0,\dots ,\epsilon \}$. Fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then $h^1({\mathcal{E}}\otimes L) \ge
-\frac{d}{6}F(n+\alpha -\zeta _0)>0$ ($F$ being defined as in Theorem \[nonstable3\]).
Observe that in Theorems \[v2\] and \[v3\] we should require $n -\alpha \notin \{0,\dots ,\epsilon \}$, but the assumption $\epsilon-\alpha-c_1+1 \le n$ implies that it is automatically fulfilled. Observe that in Theorems \[v2\] and \[v3\] we require $n -\alpha \notin \{0,\dots ,\epsilon \}$, but the assumption $\epsilon-\alpha-c_1+1 \le n$ implies that the requirement is automatically fulfilled.
The proofs of the above theorems are based on the existence of the exact sequence (\[eqa1\]) and on the properties of $\alpha$. They follow the lines of the proofs given in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$. Here and in section 4 we actually need only the Kodaira vanishing (true in characteristic 0 and assumed in characteristic $p > 0$) and no further vanishing of the first cohomology.
Also the stable case can be extended to a smooth threefold with ${\mathrm{Num}}(X) \cong {\mathbb{Z}}$. Observe that the proofs can follow the lines of the proofs given in the case ${\mathrm{Pic}}(X) \cong {\mathbb{Z}}$ and make use of Remark 7.6 (which extends \[leftvanishing\]).
More precisely we have:
\[v4\] Assumptions as in \[stable1\] and fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then, if $\alpha\ge\frac{\epsilon+5-c_1}{2}$, then $h^1({\mathcal{E}}\otimes L)\ne 0$ for $w_0\le n\le \alpha-2$.
\[v4\] Assumptions as in \[stable2\] and fix $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) = n$. Then the following hold:
1)
: $h^1({\mathcal{E}}\otimes L)\ne 0$ for $\zeta_0< n < \zeta$.
2)
: $h^1({\mathcal{E}}\otimes L)\ne 0$ for $w_0\le n \le \bar\alpha-2$, and also for $n=\bar\alpha-1$ if $\zeta\notin{\mathbb{Z}}$.
3)
: If $\zeta\in{\mathbb{Z}}$ and $\alpha<\bar\alpha$, then $h^1({\mathcal{E}}\otimes N) \ne 0$, for every $N$ such that $\eta([N]) = \bar\alpha-1$.
\[a5\] The above theorems can be applied to any $X$ such that ${\mathrm{Num}}(X)$ $\cong \mathbb {Z}$, $\epsilon =0$ and $h^1(X,L)=0$ for all $L\in {\mathrm{Pic}}(X)$ such that $\eta ([L]) \ne 0$, for instance to $X = $ an abelian threefold with ${\mathrm{Num}}(X) \cong \mathbb {Z}$.
If $X$ is any threefold (in characteristic $0$ or positive) such that $h^1(X,L) = 0, \forall L \in {\mathrm{Pic}}(X)$, then we can avoid the restriction $n-\alpha \notin \{0,...,\epsilon\}$. Not many threefolds, beside any $X \subset {\mathbb{P}}^4$, fulfil these conditions.
Observe that in Theorem \[v4\] we do not assume $\mathbf{(C4)}$ (see also remark 5.8)
Observe that also in the present case (${\mathrm{Num}}(X) \cong {\mathbb{Z}}$), we have: $\delta = 0$ if and only if ${\mathcal{E}}$ splits. Therefore Remarks 4.13 and 5.9 apply here.
[00]{} L. Chiantini, P. Valabrega, Subcanonical curves and complete intersections in projective 3-space, Ann. Mat. Pura Appl. **138** (4) (1984) 309–330. L. Chiantini, P. Valabrega, On some properties of subcanonical curves and unstable bundles, Comm. Algebra **15** (1987) 1877–1887. PH. Ellia, Sur la cohomologie de certains fibr' es de rang deux sur ${\mathbb{P}}^3$, Ann. Univ. Ferrara **38** (1992) 217–227. G. Gherardelli, Sulle curve sghembe algebriche intersezioni complete di due superficie, Atti Reale Accademia d’Italia **4** (1943) 128–132. R. Hartshorne, Algebraic Geometry, GTM 52, Springer Verlag, New York, 1977. R. Hartshorne, Stable vector bundles of rank 2 on ${\mathbb{P}}^3$, Math. Ann. **238** (1978) 229–280. R. Hartshorne, Stable reflexive sheaves, Math. Ann. **254** (1980) 121–176. R. Hartshorne, Stable Reflexive Sheaves II, Invent. Math. **66** (1982) 165–190. R. Lazarsfeld, Positivity in Algebraic Geometry I, Springer, Berlin, 2004. C. Madonna, A Splitting Criterion for Rank 2 Vector Bundles on Hypersurfaces in ${\mathbb{P}}^4$, Rend. Sem. Mat. Univ. Pol. Torino **56** (2) (1998) 43–54. D. Mumford, Abelian Varieties, Oxford University Press, 1974. S. Popescu, On the splitting criterion of Chiantini and Valabrega, Rev. Roumaine Math. Pures Appl. **33** (10) (1988) 883–887. M. Roggero, P. Valabrega, Some vanishing properties of the intermediate cohomology of a reflexive sheaf on ${\mathbb{P}}^n$, J. Algebra **170** (1) (1994) 307–321. N. Shepherd-Barron, Fano Threefolds in positive characteristic, Compositio Math. **105** (1997) 237–265. P. Valabrega, M. Valenzano, Non-vanishing theorems for non-split rank $2$ bundles on ${\mathbb{P}}^3$: a simple approach, Atti Acc. Peloritana **87** (2009) 1–18 \[DOI:10.1478/C1A0901002\]. M. Valenzano, Rank $2$ reflexive sheaves on a smooth threefold, Rend. Sem. Mat. Univ. Pol. Torino **62** (2004) 235–254.
<span style="font-variant:small-caps;">Edoardo BALLICO</span>\
Dipartimento di Matematica, Università di Trento\
via Sommarive 14, 38050 Povo (TN), Italy\
e-mail: `ballico@science.unitn.it`\
<span style="font-variant:small-caps;">Paolo VALABREGA</span>\
Dipartimento di Matematica, Politecnico di Torino\
Corso Duca degli Abruzzi 24, 10129 Torino, Italy\
e-mail: `paolo.valabrega@polito.it`\
<span style="font-variant:small-caps;">Mario VALENZANO</span>\
Dipartimento di Matematica, Università di Torino\
via Carlo Alberto 10, 10123, Torino, Italy\
e-mail: `mario.valenzano@unito.it`
[^1]: The paper was written while all authors were supported by MIUR (PRIN grant) and by local funds of their Universities, and were members of INdAM-GNSAGA.
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author:
- '[M.E. Goncharov]{}'
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[On Rota-Baxter operators of non-zero weight induced by non skew-symmetric solutions of the classical Yang-baxter equation on simple anticommutative algebras.]{}
[**M.E.Goncharov**]{}
> [Abstract. ]{} Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.
>
> [**Keywords:**]{} Rota-Baxter operator, anti-commutative algebra, Lie algebra, Malcev algebra, non-associative bialgebra, classical Yang-Baxter equation.
Given an algebra A over a field $F$ and scalar $\lambda \in
F$, a linear operator $R: A \rightarrow A$ is called a Rota—Baxter of the weight $\lambda$ if for all $x,y\in A$ the following identity holds: $$\label{el}
R(x)R(y) = R(R(x)y + xR(y) + \lambda xy).$$
As an example of a Rota-Baxter operation of weight zero one can consider the operation of integration on the algebra of continuous functions on $\mathbb{R}$: the equation follows from the integration by parts formula.
An algebra with a Rota-Baxter operation is called a Rota-Baxter algebra. These algebras first appeared in the paper of Baxter [@Br]. The combinatorial properties of Rota-Baxter algebras and operations were studied in papers of F.V. Atkinson, P. Cartier, G.-C. Rota and the others (see [@Atk]-[@Car]). For basic results and the main properties of Rota-Baxter algebras see [@Guo].
There is a standard method for constructing Rota-Baxter operations of zero weight on a simple Lie algebra $L$ from skew-symmetric solutions of the classical Yang-Baxter equations (CYBE): if $r=\sum
a_i\otimes b_i$ is a skew-symmetric solution of CYBE, then one can define an operator $R$ on $L$ by $R(a)=\sum\langle a_i,a\rangle
b_i$, where $\langle \cdot, \cdot \rangle$ is a Killing form on L. It turns out that $R$ is a Rota-Baxter operator of weight 0 ([@BD]-[@STS]).
In this paper we consider the case when $L$ is a simple anti-commutative algebra and $r\in L\otimes L$ is a non skew-symmetric solution of CYBE with $L$-invariant symmetric part. It turns out that instead of the element $r$ it is more convenient to consider structure that closely connected with the solutions of CYBE(skew-symmetric or not): the structure of bialgebra (a vector space with multiplication and comultiplication).
Lie bialgebras were introduced by Drinfeld [@Drinf] for studying the solutions of the classical Yang-Baxter equation on Lie algebras. Lie bialgebras are Lie algebras and Lie coalgebras at the same time, such that comultiplication is a 1-cocycle. Every skew-symmetric solution of the classical Yang-Baxter equation induces a structure of a Lie bialgebra on the corresponding Lie algebra $L$. It is known that in this case the Drinfeld double contain a non-zero radical.
If $r$ is not a skew-symmetric solution of CYBE then the corresponding comultiplication gives a structure of a Lie bialgebra if and only if $r+\tau(r)$ is a $L$-invariant element of $L\otimes
L$. Here $\tau$ is a switch morphism.
In section 2 we consider a structure of bialgebra on an arbitrary simple finite-dimensional algebra $A$ over a field of characteristic zero with a semisimple Drinfeld double. We prove that the structure induce on $A$ Rota-Baxter operators of a non-zero weight.
In section 3 we consider anti-commutative bialgebras with semisimple non-simple Drinfeld double constructed on simple anti-commutative algebras. We prove that in this case Rota-Baxter operators from section 2 are induced by non skew-symmetric solutions of the classical Yang-Baxter equations with $ad$-invariant symmetric part. As a corollary we obtain that if $L$ is a simple Lie algebra and $r=\sum a_i\otimes b_i$ is a non skew-symmetric solutions of the classical Yang-Baxter equations such that $r+\tau(r)$ is a $L$-invariant, then an operator $R$ on $L$ defined by $R(a)=\sum\langle a_i,a\rangle b_i$ is a Rota-Baxter operator of a non-zero weight.
In the last section we use the results from the section 3 and construct Rota-Baxter operators of non-zero weight on simple non-Lie Malcev algebra.
Definitions and preliminary results.
====================================
In the paper it is assumed that the characteristic of the ground field $F$ is 0 and all spaces are supposed to be finite-dimensional.
Given vector spaces $V$ and $U$ over a field $F$, denote by $V\otimes U$ its tensor product over $F$. Define the linear mapping $\tau$ on $V$ by $\tau(\sum\limits_ia_i\otimes
b_i)=\sum\limits_ib_i\otimes a_i$. Denote by $V^*$ the dual space of $V$.
A pair $(A, \Delta)$, where $A$ is a vector space over $F$ and $\Delta : A \rightarrow A\otimes A$ is a linear mapping, is called a coalgebra, while $\Delta$ is a comultiplication.
The following definition of a coalgebra related to some variety of algebras was given in [@ANQ].
[**Definition.**]{} Let $\mathcal{M}$ be an arbitrary variety of algebras. The pair $(A, \Delta)$ is called a $\mathcal{M}$-coalgebra if $A^*$ belongs to $\mathcal{M}$.
Given $a\in A$, put $\Delta(a) = \sum a_{(1)}\otimes a_{(2)}$.
Define a multiplication on $A^*$ by $$fg(a)=\sum f(a_{(1)})g(a_{(2)}),$$ where $f,g\in A^*$, $a\in A$ and $\Delta(a)=\sum a_{(1)}\otimes
a_{(2)}$. The algebra obtained is *the dual algebra* of the coalgebra $(A, \Delta)$.
The dual algebra $A^*$ of $(A,\Delta)$ gives rise to the following bimodule actions on $A$: $$f\rightharpoonup a=\sum a_{(1)} f(a_{(2)})\text{ and
}a\leftharpoonup f=\sum f(a_{(1)}) a_{(2)},$$ where $a\in A,\ f\in A^*$ and $\Delta(a)=\sum a_{(1)}\otimes
a_{(2)}$.
Let $A$ be an arbitrary algebra with a comultiplication $\Delta$, and let $A^*$ be the dual algebra for $(A, \Delta)$. Then $A$ induces the bimodule action on $A^*$ by the formulas $$f\leftharpoondown a(b)= f(ab)\text{ and }
b\rightharpoondown f(a)= f(ab),$$ where $a,b\in A,\ f\in A^*$.
Consider the space $D(A) = A \oplus A$ and equip it with a multiplication by putting $$(a+f)(b+g)=(ab+f\rightharpoonup b+a\leftharpoonup
g)+(fg+f\leftharpoondown b+a\rightharpoondown g).$$ Then $D(A)$ is an ordinary algebra over $F$, $A$ and $A^*$ are some subalgebras in $D(A)$. It is called *the Drinfeld double*.
Let $Q$ be a bilinear form on $D(A)$ defined by $$Q(a+f,b+g)= g(a)+ f(b)$$ for all $a,b\in A$ and $f,g\in A^*$. It is easy to check that $Q$ is a nondegenerate symmetric associative form, that is $Q(xy,z)=Q(x,yz)$.
In [@Drinf] the following definition was given:
Let $L$ be a Lie algebra with a comultiplication $\Delta$. The pair $(L,\Delta)$ is called a Lie bialgebra if and only if $(L,\Delta)$ is a Lie coalgebra and $\Delta$ is a 1-cocycle, i.e., it satisfies $$\Delta([a,b])=\sum([a_{(1)},b]\otimes
a_{(2)}+a_{(1)}\otimes[a_{(2)},b])+\sum([a,b_{(1)}]\otimes
b_{(2)}+b_{(1)}\otimes[a,b_{(2)}])$$ for all $a,b\in L$.
In [@Drinf], it was proved that the pair $(L,\Delta)$ is a Lie bialgebra if and only if its Drinfeld double $D(L)$ is a Lie algebra.
There is an important type of Lie bialgebras called coboundary bialgebras. Namely, let $L$ be a Lie algebra and $r=\sum\limits_ia_i\otimes b_i$ from $(id-\tau)(L\otimes L)$, that is, $\tau(r)=-r$. Define a comultiplication $\Delta_r$ on $L$ by $$\Delta_r(a)=\sum\limits_i[a_i,a]\otimes b_i-a_i\otimes[a,b_i]$$ for all $a\in L$. It is easy to see that $\Delta_r$ is a 1-cocycle. In [@BD] it was proved that $(L,\Delta_r)$ is a Lie coalgebra if and only if the element $$C_L(r)=[r_{12},r_{13}]-[r_{23},r_{12}]+[r_{13},r_{23}]$$ is $L$-invariant. Here $[r_{12},r_{13}]=\sum\limits_{ij}[a_i,a_j]\otimes b_i\otimes b_j$, $[r_{23},r_{12}]=\sum\limits_{ij}a_i\otimes[a_j,b_i]\otimes b_j$, and $[r_{13},r_{23}]=\sum\limits_{ij} a_i\otimes a_j\otimes [b_i,b_j]$. In particular, if $$\label{lieYB}
\sum\limits_{ij}[a_i,a_j]\otimes b_i\otimes
b_j-a_i\otimes[a_j,b_i]\otimes b_j+a_i\otimes a_j\otimes
[b_i,b_j]=0,$$ then the pair $(L,\Delta_r)$ is a Lie bialgebra. The equation is called *the classical Yang-Baxter equation*.
The equation can de considered for every variety of algebras. Let $A$ be an arbitrary algebra and $r=\sum\limits_ia_i\otimes b_i\in A\otimes A$. Then the equation $$\label{YB}
C_A(r)=r_{12}r_{13}+r_{13}r_{23}-r_{23}r_{12}=0$$ is called the classical Yang-Baxter equation on $A$. Here the subscripts specify the way of embedding $A\otimes A$ into $A\otimes
A\otimes A$, that is, $r_{12}=\sum\limits_{i} a_i\otimes b_i\otimes
1$, $r_{13}=\sum_i a_i\otimes 1\otimes b_i$, $r_{23}=\sum_i 1\otimes
a_i\otimes b_i$. Note that $C_A(r)$ is well defined even if $A$ is non-unital. The equation for different varieties of algebras was considered in [@Zhelyabin98; @Aquiar; @Zhelyabin; @Polishchuk; @Gme].
An element $r=\sum\limits_{i}a_i\otimes b_i\in A\otimes A$ induces a comultiplication $\Delta_r$ on $A$: $$\Delta_r(a)=\sum\limits a_ia\otimes b_i-a_i\otimes ab_i$$ for all $a\in A$.
In [@Zhelyabin97] the following definition of bialgebra in sense of Drinfeld for any variety of algebras was given:
[**Definition.**]{} Let $\mathcal{M}$ be an arbitrary variety of algebras and let $A$ be an algebra from $\mathcal{M}$ with a comultiplication $\Delta$. The pair $(A,\Delta)$ is called *an $\mathcal{M}$-bialgebra (in the sense of Drinfeld)* if its Drinfeld double $D(A)$ belongs to $\mathcal{M}$.
For Jordan, associative, alternative and Malcev bialgebras it is knows that if $r$ is a skew-symmetric solution of the classical Yang-Baxter equation on $A$, then $(A,\Delta_r)$ is a bialgebra of corresponding variety.
In [@Zhelyabin] it was proved that if $A$ is an arbitrary simple algebra equipped with a comultiplication $\Delta$ and $I$ is a proper ideal of $D(A)$, then $dim(I)=dim(A)$.
If $L$ is a simple finite-dimensional Lie algebra over the field of complex numbers, then the Drinfild double $D(L)$ is either contain a radical $R$ such that $D(L)=L\oplus R$ (semidirect sum) and $R^2=0$ or $D(L)=L_1\oplus L_2$ — direct sum of two simple ideals that are isomorphic to $L$ [@stolin]. In the same paper bialgebra structures on $L$ in both cases were described.
If $(L,\Delta)$ is a Lie bialgebra and the radical $R$ of $D(L)$ is not zero then the comultiplication is induced by a skew-symmetric solution of the classical Yang-Baxter equation, that is $\Delta=\Delta_r$ for some $r=\sum a_i\otimes b_i\in
(id-\tau)(L\otimes L)$ such that $C_L(r)=0$ (see [@stolin] for the case when $F$ is the field of complex numbers or [@GMM] for the more general case when $L$ is a Malcev algebra). Define an operator $R: L\rightarrow L$ by: $$\label{RB}
R(x)=\sum\limits \langle a_i,x \rangle b_i,$$ where $\langle\cdot,\cdot\rangle$ is the Killing form on $L$. It is well known that $R$ is a Rota-Baxter operator of weight 0 on $L$ (see [@BD]-[@STS]).
Bialgebras with semisimple Drinfeld double and Rota-Baxter operators of non-zero weights.
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In this section $A$ is an arbitrary simple finite-dimensional algebra over a field of characteristic zero equipped with a comultiplication $\Delta$ such that $(A,\Delta)$ is a bialgebra such that $D(A)$ is a direct sum of simple ideals.
Since $dim(I)=dim(A)$ for every proper ideal $I$ of $D(A)$ we have that $D(A)=A_1\oplus A_2$ where $A_i$ are simple algebras and $dim
A_i=dim A$ [@Zhelyabin].
\[p1\] There are two linear mapping $\phi_i:A^*\rightarrow A$, $i=1,2$ such that for all $f\in A^*:\ f-\phi_i(f)\in A_i$. Moreover, $A_i=\{f-\phi_i(f)|\ f\in A^*\}$.
Since $A$ is simple then intersections $A$ with $A_i$ are equal to zero. It means that $D(A)=A\oplus A_1=A\oplus A_2$ (as vector spaces) and for every $f\in A^*$ there are unique elements $a_i\in
A(i=1,2)$ such that $f+a_i\in A_i$. For $i=1,2$ define $\phi_i$ as $$\phi_i(f)=-a_i$$ if $f+a_i\in A_i$. Since $dimA_i=dim A^*$ we have that $A_i=\{f-\phi_i(f)|\ f\in A^*\}$.
Consider $a\in A$. There are unique $l_i\in L_i(i=1,2)$ such that $a=l_1+l_2$. By proposition \[p1\] $l_i=f_i+\phi_i(f_i)(i=1,2)$ and $a=(f_1-\phi_1(f_1))+(f_2-\phi_2(f_2))$. Thus, $f_1=-f_2$ and we proved that for every $a\in A$ there is $f\in A^*$ such that: $$\label{eq2}
a=-\phi_1(f)+\phi_2(f).$$
If for some $a\in A$ there are two elements $f_1\in A^*$ and $f_2\in
A^*$ such that $a=-\phi_1(f_i)+\phi_2(f_i)$ then we get $(f_1-\phi_1(f_1))-(f_1-\phi_2(f_1))=(f_2-\phi_1(f_2))-(f_2-\phi_2(f_2))$. Since $f-\phi_i(f)\in A_i$ we obtain that $f_1=f_2$
Define a map $\psi:L\rightarrow A^*$ as $\psi(a)=f$ if $a=-\phi_1(f)+\phi_2(f)$. It is easy to see that $\psi$ is an isomorphism of vector spaces.
We will need the following properties of the maps $\phi_i$ and $\psi$:
\[p2\] 1. For all $f,g\in A^*$ and $i=1,2$: $$\label{eq3}
\phi_i(fg)=\phi_i(f)\phi_i(g).$$
2\. For all $a,b\in A$: $$\label{eq4}
\psi(ab)=\psi(a)\leftharpoondown b=a\rightharpoondown \psi(b).$$
3\. For all $a,b\in A$: $$\label{eq5}
\psi(ab)=\psi(a)\psi(b)-\phi_1(\psi(a))\rightharpoondown\psi(b)-\psi(a)\leftharpoondown\phi_1(\psi(b)).$$ and $$\label{eq6}
\psi(ab)=-(\psi(a)\psi(b)-\phi_2(\psi(a))\rightharpoondown\psi(b)-\psi(a)\leftharpoondown\phi_2(\psi(b))).$$
Take $f,g\in A^*$ and fix $i=1,2$. From the definition of $\phi_i$ we have that $f-\phi_i(f)=p\in A_i$ and $g-\phi_i(g)=q\in A_i$. Then, since $A_i$ is an ideal of $D(A)$, $fg=\phi_i(f)\phi_i(g)+s$ for some $s\in A_i$. In means that $fg-\phi_i(f)\phi_i(g)\in A_i$ and by the definition of $\phi_i$ we conclude that $\phi_i(fg)=\phi_i(f)\phi_i(g)$.
Let us prove . Let $a,b\in A$. Then $$ab=(\psi(a)-\phi_1(\psi(a)))b-(\psi(a)-\phi_2(\psi(a)))b=$$ $$(\psi(a)\leftharpoondown b+\psi(a)\rightharpoonup
b-\phi_1(\psi(a))b)-(\psi(a)\leftharpoondown
b+\psi(a)\rightharpoonup b-\phi_2(\psi(a))b).$$ Since $A_1$ is an ideal, the expression in the first brackets lies in $A_1$. Thus, $\psi(ab)=\psi(a)\leftharpoondown b$ by the definition of the map $\psi$. Similar arguments show that $\psi(ab)=a\rightharpoondown \psi(b)$ and is proved.
In order to prove and consider elements $a,b\in A$. We have: $$a=(\psi(a)-\phi_1(\psi(a)))+(-\psi(a)+\phi_2(\psi(a)))$$ $$b=(\psi(b)-\phi_1(\psi(b)))+(-\psi(b)+\phi_2(\psi(b)))$$ Multiplying $a$ and $b$ in $D(A)$ we get: $$ab=(\psi(a)\psi(b)-\phi_1(\psi(a))\rightharpoondown\psi(b)-\psi(a)\leftharpoondown\phi_1(\psi(b))+x_1)+$$ $$((\psi(a)\psi(b)-\phi_2(\psi(a))\rightharpoondown\psi(b)-\psi(a)\leftharpoondown\phi_2(\psi(b)))+x_2)$$ where $x_i\in A$ ($i=1,2$). This proves and \[eq5\].
\[t1\] Let $R:L\rightarrow A$ be an operator defined as $$\label{d1}
R(a)=\phi_1(\psi(a)).$$ Then $R$ is a Rota-Baxter operator of weight 1 on $A$.
Using - and for all $a,b\in A$ we compute: $$R(R(a)b+aR(b)+ab)-R(a)R(b)=$$ $$=\phi_1(\psi(R(a)b+aR(b)+ab)-\phi_1(\psi(a))\phi_1(\psi(b))=$$ $$=\phi_1(R(a)\rightharpoondown \psi(b)+\psi(a)\leftharpoondown
R(b)+\psi(ab)-\psi(a)\psi(b))=0.$$
Thus, $R$ is a Rota-Baxter operator of weight 1 on $A$.
Using similar arguments one can proof the following:
Let $Q$ be an operator $Q:A\rightarrow A$ defined as $$\label{d2}
Q(a)=\phi_2(\psi(a)).$$ Then $Q$ is a Rota-Baxter operator of weight -1 on $A$.
Rota-Baxter operators and classical Yang-Baxter equation for simple anti-commutative algebras.
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Let $L$ be a simple finite-dimensional anti-commutative algebra over a field $F$ of characteristic zero and $r=\sum a_i\otimes b_i\in
L\otimes L$ is a solution of the classical Yang-Baxter equation on $L$. Recall, that if $\Delta$ is a comultiplication on $L$ then bialebra $(L,\Delta)$ is anti-commutative if and only if the Drinfeld double is anti-commutative.
For convenience we will call an algebra $A$ semisimple if $A$ is a direct sum of simple ideals.
Let $(L,\Delta)$ be a structure of an anti-commutative bialgebra on a simple anti-commutative algebra $L$ with semisimple non-simple Drinfeld double and $R$ — the Rota-Baxter operator defined as in . Then there is a non skew-symmetric solution of the classical Yang-Baxter equation $r=\sum a_i\otimes b_i$ such that $r+\tau(r)$ is $L-invariant$ and a non-degenerate associative symmetric bilinear form $\omega$ such that for all $a\in L$: $$\label{RBN}
R(a)=\sum\limits_i \omega(a_i,a)b_i.$$
Since $L$ is simple $D(L)=L_1\oplus L_2$. Consider maps $\phi_i:L^*\rightarrow L(i=1,2)$ defined in proposition 1. Since $L$ is a finite-dimensional algebra there is an element $r_1=\sum\limits_i a_i\otimes b_i\in L\otimes L$ such that $\phi_1(f)=\sum f(a_i)b_i$ for all $f\in L^*$.
Similarly, there is an element $r_2=\sum\limits_i c_i\otimes d_i\in
L\otimes L$ such that $\phi_2(f)=\sum f(c_i)d_i$. Since $\phi_1$ is a homomorphism for all $f,g\in L$ we have $$\phi_1([f,g])=-\sum [f,g](a_i)b_i=-\sum
f([a_j,a_i])g(b_j)b_i+f(a_j)g([b_j,a_i])b_i=$$ $$=[\phi_1(f),\phi_1(g)]=\sum f(a_j)g(a_i) [b_j,b_i].$$ Therefore $r_1$ is a solution of . Similar arguments show that $r_2$ is also a solution of .
Since $L_1$ and $L_2$ are simple and the form $Q$ is associative, $Q(L_1,L_2)=0$. Therefore for all $f,g\in L^*$ we have $Q(f-\phi_1(f), g-\phi_2(g))=0$. Hence, $$\sum\limits_i \langle f\otimes g, a_i\otimes b_i+d_i\otimes
c_i\rangle=0.$$ Consequently, $$\label{usl1}
r_1+\tau(r_2)=0.$$
Also we have $(f-\phi_1(f))(g-\phi_2(g))=0$. Therefore, $$fg-f\leftharpoondown \phi_2(g)-
\phi_1(f)\rightharpoondown g=0.$$ The last equality means that for all $a\in \mathbb{M}$ $$fg(a)=\sum\limits_i f(g(c_i)d_ia)+g(f(a_i)ab_i).$$ $$\langle f\otimes g, \Delta(a)\rangle=\langle f\otimes g,
\sum\limits_i d_ia\otimes c_i+a_i\otimes ab_i\rangle.$$
Using we finally obtain $$\langle f\otimes g, \Delta(a)\rangle=-\langle f\otimes g,
\sum\limits_i a_ia\otimes b_i-a_i\otimes ab_i\rangle.$$ Thus, $\Delta=-\Delta_{r_1}$.
Anti-commutativity of $D(L)$ is equivalent to $\tau(\Delta_{r_1}(a))=-\Delta_{r_1}(a)$. Therefore $[r_1+\tau(r_1),a]=0$ for all $a\in L$ and $r_1+\tau(r_1)$ is $L$-invariant.
Define a form $\omega(\cdot,\cdot)$ on $L$ by:
$$\omega(a,b)=Q(\psi(a),b)$$ for all $a,b\in L$. It is clear that $\omega$ is bilinear and non-degenerate. Let us prove that $\omega$ is associative and symmetric.
Let $a,b\in L$ and $f_1,f_2\in L^*$ such that $\psi(a)=f_1$ and $\psi(b)=f_2$. Since $Q(L_1,L_2)=0$ we have:
$$0=Q(f_1-\phi_2(f_1),f_2-\phi_1(f_2))=-Q(f_1,\phi_1(f_2))-Q(\phi_2(f_1),f_2).$$
Similarly one can prove that $$Q(f_1,\phi_2(f_2))+Q(\phi_1(f_1),f_2)=0.$$
Summing up the last two equations and using we obtain that $Q(f_1,b)-Q(a,f_2)=0$. Thus, $\omega(a,b)=\omega(b,a)$ for all $a,b\in L$.
Let $a,b,c\in L$. Using and associativity of the form $Q$ we compute: $$\omega([a,b],c)=Q(\psi([a,b]),c)=Q(\psi(a)\leftharpoondown
b,c)=Q([\psi(a),b],c)=$$ $$=Q(\psi(a),[b,c])=\omega(a,[b,c]).$$
Thus, $\omega$ is a bilinear non-degenerate symmetric associative form on $L$. And for all $a\in L$ we have $$R(a)=\phi_1(\psi(a))=\sum Q(\psi(a),a_i)b_i=\sum \omega(a,a_i)b_i.$$
Let $(L,\Delta)$ be a structure of an anti-commutative bialgebra on simple Lie algebra $L$ with semisimple non-simple Drinfeld double and $R$ — the Rota-Baxter operator defined as in . Then there is a non skew-symmetric solution of the classical Yang-Baxter equation $r=\sum a_i\otimes b_i$ such that $r+\tau(r)$ is $L-invariant$ and for all $a\in L$: $$R(a)=\sum\limits_i \langle a_i,a\rangle b_i.$$ Here $\langle\cdot,\cdot \rangle$ is the Killing form on $L$.
By theorem 3 there is a non-skew symmetric solution of CYBE $r_1$ with $L$-invariant symmetric part and $R(a)=\sum \omega(a,a_i)b_i$ for some non-degenerate associative symmetric bilinear form $\omega$ on $L$. But since $L$ is simple, there is a non-zero $\lambda\in F$ such that $\omega(a,b)=\lambda\langle a,b\rangle$. Therefore $$R(a)=\sum \omega(a,a_i)b_i=\lambda \sum \langle a, a_i\rangle b_i.$$
It is left to define $r$ as $r=\frac{1}{\lambda}r_1$ to prove the theorem.
Let $L$ be a simple anti-commutative algebra and $r=\sum\limits_ia_i\otimes b_i$ is a non skew-symmetric solution of CYBE such that $r+\tau(r)$ is $L$ invariant. Then there is a non-degenerate symmetric associative bylinear form $\omega$ on $L$ such that an operator $R:L\rightarrow L$ defined as $$R(a)=\sum\limits_i \omega(a_i,a) b_i$$ is a Rota-Baxter operator of non-zero weight.
Define a comultiplication $\Delta_r$ on $L$ as $$\Delta_r(a)=[r,a]$$ for all $a\in L$. Since $r+\tau(r)$ is $L$ invariant we have that $[r,a]=-[\tau(r),a]$ for all $a\in L$. Therefore $\tau(\Delta(a))=-\Delta(a)$ and $D(L)$ is anti-commutative.
We want to prove that Drinfeld double of the bialgebra is semisimple and non-simple. For this consider a map $\phi_1:L^*\rightarrow L$ defined as $$\label{m1}
\phi_1(f)=-\sum\limits_i f(a_i)b_i$$
Since $r$ is a solution of , $\phi_1$ is a homomorphism.
Consider a subspace $L_1=\{ f+\phi_1(f)|\ f\in L^*\}$. We want to to prove that $L_1$ is an ideal of $D(L)$. For every $a\in L$ we have: $$[f+\phi_1(f),a]=f\leftharpoondown a+f\rightharpoonup
a+[\phi_1(f),a]=$$ $$=f\leftharpoondown a +\sum f([a_i,a])b_i+\sum
f(a_i)[b_i,a]-f(a_i)[b_i,a]=f\leftharpoondown a +\sum f([a_i,a])b_i$$ On the other hand, $$\phi_1(f\leftharpoondown a)=-\sum (f\leftharpoondown
a)(a_i)b_i=-\sum f([a,a_i])b_i=\sum f([a_i,a])b_i$$ and $[f+\phi_1(f),a]\in L_1. $
Now take $g\in L^*$. We have $$[f+\phi_1(f),g]=fg+\phi_1(f)\rightharpoondown
g+\phi_1(f)\leftharpoonup g$$ Since $\phi_1$ is a homomorphism we have: $$\phi_1([f,g]+\phi_1(f)\rightharpoondown
g)=[\phi_1(f),\phi_1(g)]+\phi_1((f)\rightharpoondown g)=$$ $$=\sum
f(a_i)g(a_j)[b_i,b_j]-f(a_i)g([a_j,b_i])b_j= \phi_1(f)\leftharpoonup
g$$ and $L_1$ is a proper ideal of $D(L)$. By definition $dim
(L_1)=dim(L)$.
If $V\subset D(L)$ then by $V^{\perp}$ denote the complement of $V$ with respect to $Q$, that is $V^{\perp}=\{l\in D(L)|\ Q(l,V)=0\}$. If $V$ is a proper subalgebra in $D(L)$ then $V^{\perp}$ is a proper ideal of $D(L)$.
Consider $L_1^{\perp}$. Since $L_1$ is an ideal of $D(L)$ then $L_1^{\perp}$ is also an ideal if $D(L)$. It means that $I=L_1\cap L_1^{\perp}$ is a proper ideal of $D(L)$ and therefore $I=L_1$ or $I=0$.
Since $r+\tau(r)\neq 0$ there is $h=\sum f_j\otimes
g_j\in L^*\otimes L^*$ such that $\sum\limits_{i,j}
f_j(a_i)g_j(b_i)+f_j(b_i)g_j(a_i)\neq 0$. Then $\sum
Q(f_j+\phi_1(f_j),g_j+\phi_1(g_j))\neq0$ and $I=L_1\cap
L_1^{\perp}=0$. Therefore $D(L)=L_1\oplus L_1^{\perp}$ and $L_1^{\perp}$ is isomorphic to the quotient algebra $D(L)/L_1$.
On the other hand, $D(L)=L\oplus L_1$ (as vector spaces), therefore $L$ is also isomorphic to the quotient algebra $D(L)/L_1$. Thus, $L$ and $L_1^{\perp}$ are isomorphic. Similar arguments show that $L$ is isomorphic to $L_1$ and $D(L)$ is a sum of two simple ideals. Thus, $D(L)$ is a semisimple non-simple algebra. And the statement of the theorem follows from the definition of maps $\phi_i$ and theorems 1 and 3.
Let $L$ be a simple Lie algebra and $r=\sum\limits_ia_i\otimes b_i$ is a non skew-symmetric solution of CYBE such that $r+\tau(r)$ is $L$ invariant. Then an operator $R:L\rightarrow L$ defined as $$R(a)=\sum\limits_i \langle a_i,a\rangle b_i$$ is a Rota-Baxter operator of non-zero weight. Here $\langle\cdot,\cdot \rangle$ is the Killing form on $L$.
[**Remark.**]{} If $r$ is a non skew-symmetric solution of the classical Yang-Baxter equation such that $\tau(r)+r$ is $L$-invariant, then so is an element $r_1=\tau(r)$. Elements $r$ and $-r_1$ induce the same bialgebra structure on $L$. And if $Q$ is a Rota-Baxter operator with respect to $r_1$ then $Q$ has weight $\lambda$ and $R+Q=-\lambda id$, where $\lambda$ is a weight of operator $R$ and $id$ is the identity operator on $L$. This is not surprising since if $R$ is a Rota-Baxter operator of weight 1, then so is $-id-R$.
[**Example 1.**]{}
Let $L=sl_2$, $x,h,y$ is the standard basis of $L$, that is $hx=2x$, $hy=-2y$, $xy=h$.
Consider an element $$r=\alpha(h\otimes x-x\otimes h)+\frac{1}{4}h\otimes h+x\otimes y,$$ where $\alpha\in F$. Then for every $\alpha\in F$ $r$ is a solution of the classical Yang-Baxter equation and therefore induces on $L$ a structure of a Lie bialgebra with semisimple Drinfeld double. By theorem 4 the operator $R$ defined as is a Rota-Baxter operator of non-zero weight.
We have: $$R(x)=0,\ R(h)=2h+8\alpha x,\ \ R(y)=4(y-\alpha h).$$
Direct computations shows that $R$ is a Rota-Baxter operator of weight -4.
In order to compute the second Rota-Baxter operator $Q$ we need to consider $$r_1=\tau(r)=-\alpha(h\otimes x-x\otimes h)+\frac{1}{4}h\otimes
h+y\otimes x.$$
Thus, $$Q(x)=4x,\ Q(h)=-8\alpha x+2h,\ Q(y)=4\alpha h$$ and $Q$ is a Rota-Baxter operator of weight -4. Note that $Q+R=4id$ where $id$ is the identity operator on $L$.
The following example shows that in general a non skew-symmetric solutions of the CYBE not necessary induces a Rota-Baxter operator of non-zero weight.
[**Example 2.**]{} Let $L=sl_2(\mathbb{C})$ and let $x,h,y$ be the standard basis of $L$, that is $[h,x]=2x,\ [h,y]=-2y,\ [x,y]=h$. Consider an element $r=x\otimes x$. Obviously, $r$ is a non skew-symmetric solution of . The corresponding operator $R$ acts as follow: $$R(x)=0,\ R(h)=0,\ R(y)=4x$$ and is a Rota-Baxter operator of zero weight.
Rota-Baxter operators on the split simple Malcev algebra.
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In this section we consider simple non-Lie Malcev algebra.
Malcev algebras were introduced by A.I. Malcev [@M55] as tangent algebras for local analytic Moufang loops. The class of Malcev algebras generalizes the class of Lie algebras and has a well developed theory [@KuzSh].
An important example of a non-Lie Malcev algebra is the vector space of zero trace elements of a Caley-Dickson algebra with the commutator bracket multiplication [@S62; @K68]. In [@versh] some properties of Malcev bialgebras were studied. In particular, there were found conditions for a Malcev algebra with a comultiplication to be a Malcev bialgebra. In [@GMM] it was found a connection between solutions of the classical Yang-Baxter equation on Malcev algebras with Malcev bialgebras.
An anticommutative algebra is called a Malcev algebra if for all $x,y,z\in M$ the following equation holds: $$\label{mal1}
J(x,y,xz)=J(x,y,z)x,$$ where $J(x,y,z)=(xy)z+(yz)x+(zx)y$ is the jacobian of elements $x,y,z$.
In [@GMM] it was proved that non skew-symmetric solutions of the classical Yang-Baxter equation on simple Malcev algebra $M$ induces on $M$ a structure of Malcev biialgebra with semisimple Drinfeld double.
Let $M$ is a simple Malcev algebra. Then $M$ is a simple Lie algebra or the 7-dimentional Malcev algebra isomorphic to the commutator algebra of traceless elements of the split Caley-Dixon algebra [@Kuz].
[**Example 3**]{}. Let $\mathbb{M}$ be the simple Malcev algebra over the field of complex numbers $\mathbb{C}$. In this case $\mathbb{M}$ has a basis $h,x,x',y,y',z,z'$ with the following table of multiplication: $$hx=2x,\ hy=2y,\ hz=2z,$$ $$hx'=-2x',\ hy'=-2y',\ hz'=-2z',$$ $$xx'=yy'=zz'=h,$$ $$xy=2z',\ yz=2x',\ zx=2y',$$ $$x'y'=-2z,\ y'z'=-2x,\
z'x'=-2y.$$ The remaining products are zero. In [@GMM] it was proved that up to automorphism any non skew-symmetric solution of the classical Yang-Baxter equation $r$ has the following form: $$r=r_0+\frac{1}{4}h\otimes h+x\otimes x'+y'\otimes y+z\otimes z'$$ where $$r_0=\alpha (h\otimes x-x\otimes h)+ \beta(h\otimes y'-y'\otimes
h)+\gamma(h\otimes z-z\otimes h) +$$ $$\delta(x\otimes y'-y'\otimes x)- 2\beta(x\otimes z-z\otimes
x)+\mu(y'\otimes z-z\otimes y').$$ Scalars $\alpha,\beta,\gamma,\delta,\mu$ are arbitrary. The corresponding normalized(of weight -1) Rota-Baxter operator is the following: $$R(h)=\dfrac{1}{2}h+2\alpha x+2\beta y'+2\gamma z,\ R(x)=0,\
R(x')=x'-\alpha h+\delta y'-2\beta z$$ $$R(y)=y-\beta h-\delta x+\mu z,\ R(y')=R(z)=0,\ R(z')=z'-\gamma
h+2\beta x-\mu y'.$$
[1]{}
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---
abstract: 'The potential of large tactile arrays to improve robot perception for safe operation in human-dominated environments and of high-resolution tactile arrays to enable human-level dexterous manipulation is well accepted. However, the increase in the number of tactile sensing elements introduces challenges including wiring complexity, power consumption, and data processing. To help address these challenges, we previously developed a tactile sensing technique based compressed sensing that reduces hardware complexity and data transmission, while allowing accurate reconstruction of the full-resolution signal. In this paper, we apply tactile compressed sensing to the problem of object classification. Specifically, we perform object classification on the compressed tactile data. We evaluate our method using BubbleTouch, our tactile array simulator. Our results show our approach achieves high classification accuracy, even with compression factors up to 64.'
author:
- 'Brayden Hollis, Stacy Patterson, and Jeff Trinkle[^1]'
bibliography:
- 'ICRA17.bib'
title: '**Compressed Learning for Tactile Object Classification** '
---
INTRODUCTION
============
For robots to reliably and safely function in unstructured environments, they need to perceive and react to the environment. Sensors that sense a robot’s surroundings from a distance, such as vision sensors, are very useful and commonly employed, but when it comes to physical interactions, these sensors only offer a limited perspective of what is happening. Tactile sensors, sensors that sense the world through direct contact (e.g., force and temperature), greatly increase a robot’s understanding of its interactions. The concept of putting tactile sensors on a large portion of robots’ surfaces, which we refer to as *tactile skins*, has become its own research area.
There are numerous challenges with fully realizing tactile skins. To cover the various surfaces at useful resolutions requires a large number of individual sensing elements, called *taxels*, on the order of 1,000’s to 1,000,000’s [@dahiya10tactile]. Furthermore, the information needs to be gathered and processed at high rates, up to $1kHz$ for fine force control [@dahiya10tactile]. In conflict with these desirable features, there is limited room for the wiring of these systems, especially for skins designed separately from the robot as an add-on. In addition, tactile sensors tend to be noisy, in part from contact with the environment causing misalignments.
Even with these challenges, there are a number of potential benefits that make tactile skins worth pursuing. One application of interest is tactile object recognition. Often object recognition can be performed with other sensors, such as vision, but this is not always practical or possible. For instance, a robot trying to pull items out of a crowded cupboard may not be able to get a clear view of the item of interest due to the shelf being above or below the robot’s vision system or other objects being in the way. Using tactile sensors and tactile object recognition, the robot could find the object and retrieve it with less manipulation of the environment than grabbing items one at a time for performing visual object recognition. Another example is a rescue robot. In some disasters scenarios, visibility is limited due to smoke or dust. A rescue robot equipped with tactile sensors could still function and navigate using tactile object recognition to identify important landmarks, such as door knobs, or determine whether objects are movable or fixed, as was done by Bhattacharjee et al. in [@Bhattacharjee2012].
In previous work [@Hollis2016], we proposed using compressed sensing for data acquisition in tactile skins. Compressed sensing simultaneously samples and compresses signals. Under appropriate assumptions, the full signal can be recovered exactly or near-exactly from the compressed signal. Our approach reduces the amount of data that needs to be gathered and transferred from the tactile skin, with little to no loss of signal resolution. Fewer measurements imply increases in the signal acquisition rate, and, by performing compression in hardware, the amount of wiring can be reduced. Further, our experiments show that the reconstructed signal exhibits less noise than the raw data sampled from the noisy taxels.
In this paper, we demonstrate additional benefits of compressed sensing for tactile skins by using the compressed signals for tactile object classification. Object classification is a sub-problem of object recognition where objects are classified using a finite candidate set. We use a soft-margin support vector machine (SVM) to classify objects from their compressed tactile signals. Our tactile signals are compressed from snapshots of a tactile skin with a square array of taxels pressed onto the objects from above, similar to what might happen in the cupboard example. Direct use of the compressed signals for classification reduces processing time because the signal reconstruction phase is omitted. Further, it lowers the dimensionality of signals used for classification, which reduces processing time for both training and using the classifier. Finally, for applications where the original signal may be required after classification is performed, for example in grasping, the full tactile signals can be recovered it from the compressed signals.
Tactile object classification is an active area of research. A number of works manipulate or make multiple contacts with the objects while gathering the tactile data [@Russell2000; @Heidemann2004; @Schopfer2007; @Schopfer2009; @Takamuku2008; @Schneider2009; @Gorges2010; @Hosoda2010; @Pezzementi2011; @Schmitz2014]. These extended interactions allow them to obtain multiple tactile perspectives of the object, which reduces uncertainty. We utilize individual tactile snapshots because for some applications, like the cupboard example, extended manipulation is not practical due to space or time constraints. Additionally, for a number of these cases, extended interaction is not necessary because most objects have a limited number of stable poses.
Jimenez et al. also use single tactile snapshots for their classification [@Jimenez1997]. They still handle pose uncertainty, though, by pre-processing the full raw signal so all the observations are centered and oriented the same way in the tactile image. The pre-processed data is then used in a neural network for classification. We perform no pre-processing on our signals as this would be very challenging, if not impossible, in the compressed domain. Nonetheless, we achieve high classification accuracy.
Previous works also differ in the types of data used for classification. Some approaches use the full raw tactile data [@Jimenez1997; @Hosoda2010; @Schmitz2014]. A number of the methods reduce the dimensionality of the data before, or as part of, the machine learning process. Some manually reduce the dimension by selecting features such as average force and contact area [@Schopfer2009; @Bhattacharjee2012; @Drimus2014]. Others use automated methods to reduce the dimension, for instance, Self-Organizing Maps [@Takamuku2008; @Gorges2010] and Principle Component Analysis [@Heidemann2004; @Schopfer2007; @Pezzementi2011]. In our approach, the dimension of the data is reduced as part of the acquisition process, which may be done in hardware [@Duarte2011]. This dimension reduction is not a pre-processing step.
The rest of the paper is organized as follows. We give a general overview of SVMs in Section \[background.sect\]. We review compressed sensing theory and our work in applying it to tactile skins in Section \[cs.sect\]. Section \[methods.sect\] describes compressed learning theory and our application of it to tactile skins. Our experiments and data are explained in Section \[setup.sect\] and \[data.sect\], respectively. We present our results in Section ref[results.sect]{}, and in Section \[concl.sect\], we conclude with some final thoughts and ideas for future work.
BACKGROUND {#background.sect}
==========
Support Vector Machines are a popular classification tool in machine learning. They classify the observations by finding a hyperplane that separates the training data into its two classes such that the distance between the hyperplane and the closest observations is maximized [@Burges1998]. This distance is called the *margin* and the closest observations are called *support vectors*, as they are the observations that determine the classifier. To find the hyperplane, one solves the following minimization problem, $$\label{svm.eq}
\begin{aligned}
& \underset{b,{\ensuremath{\textbf{w}}}}{\text{minimize}}
& & \frac{1}{2}{\ensuremath{\textbf{w}}}^T{\ensuremath{\textbf{w}}} \\
& \text{subject to}
& & \ell_i({\ensuremath{\textbf{w}}}^T{\ensuremath{\textbf{x}}}_i + b) \geq 1\\
& & &\text{for }i = 1, \dots, N,
\end{aligned}$$ where $N$ is the number of observations; ${\ensuremath{\textbf{x}}}_i \in {\mathbb{R}}^n$, for $i = 1$ to $N$, are the observations; $\ell_i \in \{-1,1\}$, for $i = 1$ to $N$, are the class labels; ${\ensuremath{\textbf{w}}}$ is a vector orthogonal to the separating hyperplane; and $b=-{\ensuremath{\textbf{w}}}^T{\ensuremath{\textbf{x}}}_0$ for any point ${\ensuremath{\textbf{x}}}_0$ on the hyperplane. This optimization problem is a convex quadratic program, a well studied class of problems with many implemented solvers. Once (\[svm.eq\]) is solved for ${\ensuremath{\textbf{w}}}^*$ and $b^*$, to classify a point $\hat{{\ensuremath{\textbf{x}}}}$, one simply solves $\text{sign}({\ensuremath{\textbf{w}}}^T\hat{{\ensuremath{\textbf{x}}}}+b)$.
Often it is not possible to completely separate the data with a hyperplane, so SVMs can been modified to allow some observations to be misclassified as follows: $$\label{softsvm.eq}
\begin{aligned}
& \underset{b,{\ensuremath{\textbf{w}}}}{\text{minimize}}
& & \frac{1}{2}{\ensuremath{\textbf{w}}}^T{\ensuremath{\textbf{w}}} + C\sum_{i=1}^N\xi_i\\
& \text{subject to}
& & \ell_i({\ensuremath{\textbf{w}}}^T{\ensuremath{\textbf{x}}}_i + b) \geq 1-\xi_i\\
& & & \xi_i \geq 0\\
& & &\text{for } i = 1, \dots, N,
\end{aligned}$$ where $\xi_i$ is the amount the $i$th observation violates the margin, known as the *hinge loss*, and $C$ is a parameter to balance between maximizing the margin and reducing the margin violations. This modification is known as soft-margin SVMs.
When training soft-margin SVMs, cross-validation is used to tune the parameter $C$. Cross-validation separates the *training* set into a *development* set and a *validation* set. The development set is used to train multiple SVMs. All the trained models are then evaluated with the validation set. The model that performs the best is then retrained using the full training set and is the final learned classifier.
The SVMs discussed so far are binary classifiers. Some modification is necessary for multi-class classification. There are various extensions (for an overview see [@Hsu2002]), of which we use the Direct Acyclic Graph SVM (DAGSVM). DAGSVM trains a binary SVM for each pair of classes [@Platt2000]. During classification, an observation is classified by a binary SVM. That observation is then classified in another binary SVM with the previously assigned class as one of the two potential classes and the other class eliminated from future consideration. This is continued, sequentially eliminating classes from consideration, until a single class remains. The observation is classified as an element of the remaining class. Thus for a $M$-class problem, DAGSVM trains $\frac{M(M-1)}{2}$ binary SVMs, but classifies an observation only using $M-1$ SVMs. It was found that the order in which the binary SVMs are used for classifying the observations does not matter [@Platt2000].
While maximizing classification accuracy (the percent of classes correctly labeled) is the true objective of SVMs, SVMs can also be evaluated by the expected hinge loss $H_D({\ensuremath{\textbf{w}}}^+)$ over the problem distribution $D$, where ${\ensuremath{\textbf{w}}}^+\in{\mathbb{R}}^{n+1}$ is the vector generated by concatenating ${\ensuremath{\textbf{w}}}^*$ and $b^*$. More formally, $$H_D({\ensuremath{\textbf{w}}}^+) = \mathbb{E}_D\bigg[\max\{(0,1-y({{\ensuremath{\textbf{w}}}^*}^T{\ensuremath{\textbf{x}}} + b^*)\}\bigg].$$ $\mathbb{E}_D[\cdot]$ is the expectation over $D$.
COMPRESSED SENSING FOR TACTILE SKINS {#cs.sect}
====================================
Compressed Sensing Theory
-------------------------
In compressed sensing, a signal, for example, force readings from a tactile array at a given time, is simultaneously measured and compressed by taking linear combinations of the signal components. Specifically, the *compressed signal* ${\ensuremath{\textbf{y}}} \in{\mathbb{R}}^m$ with elements $y_i$ is obtained from the *full signal* ${\ensuremath{\textbf{x}}} \in {\mathbb{R}}^n$ as follows: $$\label{system.eq}
{\ensuremath{\textbf{y}}} = {\ensuremath{\Phi}}{\ensuremath{\textbf{x}}},$$ where ${\ensuremath{\Phi}}= [{\ensuremath{\Phi}}_{ij}]$ is the $m \times n$ *measurement matrix*. If $m < n$, (\[system.eq\]) is under-determined and, in general, ${\ensuremath{\textbf{x}}}$ cannot be recovered.
In compressed sensing, one considers a restricted set of signals that are sparse in some representation basis. Formally, a signal ${\ensuremath{\textbf{x}}}$ is $k$-sparse in a representation basis ${\ensuremath{\Psi}}\in {\mathbb{R}}^{n\times n}$ if there is a $k$-sparse vector ${\ensuremath{\textbf{s}}}$, meaning $ {\ensuremath{\textbf{s}}}$ has at most $k$ non-zero entries, such that ${\ensuremath{\textbf{x}}} = {\ensuremath{\Psi}}{\ensuremath{\textbf{s}}}$. Compressed sensing theory provides conditions under which such sparse vectors can be recovered from fewer than $n$ measurements. One such condition relates to the *restricted isometry property*, which is defined as follows.
\[RIP.def\] A matrix $A$ satisfies the $k$-*restricted isometry property* ($k$-RIP) if there exists a $\delta \in (0,1)$ such that $$\label{RIP.eq}
(1-\delta)\|{\ensuremath{\textbf{s}}}\|_2^2 \leq \|A{\ensuremath{\textbf{s}}}\|_2^2 \leq (1+\delta)\|{\ensuremath{\textbf{s}}}\|_2^2,$$ holds for all $k$-sparse ${\ensuremath{\textbf{s}}}$.
If ${\ensuremath{\textbf{x}}}$ is $k$-sparse in a representation basis ${\ensuremath{\Psi}}$, and the matrix ${\ensuremath{\Phi}}{\ensuremath{\Psi}}$ satisfies the $2k$-RIP, then ${\ensuremath{\textbf{x}}}$ can be recovered exactly from $m \in O(k \log n)$ measurements [@cs]. Many efficient recovery algorithms have been proposed. Further, it has been shown robust recovery in the presence of noise in the signal and/or measurements is feasible [@cs].
Application to Tactile Skins {#cs-tactile.sec}
----------------------------
In previous work [@Hollis2016], we investigated the application of compressed sensing in planar tactile arrays. For the measurement matrix, we used the Scrambled Block Hadamard Ensemble (SBHE), which was developed by Gan et al. [@gan08sbhe] for compressed sensing in the image domain. SBHE is a partial block Hadamard transform with randomly permuted columns and can be represented as $${\ensuremath{\Phi}}_H = Q_mWP_n,
\label{definePhi}$$ where $W$ is a $n \times n$ block diagonal matrix with each block a $B \times B$ Hadamard matrix. $P_n$ is the permutation matrix, which randomly reorders the $n$ columns of $W$, and $Q_m$ selects $m$ rows of $WP_n$ uniformly at random. The SBHE was selected for its potential hardware implementations since it separates the sensors into disjoint measurement groups. This allows simpler wiring, as each measurement samples only a limited number of taxels, and measurements within a group could share wiring. Additionally, different groups could be measured in parallel. The number of elements in a group is equal to $B$, which we set to 32 in this work.
![Example wiring schematics for a) compressed sensing measurements on a $3 \times 3$ tactile grid and b) individual sensor measurements.[]{data-label="wiring.fig"}](wiringcomp2.png){width="\linewidth"}
Figure \[wiring.fig\](a) demonstrates the wiring on a small $3 \times 3$ array. There are three measurement groups of three sensors each. The sensors in a group are daisy-chained together, with any measurement of that group able to use the same wire path. For comparison, Figure \[wiring.fig\](b) shows the same array with each element wired separately for individual taxel readings.
We used the Daubechies-2 wavelet transform [@daubechies88wavelet] for our basis ${\ensuremath{\Psi}}$. This transform is similar to the wavelet transforms used in image compression such as JPEG2000. A similar Daubechies wavelet transform was also used in Gan et al.’s experiments [@gan08sbhe].
Using these standard (i.e. pre-existing and non-optimized) compressed sensing tools, we achieved $50Hz$ reconstruction rates for a tactile array of 4096 taxels from 1365 measurements (a compression factor of 3) [@Hollis2016]. This array contains approximately the same number of sensors as the the largest existing tactile system, and the reconstruction rate is on the same order of magnitude as that system’s measurement rate [@icub]. In addition, the reconstructed signal had less noise than the raw signal, and the system has potential for wire reduction.
COMPRESSED LEARNING FOR TACTILE SKINS {#methods.sect}
=====================================
We propose a technique for applying compressed sensing to tactile object recognition. Specifically, we perform object classification on the compressed signals. This approach maintains the benefits of compressed sensing described above. In addition, it reduces processing time by avoiding signal reconstruction, and it also reduces the dimension of the signals used for classification.
We first briefly review the theory of classification using compressed signals and then give details of our method.
Compressed Learning {#cl.sect}
-------------------
Calderbank et al. first proposed classification using compressed signals, a technique that they have called *compressed learning* [@Calderbank2009]. Specifically, they show that with high probability, a soft-margin SVM trained on compressed signals has expected accuracy similar to the best linear classifier in the uncompressed data domain. This is formalized in the following theorem.
\[cl.thm\] Let $D$ be a distribution of $k$-sparse vectors ${\ensuremath{\textbf{x}}}_i \in {\mathbb{R}}^n$ such that for all $i$, ${\|{\ensuremath{\textbf{x}}}_i\|_2 \leq R}$, where $R$ is a known upper bound. Further, assume that for each ${\ensuremath{\textbf{x}}}_i$ there is a label $\ell_i \in \{-1,1\}$. Let ${\ensuremath{\Phi}}\in{\mathbb{R}}^{m~\times~n}$ be a measurement matrix that satisfies $2k$-RIP with constant $\delta$. Additionally, let $$S_{\ensuremath{\Phi}}= \{({\ensuremath{\Phi}}{\ensuremath{\textbf{x}}}_1,\ell_1),...,({\ensuremath{\Phi}}{\ensuremath{\textbf{x}}}_N,\ell_N)\}$$ be i.i.d. labeled instances compressively sampled from $D$, and let ${\ensuremath{\textbf{z}}}_{S_{\ensuremath{\Phi}}}\in{\mathbb{R}}^m$ be the linear classifier from the soft-margin SVM trained on $S_{{\ensuremath{\Phi}}}$. Finally, let ${\ensuremath{\textbf{w}}}_0\in{\mathbb{R}}^n$ be the best linear classifier in the uncompressed data domain with low expected hinge loss over $D$, $H_D({\ensuremath{\textbf{w}}}_0)$, and large margin (hence small $\|{\ensuremath{\textbf{w}}}_0\|_2$). Then with probability $1-2\rho$ over $S_{\ensuremath{\Phi}}$: $$\label{bound.eq}
H_D({\ensuremath{\textbf{z}}}_{S_{\ensuremath{\Phi}}}) \leq H_D({\ensuremath{\textbf{w}}}_0) + O\Bigg(\|{\ensuremath{\textbf{w}}}_0\|_2 \bigg(R^2 \delta + \frac{\log(\frac{1}{\rho})}{N}\bigg)^{\frac{1}{2}}\Bigg).$$
In (\[bound.eq\]), $R$, ${\ensuremath{\textbf{w}}}_0$, and $\rho$ are fixed by the problem. The variable $\delta$ is as defined in Definition \[RIP.def\] and typically increases as the amount of compression increases. Thus, (\[bound.eq\]) implies greater compression leads to less confidence in the classifier’s accuracy, which is to be expected. Also, as expected, the accuracy depends on training set size $N$. As $N$ increases, the accuracy of the compressed classifier approaches that of the optimal linear classifier.
Theorem \[cl.thm\] addresses classification with compressed signals obtained by measuring sparse signals. If the signals are themselves not sparse, but are sparse in some orthonormal basis ${\ensuremath{\Psi}}$, then Theorem 1 still applies, provided ${\ensuremath{\Phi}}{\ensuremath{\Psi}}$ satisfies $2k$-RIP [@Calderbank2009]. An important point to note is that, unlike in compressed sensing, for compressed learning it is not necessary for this basis to be known.
Application to Tactile Skins {#application-to-tactile-skins}
----------------------------
We propose to use compressed learning for object classification in tactile skins. We generate each observation as follows. The full signal is compressed using the SBHE matrix, described in Section \[cs-tactile.sec\], to generate the compressed signal of a single time instance of contact with an object. More formally, for each observation, let ${\ensuremath{\textbf{x}}}_i$ represent an $n$-vector containing the sensor readings of all taxels (which may be noisy) at the measurement time for contact with a single object. We generate a single compressed signal, an $m$-vector, from ${\ensuremath{\textbf{x}}}_i$ using the SBHE, i.e., ${\ensuremath{\textbf{y}}}_i = {\ensuremath{\Phi}}{\ensuremath{\textbf{x}}}_i$.
For classification, we use the soft-margin DAGSVM described in Section \[background.sect\]. We use the DAGSVM implemented in the MATLAB SVM Toolbox from University of East Anglia [@Cawley2000] with a validation set to perform a grid search for the parameter $C$ in (\[softsvm.eq\]).
Gan et al. [@gan08sbhe] prove that for many basis matrices ${\ensuremath{\Psi}}$ that have applications in image compression, the product ${\ensuremath{\Phi}}{\ensuremath{\Psi}}$ behaves like a Gaussian i.i.d. matrix. Further, it has been shown a Gaussian i.i.d.matrix with $m~\in~O(k \log (n/k)/\delta^2)$ rows satisfies the $k$-RIP with high probability [@baraniuk2008simple]. Therefore, by Theorem \[cl.thm\], we should obtain similar classification performance on compressed signals as we would on the full signals.
We validate this approach through simulations, which we detail in the following sections.
EXPERIMENTAL SETUP {#setup.sect}
==================
We generated tactile array data using our BubbleTouch simulator (https://github.com/bdhollis/BubbleTouch). BubbleTouch represents taxels as rigid spheres suspended in space by spring and damper pairs; one pair per sphere. Contact interactions between objects and taxels are assumed to be quasistatic to avoid simulation instabilities that commonly arise in the simulation of dynamic contact models. Tactile arrays of any shape and distribution can be created simply by creating a rigid substrate body with that shape and attaching the bases of the springs to it in the desired pattern. To explore the effect of array resolution, we created eight planar square grid arrays with the same overall dimensions ($256mm$ by $256mm$), but different resolutions ranging from $4mm$ taxels in a $64 \times 64$ array, to $256mm$ taxels in an $1 \times 1$ array.
![A union-of-spheres model generated from the Yale-CMU-Berkeley (YCB) drill.[]{data-label="drillmodel.fig"}](drillobj.jpg){width="\linewidth"}
Our objects were obtained from two sources: the YCB (Yale-CMU-Berkeley) Object Set [@Calli2015] and a few simple geometric shapes. To simplify collision detection, the objects were approximated as union of spheres. To convert a YCB object to a union-of-spheres model, we used the UC Berkley Poisson reconstructed mesh vertices as the sphere centers. For each object, all the spheres were assigned a radius equal to twice the mean distance between all vertices and their nearest neighboring vertex. An example of a union-of-spheres model for the YCB drill is shown in Figure \[drillmodel.fig\]. For primitives, for example, ellipsoids, boxes, and cylinders, we manually designed the union-of-spheres models.
-- -- -- --
-- -- -- --
To generate tactile data, each object was placed in a stable configuration on a rigid horizontal support plane and touched from above by the tactile array. Each touch was performed by initially positioning the substrate of the array far enough above the object avoid contact. From there, the substrate was translated downward, causing the taxels to contact the object and move relative to the substrate (deforming the springs and dampers). After a pre-specified downward motion of the substrate, the simulation was stopped and the spring and damper forces at all the taxels were taken as the noiseless tactile signal for this observation. To approximate the noise in real tactile sensors, random zero-mean Gaussian noise with standard deviation of $0.001N$ (equal to $5\%$ of the signal range) was added to each taxel reading. Values outside a taxel’s range (\[$0, \ 0.02$\]$N$) were clipped to the boundary. The array signal with added noise was taken as the [*raw signal*]{}. Figure \[objectlist.fig\] shows 16 images gathered from 15 objects (the mustard bottle was touched in two different orientations). The sources of the object models are in parentheses next to their names. When testing our compressed learning algorithm, we started with the raw signal from the array of the finest resolution, i.e., $64 \times 64$. The uncompressed signal was thus of length $4,096$. For each raw signal, we generated a *compressed signal* of length $m$ by left multiplying the raw signal by the appropriately-sized SBHE matrix. We use the following values for $m$: 1,024, 256, 64, 16, 4, and 1.
For comparison, we also performed classification using raw signals of the same dimensions as the compressed signals. We generated raw signals using coarser tactile arrays, consisting of 1,024, 256, 64, 16, 4, and 1 taxels. For convention, we use *signal size* to refer to the number of elements in a signal’s vector. This means for compressed signals, the signal size is $m$, the number of measurements, and for the raw signals, the signal size is the number of taxels in the array.
DATA SETS FOR CLASSIFICATION {#data.sect}
============================
For each of the 16 objects, 360 touching observations were done as described above, but with systematic off-sets from the nominal starting configuration of the array. The off-sets were $(0,2,4,6,8,10)mm$ along the rows of the array, $(0,2,4,6,8,10)mm$ along the columns of the array, and $(0^\circ,5^\circ,10^\circ,15^\circ,20^\circ,25^\circ,30^\circ,35^\circ,40^\circ,45^\circ)$ rotations about an axis normal to the array. This yielded 5,760 observations.
The development set was formed by choosing $40\%$ of the 360 perturbations uniformly at random. All observations with those 144 perturbations for all 16 objects defined the development set, which contained 2,304 observations. The validation set was formed similarly, using $20\%$ of the other observations. The test set was defined as the remaining 2,304 observations. We also tested development sets using $20\%$, $10\%$, $6\%$, $2\%$, and $0.67\%$ of the 360 perturbations with the corresponding validation sets of $10\%$, $5\%$, $3\%$, $1\%$, and $0.33\%$.
RESULTS {#results.sect}
=======
![The classification accuracy for various signal sizes using two training set sizes. For the raw signals, signal size refers to the number of taxels in the array, and for the compressed signals, signal size refers to the number of measurements. The dotted line is the accuracy of randomly assigning a label to each example.[]{data-label="signal_size.fig"}](signal_size_.pdf){width="\linewidth"}
Figure \[signal\_size.fig\] shows the overall classification accuracies for seven different signal sizes. In addition to the accuracy rates for the compressed signals, we also look at the accuracy rates for the raw signals of corresponding dimensions. Figure \[signal\_size.fig\] also shows the results for two different training set sizes of 60% and 3%. All the results are averages over 10 different splits of the data set into test and training tests. Smaller signal sizes yield less accurate classification, but even with fairly small signal sizes, the classification has a high success rate. The raw signals achieve over 85% accuracy even at a signal size of 64, with 93.3% and 87.8% for training sets of 60% and 3% respectively. The compressed signals achieve that level of accuracy for the signal size of 16, a compression factor of 256, with 93.2% and 86.7% respective accuracies. Overall, the compressed signals outperformed the corresponding raw signals of the same size. The exception is for the signal size of 4,096 for which there is no compression. These results agree with Theorem \[cl.thm\]. The compressed signals have similar accuracies to the classifier on the original signal (the raw signal of size 4096). Further, the accuracy deviates more with increased compression.
An interesting item to note is compressed signals of size less than 128 do not use every taxel value, and for the signals of size 64, at most half of the taxels are actually used. Compressed signals of larger size also may not involve every taxel.
![The classification accuracy for training sets of various sizes. For the raw signals, the number of signals refers to the number of taxels in the array, and for the compressed signals, the number of signals refers to the number of measurements. The dotted line is the accuracy of randomly assigning a label to each example.[]{data-label="training_size.fig"}](training_size_.pdf){width="\linewidth"}
Collecting large sets of training data can be inconvenient, challenging, or impractical in deployed hardware. So, we explored performance with respect to various amounts of training data to determine how much is needed for accurate classification. Our results are shown in Figure \[training\_size.fig\]. It shows the classification rates for the raw signals of size 4,096, compressed signals obtained from this raw signal, of sizes 1,024 and 16, and raw signals obtained from coarser tactile arrays of sizes 1,024 and 16. Again, the results are averaged over 10 splits of the data set.
The three larger signals all perform extremely well, with near 100% accuracy for all but the smallest training set size of 1%. Even at the 1% training set size, the three signals achieve over 90% success. For the smaller signals, the accuracy decreases as the amount of training data decreases. The raw signals of size 16 have under 60% accuracy for all training sizes. The compressed signals of size 16 perform much better, maintaining over 80% accuracy even with 3% training data. At 1% training data, the smaller-sized compressed signal classification accuracy drops to approximately 70%.
As was the case for the results shown in Figure \[signal\_size.fig\], these results also agree with compressed learning theory. The classification accuracies for the compressed signals continue to be similar to the results from the original signal, but the deviation increases as the training set size decreases in accordance with (\[bound.eq\]). This is most clearly seen in Figure \[training\_size.fig\] by comparing the deviations between original signal (raw signal of size 4,096) and the compressed signal of size 16. The deviation goes from approximately 7% to 15% between 60% and 1% training set sizes.
![The average confusion matrix over the 10 data-set splits for compressed signals of 64 elements trained on three percent of the examples of each object. The values are the percentage of the actual class examples that were label as the predicted class. Locations with no stated values have approximately zero percent of the actual class labeled as the predicted class.[]{data-label="confusion.fig"}](confusion_matrix_.pdf){width="\linewidth"}
To get a better understanding of how the classifier is performing between individual classes, we computed the confusion matrix, which shows the percentage of observations of each class that are labeled as a particular class. This helps to identify which pairs of classes are hard to discriminate. Figure \[confusion.fig\] shows the confusion matrix for the compressed signals of size 64, trained on 3% of the observations per object, averaged over the ten splits of the data-set. From the strong diagonal it is clear the classification performs well overall. The greatest confusion occurs between the volleyball and the basketball; approximately 25% of the time one is mistaken for the other. This is understandable because both are spheres with similar radii and similar tactile signals, as seen in Figures and . The gravy can and the volleyball also generate a bit of confusion. While this is less intuitive, it is not surprising. Both objects have round shapes, and while the volleyball has a much larger radius overall, Figures and show the contact radii are similar. There is also a little confusion between the upright mustard bottle and the racquetball since they also have circular contacts of similar radii. The other confusions of note is classifying the cereal box as either the cracker box or the mustard bottle on its side and the jello box as the basketball. This is a little less apparent, but the shape and dimensions are similar between the cereal box and the other two items, and basketball covers similar area as the jello box.
CONCLUSION {#concl.sect}
==========
We have developed and demonstrated an approach for tactile object classification using compressed learning. Our approach classified various objects with high accuracy, even with high levels of compression and small amounts of training data. The compressed signals generally resulted in performance similar to the full raw signal of individual taxel readings and outperformed raw signals of corresponding signal size.
Our approach offers benefits of reduced data acquisition and processing time, as well as the potential to reduce wiring complexity in hardware implementations. In addition, for tasks where the full raw signal is required after classification, this signal can be recovered from the compressed signal.
In future work, we will explore the application of compressed sensing to other tactile tasks, such as object manipulation, safe interaction with humans, and robot locomotion through rough terrain. In addition, we will investigate other compression techniques and wiring configurations. Finally, we plan to implement our approach in hardware to fully evaluate its benefits.
[^1]: B. Hollis, S. Patterson, and J. Trinkle are with the Department of Computer Science, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, USA,.
|
---
abstract: 'The growth of superconducting order after an interaction quench in a hexagonal lattice is studied. The cases of both time-reversal (TR) preserving graphene, as well as the TR broken Haldane model are explored. Spin singlet superconducting order is studied where the $s$, $d+id$, and $d-id$ wave orders are the irreducible representations of the hexagonal lattice. For small quenches, the $d$-wave order parameter grows the fastest, a result also expected when the system is in thermal equilibrium. For the TR symmetry preserving case, the growth rate of the two $d$-wave orders is identical, while the TR-broken case prefers one of the chiral $d$-wave orders over the other, leading to a TR broken topological superconductor. As the interaction quench becomes larger, a smooth crossover is found where eventually the growth rate of the $s$-wave becomes the largest. Thus for large interaction quenches, the $s$-wave is preferred over the $d$-wave for both TR preserving and TR broken systems. This result is explained in terms of the high energy quasi-particles responsible for the dynamics as the interaction quench amplitude grows. The results are relevant for time-resolved measurements that can probe the symmetry of the superconducting fluctuations in a transient regime.'
author:
- Hossein Dehghani
- Aditi Mitra
title: Dynamical generation of superconducting order of different symmetries in hexagonal lattices
---
Introduction
============
The dynamics of how superconducting order develops, and the possibility of enhancing it under nonequilibrium conditions, has become an active area of research spanning solid-state systems [@Fausti11; @Graf11; @Smallwood12; @Smallwood14; @Beck13; @Mitrano15] and cold-atomic gases [@Regal04; @Zwierlein04; @Bloch08; @Bloch12]. These modern experiments have been complemented by intense theoretical activity attempting to understand dynamics of superconducting systems following an interaction quench [@Yin16; @Yuzbashyan15; @Foster15; @Foster14] and also the possibility of driving a normal system superconducting via nonequilibrium lattice vibrations [@Sentef16; @Knap16; @Kennes17; @Sentef17b].
Most theoretical studies of nonequilibrium superconductivity, with a few exceptions [@Capone15; @Sentef17], assume that only one kind of order is relevant. However realistic systems allow for several competing superconducting orders corresponding to different irreducible representations of the crystal point group. In this paper we therefore ask the following question: is the symmetry of the superconducting order developing under nonequilibrium conditions, such as after a quench, the same or different from the one favored by the system in thermal equilibrium?
In this paper, we identify the dominant pairing symmetry following an interaction quench in a hexagonal lattice, both in the presence and absence of time-reversal (TR) symmetry. For the TR-preserving system we study doped graphene, while for the TR-broken system we consider doped graphene subjected to a high frequency circularly polarized laser. We assume that the system before the interaction quench is in the normal state, and then study dynamics of an infinitesimal superconducting fluctuation following the sudden switch on of an attractive interaction. The time-evolution of the initial superconducting fluctuation is studied within linear response, the goal being to simply identify the most unstable mode. Eventually heating effects arising due to the energy injected in the quench, and also due to the laser, could prevent true order from developing at the longest times. We do not address this long time behavior here. Thus our theoretical treatment assumes collisionless or prethermal dynamics. We are motivated by pump-probe experiments that clearly show that prethermal regimes exist, and that these transient regimes may be unstable to ordered phases different from that in thermal equilibrium [@Fausti11; @Mitrano15].
Graphene subjected to a high frequency (as compared to the bandwidth) and circularly polarized laser has been shown to be equivalent to the Haldane model [@Oka09; @Kitagawa11; @Haldane88]. In fact deviations from the Haldane model are found to be small even when accounting for how the quasi-energy bands are occupied, provided the laser is of sufficiently high frequency and weak amplitude [@Dehghani15a; @Dehghani15b; @Dehghani16]. Thus for this paper, we will model the TR-broken system as the Haldane model, and study its pairing susceptiblity following an interaction quench.
Studying the onset of superconducting instabilities in hexagonal lattices is also of experimental relevance due to the realization of such lattices using cold-atomic gases with tunable interactions [@Esslinger13; @Esslinger14]. Moreover, recent experiments in bilayer graphene have shown that a strong non-linear coupling to phonons can be achieved [@Mitrano17] further increasing the possibility of inducing superconductivity.
Superconductivity in graphene has a long history [@NetoRMP; @Roy14]. It has been predicted that graphene, with sufficient doping so that one is near the van-Hove singularities, can realize superconducting order [@Nandkishore12; @Black14]. In particular this order can be either of the $d_{x^2-y^2}$ or $d_{xy}$ kind. However unlike $d_{x^2-y^2},d_{x y}$ orders that have nodes, the spontaneously TR symmetry breaking chiral $d_{x^2-y^2}\pm id_{xy}$ order opens up a gap everywhere on the Fermi surface. Thus, from purely energetic arguments, the preferred state in thermal equilibrium is one of the two chiral $d$ wave states [@Nandkishore12; @Black14]. The $s$-wave order, while an allowed symmetry of the lattice, is always associated with a lower critical temperature $T_c$ than the chiral $d$-wave order, so that the latter is the preferred phase in thermal equilibrium. In the rest of the paper, for notational convenience we will denote $d_{x^2-y^2}\pm id_{xy}$ by $d\pm id$.
The above equilibrium results for graphene will form a backdrop for comparing our results for the superconducting order following an interaction quench. Firstly we will show that for the Haldane model realized from graphene subjected to a high frequency laser, the broken TR symmetry naturally lifts the degeneracy between the two chiral modes, $d+id$ and $d-id$. Secondly, due to band flattening caused by the laser, one need not dope the Haldane model to the same degree as graphene to achieve the same $T_c$. Thirdly, and the main result of the paper, the interaction quench amplitude can influence which superconducting order parameter is preferred by the system. Thus although for weak interaction quenches, the symmetry of the order parameter is the same as in thermal equilibrium, for larger quench amplitudes, a smooth crossover to a phase with a different superconducting order is found.
Here we discuss some subtleties both in equilibrium and out of equilibrium related to studying superconductivity in two dimensional systems. In equilibrium, Mermin-Wagner theorem does not allow for true long range order but only quasi-long range order. Nevertheless, the quasi-long range order is associated with correlation lengths that are fairly long so that from the point of view of local measurements such as the spectral density, the system appears superconducting. For this reason, the study of superconducting instabilities in two dimensional systems has been a very active area of research. In this paper, in addition to two spatial dimensions, we are exploring superconductivity in a transient regime where the system has not fully thermalized. Time-resolved measurements [@Graf11; @Smallwood12; @Smallwood14] are capable of probing the symmetry of the superconducting fluctuations, and we expect the results of the paper to be relevant for such experiments.
The paper is organized as follows. In Section \[model\] we describe the model, and outline the derivation of the pairing susceptibility and the equation of motion of the superconducting order parameter. In Section \[dopedgr\] we discuss the order parameter symmetries for graphene, while in Section \[dopedHaldane\] we do the same for the Haldane model. We present our results for the time evolution of superconducting fluctuation after an interaction quench in both graphene and the Haldane model in Section \[results\], and discuss the phase diagram in terms of the interaction quench amplitude and temperature of the initial state, for a particular choice of doping. Finally in Section \[concl\] we present our conclusions.
Model
=====
The system we will consider is a hexagonal lattice with nearest neighbor hopping, such as graphene, with and without a circularly polarized laser. In addition we will consider interactions as our goal is to explore superconductivity. Writing the full Hamiltonian as $H=H_0(t) + V$, where $H_0$ denotes the kinetic energy and $V$ the interactions, the kinetic part is $$\begin{aligned}
&&H_0(t)=-t_h\sum_{k\sigma,\alpha=1,2,3}\begin{pmatrix}a_{k\sigma}^{\dagger} & b_{k\sigma}^{\dagger}\end{pmatrix}\nonumber\\
&&\times \begin{pmatrix}0&
e^{i \vec{k}\cdot\vec{a}_{\alpha} + ia \vec{A}(t)\cdot\vec{\delta}_{\alpha}}=h_{ab}\\
e^{-i \vec{k}\cdot\vec{a}_{\alpha} - ia \vec{A}(t)\cdot\vec{\delta}_{\alpha}}=h_{ba}&0\end{pmatrix}\nonumber\\
&&\times\begin{pmatrix}a_{k\sigma} \\b_{k\sigma}\end{pmatrix}.\label{H0}\end{aligned}$$ Above $k$ is the crystal momentum, $\sigma$ denotes the electron spin, $t_h$ is the bare tunneling amplitude between the neighboring $A,B$ sites, $a_{k\sigma},b_{k\sigma}$ are electron annihilation operators for the $A,B$ sub-lattices respectively, $A(t)= A_0\left[\cos(\Omega t) \hat{x} -\sin(\Omega t)\hat{y}\right]$ denotes the circularly polarized laser of amplitude $A_0$ and frequency $\Omega$. $\vec{\delta}_i$ are the three nearest-neighbor vectors of the hexagonal lattice, which in terms of the lattice spacing $a$ are, $$\begin{aligned}
\!\!\vec{\delta}_1 = a\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right);\vec{\delta}_2 = a\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right);
\vec{\delta}_3 = a\left(-1,0\right).\end{aligned}$$ Above we have implicitly performed the replacement $b_{k\sigma}\rightarrow b_{k\sigma}e^{i\vec{k}\cdot\vec{\delta}_3}$ in order to restore the periodicity of the unit-cell. Thus $\vec{a}_i$ are the translation vectors of the hexagonal lattice, $$\begin{aligned}
&&\vec{a}_1 = \vec{\delta}_1-\vec{\delta}_3=a\left(3/2,\sqrt{3}/2\right), \nonumber\\
&&\vec{a}_2 = \vec{\delta}_2-\vec{\delta}_3=a\left(3/2,-\sqrt{3}/2\right), \nonumber\\
&&\vec{a}_3=0.\label{dela}\end{aligned}$$
The interactions correspond to density-density interactions on sites $i$,$j$, $Un_{i\uparrow}n_{j\downarrow}$. A convenient way to probe the consequences of large $U$ is to map the system to the $t$-$J$ model [@Bhaskaran02; @Schaffer07] $$\begin{aligned}
&&V = J\sum_{<ij>}\biggl[\vec{S}_i\cdot\vec{S}_j -\frac{1}{4}n_in_j\biggr],\label{Veq}\end{aligned}$$ with $J= 2t_h^2/U$. $\vec{S}_i$ and $n_i$ are respectively the spin and number operators on site $i$, and the interactions are only between nearest-neighbor sites. These operators act on a restricted Hilbert space that excludes doubly occupied sites.
Periodic drive and interactions will eventually lead to heating [@Rigol14a], however we are interested in very high frequency $\Omega > 6 t_h$ non-resonant driving, where the time scales for heating processes are exponentially long [@Alessio13; @Abanin15; @Mori16; @Bukov16]. In addition, it has been argued that an effective Hamiltonian obtained from a high-frequency expansion may be used to capture the dynamics [@Kuwahara2016; @Abanin17]. With this in mind, in the next sub-section, we derive an effective time-independent Hamiltonian by performing a high-frequency expansion. We retain the leading term that captures the effective TR symmetry breaking due to the circularly polarized laser. This leading term opens up a TR symmetry breaking gap at the Dirac points of graphene, mapping it to an interacting Haldane model.
Mapping to the interacting Haldane model
----------------------------------------
Let us Fourier transform the two off-diagonal elements of $H_0$ in Eq. , $$\begin{aligned}
&&h_{ab}^m=\frac{1}{T}\int_0^T dt e^{-im\Omega t} h_{ab}(t) \nonumber\\
&&= i^{-m}J_{-m}(A_0a)\sum_{j=1,2,3}
e^{i \vec{k}\cdot\vec{a}_{j}+i m \alpha_j},\\
&&h_{ba}^m=\frac{1}{T}\int_0^T dt e^{-im\Omega t} h_{ba}(t) \nonumber\\
&&= (-i)^{-m}J_{-m}(A_0a)\sum_{j=1,2,3}
e^{-i \vec{k}\cdot\vec{a}_{j}+i m \alpha_j}.\end{aligned}$$ Above $\alpha_1 = -\alpha_2 = \frac{\pi}{3}, \alpha_3 = \pi$ and $J_m$ denotes the Bessel function. The amplitude of the laser will be given in terms of the dimensionless quantity $A_0a$, while the laser frequency and electron temperature will be given in units of the bare tunneling amplitude $t_h$.
The high-frequency expansion for a general Hamiltonian $\hat{H}(t)$ takes the form [@Eckardt15], $$\begin{aligned}
H_{\rm eff} = \hat{H}_{m=0} + \sum_{m\neq 0} \frac{\hat{H}_m\hat{H}_{-m}}{m\hbar \Omega} + \ldots.\end{aligned}$$ Keeping only the first two terms in the above series, the kinetic energy term becomes, $$\begin{aligned}
&&H_{\rm eff}= -t_h
\sum_{k\sigma,\alpha=1,2,3}\begin{pmatrix}a_{k\sigma}^{\dagger} & b_{k\sigma}^{\dagger}\end{pmatrix}\nonumber\\
&&\times \begin{pmatrix}t_h\sum_{m\neq 0}
\frac{h^m_{ab}h^{-m}_{ba}}{\hbar m \Omega}& h_{ab}^{m=0}\\
h_{ba}^{m=0}& t_h\sum_{m\neq 0}\frac{h^m_{ba}h^{-m}_{ab}}{\hbar m \Omega} \end{pmatrix}\nonumber\\
&&\times \begin{pmatrix}a_{k\sigma} \\b_{k\sigma}\end{pmatrix}.\end{aligned}$$
Using the expressions for $h_{ab}^m, h_{ba}^m$ derived above, we obtain,
$$\begin{aligned}
&&H_{\rm eff}= -
\sum_{k\sigma}\begin{pmatrix}a_{k\sigma}^{\dagger} & b_{k\sigma}^{\dagger}\end{pmatrix}\begin{pmatrix}
2t_2\biggl[\sum_{ij=12,23,31}\sin(\vec{k}\cdot(\vec{a}_i-\vec{a}_j))\biggr]& t_1\sum_{j=1,2,3}
e^{i \vec{k}\cdot\vec{a}_{j}}\\
t_1\sum_{j=1,2,3}
e^{-i \vec{k}\cdot\vec{a}_{j}}& -2t_2\biggl[\sum_{ij=12,23,31}\sin(\vec{k}\cdot(\vec{a}_i-\vec{a}_j))\biggr]\end{pmatrix}
\begin{pmatrix}a_{k\sigma} \\b_{k\sigma}\end{pmatrix},\\
&& t_1 = t_h J_{0}(A_0a), \,\,\,\,\,\,\, \,t_2 = 2t_h^2 \sum_{m=1\ldots\infty}\frac{J_m^2(A_0a)}{\hbar m\Omega}\sin\biggl[\frac{2\pi m}{3}\biggr]\label{tren}.\end{aligned}$$
$H_{\rm eff}$ is the Haldane model for maximal flux of $\phi=\pi/2$ threading the plaquette [@Haldane88]. The above high frequency expansion shows that in general a periodic drive generates longer ranged matrix elements such as the next to nearest-neighbor tunneling terms appearing on the diagonal of $H_{\rm eff}$. In addition, these tunneling matrix elements carry non-trivial phases that depend on the polarization of the drive. The generation of the term proportional to $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ in $H_{\rm eff}$ indicates the broken TR symmetry due to the application of the circularly polarized laser. The high-frequency expansion is kept only up to the leading non-zero value as it is sufficient to lift the degeneracy at the Dirac point and open up a gap that corresponds to broken TR symmetry. Keeping higher order terms in the frequency expansion will not generate or break any further symmetries but will only renormalize the parameters of the Floquet Hamiltonian.
For the interaction it is sufficient to keep the $m=0$ term in the high-frequency limit. The interaction $V$ is given by Eq. , which we can rewrite as, [@Bhaskaran02; @Schaffer07] $$\begin{aligned}
V=-J\sum_{<ij>}h_{ij}^{\dagger}h_{ij},\end{aligned}$$ where, $$\begin{aligned}
h_{ij}^{\dagger}= \frac{1}{\sqrt{2}}\biggl(a_{i\uparrow}^{\dagger}b_{j\downarrow}^{\dagger}-a_{i\downarrow}^{\dagger}b_{j\uparrow}^{\dagger}\biggr),\end{aligned}$$ is the nearest-neighbor spin-singlet creation operator. Recall that the elimination of on-site double occupancy in the $t-J$ model excludes on site singlet formation in favor of nearest-neighbor singlet formation. As discussed further, this nearest-neighbor spin singlet formation also gives more structure to the singlet order-parameter, allowing for $s$,$d+id$,$d-id$ symmetries depending on how the order-parameter varies between the three nearest-neighbor sites.
Writing $$\begin{aligned}
\phi_k= {\rm Arg}\biggl[\sum_{\alpha}e^{i\vec{k}\cdot\vec{a}_{\alpha}}\biggr]\label{phik},\end{aligned}$$ it is convenient to parameterize the non-interacting part as, $$\begin{aligned}
&&H_{\rm eff} = \sum_{k\sigma}\begin{pmatrix}a_{k\sigma}^{\dagger} & b_{k\sigma}^{\dagger}\end{pmatrix}\nonumber\\
&&\times d_k \begin{pmatrix} \cos\theta_k & e^{i\phi_k} \sin\theta_k \\ e^{-i\phi_k} \sin\theta_k & -\cos\theta_k\end{pmatrix}
\begin{pmatrix}a_{k\sigma} \\b_{k\sigma}\end{pmatrix}.\end{aligned}$$ Using the pseudo-spin label $\tau=\pm$ to denote the eigenmodes of $H_{\rm eff}$ with energies $\pm |d_k|$, we use the standard practice of parameterizing the modes by the two angles $\phi_k,\theta_k$ as follows, $$\begin{aligned}
&&|\tau=\pm\rangle_k = \begin{pmatrix}u_{k\pm} \\v_{k\pm}\end{pmatrix},\nonumber\\
&&|\tau=+\rangle_k = \begin{pmatrix}\cos\frac{\theta_k}{2} \\ e^{-i\phi_k}\sin\frac{\theta_k}{2} \end{pmatrix} ; \,\,
|\tau=-\rangle_k = \begin{pmatrix}-\sin\frac{\theta_k}{2} \\ e^{-i\phi_k}\cos\frac{\theta_k}{2} \end{pmatrix}.\nonumber\\
\label{EVH}\end{aligned}$$
Quench dynamics
---------------
We now formulate the question as follows. Let us suppose that the interactions are initially zero so that the system is in the normal phase. An attractive interaction of magnitude $J$ is switched on at $t=0$. As the system evolves in time under the influence of this interaction quench, we would like to study the tendency of the system to become superconducting. In particular, we plan to study the time-evolution of superconducting order perturbatively in the interaction.
We consider a spin singlet superconducting order, $$\begin{aligned}
\Delta_{\alpha} = J
\biggl\langle b_{i+\alpha\downarrow}a_{i\uparrow}-b_{i+\alpha\uparrow} a_{i\downarrow} \biggr\rangle,\end{aligned}$$ where $\alpha=1,2,3$ denote the three nearest neighbor bonds to the site $i$. For a spatially uniform order parameter, we need not keep the site label $i$. The explicit dependence in $\alpha$ denotes how the order-parameter can have different phases depending on the orientation of the nearest-neighbor bonds. This piece of information is key in order to differentiate between $s$-wave ($\Delta_{\alpha}$ independent of $\alpha$) and a TR broken superconductor (where $\Delta_{\alpha}$ picks up different phases for clockwise and anti-clockwise rotations around a lattice site).
After a mean-field decoupling of the interaction, we obtain, $$\begin{aligned}
&&H(t) = H_{\rm eff}\nonumber\\
&&-\sum_{k,\alpha}\biggl[\Delta_{\alpha}(t)e^{i\vec{k}\cdot {\vec{a}}_{\alpha}}\biggl(a_{k\uparrow}^{\dagger}b^{\dagger}_{-k\downarrow}
-a_{k\downarrow}^{\dagger}b^{\dagger}_{-k\uparrow}
\biggr)+ h.c.\biggr] \label{Hmf}.\end{aligned}$$ where $\Delta_{\alpha}(t)$ is a self-consistently determined pairing field. The mean-field decoupling neglects fluctuation terms ($\Delta_i \Delta_j$), their role will be discussed later in the section.
While the interaction in Eq. is written in the sub-lattice basis, it can also be written in the energy band basis where the kinetic energy is diagonal. In that basis there are two different pairing interaction terms, one where the electrons from the same band (intra-band) are paired, and one where electrons from different bands (inter-band) are paired. It is important to keep track of both the inter- and intra- band pairing amplitudes.
The equation of motion of the order parameter is, $$\begin{aligned}
&&\Delta_{\alpha}(t)= \nonumber
J\sum_ke^{-i\vec{k}\cdot {\vec{a}}_{\alpha}}
\biggl\langle b_{-k\downarrow}(t)a_{k\uparrow}(t)-b_{-k\uparrow}(t) a_{k\downarrow}(t) \biggr\rangle,\end{aligned}$$ where the operators obey Heisenberg time-evolution for an effectively time-dependent Hamiltonian, $\hat{O}(t)=\tilde{T}e^{i\int_0^tdt' H(t')}\hat{O}(0) Te^{-i \int_0^tdt'H(t')}$. The time-dependence is due to the fluctuating pairing field $\Delta_{\alpha}(t)$ in the mean-field Hamiltonian Eq. .
Using Eq. , and to leading order in perturbation theory in the pairing amplitude $\Delta_{\alpha}$, one need only retain the commutator of the time-dependent pairing term with the observable. Thus one obtains, $$\begin{aligned}
&&\Delta_{\alpha}(t) = -iJ\sum_{kk',\beta}\int_0^t dt'\Delta_{\beta}(t')e^{-i\vec{k}\cdot {\vec{a}}_{\alpha}+ i\vec{k'}\cdot {\vec{a}}_{\beta}}\nonumber\\
&&\times \biggl\langle\biggl[\biggl(a_{k'\uparrow}^{\dagger}(t')b^{\dagger}_{-k'\downarrow}(t')
-a_{k'\downarrow}^{\dagger}(t')b^{\dagger}_{-k'\uparrow}(t')\biggr) , \nonumber\\
&&\biggl(b_{-k\downarrow}(t)a_{k\uparrow}(t)-b_{-k\uparrow}(t) a_{k\downarrow}(t) \biggr)\biggr]\biggr\rangle.\end{aligned}$$ After performing the averages with respect to the initial state denoted by $\langle \rangle$, we obtain the equation of motion, $$\begin{aligned}
\Delta_{\alpha}(t) = J \sum_{\beta}\int_0^t dt' \Pi^R_{\alpha\beta}(q=0,t,t') \Delta_{\beta}(t'). \label{eomDel2}\end{aligned}$$ Above $\Pi^R$ is the response function or pairing susceptibility of the free electron gas.
Since we assume a spatially uniform order-parameter, only momentum $q=0$ component of $\Pi^R$ appears above. For a general momentum $q$, $\Pi^R$ has the form,
$$\begin{aligned}
&&\Pi^R_{\alpha\beta}(q,t,t')=-2i\theta(t-t')\sum_{k,\tau,\tau_1=\pm}e^{-i\epsilon_{k+q\tau}(t-t')}
e^{-i\epsilon_{-k\tau_1}(t-t')}\biggl\{1-\rho_{k+q,\tau}-\rho_{-k,\tau_1}\biggr\}
\biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}|u_{k+q,\tau}|^2 |v_{-k\tau_1}|^2\nonumber\\
&&+ e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})-i \vec{q}\cdot {\vec{a}}_{\beta}}u_{k+q\tau}v_{k+q\tau}^*
v_{-k\tau_1}u^*_{-k\tau_1}\biggr],\nonumber\\\label{PiRHaldane}\end{aligned}$$
where $\rho_{k,\tau}$ is the occupation probability of the $k,\tau$ level in the absence of interactions.
We now briefly discuss the relation of Eq. to what is usually done in equilibrium to determine the superconducting phase. In equilibrium, $t=\infty$, and going into frequency space, one solves $\Delta_{\alpha} = J \sum_{\beta}\Pi^R_{\alpha \beta}(q=0,\omega=0)\Delta_{\beta}$. Secondly, the response function $\Pi^R$ is evaluated for the full interacting electron gas in the normal phase. Thus the only assumption is that the pairing amplitude $\Delta_{\alpha}$ is small. Beyond this, an additional assumption can be made, and that is of weak $J/t_h$. This is the BCS approximation, where now $\Pi^R$ is evaluated for the free Fermi gas.
In our problem, namely a quench, since the average is always with respect to the initial state which is that of a free Fermi gas, the linear-response assumption is tied to perturbation theory in $J$ as well. Strictly speaking this assumption will break down when quartic terms ($\Delta_{i}\Delta_j$) or inelastic scattering between particles become important. The time scale for this from Fermi’s Golden rule, in units of the hopping $t_h$, is $ t_{\rm in} \sim 1/(J^2T)$, where $T$ is the temperature of the electron gas, and $J,T$ are in units of the hopping $t_h$. At times longer than this time, thermalizing processes will become active.
Since finite temperatures are not detrimental to superconductivity in spatial dimensions $d>1$, including $d=2$ provided $T<T_{\rm BKT}$, where $T_{\rm BKT}$ is the Brezenskii Kosterlitz Thouless temperature, our conclusions are not completely invalidated in the inelastic scattering dominated regime. We discuss this issue in more detail in Section \[results\].
Mean-field [@Yuzbashyan15; @Capone15] and linear instability analyses [@Knap16], despite their apparent simplicity, are relevant to experiments that can probe the short time regime where quasi-particles have formed even though the bulk system has not developed a traditional Meissner effect. In particular time-resolved ARPES can pick out the dispersion of the quasi-particles, and hence the symmetries of the order-parameter [@Graf11; @Smallwood12; @Smallwood14].
In what follows we will assume that initially the free fermions were in equilibrium at temperature $T$ and chemical potential $\mu$. Thus for a free fermion dispersion $\epsilon_k$ which is also inversion symmetric, $\epsilon_{k,\tau} = \epsilon_{-k,\tau}$, we obtain, $$\begin{aligned}
&&1-\rho_{k,\tau}-\rho_{-k,\tau_1} = \delta_{\tau,\tau_1}\tanh\left(\frac{\epsilon_{k,\tau}+\mu}{2T}\right)\nonumber\\
&&+\delta_{\tau,-\tau_1}\frac{1}{2}\biggl[\tanh\left(\frac{\epsilon_{k,\tau}+\mu}{2T}\right)-
\tanh\left(\frac{\epsilon_{k,\tau}-\mu}{2T}\right)\biggr].\end{aligned}$$ A given temperature $T$ and chemical potential $\mu$ denotes a doping $\delta$ away from half-filling, $$\begin{aligned}
\delta = \frac{1}{2}\sum_k\biggl[\tanh\biggl(\frac{\epsilon_k+\mu}{2T}\biggr)
-\tanh\biggl(\frac{\epsilon_k-\mu}{2T}\biggr)\biggr].\end{aligned}$$ We will present results for a non-zero doping $\delta=0.1$. For zero doping, since the chemical potential is in the gap in equilibrium, the density of states at the Fermi energy vanishes and superconductivity does not occur in the BCS limit.
In equilibrium, in order to determine the critical temperature for some coupling $J$, one solves the gap equation, $J\Pi^R(q=0,\omega=0)=1$, where, $$\begin{aligned}
&&\Pi^R_{\alpha \beta}(q,\omega=0)= \sum_{k,\tau,\tau_1=\pm}\frac{1}{\epsilon_{k+q\tau}+ \epsilon_{-k\tau_1}}\nonumber\\
&&\times \biggl\{1-\rho_{k+q,\tau}-\rho_{-k,\tau_1}\biggr\}\biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}|u_{k+q,\tau}|^2 |v_{-k\tau_1}|^2
\nonumber\\
&&+ e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})-i \vec{q}\cdot {\vec{a}}_{\beta}}u_{k+q\tau}v_{k+q\tau}^*
v_{-k\tau_1}u^*_{-k\tau_1}\biggr]. \label{PiRHG}\end{aligned}$$ In the next sections, we will directly solve Eq. for some small, initial randomly chosen $\Delta_{\alpha}$. Since $\alpha$ takes three values, we denote the superconducting order as a vector $\vec{\Delta}$. Growing and decaying solutions in time will indicate whether the system is susceptible to pairing. In addition, how the three components $\Delta_{\alpha=1,2,3}$ of the order parameter grow will indicate the preferred symmetry of the superconducting order parameter.
It might seem that the only place the chemical potential appears is in the initial distribution function. However we will measure all energies with respect to this chemical potential. This choice is equivalent to multiplying the superconducting order-parameter by the phase $e^{-2i\mu t}$. Thus in Eq. we will denote the energy of the upper band $\epsilon_{k+}=\epsilon_k+\mu$, and that of the lower band as $\epsilon_{k-}=-\epsilon_k+\mu$, where we have also used the fact that particle-hole symmetry in the problem makes the band dispersions symmetric about zero energy.
Although Eq. was derived perturbatively in $J$, yet the solution of the integral equation is non-perturbative in $J$. To see this note that if initially at $t=0$ we had a small seed order parameter $\delta(t')\Delta_0$, then the first order correction from Eq. is $\Delta_1(t) = J\Pi^R(t,0)\Delta_0$. The second order correction is obtained from substituting this back in Eq. to obtain $\Delta_2(t) = J^2\int dt_1\Pi^R(t,t_1)\Pi^R(t_1,0)\Delta_0$, and so on. Thus the solution for Eq. can be recast as an integral equation $\Delta(t)= D^{-1}\Delta_0$, essentially the Dyson equation in real time, where $(1-J\Pi)=D$. This leads to a solution, which after a short transient $t <O(1)$ stabilizes to an exponentially growing or decaying solution, $\Delta \sim e^{(J-J_c)t}$ where $J_c$ is a critical coupling. The appearance of $J$ in the argument of the exponential shows the non-perturbative role of $J$ in determining the superconducting phase.
From complex analysis, the relation $\Delta(t)= D^{-1}\Delta_0$ implies that locations of zeros of $D$ in the complex frequency plane will determine the growth or decay rate of the order parameter. Thus for later discussions we will find it convenient to interpret the results of the time evolution in terms of the location of the zeros of $D$ in the complex frequency (denoted by $z$) plane. The explicit form for $D(q,z)$ is, $$\begin{aligned}
&&D_{\alpha\beta}(q,z) = \delta_{\alpha\beta}- J\Pi^R_{\alpha \beta}(q,z)\nonumber\\
&&= \delta_{\alpha\beta}- 2J\sum_{k,\tau,\tau_1=\pm}\biggl\{\frac{1-\rho_{k+q,\tau}-\rho_{-k,\tau_1}}{\epsilon_{k+q\tau}+ \epsilon_{-k\tau_1}-2iz}\biggr\}\nonumber\\
&&\times \biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}|u_{k+q,\tau}|^2 |v_{-k\tau_1}|^2\nonumber\\
&&+ e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})-i \vec{q}\cdot {\vec{a}}_{\beta}}u_{k+q\tau}v_{k+q\tau}^*
v_{-k\tau_1}u^*_{-k\tau_1}\biggr]. \label{Dz}\end{aligned}$$
In section \[dopedgr\] we discuss the symmetries of the superconducting order for graphene while in section \[dopedHaldane\] we discuss the same for the Haldane model. The results of the time-evolution are presented in Section \[results\].
Pairing symmetries of graphene {#dopedgr}
==============================
Since graphene has only nearest neighbor hopping, $t_2=0,t_1=t_h$, the modes are, $$\begin{aligned}
&&|\tau=\pm\rangle_k = \begin{pmatrix}u_{k\pm} \\v_{k\pm}\end{pmatrix},\nonumber\\
&&|\tau=+\rangle_k = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ e^{-i\phi_k}\end{pmatrix} ;
|\tau=-\rangle_k = \frac{1}{\sqrt{2}}\begin{pmatrix}-1 \\ e^{-i\phi_k} \end{pmatrix}.\nonumber\end{aligned}$$ $\phi_k$ is defined in Eq. .
Since under $k\rightarrow -k$, $H_k \rightarrow \tau_xH_k\tau_x$, then, $$\begin{aligned}
u_{k\tau} = u_{-k\tau}^*, v_{k\tau}= v_{-k\tau}^*.\end{aligned}$$ This gives the following expression for the pair susceptibility,
$$\begin{aligned}
&&J\Pi^R_{\alpha \beta}(t,t')=2i\theta(t-t')\frac{J}{4}\sum_k\biggl[e^{-2i(\epsilon_k+\mu)(t-t')}
\tanh\biggl(\frac{\epsilon_k+\mu}{2T}\biggr)
+e^{-2i(-\epsilon_k+\mu)(t-t')} \tanh\biggl(\frac{-\epsilon_k+\mu}{2T}\biggr)\biggr]\nonumber\\
&&\times \biggl[\cos\biggl(\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})\biggr)
+ \cos\biggl(\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})-2\phi_k\biggr)\biggr] \nonumber\\
&&+ i\theta(t-t')e^{-2i\mu(t-t')}\frac{J}{2}\sum_k\biggl[\frac{\sinh(\frac{\mu}{T})}{\cosh{(\frac{\epsilon+\mu}{2T})}\cosh{(\frac{\epsilon-\mu}{2T})}}\biggr]
\biggl[\cos\biggl(\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})\biggr)
- \cos\biggl(\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})-2\phi_k\biggr)\biggr].\end{aligned}$$
The dc component of the response function $\Pi^R(\omega=0)$ agrees with Ref. , where the first line above corresponds to intra-band pairing, and the second line to inter-band pairing.
Let us now discuss the symmetries of the $\Pi^R$ matrix. For every $\vec{k}$, there are two others oriented by $2\pi/3, 4\pi/3$ from it. We label the triad as, $$\begin{aligned}
&&\vec{k}_{m=1,2,3} \nonumber\\
&&= k\left[\cos\biggl(\theta_k+(m-1)\frac{2\pi}{3}\biggr), \sin\biggl(\theta_k+(m-1)\frac{2\pi}{3}\biggr)\right].\nonumber\end{aligned}$$ We label the nearest-neighbor vectors $\vec{\delta}_{\alpha}$ similarly, $$\begin{aligned}
&&\vec{\delta}_{j=1,2,3} \nonumber\\
&&= a \left[\cos\biggl(\frac{\pi}{3}-(j-1)\frac{2\pi}{3})\biggr), \sin\biggl(\frac{\pi}{3}-(j-1)\frac{2\pi}{3}\biggr)\right].\nonumber\end{aligned}$$ Thus, $$\begin{aligned}
\vec{k}_m\cdot\vec{\delta}_j = ka\cos\biggl(\theta_k + (m+j)\frac{2\pi}{3}+\frac{\pi}{3}\biggr).\end{aligned}$$ From above we see $\vec{k}_m\cdot\vec{\delta}_j= \vec{k}_{m\pm1}\cdot\vec{\delta}_{j\mp1}$. In addition if $j$ or $m$ change by $\pm 3$ or its multiples, the function comes back to itself. This implies the following relations, $$\begin{aligned}
&&\phi(k_m) = {\rm Arg}\biggl[\sum_{i=1,2,3} e^{i \vec{k}_m\cdot\vec{\delta}_i}\biggr] \Rightarrow \phi(k_1) \!= \!\phi(k_2)=\! \phi(k_3),\nonumber\\
&&m(k_m) = {\rm Abs}\biggl[\sum_{i=1,2,3} e^{i \vec{k}_m\cdot\vec{\delta}_i}\biggr] \Rightarrow m(k_1) \!= \!m(k_2)=\! m(k_3),\nonumber\\
&&\sum_{j=1,2,3}m(k_m)f(\vec{k}_m\cdot\vec{\delta}_j-\phi_{k_m}) = f'_m\Rightarrow f'_1=f'_2=f'_3,\nonumber\\
&&\sum_{\vec{k}}m(\vec{k})f(\vec{k}\cdot\vec{\delta}_j-\phi_k)=g_j \Rightarrow g_1=g_2=g_3.\end{aligned}$$ The above equalities simply reflect the $C_3$ symmetry of the hexagonal lattice. While the first two equalities above directly influence the energy eigenvalues and eigenvectors, the diagonal component of the pairing susceptibility $\Pi_{\alpha\alpha}$ is of the last form. Note that the vectors $\vec{\delta}, \vec{a}$ are related by a constant shift (see Eq. ), thus the arguments for the symmetries of the $\Pi$ matrix hold irrespective of whether the $\Pi$ matrix is written in the $\vec{\delta}$ or $\vec{a}$ basis.
Thus the $C_3$ symmetry implies $\Pi_{11}=\Pi_{22}=\Pi_{33}=A$. Other components of $\Pi$ may be written as, $$\begin{aligned}
\sum_{\vec{k}}m(\vec{k})l(\vec{k}\cdot\vec{\delta}_j-\phi_k)l(\vec{k}\cdot\vec{\delta}_i-\phi_k)=L_{ji}= L_{ij}.\end{aligned}$$ The above implies $\Pi_{\alpha\beta}=\Pi_{\beta\alpha}$. This together with the fact that $L_{ji}= L_{j\pm 1,i\pm 1}$ gives $\Pi_{12}= \Pi_{23}=\Pi_{31}=B$ and $\Pi_{12}=\Pi_{13}=\Pi_{23}=B$.
In fact the above symmetries hold at any instant of time $t$, so that $\Pi^R$ has the general form, $$\begin{aligned}
\Pi^R(t) = \begin{pmatrix}A(t) &B(t) & B(t) \\B(t) &A(t) & B(t) \\B(t) &B(t) &A(t)\label{PiGs}
\end{pmatrix}.\end{aligned}$$
From the structure of Eq. it follows that the eigenvalues and eigenvectors of the $\Pi^R$ matrix at any instant of time are $$\begin{aligned}
&&\lambda_s(t) = \Pi_{11}(t) + 2\Pi_{12}(t);\,\,\,\,\,\, \Delta_s = \frac{1}{\sqrt{3}}\left[1,1,1\right],\\
&&\lambda_{d_1}(t) = \lambda_{d_2}(t)= \Pi_{11}(t) - \Pi_{12}(t), \nonumber\\
&&\Delta_{d_1} = \frac{1}{\sqrt{6}}\left[2,-1,-1\right]; \,\,\,\,\,\, \Delta_{d_2} = \frac{1}{\sqrt{2}}
\left[0,1,-1\right].\end{aligned}$$ While the eigenvalues depend explicitly on time, the eigenvectors do not. Thus all throughout the time-evolution, there are three mutually orthogonal directions that remain the same. These directions correspond to a non-degenerate $s$-wave solution and a doubly degenerate $d$-wave solution. Any random initial condition for the superconducting order parameter evolves independently along these three directions. We track the time-evolution, and the fastest growing order parameter determines the nature of the superconductor at steady state. Due to the degeneracy of the two $d$-wave modes, the growth rate of the two $d$-wave orders, or any linear combination of the two $d$-wave orders will be identical.
Here we should mention that in equilibrium, the $T_c$ of the superconducting phase and its symmetry is determined from $1-J\Pi^R(\omega=0)=0$. This treatment [@Black14] gives the largest eigenvalue, and therefore the dominant instability to correspond to $d$-wave. Since this is doubly degenerate, the order parameter symmetry is not uniquely determined. Instead, energetic considerations indicate that the TR breaking combination $d_{x^2-y^2}+id_{xy}=\frac{1}{\sqrt{3}}\biggl[1,e^{2\pi i/3},e^{4\pi i/3}\biggr]$ (or its complex conjugate $d_{x^2-y^2}-id_{xy}$) will be favored. This is because the chiral order parameter has no nodes.
For the case of the quench, such energy minimization considerations no longer hold. Instead if the time-evolution shows the $d$-wave to be the fastest growing order parameter, the precise order parameter could be any linear combination in the degenerate sub-space. If the fastest growing order parameter is the $s$-wave, then the order parameter symmetry is uniquely determined. For the Haldane model, as we discuss below, the breaking of TR lifts the degeneracy in the $d$-wave sub-space.
Pairing symmetries of the Haldane model {#dopedHaldane}
=======================================
We now discuss the pairing susceptibility of the Haldane model. We again make an assumption that the fermions are in thermal equilibrium at temperature $T$ and chemical potential $\mu$ before the interaction quench. This gives,
$$\begin{aligned}
&&\Pi^R_{\alpha\beta}(q=0,t,t')=\nonumber\\
&&-2i\theta(t-t')\sum_{k}e^{-2i(\epsilon_{k}+\mu)(t-t')}
\biggl\{\tanh\left(\frac{\epsilon_k+\mu}{2T}\right)\biggr\}\biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}\cos^4\frac{\theta_k}{2}
+ e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})}e^{2i\phi_k}\cos^2\frac{\theta_k}{2}\sin^2\frac{\theta_k}{2}\biggr]\nonumber\\
&&-2i\theta(t-t')\sum_{k}e^{-2i(-\epsilon_{k}+\mu)(t-t')}
\biggl\{\tanh\left(\frac{-\epsilon_k+\mu}{2T}\right)\biggr\}\biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}\sin^4\frac{\theta_k}{2}
+ e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})}e^{2i\phi_k}\cos^2\frac{\theta_k}{2}\sin^2\frac{\theta_k}{2}\biggr]\nonumber\\
&&-2i\theta(t-t')\sum_{k}e^{-2i\mu(t-t')}
\biggl\{\tanh\left(\frac{\epsilon_k+\mu}{2T}\right)+ \tanh\left(\frac{-\epsilon_k+\mu}{2T}\right)\biggr\}
\biggl[e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}-{\vec{a}}_{\beta})}\cos^2\frac{\theta_k}{2}\sin^2\frac{\theta_k}{2}\nonumber\\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
- e^{-i\vec{k}\cdot ({\vec{a}}_{\alpha}+ {\vec{a}}_{\beta})}e^{2i\phi_k}\cos^2\frac{\theta_k}{2}\sin^2\frac{\theta_k}{2}\biggr].\end{aligned}$$
Like graphene, the Haldane model also has $C_3$ symmetry associated with invariance under rotations by $2\pi/3$. Thus following the arguments given for graphene in the previous section, we conclude, $\Pi_{11}=\Pi_{22}=\Pi_{33}=A$, $\Pi_{12}=\Pi_{23}=\Pi_{31}=B$, and $\Pi_{13}=\Pi_{21}=\Pi_{32}=C$. Thus we may write, $$\begin{aligned}
\Pi^R(t) = \begin{pmatrix}A(t) &B(t) & C(t) \\C(t) &A(t) & B(t) \\B(t) &C(t) &A(t)
\end{pmatrix}.\end{aligned}$$ Unlike graphene however, $\Pi_{\alpha\beta}\neq \Pi_{\beta \alpha}$, when $ \alpha\neq \beta$. In the dc limit $\Pi^R(\omega=0)$ is Hermitian forcing $C(\omega=0)=B^*(\omega=0)$.
The eigenvalues and eigenvectors of the $\Pi(t)$ matrix are $$\begin{aligned}
&&\lambda_s(t) = A(t)+B(t)+C(t);\,\, \Delta_s = \frac{1}{\sqrt{3}}\left[1,1,1\right],\\
&&\lambda_{d+id}(t) = A(t) + B(t) e^{2\pi i/3} + C(t) e^{4\pi i/3} ,\nonumber\\
&&\Delta_{d+id} = \frac{1}{\sqrt{3}}\begin{pmatrix}1\\ e^{2\pi i/3}\\e^{4\pi i/3}\end{pmatrix},\\
&&\lambda_{d-id}(t)=A(t) + B(t) e^{4\pi i/3} + C(t) e^{2\pi i/3},\nonumber\\
&&\Delta_{d-id} =\frac{1}{\sqrt{3}}\begin{pmatrix}1\\ e^{4\pi i/3}\\e^{2\pi i/3}\end{pmatrix}.\end{aligned}$$ Graphene corresponds to the case where $\Pi_{\alpha\beta}=\Pi_{\beta\alpha}$ making $B=C$ and $\lambda_{d+id}=\lambda_{d-id}=A-B$. Thus we find that there are no degenerate eigenvalues for the Haldane model as broken TR prefers one of the chiral $d$-wave modes over the other.
Just as for graphene, here too the eigenvalues of $\Pi^R$ are time-dependent, while the mutually orthogonal directions stay static. We again study the time-evolution of an initial random but small vector $\vec{\Delta}$, project the evolution along the three mutually orthogonal directions corresponding to the eigenvectors of $\Pi^R$, and determine the phase from the fastest growing mode.
Before we present the results for the quench, we point out that in equilibrium, the Haldane model obtained from applying a high frequency circularly polarized laser to graphene, is more susceptible to pairing than graphene. This is because the laser flattens out the band as can be seen from the renormalization of the hopping amplitudes from $t_h \rightarrow t_h J_0(A_0a)$ in Eq. . As a consequence, the same $T_c$ can be obtained at much lower doping levels in the presence of the laser than in graphene. As an example, the pairing susceptibilities derived above give that graphene for $J=0.6 t_h$ has a $T_c=0.01t_h$ at a doping of $\delta=0.11$. In contrast the Haldane model realized from a laser of amplitude $A_0a=1.0$ and frequency $\Omega =10 t_h$ shows the same $T_c$ but at a much lower doping of $\delta \sim 0.01$.
Results
=======
We now present results for the solution of the evolution equation Eq. . We start with an initial small (precise number does not matter), random vector $\vec{\Delta}$ at $t=0$, and time-evolve it forward. We project the time-evolution along the three mutually orthogonal eigenvectors of the pairing susceptibility $\Pi^R$. For graphene these correspond to the $s$-wave order parameter and a doubly degenerate sub-space with $d$-wave symmetry. For the Haldane model, the degeneracy is lifted into two chiral solutions $d+id$, and $d-id$ respectively.
The typical time-evolution of $\vec{\Delta}$ for the Haldane model is shown in Fig. \[fig1\]. The slopes are proportional to the growth rate, and for the parameters chosen, $d+id$ grows faster than $d-id$, followed by $s$-wave. The magnitude of the order-parameter at any given time depends on the initial condition, and can be rescaled away, and is therefore not of physical relevance. Only the growth rates convey the main physics. For this case we conclude that the preferred superconducting phase is $d+id$.
The time-evolution for doped graphene is similar to Fig. \[fig1\] except that the slopes for $d+id$ and $d-id$ are the same, reflecting the degenerate eigenvalues of $\Pi^R$. While Fig. \[fig1\] is for parameters where all three orders grow in time, the disordered phase is characterized by all three orders decaying exponentially in time. In addition, for $J$-values smaller than that shown in Fig. \[fig1\], we can have a situation, where only one order-parameter (typically $d$-wave for the doping levels discussed) grows, while the others decay in time. For all these scenarios, the fastest growing order-parameter determines the preferred phase of the system.
As the quench amplitude increases, the $d$-wave order parameter growth rate increases. However, after some critical value of $J$, the $s$-wave order parameter begins to grow faster, eventually out growing the $d$-wave order parameter. How the growth rate difference of the $d$-wave and $s$-wave order parameters vary with quench amplitude is shown in Fig. \[fig2\]. Initially the $d$-wave becomes more unstable as the interaction parameter increases. This result is expected as in general the growth rate of any order parameter is zero at the critical point, and grows (decreases) faster as one moves further into (away) from the ordered phase. However, we find that after a critical $J$, the difference between the growth rates of the two order parameters rapidly approaches zero, with the $s$-wave growing faster than the $d$-wave beyond a critical quench amplitude.
The phase diagram determined by the fastest growing order parameter is shown in Fig. \[fig3\] for doped graphene, and Figures \[fig4\] and \[fig5\] for the Haldane model. The two figures for the Haldane model are obtained by applying a laser of two different strengths but same frequency to graphene. We still take care that the frequency of the laser is larger than the bandwidth of graphene so that the high-frequency expansion is valid.
For all these figures, the transition from the disordered (i.e, normal) to the $d$-wave ordered phase obtained from the time-evolution, coincides with an equilibrium calculation based on identifying the $T$ and $J$ values at which $1= J\Pi^R(\omega=0)$. Thus consistent with BCS theory, the boundary between the disordered and $d$-wave phase in Figures \[fig3\], \[fig4\], \[fig5\] are strongly dependent on the density of states at the Fermi energy. For this reason, the onset of superconductivity in Fig. \[fig5\] occurs at a smaller value of the interaction as compared to the other figures because the larger amplitude laser in Fig. \[fig5\] flattens the bands more, increasing the density of states at the Fermi energy.
For larger values of $J$ (i.e., as the quench amplitude increases), eventually the $s$-wave grows faster for graphene as well as the Haldane model. This result can be understood as follows. For smaller quench amplitudes, the dynamics is primarily from quasi-particle excitations in the vicinity of the Fermi surface. Thus the equilibrium phase predicting $d$-wave order is recovered. In contrast, for larger quench amplitudes, the dynamics is governed by highly excited quasi-particles. These quasi-particles cause an effective dephasing, leading to an averaging over the entire Brillouin zone (BZ). This averaging favors an $s$-wave rather than a $d$-wave because the latter order parameter changes sign in the BZ, so that its magnitude is effectively averaged out to zero by the dephasing.
Since at short times $t\sim O(1)$, quasi-particles everywhere in the BZ participate in the dynamics, the $s$-wave component of the order-parameter grows faster than the $d$-wave (see Fig. \[fig1\]). If the quench amplitude is large, the initial impulse on the $s$-wave order-parameter is large enough to overtake the growth of the $d$-wave. For the particular case shown in Fig. \[fig1\], this initial impulse is not strong enough, and at long times the $d$-wave grows faster.
Note that the boundary between the $d$-wave and $s$-wave phases in Figures \[fig3\], \[fig4\] and \[fig5\] is rather flat as a function of temperature. This is simply reflecting the fact that since quasi-particles everywhere in the BZ are participating in the dynamics for such large quench amplitudes, these are not sensitive to the details of the initial distribution function. The location of the flat line between the $d$-wave and $s$-wave phases does however depend on the initial state through the doping level.
In all the three Figures \[fig3\], \[fig4\] and \[fig5\], we plot the phase diagram up to the temperatures where the $d$-wave and the $s$-wave growth rates are clearly different. At higher temperatures, all the order parameter growth rates are degenerate, and one has to account for effects beyond mean-field in order to lift their degeneracy, an analysis that is beyond the scope of the paper.
We make our observation regarding the favored superconducting order more formal by noting that the growth or decay rate of an initial order parameter fluctuation is determined by the location of the zeros of $D(q,z)=1-J\Pi^R(q,z)$ in the complex $z$ plane (see Eq. and discussion above it). To locate the zero $z_0$, the real and imaginary parts of $z$ should be tuned together to satisfy the two conditions $1/J = {\rm Re}[\Pi^R(z_0)]=0, {\rm Im}[\Pi^R(z_0)]=0$. The real part of the zero ${\rm Re}\left[z_0\right]$ determines the exponential growth or decay rate, while the imaginary part ${\rm Im}\left[z_0\right]$ denotes an overall oscillation.
In equilibrium, the critical temperature $T_c$ for a given $J$ is one where the zero approaches $z=0$ linearly in the deviation from the critical temperature or critical coupling. For an interaction quench deeper and deeper into the ordered phase, the pole shifts further out into the complex plane, indicating that the order parameter grows faster. This behavior can be seen in Fig. \[fig6\] where the growth rate is determined by the value of $z$ for which $1/J$ intersects with the real part of $\Pi^R(z)$. In addition, Fig. \[fig6\] clearly shows that as $J$ increases, first the $d$-wave grows faster. But for larger values of $J$, eventually the $s$-wave order grows fastest.
In Fig. \[fig6\], for simplicity we allow $z$ to be purely real, and so we assume that the zeros fall entirely on the imaginary axis, or equivalently the order-parameter has a purely exponentially growing or decaying component in time, and no oscillatory component. We look for solutions where $1/J={\rm Re}[\Pi^R(z)]$. Inspite of not doing the analysis in the full complex plane, the growth rate and critical couplings agree very well with the full time evolution. Since the full correct analysis is the real time evolution, this good agreement between the time-evolution and the zeros in the complex plane indicate that ${\rm Im}\left[z_0\right]\simeq 0$.
We now discuss the effect of inelastic scattering. This will have two effects. One is to thermalize the electron gas, giving rise to a temperature larger than the initial temperature $T$ of the electron gas. As long as the final temperature is less than $T_{\rm BKT}$, the system will show quasi-long range order at long times. The second effect of inelastic scattering is that it causes the breakdown of the mean-field approximation by causing scattering between superconducting fluctuations. For the parameters of Fig. \[fig1\], the Fermi Golden rule estimate for the inelastic scattering time (which is also the time for leaving the prethermal regime) is $t_{\rm in}=1/J^2T$ $\sim 30 t_h^{-1}$. Fig. \[fig1\] shows that the physical quantity of interest, the growth rate or slope has stabilized well before this time. In addition the growth rate itself is order of the hopping amplitude, $\nu_{s,d\pm id} \sim t_h$, and is therefore much larger than the inelastic scattering rate ($t_{\rm in}^{-1}\sim 0.03 t_h$). Thus for the parameters discussed in the paper, the transient regime is indeed dominated by the mean field estimate because the scattering rate between fluctuations is small as compared to the rate at which the order-parameter grows.
Conclusions {#concl}
===========
The growth of superconducting order under nonequilibrium conditions is an active area of research. In this paper we give an explicit example of how two competing superconducting states behave after an interaction quench. We show that tuning the interaction quench amplitude can favor one superconducting state over the other.
This result is relevant to experiments in cold-atom gases where interaction quenches into the superconducting phase can be performed. Recent experiments in pump-probe spectroscopy make these results also relevant to the solid-state. In fact a laser can modify the lattice parameters, and through it the effective interactions in the $t$-$J$ model used by us to study superconductivity. Thus even in the solid-state, a laser quench can effectively give rise to an interaction quench.
While our study was for a hexagonal lattice with and without TR symmetry, our result showing how the dominant phase can be tuned by nonequilibrium conditions, such as an external laser and interaction quenches, is rather general and applicable to a wide range of lattice models where different ordered phases compete.
Our study treated the effect of the TR symmetry breaking laser within a high frequency expansion. The dynamics in the presence of a resonant low-frequency laser is expected to be even richer, and is left for future studies. Another interesting and challenging direction of study is to go beyond mean-field by allowing for interactions between the superconducting fluctuations. Since a laser, by its spatial orientation at any given time, breaks the underlying lattice symmetries, the interactions in principle can couple different superconducting channels. This could affect the outcome of the symmetry of the dominant superconducting order-parameter at steady-state.
[*Acknowledgements:*]{} The authors thank Yonah Lemonik for many useful discussions and Daniel Yates for numerical help. This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0010821.
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|
---
address: |
Institut d’Astrophysique de Paris, C.N.R.S./Paris VI, 98$^{bis}$ Boulevard Arago, F-75014, Paris, FRANCE\
E-mail: alfred@iap.fr
author:
- 'A. VIDAL-MADJAR'
title: 'D/H MEASUREMENTS'
---
Introduction
============
During primordial Big Bang nucleosynthesis deuterium is produced in significant amounts and then destroyed in stellar interiors. It is thus a key element in cosmology and in galactic chemical evolution (see [*e.g.*]{} Audouze & Tinsley [@at]; Boesgaard & Steigman [@bs]; Olive [*et al.*]{} [@oa]; Pagel [*et al.*]{} [@pa]; Vangioni-Flam & Cassé [@vc4]$^{,~}$[@vc5]; Prantzos [@p]; Scully [*et al.*]{} [@sc]; Cassé & Vangioni-Flam [@cv]).
The [*Copernicus*]{} space observatory has provided the first direct measurement of the D/H ratio in the interstellar medium (ISM) representative of the present epoch (Rogerson & York [@ry]) :
(D/H)$^{Copernicus}_{\rm ISM}\simeq1.4\pm0.2\times10^{-5}$.
More recently D/H evaluations were made in the direction of quasars (QSOs) in low metallicity media. They were completed toward three different QSOs’ (Burles & Tytler [@bta]$^{,~}$[@btb]; O’Meara & Tytler [@ot]) leading to a possible range of $2.4-4.8\times10^{-5}$ for the primordial D/H. These values correspond to a new estimations of the baryon density of the Universe, $\Omega_{\rm b}{\rm h}^{2}=0.019\pm0.0009$, in the frame of the standard BBN model (Burles [*et al.*]{} [@bal]; Nollett & Burles [@nb]). When compared to the recent $\Omega_{\rm b}{\rm h}^{2}$ evaluation made from the Cosmic Microwave Background (CMB) observations (see [*e.g.*]{} Jaffe [*et al.*]{} [@ja]) $\Omega_{\rm b}{\rm h}^{2}=0.032\pm0.005$, this seems to lead to a possible conflict. Note that another D/H measurement made toward a low redshift QSO leading to a D/H value possibly larger than $10^{-4}$ (Webb [*et al.*]{} [@we]; Tytler [*et al.*]{} [@ty]) corresponds to an even stronger disagreement since it translates into $\Omega_{\rm b}{\rm h}^{2}\le0.01$.
It is thus important to investigate the possibility of varying D/H ratios in different astrophysical sites (see [*e.g.*]{} Lemoine [*et al.*]{} [@la9]). If variations are indeed found, their cause should be investigated before a reliable primordial D/H evaluation can be inferred from a small number of observations.
Interstellar observations
=========================
Several methods have been used to measure the interstellar D/H ratio. All will not be discussed here and for more details see [*e.g.*]{} Ferlet [@f2]. The more reliable approach is to observe in absorption, against the background continuum of stars, the atomic Lyman series of D and H in the far-UV.
Toward hot stars, with the [*Copernicus*]{} satellite, many important evaluations of D/H were obtained (see e.g. Rogerson and York [@ry]; York and Rogerson [@yr]; Vidal–Madjar [*et al.*]{} [@va1977]; Laurent [*et al.*]{} [@lvy]; Ferlet [*et al.*]{} [@fa1980]; York [@y1983]; Allen [*et al.*]{} [@aa1992]) leading to the detection of variations recently enforced by HST–GHRS (Vidal–Madjar [*et al.*]{} [@va]) toward G191–B2B showing a low value and IMAPS observations, one made toward $\delta$ Ori presenting again a low value (Jenkins [*et al.*]{} [@ja]) confirming the previous analysis made by Laurent [*et al.*]{} [@lvy] from [*Copernicus*]{} observations and the other one toward $\gamma^2$ Vel with a high value (Sonneborn [*et al.*]{} [@so]). These observations seem to indicate that in the ISM, within few hundred parsecs, D/H may vary by more than a factor $\simeq3$.
From published values, D/H ranges from
$\sim5\times10^{-6}$ $<$ (D/H)$_{ISM}$ $<$ $\sim4\times10^{-5}$.
This method also provided a precise D/H evaluation in the local ISM (LISM) in the direction of the cool star Capella (Linsky [*et al.*]{} [@la]) :
(D/H)$^{\rm GHRS}_{\rm Capella}=1.60\pm0.09^{+0.05}_{-0.10}\times10^{-5}$
Additional observations made in the LISM lead Linsky [@l98] (see references there in) to the conclusion that the D/H value within the Local Interstellar Cloud (LIC) is (compatible with 12 evaluations) :
(D/H)$^{\rm GHRS}_{\rm LIC}=1.50\pm0.10\times10^{-5}$
The nearby ISM
==============
Observations of white dwarfs (WD) in the nearby ISM (NISM) for precise D/H evaluations were first proposed and achieved in the direction of G191–B2B by Lemoine [*et al.*]{} [@la6] using the HST–GHRS spectrograph at medium resolution. Follow up observations on G191–B2B at higher resolution with the GHRS Echelle-A grating by Vidal–Madjar [*et al.*]{} [@va] (same instrument configuration used as in the Capella study) lead to a precise D/H evaluation in the NISM along this line of sight within one H[i]{} region – the Local Interstellar Cloud (LIC) also observed toward Capella (these stars are separated by $\sim7^{o}$ on the sky) – and within a more complex and ionized H[ii]{} region presenting a double velocity structure. In these two main interstellar components the D/H ratio was found to be different if one assumes that the D/H value within the LIC is the same as the one found in the direction of Capella, in which case D/H has to be lower ($\sim0.9\times10^{-5}$) in the more ionized components.
In any case a lower “average” D/H ratio is found (2$\sigma$ error) :
(D/H)$^{\rm GHRS}_{\rm G191-B2B}=1.12\pm0.16\times10^{-5}$
This result has been contested by Sahu [*et al.*]{} [@sa9] who used new HST–STIS high resolution Echelle observations. However Vidal–Madjar [@v] has showed that all data sets (GHRS and STIS) in fact converge on a same value of the D/H ratio, which furthermore agrees with that derived by Vidal–Madjar [*et al.*]{} [@va] and disagrees with that of Sahu [*et al.*]{} [@sa9].
Since the disagreement between the two analysis was on the D[i]{} column density estimation, FUSE observations were expected to clarify the situation since they give access to weaker deuterium Lyman lines that are less sensitive to saturation effects than Lyman $\alpha$. Three independent data sets were obtained corresponding to the three different FUSE entrance apertures (Vidal–Madjar [*et al.*]{} [@va1]). The fits of the D Lyman $\beta$ line in the various FUSE channels are shown in Figure 1 and compared with the estimate of Sahu [*et al.*]{} [@sa9]. These new data confirm the measurement of N(D[i]{}) of Vidal–Madjar [*et al.*]{} [@va]; the value N(D[i]{}) derived by Sahu [*et al.*]{} [@sa9] lies 6$\sigma$ away from the new result. These 6$\sigma$ are quantified in terms of $\Delta\chi^2$, including many possible systematics such as stellar continuum placement, zero level, spectral instrument shifts, line spread function profiles, all free in the fitting process (see e.g. the different stellar continuum levels in Figure 1 from left to right panels).
The H[i]{} column density toward G191–B2B is well determined. Independent measurements with EUVE (Dupuis [*et al.*]{} [@da5]), GHRS (medium[@la6] and high resolution[@v]) and STIS (high resolution[@sa9]) using several methods of evaluation (EUV, Lyman continuum opacity and Lyman $\alpha$, damping wing modelling), converge on a value of log N(H[i]{}) = 18.34 ($\pm0.03$). The error on this value includes systematic errors associated with the various measurement techniques.
Using the D[i]{} column density as measured by FUSE and the H[i]{} column density compatible with all published values, one arrives at (2$\sigma$ error) :
(D/H)$^{\rm FUSE-HST-EUVE}_{\rm G191-B2B}=1.16\pm0.24\times10^{-5}$
This value is marginally compatible ($\ge2\sigma$) with the LIC one.
The essential question remains : if D/H variations are confirmed in more sightlines, what could be their cause ?
The FUSE observatory
====================
FUSE starts to produce orders of magnitude more data on the distribution of D/H in the ISM. From the planned D/H survey, we should be able to evaluate the deuterium abundance in a wide variety of locations, possibly linked to the past star formation rate as well as to the supposed infall of less processed gas in our Galaxy, and thus better understand Galactic chemical evolution.
The FUSE sensitivity should allow evaluations of the deuterium abundance in tens of lines of sights : i) in the direction of white dwarfs and cool stars in the NISM ; ii) toward hot sub-dwarfs in the more distant ISM and nearby Galactic halo ; iii) within the Galactic disk over several kilo-parsecs in the direction of O and early B stars ; iv) in the more distant Galactic halo, within high velocity cloud complexes as well as in intergalactic clouds in the direction of low redshift QSOs, AGNs and blue compact galaxies.
The first precise D/H evaluations toward few white dwarfs were presented in early 2001 at the AAS meeting (Moos [*et al.*]{} [@ma1]; Friedman [*et al.*]{} [@fa1]; Hébrard [*et al.*]{} [@ha1]; Kruk [*et al.*]{} [@ka1]; Linsky [*et al.*]{} [@la1]; Sonneborn [*et al.*]{} [@sa1]; Vidal–Madjar [*et al.*]{} [@va1]). The deuterium Lyman lines are clearly seen toward these few WDs and, as an example, the Lyman $\beta$ line is shown in the case of G191–B2B as previously discussed (see Figure 1). Several of these D/H evaluations made in the ISM with FUSE, HST, IMAPS are shown in Figure 2 along with one made recently in the direction of one QSO from ground based observations [@ot], as a function of the line of sight average metallicity as traced by O/H when available. It seems that the D/H variation does not anti–correlate with O/H. Thus a simple mechanism as astration, able to destroy D and produce O, does not seem compatible with the observations. Other mechanisms should be investigated as the ones listed by [*e.g.*]{} Lemoine [*et al.*]{} [@la9].
Conclusion
==========
In summary the status of the different – but discordant – D/H evaluations taken with no a priori bias to select one over another could be the following.
If the variations of the D/H ratio in the NISM are illusory, one could quote an average value of (D/H)$_{\rm NISM} \simeq 1.3-1.4\times10^{-5}$ barely compatible with all observations.
More in agreement with the present observations, D/H seems to vary in the ISM. One has thus to understand why.
Until then, any single or small number of values should not be considered to represent the definitive D/H in a given region. This is particularly true for the “primordial” values found in the direction of QSOs since the physical state of the probed environment is more poorly known than the Galactic one.
Our hope is that the FUSE mission will solve these problems.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am very grateful for the entire FUSE operation team for all the impressive work they are doing to make the FUSE observatory come true. I also thank the whole FUSE team for many positive interaction and comments. This work is based on data obtained by the NASA–CNES–CSA FUSE mission operated by the Johns Hopkins University under NASA contract NAS5–32985.
[99]{} J. Audouze and B.M. Tinsley, . A.M. Boesgaard and G. Steigman, . K. Olive [*et al.*]{}, . B. Pagel [*et al.*]{}, . E. Vangioni-Flam and M. Cassé, . E. Vangioni-Flam and M. Cassé, . N. Prantzos, . S.T. Scully, [*et al.*]{}, . M. Cassé and E. Vangioni-Flam in [*Structure and Evolution of the Intergalactic Medium from QSO Absorption Line Systems*]{}, eds. P. Petitjean and S. Charlot (IAP Conference, 331, 1998). J. Rogerson and D. York, . S. Burles and D. Tytler, . S. Burles and D. Tytler, . J. O’Meara and D. Tytler, in these proceedings [*Cosmic Evolution*]{}, eds. M. Lemoine and R. Ferlet, 2001. S. Burles [*et al.*]{}, . K.M. Nollett and S. Burles, . A.H. Jaffe [*et al.*]{}, astro-ph/0007333, 2000. J.K. Webb [*et al.*]{}, . D. Tytler [*et al.*]{}, . M. Lemoine [*et al.*]{}, . R. Ferlet, in IAU$\#$150 [*Astrochemistry of Cosmic Phenomena*]{}, eds. P.D. Singh, (Kluwer, 85, 1992). D. York and J. Rogerson, . A. Vidal–Madjar [*et al.*]{}, . C. Laurent, A. Vidal–Madjar and D.G. York, . R. Ferlet [*et al.*]{}, . D.G. York, . M.M. Allen, E.B. Jenkins and T.P. Snow, . A. Vidal–Madjar [*et al.*]{}, . E.B. Jenkins [*et al.*]{}, . G. Sonneborn, [*et al.*]{}, . J. Linsky [*et al.*]{}, . J. Linsky, . M. Lemoine [*et al.*]{}, . M.S. Sahu [*et al.*]{}, . A. Vidal–Madjar, in [*The Light Elements and Their Evolution*]{}, eds. L. da Silva, M. Spite and J. R. de Medeiros (ASP Conference Series, 151, 2000). A. Vidal–Madjar [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). J. Dupuis [*et al.*]{}, . H.W. Moos [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). S.D. Friedman [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). G. Hébrard [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). J.W. Kruk [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). J.L. Linsky [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). G. Sonneborn [*et al.*]{}, [*Ap.J.*]{} in preparation, (2001). D.M. Meyer [*et al.*]{}, .
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---
abstract: 'One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved spacetime. This canonical form is needed to apply the Whitham Lagrangian method. The latter method, unlike the WKB method, places no restriction on the magnitude of Planck’s constant to obtain wave packets, and furthermore preserves the symmetries of the Dirac Lagrangian. We show using canonical Dirac fields in a curved spacetime, that the probability current has a Gordon decomposition into a convection current and a spin current, and that the spin current vanishes in the Whitham approximation, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant. We further discuss the classical-quantum correspondence in a curved spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham equations. We show that the generalized de Broglie relations in a curved spacetime are a direct consequence of Whitham’s Lagrangian method, and not just a physical hypothesis as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.'
author:
- |
Mayeul Arminjon$^{1}$ and Frank Reifler$^2$\
$^1$ *Laboratory “Soils, Solids, Structures, Risks”\
*(CNRS and Universités de Grenoble: UJF, G-INP),\
*BP 53, F-38041 Grenoble cedex 9, France.\
*$^2$ Lockheed Martin Corporation, MS2 137-205,\
*199 Borton Landing Road, Moorestown, New Jersey 08057, USA.*****
title: Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations
---
PACS numbers: 03.65.Pm Relativistic wave equations
03.75. - b Matter waves
04.62. + v Quantum fields in curved spacetime
Introduction
============
Some might argue that quantum mechanics became a universal theory of matter when, at the last Einstein-Bohr debate at the Solvay Conference in 1930, Einstein proposed a weight measurement to observe unobtrusively a particle decay in order to contradict the Heisenberg energy-time uncertainty relation [@1.]. After nearly being defeated by Einstein in the debate, Bohr surprisingly countered with a general relativistic gravitational argument. Henceforth, the relation between gravity and quantum mechanics was to become an important question in fundamental physics.
It could also have been questioned whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves six years earlier by de Broglie [@1.], were rigorously valid in a general curved spacetime. This question can be shown to be equivalent to the question of whether sufficiently small wave packets travel along classical paths consistent with the de Broglie relations. Recall that a wave packet is a wave whose amplitude, frequency, and wave vector vary slowly over a region of spacetime comparable to a period or wave length. (E.g., for an electron traveling at half the speed of light, the wave length is approximately 5 $\times$ 10${}^{-}$${}^{12}$ m, and a typical wave packet has dimensions 10${}^{-}$${}^{6}$ m [@2.].) The de Broglie relations can be observed for such wave packets.
The well known WKB approximation is commonly used to derive wave packet approximations in quantum mechanics (e.g., [@Audretsch1981A]). The WKB approximation is based on taking the limit as a physical constant, namely Planck’s constant $\hbar $, approaches zero. [^1] However, the spin connection in the Dirac equation of a curved spacetime has no effect in the first WKB approximation (i.e., the one retaining only the zero order term in $\hbar $) [@Audretsch1981A]. The assumption that the spin connection can be neglected, as it would be in a first WKB approximation, is unnecessary and is too strong for many applications in a curved spacetime, or even in a Minkowski spacetime with arbitrary coordinates. Note that throughout this paper, except for the brief description of a post Newtonian approximation in Section \[WhithamMethod\], we may set both the speed of light $c$ and Planck’s constant $\hbar $ equal to one.
The Whitham approximation [@12.], which we adopt in this paper, places no restriction on Planck’s constant. To implement the Whitham approximation and to show that it leads to propagation along classical paths, we will first show in Section \[Canonical Form\] that any Dirac equation in a curved spacetime can be transformed into an equivalent canonical form known in the literature as the “local representation" [@3.], [@4.]. In general, transformation to equivalent canonical form is a necessary step to simplify a Dirac equation so that propagation along classical paths can be derived. It will be evident in Theorem 1 of Section \[Canonical Form\] that Planck’s constant $\hbar $ does not appear in the transformations mapping Dirac equations to their equivalent canonical forms [@3.], [@4.]. Previously, these canonical forms, or “local representations" as they are called in the literature, have only been discussed in the special case of orthogonal coordinates [@3.], [@4.], [@5.].
Then in Section \[WhithamMethod\], with each Dirac equation transformed into equivalent canonical form, we apply Whitham’s Lagrangian method [@12.] to derive wave packets in general curved spacetimes. Whitham’s method preserves the symmetries of the Lagrangian, and in particular, the Whitham wave packet equations conserve the probability current. We also show that generalized de Broglie relations, as well as COW and Sagnac type terms [@10.], emerge from the Whitham equations. It will become clear in Sections 2 and 3 that for every Dirac equation transformed into equivalent canonical form, the generalized de Broglie relations have no other meaning than the fact that sufficiently small wave packets propagate along classical paths in a background of gravitational and electromagnetic fields. This is what is observed in experiments [^2] and therefore more physically precise than the statement often made that particles with a given energy and momentum possess a frequency and wave vector given by the de Broglie relations. In fact, the generalized de Broglie relations are a *direct consequence* of Whitham’s method applied to each Dirac equation transformed into equivalent canonical form, and not just a physical *hypothesis* as introduced by Einstein and de Broglie, and by many quantum mechanics textbooks.
In the WKB approximation of the standard Dirac equation in a curved spacetime, classical trajectories are derived from the Gordon decomposition of the probability current $J^\mu = J^\mu _c + J^\mu _s$ into a convection current $J^\mu _c$ and a spin current $J^\mu _s$ [@Audretsch1981A]. In Section \[WhithamMethod\], we also prove the existence of the Gordon decomposition for Dirac equations transformed into canonical form. We further show that in the Whitham approximation, the spin current $J^\mu _s$ vanishes, which explains the negligible effect of spin on wave packet solutions, independent of the size of Planck’s constant $\hbar $.
It is also clear that the canonical forms (or “local representations”) of the Dirac equations, while not unique, are the preferred representations to understand certain phenomena associated with the Dirac equations in a curved spacetime, particularly, the emergence of classical physics and its conservation laws in a quantum world. Section \[Classical-Quantum\] concludes this paper with a discussion of the classical-quantum correspondence in a curved spacetime based on both Lagrangian and Hamiltonian formulations of the Whitham equations. In this section we also include results from a previous analysis of the classical-quantum correspondence [@15.], which can be applied to the canonical forms of Dirac equations in a curved spacetime considered in this paper.
New Representations of the Dirac Equation in a Curved Spacetime and their Equivalent Canonical Forms {#Canonical Form}
====================================================================================================
Shortly after Dirac discovered his celebrated four component wave equation:
\^ \_ = - , \[GrindEQ\_\_1\_\] together with its conserved probability current: J\^ = c \^[+]{} A\^ , \[GrindEQ\_\_2\_\] his equation was studied in its widest representations for a Minkowski spacetime [@6.], [@16.].
In Eq. , the Dirac field $ \Psi $ is a four component complex function of spacetime coordinates $ x^{\mu } $, $ \mu = 0 , 1 , 2, 3 $, whose partial derivatives with respect to $ x^{\mu } $ are denoted as $ \partial _{\mu } \Psi $. The Dirac gamma matrices $ \gamma ^{\mu } $, acting on $ \Psi $, satisfy the anticommutation formula: \^ \^ + \^ \^ = 2 \^ [**1**]{}\_[[**4**]{}]{}, \[GrindEQ\_\_3\_\] where $ \eta ^{\mu \nu } $ is the inverse of the Minkowski metric tensor $ \eta _{\mu \nu } $, and $ {\bf 1}_{{\bf 4}} $ denotes the identity matrix acting on the Dirac field $ \Psi $. The mass, the speed of light, and Planck’s constant are denoted by $ m $, $ c $, and $ \hbar $, respectively. Repeated indices are summed.
In Eq. , $ \Psi ^{+} $ denotes the complex conjugate transpose (Hermitian conjugate) of the Dirac field $ \Psi $, and $ A $ is a hermitizing matrix for the Dirac gamma matrices $ \gamma ^{\mu } $. That is, [@6.], [@16.]:
[l]{} [A\^[+]{} = A,]{}\
\
[\^[+]{} = A \^ A\^[-1]{} ]{},
\[GrindEQ\_\_4\_\]
where $ \gamma ^{\mu +} $ and $ A^{+} $ denote the Hermitian conjugates of the matrices $ \gamma ^{\mu } $ and $ A $, respectively. The hermitizing matrix $ A $ is uniquely determined by the matrices $ \gamma ^{\mu } $ up to a nonzero real scalar multiple [@6.].
For a Minkowski spacetime, assuming that $ \left( \gamma ^{\mu } , A \right) $ are chosen to be constant matrices satisfying Eqs. and , every solution of Eq. satisfies the Klein-Gordon equation, and the probability current $ J^{\mu } $ in Eq. is then also conserved. That is, Eqs. and are the only conditions that the constant matrices $ \left( \gamma ^{\mu } , A \right) $ need satisfy.
The “coefficient matrices” $ \left( \gamma ^{\mu } , A \right) $ in Eqs. and are far from unique. Given a Dirac field $ \Psi $, and any set of constant coefficient matrices $ \left( \gamma ^{\mu } , A \right) $ satisfying Eqs. and , they may be transformed by any constant complex $ 4\times 4 $ matrix $ S $ as follows:
[l]{} [=S\^[-1]{} , ]{}\
\
[\^ = S\^[-1]{} \^ S,]{}\
\
[=S\^[+]{} A S.]{}
\[GrindEQ\_\_5\_\]
Such a transformation $ S $ is called a “similarity transformation” or a “spin-base transformation”, the latter referring to simply a change of basis for the four components of the Dirac field $ \Psi $. It is straightforward to see that Eqs. $-$ are invariant under all similarity transformations $ S $ by Eq. . Thus, in the widest sense no restriction to a smaller group of transformations was deemed necessary in the early development of the Dirac equation [@6.], [@16.].
Nevertheless, further choices were necessary when the Poincaré group of coordinate transformations of the Dirac equation was considered. At least three possibilities have been considered for Poincaré coordinate transformations in a Minkowski spacetime as follows:
A\) The Dirac field $ \Psi $ transforms as a quadruplet of complex scalars under a coordinate transformation. [@17.], [@18.], [@19.], [@20.]\
B\) The Dirac field $ \Psi $ transforms as a complex four-vector $ \Psi ^{\mu } $ under a coordinate transformation. [@21.]\
C\) The Dirac field $ \Psi $ transforms as a quadruplet of complex scalars under a coordinate transformation, which is then followed by a similarity transformation. (The combined transformation leaves the constant coefficient matrices $ \left(\gamma ^{\mu } , A\right) $ invariant.) [@6.]\
For a Minkowski spacetime with Poincaré coordinate transformations, all three possibilities may be considered. However, with general coordinate transformations, as required for a curved spacetime, only the first two possibilities (A) and (B) exist. Since in a general spacetime, the possibility (C) does not exist, it was replaced by the possibility (A) in what has become the standard Dirac equation, which was proposed independently by Weyl [@17.] and by Fock [@18.], hereafter called the “Dirac-Fock-Weyl” (DFW) equation. See Refs. [@19.] and [@20.]. Possibility (B) was investigated recently [@21.], which became the impetus for a more general study of Dirac equations [@7.], [@8.], [@22.].
General Dirac fields of type (A) will be said to belong to the Quadruplet Representation of the Dirac theory (or QRD theory). General Dirac fields of type (B) will be said to belong to the Tensor Representation of the Dirac theory (or TRD theory). It was recently shown that in an open neighborhood of each spacetime point, every TRD equation is in fact equivalent to a QRD equation and vice versa [@7.]. Since TRD equations are locally equivalent to QRD equations, we will only consider QRD equations in this paper. Note that there are QRD equations in a curved spacetime that are not locally equivalent to any DFW equation [@7.].
In a further evolution of the Dirac equation, which applies also to a Minkowski spacetime, the partial derivatives in the Dirac equation were replaced by covariant derivatives $ D_{\mu } = \partial _{\mu } + \Gamma _{\mu } $ where $ \Gamma _{\mu } $ are four $ 4\times 4 $ complex matrices [^3] , called “spin connection matrices”, acting on the four components of the Dirac field $ \Psi $. At the same time, the Dirac equation was generalized by substituting a general metric $ g_{\mu \nu } $ of Lorentz signature and determinant $g$ for the Minkowski metric $ \eta _{\mu \nu } $ in the anticommutation formula for the Dirac gamma matrices:
\^ \^ + \^ \^ = 2 g\^ [**1**]{}\_[[**4**]{}]{}. \[GrindEQ\_\_7\_\] For the results in this paper, mild restrictions must be placed on the metric components $ g_{\mu \nu } $, namely that $ g_{00} > 0 $ and the components $ g_{jk} $ for $ j, k = 1, 2, 3 $ form a negative definite $ 3\times 3 $ matrix. Even though these mild conditions hold for almost all spacetime metrics $ g_{\mu \nu } $ of interest, the Gödel metric is a notable exception [@23.], [@24.].
For this generalization, the coefficient matrices $ \left( \gamma ^{\mu } , A \right) $ defining the Dirac equation are augmented to become “coefficient fields” $ \left( \gamma ^{\mu } , A , \Gamma _{\mu } \right) $, which now may vary with the spacetime point. Then, in order that the Dirac equation be invariant, the transformation equations are augmented with the transformation of the spin connection matrices $ \Gamma _{\mu } $ under “local similarity transformations” $ S $ (i.e., similarity transformations $ S $ that may also vary with the spacetime point [^4] ) as follows: [@20.]
[l]{} [ \_ = S\^[-1]{} \_ S + S\^[-1]{} \_ S]{}\
\
[=S\^[-1]{} ( \_ +\_ ) S.]{}
\[GrindEQ\_\_9\_\] Indeed, for this type of transformation, we have the covariant derivatives transforming as: $\widetilde{D}_{\mu }= S^{-1} \circ D_{\mu } \circ S$. Transformations of the kind given by Eqs. and we will call local similarity transformations “of the first kind”.
Local similarity transformations “of the second kind” are defined by setting: \_ = \_. \[GrindEQ\_\_11\_\] For this second kind of transformation, given by Eqs. and , we have the covariant derivatives transforming as: $ \widetilde{D}_{\mu } = D_{\mu } $.
Two Dirac equations will be said to be “equivalent” or “classically equivalent” if there exists a local similarity transformation $ \Psi \to S^{-1} \Psi $ of any kind that transforms the solutions of one Dirac equation into the solutions of the other. [^5] From Eqs. and , the conserved probability currents for two equivalent Dirac equations are equal. Hence in any spacetime, scattering experiments will give the same results regardless of the representation of a given Dirac equation. However, in a first quantized theory, which is our concern in this paper, a local similarity transformation $ S $ may not intertwine with the quantum mechanical operators. In that case, the operators corresponding to a given observable generally will not have the same eigenvalues before and after the transformation $S$. This has been shown for the Hamiltonian (or energy) operator in previous work [@25.]. Thus, two Dirac equations that are equivalent as partial differential equations via a local similarity transformation $S$, need not be equivalent with respect to all quantum mechanical operators [@25.], [@26.], [@27.].\
The standard “Dirac Lagrangian” applies to the DFW equation [@19.] and has to be extended to include the coefficient field $A$ in the exact place of the constant hermitizing matrix valid for DFW [@25.]:
[l]{} [L= L(, \_ , x\^ )]{}\
\
[= [\[]{} \^[+]{} A\^ ( D\_ ) -( D\_ )\^[+]{} A\^ + i \^[+]{} A.]{}
\[GrindEQ\_\_8\_\] Note that the Lagrangian (\[GrindEQ\_\_8\_\]) is the local expression of a global Dirac Lagrangian based on a general Dirac operator ${\not}\mathcal{D}$ acting on the cross-sections of a spinor bundle ${\sf E}$ over the spacetime. [^6] Thus, the Lagrangian (\[GrindEQ\_\_8\_\]) gives a generalized formulation of the Dirac theory for a general Dirac operator ${\not}\mathcal{D}$ on a curved spacetime. We will see in Theorem 1 that the equivalent canonical forms of DFW equations require such a generalization.
The Euler-Lagrange equations for the Lagrangian give the following generalized Dirac equation [@7.], [@8.]:
\^ D\_ + A\^[-1]{} D\_ ( A\^ ) = - , \[GrindEQ\_\_14\_\]
where
[l]{} [D\_ \_ + \_ ]{}\
\
[D\_ \^ \_ \^ + \_ \^ - \^ \_ ]{}\
\
[D\_ A\_ A - \_\^[+]{} A - A \_, ]{}
\[GrindEQ\_\_15\_\]
and we define the Levi-Civita covariant derivatives **$ \nabla _{\mu } $** acting on the Dirac field $ \Psi $ and the coefficient fields $ \left( \gamma ^{\mu } , A \right) $ as follows:
[l]{} [\_ \_ ]{}\
\
[\_ \^ \_ \^ + {\^\_[ ]{} } \^ ]{}\
\
[\_ A\_ A]{}
\[GrindEQ\_\_16\_\]
where $\left\{^\nu _{\rho\, \mu } \right\}$ are the Christoffel symbols belonging to the Levi-Civita connection. The covariant derivatives $ D_{\mu } $ extend to $\Psi ^{+}$ by the formula $D_{\mu } \Psi ^{+}= \left( D_{\mu } \Psi \right)^{+} $, and similarly $\nabla _{\mu } \Psi ^{+}= \left(\nabla _{\mu } \Psi \right)^{+} $. As usual, covariant derivatives extend to products of fields via Leibniz’s rule for differentiating products.
If $D_{\mu } \left(A\gamma ^{\mu } \right)= 0$, then the generalized Dirac equation reduces to normal form:
\^ D\_ = - . \[GrindEQ\_\_17\_\]
Normal Dirac equations generalize the DFW property that the coefficient fields $ \left( \gamma ^{\mu } , A \right) $ be covariantly constant: $ D_{\mu } \gamma ^{\nu } = 0 $ and $ D_{\mu } A = 0 $. One can show that the weaker normality condition $ D_{\mu } \left( A\gamma ^{\mu } \right) = 0 $ is preserved under all local similarity transformations of the first kind [@7.]. The normality condition is also preserved under all coordinate transformations.
Thus we have several invariant classes of Dirac equations. First, we have the class of Dirac equations for which the coefficient fields $ \left( \gamma ^{\mu } , A \right) $ are covariantly constant $-$ that is, $ D_{\mu } \gamma ^{\nu } = 0 $ and $ D_{\mu } A = 0 $. This class contains the DFW equations as a proper subset. Second, we have the class of Dirac equations for which $ D_{\mu } \left( A\gamma ^{\mu } \right) = 0 $. This is the class of normal Dirac equations which contains the first class as a proper subset. Finally, we have the class of generalized Dirac equations which contains the other two classes. Each of these classes is invariant under all coordinate transformations and also under all local similarity transformations of the first kind [@7.].
A non-invariant class of Dirac equations that we will discuss in this paper is the class of the QRD–0 equations, in which the contracted spin connection matrix $ \Gamma \equiv \gamma ^{\mu } \Gamma _{\mu } = 0 $. See Ref. [@7.], Sect. 3.2.1. Equations in the QRD–0 class (with $ \Gamma = 0$) are generalized Dirac equations which may be written as follows: \^ \_ + A\^[-1]{} \_ ( A\^ )=- , \[GrindEQ\_\_18\_\] with the Levi-Civita covariant derivatives $ \nabla _{\mu } $ acting on the coefficient fields $ \left( \gamma ^{\mu } , A \right) $ as previously defined in Eq. . If $ \nabla _{\mu } \left( A\gamma ^{\mu } \right) = 0 $, then the QRD–0 equation reduces to the simpler normal form:
\^ \_ = - . \[GrindEQ\_\_19\_\]
Note that a QRD–0 equation which is both normal as in Eq. and equivalent to a DFW equation is called a “local representation” of the DFW equation by other authors [@3.], [@4.], [@5.]. Finding a “local representation” for a DFW equation often simplifies the process of deriving its solutions. Previously, this “local representation” of the DFW equation in a curved spacetime in the form of Eq. has only been discussed in the special case of orthogonal coordinates [@3.], [@4.], [@5.].
Given any normal Dirac equation , one can construct an equivalent normal QRD–0 equation in terms of a basis of solutions of the massless equation associated with : \^ D\_ = \^ ( \_ +\_ ) = 0. \[GrindEQ\_\_20\_\]This is useful when solutions to the massless equation are known, for which there are many examples in general relativity, including all diagonal metrics and Gödel type metrics [@29.] $-$ [@33.]. Indeed, consider a local similarity transformation $ S $ which takes the Dirac field $ \Psi $ to $ \widetilde{\Psi } $ and the coefficient fields $ \left( \gamma ^{\mu } , A , \Gamma _{\mu } \right) $ to $ \left( \widetilde{\gamma }^{\mu } , \widetilde{A} , \widetilde{\Gamma }_{\mu } \right) $, according to Eqs. and . Recall that the contracted spin connection matrix $ \widetilde{\Gamma }\equiv \widetilde{\gamma }^\mu \widetilde{\Gamma }_\mu= 0$ if the transformed equation is a QRD–0 equation. Then from Eqs. and , we get:
\^ ( \_ +\_ ) S= 0 \[GrindEQ\_\_21\_\],
whereby any four linearly independent solutions to the massless equation can be used to form the columns of the matrix-valued field $ S $. In this case, an equivalent normal QRD–0 equation can be explicitly and globally constructed.
****
We can show that any generalized Dirac equation , with very minor conditions imposed on the spacetime metric $ g_{\mu \nu } $, is equivalent to a normal QRD–0 equation , in an open neighborhood of each spacetime point, by applying local similarity transformations of the first and second kind to the Dirac field $ \Psi $. Previously, this so called “local representation” of the DFW equation in a curved spacetime has only been discussed in the special case of orthogonal coordinates [@3.], [@4.], [@5.]. Here we generalize it to essentially all Dirac equations:\
**THEOREM 1.** Let $ {\bf U} $ be any open subset of a spacetime on which local coordinates are defined. Suppose that the metric components $ g_{\mu \nu } $ for $ \mu , \nu = 0, 1, 2, 3 $ in $ {\bf U} $ satisfy $ g_{00} > 0 $ and the components $ g_{jk} $ for $ j, k = 1, 2, 3 $ form a negative definite $ 3\times 3 $ matrix. Then, for any choice of smooth coefficient fields $ \left( \gamma ^{\mu } , A \right) $ and any choice of covariant derivatives $ D_{\mu } = \partial _{\mu } + \Gamma _{\mu } $ acting on smooth Dirac fields $ \Psi $ defined on $ {\bf U} $, there exists a smooth local similarity transformation $ S $ of the first kind, composed with a smooth local similarity transformation $ T $ of the second kind, taking $ \Psi \to \left( T\circ S \right)^{-1} \Psi $, which transforms the generalized Dirac equation with smooth coefficient fields $ \left( \gamma ^{\mu } , A , \Gamma _{\mu } \right) $ into an equivalent normal QRD–0 equation, in an open neighborhood of each point $ X_{0} \in {\bf U} $.\
Thus, we may regard the normal QRD–0 equations as canonical forms for all generalized Dirac equations , in open neighborhoods of each spacetime point.
The proof of Theorem 1 relies heavily on the theory of linear hyperbolic partial differential equations [@7.], [@34.], [@35.]. Since it is not particularly enlightening beyond the explicit construction given above for the normal case, we will postpone writing out the full proof of Theorem 1 until the Appendix.
Note that every DFW equation has the normal form . By Theorem 1, every DFW equation is equivalent to a normal QRD–0 equation (called a “local representation” of the DFW equation by Schlüter, Wietschorke, and Greiner) which generally is not a DFW equation [@3.], [@4.], [@5.]. Conversely, we can show that not every normal QRD–0 equation is equivalent to a DFW equation, so that DFW equations are in fact equivalent to only a proper subset of the possible normal QRD–0 equations.\
[**EXAMPLE.**]{} Consider the flat metric $ g_{\mu \nu } $ on $ {\bf R}^{4} $ whose line element $ ds $ in rotating cylindrical coordinates $ \left( t , r , \phi , z \right) $ has the form: ds\^[2]{} = (c dt)\^[2]{} - dr\^[2]{} - r\^[2]{} ( d+ dt )\^[2]{} - dz\^[2]{} . \[GrindEQ\_\_22\_\] Using the orthonormal tetrad [@28.] as indicated by the parsing of the metric in Eq. (\[GrindEQ\_\_22\_\]), the DFW equation is given by: \^ D\_ = \^ \_ + \^[ 1]{} = -i. \[GrindEQ\_\_29\_\]
A simple local similarity transformation of the first kind: S\^[-1]{} : ’= \[GrindEQ\_\_30\_\] which is independent of Planck’s constant, time independent, and also is independent of the rotation rate $\omega $, transforms the DFW equation into a Dirac equation of the canonical form . Note that the canonical Dirac equation does not have the Mashhoon term [@9.], [@10.], whose presence or absence thus depends on the chosen representation of the Dirac field. See also Ryder [@11.].
Whitham’s Lagrangian Method — The Main Theorem {#WhithamMethod}
==============================================
In this section we apply Whitham’s Lagrangian method [@12.] to derive wave packets for Dirac equations in general curved spacetimes. Whitham’s method preserves the symmetries of the Lagrangian, and in particular, the Whitham wave packet equations conserve the probability current. We show in this section that generalized de Broglie relations, as well as COW and Sagnac type terms [@10.], emerge from the Whitham equations after transforming each Dirac equation into an equivalent canonical form. Thus, the normal QRD–0 representations (or canonical forms) of the Dirac equations are the preferred representations to express the generalized de Broglie relations in a curved spacetime.
It is noteworthy that the Whitham approximation places no restriction on Planck’s constant $\hbar $. Indeed, the transformation which takes the Dirac equation to its canonical form, is independent of Planck’s constant $\hbar $. This is obvious in the example above, as seen in Eq. (\[GrindEQ\_\_30\_\]). In fact, this independence is a general fact that follows from Eqs. (\[GrindEQ\_\_74\_\]) and (\[GrindEQ\_\_76\_\]) used in the proof of Theorem 1 in the Appendix. Thus, the Whitham approximation is not equivalent to the WKB approximation, since the latter does not require any transformation of variables. [^7]
Including both gravitational and electromagnetic external fields, the generalized Lagrangian for the Dirac equation can be written as follows:
[l]{} [L= L ( , \_ , x\^ )]{}\
\
[= [\[]{} \^[+]{} A\^ ( D\_ ) - ( D\_ )\^[+]{} A\^ + i \^[+]{} A,]{}
\[GrindEQ\_\_33\_\]with the covariant derivatives $ D_{\mu } $ defined by: D\_ = \_ + \_ + V\_, \[GrindEQ\_\_34\_\] where $ V_{\mu } $ are electromagnetic gauge potentials and $ e $ is the electric charge. We can display the interaction terms of the Lagrangian more explicitly by expressing the Lagrangian as a sum of a free and an interaction part as follows:
[l]{} [L= [\[]{} \^[+]{} A\^ ( \_ ) - ( \_ )\^[ +]{} A\^ + i \^[+]{} A]{}\
\
[+ [\[]{} \^[+]{} ( A- \^[+]{} A ) - J\^ V\_ [\]]{},]{}
\[GrindEQ\_\_35\_\] where $ J^{\mu } \equiv c \Psi ^{+} A\gamma ^{\mu } \Psi $ is the probability current, and $ \Gamma \equiv \gamma ^{\mu } \Gamma _{\mu } $ is the contracted spin connection matrix, and also noting that since $ A $ is a Hermitizing matrix for the Dirac matrices $ \gamma ^{\mu } $, we have from Eq. :
\^[+]{} A= \_ \^[+]{} \^[+]{} A= \_ \^[+]{} A\^. \[GrindEQ\_\_36\_\] Since we can first transform any Dirac equation into a normal QRD-0 equation (or canonical form) as stated in Theorem 1 of Section \[Canonical Form\], we can transform the fields in this Lagrangian so that $ \nabla _{\mu } \left( A\gamma ^{\mu } \right) = 0 $ and $ \Gamma \equiv \gamma ^{\mu } \Gamma _{\mu } = 0 $. Thus, instead of the Lagrangian , we may substitute an equivalent canonical Dirac Lagrangian:
[l]{} [L= [\[]{} \^[+]{} A\^ ( \_ ) - ( \_ )\^[ +]{} A\^ + i \^[+]{} A]{}\
\
[- J\^ V\_,]{}
\[GrindEQ\_\_37\_\] Clearly, the normal QRD-0 equation is derived from the Euler-Lagrange equations of the Lagrangian by setting the electromagnetic gauge potentials $ V_{\mu } $ equal to zero and using the normality condition: $ \nabla _{\mu } \left( A\gamma ^{\mu } \right) = 0 $.
For Whitham’s method, we set $ \Psi = \chi e^{i\theta } $ where $ \chi = \chi \left(X\right) $ is also a complex wave function with four components, and $ \theta = \theta \left(X\right) $ is a real phase at each point $ X $ in the spacetime. Whitham’s method assumes that $ \chi $ is slowly changing compared to the rapidly changing phase $ \theta $, so that we may obtain approximate wave packet solutions to the Dirac equations by neglecting $ \partial _{\mu } \chi $ with respect to $\left( \partial _{\mu } \theta \right) \chi $ in the Lagrangian. Substituting the wave function $ \Psi = \chi e^{i\theta } $ into the Lagrangian and using this approximation, we get:
L=c . \[GrindEQ\_\_38\_\] In Whitham’s method, this Lagrangian governs the wave packet motion. Clearly the Lagrangian (\[GrindEQ\_\_38\_\]) is invariant under the global gauge symmetry $\theta \rightarrow \theta + \tau $, where $\tau $ is a real constant. This leads by Noether’s theorem to the conservation of a current. In Section \[Current\] we will derive explicitly the conservation of the probability current for this Lagrangian. Thus, our goal in this section is to derive the Euler-Lagrange equations for the Lagrangian (\[GrindEQ\_\_38\_\]), which by change of field variables leads to the following main theorem of this paper.
****
**THEOREM 2.** Define a four-vector field $ u^{\mu } $ and a scalar field $ J $, related to the amplitude $ \chi $ and phase $ \theta $ of the wave function $ \Psi = \chi e^{i\theta } $ as follows:
[l]{} [u\_ - \_ - V\_, ]{}\
\
[u\^ g\^ u\_,]{}\
\
[Jc \^[+]{} A .]{}
\[GrindEQ\_\_39\_\]
i\) Then the Euler-Lagrange equations for the Whitham Lagrangian imply the following equations for the fields $ u^{\mu } $ and $ J $ in a curved spacetime:
[l]{} [g\_ u\^ u\^ = 1,]{}\
\
[\_ ( Ju\^ )= 0,]{}\
\
[u\_ = g\_ u\^ ,]{}\
\
[\_ u\_ - \_ u\_ = - F\_, ]{}
\[GrindEQ\_\_40\_\]
where $ F_{\mu \nu } \equiv \nabla _{\mu } V_{\nu } - \nabla _{\nu } V_{\mu } $ is the electromagnetic field tensor.
ii\) The four-vector field $ u^{\mu } $ is a unit velocity field, such that $ J^{\mu } = Ju^{\mu } $ is a conserved probability current.
iii\) The integral curves $ x^{\mu } \left(s\right) $ of the four-vector field $ u^{\mu } $, parameterized by arc-length $ s $, are given by the classical equations:
& = u\^ ,\
&\
& + {\^[ ]{} \_[ ]{} } u\^ u\^ = F\^ \_[ ]{} u\^, \[GrindEQ\_\_41\_\]
along which the scalar field $ J $ satisfies:
= -J \_ u\^. \[GrindEQ\_\_42\_\]
The proof and interpretation of Theorem 2 will occupy the rest of Section 3.
Euler-Lagrange Equations for the Wave Packet Lagrangian
-------------------------------------------------------
The Euler-Lagrange equations for the amplitude $ \chi = \chi \left(X\right) $ and phase $ \theta = \theta \left(X\right) $ can be derived from the wave packet Lagrangian $ L $ in Eq. as follows. First we have from Eq. :
[l]{} [ = c ,]{}\
\
[ = -c \^[+]{} A\^ .]{}
\[GrindEQ\_\_43\_\]
Then, since no derivatives of $ \chi ^{+} $ and only derivatives of $ \theta $ occur in the Lagrangian , we set equal to zero, using Eq. , the following expressions:
[l]{} [0= = ]{} [ = c ,]{}\
\
0= = \_ ( )= \_ ( -c \^[+]{} A\^ ),
which then gives the following Euler-Lagrange equations:
[l]{} [( - \_ - V\_ ) A\^ ]{} [= mc A]{},\
\
\_ ( c \^[+]{} A\^ ) = 0.
\[GrindEQ\_\_45\_\]
For the wave function $ \Psi = \chi e^{i\theta } $, with phase $ \theta $, the wave covector is $ K_{\mu } \equiv \partial _{\mu } \theta $. Thus $ \omega \equiv -K_{0} = -\partial _{0} \theta $ is the angular frequency of the wave. We define a four-vector field $ p^{\mu } $ as follows:
p\_ - \_ = -K\_. \[GrindEQ\_\_46\_\] It will be shown in Section \[Classical-Quantum\] that $ p_{\mu } =-P_{\mu } $, where $ P_{\mu } $ are canonical momentum variables.
Eq. expresses the generalized de Broglie relations $ P_{\mu } = \hbar K_{\mu } $ between the canonical momentum variables $ P_{\mu } $ and the wave covector $ K_{\mu } $. We also define a four-vector velocity field $ u^{\mu } $ from the usual classical equation with kinetic and potential terms as follows:
p\^ = mc u\^ + V\^. \[GrindEQ\_\_47\_\]
From Eq. we have $ \partial _{\mu } p_{\nu } = \partial _{\nu } p_{\mu } $. Substituting $ \Psi = \chi e^{i\theta } $ into Eq. we have $ J^{\mu } = c \chi ^{+} A\gamma ^{\mu } \chi $. We denote $ \gamma \left( u \right) \equiv u_{\mu } \gamma ^{\mu } $. Then, Eqs. become: ( u ) & = & \[gamma chi = chi\],\
\[D\_mu J\^mu = 0\] \_ ( J\^ ) & = & 0,\
\[rot p = 0\] \_ p\_ - \_ p\_ & = & 0.
The first equation is an algebraic eigenvalue equation. The second equation can be written as the covariant conservation of the probability current, $ \nabla _{\mu } J^{\mu } = 0 $, where $ \nabla _{\mu } $ denotes the Levi-Civita covariant derivatives. Since $ p_{\mu } = -\hbar {\kern 1pt} \partial _{\mu } \theta $, the third equation expresses the equality of mixed partial derivatives of $ \theta $. Furthermore, since the left-hand side of the third equation is an antisymmetric tensor, the partial derivatives $ \partial _{\mu } $ can be replaced by Levi-Civita covariant derivatives $ \nabla _{\mu } $. Thus, Eqs. – become the following covariant equations: \[gamma chi = chi-bis\] ( u ) & = & ,\
\[nabla\_mu J\^mu = 0\] \_ J\^ & = & 0,\
\[rot p = 0-bis\] \_ p\_ - \_ p\_ & = & 0. We will show below that these equations, taken together, reduce to a set of quasi-linear partial differential equations describing a scalar density $ J = c \chi ^{+} A\chi $ and the four-vector velocity field $ u^{\mu } $, whose integral curves are classical relativistic trajectories. We will further show that certain of these equations give rise to initial conditions, and of the rest, only four equations are independent. First, let us derive a dispersion relation from the algebraic equation .
Dispersion Relation
-------------------
From the algebraic equation , and the anticommutation relation of Dirac gamma matrices , we have: ( g\^ u\_ u\_ ) = ( u )\^[ 2]{} = ( u ) = . \[GrindEQ\_\_50\_\]Equation implies that at any spacetime point where the wave function $ \chi $ is not zero, the four-vector velocity field $ u^{\mu } $ satisfies $ u^{\mu } u_{\mu } = 1 $. From Eq. , this gives the dispersion relation: g\^ ( p\_ - V\_ ) ( p\_ - V\_ ) - m\^[2]{} c\^[2]{} =0. \[GrindEQ\_\_51\_\]Since $ p_{0} = \varepsilon /c $, where $ \varepsilon $ is the energy and $ P_{j} = -p_{j} $ for $ j = 1 , 2 , 3 $ are momentum variables, Eq. is a quadratic equation for the energy $ \varepsilon $.
Let us consider the dispersion relation in the absence of the electromagnetic potentials $ V_{\mu } $. We have: g\^ p\_ p\_ = g\^[00]{} ( )\^[2]{} + 2g\^[0j]{} p\_[j]{} ( ) + g\^[jk]{} p\_[j]{} p\_[k]{} = m\^[2]{} c\^[2]{}. \[GrindEQ\_\_52\_\] That is, = . \[GrindEQ\_\_53\_\]
Choosing positive energy $ \varepsilon > 0 $ and setting: g\^[00]{} = 1 - ,g\^[0j]{} = ,g\^[jk]{} = -\^[jk]{} - , \[GrindEQ\_\_54\_\]
where $ \phi $, $ \phi ^{j} $, and $ \phi ^{jk} $ are gravitational potentials, and $ \delta ^{jk} $ is the Kronecker delta, equal to one if $ j = k $ and equal to zero otherwise, then we have for a non-relativistic approximation, i.e., taking the limit of $ \varepsilon - mc^{2} $ as the speed of light $ c $ goes to infinity in Eq. :
[l]{} [- mc\^[2]{} = - mc\^[2]{} ]{}\
\
[= - mc\^[2]{} ]{}\
\
[ - mc\^[2]{} ]{}\
\
[= c p\_[j]{} + ( mc\^[2]{} + - ) - mc\^[2]{} ]{}\
\
[ \^[jk]{} p\_[j]{} p\_[k]{} + m+ \^[j]{} p\_[j]{} ]{}.
\[GrindEQ\_\_55\_\]
That is, the non-relativistic approximation gives the energy as follows;
- mc\^[2]{} \^[jk]{} p\_[j]{} p\_[k]{} + m+ \^[j]{} p\_[j]{} . \[GrindEQ\_\_56\_\]
It is straightforward to identify the three energy terms on the right hand side of Eq. as the kinetic energy, a COW potential energy, and a Sagnac potential energy, respectively.
Probability Current and Classical Trajectories {#Current}
----------------------------------------------
Recall that for Whitham’s method, we set the wave function $ \Psi = \chi e^{i\theta } $ where $ \chi = \chi \left(X\right) $ is also a wave function and $ \theta = \theta \left(X\right) $ is a real phase at each point $ X $ in the spacetime. Then the probability current $ J^{\mu } $ and scalar field $J$ are given by:
J\^ c \^[+]{} A\^ = c \^[+]{} A\^ ,\
J c \^[+]{} A= c \^[+]{} A. \[GrindEQ\_\_57\_\]
From Eq. (\[gamma chi = chi-bis\]), $\chi $ is a solution of the equation $\gamma \left( u \right) \chi = \chi$, where $u^\mu $ is a unit four-vector field satisfying $u^\mu u_\mu =1$ and where $\gamma (u)\equiv u_\mu\gamma ^\mu $. From the anticommutation relation of the Dirac gamma matrices in Eq. (\[GrindEQ\_\_3\_\]), we have: \[u vs gamma(u)\] u\^=g\^u\_=(\^\^+\^\^)u\_=. Moreover, using again the definition $\gamma (u)\equiv u_\mu\gamma ^\mu $, it follows easily from the properties of the hermitizing matrix $A$ \[Eq. (\[GrindEQ\_\_4\_\])\] that $\chi ^+ A \gamma (u)=\left[ \gamma (u)\chi \right]^+A$. We get thus from Eqs. (\[gamma chi = chi-bis\]), (\[GrindEQ\_\_57\_\]) and (\[u vs gamma(u)\]): J u\^& = & \^+A\
& = & \^+A\^+ \^+ A\^\
& = & \^+ A\^+ \^+A\^\
& = & J\^. That is, $J^\mu =J u^\mu $. Using Eqs. and together with this result, Eqs. – can be written as: \[u normed\] g\_ u\^ u\^ & = & 1,\
\[u bemol\] u\_ & = & g\_ u\^,\
\[nabla Ju=0\] \_ ( Ju\^ ) & = & 0,\
\[rot u = C F\] \_ u\_ - \_ u\_ & = & - F\_, where $ F_{\mu \nu } \equiv \nabla _{\mu } V_{\nu } - \nabla _{\nu } V_{\mu } $ is the electromagnetic field tensor. Multiply by $ u^{\nu } $ and contract the index $ \nu $ on both sides of Eq. . Then, using Eq. to set $ u^{\nu } \left( \nabla _{\mu } u_{\nu } \right) = 0 $, and finally raising the index $ \mu $, we get:
( u\^ \_ ) u\^ = F\^ \_[ ]{} u\^. \[GrindEQ\_\_59\_\]
Consider the integral curves $ x^{\mu } \left(s\right) $ of the four-vector velocity field $ u^{\mu } \left(X\right) $. Since the four-vector velocity field consists of unit vectors by Eq. , the integral curves $ x^{\mu } \left(s\right) $ are parameterized by arc-length $ s $. That is, from Eqs. and :
& = u\^ ,\
&\
& + {\^[ ]{} \_[ ]{} } u\^ u\^ = F\^ \_[ ]{} u\^. \[GrindEQ\_\_60\_\]
Note that Eqs. are precisely the classical relativistic equations of a particle of mass $ m $ and electric charge $ e $ in a gravitational field $ g_{\mu \nu } $ in the presence of an electromagnetic field $ F_{\mu \nu } $. Thus, the integral curves of the four-vector velocity field $ u^{\mu } \left(X\right) $, describing the motion of wave packets, coincide with the trajectories of classical relativistic particles. Note that in the absence of the electromagnetic field $ F_{\mu \nu } $, the classical trajectories are geodesics of the spacetime.
Finally, we note from Eq. (\[nabla Ju=0\]) that along the integral curves of $ u^{\mu } \left(X\right) $ we have: = -J \_ u\^. \[GrindEQ\_\_61\_\] This completes the proof of Theorem 2.
Note that the wave packet equations describe a certain congruence of classical trajectories together with a scalar density $ J $ that on each classical trajectory in the congruence satisfies Eq. . This congruence satisfies certain initial conditions on a spatial submanifold $ {\bf M} $ discussed in the next subsection.
Mathematical Structure of the Wave Packet Equations
---------------------------------------------------
Eq. – can be written as follows:
\[u normed-bis\] g\_ u\^ u\^ & = & 1,\
\[u bemol-bis\] u\_ & = & g\_ u\^,\
\[nabla Ju=0-2\] ( Ju\^ ) & = & 0,\
\[rot u = C F-0j\] - & = & F\_[0j]{} (j = 1 , 2 , 3),\
\[rot u = C F-jk\] - & = & F\_[jk]{}(j, k = 1 , 2 , 3). Eqs. and allow us to solve algebraically for $ u^{0} $, $ u_{0} $, and $ u_{j} $ in terms of $ u^{j} $ where $ j = 1 , 2 , 3 $. We will show below that Eq. gives merely a set of initial conditions. Thus, we are left with only four real quasi-linear partial differential equations contained in Eqs. and , for the four real fields $ u^{j} $ and $ J $. Indeed, from Eq. , we have that and are equivalent to: \[rot p 0j = 0\] & =& ,\
\[rot p jk = 0\] & = & . It follows from Eq. that Eq. remains true for all time if and only if it is true at an initial time. That is because from Eq. , we derive: ( ) = = ( ). \[GrindEQ\_\_65\_\]
It follows that Eq. gives a set of initial conditions. Thus, provided that the initial conditions are satisfied at any initial time — i.e., on a spatial submanifold $ {\bf M} $— the wave packet equations – give rise to well-defined solutions. As previously stated these solutions comprise a congruence of classical trajectories together with a scalar density.
Gordon Decomposition for Dirac Equations in Canonical Form
----------------------------------------------------------
In the WKB approximation of the DFW equation in a curved spacetime, classical trajectories are derived from the Gordon decomposition of the probability current $J^\mu = J^\mu _c + J^\mu _s$ into a convection current $J^\mu _c$ and a spin current $J^\mu _s$ [@Audretsch1981A]. In this subsection, we will prove the existence of the Gordon decomposition for all normal Dirac equations, noting that both DFW and canonical equations are normal. We will further show that in the Whitham approximation, the spin current $J^\mu _s$ vanishes, which explains the negligible effect of spin on wave packet solutions. More specifically, we will show that wave packet solutions of the form $\Psi =\chi e^{i\theta }$, where $\chi $ is slowly changing compared to a rapidly changing phase $\theta $, can only exist if the spin current $J^\mu_s$ is negligible. As discussed above, this definition of wave packet is independent of the size of Planck’s constant $\hbar $. Indeed, no assumption will be made in this subsection regarding the size of Planck’s constant $\hbar $ or the speed of light $c$, both of which we will set equal to one, $\hbar =c=1$.
For a normal Dirac equation, define the probability current $J^\mu$, the spin current $J^\mu_s$, and the convection current $J^\mu_c$ as follows:
J\^ & & \^[+]{} A\^ ,\
J\^\_s & & ,\
J\^\_c & & g\^, \[Gordon\] where $\sigma ^{\mu \nu }\equiv \frac{i}{2} \left (\gamma ^\mu \gamma ^\nu -\gamma ^\nu \gamma ^\mu \right )$ are the Dirac spin matrices. Substituting the normal Dirac equation (\[GrindEQ\_\_17\_\]) written in the following form: \[Dirac Psi=\] = \^D\_ into the formula for $J^\mu$ in Eq. (\[Gordon\]), using the gamma matrix formula $\gamma ^\mu \gamma ^\nu= g^{\mu \nu }-i\sigma ^{\mu \nu }$, and noting the asymmetry $\sigma ^{\mu \nu }=-\sigma ^{\nu \mu }$, we get from Eqs. (\[Gordon\]) and (\[Dirac Psi=\]): J\^ & & \^[+]{} A\^ =\
& = &\
& = & J\^\_c + J\^\_s. \[Gordon2\] That is, $J^\mu = J^\mu_c + J^\mu_s$, which proves the Gordon decomposition for normal Dirac equations.
Note that the probability current $J^\mu = J^\mu_c + J^\mu_s$ is covariantly conserved. Consequently, the probability density current $\sqrt{-g} J^\mu$ is conserved. In general, the currents $J^\mu_c$ and $J^\mu_s$ are not separately covariantly conserved, unless the coefficient fields $(\gamma ^\mu ,A)$ are covariantly constant. However, it is worthy to note that for Dirac equations transformed into canonical form, we may replace the covariant derivatives $D_\mu$, including electromagnetic field potentials $V_\mu $ as in Eq. (\[GrindEQ\_\_34\_\]), with the Levi-Civita covariant derivatives $\nabla _\mu +ieV_\mu $. [^8] Then the spin current $J^\mu_s$ and the convection current $J^\mu_c$ become: J\^\_s & = & ,\
J\^\_c & = & g\^. \[Gordon3\] Note that the convection current $J^{\mu }_c$ in Eq. (\[Gordon3\]) closely resembles the spin zero current of the Klein-Gordon equation in the presence of both gravitational and electromagnetic external fields [@28.], whereas the spin motion resides in the spin current $J^{\mu }_s$ [@Audretsch1981A]. Recall that we are using Whitham’s approximation, for which we set $\Psi =\chi e^{i\theta }$ and neglect $\partial _\mu \chi $ with respect to $(\partial _\mu \theta ) \chi $ \[see before Eq. (\[GrindEQ\_\_38\_\])\]. Using the definitions in Eq. (\[GrindEQ\_\_39\_\]), this gives us \[Whitham to D Psi\] (\_+ieV\_) i (\_+eV\_) =-imu\_. Then we use the definitions in Eq. (\[GrindEQ\_\_39\_\]) together with Eqs. (\[Gordon3\]) and (\[Whitham to D Psi\]) to obtain the following formulas for the spin current and the convection current: J\^\_s & & - =0,\
J\^\_c & & g\^u\_=Ju\^. \[Gordon4\] Thus, in the Whitham approximation, the spin current $J^{\mu }_s$ vanishes and the convection current $J^\mu_c = Ju^\mu$. Therefore, the probability current $J^\mu$ equals the convection current $J^\mu_c$.
The Classical-Quantum Correspondence {#Classical-Quantum}
====================================
We proved in Theorem 2 that the solutions of the Whitham approximation to the Dirac equation, Eq. , consist of a four-velocity vector field $ u^{\mu }$ whose integral curves are classical trajectories, and a scalar field $ J $ representing a conserved particle density $-$ see Eqs. and . From Whitham’s approximation of the Dirac equation, we have derived the dispersion relation , the motion of wave packets along classical trajectories , conservation of the probability current , as well as the generalized de Broglie relations $ P_{\mu } = \hbar K_{\mu } $ in Eqs. and .
We will conclude this paper by summarizing results from a previous analysis of the “classical-quantum correspondence” based on the dispersion relation alone, which can be applied to the Dirac equation [@15.]. We will interpret the fact that the integral curves of the four-velocity field $ u^{\mu } $ are classical trajectories as a “geometrical optics limit” of the Dirac equation. However, as we have seen, unlike the WKB limit, this “geometrical optics limit” is one that places no restriction on Planck’s constant. Whitham’s Lagrangian method, applied to the Dirac Lagrangian (\[GrindEQ\_\_33\_\]) transformed into equivalent canonical form (\[GrindEQ\_\_37\_\]), has also many advantages over using the dispersion relation alone as the starting point for a wave packet approximation [@12.]. Whitham’s method preserves the conservation laws inherent in the starting Lagrangian , and in particular, the Whitham wave packet equations conserve the probability current $ J^{\mu } = Ju^{\mu } $.
To any linear partial differential equation for scalar wave functions $ \Psi $, of the form:
a( X ) + \_[n = 1]{}\^[d]{} a\^[\^[\_[\_[1]{} ]{} \_[n]{} ]{} ]{} ( X ) = 0\[GrindEQ\_\_66\_\]
(summing over coordinate indices $ \mu _{r} = 0, 1, 2, 3 $ for $ r = 1, 2, 3, \cdots , n $ and over the index $ n = 1, 2, 3, \cdots , d $) where the coefficient fields $ a\left( X \right) $ and $ a^{^{\mu _{_{1} } \cdots \cdots \mu _{n} } } \left( X \right) $ depend on the spacetime point $ X $$-$ one may associate its dispersion polynomial $ \Pi _{X} \left( K \right)$. That is to say, a polynomial function of covector fields $ K_{\mu } $ at each fixed spacetime point $ X $ is given by: \_[X]{} ( K )= a( X ) + \_[n = 1]{}\^[d]{} i\^[n]{} a\^[\^[\_[[1]{} ]{} \_[ n]{} ]{} ]{} ( X ) K\_[\_1 ]{} ...... K\_[\_n]{}. \[GrindEQ\_\_67\_\]
The dispersion relation is thereby obtained from the polynomial equation $ \Pi _{X} \left( K \right)= 0 $ at each fixed spacetime point $ X $ by solving for the time component $ K_{0} $. Applications of this one-to-one correspondence are discussed in Ref. [@15.].
This applies also if the wave functions $ \Psi $ have $ m $ components and the coefficient fields $ a\left( X \right) $ and $ a^{^{\mu _{_{1} } \cdots \cdots \mu _{n} } } \left( X \right) $ are $ m\times m $ matrices, as is the case for the Dirac equation [@21.], [@36.]. Note that in the matrix case, the dispersion relation is obtained from the scalar polynomial equation $ \det \Pi _{X} \left( K \right)= 0 $. Consider a dispersion polynomial where $ a\left( X \right) $ and $ a^{^{\mu _{_{1} } \cdots \cdots \mu _{n} } } \left( X \right) $ are $ m\times m $ matrices. By solving $ \det \Pi _{X} \left( K \right)= 0 $ for the component $ K_{0} $ we get a dispersion relation:
= W( [**k**]{} , [**x**]{} [**,**]{} t ) \[GrindEQ\_\_68\_\]
expressing the angular frequency $ \omega \equiv -c\,K_{0} $ as a function of the spatial wave covector $ {\bf k} = \left( {\it K}_{{\rm 1}} , K_{{\rm 2}} , K_{{\rm 3}} \right) $, the spatial coordinates $ {\bf x} = \left( x^{1} , x^{2} , x^{3} \right) $, and time $ t $, together with the auxiliary equations [@12.], [@15.]:
+ = 0,\
\
- = 0, \[GrindEQ\_\_69\_\]
for $ j, k = 1 , 2 , 3 $. In general, none or multiple such dispersion relations can be derived as distinct real roots of the polynomial equation $ \det \Pi _{X} \left( \omega , {\bf k} \right)= 0 $ when solving for $ \omega $. Assuming at least one real root, let us choose one of them to be $ \omega = W\left( {\bf k} , {\bf x} {\bf ,} {\rm t} \right) $. Then from Eqs. and one derives the Hamiltonian system as in Ref. [@15.], Sect. 2.2: = -,= . \[GrindEQ\_\_70\_\]
Noting in Eq. that $ W \equiv -c\,K_{0} $, and recalling the generalized de Broglie relations $ P_{\mu } \equiv -p_{\mu } = \hbar K_{\mu } $ derived from wave packet motion in Eqs. and , we see that Eq. leads us to define the Hamiltonian $ H \equiv \hbar W $ and the momentum variables $ P_{j} \equiv \hbar K_{j} $ for $ j = 1 , 2 , 3 $, whereby a system of classical point particle trajectories emerges as follows: Indeed, solving the dispersion equation for the energy $ \varepsilon = H\left( {\bf p} , {\bf x} {\bf ,} {\rm t} \right) $ where $ {\bf p} = \left( P_{{\rm 1}} , P_{{\rm 2}} , P_{{\rm 3}} \right) $ is equivalent to solving it for the angular frequency $ \omega = W\left( {\bf k} , {\bf x} , t \right) $ as in Eq. . Then Eq. is equivalent to:
= -,= . \[GrindEQ\_\_71\_\]
One can show that the trajectories associated with the Hamiltonian $ H $, that is, the solution trajectories of the Hamiltonian equations , are identical to the solution trajectories of the Euler-Lagrange equations deduced from the well known Lagrangian for classical point particles in a background of electromagnetic and gravitational fields, which is given by [@37.], [@38.]:
= -mc - V\_ , \[GrindEQ\_\_72\_\]
where $ \xi $ is an arbitrary parameter for the classical trajectory $ x^{\mu } \left( \xi \right) $. To show this one first applies an inverse Legendre transformation [^9] to the Hamiltonian $ H = H\left( {\bf p} , {\bf x} {\bf ,} {\rm t} \right) $ to obtain a traditional Lagrangian $ L \left( {\bf x} , \frac{d{\bf x}}{dt} , t \right) $ and then, as in Ref. [@38.], pages 267-271, one generalizes the trajectory parameter to be an arbitrary parameter $ \xi $, instead of the coordinate time $ t $. It is straightforward to check that the equations for the classical trajectories are the Euler$-$Lagrange equations for the Lagrangian . Thus, the dispersion relation and the auxiliary equations give rise to the classical point particle trajectory equations , as is also the case for the integral curves of the Whitham equations .
Similar to the previous analysis of the classical-quantum correspondence ([@15.], Sect. 2.3), it is in the “geometrical optics limit” that the solutions of each Dirac equation transformed into equivalent canonical form satisfy the dispersion equation . Indeed, Whitham’s approximation: $ \partial _{\mu } \chi \ll \left( \partial _{\mu } \theta \right) \chi $, which we applied in Section \[WhithamMethod\] to the Dirac Lagrangian , is one way of defining precisely this limit. However, the classical Hamiltonian equations , which are based solely on the dispersion relation, give only part of the Whitham equations . In addition to providing equations equivalent to the Hamiltonian equations , the Whitham equations preserve the symmetries of the Dirac Lagrangian , and provide for the conservation of the probability current $ J^{\mu } = Ju^{\mu } $, which is a property inherited from the exact Dirac equations in a curved spacetime.
Appendix: Proof of Theorem 1
============================
To prove Theorem 1 of Section \[Canonical Form\] we will use the following corollary of a deep theorem of linear hyperbolic partial differential equations:\
**THEOREM 0.** Let $ M_{1} , M_{2} , \cdots , M_{n} $ be complex $ d\times d $ matrix functions that depend smoothly on $ n+1 $ independent real variables $ t , x_{1} , x_{2} , \cdots , x_{n} $ in a slab $ -T \le t \le T $, $ x \in {\bf R}^{{\rm n}} $, denoted as $ {\bf I}\times {\bf R}^{{\rm n}} $. Suppose that $ M_{0} , M_{1} , M_{2} , \cdots , M_{n} $ are Hermitian matrices, and furthermore assume that $ M_{0} $ is positive definite. Let $ F = F\left( S \right) $ be a homogeneous linear function of complex $ d\times d $ matrices $ S $, as well as having explicit dependence on $ t , x_{1} , x_{2} , \cdots , x_{n} $. Then the complex linear hyperbolic system: M\_[0]{} + \_[j=1]{}\^[n]{} M\_[j]{} = F( S ) \[GrindEQ\_\_73\_\] has a smooth $ d\times d $ complex matrix valued solution $ S {\bf :} {\bf I}\times {\bf R}^{{\rm n}} \to {\it M}\left( {\it C} {\it ,} {\it d} \right) $ which equals the identity matrix at $ t = 0 $, as its prescribed smooth initial data.\
**Proof:** Theorem 0 is Corollary 3 in Ref. [@7.], Sect. 6, which is based on a theorem of Lax [@34.] (see also Ref. [@35.]).\
**Proof of Theorem 1 of Section 2.**\
First note from Eq. that the matrices $ B^{\mu } \equiv A\gamma ^{\mu } $ are Hermitian matrices. Then, note that the conditions stated in Theorem 1 for the metric components $ g_{\mu \nu } $ imply that $ B^{0} \equiv A\gamma ^{0} $ is a positive definite matrix by Theorem 6 of Ref. [@22.], Appendix B. By Theorem 3 of Ref. [@8.], Sect. 3.4, Eq. (54), a local similarity transformation $ T $ of the second kind, takes a generalized Dirac equation of the form into a Dirac equation of the normal form , if and only if $ T $ satisfies the following partial differential equation: ****
B\^ D\_ T= - ( D\_ B\^ ) T . \[GrindEQ\_\_74\_\]
If we can solve Eq. for such a local similarity transformation $ T $, then the transformed coefficient fields will be as in Eqs. and :
[l]{} [\^ = T\^[-1]{} \^ T,]{}\
\
[=T\^[+]{} A T,]{}\
\
[\_ = \_ .]{}
\[GrindEQ\_\_75\_\]
Now from Eq. , a local similarity transformation $ S $ of the first kind, takes a Dirac equation of the normal form into a normal QRD–0 equation , if and only if $ S $ satisfies the following partial differential equation:
\^ \_ S= -\^ \_ S, \[GrindEQ\_\_76\_\]
where $ \widetilde{B}^{\mu } \equiv \widetilde{A}\widetilde{\gamma }^{\mu } $. The matrices $ B^{\mu } $ and $ \widetilde{B}^{\mu } $ are Hermitian, and moreover, the matrices $ B^{0} $ and $ \widetilde{B}^{0} $ are positive definite. Thus, the two systems and have same form as the complex linear hyperbolic system in Eq. .
Let $ \chi {\bf :} {\bf U} \to {\bf R} \times {\bf R}^{{\bf 3}} $, mapping $ X \to \left( t , {\bf x} \right) $, be the coordinate chart that we assume to be defined on $ {\bf U} $. Then, consider the projection map $ \pi {\bf :} {\bf R} \times {\bf R}^{{\bf 3}} \to {\bf R} $ taking $ \left( t , {\bf x} \right) \to t $. Let $ X_{0} \in {\bf U} $ and let $ t_{0} = \pi \circ \chi \left( X_{0} \right) \in {\bf R} $. Let $ {\bf M} = \left( \pi \circ \chi \right)^{-1} \left( t_{0} \right) $. Note that $ X_{0} \in {\bf M} \subset {\bf U} $. It will suffice to prove that there exist nonsingular solutions $ T $ and $ S $ of the systems and that are both defined in a common open neighborhood $ \widetilde{{\bf W}} $ of $ X_{0} \in {\bf U}$.
By Theorem 0, the Cauchy problem for with the smooth initial data $ T|_{ {\bf M}} = {\bf 1}_{{\bf 4}} $ has a smooth solution $ T $ in an open neighborhood $ {\bf W}^{{\bf \# }} $ of $ X_{0} $. Denote by $ {\bf W} $ the open subset of $ {\bf W}^{{\bf \# }} $ in which $ T $ is a nonsingular matrix so that $ T^{-1} $ exists. Note that $ X_{0} \in {\bf W} $ since $ X_{0} \in {\bf M} $ and $ T|_{ {\bf M}} = {\bf 1}_{{\bf 4}} $. Thus, $ {\bf W} $ is an open neighborhood of $ X_{0} $ such that the local similarity transformation $ T $ (and its inverse $ T^{-1} $) is well defined on $ {\bf W} $, and hence, from Eq. , the complex linear hyperbolic system is well defined on $ {\bf W} $.
By Theorem 0, the Cauchy problem for with the smooth initial data $ S|_{ {\bf M}\bigcap {\bf W}} = {\bf 1}_{{\bf 4}} $ has a smooth solution $ S $ in an open neighborhood $ \widetilde{{\bf W}}^{{\bf \# }} \subset {\bf W} $ of $ X_{0} $. Denote by $ \widetilde{{\bf W}} $ the open subset of $ \widetilde{{\bf W}}^{{\bf \# }} $ in which $ S $ is a nonsingular matrix so that $ S^{-1} $ exists. Note that $ X_{0} \in \widetilde{{\bf W}} $ since $ X_{0} \in {\bf M}\bigcap {\bf W} $ and $ S|_{ {\bf M}\bigcap {\bf W}} = {\bf 1}_{{\bf 4}} $. Thus, $ \widetilde{{\bf W}} \subset \widetilde{{\bf W}}^{{\bf \# }} \subset {\bf W} $ is an open neighborhood of $ X_{0} $ such that both the local similarity transformation $ S $ and the local similarity transformation $ T $ (and their inverses $ S^{-1} $ and $ T^{-1} $) are well defined on $ \widetilde{{\bf W}} $, and therefore the local similarity transformation $ T\circ S $ (and its inverse $ S^{-1} \circ T^{-1} $) is also well defined on $ \widetilde{{\bf W}} $. **Q. E. D.**\
[9]{}
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[^1]: More generally, a common conception is that classical physics emerges from quantum mechanics in the limit as Planck’s constant $\hbar $ approaches zero. However, the limit $\hbar \rightarrow 0$ “is not well defined mathematically unless one specifies what quantities are to be held constant during the limiting process" [@Ballentine2001]. It is interesting to note that the most classical behaving Gaussian wave functions, the coherent states of the ordinary harmonic oscillator, whose expected position and momentum obey classical equations by Ehrenfest’s theorem, do not resemble wave packets in the limit $\hbar \rightarrow 0$ [@Park1990].
[^2]: Note, however, that the electron’s magnetic moment, predicted by the Dirac equation, is not contained in the wave packet approximation. Indeed, quoting from Ref. [@13.]: “The uncertainty principle, together with the Lorentz force, prevents spin-up and spin-down electrons from being separated by a macroscopic field of the Stern-Gerlach type.” In practice, wave packet splitting in Stern-Gerlach experiments is only observed using neutral atoms or molecules, which are undisturbed by the Lorentz force [@Ballentine2001]. In Section \[WhithamMethod\], the wave packet approximation is expressed by neglecting in the Lagrangian the variation in the amplitude of the wave function as compared to the variation of its phase. This leads to wave packet equations which do not involve the electron’s magnetic moment.
[^3]: In the case of a Majorana representation of the Dirac field, the coefficient matrices $ \left( \gamma ^{\mu } , A \right) $ are pure imaginary, and the Dirac equation is real. In this case the spin connection matrices $ \Gamma _{\mu } $ are real.
[^4]: Because $ S$ depends on the spacetime point, Schlütert, Wietschorke, and Greiner call $ S $ a “local transformation” [@3.], [@4.].
[^5]: The notion of equivalence here is somewhat different than the notion used in Ref. [@25.], where equivalence was applied only to classify the coefficient fields $ \left( \gamma ^{\mu } , A \right) $ without requiring the existence of a map $ \Psi \to S^{-1} \Psi $ between the solutions of two Dirac equations.
[^6]: A globally defined generalized Dirac Lagrangian has the form: \[Lagrangian-intrinsic\]L= . where ${\not}\mathcal{D}$ is a Dirac operator acting on the cross-sections $\psi $ of a spinor bundle ${\sf E}$ over the spacetime, and $(\ ,\ )$ denotes a hermitizing metric on the fibers of ${\sf E}$. See Ref. [@7.], Sect. 2.1 and the references therein for the precise definitions. Once any coordinate chart of the spacetime and any frame field on the spinor bundle have been chosen, one gets the local expression of the global generalized Lagrangian as Eq. (\[GrindEQ\_\_8\_\]). In particular, in Eq. (\[GrindEQ\_\_8\_\]) and in the rest of this paper, $\Psi $ is the column vector made with the components of $\psi $ in the chosen frame field on ${\sf E}$. See Ref. [@7.], Sect. 2.2.
[^7]: As stated in Section \[Canonical Form\], two Dirac equations that are classically equivalent, need not be equivalent with respect to their quantum mechanical energy-momentum operators [@25.], [@26.], [@27.]. Clearly, the Whitham approximation also distinguishes them. A striking example is a Dirac equation with a Mashhoon term [@10.], [@9.], [@11.]. Applying the Whitham approximation directly to a Dirac equation with a Mashhoon term, without first transforming the Dirac field $\Psi $, does not produce wave packet motion along classical paths.
[^8]: Note that the probability current $J^\mu =\Psi ^+A\gamma ^\mu \Psi $ is invariant under local similarity transformations $S$ defined in Eq. (\[GrindEQ\_\_5\_\]), so that when transforming a Dirac equation into canonical form, it is only $J^\mu _s$ and $J^\mu _c$ which change their form.
[^9]: The Legendre transformation is its own inverse [@38.], pages 563-565. Thus, an inverse Legendre transformation is also a Legendre transformation.
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abstract: 'We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to the square root of the spectral radius of the variance matrix of $X$ when $n$ tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [@Altcirc] at the spectral edge.'
bibliography:
- 'literature.bib'
title: '[Spectral radius of random matrices with independent entries]{}'
---
Introduction
============
Girko’s celebrated *circular law* [@Girko1984; @bai1997][^1] asserts that the spectrum of an $n\times n$ random matrix $X$ with centered, independent, identically distributed (i.i.d.) entries with variance ${\mathbb{E}}|x_{ij}|^2=1/n$ converges, as $n\to \infty$, to the unit disc with a uniform limiting density of eigenvalues. The cornerstone of the proof is the *Hermitization formula* (cf. ) that connects eigenvalues of $X$ to the eigenvalues of a family of Hermitian matrices $(X-z)^*(X-z)$ with a complex parameter $z$ [@Girko1984]. The circular law for i.i.d. entries with the minimal second moment condition was established by Tao and Vu [@tao2010] after several partial results [@goetze2010; @Pan2010; @tao2008], see [@bordenave2012] for the extensive history and literature. We also refer to the recent circular law for very sparse matrices [@Rudelson2019].
The circular law establishes the weak limit of the empirical density of eigenvalues and thus it accounts for most but not all of them. In particular, it does not give information on the spectral radius $\varrho(X)$ of $X$ since the largest (in absolute value) eigenvalue may behave very differently than the bulk spectrum. In fact, such outliers do not exists but this requires a separate proof. The convergence of the spectral radius of $X$ to 1, $$\label{bai}
\varrho(X)\to 1, \qquad \mbox{almost surely as $n\to \infty$,}$$ was proven by Bai and Yin in [@Bai1986] under the fourth moment condition, ${\mathbb{E}}| n^{1/2} x_{ij} |^4 \le C$, using Wigner’s moment method. Under stronger conditions the upper bound in was independently proven in [@Geman1986], see also [@GemanHwang1982; @Nemish2018]. More recently in [@BordenaveSpectralRadius2018] the convergence $\varrho(X)\to 1$ in probability was shown assuming only finite $2+\epsilon$ moment.
Precise information on the spectral radius is available only for the Ginibre ensemble, i.e. when $x_{ij}$ are Gaussian; in this case it is known [@Rider2003; @RiderSinclair2014] that $$\label{rider}
\varrho(X) \approx 1 + \sqrt{\frac{\gamma_n}{ 4n}} + \frac{1}{ \sqrt{4n\gamma_n}} \xi, \qquad \gamma_n{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\log \frac{n}{2\pi}
-2\log\log n,$$ where $\xi$ is a Gumbel distributed random variable.
In this paper we drop the condition that the matrix elements are identically distributed and we study the spectral radius of $X$ when the variances ${\mathbb{E}}|x_{ij}|^2$ have a non-trivial profile given by the matrix ${\mathscr{S}}=({\mathbb{E}}|x_{ij}|^2)_{i,j=1}^n$. In our previous work [@Altcirc] we showed that the spectral radius of $X$ is arbitrarily close to the square root of the spectral radius of ${\mathscr{S}}$. More precisely, for any fixed $\epsilon>0$ we have $$\label{oldrho}
\sqrt{\varrho({\mathscr{S}})} - \epsilon\le \varrho(X)\le \sqrt{\varrho({\mathscr{S}})} + \epsilon$$ with very high probability for large $n$. Motivated by we expect that the precision of the approximation in can be greatly improved and the difference between $\varrho(X)$ and $\sqrt{\varrho({\mathscr{S}})}$ should not exceed $n^{-1/2}$ by much. Indeed, our first main result proves that for any $\epsilon>0$ we have $$\label{oldrho1}
\sqrt{\varrho({\mathscr{S}})} - n^{-1/2+\epsilon}\le \varrho(X)\le \sqrt{\varrho({\mathscr{S}})} + n^{-1/2+ \epsilon}$$ with very high probability for large $n$. Apart from the $n^{\epsilon}$ factor this result is optimal considering . Note that is new even for the i.i.d. case beyond Gaussian, i.e. there is no previous result on the speed of convergence in .
We remark that, compared with the spectral radius, much more is known about the largest singular value of $X$ since it is equivalent to the (square root of the) largest eigenvalue of the sample covariance matrix $XX^*$. For the top eigenvalues of $XX^*$ precise limiting behavior (Tracy-Widom) is known if $X$ has general i.i.d. matrix elements [@pillai2014], and even general diagonal population matrices are allowed [@Lee2016]. Note, however, the largest singular value of $X$ in the i.i.d. case converges to 2, i.e. it is very different from the spectral radius, indicating that $X$ is far from being normal. We stress that understanding the spectral radius is a genuinely non-Hermitian problem hence in general it is much harder than studying the largest singular value.
While the largest singular value is very important for statistical applications, the spectral radius is relevant for time evolution of complex systems. More precisely, the spectral radius controls the eigenvalue with largest real part that plays an important role in understanding the long time behavior of large systems of linear ODE’s with random coefficients of the form $$\label{ode}
\frac{{\mathrm{d}}}{{\mathrm{d}}t} u_t = - g u_t + X u_t$$ with a tunable coupling constant $g$. Such ODE system was first introduced in an ecological model to study the interplay between complexity and stability in May’s seminal paper [@may1972will], see also the recent exposition [@Allesina2015]. It has since been applied to many situations when a transience phenomenon is modelled in dynamics of complex systems; especially for neural networks, e.g. [@Sompolinsky1988; @HENNEQUIN20141394; @Grela2017]. Structured neural networks require to generalize May’s original i.i.d. model to non-constant variance profile ${\mathscr{S}}$ [@Aljadeff2015; @Muir2015; @Rajan2006; @Gudowska-Nowak2020] which we study in full generality. The long time evolution of at critical coupling $g_c{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt{\varrho({\mathscr{S}})}$ in the i.i.d. Gaussian case was computed in [@ChalkerMehlig1998] after some non-rigorous steps; the full mathematical analysis even for general distribution and beyond the i.i.d. setup was given in [@ErdosKrugerRenfrew2018; @ErdosKrugerRenfrew2019]. The time-scale on which the solution of at criticality can be computed depends on how precisely $\varrho(X)$ can be controlled by $\sqrt{\varrho({\mathscr{S}})}$. In particular, the current improvement of this precision to allows one to extend the result of [@ErdosKrugerRenfrew2018 Theorem 2.6] to very long time scales of order $n^{1/2-\epsilon}$. These applications require a separate analysis, we will not pursue them in the present work.
We now explain the key novelties of this paper, more details will be given in Section \[sec:outline\] after presenting the precise results in Section \[sec:main\_results\]. The spectral radius of $X$ is ultimately related to our second main result, the *local law* for $X$ near the spectral edges, i.e. a description of the eigenvalue density on local scales but still above the eigenvalue spacing; in this case $n^{-1/2}$. As a byproduct, we also prove the optimal $1/n$ speed of convergence in the inhomogeneous circular law [@Altcirc; @Cook2018]. Note that the limiting density has a discontinuity at the boundary of its support, the disk of radius $\sqrt{\varrho({\mathscr{S}})}$ [@Altcirc Proposition 2.4], hence the typical eigenvalue spacing at the edge and in the bulk coincide, unlike for the Hermitian problems. The local law in the bulk for $X$ with a general variance profile has been established in [@Altcirc Theorem 2.5] on scale $n^{-1/2+\epsilon}$ and with optimal error bounds. This entails an optimal local law near zero for the *Wigner-type* Hermitian matrix $$\label{eq:def_H_z_intro}
H_z {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 0 & X -z \\ (X-z)^* & 0 \end{pmatrix}$$ appearing in Girko’s formula. As long as $z$ is in the bulk spectrum of $X$, the relevant spectral parameter 0 lies in the bulk spectrum of $H_z$. Still, the local law for Wigner-type matrices [@Ajankirandommatrix] is not applicable since the *flatness condition*, that requires the variances of all matrix elements of $H_z$ be comparable, is violated by the large zero blocks in $H_z$. In fact, the corresponding Dyson equation has an unstable direction due to the *block symmetry* of $H_z$. The main achievement of [@Altcirc] was to handle this instability.
When $z$ is near the spectral edge of $X$, the density of $H_z$ develops a cusp singularity at 0. The optimal cusp local law for Wigner-type matrices with flatness condition was proven recently in [@Cusp1] relying on (i) the improved *fluctuation averaging mechanism* and (ii) the deterministic analysis of the corresponding Dyson equation in [@Altshape]. Due to the cusp, the Dyson equation has a natural unstable direction and the corresponding non-Hermitian perturbation theory is governed by a cubic equation.
The Dyson equation corresponding to the matrix $H_z$ for $z$ near the spectral edge of $X$ exhibits *both* instabilities simultaneously. This leads to the main technical achievement of this paper: we prove an *optimal local law in the cusp regime with the block instability*. Most of the paper contains our refined analysis of the Dyson equation with two instabilities, a delicate synthesis of the methods developed in [@Altcirc] and [@Altshape]. The necessary fluctuation averaging argument, however, turns out to be simpler than in [@Cusp1], the block symmetry here helps.
We remark that bulk and edge local laws for the i.i.d. case have been proven earlier [@Bourgade2014; @BYY_circular2] with the optimal scale at the edge in [@Y_circularlaw] and later with improved moment assumptions in [@Gotze2017]; see also [@tao2015] for similar results under three moment matching condition. However, these works did not provide the improved local law outside of the spectrum that is necessary to identify the spectral radius. The main difference is that the i.i.d. case results in an explicitly solvable scalar-valued Dyson equation, so the entire stability analysis boils down to analysing explicit formulas. The inhomogeneous variance profile ${\mathscr{S}}$ leads to a vector-valued Dyson equation with no explicit solution at hand; all stability properties must be obtained inherently from the equation itself. Furthermore, even in the i.i.d. case the local law for $H_z$ in [@BYY_circular2; @Y_circularlaw] was not optimal in the edge regime $|z|\approx 1$ and the authors directly estimated only the specific error terms in Girko’s formula. The optimality of our local law for $H_z$ at the edge is the main reason why the proof of the local circular law in Section \[sec:Xlaw\] is very transparent.
#### Acknowledgements.
The authors are grateful to Dominik Schröder for valuable insights, discussions on adapting the fluctuation averaging mechanism in [@Cusp1] to the current setup as well as for kindly making his graph drawing macros available to us. We also thank Gašper Tkačik for helping us with the physics literature of complex networks. The authors thank Jonas Jalowy for pointing out a step in the proof that was not explained in sufficient detail.
Main results {#sec:main_results}
============
Let $X=(x_{ij})_{i,j=1}^n\in {\mathbb{C}}^{n\times n}$ be a matrix with independent, centered entries. Let ${\mathscr{S}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathbb{E}}{\lvert x_{ij} \rvert}^2)_{i,j=1}^n$ be the matrix collecting the variances of the entries of $X$. Furthermore, our main results will require a selection of the following assumptions (we remark that the last assumption \[assum:bounded\_density\] can be substantially relaxed, see Remark \[rem:alternative\_A3\] below).
#### Assumptions
1. \[assum:flatness\] The variance matrix ${\mathscr{S}}$ of $X$ is *flat*, i.e. there are constants $s^* > s_* >0$ such that $$\label{eq:condition_flatness}
\frac{s_*}{n} \leq {\mathbb{E}}{\lvert x_{ij} \rvert}^2 \leq \frac{s^*}{n}$$ for all $i,j =1, \ldots, n$.
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1. \[assum:bounded\_moments\] The entries of $X$ have *bounded moments* in the sense that, for each $m \in {\mathbb{N}}$, there is $\mu_m>0$ such that $${\mathbb{E}}{\lvert x_{ij} \rvert}^m \leq \mu_m n^{-m/2}$$ for all $i,j = 1, \ldots, n$.
2. \[assum:bounded\_density\] Each entry of $\sqrt{n}X$ has a bounded density on ${\mathbb{C}}$ in the following sense. There are probability densities $f_{ij} \colon {\mathbb{C}}\to [0,\infty)$ such that $${\mathbb{P}}\big( \sqrt{n}\, x_{ij} \in B \big) = \int_B f_{ij}(z)\, {\mathrm{d}}^2 z$$ for all $i,j = 1, \ldots, n$ and all Borel sets $B \subset {\mathbb{C}}$ and these densities are bounded in the sense that there are $\alpha, \beta >0$ such that $f_{ij} \in L^{1+ \alpha}({\mathbb{C}})$ and $${\lVert f_{ij} \rVert}_{1 + \alpha} \leq n^\beta$$ for all $i,j = 1, \ldots, n$.
In \[assum:bounded\_density\] and in the following, ${\mathrm{d}}^2 z$ denotes the Lebesgue measure on ${\mathbb{C}}$. The main results remain valid if $X$ has all real entries, i.e. the density $f_{ij}$ of $\sqrt{n}\, x_{ij}$ in \[assum:bounded\_density\] is supported on ${\mathbb{R}}$ instead of ${\mathbb{C}}$ and we consider its $L^{1+\alpha}({\mathbb{R}})$-norm. In fact, the proofs are completely analogous. Hence, for simplicity, we only present the proofs in the complex case.
The following theorem, our first main result, provides a convergence result for the spectral radius of the random matrix $X$. For any matrix $R \in {\mathbb{C}}^{n\times n}$, we write $\varrho(R)$ for its spectral radius, i.e. $\varrho(R) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\max_{\lambda \in \operatorname*{Spec}(R)} {\lvert \lambda \rvert}$.
\[thm:spectral\_radius\_X\] Let $X$ satisfy \[assum:flatness\] – \[assum:bounded\_density\]. Then, for each (small) ${\varepsilon}>0$ and (large) $D>0$, there is $C_{{\varepsilon},D} >0$ such that $$\label{eq:spectral_radius}
{\mathbb{P}}\Big( {\left\lvert \varrho(X) - \sqrt{\varrho({\mathscr{S}})} \right\rvert} \geq n^{-1/2+{\varepsilon}} \Big) \leq \frac{C_{{\varepsilon},D}}{n^{D}}$$ for all $n \in {\mathbb{N}}$.
Here, the constant $C_{{\varepsilon}, D}$ depends only on $s_*$, $s^*$ from \[assum:flatness\], the sequence $(\mu_m)_{m \in {\mathbb{N}}}$ from \[assum:bounded\_moments\] and $\alpha$, $\beta$ from \[assum:bounded\_density\] in addition to ${\varepsilon}$ and $D$.
\[rem:upper\_bound\_spectral\_radius\] Without Assumption \[assum:bounded\_density\] our proof still implies the following upper bound on the spectral radius $\varrho(X)$ of $X$. That is, if \[assum:flatness\] and \[assum:bounded\_moments\] are satisfied then for each ${\varepsilon}>0$ and $D>0$, there is $C_{{\varepsilon},D}>0$ such that, for all $n \in {\mathbb{N}}$, we have $${\mathbb{P}}\Big( \varrho(X) \geq \sqrt{\varrho({\mathscr{S}})} + n^{-1/2 + {\varepsilon}} \Big) \leq \frac{C_{{\varepsilon},D}}{n^{D}}.$$ In particular, $X$ does not have any eigenvalue of modulus bigger than $\sqrt{\varrho({\mathscr{S}})} + n^{-1/2 + {\varepsilon}}$ with very high probability.
The next main result, Theorem \[thm:local\_law\_X\] below, shows that the eigenvalue density of $X$ is close to a deterministic density on all scales slightly above the typical eigenvalue spacing when $n$ is large. We now prepare the definition of this deterministic density. For each $\eta >0$ and $z \in {\mathbb{C}}$, we denote by $(v_1, v_2) \in (0,\infty)^n \times (0,\infty)^n$ the unique solution to the system of equations $$\label{eq:dyson_vector_equation}
\frac{1}{v_1} = \eta + {\mathscr{S}}v_2 + \frac{{\lvert z \rvert}^2}{\eta + {\mathscr{S}}^t v_1}, \qquad \frac{1}{v_2} = \eta + {\mathscr{S}}^t v_1 + \frac{{\lvert z \rvert}^2}{\eta + {\mathscr{S}}v_2}.$$ Here, any scalar is identified with the vector in ${\mathbb{C}}^n$ whose components agree all with the scalar. E.g. $\eta$ is identified with $(\eta, \ldots, \eta) \in {\mathbb{C}}^n$. Moreover, the ratio of two vectors in ${\mathbb{C}}^n$ is defined componentwise. The existence and uniqueness of $(v_1,v_2)$ has been derived in [@Altcirc Lemma 2.2] from abstract existence and uniqueness results in [@Helton01012007].
In the following, we consider $v_1=v_1(z,\eta)$ and $v_2=v_2(z,\eta)$ as functions of $\eta >0$ and $z \in {\mathbb{C}}$. In Proposition \[pro:properties\_sigma\] below, we will show that there is a probability density $\sigma \colon {\mathbb{C}}\to [0,\infty)$ such that $$\label{eq:sigma_equal_laplace_potential}
\sigma(z) = -\frac{1}{2\pi} \Delta_z \int_0^\infty \bigg( {\langle v_1(z,\eta) \rangle} - \frac{1}{1 + \eta} \bigg) {\mathrm{d}}\eta,$$ where the equality and the Laplacian $\Delta_z$ on ${\mathbb{C}}$ are understood in the sense of distributions on ${\mathbb{C}}$. Moreover, ${\langle v_1 \rangle}$ denotes the mean of the vector $v_1 \in {\mathbb{C}}^n$, i.e. ${\langle u \rangle} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{n} \sum_{i=1}^n u_i$ for any $u = (u_i)_{i=1}^n \in {\mathbb{C}}^n$. In Lemma \[lem:properties\_L\] below, we will show that the integral on the right-hand side of exists for each $z \in {\mathbb{C}}$. Proposition \[pro:properties\_sigma\] also proves further properties of $\sigma$, in particular, that the support of $\sigma$ is a disk of radius $\sqrt{\varrho({\mathscr{S}})}$ around the origin.
In order to analyze the eigenvalue density of $X$ on local scales, we consider shifted and rescaled test functions as follows. For any function $f \colon {\mathbb{C}}\to {\mathbb{C}}$, $z_0 \in {\mathbb{C}}$ and $a >0$, we define $$f_{z_0,a} \colon {\mathbb{C}}\to {\mathbb{C}}, \qquad f_{z_0,a}(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{2a} f(n^{a} (z-z_0)).$$ The eigenvalues of $X$ are denoted by $\zeta_1, \ldots, \zeta_n$. Now we are ready to state our second main result.
\[thm:local\_law\_X\] Let $X$ satisfy \[assum:flatness\] – \[assum:bounded\_density\]. Let $a \in [0,1/2]$ and $\varphi >0$. Then, for every ${\varepsilon}>0$ and $D>0$, there is $C_{{\varepsilon},D}>0$ such that $${\mathbb{P}}\bigg( {\bigg\lvert \frac{1}{n} \sum_{i =1}^n f_{z_0,a} ( \zeta_i) - \int_{{\mathbb{C}}} f_{z_0,a} (z) \sigma(z) {\mathrm{d}}^2 z \bigg\rvert} \geq \frac{{\lVert \Delta f \rVert}_{L^1}}{n^{1- 2a - {\varepsilon}}} \bigg) \leq \frac{C_{{\varepsilon},D}}{n^D}$$ uniformly for all $n \in {\mathbb{N}}$, $z_0 \in {\mathbb{C}}$ satisfying ${\lvert z_0 \rvert} \leq \varphi$ and $f \in C_0^2({\mathbb{C}})$ satisfying $\operatorname{supp}f \subset \{ z \in {\mathbb{C}}\colon {\lvert z \rvert} \leq \varphi\}$. The point $z_0$ and the function $f$ may depend on $n$. In addition to ${\varepsilon}$ and $D$, the constant $C_{{\varepsilon},D}$ depends only on $s_*, s^*$ from \[assum:flatness\], $(\mu_m)_{m \in {\mathbb{N}}}$ from \[assum:bounded\_moments\], $\alpha, \beta$ from \[assum:bounded\_density\] as well as $a$ and $\varphi$.
The bulk regime $(|z_0|<1)$ in Theorem \[thm:local\_law\_X\] has already been proven in [@Altcirc Theorem 2.5]. Choosing $a=0$ and $z_0=0$ in Theorem \[thm:local\_law\_X\] amounts to the optimal $1/n$ speed of convergence in the inhomogeneous circular law.
Finally, we state a corollary of our result showing that all normalised eigenvectors $u=(u_i)_{i=1}^n\in {\mathbb{C}}^n$ of $X$ are completely delocalized in the sense that $\max_{i=1}^{n} {\lvert u_i \rvert}\leq n^{-1/2+{\varepsilon}}$ with very high probability. Eigenvector delocalization under somewhat different conditions and with very different methods has already been established in [@rudelson2015] with recent refinements in [@Rudelson2016; @EigenvectorsLuh; @EigenvectorsTikhomirov].
\[thm:delocalization\] Let $X$ satisfy \[assum:flatness\] and \[assum:bounded\_moments\]. Then, for each ${\varepsilon}>0$ and $D>0$, there is $C_{{\varepsilon},D}>0$ such that $${\mathbb{P}}\bigg( \exists\, u\neq 0 \, \colon X u = \zeta u \text{ for some }
\zeta \in {\mathbb{C}}\text{ and } \max_{i=1}^{n} \,{\lvert u_i \rvert} \geq n^{-1/2 + {\varepsilon}} {\lVert u \rVert} \bigg) \leq \frac{C_{{\varepsilon},D}}{n^{D}}$$ for all $n \in {\mathbb{N}}$. Here, ${\lVert u \rVert}$ denotes the Euclidean norm of $u$.
\[rem:alternative\_A3\] Theorem \[thm:spectral\_radius\_X\], as well as Theorem \[thm:local\_law\_X\] (with an additional condition $\| \Delta f\|_{L^{2+\epsilon}}\le n^C\| \Delta f\|_{L^1}$, with some large constant $C$, on the test function $f$) hold if Assumption \[assum:bounded\_density\] is replaced by the following anticoncentration condition. With the Lévy concentration function $${\mathcal}{L}(Z,t) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sup_{u \in {\mathbb{R}}} {\mathbb{P}}\big( {\lvert Z-u \rvert} < t)$$ we require that $\max_{i,j} {\mathcal}{L}(\sqrt{n}\,x_{ij}, t) \leq b$ for some constants $t \geq 0$ and $b \in (0,1)$. In our main proofs, we use \[assum:bounded\_density\] for pedagogical reasons and the necessary modifications will be explained in Remark \[rem:alternative\_changes\_proof\] at the end of Section \[sec:Xlaw\] below.
Outline of the proof {#sec:outline}
--------------------
In this subsection, we outline a few central ideas of the proofs of Theorem \[thm:spectral\_radius\_X\] and Theorem \[thm:local\_law\_X\]. The spectrum of the $n\times n$-matrix $X$ can conveniently be studied by analysing the kernel of the $2n \times 2n$ Hermitian matrices $H_z$ defined through $$\label{eq:def_H_z}
H_z {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 0 & X -z \\ (X-z)^* & 0 \end{pmatrix}$$ for $z \in {\mathbb{C}}$. In fact, $z$ is an eigenvalue of $X$ if and only if the kernel of $H_z$ is nontrivial.
All spectral properties of a Hermitian matrix can be obtained from its resolvent. In fact, in many cases, the resolvent of a Hermitian random matrix becomes deterministic when its size tends to infinity and the limit is the solution to the associated Matrix Dyson equation. In our setup, the Matrix Dyson equation (MDE) for the deterministic counterpart $M=M(z,\eta)$ of the resolvent $G = G(z,\eta) = (H_z - {\mathrm{i}}\eta)^{-1}$ of $H_z$ is given by $$\label{eq:mde}
- M^{-1}(z, \eta) = \begin{pmatrix} {\mathrm{i}}\eta & z \\ \bar z & {\mathrm{i}}\eta \end{pmatrix} + {\mathcal{S}}[M(z,\eta)].$$ Here, $\eta>0$ and $z \in {\mathbb{C}}$ are parameters and ${\mathrm{i}}\eta$, $z$ and $\bar z$ are identified with the respective multiples of the $n\times n$ identity matrix. Moreover, we introduced the self-energy operator ${\mathcal{S}}\colon {\mathbb{C}}^{2n\times 2n} \to {\mathbb{C}}^{2n\times 2n}$ given by $$\label{eq:def_cS}
{\mathcal{S}}[R] = \begin{pmatrix} {\mathscr{S}}r_2 & 0 \\ 0 & {\mathscr{S}}^t r_1 \end{pmatrix}$$ for $R=(r_{ij})_{i,j=1}^{2n} \in {\mathbb{C}}^{2n \times 2n}$, where $r_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(r_{ii})_{i=1}^n$, $r_2 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(r_{ii})_{i=n+1}^{2n}$ and ${\mathscr{S}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathbb{E}}{\lvert x_{ij} \rvert}^2)_{i,j=1}^n$. The matrix on the right-hand side of denotes a $2n\times 2n$ diagonal matrix with the vector $({\mathscr{S}}r_2, {\mathscr{S}}^t r_1) \in {\mathbb{C}}^{2n}$ on its diagonal.
Two remarks about and are in order. In this paper we are interested exclusively in the kernel of $H_z$. Otherwise ${\mathrm{i}}\eta$ on the right-hand side of had to be replaced by $E+ {\mathrm{i}}\eta$ for some $E\in {\mathbb{R}}$ (see [@AjankiCorrelated; @Erdos2017Correlated; @AltEdge] for the general MDE in the random matrix setup). We also remark that the self-energy operator ${\mathcal{S}}$ in is chosen slightly differently compared to the choice of the self-energy operator for a Hermitian random matrix in [@AjankiCorrelated; @Erdos2017Correlated; @AltEdge]. Instead, we follow here the convention from [@Altcirc]. For further details, see Remark \[rem:convention\_for\_S\] below.
First, we discuss Theorem \[thm:spectral\_radius\_X\]. Suppose we already know that $G$ is very well approximated by $M$. Owing to [@Altcirc Proposition 3.2] (see also Lemma \[lem:scaling\_relation\_v\_u\] below), ${\mathrm{Im}\,}M(z,\eta)$ vanishes sufficiently fast for $\eta \downarrow 0$ as long as ${\lvert z \rvert}^2 \geq \varrho({\mathscr{S}}) + n^{-1/2+{\varepsilon}}$. Then we can immediately conclude that the kernel of $H_z$ has to be trivial. Hence, any eigenvalue of $X$ has modulus smaller than $\sqrt{\varrho({\mathscr{S}})} + n^{-1/2 + {\varepsilon}}$. Similarly, under the condition ${\lvert z \rvert}^2 < \varrho({\mathscr{S}}) - n^{-1/2 + {\varepsilon}}$, the imaginary part ${\mathrm{Im}\,}M(z,\eta)$ is big enough as $\eta \downarrow 0$ due to [@Altcirc Proposition 3.2]. This will imply that $H_z$ has a nontrivial kernel and, hence, $X$ has an eigenvalue close to $z$, thus completing the proof of .
Therefore, what remains is to prove a local law for $H_z$, i.e. that $G$ is very well approximated by $M$. The resolvent $G$ satisfies a perturbed version of the MDE , $$\label{eq:perturbed_mde}
-G^{-1} = \begin{pmatrix} {\mathrm{i}}\eta & z \\ \bar z & {\mathrm{i}}\eta \end{pmatrix} + {\mathcal{S}}[G] - DG^{-1},$$ for all $\eta >0$ and $z \in {\mathbb{C}}$. Here, we defined $D {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(H_z - {\mathbb{E}}H_z) G + {\mathcal{S}}[G] G$ which will be shown to be small in Section \[sec:local\_law\_H\] below. Consequently, we will consider as a perturbed version of the MDE, and study its stability properties under small perturbations to conclude that $G$ is close to $M$.
A simple computation starting from and yields the stability equation associated to the MDE, $$\label{eq:stability_equation}
{\mathcal{B}}[G-M] = M {\mathcal{S}}[G-M](G-M) - MD.$$ Here, ${\mathcal{B}}\colon {\mathbb{C}}^{2n\times 2n} \to {\mathbb{C}}^{2n\times 2n}$ is the *linear stability operator* of the MDE, given explicitly by $$\label{eq:def_cB}
{\mathcal{B}}[R] {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}R- M{\mathcal{S}}[R]M$$ for any $R \in {\mathbb{C}}^{2n\times 2n}$.
The stability equation is viewed as a general quadratic equation of the form $$\label{AB}
{\mathcal{B}}[Y] - {\mathcal{A}}[Y, Y] + X=0$$ for the unknown matrix $Y (= G-M)$ in the regime where $X (=MD)$ is small[^2]. Here, ${\mathcal{B}}$ is a linear map and ${\mathcal{A}}$ is a bilinear map on the space of matrices. This problem would be easily solved by a standard implicit function theorem if ${\mathcal{B}}$ had a stable (i.e. bounded) inverse; this is the case in the bulk regime. When ${\mathcal{B}}$ has unstable directions, i.e. eigenvectors corresponding to eigenvalues very close to zero, then these directions need to be handled separately.
The linear stability operator for Wigner-type matrices with a flat variance matrix in the edge or cusp regime gives rise to one unstable direction $B$ with ${\mathcal{B}}[B]\approx 0$. In this case, the solution is, to leading order, parallel to the unstable direction $B$, hence it can be written as $Y = \Theta B + \mathrm{error}$ with some complex scalar coefficient $\Theta$, determining the leading behavior of $Y$. For such $Y$ the linear term in becomes lower order and the quadratic term as well as the error term in $Y$ play an important role. Systematically expanding $Y$ up to higher orders in the small parameter $\| X\|\ll 1$, we arrive at an approximate cubic equation for $\Theta$ of the form $c_3 \Theta^3 + c_2 \Theta^2 + c_1 \Theta = \mathrm{small}$, with very precisely computed coefficients. The full derivation of this cubic equation is given in [@Cusp1 Lemma A.1]. In the bulk regime $|c_1|\sim 1$, hence the equation is practically linear. In the regime where the density vanishes, we have $c_1\approx 0$, hence higher order terms become relevant. At the edge we have $|c_2|\sim 1$, so we have a quadratic equation, while in the cusp regime $c_2\approx 0$, but $|c_3|\sim 1$, so we have a cubic equation. It turns out that under the flatness condition no other cases are possible, i.e. $|c_1|+ |c_2|+|c_3| \sim 1$. This trichotomic structural property of the underlying cubic equation was first discovered in [@AjankiSingularities], developed further in [@Altshape], and played an essential role in proving cusp local laws for Wigner-type matrices in [@Ajankirandommatrix; @Cusp1].
In our current situation, lacking flatness for $H_z$, a second unstable direction of ${\mathcal{B}}$ is present due to the specific block structure of the matrix $H_z$ which creates a major complication. We denote the unstable directions of ${\mathcal{B}}$ by $B$ and $B_*$. One of them, $B$, is the relevant one and it behaves very similarly to the one present in [@AltEdge; @Altshape; @Cusp1]. The novel unstable direction $B_*$ originates from the specific block structure of $H_z$ and ${\mathcal{S}}$ in and , respectively, and is related to the unstable direction in [@Altcirc]. We need to treat both unstable directions separately. In a generic situation, the solution to would be of the form $Y = \Theta B + \Theta_* B_* + \mathrm{error}$, where the complex scalars $\Theta$ and $\Theta_*$ satisfy a system of coupled cubic equations that is hard to analyse. Fortunately, for our applications, we have an additional input, namely we know that there is a matrix, concretely $E_-$, such that $Y=G-M$ is orthogonal to $E_-$, while $B_*$ is far from being orthogonal to $E_-$ (see below for the definition of $E_-$ and and for the orthogonality to $G$ and $M$, respectively). The existence of such matrix and a certain non-degeneracy of the two unstable directions guarantee that $\Theta_*$ is negligible and $Y$ is still essentially parallel to one unstable direction, $Y = \Theta B + \mathrm{error}$. Hence, we still need to analyse a single cubic equation for $\Theta$, albeit its coefficients, given in terms of $B$, $B_*$, $M$, $D$, ${\mathcal{B}}$ and ${\mathcal{S}}$, see Lemma \[lem:cubic\_equation\_abstract\] for their precise form, are much more complicated than those in [@Altshape; @Cusp1].
Summarizing, to understand the relationship between $G-M$ and $D$ from requires an analysis of the small eigenvalues of ${\mathcal{B}}$ in the regime, where ${\lvert z \rvert}^2$ is close to $\varrho({\mathscr{S}})$ and $\eta$ is small. This analysis is based on viewing the non-normal operator ${\mathcal{B}}$ as a perturbation around an operator of the form $1- {\mathcal{C}}{\mathcal{F}}$, where ${\mathcal{C}}$ is unitary and ${\mathcal{F}}$ is Hermitian. The unperturbed operator, $1- {\mathcal{C}}{\mathcal{F}}$, is also non-normal but simpler to analyze compared to ${\mathcal{B}}$. In fact, $1- {\mathcal{C}}{\mathcal{F}}$ has a single small eigenvalue and this eigenvalue has (algebraic and geometric) multiplicity two and we can construct appropriate eigendirections. A very fine perturbative argument reveals that after perturbation these two eigendirections will be associated to two different (small) eigenvalues $\beta$ and $\beta_*$. The distance between them is controlled from below which allows us to follow the perturbation of the eigendirections as well. Precise perturbative expansions of $B$ and $B_*$ around the corresponding eigenvectors of $1 - {\mathcal{C}}{\mathcal{F}}$ and a careful use of the specific structure of ${\mathcal{S}}$ in reveal that, up to a small error term, $B$ is orthogonal to $E_-$ while $B_*$ is far from orthogonal to $E_-$.
Moreover, we have to show that $M D$ in is sufficiently small in the unstable direction $B$ to compensate for the blow-up of ${\mathcal{B}}^{-1}$ originating from the relevant small eigenvalue $\beta$. To that end, we need to adjust the *cusp fluctuation averaging* mechanism discovered in [@Cusp1] to the current setup which will be done in Subsection \[subsec:proof\_D\_bounds\] below. This part also uses the specific block structure of $H_z$ in . We can, thus, conclude that $G-M$ is small due to which completes the sketch of the proof of Theorem \[thm:spectral\_radius\_X\].
The proof of Theorem \[thm:local\_law\_X\] also follows from the local law for $H_z$ since the observable of the eigenvalues of $X$ is related to the resolvent $G$ while the integral over $f_{z_0,a} \sigma$ is related to $M$. Indeed, [@Altcirc Eq.’s (2.10), (2.13) and (2.14)] imply that $$\label{eq:girko}
\frac{1}{n} \sum_{i=1}^n f_{z_0,a}(\zeta_i) = \frac{1}{4\pi n} \int_{{\mathbb{C}}} \Delta f_{z_0,a}(z) \log {\lvert \det H_z \rvert} {\mathrm{d}}^2 z =
-\frac{1}{4\pi n} \int_{{\mathbb{C}}} \Delta f_{z_0,a}(z) \int_0^\infty {\mathrm{Im}\,}\operatorname{Tr}G(z,\eta) {\mathrm{d}}\eta \, {\mathrm{d}}^2 z.$$ The first identity in is known as *Girko’s Hermitization formula*, the second identity (after a regularization of the $\eta$-integral at infinity) was first used in [@tao2015]. On the other hand, since the imaginary part of the diagonal of $M$ coincides with the solution $(v_1,v_2)$ of (see below), the definition of $\sigma$ in yields $$\int_{\mathbb{C}}f_{z_0,a} (z) \sigma(z){\mathrm{d}}^2 z = -\frac{1}{4\pi n} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_0^\infty {\mathrm{Im}\,}\operatorname{Tr}M(z,\eta) {\mathrm{d}}\eta\, {\mathrm{d}}^2 z.$$ Therefore, Theorem \[thm:local\_law\_X\] also follows once the closeness of $G$ and $M$ has been established as explained above.
Notations and conventions {#sec:notations}
-------------------------
In this section, we collect some notations and conventions used throughout the paper. We set $[k] {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{1, \ldots, k \} \subset {\mathbb{N}}$ for any $k \in {\mathbb{N}}$. For $z \in {\mathbb{C}}$ and $r >0$, we define the disk $D_r(z)$ in ${\mathbb{C}}$ of radius $r$ centered at $z$ through $D_r(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ w \in {\mathbb{C}}\colon {\lvert z-w \rvert} < r \}$. We use ${\mathrm{d}}^2 z$ to denote integration with respect to the Lebesgue measure on ${\mathbb{C}}$.
We now introduce some notation used for vectors, matrices and linear maps on matrices. Vectors in ${\mathbb{C}}^{2n}$ are denoted by boldfaced small Latin letters like ${\mathbf x}$, ${\mathbf y}$ etc. For vectors ${\mathbf x}= (x_a)_{a \in [2n]}, \,{\mathbf y}= (y_a)_{a \in [2n]} \in {\mathbb{C}}^{2n}$, we consider the normalized Euclidean scalar product ${\langle{{\mathbf x}} \mspace{2mu}, {{\mathbf y}}\rangle}$ and the induced normalized Euclidean norm ${\lVert {\mathbf x}\rVert}$ defined by $${\langle{{\mathbf x}} \mspace{2mu}, {{\mathbf y}}\rangle} = (2n)^{-1} \sum_{a \in [2n]} \overline{x_a} y_a, \qquad \qquad {\lVert {\mathbf x}\rVert} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\langle{{\mathbf x}} \mspace{2mu}, {{\mathbf x}}\rangle}^{1/2}.$$ Functions of vectors such as roots, powers or inverse and operations such as products of vectors are understood entrywise.
Matrices in ${\mathbb{C}}^{2n \times 2n}$ are usually denoted by capitalized Latin letters. We especially use $G$, $H$, $J$, $M$, $R$, $S$ and $T$. For a matrix $R\in {\mathbb{C}}^{2n \times 2n}$, we introduce the real part ${\mathrm{Re}\,}R$ and the imaginary part ${\mathrm{Im}\,}R$ defined through $${\mathrm{Re}\,}R {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2} \big( R + R^*\big), \qquad \qquad {\mathrm{Im}\,}R{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2{\mathrm{i}}} \big( R - R^* \big).$$ We have $R = {\mathrm{Re}\,}R + {\mathrm{i}}{\mathrm{Im}\,}R$ for all $R \in {\mathbb{C}}^{2n \times 2n}$. On ${\mathbb{C}}^{2n\times 2n}$, we consider the normalized trace ${\langle {{\,\cdot\,}}\rangle}$ and the normalized Hilbert-Schmidt scalar product ${\langle{{{\,\cdot\,}}} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}$ defined by $${\langle V \rangle} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2n} \operatorname{Tr}(V) = \frac{1}{2n} \sum_{i=1}^{2n} v_{ii}, \qquad {\langle{V} \mspace{2mu}, {W}\rangle} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2n} \operatorname{Tr}(V^* W) = {\langle V^*W \rangle}$$ for all $V=(v_{ij})_{i,j=1}^{2n}, W \in {\mathbb{C}}^{2n \times 2n}$. The norm on ${\mathbb{C}}^{2n \times 2n}$ induced by the normalized Hilbert-Schmidt scalar product is denoted by ${\lVert {{\,\cdot\,}}\rVert_{2}}$, i.e. ${\lVert V \rVert_{2}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\langle V^*V \rangle}^{1/2}$ for any $V \in {\mathbb{C}}^{2n \times 2n}$. Note that the subspaces ${{\mathcal{M}_\mathrm{d}}}$ and ${{\mathcal{M}_\mathrm{o}}}$ are orthogonal. Moreover, for $V \in {\mathbb{C}}^{2n \times 2n}$, we write ${\lVert V \rVert}$ for the operator norm of $V$ induced by the normalized Euclidean norm ${\lVert {{\,\cdot\,}}\rVert}$ on ${\mathbb{C}}^{2n}$.
We use capitalized calligraphic letters like ${\mathcal{S}}$, ${\mathcal{B}}$ and ${\mathcal{T}}$ to denote linear maps on ${\mathbb{C}}^{2n\times 2n}$. In particular, for $A, B \in {\mathbb{C}}^{2n \times 2n}$, we define the linear map ${\mathcal{C}}_{A,B} \colon {\mathbb{C}}^{2n\times 2n} \to {\mathbb{C}}^{2n\times 2n}$ through ${\mathcal{C}}_{A, B} [R] {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}A R B$ for all $R \in {\mathbb{C}}^{2n\times 2n}$. This map satisfies the identities ${\mathcal{C}}_{A,B}^* = {\mathcal{C}}_{A^*, B^*}$ and ${\mathcal{C}}_{A,B}^{-1} = {\mathcal{C}}_{A^{-1},B^{-1}}$, where the second identity requires the matrices $A$ and $B$ to be invertible. For a linear map ${\mathcal{T}}$ on ${\mathbb{C}}^{2n\times 2n}$, we consider several norms. We denote by ${\lVert {\mathcal{T}}\rVert}$ the operator norm of ${\mathcal{T}}$ induced by ${\lVert {{\,\cdot\,}}\rVert}$ on ${\mathbb{C}}^{2n\times 2n}$. Moreover, ${\lVert {\mathcal{T}}\rVert_{2\to 2}}$ denotes the operator norm of ${\mathcal{T}}$ induced by ${\lVert {{\,\cdot\,}}\rVert_{2}}$ on ${\mathbb{C}}^{2n\times 2n}$. We write ${\lVert {\mathcal{T}}\rVert_{2\to\lVert\,\cdot\,\rVert}}$ for the operator norm of ${\mathcal{T}}$ when the domain is equipped with ${\lVert {{\,\cdot\,}}\rVert_{2}}$ and the target is equipped with ${\lVert {{\,\cdot\,}}\rVert}$.
In order to simplify the notation in numerous computations, we use the following conventions. In vector-valued relations, we identify a scalar with the vector whose components all agree with this scalar. Moreover, we use the block matrix notation $$\label{eq:block_matrix_notation}
\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ exclusively for $2n\times 2n$-matrices. Here, each block is of size $n \times n$. If $a$, $b$, $c$ or $d$ are vectors (or scalars) then with a slight abuse of notations they are identified with the diagonal $n\times n$ matrices with $a$, $b$, $c$ or $d$, respectively, on the diagonal (or the respective multiple of the $n\times n$ identity matrix). Furthermore, we introduce the $2n\times 2n$ matrices $E_+$ and $E_-$ given in the block matrix notation of by $$\label{eq:def_E_pm}
E_+ {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad E_- {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ We remark that $E_+$ coincides with the identity matrix in ${\mathbb{C}}^{2n\times 2n}$. In our argument, the following sets of $2n \times 2n$-matrices appear frequently. The *diagonal* matrices ${{\mathcal{M}_\mathrm{d}}}\subset {\mathbb{C}}^{2n \times 2n}$ and the *off-diagonal* matrices ${{\mathcal{M}_\mathrm{o}}}\subset {\mathbb{C}}^{2n \times 2n}$ are defined through $${{\mathcal{M}_\mathrm{d}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\bigg\{ \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} \colon a,b \in {\mathbb{C}}^n \bigg\},
\qquad \qquad {{\mathcal{M}_\mathrm{o}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\bigg\{ \begin{pmatrix} 0 & a \\ b & 0 \end{pmatrix} \colon a,b \in {\mathbb{C}}^n \bigg\}.$$
In each section of this paper, we will specify a set of *model parameters* which are basic parameters of our model, e.g. $s_*$ and $s^*$ in . All of our estimates will hold uniformly for all models that satisfy our assumptions with the same model parameters. For $f, g \in [0,\infty)$, the comparison relation $f \lesssim g$ is true if $f \leq C g$ for some constant $C>0$ that depends only on model parameters. We also write $f \gtrsim g$ if $g \lesssim f$ and $f \sim g$ if $f \lesssim g$ and $f \gtrsim g$. If $f(i)$ and $g(i)$ depend on a further parameter $i \in I$ and $f(i) \leq C g(i)$ for all $i \in I$ then we say $f \lesssim g$ uniformly for $i \in I$. We use the same notation for nonnegative vectors and positive semidefinite matrices. Here, for vectors ${\mathbf x}= (x_a)_{a \in [2n]}, \,{\mathbf y}=(y_a)_{a \in [2n]}\in [0,\infty)^{2n}$, the comparison relation ${\mathbf x}\lesssim {\mathbf y}$ means $x_a \lesssim y_a$ uniformly for all $a \in [2n]$, i.e. the implicit constant can be chosen independently of $a$. For positive semidefinite matrices $R_1, R_2 \in {\mathbb{C}}^{2n\times 2n}$, $R_1 \lesssim R_2$ if ${\langle{{\mathbf x}} \mspace{2mu}, {R_1 {\mathbf x}}\rangle} \lesssim {\langle{{\mathbf x}} \mspace{2mu}, {R_2 {\mathbf x}}\rangle}$ uniformly for all ${\mathbf x}\in {\mathbb{C}}^{2n}$. For ${\varepsilon}>0$, scalars $f_1, f_2 \in {\mathbb{C}}$, matrices $R_1, R_2 \in {\mathbb{C}}^{2n\times 2n}$ and operators ${\mathcal{T}}_1, {\mathcal{T}}_2$ on ${\mathbb{C}}^{2n\times 2n}$, we write $f_1 = f_2 +{\mathcal{O}}({\varepsilon})$, $R_1= R_2 + {\mathcal{O}}({\varepsilon})$ and ${\mathcal{T}}_1 = {\mathcal{T}}_2 +{\mathcal{O}}({\varepsilon})$ if ${\lvert f_1 - f_2 \rvert} \lesssim {\varepsilon}$, ${\lVert R_1 - R_2 \rVert} \lesssim {\varepsilon}$ and ${\lVert {\mathcal{T}}_1 - {\mathcal{T}}_2 \rVert} \lesssim {\varepsilon}$, respectively.
Analysis of the Matrix Dyson equation {#sec:stability}
=====================================
In this section, we study the linear stability of the MDE, . According to the quadratic stability equation, , associated to the MDE the linear stability is governed by the behaviour of the stability operator ${\mathcal{B}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1 - {\mathcal{C}}_M {\mathcal{S}}$ (compare ). The main result of this section, Proposition \[pro:stability\_operator\] below, provides a complete understanding of the small, in absolute value, eigenvalues of ${\mathcal{B}}$ in the regime when $\rho= \rho(z,\eta)$ is small. Here, $\rho = \rho(z,\eta)$ is defined through $$\label{eq:def_rho}
\rho {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{\pi} {\langle {\mathrm{Im}\,}M \rangle}$$ for $\eta >0$ and $z \in {\mathbb{C}}$, where $M$ is the solution to . For the small eigenvalues and their associated eigenvectors, very precise expansions in terms of $M$ are derived in Proposition \[pro:stability\_operator\].
We warn the reader that $\rho$ should not be confused with the spectral radii $\varrho(X)$ and $\varrho({\mathscr{S}})$ used in Section \[sec:main\_results\]. The function $\rho$ is the *harmonic extension* of the *self-consistent density of states* of $H_z$ (see e.g. [@AltEdge Eq. (2)] for the definition of the self-consistent density of states).
In the remainder of the present section, we assume $\eta \in (0,1]$ and $ z \in D_\tau(0)$ for some fixed $\tau > 1$. In this section, the comparison relation $\lesssim$ introduced in Section \[sec:notations\] is understood with respect to the model parameters $\{s_*, s^*, \tau \}$. We recall that $s_*$ and $s^*$ constituted the bounds on the entries of ${\mathscr{S}}$ in .
The following proposition is the main result of the present section.
\[pro:stability\_operator\] There are (small) $\rho_* \sim 1$ and ${\varepsilon}\sim 1$ such that if $\rho + \eta/\rho \leq \rho_*$ then ${\mathcal{B}}$ has two eigenvalues $\beta$ and $\beta_*$ in $D_{\varepsilon}(0)$, i.e. $\operatorname*{Spec}({\mathcal{B}}) \cap D_{\varepsilon}(0) = \{ \beta, \beta_*\}$. Moreover, $\beta$ and $\beta_*$ have geometric and algebraic multiplicity one, $0<{\lvert \beta_* \rvert} < {\lvert \beta \rvert}$ and $$\label{eq:beta_beta_star_scaling}
{\lvert \beta_* \rvert} \sim \eta/\rho, \qquad {\lvert \beta \rvert} \sim \eta/\rho + \rho^2.$$ Furthermore, ${\mathcal{B}}$ has left and right eigenvectors ${\widehat}{B}_*$, ${\widehat}{B}$ and $B_*$, $B$ corresponding to $\beta_*$ and $\beta$, respectively, i.e., $${\mathcal{B}}[B_*] = \beta_* B_*, \qquad {\mathcal{B}}[B] = \beta B, \qquad {\mathcal{B}}^*[{\widehat}{B}_*] =\overline{\beta_*} {\widehat}{B}_*, \qquad {\mathcal{B}}^*[{\widehat}{B}] = \bar{\beta} {\widehat}{B},$$ which satisfy
\[eq:expansions\_eigenvectors\_cB\] $$\begin{aligned}
B & = \rho^{-1} {\mathrm{Im}\,}M - 2{\mathrm{i}}\rho^{-1}({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1}) ({\mathrm{Re}\,}M) + {\mathcal{O}}(\rho^2 + \eta/\rho), \label{eq:expansion_B}\\
B_* & = \rho^{-1} E_-{\mathrm{Im}\,}M + {\mathcal{O}}(\rho^2 + \eta/\rho),\label{eq:expansion_B_star} \\
{\widehat}{B} & = - \rho^{-1} {\mathrm{Im}\,}(M^{-1}) + {\mathcal{O}}(\rho^2 + \eta/\rho),\label{eq:expansion_L} \\
{\widehat}{B}_* & = - \rho^{-1} E_-{\mathrm{Im}\,}(M^{-1}) + {\mathcal{O}}(\rho^2 + \eta/\rho). \label{eq:expansion_L_star}\end{aligned}$$
For fixed $z$, the eigenvalues $\beta$ and $\beta_*$ as well as the eigenvectors $B$, $B_*$, ${\widehat}{B}$ and ${\widehat}{B}_*$ are continuous functions of $\eta$ as long as $\rho + \eta/\rho \leq \rho_*$. We also have the expansions
$$\begin{aligned}
\beta {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} & = \pi \eta \rho^{-1} + 2 \rho^2 \psi + {\mathcal{O}}(\rho^3 + \eta\rho + \eta^2/\rho^2), \label{eq:expansion_beta}\\
\beta_* {\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle} & = \pi\eta \rho^{-1} + {\mathcal{O}}(\rho^3 + \eta\rho + \eta^2/\rho^2), \label{eq:expansion_beta_star}\end{aligned}$$
where $\psi {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\rho^{-4} {\langle [({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1})]^2 \rangle}$. We have $\psi \sim 1$, ${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert}\sim 1$ and ${\lvert {\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle} \rvert} \sim 1$.
Moreover, the resolvent of ${\mathcal{B}}$ is bounded on the spectral subspace complementary to $\beta$ and $\beta_*$. That is, if ${\mathcal{Q}}$ is the spectral projection of ${\mathcal{B}}$ associated to $\operatorname*{Spec}({\mathcal{B}})\setminus \{ \beta, \beta_*\}$ then $$\label{eq:cB_inverse_Q_lesssim_one}
{\lVert {\mathcal{B}}^{-1} {\mathcal{Q}}\rVert} + {\lVert ({\mathcal{B}}^*)^{-1} {\mathcal{Q}}^* \rVert} \lesssim 1.$$
We now make a few remarks about Proposition \[pro:stability\_operator\]. First, owing to Lemma \[lem:scaling\_relation\_v\_u\] below (also note below), the condition $\rho + \eta/\rho \leq \rho_*$ with $\rho_* \sim 1$ is satisfied if ${\lvert {\lvert z \rvert}^2 - \varrho({\mathscr{S}}) \rvert} \leq \delta$ and $\eta \in (0,\delta]$ for some (small) $\delta \sim 1$. Secondly, we note that $B$, $B_*$, etc. are called eigenvectors despite that they are in fact matrices in ${\mathbb{C}}^{2n\times 2n}$. Finally, the second term on the right-hand side of is of order $\rho$, hence it is subleading compared to the first term $\rho^{-1} {\mathrm{Im}\,}M \sim 1$.
We now explain the relation between the solution $M$ to the MDE, , and the solution $(v_1, v_2)$ to . The $2n \times 2n$ matrix $M$ satisfies $$\label{eq:block_structure_M}
M(z, \eta) = \begin{pmatrix} {\mathrm{i}}v_1 & -z u \\ -\bar z u & {\mathrm{i}}v_2 \end{pmatrix},$$ where $(v_1, v_2)$ is the unique solution of and $u$ is defined through $$u {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{v_1}{\eta + {\mathscr{S}}^t v_1} = \frac{v_2}{ \eta + {\mathscr{S}}v_2}.$$ Note that $u \in (0,\infty)^n$. We remark that is the unique solution to with the side condition that ${\mathrm{Im}\,}M$ is a positive definite matrix. The existence and uniqueness of such $M$ follows from [@Helton01012007].
Throughout this section, the special structure of $M$ as presented in will play an important role. As a first instance, we see that the representation of $M$ in implies $$\label{eq:Im_M_Re_M}
{\mathrm{Im}\,}M = \begin{pmatrix} v_1 & 0 \\ 0 & v_2\end{pmatrix} , \qquad {\mathrm{Re}\,}M = \begin{pmatrix} 0 & - z u \\ -\bar z u & 0 \end{pmatrix}.$$ Therefore, ${\mathrm{Im}\,}M \in {{\mathcal{M}_\mathrm{d}}}$ and ${\mathrm{Re}\,}M \in {{\mathcal{M}_\mathrm{o}}}$. This is an important ingredient in the proof of the following corollary.
There is $\rho_* \sim 1$ such that $\rho + \eta/\rho \leq \rho_*$ implies $$\label{eq:bound_scalar_E_minus_B}
{\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert} \lesssim \rho^2 + \eta/\rho,$$ where $B$ is the right eigenvector of ${\mathcal{B}}$ from Proposition \[pro:stability\_operator\].
The expansion of $B$ in yields $${\langle{E_-} \mspace{2mu}, {B}\rangle} = \rho^{-1} {\langle{E_-} \mspace{2mu}, {{\mathrm{Im}\,}M}\rangle} - 2 {\mathrm{i}}\rho^{-1} {\langle{E_-} \mspace{2mu}, {({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1})({\mathrm{Re}\,}M)}\rangle} + {\mathcal{O}}(\rho^2 + \eta/\rho).$$ We now conclude by showing that the first two terms on the right-hand side vanish. The identity below implies ${\langle{E_-} \mspace{2mu}, {{\mathrm{Im}\,}M}\rangle} = 0$. Moreover, by , we have ${\mathrm{Re}\,}M \in {{\mathcal{M}_\mathrm{o}}}$ and ${\mathrm{Im}\,}M \in {{\mathcal{M}_\mathrm{d}}}$. Taking the imaginary part of thus yields ${\mathrm{Im}\,}M^{-1} \in {{\mathcal{M}_\mathrm{d}}}$. Therefore, ${\langle{E_-} \mspace{2mu}, {({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1})({\mathrm{Re}\,}M)}\rangle}=0$ since ${\mathrm{Re}\,}M \in {{\mathcal{M}_\mathrm{o}}}$ while $E_- ({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1}) \in {{\mathcal{M}_\mathrm{d}}}$. This completes the proof of .
Preliminaries
-------------
The MDE, , and its solution have a special scaling when ${\mathcal{S}}$ and, hence, ${\mathscr{S}}$, are rescaled by $\lambda>0$, i.e. ${\mathcal{S}}$ in is replaced by $\lambda {\mathcal{S}}$. Indeed, if $M=M(z,\eta)$ is the solution to with positive definite imaginary part then $M_\lambda(z,\eta) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lambda^{-1/2} M(z\lambda^{-1/2}, \eta \lambda^{-1/2})$ is the solution to $$- M_\lambda^{-1} = \begin{pmatrix} {\mathrm{i}}\eta & z \\ \bar z & {\mathrm{i}}\eta \end{pmatrix} + \lambda{\mathcal{S}}[M_\lambda].$$ with positive imaginary part. The same rescaling yields the positive solution of when ${\mathscr{S}}$ is replaced by $\lambda {\mathscr{S}}$ (see the explanations around (3.7) in [@Altcirc]). Therefore, by a simple rescaling, we can assume that the spectral radius is one, $$\label{eq:normalization_spectral_radius}
\varrho({\mathscr{S}}) = 1.$$ In the remainder of the paper, we will always assume .
#### Balanced polar decomposition of $M$
We first introduce a polar decomposition of $M$ that will yield a useful factorization of ${\mathcal{B}}$ which is the basis of its spectral analysis. To that end, we define $$\label{eq:def_U_Q}
U {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} {\mathrm{i}}\sqrt{\frac{v_1 v_2}{u}} & - z \sqrt{u} \\ - \bar z \sqrt{u} & {\mathrm{i}}\sqrt{\frac{v_1 v_2}{u}} \end{pmatrix}, \qquad
Q {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix} \big(\frac{uv_1}{v_2}\big)^{1/4} & 0 \\ 0 & \big(\frac{uv_2}{v_1}\big)^{1/4} \end{pmatrix},$$ where roots and powers of vectors are taken entrywise. Starting from these definitions, an easy computation shows that $M$ admits the following *balanced polar decomposition* $$\label{eq:balanced_polar_decomposition}
M = Q U Q.$$ Such polar decomposition for the solution of the Dyson equation was introduced in [@AjankiCorrelated].
The following lemma collects a few basic properties of $M$, $\rho$, $U$ and $Q$, mostly borrowed from [@Altcirc].
\[lem:scaling\_relation\_v\_u\]
1. Let $\eta \in (0,1]$ and $z \in D_\tau(0)$. We have $$\label{eq:scalar_E_minus_M_equals_0}
{\langle{E_-} \mspace{2mu}, {M}\rangle} = 0.$$ Moreover, $Q = Q^* \in {{\mathcal{M}_\mathrm{d}}}$, ${\mathrm{Im}\,}U \in {{\mathcal{M}_\mathrm{d}}}$, ${\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$ and $U$ is unitary.
2. Uniformly for all $\eta\in (0,1]$ and $z \in D_\tau(0)$, $\rho$ satisfies the scaling relations $$\label{eq:scaling_rho}
\rho \sim \begin{cases} \eta^{1/3} + (1- {\lvert z \rvert}^2)^{1/2}, & \text{ if } {\lvert z \rvert} \leq 1, \\
\frac{\eta}{{\lvert z \rvert}^2 - 1 + \eta^{2/3}}, & \text{ if } 1 \leq {\lvert z \rvert} \leq \tau, \end{cases}$$ for the matrices $U$, $Q$ and $M$, we have the estimates $$\label{eq:scaling_U_Q_M}
{\mathrm{Im}\,}U \sim \rho , \qquad Q \sim 1, \qquad {\lVert M \rVert} \lesssim 1,$$ and for the entries of $M$, we have $$\label{eq:scaling_v_1_v_2_u}
v_1 \sim v_2 \sim \rho, \qquad u \sim 1.$$
3. For fixed $z\in {\mathbb{C}}$, $M$, $\rho$, $U$ and $Q$ are continuous functions of $\eta$.
First, we remark that ${\lvert z \rvert}^2$ was denoted by $\tau$ in [@Altcirc].
The identity in follows from [@Altcirc Eq. (3.8)]. Obviously, and $v_1, v_2, u >0$ yield $Q = Q^* \in {{\mathcal{M}_\mathrm{d}}}$, ${\mathrm{Im}\,}U \in {{\mathcal{M}_\mathrm{d}}}$ and ${\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$. A simple computation reveals that $U$ is unitary as $u = v_1 v_2 + {\lvert z \rvert}^2 u^2$ due to [@Altcirc Eq. (3.32)].
From [@Altcirc Eq. (3.10), (3.11)], we conclude that $v_1$, $v_2$ and ${\mathrm{Im}\,}M$ scale as the right-hand side of . Hence, follows from the definition of $\rho$ in . Consequently, $v_1 \sim \rho \sim v_2$. Owing to [@Altcirc Eq. (3.26)], we have $u \sim 1$ uniformly for all $z \in D_\tau(0)$ and $\eta \in (0,1]$. Thus, $v_1 \sim \rho \sim v_2$ yields the first two scaling relations in . As $U$ is unitary we have ${\lVert U \rVert} = 1$. Thus, and the first two scaling relations in imply the last bound in .
For fixed $z \in {\mathbb{C}}$, the matrix $M$ is an analytic, hence, continuous function of $\eta$. Thus, $\rho$, $v_1$ and $v_2$ are continuous functions of $\eta$. Consequently, as $v_1, v_2, u>0$, the matrices $U$ and $Q$ are also continuous in $\eta$. This completes the proof of Lemma \[lem:scaling\_relation\_v\_u\].
#### Factorization of ${\mathcal{B}}$
We now present a factorization of ${\mathcal{B}}$ which will be the basis of our spectral analysis of ${\mathcal{B}}$ as a linear map on the Hilbert space $({\mathbb{C}}^{2n\times 2n}, {\langle{{{\,\cdot\,}}} \mspace{2mu}, {{{\,\cdot\,}}}\rangle})$. From , we easily obtain $$\label{eq:cB_rep_C_F}
{\mathcal{B}}= 1 - {\mathcal{C}}_M {\mathcal{S}}= {\mathcal{C}}_Q(1- {\mathcal{C}}_{U} {\mathcal{F}}) {\mathcal{C}}_Q^{-1},$$ where we introduced the positivity-preserving and Hermitian operator ${\mathcal{F}}$ on ${\mathbb{C}}^{2n \times 2n}$ defined by $$\label{eq:def_cF}
{\mathcal{F}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{C}}_Q {\mathcal{S}}{\mathcal{C}}_Q.$$
Owing to and $Q \sim 1$ by , the spectral properties of ${\mathcal{B}}$ stated in Proposition \[pro:stability\_operator\] can be obtained by analysing $1 - {\mathcal{C}}_U {\mathcal{F}}$. If $\rho$ is small then $U$ is well approximated by $P$ defined through $$\label{eq:def_P}
P {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathrm{sign}({\mathrm{Re}\,}U)} = \frac{{\mathrm{Re}\,}U}{{\lvert {\mathrm{Re}\,}U \rvert}}= \begin{pmatrix} 0 & - z/{\lvert z \rvert} \\ - \bar z /{\lvert z \rvert} & 0 \end{pmatrix}.$$ Indeed, $1- {\lvert {\mathrm{Re}\,}U \rvert} = 1- \sqrt{1 - ({\mathrm{Im}\,}U)^2} \lesssim ({\mathrm{Im}\,}U)^2 \lesssim \rho^2$ implies that $$\label{eq:P_minus_U_lesssim_rho}
{\lVert P - {\mathrm{Re}\,}U \rVert} \lesssim \rho^2, \qquad \qquad {\lVert P - U \rVert} \lesssim \rho.$$ Therefore, we will first analyse the operators ${\mathcal{F}}$ and ${\mathcal{C}}_P {\mathcal{F}}$. The proof of Proposition \[pro:stability\_operator\] will then follow by perturbation theory since implies $$\label{eq:norm_cK_minus_cL}
{\lVert 1 - {\mathcal{C}}_U {\mathcal{F}}- (1- {\mathcal{C}}_P {\mathcal{F}}) \rVert} \lesssim {\lVert P - U \rVert} \lesssim \rho.$$
#### Commutation relations
We have
$$\begin{aligned}
{3}
ME_- & = - E_-M^*,\qquad \qquad \qquad & M^* E_- & = - E_- M, && \label{eq:M_and_Emin} \\
Q E_- & = E_- Q, \qquad \qquad & Q^{-1} E_- & = E_- Q^{-1} && \label{eq:Q_Emin} \\
U E_- &= - E_- U^*,\qquad \qquad & U^* E_- &= - E_- U, && \label{eq:U_Emin} \\
PE_- &= - E_- P, \qquad \qquad &&&& \label{eq:P_Emin}\end{aligned}$$
The identities in follow by a simple computation starting from . Owing to $Q \in {{\mathcal{M}_\mathrm{d}}}$ we immediately obtain . The relations in are a direct consequence of and . The matrix representation of $P$ in directly implies .
#### Spectral properties of ${\mathcal{F}}$
\[lem:spec\_cF\] For all $\eta \in (0,1]$ and $z \in D_{\tau}(0)$, the following holds.
1. The range of ${\mathcal{F}}$ is contained in the diagonal matrices, i.e. $\operatorname{ran}{\mathcal{F}}\subset {{\mathcal{M}_\mathrm{d}}}$. Moreover, for all $R \in {\mathbb{C}}^{2n \times 2n}$, $$\label{eq:cF_Emin}
{\mathcal{F}}[RE_-] = - {\mathcal{F}}[R]E_- = - E_-{\mathcal{F}}[R] = {\mathcal{F}}[E_-R].$$
2. The top eigenvalue ${\lVert {\mathcal{F}}\rVert_{2\to 2}}$ of ${\mathcal{F}}$ is simple and satisfies $$\label{eq:normtwo_cF}
1- {\lVert {\mathcal{F}}\rVert_{2\to 2}} \sim \eta /\rho.$$
3. There is a unique positive definite eigenvector $F$ with ${\lVert F \rVert_{2}} = 1$ associated to ${\lVert {\mathcal{F}}\rVert_{2\to 2}}$. It satisfies $F \in {{\mathcal{M}_\mathrm{d}}}$.
4. The eigenvalue $-{\lVert {\mathcal{F}}\rVert_{2\to 2}}$ of ${\mathcal{F}}$ is also simple and $E_-F$ is an eigenvector corresponding to it.
5. There are $\rho_* \sim 1$ and $\vartheta \sim 1$ such that $\eta/\rho \leq \rho_*$ implies $${\lVert {\mathcal{F}}[R] \rVert_{2}} \leq {\lVert {\mathcal{F}}\rVert_{2\to 2}}(1- \vartheta) {\lVert R \rVert_{2}}$$ for all $R \in {\mathbb{C}}^{2n\times 2n}$ satisfying $R \perp F$ and $R \perp E_- F$.
Before the proof of Lemma \[lem:spec\_cF\], we introduce $F_U$ defined through $$\label{eq:def_F_U}
F_U {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\rho^{-1} {\mathrm{Im}\,}U = \rho^{-1} \begin{pmatrix} \sqrt{\frac{v_1v_2}{u}} & 0 \\ 0 & \sqrt{\frac{v_1v_2}{u}} \end{pmatrix}.$$ The importance of $F_U$ originates from the approximate eigenvector relation $$\label{eq:F_U_eigenvector_cF}
(1-{\mathcal{F}})[F_U] = \frac{\eta}{\rho} Q^2,$$ which is a consequence of the MDE, . Indeed, and imply $-U^* = Q({\mathrm{i}}\eta-A_z)Q + {\mathcal{F}}[U]$. Dividing the imaginary part of this identity by $\rho$ yields . Moreover, from , we directly deduce that $$\label{eq:F_U_sim_1}
F_U \sim 1.$$
The definition of ${\mathcal{S}}$ in implies $\operatorname{ran}{\mathcal{S}}\subset {{\mathcal{M}_\mathrm{d}}}$. Since $Q \in {{\mathcal{M}_\mathrm{d}}}$ by we deduce $\operatorname{ran}{\mathcal{F}}\subset {{\mathcal{M}_\mathrm{d}}}$. As $\operatorname{ran}{\mathcal{S}}\subset {{\mathcal{M}_\mathrm{d}}}$, we also have ${\mathcal{S}}[RE_-] = - {\mathcal{S}}[R] E_- = - E_- {\mathcal{S}}[R] = {\mathcal{S}}[E_-R]$ for all $R \in {\mathbb{C}}^{2n \times 2n}$. This completes the proof of (i) due to .
Since $\operatorname{ran}{\mathcal{F}}\subset {{\mathcal{M}_\mathrm{d}}}$, the restriction ${\mathcal{F}}|_{{\mathcal{M}_\mathrm{d}}}$ contains all spectral properties of ${\mathcal{F}}$ (apart from information about the possible eigenvalue $0$). The restriction ${\mathcal{F}}|_{{\mathcal{M}_\mathrm{d}}}$ is given by $${\mathcal{F}}\bigg[ \begin{pmatrix} r_1 & 0 \\ 0 & r_2 \end{pmatrix} \bigg] = \begin{pmatrix} {\mathscr{F}}r_2 & 0 \\ 0 & {\mathscr{F}}^t r_1 \end{pmatrix}$$ for $r_1, r_2 \in {\mathbb{C}}^n$, where we introduced the $n\times n$-matrix ${\mathscr{F}}$ defined by $$\label{eq:def_smallF}
{\mathscr{F}}r {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\bigg(\frac{u v_1}{v_2}\bigg)^{1/2} {\mathscr{S}}\bigg(r \bigg(\frac{uv_2}{v_1}\bigg)^{1/2}\bigg)$$ for $r \in {\mathbb{C}}^n$. Hence, in the standard basis of ${{\mathcal{M}_\mathrm{d}}}\cong {\mathbb{C}}^{2n}$, the restriction ${\mathcal{F}}|_{{\mathcal{M}_\mathrm{d}}}$ is represented by the $2n \times 2n$ matrix $$\boldsymbol{F} = \begin{pmatrix} 0 & {\mathscr{F}}\\ {\mathscr{F}}^t & 0 \end{pmatrix},$$ which was introduced in [@Altcirc Eq. (3.27b)] and analyzed in [@Altcirc Lemma 3.4]. The last result directly imply the simplicity of the eigenvalue ${\lVert {\mathcal{F}}\rVert_{2\to 2}}$, the existence of $F$ and $F \in {{\mathcal{M}_\mathrm{d}}}$. Owing to the second relation in , $E_-F$ is an eigenvector of ${\mathcal{F}}$ associated to $-{\lVert {\mathcal{F}}\rVert_{2\to 2}}$. For the proof of (ii), we apply ${\langle{F} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}$ to and obtain $$1- {\lVert {\mathcal{F}}\rVert_{2\to 2}} = \frac{\eta}{\rho} \frac{{\langle FQ^2 \rangle}}{{\langle FF_U \rangle}} \sim \frac{\eta}{\rho} \frac{{\langle F \rangle}}{{\langle F \rangle}} \sim \frac{\eta}{\rho}.$$ Here, we used the positive definiteness of $F$, $Q \sim 1$ by and $F_U \sim 1$ by in the second step.
In [@AltGram Lemma 3.5], the analogue of (v) for ${\mathcal{F}}|_{{\mathcal{M}_\mathrm{d}}}$ was proven by applying [@AltGram Lemma 3.3] to $\boldsymbol{F}$. Using the notation of [@AltGram Lemma 3.3], we have $L=2$ due to , $r_+ \sim r_- \sim 1$ due to and ${\widehat}{\lambda} \sim 1$ due to (ii) if $\eta/\rho \leq \rho_*$ and $\rho_* \sim1$ is chosen sufficiently small. Hence, the bound in (v) for $R \in {{\mathcal{M}_\mathrm{d}}}$ follows from [@AltGram Lemma 3.3]. Since ${\mathcal{F}}$ vanishes on the orthogonal complement of ${{\mathcal{M}_\mathrm{d}}}$, this completes the proof of Lemma \[lem:spec\_cF\].
Spectral properties of ${\mathcal{C}}_P {\mathcal{F}}$ and ${\mathcal{C}}_U {\mathcal{F}}$
------------------------------------------------------------------------------------------
For brevity we introduce the following shorthand notations for the operators in the following lemma. We define $$\label{eq:def_cK_cL}
{\mathcal{K}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1 - {\mathcal{C}}_P {\mathcal{F}}, \qquad {\mathcal{L}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1 - {\mathcal{C}}_U {\mathcal{F}}.$$ In the following lemma, we prove some resolvent bounds for these operators and show that they have at most two small eigenvalues.
\[lem:small\_eigenvalues\] There are (small) $\rho_* \sim 1$ and ${\varepsilon}\sim 1$ such that for all $z \in D_\tau(0)$ and $\eta >0$ satisfying $\rho + \eta/\rho \leq \rho_*$ and, for all ${\mathcal{T}}\in \{ {\mathcal{K}}, {\mathcal{L}}\}$, the following holds.
1. \[item:small\_eigenvalues\_resolvent\_bound\] For all $\omega \in {\mathbb{C}}$ with $\omega \notin D_{\varepsilon}(0) \cup D_{1- 2 {\varepsilon}}(1)$, we have $${\lVert ({\mathcal{T}}-\omega)^{-1} \rVert_{2\to 2}} + {\lVert ({\mathcal{T}}- \omega)^{-1} \rVert} + {\lVert ({\mathcal{T}}^*- \omega)^{-1} \rVert} \lesssim 1.$$
2. The spectral projection ${\mathcal{P}}_{\mathcal{T}}$ of ${\mathcal{T}}$, defined by $$\label{eq:cP_cT_contour_integral}
{\mathcal{P}}_{\mathcal{T}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}- \frac{1}{2\pi {\mathrm{i}}} \int_{{\partial}D_{\varepsilon}(0)} ({\mathcal{T}}- \omega)^{-1} {\mathrm{d}}\omega,$$ satisfies $\operatorname{rank}{\mathcal{P}}_{\mathcal{T}}= 2$. Moreover, for ${\mathcal{Q}}_{\mathcal{T}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1- {\mathcal{P}}_{\mathcal{T}}$, we have $$\label{eq:QT}
{\lVert {\mathcal{P}}_{\mathcal{T}}\rVert} + {\lVert {\mathcal{Q}}_{\mathcal{T}}\rVert} + {\lVert {\mathcal{P}}_{\mathcal{T}}^* \rVert} + {\lVert {\mathcal{Q}}_{\mathcal{T}}^* \rVert} + {\lVert {\mathcal{T}}^{-1}{\mathcal{Q}}_{\mathcal{T}}\rVert_{2\to 2}} + {\lVert ({\mathcal{T}}^*)^{-1} {\mathcal{Q}}_{\mathcal{T}}\rVert} + {\lVert {\mathcal{T}}^{-1}{\mathcal{Q}}_{\mathcal{T}}\rVert} \lesssim 1.$$
3. For fixed $z \in D_\tau(0)$, the spectral projections ${\mathcal{P}}_{\mathcal{T}}$ and ${\mathcal{Q}}_{{\mathcal{T}}}$ are continuous in $\eta$ as long as $\rho + \eta/\rho \leq \rho_*$.
The proof of Lemma \[lem:small\_eigenvalues\] is motivated by the proofs of [@AltEdge Lemma 4.7] and [@Altshape Lemma 5.1]. However, the additional extremal eigendirection of ${\mathcal{F}}$ requires a novel flow interpolating between $1- {\mathcal{F}}^2$ and $1- ({\mathcal{C}}_P {\mathcal{F}})^2$ instead of $1- {\mathcal{F}}$ and $1- {\mathcal{C}}_P {\mathcal{F}}$.
From , we deduce that ${\mathcal{S}}[R] \lesssim {\langle R \rangle}$ for all positive semidefinite matrices $R \in {\mathbb{C}}^{2n\times 2n}$. Thus, [@Altshape Lemma B.2(i)] implies that ${\lVert {\mathcal{S}}\rVert_{2\to\lVert\,\cdot\,\rVert}} \lesssim 1$. Therefore, for all ${\mathcal{T}}\in \{ {\mathcal{K}}, {\mathcal{L}}\}$, we have ${\lVert 1-{\mathcal{T}}\rVert_{2\to\lVert\,\cdot\,\rVert}} \lesssim 1$ due to $Q \sim 1$ and ${\lVert U \rVert} =1$ by Lemma \[lem:scaling\_relation\_v\_u\]. Hence, owing to [@Altshape Lemma B.2(ii)] and ${\lvert \omega -1 \rvert}\gtrsim 1$, it suffices to find ${\varepsilon}\sim 1$ such that
1. uniformly for all $\omega \notin D_{\varepsilon}(0) \cup D_{1-2{\varepsilon}}(1)$, we have $$\label{eq:normtwo_cT_omega}
{\lVert ({\mathcal{T}}- \omega)^{-1} \rVert_{2\to 2}} \lesssim 1,$$
2. the rank of ${\mathcal{P}}_{\mathcal{T}}$ equals 2, i.e., $\operatorname{rank}{\mathcal{P}}_{\mathcal{T}}=2$
for ${\mathcal{T}}\in \{ {\mathcal{K}}, {\mathcal{L}}\}$. Both claims for ${\mathcal{T}}= {\mathcal{K}}$ will follow from the corresponding statements for ${\mathcal{T}}= 1- ({\mathcal{C}}_P {\mathcal{F}})^2$ which we now establish by interpolating between $1 - {\mathcal{F}}^2$ and $1- ({\mathcal{C}}_P {\mathcal{F}})^2$. If ${\mathcal{T}}= 1- {\mathcal{F}}^2$ then both assertions follow from Lemma \[lem:spec\_cF\]. Moreover, a simple perturbation argument using Lemma \[lem:spec\_cF\] shows that $$\label{eq:eigenvector_cF_approx_F_U}
F = {\lVert F_U \rVert_{2}}^{-1} F_U + {\mathcal{O}}(\eta/\rho),$$ where $F$ is the eigenvector of ${\mathcal{F}}$ introduced in Lemma \[lem:spec\_cF\] (cf. the proof of [@Altcirc Lemma 3.5] for a similar argument).
In order to interpolate between $1 - {\mathcal{F}}^2$ and $1- ({\mathcal{C}}_P {\mathcal{F}})^2$ we use the following flow. For any $t \in [0,1]$, we define $${\mathcal{T}}_t {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1- {\mathcal{V}}_t {\mathcal{F}}, \qquad {\mathcal{V}}_t {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(1 - t) {\mathcal{F}}+ t {\mathcal{C}}_P {\mathcal{F}}{\mathcal{C}}_P.$$ Then ${\mathcal{T}}_0 = 1 - {\mathcal{F}}^2$ and ${\mathcal{T}}_1 = 1- ({\mathcal{C}}_P {\mathcal{F}})^2$. We now show for ${\mathcal{T}}= {\mathcal{T}}_t$ uniformly for all $t \in [0,1]$. To that end, we verify that ${\lVert ({\mathcal{T}}_t - \omega)[R] \rVert_{2}}\gtrsim 1$ uniformly for $R \in {\mathbb{C}}^{2n \times 2n}$ satisfying ${\lVert R \rVert_{2}} = 1$. If ${\lvert \omega \rvert} \geq 3$ then this follows from ${\lVert {\mathcal{V}}_t \rVert_{2\to 2}} \leq {\lVert {\mathcal{F}}\rVert_{2\to 2}} \leq 1$ by . Let ${\lvert \omega \rvert} \leq 3$ and $R \in {\mathbb{C}}^{2n \times 2n}$ satisfy ${\lVert R \rVert_{2}} =1$. We have the orthogonal decomposition $R = \alpha_+ F + \alpha_- E_- F + R_\perp$, where $R_\perp \perp E_\pm F$ (recall $E_+=1$ from ), and estimate $$\label{eq:proof_resolvent_bound_aux1}
\begin{aligned}
{\lVert ({\mathcal{T}}_t -\omega)[R] \rVert_{2}}^2 & = {\lvert \omega \rvert}^2 ({\lvert \alpha_+ \rvert}^2 + {\lvert \alpha_- \rvert}^2) + {\lVert (1-\omega - {\mathcal{V}}_t {\mathcal{F}})[R_\perp] \rVert_{2}}^2 + {\mathcal{O}}(\eta/\rho)\\
& \geq {\varepsilon}^2 ( {\lvert \alpha_+ \rvert}^2 + {\lvert \alpha_- \rvert}^2) + (\vartheta - 2{\varepsilon})^2 {\lVert R_\perp \rVert_{2}}^2 + {\mathcal{O}}(\eta/\rho).
\end{aligned}$$ We now explain how is obtained. The identity in follows from ${\mathcal{V}}_t {\mathcal{F}}[E_\pm F] = E_\pm F + {\mathcal{O}}(\eta/\rho)$ due to , , ${\mathcal{C}}_P [F_U] = F_U$, and . The lower bound in is a consequence of $${\lVert (1- \omega - {\mathcal{V}}_t {\mathcal{F}})[R_\perp] \rVert_{2}} \geq ({\lvert 1-\omega \rvert} - {\lVert {\mathcal{F}}\rVert_{2\to 2}}(1-\vartheta)) {\lVert R_\perp \rVert_{2}}
\geq (\vartheta - 2{\varepsilon}) {\lVert R_\perp \rVert_{2}},$$ where, in the first step, we used ${\lVert {\mathcal{V}}_t \rVert_{2\to 2}} \leq 1$ and ${\lVert {\mathcal{F}}[R_\perp] \rVert_{2}} \leq {\lVert {\mathcal{F}}\rVert_{2\to 2}}(1-\vartheta) {\lVert R_\perp \rVert_{2}}$ due to part (v) of Lemma \[lem:spec\_cF\]. In the second step, we employed ${\lVert {\mathcal{F}}\rVert_{2\to 2}} \leq 1$ and ${\lvert 1- \omega \rvert} \geq 1 -2 {\varepsilon}$. This shows which implies for ${\mathcal{T}}= {\mathcal{T}}_t$ if ${\varepsilon}\sim 1$ and $\rho_* \sim 1$ are chosen sufficiently small.
A similar but simpler argument to the proof of shows that ${\lVert ({\mathcal{K}}- \omega)[R] \rVert_{2}} \gtrsim 1$ uniformly for all $R \in {\mathbb{C}}^{2n \times 2n}$ satisfying ${\lVert R \rVert_{2}} =1$. This implies for ${\mathcal{T}}= {\mathcal{K}}$.
Owing to Lemma \[lem:scaling\_relation\_v\_u\] (iii) and ${\mathcal{F}}= {\mathcal{C}}_Q {\mathcal{S}}{\mathcal{C}}_Q$ (cf. ), ${\mathcal{K}}$ and ${\mathcal{L}}$ are continuous functions of $\eta$. Hence, the contour integral representation of ${\mathcal{P}}_{\mathcal{T}}$ in implies (iii).
What remains in order to complete the proof of Lemma \[lem:small\_eigenvalues\] for ${\mathcal{T}}= {\mathcal{K}}$ is showing $\operatorname{rank}{\mathcal{P}_{\mathcal{K}}}= 2$. The bound in with ${\mathcal{T}}= {\mathcal{T}}_t$ implies that ${\mathcal{P}}_{{\mathcal{T}}_t}$ is well defined for all $t \in [0,1]$. Moreover, the map $t \mapsto \operatorname{rank}{\mathcal{P}}_{{\mathcal{T}}_t}$ is continuous and, hence, constant as a continuous, integer-valued map. Therefore, $$\label{eq:rank_P_T_1}
\operatorname{rank}{\mathcal{P}}_{{\mathcal{T}}_1} = \operatorname{rank}{\mathcal{P}}_{{\mathcal{T}}_0} = 2,$$ where we used in the last step that ${\mathcal{T}}_0 = 1-{\mathcal{F}}^2$ and Lemma \[lem:spec\_cF\] (ii), (iv) and (v). Since $\operatorname*{Spec}( {\mathcal{C}}_P {\mathcal{F}}) \subset \operatorname*{Spec}(({\mathcal{C}}_P {\mathcal{F}})^2)$ the identity $\operatorname{rank}{\mathcal{P}}_{{\mathcal{T}}_1}=2$ from implies $\operatorname{rank}{\mathcal{P}_{\mathcal{K}}}\leq 2$. The following lemma provides the corresponding lower bound.
\[lem:cK\_small\_eigenvalue\] Let ${\varepsilon}$ and $\rho_*$ be chosen as in Lemma \[lem:small\_eigenvalues\]. If $\rho + \eta/\rho \leq \rho_*$ then $\operatorname*{Spec}({\mathcal{K}}) \cap D_{\varepsilon}(0)$ consists of a unique eigenvalue $\kappa$ of ${\mathcal{K}}$. This eigenvalue is positive, has algebraic and geometric multiplicity two and is a continuous function of $\eta$ for fixed $z \in D_\tau(0)$.
Owing to , the definition of ${\mathcal{T}}_1$ and $\operatorname*{Spec}( {\mathcal{C}}_P {\mathcal{F}}) \subset \operatorname*{Spec}(({\mathcal{C}}_P {\mathcal{F}})^2)$, $\operatorname*{Spec}({\mathcal{K}}) \cap D_{\varepsilon}(0)$ contains (counted with algebraic multiplicity) at most two eigenvalues of ${\mathcal{K}}$. We will now show that it contains one eigenvalue of (algebraic and geometric) multiplicity two. As $\operatorname{ran}{\mathcal{C}}_P{\mathcal{F}}\subset {{\mathcal{M}_\mathrm{d}}}$ it suffices to study the corresponding eigenvalue problem on ${{\mathcal{M}_\mathrm{d}}}$. Let $r_1, r_2 \in {\mathbb{C}}^{n}$ be vectors. We apply ${\mathcal{C}}_P {\mathcal{F}}$ to the diagonal matrix $R = \operatorname{diag}(r_1,r_2) \in {{\mathcal{M}_\mathrm{d}}}$ and obtain $$\label{eq:cC_P_cF_applied_to_diagonal}
{\mathcal{C}}_P {\mathcal{F}}[R] = {\mathcal{C}}_P {\mathcal{F}}\bigg[ \begin{pmatrix} r_1 & 0 \\ 0 & r_2 \end{pmatrix}\bigg] = \begin{pmatrix} {\mathscr{F}}^t r_1 & 0 \\ 0& {\mathscr{F}}r_2 \end{pmatrix},$$ where ${\mathscr{F}}$ denotes the $n\times n$-matrix defined in in the proof of Lemma \[lem:spec\_cF\]. The spectral radii of the matrix ${\mathscr{F}}$ and its transpose ${\mathscr{F}}^t$ agree. We denote this common spectral radius by $1- \kappa$. Since ${\lVert {\mathcal{C}}_P{\mathcal{F}}\rVert_{2\to 2}} = {\lVert {\mathcal{F}}\rVert_{2\to 2}} < 1$ by Lemma \[lem:spec\_cF\] we have $\kappa>0$. The entries of the matrices ${\mathscr{F}}$ and ${\mathscr{F}}^t$ are strictly positive. Hence, by the Perron-Frobenius theorem, there are $r_1, r_2 \in (0,\infty)^n$ such that ${\mathscr{F}}^t r_1 = (1-\kappa)r_1$ and ${\mathscr{F}}^t r_2 = (1-\kappa)r_2$. Thus, ${\mathcal{K}}[R] = \kappa R$, where we used and introduced $R {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\operatorname{diag}(r_1, r_2) \in {\mathbb{C}}^{2n \times 2n}$. Since $r_1, r_2 >0$ $E_- R$ and $R$ are linearly independent. Moreover, ${\mathcal{K}}[E_- R] = (1- {\mathcal{C}}_P {\mathcal{F}})[E_-R] = E_-{\mathcal{K}}[R]=\kappa E_- R$ due to and . Therefore, $\operatorname*{Spec}({\mathcal{K}}) \cap D_{\varepsilon}(0) = \{ \kappa \}$ and $R$ and $E_-R$ span the eigenspace of ${\mathcal{K}}$ associated to $\kappa$. Since ${\mathcal{K}}$ and ${\mathcal{P}_{\mathcal{K}}}$ are continuous functions of $\eta$, the eigenvalue $\kappa = \operatorname{Tr}({\mathcal{K}}{\mathcal{P}_{\mathcal{K}}})/2$ is also continuous with respect to $\eta$. This completes the proof of Lemma \[lem:cK\_small\_eigenvalue\].
We deduce $\operatorname{rank}{\mathcal{P}_{\mathcal{K}}}= 2$ from , $\operatorname*{Spec}( {\mathcal{C}}_P {\mathcal{F}}) \subset \operatorname*{Spec}(({\mathcal{C}}_P {\mathcal{F}})^2)$ and Lemma \[lem:cK\_small\_eigenvalue\]. This completes the proof of Lemma \[lem:small\_eigenvalues\] for ${\mathcal{T}}= {\mathcal{K}}$.
Owing to , we have ${\lVert {\mathcal{L}}- {\mathcal{K}}\rVert_{2\to 2}} \lesssim \rho$. Hence, possibly shrinking ${\varepsilon}\sim 1$ and $\rho_* \sim 1$ and a simple perturbation theory argument show the estimates in \[item:small\_eigenvalues\_resolvent\_bound\] and for ${\mathcal{T}}= {\mathcal{L}}$. Moreover, viewing ${\mathcal{L}}$ as perturbation of ${\mathcal{K}}$ and using $\operatorname{rank}{\mathcal{P}_{\mathcal{K}}}=2$ yield $\operatorname{rank}{\mathcal{P}}_{{\mathcal{L}}} =2$ for sufficiently small $\rho_* \sim 1$. This completes the proof of Lemma \[lem:small\_eigenvalues\].
Using the spectral properties of ${\mathcal{K}}$ established in Lemma \[lem:small\_eigenvalues\], we show in the following lemma that $E_\pm F_U$ are approximate eigenvectors of ${\mathcal{K}}$ associated to its small eigenvalue $\kappa$ from Lemma \[lem:cK\_small\_eigenvalue\].
\[lem:eigenvectors\_C\_S\_cF\] Let ${\varepsilon}$ and $\rho_*$ be chosen as in Lemma \[lem:small\_eigenvalues\] as well as ${\mathcal{P}}_{\mathcal{K}}$, ${\mathcal{Q}}_{\mathcal{K}}$ defined in Lemma \[lem:small\_eigenvalues\] for ${\mathcal{K}}= 1- {\mathcal{C}}_P {\mathcal{F}}$. If $\rho + \eta/\rho \leq \rho_*$ then the following holds.
1. \[item:K\_plus\] There are left and right eigenvectors ${\widehat}{K}_+$ and $K_+$ of ${\mathcal{K}}$ corresponding to $\kappa$ such that $$\label{eq:eigenvectors_1_C_S_F}
K_+ = F_U - \frac{\eta}{\rho}{\mathcal{K}}^{-1}{\mathcal{Q}}_{\mathcal{K}}{\mathcal{C}}_P[Q^2],\qquad \qquad {\widehat}{K}_+ = F_U - \frac{\eta}{\rho} ({\mathcal{K}}^*)^{-1} {\mathcal{Q}}_{\mathcal{K}}^*[Q^2].$$ They are elements of ${{\mathcal{M}_\mathrm{d}}}$, continuous functions of $\eta$ for fixed $z \in D_\tau(0)$ and satisfy $$\label{eq:scalar_L_S_R_S}
{\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle} = {\langle F_U^2 \rangle} + {\mathcal{O}}(\eta^2/\rho^2).$$ Moreover, we have $$\label{eq:expansion_lambda_S}
\kappa = \frac{\eta}{\rho} \frac{\pi}{{\langle F_U^2 \rangle}} + {\mathcal{O}}(\eta^2/\rho^2) .$$
2. \[item:K\_minus\] Furthermore, $K_- {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}E_-K_+$ and ${\widehat}{K}_- {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}E_- {\widehat}{K}_+$ are also right and left eigenvectors of ${\mathcal{K}}$ corresponding to $\kappa$ that are linearly independent of $K_+$ and ${\widehat}{K}_+$, respectively.
3. \[item:K\_projections\] The projections ${\mathcal{P}_{\mathcal{K}}}$ and ${\mathcal{P}_{\mathcal{K}}}^*$ have the representation $${\mathcal{P}_{\mathcal{K}}}= \frac{{\langle{{\widehat}{K}_+} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}}{{\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle}} K_+ + \frac{{\langle{{\widehat}{K}_-} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}}{{\langle{{\widehat}{K}_-} \mspace{2mu}, {K_-}\rangle}} K_-, \qquad
{\mathcal{P}_{\mathcal{K}}}^* = \frac{{\langle{K_+} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}}{{\langle{K_+} \mspace{2mu}, {{\widehat}{K}_+}\rangle}} {\widehat}{K}_+ + \frac{{\langle{K_-} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}}{{\langle{K_-} \mspace{2mu}, {{\widehat}{K}_-}\rangle}} {\widehat}{K}_-.$$ In particular, $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}\subset {{\mathcal{M}_\mathrm{d}}}$ and $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}^*\subset {{\mathcal{M}_\mathrm{d}}}$ as well as ${\mathcal{P}_{\mathcal{K}}}{{\mathcal{M}_\mathrm{o}}}= {\mathcal{P}_{\mathcal{K}}}^* {{\mathcal{M}_\mathrm{o}}}= \{ 0 \}$.
For the proof, we note that the definition of $F_U$ in , and the definition of $\rho$ in imply $$\label{eq:scalar_Q_2_F_U}
{\langle{Q^2} \mspace{2mu}, {F_U}\rangle} = \rho^{-1} {\langle Q {\mathrm{Im}\,}U Q \rangle} = \pi.$$
We start the proof of \[item:K\_plus\] by remarking that the eigenspace of ${\mathcal{K}}$ associated to $\kappa$ is contained in ${{\mathcal{M}_\mathrm{d}}}$ since $\operatorname{ran}{\mathcal{C}}_P {\mathcal{F}}\subset {{\mathcal{M}_\mathrm{d}}}$. Next, we apply ${\mathcal{Q}_{\mathcal{K}}}{\mathcal{C}}_P$ to , use ${\mathcal{C}}_P[F_U] = F_U$ and ${\mathcal{K}}= 1 - {\mathcal{C}}_P {\mathcal{F}}$ and obtain $${\mathcal{K}}{\mathcal{Q}_{\mathcal{K}}}[F_U] = \frac{\eta}{\rho} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{C}}_P[Q^2].$$ Hence, setting $K_+ {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}}_{{\mathcal{K}}}[F_U]$ yields $$K_+ = {\mathcal{P}}_{{\mathcal{K}}}[F_U] = F_U - {\mathcal{Q}_{\mathcal{K}}}[F_U] = F_U - \frac{\eta}{\rho}{\mathcal{K}}^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{C}}_P[Q^2].$$ This proves the expansion of $K_+$ in . For the proof of the expansion of ${\widehat}{K}_+$, we use ${\mathcal{C}}_P [F_U] = F_U$ in , apply ${\mathcal{Q}_{\mathcal{K}}}^*$ to the result and set ${\widehat}{K}_+ {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}}_{{\mathcal{K}}}^*[F_U]$. Then the expansion of ${\widehat}{K}_+$ follows similarly as the one of $K_+$. The continuity of $F_U = \rho^{-1} {\mathrm{Im}\,}U$ due to Lemma \[lem:scaling\_relation\_v\_u\] (iii) and the continuity of ${\mathcal{P}_{\mathcal{K}}}$ and ${\mathcal{P}_{\mathcal{K}}}^*$ due to Lemma \[lem:small\_eigenvalues\] (iii) imply that $K_+$ and ${\widehat}{K}_+$ are also continuous.
The relation in follows directly from since ${\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+ - F_U}\rangle} = {\langle{{\widehat}{K}_+ - F_U} \mspace{2mu}, {K_+}\rangle} = 0$ due to ${\mathcal{Q}_{\mathcal{K}}}{\mathcal{P}_{\mathcal{K}}}= 0$ and, thus, $${\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle} = {\langle{{\widehat}{K}_+} \mspace{2mu}, {F_U}\rangle} + {\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+-F_U}\rangle} = {\langle F_U^2 \rangle} + {\langle{{\widehat}{K}_+ - F_U} \mspace{2mu}, {K_+}\rangle} + {\mathcal{O}}(\eta^2/\rho^2).$$ For the proof of , we deduce from and ${\mathcal{Q}_{\mathcal{K}}}[K_+] = 0$ that $$\kappa {\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle} = {\langle{{\mathcal{K}}^*[{\widehat}{K}_+]} \mspace{2mu}, {K_+}\rangle} = {\langle{{\mathcal{K}}^*[F_U]} \mspace{2mu}, {K_+}\rangle}
= \frac{\eta}{\rho} {\langle{Q^2} \mspace{2mu}, {F_U}\rangle} + {\mathcal{O}}(\eta^2/\rho^2)
= \frac{\eta\pi}{\rho} + {\mathcal{O}}(\eta^2/\rho^2),$$ where we used ${\mathcal{C}}_P [F_U] = F_U$, and . Therefore, we obtain due to .
We now prove \[item:K\_minus\]. As in the proof of (i), we see that $E_- K_+$ and $E_-{\widehat}{K}_+$ are right and left eigenvectors of ${\mathcal{K}}$ corresponding to $\kappa$ as well. For sufficiently small $\rho_* \sim 1$ and $\eta/\rho \leq \rho_*$, $K_+$ and ${\widehat}{K}_+$ are strictly positive definite. Hence, $K_-$ and ${\widehat}{K}_-$ are linearly independent of $K_+$ and ${\widehat}{K}_+$, respectively.
Part \[item:K\_projections\] follows directly from Lemma \[lem:small\_eigenvalues\], Lemma \[lem:cK\_small\_eigenvalue\] and Lemma \[lem:eigenvectors\_C\_S\_cF\] \[item:K\_plus\], \[item:K\_minus\]. This completes the proof of Lemma \[lem:eigenvectors\_C\_S\_cF\].
Eigenvalues of ${\mathcal{L}}$ in $D_{\varepsilon}(0)$ {#subsec:eigenvalues_cL}
------------------------------------------------------
In this section, we study the small eigenvalues of ${\mathcal{L}}$ as perturbations of the small eigenvalue $\kappa$ of ${\mathcal{K}}$ (see Lemma \[lem:cK\_small\_eigenvalue\]).
\[lem:eigenvalues\_cL\] There are $\rho_* \sim 1$ and ${\varepsilon}\sim 1$ such that if $\rho + \eta/\rho \leq \rho_*$ then $\operatorname*{Spec}({\mathcal{L}}) \cap D_{\varepsilon}(0)$ consists of two eigenvalues $\beta$ and $\beta_*$. Each of these eigenvalues has algebraic and geometric multiplicity one. Moreover, they satisfy ${\lvert \beta_* \rvert} < {\lvert \beta \rvert}$ and $$\label{eq:eigenvalues_cL_expansion}
\beta_* = \kappa + {\mathcal{O}}(\rho^3 + \eta \rho), \qquad \qquad
\beta = \kappa + 2 \rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} + {\mathcal{O}}(\rho^3 + \eta \rho).$$ Furthermore, $\beta$ and $\beta_*$ are continuous functions of $\eta$ for fixed $z \in D_\tau(0)$.
We remark that the eigenvalues of ${\mathcal{L}}$ are denoted by $\beta$ and $\beta_*$ since the spectra of ${\mathcal{L}}$ and ${\mathcal{B}}$ agree. Indeed, ${\mathcal{B}}$ and ${\mathcal{L}}$ are related through the similarity transform ${\mathcal{B}}= {\mathcal{C}}_Q{\mathcal{L}}{\mathcal{C}}_Q^{-1}$ due to .
To lighten the notation in the following, we denote the difference between ${\mathcal{L}}$ and ${\mathcal{K}}$ by $$\label{eq:def_cD}
{\mathcal{D}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{L}}- {\mathcal{K}}.$$
We decompose ${\mathcal{L}}$ according to the splitting ${\mathcal{P}_{\mathcal{K}}}+ {\mathcal{Q}_{\mathcal{K}}}= 1$, i.e. we write $$\label{eq:decomposition_cL}
{\mathcal{L}}= \begin{pmatrix} {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}& {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}\\ {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}& {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}\end{pmatrix}.$$ More precisely, by we mean that we consider the (not necessarily orthogonal) decomposition ${\mathbb{C}}^{2n} = \operatorname{ran}{\mathcal{P}_{\mathcal{K}}}+ \operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$ into two complementary subspaces and the operators in the right-hand side of act among the appropriate subspaces in this decomposition, e.g. ${\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}$ is a linear operator from $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}$ to $\operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$. Notice that ${\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}-\omega) {\mathcal{Q}_{\mathcal{K}}}$ (viewed as a linear map on $\operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$) is invertible if $|\omega| \le {\varepsilon}$, where ${\varepsilon}$ was chosen as in Lemma \[lem:small\_eigenvalues\]. To see this, we use the identity $$\label{eq:triv}
(A+B) \big[ I + A^{-1} B\big]A^{-1} = I$$ for $A= {\mathcal{Q}_{\mathcal{K}}}({\mathcal{K}}-\omega) {\mathcal{Q}_{\mathcal{K}}}$, $B= {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}-{\mathcal{K}}) {\mathcal{Q}_{\mathcal{K}}}$ and $I$ being the identity map on $\operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$ and notice that $A$ (viewed as a map on $\operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$) is invertible by Lemma \[lem:small\_eigenvalues\], in fact $A^{-1} = {\mathcal{Q}_{\mathcal{K}}}({\mathcal{K}}-\omega)^{-1} {\mathcal{Q}_{\mathcal{K}}}$ with $\| A^{-1}\|\lesssim1$ by . Moreover $\| A^{-1} B \|\le \| A^{-1}\| \| {\mathcal{Q}_{\mathcal{K}}}\|^2 \| {\mathcal{L}}-{\mathcal{K}}\| \lesssim \rho$, where we used ${\lVert {\mathcal{L}}- {\mathcal{K}}\rVert} \lesssim \rho$ by and $\|{\mathcal{Q}_{\mathcal{K}}}\| \le 1 + \|{\mathcal{P}_{\mathcal{K}}}\| \lesssim 1$ by . Therefore $I+A^{-1} B$ is also invertible if $\rho$ is sufficiently small, yielding the invertibility of $A+B= {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}-\omega) {\mathcal{Q}_{\mathcal{K}}}$ from and that $$\label{eq:QLbound}
\big\| \big[ {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}-\omega) {\mathcal{Q}_{\mathcal{K}}}\big]^{-1}\big\| \lesssim 1.$$
Moreover, we use and Schur’s determinant identity to compute the determinant of ${\mathcal{L}}-\omega$ and obtain $$\label{eq:proof_eigenvalues_cL_aux1}
\det({\mathcal{L}}- \omega) = \det({\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}-\omega) {\mathcal{Q}_{\mathcal{K}}}) \det( {\mathcal{P}_{\mathcal{K}}}( {\mathcal{L}}- \omega) {\mathcal{P}_{\mathcal{K}}}- {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}( {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}- \omega){\mathcal{Q}_{\mathcal{K}}})^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}).$$ Since the first determinant on the right-hand side is not zero for $|\omega|\le {\varepsilon}$, the small eigenvalues of ${\mathcal{L}}$ are exactly those $\omega$’s for which the second determinant vanishes. Note that this is a $2\times 2$ determinant since $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}$ is two dimensional. Now we write this determinant in a convenient basis. In the basis $(K_+,\, K_-)$ of $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}$ (cf. Lemma \[lem:eigenvectors\_C\_S\_cF\]), we have $$\label{eq:PK_cL_minus_omega_PK_aux1}
{\mathcal{P}_{\mathcal{K}}}( {\mathcal{L}}- \omega) {\mathcal{P}_{\mathcal{K}}}= {\mathcal{P}_{\mathcal{K}}}( {\mathcal{K}}- \omega) {\mathcal{P}_{\mathcal{K}}}+ {\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}= \begin{pmatrix} \kappa - \omega & 0 \\ 0 &\kappa - \omega \end{pmatrix} + \Lambda,$$ where we introduce the $2 \times 2$-matrix $\Lambda$ defined through $$\label{eq:def_Lambda}
\Lambda {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{pmatrix}
\frac{{\langle{\hat{K}_+} \mspace{2mu}, {{\mathcal{D}}[K_+]}\rangle}}{{\langle{\hat{K}_+} \mspace{2mu}, {K_+}\rangle}} & \frac{{\langle{\hat{K}_+} \mspace{2mu}, {{\mathcal{D}}[K_-]}\rangle} }{{\langle{\hat{K}_+} \mspace{2mu}, {K_+}\rangle}} \\
\frac{{\langle{\hat{K}_-} \mspace{2mu}, {{\mathcal{D}}[K_+]}\rangle}}{{\langle{\hat{K}_-} \mspace{2mu}, {K_-}\rangle}} & \frac{{\langle{\hat{K}_-} \mspace{2mu}, {{\mathcal{D}}[K_-]}\rangle} }{{\langle{\hat{K}_-} \mspace{2mu}, {K_-}\rangle}}
\end{pmatrix}.$$ The following lemma which will be shown in Subsection \[subsec:proofs\_aux\_results\] provides a precise expansion of $\Lambda$ in the small $\rho$ regime.
\[lem:expansion\_Lambda\] For $\Lambda$ defined in , we have the expansion $$\Lambda = 2\rho^2 \begin{pmatrix} \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} & 0 \\ 0 &0 \end{pmatrix} + {\mathcal{O}}(\eta \rho ).$$
Lemma \[lem:expansion\_Lambda\] and imply $$\label{eq:PK_cL_minus_omega_PK_aux2}
{\mathcal{P}_{\mathcal{K}}}({\mathcal{L}}-\omega) {\mathcal{P}_{\mathcal{K}}}= \begin{pmatrix} \kappa + 2 \rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \omega & 0 \\ 0 & \kappa - \omega \end{pmatrix} + {\mathcal{O}}(\eta \rho).$$
What remains in order to compute the $2 \times 2$-determinant in is estimating ${\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}$ and ${\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}$. To that end we use the following lemma which will be proven in Subsection \[subsec:proofs\_aux\_results\] below.
\[lem:expansion\_cD\_K\] Let ${\mathcal{D}}$ be defined as in . Let $K_\pm$ and ${\widehat}{K}_\pm$ be the eigenvectors of ${\mathcal{K}}$ introduced in Lemma \[lem:eigenvectors\_C\_S\_cF\]. Then we have $${\mathcal{D}}[K_+] = {\mathcal{O}}(\rho + \eta), \qquad {\mathcal{D}}[K_-] = {\mathcal{O}}(\eta), \qquad {\mathcal{D}}^*[{\widehat}{K}_+] = {\mathcal{O}}(\rho^2 + \eta),
\qquad {\mathcal{D}}^*[{\widehat}{K}_-] = {\mathcal{O}}(\eta).$$
As ${\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}= {\mathcal{Q}_{\mathcal{K}}}{\mathcal{K}}{\mathcal{P}_{\mathcal{K}}}+ {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}= {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}$ and ${\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}= {\mathcal{P}_{\mathcal{K}}}{\mathcal{K}}{\mathcal{Q}_{\mathcal{K}}}+ {\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}= {\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}$, the representation of ${\mathcal{P}_{\mathcal{K}}}$ in Lemma \[lem:eigenvectors\_C\_S\_cF\] \[item:K\_projections\] yields
$$\begin{aligned}
{\lVert {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}\rVert} & \lesssim \max\{ {\lVert {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K_+] \rVert}, {\lVert {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K_-] \rVert} \} \lesssim \rho + \eta,
\label{eq:proof_eigenvalues_cL_aux2}\\
{\lVert {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}\rVert} & \lesssim \sup_{{\lVert R \rVert}=1} \max\big \{ {\lvert {\langle{{\widehat}{K}_+} \mspace{2mu}, {{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}[R]}\rangle} \rvert}, {\lvert {\langle{{\widehat}{K}_-} \mspace{2mu}, {{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}[R]}\rangle} \rvert} \big\} \lesssim \rho^2 + \eta,
\label{eq:proof_eigenvalues_cL_aux3}\end{aligned}$$
where the last steps follow from Lemma \[lem:expansion\_cD\_K\] and .
Therefore, we combine , , and , use $\eta \lesssim \rho$ and obtain $$\label{eq:proof_eigenvalues_cL_aux4}
{\mathcal{P}_{\mathcal{K}}}( {\mathcal{L}}- \omega) {\mathcal{P}_{\mathcal{K}}}- {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}( {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}- \omega){\mathcal{Q}_{\mathcal{K}}})^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}=
\begin{pmatrix} \kappa + 2 \rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \omega & 0 \\ 0 & \kappa - \omega \end{pmatrix} + {\mathcal{O}}(\rho^3 + \eta \rho)$$ with respect to the basis vectors $K_+$ and $K_-$.
We now analyse the small eigenvalues of ${\mathcal{L}}$. We have seen after that, for any $|\omega|\le {\varepsilon}$, we have $\det({\mathcal{L}}-\omega)=0$ if and only if $$\det\Big({\mathcal{P}_{\mathcal{K}}}( {\mathcal{L}}- \omega) {\mathcal{P}_{\mathcal{K}}}- {\mathcal{P}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{Q}_{\mathcal{K}}}( {\mathcal{Q}_{\mathcal{K}}}({\mathcal{L}}- \omega){\mathcal{Q}_{\mathcal{K}}})^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{L}}{\mathcal{P}_{\mathcal{K}}}\Big) = 0.$$ Owing to , the latter relation is equivalent to $$\label{eq:proof_eigenvalues_cL_aux5}
0 = ( \kappa - \omega + \rho^2 \gamma +\delta_{11}) (\kappa - \omega + \delta_{22}) - \delta_{12} \delta_{21},$$ where $\gamma {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}2 {\langle F_U^4 \rangle}/{\langle F_U^2 \rangle}$ and $\delta_{ij}$ are the entries of the error term on the right-hand side of . In particular, $\gamma \sim 1$ by and $\delta_{ij} = {\mathcal{O}}(\rho^3 + \eta \rho)$. The quadratic equation in has the solutions $$\omega_\pm = \kappa + \frac{\gamma}{2} \rho^2 ( 1 \pm 1) + {\mathcal{O}}(\delta + \delta^2/\rho^2),$$ where $\delta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sup_{i,j} {\lvert \delta_{ij} \rvert}$. As $\kappa$, $\rho$, $\gamma$ and $\delta_{ij}$ are continuous in $\eta$ (the continuity of $\delta_{ij}$ follows from the continuity of ${\mathcal{P}_{\mathcal{K}}}$ and ${\mathcal{L}}$), $\omega_\pm$ are continuous in $\eta$. Since $\delta = {\mathcal{O}}(\rho^3 + \eta \rho)$ and $\rho \gtrsim \eta$, that is $$\omega_+ = \kappa + 2 \rho^2\frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} + {\mathcal{O}}(\rho^3 + \eta \rho), \qquad
\omega_- = \kappa + {\mathcal{O}}(\rho^3 + \eta \rho).$$ Clearly, $\omega_+$ and $\omega_-$ are different from each other if $\rho + \eta/\rho \leq \rho_*$ and $\rho_* \sim 1$ is chosen small enough. Hence, $\omega_+$ and $\omega_-$ are two small eigenvalues of ${\mathcal{L}}$. Lemma \[lem:small\_eigenvalues\] implies that ${\mathcal{L}}$ possesses at most two small eigenvalues, thus we have fully described the spectrum of ${\mathcal{L}}$ close to zero.
Eigenvectors of ${\mathcal{L}}$ and proof of Proposition \[pro:stability\_operator\]
------------------------------------------------------------------------------------
By Lemma \[lem:eigenvalues\_cL\], there are two eigenvalues of ${\mathcal{L}}$ in $D_{\varepsilon}(0)$. The following lemma relates the corresponding eigenvectors to $F_U$ via the eigenvectors of ${\mathcal{K}}$ from Lemma \[lem:eigenvectors\_C\_S\_cF\]. The eigenvectors of ${\mathcal{L}}$ will be perturbations of those of ${\mathcal{K}}$. The main mechanism is that the two small eigenvalues of ${\mathcal{L}}$ are sufficiently separated, ${\lvert \beta- \beta_* \rvert} \sim \rho^2$ (cf. ). We will use that this separation is much larger than $\rho^3 + \eta\rho$, the effect of the perturbation ${\mathcal{D}}$ between the unperturbed spectral subspaces $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}$ and $\operatorname{ran}{\mathcal{Q}_{\mathcal{K}}}$ (see below). Hence, regular perturbation theory applies.
Owing to ${\mathcal{B}}= {\mathcal{C}}_Q {\mathcal{L}}{\mathcal{C}}_Q^{-1}$ by , we will conclude Proposition \[pro:stability\_operator\] immediately from this lemma.
\[lem:eigenvectors\_cL\] There is $\rho_* \sim 1$ such that if $\rho + \eta/\rho \leq \rho_*$ then there are right (left) eigenvectors $L$ and $L_*$ (${\widehat}{L}$ and ${\widehat}{L}_*$) of ${\mathcal{L}}$ associated to the eigenvalues $\beta$ and $\beta_*$ from Lemma \[lem:eigenvalues\_cL\], respectively, satisfying
$$\begin{aligned}
{2}
L & = F_U + 2 \rho {\mathrm{i}}F_U^2 ({\mathrm{Re}\,}U) + {\mathcal{O}}(\rho^2 + \eta/\rho), \qquad \qquad & L_* & = E_-F_U + {\mathcal{O}}(\rho^2 + \eta/\rho), \label{eq:expansion_right_eigen_cL} \\
{\widehat}{L} & = F_U + {\mathcal{O}}(\rho^2 + \eta/\rho), \qquad \qquad & {\widehat}{L}_* & = E_- F_U + {\mathcal{O}}(\rho^2 + \eta/\rho).\label{eq:expansion_left_eigen_cL} \end{aligned}$$
Moreover, $L$, $L_*$, ${\widehat}{L}$ and ${\widehat}{L}_*$ are continuous functions of $\eta$. For their scalar products, we have the expansions $$\label{eq:scalar_products_eigenvectors_cL}
{\langle{{\widehat}{L}} \mspace{2mu}, {L}\rangle} = {\langle F_U^2 \rangle} + {\mathcal{O}}(\rho^2 + \eta/\rho), \qquad \qquad {\langle{{\widehat}{L}_*} \mspace{2mu}, {L_*}\rangle} = {\langle F_U^2 \rangle} + {\mathcal{O}}(\rho^2 + \eta/\rho).$$
Before the proof of Lemma \[lem:eigenvectors\_cL\], we first conclude Proposition \[pro:stability\_operator\] from Lemma \[lem:small\_eigenvalues\], Lemma \[lem:eigenvalues\_cL\] and Lemma \[lem:eigenvectors\_cL\].
We choose ${\varepsilon}\sim 1$ as in Lemma \[lem:eigenvalues\_cL\] and $\rho_* \sim 1$ as in Lemma \[lem:eigenvectors\_cL\]. Since ${\mathcal{B}}= {\mathcal{C}}_Q {\mathcal{L}}{\mathcal{C}}_Q^{-1}$ due to , the spectra of ${\mathcal{B}}$ and ${\mathcal{L}}$ agree. Hence, $\operatorname*{Spec}({\mathcal{B}}) \cap D_{\varepsilon}(0) = \{ \beta, \beta_*\}$, with $\beta$ and $\beta_*$ as introduced in Lemma \[lem:eigenvalues\_cL\]. From Lemma \[lem:eigenvalues\_cL\], and ${\langle F_U^2 \rangle} \sim 1$ by , we obtain the scaling relations in by shrinking $\rho_* \sim 1$ if needed.
We now derive and . From , and ${\mathrm{Im}\,}U = - {\mathrm{Im}\,}U^* = - {\mathrm{Im}\,}U^{-1}$ for the unitary operator $U$ (cf. Lemma \[lem:scaling\_relation\_v\_u\] (i)), we conclude $\psi = {\langle F_U^4 \rangle}$. Moreover, $\psi \sim 1$ due to . The identity ${\mathcal{B}}= {\mathcal{C}}_Q {\mathcal{L}}{\mathcal{C}}_Q^{-1}$ and $Q = Q^*$ also imply $B ={\mathcal{C}}_Q[L]$, $B_* = {\mathcal{C}}_Q[L_*]$, ${\widehat}{B} = {\mathcal{C}}_Q^{-1}[{\widehat}{L}]$ and ${\widehat}{B}_* = {\mathcal{C}}_Q^{-1}[{\widehat}{L}_*]$. Hence, ${\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} = {\langle{{\widehat}{L}} \mspace{2mu}, {L}\rangle}$ and ${\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle} = {\langle{{\widehat}{L}_*} \mspace{2mu}, {L_*}\rangle}$ as $Q = Q^*$ (In particular, ${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert} \sim 1$ and ${\lvert {\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle} \rvert} \sim 1$). Therefore, the expansions of $\beta$ and $\beta_*$ in Lemma \[lem:eigenvalues\_cL\], the expansion of $\kappa$ in , ${\langle F_U^2 \rangle} \sim 1$ due to and yield and .
The balanced polar decomposition, , the definition of and ${\mathrm{Im}\,}U = - {\mathrm{Im}\,}U^* = - {\mathrm{Im}\,}U^{-1}$ yield that $$\label{eq:proof_stab_op_aux1}
\begin{aligned}
Q E_\pm F_U Q & = \rho^{-1} E_\pm {\mathrm{Im}\,}M, \qquad\qquad \qquad Q^{-1} E_\pm F_U Q^{-1} = - \rho^{-1} E_\pm {\mathrm{Im}\,}M^{-1}, \\
Q F_U^2 ({\mathrm{Re}\,}U) Q & = Q F_U QQ^{-1} F_U Q^{-1} Q ({\mathrm{Re}\,}U) Q = -\rho^{-2} ({\mathrm{Im}\,}M)({\mathrm{Im}\,}M^{-1})({\mathrm{Re}\,}M) .
\end{aligned}$$ Since $B ={\mathcal{C}}_Q[L]$, $B_* = {\mathcal{C}}_Q[L_*]$, ${\widehat}{B} = {\mathcal{C}}_Q^{-1}[{\widehat}{L}]$ and ${\widehat}{B}_* = {\mathcal{C}}_Q^{-1}[{\widehat}{L}_*]$, the expansions in , thus, follow from Lemma \[lem:eigenvectors\_cL\], and $Q \sim 1$ in . Moreover, the continuity of $Q$, $Q^{-1}$ and the eigenvectors of ${\mathcal{L}}$ from Lemma \[lem:eigenvectors\_cL\] yield the continuity of the eigenvectors of ${\mathcal{B}}$.
The identity ${\mathcal{B}}= {\mathcal{C}}_Q {\mathcal{L}}{\mathcal{C}}_Q^{-1}$ also implies that ${\mathcal{B}}^{-1} {\mathcal{Q}}= {\mathcal{C}}_Q {\mathcal{L}}^{-1} {\mathcal{Q}}_{\mathcal{L}}{\mathcal{C}}_Q^{-1}$. Similarly, $({\mathcal{B}}^*)^{-1} {\mathcal{Q}}^* = {\mathcal{C}}_Q^{-1} ({\mathcal{L}}^*)^{-1} {\mathcal{Q}}_{{\mathcal{L}}}^* {\mathcal{C}}_Q$ as $Q = Q^*$. Hence, the bounds in follow from in Lemma \[lem:small\_eigenvalues\]. This completes the proof of Proposition \[pro:stability\_operator\].
The remainder of this subsection is devoted to the proof of Lemma \[lem:eigenvectors\_cL\].
We fix $\lambda \in \{ \beta, \beta_*\}$. Since $\beta$ and $\beta_*$ have multiplicity one and together with ${\mathcal{L}}$ they are continuous functions of $\eta$ due to Lemma \[lem:eigenvalues\_cL\] and Lemma \[lem:scaling\_relation\_v\_u\] (iii), respectively, we find an eigenvector $L'$ of ${\mathcal{L}}$ associated to $\lambda$ such that $L'$ is a continuous function of $\eta$ and ${\lVert L' \rVert} \sim 1$.
We apply ${\mathcal{P}_{\mathcal{K}}}$ to the eigenvector relation $\lambda L' = {\mathcal{L}}[L']$, use ${\mathcal{L}}= {\mathcal{K}}+ {\mathcal{D}}$ and ${\mathcal{P}_{\mathcal{K}}}{\mathcal{K}}= {\mathcal{K}}{\mathcal{P}_{\mathcal{K}}}= \kappa {\mathcal{P}_{\mathcal{K}}}$ to obtain $$\label{eq:proof_eigenvectors_cL_aux1}
\lambda {\mathcal{P}_{\mathcal{K}}}[L'] = {\mathcal{P}_{\mathcal{K}}}({\mathcal{K}}+ {\mathcal{D}})[L'] = \kappa {\mathcal{P}_{\mathcal{K}}}[L'] + {\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}[L'] + {\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}[L'].$$
We express in the basis $(K_+,\,K_-)$ of $\operatorname{ran}{\mathcal{P}_{\mathcal{K}}}$ (cf. Lemma \[lem:eigenvectors\_C\_S\_cF\] \[item:K\_projections\]). We use that ${\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}= \Lambda$ in this basis, where $\Lambda$ is defined as in , and decompose ${\mathcal{P}_{\mathcal{K}}}[L'] = \gamma_+ K_+ + \gamma_- K_-$ for some $\gamma_+, \gamma_- \in {\mathbb{C}}$. This yields $$\label{eq:proof_eigenvectors_cL_aux3}
(\Lambda - \delta)\begin{pmatrix} \gamma_+ \\ \gamma_- \end{pmatrix} = - \begin{pmatrix} \frac{{\langle{{\widehat}{K}_+} \mspace{2mu}, {{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}[L']}\rangle}}{{\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle}} \\
\frac{{\langle{{\widehat}{K}_-} \mspace{2mu}, {{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}[L']}\rangle}}{{\langle{{\widehat}{K}_-} \mspace{2mu}, {K_-}\rangle}} \end{pmatrix}
= - \begin{pmatrix} \frac{{\langle{{\mathcal{D}}^*[{\widehat}{K}_+]} \mspace{2mu}, {{\mathcal{Q}_{\mathcal{K}}}[L']}\rangle}}{{\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle}} \\
\frac{{\langle{{\mathcal{D}}^*[{\widehat}{K}_-]} \mspace{2mu}, {{\mathcal{Q}_{\mathcal{K}}}[L']}\rangle}}{{\langle{{\widehat}{K}_-} \mspace{2mu}, {K_-}\rangle}} \end{pmatrix} = \begin{pmatrix} {\mathcal{O}}((\rho^3 + \eta \rho) {\lVert {\mathcal{P}_{\mathcal{K}}}[L'] \rVert}) \\ {\mathcal{O}}(\eta \rho {\lVert {\mathcal{P}_{\mathcal{K}}}[L'] \rVert}) \end{pmatrix},$$ where $\delta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lambda - \kappa$. Here, in the last step, we used Lemma \[lem:expansion\_cD\_K\] to estimate ${\mathcal{D}}^*[{\widehat}{K}_\pm]$. For the other factor, ${\mathcal{Q}_{\mathcal{K}}}[L']$, we use the general eigenvector perturbation result, Lemma \[lem:perturbation\_theory\_2\] in Appendix \[app:perturbation\_th\]. More precisely, applying ${\mathcal{Q}_{\mathcal{K}}}$ to and using ${\mathcal{Q}_{\mathcal{K}}}[K] = {\mathcal{Q}_{\mathcal{K}}}{\mathcal{P}_{\mathcal{K}}}[L'] = 0$, we obtain ${\lVert {\mathcal{Q}_{\mathcal{K}}}[L'] \rVert} \lesssim {\lVert {\mathcal{D}}\rVert} {\lVert {\mathcal{P}_{\mathcal{K}}}[L'] \rVert} \lesssim \rho {\lVert {\mathcal{P}_{\mathcal{K}}}[L'] \rVert}$ since ${\lVert {\mathcal{D}}\rVert} \lesssim \rho$ by . For the denominators in , we use that ${\lvert {\langle{{\widehat}{K}_s} \mspace{2mu}, {K_s}\rangle} \rvert} \sim 1$ for $s \in \{ \pm\}$ by Lemma \[lem:eigenvectors\_C\_S\_cF\] and .
From ${\lVert L' \rVert} \lesssim 1$ and , we conclude ${\lVert {\mathcal{P}_{\mathcal{K}}}[L'] \rVert} \lesssim 1$. Thus, and Lemma \[lem:expansion\_Lambda\] imply $$\label{eq:proof_eigenvectors_cL_aux2}
\begin{pmatrix} 2 \rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \delta & 0 \\ 0 & -\delta \end{pmatrix} \begin{pmatrix} \gamma_ + \\ \gamma_- \end{pmatrix}
= \begin{pmatrix} {\mathcal{O}}(\rho^3 + \eta \rho) \\ {\mathcal{O}}(\eta \rho) \end{pmatrix}.$$ In order to compute $\gamma_+$ and $\gamma_-$, we now distinguish the two cases $\lambda = \beta$ and $\lambda=\beta_*$ and apply Lemma \[lem:eigenvalues\_cL\] to estimate $\delta$. If $\lambda = \beta$ then ${\lvert \delta \rvert} \sim \rho^2$ by Lemma \[lem:eigenvalues\_cL\] and . Hence, implies ${\lvert \gamma_- \rvert} \lesssim \eta/\rho$. Thus, ${\lvert \gamma_+ \rvert} \sim 1$ as ${\lvert \gamma_+ \rvert}{\lVert K_+ \rVert} \geq {\lVert L' \rVert} -{\lvert \gamma_- \rvert} {\lVert K_- \rVert} - {\lVert {\mathcal{Q}_{\mathcal{K}}}[L'] \rVert}$, ${\lvert \gamma_- \rvert} \lesssim \eta/\rho $, ${\lVert {\mathcal{Q}_{\mathcal{K}}}[L'] \rVert} \lesssim \rho$ and ${\lVert L' \rVert}
\sim 1 \sim {\lVert K_\pm \rVert}$. In particular, $L {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}L' {\langle{{\widehat}{K}_+} \mspace{2mu}, {K_+}\rangle}/{\langle{{\widehat}{K}_+} \mspace{2mu}, {L'}\rangle} = L' /\gamma_+$ is continuous in $\eta$ and $${\mathcal{P}_{\mathcal{K}}}[L] = K_+ + {\mathcal{O}}(\eta/\rho) = F_U + {\mathcal{O}}(\eta/\rho),$$ where we used in the last step. We now apply Lemma \[lem:perturbation\_theory\_2\] to compute ${\mathcal{Q}_{\mathcal{K}}}[L]$ with $K = {\mathcal{P}_{\mathcal{K}}}[L] = F_U + {\mathcal{O}}(\eta/\rho)$. From below, we obtain $({\mathcal{K}}- \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[F_U] = -2 \rho {\mathrm{i}}F_U^2 ({\mathrm{Re}\,}U) + {\mathcal{O}}(\rho^2 + \eta)$ since ${\mathcal{Q}_{\mathcal{K}}}$ and ${\mathcal{K}}$ agree with the identity map on ${{\mathcal{M}_\mathrm{o}}}$ and $F_U^2 ({\mathrm{Re}\,}U) \in {{\mathcal{M}_\mathrm{o}}}$. Hence, Lemma \[lem:perturbation\_theory\_2\] and ${\lVert {\mathcal{D}}\rVert} \lesssim \rho$ directly imply the expansion of $L$ in .
We now consider the case $\lambda = \beta_*$. Lemma \[lem:eigenvalues\_cL\] with $\lambda = \beta_*$ implies ${\lvert \delta \rvert} \lesssim \rho^3 + \eta \rho$ and, thus, ${\lvert 2\rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \delta \rvert} \sim \rho^2$. Hence, ${\lvert \gamma_+ \rvert} \lesssim \rho + \eta/\rho$ and, similarly to the other case, we set $L_* {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}L'/\gamma_-$ and conclude $${\mathcal{P}_{\mathcal{K}}}[L_*] = E_- F_U + {\mathcal{O}}(\rho + \eta/\rho).$$
Owing to in Lemma \[lem:perturbation\_theory\_2\] with $K = {\mathcal{P}_{\mathcal{K}}}[L_*]$, we have $$\label{eq:proof_eigenvectors_cL_aux4}
{\mathcal{Q}_{\mathcal{K}}}[L_*] = - ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}[L_*] + {\mathcal{O}}(\rho^2) = - ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[E_-F_U] + {\mathcal{O}}(\rho^2 + \eta) = {\mathcal{O}}(\rho^2 + \eta),$$ where the last step follows from below. Therefore, the second identity in , Lemma \[lem:expansion\_Lambda\], Lemma \[lem:expansion\_cD\_K\] and $\rho\gtrsim \eta$ by imply $$\begin{pmatrix} 2 \rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \delta & 0 \\ 0 & -\delta \end{pmatrix} \begin{pmatrix} \gamma_ + \\ \gamma_- \end{pmatrix}
= \begin{pmatrix} {\mathcal{O}}(\rho^4 + \eta \rho) \\ {\mathcal{O}}(\eta \rho) \end{pmatrix}.$$ As ${\lvert 2\rho^2 \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} - \delta \rvert} \sim \rho^2$, we conclude ${\lvert \gamma_+ \rvert} \lesssim \rho^2 + \eta/\rho$. Hence, ${\mathcal{P}_{\mathcal{K}}}[L_*] = E_-F_U + {\mathcal{O}}(\rho^2 + \eta/\rho)$ and the expansion of $L_*$ in follows from .
A completely analogous argument starting from ${\mathcal{L}}^* [{\widehat}{L}] = \bar \lambda {\widehat}{L}$ yields the expansions of ${\widehat}{L}$ and ${\widehat}{L}_*$ in . We leave the details to the reader. From and , we directly obtain since $F_U^3({\mathrm{Re}\,}U) \in {{\mathcal{M}_\mathrm{o}}}$ implies ${\langle F_U^3 ({\mathrm{Re}\,}U) \rangle} = 0$. This completes the proof of Lemma \[lem:eigenvectors\_cL\].
Proofs of the auxiliary Lemmas \[lem:expansion\_Lambda\] and \[lem:expansion\_cD\_K\] {#subsec:proofs_aux_results}
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In this section, we show Lemma \[lem:expansion\_Lambda\] and Lemma \[lem:expansion\_cD\_K\] which were both stated in Subsection \[subsec:eigenvalues\_cL\].
Lemma \[lem:expansion\_cD\_K\] follows directly from Lemma \[lem:eigenvectors\_C\_S\_cF\] and the precise expansions of ${\mathcal{D}}[E_\pm F_U]$ and ${\mathcal{D}}^*[E_\pm F_U]$ established in below.
In the following computations, we constantly use that $P$, $U$, $U^*$ and $F_U$ commute with each other. From , and $1-U^2 = -2 {\mathrm{i}}{\mathrm{Re}\,}U {\mathrm{Im}\,}U + 2 ({\mathrm{Im}\,}U)^2$, we obtain
\[eq:cD\_F\_U\_all\_cases\] $$\label{eq:cD_F_U}
{\mathcal{D}}[F_U] = ({\mathcal{C}}_P - {\mathcal{C}}_{U}) F_U + {\mathcal{O}}(\eta) = F_U (1-U^2) + {\mathcal{O}}(\eta)= - 2 \rho {\mathrm{i}}F_U^2 ({\mathrm{Re}\,}U) + 2\rho^2 F_U^3 + {\mathcal{O}}(\eta).$$ Similarly, , , and $UE_-F_UU = - E_- U ^* F_U U = - E_- F_U$ by imply $$\label{eq:cD_Emin_F_U}
{\mathcal{D}}[E_-F_U] = - ({\mathcal{C}}_P - {\mathcal{C}}_U)E_-F_U + {\mathcal{O}}(\eta)= E_-F_U - E_-F_U + {\mathcal{O}}(\eta) ={\mathcal{O}}(\eta).$$ Since $1-(U^*)^2 = 2 {\mathrm{i}}{\mathrm{Re}\,}U {\mathrm{Im}\,}U + 2 ({\mathrm{Im}\,}U)^2$, $F_U^2 ({\mathrm{Re}\,}U) \in {{\mathcal{M}_\mathrm{o}}}$ by Lemma \[lem:scaling\_relation\_v\_u\] (i) and ${\mathcal{F}}$ vanishes on ${{\mathcal{M}_\mathrm{o}}}$, we get $$\label{eq:cD_star_F_U}
{\mathcal{D}}^*[F_U] = {\mathcal{F}}( {\mathcal{C}}_P - {\mathcal{C}}_{U^*})[F_U] = {\mathcal{F}}[F_U ( 1 - (U^*)^2)] = 2 {\mathrm{i}}\rho
{\mathcal{F}}[F_U^2 ({\mathrm{Re}\,}U)] + 2 \rho^2 {\mathcal{F}}[F_U^3] = 2 \rho^2 {\mathcal{F}}[F_U^3].$$ From $P E_- F_U P = - E_-F_U = U^* E_- F_U U^*$ by and , we deduce $$\label{eq:cD_star_Emin_F_U}
{\mathcal{D}}^*[E_- F_U] = {\mathcal{F}}[-E_- F_U + E_-F_U] = 0.$$
This completes the proof of Lemma \[lem:expansion\_cD\_K\].
We first show that, for all $s_1, s_2 \in \{ \pm \}$, we have $$\label{eq:proof_expansion_Lambda_reduction_to_leading_term}
{\langle{{\widehat}{K}_{s_1}} \mspace{2mu}, {{\mathcal{D}}[K_{s_2}]}\rangle} = {\langle{E_{s_1} F_U} \mspace{2mu}, {{\mathcal{D}}[E_{s_2} F_U]}\rangle} + {\mathcal{O}}(\eta\rho + \eta^2).$$ In fact, it is easy to see that follows from $K_\pm, {\widehat}{K}_\pm \in {{\mathcal{M}_\mathrm{d}}}$ and in Lemma \[lem:eigenvectors\_C\_S\_cF\] as well as $$\label{eq:expansion_Lambda_error_term}
{\langle{R_1} \mspace{2mu}, {({\mathcal{C}}_P - {\mathcal{C}}_U)[R_2]}\rangle} = {\mathcal{O}}(\rho^2)$$ for all $R_1, R_2 \in {{\mathcal{M}_\mathrm{d}}}$ satisfying ${\lVert R_1 \rVert}, {\lVert R_2 \rVert} \lesssim 1$. For the proof of , we expand $U = {\mathrm{Re}\,}U + {\mathrm{i}}{\mathrm{Im}\,}U$ and obtain $${\langle{R_1} \mspace{2mu}, {({\mathcal{C}}_P- {\mathcal{C}}_U)[R_2]}\rangle} = {\langle{R_1} \mspace{2mu}, {({\mathcal{C}}_P - {\mathcal{C}}_{{\mathrm{Re}\,}U})[R_2]}\rangle} - {\mathrm{i}}{\langle{R_1} \mspace{2mu}, {{\mathrm{Im}\,}U R_2 {\mathrm{Re}\,}U + {\mathrm{Re}\,}U R_2 {\mathrm{Im}\,}U}\rangle} + {\mathcal{O}}(\rho^2) = {\mathcal{O}}(\rho^2).$$ Here, we used in the first step that ${\mathrm{Im}\,}U = {\mathcal{O}}(\rho)$. For the second step, we noted that the first term is ${\mathcal{O}}(\rho^2)$ due to and the second term vanishes as ${\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$ while $R_1, R_2, {\mathrm{Im}\,}U \in {{\mathcal{M}_\mathrm{d}}}$ due to Lemma \[lem:scaling\_relation\_v\_u\] (i).
What remains is computing ${\langle{E_{s_1}F_U} \mspace{2mu}, {{\mathcal{D}}[E_{s_2} F_U]}\rangle} = {\langle{{\mathcal{D}}^*[E_{s_1}F_U]} \mspace{2mu}, {E_{s_2}F_U}\rangle}$. From and , we obtain $${\langle{{\mathcal{D}}^*[F_U]} \mspace{2mu}, {F_U}\rangle} = 2 \rho^2 {\langle F_U^4 \rangle} + {\mathcal{O}}(\eta \rho), \qquad
{\langle{{\mathcal{D}}^*[E_-F_U]} \mspace{2mu}, {F_U}\rangle} = - 2 \rho^2 {\langle F_U^4 E_- \rangle} + {\mathcal{O}}(\eta \rho) = {\mathcal{O}}(\eta \rho),$$ where we used in the very last step that ${\langle F_U^4 E_- \rangle} = 0$ since the diagonal $n$-vector components of $F_U$ are identical due to . Moreover, directly implies that ${\langle{{\mathcal{D}}^*[E_-F_U]} \mspace{2mu}, {E_\pm F_U}\rangle} = 0$.
Hence, owing to , we deduce $$\Lambda = 2\rho^2 \begin{pmatrix} \frac{{\langle F_U^4 \rangle}}{{\langle F_U^2 \rangle}} & 0 \\ 0 &0 \end{pmatrix} + {\mathcal{O}}(\eta \rho + \eta^2).$$ Using that $\rho \gtrsim \eta$ due to completes the proof of Lemma \[lem:expansion\_Lambda\].
Derivatives of $M$
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As a first application of our analysis of the stability operator ${\mathcal{B}}$ we show the following bound on the derivatives of $M$, the solution to the MDE, , with respect to $\eta$, $z$ and $\bar z$.
There is $\rho_* \sim 1$ such that $\rho + \eta/\rho \leq \rho_*$ implies $$\label{eq:pt_eta_M_bound}
{\big\lVert {\partial}_\eta M \big\rVert} + {\big\lVert {\partial}_z M \big\rVert} + {\big\lVert {\partial}_{\bar z} M \big\rVert} \lesssim \frac{1}{\rho^2 + \eta/\rho}.$$
We only show the bound on ${\partial}_\eta M$. The estimates on ${\partial}_z M$ and ${\partial}_{\bar z} M$ are shown analogously.
If $\rho_* \sim 1$ is chosen small enough and $\rho + \eta/\rho \leq \rho_*$ then ${\mathcal{B}}$ is invertible due to Proposition \[pro:stability\_operator\]. Thus, applying the implicit function theorem to yields that $M$ is differentiable with respect to $\eta$ and ${\partial}_\eta M = {\mathrm{i}}{\mathcal{B}}^{-1}[M^2]$. Hence, by Proposition \[pro:stability\_operator\], we have $$\label{eq:aux_identity_derivative_M}
-{\mathrm{i}}{\partial}_\eta M = \frac{{\langle{{\widehat}{B}} \mspace{2mu}, {M^2}\rangle}}{\beta{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}} B + \frac{{\langle{{\widehat}{B}_*} \mspace{2mu}, {M^2}\rangle}}{\beta_*{\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle}} B_* + {\mathcal{B}}^{-1} {\mathcal{Q}}[M^2].$$ Moreover, differentiating ${\langle{E_-} \mspace{2mu}, {M}\rangle} = 0$, which holds due to and , with respect to $\eta$ yields $${\langle{E_-} \mspace{2mu}, {{\partial}_\eta M}\rangle} = 0.$$ Hence, we apply ${\langle{E_-} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}$ to and obtain $$\label{eq:proof_derivative_M_aux1}
\frac{{\langle{{\widehat}{B}_*} \mspace{2mu}, {M^2}\rangle}}{\beta_*{\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle}} {\langle{E_-} \mspace{2mu}, {B_*}\rangle} = - \frac{{\langle{{\widehat}{B}} \mspace{2mu}, {M^2}\rangle}}{\beta{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}} {\langle{E_-} \mspace{2mu}, {B}\rangle} + {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1}{\mathcal{Q}}[M^2]}\rangle}.$$ The right-hand side of is ${\mathcal{O}}(1)$ since ${\widehat}{B}$, $M$ and ${\mathcal{B}}^{-1} {\mathcal{Q}}[M^2]$ are bounded, ${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert} \sim 1$ and ${\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert} \lesssim {\lvert \beta \rvert}$ by and . Since ${\lvert {\langle{E_-} \mspace{2mu}, {B_*}\rangle} \rvert} \sim 1$ and ${\lVert B_* \rVert} \lesssim 1$ by , we conclude that the second term on the right-hand side of is ${\mathcal{O}}(1)$. Thus, the leading term to ${\partial}_\eta M$ comes from the first term on the right-hand side of which can be bounded by $(\rho^2 + \eta/\rho)^{-1}$ due to the boundedness of ${\widehat}{B}$, $M$ and $B$ as well as ${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert} \sim 1$ and .
Existence and properties of $\sigma$
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The next proposition shows the existence of a probability density $\sigma$ satisfying . In particular, the logarithmic potential of the probability measure $\sigma(z) {\mathrm{d}}^2 z$ is given by the function $-2\pi L$, where, for any $z \in {\mathbb{C}}$, $L(z)$ is defined through $$\label{eq:def_L}
L(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}-\frac{1}{2\pi} \int_0^\infty \bigg( {\langle {\mathrm{Im}\,}M(z,\eta) \rangle} - \frac{1}{1 + \eta} \bigg) \, {\mathrm{d}}\eta.$$
Throughout this subsection, we use the normalization $\varrho({\mathscr{S}}) = 1$ (see ).
\[pro:properties\_sigma\] There exists an integrable function $\sigma \colon {\mathbb{C}}\to [0,\infty)$ such that $\sigma = \Delta_z L$ in the sense of distributions. Moreover, $\sigma$ can be chosen to satisfy the following properties:
1. The function $\sigma$ is a probability density on ${\mathbb{C}}$ with respect to ${\mathrm{d}}^2z$.
2. For all $z\in {\mathbb{C}}$ with ${\lvert z \rvert} >1$, we have $\sigma(z) = 0$.
3. The restriction $\sigma|_{D(0,1)}$ is infinitely often continuously differentiable and $\sigma(z) \sim 1$ uniformly for $z \in D(0,1)$.
The next lemma will directly imply Proposition \[pro:properties\_sigma\] and be proved after the proof of this proposition.
\[lem:properties\_L\] The function $L(z)$ from has the following properties:
1. The integrand on the right-hand side of is Lebesgue-integrable for every $z \in {\mathbb{C}}$.
2. $L$ is a rotationally symmetric function and continuously differentiable with respect to $z$ and $\bar z$ on ${\mathbb{C}}$.
3. $\Delta_z L(z)$ exists for all $z \in{\mathbb{C}}$ satisfying ${\lvert z \rvert} \neq 1$.
By Lemma \[lem:properties\_L\] (iii), we know that $\Delta_z L$ defines a function on ${\mathbb{C}}\setminus \{ z \in {\mathbb{C}}\colon {\lvert z \rvert} = 1 \}$. We set $\mathbf v(z,\eta)=(v_1(z,\eta),v_2(z,\eta))$ and remark that ${\langle {\mathrm{Im}\,}M \rangle} = {\langle \mathbf v \rangle}$. Moreover, [@Altcirc Eq. (4.2)] implies that $\mathbf v$ has a smooth extension to ${\mathbb{C}}\setminus \overline{D(0,1)} \times [0,\infty)$ and $\lim_{\eta \downarrow 0} \mathbf v(z,\eta) = 0$ for ${\lvert z \rvert} > 1$ due to . Therefore, we can follow the proof of [@Altcirc Eq. (4.10)] and obtain $$\int_{\tau_0 \leq {\lvert z \rvert}^2 \leq \tau_1} \Delta_z L(z) {\mathrm{d}}^2 z = 0$$ for all $\tau_1 > \tau_0 >1$. As $\tau_1$ and $\tau_0$ are arbitrary, we conclude $\sigma(z) = \Delta_z L(z) =0$ if ${\lvert z \rvert}> 1$.
Let $f \in C_0^\infty({\mathbb{C}})$ be a smooth function with compact support. We compute $$\label{eq:integration_by_parts_L_Delta_f}
\int_{\mathbb{C}}L(z) \Delta_z f(z) {\mathrm{d}}^2 z = \int_{D(0,1)} L(z) \Delta_z f(z) {\mathrm{d}}^2 z + \int_{{\mathbb{C}}\setminus \overline{D(0,1)}} L(z) \Delta_z f(z) {\mathrm{d}}^2 z = \int_{{\mathbb{C}}} \sigma(z) f(z) {\mathrm{d}}^2 z.$$ Here, we moved $\Delta_z$ to $L$ in the second step and used that $\sigma(z) = \Delta_z L(z) = 0$ for ${\lvert z \rvert}>1$ and that the boundary terms cancel each other due to the continuity of $L$, ${\partial}_z L$ and ${\partial}_{\bar z} L$ from Lemma \[lem:properties\_L\] (ii). Since $\sigma(z) = 0$ for ${\lvert z \rvert}>1$, setting $\sigma(z) = 0$ if ${\lvert z \rvert}= 1$ and using [@Altcirc Proposition 2.4] for the remaining properties complete the proof of Proposition \[pro:properties\_sigma\] .
In the proof of Lemma \[lem:properties\_L\], we will make use of the following lemma whose proof we postpone until the end of this section.
\[lem:derivatives\_M\_controlled\_by\_eta\] Uniformly for $z \in {\mathbb{C}}$ and $\eta >0$, we have $$\begin{aligned}
{\bigg\lVert M(z,\eta) - \frac{{\mathrm{i}}}{1 + \eta} \bigg\rVert} & \lesssim \frac{1}{1+\eta^{2}}, \label{eq:approximating_M_large_eta} \\
{\lVert {\partial}_z M(z,\eta) \rVert} + {\lVert {\partial}_{\bar z} M(z,\eta) \rVert} & \lesssim \frac{1}{\eta^{2/3} + \eta^{2}}.
\label{eq:control_derivative_M} \end{aligned}$$
The assertion in (i) follows immediately from . Moreover, since $M(z,\eta)$ is continuously differentiable with respect to $z$ and $\bar z$, the bound implies (ii). In [@Altcirc Proposition 2.4(i)], it was shown that $\Delta_z {\langle {\mathrm{Im}\,}M(z,\eta) \rangle} = \Delta_z {\langle \mathbf v(z,\eta) \rangle} = 4 {\langle \tau {\partial}_\tau^2 \boldsymbol{v}^\tau + {\partial}_\tau \boldsymbol{v}^\tau \rangle}|_{\tau = {\lvert z \rvert}^2}$ (the last equality uses the notation of [@Altcirc]) is integrable in $\eta>0$ on $[0,\infty)$ for all $z \in {\mathbb{C}}$ with ${\lvert z \rvert} < 1$. Completely analogously, the integrability can be shown if ${\lvert z \rvert} > 1$. This shows that $\Delta_z$ and the $\eta$-integral can be interchanged which proves (iii).
From , it is easy to get ${\lVert M \rVert} \leq \eta^{-1}$ for all $z \in {\mathbb{C}}$ and $\eta >0$. As in the proof of Lemma \[lem:scaling\_relation\_v\_u\], we see that, uniformly for ${\lvert z \rvert} \geq 2$ or $\eta \geq 1$, we have $$\label{eq:scaling_relations_large_eta_z}
\rho \sim \frac{\eta}{{\lvert z \rvert}^2 + \eta^2}, \qquad v_1 \sim v_2 \sim \rho, \qquad u \sim \frac{1}{{\lvert z \rvert}^2 + \eta^2}, \qquad {\mathrm{Im}\,}U \sim \frac{\eta}{{\lvert z \rvert} + \eta } , \qquad Q \sim \frac{1}{{\lvert z \rvert}^{1/2} + \eta^{1/2}}.$$ In particular, ${\lVert M \rVert} \lesssim (1 + \eta)^{-1}$ and follows from multiplying by ${\mathrm{i}}\eta^{-1} M$ and using ${\lVert {\mathcal}{S} \rVert} \lesssim 1$.
We remark that follows from and as well as similarly to the proof of [@Altcirc Eq. (4.2)] if ${\lvert z \rvert} \leq 2$ and $\eta \leq 1$. Hence, we assume ${\lvert z \rvert} \geq 2$ or $\eta \geq 1$ in the remainder of the proof. In this regime, we obtain $1 - {\lVert {\mathcal{F}}\rVert_{2\to 2}} \sim 1$ by following the proof of and using . Therefore, , ${\lVert {\mathcal{S}}\rVert_{2\to\lVert\,\cdot\,\rVert}} \lesssim 1$ (cf. the proof of Lemma \[lem:small\_eigenvalues\]) and imply ${\lVert {\mathcal{B}}^{-1} \rVert} \lesssim 1$. We differentiate with respect to $z$ and, thus, obtain $${\lVert {\partial}_z M(z,\eta) \rVert} = {\bigg\lVert {\mathcal{B}}^{-1}\bigg[M \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} M\bigg] \bigg\rVert} \lesssim \eta^{-2}.$$ Together with a similar argument for ${\partial}_{\bar z} M$ this completes the proof of Lemma \[lem:derivatives\_M\_controlled\_by\_eta\].
Cubic equation associated to stability equation of the MDE {#sec:cubic_equation}
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In this section, we study specific perturbations to the Matrix Dyson equation (MDE). Throughout this section, $M$ is the solution to the unperturbed MDE, . We consider solutions $G$ of the perturbed MDE, , for some $D \in{\mathbb{C}}^{2n \times 2n}$ with the additional constraint ${\langle{E_-} \mspace{2mu}, {G}\rangle} = 0$, keeping in mind that in our application the resolvent $G = (H_z - {\mathrm{i}}\eta)^{-1}$ (see below) satisfies this constraint. Since $D$ is small, we need to study the stability of the MDE, , under a small perturbation. The linear stability operator of this perturbation problem is ${\mathcal{B}}=1-{\mathcal{C}}_M {\mathcal{S}}$ (see ). When $\rho$ is small, the inverse of ${\mathcal{B}}$ blows up, hence we need to expand to the next order, i.e. study the quadratic stability equation, . The following proposition describes the stability properties of in this regime. In fact, the difference $G-M$ is dominated by the contribution $\Theta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\langle{{\widehat}{B}} \mspace{2mu}, {G-M}\rangle}/{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}$ of $G-M$ in the unstable direction $B$ of ${\mathcal{B}}$ (cf. Proposition \[pro:stability\_operator\]). The scalar quantity $\Theta$ satisfies a cubic equation. In order to control $G-M$, we will control this cubic equation via a bootstrapping argument in $\eta$ in Section \[sec:local\_law\_H\] below.
In the following proposition and the rest of the paper, we use a special matrix norm to estimate the distance between $G$ and $M$. We denote this norm by ${\lVert {{\,\cdot\,}}\rVert_{*}}$. It is slightly modified compared of those defined in [@AltEdge; @Cusp1] in order to account for the additional unstable direction of ${\mathcal{B}}$. Such norms are tailored to local law proofs by the cumulant method and have first appeared in [@Erdos2017Correlated]. We need some auxiliary notations in order to define ${\lVert {{\,\cdot\,}}\rVert_{*}}$. For $a,b,c,d \in [2n]$, we set $\kappa_c(ab, cd) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\delta_{ad}\delta_{bc} s_{ab}$ and $\kappa_d(ab,cd) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\delta_{ac} \delta_{bd} t_{ab}$, where $s_{ab} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{E}}{\lvert w_{ab} \rvert}^2$ and $t_{ab} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{E}}w_{ab}^2$ with $W = (w_{ab})_{a,b \in [2n]}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}H_z - {\mathbb{E}}H_z$ (compare for the definition of $H_z$). Moreover, for a vector ${\mathbf x}=(x_a)_{a \in [2n]}$, we write $\kappa_c({\mathbf x}b, cd) = \sum_a x_a \kappa_c(ab,cd)$ for a weighted version of $\kappa_c$. We use an analogous convention for $\kappa_d$. If we replace an index of a scalar quantity by a dot ($\cdot$) then this denotes the corresponding vector, where the omitted index runs through $[2n]$, e.g. $R_{a \cdot}$ denotes the vector $(R_{ab})_{b \in [2n]}$. For an $2n\times 2n$ matrix $R$ and vectors ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$, we use the short-hand notation $R_{{\mathbf x}{\mathbf y}}$ to denote the quadratic form ${\langle{{\mathbf x}} \mspace{2mu}, {R{\mathbf y}}\rangle}$ and $R_{{\mathbf x}a}$ to denote ${\langle{{\mathbf x}} \mspace{2mu}, {R\mathbf e_a}\rangle}$, where $\mathbf e_a$ is the $a$-th normalized standard basis vector. With these conventions, we define some sets of testvectors. For fixed vectors ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$, we define $$\begin{aligned}
I_0 & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ {\mathbf x}, {\mathbf y}\} \cup \{ \mathbf e_a, ({\widehat}{B}^*)_{a \cdot}, (({\widehat}{B}_*)^*)_{a \cdot} \colon a \in [2n] \}, \\
I_{k+1} & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}I_k \cup \{ M {\mathbf u}\colon {\mathbf u}\in I_k \} \cup \{ \kappa_c((M{\mathbf u}) a, b \cdot ),\kappa_d( (M{\mathbf u})a, b \cdot) \colon {\mathbf u}\in I_k, a,b \in [2n] \} .
\end{aligned}$$ We now introduce the ${\lVert {{\,\cdot\,}}\rVert_{*}}$-norm defined by $${\lVert R \rVert_{*}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert R \rVert_{*}}^{K,{\mathbf x},{\mathbf y}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sum_{0 \leq k < K} (2n)^{-k/2K} {\lVert R \rVert}_{I_k}+ (2n)^{-1/2} \max_{{\mathbf u}\in I_K} \frac{{\lVert R_{\cdot {\mathbf u}} \rVert}}{{\lVert {\mathbf u}\rVert}}, \qquad {\lVert R \rVert}_I {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\max_{{\mathbf u},{\mathbf v}\in I} \frac{{\lvert R_{{\mathbf u}{\mathbf v}} \rvert}}{{\lVert {\mathbf u}\rVert}{\lVert {\mathbf v}\rVert}}.$$ We remark that the norm ${\lVert {{\,\cdot\,}}\rVert_{*}}$ depends on $\eta$ and $z$ via $M = M(z,\eta)$. However, this will not play any important role in our arguments.
In this section, the model parameters for the comparison relation $\lesssim$ are given by $s_*$ and $s^*$ from as well as $\tau$ from the upper bound on ${\lvert z \rvert}$.
\[pro:cubic\_equation\] There is $\rho_* \sim 1$ such that if $\rho + \eta/\rho \leq \rho_*$ for some fixed $z \in D_\tau(0)$ and $\eta \in (0,1]$ then the following holds. We fix $K \in {\mathbb{N}}$, ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$ and set ${\lVert {{\,\cdot\,}}\rVert_{*}} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert {{\,\cdot\,}}\rVert_{*}}^{K,{\mathbf x}, {\mathbf y}}$. If $G$ and $D$ satisfy , ${\langle{E_-} \mspace{2mu}, {G}\rangle} = 0$ and ${\lVert G-M \rVert_{*}} + {\lVert D \rVert_{*}} \lesssim n^{-30/K}$ then $$\label{eq:G_minus_M_expansion_via_Theta_and_D}
G - M = \Theta B - {\mathcal{B}}^{-1} {\mathcal{Q}}[MD] + \Theta^2 {\mathcal{B}}^{-1} {\mathcal{Q}}[M {\mathcal{S}}[B]B] + E,$$ where $\Theta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\langle{{\widehat}{B}} \mspace{2mu}, {G-M}\rangle}/{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}$ and the error matrix $E$ has the upper bound $$\label{eq:normstar_E}
{\lVert E \rVert_{*}} \lesssim {\lvert \Theta \rvert}^2 (\rho + \eta/\rho) + n^{16/K}\big({\lvert \Theta \rvert}^3 + {\lvert \Theta \rvert}({\lVert D \rVert_{*}} + \rho^2 + \eta/\rho) + {\lVert D \rVert_{*}}^2 + {\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \rvert} \big)$$ with $R_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}M^* ({\mathcal{B}}^{-1} {\mathcal{Q}})^*[E_-]$. The $2n\times 2n$-matrix $R_1$ is independent of $G$ and $D$ and satisfies ${\lVert R_1 \rVert} \lesssim 1$.
Moreover, $\Theta$ fulfils the approximate cubic equation $$\label{eq:cubic_equation}
\Theta^3 + \xi_2 \Theta^2 + \xi_1 \Theta = {\varepsilon}_*$$ whose coefficients $\xi_2$ and $\xi_1$ satisfy the scaling relations $$\label{eq:scaling_xi_2_xi_1}
{\lvert \xi_2 \rvert} \sim \rho, \qquad {\lvert \xi_1 \rvert} \sim \eta/\rho + \rho^2$$ and the error term ${\varepsilon}_*$ is bounded by $$\label{eq:bound_eps_star}
{\lvert {\varepsilon}_* \rvert} \lesssim n^{62/K} \big( {\lVert D \rVert_{*}}^3 + {\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \rvert}^{3/2} + {\lvert {\langle{R_2} \mspace{2mu}, {D}\rangle} \rvert}^{3/2} \big) + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {MD}\rangle} \rvert} + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {M ({\mathcal{S}}{\mathcal{B}}^{-1} {\mathcal{Q}}[MD])({\mathcal{B}}^{-1} {\mathcal{Q}}[MD])}\rangle} \rvert}.$$ Here, the matrix $R_2\in {\mathbb{C}}^{2n \times 2n}$ is independent of $G$ and $D$ and satisfies ${\lVert R_2 \rVert} \lesssim 1$.
We note that $R_2$ has an explicit definition (see below) but its exact form will not be important.
The proof follows from an application of Lemma \[lem:cubic\_equation\_abstract\] to with the choices ${\mathcal{A}}[R,T]{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{1}{2} M( {\mathcal{S}}[R] T + {\mathcal{S}}[T]R)$, ${\mathcal{B}}=1-{\mathcal{C}}_M {\mathcal{S}}$ as in , $Y = G - M$ and $X = MD$ in Lemma \[lem:cubic\_equation\_abstract\]. Note that ${\langle{E_-} \mspace{2mu}, {G-M}\rangle} = 0$ by assumption and . We first check the conditions of Lemma \[lem:cubic\_equation\_abstract\] in with ${\lVert {{\,\cdot\,}}\rVert} \equiv {\lVert {{\,\cdot\,}}\rVert_{*}}$ and $\lambda {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{1/2K}$. Partly, they will be a consequence of the bounds
\[eq:aux\_bounds\_star\_norm\] $$\begin{aligned}
&{\lVert M {\mathcal{S}}[R]T \rVert_{*}} \lesssim n^{1/2K} {\lVert R \rVert_{*}} {\lVert T \rVert_{*}}, \quad {\lVert MR \rVert_{*}} \lesssim n^{1/2K} {\lVert R \rVert_{*}}, \quad {\lVert {\mathcal{Q}}\rVert_{\ast \to \ast}} \lesssim 1, \quad {\lVert {\mathcal{B}}^{-1} {\mathcal{Q}}\rVert_{\ast \to \ast}} \lesssim 1, \\
&{\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {R}\rangle} \rvert} + {\lvert {\langle{{\widehat}{B}_*} \mspace{2mu}, {R}\rangle} \rvert} + {\lvert {\langle{E_-} \mspace{2mu}, {R}\rangle} \rvert} \lesssim {\lVert R \rVert_{*}}, \label{eq:bounds_star_norm_scalar_products} \end{aligned}$$
for all $R, T \in {\mathbb{C}}^{2n\times 2n}$. The proof of is very similar to the one of [@AltEdge Lemma 3.4]. In particular, the bound in follows exactly as the bound on ${\lVert {\mathcal{P}}[R] \rVert_{*}}$ in the proof of [@AltEdge Lemma 3.4]. We leave the details to the reader.
Owing to and , we have ${\lVert B \rVert_{*}} + {\lVert B_* \rVert_{*}}\lesssim {\lVert B \rVert} + {\lVert B_* \rVert} \lesssim 1$. The third, sixth and ninth term in are $\sim 1$ by Proposition \[pro:stability\_operator\]. This completes the proof of with $\lambda {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{1/2K}$.
Therefore, Lemma \[lem:cubic\_equation\_abstract\] with $\delta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{-81/4K}$, ${\lvert \Theta \rvert} \lesssim {\lVert G-M \rVert_{*}} \lesssim n^{-30/K}$ and ${\lVert M D \rVert_{*}} \lesssim n^{1/2K} {\lVert D \rVert_{*}}$ imply $$\label{eq:proof_cubic_stability_aux1}
\begin{aligned}
& \mu_3 \Theta^3 + \mu_2 \Theta^2 - \beta {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \Theta = -\mu_0 + {\langle{R_2} \mspace{2mu}, {D}\rangle} \Theta \\
& \quad \qquad + {\mathcal{O}}\big( n^{-1/4K} {\lvert \Theta \rvert}^3 + n^{62/K}( {\lVert D \rVert_{*}}^3 + {\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \rvert}^{3/2})
+ n^{20/K}{\lvert \Theta \rvert}^2( {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}^2 + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} \rvert}^2) \big),
\end{aligned}$$ where $\mu_3$, $\mu_2$ and $\mu_0$ are defined as in , $R_1 = M^* ({\mathcal{B}}^{-1} {\mathcal{Q}})^*[E_-]$ and we introduced $$\label{eq:def_R_2}
R_2 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}M^* ({\mathcal{B}}^{-1} {\mathcal{Q}})^* \bigg[ {\mathcal{S}}[B^*]M^* {\widehat}{B} + {\mathcal{S}}[ M^* {\widehat}{B} B^*] - \frac{{\langle{B} \mspace{2mu}, {E_-}\rangle}}{{\langle{B_*} \mspace{2mu}, {E_-}\rangle}} \Big( {\mathcal{S}}[(B_*)^*] M^* {\widehat}{B} + {\mathcal{S}}[ M^* {\widehat}{B} (B_*)^*] \Big) \bigg].$$ Note that $R_1$ and $R_2$ are independent of $G$ and $D$ and satisfy ${\lVert R_1 \rVert} \lesssim 1$ and ${\lVert R_2 \rVert} \lesssim 1$ due to Proposition \[pro:stability\_operator\] and .
The remaining task is expanding $\mu_3$, $\mu_2$, $-\beta{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}$ and ${\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle}$ on the right-hand side of with the help of Proposition \[pro:stability\_operator\]. The coefficient $-\beta{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}$ has already been identified in . For the others, we will rewrite the expansions in Proposition \[pro:stability\_operator\] in terms of $U$, $Q$ and $F_U=\rho^{-1} {\mathrm{Im}\,}U$ defined in and via $M = QUQ$ by . In particular, $$\label{eq:Im_M_in_terms_of_Q_U}
\hspace*{-0.2cm} {\mathrm{Im}\,}M = Q ({\mathrm{Im}\,}U) Q = \rho Q F_U Q, \qquad - {\mathrm{Im}\,}M^{-1} = Q^{-1} ({\mathrm{Im}\,}U) Q^{-1} = \rho Q^{-1} F_U Q^{-1}, \qquad {\mathrm{Re}\,}M = Q ({\mathrm{Re}\,}U) Q.$$ Note that $U$, $U^*$ and $F_U$ commute. Moreover, since $U$ is unitary (cf. Lemma \[lem:scaling\_relation\_v\_u\] (i)), the estimate implies $$\label{eq:Re_U_squared_equal_1}
({\mathrm{Re}\,}U)^2 = 1- ({\mathrm{Im}\,}U)^2 = 1 + {\mathcal{O}}(\rho^2).$$ We recall that $\psi$ defined in Proposition \[pro:stability\_operator\] satisfies $\psi = {\langle F_U^4 \rangle}$ (cf. the proof of Proposition \[pro:stability\_operator\]). In the following, we will frequently use that ${\langle R \rangle} = 0$ if $R \in {{\mathcal{M}_\mathrm{o}}}$.
We now compute the coefficients from . Indeed, we now show that
\[eq:expansion\_mu\] $$\begin{aligned}
\mu_3 & = \psi + {\mathcal{O}}(\rho + \eta/\rho) , \label{eq:mu_3} \\
\mu_2 & = 3 {\mathrm{i}}\rho \psi + {\mathcal{O}}(\rho^2 + \eta/\rho).\label{eq:mu_2} \end{aligned}$$
As a preparation of the proof of , we expand ${\mathcal{A}}[B,B]$. Proposition \[pro:stability\_operator\], and the definition ${\mathcal{F}}={\mathcal{C}}_Q {\mathcal{S}}{\mathcal{C}}_Q$ from yield $$\label{eq:cA_B_B}
\begin{aligned}
{\mathcal{A}}[B,B] = M{\mathcal{S}}[B]B & = Q U {\mathcal{F}}[ F_U + 2 {\mathrm{i}}\rho F_U^2 {\mathrm{Re}\,}U] (F_U + 2 {\mathrm{i}}\rho F_U^2 {\mathrm{Re}\,}U) Q + {\mathcal{O}}(\rho^2 + \eta/\rho)\\
& = Q U F_U ( F_U + 2 {\mathrm{i}}\rho F_U^2 {\mathrm{Re}\,}U) Q + {\mathcal{O}}(\rho^2 + \eta/\rho) \\
& = Q ( F_U^2 {\mathrm{Re}\,}U + 3 {\mathrm{i}}\rho F_U^3) Q + {\mathcal{O}}(\rho^2 + \eta/\rho).
\end{aligned}$$ Here, we used that ${\mathcal{F}}$ vanishes on ${{\mathcal{M}_\mathrm{o}}}$ and $F_U^2 {\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$ as well as in the second step and $U = {\mathrm{Re}\,}U + {\mathrm{i}}{\mathrm{Im}\,}U = {\mathrm{Re}\,}U + {\mathrm{i}}\rho F_U = {\mathrm{Re}\,}U + {\mathcal{O}}(\rho)$ by in the last step.
We recall the definitions ${\mathcal{L}}=1-{\mathcal{C}}_U {\mathcal{F}}$ and ${\mathcal{K}}=1-{\mathcal{C}}_P {\mathcal{F}}$ from . Since ${\mathcal{B}}^{-1} {\mathcal{Q}}= {\mathcal{C}}_Q {\mathcal{L}}^{-1} {\mathcal{Q}}_{\mathcal{L}}{\mathcal{C}}_Q^{-1} = {\mathcal{C}}_Q {\mathcal{K}}^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{C}}_Q^{-1} + {\mathcal{O}}(\rho)$ by Lemma \[lem:small\_eigenvalues\] and we deduce from that $$\label{eq:cB_inverse_Q_cA_B_B}
{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B, B] = {\mathcal{C}}_Q {\mathcal{K}}^{-1} {\mathcal{Q}_{\mathcal{K}}}[F_U^2{\mathrm{Re}\,}U] + {\mathcal{O}}(\rho + \eta/\rho) = {\mathcal{C}}_Q [F_U^2{\mathrm{Re}\,}U] + {\mathcal{O}}(\rho + \eta/\rho).$$ The last step follows since ${\mathcal{K}}^{-1} {\mathcal{Q}_{\mathcal{K}}}$ acts as the identity map on ${{\mathcal{M}_\mathrm{o}}}$ and $F_U^2 {\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$.
For the first term in the definition of $\mu_3$ of , we use Proposition \[pro:stability\_operator\], and to obtain $$\label{eq:first_term_mu3}
2{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]}\rangle} = {\langle F_U U{\mathcal{F}}[F_U] F_U^2 ({\mathrm{Re}\,}U) \rangle} + {\mathcal{O}}(\rho + \eta/\rho) = \psi + {\mathcal{O}}(\rho + \eta/\rho).$$ In the first step, we also employed that ${\mathcal{S}}$ vanishes on ${{\mathcal{M}_\mathrm{o}}}$ and ${\mathcal{Q}}[F_U^2 {\mathrm{Re}\,}U] \in {{\mathcal{M}_\mathrm{o}}}$. The second step is a consequence of , and $\psi = {\langle F_U^4 \rangle}$.
To estimate the second term in the definition of $\mu_3$, we now estimate ${\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle}$. Proposition \[pro:stability\_operator\] and imply $$\label{eq:scalar_L_cA_B_B_star}
\begin{aligned}
2 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} & = {\langle F_U^2 U {\mathcal{F}}[F_U] E_- \rangle} + {\langle U {\mathcal{F}}[E_-F_U] F_U^2 (1 + 2 {\mathrm{i}}\rho F_U {\mathrm{Re}\,}U) \rangle} + {\mathcal{O}}(\rho^2 + \eta/\rho) \\
& = {\langle F_U^3 {\mathrm{Re}\,}U E_- \rangle} + {\mathrm{i}}\rho {\langle F_U^4 E_- \rangle} - {\langle E_- {\mathrm{Re}\,}U F_U^3 \rangle} - 2 {\mathrm{i}}\rho {\langle E_- F_U^4({\mathrm{Re}\,}U)^2 \rangle} +{\mathcal{O}}(\rho^2 + \eta/\rho) \\
& = {\mathcal{O}}(\rho^2 + \eta/\rho).
\end{aligned}$$ Here, we used that ${\mathcal{F}}$ vanishes on $F_U^2 {\mathrm{Re}\,}U \in {{\mathcal{M}_\mathrm{o}}}$ in the first step. The second step follows from and . In the last step, after cancelling the first and third terms, we employed and ${\langle F_U^4 E_- \rangle} = 0$ by .
The expansion in and $E_- Q ({\mathrm{Re}\,}U) F_U^2 Q \in {{\mathcal{M}_\mathrm{o}}}$ imply $$\label{eq:scalar_Emin_cB_inverse_Q_cA_B_B}
{\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]}\rangle} = {\langle E_- Q ({\mathrm{Re}\,}U) F_U^2 Q \rangle} + {\mathcal{O}}(\rho +\eta/\rho) = {\mathcal{O}}(\rho + \eta/\rho).$$ From , , ${\lvert {\langle{E_-} \mspace{2mu}, {B_*}\rangle} \rvert} \sim 1$ by and we conclude that $\mu_3$ defined in satisfies .
We now turn to the expansion of $\mu_2$. From Proposition \[pro:stability\_operator\], and , we conclude $$\label{eq:expansion_mu_2}
\mu_2= {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B]}\rangle}
= {\langle F_U^3 {\mathrm{Re}\,}U + 3 {\mathrm{i}}\rho F_U^4 \rangle} + {\mathcal{O}}(\rho^2 + \eta/\rho)
= 3 {\mathrm{i}}\rho \psi + {\mathcal{O}}(\rho^2 + \eta/\rho).$$ Here, we used $F_U^3 ({\mathrm{Re}\,}U) \in {{\mathcal{M}_\mathrm{o}}}$ and $\psi = {\langle F_U^4 \rangle}$ in the last step. This completes the proof of .
We now continue to estimate the right-hand side of . Young’s inequality implies that ${\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \Theta \rvert} \leq n^{-1/4K}{\lvert \Theta \rvert}^3 + n^{1/8K} {\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \rvert}^{3/2}$. Then, we incorporate the error terms on the right-hand side of bounded by $n^{-1/4K} {\lvert \Theta \rvert}^3$ and introduce ${\widetilde}{\mu}_3$ such that ${\widetilde}{\mu}_3 \Theta^3 = \mu_3 \Theta^3 + {\mathcal{O}}(n^{-1/4K}{\lvert \Theta \rvert}^3)$. Hence, ${\lvert {\widetilde}{\mu}_3 \rvert} \sim 1$ by and $\psi \sim 1$ by Proposition \[pro:stability\_operator\]. After this rearrangement, we divide by ${\widetilde}{\mu}_3$ and set $$\xi_2 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\mu_2 / {\widetilde}{\mu}_3, \qquad
\xi_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big( - \beta{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} + {\mathcal{O}}(n^{20/K} {\lvert \Theta \rvert}( {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}^2 + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} \rvert}^2)) \big) /{\widetilde}{\mu}_3.$$ Since ${\lvert {\widetilde}{\mu}_3 \rvert} \sim 1$, we conclude ${\lvert \xi_2 \rvert} \sim {\lvert \mu_2 \rvert} \sim \rho$ due to and $\psi \sim 1$. For the scaling relation of $\xi_1$, we note that ${\lvert \beta {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert} \sim \rho^2 + \eta/\rho$ by and ${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert} \sim 1$ from Proposition \[pro:stability\_operator\]. Moreover, from and , we obtain $n^{20/K} {\lvert \Theta \rvert}( {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}^2 + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} \rvert}^2)) \lesssim n^{-10/K} ( \rho^4 + \eta^2/\rho^2)$. Hence, ${\lvert \xi_1 \rvert} \sim \rho^2 + \eta/\rho$. This completes the proof of , the scaling relations and the bound on ${\varepsilon}_*$ in .
Finally, the expansion of $G-M$ in and the error estimate in follow from in Lemma \[lem:cubic\_equation\_abstract\] together with , ${\lVert MD \rVert_{*}} \lesssim n^{1/2K} {\lVert D \rVert_{*}}$, and $R_1=M^* ({\mathcal{B}}^{-1} {\mathcal{Q}})^*[E_-]$. This completes the proof of Proposition \[pro:cubic\_equation\].
Local law for $H_z$ {#sec:local_law_H}
===================
The main result of this section, Theorem \[thm:local\_law\_H\], is a precise expansion of the resolvent of $H_z$ at ${\mathrm{i}}\eta$ when $\eta>0$ is sufficiently small and the modulus of $z\in {\mathbb{C}}$ is close to 1. We recall that we assume $\varrho({\mathscr{S}}) = 1$ (cf. and the associated explanations). For the formulation of Theorem \[thm:local\_law\_H\] as well as the subsequent statements and arguments, we use the following notion for high probability estimates.
\[def:stochastic\_domination\] Let $\Phi= (\Phi^{(n)})_n$ and $\Psi =(\Psi^{(n)})_n$ be two sequences of nonnegative random variables. We say $\Phi$ is *stochastically dominated* by $\Psi$ and write $\Phi \prec \Psi$ if, for all (small) ${\varepsilon}>0$ and (large) $D>0$, there is a constant $C_{{\varepsilon},D}>0$ such that $$\label{eq:def_stochastic_domination}
{\mathbb{P}}\Big( \Phi^{(n)} > n^{\varepsilon}\Psi^{(n)} \Big) \leq \frac{ C_{{\varepsilon},D}}{n^D}$$ for all $n \in {\mathbb{N}}$. If $\Phi^{(n)}$ and $\Psi^{(n)}$ depend on some parameter family $U^{(n)}$ and holds for all $u \in U^{(n)}$ then we say that $\Phi \prec \Psi$ uniformly for all $u \in U^{(n)}$.
In the following, let $\rho = {\langle {\mathrm{Im}\,}M \rangle}/\pi$ (cf. ), $H_z$ be defined as in and $G {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(H_z - {\mathrm{i}}\eta)^{-1}$. Moreover, $M$ denotes the solution of the MDE, . For each $z\in {\mathbb{C}}$, we now introduce the *fluctuation scale* ${\eta_{\mathrm{f}}}={\eta_{\mathrm{f}}}(z)$ of eigenvalues of $H_z$ around zero: We set $$\label{eq:def_etaf}
{\eta_{\mathrm{f}}}(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\begin{cases} (1-{\lvert z \rvert}^2)^{-1/2} n^{-1}, & \text{ if } {\lvert z \rvert}^2 \leq 1 - n^{-1/2}, \\
n^{-3/4}, & \text{ if } 1 - n^{-1/2} < {\lvert z \rvert}^2 \leq 1 + n^{-1/2}, \\
({\lvert z \rvert}^2 - 1)^{1/6} n^{-2/3}, & \text{ if } 1 + n^{-1/2} < {\lvert z \rvert}^2 \leq 2,\\
n^{-2/3}, & \text{ if } {\lvert z \rvert}^2 > 2.
\end{cases}$$ The fluctuation scale describes the typical eigenvalue spacing of $H_z$ at zero (first two cases) and at the spectral edges of the eigenvalue density of $H_z$ close to zero (last two cases). The definition of ${\eta_{\mathrm{f}}}$ in is motivated by the definition of the fluctuation scale in [@Cusp1] and the scaling relations of $\rho$ from . For ${\lvert z \rvert} > 1$, the eigenvalue density of $H_z$ has a gap of size $\Delta \sim ({\lvert z \rvert}^2-1)^{3/2}$ around zero, hence, is analogous to [@Cusp1 Eq. (2.7)]. If ${\lvert z \rvert} \leq 1$ then the eigenvalue density of $H_z$ has a small local minimum of height $\rho_0 \sim (1-{\lvert z \rvert}^2)^{1/2}$ at zero. So should be compared to [@Cusp1 Eq. (A.8a)].
\[thm:local\_law\_H\] Let $X$ satisfy \[assum:flatness\] and \[assum:bounded\_moments\]. Then there is $\tau_* \sim 1$ such that, for each $\zeta>0$, the estimates $$\label{eq:G_minus_M_standard}
{\lvert {\langle{{\mathbf x}} \mspace{2mu}, {(G- M){\mathbf y}}\rangle} \rvert} \prec {\lVert {\mathbf x}\rVert}{\lVert {\mathbf y}\rVert}\bigg(\sqrt{\frac{\rho}{n\eta}} + \frac{1}{n\eta} \bigg), \qquad \qquad
{\lvert {\langle R(G-M) \rangle} \rvert} \prec \frac{{\lVert R \rVert}}{n \eta}$$ hold uniformly for all $z \in {\mathbb{C}}$ satisfying ${\lvert {\lvert z \rvert} - 1 \rvert} \leq \tau_*$, for all $\eta \in [n^{\zeta}{\eta_{\mathrm{f}}}(z),n^{100}]$, for any deterministic vectors ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$ and deterministic matrix $R \in {\mathbb{C}}^{2n\times 2n}$.
Moreover, outside the spectrum, for each $\zeta>0$ and $\gamma>0$, we have the improved bound $$\label{eq:G_minus_M_average_improved}
{\lvert {\langle R (G-M) \rangle} \rvert} \prec {\lVert R \rVert} \frac{n^{-\gamma/3}}{n\eta}$$ uniformly for all $z \in {\mathbb{C}}$ and $\eta \in {\mathbb{R}}$ satisfying ${\lvert z \rvert}^2 \geq 1 + (n^\gamma \eta)^{2/3}$, $ {\lvert z \rvert} \leq 1 + \tau_*$, $n^{\zeta} {\eta_{\mathrm{f}}}(z) \leq \eta \leq \tau_*$ and $R \in {\mathbb{C}}^{2n\times 2n}$.
We stress that the spectral parameter of the resolvent $G$ in the previous theorem and throughout the entire paper lies on the imaginary axis and is given by ${\mathrm{i}}\eta$. With additional efforts our method can be extended to spectral parameters near the imaginary axis, but the Hermitization formula requires to understand the resolvent of $H_z$ only on the imaginary axis, so we restrict ourselves to this case. After the proof of Theorem \[thm:local\_law\_H\], we will establish the following corollary that will directly imply Corollary \[thm:delocalization\].
\[cor:delocalization\] Let $\tau_* \sim 1$ be chosen as in Theorem \[thm:local\_law\_H\]. Let ${\mathbf x}\in {\mathbb{C}}^{2n}$ be a fixed deterministic vector. If ${\mathbf u}\in {\mathbb{C}}^{2n}$ is contained in the kernel of $H_z$ for some $z \in {\mathbb{C}}$ satisfying ${\lvert {\lvert z \rvert} - 1 \rvert} \leq \tau_*$, i.e. $H_z {\mathbf u}= 0$ then $${\lvert {\langle{{\mathbf x}} \mspace{2mu}, {{\mathbf u}}\rangle} \rvert} \prec n^{-1/2} {\lVert {\mathbf x}\rVert} {\lVert {\mathbf u}\rVert}.$$
We remark that the conclusion of Corollary \[cor:delocalization\] is also true if ${\lvert {\lvert z \rvert} - 1 \rvert} >\tau_*$. This can easily be shown following the proof of [@Altcirc Theorem 5.2], where certain steps of the proof of the local law from [@AjankiCorrelated] have been used except that now analogous inputs from [@Erdos2017Correlated] are needed instead of [@AjankiCorrelated].
Let $u\in {\mathbb{C}}^n$ be an eigenvector of $X$, i.e. $X u = \zeta u$ for some $\zeta \in {\mathbb{C}}$. If ${\lvert \zeta \rvert} \leq 1 - \tau_*$ then the claim follows from [@Altcirc Corollary 2.6]. Otherwise, we can assume that ${\lvert {\lvert \zeta \rvert} - 1 \rvert} \leq \tau_*$ by [@Altcirc Theorem 2.5 (ii)]. We set ${\mathbf u}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}(0,u)^t$ and obtain $H_\zeta {\mathbf u}= 0$. Hence, we choose ${\mathbf x}= \mathbf{e}_i$ in Corollary \[cor:delocalization\] and obtain a bound on ${\lvert u_i \rvert}$. Finally, taking the maximum over $i \in [n]$ and a simple union bound complete the proof of Corollary \[thm:delocalization\].
\[rem:convention\_for\_S\] We warn the reader that in this paper the definition of the self-energy operator ${\mathcal{S}}$ given in coincides with the definition in [@Altcirc Eq. (3.3)]. However, this convention differs from the more canonical choice, $R \mapsto {\mathbb{E}}[(H- {\mathbb{E}}H)R(H- {\mathbb{E}}H)]$, typically used for a Hermitian random matrix $H$ in several other works (e.g. [@AjankiCorrelated; @Erdos2017Correlated; @AltEdge]). The present choice of ${\mathcal{S}}$ substantially simplifies the analysis of the associated MDE. As a price for this, we will need a simple adjustment when estimating the error term $D$ in the perturbed Dyson equation, , since the convenient estimate on $D$ builds upon the canonical choice of ${\mathcal{S}}$. As Proposition \[pro:D\_bounds\] below shows, nevertheless the very same estimates on $D$ as for the canonical choice [@Cusp1] can be obtained for the current choice.
We will establish Theorem \[thm:local\_law\_H\] in Subsection \[subsec:proof\_local\_law\_H\] below. The proof will consist of a bootstrapping argument using the stability properties of the MDE in the previous sections, in particular, Proposition \[pro:cubic\_equation\], and the following bounds on the error term $D$ in the perturbed MDE for $G$, . To formulate these bounds, we now introduce some norms for random matrices and a spectral domain. For $p \geq 1$, a scalar-valued random variable $Z$ and a random matrices $Y\in {\mathbb{C}}^{2n\times 2n}$, we define the $p$th-moment norms $${\lVert Z \rVert}_p {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big( {\mathbb{E}}{\lvert Z \rvert}^p \big)^{1/p}, \qquad {\lVert Y \rVert}_p {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sup_{{\mathbf x},{\mathbf y}} \frac{{\lVert {\langle{{\mathbf x}} \mspace{2mu}, {Y{\mathbf y}}\rangle} \rVert}_p}{{\lVert {\mathbf x}\rVert}{\lVert {\mathbf y}\rVert}}.$$ For $\zeta>0$, we introduce the spectral domain $${\mathbb{D}}_\zeta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ (z,\eta) \in {\mathbb{C}}\times {\mathbb{R}}\colon n^{-1+ \zeta}\leq \eta \leq \tau_*, \, {\lvert {\lvert z \rvert} - 1 \rvert} \leq \tau_* \},$$ where $\tau_*\sim 1$ is chosen such that implies $\rho + \eta/\rho\leq \rho_*$ for all $(z,\eta) \in {\mathbb{D}}_\zeta$ with $\rho_*$ from Proposition \[pro:cubic\_equation\].
\[pro:D\_bounds\] Under the assumptions of Theorem \[thm:local\_law\_H\], there is a constant $C>0$ such that for any $p \geq 1$, ${\varepsilon}>0$, $(z,\eta) \in {\mathbb{D}}_0$, and any deterministic ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$ and $R \in {\mathbb{C}}^{2n \times 2n}$, we have the moment bounds
$$\begin{aligned}
{\lVert {\langle{{\mathbf x}} \mspace{2mu}, {D{\mathbf y}}\rangle} \rVert}_p & \leq_{{\varepsilon},p} {\lVert {\mathbf x}\rVert} {\lVert {\mathbf y}\rVert} N^{\varepsilon}\psi_q' \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{N}} \bigg)^{Cp}, \label{eq:D_bound_isotropic} \\
{\lVert {\langle RD \rangle} \rVert}_p & \leq_{{\varepsilon},p} {\lVert R \rVert} N^{\varepsilon}\Big[\psi_q'\Big]^2 \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{N}} \bigg)^{Cp}. \label{eq:D_bound_average} \end{aligned}$$
Moreover, if $R \in {{\mathcal{M}_\mathrm{o}}}$ then we have the improved estimate $$\label{eq:D_bound_cusp_FA}
{\lVert {\langle R D \rangle} \rVert}_p \leq_{{\varepsilon},p} {\lVert R \rVert} N^{\varepsilon}\sigma_q \Big[\psi + \psi_q'\Big]^2 \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{N}} \bigg)^{Cp}.$$
Here, we used the $z$-dependent control parameters $$\psi {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt{\frac{\rho}{n\eta}}, \quad \psi_q' {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\sqrt{\frac{{\lVert {\mathrm{Im}\,}G \rVert}_q}{n\eta}}, \quad \psi_q'' {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\lVert G-M \rVert}_q, \quad \sigma_q {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\rho + \psi + \sqrt{\frac{\eta}{\rho}} + \psi_q' + \psi_q''$$ with $q {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}Cp^3/{\varepsilon}$.
\[rem:sigma\] This proposition is the exact counterpart of the cusp fluctuation averaging in [@Cusp1 Proposition 4.12] with $\sigma=0$, hence the definition of $\sigma_q$ does not contain $\sigma$. Notice that $\sigma=0$ in our case following from the fact that the spectral parameter $i\eta$ lies on the imaginary axis to which the spectrum is symmetric.
We remark that $\psi$ in Proposition \[pro:D\_bounds\] is different from the $\psi$ defined in Proposition \[pro:stability\_operator\]. This should not lead to any confusion since the latter notation is used only in Section \[sec:stability\] and \[sec:cubic\_equation\] while the former is used exclusively in Proposition \[pro:D\_bounds\] and Section \[subsec:proof\_D\_bounds\]. We prefer to stick to these notations for compatibility with the publications [@Ajankirandommatrix; @Altshape; @Cusp1]. We will show Proposition \[pro:D\_bounds\] in Subsection \[subsec:proof\_D\_bounds\] below.
Proof of Theorem \[thm:local\_law\_H\] {#subsec:proof_local_law_H}
--------------------------------------
This subsection is devoted to the proof of Theorem \[thm:local\_law\_H\]. To that end, we follow the arguments from [@Cusp1 Sections 3.2, 3.3], where the local law for a general Hermitian matrix close to a cusp regime was deduced from estimates on $D$ as provided in Proposition \[pro:D\_bounds\]. We will present the main steps of the proof, focusing on the differences, but for arguments that require only simple (mostly notational) adjustments we will refer the reader to [@Cusp1]. When comparing to [@Cusp1], the reader should think of the following cases described in the notation of [@Cusp1 Eq. (3.7b)]. For ${\lvert z \rvert} \leq 1$, the eigenvalue density of $H_z$ has a local minimum of size $\rho(\tau_0) \sim (1-{\lvert z \rvert}^2)^{1/2}$ at $\tau_0 = 0$ and $\omega = 0$. For ${\lvert z \rvert} > 1$, the spectrum of $H_z$ has a symmetric gap of size $\Delta
\sim ({\lvert z \rvert}^2-1)^{3/2}$ around zero and we study the resolvent of $H_z$ at the middle of this gap, ${\lvert \omega \rvert} = \Delta/2$. For a random matrix $Y \in {\mathbb{C}}^{2n\times 2n}$ and a deterministic control parameter $\Lambda = \Lambda(z)$, we define the notations ${\lvert Y \rvert} \prec \Lambda$ and ${\lvert Y \rvert}_\mathrm{av} \prec \Lambda$ as follows $$\begin{aligned}
{4}
{\lvert Y \rvert} & \prec \Lambda \qquad &\Longleftrightarrow&& \qquad {\lvert Y_{{\mathbf x}{\mathbf y}} \rvert} &\prec \Lambda {\lVert {\mathbf x}\rVert}{\lVert {\mathbf y}\rVert} \quad &&\text{ uniformly for all } {\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}, \\
{\lvert Y \rvert}_\mathrm{av} & \prec \Lambda \qquad & \Longleftrightarrow &&\qquad {\lvert {\langle RY \rangle} \rvert} &\prec \Lambda {\lVert R \rVert} \quad &&\text{ uniformly for all } R \in {\mathbb{C}}^{2n\times 2n}. \end{aligned}$$ We recall that by definition $Y_{{\mathbf x}{\mathbf y}}={\langle{{\mathbf x}} \mspace{2mu}, {Y {\mathbf y}}\rangle}$ for ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$. The following lemma relates this notion of high probability bounds to the high moments estimates introduced above. We leave the simple adjustments of the proof of [@AltEdge Lemma 3.7] to the reader.
\[lem:high\_moment\_bound\_to\_high\_prob\_bound\] Let $Y$ be a random matrix in ${\mathbb{C}}^{2n\times 2n}$, $\Phi$ a deterministic control parameter such that $\Phi\geq n^{-C}$ and ${\lVert Y \rVert} \leq n^C$ for some $C >0$. Let $K \in {\mathbb{N}}$ be fixed. Then we have $${\lVert Y \rVert_{*}}^{K,{\mathbf x},{\mathbf y}} \prec \Phi \text{ uniformly for } {\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n} \quad \Longleftrightarrow \quad {\lvert Y \rvert} \prec \Phi \quad \Longleftrightarrow \quad {\lVert Y \rVert}_p \leq_{{\varepsilon},p} n^{\varepsilon}\Phi \text{ for all } {\varepsilon}>0, p \geq 1.$$
The next lemma adapts Proposition \[pro:cubic\_equation\] to the random matrix setup with the help of Proposition \[pro:D\_bounds\]. The lemma is the analogue of [@Cusp1 Lemma 3.8] in our setup.
\[lem:cubic\_inequality\_refined\] Let $\zeta, c>0$ be fixed and sufficiently small. We assume that ${\lvert G-M \rvert} \prec \Lambda$, ${\lvert {\mathrm{Im}\,}(G-M) \rvert}\prec \Xi$ and ${\lvert \Theta \rvert} \prec \theta$ at some fixed $(z,\eta) \in{\mathbb{D}}_{\zeta}$ for some deterministic control parameters $\Lambda$, $\Xi$ and $\theta$ such that $\Lambda+ \Xi + \theta \lesssim n^{-c}$. Then, for any sufficiently small ${\varepsilon}>0$, the estimates $$\label{eq:cubic_equation_refined}
{\lvert \Theta^3 + \xi_2 \Theta^2 + \xi_1 \Theta \rvert} \prec n^{2{\varepsilon}} \bigg( \rho + \frac{\eta^{1/2}}{\rho^{1/2}} + \bigg( \frac{\rho + \Xi}{n\eta} \bigg)^{1/2} \bigg) \frac{\rho + \Xi}{n\eta} + n^{-{\varepsilon}} \theta^3$$ and $$\label{eq:G-M_theta}
{\lvert G-M \rvert} \prec \theta + \sqrt{\frac{\rho + \Xi}{n \eta}}, \qquad \qquad {\lvert G-M \rvert}_\mathrm{av} \prec \theta + \frac{\rho + \Xi}{n\eta}$$ hold, where $\xi_2$ and $\xi_1$ are chosen as in Proposition \[pro:cubic\_equation\] and $\Theta = {\langle{{\widehat}{B}} \mspace{2mu}, {G-M}\rangle}/{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}$.
Moreover, for fixed $z$, $\Theta$ is a continuous function of $\eta$ as long as $(z,\eta) \in {\mathbb{D}}_\zeta$.
Owing to Lemma \[lem:high\_moment\_bound\_to\_high\_prob\_bound\] and ${\lvert G \rvert} \prec {\lVert M \rVert} + \Lambda \lesssim 1$, the high-moment bounds in and imply $$\label{eq:aux_bounds_D}
{\lvert D \rvert} \prec \sqrt{\frac{\rho + \Xi}{n \eta}}, \qquad \qquad {\lvert D \rvert}_\mathrm{av} \prec \frac{\rho + \Xi}{n\eta}.$$ We conclude that the assumption on ${\lVert G-M \rVert_{*}} + {\lVert D \rVert_{*}}$ in Proposition \[pro:cubic\_equation\] is satisfied for sufficiently large $K$ depending on $c$ and ${\varepsilon}$ in the definition of $\prec$ in Definition \[def:stochastic\_domination\]. What remains to ensure the applicability of Proposition \[pro:cubic\_equation\] is checking ${\langle{E_-} \mspace{2mu}, {G}\rangle} = 0$. In fact, we now prove that, for each $z \in {\mathbb{C}}$ and $\eta >0$, the resolvent $G = (H_z - {\mathrm{i}}\eta)^{-1}$ satisfies $$\label{eq:scalar_E_minus_G_equals_0}
{\langle{E_-} \mspace{2mu}, {G}\rangle} = 0.$$ For the proof of , we denote by $G_{11}, G_{22} \in {\mathbb{C}}^{n\times n}$ the upper-left and lower-right $n\times n$-minor of the resolvent $G = (H_z- {\mathrm{i}}\eta)^{-1} \in {\mathbb{C}}^{2n\times 2n}$. Then the block structure of $H_z$ from yields $$G_{11} = \frac{{\mathrm{i}}\eta }{(X-z)(X-z)^* + \eta^2}, \qquad G_{22} = \frac{{\mathrm{i}}\eta}{(X-z)^* (X-z) + \eta^2} .$$ Since $(X-z)(X-z)^*$ and $(X-z)^*(X-z)$ have the same eigenvalues we obtain $(2n) {\langle{E_-} \mspace{2mu}, {G}\rangle} = \operatorname{Tr}G_{11} - \operatorname{Tr}G_{22} = 0$. This shows and, thus, ensures the applicability of Proposition \[pro:cubic\_equation\].
The first bound in , the bounds on ${\mathcal{B}}^{-1} {\mathcal{Q}}$ and $MR$ in and Lemma \[lem:high\_moment\_bound\_to\_high\_prob\_bound\] yield $$\label{eq:proof_cubic_refined_aux4}
{\lvert {\mathcal{B}}^{-1} {\mathcal{Q}}[MD] \rvert}\prec \frac{\rho + \Xi}{n \eta}$$ by choosing $K$ sufficiently large to absorb various $n^{1/K}$-factors into $\prec$. Similarly, we use , , the assumption ${\lvert \Theta \rvert} \prec \theta$ and Lemma \[lem:high\_moment\_bound\_to\_high\_prob\_bound\] to estimate the other terms in and and deduce .
What remains is estimating the right-hand side of to obtain . Incorporating the $n^{1/K}$ factors into $\prec$, we see that ${\lVert D \rVert}_*^3$, ${\lvert {\langle{R_1} \mspace{2mu}, {D}\rangle} \rvert}^{3/2}$ and ${\lvert {\langle{R_2} \mspace{2mu}, {D}\rangle} \rvert}^{3/2}$ are dominated by the right-hand side of due to and Lemma \[lem:high\_moment\_bound\_to\_high\_prob\_bound\]. Recall $M = {\mathrm{Re}\,}M + {\mathcal{O}}(\rho)$ with ${\mathrm{Re}\,}M \in {{\mathcal{M}_\mathrm{o}}}$ and ${\widehat}{B}^* = -\rho^{-1} {\mathrm{Im}\,}M^{-1} + {\mathcal{O}}(\rho + \eta/\rho)$ (cf. ) with $-\rho^{-1} {\mathrm{Im}\,}M^{-1} \sim 1$ and $-\rho^{-1} {\mathrm{Im}\,}M^{-1} \in {{\mathcal{M}_\mathrm{d}}}$. Therefore, $M$ is almost off-diagonal while ${\widehat}{B}^*$ is almost diagonal, so we find $B_1 \in {{\mathcal{M}_\mathrm{o}}}$, $B_2 \in {\mathbb{C}}^{2n\times 2n}$ such that ${\widehat}{B}^*M = B_1 + B_2$ and ${\lVert B_1 \rVert} \lesssim 1$, ${\lVert B_2 \rVert} \lesssim \rho + \eta/\rho$. Hence, and imply $${\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {MD}\rangle} \rvert} \prec \bigg( \rho + \frac{\eta^{1/2}}{\rho^{1/2}} + \theta + \bigg( \frac{\rho + \Xi}{n \eta} \bigg)^{1/2} \bigg) \frac{ \rho + \Xi}{n \eta} \prec n^{\varepsilon}\bigg( \rho + \frac{\eta^{1/2}}{\rho^{1/2}}
+ \bigg( \frac{\rho + \Xi}{n \eta} \bigg)^{1/2} \bigg) \frac{ \rho + \Xi}{n \eta} + n^{-{\varepsilon}} \theta^3,$$ where we used the bound on ${\lvert G-M \rvert}$ from in the second step and Young’s inequality in the last step.
We now conclude the proof of by showing that $$\label{eq:proof_cubic_refined_aux2}
{\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {M ({\mathcal{S}}{\mathcal{B}}^{-1} {\mathcal{Q}}[MD]){\mathcal{B}}^{-1} {\mathcal{Q}}[MD]}\rangle} \rvert} \prec \bigg( \frac{\rho + \Xi}{n\eta} \bigg)^{3/2}.$$ Since $\operatorname{ran}{\mathcal{S}}\subset {{\mathcal{M}_\mathrm{d}}}$ and ${\lvert ({\widehat}{B}^* M)_{aa} \rvert} \leq {\lVert {\widehat}{B}^* M \rVert} \lesssim 1$, we have $$\label{eq:proof_cubic_refined_aux3}
{\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {M ({\mathcal{S}}{\mathcal{B}}^{-1} {\mathcal{Q}}[MD]){\mathcal{B}}^{-1} {\mathcal{Q}}[MD]}\rangle} \rvert} = \frac{1}{2n} {\bigg\lvert \sum_{a \in [2n]} ({\widehat}{B}^* M)_{aa} {\mathcal{S}}[A]_{aa} A_{aa} \bigg\rvert} \lesssim \max_{a \in [2n]} {\lvert {\mathcal{S}}[A]_{aa} \rvert} \max_{a \in [2n]} {\lvert A_{aa} \rvert},$$ where $A {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{B}}^{-1}{\mathcal{Q}}[MD]$. Owing to the second bound in , the definition of ${\lvert {{\,\cdot\,}}\rvert}_\mathrm{av}$ and ${\lVert M^* ({\mathcal{B}}^*)^{-1} {\mathcal{Q}}^* {\mathcal{S}}[{{\mathbf e}_a}{\mathbf e}_a^*] \rVert} \lesssim 1$, we obtain $$\label{eq:proof_cubic_refined_aux1}
{\lvert {\mathcal{S}}[A]_{aa} \rvert} = {\lvert {\langle{{\mathbf e}_a} \mspace{2mu}, {{\mathcal{S}}[A] {\mathbf e}_a}\rangle} \rvert} = {\lvert {\langle{{\mathcal{S}}[{\mathbf e}_a{\mathbf e}_a^*]} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[MD]}\rangle} \rvert} = {\lvert {\langle{M^* ({\mathcal{B}}^*)^{-1} {\mathcal{Q}}^*{\mathcal{S}}[{{\mathbf e}_a}{\mathbf e}_a^*]} \mspace{2mu}, {D}\rangle} \rvert} \prec \frac{\rho + \Xi}{n\eta}.$$ Therefore, follows by using and in . This completes the proof of .
Finally, we note that $\Theta$ is a continuous function of $\eta$ as ${\widehat}{B}$, $B$, $G$ and $M$ are continuous with respect to $\eta$. This completes the proof of Lemma \[lem:cubic\_inequality\_refined\].
We now introduce ${\widetilde}{\xi}_2$ and ${\widetilde}{\xi}_1$ which will turn out to be comparable versions of the coefficients $\xi_2$ and $\xi_1$, respectively, (see above and Lemma \[lem:inequality\_coefficients\] \[item:xi\_and\_tilde\_xi\] below). Moreover, they depend explicitly and monotonically on $\eta$ which will be important for our arguments. We define $$\label{eq:def_wt_xi_2_1}
{\widetilde}{\xi}_2 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\big\lvert 1-{\lvert z \rvert}^2 \big\rvert}^{1/2} + \eta^{1/3}, \qquad {\widetilde}{\xi}_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big({\widetilde}{\xi}_2\big)^2.$$ These definitions are chosen in analogy to [@Cusp1 Eq. (3.7e)], where in the first case we chose ${\lvert \omega \rvert} \sim \Delta \sim ({\lvert z \rvert}^2 - 1)^{3/2}$ and, in the second case, $\rho(\tau_0) \sim (1- {\lvert z \rvert}^2)^{1/2}$ and $\omega = 0$.
\[lem:inequality\_coefficients\]
1. \[item:xi\_and\_tilde\_xi\] For all $z \in D_\tau(0)$ and $\eta \in (0,1]$, we have $\rho^2 + \eta/\rho \sim {\widetilde}{\xi}_1$. For any $\eta \in (0,1]$, we have ${\widetilde}{\xi}_2 \sim \rho$ if $z \in {\mathbb{C}}$ satisfies ${\lvert z \rvert} \leq 1$ and ${\widetilde}{\xi}_2 \gtrsim \rho$ if $z \in D_\tau(0)\setminus \overline{D_1(0)}$.
2. \[item:comparsion\_tilde\_xi\_rho\] Uniformly for all $z \in D_\tau(0)$ and $\eta \geq {\eta_{\mathrm{f}}}$, we have $${\widetilde}{\xi}_2 \gtrsim \frac{1}{n\eta} + \bigg( \frac{\rho}{n \eta} \bigg)^{1/2}, \qquad {\widetilde}{\xi}_1 \gtrsim {\widetilde}{\xi}_2 \bigg( \rho + \frac{1}{n \eta} \bigg).$$
The scaling relations in \[item:xi\_and\_tilde\_xi\] follow easily from the scaling relations for $\rho$ in by distinguishing the regimes ${\lvert z \rvert} \leq 1$ and ${\lvert z \rvert} >1$.
The first bound in \[item:comparsion\_tilde\_xi\_rho\] follows once ${\widetilde}{\xi}_2 \gtrsim 1/(n\eta)$ and $({\widetilde}{\xi}_2)^2 \gtrsim \rho/(n\eta)$ are proven. For ${\lvert z \rvert}^2 \leq 1 - n^{-1/2}$, we have $(1-{\lvert z \rvert}^2)^{1/2} \gtrsim 1/(n\eta)$ if $\eta \geq {\eta_{\mathrm{f}}}(z)$. If ${\lvert z \rvert}^2 > 1 - n^{-1/2}$ then ${\eta_{\mathrm{f}}}(z) \geq n^{-3/4}$ by definition and, hence, $\eta^{1/3} \gtrsim 1/(n\eta)$. This shows ${\widetilde}{\xi}_2 \gtrsim 1/(n\eta)$ in all regimes. If ${\lvert z \rvert} \leq 1$ then ${\widetilde}{\xi}_2 \sim \rho$ by \[item:xi\_and\_tilde\_xi\] and $\rho^2 \gtrsim \rho/(n\eta)$ is easily verified due to . For ${\lvert z \rvert}>1$, $({\widetilde}{\xi}_2)^2 \gtrsim \rho/(n\eta)$ is equivalent to $({\lvert z \rvert}^2- 1)^{1/2} + \eta^{1/3} \gtrsim n^{-1/4}$ which follows directly from ${\eta_{\mathrm{f}}}(z) \gtrsim n^{-3/4}$ in this regime. This shows the first bound in \[item:comparsion\_tilde\_xi\_rho\].
We note that, owing to ${\widetilde}{\xi}_1 = ({\widetilde}{\xi}_2)^2$, the second bound in \[item:comparsion\_tilde\_xi\_rho\] is equivalent to ${\widetilde}{\xi}_2 \gtrsim \rho + 1/(n \eta)$. But we know ${\widetilde}{\xi}_2 \gtrsim \rho$ from \[item:xi\_and\_tilde\_xi\]. This completes the proof of Lemma \[lem:inequality\_coefficients\].
We will only consider the bounds in for $\eta \leq \tau_*$ since the opposite regime is covered by [@Erdos2017Correlated Theorem 2.1] due to $\rho \sim \eta^{-1}$ for $\eta \geq \tau_*$ by and [@Altcirc Eq. (3.9)]. The bounds and are the analogues of (3.28) and (3.30) in [@Cusp1], respectively. Given the preparations presented above, the proofs of and , thus, the one of Theorem \[thm:local\_law\_H\], are identical to the proofs of [@Cusp1 Eq.’s (3.28) and (3.30)] in [@Cusp1 Section 3.3]. Therefore, we only describe the main strategy here and explain the applicability of certain inputs.
The proof of Theorem \[thm:local\_law\_H\] starts with the following (isotropic) rough bound on $G-M$.
\[lem:rough\_bound\] For any $\zeta >0$, there exists a constant $c>0$ such that the rough bounds $${\lvert G-M \rvert} \prec n^{-c}$$ holds on the spectral domain ${\mathbb{D}}_\zeta$.
The proof of Lemma \[lem:rough\_bound\] is identical to the one of [@Cusp1 Lemma 3.9]. We explain the main idea. From [@Erdos2017Correlated Theorem 2.1], an initial bound on ${\lvert G-M \rvert}$ at $\eta = \tau_*$ is deduced. We remark that [@Erdos2017Correlated Theorem 2.1] is also applicable in our setup. Using the monotonicity of the map $\eta \mapsto \eta {\lVert G(z,\eta) \rVert}_p$ (which is shown in the same way as in [@Erdos2017Correlated Eq. (5.11)]), the boostrapping result [@Cusp1 Lemma 3.10] for cubic inequalities and Lemma \[lem:cubic\_inequality\_refined\], this initial bound is strengthened and propagated down to all (small) values of $\eta$ in ${\mathbb{D}}_\zeta$. Moreover, the assumptions in (ii) of [@Cusp1 Lemma 3.10] are easily checked by using the definitions of ${\widetilde}{\xi}_2$ and ${\widetilde}{\xi}_1$ in and Lemma \[lem:inequality\_coefficients\] \[item:xi\_and\_tilde\_xi\]. This completes the proof of Lemma \[lem:rough\_bound\].
As in the proof of [@Cusp1 Theorem 2.5], we deduce the following proposition by employing Lemma \[lem:cubic\_inequality\_refined\] instead of [@Cusp1 Lemma 3.8].
\[pro:local\_law\_all\_scales\] Let $\zeta>0$. On ${\mathbb{D}}_\zeta$, we have the bounds $$\label{eq:local_law_n_minus_1}
{\lvert G-M \rvert} \prec \theta_* + \sqrt{\frac{\rho}{n \eta}} + \frac{1}{n\eta}, \qquad {\lvert G-M \rvert}_\mathrm{av} \prec \theta_* + \frac{\rho}{n \eta} + \frac{1}{(n\eta)^2},$$ where $\theta_*$ is defined through $$\theta_* {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\min \{ d_*^{1/3}, d_*^{1/2}/{\widetilde}{\xi}_2^{1/2}, d_*/{\widetilde}{\xi}_1 \}, \qquad d_* {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\widetilde}{\xi}_2 \bigg( \frac{{\widetilde}{\rho}}{n \eta} + \frac{1}{(n \eta)^2} \bigg) + \frac{1}{(n \eta)^3} + \bigg( \frac{{\widetilde}{\rho}}{n\eta}
\bigg)^{3/2}$$ and ${\widetilde}{\rho}$ denotes the right-hand side of , i.e. $\rho \sim {\widetilde}{\rho}$.
Now, we follow the proof of [@Cusp1 Eq. (3.28)] and use Lemma \[lem:inequality\_coefficients\] \[item:comparsion\_tilde\_xi\_rho\] instead of [@Cusp1 Lemma 3.3] to obtain both bounds in .
We now strengthen to outside of the spectrum. If ${\lvert z \rvert}^2 > 1 + (n^\gamma \eta)^{2/3}$ then $\theta_* \leq d_*/{\widetilde}{\xi}_1$, Lemma \[lem:inequality\_coefficients\] \[item:comparsion\_tilde\_xi\_rho\], $({\widetilde}{\xi}_2)^2 = {\widetilde}{\xi}_1$ and imply $$\label{eq:proof_local_law_H_aux1}
\theta_* + \frac{{\widetilde}{\rho}}{n \eta} + \frac{1}{(n\eta)^2} \lesssim \frac{{\widetilde}{\xi}_2}{{\widetilde}{\xi}_1} \bigg( \frac{\rho}{n\eta} + \frac{1}{(n\eta)^2} \bigg)
\lesssim \frac{1}{({\lvert z \rvert}^2 - 1)^{1/2}} \bigg( \frac{\eta}{{\lvert z \rvert}^2 - 1} + \frac{1}{n \eta} \bigg)\frac{1}{n\eta} \lesssim \frac{n^{-\gamma/3}}{n\eta}$$ for $\eta \geq {\eta_{\mathrm{f}}}(z) \gtrsim n^{-3/4}$ (cf. [@Cusp1 Eq. (3.30)]). Applying to the second bound in yields the improved bound and, thus, completes the proof of Theorem \[thm:local\_law\_H\].
From Proposition \[pro:local\_law\_all\_scales\], we now conclude Corollary \[cor:delocalization\].
The bound on ${\lvert G-M \rvert}$ in and the argument from [@Ajankirandommatrix Corollary 1.14] directly imply Corollary \[cor:delocalization\].
We conclude this subsection by collecting two simple consequences of the previous results and [@Altcirc]. For the remainder of Section \[subsec:proof\_local\_law\_H\], $\tau>0$ will be a parameter bounding the spectral parameter $z$ from above. The implicit constant in $\prec$-estimates is allowed to depending on $\tau$.
\[cor:im\_local\_law\_everywhere\] Let $X$ satisfy \[assum:flatness\] and \[assum:bounded\_moments\]. Let $\zeta>0$. Then we have $${\lvert {\mathrm{Im}\,}{\langle G(z,\eta)-M(z,\eta) \rangle} \rvert} \prec \frac{1}{n\eta}$$ uniformly for all $z \in D_{\tau}(0)$ and all $ \eta \in [n^\zeta {\eta_{\mathrm{f}}}(z), n^{100}]$.
The corollary is a direct consequence of the local law near the edge, , the local law away from the edge, [@Altcirc Eq. (5.4)], and the definition of ${\eta_{\mathrm{f}}}$ in .
We denote the eigenvalues of $H_z$ by $\lambda_1(z), \ldots, \lambda_{2n}(z)$. The following lemma provides a simple bound on the number of eigenvalues of $H_z$ in the interval $[-\eta,\eta]$. It is an extension of [@Altcirc Eq. (5.22)] to the edge regime.
\[lem:number\_eigenvalues\] Let $X$ satisfy \[assum:flatness\] and \[assum:bounded\_moments\]. Let $\zeta>0$. Then we have $${\lvert \{ i \in [2n] \colon {\lvert \lambda_i(z) \rvert} \leq \eta \} \rvert} \prec n \eta \rho + 1$$ uniformly for all $z \in D_\tau(0)$ and for all $\eta \in [n^\zeta{\eta_{\mathrm{f}}}(z),n^{100}]$.
We define $\Lambda_\eta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\{ i \in [2n] \colon {\lvert \lambda_i(z) \rvert} \leq \eta \}$. For $\eta \geq n^{\zeta}{\eta_{\mathrm{f}}}$, we obtain from Corollary \[cor:im\_local\_law\_everywhere\] and $\rho = {\langle {\mathrm{Im}\,}M \rangle}/\rho$ that $$\frac{{\lvert \Lambda_\eta \rvert}}{2\eta} \leq \sum_{i \in \Lambda_\eta} \frac{\eta}{\eta^2 + \lambda_i(z)^2} \leq 2n {\mathrm{Im}\,}{\langle G(z,\eta) \rangle} \prec n \bigg(\rho + \frac{1}{n \eta} \bigg) \lesssim n \rho + \frac{1}{\eta}.$$ This completes the proof of Lemma \[lem:number\_eigenvalues\].
Cusp fluctuation averaging – Proof of Proposition \[pro:D\_bounds\] {#subsec:proof_D_bounds}
-------------------------------------------------------------------
In this subsection we will provide the proof of Proposition \[pro:D\_bounds\]. Since the self-consistent density of states of the Hermitian matrix $H=H_z$ develops a cusp singularity at the origin in the regime ${\lvert {\lvert z \rvert}-1 \rvert}\ll 1$, this result is analogous to [@Cusp1 Theorem 3.7], which provides an improved bound for specific averages of the random error matrix in the MDE. This improved bound is called *cusp fluctuation averaging* and takes the form in the current work. In [@Cusp1] the expectation ${\mathbb{E}}H$ was diagonal and the self-energy operator was assumed to satisfy the flatness condition [@Cusp1 Eq. (3.6)]. Both conditions are violated in our current setup and thus the result from [@Cusp1] is not directly applicable. However, with minor modifications the proof of [@Cusp1 Theorem 3.7] can be adjusted to yield Proposition \[pro:D\_bounds\]. In fact, the cancellation that underlies the cusp fluctuation averaging is simpler and more robust for $H$ with the bipartite structure . An indication of this fact is that holds for any $R \in {{\mathcal{M}_\mathrm{o}}}$ while the corresponding bound in [@Cusp1 Theorem 3.7] only holds when the error matrix is averaged against a specific vector that depends on $M$, the solution to the MDE.
For the purpose of being able to follow the strategy from [@Cusp1] very closely we define $$\label{eq:quantities from Cusp1}
W{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}H_z-{\mathbb{E}}H_z\,, \qquad {\widetilde}{{\mathcal}{S}}[R]{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathbb{E}}W RW\,, \qquad {\widetilde}{D} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}WG+ {\widetilde}{{\mathcal}{S}}[G]G\,.$$ The modified self-energy operator ${\widetilde}{{\mathcal}{S}}$ is introduced to match the convention of [@Cusp1] (cf. Remark \[rem:convention\_for\_S\]). This differs from the self-energy operator ${\mathcal}{S}$ defined in this paper (see and Remark \[rem:convention\_for\_S\]), which is the block diagonal part of ${\widetilde}{{\mathcal}S}$, consisting of the blocks $ {\mathbb{E}}X R_{22} X^*$ and $ {\mathbb{E}}X^* R_{11} X$, both themselves being diagonal since $X$ has independent entries. The difference between the two versions of the self-energy is $$\label{def of cal T}
{\mathcal}{T}[ R ] {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\widetilde}{{\mathcal}{S}}-{\mathcal}{S} )[R]=
\left(
\begin{array}{cc}
0 & {\mathbb{E}}X R_{21} X
\\
{\mathbb{E}}X^* R_{12}X^*&0
\end{array}
\right)
= \left(
\begin{array}{cc}
0 & {\mathscr{T}}\odot R_{21}^t
\\
{{\mathscr{T}}}^* \odot R_{12}^t &0
\end{array}
\right)
= T \odot R^t
\,,$$ where $\odot$ indicates the entrywise (Hadamard) matrix product and we introduced the matrices ${\mathscr{T}}=({\mathbb{E}}x_{ij}^2)_{i,j=1}^n \in {\mathbb{C}}^{n\times n}$ and $T = (t_{ij})_{i,j=1}^{2n}\in{\mathbb{C}}^{2n\times 2n}$ with entries $t_{ij} = {\mathbb{E}}w_{ij}^2$ being the second moments of the entries of $W =(w_{ij})_{i,j=1}^{2n}$ from . The modified error matrix ${\widetilde}{D}$ was labelled $D$ in [@Cusp1] and is the natural error when considering the MDE with self-energy ${\widetilde}{{\mathcal}{S}}$ and corresponding solution ${\widetilde}{M}$. In the current work we will stick to the convention from [@Altcirc] with respect to the definition of ${\mathcal}{S},D,M$ in order to keep the MDE and its solution simple and thus we indicate the corresponding quantities ${\widetilde}{{\mathcal}{S}},{\widetilde}{D},{\widetilde}{M}$ from [@Cusp1] by a tilde. Another notational difference is that the dimension of $H$ was denoted by $N$ in [@Cusp1], that corresponds to $N=2n$ in this paper.
We start the proof of Proposition \[pro:D\_bounds\] by showing that it suffices to establish its statement for $D$ replaced by ${\widetilde}{D}$. Let us therefore assume the following proposition whose proof is the main content of this subsection.
\[pro:wtD\_bounds\] The statement of Proposition \[pro:D\_bounds\] holds with $D$ replaced by ${\widetilde}{D}$, i.e. under the same assumptions and with the same constants we have the estimates
$$\begin{aligned}
{\lVert {\langle{{\mathbf x}} \mspace{2mu}, { {\widetilde}{D}{\mathbf y}}\rangle} \rVert}_p & \leq_{{\varepsilon},p} {\lVert {\mathbf x}\rVert} {\lVert {\mathbf y}\rVert} n^{\varepsilon}\psi_q' \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{n}} \bigg)^{Cp}, \label{eq:wtD_bound_isotropic} \\
{\lVert {\langle R {\widetilde}{D} \rangle} \rVert}_p & \leq_{{\varepsilon},p} {\lVert R \rVert} n^{\varepsilon}\Big[\psi_q'\Big]^2 \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{n}} \bigg)^{Cp},\label{eq:wtD_bound_average} \end{aligned}$$
and for $R \in {{\mathcal{M}_\mathrm{o}}}$ the improved estimate $$\label{eq:wtD_bound_cusp_FA}
{\lVert {\langle R {\widetilde}{D} \rangle} \rVert}_p \leq_{{\varepsilon},p} {\lVert R \rVert} n^{\varepsilon}\sigma_q \Big[\psi + \psi_q'\Big]^2 \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{n}} \bigg)^{Cp}.$$ Furthermore, with ${\mathcal}{T}$ from and for an arbitrary deterministic matrix $R$, $$\label{eq:cal T GG}
{\lVert {\langle R{\mathcal}{T}[G]G \rangle} \rVert}_p \leq_{{\varepsilon},p} {\lVert R \rVert} n^{\varepsilon}\sigma_q \Big[\psi + \psi_q'\Big]^2 \Big( 1 + {\lVert G \rVert}_q \Big)^C \bigg( 1 + \frac{{\lVert G \rVert}_q}{\sqrt{n}} \bigg)^{Cp}.$$
Given the bounds from Proposition \[pro:wtD\_bounds\] it suffices to estimate the difference ${\widetilde}{D}-D = {\mathcal}{T}[G]G$. First follows from because for normalized vectors ${\mathbf x}, {\mathbf y}\in {\mathbb{C}}^{2n}$ we have $${\lVert {\langle{{\mathbf x}} \mspace{2mu}, {{\mathcal}{T}[G]G {\mathbf y}}\rangle} \rVert}_p = {\big\lVert {\textstyle \sum_i} G_{i {\mathbf v}_i }G_{i{\mathbf y}} \big\rVert}_p
\lesssim n^{\varepsilon}{\lVert G \rVert}_{1/{\varepsilon}}{\biggl({\frac{{\lVert \operatorname{Im}G \rVert}_p}{n \eta}}\biggr)}^{1/2}\le n^{\varepsilon}{\lVert G \rVert}_{q}\psi_q',$$ where in the equality we introduced the vectors ${\mathbf v}_i = (t_{ji}{\overline{x} \!\,}_j)_j $ with ${\lVert {\mathbf v}_i \rVert}_\infty \lesssim \frac{1}{n} $ and in the first inequality we used the Ward identity in the second factor after applying the general bound ${\lVert \sum_i X_i Y_i \rVert}_p \le n^{\varepsilon}\sup_i{\lVert X_i \rVert}_{1/{\varepsilon}}{\lVert \sum_i {\lvert Y_i \rvert} \rVert}_{2p}$ for any random variables $(X_i,Y_i)_{i=1}^{2n}$ and ${\varepsilon}\in (0,1/2p)$. Then implies by taking the ${\lVert {\mspace{2 mu}}\cdot{\mspace{2 mu}}\rVert}_p$-norm on both sides of $${\lvert {\langle R{\mathcal}{T}[G]G \rangle} \rvert} \lesssim \frac{{\lVert R \rVert}}{n} {\langle G^*G \rangle} = {\lVert R \rVert}\frac{{\langle \operatorname{Im}G \rangle}}{n\eta} \,,$$ where we used ${\lVert {\mathcal}{T} \rVert_{2\to 2}}\lesssim \frac{1}{n}$. Finally, immediately follows from and .
The remainder of this subsection is dedicated to proving Proposition \[pro:wtD\_bounds\]. To avoid repetition we will only point out the necessary modifications to the proof of [@Cusp1 Theorem 3.7].
The proof of and is exactly the same as the proof of [@Cusp1 Eq.’s (3.11a) and (3.11b)], which, in turn directly follow from [@Erdos2017Correlated Theorem 4.1]. Note that this latter theorem does not assume flatness, i.e. lower bound on ${\widetilde}{{\mathcal{S}}}$, hence it is directly applicable to our $H$ as well. We also remark that the proof of and requires only double index graphs (in the sense of [@Erdos2017Correlated]) and their estimates rely only on the power counting of Wardable edges. A self-contained summary of the necessary concepts can be found in [@Cusp1 Section 4.1-4.7], where the quite involved cumulant expansion from [@Erdos2017Correlated], originally designed to handle any correlation, is translated into the much simpler independent setup. This summary in [@Cusp1] has the advantage that it also introduces the single index graphs as a preparation for the more involved $\sigma$-cell estimates needed for the cusp fluctuation averaging.
In the rest of the proof we focus on and and we assume that the reader is familiar with [@Cusp1 Section 4], but no familiarity with [@Erdos2017Correlated] is assumed. We will exclusively work with single index graphs as defined in [@Cusp1 Section 4.2]. In the rest of this section we use $N=2n$ for easier comparison with [@Cusp1].
We start with the proof of . We write $R \in {{\mathcal{M}_\mathrm{o}}}$ as $R = J\operatorname{diag}({\mathbf r})$ for some ${\mathbf r}\in {\mathbb{R}}^{2n}$, where we can without loss of generality assume that $R$ has real entries and the matrix $J$ that exchanges ${{\mathcal{M}_\mathrm{o}}}$ and ${{\mathcal{M}_\mathrm{d}}}$ is $$J {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\left(
\begin{array}{cc}
0 & 1
\\
1& 0
\end{array}
\right)\,.$$ With this notation the left-hand side of takes the form ${\langle \operatorname{diag}({\mathbf r}) {\widetilde}{D}J \rangle}= {\langle \operatorname{diag}({\mathbf r})(W + {\widetilde}{{\mathcal}{S}}[G])GJ \rangle}$. This form exactly matches the left-hand side of [@Cusp1 Eq. (3.11c)] with ${\mathbf r}= \mathbf{pf}$, except that the last factor inside the trace is $GJ$ instead of just $G$. To accommodate this change we slightly extend the set of single index graphs $\Gamma \in {\mathcal}{G}$ defined in [@Cusp1 Section 4.2] by allowing two additional types of $G$-edges in ${\mathrm}{GE}={\mathrm}{GE}(\Gamma)$. We call the original $G$-edges from [@Cusp1], that encode the entries of $G$ and $G^*$, *straight $G$-edges* and add new *twisted $G$-edges* that represent the entries of $GJ$ and $(GJ)^* = JG^*$, respectively. Graphically $(GJ)_{ab}$ will be encoded by a solid directed line from vertex $a$ to vertex $b$ and with a superscript $J$ on the line. Similarly, $(GJ)^*_{ab}$ is a dashed line from $a$ to $b$ with a superscript $J$. Hence, the new twisted $G$-edges are represented by $$GJ = \ssGraph{a --[g,J] b;}\, , \qquad (GJ)^* = \ssGraph{a --[s,J] b;}\,.$$ The terminology $G$-edge will refer to all four types of edges. In particular, all of them are taken into account for the $G$-edge degree of vertices and when determining whether a subset ${\mathrm}{GE}_W \subset {\mathrm}{GE}$ is classified as Wardable (cf. [@Cusp1 Definition 4.6]). The latter is justified since the Ward estimates (cf. [@Cusp1 Eq. (4.12b)]) [ $$\label{Ward estimates} \begin{split}
\sum_a {\lvert (GJ)_{ab} \rvert}^2 = \frac{\big(J(\operatorname{Im}G)J\big)_{bb}}{\eta}\lesssim N \psi^2\,, \quad \sum_b {\lvert (GJ)_{ab} \rvert}^2 = \frac{\operatorname{Im}G_{aa}}{\eta}\lesssim N \psi^2
\,, \quad \psi= \bigg(\frac{\rho}{N \eta}\bigg)^{1/2}
\end{split}$$ ]{} are valid for twisted $G$-edges as well. As in [@Cusp1] the inequalites are meant in a high moment sense. Similarly, the derivatives of the twisted edges follow the same rules as the derivatives of the untwisted edges with respect to the matrix elements of $W$, for example $$\frac{\partial}{\partial w_{ab}} (GJ) = G \Delta^{ab} (GJ), \qquad \mbox{with}\quad (\Delta^{ab})_{ij} = \delta_{ia}\delta_{jb},$$ i.e., simply one of the resulting two $G$-edges remains twisted. In particular, the number of twisted edges remains unchanged.
According to the single index graph expansion (cf. [@Cusp1 Eq. (4.8)]) the $p$-th moment of ${\langle R {\widetilde}{D} \rangle}$ can now be written as a sum over the values ${\mathrm}{Val}(\Gamma)$ associated to the graphs $\Gamma$ within the subset ${\mathcal}{G}(p) \subset {\mathcal}{G}$ of single index graphs. This subset originates from the single index resolution (cf. [@Cusp1 Definition 4.2]) of double index graphs, i.e. $$\label{eq:graph expansion}
{\mathbb{E}}{\lvert {\langle \operatorname{diag}({\mathbf r}) {\widetilde}{D}J \rangle} \rvert}^p = N^{-p} \sum_{\Gamma \in {\mathcal}{G}(p)}{\mathrm}{Val}(\Gamma)+ {\mathcal{O}}(N^{-p})\,.$$ The twisted $G$-edges enter into the graphs ${\mathcal}{G}(p)$ through the following simple modification (iv)’ of [@Cusp1 (iv) from Definition 4.2] that originates from the fact that a wiggled $G$-edge in double index graphs is now associated to the matrix $GJ\operatorname{diag}({\mathbf r})$ and its adjoint instead of $G\operatorname{diag}({\mathbf r})$ with ${\mathbf r}= \mathbf{pf}$ as in [@Cusp1]:
- If a wiggled $G$-edge is mapped to an edge $e$ from $u$ to $v$, then $v$ is equipped with a weight of ${\mathbf r}$ and $e$ is twisted. If a wiggled $G^*$-edge is mapped to an edge ${\widetilde}{e}$ from $u$ to $v$, then $u$ is equipped with weight ${\mathbf r}$ and ${\widetilde}{e}$ is twisted. All vertices with no weight specified in this way are equipped with constant weight $\mathbf 1$.
The above changes reveal a one-to-one correspondence between the set of graphs ${\mathcal}{G}(p)$ in [@Cusp1] and its modification in the current work. This correspondence shows that the single index graph expansion is entirely unaffected by the presence of the off-diagonal matrix $J$ apart from replacing each weight $\mathbf{pf}$ in ${\mathcal}{G}(p)$ from [@Cusp1] by a weight ${\mathbf r}$ and replacing $p$ straight $G$-edges by twisted ones. More precisely, if in a graph from [@Cusp1] a vertex $v$ had a weight $(\mathbf{pf})^k$, then in its corresponding graph the vertex $v$ is adjacent to exactly $k_1$ twisted $G$-edges that end at $v$ and $k_2$ twisted $G^*$-edges that start from $v$ such that $k_1 +k_2=k$.
Since the graphs contained in the sets ${\mathcal}{G}$ and ${\mathcal}{G}(p)$ do not differ between the current work and [@Cusp1] once the distinction between straight and twisted edges is dropped and the exact form of the weight ${\mathbf r}$ is irrelevant, any result from [@Cusp1 Section 4] that is insensitive to these distinctions can be directly applied here. In particular, [@Cusp1 Lemmas 4.7, 4.8 and 4.11] remain valid.
The most relevant difference between our setup and [@Cusp1] concerns the specific mechanism behind the cusp fluctuation averaging. This mechanism is revealed by exploiting a local cancellation within the graphs ${\mathcal}{G}$ appearing along the expansion that is associated to specific edges, called $\sigma$-cells (cf. [@Cusp1 Definition 4.10]). For the following discussion we recall the definition of $\sigma$-cells and rephrase it so that it fits our setup.
A $\sigma$-cell is an interaction edge $e=(a,b)$ inside a graph $\Gamma \in {\mathcal}{G}$ such that there are exactly two $G$-edges adjacent to each endpoint of $e$, loops are not allowed, and to precisely one of the endpoints (say $a$) an additional weight ${\mathbf r}$ is attached. In a typical $\sigma$-cell there are four $G$-edges that connect external vertices $x,y,u,v$ with $(a,b)$ and the adjacent $G$-edges encode the expression $$\label{eq:sigma cell}
{\mathbb{E}}{\mspace{2 mu}}\sum r_a ({\widetilde}{G}J)_{xa}G^{(1)}_{ay}K_{ba}G^{(2)}_{ub}G^{(3)}_{vb}f_{xyuv}\,.$$ Here $f_{xyuv}$ represents the rest of the graph and is independent of $a,b$; the sum runs over all vertex indices and $K_{ba}$ is either ${\mathbb{E}}{\lvert w_{ba} \rvert}^2$, ${\mathbb{E}}w_{ba}^2$ or ${\mathbb{E}}{w}_{ab}^2$. Furthermore, ${\widetilde}{G} \in \{G, {\overline{G} \!\,}\}$ and $G^{(i)} \in \{G,G^t,G^*,{\overline{G} \!\,}\}$ (here $G^t$ and ${\overline{G} \!\,}$ just denote a $G$- or a $G^*$-edge with opposite orientation to save us from writing out all possibilities in ). Some $G$’s in may coincide giving rise to two additional options for a $\sigma$-cell, the first one with two external indices, and the second one with no external index: $$\label{eq:sigma cell1}
{\mathbb{E}}{\mspace{2 mu}}\sum r_a ({\widetilde}{G}J)_{xa}G^{(1)}_{ab}K_{ba}G^{(2)}_{ub}f_{xu}\,, \qquad \mbox{and}\qquad
{\mathbb{E}}{\mspace{2 mu}}\sum r_a ({\widetilde}{G}J)_{xa}G^{(1)}_{ab}K_{ba}f\,.$$
The graphical representation of these three types of $\sigma$-cells is the same as drawn in [@Cusp1 Definition 4.10] with weight ${\mathbf r}=\mathbf{pf}$, except that one $G$-edge adjacent to $a$ is twisted. For example, the $\sigma$-cell with four external indices is represented by $$\ssGraph{ a[label=$a$,lbr] --[K] b[label=$b$,r1]; x[label=below:$x$,o] --[J] a -- y[label=$y$,o]; v[label=below:$v$,o] -- b --u[label=$u$,o]; },$$ where the solid lines are $G$-edges, exactly one of them twisted (indicated by $J$), and without indicating their orientation. The interaction edge $K$ is depicted by the dotted line, while the weights ${\mathbf r}$ and $\mathbf{1}$ attached to vertices are indicated by arrows pointing to these vertices. The weight $\mathbf{1}$ could be ignored, it plays no specific role in the current paper; we drew it only for consistency with the picture in [@Cusp1] where it was essential that exactly one edge of the $\sigma$-cell receives a specific weight. The graphical picture of the other two types of $\sigma$-cells are analogous.
Exactly as in [@Cusp1] the cusp fluctuation mechanism will allow us to gain a factor $\sigma_q$ as defined in Proposition \[pro:D\_bounds\] for every $\sigma$-cell inside each graph. In [@Cusp1] this gain is stated as [@Cusp1 Proposition 4.12]. The statement of this proposition remains valid in our current setup without any changes except for the modified definition of $\sigma_q$ (cf. Remark \[rem:sigma\]). Up to the proof of [@Cusp1 Proposition 4.12] we have now verified all ingredients in the proof of [@Cusp1 Theorem 3.7] and thus its analog Proposition \[pro:wtD\_bounds\]. Therefore, we will finish this subsection by pointing out the necessary modifications to the proof of [@Cusp1 Proposition 4.12].\
*Modification of the proof of [@Cusp1 Proposition 4.12]:* The proof of this proposition in [@Cusp1] has two ingredients. The first is an explicit computation that involves the projections on stable and unstable directions of the stability operator $B$ of the MDE (cf. [@Cusp1 Eq. (4.28)]). This computation is extremely delicate and involves the precise choice for $K_{ba}$, $r_a$, $G^{(i)}$ and their relation in the $\sigma$-cell . Its outcome is that up to a sufficiently small error term it is possible to act with the stability operator on any vertex $a$ of the $\sigma$-cell. This action of $B$ on $a$ leads to an improvement of the bound on the corresponding graph that is stated as [@Cusp1 Lemma 4.13].
In our current setup the gain $\sigma_q$ for every $\sigma$-cell inside a graph $\Gamma$ is much more robust than in [@Cusp1], it is basically a consequence of the almost off-diagonality of $M$. There is no need to act with the stability operator and also the specific weights attached to the vertices of the $\sigma$-cells are not important. Instead, the value of any graph containing a $\sigma$-cell can be estimated directly by $\sigma_q$ times the sum of values of graphs with one sigma cell locally resolved (removed). For this gain the concrete choice of $K_{ab}$, $r_a$ and $G^{(i)}$ does not matter as long as ${\lvert K_{ba} \rvert} \lesssim \frac{1}{N}$ and $r_a \lesssim 1$. Furthermore, we will not make use of resolvents $G^{(2)}$ and $G^{(3)}$ in the corresponding calculations. Thus in the following, instead of , we will only consider the simplified expression $$\label{halfcell}
{\mathbb{E}}\sum ({\widetilde}{G}J K^{(b)} G^{(1)})_{xy}f_{xyb} ={\mathbb{E}}\sum_a ({\widetilde}{G}J)_{xa} k^{(b)}_a G^{(1)}_{ay}f_{xyb}\, ,$$ where $K^{(b)} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\operatorname{diag}({\mathbf k}^{(b)})$ is a diagonal matrix whose diagonal ${\mathbf k}^{(b)}$ has components $k^{(b)}_a{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}r_a K_{ba}$, and $f = f_{xyb}$ encodes the rest of the graph. This is exactly the reference graph $\Gamma$ at the beginning of the proof of [@Cusp1 Lemma 4.13], but now the left edge is twisted and a weight ${\mathbf k}^{(b)}$ is attached to the vertex $a$. With the choice ${\widetilde}{G}=G$, $G^{(1)} = G^*$ we have $$\Gamma {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\ssGraph{x[label=below:$x$,o] --[g,J] a[label=below:$a$,tbk]; a -- [s] y[label=below:$y$,o];},$$ which corresponds to the case $GJK^{(b)}G^*$ in . Since for all possible choices ${\widetilde}{G} \in \{G, {\overline{G} \!\,}\}$ and $G^{(1)} \in \{G,G^t,G^*,{\overline{G} \!\,}\}$ the discussion is analogous, we will restrict ourselves to the case $GJK^{(b)}G^*$.
In complete analogy to [@Cusp1 Eq. (4.22)], but using simply the identity operator instead of the stability operator $B$ and inserting the identity $$G=M -G{\mathcal}{S}[M]M-GWM$$ for the resolvent factor on the left into , we find by that $$\label{eq:sigma cell expansion}
{\mathbb{E}}(GJK^{(b)}G^*)_{xy}f = {\mathbb{E}}\Big( {\bigl[{1+G{\mathcal}{S}[G-M]-G({\widetilde}{{\mathcal}{S}}[G]+W)+G{\mathcal}{T}[G-M]+G{\mathcal}{T}[M]}\bigr]}(MJK^{(b)}G^*)\Big)_{xy} f\,.$$ Notice that the twisted $G$-edge corresponding to $GJ$ disappeared and $J$ now appears only together with $M$ in the form $MJK^{(b)}$.
We estimate the five summands inside the square brackets of . This means to show that their power counting estimate (defined as W-Est in [@Cusp1 Lemma 4.8]) is smaller than the W-Est of the left-hand side of , $\mbox{W-Est}(\Gamma)$, at least by a factor $\sigma_q$, i.e. we have [ $$\label{graph expansion} \begin{split}
{\mathrm}{Val}(\Gamma) = \sum_{\Gamma' \in {\mathcal}{G}_\Gamma}{\mathrm}{Val}(\Gamma') + {\mathcal{O}}(N^{-p}) \quad \text{with} \quad \mbox{W-Est}(\Gamma')\le_p \sigma_q\mbox{W-Est}(\Gamma)\,,
\end{split}$$ ]{} where all graphs $\Gamma' \in {\mathcal}{G}_\Gamma$ have one $\sigma$-cell less than $\Gamma$. Note that in contrast to [@Cusp1 Lemma 4.13] no insertion of the stability operator $B$ is needed for to hold and that in contrast to [@Cusp1 Proposition 4.12] the additional graph $\Gamma_\sigma$ is absent from the right hand side. In this sense combines these two statements from [@Cusp1] in a simplified fashion.
The first, second and fourth summands in correspond to the graphs treated in parts (a), (b) and (c) inside the proof of [@Cusp1 Lemma 4.13] and their estimates follow completely analogously. We illustrate this with the simplest first and the more complex fourth term. The first term gives $ {\mathbb{E}}(MJK^{(b)}G^*)_{xy} f$, which would exactly be case (a) in the proof of [@Cusp1 Lemma 4.13] if $MJK^{(b)}$ were diagonal. However, even if it has an offdiagonal part (in the sense of ${{\mathcal{M}_\mathrm{o}}}$), the same bound holds, i.e. we still have $$\label{aterm}
\mbox{W-Est}\Big( \sum_a (MJK^{(b)})_{xa} G^*_{ay} f\Big) \le \frac{1}{N\psi^2} \mbox{W-Est}\Big( \sum_a (GJK^{(b)} G^*)_{xy}f \Big) =
\sigma_q \mbox{W-Est}(\Gamma)\,,$$ where $1/N$ comes from the fact that the summation over $a$ collapses to two values, $a=x$ and $a=\hat x$ and $\psi^2$ accounts for the two Wardable edges in $\Gamma$. Here we defined $\hat x{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}x+n \mbox{(mod $2n$)}$ to be the complementary index of $x$. In order to see more systematically, set ${\mathbf m}_d {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\operatorname{diag}(M)$ and ${\mathbf m}_o{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\operatorname{diag}(MJ)$ to be the vectors representing the diagonal and offdiagonal parts of $M$. Then $$\label{MJ}
MJK^{(b)} = \operatorname{diag}( {\mathbf m}_o {\mathbf k}^{(b)}) + J\operatorname{diag}( \widetilde {{\mathbf m}_d} {\mathbf k}^{(b)})
= \operatorname{diag}( {\mathbf m}_o {\mathbf k}^{(b)}) + \operatorname{diag}( {\mathbf m}_d \widetilde {{\mathbf k}^{(b)} })J\,,$$ where for any $2n$ vector ${\mathbf v}=({\mathbf v}_1, {\mathbf v}_2)$ with ${\mathbf v}_i\in {\mathbb{C}}^n$ we define $\widetilde{{\mathbf v}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathbf v}_2, {\mathbf v}_1)$. Thus, graphically, the factor $MJK^{(b)}$ can be represented as a sum of two graphs with a weight assigned to one vertex and for one of them there is an additional twist operator $J$ which one may put on either side. Therefore, the graph on the left-hand side of can be represented by $$\label{Mres}
\ssGraph{x[label=below:$x$,o] -- [eq] a[label=below:$a$,tmok] --[s] y[label=below:$y$,o];} + \ssGraph{x[label=below:$x$,o] -- [eq] a[label=below:$a$,mdtk]--[s,J] y[label=below:$y$,o];}.$$ Here the double solid line depicts the identity operator as in [@Cusp1]. Both graphs in are exactly the same as the one in case (a) within the proof of [@Cusp1 Lemma 4.13] with the only changes being the modified vertex weights ${\mathbf m}_o {\mathbf k}^{(b)}$ and $ {\mathbf m}_d\widetilde{{\mathbf k}}^{(b)}$ instead of just ${\mathbf m}$ and the twisted $G^*$-edge in the second graph of . To justify the Ward estimate we use only the fact that ${\lVert {\mathbf m}_o \rVert}_\infty \lesssim 1$ and ${\lVert {\mathbf m}_d \rVert}_\infty \lesssim 1$, although the latter estimate can be improved to ${\lVert {\mathbf m}_d \rVert}_\infty \lesssim \rho$. Here we denote by ${\lVert {\mathbf x}\rVert}_\infty :=\max_{a \in [2n]}{\lvert x_a \rvert}$ the maximum norm of a vector ${\mathbf x}$.
The decomposition of $MJK^b$ into a sum of two terms, each with usual weights ${\mathbf m}$ and one of them with a twisted edge, can be reinterpreted in all other cases. Thus, the second and fourth term in can be treated analogously to the cases (b) and (c) within the proof of [@Cusp1 Lemma 4.13]. In particular, the fourth term is split as $$\label{spl}
{\mathbb{E}}(G {\mathcal}{T}[G-M]MJK^{(b)}G^*)_{xy} f = {\mathbb{E}}\sum {\bigl({t_{ba}G_{xb} (G-M)_{ab}u_a G^*_{ay} + t_{ba}G_{xb} (G-M)_{ab}v_a (GJ)^*_{ay}}\bigr)} f \,,$$ for some bounded vectors ${\mathbf u},{\mathbf v}$ with ${\lVert {\mathbf u}\rVert}_\infty+{\lVert {\mathbf v}\rVert}_\infty\lesssim 1$. Thus the corresponding two graphs exactly match the first graph depicted in (c) of the proof of [@Cusp1 Lemma 4.13] with locally modified weights and the second one having a twisted $G^*$-edge.
For the third term in we use a cumulant expansion and find $$\label{eq:cumulant expansion}
\begin{split}
{\mathbb{E}}{\mspace{2 mu}}(G({\widetilde}{{\mathcal}{S}}[G]+W)MJK^{(b)}G^*)_{xy} f &= - {\mathbb{E}}{\mspace{2 mu}}(G{\widetilde}{{\mathcal}{S}}[MJK^{(b)}G^*]G^*)_{xy} f
\\
&\qquad+ \sum_{a,c}\sum_{k=2}^{6p} \sum_{\mathbf{\beta} \in I^k} \kappa(ac,\underline{\beta}) {\mathbb{E}}\partial_{\underline{\beta}}[G_{xa}(MJK^{(b)}G^*)_{cy}f]
\\
&\qquad+ \sum_{a,c} {\mathbb{E}}\, G_{xa}(MJK^{(b)}G^*)_{cy}({\mathbb{E}}{\lvert w_{ac} \rvert}^2\partial_{ac}+{\mathbb{E}}w_{ac}^2\partial_{ca}) f+ {\mathcal{O}}(N^{-p})\,,
\end{split}$$ where $\mathbf{\beta}$ is a $k$-tuple of double indices from $I=[N]\times[N]$, $\underline{\beta}$ is the multiset formed out of the entries of $\mathbf{\beta}$ (multisets allow repetitions) and $\partial_{\underline{\beta}} = \prod_{(ij)\in\underline{\beta}} \partial_{w_{ij}}$. The notation $\kappa(ac,\underline{\beta})$ denotes the higher order cumulants of $w_{ab}$ and $\{ w_\beta\; : \; \beta\in \underline{\beta}\}$. The second and third summands on the right-hand side of correspond to the graphs treated in (e) and (d) of the proof of [@Cusp1 Lemma 4.13], respectively, with the factor $MJK^{(b)}$ reinterpreted as sum of two terms with some weight ${\mathbf m}$ as explained in . Thus we focus our attention on the first summand which reflects the the main difference between our setup and that in [@Cusp1].
This term is expanded further using $$\label{SM}
{\widetilde}{{\mathcal}{S}}[MJK^{(b)}G^*] = {\mathcal}{S}[MJK^{(b)}M^*]+ {\mathcal}{S}[MJK^{(b)}(G-M)^*]+ {{\mathcal}{T}}[MJK^{(b)}G^*]\,.$$ At this point the main mechanism behind the cusp fluctuation averaging is revealed by the fact that the leading term ${\mathcal}{S}[MJK^{(b)}M^*] = \operatorname{diag}({\mathbf x})$ for some vector ${\mathbf x}$ with ${\lvert x_i \rvert} \lesssim \frac{\rho}{N}$, because the diagonal elements of $M$ are of order ${\mathcal{O}}(\rho)$. This is the only place where the smallness of the diagonal elements of $M$ is used, in all other estimates we used only the block diagonal structure of $M$ and the boundedness of its matrix elements. The other two terms in are smaller order. In fact, the second term exactly corresponds to the term encoded by the fourth graph on the right hand side of [@Cusp1 Eq. (4.27)], taking into account that this graph now splits into two due to the decomposition with an extra twisted edge on one of the resulting graphs similarly to . Therefore this term is estimated as explained in (b) of the proof of [@Cusp1 Lemma 4.13]. The term corresponding to the last summand in is analogous to the sixth graph in [@Cusp1 Eq. (4.27)] and thus treated as explained in (c) within the proof of [@Cusp1 Lemma 4.13].
Finally, we treat the last $G{\mathcal}{T}[M]$ term in which was absent in [@Cusp1] and stems from the difference between the two self-energy operators ${\mathcal}{S}$ and ${\widetilde}{{\mathcal}{S}}$. This term leads to a contribution of the form ${\mathbb{E}}(GKG^*)_{xy} f $ with the block diagonal matrix $K = {\mathcal}{T}[M]MJ K^{(b)}$ that satisfies ${\lVert K \rVert}\lesssim N^{-1}$ in . Thus the ensuing two graphs $\Gamma' \in {\mathcal}{G}_\Gamma$ have the same Ward estimates as $\Gamma$ but an additional $N^{-1}$-edge weight. This completes the proof of Proposition \[pro:wtD\_bounds\].
Local law for $X$ {#sec:Xlaw}
=================
In this section, we provide the proofs of Theorem \[thm:spectral\_radius\_X\] and Theorem \[thm:local\_law\_X\]. We start with the proof of Theorem \[thm:local\_law\_X\] which, given Theorem \[thm:local\_law\_H\], will follow a similar argument as the proof of [@Altcirc Theorem 2.5(i)].
In this section, the model parameters consist of $s_*$, $s^*$ from \[assum:flatness\], the sequence $(\mu_m)_m$ from \[assum:bounded\_moments\] and $\alpha$, $\beta$ from \[assum:bounded\_density\] as well as $a$ and $\varphi$ from Theorem \[thm:local\_law\_X\]. Therefore, the implicit constants in the comparison relation and the stochastic domination are allowed to depend on these parameters.
Let $T>0$. From [@Altcirc Eq. (2.15)] and in the proof of Proposition \[pro:properties\_sigma\], we get that $$\label{eq:main_decomposition}
\begin{aligned}
\frac{1}{n} \sum_{i=1}^n f_{z_0,a}(\zeta_i) - \int_{\mathbb{C}}f_{z_0,a}(z) \sigma(z) {\mathrm{d}}^2 z = \, & \frac{1}{4\pi n} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \log {\lvert \det(H_z - {\mathrm{i}}T) \rvert} {\mathrm{d}}^2 z \\
& - \frac{1}{2\pi} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_0^T {\mathrm{Im}\,}{\langle G(z,\eta)- M(z,\eta) \rangle} {\mathrm{d}}\eta \, {\mathrm{d}}^2 z \\
& + \frac{1}{2\pi} \int_{\mathbb{C}}\Delta f_{z_0,a}(z) \int_T^\infty \bigg( {\mathrm{Im}\,}{\langle M(z,\eta) \rangle} - \frac{1}{1 + \eta } \bigg) {\mathrm{d}}\eta \, {\mathrm{d}}^2 z.
\end{aligned}$$ Here, we used that ${\langle v_1^\tau(\eta)|_{\tau = {\lvert z \rvert}^2} \rangle} = {\mathrm{Im}\,}{\langle M(z,\eta) \rangle}$ and $m^z({\mathrm{i}}\eta) = {\langle G(z,\eta) \rangle}$, where $m^z$ is the Stieltjes transform of the empirical spectral measure of $H_z$ (see [@Altcirc Eq. (2.12)]). We also employed $\int_{{\mathbb{C}}} \Delta f_{z_0,a}(z)\int_0^T (1+ \eta)^{-1} {\mathrm{d}}\eta {\mathrm{d}}^2 z = 0$ as $f \in C_0^2({\mathbb{C}})$.
We remark that $\operatorname{supp}f_{z_0,a} \subset z_0 + \operatorname{supp}f \subset D_{2\varphi}(0)$. For the remainder of the proof, we choose $T {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}n^{100}$. The same arguments used in the proof of [@Altcirc Theorem 2.5] to control the first and third term on the right-hand side of for ${\lvert z_0 \rvert} \leq 1$ imply that those terms are stochastically dominated by $n^{-1 + 2a} {\lVert \Delta f \rVert}_1$ for all $z_0 \in {\mathbb{C}}$ such that ${\lvert z_0 \rvert} \leq \varphi$.
What remains is bounding the second term on right-hand side of . To that end, we fix $z$ and estimate the ${\mathrm{d}}\eta$-integral by $$I(z) {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\int_0^T {\mathrm{Im}\,}{\langle G(z,\eta)- M(z,\eta) \rangle} {\mathrm{d}}\eta$$ for $z \in D_{2\varphi}(0)$ via the following lemma which is an extension of [@Altcirc Lemma 5.8].
\[lem:moment\_bound\_I\] For every $\delta >0$ and $p \in {\mathbb{N}}$, there is a positive constant $C$, depending only on $\delta$ and $p$ in addition to the model parameters, such that $$\sup_{z \in D_{2\varphi}(0)} {\mathbb{E}}{\lvert I(z) \rvert}^p \leq C \frac{n^{\delta p}}{n^p}.$$
We postpone the proof of Lemma \[lem:moment\_bound\_I\] to the end of this section. Now, Lemma \[lem:moment\_bound\_I\] implies Theorem \[thm:local\_law\_X\] along the same steps used to conclude [@Altcirc Theorem 2.5(i)] from [@Altcirc Lemma 5.8]. This completes the proof of Theorem \[thm:local\_law\_X\].
Before we prove Lemma \[lem:moment\_bound\_I\], we first conclude Theorem \[thm:spectral\_radius\_X\] from Theorem \[thm:local\_law\_X\] and the improved bound on $G-M$ in in Theorem \[thm:local\_law\_H\].
We first show the upper bound on $\varrho(X)$ as detailed in Remark \[rem:upper\_bound\_spectral\_radius\]. Throughout the proof, we say that an event $\Xi =\Xi_n$ (in the probability space of $n\times n$ random matrices $X$) holds with *very high probability* if for each $D>0$, there is $C>0$ such that ${\mathbb{P}}( \Xi) \geq 1 - C n^{-D}$ for all $n \in {\mathbb{N}}$.
Let $\tau_* \sim 1$ be chosen as in Theorem \[thm:local\_law\_H\]. From [@Altcirc Theorem 2.5(ii)], we conclude that the event $$\Xi_* {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big\{ \operatorname*{Spec}(X) \subset D_{1 + \tau_*} (0) \big \}$$ holds with very high probability. Fix ${\varepsilon}>0$. We set $\eta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\eta_{\mathrm{f}}}n^{{\varepsilon}/12}$. We use for a sufficiently fine grid of values of $z$, ${\lvert {\langle {\mathrm{Im}\,}M \rangle} \rvert}\lesssim n^{-{\varepsilon}/6} /(n\eta)$ by if ${\lvert z \rvert} -1 \geq n^{-1/2+{\varepsilon}}$, a union bound over the grid elements and the Lipschitz-continuity of $G(z,\eta)$ as a function of $z$ to obtain that the event $$\Xi_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big\{{\langle {\mathrm{Im}\,}G(z,\eta) \rangle} \leq n^{-{\varepsilon}/8}/(n\eta) \text{ for all } z \in {\mathbb{C}}\text{ satisfying } n^{-1/2 + {\varepsilon}} \leq {\lvert z \rvert} -1 \leq \tau_* \big \}$$ holds with very high probability. Let $\lambda_1, \ldots, \lambda_{2n}$ denote the eigenvalues of $H_z$. If $H_z$ has a nontrivial kernel, i.e. $\lambda_i = 0$ for some $i \in \{1, \ldots, 2n\}$ then we conclude ${\langle {\mathrm{Im}\,}G(z,\eta) \rangle} \geq 1/(2n\eta)$ for any $\eta >0$. Hence, $\Xi_1 \subset \Xi_2$, where we defined $$\Xi_2 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\big \{ 0 \notin \operatorname*{Spec}(H_z) \text{ for all } z \in {\mathbb{C}}\text{ satisfying } n^{-1/2 + {\varepsilon}} \leq {\lvert z \rvert} -1 \leq \tau_* \big \}.$$ On the other hand, $0 \in \operatorname*{Spec}(H_z)$ if and only if $z \in \operatorname*{Spec}(X)$. Since $\Xi_*$ and $\Xi_2$ hold with very high probability, this completes the proof of the upper bound on $\varrho(X)$, i.e. the one of Remark \[rem:upper\_bound\_spectral\_radius\].
The corresponding lower bound on $\varrho(X)$ follows directly from Theorem \[thm:local\_law\_X\] with ${\lvert z_0 \rvert} = 1 - n^{-1/2 + {\varepsilon}}$, $a = 1/2 - {\varepsilon}$ and a suitably chosen test function $f$ which completes the proof of Theorem \[thm:spectral\_radius\_X\].
The proof proceeds analogously to the proof of [@Altcirc Lemma 5.8]. However, we have to replace the fluctuation scale in the bulk, $n^{-1}$, by the $z$-dependent fluctuation scale ${\eta_{\mathrm{f}}}$ defined in . Throughout the proof, we will omit the dependence of $G(z,\eta)$ and $M(z,\eta)$ on $z$ and $\eta$ from our notation and write $G= G(z,\eta)$ and $M = M(z,\eta)$. Similarly, we denote the $2n$ eigenvalues of $H_z$ by $\lambda_1, \ldots, \lambda_{2n}$.
We will have to convert a few $\prec$-bounds into moment bounds. In order to do that, we will use the following straightforward estimate. Let $c>0$. Then, for each $\delta>0$ and $p \in {\mathbb{N}}$, there is $C$, depending on $c$, $\delta$ and $p$, such that any random variable $Y \geq 0$ satisfies $$\label{eq:estimate_prec_to_moment}
Y \prec n^{-1}, ~ Y \leq n^c \qquad \Longrightarrow \qquad {\mathbb{E}}Y^p \leq C n^{p(-1 + \delta)}.$$
We fix ${\varepsilon}>0$, choose $l >0$ sufficiently large and decompose the integral in the definition of $I$ to obtain $$\label{eq:decomposition_I}
I(z) = \frac{1}{n} \sum_{{\lvert \lambda_i \rvert} < n^{-l}} \log \bigg(1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2}\bigg) + \frac{1}{n} \sum_{{\lvert \lambda_i \rvert} \geq n^{-l} } \log \bigg( 1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2} \bigg)
- \int_0^{{\eta_{\mathrm{f}}}n^{\varepsilon}} {\langle {\mathrm{Im}\,}M \rangle} {\mathrm{d}}\eta + \int_{{\eta_{\mathrm{f}}}n^{\varepsilon}}^T {\mathrm{Im}\,}{\langle G- M \rangle} {\mathrm{d}}\eta.$$
We now estimate the $p$th moment of each term on the right-hand side of individually. Exactly as in the proof of [@Altcirc Lemma 5.8], we choose $l>0$ sufficiently large, depending on $\alpha$, $\beta$ from \[assum:bounded\_density\] and on $p$ such that the estimate on the smallest singular value of $X-z$ in [@Altcirc Proposition 5.7], which holds uniformly for $z \in D_{2\varphi}(0)$, implies $${\mathbb{E}}{\bigg\lvert \frac{1}{n} \sum_{{\lvert \lambda_i \rvert} < n^{-l}} \log \bigg( 1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2} \bigg) \bigg\rvert}^p \leq n^{-p}.$$ For the second term on the right-hand side of , we distinguish the three regimes, ${\lvert \lambda_i \rvert} \in [n^{-l}, {\eta_{\mathrm{f}}}n^{{\varepsilon}}]$, ${\lvert \lambda_i \rvert} \in [{\eta_{\mathrm{f}}}n^{{\varepsilon}},{\eta_{\mathrm{f}}}n^{1/2}]$ and ${\lvert \lambda_i \rvert} > {\eta_{\mathrm{f}}}n^{1/2}$. In the first regime, we use ${\eta_{\mathrm{f}}}^2 n^{2 {\varepsilon}} \lambda_i^{-2} \leq n^{2l +2}$ and Lemma \[lem:number\_eigenvalues\] with $\eta = {\eta_{\mathrm{f}}}n^{\varepsilon}$ and obtain $$\frac{1}{n} \sum_{{\lvert \lambda_i \rvert} \in [n^{-l}, {\eta_{\mathrm{f}}}n^{\varepsilon}]} \log \bigg(1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2} \bigg) \leq \frac{C \log n}{n} {\lvert \{ i \colon {\lvert \lambda_i \rvert} \leq {\eta_{\mathrm{f}}}n^{\varepsilon}\} \rvert} \prec \frac{n^{2{\varepsilon}}}{n}.$$ We decompose the second regime, ${\lvert \lambda_i \rvert} \in [{\eta_{\mathrm{f}}}n^{{\varepsilon}}, {\eta_{\mathrm{f}}}n^{1/2}]$, into the union of the intervals $[\eta_k, \eta_{k+1}]$, where $\eta_k {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\eta_{\mathrm{f}}}n^{\varepsilon}2^k$, $k=0, \ldots, N$ and $N \lesssim \log n$. Hence, $\log(1+ x) \leq x$ for $x >0$ yields $$\frac{1}{n} \sum_{{\lvert \lambda_i \rvert} \in [{\eta_{\mathrm{f}}}n^{{\varepsilon}}, {\eta_{\mathrm{f}}}n^{1/2}]} \log \bigg( 1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2} \bigg) \leq \frac{2}{n} \sum_{k=0}^N \sum_{\lambda_i \in [\eta_k, \eta_{k+1}]} {\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}} \lambda_i^{-2} \prec \frac{n^{3{\varepsilon}}}{n}.$$ In the third regime, ${\lvert \lambda_i \rvert} > {\eta_{\mathrm{f}}}n^{1/2}$, we conclude from ${\lvert \lambda_i \rvert} > {\eta_{\mathrm{f}}}n^{1/2}$ that $$\frac{1}{n}\sum_{{\lvert \lambda_i \rvert} >{\eta_{\mathrm{f}}}n^{1/2}} \log \bigg( 1 + \frac{{\eta_{\mathrm{f}}}^2 n^{2{\varepsilon}}}{\lambda_i^2} \bigg) \leq \frac{1}{n} \sum_{{\lvert \lambda_i \rvert} > {\eta_{\mathrm{f}}}n^{1/2}} \log \big( 1 + n^{-1 + 2{\varepsilon}}\big) \leq \frac{2n^{2{\varepsilon}}}{n}.$$ Therefore, implies that the second term on the right-hand side of satisfies the bound in Lemma \[lem:moment\_bound\_I\].
For the third term in , we use and distinguish the regimes in the definition of ${\eta_{\mathrm{f}}}$ in and obtain $$\int_0^{{\eta_{\mathrm{f}}}n^{\varepsilon}} {\langle {\mathrm{Im}\,}M \rangle} \,{\mathrm{d}}\eta \sim \int_0^{{\eta_{\mathrm{f}}}n^{\varepsilon}} \rho\, {\mathrm{d}}\eta \lesssim n^{-1 + 4{\varepsilon}/3}.$$
Before estimating the last term in , we conclude $\sup_{\eta \in [{\eta_{\mathrm{f}}}n^{{\varepsilon}},T]} \eta {\lvert {\mathrm{Im}\,}{\langle G-M \rangle} \rvert} \prec n^{-1}$ from Corollary \[cor:im\_local\_law\_everywhere\]. Here, we also used a union bound and the Lipschitz-continuity of $G$ and $M$ as functions of $\eta$ in the following sense: There is $c\geq 2$ such that ${\lVert G(z,\eta_1) - G(z,\eta_2) \rVert} + {\lVert M(z,\eta_1) - M(z,\eta_2) \rVert} \lesssim n^{c} {\lvert \eta_1-\eta_2 \rvert}$ for all $\eta_1, \eta_2 \geq n^{-1}$ and $z \in D_{2\varphi}(0)$. For $G$ this follows from resolvent identities. For $M$ this was shown in [@Kronecker Corollary 3.8][^3]. The bound $\sup_{\eta \in [{\eta_{\mathrm{f}}}n^{\varepsilon}, T]} \eta {\lvert {\mathrm{Im}\,}{\langle G- M \rangle} \rvert} \prec n^{-1}$ implies $$\int_{{\eta_{\mathrm{f}}}n^{\varepsilon}}^T {\lvert {\mathrm{Im}\,}{\langle G-M \rangle} \rvert} {\mathrm{d}}\eta \prec n^{-1}.$$ Owing to this implies the desired estimate on the $p$th moment of the last term on the right-hand side of . This completes the proof of Lemma \[lem:moment\_bound\_I\].
\[rem:alternative\_changes\_proof\] Instead of Assumption \[assum:bounded\_density\] we now assume the condition $\max_{i,j} {\mathcal}{L}(\sqrt{n}\, x_{ij}, t) \leq b$ from Remark \[rem:alternative\_A3\] and we explain the necessary modifications in the proof of Theorem \[thm:local\_law\_X\]. The only place where Assumption \[assum:bounded\_density\] is used in our entire argument was in estimating the first term in . Under this new condition, it is a simple consequence of [@LTV_singular_value2019 Theorem 1.1] that $$\label{eq:abs_log_lambda_1_prec_1}
{\lvert \log \lambda_1(z) \rvert} \prec 1,$$ where we denoted by $\lambda_1(z)$ the smallest, nonnegative eigenvalue of $H_z$. With this bound at hand, the sampling method from [@tao2015 Lemma 36] can be used to approximate the first $z$-integral in (see also ) by an average over $n^C$ many evaluations of the integrand in $z$ (the sampling method requires the additional condition $\|\Delta f\|_{L^{2+\epsilon}}\le n^C\| \Delta f\|_{L^1}$). Applying a union bound reduces Theorem \[thm:local\_law\_X\] to the local law for $H_z$, Theorem \[thm:local\_law\_H\], with one fixed $z$. Note that $I(z)$ does not have finite expectation in general, e.g. if the distribution of the matrix elements of $X$ has atoms. Thus, Lemma \[lem:moment\_bound\_I\] as it is stated cannot be correct in general. Instead, we need to split the $\eta$-integration in the definition of $I(z)$ and we use to control the regime $\eta \in [0,n^{-l}]$. For the remaining regime the very small singular values do not play any role and the above proof directly applies.
Derivation of cubic equations {#app:cubic_equation}
=============================
The following general lemma determines the first few terms in the perturbative expansion of the solution $Y$ to the matrix-valued quadratic equation ${\mathcal{B}}[Y] - {\mathcal{A}}[Y,Y] + X =0$ in the regime, where $X$ is small. Here, ${\mathcal{B}}$ is a linear map and ${\mathcal{A}}$ is a bilinear map on the space of matrices. As explained in Section \[sec:outline\], the unstable directions of ${\mathcal{B}}$ play a particular role. The case of one unstable direction was treated in [@Cusp1 Lemma A.1] which was sufficient for analysing Wigner-type matrices with a flat variance matrix. In our current situation, a second unstable direction is present due to the specific block structure of the matrix $H_z$. Accordingly, in the next lemma, we need to treat both unstable directions, $B$ and $B_*$, separately. We are, however, in the special situation, where $Y$ is orthogonal to $E_-$ and $B_*$ is far from being orthogonal to $E_-$. This allows us to neglect the component of $Y$ in the direction of $B_*$ and arrive at a single cubic equation for $\Theta$, the coefficient of $Y$ in the $B$ direction, $Y=\Theta B+ \mathrm{error}$. The main result of this lemma is to determine the coefficients of the cubic equation for $\Theta$.
\[lem:cubic\_equation\_abstract\] Let ${\langle{{{\,\cdot\,}}} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}$ be the Hilbert-Schmidt scalar product on ${\mathbb{C}}^{2n \times 2n}$ and ${\lVert {{\,\cdot\,}}\rVert}$ an arbitrary norm on ${\mathbb{C}}^{2n\times 2n}$. Let ${\mathcal{A}}\colon {\mathbb{C}}^{2n\times 2n}\times {\mathbb{C}}^{2n \times 2n} \to {\mathbb{C}}^{2n \times 2n}$ be a bilinear map such that ${\mathcal{A}}[R,T] = {\mathcal{A}}[T,R]$ for all $R, T \in {\mathbb{C}}^{2n \times 2n}$. Let ${\mathcal{B}}\colon {\mathbb{C}}^{2n \times 2n} \to {\mathbb{C}}^{2n \times 2n}$ be a linear operator with two simple eigenvalues $\beta$ and $\beta_*$ with associated left and right eigenvectors ${\widehat}{B}$, $B$ and ${\widehat}{B}_*$, $B_*$, respectively.
For some $\lambda \geq 1$, we assume that $$\label{eq:conditions_abstract_cubic}
{\lVert {\mathcal{A}}\rVert} + {\lVert {\mathcal{B}}^{-1} {\mathcal{Q}}\rVert} + \frac{1}{{\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} \rvert}} + {\lVert {\langle{{\widehat}{B}} \mspace{2mu}, {{{\,\cdot\,}}}\rangle} \rVert} + {\lVert B \rVert} +
\frac{1}{{\lvert {\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle} \rvert}} + {\lVert {\langle{{\widehat}{B}_*} \mspace{2mu}, {{{\,\cdot\,}}}\rangle} \rVert} + {\lVert B_* \rVert} + \frac{1}{{\lvert {\langle{E_-} \mspace{2mu}, {B_*}\rangle} \rvert}} + {\lVert {\langle{E_-} \mspace{2mu}, {{{\,\cdot\,}}}\rangle} \rVert} \leq \lambda,$$ where ${\lVert {{\,\cdot\,}}\rVert}$ denotes the norm on bilinear maps, linear operators and linear forms induced by the norm on ${\mathbb{C}}^{2n \times 2n}$.
Then there is a universal constant $c>0$ such that for any $Y, X \in {\mathbb{C}}^{2n \times 2n}$ with ${\lVert Y \rVert} + {\lVert X \rVert} \leq c\lambda^{-12}$ that satisfy the quadratic equation $$\label{eq:quadratic_equation}
{\mathcal{B}}[Y] - {\mathcal{A}}[Y,Y] + X = 0$$ with the constraint ${\langle{E_-} \mspace{2mu}, {Y}\rangle} = 0$ the following holds: For any $\delta \in (0,1)$, the coefficient $$\Theta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{{\langle{{\widehat}{B}} \mspace{2mu}, {Y}\rangle}}{{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}}$$ fulfills the cubic equation $$\label{eq:abstract_cubic}
\begin{aligned}
& \mu_3 \Theta^3 + \mu_2 \Theta^2 + \mu_1 \Theta + \mu_0 \\
& \qquad \quad = \lambda^{40}
{\mathcal{O}}\big(\delta {\lvert \Theta \rvert}^3 + {\lvert \Theta \rvert}^4 + \delta^{-2} ( {\lVert X \rVert}^3 + {\lvert {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[X]}\rangle} \rvert}^{3/2}) + {\lvert \Theta \rvert}^2( {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}^2 + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} \rvert}^2) \big)
\end{aligned}$$ whose coefficients are given by $$\label{eq:coefficients_abstract_cubic}
\begin{aligned}
\mu_3 & = 2{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]]}\rangle}- \frac{2 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]}\rangle}}{{\langle{E_-} \mspace{2mu}, {B_*}\rangle}},
\\ \mu_2 & = {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B]}\rangle},
\\ \mu_1 & = -\beta {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} - 2 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B, {\mathcal{B}}^{-1} {\mathcal{Q}}[X]]}\rangle} + \frac{2 {\langle{E_-} \mspace{2mu}, {B}\rangle}}{{\langle{E_-} \mspace{2mu}, {B_*}\rangle}} {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[{\mathcal{B}}^{-1} {\mathcal{Q}}[X],B_*]}\rangle},
\\ \mu_0 & = {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[{\mathcal{B}}^{-1}{\mathcal{Q}}[X], {\mathcal{B}}^{-1} {\mathcal{Q}}[X]] - X}\rangle}.
\end{aligned}$$ Moreover, $Y$ can be expressed by $\Theta$ and $X$ via $$\label{eq:expansion_Y_abstract_cubic}
\begin{aligned}
& Y = \Theta B - {\mathcal{B}}^{-1} {\mathcal{Q}}[X] + \Theta^2 {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B] - \Theta^2 \frac{{\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]}\rangle}}{{\langle{E_-} \mspace{2mu}, {B_*}\rangle}} B_* \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \lambda^{30} {\mathcal{O}}\big( {\lvert \Theta \rvert}^3 + {\lvert \Theta \rvert}({\lVert X \rVert} + {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}) + {\lVert X \rVert}^2 + {\lvert {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[X]}\rangle} \rvert} \big).
\end{aligned}$$
We decompose $Y$ according to the spectral subspaces of ${\mathcal{B}}$. This yields $$\label{eq:proof_abstract_cubic_Y_decom_spectral}
Y = \Theta B + \Theta_* B_* + {\mathcal{Q}}[Y], \qquad \qquad
\Theta{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{{\langle{{\widehat}{B}} \mspace{2mu}, {Y}\rangle}}{{\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle}}, \quad \Theta_*{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{{\langle{{\widehat}{B}_*} \mspace{2mu}, {Y}\rangle}}{{\langle{{\widehat}{B}_*} \mspace{2mu}, {B_*}\rangle}}.$$ We define $$\label{eq:proof_abstract_cubic_decomposition}
\begin{aligned}
Y_1 & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\Theta B - {\mathcal{B}}^{-1} {\mathcal{Q}}[X], & Y_3 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\frac{{\langle{E_-} \mspace{2mu}, { {\mathcal{B}}^{-1} {\mathcal{Q}}[X] -\Theta {B} - {{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y_1, Y_1]} }\rangle}}{{\langle{E_-} \mspace{2mu}, {B_*}\rangle}} B_*,\\
Y_2 & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{Q}}[Y] + {\mathcal{B}}^{-1} {\mathcal{Q}}[X] + \Theta_* B_* - Y_3. \quad &
\end{aligned}$$ Obviously, $Y = Y_1 + Y_2 + Y_3$ as well as $Y_1 = \lambda{\mathcal{O}}_1$ and $Y_3 = \lambda^7{\mathcal{O}}_2$, where we introduced the notation $${\mathcal{O}}_k = {\mathcal{O}}\Big( {\lvert \Theta \rvert}^k + {\lVert X \rVert}^k + {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \Theta \rvert}^{k/2} + {\lvert {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[X]}\rangle} \rvert}^{k/2}\Big)$$ and used the convention that $Z ={\mathcal{O}}_k$ means ${\lVert Z \rVert} = {\mathcal{O}}_k$. Here and in the following, the implicit constant in ${\mathcal{O}}$ will always be independent of $\lambda$.
From ${\langle{E_-} \mspace{2mu}, {Y}\rangle} = 0$ and , we obtain $$\Theta_* {\langle{E_-} \mspace{2mu}, {B_*}\rangle} = - \Theta {\langle{E_-} \mspace{2mu}, {B}\rangle} - {\langle{E_-} \mspace{2mu}, {{\mathcal{Q}}[Y]}\rangle} = {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[X] - \Theta B- {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y, Y]}\rangle}.$$ Here, we used that ${\mathcal{Q}}[Y] = {\mathcal{B}}^{-1} {\mathcal{A}}[Y,Y] - {\mathcal{B}}^{-1} {\mathcal{Q}}[X]$ by in the second step. This shows that $\Theta_*$ is the coefficient of $B_*$ in the definition of $Y_3$ up to replacing $Y$ by $Y_1$. Thus, we deduce that $$\label{eq:proof_abstract_cubic_8}
{\lVert \Theta_* B_* - Y_3 \rVert} = \lambda^5{\mathcal{O}}\big({\lVert Y_1 \rVert}( {\lVert Y_2 \rVert} + {\lVert Y_3 \rVert}) + {\lVert Y_2 \rVert}^2 + {\lVert Y_3 \rVert}^2\big).$$
We insert $Y =Y_1 + Y_2 + Y_3$ into and obtain $$\label{eq:proof_abstract_cubic_1}
\Theta \beta B + \Theta_* \beta_* B_* + {\mathcal{B}}{\mathcal{Q}}[Y_2] + (1 - {\mathcal{Q}})[X] = {\mathcal{A}}[Y,Y].$$ Applying ${\mathcal{B}}^{-1}{\mathcal{Q}}$ to the previous relation implies $$\label{eq:proof_abstract_cubic_3}
{\mathcal{Q}}[Y_2] = {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y,Y].$$ Hence, ${\lVert {\mathcal{Q}}[Y_2] \rVert} = \lambda^2{\mathcal{O}}({\lVert Y_1 \rVert}^2 + {\lVert Y_2 \rVert}^2 + {\lVert Y_3 \rVert}^2)$. As $Y_2 = {\mathcal{Q}}[Y_2] + \Theta_* B_* - Y_3$, we get from that $$\label{eq:proof_abstract_cubic_9}
{\lVert Y_2 \rVert} = \lambda^5{\mathcal{O}}({\lVert Y_1 \rVert}^2 + {\lVert Y_2 \rVert}^2 + {\lVert Y_3 \rVert}^2).$$ The definition of $Y_1$, $Y_2$ and $Y_3$ as well as the conditions and ${\lVert X \rVert} + {\lVert Y \rVert}\leq c \lambda^{-12}$ yield ${\lVert Y_2 \rVert} \leq cC \lambda^{-5}$ for some universal constant $C>0$. Hence, implies $Y_2 = \lambda^{19}{\mathcal{O}}_2$ as $Y_1 = \lambda {\mathcal{O}}_1$ and $Y_3 = \lambda^7 {\mathcal{O}}_2$ if $c$ is chosen sufficiently small independently of $\lambda$. Therefore, we conclude from , and ${\lvert \Theta \rvert} + {\lVert X \rVert} = {\mathcal{O}}(\lambda^{-10})$ that $$\label{eq:proof_abstract_cubic_4}
Y_2 = {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y_1, Y_1] + \lambda^{30}{\mathcal{O}}_3.$$ In particular, this implies as $Y = Y_1 + Y_2 + Y_3$.
Applying ${\langle{{\widehat}{B}} \mspace{2mu}, {{{\,\cdot\,}}}\rangle}$ to yields $$\label{eq:proof_abstract_cubic_6}
\Theta \beta {\langle{{\widehat}{B}} \mspace{2mu}, {B}\rangle} + {\langle{{\widehat}{B}} \mspace{2mu}, {X}\rangle} = {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y,Y]}\rangle}.$$ Therefore, we now show by computing ${\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y,Y]}\rangle}$. Using $Y_1 = \lambda{\mathcal{O}}_1$, $Y_2 = \lambda^{19} {\mathcal{O}}_2$, $Y_3 = \lambda^{7}{\mathcal{O}}_2$ and , we deduce $$\label{eq:proof_abstract_cubic_2}
{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y,Y]}\rangle} = {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y_1, Y_1]}\rangle} + 2{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y_1, Y_3]}\rangle} + 2 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y_1, {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y_1, Y_1]]}\rangle} + \lambda^{40}{\mathcal{O}}_4.$$ For a linear operator ${\mathcal}{K}_1$ and a bilinear operator ${\mathcal}{K}_2$ with ${\lVert {\mathcal}{K}_1 \rVert} + {\lVert {\mathcal}K_2 \rVert} \leq 1$, we have $${\lVert \Theta {\mathcal}{K}_2[R,R] \rVert} \leq \delta {\lvert \Theta \rvert}^3 + \delta^{-1/2} {\lVert R \rVert}^3, \qquad {\lVert \Theta^2 {\mathcal}{K}_1[R] \rVert} \leq \delta {\lvert \Theta \rvert}^3 + \delta^{-2} {\lVert R \rVert}^3$$ for any matrix $R \in {\mathbb{C}}^{2n \times 2n}$ since $\delta >0$. Therefore, as $\delta \in (0,1)$, we obtain $$\label{eq:proof_abstract_cubic_5}
{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y_1,{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[Y_1,Y_1]]}\rangle} = \Theta^3 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B, {\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]]}\rangle} + \lambda^7{\mathcal{O}}(\delta{\lvert \Theta \rvert}^3 + \delta^{-2} {\lVert X \rVert}^3).$$ Similarly, we conclude $$\label{eq:proof_abstract_cubic_7}
\hspace*{-0.33cm}
\begin{aligned}
{\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[Y_1, Y_3]}\rangle} = \, & \frac{\Theta {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[{\mathcal{B}}^{-1} {\mathcal{Q}}[X], B_*]}\rangle} {\langle{E_-} \mspace{2mu}, {B}\rangle} - \Theta^3 {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle}{\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}{\mathcal{A}}[B,B]}\rangle} }{{\langle{E_-} \mspace{2mu}, {B_*}\rangle}} \\
& + \lambda^{10}{\mathcal{O}}\big(\delta {\lvert \Theta \rvert}^3 + \delta^{-2} {\lVert X \rVert}^3 + \delta^{-2} {\lvert {\langle{E_-} \mspace{2mu}, {{\mathcal{B}}^{-1} {\mathcal{Q}}[X]}\rangle} \rvert}^{3/2} + {\lvert \Theta \rvert}^2( {\lvert {\langle{E_-} \mspace{2mu}, {B}\rangle} \rvert}^2 + {\lvert {\langle{{\widehat}{B}} \mspace{2mu}, {{\mathcal{A}}[B,B_*]}\rangle} \rvert}^2) \big).
\end{aligned}$$
Finally, we expand the first term on the right-hand side of using the definition of $Y_1$ from and insert as well as into to compute the second and third term. We apply the result to and obtain the cubic equation in with the coefficients detailed in .
Non-Hermitian perturbation theory {#app:perturbation_th}
=================================
In this section, we present for the reader’s convenience the perturbation theory for a non-Hermitian operator ${\mathcal{K}}$ on ${\mathbb{C}}^{2n\times 2n}$ with an isolated eigenvalue $\kappa$. We denote by ${\mathcal{P}_{\mathcal{K}}}$ the spectral projection of ${\mathcal{K}}$ associated to $\kappa$ and set ${\mathcal{Q}_{\mathcal{K}}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}1- {\mathcal{P}_{\mathcal{K}}}$. We assume that the algebraic multiplicity of $\kappa$ coincides with its geometric multiplicity. In particular, this condition ensures that, for any $L \in {\mathbb{C}}^{2n\times 2n}$, we have $${\mathcal{K}}[K] = \kappa K, \qquad {\mathcal{K}}^*[{\widehat}{K}] = \bar \kappa {\widehat}{K},$$ where $K {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}_{\mathcal{K}}}[L]$ and ${\widehat}{K} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}_{\mathcal{K}}}^*[L]$. That is, $K$ and ${\widehat}{K}$ are right and left eigenvectors of ${\mathcal{K}}$ corresponding to $\kappa$, respectively.
Throughout this section, we suppose that there is a constant $C>0$ such that $$\label{eq:assumptions_cK}
{\lVert {\mathcal{K}}\rVert} + {\lVert ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}\rVert} +{\lVert {\mathcal{P}_{\mathcal{K}}}\rVert} \leq C.$$ Here and in the following, ${\lVert {{\,\cdot\,}}\rVert}$ denotes the operator norm of operators on ${\mathbb{C}}^{2n\times 2n}$ induced by some norm ${\lVert {{\,\cdot\,}}\rVert}$ on the matrices in ${\mathbb{C}}^{2n\times 2n}$.
\[lem:perturbation\_theory\_2\] There is ${\varepsilon}>0$, depending only on $C$ from , such that the following holds.
If ${\mathcal{L}}$ is a linear map on ${\mathbb{C}}^{2n\times 2n}$ satisfying ${\lVert {\mathcal{K}}- {\mathcal{L}}\rVert} \leq {\varepsilon}$ and $\lambda$ is an eigenvalue of ${\mathcal{L}}$ satisfying ${\lvert \kappa - \lambda \rvert} \leq {\varepsilon}$ then, for any right and left normalized eigenvectors $L$ and ${\widehat}{L}$ of ${\mathcal{L}}$ associated to $\lambda$, we have
$$\begin{aligned}
\lambda {\langle{{\widehat}{L}} \mspace{2mu}, {L}\rangle} & = \kappa{\langle{{\widehat}{K}} \mspace{2mu}, {K}\rangle} + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}[K]}\rangle} + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}{\mathcal{Q}_{\mathcal{K}}}(2 \kappa - {\mathcal{K}}) ({\mathcal{K}}- \kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K]}\rangle} + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3),\label{eq:pert_th_eigenvalue}\\
L & = K -({\mathcal{K}}- \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] + L_2 + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3) ,\label{eq:pert_th_eigenvector_right} \\
{\widehat}{L} &= {\widehat}{K} -({\mathcal{K}}^*- \bar \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}^* {\mathcal{D}}^* [{\widehat}{K}] + {\widehat}{L}_2 + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3)\label{eq:pert_th_eigenvector_left} , \end{aligned}$$
where we used the definitions ${\mathcal{D}}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{L}}- {\mathcal{K}}$, $K {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}_{\mathcal{K}}}[L]$ and ${\widehat}{K} {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}{\mathcal{P}_{\mathcal{K}}}^*[{\widehat}{L}]$ as well as $$\begin{aligned}
L_2 & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] - ({\mathcal{K}}- \kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}[K], \\
{\widehat}{L}_2 & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}({\mathcal{K}}^*-\bar \kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}^* {\mathcal{D}}^* ({\mathcal{K}}^*-\bar \kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}^* {\mathcal{D}}^* [{\widehat}{K}] - ({\mathcal{K}}^*- \bar\kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}^* {\mathcal{D}}^*{\mathcal{P}_{\mathcal{K}}}^*{\mathcal{D}}^* [{\widehat}{K}]. \end{aligned}$$
In the previous lemma and in the following, the implicit constants in the comparison relation $\lesssim$ and in ${\mathcal{O}}$ depend only on $C$ from .
We first establish the relations and for the eigenvectors. The eigenvector relation ${\mathcal{L}}[L] = \lambda L$ together with the definition $\delta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}\lambda - \kappa$ yield $$\label{eq:delta_a}
{\mathcal{K}}{\mathcal{Q}_{\mathcal{K}}}[L] + {\mathcal{D}}[L] = \delta K + \beta {\mathcal{Q}_{\mathcal{K}}}[L ].$$ Here, we also employed ${\mathcal{K}}[K]= \kappa K$ and $ L = K + {\mathcal{Q}_{\mathcal{K}}}[L]$.
By applying $({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}$ in , we get $$\label{eq:Q_b}
{\mathcal{Q}_{\mathcal{K}}}[L] = - ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[L ] + \delta ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}[L].$$ This relation immediately implies $${\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert} \lesssim {\lVert {\mathcal{D}}\rVert} + {\lvert \delta \rvert}{\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert}.$$ Thus, we obtain ${\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert} \lesssim {\lVert {\mathcal{D}}\rVert}$ by choosing ${\varepsilon}$ sufficiently small. Hence, implies $${\lvert \delta \rvert} {\lVert K \rVert} \leq {\lvert \beta \rvert}{\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert} + {\lVert {\mathcal{K}}{\mathcal{Q}_{\mathcal{K}}}[L] \rVert} + {\lVert {\mathcal{D}}[L] \rVert} = {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}).$$ Since ${\lVert K \rVert} \geq {\lVert L \rVert} - {\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert} \geq 1/2$ for sufficiently small ${\varepsilon}$, we conclude $${\lvert \delta \rvert} \lesssim {\lVert {\mathcal{D}}\rVert}.$$
We start from $ L = K + {\mathcal{Q}_{\mathcal{K}}}[L]$ and iteratively replace ${\mathcal{Q}_{\mathcal{K}}}[L]$ by using to obtain $$\label{eq:proof_expansion_b}
\hspace*{-0.20cm}
\begin{aligned}
L & = K - ({\mathcal{K}}- \kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[L] + \delta ({\mathcal{K}}- \kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}[L] \\
& = K - ({\mathcal{K}}- \kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[L] - \delta ({\mathcal{K}}- \kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[L] + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3) \\
& = K -({\mathcal{K}}- \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] + ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] - \delta ({\mathcal{K}}- \kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3)\\
& = K -({\mathcal{K}}- \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] + ({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}({\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] - ({\mathcal{K}}- \kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}{\mathcal{P}_{\mathcal{K}}}{\mathcal{D}}[K] + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3) .
\end{aligned}$$ Here, we also used that ${\lvert \delta \rvert} + {\lVert {\mathcal{Q}_{\mathcal{K}}}[L] \rVert} ={\mathcal{O}}({\lVert {\mathcal{D}}\rVert})$. The last step in follows from $$\delta K = {\mathcal{D}}[L] + {\mathcal{K}}{\mathcal{Q}_{\mathcal{K}}}[L] - \beta {\mathcal{Q}_{\mathcal{K}}}[L] = {\mathcal{D}}[K] + ({\mathcal{K}}- \kappa){\mathcal{Q}_{\mathcal{K}}}[L] + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^2) = {\mathcal{D}}[K] - {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] +{\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^2),$$ which is a consequence of and . This completes the proof of . A completely analogous argument yields .
For the proof of , we first define $L_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}-({\mathcal{K}}- \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K]$ and ${\widehat}{L}_1 {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt\hbox{\scriptsize.}\hbox{\scriptsize.}}}=}-({\mathcal{K}}^*- \bar \kappa)^{-1}{\mathcal{Q}_{\mathcal{K}}}^* {\mathcal{D}}^* [{\widehat}{K}]$. Using and in the relation $\beta {\langle{{\widehat}{L}} \mspace{2mu}, {L}\rangle} = {\langle{{\widehat}{L}} \mspace{2mu}, {({\mathcal{K}}+{\mathcal{D}})[L]}\rangle}$ yields $$\begin{aligned}
\beta {\langle{{\widehat}{L}} \mspace{2mu}, {L}\rangle} =\, & \kappa {\langle{{\widehat}{K}} \mspace{2mu}, {K}\rangle} + \kappa {\langle{{\widehat}{L}_1} \mspace{2mu}, {K}\rangle} + \kappa {\langle{{\widehat}{L}_2} \mspace{2mu}, {K}\rangle} + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{K}}[L_1]}\rangle} + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{K}}[L_2]}\rangle} + {\langle{{\widehat}{L}_1} \mspace{2mu}, {{\mathcal{K}}[L_1]}\rangle} \\
& \qquad \qquad \qquad + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}[K]}\rangle} + {\langle{{\widehat}{L}_1} \mspace{2mu}, {{\mathcal{D}}[K]}\rangle} + {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}[L_1]}\rangle} + {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^3)
\end{aligned}$$ since $L_1, {\widehat}{L}_1 = {\mathcal{O}}({\lVert {\mathcal{D}}\rVert})$ and $L_2, {\widehat}{L}_2 = {\mathcal{O}}({\lVert {\mathcal{D}}\rVert}^2)$. We remark that ${\langle{{\widehat}{L}_1} \mspace{2mu}, {K}\rangle} = {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{K}}[L_1]}\rangle} = {\langle{{\widehat}{L}_2} \mspace{2mu}, {K}\rangle} = {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{K}}[L_2]}\rangle} = 0$ since ${\mathcal{Q}_{\mathcal{K}}}[K] = 0$ and ${\mathcal{Q}_{\mathcal{K}}}^*[{\widehat}{K}] = 0$. For the remaining terms, we get $${\langle{{\widehat}{L}_1} \mspace{2mu}, {{\mathcal{K}}[L_1]}\rangle} = {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}{\mathcal{K}}( {\mathcal{K}}-\kappa)^{-2} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K] }\rangle}, \qquad \qquad
{\langle{{\widehat}{L}_1} \mspace{2mu}, {{\mathcal{D}}[K]}\rangle} = {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}[L_1]}\rangle} = - {\langle{{\widehat}{K}} \mspace{2mu}, {{\mathcal{D}}( {\mathcal{K}}-\kappa)^{-1} {\mathcal{Q}_{\mathcal{K}}}{\mathcal{D}}[K]}\rangle}.$$ Therefore, a simple computation yields . This completes the proof of Lemma \[lem:perturbation\_theory\_2\].
[^1]: The original proof in [@Girko1984] was not considered complete and Bai published a clean version under more restrictive conditions [@bai1997]. Girko informed us that an extended version of his original proof with explanations and corrections will appear soon [@GirkoBookSoon Chapter 6].
[^2]: $X$ here should not be confused with the original random matrix.
[^3]: Note that the operator norm on ${\mathbb{C}}^{2n\times 2n}$ induced by the Euclidean norm on ${\mathbb{C}}^{2n}$ was denoted by ${\lVert {{\,\cdot\,}}\rVert}_2$ in [@Kronecker].
|
---
abstract: 'In recent years, mono-layers and multi-layers of hexagonal boron nitride (hBN) have been demonstrated as host materials for localized atomic defects that can be used as emitters for ultra-bright, non-classical light. The origin of the emission, however, is still subject to debate. Based on measurements of photon statistics, lifetime and polarization on selected emitters we find that these atomic defects do not act as pure single photon emitters. Our results strongly and consistently indicate that each zero phonon line of individual emitters is comprised of two independent electronic transitions. These results give new insights into the nature of the observed emission and hint at a double defect nature of emitters in multi-layer hBN.'
author:
- Alexander Bommer
- Christoph Becher
title: |
Origin of nonclassical light emission\
from defects in multi-layer hexagonal boron nitride
---
Introduction
============
Recently, two dimensional van der Waals materials have emerged as promising platforms for optoelectronics [@Srivastava2015; @Palacios-Berraquero2016; @Chakraborty2015], candidates for future UV-LEDs [@Kenji2011; @Watanabe2009] and host materials for emitters of non-classical light [@Tonndorf2015; @Koperski2015; @Tran2015; @Tran2016a; @Tran2016b; @Martinez2016; @Chejanovsky2016; @Schell2016; @Jungwirth2016; @Exarhos2017]. Especially atomic defects in hexagonal boron nitride have shown to belong to the brightest emitters of non-classical light ever reported. hBN is a semiconductor with a large band gap of around 6eV [@Cassabois2016]. Therefore, it is widely believed, that at the origin of the emission are localized defects in the host material that give rise to electronic transitions between discrete energy levels within the band gap, as it is the case for color centers in diamond [@Doherty2013; @Neu2011]. However, the exact nature of the defects still remains unclear and is subject of ongoing experimental and theoretical investigations [@Tawfik2017; @Abdi2018; @Sajid2018; @Lopez2018]. For application in quantum information one needs narrowband and background free emission lines. The emitters selected in this work fulfill these criteria and exhibit spectra consisting of an asymmetric zero phonon line (ZPL) and a phonon side band 165meV red shifted from the ZPL. This energy shift corresponds to a well-known phonon mode in hBN [@Geick1966; @Reich2005; @Nemanich1981]. The asymmetry of the ZPL is commonly attributed to phonon interaction and the ZPL wavelengths have been shown to spread across a range from 500-800nm [@Dietrich2018], which is attributed to strain inside the host crystal [@Tran2016a; @Grosso2017]. Independent of the emission wavelength, the ZPL is assumed to consist of a single, linearly polarized dipole transition giving rise to single photon emission. In this publication, on the contrary, we provide strong evidence for the presence of two independent emitters in each defect and show that the second line causing the asymmetry of the ZPL indeed is a second electronic transition. By carefully evaluating photon correlation measurements we see that we only are able to fully reproduce our data by using an extended g$^{(2)}$-function, that takes into account two independent transitions. We gain full access to the parameters of the g$^{(2)}$-function via independently measuring the spectra and the excitation power dependent photon emission rates of the corresponding emitters. We further confirm the existence of double defects via measuring polarization dependent spectra and performing time correlated single photon counting (TCSPC) measurements.
Investigation of emission from point defects in hBN
===================================================
We spectroscopically investigate micrometer sized multi-layer flakes of hexagonal boron nitride in a home built laser scanning confocal microscope under continuous wave excitation at $\lambda$ = 532nm. The commercially available flakes (*Graphene Supermarket*) are diluted in a solution (50% water, 50% ethanol) with a concentration of 5.5mg/ml and put in an ultrasonic bath to break up agglomerates. The solution is drop cast (5-10$\mu$l) onto a silicon wafer with an iridium layer for enhanced photon collection efficiency. The substrate is heated on a hotplate to 70$^{\circ}$C to evaporate the liquid. After drop casting, individual flakes can be addressed in the confocal microscope.
![a,d,g) Typical spectra of three defects (E1,E2,E3) in hBN consisting of four Lorentzian lines; b,e,f) Saturation measurements on the defects from a),d) and g) with no significant background contribution; c,f,i) g$^{(2)}$-intensity correlation measurements on the defects shown in a),d) and g). Photons are collected from the spectral regions enclosed in dashed lines in a),d) and g), respectively. See main text for details.[]{data-label="fig:g"}](fig1_opt.pdf){width="50.00000%"}
Fig.\[fig:g\]a,d,g) show typical spectra of point emitters inside the flakes. Although they differ in their central wavelengths, their spectral shapes are very similar. The spectra are fit with four Lorentzian lines which we will discuss later in closer detail. Saturation measurements in Fig.\[fig:g\]b,e,h) show typical saturation count rates ($\approx$ 1-2Mcts/s) and saturation powers ($\approx$ 1mW) of these emitters, in good agreement to previous reports [@Tran2015; @Tran2016a; @Tran2016b]. The red lines are fits according to $$I(P) = \frac{I_{\text{sat}}\cdot P}{P_{\text{sat}}+P} + C_{\text{back}}\cdot P.$$ Here, $I_{\text{sat}}$ and $P_{\text{sat}}$ are the saturation count rates and saturation powers of the emitters, whereas $C_{\text{back}}$ describes a potential contribution due to linear background emission stemming from the host material. Note, that this contribution is negligible in the presented data. This is in accordance with the very clean spectra presented in Fig.\[fig:g\]a,d,g), where also no significant background contribution is visible. Contrary to these findings, Fig.\[fig:back\]a) shows a spectrum, which clearly contains additional background emission. In approximately one out of fifty flakes background-free emission can be found. This background emission is also visible as a prominent linear increase in a corresponding saturation measurement in Fig.\[fig:back\]b). Note, that the saturation measurements are always taken including all four lines. As a last step, we perform g$^{(2)}$-photon correlation measurements (Fig.\[fig:g\]c,f,i) to get information about the photon statistics. Even though we did not observe any background emission in all previous measurements, we further reduce the spectral window from which we collect photons for the g$^{(2)}$-measurements to the region of the ZPL (regions enclosed by dashed lines in the corresponding spectra) and take the measurements at excitation powers far lower than the emitters’ saturation powers. It strikes the eye, that despite vanishing background fluorescence, the g$^{(2)}$-functions do not vanish at all at zero time delay as one would expect for an ideal single photon source. Fig.\[fig:back\]d,e,f) further shows an example, where background emission from the host material is present but becomes relevant only at about 20xP$_{\text{sat}}$. Nevertheless, for almost vanishing excitation power (P=3,5$\mu$W), the value of g$^{(2)}$(0) is still much larger than zero. As we show below also the timing jitter of the photon detectors does not explain the deviation from ideal single photon statistics as the emitter fluorescence lifetime is larger than the jitter. Instead, we have to assume that the asymmetric shape of the ZPL is due to the presence of two independent emission lines.\
In the following we develop a model model for the photon correlation functions that, besides background emission and the timing jitter of the photon detector, accounts for the presence of a second emitter and prove that this model fully reproduces the measurements. We start with the well-known g$^{(2)}$-function for a three level system: $$g_i^{(2)}(\tau) = 1-(1+a)\cdot e^{-\frac{|\tau|}{\tau_1}}+a\cdot e^{-\frac{|\tau|}{\tau_2}}
\label{g2-dreiniveau}$$ We now, step by step, include all experimental parameters, that influence the shape of the g$^{(2)}$-function: Although negligible in the presented data (but not in general), we start with uncorrelated background emission, that can be extracted from saturation measurements. Including this into the model, the g$^{(2)}$-function reads [@Brouri2000]: $$g_p^{(2)}(\tau) =
\frac{1}{p^2}\cdot\left[g_i^{(2)}(\tau)-(1-p^2)\right]
\label{g2pf}$$ Here, $p$ is the fraction of measured photons stemming from the emitter compared to the measured total count rate. Note, that one should also consider dark counts of the detector in the description. In our case, these dark counts ($\approx$ 100-200 cts/s) are negligible compared to the signal from the emitters. Second, we include the timing jitter $\sigma$ of the counting electronics. This jitter is an uncertainty in the time between the arrival and the detection of a single photon and has been measured via ultra-fast laser pulses ($\sigma \approx 490$ps). It is included via the convolution of equation \[g2pf\] with the Gaussian-shape of the instrument response function IRF(t). $$g_{p,j}^{(2)}(\tau) = \text{IRF}(\tau)*g_p^{(2)}(\tau) =
\int_{-\infty}^{\infty}\text{IRF}(\tau)\cdot g_p^{(2)}(\tau-t) dt
\label{eq:pj}$$ Equation \[eq:pj\] is the final description for the case that we collect emission from exactly one single emitter. The blue solid lines in Fig.\[fig:g\]c,f,i) are fits to the data according to this model. It strikes the eye that this function is not able to reproduce the data. In particular, the model demands a much lower value for g$^{(2)}$(0) than it is provided by the data. We want to stress that we also can reproduce the data by taking the signal to background ratio $p$ as a fit parameter. This, however, strongly contradicts our findings of vanishing background in the spectrum and the saturation measurement.\
Therefore, as a last step, we also take into account the influence of a second emitter in the detection focal volume.
![a) Spectrum of an emitter in hBN (E4) with a clear background contribution; b) Saturation measurement on the emitter in a). The background contribution is visible as prominent linear increase in the emission rate at increasing excitation powers; c) g$^{(2)}$-function on an hBN emitter (E3) showing typical bunching timescales of several hundreds of microseconds; d,e,f) Spectrum, Saturation measurement and g$^{(2)}$-function of an emitter (E5) with a clean spectrum. Background contribution becomes relevant at about 20xP$_{\text{sat}}$. Still, g$^{(2)}$(0) is strongly limited even at almost vanishing excitation powers.[]{data-label="fig:back"}](fig2_opt.pdf){width="50.00000%"}
Let $I_{\text{tot}}=I_1+I_2$ be the total detected emission with $I_1=z\cdot I_{\text{tot}}$ and $I_2=(1-z)\cdot I_{\text{tot}}$ being the relative fractions of the emission of emitter 1 and emitter 2 respectively. This leads to $$\begin{aligned}
g^{(2)}(\tau)=\frac{\langle
I_{\text{tot}}(t)I_{\text{tot}}(t+\tau)\rangle}{\langle
I_{\text{tot}}(t)\rangle^2}\nonumber\\
= z^2\cdot g_1^2(\tau)+(1-z)^2\cdot g_2^2(\tau) + \\
\underbrace{\frac{\langle I_2(t)I_1(t+\tau)\rangle}{\langle
I_{\text{ges}}(t)\rangle^2}+\frac{\langle I_1(t)I_2(t+
\tau)\rangle}{\langle
I_{\text{ges}}(t)\rangle^2}}_{g^2_{\text{mix}}}\nonumber.\end{aligned}$$ In order to reduce the number of fit parameters, we assume g$_1^2(\tau) =$ g$_2^2(\tau)$. Because of the independence of $I_1$ and $I_2$, the two mixing terms will be constant for all $\tau$ and by making the assumption, that g$_1^2(0) =$ g$_2^2(0)$=$0$, we find g$^2_{\text{mix}}=2z(1-z)$. We eventually arrive at $$g^{(2)}(\tau) = (1-2z(1-z))g_{p,j}^{(2)}(\tau) + 2z(1-z).
\label{eq:Full}$$ In contrast to reports in literature, where the asymmetry of the ZPL is attributed to phonon interaction [@Tran2015], we here fully reproduce the lineshape by fitting two Lorentzian lines representing two independent electronic transitions. By calculating the areas under the individual Lorentzians, we get information about the relative oscillator strengths of both emitters, corresponding to the parameter $z$ in equation \[eq:Full\] (numbers also given in the spectra in Fig.\[fig:g\] and Fig.\[fig:back\]). By taking into account the double emission spectrum within the model for the g$^{(2)}$-function we are able to perfectly describe the measured photon correlation data (solid red lines in Fig.\[fig:g\]c,f,i and Fig.\[fig:back\]c,f).
![Spectral line position data extracted from 30 emitters that show comparable spectral fingerprints as explained in the main text. Due to strain in the material, the central wavelengths of peak 1 range from 600nm - 720nm. However, independent of the wavelengths, the energy distance between the lines roughly remain constant. $\Delta$E$_{12}$ = 12(10)meV, $\Delta$E$_{13}$ = 158(17)meV, $\Delta$E$_{14}$ = 187(15)meV, $\Delta$E$_{24}$ = 174(15)meV. The semi-transparent, horizontal lines are a guide to the eye.[]{data-label="fig:spec"}](fig3_opt.pdf){width="50.00000%"}
Interestingly, our photon correlation measurements correspond perfectly to reports in literature in terms of bunching dynamics and dips in the g$^{(2)}$-function at zero time delay [@Tran2015; @Tran2016a; @Tran2016b]. Non-vanishing values of g$^{(2)}$(0) in these reports were always attributed to residual background fluorescence which, however, is not further defined or shown. To our knowledge, the full set of information needed to accurately describe the situation has never been reported [@Tran2015; @Tran2016b; @Mendelson2018; @Choi2016]. Furthermore, we want to point out that most of the emitters measured in this work show very strong bunching on a timescale of several hundreds of microseconds up to milliseconds as it has been shown in previous work (see for example Fig.\[fig:back\]c) [@Tran2016a; @Tran2016b]. Therefore, a proper normalization of the g$^{(2)}$-function to the constant number of events for long time delays $\tau$ or to the recorded photon count rates is imperative. The absence of satisfactorily explanations in literature and the excellent agreement of measured photon correlation functions with the double defect model suggest that most probably the majority of *single* emitters in literature are indeed double defects.\
We now turn in closer detail to the emitters optical spectra providing further evidence for our model. Fig.\[fig:spec\] shows normalized emission spectra of a collection of emitters in the multi-layer flakes under investigation. The central wavelength $\lambda_1$ of the highest energy line (line 1) ranges from 600-720nm. This wavelength range is limited by the spectral filter window in which we collect fluorescence. On the y-axis the energy separations between all lines are shown, where the energy of line 1 (black) is always set to zero. It strikes the eye, that the energy distances between the lines remain approximately constant independent of the central wavelength of line 1 in the spectrum. In literature, the spectrum is described as an asymmetric zero phonon line with a red shifted (165meV) phonon side band the energy of which belongs to a well-known phonon mode in hBN [@Geick1966; @Reich2005; @Nemanich1981]. We here first state that there are actually two ZPLs (line 1, black and line 2, red) with two phonon side bands (line 3, blue and line 4, orange). Averaged over all observed emitters with this particular spectral fingerprint, the energy difference between line 1 (black) and line 3 (blue) amounts to $\Delta$E$_{13}$=158(17)meV, whereas the distance between line 2 (red) and line 4 (orange) yields $\Delta$E$_{24}$=174(15)meV. Within the error bars, both values match the phonon mode at 165meV. Staying in our picture of line 1 and 2 being electronic transitions, we thus attribute the lines 3 and 4 to be their respective phonon side bands.
![a,c) Spectra of two typical emitter in hBN. The solid, colored lines are Lorentzian fits to the data. b,d) Lifetime measurements on the emitters shown in a) and c). The solid lines are fits according to a mono-exponential (blue) and bi-exponential (red) decay including the instrument response function of the setup. Both measurements follow a bi-exponential decay with time constants $t_1$=0.82ns, $t_2$=4.0ns and $t_1$=1.7ns, $t_2$=4.5ns. e,f) Polarization dependent optical spectra of two typical hBN emitter in emission. The line shape strongly depends on the angle of an polarizer in the detection path.[]{data-label="fig:LP"}](fig4_opt.pdf){width="50.00000%"}
Next, we perform TCSPC-measurements to gain further information about the lifetimes of the excited states of the investigated emitters. Two electronic transitions with potentially differing lifetimes should be visible as a bi-exponential decay. For the TCSPC-measurements we use a white light laser filtered to 532nm, with a pulse duration of 200ps and a pulse repetition rate of 10MHz. Measurements on two typical hBN emitters (Fig.\[fig:LP\]a,c) are shown in Fig.\[fig:LP\]b,d) (black dots). The solid lines are fits according to $$L(t) =
y_0+\left(1-\text{erf}\left(-\frac{t-t_0}{\sigma}\right)\right)\cdot\sum_{i=1}^n
A_i\cdot \displaystyle{\text{e}^{-\frac{t-t_0}{t_i}}}$$ with one (blue, $n$=1) and two (red, $n$=2) time constants. As for the g$^{(2)}$-functions the instrument response function IRF(t) with a timing jitter of $\sigma$=490ps is included into the fit function via convoluting the Gaussian IRF(t) with the exponential decay of the electronic transition. In both measurements, the data points clearly follow a bi-exponential decay. The observed time constants ($t_1$=0.82ns, $t_2$=4.0ns and $t_1$=1.7ns, $t_2$=4.5ns) correspond to the range of typical lifetimes observed for this type of emitters [@Tran2015; @Tran2016a; @Tran2016b]. In literature, however, usually just a single exponential decay is used to fit the data in a regime between 2-10ns and the full information about the timing resolution of the setup is not considered. The presence of two time constants of the same order of magnitude further indicates the existence of two excited states in the defect and corresponds perfectly with the assumption of two independent emitters in the same defect.\
As a last step we now turn to the polarization of the defect emission. Linear excitation dipoles with visibilities between 20% and 80% have been reported whereas the emission dipole are supposed to show close to unity visibility and are linearly polarized [@Tran2015; @Exarhos2017]. There is, however, a difference in the relative orientations of the excitation and emission dipoles between 30$^{\circ}$ to 90$^{\circ}$ [@Exarhos2017; @Choi2016]. However, to our knowledge, polarization dependent spectra have not been investigated in literature yet. Fig. \[fig:LP\]e,f) show normalized emission spectra of two different hBN emitters with ZPL (line 1) at around 790nm and 650nm where different curves correspond to different settings of the polarization analyzer in the detection path. One can clearly see that dependent on the angle of a linear polarizer in the detection path, the line shape of the dominant line in the spectrum strongly varies. This indicates that here the two dipoles contributing to the ZPL have different relative polarizations. Note, that we also can find spectra in which the line shape does not change significantly upon changing the detection angle of the polarization analyzer.
Conclusions
===========
In summary, we presented new insights into the nature of non-classical light emission from defects in multi-layer flakes of hexagonal boron nitride. Via careful evaluation of g$^{(2)}$-photon correlation measurements, TCSPC-measurements and polarization dependent emission spectra we gather strong evidence that, in contrast to previous reports, these atomic defects are no single emitter systems but are comprised of two independent emitting systems here coined as “double defect”. We draw this conclusion via collecting all necessary information to describe the photon statistics through independent measurements of the background contribution, the timing jitter of the counting electronic and the spectra of the emitters. Interestingly, our photon correlation measurements correspond perfectly to reports in literature in terms of bunching dynamics and dips in the g$^{(2)}$-function at zero time delay.\
Our assumptions are corroborated by the decomposition of the asymmetric ZPL into two Lorentzian lines, both describing one individual electronic transition. Based on the existence of a characteristic phonon mode of hBN at 165meV, we were able to assign the two dominant lines in the phonon side band to each of the electronic transitions. Eventually, the presence of a bi-exponential decay in TCSPC measurements and polarization dependent emission spectra further support our model. We want to point out that our measured photon correlation functions perfectly correspond to the ones previously reported in literature. Based on these results we have to assume that many of the reported single photon emitters consist of “double defects”’ as described in this publication.
Acknowledgement
===============
The authors want to thank Johannes Görlitz, Benjamin Kambs, Dennis Herrmann, Igor Aharonovich, Dirk Englund, Lee Bassett and Adam Gali for helpful discussions. This work was partially funded by the European Union 7th Framework Program under Grant Agreement No. 61807 (WASPS).
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abstract: 'Assuming conformally flat metric we obtain inhomogeneous solutions of Einstein equations with the energy-momentum of a viscous fluid. We suggest that the viscous solution can be applied as a model of an expanding inhomogeneous dark energy.'
author:
- |
Z. Haba\
Institute of Theoretical Physics, University of Wroclaw,\
50-204 Wroclaw, Plac Maxa Borna 9, Poland
title: An inhomogeneous solution of Einstein equations with viscous fluids
---
Introduction
============
Einstein equations are studied with various assumptions on the energy-momentum of “matter”. The quantum form of the equations with the energy-momentum of the fields of the standard model cannot be realized because of the difficulties with the quantization of the gravitational field. We must use approximations for the rhs of Einstein equations. The most studied models involve a perfect fluid on the rhs of Einstein equations. The solutions can describe an expanding universe or compact objects like the static as well as expanding and collapsing stars [@gravitation][@weinberg] (for inhomogeneous solutions and their relevance in gravity see [@solutions][@bolejko][@flan][@mukh][@ellis]). However, it is obvious that the assumption of a perfect fluid on the rhs of Einstein equations, usually applied in gravitational models, is an idealization. All physical fluids at high temperature have a non-zero viscosity. It is not simple to include viscosity in the solutions of Einstein equations. Concerning the homogeneous solutions, it is known that the FLWR form of the solution admits only the bulk viscosity. The effect of the bulk viscosity on the expansion of the universe has been discussed in [@weinberg][@klimek] [@odintsov][@odintsov2]. The introduction of the bulk viscosity modifies the formula for the pressure which leads to interesting reformulations in the dynamics and thermodynamics depending on the change of the equation of state [@odintsov][@odintsov2][@brevik0]. There are many examples of inhomogeneous solutions of Einstein equations with a perfect fluid on the rhs [@solutions][@bolejko]. However, it seems that no examples are known of inhomogeneous solutions with a shear viscosity of the fluid. Recently the relativistic fluid equations have been studied in the heavy-ion physics (with a suggestion of a possible simulation of the Big Bang, see the review [@hunperfect]). Some solutions of the hydrodynamics equations for the perfect fluid have been obtained, (see ,e.g., [@hunperfect2]). Little is known about solutions with viscosity (see however [@hungarian]). Another approach to such models is developed in refs.[@ion1][@ion2][@ion3] where the Einstein tensor for various metrics is interpreted as an energy-momentum tensor of a fluid (including possibly a viscous fluid).
In this paper we assume that the metric is conformally flat. A large class of models can be expressed in a conformally flat form including the FLWR solutions [@infeld] [@taubes]. However, the FLWR metric admits only bulk viscosity of the fluid. We obtain conformally flat inhomogeneous solutions with a non-zero shear and bulk viscosity which satisfies the dark energy equation of state $\rho=-p$. It is known that if the equation of state for the energy-momentum of a perfect fluid has the form $\rho=-p$ then from the conservation law it follows that $\rho=const$. In such a case the dark energy is just the cosmological constant. We obtain a solution in the form of an expanding viscous fluid. The expansion is at lower rate than the expansion of radiation and of the dust. The resulting viscous fluid could be a candidate for a dark energy dominating the energy-momentum at large time.
The energy-momentum tensor
==========================
We consider Einstein equations $$G_{\mu\nu}=8\pi G T_{\mu\nu},$$ where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor and $T_{\mu\nu}$ is the energy-momentum. We set the velocity of light c=1 and the length will be measured in Planck units $\sqrt{8\pi G}$ (so coordinates will be dimensionless), hence we set $\sqrt{8\pi G}=1$ from now on. In the case of a dust with a density $\rho$ the energy-momentum $T_{\mu\nu}=\rho v_{\mu}v_{\nu}$ is conserved $\nabla_{\mu}T^{\mu\nu}=0$ (together with the current $\nabla_{\mu}(\rho v^{\mu})=0$), where the relativistic velocity $v_{\mu}$ satisfies the equation $$g^{\mu\nu}v_{\mu}v_{\nu}=1.$$ We can apply the Hamilton-Jacobi theory to express the velocity by the action S, $v_{\mu}=\frac{\partial S}{\partial
x^{\mu}}$. Then, equation (2) reads $$g^{\mu\nu}\frac{\partial S}{\partial x^{\mu}}\frac{\partial
S}{\partial x^{\nu}}=1$$ Eqs.(1) and (3) can be considered as a system of equations which determine the metric $g_{\mu\nu}$ and the fluid velocities $v_{\mu}$. We extend this scheme to the viscous energy-momentum $$\begin{array}{l}T_{\mu\nu}=(\rho+p)v_{\mu}v_{\nu}
-g_{\mu\nu}p-\eta_{g}
(\nabla_{\mu}v_{\nu}+\nabla_{\nu}v_{\mu})\cr-\gamma_{g}
g_{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}v_{\beta},
\end{array}$$where $p$ is the pressure. In physical fluids the shear $\eta$ and bulk $\gamma$ viscosities depend in an involved way on the temperature and density of the fluids. In our solution the viscosities depend on the metric through its scale factor (the viscosities are decreasing with an expansion of the fluid).
We use the decomposition $$g^{\alpha\beta}=v^{\alpha}v^{\beta}+H^{\alpha\beta}$$ where $$H^{\alpha\beta}=g^{\alpha\beta}-v^{\alpha}v^{\beta}$$ in order to rewrite the energy-momentum (4) in the standard form of Landau-Lifshitz [@LL] and Weinberg [@weinberg][@weinberg2] $$\begin{array}{l}
T^{\mu\nu} =(p_{e}+\rho_{e})v^{\mu}v^{\nu}-p_{e}g^{\mu\nu}\cr
-\eta_{g}
H^{\mu\alpha}H^{\nu\beta}(\nabla_{\alpha}v_{\beta}+\nabla_{\beta}v_{\alpha}
-\frac{2}{3}g_{\alpha\beta}g^{\sigma\lambda}\nabla_{\sigma}v_{\lambda})\cr
-\gamma_{g}H^{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}v_{\beta}-
\kappa_{g}
(H^{\mu\lambda}v^{\nu}+H^{\nu\lambda}v^{\mu})Q_{\lambda},
\end{array}$$ where the heat current $Q$ is $$Q_{\lambda}=\partial_{\lambda}T_{g}+T_{g}v^{\alpha}\nabla_{\alpha}v_{\lambda}$$ with the heat conductivity $\kappa_{g}$ and temperature $T_{g}$ satisfying the relations $\partial_{\lambda}T_{g}=0$ and $\kappa_{g} T_{g}= \eta_{g}$. So, $$\kappa_{g} Q_{\lambda}=\eta_{g}
v^{\alpha}\nabla_{\alpha}v_{\lambda}.$$ The effective pressure and density are $$p_{e}=p-\frac{2}{3}\eta_{g}
g^{\sigma\lambda}\nabla_{\sigma}v_{\lambda},$$$$\rho_{e}=\rho-\gamma_{g}
g^{\sigma\lambda}\nabla_{\sigma}v_{\lambda}.$$ We shall require for our solution in sec.4 that $\rho+p=0$ then $$\rho_{e}+p_{e}=-(\frac{2}{3}\eta_{g}+\gamma_{g})
g^{\sigma\lambda}\nabla_{\sigma}v_{\lambda}.$$ As is well-known [@LL][@weinberg] the form (5) of the energy-momentum with positive $\eta, \kappa$ and $\gamma$ ensures the increase of the entropy.
If eqs.(1) are to be non-contradictory we must have $$g^{\alpha\mu}\nabla_{\alpha}T_{\mu\nu}=0.$$ Eqs.(10) can be considered as differential equations (relativistic Navier-Stokes equations) relating $\rho,p$ and $v$. Eqs.(10) with the metric $g_{\mu\nu}$ solving Einstein equations (1) follow from the Einstein equations. However, when $v_{\alpha}$ satisfy (10) on a certain manifold with the metrics $g_{\mu\nu}$ then only a subset of $(g_{\mu\nu},v_{\alpha})$ will satisfy both eq.(10) and eq.(1).
The conformally flat metric
===========================
We consider the conformally flat metric in four space-time dimensions $$ds^{2}=a(x)^{2}(d\tau^{2}-d{\bf
x}^{2})=a(x)^{2}\eta^{\mu\nu}dx_{\mu}dx_{\nu},$$where $\eta^{\mu\nu}$ is the Minkowski metric. Then, the Einstein tensor can be expressed in the form [@blaschke][@solutions][@ijgm]
$$G_{\mu\nu}=(\tilde{\rho}+\tilde{p})\tilde{u}_{\mu}\tilde{u}_{\nu}-\tilde{p}g_{\mu\nu}+\Pi_{\mu\nu},$$
where the fluid velocity is defined by $$\tilde{u}_{\mu}=\partial_{\mu}a
\Big(g^{\mu\nu}\partial_{\mu}a\partial_{\nu}a\Big)^{-\frac{1}{2}}.$$ The energy density is $$\tilde{\rho}=3a^{-2}g^{\mu\nu}\partial_{\mu}a\partial_{\nu}a-g^{\mu\nu}\Pi_{\mu\nu}$$ the pressure $$\tilde{p}=a^{-2}g^{\mu\nu}\partial_{\mu}a\partial_{\nu}a+g^{\mu\nu}\Pi_{\mu\nu},$$ where $$\Pi_{\mu\nu}=-2a^{-1}\partial_{\mu}\partial_{\nu}a.$$
Inserting the velocities (13) in eq.(12) we can also express $G_{\mu\nu}$ in the form $$G_{\mu\nu}=4a^{-2}\partial_{\mu}a\partial_{\nu}a-\tilde{p}g_{\mu\nu}+\Pi_{\mu\nu},$$ Eqs.(12)-(16) suggest that $\tilde{u}_{\mu}\simeq \partial_{\mu}a$ can be a solution of Einstein equations (1). In the next section we show that this is really the case.
Einstein equations with a viscous fluid
========================================
In eq.(4) the covariant derivative is expressed by the Christoffel connection in conformally flat space [@blaschke](from now on the indices will be raised by means of the Minkowski metric) $$\Gamma^{\lambda}_{\mu\nu}=\delta^{\lambda}_{\mu}\partial_{\nu}\ln
a+\delta^{\lambda}_{\nu}\partial_{\mu}\ln a
-\eta_{\mu\nu}\partial^{\lambda}\ln a$$
It is easy to see that the equation $$G_{0j}=T_{0j}=-\eta_{g}(\nabla_{j}v_{0}+\nabla_{0}v_{j})$$ is satisfied if the equation of state is $$\rho+p=0$$the velocity $$v_{\mu}=\frac{\partial}{\partial x^{\mu}}a$$ and$$\eta_{g}=a^{-1}.$$ However, if the solution (21) of eq.(19) is to be the relativistic velocity then the normalization (2) must be satisfied i.e. $g^{\mu\nu}\partial_{\mu}a\partial_{\nu}a=1$. We write $a=\exp(\psi)$. Then, on the basis of eqs.(2) and (21) $\psi$ satisfies the Hamilton-Jacobi equation $$\eta^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi=1.$$ This is the Hamilton-Jacobi equation (as described at eq.(3)) for a free particle (with the unit mass) in the Minkowski space (its velocity is $v_{\mu}=\partial_{\mu}\psi$).
The remaining components of the energy-momentum tensor are $$T_{00}=-pa^{2}-2 a^{-1}\nabla_{0}v_{0}-\gamma_{g}
(\nabla_{0}v_{0}-\nabla_{k}v_{k}),$$where
$$\begin{array}{l}
\nabla_{0}v_{0}-\nabla_{k}v_{k}\cr=\partial_{0}v_{0}-\partial_{k}v_{k}+2a^{-1}v_{0}\partial_{0}a
-2a^{-1}v_{k}\partial_{k}a\end{array}$$
and $$\begin{array}{l}
T_{jk}=\delta_{jk}\Big(pa^{2} -2(a^{-2}\partial_{r}a
v_{r}-a^{-2}\partial_{0}av_{0})\cr -\gamma_{g}
(\partial_{0}v_{0}-\partial_{j}v_{j}+2a^{-1}v_{0}a-2\partial_{r}v_{r})\Big)
\cr
-a^{-1}\Big(\partial_{j}v_{k}+\partial_{k}v_{j}-2a^{-1}\partial_{j}av_{k}
-2a^{-1}\partial_{k}av_{j}\Big).
\end{array}$$ It can be seen that the non-diagonal parts of $G_{jk}$ (17) and $T_{jk}$ (26) coincide if the velocity is determined by eq.(21) and $\eta_{g}=a^{-1}$ (eq.(22)). There remain the diagonal ($\delta_{jk}$) parts of eqs.(26) and (17). They are equal if $$p=(-3(a\gamma_{g}+1)-(a\gamma_{g}+2)\partial^{\mu}\partial_{\mu}\psi)\exp(-2\psi).$$ The 00-equation as derived from eqs.(24) and (17) is satisfied if $$-p=(1+2a\gamma_{g})a^{-2}\eta^{\mu\nu}\partial_{\nu}a\partial_{\mu}a
+(2+a\gamma_{g})a^{-1}\eta^{\mu\nu}\partial_{\nu}\partial_{\mu}a$$ It coincides with eq.(27) (which followed from the spatial diagonal part of eq.(4)). So Einstein equations determine $p$ and $v_{\mu}$ in terms of $\psi$ satisfying the Hamilton-Jacobi equation (23). The relativistic Navier-Stokes equations for $v_{\mu}$ follow from eq.(10). These equations can be obtained simply by differentiation of eq.(28) and an insertion of $\partial_{\mu}a=v_{\mu}$ because such a velocity is the solution of these equations.
The energy and pressure are functions of the solution $\psi$ of eq.(23). A construction of the general solution in terms of characteristics is discussed in mathematical literature [@HJ]. The solution can also be obtained by a calculation of the action for the relativistic Hamiltonian $\sqrt{1+{\bf v}^{2}}$ [@arnold]. Let us consider some special cases. $\psi(x)=q_{\mu}x^{\mu}$ with $q_{\mu}q^{\mu}=1$ is the solution of eq.(23). The next example is $\psi(x)=\sqrt{x^{2}}$ where $x^{2}=x_{\mu}x^{\mu}$. It is the solution of eq.(23) for time-like $x$. However, let us note that in these cases the viscosity term in eq.(17) (as well as eq.(2))can be written in the form $$\Pi_{\mu\nu}=(\delta\rho+\delta p)v_{\mu}v_{\nu}-\delta p
g_{\mu\nu}$$with certain $\delta\rho$ and $\delta p$. Hence, the viscosity terms only modify the definition of $\rho$ and $p$. In the first case $\nabla_{\mu}v_{\nu}+\nabla_{\nu}v_{\mu}\simeq
q_{\mu}q_{\nu}\simeq v_{\mu}v_{\nu}$ (the perfect fluids of ref.[@ijgm]). In the second case $\nabla_{\mu}v_{\nu}+\nabla_{\nu}v_{\mu}\simeq v_{\mu}v_{\nu}
+\eta_{\mu\nu}\delta p$ .
A non-trivial viscosity results from the solution $$\psi=\sqrt{C_{2}-\tau-r}\sqrt{C_{1}-\tau+r},$$ where $r=\vert {\bf x}\vert$. It solves eq.(23) if $\tau<C_{1}+r$ and $\tau<C_{2}-r$. We can get a solution for a large time $\tau>C_{1}+r$ and $\tau>-C_{2}+r$ if we write $\psi$ in the form$$\psi=\sqrt{-C_{2}+\tau+r}\sqrt{-C_{1}+\tau-r}.$$If $C_{1}=C_{2}=0$ then we return to the solution $\sqrt{x^{2}}$.
We obtain a local solution of eq.(23) depending on all variables if we treat eq.(23) as the Hamilton-Jacobi equation for the Hamiltonian $\sqrt{1+{\bf v}^{2}}$. Through the separation of variables in cylindrical coordinates $(r,\phi,z)$ we obtain $$\psi=-E\tau+L\phi+f(r)+\gamma z$$with the constant angular velocity $v_{\phi}=L$ and radial velocity $$v_{r}=\frac{df}{dr}=\sqrt
{E^{2}-1-\gamma^{2}-\frac{L^{2}}{r^{2}}}.$$ Hence, $$\begin{array}{l}
f(r)=L\sqrt
{E^{2}-1-\gamma^{2}-\frac{L^{2}}{r^{2}}}\cr+L\arctan\Big(\frac{1}{\sqrt
{E^{2}-1-\gamma^{2}-\frac{L^{2}}{r^{2}}}}\Big).\end{array}$$ $\psi$ of eq.(30) solves eq.(23) if $r^{2}\geq
L^{2}(E^{2}-1-\gamma^{2})^{-1}\geq 0$. This local solution of eq.(23) does not define $a(\psi)=\exp(\psi)$ for all $0\leq\phi
\leq 2\pi$ because $\psi(\phi+2\pi)\neq \psi(\phi)$. However, the local coordinates define the Christoffel symbols (18) depending on $\partial_{\nu}\psi$ which are functions only on $r$.As a consequence the Riemann tensor depends solely on $r$ and is well-defined if $r^{2}\geq L^{2}(E^{2}-1-\gamma^{2})^{-1}\geq 0$. $v_{\nu}=\partial_{\nu}\psi$ of eq.(30) solve Einstein equations (1) (which depend only on derivatives of $\psi$). We expect that there exists another system of neighborhoods with coordinates which extend the cylindrical coordinates $(r,\phi,z)$. However, the range of small $r$ seems to be an essential singularity of the solution (like $r=0$ of the Schwarzschild solution). The function (30) describes the solution of the Hamilton-Jacobi equation corresponding to a relativistic particle moving in the Minkowski space with a forbidden region of large $L^{2}r^{-2}$ [@landau].
Non-singular solution with the viscous fluid
=============================================
We consider a special case of eq.(30) ($L=\gamma=0$, spherical coordinates with $\theta =\frac{\pi}{2}$ and $r=\sqrt{x^{2}+y^{2}+z^{2}}$). A spherically invariant solution of eq.(23) which makes sense for arbitrary $(\tau,r)$ can be expressed as $$\psi=\cosh(\alpha)\tau+\sinh(\alpha)r$$ with an arbitrary real $\alpha$.
For a better physical interpretation let us change coordinates $(\tau,r)\rightarrow (t,R)$ $$t=\exp(\tau
\cosh(\alpha)+r\sinh(\alpha))=a$$ $$R=\tau\cosh(\alpha)+r\frac{\cosh^{2}(\alpha)}{\sinh(\alpha)}.$$ Then, the metric is $$\begin{array}{l}
ds^{2}=dt^{2}-t^{2}\frac{\cosh^{2}(\alpha)}{\sinh^{2}(\alpha)}dR^{2}
\cr-t^{2}\sinh^{2}(\alpha)(R-\ln(t))^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}). \end{array}$$ We can imbed this universe in the Minkowski space-time ($(\tilde{\tau},\tilde{\bf
x})$ with $ \tilde{\tau}\geq \vert\tilde{\bf x}\vert $) introducing the coordinates $$\tilde{\tau}=t\cosh(R\coth(\alpha)),$$ $$\tilde{r}=t\sinh(R\coth(\alpha)).$$ Then $$\begin{array}{l}
ds^{2}=d\tilde{\tau}^{2}-d\tilde{r}^{2}\cr-\frac{1}{4}(\tilde{\tau}^{2}-\tilde{r}^{2})
\sinh^{2}(\alpha)\ln^{2}D(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\end{array}$$ where $$D=(\tilde{\tau}-\tilde{r})^{1+q}(\tilde{\tau}+\tilde{r})^{1-q}=
(\tilde{\tau}^{2}-\tilde{r}^{2})\exp(2q\tilde{\eta}),$$where we introduced the space-time rapidity $\tilde{\eta}$( often applied in a description of heavy ion-collisions [@hunperfect]) $$\tilde{\eta}=\frac{1}{2}\ln\Big((\tilde{\tau}-\tilde{r})(\tilde{\tau}+\tilde{r})^{-1}\Big)$$ and $$q=\tanh(\alpha).$$ In these coordinates the radial velocity (the remaining components of the velocity are zero) has the simple form $$v_{r}=\sqrt{\tilde{\tau}^{2}-\tilde{r}^{2}}\sinh(\alpha).$$ From eq.(27) the energy density of the fluid is $$\rho=(\tilde{\tau}^{2}-\tilde{r}^{2})^{-1}\Big(3(1+a\gamma_{g})+4(2+a\gamma_{g})(\ln(D))^{-1}\Big).$$(the space-time dependence of $\gamma_{g}$ is not determined by Einstein equations (1)). The gravitational energy density can be defined as $$\begin{array}{l}
T^{0}_{0}=G^{0}_{0}=a^{-4}(3(\partial_{0}a)^{2}+(\nabla
a)^{2}-2a\triangle a)\cr
=(\tilde{\tau}^{2}-\tilde{r}^{2})^{-1}\Big(3+2\sinh^{2}(\alpha)+8(\sinh(\alpha)\ln(D))^{-1}\Big)
\end{array}$$ The density of the gravitational momentum (only the component with radial index is different from zero) is $$\begin{array}{l}
T^{0}_{r}=a^{-4}(4\partial_{0}a\partial_{r}a-2a\partial_{0}\partial_{r}
a)\cr=(\tilde{\tau}^{2}-\tilde{r}^{2})^{-1}4\cosh(\alpha)\sinh(\alpha)\simeq
v_{r}^{-2}\end{array}$$
Summary
=======
We have derived a solution of Einstein equations with a viscous fluid which is different from the well-known homogeneous solutions with a perfect fluid. This kind of matter could be a constituent of the models of the universe or could exist in the form of (expanding) galaxies. The fluid density is decreasing like $a^{-2}
$(if we assume that the bulk viscosity behaves as $\gamma_{g}\simeq a^{-1}$ like the shear viscosity ) with logarithmic corrections increasing the decay in comparison to the coasting cosmology [@coasting] . A contribution of such a fluid to the total energy density (consisting of radiation, dark matter and “baryons” ) becomes relevant for a large time. It can be applied in models attempting to explain the coincidence problem.
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---
abstract: 'This survey presents an overview of the advances around Tverberg’s theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg’s theorem and its applications. The survey contains several open problems and conjectures.'
author:
- Imre Bárány
- Pablo Soberón
title: 'Tverberg’s theorem is 50 years old: a survey'
---
Introduction {#section-introduction}
============
Tverberg’s theorem has been a cornerstone of combinatorial convexity for over fifty years. Its impact and influence is only comparable to that of the famous and classic theorems of Carathéodory and Helly. This gem lies at the crossroads of combinatorics, topology, and linear algebra, and continues to yield challenging and interesting open problems. Its states the following.
Given $(r-1)(d+1)+1$ points in ${\mathds{R}}^d$, there is a partition of them into $r$ parts whose convex hulls intersect.
More formally, given $X\subset {\mathds{R}}^d$ of $(r-1)(d+1)+1$ points, there is a partition $X=X_1\cup \dots \cup X_r$ such that $\bigcap_{j=1}^r \operatorname{conv}X_j \ne \emptyset$. Such a partition is called a *Tverberg partition*. The number of points in this result is optimal, as a dimension-counting argument shows. In fact, if $X$ is in general enough position and in the partition $X=X_1\cup \ldots \cup X_r$ we have $1\le |X_j|\le d+1$ for every $j$, then $\bigcap_{j=1}^r \operatorname{aff}X_j$ is a single point if $|X|= (r-1)(d+1)+1$, and is empty if $|X|\le (r-1)(d+1)$.
The last decade has seen an impressive sequence of results around Tverberg’s theorem. The purpose of this survey is to give a broad overview of the current state of the field and point out key open problems. Other surveys covering different aspects of Tverberg’s theorem can be found in [@Eckhoff:1979bi; @Eck93survey; @Matousek:2002td; @BBZ17survey; @de2017discrete; @BZ17].
The paper is organized as follows. In sections \[section-topological\] and \[section-colored\] we describe the topological and colorful versions of Tverberg’s theorem, which have received the most attention in recent years. In sections \[section-intersection\] and \[section-universal\] we discuss a large number of variations and conjectures around Tverberg’s theorem. In Section \[section-applications\] we describe some applications of Tverberg’s theorem. Finally, in Section \[section-spaces\] we present Tverberg-type results where the settings have changed dramatically, such as Tverberg for convexity spaces or quantitative versions. In that last section, we focus mostly on results which are related to geometry.
Interlude: a short history of Tverberg’s theorem
------------------------------------------------
An early predecessor of Tverberg’s theorem is Radon’s lemma from 1921 [@Radon:1921vh; @Eckhoff:1979bi]. Radon used it in his proof of Helly’s theorem. It says that *any set $X$ of $d+2$ points in ${\mathds{R}}^d$ can be split into two sets whose convex hulls intersect*. So it is the case $r=2$ of Tverberg’s theorem. Its proof is simple: the $d+2$ vectors in $X$ have a nontrivial affine dependence $\sum_{x \in X}{\alpha}(x)x=0$ and $\sum_{x \in X}{\alpha}(x)=0$. The sets $X_1=\{x \in X: {\alpha}(x)\ge 0\}$ and $X_2=\{x \in X: {\alpha}(x) < 0\}$ form a partition of $X$ and their convex hulls intersect, as one can easily check.
Another result linked to this theorem is Rado’s centerpoint theorem. This states that *for any set $X$ of $n$ points in ${\mathds{R}}^d$, there is a point $p$ such that any closed half-space that contains $p$ also contains at least $\left\lceil \frac{n}{d+1}\right\rceil$ points of $X$*. The standard proof of this result uses Helly’s theorem. Tverberg’s theorem implies it in few lines: setting $r=\left\lceil \frac{n}{d+1}\right\rceil$, there is a partition of $X$ into $r$ parts $X_1,\ldots,X_r$ and a point $p\in {\mathds{R}}^d$ such that $p \in \bigcap_{j=1}^r \operatorname{conv}X_j$. Then $p$ is a centerpoint of $X$: every closed halfspace containing $p$ contains at least one point from each $X_j$.
In a paper entitled “On $3N$ points in a plane” Birch [@Birch:1959] proves that any $3N$ points in the plane determine $N$ triangles that have a point in common. His motivation was the (planar) centerpoint theorem. Actually, he proves more, namely the case $d=2$ of Tverberg’s theorem and states the general case as a conjecture.
Tverberg’s original motivation was also the centerpoint theorem and he learned about Birch’s result and conjecture only later. He proved it first for $d=3$ in 1963, and in full generality in 1964. Here is, in his own words, how he found the proof: “I recall that the weather was bitterly cold in Manchester. I awoke very early one morning shivering, as the electric heater in the hotel room had gone off, and I did not have an extra shilling to feed the meter. So, instead of falling back to sleep, I reviewed the problem once more, and then the solution dawned on me!” [@tve:recollections].
Proof methods
-------------
By now there are several proofs of Tverberg’s theorem, two by Tverberg himself [@Tverberg:1966tb; @Tve81], one by Tverberg and Vrećica [@TV93], by Roudneff [@Roudneff:2001cl], by Sarkaria [@Sarkaria:1992vt], and by Zvagelskii [@Zva08]. We explain here two of them. The first (due to Roudneff) cleverly chooses a function whose minimum is taken on a Tverberg partition.
by Roudneff. We assume that the points of $X$ are in general position (the coordinates are algebraically independent, say). Assume ${{\mathcal P}}=\{X_1,\ldots,X_r\}$ is an $r$-partition of $X$ with $1\le |X_j|\le d+1$ and define the function $$f(x,{{\mathcal P}})=\sum_{j=1}^r \operatorname{dist}^2(x,\operatorname{conv}X_j).$$ Here $\operatorname{dist}$ is the distance given by the Euclidean norm, which is denoted by $\|\cdot\|$. For fixed ${{\mathcal P}}$ the function $f$ is convex on ${\mathds{R}}^d$. It tends to infinity as $\|x\| \to \infty$ so it attains its minimum. Then there is a partition, say ${{\mathcal P}}$, where the minimum of the function $f(x,{{\mathcal P}})$ is the smallest; let it be $\mu$. We are going to show that $\mu=0$, which clearly suffices. Assume on the contrary, that $\mu >0$ and is reached at $z \in {\mathds{R}}^d$. Denote by $y_j$ the (unique) point in $\operatorname{conv}X_j$ with $\operatorname{dist}(z,\operatorname{conv}X_j)=\|z-y_j\|$. The function $x \mapsto \sum_1^r \|x-y_j\|^2$ takes its minimum also at $x=z$ so its gradient at $x=z$ is zero: $\sum_1^r (z-y_j)=0$. Note that $z=y_j$ is possible but cannot hold for all $j$ since $\mu>0$.
Define $Y_j\subset X_j$ for $j=1,\ldots,r$ via $y_j \in \operatorname{relint}\operatorname{conv}Y_j$. We [**claim**]{} that $\bigcap _1^r \operatorname{aff}Y_j=\emptyset$. Otherwise there is a point $v \in \bigcap _1^r \operatorname{aff}Y_j$. Let $\langle \cdot, \cdot \rangle$ denote the standard scalar product, so $\langle x,x \rangle=\|x\|^2$, for instance. Then $\langle z-v,z-y_j\rangle>0$ if $y_i \ne z$ (because $y_j$ is the closest point to $z$ in $\operatorname{conv}Y_j$) and $\langle z-v, z-y_j\rangle =0$ if $y_j = z$. Summing these inequalities and equalities gives $\langle z-v,\sum_1^r (z-y_j)\rangle >0$, contradicting $\sum_1^r (z-y_j)=0$.
The dimension counting argument mentioned in the introduction shows now that $\sum_1^r |Y_j|\le (r-1)(d+1)$ so one point of $X$, say $x$, is not used in any $Y_j$. This is the point where the general position of $X$ is used. We can decrease the value $\mu$ if $\langle x-y_j,z-y_j \rangle>0$ for some $j$ with $y_j\ne z$ because by adding $x$ to $Y_j$ there appears a point on the segment $[x,y_j]\subset \operatorname{conv}(Y_j\cup \{x\})$ that is closer to $z$ than $y_j$. Thus $\langle x-y_j, z-y_j\rangle \le 0$ must hold for every $j$. Summing these inequalities gives $$\begin{aligned}
0 &\ge& \sum_1^r \langle x-y_j, z-y_j\rangle = \sum_1^r \Big\langle(x-z)+(z-y_j),z-y_j\Big\rangle \\
&=& \Big\langle x-z, \sum_1^r (z-y_j)\Big\rangle+ \sum_1^r \langle z-y_j, z-y_j \rangle =0+\mu >0,\end{aligned}$$ a contradiction.
by Sarkaria. This proof has two ingredients. One is the so-called Colorful Carathéodory theorem of the first author [@Barany:1982va]. Carathéodory’s classical theorem [@Carath1907] says in essence that being in the convex hull has a very finite reason. Precisely, if $A \subset {\mathds{R}}^d$ and $a \in \operatorname{conv}A$, then $a \in \operatorname{conv}B$ for some $B \subset A$ with $|B| \le d+1$. In the colorful version there are $d+1$ sets or “colors” $A_1,\dots,A_{d+1} \subset {\mathds{R}}^d$ and $a \in \bigcap_{i=1}^{d+1} \operatorname{conv}A_i$. A [*transversal*]{} of the sets $A_1,\dots,A_{d+1}$ is simply a set with a point $a_i \in A_i$ for every $i$.
\[th:Car\] Assume $A_1,\dots,A_{d+1} \subset {\mathds{R}}^d$ and $a \in \bigcap_{i=1}^{d+1} \operatorname{conv}A_i$. Then there is a transversal $\{a_i \in A_i: i \in [d+1]\}$, such that $a \in \operatorname{conv}\{a_1,\dots,a_{d+1}\}$.
The colorful version contains the original one: simply take $A_i=A$ for every $i$.
The second ingredient is Sarkaria’s tensor trick [@Sarkaria:1992vt]. We explain it in the form given in [@BaranyOnn]. It begins with an artificial tool: choose vectors $v_1,\dots,v_r \in {\mathds{R}}^{r-1}$ so that their unique (up to a multiplier) linear dependence is $v_1+\dots+v_r=0$. Now let $X=\{x_0,x_1,\ldots,x_n\}$ be the set of $(r-1)(d+1)+1$ points given in Tverberg’s theorem, so $n=(r-1)(d+1)$. With $x_i$ and $v_j$ we associate the tensor $$\overline x_{i,j}= v_j \otimes (x_i,1) \in {\mathds{R}}^n,$$ the tensor $\overline x_{i,j}$ can be thought of as an $(r-1)\times (d+1)$ matrix as well. Note that we moved to the $n$-dimensional space because $\overline x_{i,j}\in {\mathds{R}}^n$, while the original points $x_i$ are in ${\mathds{R}}^d$. Observe that the origin is in the convex hull of the set $$A_i=\{\overline x_{i,1},\overline x_{i,2},\ldots,\overline x_{i,r}\}$$ for every $i$. The Colorful Carathéodory theorem applies now in ${\mathds{R}}^n$ and gives, for every $x_i$, a tensor $\overline x_{i,j(i)}$ with $0\in \operatorname{conv}\{\overline x_{0,j(0)},\overline x_{1,j(1)},\ldots,\overline x_{n,j(n)}\}$. Thus $0 \in {\mathds{R}}^n$ can be written as a convex combination of the tensors $\overline x_{i,j(i)}$: $$\begin{aligned}
0&=&\sum_{i=0}^n {\alpha}_i \overline x_{i,j(i)}= \sum_{i=0}^n {\alpha}_i v_{j(i)} \otimes (x_i,1)\\
&=& \sum_{j=1}^r v_j \otimes \left( \sum_{i:j=j(i)}{\alpha}_i(x_i,1)\right) = \sum_{j=1}^r v_j \otimes \left( \sum_{x_i\in X_j}{\alpha}_i(x_i,1)\right),\end{aligned}$$ where $X_j:=\{x_i\in X: j(i)=j\}$. These sets form a partition of $X$ into $r$ parts. There is a vector $u \in {\mathds{R}}^{r-1}$ orthogonal to $v_3,\ldots,v_r$ such that $\langle u, v_1\rangle =1$. Then $\langle u,v_2\rangle =-1$ because of the condition $v_1+\dots+v_r=0$. Multiplying the last equation by $u$ from the left gives $\sum_{x_i\in X_1}{\alpha}_i(x_i,1)=\sum_{x_i\in X_2}{\alpha}_i(x_i,1)$. It follows then that $$\sum_{x_i\in X_1}{\alpha}_i(x_i,1)=\sum_{x_i\in X_2}{\alpha}_i(x_i,1)=\ldots=\sum_{x_i\in X_r}{\alpha}_i(x_i,1).$$ Reading the last coordinate here shows that ${\alpha}:=\sum_{x_i\in X_1}{\alpha}_i=\sum_{x_i\in X_2}{\alpha}_i=\ldots=\sum_{x_i\in X_r}{\alpha}_i >0$. (Actually ${\alpha}=1/r$.) Then $$p:=\frac 1{{\alpha}}\sum_{x_i\in X_1}{\alpha}_ix_i=\frac 1{{\alpha}}\sum_{x_i\in X_2}{\alpha}_ix_i=\ldots=\frac 1{{\alpha}}\sum_{x_i\in X_r}{\alpha}_ix_i$$ is a point in the convex hull of each $X_j$: $X_1,\ldots,X_r$ is the required partition.
There is more to Sarkaria’s method than just this proof. To see this let $X_1,\ldots,X_r$ be finite (or compact) sets in ${\mathds{R}}^d$. What condition guarantees that $\bigcap_1^r \operatorname{conv}X_j = \emptyset$? There is a classical necessary and sufficient condition:
\[th:halfspace\] Under the above conditions, $\bigcap_1^r \operatorname{conv}X_j=\emptyset$ if and only if there are closed halfspaces $D_1,\dots,D_r$ with $X_j \subset D_j$ for every $j\in [r]$ such that $\bigcap_1^r D_j=\emptyset$.
The [**proof**]{} is easy. One direction is trivial. In the other direction the case $r=2$ is just the separation theorem for convex sets, and induction on $r$ works for $r>2$.
Here comes another necessary and sufficient condition from Arocha et al [@Arocha:2009ft]. First define $X=\bigcup_1^r X_j$, here either $X$ is a multiset or we assume that the sets $X_j$ are disjoint. For $x \in X$ denote, as before, $$\overline{x} = v_j \otimes (x,1) \mbox{ if } x\in X_j \mbox{ and set } \overline{X} = \{\overline{x} : x \in X\}.$$ Here the vectors $v_j \in {\mathds{R}}^{r-1}$ are the same as before.
\[th:sark\] Under the above conditions, $\bigcap_1^r \operatorname{conv}X_j \ne \emptyset$ if and only if $0 \in \operatorname{conv}\overline X$.
The [**proof**]{} is essentially the same as above, starting with the convex combination of the vectors in $\overline{X}$ representing the origin: $$\begin{aligned}
0&=&\sum_{x \in X} {\alpha}(x)\overline{x}=\sum_{j=1}^r\sum_{x \in X_j}{\alpha}(x)v_j\otimes (x,1)\\
&=&\sum_{j=1}^rv_j\otimes \sum_{x \in X_j}{\alpha}(x)(x,1).\end{aligned}$$ After this factorization the arguments are analogous to the previous proof.
Acknowledgments
===============
This work was partly supported by the National Science Foundation under Grant No. DMS-1440140 while the first author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The first author was also supported by Hungarian National Research, Development and Innovation Office Grants no K111827 and K116769. The authors would like to thank Frédéric Meunier, Uli Wagner, Günter M. Ziegler, and an anonymous referee for their careful revision and helpful comments.
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Imre Bárány\
<span style="font-variant:small-caps;">Alfréd Rényi Institute of Mathematics,\
Hungarian Academy of Sciences\
H-1364 Budapest Pf. 127 Hungary\
and\
Department of Mathematics\
University College London\
Gower Street, London, WC1E 6BT, UK</span>\
*E-mail address:* `barany.imre@renyi.mta.hu`
Pablo Soberón\
<span style="font-variant:small-caps;">Mathematics Department\
Northeastern University\
Boston, MA 02445, USA</span>\
*E-mail address:* `pablo.soberon@ciencias.unam.mx`
|
---
abstract: 'Here we present a kinematic study of the Galactic halo out to a radius of $\sim60$ kpc, using 4664 blue horizontal branch (BHB) stars selected from the SDSS/SEGUE survey, to determine key dynamical properties. Using a maximum likelihood analysis, we determine the velocity dispersion profiles in spherical coordinates ($\rsigma$, $\tsigma$, $\psigma$) and the anisotropy profile ($\beta$). The radial velocity dispersion profile ($\rsigma$) is measured out to a galactocentric radius of $r \sim 60$ kpc, but due to the lack of proper-motion information, $\tsigma$, $\psigma$ and $\beta$ could only be derived directly out to $r \sim 25$ kpc. From a starting value of $\beta\approx 0.5$ in the inner parts ($9<r/\kpc<12$), the profile falls sharply in the range $r \approx 13-18$ kpc, with a minimum value of $\beta=-1.2$ at $r=17$ kpc, rising sharply at larger radius. In the outer parts, in the range $25<r/\kpc<56$, we predict the profile to be roughly constant with a value of $\beta\approx 0.5$. The newly discovered kinematic anomalies are shown [*not*]{} to arise from halo substructures. We also studied the anisotropy profile of simulated stellar halos formed purely by accretion and found that they [*cannot*]{} reproduce the sharp dip seen in the data. From the Jeans equation, we compute the stellar rotation curve ($\vcirc$) of the Galaxy out to $r \sim 25$ kpc. The mass of the Galaxy within $r \lesssim 25$ kpc is determined to be $2.1 \times 10^{11}$ $\msun$, and with a 3-component fit to $\vcirc(r)$, we determine the virial mass of the Milky Way dark matter halo to be $M_{\rm vir} = 0.9 ^{+0.4}_{-0.3} \times 10^{12}$ $\msun$ ($R_{\rm vir} = 249^{+34}_{-31}$ kpc).'
author:
- 'PRAJWAL R. KAFLE$^{1}$, SANJIB SHARMA, GERAINT F. LEWIS & JOSS BLAND-HAWTHORN'
bibliography:
- '\\path/stellar\_halo.bib'
- '\\path/dispersion\_profile.bib'
- '\\path/anisotropic\_model.bib'
- '\\path/misc\_reference.bib'
- '\\path/mass\_estimator.bib'
- '\\path/reference\_veldisp\_paper.bib'
- '\\path/msto.bib'
title: Kinematics of the Stellar Halo and the Mass Distribution of the Milky Way using BHB stars
---
INTRODUCTION
============
Understanding the formation of stellar halos gives vital insight into the formation history and the evolution of galaxies . Under the currently favored $\Lambda$CDM model of galaxy formation, the stellar halos are thought to have been built up, at least in part, by accretion of satellite galaxies [@1978MNRAS.183..341W; @1978ApJ...225..357S; @1999MNRAS.307..495H]. Discoveries of structures like the Sagittarius dwarf galaxy [@1994Natur.370..194I; @1995MNRAS.277..781I; @2003ApJ...599.1082M; @2006ApJ...642L.137B], the Virgo over-density [@2008ApJ...673..864J], the Triangulum-Andromeda structure [@2004ApJ...615..732R; @2004ApJ...615..738M; @2007ApJ...668L.123M] and the low-latitude Monoceros ring [@2002ApJ...569..245N] lend further support to the idea of the stellar halo being formed by accretion. Other than accretion, a part of the stellar halo could also be formed by in-situ stars. Recent hydro-dynamical simulations [@2006MNRAS.365..747A; @2009ApJ...702.1058Z; @2011MNRAS.417.1260F; @2012MNRAS.420.2245M] of galaxy formation suggest that in the inner regions the stellar halo might be dominated by in-situ stars, whose kinematic properties are distinct from the accreted stars.
Observational evidences of multi-component halo have been found in dynamical studies of SDSS calibration stars [@2007Natur.450.1020C; @2010ApJ...712..692C; @2012ApJ...746...34B], in rotational behavior in metallicity bins [@2011MNRAS.411.1480D], in kinematics of Galactic anti-center and North Galactic Pole population [@2007MNRAS.375.1381K], in chemical properties [@2010ApJ...714..663D], and also in age difference between in-situ and accreted halo [@2012Natur.486...90K]. There also exists a counter-claim by [@2011MNRAS.415.3807S], who demonstrate that the evidence of retrograde signal in the outer halo in [@2010ApJ...712..692C] sample is weak and is because of a manifestation of incorrect distance estimates. Investigating, whether any signal of multi-component halo could also be seen in the dispersions of the velocity of the halo population is important. Ultimately, studying the velocity dispersion profiles of the halos and comparing them with simulations might help to constrain the model of galaxy formation.
Lack of proper motions and the slightly off centered position of the Sun with respect to the galactic center poses a unique challenge in studying the kinematics of the stellar halo. At distances much larger than $R_{\sun}$, line of sight velocity is same as radial velocity with respect to the galactic center. Hence the $\rsigma$ profile is easy to compute at large distances and this has been well studied; [@2005MNRAS.364..433B] studied the line-of-sight velocity dispersion ($\vlossigma$) of a mixture of 240 halo objects and found that $\vlossigma$ decreases monotonically beyond $r$ $\sim$ 30 kpc. In the outer most halo at $r\sim100$ kpc [@2005MNRAS.364..433B] and recently, [@2012MNRAS.425.2840D] both see a significant drop in $\sigmalos$ value to $\sim50$ kms$^{-1}$.
Conversely, [@2010ApJ...719.1582D] studied 666 BHB stars from the 2QZ Redshift Survey and found the velocity dispersion profile increases at large distances. But using 910 distant halo A-type stars, [@2010AJ....139...59B] found that there is a mean decline of ${\rm -}0.38$ $\pm$ 0.12 kms$^{-1}$ kpc$^{-1}$ in $\rsigma$ over 15 $< r/\kpc < 75$. More recently, [@2008ApJ...684.1143X] used 2401 BHB halo stars within 60 kpc and measured a slower decline in $\vlossigma$ compared to earlier studies. At small $r$ it has been difficult to derive the $\rsigma$ profile, and the only attempt to measure the $\rsigma$ in inner-halo was undertaken by [@1997ApJ...481..775S]. They find a sharp fall in $\rsigma$ from $\sim140$ kms$^{-1}$ to $\sim100$ kms$^{-1}$ at $r\approx12$ kpc although they assume the circular velocity to be constant.
In a solar neighborhood one can get useful proper motions and with this all the three velocity dispersions ($\rsigma$, $\tsigma$, $\psigma$) can be measured. [@2009ApJ...698.1110S] used the full phase space information of $\sim 1700$ halo subdwarfs from the solar neighborhood ($<5$ kpc) and determined the velocity dispersions to be ($\rsigma$, $\tsigma$, $\psigma$) = ($143\pm2$, $82\pm2$, $77\pm2$) kms$^{-1}$. Also, [@2010ApJ...716....1B] analyzed the proper motions of a large sample of main-sequence stars within the solar neighborhood ($<10$ kpc) and found $\rsigma=141$ kms$^{-1}$, $\tsigma=75$ kms$^{-1}$, $\psigma=85$ kms$^{-1}$. A summary of the estimated values of the velocity dispersions found in the literature is given in Table \[table:dispersion\_table\].
Sample (number) distance (kpc) $\rsigma, \tsigma, \psigma$ (kms$^{-1}$) anisotropy ($\beta$) Reference
------------------------ -------------------------------- ---------------------------------------------- ------------------------------------ ---------------------- -- -- -- --
BHB ($\sim700$) ($r\lesssim 20$, $r\gtrsim45$) $\rsigma = (140, 90-110)$ (radial(0.5),tangential(-1.3)) @1997ApJ...481..775S
BHB (1170) $5\lesssim d \lesssim 96$ $101.4\pm2.8$, $97.7\pm16.4$, $107.4\pm16.6$ nearly isotropic @2004AJ....127..914S
BHB (1933) $16<r<48$ - radial ($0.5^{+0.08}_{-0.2}$) @2012MNRAS.tmpL.469D
Subdwarfs ($\sim1700$) $d<5$ $143\pm2$, $82\pm2$, $77\pm2$ radial ($\sim0.69$) @2009MNRAS.399.1223S
MS (10$^{5}$) $d\backsimeq10$ 141,75,85 radial ($\sim0.67$) @2010ApJ...716....1B
BHB (3549) ($10<r<25$, $25<r<50$) - (tangential($-0.6$),radial($0.5$)) @2011MNRAS.411.1480D
\[table:dispersion\_table\]
Due to the lack of reliable proper motions of the halo field stars, even the fundamental quantities like the tangential components of the velocity dispersions as well as the anisotropy ($\beta$) are still badly understood beyond solar neighborhood. The situation is however not as hopeless as it might seem. At small $r$, using only line of sight velocity it is possible to put some constraints on these quantities using maximum likelihood techniques, where a model or a distribution function needs to be specified [*a priori*]{}. In an analysis of $1170$ BHB stars ranging from $5\lesssim d/\kpc \lesssim96$, where $d$ is now the stellar distance rather than a radius, [@2004AJ....127..914S] fit an ellipsoidal distribution of velocities and find that the halo is isotropic. Similarly, @2011MNRAS.411.1480D [hereafter D11] fit a constant anisotropy model (power law Distribution Function) to 3549 BHB stars constructed from the SDSS Data Release 7 (DR7) and find that the halo between $r = 10-25$ kpc is tangential whereas the distant halo within $25<r/\kpc<50$ is radial. D11 claims for the tangential inner halo are in contrast with the result by [@2010ApJ...716....1B] who found the inner halo to be radial in the similar region ($d\approx 10$ kpc). D11 assume the distribution function (DF) to be such that the tracer density and the potential both are power laws. Without [*a priori*]{} knowledge of the density slope their estimates will have some systematics. Additionally, the potential was also kept constant in their analysis and thus can bias the results due to the degeneracy between the potential and the anisotropy.
In their most recent work, @2012MNRAS.tmpL.469D [hereafter D12] allow both the potential and $\beta$ to vary and thereby break the degeneracy, and find that the outer halo within $16< r/\kpc<48$ is radial with $\beta =0.5^{+0.1}_{-0.2}$. Previous estimates of the velocity anisotropy ($\beta$) in the solar neighborhood and the nearby halo are summarized in Table \[table:dispersion\_table\]. All the above estimates of an anisotropy of the distant halo is done in a large radial bins. Their results might be accurate for the given radial bins and could also be the actual anisotropy of the halo provided the anisotropy remains nearly constant or monotonic through out. On the other hand, if the actual $\beta$ of the system is not monotonic but has a more complex shape, then estimating it in the larger bin will just capture an overall property of that bin.
Theoretically there are families of the distribution function of type which result in a constant anisotropic system, as well with the families of models those have their own anisotropy profiles, which include Osipkov-Merritt model ([@1979SvAL....5...42O; @1985MNRAS.214P..25M; @1985AJ.....90.1027M]), [@1991MNRAS.250..812G], [@1991MNRAS.253..414C], and few more with the Hernquist potential-density model in . The question we ask is how well do these anisotropy profiles match the anisotropy of the Galaxy? More fundamentally, how well do we know the anisotropy of the halo? To this end we thus compute the beta profile with much finer spatial resolution and without any prior assumptions about density or potential.
Another use of studying the kinematics of the stellar halo is to constrain the mass and the potential of the Milky Way. Most of the methods to estimate the mass require the knowledge of the anisotropy parameter $\beta$. Without the unbiased estimate of the velocity anisotropy, constraining the mass of the system via the Jeans equation could be uncertain by 73% (for $-4.5<\beta<0.44$) as found by @2010MNRAS.406..264W in their estimates of mass of the Galaxy. Several other authors have also used this assumption to estimate the mass of the galaxy [@2006MNRAS.369.1688D; @2010ApJ...720L.108G; @2012MNRAS.425.2840D]. Precaution needs to be taken while making an arbitrary assumption about the anisotropy. It has been found that the halo stars, the satellites and the dark matter halo have different orbital properties [@2006MNRAS.365..747A; @2007MNRAS.379.1464S]. Hence assuming a constant anisotropy for both field stars as well as satellites [@2005MNRAS.364..433B; @2006MNRAS.369.1688D] could introduce systematic uncertainties in the mass estimate. Ideally, in order to break the degeneracy we must have a separate estimation of the radial velocity dispersion, the velocity anisotropy and underlying density of the population, as pointed out by @2006MNRAS.369.1688D.
The orbital evolution of the Magellanic clouds [@1982MNRAS.198..707L; @2007ApJ...668..949B], the local escape speed [@2007MNRAS.379..755S], the timing argument [@2008MNRAS.384.1459L] and the study of the kinematic of the tracers population [@1996ApJ...457..228K; @2008ApJ...684.1143X; @2010ApJ...720L.108G; @2010MNRAS.406..264W] are the methods undertaken in order to constrain the mass of the Galaxy. Summarizing all these attempts to constrain the mass of the Galaxy, the consensus can be found between $0.5-3.5\times10^{12}$ $\msun$. Recently using BHB stars [@2008ApJ...684.1143X] estimate the mass of the Milky Way to be $0.91_{-0.18}^{+0.27} \times 10^{12} M_{\sun}$. However, they make an assumption that the variation of $(\vcirc/\vlos)$, the ratio of the circular to the line of sight velocity, with radius is same as that in simulations. In this paper, we estimate $\vcirc$ as far out as possible without any assumption and then use it to estimate the dark matter halo mass.
This work focuses mainly on the study of the kinematics of the stellar halo in order to present the unbiased estimation of the velocity dispersions, anisotropy parameter and circular velocity as a function of radius to the extent data supports. We use a DF which does not require any assumption to be made [*a priori*]{} about the density profile or the potential. We then use our measurements of velocity dispersions to estimate the rotation curve of the Galaxy. The disc and bulge mass already being constrained from [@2009PASJ...61..227S], we focus on constraining the dark matter halo mass. Using the circular velocity curve ($\vcirc(r)$) we can estimate $\beta(r)$ out to as far as $\rsigma(r)$ is available. Finally, we compare our results with simulations in which the halo is formed purely by accretion.
This paper is organized as follows: in Section \[sec:theory\] we discuss the theoretical aspect of our analysis, the methodology adopted, and the details about the sample; in Section \[sec:dispersion\_profile\] we present our result for the velocity dispersions, confer the results between the alternative models, and investigate the contribution of the halo substructures ; in Section \[sec:circ\_velocity\] we present our estimation of the mass of the Galaxy. Results are then compared with the simulations in Section \[sec:simulated\_halo\]. In Section \[sec:conclusion\] we present our conclusion and discuss our result.
THEORY AND METHOD {#sec:theory}
=================
We are interested in calculating the velocity dispersions $(\rsigma,\tsigma,\psigma)$ for the stellar halo. However, the data we have is line of sight velocities. To proceed we need to make some assumptions about the position and the velocity of the Sun with respect to the galactic center. We assume ${\rm R}_{\sun}=-8.5$ kpc, the velocity of the local standard of rest (LSR), v$_{\rm LSR}$, is taken to be IAU adopted value = 220 kms$^{-1}$, and the solar motion with respect to LSR (U, V, W)$_{\sun}$ = +11.1, +12.24, +7.25 in kms$^{-1}$ [@2010MNRAS.403.1829S]. Spherical and heliocentric coordinate system are expressed in terms of ($r$, $\theta$, $\phi$), and ($d$, $l$, $b$) respectively.
Distribution Function
---------------------
The distribution function (DF), f, is defined such that f([**x**]{},[**v**]{}) d$^{3}$[**x**]{} d$^{3}$[**v**]{} is the probability of finding a randomly picked star in a phase-space volume d$^{3}$[**x**]{} d$^{3}$[**v**]{}. In general, we consider the stellar halo as an anisotropic spherical system. The anisotropy is defined as $$\label{eqn:anisotropy}
\beta = 1 - \frac{\tsigma^{2} + \psigma^{2}}{2\rsigma^{2}},$$ $\rsigma,\tsigma$ and $\psigma$ being the velocity dispersions in spherical coordinates, and it describes the orbital structure of the system. The values of this parameter range from –$\infty$ for purely circular trajectories to 1 for purely radial orbits.
Families of DFs that generate the collisionless anisotropic spherical systems with constant or varying velocity anisotropy can be found in detail in [@2008gady.book.....B]. One such distribution function with constant anisotropy is $$\label{eqn:constant_ansitropy}
f(E,L) = f(E) L^{-2\beta}.$$ Here $E=\Phi(r)-(v^{2}/2)$ is the relative energy per unit mass and $L$ is the modulus of the angular momentum vector per unit mass. Recently, a DF given by Equation (\[eqn:constant\_ansitropy\]) with the energy term from [@1997MNRAS.286..315E]: $$\label{eqn:Evansfe}
f(E,L)\propto E^{(\beta(\gamma-2)/\gamma)+(\alpha/\gamma)-3/2} L^{-2\beta},$$ was used by @2011MNRAS.411.1480D and @2012MNRAS.tmpL.469D to study the rotation, anisotropy and mass of the Galactic halo. The parameters $\alpha$ and $\gamma$ are the logarithmic slopes of the density ($\rho \propto r^{-\alpha}$) and potential ($\Phi \propto r^{-\gamma}$) respectively. Hereafter, we refer to this function as D11 DF.
If one is interested in deriving the dispersion profiles, a simple distribution function that one can use is the Gaussian velocity ellipsoidal distribution function (GVE DF). The GVE DF has been used in the context of the stellar halo by [@1980MNRAS.193..295F] using globular clusters as tracers and by [@2004AJ....127..914S] and [@2009MNRAS.399.1223S] using halo stars. A GVE DF with rotation about $z$-axis is given by
$$\label{eqn:veDF}
f(r, {\bf v}) = \frac{\rho(r)}{(2\pi)^{3/2}\rsigma\tsigma\psigma} \text{exp}\left[-\frac{1}{2}\left(\frac{v_{r}^{2}}{\sigma_{r}^{2}}+
\frac{v_{\theta}^{2}}{\sigma_{\theta}^{2}}+\frac{(v_{\phi}-\vrot)^{2}}{\sigma_{\phi}^{2}}\right)\right]$$
The DF as given by Equation (\[eqn:veDF\]) assumes that the velocity ellipsoid is perfectly aligned with the spherical coordinates, but in general the velocity ellipsoid can have a tilt. Using halo subdwarf stars @2009ApJ...698.1110S and @2010ApJ...716....1B have found that the tilt is small and consistent with zero. Hence, it is safe to ignore the tilt while computing velocity dispersions.
Parameter estimation
--------------------
The proper motion information of the stars in the stellar halo beyond solar neighborhood ($r\gtrsim10$ kpc) is not accurate enough to properly constrain the tangential motions. Nevertheless, our position in the Galaxy still makes it possible to constrain these quantities by utilizing the tangential information carried by the line-of-sight velocities of the stars. However, for that we need to marginalize the distribution function over the unknown quantities, which in this case are the tangential components ($v_{l}$, $v_{b}$). The expression for the marginalized DF is given by,
$$\label{eqn:vlosd}
F(l, b, d, \vlos|\rsigma,\tsigma,\psigma,\vrot) = \iint f \text{d}v_{l} \text{d}v_{b}.$$
We use maximum likelihood method to estimate the model parameters. The log-likelihood function which we maximize is given by $$\label{eqn:likelihood}
\mathcal{L} (l, b, d ,\vlos|\rsigma,\tsigma,\psigma,\vrot) =
\sum_{i}^{n}\text{log}F(l_{i}, b_{i}, d_{i}, v_{los_{i}}),$$ where $n$ is the number of stars in the system under study. We use Markov Chain Monte-Carlo (MCMC) with the Metropolis Hasting algorithm (MHA) to obtain the posterior distribution. We quote the central values of the velocity dispersions ($\rsigma, \tsigma, \psigma$) as our initial estimates and 16 and 84 percentiles as the error associated. Note, for the GVE DF the density term $\rho(r)$ in Equation (\[eqn:veDF\]) is not a function of model parameters and hence it does not have any effect on the likelihood analysis.
Once the radial velocity dispersion $\rsigma$ and the anisotropy parameter $\beta$ are evaluated, Jeans equation [@1915MNRAS..76...70J] can be used to estimate the circular velocity $v_{\rm circ}$ of the spherical system in equilibrium using the relation $$\label{eqn:Jeanseq}
\vcirc^{2}(r) = -\rsigma^{2} \left[\frac{{\rm d\ln}\rho}{{\rm d\ln}
r}+\frac{{\rm d\ln}\rsigma^{2}}{{\rm d\ln}r}+2\beta\right],$$ where $\rho \propto r^{-\alpha}$ is the density of the tracer population, which implies ${\rm d} \ln \rho / {\rm d} \ln r = -\alpha$. Through out the analysis we assume the density to be double power law with $\alpha = 2.4$ ($r \leqslant 27$ kpc) and $\alpha = 4.5$ ($r > 27$ kpc) in agreement with the recent works by [@2011MNRAS.416.2903D] and [@2009MNRAS.398.1757W].
For systems with constant anisotropy and a given $v_{\rm circ}$, the solution to the differential equation (\[eqn:Jeanseq\]) subject to the boundary condition $\lim_{r \to \infty}$ $\rsigma^{2}$ = 0 reads $$\label{eqn:Jeanseq_rsig}
\rsigma^{2}(r) = \frac{1}{r^{2\beta}\rho(r)} \int_{r}^{\infty} dr' r'^{2\beta} \rho(r') ({\rm d}\Phi/{\rm d}r')$$ Assuming density and anisotropy are known we can use this solution to estimate $\rsigma$ as a function of $r$.
DATA: BHB {#sec:data}
----------
![Radial distribution of BHB stars in galactocentric coordinates. The distribution has a peak at around 16 kpc. Most of the stars are found to lie in the range $10<r/\kpc<25$.[]{data-label="fig:rgc"}](f1){width="47.50000%"}
Being luminous and having nearly constant magnitude BHB stars are ideal for studying the stellar halo, and this is what we use in our study. We use the BHB catalog published by X11 for our analysis. The catalog comprises of 4985 BHB stars obtained from Sloan Digital Sky Survey (SDSS) Data Release 8 [@2011ApJS..193...29A]. The stars were selected by imposing limits on color and Balmer line profile measurements. Imposing limits on Balmer line profile measurements allow one to remove the main sequence stars and Blue Stragglers. Further details on BHB candidate selection can be found in @2008ApJ...684.1143X and references therein. To avoid contamination from the disk stars, we restrict our analysis to stars having a distance $|z|>4$ kpc from the galactic mid-plane. As mentioned earlier, the way our likelihood function (Equation \[eqn:vlosd\]) is laid out, this cut in distance above the plane will not introduce any bias. No velocity limits have been imposed to obtain the sample and thus for the purpose of kinematic studies the population of BHB stars we select can be considered to be kinematically unbiased.
For the stars that we study the angular position is known very accurately, but the distance and radial velocity have some uncertainty associated with them. To get more accurate distances we recalibrate X11 distances using a color-magnitude relation derived for the same population from @2011MNRAS.416.2903D. The estimated dispersion in g-band magnitudes is 0.13, equivalent to a distance uncertainty of 6$\%$. For the SEGUE [@2009AJ....137.4377Y] radial velocity measurements, 94% of our sample have an uncertainty of less than 8 kms$^{-1}$.
The galactocentric radial distribution of the final BHB samples is shown in Figure \[fig:rgc\]. It can be seen that the distribution peaks at around 16 kpc. Most of the stars are found to lie in the range $10<r/\kpc<25$.
Velocity Dispersion profile of the halo {#sec:dispersion_profile}
=======================================
We study the kinematics of the halo in radial bins to obtain radial profile of the model parameters ($\rsigma, \tsigma, \psigma$) and also $\beta$. Using only line-of-sight velocity information the tangential components, $\tsigma$ and $\psigma$, are difficult to constrain except in the very inner regions of the halo. However, $\rsigma$ can be well constrained both in the inner and the outer halo. This means that relatively larger number of stars ($>$1000) per bin are required to estimate $\tsigma$ and $\psigma$ as compared to $\rsigma$. Given that we only have about 4000 stars this means that we cannot measure the $\tsigma$ and $\psigma$ profiles with sufficient spatial resolution. Hence, we employ two different binning schemes or estimators, one for radial velocity and the other for tangential velocity. The estimators are; the equi-populated estimator (hereafter EPE) and central moving estimator (CME). In EPE the data is binned radially with each bin containing equal number of particles and this is used for computing $\rsigma(r)$. In CME a set of equi-spaced positions in $r$ are chosen and then at each position an equal number of points either side of the chosen central value are used to estimate the desired quantity. We use the CME for computing $\tsigma(r)$ and $\psigma(r)$. The crucial difference between the two schemes is that while the bins are non-overlapping in the former, in the latter they can be overlapping. In EPE the spacing between the bins is directly proportional to the number of particles in each bin $n_{\rm bin}$. Hence, if the desired quantity can be estimated with sufficient accuracy employing small $n_{\rm bin}$, then EPE is the desired method. However, if this is not the case then it is better to use the CME method as the spatial resolution is not directly dependent on $n_{\rm bin}$ [^1].
Finally, for our data the number density of points in $r$ is highly non-uniform, and hence it is not accurate to assume that the desired quantity has been estimated at the center of the bin. To alleviate this number density bias, for both schemes we compute the final position of the bin as the mean $r$ of the points in the bin.
Radial velocity dispersion profile ($\rsigma$(r)) {#subsec:radialdis}
-------------------------------------------------
Here we focus on the nature of the $\rsigma(r)$ profile of the halo for which we adopt the EPE method with $n_{\rm bin} = 400$. As explained previously, $\rsigma$ can be measured out to the extent of the data ($r\sim60$ kpc). The values of $\rsigma$ obtained from the likelihood analysis are given in Figure \[fig:sigmar\_profile\] and the error bars represent the 1$\sigma$ confidence interval and are determined from the likelihood fitting.
![Radial velocity dispersion in radial bins. The black dashed line is the $\sigmalos$ profile from [@2008ApJ...684.1143X]. Black dashed-dotted and red dashed lines are the Sommer-Larsen profiles given by Equation (\[eqn:rprofile\]) for the fitting parameters taken from [@1997ApJ...481..775S] and from the fit to our estimated values of $\rsigma$ respectively.[]{data-label="fig:sigmar_profile"}](f2){width="47.50000%"}
We find that the radial velocity dispersion, $\rsigma$ at the Sun’s position (R$_{\sun}$ = 8.5) is 145.6 kms$^{-1}$. However, beyond the solar neighborhood $\rsigma$ sharply decreases until $r\sim15$ kpc, after which it decreases much slowly and approaches a value of around $\sim$100 kms$^{-1}$ at 56 kpc. The error bar in $\rsigma$ for the inner halo population is large and is mainly because the $\vlos$ contains less radial velocity information as compared to outer parts. The overlaid black dashed line is the linear approximation ($\approx 111-0.31r$) for $\sigmalos$ profile from [@2008ApJ...684.1143X]. Additionally, other previous attempts to fit the profile for $\rsigma$ in the outer parts of the halo [@2005MNRAS.364..433B; @2010ApJ...720L.108G; @2010AJ....139...59B] also found profiles similar to [@2008ApJ...684.1143X], with slightly varying slope and normalization. All these profiles are reasonable estimates of $\rsigma$ for the outer halo ($d\gg$R$_{\sun}$) where the assumption $\rsigma \approx
\sigmalos$ holds. In the inner halo ($r\lesssim 15$ kpc) however the approximation breaks down and $\rsigma$ strongly deviates from $\sigmalos$. It can be seen that the deviation of $\rsigma$ from $\sigmalos$ increases as one approaches the center and at R$_{\sun}$ the deviation is as high as $\sim 40$ kms$^{-1}$.
[@1997ApJ...481..775S] provide a functional form for fitting the $\rsigma$ profiles which is given by $$\label{eqn:rprofile}
\rsigma^{2} = \sigma_{0}^{2} + \frac{\sigma_{+}^2}{\pi}\left[\frac{\pi}{2}-{\rm tan}^{-1} \left(\frac{r-r_{0}}{l}\right)\right].$$ This has a shape which is similar to our $\rsigma(r)$ profile and we fit it to find $\sigma_0=94.5$ kms$^{-1}$, $\sigma_+=122.3$ kms$^{-1}$, $r_0=13.2$ kpc, and $l=2.6$ kpc. In this function the fit parameter $\sigma_{0}$ gives the asymptotic value that $\rsigma$ achieves in the outer halo, whereas ($\sigma_0^2 + \sigma_+^2)^{1/2}$ gives the approximate value for $\rsigma$ in the inner halo. The fit parameters $r_0$ and $l$ determine the turn-off point and the steepness of the transition of the profile respectively. A lower value of $l$ gives a steeper transition, this can be seen from the comparison between the Sommer-Larsen fit ($l= 7.5$ kpc) and the red line in Figure \[fig:sigmar\_profile\] which is our fit having smaller $l$.
Tangential velocity dispersion and anisotropy profiles ($\tsigma(r)$, $\psigma(r)$ and $\beta(r)$) {#subsec:tangentialdis}
--------------------------------------------------------------------------------------------------
![Velocity Dispersions and Anisotropy profiles: From top to bottom is $\rsigma,
\tsigma, \psigma$, and $\beta$ profiles of the stellar halo estimated in the radial bins. The diamond and the round markers are the results for the two binning schemes , namely the EPE and the CME respectively. Note that the last radial bin marked with the open diamond contains the remaining stars. The diamond markers in plot (a) is just shown for the ease of comparison. []{data-label="fig:aniso"}](f3){width="47.50000%"}
There have been few attempts to constrain the tangential velocity dispersions and most of the studies are either restricted to solar neighborhood [@2009MNRAS.399.1223S; @2010ApJ...716....1B] or for the overall system [@2004AJ....127..899S]. The tangential velocity dispersions not only provide the information of the anisotropy of the stellar population (through Equation \[eqn:anisotropy\]) but together with $\rsigma$ also helps to measure the mass distribution of the Galaxy.
We estimate the tangential velocity dispersions ($\tsigma$ and $\psigma$) using CME with $n_{\rm bin} = 1200$ stars. Given the quality of the data, we are only able to measure $\tsigma$ and $\psigma$ out to $\sim 25$ kpc. Our estimates of $\tsigma$ and $\psigma$ are shown in Figure \[fig:aniso\][b,c]{} by the black dots with error bars. For uniformity, we also estimate the $\rsigma$ with this binning scheme and this is shown (only out to $\sim 25$ kpc) in Figure \[fig:aniso\][a]{} by the black dots with error bars.
In general the tangential components $\tsigma$ and $\psigma$ near the solar neighborhood are found to be comparatively lower $(\tsigma = 85_{-9}^{+8}$ kms$^{-1}$, $\psigma = 95_{-8}^{+8}$ kms$^{-1}$) than the radial dispersion $\rsigma$. It can be seen in the Figure \[fig:aniso\][b and c]{} that there is a sharp rise in the values of $\tsigma$ and $\psigma$ at $r=17$ kpc. Beyond this $\psigma$ falls whereas $\tsigma$ rises, given the large uncertainties and low number of independent bins it is unclear if this is a real or a spurious trend.
By substituting the estimates of the tangential and the radial velocity dispersions obtained using CME $(n_{\rm bin} = 1200)$ from the above analysis into Equation (\[eqn:anisotropy\]) we compute the corresponding values of the anisotropy constant in the respective bins. As shown in Figure \[fig:aniso\][d]{}, the halo within 12 kpc has $\beta \sim 0.5$ whereas the outer halo beyond the turn-off point is nearly isotropic. We discover a significant drop in the $\beta$ profile at $r=17$ kpc. Here the halo is strongly tangential with $\beta = -1.2$. We later confirm that the trend observed in anisotropy is neither due to the manifestation of the systematics introduced by the chosen model ( §\[sec:alternative\_model\]) nor due to presence of the halo substructures (§\[sec:halo\_substructures\]). It is also shown in the appendix \[sec:vlsrrsun\_effect\] that assumed v$_{\rm LSR}$ and R$_\sun$ have negligible effects upon these estimates. The probable reasons for this sudden turn-over in the properties of the halo are discussed in the conclusion.
We know that the consecutive CME bins overlap in radius and thus the dispersion profiles demonstrated in Figure \[fig:aniso\] is a smoothed version of the actual dispersion profiles of the halo. However, to check for any systematic associated with the choice of the binning scheme, we also estimate $\rsigma, \tsigma, \psigma$, and $\beta$ in traditional equi-populated (EPE) bins $(n_{\rm bin} = 700)$. The measured values in these bins are shown by the diamond points in Figure \[fig:aniso\]. If the number of stars per bin is less than 700 it is difficult to constrain $\tsigma$ and $\psigma$. Even with $n_{\rm bin} = 700$ we were able to constrain the tangential motion only till $16$ kpc (first three diamond points). Hence, we construct the last bin by grouping all the stars beyond $16$ kpc into one bin (rightmost diamond point). More importantly, with this binning scheme we are only interested to see whether we obtain the corresponding dip or rise (depending on the parameter of interest) seen in Figure \[fig:aniso\] or not. We find that except for the right-most diamond points all of other diamond points in the figure are in agreement with our previous estimates of the dispersions (given by black dots). The rightmost diamond points are calculated in a huge bin with more than 50% of the total sample. Particularly for $\tsigma$, $\psigma$ and $\beta$ given the non-monotonic trend they have, hence we do not claim the last diamond point is the correct estimate of anisotropy at $r\sim35$ kpc.
None of the uncertainties quoted in the above estimates of $\rsigma$, $\tsigma$ and $\psigma$ include the uncertainties in distances and radial velocities. As mentioned earlier, the errors in distance and radial velocity are quite small, and convolving the model (Equation \[eqn:veDF\]) with the error functions should not change the results.
$\beta(r)$ from fitting Distribution Function (f(E,L)) {#sec:alternative_model}
------------------------------------------------------
It would be interesting to see whether the $\beta$ profile presented above, in particular the dip seen at $r=17$ kpc, is an artifact of our chosen model (GVE) or a real inherent feature of the Galactic halo. In order to pursue it, here we explore the effect of the chosen model on the determination of the velocity anisotropy ($\beta$). For the comparative study, the alternative model we choose is the D11 DF (Equation \[eqn:Evansfe\]). We consider anisotropy ($\beta$) and potential $(= \Phi_{0}r^{-\gamma})$ as free model parameters and constrain them using the maximum likelihood method.
{width="95.00000%"}
First, we compare the theoretical properties of the DFs at our disposal namely, GVE and D11 DF. In Figure \[fig:VLOSD\] we plot the theoretical LOSVDs of models along two separate line-of-sights. We also show the LOSVDs in Figure \[fig:VLOSD\] for two different distances representing the inner halo ($d$ = 15 kpc, given by red lines) and the outer halo ($d$ = 50 kpc, given by blue lines). For the inner-halo we assume $\alpha$ = 2.4 and assign radially biased anisotropy ($\beta$ = 0.4) whereas for the outer-halo we assume $\alpha$ = 4.5 and assign tangentially biased anisotropy ($\beta$ = -2.5) in accordance to [@2009MNRAS.398.1757W] and [@2011MNRAS.416.2903D] estimates for $\alpha$. Note, the density normalization at the break radius ($r = 27$ kpc) is assumed to be equal. For the assumed constant potential the solid lines in Figure \[fig:VLOSD\] (both left, right panels) are the LOSVDs obtained by adopting the D11 model and dashed lines are the LOSVDs of our GVE model. Recall that our model does not take $\beta$ directly but demands the information of the velocity dispersion components ($\rsigma, \tsigma, \psigma$) individually. To make the LOSVDs obtained from both the models comparable we estimate $\rsigma, \tsigma, \psigma$ from the set of values of $\beta, \alpha,
\Phi_{\circ}$ and $\gamma$ chosen to obtain LOSVDs of D11 DF. For an assumed potential power law we use Equation (\[eqn:Jeanseq\_rsig\]) to first calculate $\rsigma$, to put in our model. For an assumed $\beta$, substituting this $\rsigma$ in Equation (\[eqn:anisotropy\]) gives the corresponding value for $\sigma_{\rm t}$ (= $\sqrt{\tsigma^{2} + \psigma^{2}}$). It can be seen in the figure that for all the four cases LOSVDs obtained from both the models match well. Naively, from these perfect matches of the LOSVDs at different line-of-sights and distances it can be anticipated that the estimation of $\beta(r)$ with both models should also match.
![Anisotropy ($\beta$) estimates in the CME using D11 model (blue points) and GVE model (black points). Each bin consists of 1200 stars. Anisotropy estimates with GVE distribution is done with MCMC technique whereas estimates with D11 model is done with the brute-force grid based analysis. Assigned asymmetric uncertainties are 1$\sigma$ confidence intervals obtained from likelihood fitting.[]{data-label="fig:beta_2models"}](f5){width="47.50000%"}
Now we estimate the $\beta(r)$ using exactly the same sample of BHB stars in same radial bins as in §\[subsec:tangentialdis\] (CME, $n_{\rm bin} = 1200$) but with D11 DF. Blue points in Figure \[fig:beta\_2models\] demonstrate the $\beta$ profile obtained by fitting D11 DF. Here we use brute-force grid based analysis to constrain the model parameters $\beta$, $\Phi_{\circ}$ and $\gamma$. To facilitate the comparison our estimates from Figure \[fig:aniso\][d]{} are over plotted in Figure \[fig:beta\_2models\] and are shown by the black points. From Figure \[fig:beta\_2models\] it can be seen that within the range of uncertainties the measured values of $\beta$ with both models (D11 DF and GVE model) agree. However, a slight bias can be seen in the sense that $\beta$ obtained from GVE DF is in general higher. The reason for this discrepancy lies in the fact that $\beta$ obtained from the D11 DF has a dependence on $\alpha$. Hence, unless the underlying $\alpha$ value of the sample is exactly known, a mismatch is expected. The estimated value of $\beta$ increases with the adopted value of $\alpha$ (see Fig-3 D11). This suggests that the actual value of $\alpha$ is even higher than the one that is adopted here (2.4).
![The posterior distributions of the parameters for bin centered at r = 16.93 kpc. Upper Right: The posterior distributions of the GVE model parameters, $\rsigma$, $\tsigma$ and $\psigma$, obtained with 5$\times$10$^{5}$ MCMC random walks. In the inset is the derived distribution of the $\beta$ parameter. Upper Left and Lower panels: The joint likelihood contours of the D11 model parameters $\beta$, potential normalization ($\Phi_{\circ}$) and potential slope ($\gamma$) are obtained with brute force analysis. The outer contour displays 1 $\sigma$ region whereas the inner contour demonstrates a region of $50\%$ confidence interval. Cross hair corresponds to the point where the likelihood is maximum.[]{data-label="fig:r16.93_D11DF_&_veDF"}](f6){width="47.50000%"}
In order to give an estimate of the quality of constraints obtained from the likelihood analysis we display the likelihood contours in the parameter space in the top left and bottom two plots in Figure \[fig:r16.93\_D11DF\_&\_veDF\] for a bin centered at $r$ = 16.93 kpc where the maximum dip in the $\beta$ profile was seen in Figure \[fig:beta\_2models\]. Additionally, in the corresponding bin, the top right plot in figure demonstrates the posterior distributions of the model parameters $\rsigma$, $\tsigma$ and $\psigma$ of the GVE model obtained from $5 \times 10^5$ MCMC random walks. It can be seen that even at a distance of just twice of R$_{\sun}$, $\tsigma$ and $\psigma$ distributions are quite broad as compared to $\rsigma$ distribution; this is the reason for the large uncertainty in the value of $\beta$ as we move outwards in $r$.
Effect of the halo substructures {#sec:halo_substructures}
--------------------------------
There is now enough observational evidence to suggest that the stellar halo is highly structured particularly as one moves outwards into the halo [@2008ApJ...680..295B]. Using clustering algorithms on simulated N-body stellar halos, [@2011ApJ...728..106S] find that the fraction of material in substructures increases monotonically as a function of distance from the center and at around $65$ kpc can be as high $50\%$. Hence, while studying the kinematic properties of the halo should we include the substructures or exclude them? If the kinematic properties of a sample are dominated by a few massive accretion events then one should exclude them. However, in-spite of being highly structured if the sample is a superposition of large number of events with none of them being individually too dominant then it is best to include them. Results of [@2011ApJ...728..106S] on simulated halos show that for the range of radii that spans our BHB stars ($r<40 \ \kpc$) the amount of material in substructure should be less than 20%. So we expect the substructures to have a minor effect on the kinematic properties that we have derived. However, it is still important to check if this is true.
To study the effect of substructures on our estimation of the dispersions we mask two prominent features of the halo that contaminate our sample, namely, the Sagittarius stellar stream and the Virgo over-density. Cuts we impose include the Lambert Equal-Area projection cut as given in @2008ApJ...680..295B and an additional cut in equatorial coordinates (RA and DEC) suggested by @2011MNRAS.416.2903D. We mask the region within $0<$ X (abscissa of the equal-area projection)$<30$, where X is given by 63.63961\[2(1–sin $b$)\]$^{1/2}$; and $0^{\circ}< {\rm RA} <50^{\circ}$, and $-30^{\circ}< {\rm DEC} <0^{\circ}$, which is purely a geometric cut. These stringent cuts reduce our final sample to 2975 stars. A proper phase-space masking of these structures will be revised in the future work.
![Effect of the halo substructures, namely, the Sagittarius stellar stream and the Virgo over-density. Each CME bins contain 500 stars. Black and red points are our result for the masked and the unmasked halo respectively. Error bars quoted are 1$\sigma$ credibility interval.[]{data-label="fig:saggi_virgo"}](f7){width="47.50000%"}
In Figure \[fig:saggi\_virgo\] we present our result obtained after masking the substructures. As masking reduces the sample size almost by half, we employ CME with $n_{\rm bin} = 500$ stars only, instead of $n_{\rm bin} = 1200$ as was done earlier with unmasked data, to avoid excessive smoothing of the estimated profiles. Figure \[fig:saggi\_virgo\] shows that the masking of substructures have little effect on the estimation of velocity dispersion profiles, the $\beta$ profile is almost unchanged. This alleviates the concern that perhaps the turn-over points in the velocity dispersion profiles and the dip in $\beta$ profile, seen in Figure \[fig:aniso\], in the region $r=13-18$ kpc could be due to the dominance of halo substructures.
Comparison of anisotropy estimates in D11 and D12 radial bins
=============================================================
In their recent work D11 and D12 fit a distribution function of the form given by Equation (\[eqn:Evansfe\]), to the BHB samples obtained from SEGUE survey in order to estimate the model parameters. The models adopted are constant anisotropy models given by Equation (\[eqn:constant\_ansitropy\]). In D11, the potential is assumed to be a power law ($\propto r^{-\gamma}$) and with a constant index ($\gamma = 0.35$). Later in D12, they break the degeneracy present in their model and consider the potential normalization ($\Phi_{0}$), potential slope ($\gamma$) and anisotropy ($\beta$) as free parameters. Note, in D11 there is an additional parameter specifying rotation (odd part of the DF) but this was dropped in D12, as they were not focusing on rotation. The methodology applied to measure the model parameters is similar to ours which involves marginalizing the DF over the tangential velocities to derive the line-of-sight velocity distribution (LOSVD); fitting the LOSVD to the data using the maximum likelihood method and in return obtaining the best estimates of the model parameters.
Anisotropies of inner and outer halo by D11 {#sec:D11comp}
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D11 study the rotation and the anisotropy of the BHB samples taken from SDSS Data Release 7 [@2009ApJS..182..543A]. First, in order to construct the sample used by them we query SDSS DR7 database to select the candidate BHB stars using the color and the stellar parameters ranges given in D11. Like them we also mask the Sagittarius dwarf galaxy which reduces the original sample size by 40$\%$ to $\sim 3500$.
D11 measure the anisotropy of the halo in radial and metallicity bins. In the inner halo ($10 < r/\kpc< 25$) both metal-rich (\[Fe/H\]$>-2$) and metal poor (\[Fe/H\]$<-2$) stars are found to be tangential with $\beta$ $\sim-1.2$ and $\sim-0.2$ respectively. In the corresponding metallicity bins, the outer halo ($25 < r/\kpc < 50$) is found to be radial with $\beta$ of $\sim0.4$ and $\sim0.5$ respectively. Since they do not give the estimates of $\beta$ in combined metallicity bins for the inner and outer halo, we estimate them here using the same methodology as adopted by them. For the inner halo we find $\beta=-0.62$ (tangential) and for the outer halo we find $\beta=0.41$ (radial). These estimates are consistent with D11 results, if we combine their low and high metallicity $\beta$ values in each radial bin.
The inner halo within the solar vicinity has been found to be radial in studies of halo subdwarfs by @2009MNRAS.399.1223S and in studies of $10^{5}$ main sequence stars by @2010ApJ...716....1B (see also Figure \[fig:aniso\][d]{}). Hence, a tangential inner-halo as predicted by D11 is surprising. The first thing to check is if the D11 result is due to some of the assumptions made by them. For example, in D11 the logarithmic density slope was assumed to be constant and equal to $-3.5$. Later on [@2011MNRAS.416.2903D] conducted a detailed analysis of the BHB stars to estimate their density profile and found that the profile is of the form of a broken power law, the inner-halo ($<27$ kpc) having a profile index of $-2.3$ and the outer halo having a profile index of $-4.6$. This is in good agreement with the findings of @2009MNRAS.398.1757W that the halo within 25 kpc has the profile index $-2.4$ and beyond which it is steeper with an index $-4.5$.
If we update the density profile index in the D11 case with the above values then we expect the inner halo which is already tangential to become even more tangential and the outer halo which is already radial to become even more radial. This is because $\beta$ has a dependence on the adopted value of $\alpha$ as shown by D11 (Figure 5). In general $\beta$ increases with an increase in the adopted value of $\alpha$. Another effect that can potentially bias the results is the fact that the potential parameters ($\Phi_{0}, \gamma$) have been kept fixed in the D11 analysis, i.e., the degeneracy between $\beta$ and potential has not been broken. After analyzing data in finer bins and breaking the degeneracy among the model parameters we do see that in Figure \[fig:beta\_2models\] the inner halo is radial (from §\[sec:alternative\_model\]) as was also found with the GVE model. Hence the assumption of a fixed potential can bias the estimation of $\beta$. However, a much more apparent reason for the discrepancy is as follows. Clearly, from Figure \[fig:aniso\] the radial bin from 10-25 kpc will encompass the stars within 13-17 kpc which are predominantly tangential. Since the probability density of stars in radius also peaks at around 16 kpc (see Figure \[fig:rgc\]) we anticipate the overall $\beta$ to be tangential. To conclude, the tangential behavior of the inner-halo seen by D11 is most likely due to the large radial bin size adopted by them.
Anisotropy at $16 < r/\kpc< 48$ seen by D12 {#subsec:r1648}
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![Posterior distribution of velocity dispersions for D12 data set within $16 < r/\kpc< 48$ using a GVE model. The value of v$_{\rm LSR}$ here is taken to be 240 kms$^{-1}$ to keep it same as in D12. []{data-label="fig:vel_dispersion_r1648"}](f8){width="40.00000%"}
D12 re-calibrate the distances of BHB stars using the color-magnitude relation given by [@2011MNRAS.416.2903D] and then select stars within $16 \lesssim r/\kpc \lesssim 48$ ( 1933 stars) from @2011ApJ...738...79X BHB samples. They fit the D11 model to study the nature of $\beta$ in the outer halo. Unlike D11, as mentioned previously here they break the degeneracy present in their model and consider the $\beta$, $\Phi_{o}$ and $\gamma$ as free parameters. Thus while they fit the model they simultaneously estimates all three parameters. They find $\beta=0.4^{+0.2}_{-0.2}$ for $\alpha=4.6$. Using a model allowing for oblateness ($q=0.59$) they find $\beta=0.5^{+0.1}_{-0.2}$. If we apply the GVE distribution function to stars in the range $16<r/\kpc<48$ we find $\beta=-0.14^{+0.52}_{-0.66}$ ($\rsigma=97.3^{+2.9}_{-3.0}$ kms$^{-1}$, $\tsigma=122.7^{+26.4}_{-33.0}$ kms$^{-1}$ and $\psigma=78.5^{+34.3}_{-40.7}$ kms$^{-1}$). This more or less looks like the mean value of $\beta$ in this range (see Figure \[fig:beta\_2models\]) provided we take into account the fact that the number density of stars peak at around $r=16$ kpc. Although the D12 value is still within our 1 $\sigma$ region, our predicted value is lower than D12. It can be seen from Figure \[fig:beta\_2models\] that $\beta$ is not constant in the range $16<r/\kpc<23$. It increases from being tangential to isotropic. Using the D11 DF also gives similar result (§\[sec:alternative\_model\]). Beyond this range we cannot directly measure $\beta$, but by deriving a best fit circular velocity profile and making use of $\rsigma$ profile which is available till $r=56$ kpc we can predict $\beta$, and this is returned to in section \[sec:circ\_velocity\]. However, an assumption about $\alpha$ also has to be made. Beyond, $r>27$ kpc the density slope has been shown to change from $-2.4$ to $-4.5$. Adopting a steeper density slope increases the value of $\beta$. For $\alpha=4.5$ we find $\beta \sim 0.5$ for $r>27$ kpc; this is more in agreement with D12. To conclude, the D12 value of $\beta=0.4$ for $16<r/\kpc<48$, although derived for a sample which is dominated by stars within $r<27$ kpc, is not appropriate for the range $18<r/\kpc<23$, instead it is correct for the range $23<r/\kpc<48$.
![Maximum likelihood analysis of the anisotropy parameter, $\beta$, for stars in the radial bin $35 < r/\kpc < 84$. Solid black lines are the 1$\sigma$ confidence region. Density and potential slopes are taken to be 4.5 and 0.35.[]{data-label="fig:r35_84_D11"}](f9){width="45.00000%"}
Finally, we check how best we can measure $\beta$ in the outer most parts, $35 < r/\kpc < 84$, using the D11 DF and the data in hand. This region consists of $762$ stars, and we assume $\alpha=4.5$ and $\gamma=0.35$ and repeat the analysis as in D12. The likelihood distributions of model parameters are shown in Figure \[fig:r35\_84\_D11\]. The mass-anisotropy degeneracy is clearly visible here, suggesting that it is very difficult to directly measure $\beta$ unless an explicit or implicit assumption about the potential is made.
Circular Velocity curve of the Galaxy {#sec:circ_velocity}
=====================================
Here we use the measured values of $\rsigma(r)$ and $\beta(r)$ from our analysis given in Figure \[fig:aniso\] to determine the circular velocity curve of the Galaxy ($\vcirc$) through the Jeans equation (Equation \[eqn:Jeanseq\]). Besides anisotropy and radial velocity dispersion information we also need to adopt some density profile for the tracer population but not of the spectroscopic sample. We adopt a value of $\alpha$ = 2.4 as suggested by recent works of [@2009MNRAS.398.1757W] and [@2011MNRAS.416.2903D], for the range of distance explored here ($r<25$ kpc). The blue dots with error bars in Figure \[fig:vcirc\] are our estimates of $\vcirc$ using CME with $n_{\rm bin} = 1200$. The uncertainties on $\vcirc$ were computed using a Monte Carlo based scheme from uncertainties in $\rsigma$ and $\beta$. The leftmost and rightmost points have comparatively larger error bars as compared to intermediate points. For the leftmost point the large error bar is mainly due to large error in the value of $\rsigma$. On the other hand, for the right-most point the large error bar is mainly due to large error in the values of $\tsigma$ and $\psigma$.
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In Figure \[fig:vcirc\] it can be seen that the circular velocity profile derived from our analysis (blue points) display prominent features. We now check if such features are also observed in other studies using tracers other than BHB stars. For this we over-plot $\vcirc$ compiled by [@2009PASJ...61..227S] as black dots, obtained from several references . Further details about the source of each individual point can be found in @2009PASJ...61..227S and references therein. Note, $\vcirc$ values of [@2009PASJ...61..227S] are computed for (R$_{\sun}$, v$_{\rm LSR}$) = (8.0 kpc, 200.0 kms$^{-1}$). Correcting the $\vcirc$ for our adopted values of (R$_{\sun}$, v$_{\rm LSR}$) = (8.5 kpc, 220.0 kms$^{-1}$) is beyond the scope of this work and hence we simply over-plot these published values in Figure \[fig:vcirc\]. It can be seen there is a prominent dip at 9 kpc in the [@2009PASJ...61..227S] compiled $\vcirc$ profile. They explain this dip by introducing massless rings on top of a disk with exponential surface density. We also find a similar dip in our $\vcirc$ profile but at around 10-12 kpc. The slight shift in the position of dip could be due to large width of our bins and also due to the fact that unlike [@2009PASJ...61..227S], who measure $\vcirc$ in the mid-plane of the Galaxy, we measure $\vcirc$ over a spherical shell that intersects with the SDSS footprint.
We now estimate the mass of the dark matter halo of the Milky Way, assuming a three component model of the Galaxy consisting of the bulge, the disk, and the halo. The bulge is modeled as a Hernquist sphere and the disk is assumed to follow an exponential profile [@2008ApJ...684.1143X]. The parameters for the bulge and disk are taken from [@2009PASJ...61..227S] and are kept fixed. Although [@2009PASJ...61..227S] use massless rings, we here have ignored them since our main aim is to fit the dark matter halo. We model the dark matter halo using the NFW [@1996ApJ...462..563N] density profile. Here we consider both the halo and the bulge to be spherically symmetric. The non-axisymmetric effect due to a bar shaped bulge is neglected here. Potentials for bulge, halo, and exponential disk can be expressed as, $$\begin{aligned}
\label{eqn:NFWpot}
\Phi_{\rm disk} (r)& = &-\frac{GM_{\rm disk}(1-e^{-r/b})}{r} \label{eqn:diskpot},\\
\Phi_{\rm bulge} (r)& = &-\frac{GM_{\rm bulge}}{r+a} \label{eqn:bulgepot},{\rm and}\\
\Phi_{\rm NFW} (r) & = &-\frac{GM_{\rm vir}\ln(1+r c/R_{\rm vir})}{g(c)r} \end{aligned}$$ where $M_{\rm disk}$ = 6.5$\times$10$^{10}$ $\msun$, $b$ = 3.5 kpc, $M_{\rm bulge}$ = 1.80$\times$10$^{10}$ $\msun$, and $a$ = 0.5 kpc [@2009PASJ...61..227S].
Note, the disk potential as given by Equation (\[eqn:diskpot\]) is spherically symmetric. It means the disk is considered to be a spherical body with exponential surface density fall-off. To get an idea on the error that is incurred due to the assumption of the disk as a spherical body with the mass same as the flattened disk we refer reader to [@2008gady.book.....B](Figure 2.17). Roughly the maximum error in $\vcirc$ is $13\%$, which is at a distance about twice of the disk scale length. However, at the larger distances along the mid-plane the discrepancy is smaller. In the general case of triplanar symmetry (elliptic disk), in reality, the disk potential has to be the function of both polar and azimuthal coordinates and in the special case of axial symmetry (circular disk) it has to be the function of sole polar coordinates, in addition to the radial coordinates ($r$). We here use the spherically symmetric form for two reasons. Firstly, we make use of the the spherical form of the Jeans equation given by Equation (\[eqn:Jeanseq\]) which demands a spherical potential. Secondly, it is to ease the comparison with earlier studies , e.g., @2008ApJ...684.1143X, that adopt a similar definition. However, later on we consider a 3D disk potential and discuss its consequences on the estimation of mass. The function $g$ in NFW potential is given by $$g(c) = \ln(1+c) - \frac{c}{1+c}$$ and $$R_{\rm vir} =\left(\frac{2 M_{\rm vir}G}{H_0^2 \Omega_m \Delta_{\rm th}}\right)^{1/3}.$$
The total potential $\Phi(r)$ of the Galaxy is then simply $$\label{eqn:totalpot}
\Phi(r) = \Phi_{\rm bulge}(r) + \Phi_{\rm disk}(r) + \Phi_{\rm NFW}(r)$$
We adopt the value of Hubble constant, H$_{0}$ = 70.4 kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$ [@2011ApJS..192...18K], and $\Delta_{\rm th}=340$ [@1998ApJ...495...80B].
A NFW halo has two free parameters the mass $M_{\rm vir}$ and the concentration $c$. Since we do not have enough data points spanning a wide range in radius, we avoid fitting both the parameters simultaneously. Instead we use the concentration mass relation, $$c = 327.3 M_{\rm vir}^{-0.12}, \ \ 10^{11} \leqslant M_{\rm vir}/M_{\sun} \leqslant 10^{13},$$ as has been reported in N body simulations of dark matter halos [@2007MNRAS.378...55M].
Finally, we can derive the resultant circular velocity ($\vcirc$) from the total potential (Equation \[eqn:totalpot\]) by computing $(r d\Phi/dr)^{1/2}$. We fit the obtained theoretical rotation velocity curve to our observed values of $\vcirc$ shown by the blue dots in Figure \[fig:vcirc\], and the red line is our best fit circular velocity curve. The $\vcirc$ profiles for the different components are also shown separately. The dashed black line is the corresponding best fit NFW halo profile. The best fit value for the fit parameter, $M_{\rm vir}$, for our three component baryon and dark matter mass distribution is 0.9$^{+0.4}_{-0.3}$ $\times$ 10$^{12}$ $\msun$. The corresponding values of $R_{\rm vir}$ and $c$ derived from the best fit value of $M_{\rm vir}$ are 249$^{+34}_{-31}$ kpc and 12.0$^{+0.6}_{-0.5}$ respectively. We estimate the mass of the Galaxy within $r \lesssim 25$ kpc to be 2.1 $\times$ 10$^{11}$ $\msun$. Assuming a functional form for $P(v_{los}/\vcirc)$ obtained from simulations [@2008ApJ...684.1143X] derive the $\vcirc$ from the $v_{los}$ of BHB stars. The derived $\vcirc$ is then used to estimate the virial mass of the dark matter halo. They find $M_{\rm vir} = 0.91^{+0.27}_{-0.18} \times 10^{12}$ $\msun $ which is in good agreement with our result, note uncertainties are however, slightly larger in our estimates. Since unlike them we do not make any assumption about the functional form of $P(v_{los}/\vcirc)$.
Here we study the effect of chosen disk models for which we consider more realistic three-dimensional potential for the disk by [@1975PASJ...27..533M], which holds for the special case of a circular disk. The expression for this potential is given by $$\label{eqn:miyamotondisk}
\Phi_{\rm disk} (R,z) = -\frac{G M_{\rm disk}} {\sqrt{R^2+(a+\sqrt{z^2+b^2})^2}}.$$ Here again, the disk parameters are obtained from the best fit values along the galactic mid-plane $(z=0)$ which reproduces $\vcirc(R)$ profile for [@2009PASJ...61..227S] razor-thin exponential disk (@1970ApJ...160..811F) i.e. $b = 0.0\:\kpc$ [^2]. Hence the best fit value for the disk parameter $a = 2.5$ kpc whereas mass $M_{\rm disk} = 6.5 \times 10^{10}$ $\msun$ is taken same as in [@2009PASJ...61..227S]. Since we assume the disk potential to be three-dimensional here, for the purpose of computing the total $\vcirc$ we consider the component of force along the radial direction ($r$) only. The bulge and the halo models are kept same as earlier. The best fit values for the NFW halo parameters with the updated disk model is found to be $M_{\rm vir} = 1.2^{+0.5}_{-0.4} \times 10^{12}$ $\msun$ with $R_{\rm vir} = 274^{+35}_{-30} \: \kpc$. Instead if we consider the total magnitude of the force in order to compute the $\vcirc$ we estimate $M_{\rm vir} (R_{\rm vir} = 269^{+34}_{-32}\:\kpc) = 1.1^{+0.5}_{-0.4}
\times 10^{12}$ $\msun$. Historically, large values of $a$ have been assumed while modeling the disk ([@1995ApJ...453..673W; @2009IAUS..254..241B]). We find that this leads to a much more massive dark matter halo– for $a = 4.5 \: \kpc$ we find $M_{\rm vir} (R_{\rm vir} = 299^{+36}_{-33}\ \kpc) = 1.6^{+0.6}_{-0.5} \times 10^{12}$ $\msun$ and for $a=6.5\:\kpc$ we find $M_{\rm vir} (R_{\rm vir} = 321^{+35}_{-34}\:\kpc) = 1.9^{+0.7}_{-0.6} \times 10^{12}$ $\msun$.
There are few things which have insignificant or unexplored effects on our mass estimation e.g. we do not take into account the mass of the super-massive black hole, which is $\sim 4\times10^6$ $\msun$ [@2002Natur.419..694S; @2005ApJ...628..246E; @2009ApJ...692.1075G], and is approximately 1/1000 of the mass of the bulge. Moreover, its effect is like that of a point mass and can be easily absorbed into the bulge mass. Another effect that is not considered is the tidal effect on the primary object that has been qualitatively studied to find the mass ratio between the Galaxy and M31 [@2009MNRAS.397.1990B] and depends strongly on different impact parameters . We find that R$_{\rm vir}$ of the Galaxy is $\sim250$ kpc and for M31 it is $\sim 260$ kpc [@2008MNRAS.389.1911S; @2007ApJ...670L...9M]. Given that the distance between these two galaxies is $\sim780$ kpc [@2005ApJ...635L..37R; @2005MNRAS.356..979M; @2006Ap.....49....3K], which is more than the double of the sum of their virial radii, we believe the tidal effect of M31 on overall mass estimate of the Galaxy, if any, should be negligible. The tidal effects of LMC and SMC on the Galaxy have not been explored in this paper.
![Dashed black line is our best fit model of $\vcirc$ given by red line in Figure \[fig:vcirc\]. Black dots are the values from the literatures labeled as W99, X08, S09, W10, G10, S11, and D12b respectively. To make the plot less obscure we do not include similar findings from the literature. For details about the similar results refer to the text.[]{data-label="fig:vcirc_all"}](f11){width="47.50000%"}
Note, our mass modeling of the Milky Way, does not make any assumption about the value of $\beta$, instead we use the value of $\beta$ directly computed from the data. The only assumption that we make is that the density of the dark matter halo follows an NFW profile. As long as that assumption holds our estimates for $\vcirc$ and mass of the Milky Way should also be valid in the outer parts $r>25$ kpc where we cannot directly measure $\beta$. If one wants to directly measure $\vcirc$ in the outer parts using only line of sight velocities then one has to make an assumption about the underlying $\beta$. Several attempts have been made in this regard, with each of them making different assumptions about $\beta$ and hence introducing a bias in the estimated mass. Below we compare these with our prediction for the mass of the Galaxy. Dashed line in Figure \[fig:vcirc\_all\] is our best fit model of $\vcirc$. Since literature sources mostly report mass within a certain radius, to facilitate comparison we convert it to $\vcirc$ using the relation $$M(<r) = \vcirc^{2}r/{\rm G}.$$ In Figure \[fig:vcirc\_all\] the plotted $v_{\rm circ}$ from different sources span a wide range in radii and were computed using different types of tracer populations. By fitting a model to the kinematics of the satellite galaxies and the globular clusters, [@1999MNRAS.310..645W] measure the mass to be $M(50 \:\kpc) \sim 5.4^{+0.2}_{-3.6} \times 10^{11}$ $\msun$. This agrees with estimates by [@1996ApJ...457..228K] ($M(50 \ \kpc) = (4.9 \pm 1.1) \times 10^{11}$ $\msun$) and ($M(50 \ \kpc) = 1.8-2.5 \times 10^{11}$ $\msun$). [@2010MNRAS.406..264W] apply the tracer mass estimator formalism to 26 satellite galaxies and find that $M(300 \ \kpc) = (0.9 \pm 0.3) \times 10^{12}$ $\msun$. Their mass estimate is however prone to the systematics introduced by assumed $\beta$, as duly mentioned by them. It can be seen in the figure that at $r$ = 100 kpc, depending upon the chosen anisotropy, their mass could vary anywhere between $0.3 \times 10^{12}$ $\msun$ and $1.4 \times 10^{12}$ $\msun$. Studying the hyper-velocity stars within 80 kpc, and assuming $\beta=0.4$, [@2010ApJ...720L.108G] estimate $M = 6.9 _{+3.0}^{-1.2} \times 10^{11}$ $\msun$, which is slightly higher than our estimate. Using BHB stars and [@2010MNRAS.406..264W] tracer formalism estimate the mass at $r = 85\ \kpc$ to be $8.83\pm{0.73}\times 10^{11}$ $\msun$ . This is slightly higher than our estimate, probably because they assume $\beta=0$. With the mixed sample of tracers (BHB and CN stars) populating the outer-most halo ($r\sim50-150\:\kpc$), [@2012MNRAS.425.2840D] estimated mass of the Galaxy to be $M(150 \: \kpc)=(5-10) \times 10^{11}$ $\msun$. The variation is mainly due to uncertainty on the adopted potential and density slopes, and anisotropy. Their range of mass at the outer-most halo falls within our estimation.
Kinematics of the simulated stellar halo {#sec:simulated_halo}
========================================
![Velocity dispersions and anisotropy profiles of BHB stars in simulated stellar halo: From top to bottom are $\rsigma$, $\tsigma$, $\psigma$, and $\beta$ profiles of the 11 instances of simulated halo taken from [@2005ApJ...635..931B]. The thick red lines are the mean profiles of the 11 halos.[]{data-label="fig:kinematics_simulation"}](f12){width="47.50000%"}
![The mean $\beta$ profile of BHB stars in 11 simulated $\Lambda$CDM stellar halos of [@2005ApJ...635..931B]. Shown alongside as dashed line is the best fit analytic function of form given by [@1991MNRAS.253..414C].[]{data-label="fig:beta_lcdm"}](f13){width="47.50000%"}
![The $\beta$ profile of simulated stellar halos having non-standard accretion history.[]{data-label="fig:beta_new"}](f14){width="47.50000%"}
We now study the kinematic properties of simulated stellar halos in which the halos are formed purely by accretion of satellite galaxies. For this we use the simulations of [@2005ApJ...635..931B]. In order to construct a synthetic sample of BHB stars from these simulations we use the code GALAXIA [@2011ApJ...730....3S]. Figure \[fig:kinematics\_simulation\] shows the velocity dispersion and anisotropy profiles of 11 different $\Lambda$CDM halos as a function of galactocentric radius $r$. The mean of all the halos is also shown alongside as thick red line. In general the velocity dispersions fall off with radius. At small $r$ the fall is rapid but at large $r$ it is much slower. Asymptotically the ratio $\rsigma/V_{\rm vir}$ approaches a value of around 0.8. The $\beta$ rises rapidly from a value of zero in the center to about 0.8 at $r\sim 10 \:\kpc$, and thereafter shows very little change. These results are in good agreement with results of [@2006MNRAS.365..747A; @2007MNRAS.379.1464S], who study the stellar halo formed in cosmological hydro-dynamical simulations including star formation and feedback.
Firstly notice that $\Lambda$CDM halos are rarely tangential for any given radius. For most of the range of $r$, $\beta$ is in general greater than 0.5. We fit an analytic function of [@1991MNRAS.253..414C] form given by $$\beta(r)=\beta_0 \frac{r^2}{r^2+r_0^2}
\label{eqn:cford_profile}$$ to the mean $\beta$ profile of the 11 $\Lambda$CDM halos. The best fit values of the free parameters were found to be $\beta_0=0.765$ and $r_0=0.00843 R_{\rm vir}$. Figure \[fig:beta\_lcdm\] shows that the fit is quite good for a wide range of $r$. The slight mismatch at $r<1$ kpc could be due to issues related to force resolution. In the outer parts most of the mass is in bound structures and hence is not smoothly distributed. This is probably responsible for the non-smooth behavior in $\beta$ in the outer parts.
Figure \[fig:beta\_new\] presents the beta profiles for halos having non-$\Lambda$CDM accretion history, i.e., halos having accretion history significantly different from that predicted by the $\Lambda$CDM model of galaxy formation. Six halos that we consider are with accretion events being dominated by 1) radial orbits 2) circular orbits 3) old events 4) recent events 5) higher luminosity and 6) low luminosity and these are from simulations of [@2008ApJ...689..936J]. Signatures of different accretion events can be seen in the $\beta$ profiles. The most significant difference is between the radial and the circular halo, which is expected since the orbital properties of the satellites were different to begin with. It is interesting to note that the circular halo is the only one among all the simulated halos that can have $\beta<0$. The old halo almost perfectly follows the mean profile that we had derived for the 11 $\Lambda$CDM halos and has the smoothest profile. This is due to the fact that the stars in this halo are completely phase mixed and have no structures of any kind. The young halo has very few stars in inner regions and shows non smooth behavior due to presence of significant amount of structures. The high luminosity halo is also very similar to old halo. However, the low-luminosity halo has higher $\beta$ for $r<0.01 R_{\rm vir}$. This is most likely due to circularization of orbits when acted upon by dynamical friction. Orbit circularization has also been reported by [@2007MNRAS.379.1464S] in their simulations. Note, the effect of dynamical friction is strongest for high luminosity events and weakest for low luminosity events. Moreover, satellites when acted upon by dynamical friction loose energy and move towards the inner regions of the halo. This partly explains as to why the high luminosity halo has low $\beta$ in the inner regions as compared to the low luminosity halo.
{width="85.00000%"}
In §\[subsec:tangentialdis\] we measured $\beta$ until $r=23$ kpc. Beyond this we can measure $\rsigma$ out to $r=56$ kpc, but not $\psigma$ and $\tsigma$. By using the circular velocity curve that we derived in §\[sec:circ\_velocity\] we can predict $\beta$ beyond $r>23$ kpc making use of the Jeans equation (Equation \[eqn:Jeanseq\]). To proceed we need to make an assumption about the density slope ($\alpha$); beyond $r>27$ kpc it has been shown that $\alpha$ is around 4.5. Assuming this, the predicted $\beta$ is plotted in Figure \[fig:beta\_prediction\]. It can be seen that there is a slight jump in the value of $\beta$ passing from $r = 23$ kpc to $r = 27$ kpc and beyond this the value of $\beta$ is around 0.4. The sudden jump in $\beta$ occurs via the Jeans equation (Equation \[eqn:Jeanseq\]), due to the discontinuity in $\alpha$ ($= - {\rm d} \ln \rho/ {\rm d} \ln {\rm r}$). Note, an assumption of steeper density slope would increase the $\beta$ and vice versa. The red line in figure is the anisotropy profile fitted to the simulated $\Lambda$CDM halos given by Equation (\[eqn:cford\_profile\]) from §\[sec:simulated\_halo\]. It can be seen that accretion based models cannot explain the dip that is in observations, specially the profile in the region $12<r/\kpc<23$. However, outside this region the simulations are roughly in agreement with observations. For $r<12$ kpc the observations match the value of around $\beta=0.5$ seen in simulations. For $r>23$ kpc the overall value of $\beta$ in observations is slightly low but the profile is flat as in simulations. The outer halo at $r=56$ kpc is radial with $\beta=0.55$.
CONCLUSION and DISCUSSION {#sec:conclusion}
=========================
We study the kinematics of $\sim4500$ BHB stars to obtain the velocity dispersion profiles along three orthogonal axes in spherical polar coordinates using the gaussian velocity ellipsoidal (GVE) DF. GVE as an estimator of the velocity dispersion has the advantage that no assumptions about potential or density are needed [*a priori*]{}. From the estimated velocity dispersion profiles using maximum likelihood analysis, we also derive the anisotropy profile of the Galactic halo and compare it to the simulated $\Lambda$CDM halos. Finally, using radial velocity dispersion profile, anisotropy profile and density power law we constrain the mass of the Milky Way Galaxy using the Jeans equation.
We measure the $\rsigma$ profile of the halo out to $r\sim60$ kpc. At large distance ($d\gg R_{\sun}$), $\rsigma$ can be approximated by $\sigmalos$. Thus in outskirt, $\sigmalos$ profile given by [@2008ApJ...684.1143X] converges to our $\rsigma$ profile. At $r\sim60$ kpc $\rsigma$ attains $\backsimeq 100$ kms$^{-1}$. However, in the inner halo the approximation ($\rsigma\approx\sigmalos$) is invalid and we find that the deviation of $\sigmalos$ from $\rsigma$ is as high as $\sim40$ kms$^{-1}$. We obtain a $\rsigma(r)$ profile with plateau in the inner halo, a sudden fall at $r\sim15$ kpc and a gradual decline outwards. Qualitatively, similar profile is also found by @1997ApJ...481..775S. However, our $\rsigma$ profile sharply falls at $r\sim15$ kpc whereas they find a gentle transition. The is probably due to the fact that they make an assumption that $\vcirc(r)$ is constant which we find is not completely true.
Next we estimate the tangential velocity dispersions, $\tsigma$ and $\psigma$. Using these estimates we are able to measure the $\beta(r)$ till $r = 25$ kpc. Astonishingly, we discover a dip in the $\beta$ profile at $r = 17$ kpc, where $\beta\backsimeq-1.2$. We find that the inner halo ($r<12$ kpc) is radial with $\beta \backsimeq 0.5$. This result of radially biased inner halo concurs with the recent results by @2009MNRAS.399.1223S [@2010ApJ...716....1B] using the proper motions. Beyond the switch over point in the range $18 \lesssim r/\kpc \lesssim 25$ the anisotropy rises slightly and becomes isotropic to mild radial. We also verify the result using an alternative DF, namely the D11 DF. A small systematic in the $\beta(r)$ profiles is seen from these two models which is mainly due to the assumption about density slope ($\alpha$) that needs to be made [*a priori*]{} for D11 DF. We check for the contribution of the halo substructures, namely Virgo Over-density and Sagittarius stellar stream, and find that they have little effect in the anisotropy profile. The effects of v$_{\rm LSR}$ and R$_{\sun}$ upon our velocity dispersions and anisotropy estimates are also found to be negligible.
D11 study the BHB stars in the radial bin ($10<r/\kpc<25$) and find the halo to be tangential. After re-analyzing the D11 sample in this bin, we find that this is mainly because of the choice of their bin size that encompasses the transition region ($13<r/\kpc<17$) where we detected a dip in the $\beta(r)$. Possibly it could be also because of the degeneracy between potential and anisotropy in their model. However, in their recent work D12 break the degeneracy among their model parameters and measure $\beta = 0.4_{-0.2}^{+0.15}$ with $\alpha=4.6$ in the region $16\lesssim r/\kpc \lesssim48$. Within the range of uncertainty our value for $\beta$ ($-0.14^{+0.52}_{-0.66}$) using GVE model agrees with them. We find that D12 value of $\beta\approx0.4$ in this bin, although derive from a sample which is dominated by stars within $r<27$ kpc, is not appropriate for the range $18<r/\kpc<23$, instead it is appropriate for the range $23<r/\kpc<48$. Finally, we check how well we can estimate the mass and anisotropy together using D11 DF in the outer-most region ($35<r/\kpc<84$) We find that due to the lack of tangential information, a degeneracy between mass-anisotropy cannot be broken.
Substituting the estimates of $\rsigma(r)$ and $\beta(r)$ in the Jeans equation, we then calculate the circular velocity profile of the Galaxy ($\vcirc(r)$). We detect the dip in the $\vcirc$ profile at 10-12 kpc, also seen by [@2009PASJ...61..227S] at 9 kpc, which is attributed to the massless ring as a perturbation to the disk. Finally, we fit the three component (exponential disk, Hernquist bulge, and NFW halo) galaxy model to the observed $\vcirc$ profile in order to obtain the mass distribution of the Galaxy. From our best fit model, we calculate $M_{\rm vir}$ of the halo to be $0.9^{+0.4}_{-0.3} \times 10^{12}$ $\msun$ with $R_{\rm vir} = 249^{+34}_{-31}\:\kpc$ and concentration parameter, $c = 12.0^{+0.6}_{-0.5}$. The mass of the Galaxy, within the extent we are able to constrain all the three components of velocity dispersions, is estimated to be $M(r \lesssim 25 \:\kpc) = 2.1 \times 10^{11}$ $\msun$. Our estimate for $M_{\rm vir}$ is in good agreement with the most of the recent estimates namely by as demonstrated in Figure \[fig:vcirc\_all\]. In their studies of same population of stars (BHB), [@2008ApJ...684.1143X] also fit a three component galaxy model and calculates $M_{\rm vir}$ to be $0.91^{+0.27}_{-0.18}\times 10^{12}$ $\msun$. Our result for $M_{\rm vir}$ is in very good agreement with their estimate but our uncertainty is slightly larger. Note that they make an assumption about the ($\vcirc/\vlos$) with radius from the simulations whereas we do not make any such assumption. Additionally, we also consider a more realistic three-dimensional disk model which is found to predict slightly higher Galactic mass $M_{\rm vir} = 1.2 ^{+0.5}_{-0.4}\times 10^{12}$ $\msun$ with $R_{\rm vir} = 274^{+35}_{-30}\:\kpc$ for flattening constant $a = 2.5$.
In the end, we used the measured quantities $\rsigma$ and $\vcirc$ to extend the $\beta$ profile beyond $r\sim25$ kpc up to the distance where $\rsigma(r)$ can be confidently measured ($r\sim60$ kpc). The only assumption that we make here is about the density profile which we choose to be $\alpha = 4.5$ in consent to the recent results by [@2009MNRAS.398.1757W; @2011MNRAS.416.2903D]. We find that the outer halo is radial and attains $\beta = 0.55$ at $r\sim60$ kpc.
We also compare our result with simulated stellar halo which are formed purely by accretion [@2005ApJ...635..931B]. This simulated halo is found to be in rough agreement to the observed halo in the inner region $r<12$ kpc. It is seen that in none of the instances of simulations the $\beta$ profiles obtained could predict tangential halo at any distance and thus fails to explain a dip seen at $r=17$ kpc in observed $\beta$ profile. In contrast, in the outer region ($r>25$ kpc) simulations and observations both agree in overall sense of the anisotropy and predict a flat anisotropy profile.
In all the observed quantities $\rsigma, \tsigma, \psigma$ and $\beta$ we see a dramatic shift in properties at $r\sim17$ kpc. We noticed that these undulations in the profiles are translated into our $\vcirc$ estimation resulting a varying $\vcirc$ profile. It could be true other way around, in a sense that the non-monotonic trends seen in all of our kinematic profiles could be due to the presence of so far unaccounted features in the Milky Way potential.
Alternatively, the shift in the properties seen in the observed profiles could possibly be an indication of a complex multi-component halo. Recently there have been series of works advocating multi-component halo. The studies of the calibration stars by @2007Natur.450.1020C and @2010ApJ...712..692C from the SDSS survey and the follow-up studies by @2012ApJ...746...34B have shown that the halo has at-least two distinct components. They associated the inner-halo to be formed in-situ whereas the outer halo are considered to be formed by accretion. @2010ApJ...712..692C and @2010ApJ...714..663D have found that the population fraction inversion point between the inner and outer halo lies between $\sim15-20$ kpc. @2012MNRAS.422.2116K studied the BHB and RR Lyrae stars towards the galactic-anticenter and North Galactic Pole and found that the retrograde component of the halo dominates for $r>12.5$ kpc. It seems that this transition between the inner to the outer halo is recorded in the $\beta$ of the halo as well. Additionally, duality in the formation history of the halo has also been seen in the recent smooth particle hydrodynamics and N-body simulations by @2009ApJ...702.1058Z [@2012MNRAS.420.2245M; @2011MNRAS.417.1260F]. @2012Natur.486...90K recently attribute the age difference of 2 billion years in the halo components to in-situ and accretion. On the contrary, @2011MNRAS.415.3807S reanalyzed the calibration stars from @2010ApJ...712..692C and find no reliable evidence of the existence of outer retrograde halo.
In a nutshell, the stellar halo is a test bed to understand the formation history of the galaxy. Even in this era where we have access to huge volume of spectroscopic and photometric data, the crucial physical quantities like velocity dispersions and anisotropy are not completely understood due to the lack of proper motions. With the advent of data inflowing in the coming decades through the magnificent next generation of spectroscopic survey like LEGUE [@2012RAA....12..735D] and specially, unprecedented proper motions from an astrometric mission like GAIA will help to put strong constrains on these fundamental quantities. Additionally, to see a bigger picture, confirming the results with the different stellar types or an alternative tracers is also crucial. Moreover, exploring the southern sky is equally important to complete the picture, for which up-coming spectroscopic survey like GALAH [^3] will also play an important role.
ACKNOWLEDGEMENT {#acknowledgement .unnumbered}
===============
We sincerely thank the anonymous referee for comments those helped to improve the paper. We also thank Dr. Ralph Schonrich and Francesco Fermani for their comments on the manuscript, particularly on the effect of (v$_{\rm LSR}$, R$_{\sun}$). Dr. Xiang Xiang Xue is thanked for the on-line publication of the clean sample of BHB stars. Mr. Tim White is also thanked for the comments on original manuscript. PRK acknowledges University of Sydney International Scholarship (USydIS) for the support of his candidature. GFL acknowledges support from ARC Discovery Project DP0665574. J.B.H. is funded through a Federation Fellowship from the Australian Research Council (ARC) and S.S. is funded through ARC DP grant 0988751 which supports the HERMES project.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is <http://www.sdss.org/.>
A. Binning and Effect of the bin size {#sec:binning}
=====================================
Here we investigate an effect of $n_{\rm bin}$ to our analysis. Figure \[fig:effect\_of\_the\_binning\] shows the velocity dispersion profiles and the anisotropy profile for the same sample of stars but with different particles in each bin. We see that with the decrease in the number of stars in each bin the uncertainties in the result increases. But the overall trend remains unaltered.
![Effect of the number of particles in each CME bin. Red, green and blue points are the estimates with 750, 1000 and 1200 number of stars in each bin respectively. From top to bottom are the $\rsigma$, $\tsigma$, $\psigma$ and anisotropy profile.[]{data-label="fig:effect_of_the_binning"}](f16){width="95.00000%"}
B. Effect of $_{\rm LSR}$ and R$_\sun$ {#sec:vlsrrsun_effect}
======================================
In literatures there are varied claims about the value of v$_{\rm LSR}$ ranging from 184–270 kms$^{-1}$ [@1998MNRAS.297..943O; @1999ApJ...524L..39M; @2009ApJ...700..137R; @2009ApJ...704.1704B; @2010ApJ...712..260K]. Similarly, the value of R$_\sun$ is also disputable within 8–8.5 kpc . [@2010MNRAS.402..934M] found that the ratio v$_{\rm LSR}$/R$_\sun$ can be better constrained than each of them alone and should range between $29.9-31.6$ kms$^{-1}$kpc$^{-1}$. Distressingly, there is still no consensus upon the values of (v$_{\rm LSR}$, R$_\sun$). To study the effect of chosen values of (v$_{\rm LSR}$, R$_\sun$) upon our estimates of dispersion profiles we repeat the same analysis done to obtain the black diamond points in §\[sec:dispersion\_profile\] (Figure \[fig:aniso\]). In Figure \[fig:effect\_vlsrRsun\] we show the results for different values of (v$_{\rm LSR}$, R$_\sun$). Here again the black diamond markers are obtained for a case (v$_{\rm LSR}$, R$_\sun$) = (220.0 kms$^{-1}$, 8.5 kpc) and is thus a replica of diamond points from Figure \[fig:aniso\] put again for the ease of comparison. The red and black markers in the figure show the effect of chosen R$_{\sun}$ values upon our estimates whereas the red, blue and cyan markers demonstrate the effect of chosen v$_{\rm LSR}$. Within the range of (v$_{\rm LSR}$, R$_\sun$) investigated, the figure clearly demonstrates a negligible effect of them upon our estimates, given the range of uncertainties.
![Velocity dispersion and anisotropy profiles for different combinations of v$_{\rm LSR}$ and R$_\sun$.[]{data-label="fig:effect_vlsrRsun"}](f17){width="95.00000%"}
[^1]: For the effect of the bin size, see appendix \[sec:binning\].
[^2]: The disk model with $b=0$ is also otherwise known as Toomre’s model or Kuzmin disk [@1963ApJ...138..385T; @2008gady.book.....B]
[^3]: <http://www.aao.gov.au/HERMES/GALAH/Home.html>
|
---
abstract: 'A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case, systematically studied in the present paper it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell’s equation. This new approach also sheds light on some classical results. In particular we connect search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics. As a byproduct, an intriguing connection between Cayle type conditions and discriminantly separable polynomials has been observed for all those small periods.'
author:
- Vladimir Dragović
- Milena Radnović
title: Caustics of Poncelet polygons and classical extremal polynomials
---
MSC2010: 14H70, 41A10 (70H06, 37J35, 26C05)
Key words: Poncelet polygons, elliptical billiards, Cayley conditions, extremal polynomials, elliptic curves, periodic trajectories, caustics, Pell’s equations, Chebyshev polynomials, Zolotarev polynomials, Akhiezer polynomials, discriminantly separable polynomials.
Introduction
============
In our recent paper [@DragRadn2018] we have developed a strong link between the theory of billiards within quadrics in $d$-dimensional space and the theory of extremal polynomials on the systems of $d$ intervals on the real line. Using this link, we proved some fundamental properties of the billiard dynamics and paved a road to a comprehensive study of periodic trajectories of the billiards within ellipsoids in the $d$-dimensional Euclidean space.
The goal of the present paper is to provide the case study of the basic, planar case. It is well known that a geometric manifestation of integrability of elliptical billiards is the existence of a caustic, a conic confocal with the boundary of a billiard table, which is tangent to every segment of a given billiard trajectory. It is also well known that the elliptical billiard dynamics is equivalent to projective-geometry situation considered by Poncelet in 1813/14, when the boundary conic and the caustic conic are in arbitrary position, not necessarily confocal. It should also be mentioned that it is possible to redefine the billiard reflection in projective-geometric terms of harmonic conjugation of four lines in a pencil, to associate a generalized billiard system to any pair of conics, a boundary and a caustic. This is the reason why we call periodic trajectories also the Poncelet polygons, or if the period is $n$, the Poncelet $n$-gons. The Poncelet theorem ([@Poncelet1822], see also [@LebCONIQUES; @GrifHar1978; @BergerGeometryII; @DragRadn2011book]) states that if such a polygon exists for the fixed boundary and the caustic, then there are infinitely many such polygons, sharing the boundary and the caustic. Cayley around 1853 derived a criterion which answers the question for two given conics, a boundary and a caustic, whether the dynamics they generate is periodic with a given period or not. Cayley answered the question by translating it to an equivalent question whether a given point of an elliptic curve is of a given order in the group structure defined by the elliptic curve. Some 35 years later, Halphen established a relationship between the Poncelet polygons and continued fractions and approximation theory, see [@Hal1888]\*[Part 2, page 600]{}.
Using the fact that the classical Cayley condition can be reformulated as a polynomial functional equation of Pell’s type, we are going to show that search for caustics which generate periodic trajectories of a given period is intimately related to the classical extremal polynomials, namely to the Zolotarev polynomials and Akhiezer polynomials. Zolotarev was a talented student of Chebyshev, the founding father of a famous Sankt Petersburg school in the second half of the XIX century. Chebyshev introduced celebrated Chebyshev polynomials in his study of extremal problems on an interval. It is interesting to point out that Chebyshev came to the extremal questions from the engineering problems from the theory of mechanisms. These problems were intimately related to the main technological questions brought by the industrial revolution. Similar questions were also studied by Poncelet. Akhiezer, one of the prominent figures of XX century mathematics, significantly developed further the ideas of Chebyshev and his school. Based upon his deep results on orthogonal polynomials and their continuous analogues, the notion of Baker-Akhiezer functions emerged in work of I. M. Krichever forty year ago. Soon it became one of the main tools of modern algebro-geometric theory of integrable systems, established by Novikov, Dubrovin, and others in 1970’s, see [@DKN] and references therein.
Apart from the connections between the Cayley condition and Zolotarev and Akhiezer extremal polynomials, we also establish in this paper an unexpected relationship between the Cayley condition and another class of polynomials, so-called discrimintly separable polynomials. This class has been introduced quite recently, less than a decade ago, in [@Drag2010], related to the celebrated Kowalevski top and the Kowalevski integration procedure for equations of motion of the top [@Kow1889]. Later, such polynomials were related to other continuous and discrete integrable systems, see [@DragKuk2014jgm; @DragKuk2014rcd; @DragKuk2014steklov; @DragKuk2017].
### Organization of this paper {#organization-of-this-paper .unnumbered}
The next Section \[sec:periodic\] introduces the basic notions related to elliptical billiards, derives an algebro-geometric criterion for periodicity with a given period $n$ for billiard trajectories within given ellipse and a confocal caustic in terms of an elliptic curve which is isomorphic, but different then so called Cayley’s cubic, originally studied by Cayley. Then a detailed study and two characterizations of $n$ elliptic periodic trajectories are provided. Let us recall that a $n$ elliptic periodic trajectory has the period $n$ in elliptic coordinates, but is not necessarily periodic in Cartesian coordinates. One of the characterizations is given in terms of the underlying elliptic curve. The second one is done in Section \[sec:poly\] in terms of extremal polynomials on two intervals and solutions of polynomial functional equations, like the Pell equation. This Section concludes with a detailed analysis of winding numbers and their different appearances. The derived properties of winding numbers serve as a main classifying tool for the caustics which generate periodic trajectories of given period. They also lead to another proof of the monotonicity of rotation numbers. Section \[sec:examples\] employs the derived criteria for periodicity and winding numbers to study in detail the cases of small periods $n=2, 3, 4, 5, 6$. Section \[sec:classicalextrema\] related the results from the previous Section to three classes of classical extremal polynomials on two intervals introduced by Zolotarev and Akhiezer. The last Section \[sec:separable\] collects come intriguing, yet not fully understood, experimental observations that Cayley type conditions produce so called discriminantly seaparable polynomials.
Elliptical billiards and periodic trajectories {#sec:periodic}
==============================================
Elliptical billiard and confocal families {#sec:confocal}
-----------------------------------------
*Mathematical billiard* within a plane domain is a dynamical system where a particle moves without constraints within the domain, and obeys the billiard reflection law on the boundary [@KozTrBIL], see Figure \[fig:reflection\]. Billiard trajectories are polygonal lines with verteces on the boundary.
(,) – ([-/2]{},[-/2]{}) arc ([atan(/ )]{}:5:[-sqrt(+)/2]{}) – (,);
(,) – ([+/2]{},[+/2]{}) arc ([atan(/ )]{}:85:[sqrt(+)/2]{}) – (,);
(0,0) circle \[x radius=, y radius=\];
([-]{}, [-]{}) to ([+]{}, [+]{});
(, ) to (0.5,-1.2); (2.7,0.3) to (, ) ;
Mathematical billiard is an idealised model, where the billiard ball is replaced by a material point, and the friction and spin are neglected. Such billiard system in the Euclidean space is a good model for the motion of light rays, with mirror boundary. The dynamics has two different regimes: inside the billiard domain, and the impacts. We assume that the impacts are *absolutely elastic*, which means that the geometric billiard law is satisfied, i.e. the impact and reflection angles are congruent to each other, and the speed remains unchanged. Here, we assume that the material point is travelling under inertia between the impacts, although motion in a force field can also be considered, for example in a gravitational field [@KL1991], with Hooke’s potential [@Fed2001; @Radn2015], or with integrable potentials [@Drag2002].
In this work, we will study *elliptical billiards* – the mathematical billiards within an ellipse in the Euclidean plane: $${\pazocal{E}}\ :\ \frac{\mathsf{x}^2}{a}+\frac{\mathsf{y}^2}{b}=1,
\quad
a>b>0.$$ Each trajectory of the elliptic billiard has *a caustic*: a curve such that each segment of the trajectory lies on its tangent line. Moreover, the caustics of billiard trajectories within ${\pazocal{E}}$ belong to the family of conics confocal with the boundary: $$\label{eq:confocal}
{\pazocal{C}}_{\lambda}\ :\ \frac {\mathsf{x}^2}{a-\lambda}+ \frac{\mathsf{y}^2}{b-\lambda }=1.$$
We notice that the family contains two types of smooth conics: ellipses, corresponding to $\lambda<b$, and hyperbolas, corresponding to $\lambda\in(b,a)$, see Figure \[fig:confocal\].
(0,0) circle \[x radius=[sqrt(8)]{}, y radius=[sqrt(5)]{}\]; (0,0) circle \[x radius=2, y radius=1\]; (0,0) circle \[x radius=[sqrt(6)]{}, y radius=[sqrt(3)]{}\];
([sqrt(3)]{},0) circle \[x radius=0.1,y radius=0.1\]; ([-sqrt(3)]{},0) circle \[x radius=0.1,y radius=0.1\];
plot ([sqrt(2\*(+1))]{},); plot ([-sqrt(2\*(+1))]{},);
plot ([sqrt(1+/2))]{},); plot ([-sqrt(1+/2))]{},);
plot ([0.3\*sqrt(1+/2.7))]{},); plot ([-0.3\*sqrt(1+/2.7))]{},);
In addition, there are two degenerated conics and the family: the $\mathsf{x}$-axis, corresponding to $\lambda=b$; and the $\mathsf{y}$-axis, corresponding to $\lambda=a$.
Each point in the plane, which is not a focus of the confocal family, lies on exactly two conics ${\pazocal{C}}_{\lambda_1}$ and ${\pazocal{C}}_{\lambda_2}$ from – one ellipse and one hyperbola, which are orthogonal to each other at the intersection point. In other words, we can join to such a point a unique pair of parameters $(\lambda_1,\lambda_2)$, $\lambda_1<\lambda_2$, corresponding to the two confocal conics which contain the point. The pair $(\lambda_1,\lambda_2)$ is called *elliptic coordinates* of the point. We note that points symmetric with respect to the $\mathsf{x}$- and $\mathsf{y}$-axes have the same elliptic coordinates.
All segments of the billiard trajectories within ${\pazocal{E}}$ with a fixed caustic ${\pazocal{C}}_{\lambda_0}$ lie in a domain $\Omega_{\lambda_0}$, that is:
- if ${\pazocal{C}}_{\lambda_0}$ is an ellipse, that is $\lambda_0\in(0,b)$, $\Omega_{\lambda_0}$ is the annulus between the billiard boundary ${\pazocal{E}}$ and the caustic;
- if ${\pazocal{C}}_{\lambda_0}$ is a hyperbola, that is $\lambda_0\in(b,a)$, $\Omega_{\lambda_0}$ is the part within the billiard boundary ${\pazocal{E}}$ which is between the branches of the caustic.
In the elliptic coordinates, $\Omega_{\lambda_0}$ is given by: $$(\lambda_1,\lambda_2)\in[0,\alpha_1]\times[\alpha_2,a],
\quad
\alpha_1=\min\{b,\lambda_0\},
\
\alpha_2=\max\{b,\lambda_0\}.$$ On any billiard trajectory, the value $\lambda_1=0$ is achieved at the reflection points on the boundary, value $\lambda_2=a$ at the intersection points with the $\mathsf{y}$-axis, while corresponding elliptic coordinate has value $\lambda_0$ at the touching points with the caustic, and value $b$ at the intersection points with $\mathsf{x}$-axis. Between these points, the elliptic coordinates change monotonously.
For a periodic billiard trajectory, we introduce *the winding numbers* $(m_0,m_1)$: $m_0$ is the number of its reflection points, and $m_1$ the number of its intersection with the $\mathsf{y}$-axis. The number $m_1$ is always even, since the $\mathsf{y}$-axis must be crossed even number of times along the period.
Periodic trajectories and Cayley’s cubic {#sec:cayley}
----------------------------------------
It is of particular interest to condsider periodic billiard trajectories – the trajectories that become closed after certain number of reflections. In the next Lemma, we note that the type of the caustic may be determined by the period of a trajectory.
\[lemma:hyperbola\] The period of a closed billiard trajectory with a hyperbola as caustic is always even.
Denote by $F_1, F_2$ the focal points of the boundary ellipse. Then every segment of the billiard trajectory intersects the segment $F_1F_2$. Thus, for a periodic trajectory there should be an even number of intersections of the trajectory with $F_1F_2$ and the period is even.
The Poncelet theorem [@Poncelet1822; @LebCONIQUES; @DragRadn2011book] implies that all trajectories sharing the same caustic with a periodic elliptical billiard trajectory are also periodic, and moreover all these trajectories have the same period. It is natural to ask about an analytic condition that would determine if elliptical billiard trajectories with a given caustic will become closed after certain number of bounces. Such condition was derived by Cayley in the mid XIXth century.
Here, in Theorems \[th:curve-billiard\] and \[th:cayley-billiard\], we present the derivation of the analytic conditions following the ideas Jacobi and Darboux [@JacobiGW; @DarbouxSUR], see also [@DragRadn2004]. In Theorem \[th:cayley\], we will present the classical Cayley’s conditions, since it has a slightly different form, and show they are equivalent to the ones presented in Theorem \[th:cayley-billiard\].
\[th:curve-billiard\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $n$-periodic if and only if $nQ_{0}\sim nQ_{\lambda_0}$ on the elliptic curve: $$\label{eq:billiard-cubic}
{\mathcal{C}}\ :\ y^2=(a-x)(b-x)(\lambda_0-x),$$ with $Q_0$ being a point of ${\mathcal{C}}$ corresponding to $x=0$, and $Q_{\lambda_0}$ the point corresponding to $x=\lambda_0$.
Consider the integral $$\label{eq:integral}
\frac{d\lambda_1}{\sqrt{(a-\lambda_1)(b-\lambda_1)(\lambda_0-\lambda_1)}}
+
\frac{d\lambda_2}{\sqrt{(a-\lambda_2)(b-\lambda_2)(\lambda_0-\lambda_2)}}$$ along the polygonal line, which represents $n$ consecutive segments of a billiard trajectory within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$. That integral equals to zero along each line touching ${\pazocal{C}}_{\lambda_0}$. Thus, considering the behaviour of the elliptic coordinates along the trajectory with winding numbers $(m_0,m_1)$, $m_0=n$, we get that the first vertex of the polygonal line will coincide with the last one if and only if $$n(Q_0-Q_{\alpha_1})+m_1(Q_{\alpha_2}-Q_a)\sim0$$ on ${\mathcal{C}}$. Here, we denoted by $Q_{\alpha_1}$, $Q_{\alpha_2}$, $Q_{a}$ the points of ${\mathcal{C}}$ corresponding to $x$ equal to $\alpha_1$, $\alpha_2$, $a$ respectively. Since $2Q_{\alpha_2}\sim2Q_a$ and $m_1$ is even, we obtained that the periodicity condition reduced to: $$n(Q_0-Q_{\alpha_1})\sim0$$ Now, if $n$ is even, $2{\pazocal{Q}}_{\alpha_1}\sim2Q_{\lambda_0}$ implies $n(Q_0-Q_{\lambda_0})\sim0$. If $n$ is odd, then, according to Lemma \[lemma:hyperbola\], the caustic of the trajectory must be an ellipse, so $\alpha_1=\lambda_0$.
\[th:cayley-billiard\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $n$-periodic if and only if: $$\begin{gathered}
C_2=0,
\quad
\left|
\begin{array}{cc}
C_2 & C_3
\\
C_3 & C_4
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
C_2 & C_3 & C_4
\\
C_3 & C_4 & C_5
\\
C_4 & C_5 & C_6
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=3,5,7,\dots
\\
B_3=0,
\quad
\left|
\begin{array}{cc}
B_3 & B_4
\\
B_4 & B_5
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
B_3 & B_4 & B_5
\\
B_4 & B_5 & B_6
\\
B_5 & B_6 & B_7
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=4,6,8,\dots.
\end{gathered}$$ Here, we denoted: $$\begin{gathered}
\sqrt{(a-x)(b-x)(\lambda_0-x)}=B_0+B_1x+B_2x^2+\dots,
\\
\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{\lambda_0-x}=C_0+C_1x+C_2x^2+\dots,
\end{gathered}$$ the Taylor expansions around $x=0$.
Denote by $Q_{\infty}$ the point of ${\mathcal{C}}$ [eq:billiard-cubic]{} corresponding to $x=\infty$ and notice that $$\label{eq:2Q}
2Q_{\lambda_0}\sim2 Q_{\infty}.$$
Consider first $n$ even. Because of [eq:2Q]{}, the condition $nQ_{0}\sim nQ_{\lambda_0}$ is equivalent to $nQ_{0}\sim nQ_{\infty}$, which is equivalent to the existence of a meromorphic function of ${\mathcal{C}}$ with the unique pole at $Q_{\infty}$ and unique zero at $Q_{0}$, such that the pole and the zero are both of the multiplicity $n$. The basis of $\mathcal{L}(nQ_{\infty})$ is: $$\label{eq:basis-even}
1,x,x^2,\dots,x^{n/2},y,xy, x^{n/2-2}y,$$ thus a non-trivial linear combination of those functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left|
\begin{array}{llll}
B_{n/2+1} & B_{n/2} & \dots & B_3\\
B_{n/2+2} & B_{n/2+1} &\dots & B_4\\
\dots\\
B_{n-1} & B_n &\dots & B_{n/2+1}
\end{array}
\right|=0.$$
Now, suppose $n$ is odd. Because of [eq:2Q]{}, the condition $nQ_{0}\sim nQ_{\lambda_0}$ is equivalent to $nQ_{0}\sim (n-1)Q_{\infty}+Q_{\lambda_0}$, which is equivalent to the existence of a meromorphic function of ${\mathcal{C}}$ with only two poles: of order $n-1$ at $Q_{\infty}$ and a simple pole at $Q_{\lambda_0}$, and unique zero at $Q_{0}$. The basis $\mathcal{L}( (n-1)Q_{\infty}+Q_{\lambda_0})$ is: $$\label{eq:basis-odd}
1,x,x^2,\dots,x^{(n-1)/2},\frac{y}{\lambda_0-x}, \frac{xy}{\lambda_0-x}, \dots, \frac{x^{(n-1)/2-1}y}{\lambda_0-x},$$ thus a non-trivial linear combination of those functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left|
\begin{array}{llll}
C_{(n-1)/2+1} & C_{(n-1)/2} & \dots & C_2\\
C_{(n-1)/2+2} & C_{(n-1)/2+1} &\dots & C_3\\
\dots\\
C_{n-1} & C_n &\dots & C_{(n-1)/2+1}
\end{array}
\right|=0.$$
We can rewrite the equation (\[eq:confocal\]) of the family of confocal conics in a matrix form: $$(x\, y\, 1) M_ {\lambda}(x\, y \,1)^T=0,$$ where $$M_{\lambda}=\left(
\begin{array}{ccc}
\dfrac1{a-\lambda} & 0 & 0\\
0 & \dfrac1{b-\lambda} & 0\\
0& 0& -1
\end{array}
\right).$$ Following Cayley, we can use the matrices $M_{\lambda}$ to define another elliptic curve, so called *Cayley’s cubic* $$\label{eq:cayley-cubic}
{\mathcal{C}}^*\ :\ y^2=\det(M_0+xM_{\lambda_0}).$$ The curve [eq:cayley-cubic]{} can be explicitly written as: $$y^2=-(x+1) \left(\frac{1}{a-\lambda_0 }+\frac{x}{a}\right) \left(\frac{1}{b-\lambda_0 }+\frac{x}{b}\right).$$ This is an elliptic curve with branching points corresponding to: $$x\in\left\{-1,\frac{a}{\lambda_0-a},\frac{b}{\lambda_0-b},\infty\right\}.$$ Using the bilinear transformation: $$\begin{gathered}
(x,y)\mapsto(x_1,y_1),
\\
x_1=\frac{\lambda_0 x}{1+x},
\quad
y_1=\const\cdot\frac{y}{1+x},\end{gathered}$$ The branching points are transformed to $\{\infty,a,b,\lambda_0\}$ respectively and the Cayley’s cubic into $y_1^2=(a-x_1)(b-x_1)(\lambda_0-x_1)$.
Thus, $nP_0\sim nP_{\infty}$ on ${\mathcal{C}}^*$ is equivalent to $nQ_0\sim nQ_{\lambda_0}$ on ${\mathcal{C}}$. We conclude that previous Theorem is equivalent to the following formulation of the classical Cayley condition.
\[th:cayley\] There is a closed polygonal line with $n$ vertices, inscribed in ${\pazocal{E}}$ and circumscribed about ${\pazocal{C}}_{\lambda_0}$ if and only if $nP_{0}\sim nP_{\infty}$ on the Cayley’s cubic ${\mathcal{C}}^*$, given by [eq:cayley-cubic]{}, with $P_0$ being one of the points of the curve ${\mathcal{C}}^*$ corresponding to $x=0$, and $P_{\infty}$ the point corresponding to $x=\infty$.
Further, $nP_{0}\sim nP_{\infty}$ is equivalent to: $$\begin{gathered}
A_2=0,
\quad
\left|
\begin{array}{cc}
A_2 & A_3
\\
A_3 & A_4
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
A_2 & A_3 & A_4
\\
A_3 & A_4 & A_5
\\
A_4 & A_5 & A_6
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=3,5,7,\dots
\\
A_3=0,
\quad
\left|
\begin{array}{cc}
A_3 & A_4
\\
A_4 & A_5
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
A_3 & A_4 & A_5
\\
A_4 & A_5 & A_6
\\
A_5 & A_6 & A_7
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=4,6,8,\dots,
\end{gathered}$$ with $$\sqrt{\det(M_0+xM_{\lambda_0})}=A_0+A_1x+A_2x^2+A_3x^3+\dots,$$ being the Taylor expansion about $x=0$.
Elliptic periodic trajectories {#sec:elliptic-periodic}
------------------------------
Points of the plane which are symmetric with respect to the coordinate axes share the same elliptic coordinates, thus there is no bijection between the elliptic and the Cartesian coordinates. Thus, we introduce a separate notion of periodicity in elliptic coordinates.
A billiard trajectory is *$n$-elliptic periodic* is it is $n$-periodic in elliptic coordinates joined to the confocal family [eq:confocal]{}.
\[th:even-periodic\] A billiard trajectory within ${\pazocal{E}}$ is $2n$-periodic if and only if it is $n$-elliptic periodic.
The symmetry implies that each $n$-elliptic periodic trajectory is symmetric with respect to one of the axes or the origin. Consequently, such a trajectory is always $2n$-periodic.
Along any $2n$-periodic trajectory, the elliptic coordinate $\lambda_1$ will trace the segment $[0,\alpha_1]$ $2n$ times in each direction. Similarly, $\lambda_2$ will trace the segment $[\alpha_2,a]$ an even number of times in each direction, since each periodic trajectory intersects the $\mathsf{y}$-axis even number of times. Each of these intersections corresponds to $\lambda_2 = a$. This all together implies that the given trajectory will be elliptic periodic with period $n$.
Thus, we see that in the Euclidean plane, there are no $2n$-periodic trajectories which are not $n$-elliptic periodic. The situation is different in higher-dimensional spaces, as is shown in [@DragRadn2018].
Now, we will derive algebro-geometric conditions for elliptic periodic trajectories.
\[th:elliptic-periodic\] A billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{Q}}_{\lambda_0}$ is $n$-elliptic periodic without being $n$-periodic if and only if one of the following conditions is satisfied on ${\mathcal{C}}$:
- ${\pazocal{Q}}_{\lambda_0}$ is an ellipse and $n(Q_{0}-Q_{\lambda_0})+Q_b-Q_a\sim0$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola and $n(Q_{0}-Q_{b})+Q_{\lambda_0}-Q_a\sim0$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ is odd, and $n(Q_{0}-Q_{b})\sim0$.
Moreover, such trajectories are always symmetric with respect to the origin in Case (a) and, when $n$ is odd, in Case (b). They are symmetric with respect to the larger axis in Case (b) for even $n$, and with respect to the smaller axis in Case (c).
Let $M_0$ be the initial point of a given $n$-elliptic periodic trajectory, and $M_1$ the next point on the trajectory with the same elliptic coordinates. Then, integrating from $M_0$ to $M_1$ along the trajectory, we get: $$n(Q_0-Q_{\alpha_1})+m(Q_{\alpha_2}-Q_{a})\sim0,$$ where $m$ is the number of times that the particle crossed the $\mathsf{y}$-axis. Along that part of the trajectory, the particle crossed the $\mathsf{x}$-axis $m'$ times, where $m'=n$ if $b=\alpha_1$, and $m'=m$ if $b=\alpha_2$.
To conclude the proof, we notice that at least one of $m$, $m'$ must be odd if the trajectory is not $n$-periodic.
The explicit Cayley-type conditions for elliptic periodic trajectories are:
\[th:elliptic-cayley\] A billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{Q}}_{\lambda_0}$ is $n$-elliptic periodic without being $n$-periodic if and only if one of the following conditions is satisfied:
- ${\pazocal{Q}}_{\lambda_0}$ is an ellipse and $$\begin{gathered}
C_1=0,
\quad
\left|
\begin{array}{cc}
C_1 & C_2
\\
C_2 & C_3
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
C_1 & C_2 & C_3
\\
C_2 & C_3 & C_4
\\
C_3 & C_4 & C_5
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=2,4,6,\dots
\\
B_2=0,
\quad
\left|
\begin{array}{cc}
B_2 & B_3
\\
B_3 & B_4
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
B_2 & B_3 & B_4
\\
B_3 & B_4 & B_5
\\
B_4 & B_5 & B_6
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=3,5,7,\dots;
\end{gathered}$$
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola and $$\begin{gathered}
D_1=0,
\quad
\left|
\begin{array}{cc}
D_1 & D_2
\\
D_2 & D_3
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
D_1 & D_2 & D_3
\\
D_2 & D_3 & D_4
\\
D_3 & D_4 & D_5
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=2,4,6,\dots
\\
B_2=0,
\quad
\left|
\begin{array}{cc}
B_2 & B_3
\\
B_3 & B_4
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
B_2 & B_3 & B_4
\\
B_3 & B_4 & B_5
\\
B_4 & B_5 & B_6
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=3,5,7,\dots;
\end{gathered}$$
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ is odd, and $$D_2=0,
\quad
\left|
\begin{array}{cc}
D_2 & D_3
\\
D_3 & D_4
\end{array}
\right|=0,
\quad
\left|
\begin{array}{ccc}
D_2 & D_3 & D_4
\\
D_3 & D_4 & D_5
\\
D_4 & D_5 & D_6
\end{array}
\right|=0,
\dots
\quad\text{for}\quad n=3,5,7,\dots.$$
Here, we denoted: $$\begin{gathered}
\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{b-x}=D_0+D_1x+D_2x^2+\dots,
\end{gathered}$$ the Taylor expansion around $x=0$, while $B$s and $C$s are as in Theorem \[th:cayley-billiard\].
\(a) In this case, the caustic ${\pazocal{Q}}_{\lambda_0}$ is an ellipse.
Take first $n$ even. Using Theorem \[th:elliptic-periodic\], we have: $$nQ_0
\sim
nQ_{\lambda_0}-Q_b+Q_a\sim nQ_{\infty}-Q_b+Q_a
\sim
n Q_{\infty}-Q_{\infty}+Q_{\lambda_0}
\sim
(n-1)Q_{\infty}+Q_{\lambda_0}.$$ The basis of $\mathcal{L}((n-1)Q_{\infty}+Q_{\lambda_0})$ is: $$1,x,x^2,\dots,x^{n/2-1},\frac{y}{x-\lambda_0},\frac{xy}{x-\lambda_0},
\frac{x^{n/2-1}y}{x-\lambda_0},$$ thus a non-trivial linear combination of these functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left|
\begin{array}{llll}
C_{n/2} & C_{n/2-1} & \dots & C_1\\
C_{n/2+1} & C_{n/2} & \dots & C_2\\
\dots\\
C_{n-1} & C_{n-2} & \dots & C_{n/2}
\end{array}
\right|
=0.$$ For odd $n$, we have: $$nQ_0
\sim
nQ_{\lambda_0}-Q_b+Q_a
\sim
(n-1)Q_{\infty}+Q_{\lambda_0}-Q_b+Q_a
\sim
(n-1) Q_{\infty}+Q_{\infty}
\sim
nQ_{\infty}.$$ The basis of $\mathcal{L}(nQ_{\infty})$ is: $$1,x,x^2,\dots,x^{(n-1)/2},y,xy, x^{(n-1)/2-1}y,$$ thus a non-trivial linear combination of these functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left|
\begin{array}{llll}
B_{(n-1)/2+1} & B_{(n-1)/2} & \dots & B_2\\
B_{(n-1)/2+2} & B_{(n-1)/2+1} & \dots & B_3\\
\dots\\
B_{n-1} & B_{n-2} & \dots & B_{(n-1)/2+1}
\end{array}
\right|
=0.$$
\(b) In this case, the caustic ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola.
Take first $n$ even. Using Theorem \[th:elliptic-periodic\], we have: $$nQ_0
\sim
nQ_{b}-Q_{\lambda_0}+Q_a\sim nQ_{\infty}-Q_{\lambda_0}+Q_a
\sim
n Q_{\infty}-Q_{\infty}+Q_{b}
\sim
(n-1)Q_{\infty}+Q_{b}.$$ The basis of $\mathcal{L}((n-1)Q_{\infty}+Q_{b})$ is: $$1,x,x^2,\dots,x^{n/2-1},\frac{y}{x-b},\frac{xy}{x-b},
\frac{x^{n/2-1}y}{x-b},$$ thus a non-trivial linear combination of these functions with a zero of order $n$ at $x=0$ exists if and only if: $$\left|
\begin{array}{llll}
D_{n/2} & D_{n/2-1} & \dots & D_1\\
D_{n/2+1} & D_{n/2} & \dots & D_2\\
\dots\\
D_{n-1} & D_{n-2} & \dots & D_{n/2}
\end{array}
\right|
=0.$$ For odd $n$, we have: $$nQ_0
\sim
nQ_{b}+Q_{\lambda_0}-Q_a
\sim
(n-1)Q_{\infty}+Q_{b}+Q_{\lambda_0}-Q_a
\sim
(n-1) Q_{\infty}+Q_{\infty}
\sim
nQ_{\infty}.$$ The determinant conditions are then obtained as in part (a).
\(c) We have $nQ_0\sim nQ_b\sim(n-1)Q_{\infty}+Q_b$. The conditions are derived as in the proof of Theorem \[th:cayley-billiard\], just replacing $\lambda_0$ by $b$.
We notice that whenever a trajectory is $n$-elliptic periodic, the divisor $n(Q_0-Q_{\lambda_0})$ is of order $2$ on the elliptic curve [eq:billiard-cubic]{}, which means it is equivalent to one of the following: $Q_{\lambda_0}-Q_{\infty}$, $Q_{a}-Q_{\infty}$, or $Q_{b}-Q_{\infty}$.
More precisely, a billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{Q}}_{\lambda_0}$ is $n$-elliptic periodic without being $n$-periodic if and only if:
- ${\pazocal{Q}}_{\lambda_0}$ is an ellipse and $n(Q_{0}-Q_{\lambda_0})\sim Q_{\lambda_0}-Q_{\infty}$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ even, and $n(Q_{0}-Q_{\lambda_0})\sim Q_{b}-Q_{\infty}$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ odd, and $n(Q_{0}-Q_{\lambda_0})$ is equivalent to $ Q_{a}-Q_{\infty}$ or $Q_{\lambda_0}-Q_{\infty}$.
Theorem \[th:cayley-billiard\] implies that a billiard trajectory within ${\pazocal{E}}$ with the caustic ${\pazocal{Q}}_{\lambda_0}$ is $2n$-periodic if and only if $n(Q_0-Q_{\lambda_0})$ is of order $2$ or equivalent to the zero divisor. Thus, such a trajectory would be $2n$-periodic without being $n$-elliptic periodic if and only if:
- ${\pazocal{Q}}_{\lambda_0}$ is an ellipse and $n(Q_{0}-Q_{\lambda_0})$ is equivalent to $Q_{a}-Q_{\infty}$ or $ Q_{b}-Q_{\infty}$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ even, and $n(Q_{0}-Q_{\lambda_0})$ is equivalent to $Q_{a}-Q_{\infty}$ or $ Q_{\lambda_0}-Q_{\infty}$;
- ${\pazocal{Q}}_{\lambda_0}$ is a hyperbola, $n$ odd, and $n(Q_{0}-Q_{\lambda_0})\sim Q_{b}-Q_{\infty}$.
Theorem \[th:even-periodic\] shows that none of these scenarios can be realized.
Polynomial functional equations and periodicity conditions {#sec:poly}
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Pell’s equations, extremal polynomials, and periodicity
-------------------------------------------------------
The matrix conditions for periodic trajectories, presented in Sections \[sec:cayley\] and \[sec:elliptic-periodic\], can be equivalently written in a form of polynomial functional equations. We will derive those equations in this section, and relate them to the so called *Pell equations*. For discussion including higher-dimensional cases, see [@DragRadn2018] and references therein.
\[th:polynomial\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $n$-periodic if and only if there exists a pair of real polynomials $p_{d_1}$, $q_{d_2}$ of degrees $d_1$, $d_2$ respectively, and satisfying the following:
- if $n=2m$ is even, then $d_1=m$, $d_2=m-2$, and $$p_{m}^2(s)
-
s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)
{q}_{m-2}^2(s)=1;$$
- if $n=2m+1$ is odd, then $d_1=m$, $d_2=m-1$, and $$\left(s-\frac1{\lambda_0}\right)p_m^2(s)
-
s\left(s-\frac1a\right)\left(s-\frac1b\right)q_{m-1}^2(s)=-1.$$
We note first that the proof of Theorem \[th:cayley-billiard\] implies that there is a non-trivial linear combination of the bases for $n$ even, or for $n$ odd, with the zero of order $n$ at $x=0$.
\(a) For $n=2m$, from there we get that there are polynomials $p_m^*(x)$ and $q_{m-2}^*(x)$ of degrees $m$ and $m-2$ respectively, such that the expression $$p_{m}^*(x)-q_{m-2}^*(x)\sqrt{(a-x)(b-x)(\lambda_0-x)}$$ has a zero of order $2m$ at $x=0$. Multiplying that expression by $$p_{m}^*(x)+q_{m-2}^*(x)\sqrt{(a-x)(b-x)(\lambda_0-x)},$$ we get that the polynomial $(p_{m}^*(x))^2-(a-x)(b-x)(\lambda_0-x)(q_{m-2}^*(x))^{2}$ has a zero of order $2m$ at $x=0$. Since the degree of that polynomial is $2m$, is follows that: $$(p_{m}^*(x))^2-(a-x)(b-x)(\lambda_0-x)(q_{m-2}^*(x))^{2}=cx^{2m},$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $p_m^*$. Dividing the last relation by $cx^{2m}$ and introducing $s=1/x$, we get the requested relation.
\(b) On the other hand, for $n=2m+1$, we get that there are polynomials $p_m^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$ respectively, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{\lambda_0-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(\lambda_0-x)
\left(
p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{\lambda_0-x}
\right)
,$$ we get that the polynomial $(\lambda_0-x)(p_{m}^*(x))^2-(a-x)(b-x)(q_{m-1}^*(x))^{2}$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(\lambda_0-x)(p_{m}^*(x))^2-(a-x)(b-x)(q_{m-1}^*(x))^{2}=cx^{2m+1},$$ for some constant $c$. Notice that $c$ is positive, since it equals the opposite of the square of the leading coefficient of $p_m^*$. Dividing the last relation by $-cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation.
\[cor:pell-periodic\] If the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $n$-periodic, then there exist real polynomials $\hat{p}_n$ and $\hat{q}_{n-2}$ of degrees $n$ and $n-2$ respectively, which satisfy the Pell equation: $$\label{eq:pell}
\hat{p}_{n}^2(s)-s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)\hat{q}_{n-2}^2(s)=1.$$
For $n=2m$, take $\hat{p}_n=2p_{m}^2-1$ and $\hat{q}_{n-2}=2p_mq_{m-2}$. For $n=2m+1$, we set $\hat{p}_n=2\left(s-\dfrac1{\lambda_0}\right)p_{m}^2+1$ and $\hat{q}_{n-2}=2p_mq_{m-1}$.
\[th:polynomial-elliptic\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are elliptic $n$-periodic without being $n$-periodic if and only if there exists a pair of real polynomials $p_{d_1}$, $q_{d_2}$ of degrees $d_1$, $d_2$ respectively, and satisfying the following:
- ${\pazocal{C}}_{\lambda_0}$ is an ellipse and
- $n=2m$ is even, $d_1=d_2=m-1$, $$s\left(s-\frac1{\lambda_0}\right)p_{m-1}^2(s)
-\left(s-\frac1a\right)\left(s-\frac1b\right)q_{m-1}^2(s)=-1;$$
- $n=2m+1$ is odd, $ d_1=m$, $d_2=m-1$, $$s p_{m}^2(s)
-\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)q_{m-1}^2(s)=1;$$
- ${\pazocal{C}}_{\lambda_0}$ is a hyperbola and
- $n=2m$ is even, $d_1=d_2=m-1$, $$s\left(s-\frac1{b}\right)p_{m-1}^2(s)
-\left(s-\frac1a\right)\left(s-\frac1{\lambda_0}\right)q_{m-1}^2(s)=-1;$$
- $n=2m+1$ is odd, $ d_1=m$, $d_2=m-1$, $$s p_{m}^2(s)
-\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)q_{m-1}^2(s)=1;$$
- ${\pazocal{C}}_{\lambda_0}$ is a hyperbola, $n=2m+1$ is odd, $d_1=m$, $d_2=m-1$, $$\left(s-\frac1b\right) p_{m}^2(s)
-s\left(s-\frac1a\right)\left(s-\frac1{\lambda_0}\right)q_{m-1}^2(s)=-1.$$
\(a) For $n=2m$, the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m-1}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m-1$, such that the expression $$p_{m-1}^*(x)-q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{\lambda_0-x}$$ has a zero of order $2m$ at $x=0$. Multiplying that expression by $$(\lambda_0-x)\left(p_{m-1}^*(x)+q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{\lambda_0-x}\right),$$ we get that the polynomial $(\lambda_0-x)(p_{m-1}^*(x))^2-(a-x)(b-x)(q_{m-1}^*(x))^2$ has a zero of order $2m$ at $x=0$. Since the degree of that polynomial is $2m$, is follows that: $$(\lambda_0-x)(p_{m-1}^*(x))^2-(a-x)(b-x)(q_{m-1}^*(x))^2=cx^{2m},$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $q_{m-1}^*$. Dividing the last relation by $-cx^{2m}$ and introducing $s=1/x$, we get the requested relation.
For $n=2m+1$, the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\sqrt{(a-x)(b-x)(\lambda_0-x)}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$p_{m}^*(x)+q_{m-1}^*(x)\sqrt{(a-x)(b-x)(\lambda_0-x)}
,$$ we get that the polynomial $(p_{m}^*(x))^2-(a-x)(b-x)(\lambda_0-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(p_{m}^*(x))^2-(a-x)(b-x)(\lambda_0-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is positive, since it equals the square of the leading coefficient of $q_{m-1}^*$. Dividing the last relation by $cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation.
The result (b) is obtained in a similar way.
For (c), the proof of Theorem \[th:elliptic-cayley\] implies that there are polynomials $p_{m}^*(x)$ and $q_{m-1}^*(x)$ of degrees $m$ and $m-1$, such that the expression $$p_{m}^*(x)-q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{b-x}$$ has a zero of order $2m+1$ at $x=0$. Multiplying that expression by $$(b-x)\left(p_{m}^*(x)+q_{m-1}^*(x)\frac{\sqrt{(a-x)(b-x)(\lambda_0-x)}}{b-x}\right)
,$$ we get that the polynomial $(b-x)(p_{m}^*(x))^2-(a-x)(\lambda_0-x)(q_{m-1}^*(x))^2$ has a zero of order $2m+1$ at $x=0$. Since the degree of that polynomial is $2m+1$, is follows that: $$(b-x)(p_{m}^*(x))^2-(a-x)(\lambda_0-x)(q_{m-1}^*(x))^2=cx^{2m+1}$$ for some constant $c$. Notice that $c$ is negative, since it is opposite to the square of the leading coefficient of $p_{m}^*$. Dividing the last relation by $-cx^{2m+1}$ and introducing $s=1/x$, we get the requested relation.
From Corollary \[cor:pell-periodic\] and Theorem \[th:polynomial-elliptic\] we get:
\[cor:pell-elliptic\] The billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $n$-elliptic periodic if and only if there exist a pair of real polynomials $\hat{p}_n$, $\hat{q}_{n-2}$ of degrees $n$ and $n-2$ respectively, such that the Pell’s equation holds.
If the trajectories are $n$-periodic, the statement is true according to Corollary \[cor:pell-periodic\].
If the trajectories are $n$-elliptic periodic without being $n$-periodic, then one of the cases of Theorem \[th:polynomial-elliptic\] is satisfied.
In the case (a), $n=2m$, we have $\hat{p}_n=2s\left(s-\dfrac1{\lambda_0}\right)p_{m-1}^2(s)+1$, $\hat{q}_{n-2}(s)=2p_{m-1}(s)q_{m-1}(s)$. For $n=2m+1$, we have $\hat{p}_n=2sp_{m}^2(s)-1$, $\hat{q}_{n-2}(s)=2p_{m}(s)q_{m-1}(s)$.
In the case (b), $n=2m$, we have $\hat{p}_n=2s\left(s-\dfrac1b\right)p_{m-1}^2(s)+1$, $\hat{q}_{n-2}(s)=2p_{m-1}(s)q_{m-1}(s)$. For $n=2m+1$, it is the same as in (a).
In the case (c), we have $\hat{p}_n=2\left(s-\dfrac1b\right)p_{m}^2(s)+1$, $\hat{q}_{n-2}(s)=2p_{m}(s)q_{m-1}(s)$.
Now, suppose that the Pell equation is satisfied. Then, according to part (a) of Theorem \[th:polynomial\], the trajectories are $2n$-periodic, and by Theorem \[th:even-periodic\] they are $n$-elliptic periodic.
The next two theorems address the question how to distinguish $n$-elliptic periodic trajectories which are not $n$-periodic from those that are.
\[th:odd-pell\] Suppose that real polynomials $\hat{p}_{2n+1}$, $\hat{q}_{2n-1}$ of degrees $2n+1$, $2n-1$ respectively satisfy the Pell equation: $$\label{eq:pell-odd}
\hat{p}_{2n+1}^2(s)
-
s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)
\hat{q}_{2n-1}^2(s)=1.$$ Then there exist unique real polynomials $p_n$, $q_{n-1}$, $S_1$, $S_3$ of degrees $n$, $n-1$, $1$, $3$ respectively, such that:
- $S_1$, $S_3$ are monic and $
S_1S_3=s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right);
$
- $S_1p_n^2-S_3q_{n-1}^2$ equals $1$ or $-1$.
Moreover, the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are all $(2n+1)$-elliptic periodic. They are $(2n+1)$-periodic if and only if $S_1(s)=s-\dfrac1{\lambda_0}$
Equation is equivalent to: $$(\hat{p}_{2n+1}(s)-1)(\hat{p}_{2n+1}(s)+1)=s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)\hat{q}_{2n-1}^2(s).$$ Notice that the two factors, $\hat{p}_{2n+1}(s)-1$ and $\hat{p}_{2n+1}(s)+1$, of the lefthand side are mutually prime, which implies that polynomial $\hat{p}_{2n+1}$ takes one of the values $1$ and $-1$ at three of the points from the set $\{0,1/a,1/b,1/\lambda_0\}$, and the opposite value at the remaining point of that set. Replacing $\hat{p}_{2n+1}$ by $-\hat{p}_{2n+1}$ if needed, we can assume that $\hat{p}_{2n+1}(s_1)=1$ at only one point $s_1\in\{0,1/a,1/b,1/\lambda_0\}$.
Now we set $S_1$ and $S_3$ as: $$S_1(s)=s-s_1,
\quad
S_1(s)S_3(s)=s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right).$$ Polynomials $p_n$, $q_{n-1}$ are such that: $$\hat{p}_{2n+1}-1=\sigma \cdot 2 S_1 p_n^2,
\quad
\hat{p}_{2n+1}+1=\sigma \cdot 2 S_3 q_{n-1}^2,
\quad
\hat{q}_{2n-1}=2p_nq_{n-1},
\quad
\sigma\in\{-1,1\}.$$ From there, we get: $$\label{eq:sigma}
S_1p_n^2-S_3q_n^2=-\sigma.$$
Denote $\{c_1,c_2,c_3,c_4\}=\{0,1/a,1/b,1/\lambda_0\}$, so that $c_4<c_3<c_2<c_1$. Since $0<b<a$ and $0<\lambda_0<a$, we have $c_4=0$, and $c_1\in\{1/b,1/\lambda_0\}$. If $c_2$ or $c_3$ is the root of $S_1$, then $S_1$ is negative on $[c_4,c_3]$ and positive on $[c_2,c_1]$, while $S_3$ has the opposite signs on these two segments. Thus $S_1p_n^2-S_3q_{n-1}^2$ is negative on $[c_4,c_3]$, and positive on $[c_2,c_1]$, which is not possible. We conclude that the root of $S_1$ can be only $c_1$ or $c_4$.
If $S_1(s)=s-c_4=s$, then $-\sigma$ equals the free coefficient of $-S_3q_n^2$, which is positive, thus $-\sigma=1$, and becomes: $$sp_n^2-\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right)q_n^2=1.$$ Now applying cases (a) and (b) of Theorem \[th:polynomial-elliptic\], we get that the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $(2n+1)$-elliptic periodic without being $(2n+1)$-periodic.
If $S_1(s)=s-c_1$, then $-\sigma$ equals the free coefficient of $S_1p_n^2$, which is negative, thus $-\sigma=-1$. Thus, when $c_1=1/b$, i.e. when ${\pazocal{C}}_{\lambda_0}$ is a hyperbola, becomes: $$\left(s-\frac1b\right)p_n^2-s\left(s-\dfrac1a\right)\left(s-\dfrac1{\lambda_0}\right)q_n^2=1,$$ so the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are elliptic $n$-periodic without being $n$-periodic, according to case (c) of Theorem \[th:polynomial-elliptic\].
If $c_1=1/\lambda_0$, which means that ${\pazocal{C}}_{\lambda_0}$ is an ellipse, becomes: $$\left(s-\frac1{\lambda_0}\right)p_n^2-s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)q_n^2=1,$$ so the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $(2n+1)$-periodic, according to case (b) of Theorem \[th:polynomial\].
It is interesting here to notice, that the proof of Theorem \[th:odd-pell\] represents an alternative, algebraic, proof of Lemma \[lemma:hyperbola\].
\[th:even-pell\] Suppose that real polynomials $\hat{p}_{2n}$, $\hat{q}_{2n-2}$ of degrees $2n$, $2n-2$ respectively satisfy the Pell’s equation: $$\label{eq:pell-even}
\hat{p}_{2n}^2(s)
-
s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)
\hat{q}_{2n-2}^2(s)=1.$$ Then there exist unique real polynomials $p_{d_1}$, $q_{d_2}$, $S'$, $S''$ of degrees $d_1$, $d_2$, $d'$, $d''$ respectively, such that either $d_1=d_2=n-1$, $d'=d''=2$ or $d_1=n$, $d_2=n-2$, $d'=0$, $d''=4$, and:
- $S'$, $S''$ are monic and $
S'S''=s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right);
$
- $S'p_{d_1}^2-S''q_{d_2}^2$ equals $1$ or $-1$.
Moreover, the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are all $2n$-elliptic periodic. They are $2n$-periodic if and only if $S'=1$.
Equation is equivalent to: $$(\hat{p}_{2n}(s)-1)(\hat{p}_{2n}(s)+1)=s\left(s-\frac1a\right)\left(s-\frac1b\right)\left(s-\frac1{\lambda_0}\right)\hat{q}_{2n-2}^2(s).$$ Notice that the two factors, $\hat{p}_{2n}(s)-1$ and $\hat{p}_{2n}(s)+1$, of the lefthand side are mutually prime, which implies that polynomial $\hat{p}_{2n}$ takes the value $1$ at even number of points from the set $\{0,1/a,1/b,1/\lambda_0\}$, and the opposite value $-1$ at the remaining points of that set.
First, suppose that $\hat{p}_{2n}(s)$ takes the same value at each point of those for points. Replacing $\hat{p}_{2n}$ by $-\hat{p}_{2n}$ if needed, we can assume that value is $-1$. Set $S'$ and $S''$ as: $$S'(s)=1,
\quad
S''(s)=s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right).$$ Polynomials $p_n$, $q_{n-2}$ are such that: $$\hat{p}_{2n}-1=\sigma\cdot 2 p_n^2,
\quad
\hat{p}_{2n}+1=\sigma\cdot 2 S'' q_{n-2}^2,
\quad
\hat{q}_{2n-2}=2p_nq_{n-2},
\quad
\sigma\in\{-1,1\},$$ from where we get $p_n^2-S''q_{n-2}^2=-\sigma$. We can conclude that $-\sigma=1$, since it equals the square of the free coefficient of $p_n$. Finally, applying case (a) of Theorem \[th:polynomial\], we can conclude that the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $2n$-periodic.
Second, we will suppose that $\hat{p}_{2n}(s)$ takes each of the values $-1$, $1$ at two points from the set $\{0,1/a,1/b,1/\lambda_0\}$. Replacing $\hat{p}_{2n}$ by $-\hat{p}_{2n}$ if needed, we can assume that $\hat{p}_{2n}(0)=\hat{p}_{2n}(s_1)=1$, for $s_1\in\{1/a,1/b,1/\lambda_0\}$. Set $S'$, $S''$ as: $$S'(s)=s(s-s_1),
\quad
S'(s)S''(s)=s\left(s-\dfrac1a\right)\left(s-\dfrac1b\right)\left(s-\dfrac1{\lambda_0}\right).$$ Polynomials $p_{n-1}$, $q_{n-1}$ are such that: $$\hat{p}_{2n}-1=\sigma\cdot2S'p_{n-1}^2,
\quad
\hat{p}_{2n}+1=\sigma\cdot2S''q_{n-1}^2,
\quad
\hat{q}_{2n-2}=2p_{n-1}q_{n-1},
\quad
\sigma\in\{-1,1\},$$ so $S'p_{n-1}^2-S''q_{n-1}^2=-\sigma$. From there, $-\sigma$ equals the free coefficient of $-S''q_{n-1}^2$, which is negative, thus $-\sigma=-1$: $$\label{eq:sigma2}
S' p_n^2-S''q_{n-2}^2=-1.$$ As in the proof of Theorem \[th:odd-pell\], we denote by $c_4<c_3<c_2<c_1$ the elements of $\{0,1/a,1/b,1/\lambda_0\}$. If $c_2$ or $c_3$ is the root of $S'$, then $S'$ is negative on $[c_4,c_3]$ and positive on $[c_2,c_1]$, while $S''$ has the opposite signs on these two segments. Thus $S'p_{n-1}^2-S''q_{n-1}^2$ is negative on $[c_4,c_3]$, and positive on $[c_2,c_1]$, which is not possible. We conclude that $s_1=c_1$, which can be either $1/\lambda_0$ or $1/b$.
If $c_1=1/\lambda_0$, that means ${\pazocal{C}}_{\lambda_0}$ is an ellipse, while it is a hyperbola for $c_1=1/b$. Thus, implies that cases (a) and (b) respectively of Theorem \[th:polynomial\] are satisfied, i.e. the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda_0}$ are $2n$-elliptic periodic without being $2n$-periodic.
Generalized Chebyshev polynomials and rotation function
-------------------------------------------------------
From previous consideration we see that a caustic ${\pazocal{C}}_{\lambda_0}$ generates $n$-elliptic periodic trajectories within ${\pazocal{E}}$ if and only if there exist a pair of real polynomials $\hat{p}_n$, $\hat{q}_{n-2}$ of degrees $n$ and $n-2$ respectively such that the Pell equation holds: $$\hat{p}_n^2(s)-\hat{\mathcal{P}}_{4}(s)\hat{q}_{n-2}^2(s)=1.$$ Here $\hat{\mathcal{P}}_{4}(s)=\prod_{i=1}^4(s-c_i)$, assuming that $\{c_1, c_2, c_3, c_4\}=\{1/\lambda_0, 1/a, 1/b, 0\}$ are, as in the proofs of Theorems \[th:odd-pell\] and \[th:even-pell\], ordered by the condition $
c_4=0<c_3<c_2<c_1.
$
The polynomials $\hat {p}_n$ are so called *generalized Chebyshev polynomials* on two intervals $[c_4, c_3]\cup [c_2, c_1]$, with an appropriate normalization. Namely, one can consider the question of finding the monic polynomial of certain degree $n$ which minimizes the maximum norm on the union of two intervals. Denote such a polynomial as $\hat P_n$ and its norm $L_n$. The fact that polynomial $\hat {p}_n$ is a solution of the Pell equation on the union of intervals $[c_4, c_3]\cup [c_2, c_1]$ is equivalent to the following conditions:
- $\hat {p}_n=\hat {P}_n/\pm L_n$
- the set $[c_4, c_3]\cup [c_2, c_1]$ is the maximal subset of $\mathbf R$ for which $\hat {P}_n$ is the minimal polynomial in the sense above.
Chebyshev was the first who considered a similar problem on one interval, and this was how celebrated Chebyshev polynomials emerged in XIXth century. We are going to say a bit more about original Chebyshev polynomials in Section \[sec:classicalextrema\]. Now, following the principles formulated by Chebyshev and his school and also Borel (see [@AhiezerAPPROX]), we are going to study the structure of extremal points of $\hat{p}_n$, in particular the set of points of alternance.
Notice that the roots of $\hat{\mathcal{P}}_{4}(s)$ are simple solutions of the equation $\hat{p}_n^2(s)=1$, while the roots of $\hat{q}_{n-2}(s)$ are double solutions of the equation $\hat{p}_n^2(s)=1$. Because of the degrees of the polynomials, these are all points where $\hat{p}_n^2(s)$ equals to unity.
Let us recall that a set of *points of alternance* is, by definition, a subset of the solutions of the equation $\hat{p}_n^2(s)=1$, with the maximal number of elements, such that the signs of $\hat{p}_n$ alter on it. Such a set is not uniquely determined, however the number of its elements if fixed and equal to $n+1$.
If we denote the number of points of alternance of the polynomial $\hat{p}_n$ on the segment $[c_{4},c_{3}]$ as $1+m_1$ and on the segment $[c_{4},c_{1}]$ as $1+m_0$, we see that the difference $m_{0}-m_1$ is thus equal to the number of points of alternance on the interval $[c_{3},c_{1}]$. According to the structure of the sets of the alternance, that number equals the sum of the numbers of the double points of alternance from the interval $(c_{2},c_{1})$ and one simple point of alternance at one of the endpoints of the interval. Thus we get $$m_{0}=m_{1}+\tau_1 +1,$$ and $$m_0> m_1.$$ Here $\tau_1$ is the number of zeros of the polynomial $\hat {q}_{n-2}$ on the interval $(c_{2},c_{1})$. The pair $(\tau_1, \tau_2)$, with $\tau_2 = m_1-1$ is called *the signature*, see [@RR2014].
From [@KLN1990] it follows that the numbers $(n=m_0, m_1)$ satisfy the following condition: $$\label{eq:windingKLN}
n\int_{c_1}^{\infty}\frac{1}{\sqrt{\hat{\mathcal{P}}_{4}(s)}}ds
=
m_1\int_{c_{3}}^{c_{2}}\frac{1}{\sqrt{\hat{\mathcal{P}}_{4}(s)}}ds.$$ Thus $(m_0, m_1)$ will represent exactly the winding numbers, which we introduced at the end of Section \[sec:confocal\], of the corresponding billiard trajectories.
For the reader’s sake, we are going to review briefly the results about the winding numbers from [@DragRadn2018] about winding numbers, specialized for the planar case.
\[lemma:pell-unique\] Let $p_n$, $p_n^*$ be two polynomials of degree $n$, which solve the Pell’s equations. Denote by $${\pazocal{I}}=[c_4,c_3]\cup[c_2,c_1]
\quad\text{and}\quad
{\pazocal{I}}^*=[c_4^*,c_3^*]\cup[c_2^*,c_1^*]$$ respectively the sets $\{x\mid |p_n(x)|\le 1\}$ and $\{x\mid |p_n^*(x)|\le 1\}$. Suppose that:
- at least one of the segments from ${\pazocal{I}}$ coincides with a segment from ${\pazocal{I}}^*$;
- the other pair of segments from ${\pazocal{I}}$ and ${\pazocal{I}}^*$ have joint either left or right endpoint;
- in each pair of the corresponding segments $[c_4,c_3]$, $[c_4^*,c_3^*]$ and $[c_2,c_1]$, $[c_2^*,c_1^*]$, the polynomials $p_n$, $p_n^*$ have the same number of extreme points.
Then the polynomials $p_n$, $p_n^*$ coincide up to a constant multiplier and sets ${\pazocal{I}}$ and ${\pazocal{I}}^*$ coincide.
\[th:signature-caustics\] Let $m_0$, $m_1$ be given integers. Then there is at most one ellipse ${\pazocal{E}}'$ and at most one hyperbola $\pazocal{H}$ from the confocal family , such that the billiard trajectories within ${\pazocal{E}}$ and with caustics ${\pazocal{E}}'$ and $\pazocal{H}$ are periodic with winding numbers $(m_0,m_1)$.
All assumptions of Lemma \[lemma:pell-unique\] are satisfied.
\[cor:rotation2\] The rotation number $$\rho(\lambda)
=
\rho(\lambda, a, b)
=
\frac
{\int_0^{\min\{b, \lambda\}}\frac{dt}{\sqrt{(\lambda-t)(b-t)(a-t)}}}
{\int^0_{\max\{b, \lambda\}}\frac{dt}{\sqrt{(\lambda-t)(b-t)(a-t)}}}$$ is a strictly monotone function on each of the intervals $(-\infty, b)$ and $(b, a)$.
First, observe that the rotation number is rational if and only if the billiard trajectories within ${\pazocal{E}}$ with caustic ${\pazocal{C}}_{\lambda}$ are periodic, when $\rho(\lambda)=m_1/m_0$, see equation (\[eq:windingKLN\]). According to Theorem \[th:signature-caustics\], $\rho$ can take any rational value at most once on each of the intervals $(-\infty, b)$ and $(b, a)$. In addition, the rotation map for the billiard within ellipse is a diffeomorphism at an open dense subset of each of the intervals $(-\infty, b)$ and $(b, a)$, which can be proved by using the same arguments as in [@PT2011]. From there, $\rho$ will be one-to-one on the whole intervals, thus, since it is continuous, also strictly monotone.
In [@DuistermaatBOOK], there is another proof of Corollary \[cor:rotation2\], which uses the theory of algebraic surfaces. A different proof of injectivity of rotation map, which works for higher-dimensional situations as well was presented recently in [@DragRadn2018].
Trajectories with small periods: $n=3, 4, 5, 6$ {#sec:examples}
===============================================
$3$-periodic trajectories
-------------------------
There is a $3$-periodic trajectory of the billiard within ${\pazocal{E}}$, with a non-degenerate caustic ${\pazocal{C}}_{\lambda_0}$ if and only if the following conditions are satisfied:
- the caustics is an ellipse, i.e. $\lambda_0\in(0,b)$; and
- $C_2=0$.
We can calculate: $$\label{eq:C2}
C_2=\frac{(a-b)^2\lambda_0 ^2 + 2ab(a+b)\lambda_0-3 a^2 b^2}{8( a b)^{3/2}\lambda_0^{5/2}} ,$$ so $C_2=0$ is equivalent to: $$\lambda_0=-\frac{ab(a+b)\pm2ab\sqrt{a^2-ab+b^2}}{(a-b)^2}.$$ Both solutions are always real, one of them being negative, and the other positive and smaller than $b$. Thus there is a unique caustic giving $3$-peroidic trajectories that corresponds to real motion, the ellipse defined with: $$\label{eq:lambda3}
\lambda_0=-\frac{ab(a+b)+2ab\sqrt{a^2-ab+b^2}}{(a-b)^2}.$$ The winding numbers of such trajectories satisfy $m_0>m_1$, with $m_0=3$ and $m_1$ being even. Thus, $(m_0,m_1)=(3,2)$, $(\tau_1,\tau_2)=(0,1)$. The graph of the corresponding polynomial $\hat{p}_3(s)$ is shown in Figure \[fig:p3\].
(0,1)–(5,1); (0,-1)–(5,-1);
(0.95,0)–(0.95,1); (5,0)–(5,1); (2.25,0)–(2.25,-1); (3.88,0)–(3.88,-1);
plot \[smooth, tension=1\] coordinates [ (0,-1) (1,0.99) (3,-1.5) (5,1)]{};
(-0.5,0)–(5.5,0); (0,-2)–(0,2);
(0, 1) circle (2pt); (-0.2,1) node [$1$]{};
(0, -1) circle (2pt); (-0.4,-1) node [$-1$]{};
(0.95, 1) circle (2pt); (0.95, 0) circle (2pt); (0.95,-0.35) node [$\gamma$]{};
(5, 1) circle (2pt); (5, 0) circle (2pt); (5,-0.35) node [$c_1$]{};
(2.25, -1) circle (2pt); (2.25, 0) circle (2pt); (2.25,0.35) node [$c_3$]{};
(3.88, -1) circle (2pt); (3.88, 0) circle (2pt); (3.88,0.35) node [$c_2$]{};
$4$-periodic trajectories
-------------------------
There is a $4$-periodic trajectory of the billiard within ${\pazocal{E}}$, with non-degenerate caustic ${\pazocal{C}}_{\lambda_0}$ if and only if $B_3=0$.
We can calculate: $$\label{eq:B3}
B_3=
\frac{(a \lambda_0 -a b+\lambda_0 b) (a \lambda_0 +a b-\lambda_0 b) (-a \lambda_0 +a b+\lambda_0 b)}{16(ab\lambda_0)^{5/2}}
,$$ so the coefficient equals zero if and only if: $$\label{eq:c4lambda}
\lambda_0
\in
\left\{
\frac{ab}{a+b},\frac{ab}{a-b},\frac{ab}{b-a}
\right\}
.$$ The first solution $\lambda_0=ab/(a+b)$ is positive and smaller than $b$, so it corresponds to a confocal ellipse as a caustic. The second solution $\lambda_0=ab/(a-b)$ is always bigger than $b$. It is also smaller than $a$ if and only if $b<a/2$, so this is when a confocal hyperbola as a caustic for a $4$-periodic trajectory exists. The third solution is negative, so it does not correspond to any real trajectories.
The winding numbers of $4$-periodic trajectories satisfy $m_0>m_1$, with $m_0=4$ and $m_1$ being even. Thus, $(m_0,m_1)=(4,2)$, $(\tau_1,\tau_2)=(1,1)$. The graph of the corresponding polynomial $\hat{p}_4(s)$ is shown in Figure \[fig:p4\].
(0,1)–(4.9,1); (0,-1)–(6,-1);
(0.95,0)–(0.95,1); (4.9,0)–(4.9,1); (2.26,0)–(2.26,-1); (3.69,0)–(3.69,-1); (6,0)–(6,-1);
plot \[smooth, tension=1\] coordinates [ (0,-1) (1,0.99) (3,-1.5) (4.8,0.99)(6,-1)]{};
(-0.5,0)–(6.5,0); (0,-2)–(0,2);
(0, 1) circle (2pt); (-0.2,1) node [$1$]{};
(0, -1) circle (2pt); (-0.4,-1) node [$-1$]{};
(0.95, 1) circle (2pt); (0.95, 0) circle (2pt);
(4.9, 1) circle (2pt); (4.9, 0) circle (2pt);
(2.26, -1) circle (2pt); (2.26, 0) circle (2pt); (2.26,0.35) node [$c_3$]{};
(3.69, -1) circle (2pt); (3.69, 0) circle (2pt); (3.69,0.35) node [$c_2$]{};
(6,-1) circle (2pt); (6,0) circle (2pt); (6,0.35) node [$c_1$]{};
$5$-periodic trajectories
-------------------------
There is a $5$-periodic trajectory of the billiard within ${\pazocal{E}}$, with non-degenerate caustic ${\pazocal{C}}_{\lambda_0}$ if and only if the following conditions are satisfied:
- the caustics is an ellipse, i.e. $\lambda_0\in(0,b)$; and
- $C_2C_4=C_3^2$.
We can calculate: $$\begin{aligned}
C_3^2-C_2C_4=&\
\frac{1}{1024 a^5 b^5 \lambda_0 ^7}
\times
\\
&\times
\left(
-(a-b)^6\lambda_0^6
-2 a b (a-b)^2 (a+b) (3 a+b) (a+3 b)\lambda_0^5
\right.
\\
&\qquad\left.
+\ a^2 b^2 (a-b)^2 \left(29 a^2+54 a b+29 b^2\right)\lambda_0^4
-36 a^3 b^3 (a-b)^2 (a+b)\lambda_0^3
\right.
\\
&\qquad\left.
+\ a^4 b^4 \left(9 a^2-34 a b+9 b^2\right)\lambda_0^2
+10 a^5 b^5 (a+b)\lambda_0
-5 a^6 b^6
\right).
\end{aligned}$$
The winding numbers of $5$-periodic trajectories satisfy $m_0>m_1$, with $m_0=5$ and $m_1$ being even. Thus, $(m_0,m_1)\in\{(5,2),(5,4)\}$, with $(\tau_1,\tau_2)\in\{(2,1),(0,3)\}$ respecitvely. The graph of the corresponding polynomial $\hat{p}_5(s)$ is shown in Figures \[fig:p52\] and \[fig:p54\].
(0,1)–(7.5,1); (0,-1)–(6.3,-1);
(0.95,0)–(0.95,1); (4.9,0)–(4.9,1); (2.26,0)–(2.26,-1); (3.69,0)–(3.69,-1); (6.3,0)–(6.3,-1); (7.5,0)–(7.5,1);
plot \[smooth, tension=1\] coordinates [ (0,-1) (1,0.99) (3,-1.5) (4.8,0.99)(6.3,-1)(7.5,1)]{};
(-0.5,0)–(8,0); (0,-2)–(0,2);
(0, 1) circle (2pt); (-0.2,1) node [$1$]{};
(0, -1) circle (2pt); (-0.4,-1) node [$-1$]{};
(0.95, 1) circle (2pt); (0.95, 0) circle (2pt);
(4.9, 1) circle (2pt); (4.9, 0) circle (2pt);
(2.26, -1) circle (2pt); (2.26, 0) circle (2pt); (2.26,0.35) node [$c_3$]{};
(3.69, -1) circle (2pt); (3.69, 0) circle (2pt); (3.69,0.35) node [$c_2$]{};
(6.3,-1) circle (2pt); (6.3,0) circle (2pt);
(7.5, 1) circle (2pt); (7.5, 0) circle (2pt); (7.5,-0.35) node [$c_1$]{};
(0,1)–(8,1); (0,-1)–(7.35,-1);
(1,0)–(1,1); (4.75,0)–(4.75,1); (3,0)–(3,-1); (6.1,0)–(6.1,-1); (7.35,0)–(7.35,-1); (8,0)–(8,1);
plot \[smooth, tension=1\] coordinates [ (0,-1) (1,0.99) (3,-0.99) (4.8,0.99)(6.7,-1.5)(8,1)]{};
(-0.5,0)–(8.5,0); (0,-2)–(0,2);
(0, 1) circle (2pt); (-0.2,1) node [$1$]{};
(0, -1) circle (2pt); (-0.4,-1) node [$-1$]{};
(1, 1) circle (2pt); (1, 0) circle (2pt);
(4.75, 1) circle (2pt); (4.75, 0) circle (2pt);
(3, -1) circle (2pt); (3, 0) circle (2pt);
(6.1,-1) circle (2pt); (6.1,0) circle (2pt); (6.1,0.35) node [$c_3$]{};
(7.35,-1) circle (2pt); (7.35,0) circle (2pt); (7.35,0.35) node [$c_2$]{};
(8, 1) circle (2pt); (8, 0) circle (2pt); (8,-0.35) node [$c_1$]{};
$6$-periodic trajectories
-------------------------
There is a $6$-periodic trajectory of the billiard within ${\pazocal{E}}$, with non-degenerate caustic ${\pazocal{C}}_{\lambda_0}$ if and only if $B_4^2-B_3 B_5=0$.
We calculate: $$\begin{aligned}
B_4^2-B_3 B_5
=
-\frac{B_2C_2}{256 (ab)^4\lambda_0^{3} }
\ &\times
\left((a-b)(a+3b)\lambda_0 ^2-2 a b (a-b)\lambda_0 +a^2 b^2\right)
\\
&\times
\left(
(a-b)(3a+b)\lambda_0 ^2 -2ab(a-b)\lambda_0 -a^2 b^2
\right).
\end{aligned}$$
First, let us consider the condition $B_2=0$. We have $$\label{eq:B2}
B_2=-\frac{a^2 b^2-2 a b \lambda _0 (a+b)+\lambda _0^2 (a-b)^2}{8 (a b \lambda _0)^{3/2}}$$ so $B_2=0$ is equivalent to: $
\lambda_0=ab/(\sqrt{a}\pm\sqrt{b})^2.
$ From the condition $\lambda_0<b$, we have a unique solution which gives an ellipse as the caustic: $$\lambda_0=\frac{ab}{(\sqrt{a}+\sqrt{b})^2}.$$ For $a>4b$, the option $\lambda_0=ab/(\sqrt{a}-\sqrt{b})^2$ will provide a hyperbola as the caustic, see Figure \[fig:traj62h\].
(0,0) ellipse (2.1213cm and 1cm);
plot ([0.9597\*sqrt(1+/2.5789)]{}, );
plot ([-0.9597\*sqrt(1+/2.5789)]{}, );
(1.03238, 0.873586)–(0.857015, -0.914759)–(-0.207025, 0.995226)–(-1.03238, -0.873586)–(-0.857015, 0.914759)–(0.207025, -0.995226)–(1.03238, 0.873586);
Notice that the winding numbers of $6$-periodic trajectories obtained from the condition $B_2=0$ are $(m_0,m_1)=(6,2)$. This is because $B_2=0$ is equivalent to $3P_0\sim P_{\infty}$ on the elliptic curve. On the other hand, the obtained trajectories are $3$-periodic in elliptic coordinates, and they transverse the $\mathsf{y}$-axis only once along the period, so the condition of their periodicity is equivalent to: $$\begin{aligned}
&3(P_0-P_b) + (P_{\lambda_0}-P_a)\sim0,\quad &\text{for}\quad& \lambda_0\in(b,a),
\\
&3(P_0-P_{\lambda_0}) + (P_b-P_a)\sim0,\quad &\text{for}\quad& \lambda_0\in(0,b).
\end{aligned}$$ Both of the obtained relations are equivalent to $3P_0\sim 3P_{\infty}$, since $2P_{\infty}\sim2P_a\sim2P_b\sim2P_{\lambda_0}$ and $3P_{\infty}\sim P_a+P_b+P_{\lambda_0}$.
Next, $C_2=0$ gives $3$-periodic trajectories.
The discriminant of $(a-b)(a+3b)\lambda_0 ^2-2 a b (a-b)\lambda_0 +a^2 b^2$ is $-16 a^2 b^3 (a-b)$, which is negative, so the expression has no real roots in $\lambda_0$.
On the other hand, $(a-b)(3a+b)\lambda_0 ^2 -2ab(a-b)\lambda_0 -a^2 b^2$ has two real roots: $$\lambda_0=\frac{ab(a-b)\pm2ab \sqrt{a(a - b)}}{(a-b)(3a+b)}.$$ The smaller one is always negative, so it does not correspond to a real billiard trajectory.
Let us analyze the bigger one: $$\lambda_0=\frac{ab(a-b)+2ab \sqrt{a(a - b)}}{(a-b)(3a+b)}.$$ We have that always $\lambda_0>b$, since: $$\begin{gathered}
\frac{ab(a-b)+2ab \sqrt{a(a - b)}}{(a-b)(3a+b)}>b
\qquad\Leftrightarrow\qquad
b^2(3a+b)>0.\end{gathered}$$ We need to check the condition $\lambda_0<a$, which is equivalent to: $$\begin{gathered}
\frac{ab(a-b)+2ab \sqrt{a(a - b)}}{(a-b)(3a+b)}<a
\qquad\Leftrightarrow\qquad
a>\frac43b.\end{gathered}$$
Thus for $a>4b/3$, there is another class of $6$-periodic trajectories which have a hyperbola as a caustic, see Figure \[fig:traj64h\].
(0,0) ellipse (1.4142cm and 1cm);
plot ([0.9519\*sqrt(1+10.6569)]{}, );
plot ([-0.9519\*sqrt(1+10.6569)]{}, );
(1.2041, 0.52447)–(0.462044, -0.945123)–(-1.32834, 0.343164)–(1.2041, -0.52447)–(0.462044, 0.945123)–(-1.32834, -0.343164)–(1.2041, 0.52447);
The winding numbers of those $6$-periodic trajectories are $(m_0,m_1)=(6,4)$. The obtained trajectories are $3$-periodic in elliptic coordinates, and they transverse the $\mathsf{y}$-axis twice along the period, so the condition of their periodicity is equivalent to: $$\begin{aligned}
&3(P_0-P_b) + 2(P_{\lambda_0}-P_a)\sim0,
\end{aligned}$$ which is equivalent to $3P_0\sim 3P_b$.
Periodic billiard trajectories and classical extremal polynomials {#sec:classicalextrema}
=================================================================
We want to employ the classical theory of extremal polynomials on two intervals to get explicit formulas for the polynomials $\hat p_n$. As it is well know, the extremal polynomials don’t exist on any union of two disjoint real intervals. Thus, we want to exploit the relationship between the end points of such configurations of two intervals and to relate it to the formulas for the caustics which generate periodic billiard trajectories. To illustrate the main idea, we are going to start with families of polynomials, introduced by Zolotarev in 1870’s, [@AhiezerAPPROX; @Akh4]. Later on, we will use two families of polynomials derived by Akhiezer, first in 1928 and second, more general, in 1930’s to get the general formulae for our polynomials $\hat p_n$, [@Akh1; @Akh2; @Akh3].
Let us recall that the celebrated Chebyshev polynomials $T_n(x), n= 0, 1, 2,\dots$ defined by the recursion: $$\label{eq:cheb1}
T_0(x)=1, \, T_1(x)=x,\, T_{n+1}(x)+T_{n-1}(x)=2xT_n(x),$$ for $n=1, 2\dots$ can be parameterized as $$\label{eq:cheb2}
T_n(x)=\cos n\phi,\, x=\cos\phi,$$ or, alternatively: $$\label{eq:cheb3}
T_n(x)=\frac{1}{2}\left(v^n+\frac{1}{v^n}\right), \quad x=\frac{1}{2}\left(v+\frac{1}{v}\right).$$ Denote $L_0=1$ and $L_n=2^{1-n}, n=1, 2,\dots$. Then the Chebyshev Theorem states that the polynomials $L_nT_n(x)$ are characterized as the solutions of the following minmax problem:
*find the polynomial of degree $n$ with the leading coefficient equal 1 which minimizes the uniform norm on the interval $[-1, 1]$.*
Zolotarev polynomials:
----------------------
Following the ideas of Chebyshev, his student Zolotarev posed and solved a handful of problems, including the following:
*For the given real parameter $\sigma$ and all polynomials of degree $n$ of the form: $$\label{eq:zol1}
p(x)=x^n-n\sigma x^{n-1} + p_2x^{n-2}+\dots p_n,$$ find the one with the minimal uniform norm on the interval $[-1, 1]$.*
Denote this minimal uniform norm as $L_n=L(\sigma, n)$.
For $\sigma>\tan^2(\Pi/2n)$, the solution $z_n$ has the following property ([@AhiezerAPPROX], p. 298), see Figure \[fig:zn\]:
(0,1)–(8,1); (0,-1)–(7.35,-1);
(1,0)–(1,1); (4.73,0)–(4.73,1); (3,0)–(3,-1); (6.1,0)–(6.1,-1); (7.35,0)–(7.35,-1); (8,0)–(8,1); (0,-1)–(0,1);
plot \[smooth, tension=1\] coordinates [ (0,-1) (1,0.99) (3,-0.99) (4.8,0.99)(6.7,-1.5)(8,1)]{};
(-0.5,0)–(8.5,0);
(0, 0) circle (2pt); (-0.4,-0.3) node [$-1$]{};
(0, 1) circle (2pt); (-0.4,1) node [$L_n$]{};
(0, -1) circle (2pt); (-0.6,-1) node [$-L_n$]{};
(1, 1) circle (2pt); (1, 0) circle (2pt);
(4.73, 1) circle (2pt); (4.73, 0) circle (2pt);
(3, -1) circle (2pt); (3, 0) circle (2pt);
(6.1,-1) circle (2pt); (6.1,0) circle (2pt); (6.1,0.35) node [$1$]{};
(7.35,-1) circle (2pt); (7.35,0) circle (2pt); (7.35,0.35) node [$\alpha$]{};
(8, 1) circle (2pt); (8, 0) circle (2pt); (8,-0.35) node [$\beta$]{};
The polynomials $z_n$ are given by the following explicit formulae: $$\label{eq:zn}
z_n=\ell_n\left(v(u)^n+\frac1{v(u)^n}\right), \, x=\frac{\sn^2u +\sn^2\frac{K}{n}}{\sn^2u -\sn^2\frac{K}{n}},$$ where $$\ell_n=\frac1{2^n}\left(\frac{\sqrt{\kappa}\theta_1^2(0)}{H_1\left(\frac{K}{n}\right)\theta_1\left(\frac{K}{n}\right)}\right)^{2n}, \quad v(u)=\frac{H\left(\frac{K}{n}-u\right)}{H\left(\frac{K}{n}+u\right)}$$ and $$\sigma=\frac{2\sn\frac{K}{n}}{\cn\frac{K}{n}\dn\frac{K}{n}}\left(\frac1{\sn\frac{2K}{n}}-\frac {\theta'\left(\frac{K}{n}\right)}{\theta\left(\frac{K}{n}\right)}\right)-1.$$ Formulae for the endpoints of the second interval are: $$\label{eq:alphabetan}
\alpha =\frac{1+\kappa^2\sn^2\frac{K}{n}}{\dn^2\frac{K}{n}}, \quad \beta =\frac{1+\sn^2\frac{K}{n}}{\cn^2\frac{K}{n}},$$ with $$\kappa^2=\frac{(\alpha-1)(\beta+1)}{(\alpha+1)(\beta-1)}.$$ In order to derive the formulas for $\hat p_3$ in terms of $z_3$, let us construct an affine transformation: $$h:[-1, 1]\cup [\alpha, \beta]\rightarrow [0, a^{-1}]\cup[b^{-1}, \lambda_0^{-1}], \, h(x) = \hat a x +\hat b.$$ According to Cayley’s condition (\[eq:lambda3\]) $$\lambda_0=-\frac{ab(a+b)+2ab\sqrt{a^2-ab+b^2}}{(a-b)^2}.$$ We immediately get $$\hat a = \hat b, \, \hat a = \frac1{2a}$$ and $$\begin{gathered}
\label{eq:alphabetacayley3}
\alpha = 2t-1,\\
\beta = -\frac{2t^2-3t+3+2\sqrt{t^2-t+1}}{t+1+2\sqrt{t^2-t+1}},\\
\beta = \frac{4}{3} \sqrt{t^2-t+1} +\frac {2}{3}t +\frac {5}{3},\end{gathered}$$ where $t=a/b$.
There is a relation between $\alpha$ and $\beta$ as defined by the formulae above: $$\label{eq:criterion3}
9\beta^2-3\alpha^2-6\alpha\beta+12\alpha - 36 \beta +56=0.$$ Now we get the following
The polynomial $\hat p_3$ can be expressed through the Zolotarev polynoamil $z_3$ up to a nonessential constant factor: $$\hat p_3 (s) \sim z_3( 2as-1).$$
To verify the proposition, we should certify that $\alpha$ and $\beta$ defined in (\[eq:alphabetan\]) for $n=3$ satisfy the relation (\[eq:alphabetacayley3\]).
In order to do that we will use well-known identities for the Jacobi elliptic functions:
$$\begin{gathered}
\sn^2u+\cn^2u+1,
\\
\kappa^2\sn^2u+\dn^2u=1,
\\
\sn(u+v)=\frac{\sn\, u\cn\, v\dn \,v + \sn\, v\cn\, u\dn\, u}{1-\kappa^2\sn^2 u \sn ^2 v},
\\
\sn (K-u)=\frac{\cn\, u}{\dn\, u}.\end{gathered}$$
In particular, we get $$\begin{gathered}
\sn \left(\frac {2K}{3}\right)= \frac{2\sn\, \frac{K}{3}\cn \,\frac{K}{3}\dn \,\frac{K}{3}}{1-\kappa^2\sn^4\frac{K}{3}},
\\
\sn \left(\frac {2K}{3}\right)= \sn \left(K-\frac {K}{3}\right)=\frac{\cn \,\frac{K}{3}}{\dn\, \frac{K}{3}}.\end{gathered}$$ Let us denote $$Y=\sn \left(\frac {K}{3}\right),$$ then from the previous two relations we get: $$1-2Y+2\kappa^2Y^3 -\kappa^2Y^4=0.$$ We can express $\kappa$ in terms of $Y$ and get: $$\kappa^2=\frac{2Y-1}{Y^3(2-Y)}.$$ By plugging the last relation into (\[eq:alphabetan\]) for $n=3$ we get $$\alpha=\frac{Y^2-4Y+1}{Y^2-1}.$$ Since, at the same time from the Cayley condition we have $\alpha = 2t-1$, with $t=a/b$, we can express $Y$ in terms of $t$: $$(t-1)Y^2+2Y-t=0,$$ and $$\label{eq:Yt}
Y=\frac{-1\pm \sqrt{1-t+t^2}}{t-1}.$$ We plug the last relation into the formula for $\beta$ from (\[eq:alphabetan\]) for $n=3$ and we get a formula for $\beta$ in terms of $t$: $$\begin{gathered}
\label{eq:betat}
\beta=\frac{(t-1)^2+(-1\pm\sqrt{t^2-t+1})^2}{(t-1)^2-(-1\pm\sqrt{t^2-t+1})^2},
\\
\beta = -\frac{2t^2-3t+3\pm2\sqrt{t^2-t+1}}{t+1\pm 2\sqrt{t^2-t+1}}.\end{gathered}$$ We see that the last formula with the choice of the $+$ sign corresponds to a formula for $\beta$ from (\[eq:alphabetacayley3\]). This finalizes the verification. One can observe that the $-$ sign option from the formula above would correspond to the $-$ sign in the formula for $\lambda_0$ above the formula (\[eq:lambda3\]).
Among the polynomials $\hat p_n$ the property of type $\Pi1$ can be attributed only to those with $n=2k+1$ and winding numbers $(2k+1, 2k)$, in other words to those with the signature $(0, 2k-1)$. For example, this is satisfied for the polynomial $\hat p_5$ presented in the Fig. \[fig:p54\] while it is not true for the polynomial $\hat p_5$ presented in the Fig. \[fig:p52\].
Akhiezer polynomials on symmetric intervals $[-1, -\alpha]\cup[\alpha, 1]$
--------------------------------------------------------------------------
The problem of finding polynomials of degree $n$ with the leading coefficient 1 and minimizing the uniform norm on the union of two symmetric intervals $[-1, -\alpha]\cup[\alpha, 1]$, for given $0<\alpha <1$ appeared to be of a significant interest in radio-techniques applications. Following the ideas of Chebyshev and Zolotarev, Akhiezer derived in 1928 the explicit formulae for such polynomials $A_n(x;\alpha)$ with the deviation $L_n(\alpha)$, [@AhiezerAPPROX; @Akh4].
These formulas are specially simple in the case of even degrees $n=2m$, when Akhiezer polynomials $A_{2m}$ are obtained by a quadratic substitution from the Chebyshev polynomial $T_m$:
$$\label{eq:A2m}
A_{2m}(x;\alpha)=\frac{(1-\alpha^2)^m}{2^{2m-1}}T_m\left(\frac{2x^2-1-\alpha^2}
{1-\alpha^2}\right),$$
with $$L_{2m}(\alpha)=\frac{(1-\alpha^2)^m}{2^{2m-1}}.$$ We are going to construct $\hat p_4(s)$ up to a nonessential constant factor as a composition of $A_4(x;\alpha)$ for certain $\alpha$ and an affine transformation. We are going to study the possibility to have an affine transformation $$g:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [0, c_3]\cup [c_2, c_1],\quad g(x)=\hat a x +\hat b$$ in two versions, depending if the caustic corresponds to the reciprocical value of $c_1$ or of $c_2$. The former case corresponds to the case of caustic being ellipse and the latter of being hyperbola. This we will denote these two cases (E) and (H) respectively.
[**Case (E).**]{}
For $$g:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [0, a^{-1}]\cup [b^{-1}, \lambda^{-1}],\quad g(x)=\hat a x +\hat b$$ we get $$\hat a =\hat b,\quad \alpha =\frac {a-b}{a+b}, \quad \hat a= \frac {a+b}{2ab}.$$ Thus: $$g(1)=2\hat a= \frac{a+b}{ab}=\frac1{\lambda}$$ implies $$\lambda=\frac{ab}{a+b},$$ which coincides with the formula for the caustic -ellipse for $4$ periodic billiard trajectories derived from the Cayley condition.
In this case the polynomial $\hat p_4(s)$ is equal up to a constant multiplier to $$\label{eq:p4t2}
\hat p_4(s) ~\sim \frac{2a^2b^2}{(a+b)^4}T_2(2abs^2 -2(a+b)s+1),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial.
[**Case (H).**]{}
For $$g:[-1,-\alpha]\cup[\alpha, 1]\rightarrow [0, a^{-1}]\cup [\lambda^{-1}, b^{-1}],\quad g(x)=\hat a x +\hat b$$ we get $$\hat a =\hat b,\quad \alpha =\frac {a-2b}{a}, \quad \hat a= \frac {1}{2b}.$$ Thus: $$g(\alpha)=\hat a(1+\alpha)= \frac{a-b}{ab}=\frac1{\lambda}$$ implies $$\lambda=\frac{ab}{a-b},$$ which coincides with the formula for the caustic - hyperbola for $4$ periodic billiard trajectories derived from the Cayley condition.
In this case the polynomial $\hat p_4(s)$ is equal up to a constant multiplier to $$\label{eq:p4t2bis}
\hat p_4(s) \sim \frac{2b^2(a-b)^2}{a^4}T_2\left(\frac{a^2s^2-4a^2bs +8b^3(a-b)}{8b^3(a -b)}\right),$$ where $T_2(x)=2x^2-1$ is the second Chebyshev polynomial.
The polynomials $\hat p_4$ from (\[eq:p4t2\]) and (\[eq:p4t2bis\]) can be rewritten in the canonical form respectively as: $$\hat p_4 (s) \sim 8a^2b^2 s^4 -16ab(a+b)s^3 + 8 (a^2+3ab +b^2) s^2-8(a+b) s+1$$ and
$$\hat p_4 (s) \sim a^4s^4-8a^4bs^3+16a^2b^2(a^2+ab-b^2)s^2 +64 a^2b^4(b-a)s +32b^9(b-2a).$$
By analyzing the structure of the extremal points of the polynomials $A_n$, we generalize the last two propositions as follows:
For trajectories with period $4k$ and winding numbers $(4k, 2k)$, the corresponding polynomials $\hat p_{4k}$ are equal up to a nonessential constant factor to: $$\hat p_{4k}(s) ~\sim T_{2k}(2abs^2 -2(a+b)s+1),$$ for the caustic being ellipse, and to $$\hat p_{4k}(s) \sim T_{2k}\left(\frac{a^2s^2-4a^2bs +8b^3(a-b)}{8b^3(a -b)}\right),$$ in the case of hyperbola as the caustic, where $T_{2k}$ is the $2k$-th Chebyshev polynomial.
Since the polynomials $A_n$ have symmetrically distributed extremal points in the intervals $[-1,-\alpha]$ and $[\alpha, 1]$, they can’t serve as models for polynomials $\hat p_k$ except in the cases listed in the above Proposition.
General Akhiezer polynomials on unions of two intervals
-------------------------------------------------------
So far, we managed to express polynomials $\hat p_3, \hat p_4$, $\hat p_5$ only in the case of the signature $(0,3)$, $\hat p_{2n+1}$ with the signature $(0, 2n-1)$ and $\hat p_{4k}$ only in the case of the signature $(2k-1, 2k-1)$, by use of polynomials of Zolotarev and Akhiezer. However, we were unable to get for example the polynomial $\hat p_5$ with the signature $(2,1)$, as well as polynomials $\hat p_{4k+2}$ and $\hat p_{4k}, \hat p_{2k+1}$ with signatures different that those listed above. Thus, in order to get the general formulae for the general polynomials $\hat p_n$, we need to employ a more general theory of extremal polynomials on two intervals, developed by Akhiezer [@Akh1; @Akh2; @Akh3].
Following Akhiezer, let us consider the union of two intervals $
[-1, \alpha]\cup[\beta, 1],
$ where $$\alpha = 1-2\sn^2\frac {m}{n}K, \quad \beta = 2\sn^2\frac {n-m}{n}K-1.$$ Define $$\label{eq:genakhiezer}
TA_n(x, m, k)=L\left(v(u)^n + \frac1{v(u)^n}\right),$$ where $$v(u)=\frac{H\left(u-\frac{mK}{n}\right)}{H\left(\frac{mK}{n}+u\right)},$$ $$x=\frac{\sn^2u\quad\cn^2\frac{m}{n}K+\cn^2u\quad\sn^2\frac{m}{n}K}{\sn^2u-\sn^2\frac{m}{n}K},$$ and $$L=\frac1{2^{n-1}}\left(\frac{\theta(0)\theta_1(0)}
{\theta(\frac{m}{n}K)\theta_1(\frac{m}{n}K)}\right),\quad k^2=\frac{2(\beta-\alpha)}{(1-\alpha)(1-\beta)}.$$ Akhiezer proved the following results:
\[th:akhiezer\]
- The function $TA_n(x, m, k)$ is a polynomial of degree $n$ in $x$ with the leading coefficient 1 and the second coefficient equal to $-n\tau_1$, where $$\tau_1=-1 + 2\frac{\sn\frac{m}{n}K\quad\cn{\frac{m}{n}K}}{\dn{\frac{m}{n}K}}
\left(\frac1{\sn\frac{2m}{n}K}-\frac{\theta'(\frac{m}{n}K)}{\theta(\frac{m}{n}K)}\right).$$
- The maximum of the modulus of $T_n$ on the union of the two intervals $[-1, \alpha]\cup[\beta, 1]$ is $L$.
- The function $T_n$ takes the values $\pm L$ with alternating signs at $\mu=n-m+1$ consecutive points of the interval $[-1, \alpha]$ and at $\nu=m+1$ consecutive points of the interval $[\beta, 1]$. In addition $$T_n(\alpha, m, k)=T_n(\beta, m, k)=(-1)^mL,$$ and for any $x\in (\alpha, \beta)$ it holds: $$(-1)^mT_n(x, m, k)>L.$$
- Let $F$ be a polynomial of degree $n$ in $x$ with the leading coefficient equal 1, such that:
- $\max |F(x)| = L$ for $x\in [-1, \alpha]\cup[\beta, 1]$;
- $F(x)$ takes values $\pm L$ with alternating signs at $n-m+1$ consecutive points of the interval $[-1, \alpha]$ and at $m+1$ consecutive points of the interval $[\beta, 1]$.
Then $F(x)=T_n(x, m, k)$.
Let us observe that the polynomials $\hat p_n(s)$ satisfy the conditions of the item (d) of the Akhiezer Theorem, up to an affine change of variables and up to a nonessential constant factor. Indeed, let us denote by $$\hat p_n(s, 2l)$$ such a polynomial $\hat p_n$ which corresponds to the winding numbers $(n, 2l), 2l<n$. Then the corresponding signature is $(n-2l-1, 2l-1)$. The number of alternating points on the interval $[0, c_3]$ of the polynomial $\hat p_n(s, 2l)$ is equal to $2l+1$, while the number of its alternating points on the interval $[c_2, c_1]$ is equal to $n-2l+1$. Thus $m$ from the Akhiezer Theorem is $$m=n-2l.$$ Now, let us determine the affine transformations in two cases: (E) when the caustic is an ellipse and (H) when the caustic is hyperbola.
[**Case (E)**]{}.
For $$h:[-1,\alpha]\cup[\beta, 1]\rightarrow [0, a^{-1}]\cup [b^{-1}, \lambda^{-1}],\quad h(x)=\hat a x +\hat b$$ we get $$\hat a =\hat b,\quad \hat a= \frac {1}{a(\alpha+1)},\quad \frac {\beta+1}{\alpha +1}=\frac{a}{b}.$$ Thus: $$\lambda=\frac{a(\alpha+1)}{2}= \frac{b(\beta+1)}{2}.$$ We have proved the following theorem.
The polynomials $\hat p_n$ can be expressed up to a nonessential multiplier as a composition of a $TA_n$ polynomial and an affine transformation: $$\label{eq:pntan2}
\hat p_n(s, 2l)\sim TA_n(a(\alpha+1)s-1; n-2l, k).$$
[**Case (H)**]{}.
For $n$ even there is one more option, with the caustic being a hyperbola.
For $$h_1:[-1,\alpha]\cup[\beta, 1]\rightarrow [0, a^{-1}]\cup [\lambda^{-1}, b^{-1}],\quad h_1(x)=\hat a x +\hat b$$ we get $$\hat a =\hat b,\quad \hat a= \frac {1}{2b}, \quad \alpha =\frac{2b}{a}-1.$$ Thus: $$\lambda=\frac{2b}{(\beta+1)}.$$
The polynomials $\hat p_n$ can be expressed up to a nonessential multiplier as a composition of a $TA_n$ polynomial and an affine transformation: $$\label{eq:pntan}
\hat p_n(s, 2l)\sim TA_n(2bs-1; n-2l, k).$$
The relation between $\alpha$ and $\beta$ given by: $$\alpha = 1-2\sn^2\frac {m}{n}K, \quad \beta = 2\sn^2\frac {n-m}{n}K-1,$$ can be seen more clearly if we introduce $$Z=\sn\frac{mK}{n}.$$ We have then $$\sn^2\left(\frac{n-m}{n}K\right)=\frac{1-Z^2}{1-\kappa^2Z^2},$$ and $$\alpha =1-2Z^2, \quad \beta=\frac{1+(\kappa^2-2)Z^2}{1-\kappa^2Z^2}.$$
Let us illustrate the last two Theorems for $n=4$. Then $l=1,\quad m=n-2l=2$. It is well-known that $$\sn^2\frac{K}{2}=\frac{1}{1+\sqrt{1-\kappa^2}}.$$ We have $$\alpha=-\beta=1-2\sn^2\frac{K}{2}.$$
[**Case (E)**]{}.
Plugging $\beta=-\alpha$ into $$\frac {\beta+1}{\alpha +1}=\frac{a}{b}=t,$$ we get $$\alpha=\frac{1-t}{1+t}.$$ From $$\lambda=\frac{a(\alpha+1)}{2}=\frac{a}{1+t},$$ we get $$\lambda=\frac{ab}{a+b},$$ which coincides with one of the values obtained from the Cayley-type condition in (\[eq:c4lambda\]). From the relation $$\alpha= 1- 2 \sn^2\frac{K}{2}=\frac{1-t}{1+t},$$ we get a relation between $t$, the ratio of the squares of the semi-axes of the ellipse and the elliptic modulus $\kappa$: $$t=\frac{1}{\sqrt{1-\kappa^2}}.$$
Finally, we get: $$\hat p_4(s, 2)\sim TA_4\left(2\frac{ab}{a+b}s-1; 2, \sqrt{\frac{a^2-b^2}{a^2}}\right).$$
[**Case (H)**]{}.
From $$\beta=-\alpha=1-\frac{2}{t}$$ we get $$\lambda=\frac{bt}{t-1},$$ which gives $$\lambda=\frac{ab}{a-b},$$ which is again one of the values obtained from the Cayley-type condition in (\[eq:c4lambda\]).
From the relation $$\alpha= 1- 2 \sn^2\frac{K}{2}=\frac{2}{t}-1,$$ we get a relation between $t$, the ratio of the squares of the semi-axes of the ellipse and the elliptic modulus $\kappa$: $$t=\frac{1+\sqrt{1-\kappa^2}}{\sqrt{1-\kappa^2}}.$$
Finally, we get: $$\hat p_4(s, 2)\sim TA_4\left(2bs-1; 2, \sqrt{\frac{a^2-2ab}{(a-b)^2}}\right).$$
Let us consider now the case $n=5$. Let us denote $$\sn\quad \frac{K}{5}=Y.$$ Then, from the addition formulas we get: $$\sn^2\quad \frac{2K}{5}=\frac{4Y^2(1-Y^2)(1-\kappa^2Y^2)}{1-\kappa^2Y^4}.$$ Similarly, we get $$\sn^2 \frac{4K}{5}=\frac{16(1-\kappa^2Y^4)^8Y^2(1-Y^2)(1-\kappa^2Y^2)(1-\sn^2\quad \frac{2K}{5})(1-\kappa^2\sn^2\quad \frac{2K}{5})}{(1-\kappa^2Y^4)^2((1-\kappa^2Y^4)^4-16Y^4(1-Y^2)^2(1-\kappa^2Y^2)^2)^2}.$$ From the last relation and $$\sn^2 \frac{4K}{5}=\frac{1-Y^2}{1-\kappa^2Y^2}$$ we get: $$0=P(Z, s)=\sum_{p, q}P_{p,q}Z^ps^q=\sum_{p=0}^{16}E_p(s)Z^{p}=\sum_{q=0}^{8}F_q(Z)s^q,$$ where $$Z=Y^2,\quad s=\kappa^2$$ and $$\begin{aligned}
&F_0=-1+16Z-64Z^2+64 Z^3,
\\
& F_1=-56Z^2+352Z^3-416Z^4,
\\
&F_2=144Z^3-1244Z^4+2160Z^5-1280Z^6+896Z^7-256Z^8,
\\
&F_3 =-160Z^4+2144Z^5-4744Z^6+4160Z^7-3264Z^8+1024Z^9,
\\
& F_4=64Z^5-1984Z^6+5360Z^7-5830Z^8+5360Z^9-1984Z^{10}+64Z^{11},
\\
& F_5 =-160Z^{12}+2144Z^{11}-4744Z^{10}+4160Z^9-3264Z^8+1024Z^7,
\\
& F_6=144Z^{13}-1244Z^{12}+2160Z^{11}-1280Z^{10}+896Z^{9}-256Z^8,
\\
& F_7= -56Z^{14}+352Z^{13}-416Z^{12},
\\
&F_8= -Z^{16}+16Z^{15}-64Z^{14}+64 Z^{13}.
\end{aligned}$$
Observe the symmetry: $$\label{eq:summP}
P_{p,q}=P_{16-p, 8-q}.$$
[**Case $l=1$.**]{} From the Akhiezer formulae we get
$$\alpha = 1-2\sn^2\frac {3}{5}K, \quad \beta = 2\sn^2\frac {2}{5}K-1,$$ and $$\label{eq:tab52}
\frac{a}{b}=t=\frac{\sn^2\frac {2}{5}K}{1-\sn^2\frac {3}{5}K}.$$ Thus, $$\lambda_0=a\left(1-\sn^2\frac {3}{5}K\right),$$ and $$\hat p_5(s, 2)\sim TA_4\left(2a\left(1-\sn^2\frac {3}{5}K\right)s-1; 3, \kappa\right).$$
From the equation (\[eq:tab52\]) and one of the addition formulas: $$\sn^2 \frac {3}{5}K=\frac{1-\sn^2\frac{2}{5}K}{1-\kappa^2\sn^2\frac{2}{5}K}$$ we get: $$\kappa^2=\frac{t-1}{Wt},$$ where $W=\sn^2 \frac{3}{5}K.$
[**Case $l=2$.**]{}
Similarly, from $$\alpha = 1-2\sn^2\frac {1}{5}K, \quad \beta = 2\sn^2\frac {4}{5}K-1,$$ and $$\frac{a}{b}=t=\frac{\sn^2\frac {4}{5}K}{1-\sn^2\frac {1}{5}K}.$$ we get $$\lambda_0=a\left(1-\sn^2\frac {1}{5}K\right),$$ and $$\hat p_5(s, 4)\sim TA_4\left(2a\left(1-\sn^2\frac {1}{5}K\right)s-1; 1, \kappa\right).$$ Here $$\kappa^2=\frac{t-1}{W_1},$$ where $W_1=\sn^2\frac{4}{5}K.$
Discriminantly separable and discriminantly factorizable polynomials {#sec:separable}
====================================================================
Definition of discriminantly separable polynomials
--------------------------------------------------
We briefly review the basic notions and indicate several relationships and applications to different areas of mathematics and mechanics, we provide a general definition of the discriminantly separable polynomials. By $\mathcal{P}_m^n$ denote the polynomials of $m$ variables of degree $n$ in each variable.
\[[@Drag2010]\] A polynomial $F(x_1,\dots,x_n)$ is *discriminantly separable* if there exist polynomials $f_1(x_1)$, …, $f_n(x_n)$ such that the discriminant $\mathcal{D}_{x_i}F$ of $F$ with respect to $x_i$ satisfies: $$\mathcal{D}_{x_i}F(x_1,\dots, \hat x_i, \dots, x_n)=\prod_{j\ne
i}f_j(x_j),$$ for each $i=1,\dots,n$. $F$ is *symmetrically discriminantly separable* if $f_2=f_3=\dots = f_n$, while it is *strongly discriminantly separable* if $f_1=f_2=f_3=\dots = f_n.$ It is *weakly discriminantly separable* if there exist polynomials $f^j_i(x_i)$ such that for every $i=1,\dots , n$ $$\mathcal{D}_{x_i}F(x_1,\dots, \hat x_i, \dots, x_n)=\prod_{j\ne
i}f^i_j(x_j).$$
$n$-valued groups {#sec:dvg}
-----------------
The idea of $n$-valued groups, on a local level, goes back to Buchstaber and Novikov, to their 1971 study of characteristic classes of vector bundles. That concept was significantly developed further by Buchstaber and his collaborators ([@Buch2006] and references therein). An $n$-valued group on $X$ can be defined as a map: $$\aligned
&m:\quad X\times X \rightarrow (X)^n\\
&m(x,y)=x*y=[z_1,\dots, z_n],
\endaligned$$ where $(X)^n$ denotes the symmetric $n$-th power of $X$ and $z_i$ coordinates therein. Such a map should satisfy the following axioms.
Associativity:
: The condition of equality of two $n^2$-sets $$\aligned &[x*(y*z)_1,\dots, x*(y*z)_n]\\
&[(x*y)_1*z,\dots, (x*y)_n*z]
\endaligned$$ for all triplets $(x,y,z)\in X^3$.
Unit element:
: An element $e\in X$ is *a unit* if $
e*x=x*e=[x,\dots,x],
$ for all $x\in X$.
Inverse:
: A map $\inv: X\rightarrow X$ is *an inverse* if $e\in \inv(x)*x$ and $e\in x*\inv(x)$, for all $x\in X$.
Buchstaber says that $m$ defines *an $n$-valued group structure* $(X, m, e, \inv)$ if it is associative, with a unit and an inverse.
An $n$-valued group $X$ acts on a set $Y$ if there is a mapping $$\phi\ :\ X\times Y \rightarrow (Y)^n,
\quad
\phi (x,y)=x\circ y,$$ such that the two $n^2$-multisubsets $x_1\circ (x_2\circ y)$ and $(x_1*x_2)\circ y$ of $Y$ are equal for all $x_1, x_2\in X$, $y\in Y$. It is also assumed $
e\circ y=[y,\dots, y]
$ for all $y\in Y$.
The list of elementary $n$-valued groups has been done in [@Buch2006]. For a fixed $n$, the corresponding $n$-valued group is defined by a symmetric polynomial $p_n \in \mathcal P_3^n$.
We recall that the elementary symmetric functions of three variables are denoted as $s_1, s_2, s_3$: $$s_1=x+y+z, \quad s_2=xy+xz+yz, \quad s_3=xyz.$$
Let us consider a few simple examples.
\[lemma:p2\] Two-valued group $p_2$ is defined by the relation $$\begin{gathered}
m_2:\quad \mathbf C \times \mathbf C \rightarrow (\mathbf
C)^2,\\
x *_2 y =[(\sqrt{x}+\sqrt{y})^2, (\sqrt{x}-\sqrt{y})^2].\end{gathered}$$ The product $x *_2 y$ is given by the solutions of the polynomial equation $
p_2(z, x, y)=0
$ in $z$, where $$p_2(z, x, y)= (x+y+z)^2-4(xy+yz+zx).$$ As observed in [@Drag2010], the polynomial $p_2(z, x,
y)$ is strongly discriminantly separable: $$\mathcal D_z(p_2)(x,y)=2x\cdot2y, \quad \mathcal
D_x(p_2)(y,z)=2y\cdot2z, \quad \mathcal D_y(p_2)(x,z)=2x\cdot2z,$$ so it generates a case of generalized Kowalevski system of differential equations from [@Drag2010].
Now, we can reproduce a small mathematical experiment from [@Drag2012] with the next cases of elementary $n$ valued groups, with small $n$.
\[example:p3\] $p_3= s_1^3 - 3^3 s_3$, $\mathcal D_z p_3 = y^2x^2(x-y)^2$.
\[example:p4\] $$p_4 = s_1^4 - 2^3 s_1^2s_2 + 2^4s_2^2 - 2^7 s_1s_3,\quad
\mathcal D_z p_4 = y^3x^3(x-y)^2(y+4x)^2(4y+x)^2.$$
\[example:p5\] $$p_5 =s_1^5 - 5^4 s_1^2s_3 + 5^5s_2s_3,\quad
\mathcal D_zp_5 = y^4x^4(x-y)^4(x^2-y^2- 11xy )^2( x^2-y^2+11xy )^2.$$
We see that the polynomials $p_3$, $p_4$, $p_5$ are not any more discriminantly separable. Nevertheless, following [@Drag2012], we can observe amazing factoriziblity property of their discriminants. Unexpectedly, the Hadamard-Hankel determinants which appeared in the study of the Cayley-type conditions in previous sections, have the same algebraic properties.
Cayley-type conditions and discriminantly factorizible polynomials
------------------------------------------------------------------
\[example:c1\] Denote the numerator in the expression as $F_2(\lambda_0,a,b)$: $$F_2(\lambda_0,a,b)=\lambda_0 ^2 (a-b)^2+2\lambda_0 ab(a+b)-3a^2 b^2.$$ $F_2$ is a discriminantly factorizible polynomial. Its discriminant with respect to $\lambda_0$ is: $$\mathcal D_{\lambda_0}F_2=16a^2b^2(a^2-ab+b^2).$$
\[example:c2\] Denote the numerator in as $F_3(\lambda_0,a,b)$: $$\begin{aligned}
F_3(\lambda_0,a,b)\ &=(a \lambda_0 -a b+\lambda_0 b) (a \lambda_0 +a b-\lambda_0 b) (-a \lambda_0 +a b+\lambda_0 b)
\\
&=
-(a-b)^2(a+b)\lambda_0 ^3 +ab(a-b)^2\lambda_0 ^2+a^2b^2(a+b) \lambda_0
-a^3 b^3.
\end{aligned}$$ Its discriminant with respect to $\lambda_0$ is: $$\mathcal D_{\lambda_0}F_3=64 a^8 b^8 (a-b)^2.$$
\[example:c3\] The discriminant with respect to $\lambda_0$ of the polynomial numerator of $B_4^2-B_3 B_5$ equals: $$-309485009821345068724781056\cdot a^{74} b^{74} (a-b)^{18} \left(a^2-a b+b^2\right).$$
\[example:c4\] The discriminant of the numerator of $C_3^2-C_2C_4$ with respect to $\lambda_0$ is: $$\begin{gathered}
-87960930222080\cdot a^{38} b^{38} (a-b)^8
\times\\ \times
\left(27 a^6-81 a^5 b+322 a^4 b^2-509 a^3 b^3+322 a^2 b^4-81 a b^5+27 b^6\right).
\end{gathered}$$
\[example:c5\] Denote the numerator in the expression as $F_2(\lambda_0,a,b)$: $$F_2(\lambda_0,a,b)=\lambda_0 ^2 (a-b)^2-2\lambda_0 ab(a+b)+a^2 b^2.$$ $F_2$ is a strongly discriminantly separable polynomial. Its discriminant with respect to $\lambda_0$ is: $$\mathcal D_{\lambda_0}F_2=16a^3b^3.$$
We observe that in the Examples \[lemma:p2\] and \[example:c5\] we are getting disciminantly separable polynomials. In the rest of the Examples \[example:p3\], \[example:p4\], \[example:p5\], \[example:c1\], \[example:c2\], \[example:c3\], \[example:c4\], the polynomials are not discriminantly separable, but discriminantly factorizable. However, it is important to note that the factors in all these latter examples are homogeneous. Thus, by a change of variables in the polynomials $p_3$, $p_4$, $p_5$ $(x, y)\mapsto (x, z)$, $z=x/y$ we are getting discriminantly separable polynomials in the new coordinates $(x, z)$. Similarly, in Examples \[example:c1\], \[example:c2\], \[example:c3\], \[example:c4\], the change of variables $(a, b)\mapsto (a, \hat b)$, with $\hat b=a/b$, transforms the polynomials into discriminantly separable polynomials in new variables $(a, \hat b)$. It would be very interesting to study further the observed relationship between $n$-valued groups and Cayley-type conditions with the discriminantly separable polynomials.
2-valued group on $\mathbf {CP}^1$, disciminantly separable polynomials and Great Poncelet theorem for triangles
----------------------------------------------------------------------------------------------------------------
It appears that the general equation of pencil of conics corresponds to an action of a two valued group. We used this correspondence to provide a novel interpretation of ’the mysterious Kowalevski change of variables’. It turned out that the associativity condition for this action is equivalent to the Great Poncelet Theorem for a triangle, as it was observed in [@Drag2010]. We are going to close the loop in the paper with a brief reminder at the end about the Great Poncelet Theorem for triangles and this relationship to the associativity of the two-valued group, and thus, with the Kowalevski change of variables.
Consider the general pencil equation $\mathcal F(s,x_1, x_2)=0$, with $s$ being the parameter of the pencil, and $x_1$, $x_2$ the Darboux coordinates (see [@Drag2010]). That pencil is related to two elliptic curves: $\tilde\Gamma_1: y^2=P(x)$ and $\tilde\Gamma_2: t^2=J(s),$ where the polynomials $P$, $J$ are of degrees four and three respectively. These two curves appear to be isomorphic. Rewrite the cubic one $\tilde\Gamma_2$ in the canonical form $\tilde\Gamma_2: t^2=J'(s)=4s^3-g_2s-g_3$. Let $\psi: \tilde\Gamma_2\rightarrow \tilde\Gamma_1$ be a birational morphism between the curves induced by a fractional-linear transformation $\hat \psi$ which maps the three zeros of $J'$ and $\infty$ to the four zeros of the polynomial $P$.
The curve $\tilde\Gamma_2$ as a cubic has a group structure with the neutral element at infinity. With the subgroup $\mathbf {Z}_2$, it defines the standard two-valued group structure on $\mathbf {CP}^1$ (see [@Buch1990]): $$\label{eq:G2Z2}
s_1 *_c s_2 =
\left[-s_1-s_2+\left(\frac{t_1-t_2}{2(s_1-s_2)}\right)^2,-s_1-s_2+\left(\frac{t_1+t_2}{2(s_1-s_2)}\right)^2\right],$$ where $t_i=J'(s_i)$, $i=1,2.$
\[th:G2Z21\][@Drag2010] The general pencil equation after fractional-linear transformations $$\mathcal F(s, \hat \psi^{-1}(x_1), \hat \psi^{-1}(x_2))=0$$ induces the two-valued coset group structure $(\tilde\Gamma_2, \mathbf
Z_2)$ defined by .
A proof is given in [@Drag2010].
The geometric meaning of the pencil equation and algebraic structure of the two valued group give together a connection observed in [@Drag2010] with the Great Poncelet Theorem ([@Poncelet1822], see also [@BergerGeometryII; @DragRadn2011book]). We recall the formulation of the Great Poncelet Theorem for triangles in the form we need below.
Let ${\pazocal{C}}_1$, ${\pazocal{C}}_2$, ${\pazocal{C}}_3$, ${\pazocal{C}}$ be conics from a pencil and $a_1$, $a_2$, $a_3$ tangent lines to ${\pazocal{C}}$, such that $a_1$, $a_2$ intersect on ${\pazocal{C}}_1$, $a_2$, $a_3$ intersect on ${\pazocal{C}}_2$ and $a_1$, $a_3$ intersect on ${\pazocal{C}}_3$. Moreover, we suppose that the tangents to ${\pazocal{C}}_1$, ${\pazocal{C}}_2$, ${\pazocal{C}}_3$ at the intersection points are not concurrent. Suppose that $b_1$, $b_2$ are tangents to ${\pazocal{C}}$ which intersect on ${\pazocal{C}}_1$. Then there exists $b_3$, a tangent to ${\pazocal{C}}$ such that the triplet $(b_1,b_2,b_3)$ satisfies all conditions as $(a_1,a_2,a_3)$.
The associativity condition for the action of the two-valued group $(\Gamma_2,\mathbf Z_2)$ is as follows.
\[th:G2Z23\][@Drag2010] Associativity conditions for the group structure of the two-valued coset group $(\Gamma_2,\mathbf Z_2)$ and for its action on $\mathbf {CP}^1$ are equivalent to the great Poncelet theorem for a triangle.
The proof is given in [@Drag2010].
Acknowledgement {#acknowledgement .unnumbered}
---------------
This research was supported by the Serbian Ministry of Education, Science, and Technological Development, Project 174020 *Geometry and Topology of Manifolds, Classical Mechanics, and Integrable Dynamical Systems*; and the Australian Research Council, Project DP190101838 *Billiards within confocal quadrics and beyond*.
The authors are grateful to Andrey Mironov for inspiring discussions during GDIS 2018.
|
---
abstract: 'The very first galaxies that started the cosmic dawn likely resided in so-called “minihaloes’’, with masses of ${\sim}10^5$–$10^8{{{\rm M}_\odot}}$, accreting their gas from the intergalactic medium through H$_2$ cooling. Such molecularly cooled galaxies (MCGs) mostly formed in pristine environments, hosted massive, metal-free stars, and were eventually sterilized by the build-up of a disassociating (Lyman–Werner; LW) background. Therefore, their properties might be very different from the galaxies we see in the later Universe. Although MCGs are probably too faint to be observed directly, we could nevertheless infer their properties from the imprint they leave in the cosmic 21-cm signal. Here we quantify this imprint by extending the public simulation code [[<span style="font-variant:small-caps;">FAST</span>]{}]{} to allow for a distinct population of MCGs. We allow MCGs to have different properties from other galaxies, including unique scaling relations for their stellar-to-halo mass ratios, ionizing escape fractions, and spectral energy distributions. We track inhomogeneous recombinations, disassociative LW feedback, and photoheating from reionization. After demonstrating how MCGs can shape the 21-cm signal, we explore to what extent current observations can already place constraints on their properties. The cosmic microwave background optical depth from [*Planck*]{} sets an upper limit on the product of the ionizing escape fraction and the stellar mass in MCGs. When including also the timing of the putative EDGES absorption signal, we find an additional strong degeneracy between the stellar mass and the X-ray luminosity of MCGs. If proven to be of cosmic origin, the timing of the EDGES signal would have been set by MCGs.'
author:
- |
Yuxiang Qin$^{1}$[^1], Andrei Mesinger$^1$, Jaehong Park$^1$, Bradley Greig$^{2,3}$, and\
$^{1}$Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy\
$^{2}$School of Physics, University of Melbourne, Parkville, VIC 3010, Australia\
$^{3}$ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)\
$^{4}$Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138, USA
bibliography:
- 'reference.bib'
title: 'A tale of two sites - : Inferring the properties of minihalo-hosted galaxies from current observations'
---
\[firstpage\]
galaxies: high-redshift; intergalactic medium; dark ages, reionization, first stars; diffuse radiation; early Universe; cosmology: theory
Introduction
============
The hyperfine spin-flip transition of neutral hydrogen, corresponding to a photon with a wavelength of 21 cm, promises to revolutionize our understanding of the first billion years of the Universe. The cosmic 21-cm signal is typically expressed as the brightness temperature contrast of the cosmic hydrogen against the cosmic microwave background (CMB), at a redshifted frequency $\nu$ and spatial position ${\bf r}$ (e.g. @Furlanetto2006PhR...433..181F) $$\label{eq:dtb_define}
\begin{split}
&\delta T_\mathrm{b}\left(\nu, {\bf r}\right) = \left(T_\mathrm{S} - T_\mathrm{CMB}\right)
\left({1{-}e^{{-}\tau_\mathrm{\nu_0}}}\right){\left(1+z\right)}^{-1} \\
&\approx 20\mathrm{mK} \left(1{-}\frac{\mathrm{T_\mathrm{CMB}}}{T_\mathrm{S}}\right)
\frac{x_{\mathrm{HI}} \left(1+\delta\right)}{1+\frac{\mathrm{d}v_r}{\mathrm{d}r}/H}
\sqrt{\frac{1{+}z}{10}\frac{0.15}{\Omega_\mathrm{m}h^2}}\frac{\Omega_\mathrm{b}h^2}{0.023},
\end{split}$$ where $T_\mathrm{CMB}$ is the CMB temperature; $T_{\rm S}$ is the spin temperature denoting the relative level populations of the hyperfine transition; [$H$ is the Hubble constant at redshift $z$;]{} and $\tau_\mathrm{\nu_0}$ is the optical depth and is a function of the spin temperature, neutral hydrogen fraction ($x_{{\mathrm{H}\textsc{i}}}$), local overdensity ($\delta \equiv \rho_{\rm b}/\bar{\rho}_{\rm b} - 1$ with $\rho_{\rm b}$ and $\bar{\rho}_{\rm b}$ being the baryonic density and its cosmic mean, respectively), and the line-of-sight velocity gradient (d$v_r$/d$r$), as well as cosmological parameters, such as the present baryon and matter abundances, $\Omega_{\mathrm{b}}$ and $\Omega_{\mathrm{m}}$, and the Hubble constant, $h$.
In addition to physical cosmology, the signal is sensitive to the ionization and thermal state of the intergalactic medium (IGM), which are governed by the ionizing, X-ray and soft UV radiation fields during the cosmic dawn (CD) and subsequent epoch of reionization (EoR). These radiation fields are established by stars and black holes inside the first generations of galaxies (though exotic sources such as dark matter annihilation or primordial black holes might contribute; e.g. @Evoli2014JCAP...11..024E [@Lopez-Honorez2016JCAP...08..004L; @Hektor2018PhRvD..98b3503H]). Thus, the cosmic 21-cm signal encodes the properties of unseen galaxies during the first billion years. Current interferometers, including the Low-Frequency Array (LOFAR[^2]; ) and the Murchison Widefield Array (MWA[^3]; @Tingay2013PASA...30....7T [@Beardsley2016ApJ...833..102B]), are aiming for a statistical detection of the EoR; however, next-generation instruments, such as the Hydrogen Epoch of Reionization Arrays (HERA[^4]; @DeBoer2017PASP..129d5001D [@Kohn2019ApJ...882...58K]) and the Square Kilometre Array (SKA[^5]; @Mellema2013ExA....36..235M [@Koopmans2015aska.confE...1K]), are expected to characterize the topology of the CD, allowing us to indirectly study the very first galaxies out to $z{\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}20$–30.
The first generations of galaxies are expected to reside in so-called minihaloes, with virial temperatures below $\sim10^4$K. At these low temperatures, cooling through atomic hydrogen (${{\mathrm{H}\textsc{i}}}$) and helium (He) is inefficient; therefore, minihaloes obtain gas from the IGM and [cool]{} predominately through molecular hydrogen (${{\mathrm{H}_2}}$) cooling (e.g. @Haiman1996ApJ...467..522H [@Haiman1997ApJ...476..458H; @Yoshida2003ApJ...598...73Y; @Yoshida2006]). As they form out of pristine (unpolluted) gas, these minihaloes or molecular-cooling galaxies (MCGs) are expected to host metal-free (so-called PopIII) stars and associated remnants [@Wise2012ApJ...745...50W; @Xu2016ApJ...833...84X]. Their shallow potential wells suggest that they are sensitive to supernova and photoheating feedback [@Haiman2000ApJ...534...11H; @Wise2007; @Kimm2016]. Moreover, star formation inside MCGs is expected to be transient since an H$_2$-disassociating (Lyman–Werner; LW) background becomes established soon after the first stars form (e.g. @Johnson2007ApJ...665...85J [@Ahn2009ApJ...695.1430A; @Holzbauer2012MNRAS.419..718H; @Fialkov2013MNRAS.432.2909F; @Jaacks2018; @Schauer2019MNRAS.484.3510S]). Because of this uniqueness (pristine environment, top-heavy initial mass function (IMF), transient star formation, peculiar energetics), it is doubtful that the typical properties of MCGs can be adequately captured by simply extending the scaling relations inferred from observations of their massive counterparts at lower redshifts (e.g. @Mirocha2019MNRAS.483.1980M [@Mebane2019arXiv191010171M]).
Unfortunately, studying MCGs through direct observations is unlikely in the foreseeable future. Most are expected to have UV magnitudes in the range of $M_{\rm UV}{\sim}$-5 to -12 (e.g. @OShea2015ApJ...807L..12O [@Xu2016ApJ...833...84X]), below the observational limit of even the next-generation infrared instrument, the [*James Webb Space Telescope*]{} ([*JWST*]{}; @Gardner2006SSRv..123..485G). Gravitational lensing has allowed us to push UV luminosity functions (LFs) down to $M_{\rm UV}{\sim}{-}12.5$ at $z\approx6$ (e.g. @Bouwens2017ApJ...843..129B [@Livermore2017ApJ...835..113L; @Atek2018MNRAS.479.5184A; @Ishigaki2018ApJ...854...73I]). However, even if the associated large systematic uncertainties can be mitigated, MCGs seem unlikely to persist at these low redshifts and few if any might be found in the effective lensing volume (e.g. @Atek2018MNRAS.479.5184A).
Thankfully, these transient first galaxies leave an imprint in the timing and topology of the 21-cm signal. For example, massive stars in MCGs could be responsible for a tail in the reionization history extending towards high redshifts (e.g. @Ahn2012ApJ...756L..16A [@Visbal2015MNRAS.453.4456V; @Miranda2017MNRAS.467.4050M]), and imprint more small-scale structure in the reionization topology (e.g. @Mesinger2012MNRAS.422.1403M [@Koh2018MNRAS.474.3817K]), while the neutral gas inside minihaloes can act as ionizing photon sinks, delaying reionization and further affecting the topological features (e.g. @Ciardi2006MNRAS.366..689C [@McQuinn2007MNRAS.377.1043M]). MCGs could play an even more prominent role in the timing and morphology of the earlier epochs when X-rays and soft UV photons drive the 21-cm signal (e.g. @Fialkov2013MNRAS.432.2909F [@Fialkov2014MNRAS.445..213F; @Mirocha2017; @Munoz2019PhRvD.100f3538M; @Mebane2019arXiv191010171M]). Indeed, these early epochs recently received attention because of the claimed detection of the globally averaged 21-cm absorption feature at $z {\sim}17$ by the Experiment to Detect the Global Epoch of Reionization Signature (EDGES; @Bowman2018). Although the cosmological interpretation of the EDGES result remains controversial (e.g. @Hills2018arXiv180501421H [@Bradley2019ApJ...874..153B; @Sims2019MNRAS.488.2904S]), if the signal is indeed proven to be of cosmic origins, X-rays and soft UV radiation from MCGs were likely responsible for its timing (e.g. @Mirocha2019MNRAS.483.1980M).
In this work, we explore the imprints of MCGs in the 21-cm signal from the EoR and CD, introducing a distinct population in the public code [<span style="font-variant:small-caps;">FAST</span>]{}[@Mesinger2007ApJ...669..663M; @Mesinger2011MNRAS.411..955M], whose abundance is regulated by an H$_2$-disassociating background (see also @Fialkov2013MNRAS.432.2909F [@Munoz2019PhRvD.100f3538M]). We build upon the model of @Park2019MNRAS.484..933P, whose parametrization allows star formation rates (SFRs) to scale non-linearly with the mass of the host halo, thus allowing the source models to be consistent with current UV LF observations (e.g. @Hassan2016 [@Mirocha2016; @Mutch2016]). We extend this model, allowing MCGs to have their own unique properties, including star formation efficiencies, ionizing escape fractions, and X-ray and soft UV emissivities. By varying the free parameters in our model, we quantify how the diverse properties of two galaxy populations (atomic and molecular cooling) are imprinted in the global and interferometric 21-cm signals. As a proof of concept, we confront this extended two-population model with the putative EDGES detection, using its timing to constrain the properties of minihalo-hosted galaxies within a Bayesian framework, 21CMMC[^6] [@Greig2015MNRAS.449.4246G]. The code developed for this work will be part of the upcoming v3.0.0 release of [<span style="font-variant:small-caps;">FAST</span>]{}[^7].
This paper is organized as follows. We present our model in Section \[sec:models\]. In Section \[sec:xHandTs\], we investigate the impact of the physical properties of our model on the 21-cm signal. In Section \[sec:constraint\] we perform a Monte Carlo Markov Chain (MCMC) with a subset of model parameters, showing constraints available with and without the EDGES result. Finally, we conclude in Section \[sec:conclusion\]. In this work, we use a $\Lambda$CDM cosmology with parameters $\Omega_{\mathrm{m}}, \Omega_{\mathrm{b}}, \Omega_{\mathrm{\Lambda}}, h, \sigma_8$ and $n_s$ = (0.31, 0.048, 0.69, 0.68, 0.81, and 0.97, consistent with results from the [*Planck*]{} satellite (e.g. ).
Modelling galaxies, the IGM, and cosmic radiation fields {#sec:models}
========================================================
Star formation and galaxy evolution {#subsec:sf_models}
-----------------------------------
As gas from the IGM accretes onto dark matter halos, its gravitational potential energy is converted into heat. In order to avoid becoming pressure supported and continue collapsing onto the galaxy at the centre of the halo, gas needs to cool by emitting radiation that must escape the system. Galaxies can be classified by the dominant cooling channel through which the IGM gas has been accreted onto the halo: (i) atomic-cooling galaxies (ACGs), which predominantly obtained their gas through ${{\mathrm{H}\textsc{i}}}$ (and $\textsc{H}$e) line transitions efficient at virial temperatures ($T_\mathrm{vir}$) above $10^4 \mathrm{K}$; and (ii) MCGs, in which the gas cools mainly through the $\textsc{H}_2$ rotational–vibrational transitions efficient at $T_\mathrm{vir}{\sim}10^3 {-} 10^4 \mathrm{K}$; most ACGs at high redshifts are “second-generation” galaxies, forming out of MCG building blocks. The pre-enrichment by metals as well as the different energetics and cooling processes suggests that the stellar component and interstellar medium (ISM) of ACG and MCG should have different properties. As these properties are currently poorly understood, we describe them through relatively generic and flexible parametric models. Next, we introduce these for both ACGs and MCGs.
### Atomic-cooling galaxies {#subsec:source model}
“Massive” ACGs ($T_\mathrm{vir}{>}10^5\mathrm{K}$; e.g. @Kuhlen2012 [@Mason2015; @Liu2016]) have been observed by the [*Hubble Space Telescope*]{} ([*HST*]{}). The resulting non-ionizing UV LFs (e.g. @Bouwens2015a [@Bouwens2016; @Finkelstein2015ApJ...810...71F; @Livermore2017ApJ...835..113L; @Atek2018MNRAS.479.5184A; @Oesch2018ApJ...855..105O; @Bhatawdekar2019MNRAS.tmp..843B]) give invaluable insight into star formation processes inside these galaxies, ruling out the constant mass-to-light ratio models commonly found in early 21-cm forecasts (e.g. @Mesinger2011MNRAS.411..955M [@Fialkov2013MNRAS.432.2909F]).
Here we build upon the model of @Park2019MNRAS.484..933P [hereafter ], which is flexible enough to reproduce observed high-redshift LFs. This simple model describes the ACG population through power-law scaling relations with the halo mass function (HMF; see also @Kuhlen2012 [@Mitra2015; @Sun2015; @Behroozi2019MNRAS.488.3143B]) Although individual galaxies have much more complicated and stochastic evolution of properties (e.g. @Mutch2016 [@Ma2018; @Yung2019MNRAS.490.2855Y]), the 21-cm signal on observable scales (${{\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}} 10 \mathrm{Mpc}$) is sourced by ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}$ hundreds of galaxies, motivating the use of simple and computationally efficient average scaling relations.
Specifically, we describe the stellar mass of an ACG, $M_*^{\rm atom}$, hosted in a halo with a mass of ${{M_{\rm vir}}}$ by $$\label{eq:m*_atom}
M_*^{\rm atom} = \min\left[1, ~f_{*,10}^{\rm atom}\left(\frac{{M_{\rm vir}}}{10^{10}{{\rm M}_\odot}}\right)^{\alpha_*}\right] \frac{\Omega_{\mathrm{b}}}{\Omega_{\mathrm{m}}} {{M_{\rm vir}}},$$ where $f_{*,10}^{\rm atom}$ and ${\alpha_*}$ are the normalization factor and power-law index. More detailed models recover such scaling relations for the bulk of the high-redshift galaxy population (e.g. @Moster2013MNRAS.428.3121M [@Mutch2016; @Sun2015; @Tacchella2018ApJ...868...92T; @Behroozi2019MNRAS.488.3143B; @Yung2019MNRAS.490.2855Y]). Note that we do not consider AGN feedback, which is thought to dominate in the most massive galaxies, as these are too rare at high redshifts to shape the 21-cm signal (e.g. @Mitra2015 [@Manti2017; @Parsa2017; @Qin2017a; @Ricci2017MNRAS.465.1915R; @Garaldi2019MNRAS.483.5301G]). We also do not include a redshift evolution in this stellar-to-halo mass relation, which is supported by some simulation results (e.g. @Mutch2016 [@Xu2016ApJ...833...84X]), although generalizing the model to include a redshift evolution is trivial.
The corresponding average SFR is assumed to be $$\label{eq:sfr}
{\rm SFR}^{\rm atom} = \frac{M_*^{\rm atom}}{t_* H^{-1}\left(z\right)},$$ where $t_*$ is a free parameter corresponding to the typical star formation time-scale, defined as a fraction of the Hubble time. Since the dynamical time of a halo scales with the Hubble time during matter domination, this is analogous to assuming the star formation time scales with the dynamical time.
We include an exponential duty cycle[^8] to describe the mass function of halos (HMF) that host star-forming ACGs $$\label{eq:hmf_atom}
\phi^{\rm atom} = \frac{{{\rm d}}n}{{{\rm d}}{M_{\rm vir}}} \exp\left({-\dfrac{M_{\rm crit}^{\rm atom}}{{M_{\rm vir}}}}\right),$$ where ${{{\rm d}}n}/{{{\rm d}}{M_{\rm vir}}}$ is the mass function of all halos. The exponential term in equation (\[eq:hmf\_atom\]) accounts for inefficient star formation in halos below a characteristic mass scale (i.e. turnover mass) $$\label{eq:m_crit_atom}
M_{\rm crit}^{\rm atom} = \max\left[ M_{\rm crit}^{\rm cool}, M_{\rm crit}^{\rm ion}, M_{\rm crit}^{\rm SN} \right] ~ .$$ As can be seen from equation (\[eq:m\_crit\_atom\]), we assume that star formation in small ACGs can be limited by three physical processes: (i) inefficient cooling, $M_{\rm crit}^{\rm cool}$; (ii) photoheating feedback from reionization, $M_{\rm crit}^{\rm ion}$; and (iii) supernova feedback, $M_{\rm crit}^{\rm SN}$.
For (i), we assume the ${{\mathrm{H}\textsc{i}}}$ cooling threshold to be 10$^4$K. The corresponding halo mass can be expressed as (e.g. @Barkana2000) $$\label{eq:mcrit_cool}
\frac{M_{\rm crit}^{\rm cool}}{5{\times}10^7 {{\rm M}_\odot}} = \left(\frac{0.678}{h}\right) \left(\frac{0.59}{\mu}\frac{10}{1{+}z}\right)^{1.5}
\left(\frac{{\Omega_{\rm m}}^{\rm z}}{{\Omega_{\rm m}}}\frac{18 {\rm {\rm \pi}}^2}{\Delta_{\rm c}}\right)^{0.5},$$ where $\mu$ is the mean molecular weight, $\Delta_c$ is the critical overdensity of halos at collapse in the spherical collapse model, and ${\Omega_{\rm m}}^{\rm z}$ is the matter density [in units of the critical density at redshift $z$]{}.
For photoheating feedback inside reionized regions of the Universe (ii; see @Efstathiou1992MNRAS.256P..43E [@Shapiro1994ApJ...427...25S; @Thoul1996ApJ...465..608T; @Hui1997MNRAS.292...27H]; [@Dijkstra2004ApJ...601..666D]{}), we take the functional form from @Sobacchi2014MNRAS.440.1662S $$\label{eq:mcrit_ion}
\dfrac{M_{\rm crit}^{\rm ion}}{2.8{\times}10^9 {{\rm M}_\odot}} {=} \left(\frac{f_{\rm bias}\bar{\Gamma}_{\rm ion}}{10^{-12}{\rm s^{-1}}}\right)^{0.17} \left(\frac{10}{1{+}z}\right)^{2.1}\left[1 {-} \left(\frac{1{+}z}{1{+}z^{\rm ion}}\right)^{2}\right]^{2.5},$$ where $\bar{\Gamma}_{\rm ion}$ and $z^{\rm ion}$ are the local photoionization rate and the redshift at which the local patch of the IGM was reionized, respectively; the factor $f_{\rm bias}{\approx2}$ accounts for the enhanced photoionization rate at galaxy locations due to their clustering [@Mesinger2008MNRAS.390.1071M]. We note that although equation (\[eq:mcrit\_ion\]) is obtained from 1D collapse simulations, it is consistent with results from more detailed 3D simulations at the relevant redshifts (e.g. @Noh2014MNRAS.444..503N [@Katz2019arXiv190511414K]). We will discuss how to calculate the ionizing background and determine the redshift of ionization in Section \[subsubsec:UVionizing\].
Supernova feedback (iii) is probably the least well-understood feedback process. The dynamic range required to study supernova feedback is enormous. Thus, its implementation in cosmological simulations is resolution dependent and relies on the choice of subgrid prescription (e.g. @DallaVecchia2008 [@DallaVecchia2012; @Hopkins2014MNRAS.445..581H; @Keller2014; @Hopkins2017]; Gillet et al., in prep.; Pallottini et al., in prep.). Although it is a free parameter in our model, for this work we assume it to be subdominant compared to inefficient cooling and photoheating in regulating star formation, i.e. taking $M_{\rm crit}^{\rm SN} \le \max\left[M_{\rm crit}^{\rm cool}, M_{\rm crit}^{\rm ion} \right]$ [and ignore $M_{\rm crit}^{\rm SN}$ in equation (\[eq:m\_crit\_atom\])]{}. This is a conservative choice in that it maximizes the importance of star formation in small ACGs and minihaloes, which is the focus of this work. Note that supernova feedback is still expected to determine the scaling of the stellar-to-halo mass relation (e.g. @Moster2013MNRAS.428.3121M [@Wyithe2013MNRAS.428.2741W; @Dayal2014MNRAS.445.2545D; @Mutch2016; @Sun2015; @Tacchella2018ApJ...868...92T]), even if it is not responsible for a faint end turnover of the LFs.
### Molecular-cooling galaxies {#subsec:source model_mcg}
Since star formation can proceed differently in MCGs compared to ACGs, we allow them to have a different stellar-to-halo mass normalization $$\label{eq:m*_mol}
M_*^{\rm mol} = \min\left[1, f_{*,7}^{\rm mol}\left(\frac{{M_{\rm vir}}}{10^{7}{{\rm M}_\odot}}\right)^{\alpha_*}\right] \frac{\Omega_{\mathrm{b}}}{\Omega_{\mathrm{m}}} {{M_{\rm vir}}} ~ ,$$ and calculate their SFRs analogously to equation (\[eq:sfr\]). We define the mass function of MCG hosts as $$\label{eq:hmf_mol}
\phi^{\rm mol} =\frac{{{\rm d}}n}{{{\rm d}}{M_{\rm vir}}} \exp\left({-\dfrac{M_{\rm crit}^{\rm mol}}{{M_{\rm vir}}}}\right)\exp\left({-\dfrac{{M_{\rm vir}}}{M_{\rm crit}^{\rm cool}}}\right) ~ .$$ The two exponential terms in equation (\[eq:hmf\_mol\]) correspond to duty cycles of halos hosting MCGs, setting both a lower and an upper mass threshold. The upper mass threshold, $M_{\rm crit}^{\rm cool}$, corresponds to the transition between MCG and ACG, at around $T_\mathrm{vir}\sim10^4{\rm K}$ (see equation \[eq:mcrit\_cool\]).
It is worth noting that our duty cycles are exponential functions of halo mass, which is a somewhat arbitrary choice. One impact of this is that the transition from MCGs and ACGs is not a step function at $T_{\rm vir}\sim10^4$ K, as is commonly assumed due to the rapid drop in the atomic cooling curve. However, it is plausible to expect a transition smoother than a step function [from the large scatter in the gas temperature-to-halo mass relation (e.g. @Shang2010MNRAS.402.1249S). Additionally]{}, a fraction of galaxies with $T_\mathrm{vir}{>}10^4{\rm K}$ could have obtained most of their gas at earlier times when the H$_2$ cooling channel was dominant. Similarly, one could have some rare galaxies with ACG-like properties below the cooling threshold, if they occur in pre-enriched dense environments with rapid accretion of cold streams (e.g. @Qin_2019). In practice, these duty cycles serve as window functions over the HMFs to encapsulate two distinct galaxy populations, and our results are not sensitive to the specific choice of window function.
The lower mass threshold for star-forming MCGs is set by cooling and feedback, analogously to equation (\[eq:m\_crit\_atom\]) for ACGs $$\label{eq:m_crit_mol}
M_{\rm crit}^{\rm mol} = \max\left[ M_{\rm crit}^{\rm diss}, M_{\rm crit}^{\rm ion}, M_{\rm crit}^{\rm SN} \right] ~ .$$ The efficiency of ${{\mathrm{H}_2}}$ cooling depends on the strength of the dissociating (LW) background, in the energy range 11.2 – 13.6 eV. We quantify this using the fitting formulae from [@Visbal2015MNRAS.453.4456V]{} $$\label{eq:mcrit,diss}
\frac{M_{\rm crit}^{\rm diss}}{2.5\times10^5{{\rm M}_\odot}} = \left(\frac{26}{1{+}z}\right)^{1.5}\left[1+ 22.87\times{J^{\rm 21}_{\rm LW,eff}}^{0.47}\right] ~.$$ Here the unitless quantity, $J^{\rm 21}_{\rm LW,eff}$, represents the (local) LW intensity impinging on the MCG $$\label{eq:fshiled}
J^{\rm 21}_{\rm LW,eff} = \frac{J_{\rm LW}}{{\rm 10^{{-}21}{\rm erg\ s^{{-}1}\ Hz^{-1}\ cm^{{-}2}\ sr^{{-}1}}}} \left(1-f_{\rm H_2}^{\rm shield}\right)$$ with $J_{\rm LW}$ corresponding to the local (inhomogeneous) LW background (LWB; discussed in Section \[subsubsec:LW\]), and the factor $f_{\rm H_2}^{\rm shield}$ accounting for self-shielding of star-forming regions by the ISM and the circumgalactic medium of the host galaxy[^9] (e.g. @Draine1996ApJ...468..269D [@Wolcott-Green2011MNRAS.418..838W]).
In this work, we do not account for a possible additional suppression of star formation in minihaloes due to the relative velocities of dark matter and baryons, imprinted at recombination [@Tseliakhovich2010PhRvD..82h3520T]. The root-mean-square velocity offset at $z\sim20$ is roughly $\sigma_{\rm vb} \sim 0.5{\rm km\ s^{-1}}$ [@Munoz2018Natur.557..684M], which is smaller than the typical circular velocity of minihaloes, $v_{\rm circ} \sim 4{\rm km\ s^{-1}}$. Therefore, relative velocities are unlikely to be the main bottleneck in feeding gas to MCGs at observable redshifts (e.g. @Fialkov2012MNRAS.424.1335F). Nevertheless, they do somewhat suppress their cold gas reservoir (e.g. @Dalal:2010yt [@Greif2011ApJ...736..147G; @Oleary2012ApJ...760....4O; @Schauer2019MNRAS.484.3510S]), which can in turn suppress their SFRs. Although modest, such a decrease in SFRs is correlated on fairly large scales, set by acoustic oscillations prior to recombination. This might spatially modulate the 21-cm signal in a way that could be detectable with next-generation interferometers (e.g. @Fialkov2012MNRAS.424.1335F [@Munoz2019PhRvD.100f3538M]), providing a standard ruler at CD [@Munoz:2019fkt]. We postpone a detailed investigation of this claim to future work (Muñoz et al. in prep.).
### UV LFs {#subsubsec:UVLF}
{width="54.00000%"}\
{width="92.00000%"}
In order to compare our models with observed LFs from [*HST*]{}, we convert the SFR to a corresponding intrinsic UV luminosity at $1500\rm{\AA}$ via ${L_{\rm 1500}}/{{\rm SFR}} = 8.7\times10^{27}{\rm erg\ s^{-1} Hz^{-1} {{{\rm M}_\odot}}^{-1} {\rm yr}}$ (e.g. ). This conversion factor can vary by up to a factor of $\sim$2, depending on the IMF, [metallicity]{}, and recent star formation history (e.g. @Tumlinson2000ApJ...528L..65T [@Bromm2001ApJ...552..464B; @Eldridge2017PASA...34...58E; @Wilkins2019arXiv191005220W]). As it is degenerate with the stellar fraction, a misestimate would imply a bias in constraints on $f_\ast$ from LF observations. For simplicity, here we use the same conversion factor for both MCGs and ACGs; however, MCGs are generally too faint to be constrained by LF observations (see the bottom right panel of Figure \[fig:example\]) making our results insensitive to this choice. In future work, we will expand on this conversion, including the relevant uncertainties in our forward modelling.
### An illustration of our two-population source model
We illustrate the updated two-population source model in Fig. \[fig:example\], assuming MCGs and ACGs follow the same star formation efficiency – halo mass relation ($f_{*,7}^{\rm mol}={{1000^{-\alpha_*}}f_{*,10}^{\rm atom}}$). In the top panel, we show the evolution of critical masses determined by: (i) the atomic cooling efficiency, $M_{\rm crit}^{\rm cool}$; (ii) photodissociation of ${\mathrm{H}_2}$, $M_{\rm crit}^{\rm diss}$; and (iii) photoheating from reionization, $M_{\rm crit}^{\rm ion}$. Also shown are the corresponding halo mass scales below which star formation in ACGs ($M_{\rm crit}^{\rm atom}$) and MCGs becomes inefficient ($M_{\rm crit}^{\rm mol}$), determined by effects (i) – (iii). Note that photoheating only becomes dominant in the advanced stages of reionization (e.g. @Mesinger2008MNRAS.390.1071M [@Ocvirk2018arXiv181111192O; @Katz2019arXiv190511414K]).
We select three representative epochs and show the mass functions of all halos (black dotted line), as well as those hosting ACGs (red solid line) and MCGs (blue solid line) in the lower left panel of Fig. \[fig:example\]. As expected, star formation in ACGs and MCGs is regulated by inefficient cooling in the very early universe. As the intensity of the LWB increases with time, it becomes increasingly difficult for gas to cool [in]{} minihaloes, as denoted by the shift of $M_{\rm crit}^{\rm diss}$ towards higher masses (c.f. equation \[eq:mcrit,diss\]). At early times before the bulk of the EoR, the cosmic ${{\mathrm{H}\textsc{ii}}}$ regions are still confined to be proximate to the nascent galaxies; therefore photoheating feedback does not affect most of the volume. Moreover, since the gas responds to the radiation background on roughly the sound-crossing time-scale, photoheating feedback only becomes the dominant negative feedback for both galaxy populations towards the end of the EoR (see the late rise in the dotted gray curve in the top panel).
In the lower right panel of Fig. \[fig:example\], we show the corresponding 1500Å UV LFs together with an extreme model in which the star formation efficiency of MCGs is increased by a factor of $1000^{\alpha_*}$ (i.e. $f_{*,7}^{\rm mol}=f_{*,10}^{\rm atom}$; [*dashed blue curves*]{}). We see that MCGs only dominate the UV LFs at magnitudes fainter than $M_{1500} {\sim}{-}10$ and redshifts higher than $z{\sim}10$. Thus direct observations of individual MCGs are unlikely even with [*JWST*]{}, which can extend current [*HST*]{} observations by $\sim$1–2 mag (i.e. $M_{1500}{\sim}-13$ at $z\sim6$; @Finkelstein2016PASA...33...37F; R. Bouwens and P. Oesch, private communication).
Fig. \[fig:example\] demonstrates the (parametrized) impact of various feedback mechanisms on star formation in MCGs and ACGs. In the next sections, we describe the calculation of IGM properties as well as the ionizing, LW, X-ray and [[[Lyman-$\alpha$]{}]{}]{} radiation fields – the essential ingredients that govern these feedback scales and regulate the gas properties responsible for the 21-cm signal.
IGM evolution {#subsec:igm}
-------------
The IGM density and velocity fields are computed at the desired redshift by evolving an initial Gaussian realization with second-order Lagrangian perturbation theory (e.g. @Scoccimarro1998MNRAS.299.1097S). The ionization field of the IGM is assumed to be bi-modal – due to the short mean free path of UV ionizing photons in the neutral IGM and the long average recombination time-scale in the ionized IGM, (almost) fully ionized regions begin appearing and spreading into (almost) fully neutral regions[^10] (e.g. @Trac2011ASL.....4..228T [@Zahn2011MNRAS.414..727Z]).
Ionized regions of the IGM are identified using the excursion-set procedure described in Section \[subsubsec:UVionizing\]. Inside these cosmic HII regions, the temperature is assumed to be ${\sim}10^4$K, while a small amount of residual ${\mathrm{H}\textsc{i}}$ remains according to photoionization equilibrium with the local (inhomogeneous) UV background (see Section \[subsubsec:UVionizing\]).
Outside of the cosmic HII regions, the neutral IGM is still impacted by X-ray photons from galaxies [[@Mesinger2013MNRAS.431..621M]]{}, which have long mean free paths. In the neutral IGM, the temperature, $T_{\rm g}$, and ionized fraction, $x_e$, of the gas are evolved from initial conditions computed with <span style="font-variant:small-caps;">recfast</span> [@Seager1999ApJ...523L...1S], according to the following differential equations: $$\label{eq:x_e}
\dot{x}_e = {-} \alpha_{\rm A} C_{\rm sub} x_e^2 n_{\rm b}f_{\rm H} + \Lambda_{X}$$ with $\alpha_{\rm A}$, $C_{\rm sub}$, $n_{\rm b}$, $f_{\rm H}$ and $\Lambda_{X}$ representing the case-A recombination coefficient, subgrid clumping factor, number density of baryons in the simulation cell, number fraction of hydrogen, and the X-ray ionization rate per baryon, respectively, and $$\label{eq:T_g}
\frac{3}{2} \left(1{+}x_e\right) \dot{T}_{\rm g}= \left(1{+}x_e\right) \frac{\dot{n}_{\rm b}}{n_{\rm b}} {T}_{\rm g} {-} \frac{3}{2} {\dot{x}_e} {T}_{\rm g}{+} k_{\rm B}^{-1}\left(\varepsilon_{X} {+} \varepsilon_{\rm CMB}\right)$$ with $k_{\rm B}$ being the Boltzmann constant, $\varepsilon_{X}$ and $\varepsilon_{\rm CMB}$ (in units of ${\rm erg\ s^{-1}}$) correspond to the heating rate per baryon due to X-rays and CMB photons, respectively. Note that the terms on the right side of equation (\[eq:x\_e\]) refer to recombinations and ionization with X-rays while those of equation (\[eq:T\_g\]) correspond to heating/cooling due to structure formation, changing species, X-ray and Compton heating [@Seager1999ApJ...523L...1S], respectively. We ignore other heating processes that are expected to be subdominant at the relevant redshifts, such as dark matter annihilation or shock heating (e.g. @Furlanetto2006PhR...433..181F [@McQuinn2012ApJ...760....3M; @Evoli2014JCAP...11..024E; @Lopez-Honorez2016JCAP...08..004L]). We describe the calculation of X-ray ionization and heating rates in Section \[subsubsec:xraysandlya\].
Radiation fields {#subsec:photon model}
----------------
Cosmic radiation fields regulate the ionization and thermal state of the IGM, as well as the star formation feedback processes described previously. Here we summarize how we calculate the ionizing, LW, X-ray and [[[Lyman-$\alpha$]{}]{}]{} radiation fields.
### UV ionizing photons {#subsubsec:UVionizing}
We follow an excursion-set approach [@Furlanetto2004ApJ...613....1F] to identify cosmic ${{\mathrm{H}\textsc{ii}}}$ regions – counting the number of ionizing photons in spheres of decreasing radius around each IGM parcel. A cell, centred at $({\bf r}, z)$, is considered ionized if, at any radius $R$, $$\label{eq:ionization}
\bar{n}_{\rm ion} \ge \left(1+ \bar{n}_{\rm rec}\right)\left(1 - \bar{x}_e\right).$$ Here, $\bar{n}_{\rm ion}$ is the cumulative number of ionizing photons per baryon, $\bar{n}_{\rm rec}$ is the cumulative number of recombinations per baryon, and $\bar{x}_e$ accounts for X-ray ionizations as described in the previous section. The averaging is performed over the spherical region with a radius of $R$ and a corresponding overdensity of $\delta_{\rm R|_{{\bf r},z}} = \rho_{\rm b} / \bar{\rho}_{\rm b} - 1$.
The left-hand side of equation (\[eq:ionization\]) is calculated using an updated form from equation (15) of , accounting for both galaxy populations (i.e. MCG and ACG; $i \in \left\{{\rm mol}, {\rm atom}\right\}$). Specifically, the cumulative number of ionizing photons per baryon in a spherical IGM patch is $$\label{eq:dotn_ion}
\bar{n}_{\rm ion}\left({\bf r},z|R,\delta_{\rm R|_{{\bf r},z}}\right) {=} \rho_{\rm b}^{-1}\sum_{i\in\left\{\substack{{\rm mol,}\\{\rm atom}}\right\}} \int {\rm d}M_{\rm vir} {\phi}^{i} M_*^{i} n_{\gamma}^{i} f_{\rm esc}^{i}.$$ In this equation[^11]
1. ${\phi}^{i} \left(M_{\rm vir}, {\bf r}, z | R, \delta_{\rm R|_{{\bf r},z}} \right)$ represents the differential number density of halos of mass $M_{\rm vir}$ that host ACGs or MCGs (see equations \[eq:hmf\_atom\] and \[eq:hmf\_mol\]), in a spherical volume centred at $({\bf r}, z)$ of radius $R$ and overdensity $\delta_{\rm R|_{{\bf r},z}}$, computed using the hybrid conditional mass function suggested by @Barkana2005ApJ...626....1B, and adapted to quasi-linear density fields in @Mesinger2011MNRAS.411..955M;
2. $M_*^{i}$ refers to the stellar mass of ACGs and MCGs following equations (\[eq:m\*\_atom\]) and (\[eq:m\*\_mol\]);
3. $n_{\gamma}^{i}$ corresponds to the number of ionizing photons emitted per stellar baryon. Following , $n_{\gamma}^{\rm atom}=5\times10^3$ is chosen for ACGs. We note that, similarly to the SFR-$L_{\rm 1500}$ conversion factor (see Section \[subsubsec:UVLF\]), $n_{\gamma}$ depends on the IMF and this value is close to a PopII star-dominated galaxy assuming a Salpeter IMF. We instead choose $n_{\gamma}^{\rm mol} = 5\times10^4$ for MCGs since they should preferentially host metal-free, PopIII stars, expected to have a higher ionizing photon emissivity (e.g. ). It is worth noting that the large degeneracy between the efficiency of ionizing photon production and the ionizing escape fraction (see below) means that uncertainties in the former (which we hold fixed in this work) can be subsumed in the inferred constraints on the latter.
4. $f_{\rm esc}^{i}$ is the escape fraction defined as the number ratio of ionizing photons that reach the IGM to those intrinsically emitted. It is determined by the ISM properties, such as the ${{\mathrm{H}\textsc{i}}}$ filling factor, dust, and their distribution with respect to star formation sites. In low-mass halos, the gravitational potential is shallow, facilitating the creation of low column density channels through which ionizing photons can escape. This is expected to result in a negative correlation between $f_{\rm esc}$ and the host halo mass, $M_{\rm vir}$ (e.g. @Ferrara2013MNRAS.431.2826F [@Kimm2014; @Paardekooper2015MNRAS.451.2544P; @Xu2016ApJ...833...84X], but also see @Ma2015MNRAS.453..960M [@Naidu2019arXiv190713130N]). We adopt a power-law relation for the escape fraction to halo mass, allowing both the normalization and scaling to be different between MCGs and ACGs (see e.g. fig. 15 in @Xu2016ApJ...833...84X) $$\label{eq:f_esc}
f_{\rm esc}^{{\rm atom}({\rm mol})} =\min\left[1, {\color{black}f_{\rm esc,{{10}({7})}}^{{\rm atom}({\rm mol})}} \left(\frac{M_{\rm vir}}{10^{{10}({7})}{{{\rm M}_\odot}}}\right)^{\alpha_{\rm esc}^{{\rm atom}({\rm mol})}}\right].$$
To account for inhomogeneous recombinations, we follow @Sobacchi2014MNRAS.440.1662S and calculate the number of recombinations per baryon by $$\label{eq:nrec}
\begin{split}
n_{\rm rec}\left({\bf r},z \right) {=} \int_{z_{\rm ion}}^{z} {\rm d z^\prime}\frac{\rm d t}{\rm d z^\prime} \int_{0}^{18{\rm \pi}^2} {\rm d}\Delta_{\rm sub} \frac{\rm d n}{\rm d \Delta_{\rm sub}}\times \\
\alpha_{\rm B} \bar{n}_{\rm b} f_{\rm H} \Delta_{\rm cell}^{-1}\Delta_{\rm sub}^2 \left(1{-}x_{\rm {\mathrm{H}\textsc{i}}, sub}\right)^2,
\end{split}$$ where $z_{\rm ion}({\bf r})$ is the reionization redshift of the cell; the upper limit of integration, $18{\rm \pi}^2$, corresponds to the overdensities of halos in the spherical collapse model; ${\rm d n}/{\rm d \Delta_{\rm sub}}\left(z^\prime, \Delta_{\rm sub}| \Delta_{\rm cell}\right)$ is the probability distribution function (PDF) of the subgrid (unresolved) overdensities, $\Delta_{\rm sub}$, taken from @Miralda2000ApJ...530....1M and adjusted for the mean overdensity of the cell, $\Delta_{\rm cell}\equiv n_{\rm b}/\bar{n}_{\rm b}$, according to @Sobacchi2014MNRAS.440.1662S; $\alpha_{\rm B}$ is the case-B recombination coefficient evaluated at $T_g = 10^4{\rm K}$; and the fraction of residual neutral hydrogen inside the ionized region, $x_{{\mathrm{H}\textsc{i}}, {\rm sub}}\left(z^{\prime}, \Delta_{\rm cell}, T_{\rm g}, \bar{\Gamma}_{\rm ion}\right)$, is evaluated assuming photoionization equilibrium and accounting for attenuation of the local ionizing background according to @Rahmati2013MNRAS.430.2427R.
Inside each cosmic [${\mathrm{H}\textsc{ii}}$]{} region, we compute the local, average photoionization rate following @Sobacchi2014MNRAS.440.1662S $$\label{eq:gamma}
{\bar{\Gamma}_{\rm ion}\left({\bf r},z\right)} = \left(1+z\right)^2 R\sigma_{\rm H}\frac{\alpha_{\rm UVB}}{\alpha_{\rm UVB}+\beta_{\rm H}}\bar{n}_{\rm b}{\dot{\bar{n}}_{\rm ion}},$$ where $\alpha_{\rm UVB}$ corresponds to the UVB spectral index, $\beta_{\rm H} \approx 2.75$ is the HI photoionization cross-section spectral index, $R$ is the local mean free path – approximated by the largest radius at which equation (\[eq:ionization\]) is satisfied, and $\dot{\bar{n}}_{\rm ion}$ represents the ionizing photon production rate following equation (\[eq:dotn\_ion\]) with $M_*$ being replaced by the SFR (see equation \[eq:sfr\]). The ionizing background inside cosmic HII regions is used to calculate the critical mass below which photoheating quenches star formation (c.f. equation \[eq:mcrit\_ion\]), as well as for computing subgrid recombinations (c.f. equation \[eq:nrec\]).
### LW photons {#subsubsec:LW}
Compared to ionizing photons, LW photons have much longer mean free paths in the high-redshift Universe. Therefore, to calculate the LW radiation field at $z$, we must account for distant galaxies, integrating back along the light-cone to include galaxies at higher redshifts, $z^\prime\ge z$, and redshifting the emitted spectrum, $\nu^\prime = \nu \frac{1+z\prime}{1+z}$.
The large resonant cross-section in the Lyman series $\nu_{\rm n}\equiv \nu_{\rm H}\left(1-n^{-2}\right)$ with $\nu_{\rm H}{=}3.29\times10^{15}{\rm GHz}$ being the Lyman limit frequency and $n\in\left[2, 23\right]$[^12] provides a barrier for LW photons – setting a maximum redshift, $z_{{\rm max}}$, from which they can reach $z$ $$\frac{1 + z_{{\rm max}}\left(n\right)}{1+z} = \frac{1-(n+1)^{-2}}{1-n^{-2}}.$$ Equivalently, there is a highest order of Lyman transition, $n_{\rm max}\left(z\right)$, for a given redshift ($z^\prime$), above which photons will redshift into the $n$th-order Lyman transition and be absorbed in the IGM. [Note that absorptions of the LW radiation due to the presence of ${\mathrm{H}_2}$ in the IGM are not considered in this work [@Haiman2000ApJ...534...11H; @Ricotti2001ApJ...560..580R]]{}.
![\[fig:spectra\]*Top panel:* PopII- and PopIII-dominated stellar spectra [@Barkana2005ApJ...626....1B] used in this work for ACGs and MCGs, respectively. *Bottom panel:* normalized emissivity of the LWB at $z=15$. Photons from higher redshifts ($z^\prime=15.5$ and 16) that contribute to the $z=15$ background are shown in the received frame (i.e. $z=15$) for comparison – for a given $z^\prime$, there is a maximum energy level in the Lyman series above which photons will be absorbed in the IGM before reaching $z$.](./figs/Spectra.pdf){width="\textwidth"}
Similarly to the [[[Lyman-$\alpha$]{}]{}]{} background calculation of direct stellar emission in @Mesinger2011MNRAS.411..955M, the LWB is evaluated with a sum over the Lyman series (see also e.g. @Pritchard2007MNRAS.376.1680P [@Ahn2009ApJ...695.1430A; @Fialkov2013MNRAS.432.2909F; @Munoz2019PhRvD.100f3538M]). After rearranging the integral over redshift and the sum over Lyman series, we obtain the LW radiation intensity, $J_{\rm LW}$ in units of ${\rm{\rm erg\ s^{-1}Hz^{-1}cm^{-2}sr^{-1}}}$, by $$\label{eq:J_LW}
J_{\rm LW} \left({\bf r},z | R, \delta_{\rm R|_{{\bf r},z}} \right) =\frac{\left(1+z\right)^3}{\rm 4{\rm \pi}} \int_{z}^{\infty} {{{\rm d}}}z^\prime\frac{{c {{\rm d}}}t}{{{{\rm d}}}z^\prime} {\epsilon}_{\rm LW}e^{-\tau_{\rm LW}},$$ where we assume that the LW photons are only attenuated at resonance, and the corresponding emissivity becomes $$\label{eq:e_LW}
\begin{split}
{\epsilon}_{\rm LW}\left({\bf r},z^\prime|z,R,\delta_{\rm R|_{{\bf r},z}}\right)& = \sum_{i\in\left\{\substack{{\rm mol,}\\{\rm atom}}\right\}} \int {\rm d}M_{\rm vir} {\phi}^{i} {\rm SFR}^i \times \\
& \sum_{n=2}^{n_{\rm max}(z)} \int_{\max\left(\nu_{\rm n}^\prime, \nu_{\rm LW}\right)}^{\nu_{{\rm n+1}}}\frac{{{{\rm d}}}n_{\gamma/\odot}^{i}}{{{{\rm d}}}\nu^{\prime\prime}} h {{{\rm d}}}\nu^{\prime\prime}.
\end{split}$$ When computing the LW emissivity[^13], we use the PopII- and PopIII-dominated spectral energy distributions (SEDs), ${{{{\rm d}}}n_{\gamma/\odot}^{i}}/{{{{\rm d}}}\nu^{\prime\prime}}$ (number of photons per mass in stars per unit frequency), from @Barkana2005ApJ...626....1B for ACGs and MCGs, respectively. These are shown in the top panel of Fig. \[fig:spectra\]. They follow piece-wise power laws between pairs of $\nu_{\rm n}$ and $\nu_{\rm n+1}$ with normalizations and scaling indices chosen to reproduce results from stellar-population synthesis models [@Leitherer1999ApJS..123....3L; @Bromm2001ApJ...552..464B]. We also present the integral in equation (\[eq:e\_LW\]) with the current and higher redshifts being $z=15$ and $z^\prime=15.5$ and 16 in the lower panel. We see that only a fraction of high-redshift photons between several low-order Lyman transitions can make a contribution to the radiation background of LW at lower redshifts because of resonant scattering – the so-called “picket fence” absorption (e.g. @Haiman1997ApJ...476..458H [@Ahn2009ApJ...695.1430A; @Fialkov2013MNRAS.432.2909F]). We then use equations (\[eq:mcrit,diss\]) and (\[eq:fshiled\]) to calculate the corresponding LW feedback on MCG star formation.
### X-rays and [[[Lyman-$\alpha$]{}]{}]{} photons {#subsubsec:xraysandlya}
We extend @Mesinger2011MNRAS.411..955M and when estimating the X-ray heating and [ionization as well as]{} [[[Lyman-$\alpha$]{}]{}]{} coupling to allow for both galaxy populations. We give a brief review of the relevant calculation and encourage readers to follow these two papers for more details.
We start with an assumption that the X-ray emission from all galaxies (MCG and ACG; $i \in \left\{{\rm mol}, {\rm atom}\right\}$) follows a power law with an energy index of $\alpha_{\rm X}$ and a specific luminosity of $$\label{eq:xspec}
\frac{{\rm d}L_{\rm X/\dot{\odot}}^{i}}{{\rm d}E}\Big(E\Big) {=} L_{\rm X<2keV/\dot{\odot}}^{i} \left(\int_{E_0}^{\rm 2keV}{\rm d}EE^{-\alpha_{\rm X}}\right)^{-1}E^{-\alpha_{\rm X}},$$ where $E_0$ represents the minimum energy that an X-ray photon needs to escape from the host galaxy into the IGM \[for reference, @Das_2017 estimate $E_0 \sim$ 0.5 keV\] while $L_{\rm X<2keV/\dot{\odot}}^{i}$ is the total luminosity between $E_0$ and 2keV.
At these redshifts, the dominant source of soft X-rays (which is relevant for heating/ionizing the IGM) are expected to be High-Mass X-ray Binaries (HMXBs) and potentially also the hot ISM (e.g. @Fragos2013ApJ...764...41F [@Sanderbeck_2018]). Both of these have luminosities that scale with the SFR of the host galaxy (e.g. @Mineo2012 [@Fragos2013ApJ...764...41F; @Pacucci2014]). Thus, the “$/\dot{\odot}$” in equation (\[eq:xspec\]) indicates the quantity is per unit SFR – e.g. $L_{\rm X<2keV/\dot{\odot}}^{\rm mol}$ and $L_{\rm X<2keV/\dot{\odot}}^{\rm atom}$ represent the soft-band X-ray luminosities per SFR for MCGs and ACGs, respectively, which are considered free parameters in our model. Next, we link the X-ray radiation intensity, $J_X$ in units of ${\rm{\rm erg\ s^{-1}keV^{-1}cm^{-2}sr^{-1}}}$, to star formation following[^14] equation (\[eq:J\_LW\]) with the emissivity term (i.e. ${\epsilon}_{\rm LW}$) being replaced by [ $$\label{eq:e_X}
{\epsilon}_{X}\left({\bf r},z^\prime\right) = \sum_{i\in\left\{\substack{{\rm mol,}\\{\rm atom}}\right\}} \int {\rm d}M_{\rm vir} {\phi}^{i} {\rm SFR}^i \frac{{\rm d}L_{\rm X/\dot{\odot}}^{i}}{{\rm d}E}.$$ ]{} Note that the emissivity is evaluated in the rest frame, $E^\prime = E \left(1+z^\prime\right)/\left(1+z\right)$. The ionization (see equation \[eq:x\_e\]) and heating rates per baryon by X-rays (see equation \[eq:T\_g\]) are then computed as follows $$\Lambda_{X}\left({\mathbf r}, z\right) = \int_{E_0}^{\infty} {\rm d}E \frac{4{\rm \pi} J_X}{E} \sum_{j} x^{j} \sigma^{j} f^{j} \left[{(E{-}E_{\rm th}^{j})}\sum_{k} \frac{f_{\rm ion}^{k}}{E_{\rm th}^{k}} + 1 \right]$$ and $$\label{eq:xheating}
\varepsilon_{X}\left({\mathbf r}, z\right) = \int_{E_0}^{\infty} {\rm d}E \frac{4{\rm \pi} J_X}{E} \sum_{j} x^{j} \sigma^{j} f^{j}{(E{-}E_{\rm th}^{j})} f_{\rm heat}$$ where $f^{j}$ is the number fraction of each species, $j$, with $j\in\left[\mathrm{H}{\textsc{I}}, \mathrm{He}{\textsc{I}}, \mathrm{He}{\textsc{II}}\right]$, $\sigma^{j}$ is the ionization cross-section, and $E_{\rm th}^{j}$ is the corresponding energy; $f_{\rm heat}$ and $f_{\rm ion}^{k}$ represent the fraction of the electron energy after ionization, $E{-}E_{\rm th}^{j}$, that contributes to heating or secondary ionization of each species [[@Furlanetto2010MNRAS.404.1869F]]{}; and $x^{j} \equiv 1{-} x_e$ when $j\in\left[\mathrm{H}{\textsc{I}}, \mathrm{He}{\textsc{I}}\right]$ or $x_e$ for $\mathrm{He}{\textsc{II}}$ [represents the secondary ionization fractions (see Section \[subsec:igm\])]{}.
The [[[Lyman-$\alpha$]{}]{}]{} background component coming from direct stellar emission is computed by integrating the emissivity along the light-cone. The evaluation of this background, $J_{\alpha}^{*}$ in units of ${\rm s^{-1}Hz^{-1}cm^{-2}sr^{-1}}$, follows equation (\[eq:J\_LW\]) with the emissivity term (i.e. ${\epsilon}_{\rm LW}$) being replaced by the effective photon number emissivity $${\epsilon}_{\alpha}^{*}\left({\bf r},z^\prime\right) {=} \sum_{i\in\left\{\substack{{\rm mol,}\\{\rm atom}}\right\}} \int {\rm d}M_{\rm vir} {\phi}^{i} {\rm SFR}_*^{i} \sum_{n=2}^{n_{\rm max}\left(z\right)}\frac{{{{\rm d}}}n_{\gamma/\dot{\odot}}^{i}}{{{{\rm d}}}\nu^{\prime\prime}} f_{\rm recycle}\left(n\right),$$ where $f_{\rm recycle}$ is the fraction of absorbed photons at the n-th Lyman resonance level that are re-emitted at [[[Lyman-$\alpha$]{}]{}]{} [@Hirata2006MNRAS.367..259H; @Pritchard2007MNRAS.376.1680P].
On the other hand, the [[[Lyman-$\alpha$]{}]{}]{} background due to X-ray excitation (in units of ${\rm s^{-1}Hz^{-1}cm^{-2}sr^{-1}}$) can be linked to the heating rate (see equation \[eq:xheating\]) following $$$$ where $f_{\alpha}$ is the fraction of the electron energy ($E-E_{\rm th}^{j}$) that contributes to emitting [[[Lyman-$\alpha$]{}]{}]{} photons with a frequency of ${\nu_\alpha}\equiv 2.47\times10^{15}{\rm Hz}$. The total [[[Lyman-$\alpha$]{}]{}]{} background that is used to evaluate the [[[Lyman-$\alpha$]{}]{}]{} coupling coefficient (see the following section) is the sum of both X-ray and stellar contribution $$J_{\alpha,{\rm eff}} \times {\rm s^{-1}Hz^{-1}cm^{-2}sr^{-1}} = \left(J_\alpha^{\rm X} + J_\alpha^*\right) \times S_{\alpha}$$ where $S_\alpha$ is a quantum mechanical correction factor of order unity [@Hirata2006MNRAS.367..259H].
Modelling the 21-cm signal {#sec:xHandTs}
==========================
We compute the inhomogeneous 21-cm brightness temperature according to equation (\[eq:dtb\_define\]), albeit with the subgrid non-linear treatment of redshift space distortions and without assuming the optically thin limit (@Greig2018MNRAS.477.3217G; see also @Datta2012MNRAS.424.1877D [@Datta2014MNRAS.442.1491D; @Mao2012MNRAS.422..926M; @Jensen2013MNRAS.435..460J]). The ionization and density fields were discussed previously. The spin temperature is computed according to $$\label{eq:Tspin}
T_{\rm s}^{-1} = \frac{T_{\rm CMB}^{-1} + \left(x_\alpha + x_{\rm c}\right) T_{\rm g}^{-1}}{1+x_\alpha + x_{\rm c}},$$ with the collisional, $x_{\rm c}$, and [[[Lyman-$\alpha$]{}]{}]{} coupling coefficients [@Wouthuysen1952AJ.....57R..31W], $x_{\alpha}$, being calculated by $$\label{eq:x_c}
{x_{\rm c}}= \left(\frac{T_{\rm CMB}}{0.0628{\rm K}}\right)^{-1} \sum_{i\in\left\{\substack{{\rm e,p,{\mathrm{H}\textsc{i}}}}\right\}} \frac{n_{\rm b} f^\prime_{i} \kappa_{i}}{2.85\times10^{-15}{\rm s}^{-1}}$$ and $$\label{eq:x_a}
x_\alpha = 1.7\times 10^{11} \left(1+z\right)^{-1} J_{\alpha,{\rm eff}},$$ where $f^\prime_{i}$ and $\kappa_{i}$ with $i\in\left[e, p, {\mathrm{H}\textsc{i}}\right]$ represent the number fractions of free electrons, protons and neutral hydrogen and their cross-sections with ${\mathrm{H}\textsc{i}}$ taken from @Zygelman2005ApJ...622.1356Z and @Furlanetto2007MNRAS.374..547F. The IGM only becomes visible in contrast against the CMB if (at least) one of the coupling coefficients in equation (\[eq:Tspin\]) is non-negligible.
Building physical intuition – general trends of the reference model
-------------------------------------------------------------------
We summarize the relevant model parameters in Table \[tab:parameters\] together with the values chosen for a reference model. We present this reference model, including slices through various fields in Fig. \[fig:reference\] and the 21-cm power spectra in Fig. \[fig:reference\_PS\]. Simulations presented in this section share the same initial conditions and are performed within periodic boxes that have a comoving length of 300Mpc and a cell resolution of 1.17Mpc (300Mpc/256). Unless otherwise specified, values are consistent with those in , for the parameters the two works have in common. We will demonstrate below how current observations can constrain a subset of these parameters in Section \[sec:constraint\].
Looking at the light-cones in Fig. \[fig:reference\], we see immediately that the structure of the 21-cm signal ([*rightmost panel*]{}) is governed by various radiation fields, with specific fields dominating at different epochs. The early 21-cm structures ($z\sim20-30$) are imprinted by the [[[Lyman-$\alpha$]{}]{}]{} background ([*second panel*]{}), which is fairly uniform. However, regions around the nascent galaxies, which are hosted by large-scale matter overdensities ([*first panel*]{}), see enhanced fluxes by factors of up to a few. These regions also have a higher LW flux ([*third panel*]{}), with intensities reaching values large enough for negative feedback on MCGs ([*seventh panel*]{}) during the [[[Lyman-$\alpha$]{}]{}]{} coupling epoch. By $z\sim 20$, the LW feedback is significant through the IGM – the median $J_{\rm LW,eff}^{\rm 21}$ exceeds $10^{-2}$ and the critical mass, $M_{\rm crit}^{\rm diss}$, becomes more than three times the molecular-cooling threshold (see equation \[eq:mcrit,diss\]), leading to a factor of ${\sim}2$ suppression on the number density of low-mass MCGs.
Shortly thereafter, X-rays from the first galaxies begin to dominate the thermal evolution of the IGM ([*fourth panel*]{}). By $z\sim18$, $\varepsilon_{X}$ exceeds ${\rm}100 k_{\rm B}{\rm K/Gyr}$ in most parts of the simulation box, overcoming adiabatic cooling of the gas (see equation \[eq:T\_g\]). With $\dot{T}_{\rm g}$ becoming positive, $\delta T_{\rm b}$ reaches its minimum and we see an absorption feature in the 21-cm light-cone, which fades away at $z{\sim}12$ (see also Fig. \[fig:reference\_PS\]). After that, gas becomes hotter than the CMB and the signal is in emission.
{width="105.00000%"}
\[tab:parameters\]
Parameter Sec. Eq. Description Ref. MCMC
-------------------------------------------------- ------------------------------------------------------ ------------------------------------------ --------------------------------------------------------------- ----------------- -----------------
$\log_{10}f_{*,10}^{\rm atom}$ \[subsec:source model\] \[eq:m\*\_atom\] -1.25 -1.25
$\log_{10}f_{*,7}^{\rm mol}$ \[subsec:source model\_mcg\] \[eq:m\*\_mol\] -1.75 \[-3, 0\]
$\alpha_*$ \[subsec:source model\],\[subsec:source model\_mcg\] \[eq:m\*\_atom\],\[eq:m\*\_mol\] Stellar-to-halo mass power-law index 0.5 0.5
$t_*$ \[subsec:source model\] \[eq:sfr\] Star formation time-scale in units of $H^{-1}(z)$ 0.5 0.5
$M_{\rm crit}^{\rm SN}$ \[subsec:source model\],\[subsec:source model\_mcg\] \[eq:m\_crit\_atom\],\[eq:m\_crit\_mol\] Critical halo mass for supernova feedback - -
$f_{{\mathrm{H}_2}}^{\rm shield}$ \[subsec:source model\] \[eq:fshiled\] Self-shielding factor of ${\mathrm{H}_2}$ for LW dissociation 0.0 0.0
$n_{\gamma}^{\rm atom}$ \[subsubsec:UVionizing\] \[eq:dotn\_ion\] $5{\times}10^3$ $5{\times}10^3$
$n_{\gamma}^{\rm mol}$ \[subsubsec:UVionizing\] \[eq:dotn\_ion\] $5{\times}10^4$ $5{\times}10^4$
$\log_{10}f_{\rm esc,10}^{\rm atom}$ -1.22 \[-3, 0\]
$\log_{10}f_{\rm esc,7}^{\rm mol}$ -2.22 \[-3, 0\]
$\alpha_{\rm esc}^{\rm atom}$ 0 0
$\alpha_{\rm esc}^{\rm mol}$ 0 0
$\alpha_{\rm UVB}$ \[subsubsec:UVionizing\] \[eq:gamma\] Spectral index of the ionizing background 5 5
$E_0/{\rm eV}$ \[subsubsec:xraysandlya\] \[eq:xspec\] Minimum X-ray energy escaping the galaxies into the IGM 500 \[100, 1500\]
$\alpha_{\rm X}$ \[subsubsec:xraysandlya\] \[eq:xspec\] Spectral index of X-ray sources 1.0 1.0
$\log_{10}L_{\rm X<2keV/\dot{\odot}}^{\rm atom}$ 40.5
$\log_{10}L_{\rm X<2keV/\dot{\odot}}^{\rm mol}$ 40.5
Although it is a free parameter, for this work we maximize the importance of small galaxies by assuming supernova feedback is subdominant compared to inefficient cooling and photoheating in determining the faint turnover, i.e. $M_{\rm crit}^{\rm SN} {\le} \max\left[M_{\rm crit}^{\rm cool}, M_{\rm crit}^{\rm ion} \right]$;
In this work, we set $\alpha_{\rm X}=1$, motivated by observations of (population-averaged) spectra of high-mass X-ray binaries in local galaxies (e.g. @Mineo2012 [@Fragos2013ApJ...764...41F; @Pacucci2014]).
We assume that ACGs and MCGs possess similar X-ray luminosities during MCMC, i.e. $L_{\rm X<2keV/\dot{\odot}}^{\rm atom} {=}L_{\rm X<2keV/\dot{\odot}}^{\rm mol}{\equiv} L_{\rm X<2keV/\dot{\odot}}$.
![\[fig:reference\_PS\]Dimensionless power spectra of 21-cm ($\delta T_{\rm b}^2 \Delta_{21}^2$) for the reference model as a function of wave number and redshift (or observed frequency) on the top panel and for two typical scales in the observable range, $k=0.1$ and 0.5 Mpc$^{-1}$ in the bottom panels. We also present the result from a model without minihaloes (*noMCGs*) for comparison. [The kinks at high redshift are numerical due to very rare sources and can be fixed by running the simulation with a higher cadence.]{}](./figs/reference_PS.pdf){width="\textwidth"}
The EoR ([*second to last panel*]{}), as well as the associated photoheating feedback ([*sixth panel*]{}) and recombinations fields ([*eight panel*]{}), is driven by short mean free path ionizing photons. Therefore, their evolution is not sensitive to a diffuse, increasing background (as is the case for X-rays and LW photons) but proceeds in a “percolating fashion” (e.g. @furlanetto2016reionization) with medium-to-large scales being closely tied to the underlying density field (e.g. @Zahn2011MNRAS.414..727Z [@battaglia2013reionization; @mcquinn2018observable]). The EoR history of this model is chosen to agree with current observational constraints, finishing by $z\sim6$ (e.g. @McGreer2015MNRAS.447..499M), having a mid-point of around $z\sim7-8$ (), and a small tail extending to higher redshifts corresponding to small HII regions around the nascent first galaxies (e.g. @Mitra2015 [@Greig2016]).
The 21-cm PS of this model, presented in Fig. \[fig:reference\_PS\], shows the characteristic triple peak structure of the large-scale power evolution, driven by fluctuations in the [[[Lyman-$\alpha$]{}]{}]{} coupling, X-ray heating, and reionization fields. On smaller scales, the first two peaks merge due to a larger negative contribution of the cross-terms of the component fields (see discussions in ). For reference, we also show in grey the same astrophysical model but with no minihaloes. We see in general that the astrophysical epochs in this model are delayed, especially the earliest ones, and there is more power on large scales. We will return to this below.
{width="98.00000%"}
Parameter dependence
--------------------
Here we vary some of the astrophysical parameters characterizing MCGs, illustrating the resulting impact on cosmic fields. In Fig. \[fig:showcases\_signal\], we show the redshift evolution of the median values of various fields in the top rows, together with the 21-cm PS in the bottom rows. Along the columns, we vary the normalization of the stellar-to-halo mass relation (i.e. ratio at $M_{\rm vir}{=}10^7{{\rm M}_\odot}$; $f_{*,7}^{\rm mol}$), the ionizing escape fraction ($f_{\rm esc}^{\rm mol}$), the soft-band X-ray luminosity per SFR ($L_{\rm X<2keV/\dot{\odot}}^{\rm mol}$) and self-shielding factor of [${\mathrm{H}_2}$]{} ($f_{{\mathrm{H}_2}}^{\rm shield}$). We only vary one parameter at a time, fixing the remaining parameters to those of the reference model. The general trends are the following:
1. varying $f_{*,7}^{\rm mol}$ leads to different production rates of photons in all wavelengths of interest. An increasing stellar mass in MCGs shifts all astrophysical epochs to earlier times. Understandably, the earlier epochs of [[[Lyman-$\alpha$]{}]{}]{} pumping and X-ray heating are most [affected]{}, as MCGs have a larger relative contribution at higher redshifts. However increasing the efficiency to 10 times our fiducial one (or analogously increasing the ionizing escape fraction) shifts reionization to earlier times. In this case, MCGs can contribute more ionizing photons than ACGs well into the EoR ($z{\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}6$; comparing grey and red dotted curves in the fourth row), and the midpoint of the EoR shifts to $z\sim9$;
2. varying $f_{\rm esc}^{\rm mol}$ around our fiducial model only has a minor impact on the timing of the EoR. Because the overall emission of ionizing photons depends on the product of the escape fraction and the SFR, increasing the escape fraction by a factor of 10 results in a shift of the EoR to earlier times, as seen in the previous column. Note that the stellar mass and the escape fraction do not have a completely degenerate impact on the EoR timing as the star formation time-scale evolves with redshift, and radiative feedback can regulate star formation;
3. varying $L_{\rm X<2keV/\dot{\odot}}^{\rm mol}$ impacts almost exclusively the Epoch of Heating (EoH), as X-rays are inefficient at reionizing the Universe. Increasing the X-ray luminosity shifts the EoH to earlier times. As a result, the EoR and [[[Lyman-$\alpha$]{}]{}]{} pumping epochs increasingly overlap, which dramatically reduces the maximum contrast between the gas and CMB temperatures, and the corresponding minima in the global signal. Moreover, the resulting 21-cm power on small and medium scales is also reduced due to the increased negative contribution of the cross-correlation between the temperature and [[[Lyman-$\alpha$]{}]{}]{} coupling fields;[^15]
4. varying $f_{{\mathrm{H}_2}}^{\rm shield}$ changes how sensitive MCGs respond to negative feedback from the LWB (e.g. @Schauer2015MNRAS.454.2441S). As our reference model assumes no self-shielding, $f_{{\mathrm{H}_2}}^{\rm shield} = 0$, here we only increase this parameter to 0.5 and 0.9. A larger $f_{{\mathrm{H}_2}}^{\rm shield}$ decreases the effective LW radiation penetrating the ISM of the galaxies ([*second row*]{}), decreasing the impact of LW feedback. With a correspondingly higher contribution of MCGs when self-shielding is increased, astrophysical epochs are shifted earlier; however, the effect is extremely small, indicating that negative LW feedback in our model is not very important.
We also present a model (*noMCGs*) where contribution from minihaloes is excluded. Comparing with *noMCGs*, we see that, depending on the values used for the aforementioned parameters, MCGs can be the dominant source of radiation in the early universe, governing the global evolution of 21-cm signal, and altering its morphology. Therefore, 21-cm observables can be a powerful tool to probe the properties of first galaxies. In the next section, we will quantify how current high-redshift observations can jointly constrain the properties of MCGs and ACGs within a Bayesian analysis framework.
Inferring the astrophysics of minihaloes {#sec:constraint}
========================================
The previous section illustrates how varying galaxy properties can impact the 21-cm signal. However, our model has many free parameters which characterize both ACGs and MCGs. Can these parameters be constrained by current and upcoming observations? In a companion paper, we will quantify the parameter constraints and degeneracies available with future 21-cm interferometric observations. Here we focus on current observations of the EoR and CD, seeing if these can already be used to inform our model and infer the astrophysics of minihaloes. These observations [^16] include
1. the galaxy UV LF at $z{=}6{-}10$ from @Bouwens2015a [@Bouwens2016] and @Oesch2018ApJ...855..105O;
2. the upper limit on the neutral hydrogen fraction at $z\sim5.9$, $x_{{\mathrm{H}\textsc{i}}}<0.06{+}0.05(1\sigma)$, measured using the dark fraction of QSO spectra [@McGreer2015MNRAS.447..499M];
3. the Thomson scattering optical depth of CMB photons reported by , $\tau_e=0.058{\pm}0.012(1\sigma)$; and
4. the timing[^17] of the 21-cm global absorption feature reported by EDGES, which has a minimum at a frequency of 78$\pm1(1\sigma)$MHz [@Bowman2018].
To quantify parameter constraints implied by these observations, we use the MCMC module, 21CMMC [@Greig2015MNRAS.449.4246G; @Greig2017MNRAS.472.2651G; @Greig2018MNRAS.477.3217G], which forward-models 21-cm light-cones using the EMCEE sampler [@Goodman2010CAMCS...5...65G; @Foreman2013PASP..125..306F]. Unfortunately, varying all of the model parameters listed in Table 1 is computationally challenging, and would require high-performance computing resources. We defer a larger parameter space exploration to future work.
For this introductory work, we limit our parameter space. Specifically, we fix the stellar-to-halo mass relation of ACGs to the recovered median values in , as current LFs already provide reasonable constraints on these parameters. These include: (i)) the stellar-to-halo mass ratio at $M_{\rm}=10^{10}{\rm M}_\odot$ for ACGs, $\log_{10} f_{*,10}^{\rm atom}{=}{-}1.25$; (ii) the power-law index of the ACG stellar-to-halo mass relation, $\alpha_{*}{=}0.5$; and (iii) the star formation time-scale, $t_* {=} 0.5$. Fixing these values ensures that the modelled galaxy UV LFs are in agreement with high-redshift observations at the bright end (see also Fig. \[fig:example\]).
{width="\textwidth"} {width="52.00000%"}
Additionally, we consider constant escape fractions for each population (i.e. ${\alpha_{\rm esc}^{{\rm atom}({\rm mol})}}=0$), ignore self-shielding of ${\mathrm{H}_2}$ (i.e. $f_{{\mathrm{H}_2}}^{\rm shield}=0$), and further assume ACGs and MCGs possess a similar X-ray luminosity per SFR[^18] (i.e. $L_{\rm X<2keV/\dot{\odot}}^{\rm atom} = L_{\rm X<2keV/\dot{\odot}}^{\rm mol}\equiv L_{\rm X<2keV/\dot{\odot}}$).
We thus explore the following parameters with flat priors in linear or logarithmic scale:
1. the normalization of the MCG stellar-to-halo mass ratio, $\log_{10} f_{*,7}^{\rm mol}\in\left[-3,0\right]$;
2. the escape fraction of ionizing photons for ACGs, $\log_{10} f_{\rm esc}^{\rm atom}\in\left[-3,0\right]$;
3. the escape fraction of ionizing photons for MCGs, $\log_{10} f_{\rm esc}^{\rm mol}\in\left[-3,0\right]$;
4. the minimum energy for X-rays to reach the IGM, $E_0\in\left[100,1500\right]{\rm eV}$; and
5. the soft-band X-ray luminosity per SFR, $\log_{10}\left[L_{\rm X<2keV/\dot{\odot}}/{\rm erg\ s^{-1}{{\rm M}_\odot}^{-1} yr}\right]\in\left[38, 44\right]$.
For the sake of computing efficiency, we have chosen a slightly smaller box with a comoving length of 250Mpc and a cell resolution of 1.95Mpc (250Mpc/128) when performing the MCMC.
Fig. \[fig:MCMC\] shows the marginalized posterior distributions together with the corresponding marginalized \[14, 86\] percentiles of the average EoR and 21-cm redshift evolutions. We also identify the timing when $\delta T_{\rm b}$ reaches its minimum as well as the full width at half-maximum (FWHM) of $\delta T_{\rm b}-$frequency, and show their PDFs in the right-hand subpanels. The red curves and shaded areas correspond to constraints using all of the above observations, [*except*]{} EDGES (*noEDGES*). Even without EDGES, we see a strong degeneracy between the allowed SFR and the ionizing escape fractions in MCGs – high values of either $f_{*,7}^{\rm mol}$ or $f_{\rm esc}^{\rm mol}$ are excluded, as they would reionize the Universe too early to be consistent with [*Planck*]{} observations (see also, e.g. @Visbal2015MNRAS.453.4456V). On the other hand, an escape fraction of ionizing photons in ACGs of $f_{\rm esc}^{\rm atom}{\sim}3 {-}15$% is required to ensure a sufficiently ionized universe at $z{\sim}6$. As expected, without any information of 21cm, the X-ray properties cannot be constrained by any of these measurements.
We then add in the constraints from EDGES timing (*EDGES\_FREQ*). The corresponding marginalized PDFs are shown with blue curves and shaded regions. Most constraints tighten only slightly when including the timing of the EDGES signal. In particular, the aforementioned degeneracy between the stellar-halo mass ratio and ionizing escape fraction is mostly unchanged. We fit this degeneracy in both cases to obtain the following relations (median with \[14, 86\] percentiles): $$\label{eq:f*fesc}
\log_{10}\left({f_{*,7}^{\rm mol}f_{\rm esc}^{\rm mol}}\right) =
\begin{cases}
-4.23_{-0.74}^{+0.58} ~ ({\it noEDGES})\\
-4.07_{-0.73}^{+0.50} ~ ({\it EDGES\_FREQ}).
\end{cases}$$ However, the most striking change is in the $f_{*,7}^{\rm mol}$ – $L_{\rm X<2keV/\dot{\odot}}$ plane. We see that a strong degeneracy emerges between these two parameters $$\label{eq:f*lx}
\log_{10}\left(f_{*,7}^{\rm mol}\frac{L_{\rm X<2keV/\dot{\odot}}}{ {\rm erg\ s^{-1}\ M_{\odot}^{-1}\ yr}} \right) = 38.92_{-0.15}^{+0.24} ~ ({\it EDGES\_FREQ}).$$ If the EDGES signal at $78\pm1$MHz is cosmological, soft UV and X-ray photons from galaxies are needed at $z\sim$17–20 to source the [[[Lyman-$\alpha$]{}]{}]{} coupling and subsequent X-ray heating transitions, regardless of the physical explanation of the depth of the signal [@Madau2018; @Mirocha2019MNRAS.483.1980M]. However, the stellar-to-halo mass relation implied by observations of high-redshift UV LFs is insufficient to heat the IGM at such high redshifts [@Mirocha2016; @Mirocha2019MNRAS.483.1980M; @Park2020MNRAS.491.3891P]. This is in contrast with early estimates of X-ray heating, based on assumptions of a constant stellar-to-halo mass ratio (e.g. @Mesinger2016 [@Fialkov2018PhRvL.121a1101F]). As a result, the cosmological explanation of the EDGES signal requires MCGs to set the timing of the signal.[^19]
We quantify this claim further in Fig. \[fig:MCMC\_photonbudget\], which shows the relative contribution of MCGs and ACGs in the LWB, [[[Lyman-$\alpha$]{}]{}]{} background, cumulative ionizing photon number, and X-ray heating rates, corresponding to the *EDGES\_FREQ* posterior[^20]. We see that MCGs dominate the LW, [[[Lyman-$\alpha$]{}]{}]{}, UV ionizing and X-ray radiation at $z{\gtrsim}15$, ${\gtrsim}15$, ${\gtrsim}11$ and ${\gtrsim}13$, respectively, showing that they are the dominant population during the cosmological interpretation of the EDGES signal.
{width="65.00000%"}
{width="60.00000%"}
Nevertheless, we note that the MCGs required for explaining EDGES timing are still “unusual”. The degeneracies quantified in equations (\[eq:f\*fesc\]) and (\[eq:f\*lx\]) mean that MCGs must have been inefficient at producing ionizing photons but efficient in emitting X-ray photons. For example, if the ionizing escape fraction of MCGs was above 10 percent (e.g. @Xu2016ApJ...833...84X), then in order to match both [*Planck*]{} and EDGES observations, the star formation (X-ray production) efficiencies of MCGs must have been a factor of ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}$ 10 smaller (${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}$100-1000 greater) than those of ACGs. In Fig. \[fig:MCMC\], we indicate the expected range for $L_{\rm X<2keV/\dot{\odot}}$ if HMXBs were similar to those in local galaxies [@Mineo2012] extrapolated to low-metallicity environments [@Fragos2013ApJ...764...41F]. We see qualitatively that most MCGs require higher X-ray efficiencies than are theoretically expected even extrapolating to metal-free environments.
Finally, in Fig. \[fig:MCMC\_LF\], we show the marginalized UV LFs of ACGs[^21] and MCGs corresponding to the [*EDGES\_FREQ*]{} posterior ([*blue shaded regions*]{}; the [*noEDGES*]{} LFs are comparable due to the similar distributions of $f_{*,7}^{\rm mol}$ and $f_{\rm esc}$ between these two results). Comparing between LFs of MCGs and ACGs, we recover the result from Fig. \[fig:example\] and find that MCGs likely only dominate at very high redshifts and faint magnitudes.
Conclusions {#sec:conclusion}
===========
In this work, we include the first, molecularly cooled galaxies that are expected to start the CD in the public [[<span style="font-variant:small-caps;">FAST</span>]{}]{} simulation code. We consider atomic- (ACGs) and molecular-cooled galaxies (MCGs) as two different populations that source the underlying radiation backgrounds and drive the structure of 21-cm brightness temperature. We allow the stellar mass and SEDs of the two galaxy populations to scale differently with the host halo mass. We track inhomogeneous recombinations and self-consistently follow the relevant radiative feedback mechanisms, including inhomogeneous LW feedback and photoheating feedback on star formation.
We demonstrate how 21-cm observables vary with MCG properties. These include their star formation efficiencies, UV ionizing escape fractions, X-ray luminosities and ${{\mathrm{H}_2}}$ self-shielding factors against LWB. We then use the Bayesian analysis tool, 21CMMC, to quantify what current observations imply for the MCG population. We use constraints from: (i) current galaxy luminosity functions at $z{\sim}6{-}10$ [@Bouwens2015a; @Bouwens2016; @Oesch2018ApJ...855..105O]; (ii) the dark fraction upper limit on the neutral hydrogen fraction at $z{\sim}5.9$ [@McGreer2015MNRAS.447..499M]; and (iii) the Thomson scattering optical depth of the CMB . We find that the optical depth already rules out models with a high stellar-to-halo mass ratio and high escape fractions in MCGs [@Visbal2015MNRAS.453.4456V].
We also consider the timing of the first claimed detection of the sky-average 21-cm brightness temperature, from EDGES, as an ancillary data set. We find that MCGs are required to produce a global absorption signal around 78MHz. Moreover, the resulting strong degeneracy between the SFRs and X-ray luminosities of MCGs implies that they would have unexpected properties. For example, if the ionizing escape fraction of MCGs was above 10 per cent (e.g. @Xu2016ApJ...833...84X), then in order to match both [*Planck*]{} and EDGES observations, the star formation (X-ray production) efficiencies of MCGs must have been a factor of ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}$ 10 smaller (${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}$100-1000 greater) than those of ACGs. We conclude that the high-redshift 21-cm signal can be a powerful probe of the properties of the first galaxies, which are too faint to be studied using direct observations even with [*JWST*]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
[The authors thank Zoltan Haiman and the anonymous referee for their comprehensive review and positive comments.]{} This work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (AIDA – \#638809). Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project \#CE170100013 as well as the NSF grant AST-1813694.
\[lastpage\]
[^1]: E-mail: Yuxiang.L.Qin@gmail.com
[^2]: <http://www.lofar.org/>
[^3]: <http://www.mwatelescope.org/>
[^4]: <http://reionization.org/>
[^5]: <https://www.skatelescope.org/>
[^6]: <https://github.com/BradGreig/21CMMC>
[^7]: <https://github.com/21cmfast/21cmFAST>
[^8]: [The duty cycle is defined as the fraction of halos that harbour star-forming galaxies for a given halo mass. It is used to describe the stochasticity of star formation and can be considered as an occupation fraction (e.g. @Lippai2009ApJ...701..360L [@Miller_2015]).]{}
[^9]: Since the column density ratio between ${\mathrm{H}\textsc{i}}$ and ${\mathrm{H}_2}$ ($N_{{\mathrm{H}\textsc{i}}}/N_{{\mathrm{H}_2}}$) at high redshift is poorly understood (e.g. @Cen2003ApJ...591...12C), and self-shielding also depends on the temperature and velocity structure of the ISM (e.g. @Wolcott-Green2011MNRAS.418..838W), here we allow $f_{\rm H_2}^{\rm shield}$ to be a free parameter instead of relating it to the typical ${\mathrm{H}\textsc{i}}$ column density.
[^10]: As described below, we also account for partial ionization by X-rays, which blurs this distinction for extreme models (e.g. @RO04 [@Mesinger2013MNRAS.431..621M]).
[^11]: The cumulative photon number density, $\bar{n}_{\rm ion}$, is computed via trapezoidal integration over redshift snapshots in each region. Our approximate treatment of photoheating feedback has a somewhat too rapid evolution at the final stages of reionization (e.g. @Noh2014MNRAS.444..503N [@Katz2019arXiv190511414K]). To compensate for this, we compute the ionizing photon number assuming the same the critical mass threshold (i.e. $M_{\rm crit}^{\rm atom}$ and [$M_{\rm crit}^{\rm mol}$]{}) between two consecutive snapshots, which also ensures $\bar{n}_{\rm ion}$ to increase monotonically with time.
[^12]: Following @Barkana2005ApJ...626....1B, higher order (${>}23$) Lyman transitions are ignored as their contribution is negligible.
[^13]: When estimating the radiation background of LW (as well as X-ray and [[[Lyman-$\alpha$]{}]{}]{}), we assume that $M_{\rm crit}^{\rm ion}$ is less than $M_{\rm crit}^{\rm diss}$ for the sake of computational efficiency. This is a valid assumption for the very high redshifts at which the MCG contribution is non-negligible, since photoheating feedback is only dominant after the bulk of reionization (see Fig. 1 and associated discussion).
[^14]: For the sake of computing efficiency, we follow @Mesinger2011MNRAS.411..955M and approximate $e^{-\tau_{\rm X}}{=}0$ when ${\tau_{\rm X}}{\ge}1$ and 1 otherwise. In practice, this approximation makes virtually no impact on the 21-cm power spectrum evolution (e.g. @Das_2017).
[^15]: In other words, during [[[Lyman-$\alpha$]{}]{}]{} pumping, the regions close to galaxies have the strongest coupling, with their spin temperatures approaching the gas temperature ($T_s \sim T_g \ll T_{\rm cmb}$) while most of the IGM has a spin temperature close to that of the CMB ($T_s \sim T_{\rm cmb} \gg T_g$). Thus regions close to galaxies appear as cold spots in the 21-cm signal during this early stage when the IGM is still colder than the CMB. However, if X-ray heating is more efficient, the gas surrounding the first galaxies can heat up before coupling is completed. In such a case of strong overlap of the EoH and epoch of [[[Lyman-$\alpha$]{}]{}]{} coupling, regions close to galaxies can be heated and coupled ($T_s = T_g \sim T_{\rm cmb}$), while those regions distant from galaxies are still cold but not coupled ($T_s \sim T_{\rm cmb} \gg T_g$). In this case most of the IGM can have spin temperatures that are closer to the CMB temperature, reducing the mean 21-cm signal and spatial fluctuations.
[^16]: We assume the corresponding uncertainties to be Gaussian or one-sided Gaussian (for upper limits).
[^17]: If the EDGES signal is indeed cosmological, its amplitude could only be explained with non-standard models (e.g. @Ewall-Wice2018ApJ...868...63E [@Fialkov2018PhRvL.121a1101F; @Munoz2018Natur.557..684M; @Mebane2019arXiv191010171M]). We do not go into the physical sources of the unexpectedly deep absorption signal in this work. Nevertheless, current explanations still rely on X-rays and soft UV radiation from galaxies to govern its [*timing*]{}.
[^18]: Although the X-ray luminosity of HMXBs scales with decreasing metallicity (), theoretically this trend is expected to saturate for metallicities below roughly 10 per cent solar (e.g. @Fragos2013ApJ...764...41F). Thus, assuming similar X-ray luminosity to SFRs for ACGs and MCGs could be reasonable if the level of metal enrichment in early ACGs is fairly modest. In any case, our results can be treated as a lower limit on the contribution of MCGs to the X-ray background.
[^19]: One could get around this claim if the ACGs were allowed to have higher values for the X-ray luminosity-to-SFR relation, i.e. $L_{\rm X<2keV/\dot{\odot}}^{\rm atom} > L_{\rm X<2keV/\dot{\odot}}^{\rm mol}$. However, this is contrary to the expected trend, since $L_{\rm X<2keV/\dot{\odot}}$ for HMXBs should [*increase*]{} with [*decreasing*]{} metallicity (e.g. ).
[^20]: The contribution of ACGs to $J^{\rm 21}_{\rm LW,eff}$ and $J_{\alpha,{\rm eff}}$ does not have a spread for our posterior, since we are fixing $f_{*,10}^{\rm atom}$ and $\alpha_*$ (motivated by the comparably tight, ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}0.3$ dex constraints on these parameters from current LF observations; ), and since we are assuming that SN feedback is subdominant in setting the turnover mass (which maximizes the abundances of ACGs). Varying the X-ray luminosity and escape fraction, however, does result in a spread in the X-ray heating and cumulative number of ionizing photos per baryon for ACGs.
[^21]: We see that, by fixing $f_{*,10}^{\rm atom}$, ACG UV LFs only show large uncertainties at low redshifts and faint galaxies, when photoheating feedback from reionization becomes significant. As stated previously, this is due to our assumption of $M_{\rm crit}^{\rm SN} {\le} \max\left[M_{\rm crit}^{\rm cool}, M_{\rm crit}^{\rm ion} \right]$, so as to maximize the star formation in faint galaxies, which are the focus of this work. Thus, we caution that the scatter in the ACG LFs is underestimated.
|
---
abstract: 'We study the average distortion introduced by scalar, vector, and entropy coded quantization of compressive sensing (CS) measurements. The asymptotic behavior of the underlying quantization schemes is either quantified exactly or characterized via bounds. We adapt two benchmark CS reconstruction algorithms to accommodate quantization errors, and empirically demonstrate that these methods significantly reduce the reconstruction distortion when compared to standard CS techniques.'
author:
- |
Wei Dai, Hoa Vinh Pham, and Olgica Milenkovic\
Department of Electrical and Computer Engineering\
University of Illinois at Urbana-Champaign
bibliography:
- 'Quantization.bib'
title: Quantized Compressive Sensing
---
\[sec:Introduction\]Introduction
================================
Compressive sensing (CS) is a linear sampling method that converts unknown input signals, embedded in a high dimensional space, into signals that lie in a space of significantly smaller dimension. In general, it is not possible to uniquely recover an unknown signal using measurements of reduced-dimensionality. Nevertheless, if the input signal is sufficiently sparse, exact reconstruction is possible. In this context, assume that the unknown signal $\mathbf{x}\in\mathbb{R}^{N}$ is $K$-sparse, i.e., that there are at most $K$ nonzero entries in $\mathbf{x}$. A naive reconstruction method is to search among all possible signals and find the sparsest one which is consistent with the linear measurements. This method requires only $m=2K$ random linear measurements, but finding the sparsest signal representation is an NP-hard problem. On the other hand, Donoho and Candès et. al. demonstrated in [@Donoho_IT2006_CompressedSensing; @Candes_Tao_IT2005_decoding_linear_programming; @Candes_Tao_FOCS05_Error_Correction_Linear_Programming; @Candes_Tao_IT2006_Near_Optimal_Signal_Recovery] that sparse signal reconstruction is a polynomial time problem if more measurements are taken. This is achieved by casting the reconstruction problem as a linear programming problem and solving it using the *basis pursuit (BP)* method. More recently, the authors proposed the *subspace pursuit (SP)* algorithm in [@Dai_2008_Subspace_Pursuit] (see also the independent work [@Tropp2008_CoSamp] for a closely related approach). The computational complexity of the SP algorithm is linear in the signal dimension, and the required number of linear measurements is of the same order as that for the BP method.
For most practical applications, it is reasonable to assume that the measurements are quantized and therefore do not have infinite precision. When the quantization error is bounded and known in advance, upper bounds on the reconstruction distortion were derived for the BP method in [@Candes_Tao_ApplMath2006_Stable_Signal_Recovery] and the SP algorithm in [@Dai_2008_Subspace_Pursuit; @Tropp2008_CoSamp], respectively. For bounded compressible signals, which have transform coefficients with magnitudes that decay according to a power law, an upper bound on the reconstruction distortion introduced by a uniform quantizer was derived in [@Candes2006_DCC_encoding_lp_ball]. The same quantizer was studied in [@Baraniuk2008_quantization_sparse_representations] for exactly $K$-sparse signals and it was shown that a large fraction of quantization regions is not used [@Baraniuk2008_quantization_sparse_representations]. All of the above approaches focus on the worst case analysis, or simple one-bit quantization [@Braniuk2008_ciss_1bitquantization]. An exception includes the overview paper [@Goyal2008_SPM_quantized_CS], which focuses on the average performance of uniform quantizers, assuming that the support set of the sparse signal is available at the quantizer.
As opposed to the worst case analysis, we consider the average distortion introduced by quantization. We study the asymptotic distortion rate functions for scalar quantization, entropy coded scalar quantization, and vector quantization of the measurement vectors. Exact asymptotic distortion rate functions are derived for scalar quantization when both the measurement matrix and the sparse signals obey a certain probabilistic model. Lower and upper bounds on the asymptotic distortion rate functions are also derived for other quantization scenarios, and the problem of compressive sensing matrix quantization is briefly discussed as well. In addition, two benchmark CS reconstruction algorithms are adapted to accommodate quantization errors. Simulations show that the new algorithms offer significant performance improvement over classical CS reconstruction techniques that do not take quantization errors into consideration.
This paper is organized as follows. Section \[sec:Preliminaries\] contains a brief overview of CS theory, the BP and SP reconstruction algorithms, and various quantization techniques. In Section \[sec:Distortion-Analysis\], we analyze the CS distortion rate function and examine the influence of quantization errors on the BP and SP reconstruction algorithms. In Section \[sec:Modified-Algorithm\], we describe two modifications of the aforementioned algorithms, suitable for quantized data, that offer significant performance improvements when compared to standard BP and SP techniques. Simulation results are presented in Section \[sec:Empirical-Results\].
\[sec:Preliminaries\]Preliminaries
==================================
\[sub:Compressive-Sensing\]Compressive Sensing (CS)
---------------------------------------------------
In CS, one encodes a signal $\mathbf{x}$ of dimension $N$ by computing a measurement vector $\mathbf{y}$ of dimension of $m\ll N$ via linear projections, i.e., $$\mathbf{y}=\mathbf{\Phi}\mathbf{x},$$ where $\mathbf{\Phi}\in\mathbb{R}^{m\times N}$ is referred to as the *measurement matrix*. In this paper, we assume that $\mathbf{x}\in\mathbb{R}^{N}$ is exactly $K$-sparse, i.e., that there are exactly $K$ entries of $\mathbf{x}$ that are nonzero. The reconstruction problem is to recover $\mathbf{x}$ given $\mathbf{y}$ and $\mathbf{\Phi}$.
The BP method is a technique that casts the reconstruction problem as a $l_{1}$-regularized optimization problem, i.e., $$\min\;\left\Vert \mathbf{x}\right\Vert _{1}\;\mathrm{subject\; to}\;\mathbf{y}=\mathbf{\Phi}\mathbf{x},\label{eq:BP-standard}$$ where $\left\Vert \mathbf{x}\right\Vert _{1}=\sum_{i=1}^{N}\left|x_{i}\right|$ denotes the $l_{1}$-norm of the vector $\mathbf{x}$. It is a convex optimization problem and can be solved efficiently by linear programming techniques. The reconstruction complexity equals $O\left(m^{2}N^{3/2}\right)$ if the convex optimization problem is solved using interior point methods [@Nesterov_book1994_Interior_point_Convex_Programming].
The computational complexity of CS reconstruction can be further reduced by the SP algorithm, recently proposed by two research groups [@Dai_2008_Subspace_Pursuit; @Tropp2008_CoSamp]. It is an iterative algorithm drawing on the theory of list decoding. The computational complexity of this algorithm is upper bounded by $O\left(Km(N+K^{2})\right)$, which is significantly smaller than the complexity of the BP method whenever $K\ll N$. See [@Dai_2008_Subspace_Pursuit] for a detailed performance and complexity analysis of this greedy algorithm.
A sufficient condition for both the BP and SP algorithms to perform exact reconstruction is based on the so called restricted isometry property (RIP) [@Candes_Tao_IT2005_decoding_linear_programming], formally defined as follows.
\[def:RIP\]*(RIP).* A matrix $\mathbf{\Phi}\in\mathbb{R}^{m\times N}$ is said to satisfy the Restricted Isometry Property (RIP) with coefficients $\left(K,\delta\right)$ for $K\le m$, $0\leq\delta\leq1$, if for all index sets $I\subset\left\{ 1,\cdots,N\right\} $ such that $\left|I\right|\le K$ and for all $\mathbf{q}\in\mathbb{R}^{\left|I\right|}$, one has $$\left(1-\delta\right)\left\Vert \mathbf{q}\right\Vert _{2}^{2}\le\left\Vert \mathbf{\Phi}_{I}\mathbf{q}\right\Vert _{2}^{2}\le\left(1+\delta\right)\left\Vert \mathbf{q}\right\Vert _{2}^{2}.$$
The RIP parameter $\delta_{K}$ is defined as the infimum of all parameters $\delta$ for which the RIP holds, i.e., $$\begin{aligned}
\delta_{K} & :=\inf\left\{ \delta:\;\left(1-\delta\right)\left\Vert \mathbf{q}\right\Vert _{2}^{2}\le\left\Vert \mathbf{\Phi}_{I}\mathbf{q}\right\Vert _{2}^{2}\le\left(1+\delta\right)\left\Vert \mathbf{q}\right\Vert _{2}^{2},\right.\nonumber \\
& \quad\quad\quad\quad\left.\;\forall\left|I\right|\le K,\;\forall\mathbf{q}\in\mathbb{R}^{\left|I\right|}\right\} .\label{eq:def-RIP-parameter}\end{aligned}$$
It was shown in [@Candes_Tao_ApplMath2006_Stable_Signal_Recovery; @Dai_2008_Subspace_Pursuit] that both BP and SP algorithms lead to exact reconstructions of $K$-sparse signals if the matrix $\mathbf{\Phi}$ satisfies the RIP with a constant parameter, i.e., $\delta_{c_{1}K}\le c_{0}$ where both $c_{1}\in\mathbb{R}^{+}$ and $c_{0}\in\left(0,1\right)$ are constants independent of $K$ (although different algorithms may have different parameters $c_{0}$s and $c_{1}$s). Most known families of matrices satisfying the RIP property with optimal or near-optimal performance guarantees are random, including Gaussian random matrices with i.i.d. $\mathcal{N}\left(0,1/m\right)$ entries, where $m\ge O\left(K\log N\right)$.
For completeness, we briefly describe the SP algorithm. For an index set $T\subset\left\{ 1,2,\cdots,N\right\} $, let $\mathbf{\Phi}_{T}$ be the “truncated matrix” consisting of the columns of $\mathbf{\Phi}$ indexed by $T$, and let $\mathrm{span}\left(\mathbf{\Phi}_{T}\right)$ denote the subspace in $\mathbb{R}^{m}$ spanned by the columns of $\mathbf{\Phi}_{T}$. Suppose that $\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}$ is invertible. For any given $\mathbf{y}\in\mathbb{R}^{m}$, the projection of $\mathbf{y}$ onto $\mathrm{span}\left(\mathbf{\Phi}_{T}\right)$ is defined as$$\begin{aligned}
& \mathbf{y}_{p}=\mathrm{proj}\left(\mathbf{y},\mathbf{\Phi}_{T}\right):=\mathbf{\Phi}_{T}\left(\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}\right)^{-1}\mathbf{\Phi}_{T}^{*}\mathbf{y},\label{eq:def-proj}\end{aligned}$$ where $\mathbf{\Phi}^{*}$ denotes the conjugate transpose of $\mathbf{\Phi}$.
The corresponding projection residue vector $\mathbf{y}_{r}$ and projection coefficient vector $\mathbf{x}_{p}$ are defined as $$\mathbf{y}_{r}=\mathrm{resid}\left(\mathbf{y},\mathbf{\Phi}_{T}\right):=\mathbf{y}-\mathbf{y}_{p},\label{eq:def-proj-residue}$$ and$$\begin{aligned}
& \mathbf{x}_{p}=\mathrm{pcoeff}\left(\mathbf{y},\mathbf{\Phi}_{T}\right):=\left(\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}\right)^{-1}\mathbf{\Phi}_{T}^{*}\mathbf{y}.\label{eq:def-proj-coeff}\end{aligned}$$ The steps of the SP algorithm are summarized below.
**Input**: $K$, $\mathbf{\Phi}$, $\mathbf{y}$
**Initialization**: Let $T^{0}=\left\{ K^{\phantom{*}}\right.$indices corresponding to entries of largest magnitude in $\left.\mathbf{\Phi}^{*}\mathbf{y}\right\} $ and $\mathbf{y}_{r}^{0}=\mathrm{resid}\left(\mathbf{y},\mathbf{\Phi}_{\hat{T}^{0}}\right)$.
**Iteration**: At the $\ell^{\mathrm{th}}$ iteration, go through the following steps.
1. $\tilde{T}^{\ell}=T^{\ell-1}\bigcup$$\left\{ K^{\phantom{*}}\right.$indices corresponding to entries of largest magnitude in $\left.\mathbf{\Phi}^{*}\mathbf{y}_{r}^{\ell-1}\right\} $.
2. Let $\mathbf{x}_{p}=\mathrm{pcoeff}\left(\mathbf{y},\mathbf{\Phi}_{\tilde{T}^{\ell}}\right)$ and $T^{\ell}=\left\{ K^{\phantom{*}}\right.$indices corresponding to entries of largest magnitude in $\left.\mathbf{x}_{p}\right\} $.
3. $\mathbf{y}_{r}^{\ell}=\mathrm{resid}\left(\mathbf{y},\mathbf{\Phi}_{T^{\ell}}\right).$
4. If $\left\Vert \mathbf{y}_{r}^{\ell}\right\Vert _{2}>\left\Vert \mathbf{y}_{r}^{\ell-1}\right\Vert _{2}$, let $T^{\ell}=T^{\ell-1}$ and quit the iteration.
**Output**: The vector $\hat{\mathbf{x}}$ satisfying $\hat{\mathbf{x}}_{\left\{ 1,\cdots,N\right\} -T^{\ell}}=\mathbf{0}$ and $\hat{\mathbf{x}}_{T^{\ell}}=\mathrm{pcoeff}\left(\mathbf{y},\mathbf{\Phi}_{T^{\ell}}\right)$.
In what follows, we study the performance of the SP and BP reconstruction algorithms when the measurements are subjected to three different quantization schemes. We also discuss the issue of quantizing the measurement matrix values.
\[sub:Scalar-Quantization\]Scalar and Vector Quantization
---------------------------------------------------------
Let $\mathcal{C}\subset\mathbb{R}^{m}$ be a finite discrete set, referred to as a codebook. A quantizer is a mapping from $\mathbb{R}^{m}$ to the codebook $\mathcal{C}$ with the property that $$\begin{aligned}
\mathfrak{q}:\;\mathbb{R}^{m} & \rightarrow\mathcal{C}\nonumber \\
\mathbf{y} & \mapsto\bm{\omega}\in\mathcal{C}\;\mathrm{if}\;\mathbf{y}\in\mathcal{R}_{\bm{\omega}},\label{eq:scalar-quant}\end{aligned}$$ where $\bm{\omega}$ is referred to as a *level* and $\mathcal{R}_{\bm{\omega}}$ is the *quantization region* corresponding to the level $\bm{\omega}$. The performance of a quantizer is often described by its distortion-rate function, defined as follows. Let the distortion measure be the squared Euclidean distance (i.e., mean squared error (MSE)). For a random source $\mathbf{Y}\in\mathbb{R}^{m}$, the distortion associated with a quantizer $\mathfrak{q}$ is $D_{\mathfrak{q}}:=\mathrm{E}\left[\left\Vert \mathbf{Y}-\mathfrak{q}\left(\mathbf{Y}\right)\right\Vert _{2}^{2}\right]$. For a given codebook $\mathcal{C}$, the optimal quantization function that minimizes the Euclidean distortion measure is given by $$\mathfrak{q}^{*}\left(\mathbf{Y}\right)=\underset{\bm{\omega}\in\mathcal{C}}{\arg\;\min}\;\left\Vert \mathbf{Y}-\bm{\omega}\right\Vert _{2}^{2}.$$ As a result, the corresponding quantization region is given by $$\mathcal{R}_{\bm{\omega}}:=\left\{ \mathbf{y}\in\mathbb{R}^{m}:\;\left\Vert \mathbf{y}-\bm{\omega}\right\Vert _{2}^{2}\le\left\Vert \mathbf{y}-\bm{\omega}^{\prime}\right\Vert _{2}^{2},\;\forall\bm{\omega}^{\prime}\in\mathcal{C}\right\} ,\label{eq:def-quant-region}$$ and the distortion associated with this codebook $\mathcal{C}$ equals $$D\left(\mathcal{C}\right):=\mathrm{E}\left[\left\Vert \mathbf{Y}-\mathfrak{q}^{*}\left(\mathbf{Y}\right)\right\Vert _{2}^{2}\right].$$ Let $R:=\frac{1}{m}\log_{2}\left|\mathcal{C}\right|$ be the rate of the codebook $\mathcal{C}$. For a given code rate $R$, the distortion rate function is given by
$$D^{*}\left(R\right):=\underset{\mathcal{C}:\;\frac{1}{m}\log_{2}\left|\mathcal{C}\right|\le R}{\inf}\; D\left(\mathcal{C}\right).\label{eq:def-DRF-vector}$$
For simplicity, assume that the random source $\mathbf{Y}$ does not have mass points, and that the levels in the quantization codebook are all distinct. With these assumptions, though different quantization regions (\[eq:def-quant-region\]) may overlap, the ties can be broken arbitrarily as they happen with probability zero.
We study both vector quantization and scalar quantization. Scalar quantization has lower computational complexity than vector quantization. It is a special case of vector quantization when $m=1$. To distinguish the two schemes, we use the subscripts $SQ$ and $VQ$ to refer to scalar and vector quantization, respectively. For quantized compressive sensing, we assume that the quantization functions for all the coordinate of $\mathbf{Y}$ are the same. The corresponding distortion rate function is therefore of the form $$D_{SQ}^{*}\left(R\right):=\underset{\mathcal{C}_{SQ}:\;\log_{2}\left|\mathcal{C}_{SQ}\right|\le R}{\inf}\mathrm{E}_{\mathbf{Y}}\left[\sum_{i=1}^{m}\left|Y_{i}-\mathfrak{q}_{SQ}\left(Y_{i}\right)\right|^{2}\right].\label{eq:def-DRF-scalar}$$
Necessary conditions for optimal scalar quantizer design can be found in [@Lloyd1982_IT_quantization]. The quantization region for the level $\omega_{i}\in\mathcal{C}$, $i=1,2,\cdots,2^{R}$, can be written in the form $\mathcal{R}_{\omega_{i}}=\overline{\left(t_{i-1},t_{i}\right)}$, where $t_{i-1},t_{i}\in\mathbb{R}\bigcup\left\{ -\infty\right\} \bigcup\left\{ +\infty\right\} $ and $\overline{\left(t_{i-1},t_{i}\right)}$ is the closure of the open interval $\left(t_{i-1},t_{i}\right)$. An optimal quantizer satisfies the following conditions:
1. If the optimal quantizer has levels $\omega_{i-1}$ and $\omega_{i}$, then the threshold that minimizes the mean square error (MSE) is $$t_{i}=\frac{1}{2}\left(\omega_{i}+\omega_{i+1}\right).\label{eq:thresh-update}$$
2. If the optimal quantizer has thresholds $t_{i-1}$ and $t_{i}$, then the level that minimizes the MSE is $$\omega_{i}=\mathrm{E}\left[Y|Y\in\overline{\left(t_{i-1},t_{i}\right)}\right].\label{eq:center-update}$$
Lloyd’s algorithm [@Lloyd1982_IT_quantization] for quantizer codebook design is based on the above necessary conditions. Lloyd’s algorithm starts with an initial codebook, and then in each iteration, computes the thresholds $t_{i}$s according to (\[eq:thresh-update\]) and updates the codebook via (\[eq:center-update\]). Although Lloyd’s algorithm is not guaranteed to find a global optimum for the quantization regions, it produces locally optimal codebooks.
As a low-complexity alternative to non-uniform quantizers, uniform scalar quantizers are widely used in practice. A uniform scalar quantizer is associated with a “uniform codebook” $\mathcal{C}_{u,SQ}=\left\{ \omega_{1}<\omega_{2}<\cdots<\omega_{M}\right\} ,$ for which $\omega_{i}-\omega_{i-1}=\omega_{j}-\omega_{j-1}$ for all $1<i\ne j\le M$. The difference between adjacent levels is often referred to as the step size, and denoted by $\Delta_{u,SQ}$. The corresponding distortion rate function is given by $$\begin{aligned}
D_{u,SQ}^{*}\left(R\right) & :=\underset{\mathcal{C}_{u,SQ}:\;\log_{2}\left|\mathcal{C}_{u,SQ}\right|\le R}{\inf}\nonumber \\
& \qquad\quad\mathrm{E}_{\mathbf{Y}}\left[\sum_{i=1}^{m}\left|Y_{i}-\mathfrak{q}_{SQ}\left(Y_{i}\right)\right|^{2}\right].\label{eq:def-DRF-uniform-scalar}\end{aligned}$$ where $\mathcal{C}_{SQ}$ in (\[eq:def-DRF-scalar\]) is replaced by $\mathcal{C}_{u,SQ}$.
Definitions (\[eq:def-DRF-scalar\]) and (\[eq:def-DRF-uniform-scalar\]) are consistent with (\[eq:def-DRF-vector\]) as a Cartesian product of scalar quantizers can be viewed as a special form of a vector quantizer.
\[sec:Distortion-Analysis\]Distortion Analysis
==============================================
We analyze the asymptotic behavior of the distortion rate functions introduced in the previous section. We assume that the quantization codebook $\mathcal{C}$, for both scalar and vector quantization, is designed offline and fixed when the measurements are taken.
Distortion of Scalar Quantization
---------------------------------
For scalar quantization, we consider the following two CS scenarios.
**Assumptions I**:
1. Let $\mathbf{\Phi}=\frac{1}{\sqrt{m}}\mathbf{A}\in\mathbb{R}^{m\times N}$, where the entries of $\mathbf{A}$ are i.i.d. Subgaussian random variables[^1] with zero mean and unit variance.
2. Let $\mathbf{X}\in\mathbb{R}^{N}$ be an exactly $K$-sparse vector, that is, a signal that has exactly $K$ nonzero entries. We assume that the nonzero entries of $\mathbf{X}$ are i.i.d. Subgaussian random variables with zero mean and unit variance, although more general models can be analyzed in a similar manner.
**Assumptions II**: Assume that $\mathbf{X}\in\mathbb{R}^{n}$ is exactly $K$-sparse, and that the nonzero entries of $\mathbf{X}$ are i.i.d. standard Gaussian random variables.
The asymptotic distortion-rate function of the measurement vector under the first CS scenario is characterized in Theorem \[thm:DRF-Gaussian-Matrix\].
\[thm:DRF-Gaussian-Matrix\]Suppose that Assumptions I hold. Then $$\underset{R\rightarrow\infty}{\lim}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\;\frac{2^{2R}}{K}D_{SQ}^{*}\left(R\right)=\frac{\pi\sqrt{3}}{2},\label{eq:DRF-Gaussian-non-uniform}$$ and $$\underset{R\rightarrow\infty}{\lim}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\;\frac{2^{2R}}{KR}D_{u,SQ}^{*}\left(R\right)=\frac{4}{3}\ln2.\label{eq:DRF-Gaussian-uniform}$$
The proof is based on the fact that the distributions of $\sqrt{\frac{m}{K}}Y_{i}$, $1\le i\le m$, weakly converge to standard Gaussian distributions. The detailed description is given in Appendix \[sub:Pf-Thm-Gaussian-scalar\].
To study the scenario described by Assumptions II, we need the following definitions. For a given matrix $\mathbf{\Phi}$, let $$\mu_{1}:=\frac{1}{N}\sum_{i\in\left[m\right],j\in\left[N\right]}\varphi_{i,j}^{2},\label{eq:def-mu-1}$$ and $$\mu_{2}:=\underset{i\in\left[m\right],T\in{\left[N\right] \choose K}}{\max}\frac{m}{K}\sum_{j\in T}\varphi_{i,j}^{2},\label{eq:def-mu-2}$$ where $\left[m\right]=\left\{ 1,2,\cdots,m\right\} $ and ${\left[N\right] \choose K}$ denotes the set of all subsets of $\left[N\right]$ with cardinality $K$. Note that if the matrix $\mathbf{\Phi}$ is generated from the random ensemble described in Assumption I.1), then $\mu_{1}\in\left(1-\epsilon,1+\epsilon\right)$ with high probability, for all $\epsilon>0$, and whenever $m$ and $N$ are sufficiently large. It is straightforward to verify that $\mu_{1}\le\mu_{2}$.
With these definitions at hand, bounds on the distortion rate function can be described as below.
\[thm:DRF-Matrix\]Suppose that Assumption II holds. Then$$\begin{aligned}
& \frac{\pi\sqrt{3}}{2}\mu_{1}\le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}D_{SQ}^{*}\left(R\right)\nonumber \\
& \quad\le\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}D_{SQ}^{*}\left(R\right)\le\frac{\pi\sqrt{3}}{2}\mu_{2},\label{eq:DRF-non-uniform}\end{aligned}$$ and $$\frac{4\ln2}{3}\mu_{1}\le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{KR}D_{u,SQ}^{*}\left(R\right).\label{eq:DRF-uniform}$$
The detailed proof is postponed to Appendix \[sub:Pf-lb-ub\]. Here, we sketch the basic ideas behind the proof. In order to construct a lower bound, suppose that one has prior information about the support set $T$ before taking the measurements. For a given value of $i$ and for a given $T$, we calculate the corresponding asymptotic distortion-rate function. The lower bound is obtained by taking the average of these distortion-rate functions over all possible values of $i$ and $T$. For the upper bound, we design a sequence of sub-optimal scalar quantizers, then apply them to all measurement components, and finally construct a uniform upper bound on their asymptotic distortion-rate functions, valid for all $i$ and $T$. The uniform upper bound is given in (\[eq:DRF-non-uniform\]).
Our results are based on the fundamental assumption that the sparsity level $K$ is known in advance and that the statistics of the sparse vector $\mathbf{x}$ is specified. Very frequently, however, this is not the case in practice. If we relax Assumptions I and II further by assuming that $K$ is sufficiently large, it will often be the case that the statistics of the measurement $Y_{i}$ is well approximated by a Gaussian distribution. Here, note that different $Y_{i}$ variables may have different variances and these variances are generally unknown in advance. The problem of statistical mismatch has been analyzed in the proof of the upper bound (\[eq:DRF-non-uniform\]) (see Proposition \[pro:ub-mismatch\] of Appendix \[sub:Pf-lb-ub\] for details). In particular, non-uniform quantization with slightly over-estimated variance performs better than that with under-estimated variance [@Sayood2005_book_data_compression Chapter 8.6].
According to Theorem \[thm:DRF-Gaussian-Matrix\], if the quantization rate $R$ is sufficiently large, the distortion of the optimal non-uniform quantizer is approximately only $1/R$ of that of the optimal uniform quantizer. This gap can be closed by using entropy coding techniques in conjunction with uniform quantizers.
Uniform Scalar Quantization with Entropy Encoding
-------------------------------------------------
Let $\mathcal{B}_{enc}=\left\{ \bm{v}_{1},\bm{v}_{2},\cdots,\bm{v}_{M}\right\} $ be a binary codebook, where the codewords $\bm{v}_{i}$, $1\le i\le M$, are finite-length strings over the binary field with elements $\left\{ 0,1\right\} $. The codebook $\mathcal{B}_{enc}$ can, in general, contain codewords of variable length - i.e., the lengths of different codewords are allowed to be different. Let $\ell_{i}$ be the length of codeword $\bm{v}_{i}$, $i=1,2,\cdots,M$. Then $\bm{v}_{i}\in\left\{ 0,1\right\} ^{\ell_{i}\times1}$. For a given quantization codebook $\mathcal{C}=\left\{ \omega_{1},\omega_{2},\cdots,\omega_{M}\right\} $, the encoding function $\mathfrak{f}_{enc}$ is a mapping from the quantization codebook $\mathcal{C}$ to the binary codebook $\mathcal{B}_{enc}$, i.e., $\mathfrak{f}_{enc}\left(\omega\right)=\bm{v}\in\mathcal{B}_{enc}$. The extension $\mathfrak{f}_{enc}^{*}$ is a mapping from finite length strings of $\mathcal{C}$ to finite length strings of $\mathcal{B}_{enc}$ (a concatenation of the corresponding binary codewords): $$\mathfrak{f}_{enc}^{*}\left(\omega_{i_{1}}\omega_{i_{2}}\cdots\omega_{i_{s}}\right)=\mathfrak{f}_{enc}\left(\omega_{i_{1}}\right)\mathfrak{f}_{enc}\left(\omega_{i_{2}}\right)\cdots\mathfrak{f}_{enc}\left(\omega_{i_{s}}\right).$$ The code $\mathcal{B}_{enc}$ is called *uniquely decodable* if any concatenation of binary codewords $\bm{v}_{i_{1}}\bm{v}_{i_{2}}\cdots\bm{v}_{i_{s}}$ has only one possible preimage string $\omega_{j_{1}}\omega_{j_{2}}\cdots\omega_{j_{s}}$ producing it. In practice, the code $\mathcal{B}_{enc}$ is often chosen to be a *prefix* code, that is, no codeword is a prefix of any other codeword. A prefix code can be uniquely decoded as the end of a codeword is immediately recognizable without checking future encoded bits.
We consider the case in which scalar quantization is followed by variable-length encoding. The corresponding expected encoding length $\bar{L}$ is defined by $$\bar{L}=\mathrm{E}_{Y}\left[\mathfrak{L}\circ\mathfrak{f}_{enc}\circ\mathfrak{q}_{SQ}\left(Y\right)\right],$$ where $\mathfrak{L}\left(\bm{v}\right)$ outputs the length of the encoding codeword $\bm{v}\in\mathcal{B}_{enc}$. The goal is to *jointly* design $\mathfrak{q}_{SQ}$ and $\mathfrak{f}_{enc}$ to minimize the expected encoding length $\bar{L}$. We are interested in the distortion rate function defined by $$D_{enc}^{*}\left(R\right):=\underset{\bar{L}\le R}{\inf}\;\mathrm{E}_{\mathbf{Y}}\left[\sum_{i=1}^{m}\left|Y_{i}-\mathfrak{q}_{SQ}\left(Y_{i}\right)\right|^{2}\right].\label{eq:DRF-encoding}$$
\[thm:Encoding\]Suppose that Assumptions I hold. Then$$\begin{aligned}
\frac{\pi e}{6} & \le\underset{R\rightarrow\infty}{\lim\inf}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}D_{enc}^{*}\left(R\right)\\
& \le\underset{R\rightarrow\infty}{\lim\sup}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}D_{enc}^{*}\left(R\right)\le\frac{\pi e}{3},\end{aligned}$$ and the upper bound is achieved by a uniform scalar quantizer with $$\underset{R\rightarrow\infty}{\lim}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\sqrt{\frac{m}{2\pi eK}}2^{R}\Delta_{u,SQ}=1,$$ followed by Huffmann encoding.
Given a quantization function, Huffmann encoding gives an optimal prefix code that minimizes $\bar{L}$ [@Cover1991_book_information_theory Chapter 5]. Let $p_{i}=\Pr\left(Y:\;\mathfrak{q}\left(Y\right)=\omega_{i}\right)$ and let $\ell_{i}$ be the length of encoded codeword $\mathfrak{f}_{enc}\left(\omega_{i}\right)$. Let $H:=\sum_{i=1}^{M}-p_{i}\log_{2}p_{i}$. Then $H\le\bar{L}=\sum_{i}p_{i}\ell_{i}\le H+1$. In addition, it is well known that the distortion of scalar quantization of a Gaussian source is lower bounded by $\frac{1}{12}2^{2\left(h-H\right)}\left(1+o_{H}\left(1\right)\right)$, where $h$ denotes the differential entropy of the source, and the lower bound is achieved by a uniform quantizer. Calculating $h$ and interpreting $H$ as a function of $\bar{L}$ establish the claimed result.
As expected, for a given average description length, the average distortion of uniform scalar quantization and Huffmann encoding is smaller than that of an optimal scalar quantizer with fixed length encoding.
Distortion of Vector Quantization
---------------------------------
For the purpose of analyzing vector quantization schemes, we make the following assumptions.
**Assumptions III**:
1. Let $\mathbf{\Phi}\in\mathbb{R}^{m\times N}$ be a matrix satisfying the RIP with parameter $\delta_{K}\in\left(0,1\right)$.
2. Assume that $\mathbf{X}\in\mathbb{R}^{n}$ is exactly $K$-sparse, and that the nonzero entries of $\mathbf{X}$ are i.i.d. standard Gaussian random variables.
\[thm:DRF-vector\]Suppose that Assumptions III hold. Then$$\begin{aligned}
& \left(1-\delta_{K}\right)\left(1+o_{K}\left(1\right)\right)\le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2Rm/K}}{K}D_{VQ}^{*}\left(R\right)\label{eq:lb-DRF-VQ}\\
& \quad\le\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{m}D_{VQ}^{*}\left(R\right)\le\left(1+\delta_{K}\right)\left(1+o_{m}\left(1\right)\right),\label{eq:ub-DRF-VQ-1}\end{aligned}$$ where $o_{K}\left(1\right)\overset{K\rightarrow\infty}{\rightarrow}0$ and $o_{m}\left(1\right)\overset{m\rightarrow\infty}{\rightarrow}0$. Another upper bound on $D_{VQ}^{*}\left(R\right)$ is given by $$\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}D_{VQ}^{*}\left(R\right)\le\frac{\pi\sqrt{3}}{2}\mu_{2},\label{eq:ub-DRF-VQ-2}$$ where $\mu_{2}$ is as defined in (\[eq:def-mu-2\]).
The comparison of the two upper bounds in (\[eq:ub-DRF-VQ-1\]) and (\[eq:ub-DRF-VQ-2\]) depends on the ratio between $m$ and $K$. Consider the case where $N=\beta K$, $m=\Theta\left(K\log\left(N/K\right)\right)=\alpha K$ for some $\alpha,\beta>1$. The first upper bound becomes $$\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}D_{VQ}^{*}\left(R\right)\le\alpha\left(1+\delta_{K}\right)\left(1+o_{m}\left(1\right)\right).$$ It is smaller than the second upper bound if and only if $$\delta_{K}<\frac{\pi\sqrt{3}}{2\alpha}\mu_{2}-1.$$
The upper bound (\[eq:ub-DRF-VQ-2\]) is obtained by using the Cartesian product of scalar quantizers and invoking the result in (\[eq:DRF-non-uniform\]). The bounds (\[eq:lb-DRF-VQ\]) and (\[eq:ub-DRF-VQ-1\]) are proved in Appendix \[sub:Pf-lb-ub\]. The basic ideas behind the proof are similar to those used for proving Theorem \[thm:DRF-Matrix\]: the lower bound is obtained by averaging the distortions of optimal quantizers for every $T\in{\left[N\right] \choose K}$, while the upper bound is a uniform upper bound on the distortions of quantizers constructed for all $T\in{\left[N\right] \choose K}$.
Note that the lower bound in (\[eq:lb-DRF-VQ\]) is not achievable when $K<m$. The upper bounds (\[eq:ub-DRF-VQ-1\]) and (\[eq:ub-DRF-VQ-2\]) do not guarantee significant distortion reduction of vector quantization compared with scalar quantization. Due to their inherently high computational complexity, vector quantizers do not offer clear advantages that justify their use in practice.
CS Measurement Matrix Quantization Effects
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In CS theory, the measurement matrix is generated either randomly or by some deterministic construction. Examples include Gaussian random matrices and the deterministic construction based on Vandermonde matrices [@Tarokh2007_ISIT_CS_ReedSolomon; @Shokrollahi2009_bit_precision_CS]. In both examples, the matrix entries typically have infinite precision, which is not the case in practice. It is therefore also plausible to study the effect of quantization of CS measurement matrix.
Consider Assumption I where the measurement matrix is randomly generated. Let us assume that every entry $\varphi_{i,j}$, $1\le i\le m$ and $1\le j\le N$, is quantized using a finite number of bits. Note that $\hat{\varphi}_{i,j}=\mathfrak{q}\left(\varphi_{i,j}\right)$ is a bounded random variable and therefore Subgaussian distributed. The results in Theorem \[thm:DRF-Gaussian-Matrix\] are therefore automatically valid for quantized matrices as well.
Suppose that the measurement matrix is constructed deterministically and then quantized using a finite number of bits. The parameters $\mu_{1}$, $\mu_{2}$ and $\delta_{K}$ of the quantized measurement matrix can be computed according to (\[eq:def-mu-1\]), (\[eq:def-mu-2\]) and (\[eq:def-RIP-parameter\]), respectively. The results regarding scalar quantization and vector quantization described in Theorems \[thm:DRF-Matrix\] and \[thm:DRF-vector\] can be easily seen to hold in this case as well.
Reconstruction Distortion
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Based on the results of the previous section, we are ready to quantify the reconstruction distortion of BP and SP methods introduced by quantization error.
It is well known from CS literature that the reconstruction distortion is dependent on the distortion in the measurements. Consider the quantized CS given by $$\hat{\mathbf{Y}}=\mathfrak{q}\left(\mathbf{Y}\right)=\mathbf{\Phi}\mathbf{X}+\mathbf{E},$$ and where $\mathbf{E}\in\mathbb{R}^{m}$ denotes the quantization error. Let $\hat{\mathbf{X}}$ be the reconstructed signal based on the quantized measurements $\hat{\mathbf{Y}}$. Then the reconstruction distortion can be upper bounded by $$\left\Vert \mathbf{X}-\hat{\mathbf{X}}\right\Vert _{2}^{2}\le c^{2}\left\Vert \mathbf{E}\right\Vert _{2}^{2},\label{eq:ub-reconst-dist}$$ where the constant $c$ differs for different reconstruction algorithms. The best bounding constant for the BP method was given in [@Candes_Tao_ApplMath2006_Stable_Signal_Recovery], and equals $$c_{bp}=\frac{4}{\sqrt{3-3\delta_{4K}}-\sqrt{1+\delta_{4K}}},$$ while for the SP algorithm, the constant was estimated in [@Dai_2008_Subspace_Pursuit] $$c_{sp}=\frac{1+\delta_{3K}+\delta_{3K}^{2}}{\delta_{3K}\left(1-\delta_{3K}\right)}.$$
A lower bound on the reconstruction distortion is given as follows. Suppose that the support set $T$ of the sparse signal $\mathbf{x}$ is perfectly reconstructed. The reconstructed signal $\hat{\mathbf{X}}$ is given by $$\hat{\mathbf{X}}=\left(\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}\right)^{-1}\mathbf{\Phi}_{T}\hat{\mathbf{Y}},$$ and the reconstruction distortion is lower bounded by $$\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\ge\left(\frac{\sqrt{1-\delta_{K}}}{1+\delta_{K}}\right)^{2}\left\Vert \hat{\mathbf{Y}}-\mathbf{Y}\right\Vert _{2}^{2}=\frac{1-\delta_{K}}{\left(1+\delta_{K}\right)^{2}}\left\Vert \mathbf{E}\right\Vert _{2}^{2}.\label{eq:lb-reconst-dist}$$ For short, let $$c_{lb}=\frac{\sqrt{1-\delta_{K}}}{1+\delta_{K}}.$$
Combining the bounds (\[eq:ub-reconst-dist\],\[eq:lb-reconst-dist\]) and the results in Theorems \[thm:DRF-Gaussian-Matrix\]-\[thm:DRF-vector\], we summarize the asymptotic bounds on the reconstruction distortion as follows. Under Assumptions I, the reconstruction distortion of scalar quantization is bounded by $$\begin{aligned}
c_{lb}^{2}\frac{\pi\sqrt{3}}{2} & \le\underset{R\rightarrow\infty}{\lim}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\frac{\pi\sqrt{3}}{2} & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\frac{\pi\sqrt{3}}{2} & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases},\end{aligned}$$ and the reconstruction distortion of uniform scalar quantization is bounded by $$\begin{aligned}
c_{lb}^{2}\frac{4\log2}{3} & \le\underset{R\rightarrow\infty}{\lim}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\frac{2^{2R}}{KR}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\frac{4\log2}{3} & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\frac{4\log2}{3} & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases}.\end{aligned}$$ Suppose that Assumption II holds. The reconstruction distortions for scalar quantization and uniform scalar quantization are respectively bounded by$$\begin{aligned}
c_{lb}^{2}\frac{\pi\sqrt{3}}{2}\mu_{1} & \le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\frac{\pi\sqrt{3}}{2}\mu_{2} & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\frac{\pi\sqrt{3}}{2}\mu_{2} & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases}\end{aligned}$$ and $$c_{lb}^{2}\frac{4\log2}{3}\mu_{1}\le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{KR}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right].$$ Given the encoding rate $R$ per measurement, the reconstruction distortion of the optimal scalar quantizer is bounded as $$\begin{aligned}
c_{lb}^{2}\frac{\pi e}{6} & \le\underset{R\rightarrow\infty}{\lim\inf}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\underset{R\rightarrow\infty}{\lim\sup}\underset{\left(K,m,N\right)\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\frac{\pi e}{3} & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\frac{\pi e}{3} & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases}.\end{aligned}$$ The bounds for reconstruction distortion associated with vector quantization are given by $$\begin{aligned}
& c_{lb}^{2}\left(1-\delta_{K}\right)\left(1+o_{K}\left(1\right)\right)\\
& \le\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2Rm/K}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{m}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\left(1+\delta_{K}\right)\left(1+o_{m}\left(1\right)\right) & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\left(1+\delta_{K}\right)\left(1+o_{m}\left(1\right)\right) & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases},\end{aligned}$$ and $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{X}}-\mathbf{X}\right\Vert _{2}^{2}\right]\\
& \le\begin{cases}
c_{sp}^{2}\frac{\pi\sqrt{3}}{2}\mu_{2} & \mathrm{for\; subspace\; algorithm}\\
c_{bp}^{2}\frac{\pi\sqrt{3}}{2}\mu_{2} & \mathrm{for\; basis\; pursuit\; algorithm}\end{cases}.\end{aligned}$$
It is worth noting that the upper bound (\[eq:ub-reconst-dist\]) on reconstruction distortion may not be tight. Empirical experiments show that this upper bound often significantly over-estimates the reconstruction distortion [@Candes_Tao_ApplMath2006_Stable_Signal_Recovery; @Dai_2008_Subspace_Pursuit].
\[sec:Modified-Algorithm\]Reconstruction Algorithms for Quantized CS
====================================================================
We present next modifications of BP and SP algorithms that take into account quantization effects.
To describe these algorithms, we find the following notation useful. Let $\hat{\mathbf{Y}}$ be the quantized measurement vector. Given a vector $\hat{\mathbf{Y}}$, the corresponding quantization region can be easily identified: the quantization region of vector quantization $\mathcal{R}_{\hat{\mathbf{Y}}}$ is defined in (\[eq:def-quant-region\]); that of scalar quantization is given by the Cartesian product of the quantization regions for each coordinate, i.e., $\mathcal{R}_{\hat{\mathbf{Y}}}=\prod_{i=1}^{m}\mathcal{R}_{\hat{Y}_{i}}$ where $\mathcal{R}_{\hat{Y}_{i}}$ is the quantization region of $\hat{Y}_{i}$.
Similar to the standard BP method, the reconstruction problem can be now casted as $$\min\left\Vert \mathbf{x}\right\Vert _{1}\;\mathrm{subject\; to}\;\mathbf{\Phi}\mathbf{x}\in\mathcal{R}_{\hat{\mathbf{Y}}}.\label{eq:BP-modified}$$ It can be verified that $\mathcal{R}_{\hat{\mathbf{Y}}}$ is a closed convex set and therefore (\[eq:BP-modified\]) is a convex optimization problem and can be efficiently solved by linear programming techniques.
In order to adapt the SP algorithm to the quantization scenario at hand, we describe first a geometric interpretation of the projection operation in the SP algorithm. Given $\mathbf{y}\in\mathbb{R}^{m}$ and $\mathbf{\Phi}_{T}\in\mathbb{R}^{m\times\left|T\right|}$, suppose that $\mathbf{\Phi}_{T}$ has full column rank, in other words, suppose that the columns of $\mathbf{\Phi}_{T}$ are linearly independent. The projection operation in (\[eq:def-proj\]) is equivalent to the optimization problem $$\underset{\mathbf{x}\in\mathbb{R}^{\left|T\right|}}{\min}\left\Vert \mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}\right\Vert _{2}^{2}.\label{eq:proj-optim}$$ Let $\mathbf{x}^{*}$ be the solution of the quadratic optimization problem (\[eq:proj-optim\]). Then functions (\[eq:def-proj\]-\[eq:def-proj-coeff\]) are equivalent to $\mathrm{proj}\left(\mathbf{y},\mathbf{\Phi}_{T}\right)=\mathbf{\Phi}_{T}\mathbf{x}^{*}$, $\mathrm{resid}\left(\mathbf{y},\mathbf{\Phi}_{T}\right)=\mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}^{*}$ and $\mathrm{pcoeff}\left(\mathbf{y},\mathbf{\Phi}_{T}\right)=\mathbf{x}^{*}$.
The modified SP algorithm is based on the above geometric interpretation. More precisely, we use the following definition.
\[def:x-y-stars\]For given $\mathbf{\Phi}_{T}\in\mathbb{R}^{m\times\left|T\right|}$, $\hat{\mathbf{Y}}$ and $\mathcal{R}_{\hat{\mathbf{Y}}}$, define$$\begin{aligned}
\mathcal{Q} & :=\left\{ \left(\mathbf{x},\mathbf{y}\right)\in\mathbb{R}^{\left|T\right|}\times\mathcal{R}_{\hat{\mathbf{Y}}}:\right.\nonumber \\
& \left.\left\Vert \mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}\right\Vert _{2}\le\left\Vert \mathbf{y}^{\prime}-\mathbf{\Phi}_{T}\mathbf{x}^{\prime}\right\Vert _{2}\;\forall\left(\mathbf{x}^{\prime},\mathbf{y}^{\prime}\right)\in\mathbb{R}^{\left|T\right|}\times\mathcal{R}_{\hat{\mathbf{Y}}}\right\} ,\label{eq:def-proj-q-plane}\end{aligned}$$ and $$\left(\tilde{\mathbf{x}},\tilde{\mathbf{y}}\right)=\underset{\left(\mathbf{x},\mathbf{y}\right)\in\mathcal{Q}}{\arg\min}\left\Vert \mathbf{y}-\hat{\mathbf{Y}}\right\Vert _{2}.\label{eq:def-proj-q-solution}$$
It can be verified that the pair $\left(\tilde{\mathbf{x}},\tilde{\mathbf{y}}\right)$ is well defined. See Appendix \[sub:pf-well-define\] for details.
This definition is introduced to identify the best approximation for $\hat{\mathbf{Y}}$ among multiple points in $\mathcal{R}_{\hat{\mathbf{Y}}}$ that minimize $\left\Vert \mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}\right\Vert _{2}$. Based on this definition, we replace the $\mathrm{resid}$ and $\mathrm{pcoeff}$ functions in Algorithm \[alg:Subspace-Pursuit-Algorithm\] with new functions $$\mathrm{resid}^{\left(q\right)}\left(\hat{\mathbf{Y}},\mathbf{\Phi}_{T}\right):=\tilde{\mathbf{y}}-\mathbf{\Phi}_{T}\tilde{\mathbf{x}}$$ and $$\mathrm{pcoeff}^{\left(q\right)}\left(\hat{\mathbf{Y}},\mathbf{\Phi}_{T}\right):=\tilde{\mathbf{x}},$$ where the superscript $\left(q\right)$ emphasizes that these definitions are for the quantized case. This gives the modified SP algorithm.
The advantage of the modified algorithms are verified by the simulation results presented in the next section.
\[sec:Empirical-Results\]Empirical Results
==========================================
We performed extensive computer simulations in order to compare the performance of different quantizers and different reconstruction algorithms empirically. The parameters used in our simulations are $m=128$, $N=256$ and $K=6$. Given these parameters, we generated realizations of $m\times N$ sampling matrices from the i.i.d. standard Gaussian ensemble and normalize the columns to have unit $l_{2}$-norm. We also selected a support set $T$ of size $\left|T\right|=K$ uniformly at random, generated the entries supported by $T$ from the standard i.i.d. Gaussian distribution and set all other entries to zero. We let the quantization rates vary from two to six bits. For each quantization rate, we used Lloyd’s algorithm (Section \[sub:Scalar-Quantization\]) to obtain a nonuniform quantizer and employed brute-force search to find the optimal uniform quantizer. To test different quantizers and reconstruction algorithms, we randomly generated $\mathbf{\Phi}$ and $\mathbf{x}$ independently a thousand times. For each realization, we calculated the measurements $\mathbf{Y}$, the quantized measurements $\hat{\mathbf{Y}}$ and the reconstructed signal $\hat{\mathbf{X}}$.
Fig. \[fig:dist-measurements\] compares uniform and uniform quantizers with respect to measurement distortion. Though the quantization rates in our experiments are relatively small, the simulation results are consistent with the asymptotic results in Theorem \[thm:DRF-Gaussian-Matrix\]: nonuniform quantization is better than uniform quantization and the gain increases with the quantization rate. Fig. \[fig:Rec-Dist-Standard\] compares the reconstruction distortion of the standard BP and SP algorithms. The comparison of the modified algorithms is given in Fig. \[fig:dist-reconstruction\]. The modified algorithms reduce the reconstruction distortion significantly. When the quantization rate is six bits, the reconstruction distortion of the modified algorithms is roughly one tenth of that of the standard algorithms. Furthermore, for both the standard and modified algorithms, the reconstruction distortion given by SP algorithms is much smaller than that of BP methods. Note that the computational complexity of the SP algorithms is also smaller than that of the BP methods, which shows clear advantages for using SP algorithms in conjunction with quantized CS data. An interesting phenomenon occurs for the case of the modified BP method: although nonuniform quantization gives smaller measurement distortion, the corresponding reconstruction distortion is actually slightly larger than that of uniform quantization. We do not have solid analytical arguments to completely explain this somewhat counter-intuitive fact.
\[sub:Pf-Thm-Gaussian-scalar\]Proof of Theorem \[thm:DRF-Gaussian-Matrix\]
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Let $T=\left\{ 1\le j\le N:\; X_{j}\ne0\right\} $ be the support set of $\mathbf{x}$, i.e., $x_{i}\ne0$ for all $i\in T$ and $x_{j}=0$ for all $j\notin T$. It is easy to show that for all $1\le i\le m$ and $T\subset\left\{ 1,\cdots,N\right\} $ such that $\left|T\right|=K$, $$\mathrm{E}\left[\sum_{j\in T}A_{i,j}X_{j}\right]=0$$ and $$\mathrm{E}\left[\left(\sum_{j\in T}A_{i,j}X_{j}\right)^{2}\right]=K.$$
According to the Central Limit Theorem, the distribution of $\frac{1}{\sqrt{K}}\sum_{j\in T}A_{i,j}X_{j}$ converges weakly to the standard Gaussian distribution as $K\rightarrow\infty$. This can be verified by the facts that $A_{i,j}X_{j}$s are independent and identically distributed, and that the moment generating function of $A_{i,j}X_{j}$ is well defined. As a result, the distribution of $\sqrt{\frac{m}{K}}Y_{i}$ converges weakly to the standard Gaussian distribution as $K,m,N\rightarrow\infty$.
We apply a scalar quantizer with $2^{R}$ levels to the random variable $\sqrt{\frac{m}{K}}Y_{i}$. In this case, one has $$\begin{aligned}
& \frac{1}{K}\mathrm{E}\left[\left\Vert \hat{\mathbf{Y}}-\mathbf{Y}\right\Vert _{2}^{2}\right]\nonumber \\
& =\frac{1}{m}\frac{m}{K}\mathrm{E}\left[\sum_{i=1}^{m}\left(\hat{Y}_{i}-Y_{i}\right)^{2}\right]\nonumber \\
& =\frac{1}{m}\sum_{i=1}^{m}\mathrm{E}\left[\left(\sqrt{\frac{m}{K}}\hat{Y}_{i}-\sqrt{\frac{m}{K}}Y_{i}\right)^{2}\right]\nonumber \\
& =\mathrm{E}\left[\left(\sqrt{\frac{m}{K}}\hat{Y}_{i}-\sqrt{\frac{m}{K}}Y_{i}\right)^{2}\right],\label{eq:T1-02}\end{aligned}$$ where the last line represents the distortion of quantizing $\sqrt{\frac{m}{K}}Y_{i}$. Note that the distortion-rate function for scalar quantization of a Gaussian random variable is given by $$\underset{R\rightarrow\infty}{\lim}2^{2R}D_{g}^{*}\left(R\right)=\frac{\pi\sqrt{3}}{2}\sigma^{2},\label{eq:DRF-scalar-Gaussian}$$ where $\sigma^{2}$ is the variance of the underlying Gaussian source (see [@Zadar1964_thesis_nonuniform_quantization] for a detailed proof of this result). We then have $$\underset{R\rightarrow\infty}{\lim}\;\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\frac{2^{2R}}{K}D^{*}\left(R\right)=\underset{R\rightarrow\infty}{\lim}2^{2R}D_{g}^{*}\left(R\right)=\frac{\pi\sqrt{3}}{2},$$ which completes the proof of (\[eq:DRF-Gaussian-non-uniform\]).
Consider a uniform quantizer with codebook $\mathcal{C}_{u}$, such that $\left|\mathcal{C}_{u}\right|=2^{R}$, and apply the corresponding uniform quantizer to the random variable $\sqrt{\frac{m}{K}}Y_{i}$. It was shown in [@Neuhoff2001_IT_optimal_uniform_scalar_quantization] that the distortion-rate function of uniform scalar quantizers of a Gaussian random variable equals $$\underset{R\rightarrow\infty}{\lim}\frac{2^{2R}}{R}D_{u,g}^{*}\left(R\right)=\frac{4}{3}\sigma^{2}\log2.\label{eq:DRF-uniform-scalar-Gaussian}$$ It is clear that $$\underset{R\rightarrow\infty}{\lim}\;\underset{\left(K,m,N\right)\rightarrow\infty}{\lim}\frac{2^{2R}}{KR}D_{u}^{*}\left(R\right)=\underset{R\rightarrow\infty}{\lim}\frac{2^{2R}}{R}D_{u,g}^{*}\left(R\right)=\frac{4}{3}\log2,$$ This proves Theorem \[thm:DRF-Gaussian-Matrix\].
\[sub:Pf-lb-ub\]Proof of Theorems \[thm:DRF-Matrix\] and \[thm:DRF-vector\]
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For completeness, let us first briefly review the key results used for deriving the asymptotic distortion-rate function for CS vector quantization. Suppose the source $\mathbf{Y}\in\mathbb{R}^{k}$ has probability density function $f\left(\mathbf{y}\right)$. Let $\mathcal{R}\subset\mathbb{R}^{k}$ be a quantization region and $\bm{\omega}\in\mathcal{C}$ be the corresponding quantization level. The corresponding normalized moment of inertia (NMI) is defined as $$m\left(\mathcal{R}\right)=\frac{\frac{1}{k}\int_{\mathcal{R}}\left\Vert \mathbf{y}-\bm{\omega}\right\Vert _{2}^{2}f\left(\mathbf{y}\right)d\mathbf{y}}{\left(\int_{\mathcal{R}}d\mathbf{y}\right)^{1+2/k}}.$$ The optimal NMI equals $$m_{k}^{*}=\underset{\mathcal{R}\subset\mathbb{R}^{k}}{\inf}m\left(\mathcal{R}\right),$$ only depends on the number of dimensions: $m_{k}^{*}=c_{k}$ with $c_{k}=\frac{1}{12}$ when $k=1$ and $c_{k}\rightarrow\frac{1}{2\pi e}$ when $k\rightarrow\infty$. Thus the distortion rate function satisfies $$\begin{aligned}
\underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D\left(R\right) & =\int\frac{f\left(\mathbf{y}\right)}{\lambda_{k}^{2/k}\left(\mathbf{y}\right)}m_{k}^{*}d\mathbf{y},\label{eq:distortion-point-density-fn}\end{aligned}$$ where $R$ is the quantization rate per dimension, and $\lambda_{k}\left(\mathbf{y}\right)$ denotes the point density function. In this case, the integral $$\int_{\mathcal{M}}\lambda_{k}\left(\mathbf{y}\right)d\mathbf{y}$$ gives the fraction of quantization levels belonging to $\mathcal{M}$ for all measurable sets $\mathcal{M}\subset\mathbb{R}^{k}$. For simplicity, we have assumed that $\lambda_{k}\left(\mathbf{y}\right)$ is continuous on $\mathbb{R}^{k}$. For fixed $m_{k}^{*}$, the problem of designing an asymptotically optimal quantizer can be reduced to the problem of finding the point density function $\lambda_{k}^{*}\left(\mathbf{y}\right)$ that minimizes (\[eq:distortion-point-density-fn\]). By Hölder’s inequality, the optimal point density function is given by $$\lambda_{k}^{*}\left(\mathbf{y}\right)=\frac{f^{k/\left(k+2\right)}\left(\mathbf{y}\right)}{\int f^{k/\left(k+2\right)}\left(\mathbf{y}\right)\cdot d\mathbf{y}},$$ and the asymptotic distortion rate function is therefore $$\begin{aligned}
\underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D^{*}\left(R\right) & =c_{k}\left(\int f^{k/\left(k+2\right)}\left(\mathbf{y}\right)\cdot d\mathbf{y}\right)^{\frac{k+2}{k}}.\label{eq:DRF-in-point-density-fn}\end{aligned}$$ If the source $\mathbf{Y}$ is Gaussian distributed with covariance matrix $\mathbf{\Sigma}>0$, then the asymptotic distortion rate function (\[eq:DRF-in-point-density-fn\]) can be explicitly evaluated as $$\begin{aligned}
\underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D^{*}\left(R\right) & =c_{k}\left|2\pi\mathbf{\Sigma}\right|^{\frac{1}{k}}\left(\frac{k+2}{k}\right)^{\frac{k+2}{2}}\label{eq:DRF-point-density-Gaussian}\\
& =\left|\mathbf{\Sigma}\right|^{\frac{1}{k}}\left(1+o_{K}\left(1\right)\right),\nonumber \end{aligned}$$ where $o_{K}\left(1\right)\rightarrow0$ as $K\rightarrow\infty$, and the last equality follows from the fact that $c_{k}\rightarrow\frac{1}{2\pi e}$ and $\left(\frac{k+2}{2}\right)^{\frac{k+2}{2}}\rightarrow e$ as $k\rightarrow\infty$.
We present next the key results used for proving the upper bounds in (\[eq:DRF-non-uniform\]) and (\[eq:ub-DRF-VQ-1\]).
\[pro:ub-mismatch\] Let $\mathbf{Y}_{0}\in\mathbb{R}^{k}$ be a Gaussian random vector with zero mean and covariance matrix $\mathbf{\Sigma}_{0}$. Let $\left\{ \mathfrak{q}_{R}\left(\cdot\right)\right\} $, where the subscript $R$ denotes the quantization rate, be a sequence of quantizers designed to achieve the asymptotic distortion rate function for Gaussian source $\mathcal{N}\left(\mathbf{0},\mathbf{\Sigma}_{1}\right)$ with $\mathbf{0}<\mathbf{\Sigma}_{1}\in\mathbb{R}^{k\times k}$. Apply $\mathfrak{q}_{R}\left(\cdot\right)$ to $\mathbf{Y}_{0}$. If $\mathbf{\Sigma}_{0}<\mathbf{\Sigma}_{1}$, then$$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{2^{2R}}{k}\mathrm{E}_{Y_{0}}\left[\left\Vert \mathbf{Y}_{0}-\mathfrak{q}_{R}\left(\mathbf{Y}_{0}\right)\right\Vert _{2}^{2}\right]\nonumber \\
& \le c_{k}\left(2\pi\mathbf{\Sigma}_{1}\right)^{\frac{1}{k}}\left(\frac{k+2}{k}\right)^{\frac{k+2}{2}}.\label{eq:ub-03}\end{aligned}$$
First assume that $\mathbf{0}<\mathbf{\Sigma}_{0}$. Let $f_{0}\left(\mathbf{y}\right)$ and $f_{1}\left(\mathbf{y}\right)$ be the probability density functions for $\mathbf{Y}_{0}$ and $\mathbf{Y}_{1}$, respectively. Denote $\mathrm{E}_{Y_{0}}\left[\left\Vert \mathbf{Y}_{0}-\mathfrak{q}_{R}\left(\mathbf{Y}_{0}\right)\right\Vert _{2}^{2}\right]$ by $D\left(R\right)$. It is clear that $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D\left(R\right)\nonumber \\
& =c_{k}\int\frac{f_{0}\left(\mathbf{y}\right)}{\left(\lambda_{k,1}^{*}\left(\mathbf{y}\right)\right)^{2/k}}d\mathbf{y}\nonumber \\
& =c_{k}\int\frac{f_{0}\left(\mathbf{y}\right)}{f_{1}^{2/\left(k+2\right)}\left(\mathbf{y}\right)}d\mathbf{y}\cdot\left(\int f_{1}^{k/\left(k+2\right)}\left(\mathbf{y}\right)d\mathbf{y}\right)^{\frac{2}{k}}.\label{eq:ub-01}\end{aligned}$$ We upper bound the first integral as follows $$\begin{aligned}
& \int\frac{f_{0}\left(\mathbf{y}\right)}{f_{1}^{2/\left(k+2\right)}\left(\mathbf{y}\right)}d\mathbf{y}\nonumber \\
& =\frac{\left|2\pi\mathbf{\Sigma}_{1}\right|^{\frac{1}{k+2}}}{\left|2\pi\mathbf{\Sigma}_{0}\right|^{1/2}}\int\exp\left\{ -\frac{1}{2}\mathbf{y}^{*}\left(\mathbf{\Sigma}_{0}^{-1}-\frac{2}{k+2}\mathbf{\Sigma}_{1}^{-1}\right)\mathbf{y}\right\} d\mathbf{y}\nonumber \\
& \overset{\left(a\right)}{=}\frac{\left|2\pi\mathbf{\Sigma}_{1}\right|^{\frac{1}{k+2}}}{\left|2\pi\mathbf{\Sigma}_{0}\right|^{1/2}}\frac{\left|2\pi\bm{\Sigma}_{0}\right|^{1/2}}{\left|\mathbf{I}_{k}-\frac{2}{k+2}\bm{\Sigma}_{0}\bm{\Sigma}_{1}^{-1}\right|^{1/2}}\nonumber \\
& \overset{\left(b\right)}{\le}\left|2\pi\bm{\Sigma}_{1}\right|^{\frac{1}{k+2}}\left(\frac{k+2}{k}\right)^{\frac{k}{2}}\nonumber \\
& =\int f_{1}^{\frac{k}{k+2}}\left(\bm{x}\right)d\bm{x},\label{eq:ub-02}\end{aligned}$$ where $\left(a\right)$ holds because $$\begin{aligned}
& \mathbf{\Sigma}_{0}^{-1}-\frac{2}{k+2}\bm{\Sigma}_{1}^{-1}\\
& =\bm{\Sigma}_{0}^{-1}\left(\mathbf{I}_{k}-\frac{2}{k+2}\bm{\Sigma}_{0}\bm{\Sigma}_{1}^{-1}\right)\\
& =\left[\left(\mathbf{I}_{k}-\frac{2}{k+2}\bm{\Sigma}_{0}\bm{\Sigma}_{1}^{-1}\right)^{-1}\bm{\Sigma}_{0}\right]^{-1},\end{aligned}$$ and $\left(b\right)$ follows from the assumption $\mathbf{\Sigma}_{0}<\mathbf{\Sigma}_{1}$. Substituting (\[eq:ub-02\]) into (\[eq:ub-01\]), one obtains$$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D\left(R\right)\\
& \le c_{k}\left(\int f_{1}^{k/\left(k+2\right)}\left(\mathbf{y}\right)d\mathbf{y}\right)^{\frac{k+2}{k}}\\
& =c_{k}\left|2\pi\mathbf{\Sigma}_{1}\right|^{\frac{1}{k}}\left(\frac{k+2}{k}\right)^{\frac{k+2}{2}},\end{aligned}$$ which will be used to prove the upper bounds in (\[eq:DRF-non-uniform\]) and (\[eq:ub-DRF-VQ-1\]).
Suppose that $\left|\mathbf{\Sigma}_{0}\right|=0$ (some of the eigenvalues of $\mathbf{\Sigma}_{0}$ are zero). Since $\mathbf{\Sigma}_{0}<\mathbf{\Sigma}_{1}$, when $\epsilon>0$ is sufficiently small, we have $\mathbf{0}<\mathbf{\Sigma}_{\epsilon}:=\mathbf{\Sigma}_{0}+\epsilon\mathbf{I}<\mathbf{\Sigma}_{1}$. Let $f_{\epsilon}\left(\mathbf{y}\right)$ be the probability density function of Gaussian vector with zero mean and variance $\mathbf{\Sigma}_{\epsilon}$. Then, $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{2^{R}}{k}D\left(R\right)\\
& =c_{k}\int\frac{f_{0}\left(\mathbf{y}\right)}{\left(\lambda_{k,1}^{*}\left(\mathbf{y}\right)\right)^{2/k}}d\mathbf{y}\\
& =c_{k}\int\frac{\underset{\epsilon\rightarrow0}{\lim}f_{\epsilon}\left(\mathbf{y}\right)}{\left(\lambda_{k,1}^{*}\left(\mathbf{y}\right)\right)^{2/k}}d\mathbf{y}\\
& \overset{\left(c\right)}{\le}c_{k}\underset{\epsilon\rightarrow0}{\lim\inf}\int\frac{f_{\epsilon}\left(\mathbf{y}\right)}{\left(\lambda_{k,1}^{*}\left(\mathbf{y}\right)\right)^{2/k}}d\mathbf{y}\\
& \overset{\left(d\right)}{\le}c_{k}\left|2\pi\mathbf{\Sigma}_{1}\right|^{\frac{1}{k}}\left(\frac{k+2}{k}\right)^{\frac{k+2}{2}},\end{aligned}$$ where $\left(c\right)$ follows from Fatou’s lemma [@Royden1988_real_analysis], and $\left(d\right)$ follows from the first part of this proof. This proves the proposition.
### Lower Bounds for Scalar Quantization
$ $
We prove the lower bound in (\[eq:DRF-non-uniform\]). Given Assumptions II, each $Y_{i}$, $1\le i\le m$, is a linear combination of Gaussian random variables, and therefore each $Y_{i}$ is a Gaussian random variable itself. For a given $i$ and a given $T$, the mean and the variance of $Y_{i}$ are $\mathrm{E}\left[Y_{i}\right]=0$ and $\sigma_{i,T}^{2}=\mathrm{E}\left[Y_{i}^{2}\right]=\sum_{j\in T}\varphi_{i,j}^{2}$, respectively. The variance depends on the row index $i$ and the support set $T$. We calculate the average variance across all rows and all support sets as $$\begin{aligned}
\bar{\sigma}^{2} & =\frac{1}{m}\sum_{i=1}^{m}\left(\frac{1}{{N \choose K}}\sum_{T}\;\sum_{j\in T}\varphi_{i,j}^{2}\right)\nonumber \\
& =\frac{1}{m}\frac{1}{{N \choose K}}\sum_{T}\;\sum_{j\in T}\left(\sum_{i=1}^{m}\varphi_{i,j}^{2}\right)\nonumber \\
& \overset{\left(a\right)}{=}\frac{1}{m}\frac{1}{{N \choose K}}\sum_{j=1}^{N}\left(\sum_{T:\; j\in T}\left\Vert \bm{\varphi}_{j}\right\Vert _{2}^{2}\right)\nonumber \\
& \overset{\left(b\right)}{=}\frac{1}{m}\frac{1}{{N \choose K}}\sum_{j=1}^{N}{N-1 \choose K-1}\left\Vert \bm{\varphi}_{j}\right\Vert _{2}^{2}\nonumber \\
& \overset{\left(c\right)}{=}\frac{K}{m}\frac{1}{N}\sum_{j=1}^{N}\left\Vert \bm{\varphi}_{j}\right\Vert _{2}^{2}\nonumber \\
& \overset{\left(d\right)}{=}\frac{K}{m}\mu_{1},\label{eq:sigma-bar}\end{aligned}$$ where
[[$\left(a\right)$]{}]{}
: is obtained by exchanging the sums over $T$ and $j$,
[[$\left(b\right)$]{}]{}
: holds because for any given $1\le j\le N$, there are ${N-1 \choose K-1}$ many subsets $T$ containing the index $j$,
[[$\left(c\right)$]{}]{}
: is due to the fact that ${N-1 \choose K-1}/{N \choose K}=K/N$,
[[$\left(d\right)$]{}]{}
: follows from the definition (\[eq:def-mu-1\]).
Suppose that one deals with the ideal case: the support set $T$ is known before taking the measurements; and for different values of $i$ and $T$, we are allowed to use different quantizers. Given $i$ and $T$, we apply the optimal quantizer for the Gaussian random variable $\sqrt{\frac{m}{K}}Y_{i}$, so that the quantization distortion of $Y_{i}$ satisfies $$\underset{R\rightarrow\infty}{\lim}2^{2R}D_{i,T}^{*}\left(R\right)=\frac{\pi\left(\frac{m}{K}\sigma_{i,T}^{2}\right)}{2}\sqrt{3},$$ which is a direct application of (\[eq:DRF-in-point-density-fn\]) with $k=1$. Taking the average over all $i$ and all $T$ gives $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{1}{m}\sum_{i=1}^{m}\mathrm{E}_{T}\left[2^{2R}D_{i,T}^{*}\left(R\right)\right]\\
& =\frac{1}{m}\sum_{i=1}^{m}\frac{1}{{T \choose K}}\sum_{T}\left(\underset{R\rightarrow\infty}{\lim}2^{2R}D_{i,T}^{*}\left(R\right)\right)\\
& =\frac{1}{m}\frac{1}{{T \choose K}}\sum_{i=1}^{m}\sum_{T}\left(\frac{\pi\left(\frac{m}{K}\sigma_{i,T}^{2}\right)}{2}\sqrt{3}\right)\\
& =\frac{\pi\mu_{1}}{2}\sqrt{3},\end{aligned}$$ where the last equality follows from (\[eq:sigma-bar\]).
However, the support set $T$ is unknown before taking the measurements. Furthermore, the same quantizer has to be employed for different choices of $i$ and $T$. Thus, for every $R$, $i$ and $T$, $\mathrm{E}_{Y_{i}}\left[\frac{m}{K}\left|Y_{i}-\hat{Y}_{i}\right|^{2}\right]\ge D_{i,T}^{*}\left(R\right)$. As a result,$$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}\mathrm{E}_{T}\left[\mathrm{E}_{Y}\left[\left\Vert \hat{\mathbf{Y}}-\mathbf{Y}\right\Vert _{2}^{2}\right]\right]\\
& =\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{{N \choose T}}\sum_{T}\frac{1}{m}\sum_{i=1}^{m}\frac{m}{K}\mathrm{E}_{Y}\left[\left(\hat{y}_{i}-y_{i}\right)^{2}\right]\\
& \ge\underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{{N \choose T}}\sum_{T}\frac{1}{m}\sum_{i=1}^{m}D_{i,T}^{*}\left(R\right)\\
& =\frac{\pi\mu_{1}}{2}\sqrt{3}.\end{aligned}$$ Since the above derivation is valid for all $K$, $m$ and $N$, the claim in (\[eq:DRF-non-uniform\]) holds.
The result in (\[eq:DRF-uniform\]) for uniform quantizers can be proved using similar arguments. For the ideal case, given $i$ and $T$, apply the optimal *uniform* quantizer for the standard Gaussian random variable to $\sqrt{\frac{m}{K}}y_{i}$. The corresponding distortion rate function for this case was characterized in [@Neuhoff2001_IT_optimal_uniform_scalar_quantization] and s given by $$\underset{R\rightarrow\infty}{\lim}2^{2R}D_{u,i,T}^{*}\left(R\right)=\frac{4}{3}\sigma_{i,T}^{2}\ln2.$$ Therefore, $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim\inf}\frac{2^{2R}}{K}\mathrm{E}_{T}\left[\mathrm{E}_{Y}\left[\left\Vert \hat{\mathbf{Y}}-\mathbf{Y}\right\Vert _{2}^{2}\right]\right]\\
& \ge\frac{4}{3}\mu_{1}\ln2,\end{aligned}$$ which completes the proof of (\[eq:DRF-uniform\]).
### The Upper Bound for Scalar Quantization
$ $
By the definition of $\mu_{2}$ in (\[eq:def-mu-2\]), the variance of the Gaussian random variable $\sqrt{\frac{m}{K}}Y_{i}$ is upper bounded by $\mu_{2}$ uniformly for all $i$ and all $T$. For each quantization rate $R$, we design the optimal quantizer for a Gaussian source with variance $\mu_{2}$ and apply this quantizer to quantize all components of $\mathbf{Y}$. Using (\[eq:ub-03\]), one can show that the quantization distortion for all $i$ and $T$ satisfies$$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim\sup}\frac{2^{2R}}{K}\mathrm{E}_{T}\mathrm{E}_{\mathbf{Y}}\left[\left\Vert \hat{\mathbf{Y}}-\mathbf{Y}\right\Vert _{2}^{2}\right]\\
& \le\frac{\pi}{2}\mu_{2}\sqrt{3},\end{aligned}$$ which proves the upper bound in (\[eq:DRF-non-uniform\]).
### The Lower Bound for Vector Quantization
$ $
The basic idea for proving the lower bound in (\[eq:lb-DRF-VQ\]) is similar to that behind (\[eq:DRF-non-uniform\]). For each $T$, a lower bound on the minimum achievable distortion is derived. The average distortion taken over all the sets $T$ serves as a lower bound of the overall distortion-rate function.
Suppose the ideal case where we have prior knowledge of $T\in{\left[N\right] \choose K}$. We study the distortion rate function for every given $T$. The measurement vector $\mathbf{Y}$ is Gaussian distributed with zero mean and covariance matrix $\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}$, where $\mathbf{\Phi}_{T}$ consists of the columns of $\mathbf{\Phi}$ indexed by $T$. The singular value decomposition of $\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}$ gives $\mathbf{U}_{T}\mathbf{\Lambda}_{T}\mathbf{U}_{T}^{*}$, where $\mathbf{U}_{T}\in\mathbb{R}^{m\times m}$ has orthonormal columns and $\mathbf{\Lambda}_{T}=\mathrm{diag}\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{m}\right)$ is the diagonal matrix formed by the singular values $\lambda_{1}\ge\lambda_{2}\ge\cdots\ge\lambda_{m}$. Note that $\lambda_{i}\left(\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}\right)=\lambda_{i}\left(\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}\right)$ for $1\le i\le K$. According to Assumption III.1, the measurement matrix $\mathbf{\Phi}$ satisfies the RIP with constant parameter $\delta_{K}$ , which implies that $1-\delta_{K}\le\lambda_{i}\left(\mathbf{\Phi}_{T}^{*}\mathbf{\Phi}_{T}\right)\le1+\delta_{K}$ for all $1\le i\le K$. It can be concluded that $1-\delta_{K}\le\lambda_{i}\le1+\delta_{K}$ for $1\le i\le K$ and $\lambda_{i}=0$ for $K+1\le i\le m$. As a result, $\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}=\mathbf{U}_{T,K}\mathbf{\Lambda}_{T,K}\mathbf{U}_{T,K}^{*}$ where $\mathbf{U}_{T,K}\in\mathbb{R}^{m\times K}$ contains the first $K$ columns of $\mathbf{U}_{T}$ and $\mathbf{\Lambda}_{T,K}\in\mathbb{R}^{K\times K}$ is the diagonal matrix formed by the $K$ largest singular values. Denote the matrix formed by the last $m-K$ columns of $\mathbf{U}$ by $\mathbf{U}_{T,K}^{\perp}$: clearly, $\mathbf{U}_{T}=\left[\mathbf{U}_{T,K}|\mathbf{U}_{T,K}^{\perp}\right]$.
The best quantization strategy is to quantize $\bar{\mathbf{Y}}=\mathbf{U}_{T,K}^{*}\mathbf{Y}$ so that no quantization bit is used for the “trivial signal” $\left(\mathbf{U}_{T,K}^{\perp}\right)^{*}\mathbf{Y}$. It is clear that $\bar{\mathbf{Y}}\sim\mathcal{N}\left(\mathbf{0},\mathbf{\Lambda}_{T,K}\right)$ and $\mathbf{0}<\mathbf{\Lambda}_{T,K}$. The corresponding asymptotic distortion rate function is therefore $$\begin{aligned}
\underset{R\rightarrow\infty}{\lim}\frac{2^{2mR/K}}{K}D_{T}^{*}\left(R\right) & \overset{\left(\ref{eq:DRF-point-density-Gaussian}\right)}{=}c_{K}\left(2\pi\mathbf{\Lambda}_{T,K}\right)^{\frac{1}{K}}\left(\frac{K+2}{K}\right)^{\frac{K+2}{2}}\\
& \ge\left(1-\delta_{K}\right)\left(1+o_{K}\left(1\right)\right),\end{aligned}$$ where the $2^{2mR/K}$ term comes from the fact that the total quantization rate $mR$ is used to quantize a $K$-dimensional signal. Since this lower bound is valid for all $T\in{\left[N\right] \choose K}$, we have proved the lower bound in (\[eq:lb-DRF-VQ\]).
### The Upper Bound for Vector Quantization
$ $
Let $\epsilon>0$ be a small constant. Let $\left\{ \mathfrak{q}_{R}\left(\cdot\right)\right\} $ be a sequence of quantizers that approaches the asymptotic distortion rate function for quantizing $\bar{\mathbf{Y}}\sim\mathcal{N}\left(\mathbf{0},\left(1+\delta_{K}+\epsilon\right)\mathbf{I}_{m}\right)$. To prove the upper bound in (\[eq:ub-DRF-VQ-1\]), apply the quantizer sequence $\left\{ \mathfrak{q}_{R}\left(\cdot\right)\right\} $ to $\mathbf{Y}$. For every $T\in{\left[N\right] \choose K}$, $\mathbf{Y}\sim\mathcal{N}\left(\mathbf{0},\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}\right)$. According to the Assumption III.1, $\mathbf{\Phi}_{T}\mathbf{\Phi}_{T}^{*}<\left(1+\delta_{K}+\epsilon\right)\mathbf{I}_{m}$. Applying Proposition \[pro:ub-mismatch\], we have $$\begin{aligned}
& \underset{R\rightarrow\infty}{\lim}\frac{2^{2R}}{m}\mathrm{E}_{\mathbf{Y}}\left[\left\Vert \mathbf{Y}-\mathfrak{q}_{R}\left(\mathbf{Y}\right)\right\Vert _{2}^{2}\right]\\
& \le\left(1+\delta_{K}+\epsilon\right)\left(1+o_{M}\left(1\right)\right).\end{aligned}$$ The upper bound in (\[eq:ub-DRF-VQ-1\]) is proved by taking the limit $\epsilon\downarrow0$.
\[sub:pf-well-define\]The Existence and Uniqueness of $\left(\tilde{\mathbf{x}},\tilde{\mathbf{y}}\right)$ in Equation (\[eq:def-proj-q-solution\])
---------------------------------------------------------------------------------------------------------------------------------------------------
Consider the optimization problem $$\underset{\left(\mathbf{x},\mathbf{y}\right)\in\mathbb{R}^{\left|T\right|}\times\mathcal{R}_{\hat{\mathbf{Y}}}}{\min}\;\left\Vert \mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}\right\Vert _{2},\label{eq:well-def-01}$$ which is equivalent to$$\underset{\left(\mathbf{x},\mathbf{y}\right)\in\mathbb{R}^{\left|T\right|}\times\mathcal{R}_{\hat{\mathbf{Y}}}}{\min}\;\left\Vert \left[-\mathbf{\Phi}_{T}\;\mathbf{I}\right]\left[\begin{array}{c}
\mathbf{x}\\
\mathbf{y}\end{array}\right]\right\Vert _{2}^{2}.\label{eq:well-def-02}$$ Note that the objective function is convex and the constraint set is convex and closed. The optimization problem (\[eq:well-def-02\]) has at least one solution. Note that the matrix $\left[-\mathbf{\Phi}_{T}\;\mathbf{I}\right]$ does not have full row-rank. Hence, the solution may not be unique: the set $\mathcal{Q}$ defined in (\[eq:def-proj-q-plane\]) gives all the possible solutions, and is convex and closed.
Let $\mathfrak{P}$ be the projection function from $\mathbb{R}^{\left|T\right|}\times\mathbb{R}^{m}$ to $\mathbb{R}^{m}$, i.e., $\mathfrak{P}\left(\left(\mathbf{x},\mathbf{y}\right)\right)=\mathbf{y}$. Since the set $\mathcal{Q}$ is convex, the set $\mathfrak{P}\left(\mathcal{Q}\right)$ is also convex. The quadratic optimization problem $$\underset{\mathbf{y}\in\mathfrak{P}\left(\mathcal{Q}\right)}{\min}\left\Vert \hat{\mathbf{Y}}-\mathbf{y}\right\Vert _{2}$$ has a unique solution. Denote this unique solution by $\tilde{\mathbf{y}}$. Furthermore, recall our assumption that $\mathbf{\Phi}_{T}$ has full column rank. For any given $\mathbf{y}\in\mathbb{R}^{m}$, the solution of $$\underset{\mathbf{x}\in\mathbb{R}^{\left|T\right|}}{\min}\;\left\Vert \mathbf{y}-\mathbf{\Phi}_{T}\mathbf{x}\right\Vert _{2}$$ is therefore unique. As a result, there exists a unique $\tilde{\mathbf{x}}\in\mathbb{R}^{\left|T\right|}$ such that $\left(\tilde{\mathbf{x}},\tilde{\mathbf{y}}\right)\in\mathcal{Q}$. This establishes the existence and uniqueness of the point $\left(\tilde{\mathbf{x}},\tilde{\mathbf{y}}\right)$.
![\[fig:dist-measurements\]Distortion in the measurements.](figures/MeasurementDistortion)
[^1]: A random variable $X$ is said to be *Subgaussian* if there exist positive constants $c_{1}$ and $c_{2}$ such that $$\Pr\left(\left|X\right|>x\right)\le c_{1}e^{-c_{2}x^{2}}\quad\forall x>0.$$
One property of Subgaussian distributions is that they have a well defined moment generating function. Note that the Gaussian and Bernoulli distributions are special cases of the Subgaussian distribution.
|
---
abstract: 'Electron and ion energization (i.e., heating and nonthermal acceleration) is a fundamental, but poorly understood, outcome of plasma turbulence. In this work, we present new results on this topic from particle-in-cell simulations of driven turbulence in collisionless, relativistic electron-ion plasma. We focus on temperatures such that ions (protons) are sub-relativistic and electrons are ultra-relativistic, a regime relevant for high-energy astrophysical systems such as hot accretion flows onto black holes. We find that ions tend to be preferentially heated, gaining up to an order of magnitude more energy than electrons, and propose a simple empirical formula to describe the electron-ion energy partition as a function of the ratio of electron-to-ion gyroradii (which in turn is a function of initial temperatures and plasma beta). We also find that while efficient nonthermal particle acceleration occurs for both species in the ultra-relativistic regime, nonthermal electron populations are diminished with decreasing temperature whereas nonthermal ion populations are essentially unchanged. These results have implications for modeling and interpreting observations of hot accretion flows.'
author:
- Vladimir Zhdankin
- 'Dmitri A. Uzdensky'
- 'Gregory R. Werner'
- 'Mitchell C. Begelman'
title: Electron and ion energization in relativistic plasma turbulence
---
[^1]
[*Introduction.*]{}— Plasma energization via turbulent dissipation is a fundamental topic in plasma physics. It involves a number of important questions that are difficult to address with analytic theory, including: What fraction of injected energy is dissipated into each of the constituent particle species (electrons, ions, etc.)? Are there collisionless mechanisms of thermal coupling between electrons and ions, sufficient to keep their corresponding temperatures comparable? Is there nonthermal particle acceleration (NTPA), and if so, what are the characteristics of the resulting energetic electron and ion populations?
The answers to these questions have important implications for myriad space and astrophysical systems. For motivation in this paper, we consider the example of radiatively inefficient accretion flows (RIAFs) around black holes. RIAFs comprise tenuous, collisionless plasma with relativistic electrons subject to radiative cooling. Such an accretion flow is at risk of collapsing into a collisional thin disk. To explain the survival of RIAFs, models require the ambient turbulence to heat ions preferentially, which establishes a “two-temperature” plasma with sufficient ion pressure to avoid collapse [@shapiro_lightman_eardley_1976; @ichimaru_1977; @rees_etal_1982; @narayan_yi_1995; @quataert_gruzinov_1999]. An accurate, comprehensive theoretical prescription for the electron and ion heating rates would be extremely valuable for phenomenological models or global magnetohydrodynamic (MHD) simulations of RIAFs [@ressler_etal_2015; @foucart_etal_2015; @ball_etal_2018b; @ryan_etal_2018]. However, so far, only simplified (linearized, non-radiative, non-relativistic) analytic models [@howes_2010; @quataert_gruzinov_1999] and empirical fitting formulae from idealized kinetic simulations [e.g., @matthaeus_etal_2016; @rowan_sironi_narayan_2017; @werner_etal_2018; @kawazura_etal_2018] exist. Furthermore, the conceivable existence of collective plasma phenomena that thermally couple electrons and ions could complicate the establishment of high ion temperatures [@begelman_chiueh_1988; @sironi_narayan_2015].
RIAFs are also notable for their highly nonthermal radiative signatures [e.g., @yuan_etal_2003; @remillard_mcclintock_2006; @yuan_narayan_2014]. Understanding the physical processes responsible for the underlying NTPA is essential for interpreting observations. Collisionless plasma turbulence driven by the magnetorotational instability [@balbus_hawley_1991] is a primary candidate source of NTPA, with supporting evidence from local (shearing-box) kinetic simulations [e.g., @riquelme_etal_2012; @hoshino_2013; @hoshino_2015; @kunz_etal_2016; @inchingolo_etal_2018] and MHD test-particle simulations [@kimura_etal_2016]. An essential next step is to systematically determine the properties of NTPA in realistic parameter regimes at large system size.
First-principles kinetic simulations offer empirical insights necessary to build a rigorous understanding of electron and ion heating, thermal coupling, and NTPA in parameter regimes relevant for RIAFs. In this work, we use particle-in-cell (PIC) simulations of driven turbulence to study electron and ion energization (i.e., heating and NTPA) in relativistic plasmas. We focus on the regime where ions (protons) are sub-relativistic and electrons are ultra-relativistic, taking temperatures in the range $m_e c^2 \lesssim T \lesssim m_i c^2$, which we refer to as the [*semirelativistic*]{} regime [@werner_etal_2018]. This regime is amenable to fully kinetic simulations using the real electron-proton mass ratio, as demonstrated by recent PIC studies of magnetic reconnection [@guo_etal_2016; @rowan_sironi_narayan_2017; @werner_etal_2018; @ball_etal_2018], due to the large relativistic mass of electrons reducing the kinetic scale separation with ions. Our results indicate that turbulence in this physical regime can efficiently energize ions, while electron energization becomes less efficient with decreasing temperature.
[ ]{}\
[*Method.*]{}— The simulation set-up is similar to our previous work on pair (electron-positron) plasma turbulence [e.g., @zhdankin_etal_2018a]. We perform the simulations with the explicit electromagnetic PIC code [Zeltron]{} [@cerutti_etal_2013] using charge-conserving current deposition [@esirkepov_2001]. The domain is a periodic cubic box of size $L^3$ with uniform mean magnetic field $\boldsymbol{B}_0 = B_0 \hat{\boldsymbol{z}}$. We initialize particles from a uniform Maxwell-Jüttner distribution with particle density per species $n_0$ and equal electron and ion temperatures, $T_e = T_i = T_0$. We then drive strong turbulence (with rms fluctuating magnetic field $\delta B_{\rm rms} \sim B_0$) at low wavenumber modes ($k = 2\pi/L$) by applying a randomly fluctuating external current density [@tenbarge_etal_2014]. We set $32$ particles per cell per species in all production runs.
There are then three free dimensionless parameters: the initial temperature relative to ion rest mass energy, $\theta_{i0} = T_0/m_i c^2$, the initial plasma beta, $\beta_0 = 16 \pi n_0 T_0 / B_0^2$, and the ratio of the driving scale to the ion Larmor radius, $L/2\pi\rho_{i0}$ (subscript zero refers to initial values of parameters). The characteristic Larmor radii are given by $\rho_{s} = (\gamma_{s}^2 - 1)^{1/2} m_s c^2/e B_{\rm rms}$, where $\gamma_{s} = 1 + E_s/m_s c^2$ (for species $s \in \{e, i\}$) are the mean particle Lorentz factors, $E_s$ are the mean particle kinetic energies (for species $s$), and $B_{\rm rms}$ is the rms total magnetic field. In the semirelativistic regime ($m_e/m_i \ll \theta_{i0} \ll 1$), the separation between the electron and ion Larmor radii is given by $\rho_{e0}/\rho_{i0} \sim \theta_{i0}^{1/2}$, and the separation between the Larmor radius and skin depth scales as $\rho_{e0}/d_{e0} \sim \rho_{i0}/d_{i0} \sim \beta_0^{1/2}$. In the fully relativistic limit ($\theta_{i0} \gg 1$), the particle inertia is set by the relativistic mass, making the system similar to a pair plasma ($\rho_{e0} = \rho_{i0}$ and $d_{e0} = d_{i0}$). Thus, $\theta_{i0}$ controls the electron-ion scale separation. In our simulations, we fix the cell size to $\delta x = \min{(\rho_e/2, d_e/2)}$, i.e., relative to electron scales. For given plasma parameters, $L$ is thus proportional to the number of cells in each direction. For a fixed number of cells, obtaining a large ion kinetic range ($\rho_i/\rho_e$) comes at the expense of the inertial range ($L/2\pi\rho_i$), and vice versa. Finally, we note that in the semirelativistic regime, the initial Alfvén velocity scales as $v_{A0}/c \sim (\theta_{i0}/\beta_0)^{1/2}$; thus, the turbulent motions become increasingly sub-relativistic with decreasing $\theta_{i0}$.
Our primary scan is performed with $256^3$-cell and $512^3$-cell simulations with $\theta_{i0}$ varying in the range $[1/2048, 10]$ at fixed $\beta_0 = 4/3$; we do a secondary scan with $\beta_0$ varying in the range $[1/12, 64/3]$ at fixed $\theta_{i0} = 1/16$. In addition, we performed three $768^3$ simulations with $\theta_{i0} \in \{ 1/1024, 1/256, 1/64 \}$ (at $\beta_0 = 4/3$) and one $1024^3$ simulation with $\theta_{i0} = 1/256$ and $\beta_0 = 4/3$ (and $L/2\pi\rho_{i0} = 8.8$ and $\rho_{i0}/\rho_{e0} = 9.3$).
![\[fig:spec\] Top: Magnetic energy spectrum, compensated by $k_\perp^{5/3}$, for $1024^3$ electron-ion simulation (red) compared to a similar $1024^3$ pair-plasma simulation (blue). Power-law scalings are shown for reference (green). Bottom: Evolution of plasma parameters in the $1024^3$ simulation, including $\delta B_{\rm rms}/B_0$ (magenta), $\beta$ (red), and $\theta_i$ (blue; computed from particle energy assuming a thermal distribution). Also shown is the evolution of the electron-ion energy ratio $E_e/E_i$ (yellow), the electron-ion energy gain ratio $\Delta E_e/\Delta E_i$ (black), and ratio of electron-to-ion Larmor radii $\rho_e/\rho_i$ (green).](spec_mag_1024cube_pairtoo_prod.png "fig:"){width="0.95\columnwidth"} ![\[fig:spec\] Top: Magnetic energy spectrum, compensated by $k_\perp^{5/3}$, for $1024^3$ electron-ion simulation (red) compared to a similar $1024^3$ pair-plasma simulation (blue). Power-law scalings are shown for reference (green). Bottom: Evolution of plasma parameters in the $1024^3$ simulation, including $\delta B_{\rm rms}/B_0$ (magenta), $\beta$ (red), and $\theta_i$ (blue; computed from particle energy assuming a thermal distribution). Also shown is the evolution of the electron-ion energy ratio $E_e/E_i$ (yellow), the electron-ion energy gain ratio $\Delta E_e/\Delta E_i$ (black), and ratio of electron-to-ion Larmor radii $\rho_e/\rho_i$ (green).](params_1024cube_prod2.png "fig:"){width="0.95\columnwidth"}
![\[fig:partition\] Top: Ratio of electron-to-ion energy gain, $\Delta E_e/\Delta E_i$, for varying $\theta_{i0}$ at $t v_{A0}/L = 6$ and $\beta_{0} = 4/3$; two sizes are compared, $256^3$ (blue) and $512^3$ (red). Inset: similar for varying $\beta_{0}$ (at fixed $\theta_{i0} = 1/16$). The fit by $(\rho_e/\rho_i)^{2/3}$ (measured at $t v_{A0}/L = 6$) is also shown (dashed). Bottom: $\Delta E_e/\Delta E_i$ versus mean $\rho_e/\rho_i$ for all simulations (ignoring $256^3$ cases that have not converged with system size), measured over intervals of duration $L/v_{A0}$ starting at $t v_{A0}/L \in \{ 3,4,5 \}$.](partition_inset_relabel.png "fig:"){width="0.95\columnwidth"} ![\[fig:partition\] Top: Ratio of electron-to-ion energy gain, $\Delta E_e/\Delta E_i$, for varying $\theta_{i0}$ at $t v_{A0}/L = 6$ and $\beta_{0} = 4/3$; two sizes are compared, $256^3$ (blue) and $512^3$ (red). Inset: similar for varying $\beta_{0}$ (at fixed $\theta_{i0} = 1/16$). The fit by $(\rho_e/\rho_i)^{2/3}$ (measured at $t v_{A0}/L = 6$) is also shown (dashed). Bottom: $\Delta E_e/\Delta E_i$ versus mean $\rho_e/\rho_i$ for all simulations (ignoring $256^3$ cases that have not converged with system size), measured over intervals of duration $L/v_{A0}$ starting at $t v_{A0}/L \in \{ 3,4,5 \}$.](partition_vs_rhoratio_relabel.png "fig:"){width="0.95\columnwidth"}
[ ]{}\
[*Results.*]{}— We first consider the magnetic energy spectrum, integrated over wavenumbers parallel to the guide field $\boldsymbol{B}_0$ and directions perpendicular to $\boldsymbol{B}_0$, which we denote $E_{\rm mag}(k_\perp)$, where $k_\perp$ is the wavenumber perpendicular to $\boldsymbol{B}_0$. We show $E_{\rm mag}(k_\perp)$ compensated by $k_\perp^{5/3}$ and time-averaged from $4.3 L/v_{A0}$ to $5.7 L/v_{A0}$, for the $1024^3$ case ($\theta_{i0} = 1/256$, $\beta_0 = 4/3$) in the top panel of Fig. \[fig:spec\]. To illustrate the effects of ions, we compare this to the spectrum from a similar $1024^3$ relativistic pair-plasma simulation (taken from our previous work [@zhdankin_etal_2018b]). Both simulations are consistent with a power-law spectrum with index near $-5/3$ at large scales ($k_\perp \rho_i \lesssim 1$ for electron-ion and $k_\perp \rho_e \lesssim 1$ for pair), broadly consistent with inertial-range MHD turbulence phenomenology [e.g., @goldreich_sridhar_1995]. The spectrum for the electron-ion case is significantly steeper in the ion kinetic range (between $k_\perp \rho_i =1$ and $k_\perp \rho_e = 1$), although not a clear power law; for reference, we show a comparison to a power law with index $-2.8$, often measured in the ion kinetic range for non-relativistic plasmas (including the solar wind [e.g., @alexandrova_etal_2009; @sahraoui_etal_2009; @kiyani_etal_2009; @alexandrova_etal_2012; @kiyani_etal_2015] and simulations [e.g. @boldyrev_perez_2012; @told_etal_2015; @cerri_etal_2017; @groselj_etal_2018]). A definitive measurement of the spectrum in the ion kinetic range requires larger simulations with lower $\theta_{i0}$ (larger $\rho_i/\rho_e$), in order to simultaneously resolve a long inertial range and ion kinetic range. In the electron kinetic range ($k_\perp \rho_e \gtrsim 1$), there appears to be a power law with index near $-4$, similar to the sub-Larmor spectrum in the pair-plasma case [@zhdankin_etal_2018a].
In the bottom panel of Fig. \[fig:spec\], we show the evolution of the physical parameters $\beta$, $\delta B_{\rm rms}/B_0$, and $\theta_i$ (defined as $2 E_i/ 3 m_i c^2$) for the $1024^3$ simulation. Due to continuous energy injection, $\theta_i$ and $\beta$ both steadily increase over the simulation. We also show the electron-to-ion ratios of kinetic energy gains $\Delta E_e/\Delta E_i$, total kinetic energies $E_e/E_i$, and Larmor radii $\rho_e/\rho_i$. We find that these measured quantities vary only weakly with time after $\sim 3 L/v_{A0}$; this is also true in most of our other simulations (not shown). The approximate constancy of these quantities with time allows us to perform robust measurements of energy partition, as we discuss next.
In Fig. \[fig:partition\], we show the ratio of electron-to-ion energy gain, $\Delta E_e/\Delta E_i$, as a function of various parameters. In the top panel and inset, we show $\Delta E_e/\Delta E_i$ as a function of $\theta_{i0}$ (at fixed $\beta_0 = 4/3$) and as a function of $\beta_0$ (at fixed $\theta_{i0} = 1/16$), measured from the initial time to $t v_{A0}/L = 6$ (arbitrarily chosen; other times give similar results). We find that $\Delta E_e/\Delta E_i \approx 1$ at $\theta_{i0} \gtrsim 1$ (i.e., in the relativistic regime, as expected theoretically) and decreases with lower $\theta_{i0}$, reaching $\Delta E_e/\Delta E_i \sim 0.1$ at $\theta_{i0} = 1/2048$ (near the transition between the semirelativistic regime and the fully sub-relativistic regime, where $T_0/m_e c^2 = \theta_{i0} m_i/m_e \sim 1$). Comparison of the $256^3$ and $512^3$ simulations indicates that results are converged with respect to system size except at sufficiently low $\theta_{i0}$, low $\beta_0$, or high $\beta_0$ (in these exceptions, $L/2\pi\rho_i \sim 1$ at late times so the driving interferes with kinetic processes).
![\[fig:distributions\] Top: evolution of ion four-velocity distribution $f_i(u/c)$ to a power law with fitted index $\alpha_i \approx 2.9$ (black dashed), spanning up to $u_{{\rm max},i}$ (green dashed). Middle: similar for electron four-velocity distribution $f_e(u/c)$, which evolves to a power law with fitted index $\alpha_e \approx 3.8$. Bottom: evolution of the fitted power-law indices $\alpha_s$ for ions (solid) and electrons (dashed) for $512^3$ simulations with varying $\theta_{i0}$.](disti_prod.png "fig:"){width="0.95\columnwidth"} ![\[fig:distributions\] Top: evolution of ion four-velocity distribution $f_i(u/c)$ to a power law with fitted index $\alpha_i \approx 2.9$ (black dashed), spanning up to $u_{{\rm max},i}$ (green dashed). Middle: similar for electron four-velocity distribution $f_e(u/c)$, which evolves to a power law with fitted index $\alpha_e \approx 3.8$. Bottom: evolution of the fitted power-law indices $\alpha_s$ for ions (solid) and electrons (dashed) for $512^3$ simulations with varying $\theta_{i0}$.](diste_prod.png "fig:"){width="0.95\columnwidth"} ![\[fig:distributions\] Top: evolution of ion four-velocity distribution $f_i(u/c)$ to a power law with fitted index $\alpha_i \approx 2.9$ (black dashed), spanning up to $u_{{\rm max},i}$ (green dashed). Middle: similar for electron four-velocity distribution $f_e(u/c)$, which evolves to a power law with fitted index $\alpha_e \approx 3.8$. Bottom: evolution of the fitted power-law indices $\alpha_s$ for ions (solid) and electrons (dashed) for $512^3$ simulations with varying $\theta_{i0}$.](index_vs_time_prod.png "fig:"){width="0.95\columnwidth"}
Intriguingly, we find that the results can be well fit by the time-dependent empirical formula, $$\begin{aligned}
\Delta E_e/ \Delta E_i \sim (\rho_e/\rho_i)^{2/3} \, . \label{eq:emp}\end{aligned}$$ Note that $\rho_e/\rho_i$ is a nontrivial function of $\theta_{i0}$, $\beta_0$, and, to a lesser extent, time. The bottom panel of Fig. \[fig:partition\] explicitly shows the scaling of Eq. \[eq:emp\] compared to all simulations in our scan, where $\Delta E_e/\Delta E_i$ is now measured over intervals of duration $L/v_{A0}$ starting at $t v_{A0}/L \in \{ 3, 4, 5 \}$, to represent the short-term heating during fully developed turbulence.
We next describe results on NTPA. In Fig. \[fig:distributions\], we show the time evolution of the distributions $f_s(u)$ of four-velocities $u$, for ions and electrons ($s \in \{e, i\}$) in a representative semirelativistic case ($768^3$, $\theta_{i0} = 1/64$, $\beta_0 = 4/3$). We find that power-law tails gradually form over a number of dynamical times ($\sim 15 L/v_{A0}$, in this case), and become fully developed when the most energetic particles begin to accumulate at the system-size limited velocity, $u_{{\rm max},s} = L e B_0/2 m_s c$. To characterize the distributions, we measure the power-law indices $-\alpha_s = d\log{f_s}/d\log{u}$ at the geometric mean of the peak of the distribution and $u_{{\rm max},s}$. The ion distribution attains a fitted power-law index $\alpha_i \approx 2.9$, while the electrons attain $\alpha_e \approx 3.8$, indicating that ion acceleration is more efficient in this regime. Intriguingly, the late-time power-law index for ions is similar to that for the relativistic pair-plasma case at the same plasma beta [@zhdankin_etal_2018b]. Note that nonthermal ions are essentially relativistic in this example, despite being initialized well within the sub-relativistic regime; studying the transition of the power law through $u/c \sim 1$ will require even larger simulations with lower $\theta_{i0}$.
To illustrate the parameter dependence of the nonthermal distributions, we show the time evolution of $\alpha_s$ for simulations with varying $\theta_{i0}$ (fixed $\beta_0 = 4/3$) in the bottom panel of Fig. \[fig:distributions\]. Note that $\alpha_s$ decreases in time and does not saturate at a well-defined value, due to the pile-up of particles near $u_{{\rm max},s}$ influencing the measurement of $\alpha_s$ at late times [c.f., @zhdankin_etal_2018b]. We find that ions always reach $\alpha_i \sim 3$ before the pile-up becomes significant; this is a similar value to that in the fully relativistic case ($\theta_{i0} = 10$), although it takes a longer time to reach this value at low $\theta_{i0}$ (consistent with the diffusive particle acceleration timescale increasing with decreasing $v_{A0}/c$ [@zhdankin_etal_2018b]). The electron distributions, however, become softer (i.e., larger $\alpha_e$) when $\theta_{i0}$ is decreased. Hence, our results indicate that NTPA for ions remains as efficient in the semirelativistic regime as in the ultrarelativistic regime, while it becomes inefficient for electrons in the limit of low $\theta_{i0}$.
To further characterize the NTPA, we decompose the particle distributions into thermal and nonthermal components. To do this, we define the thermal part to be a Maxwell-Jüttner distribution with temperature and normalization such that the corresponding peak coincides with the peak of the measured distribution; we also consider any excess of the measured distribution at energies below the peak value to be part of the thermal component. The nonthermal population is then defined to be the difference between the measured distribution and the thermal fit.
We show the fraction of the kinetic energy in the nonthermal population, $E_{{\rm nth},s}$, and the fraction of particles in the nonthermal population, $N_{{\rm nth},s}$, for electrons and ions ($s\in\{e,i\}$) as functions of $\theta_{i0}$ (fixed $\beta_0 = 4/3$ and $t v_{A0}/L = 6$) in Fig. \[fig:decomp\]. We find that the nonthermal energy fraction is roughly constant for ions (between $60-70\%$), but declines with decreasing $\theta_{i0}$ for electrons (from $\sim 70\%$ to $\sim 6\%$). The nonthermal number fraction is qualitatively similar to this, with $\sim 30\%$ of both particle species being nonthermal for $\theta_{i0} \gtrsim 1$ and the fraction of nonthermal electrons declining to $\sim 2\%$ at low $\theta_{i0}$. This confirms that nonthermal ion energization is significant in the semirelativistic regime, while electron energization is diminished. Interestingly, we find that the dependence of $E_{{\rm nth},e}$ on $\theta_{i0}$ mirrors the overall energy partition, being well fit by $0.7 (\rho_e/\rho_i)^{2/3}$ (similar to Eq. \[eq:emp\]), suggesting that NTPA is linked to the available energy budget. The nonthermal fractions also have a $\beta_0$ dependence (not shown), such that the fractions increase (decrease) with decreasing (increasing) $\beta_0$.
![\[fig:decomp\] Nonthermal energy (solid) and number (dashed) fractions for ions (red) and electrons (blue). The scaling $0.7 (\rho_e/\rho_i)^{2/3}$, tracing energy partition, is also shown (black).](decomp_nonthermal_prod.png){width="0.95\columnwidth"}
[ ]{}\
[*Conclusions.*]{}— In this Letter, we investigated electron and ion energization in collisionless plasma turbulence in the semirelativistic regime ($m_e c^2 \lesssim T \lesssim m_i c^2$), where electrons are ultra-relativistic and ions are sub-relativistic. We used PIC simulations to perform a parameter scan in initial temperature $T_0 = \theta_{i0} m_i c^2$ that covered nearly the entire semirelativistic regime. This work thus fills a void between previous studies of turbulence in the non-relativistic regime (typically studied with reduced plasma models) and in the fully relativistic (i.e., pair plasma) regime. This study is primarily relevant for high-energy astrophysical systems with relativistic and nonthermal components, such as RIAFs.
Our results support the prevailing view that turbulent electron-ion plasmas evolve toward a non-equilibrium, “two-temperature” state. In particular, we find that ions reach a higher temperature than electrons in most of the explored parameter space. At a glance, this result appears to differ from non-relativistic analytical and numerical studies that find preferential electron heating at low $\beta$ [e.g., @howes_2010; @kawazura_etal_2018]. Aside from relativistic effects, this difference can be attributed to the fact that our numerical set-up lacks an energy sink: the absence of cooling prevents low $\beta$ from being sustained for more than a few large-scale dynamical times, so simulations tend to be in the $\beta \gtrsim 1$ regime at late times, where preferential ion heating may be expected based on those previous works. A more detailed comparison of our results to non-relativistic plasmas is deferred to future work.
Our results indicate that NTPA is efficient for both species in the fully relativistic regime, but becomes inefficient for electrons when temperature is decreased through the semirelativistic regime (in contrast to ions, which continue to be efficiently accelerated). To produce hard nonthermal electron radiative signatures, astrophysical systems then require either low $\beta$ or ions with near-relativistic temperature. Cosmic ray acceleration, on the other hand, can occur even if ions are initially sub-relativistic. We caution, however, that extrapolating these conclusions regarding NTPA (and, to some extent, energy partition) to large system size is nontrivial and requires a separate scaling study [@zhdankin_etal_2017; @zhdankin_etal_2018b], which can perhaps be connected to MHD test-particle approaches [e.g., @dmitruk_etal_2004; @dalena_etal_2014; @lynn_etal_2014].
This paper constitutes the first numerical investigation of plasma turbulence in the semirelativistic regime using first-principles PIC simulations. Our results, including the empirical formula for the energy partition (Eq. \[eq:emp\]), will be useful for modeling RIAFs and for guiding future theoretical efforts toward understanding turbulent particle energization. It is tempting to connect this empirical formula to the scaling of the turbulent fluctuations in the inertial range [as in, e.g., @matthaeus_etal_2016] or kinetic range, but this requires a careful analysis of the turbulence statistics and dissipation mechanisms in this regime. Hence, we leave a physical basis for this formula to future work.
The authors acknowledge support from NSF grant AST-1411879 and NASA ATP grants NNX16AB28G and NNX17AK57G. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. This work also used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. This work used the XSEDE supercomputer Stampede2 at the Texas Advanced Computer Center (TACC) through allocation TG-PHY160032 [@xsede].
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{}, @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1109/MCSE.2014.80)
[^1]: Einstein fellow
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[ ]{}\
Rei <span style="font-variant:small-caps;">Inoue</span>
\
*Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo\
Komaba 3–8–1, Meguro, Tokyo 153-8902, Japan.*\
**Abstract:** We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_F$ of polynomial matrices. Let $X$ be the algebraic curve given by the common characteristic equation for $\boldsymbol{\mathcal{M}}_F$. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of $X$. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As the application, we discuss the algebraic completely integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.\
Introduction
============
The algebro-geometric structure of the completely integrable Hamiltonian systems was unveiled around 1980 (see [@DubMatNov76; @Krichever78; @MoerMum79; @AdlerMoer80; @ReySemenov94] and references therein), and has been extensively studied. It was a remarkable discovery that the Hamiltonian flows of the systems are linearized on algebraic varieties like the Jacobi variety $J(X)$ of an algebraic curve $X$. Many of the systems are described by the Lax equation of (Laurent) polynomial matrices of a spectral parameter, and $X$ comes from its fixed characteristic equation which gives the level set of the Lax matrix. Typically the flows are linearized by the following procedure; $$\begin{aligned}
\label{procedure}
\begin{split}
\text{System}
&~~ \stackrel{\text{(I)}}{\longrightarrow} ~~ \text{Lax matrix} ~( \rightarrow ~ X)
\\
&~~ \stackrel{\text{(II)}}{\longrightarrow} ~~ \text{Div}_{\text{eff}}(X)
\\
&~~ \stackrel{\text{(III)}}{\longrightarrow} ~~ J(X)
\end{split}\end{aligned}$$ where Div$_{\text{eff}}(X)$ is the set of effective divisors. The arrows (II) and (III) are respectively induced by the eigenvector map and the Abel map. In many cases, the linearization of the flows are related to the Lie algebraic symmetry of Lax matrices [@ReySemenov94]. On the other hand, in [@Griffiths85] the condition of the linearization was discussed based on a cohomological interpretation of the Lax equation.
Roughly speaking, (I) is heuristic, and (II) and (III) are systematic. By the Abel-Jacobi theorem (III) is understood in general framework, but (II) depends on the Lax matrix very much. Beauville showed that if we replace the Lax matrix with a set of gauge equivalence classes of polynomial matrices, (II) becomes an isomorphism [@Beauville90]. He further proved that over the tangent space of the set there exists the $g$ dimensional invariant vector field linearized on $J(X)$, where $g$ is the genus of $X$. As claimed in [@Smirnov-Zeitlin0203], to study concrete integrable systems we need to choose the orbit which gives the representative of the gauge equivalence class. In fact, Mumford already gave an important example when $X$ was a hyperelliptic curve [@Mumford-Book], and introduced the set of representatives with the explicit isomorphic maps (II) and (III). The dynamical system he introduced is called Mumford system, and has been studied from many points of view [@DonagiMarkman96; @Medan99; @SmirnovNakayashiki00; @Van1638; @Vivolo03]. Recently Smirnov and Zeitlin constructed the representative of the wider class of gauge equivalence classes, by starting with $N$ by $N$ monodromy matrices of some special forms [@Smirnov-Zeitlin0111; @Smirnov-Zeitlin0203]. They constructed the isomorphism (II) by making use of the separation of variables (SoV) a lá Sklyanin [@Sklyanin95].
In this paper, we consider the extension of [@Smirnov-Zeitlin0111], and construct the isomorphic map (II) for a certain class of monodromy matrices. We introduce $N$ by $N$ monodromy matrices $\mathbf{T}_{m;n_1,n_2}(z)$ ($n_1 = 1,\cdots, N-1,~ n_2 = 1, \cdots, N$), whose entries are polynomials of a spectral parameter $z$ of degree $m$. We fix a level set of $\mathbf{T}_{m;n_1,n_2}(z)$, where the characteristic polynomial of $\mathbf{T}_{m;n_1,n_2}(z)$ is fixed to be $F_{m;n_1,n_2}(z,w) \in \mathbb{C}[z,w]$. We write this set as $\{\mathbf{T}_{m;n_1,n_2}(z)\}_{F_{m;n_1,n_2}}$. The characteristic equation $F_{m;n_1,n_2}(z,w) = 0$ defines the complete algebraic curve $X$ and the set of gauge equivalence classes $\boldsymbol{\mathcal{M}}_{F_{m;n_1,n_2}}$. Let $\{\mathbf{M}(z)\}_{F_{m;n_1,n_2}}$ be the set of representatives of $\boldsymbol{\mathcal{M}}_{F_{m;n_1,n_2}}$. Starting with the level set $\{\mathbf{T}_{m;n_1,n_2}(z)\}_{F_{m;n_1,n_2}}$, we study the following diagram $$\begin{aligned}
\label{eigenvector-map}
\begin{split}
&\{\mathbf{T}_{m;n_1,n_2}(z)\}_{F_{m;n_1,n_2}}
~ \stackrel{\text{(b)}}{\longrightarrow}~
X(g)
\\
&~~~~~~~~~~~~~~~~~~ \text{\scriptsize (a)} \downarrow \hspace{1.3cm}
\nearrow \text{\scriptsize (c)}
\\
&\hspace{2.5cm}\{\mathbf{M}(z)\}_{F_{m;n_1,n_2}}
\end{split} \end{aligned}$$ where $X(g) \subset \text{Div}_{\text{eff}}(X)$ is the set of effective divisors of degree $g$. The map (a) is the gauge transformation, (b) is based on SoV, and we construct these two so as to make the diagram commutative. Then the map (c) produces nothing but the case that (II) is isomorphic. In [@Smirnov-Zeitlin0111], the maps in were given for $\{\mathbf{T}_{m;1,1}(z)\}_{F_{m;1,1}}$ of general $N$. We study in detail for general cases of $N=2$ and $3$ here.
Next, as an application we study the integrable Hamiltonian structure of the extended Lotka-Volterra lattice. This is defined by the differential-difference equation $$\label{Bogo}
\frac{\mathrm{d} V_n}{\mathrm{d} t}
=
2 \,V_n \,
\sum_{k=1}^{N-1} \left(V_{n+k} - V_{n-k} \right),$$ where $V_n \equiv V_n(t) \in \mathbb{C}$ for $n \in \mathbb{Z}$. This model has the Hamiltonian structure and a family of integrals of motion in involution [@Bogo88; @Suris94; @InoueHikami98-Bogo]. When the model is infinite dimensional, the $N=2$ case is known as the lattice KdV hierarchy [@FadTak86], and the general $N>2$ case is related to the lattice $N$-reduced KP hierarchy [@BonoraColCon96; @AntoBelovChal97; @FrenkelReshSemenov97; @HikamiSogoInoue97]. We set a periodic boundary condition $V_{n+L} = V_n$ for $L \in \mathbb{Z}_{\geq 2N-1}$, and write LV($N,L$) for this finite dimensional model. In [@Vanhae01] the algebraic completely integrability of LV($2,L$) was shown based on the analogues of the Mumford system, and its invariant manifold is associated with an affine part of the Prym variety. Now, as the sequel of [@Inoue02-JBogo] we study the integrability of LV($N,L$) by applying the structure . We show that the monodromy matrix of LV($N,L$) is related to $\mathbf{T}_{m;n_1,n_2}(z)$ where the correspondence $L \leftrightarrow (m,n_1,n_2)$ is determined uniquely, and that the Poisson structure over $\{\mathbf{M}(z)\}_{F_{m;n_1,n_2}}$ is nicely embedded in that of LV($N,L$). These enable SoV to describe explicitly the map (c) as algebraic relations between the divisors in $X(g)$ and the dynamical variables $V_n$’s. Finally we give another proof of the algebraic completely integrability of $N=2$ case, and newly show the $N=3$ case;
\[th:N=2-3\] LV($N,L$) is algebraic completely integrable for $L \in \mathbb{Z}_{\geq 2N-1}$, $N=2$ and $3$.
We believe that it is true for general $N$.
This paper is arranged as follows; in §2, after a preliminary we introduce a class of $N$ by $N$ monodromy matrices $\mathbf{T}_{m;n_1,n_2}(z)$ which satisfy the fundamental Poisson relation with the classical $r$-matrix. By starting with these matrices we explain how to construct the maps in . In §3, we study the $N=2,3$ cases, where the set of representatives $\{\mathbf{M}(z)\}_{F_{m;n_1,n_2}}$ and the eigenvector map (c) are explicitly obtained. In §4, we discuss the Hamiltonian structure of LV($N,L$) and prove Theorem \[th:N=2-3\].
The advantage of our way to investigate LV($N,L$) is that we obtain the isomorphic eigenvector map explicitly written as algebraic relations between the divisor and the dynamical variables. On the other hand, as discussed in [@Vanhae01], for a model given by homogeneous evolution equations like , the Painlevé analysis [@AdlerMoer89] becomes a powerful tool to construct the associated invariant manifold. It may be interesting to study the invariant manifold for LV($N,L$) based on these two viewpoints.
Representatives for $\boldsymbol{\mathcal{M}}_F$ and eigenvector map
====================================================================
Preliminary
-----------
Fix a polynomial $F(z,w)$ of the form $$\begin{aligned}
\label{curve-general}
F(z,w)
\equiv
w^N - f_1(z) w^{N-1} + f_2(z) w^{N-2} - \cdots
+ (-1)^N f_N(z),\end{aligned}$$ where each polynomial $f_i(z)$ satisfies $\text{deg} f_i(z) \leq i m$. Let $X$ be the complete algebraic curve defined by $F(z,w) = 0$. We assume $X$ is smooth, and let $g$ be its genus. Let $\boldsymbol{\mathcal{M}}_F$ be the set of gauge equivalence classes of $N$ by $N$ matrices whose matrix elements are polynomials of $z$ of degree $m \in \mathbb{Z}_{>0}$; $$\begin{aligned}
\label{gauge-equiv}
\boldsymbol{\mathcal{M}}_F
=
\{ \mathbf{M}(z) ~|~
&\deg(\mathbf{M}(z)_{i,j}) \leq m \text{ for all } i,j, ~
\nonumber \\
&\operatorname{Det}\bigl( w \openone - \mathbf{M}(z) \bigr) = F(z,w) \}
~/~ \mathbf{GL}_N(\mathbb{C}).\end{aligned}$$
For $\boldsymbol{\mathcal{M}}_F$ Beauville introduced the isomorphism [@Beauville90] $$\label{M-Div}
\boldsymbol{\mathcal{M}}_F \simeq X(g) - D.$$ Here $X(g)$ is the set of effective divisors $X(g) = X^g / \mathfrak{S}_g \subset
\mathrm{Div}_{\mathrm{eff}}(X)$, $\mathfrak{S}_g$ is the symmetric group and $D$ is a subset of $X(g)$. The Abel map induces the isomorphism, $$\label{Div-Jac}
X(g) - D \simeq J(X) - \Theta,$$ where $D$ is mapped to a $(g-1)$-dimensional subvariety $\Theta$ called the theta divisor of the Jacobi variety $J(X)$.
We call $J(X) - \Theta$ the affine Jacobi variety of $X$ and write $J_{\text{aff}}(X)$ for it. We denote the set of representatives of $\boldsymbol{\mathcal{M}}_F$ using $\{\mathbf{M}(z)\}_F$. Due to and $\{\mathbf{M}(z)\}_F$ gives the matrix realization of $J_{\text{aff}}(X)$. Herewith the arrows (II) and (III) in the procedure becomes isomorphisms (II$^\prime$) and (III$^\prime$); $$\begin{aligned}
\label{iso-procedure}
\begin{split}
\{\mathbf{M}(z)\}_F
~~ &\stackrel{\text{(II$^\prime$)}}{\longrightarrow}
~~ X(g) - D
\\
&\stackrel{\text{(III$^\prime$)}}{\longrightarrow}
~~ J_{\text{aff}}(X).
\end{split}\end{aligned}$$
In this article, we let $\mathbf{M}_N(\mathbb{C})$ be a set of $N$ by $N$ complex matrices, $\mathbf{E}_{i,j}$ be an $N$ by $N$ basic matrix as $(\mathbf{E}_{i,j})_{m,n} = \delta_{m,i}\delta_{n,j}$, and $\vec{e}_i$ be an $N$ dimensional low vector whose entries are zero but $i$-th is one.
Classification of monodromy matrices and $\boldsymbol{\mathcal{M}}_F$
---------------------------------------------------------------------
We introduce lower/upper triangular $N$ by $N$ matrices, $$\begin{aligned}
\label{mu-}
\begin{split}
&\boldsymbol{\mu}^{(i)}_-
=
\text{\tiny $i+1 \rightarrow$}
\begin{pmatrix}
0 & \cdots & & & & & \cdots & 0\\
\vdots & & & & & & & \vdots\\
0 & \cdots & & & & & \cdots & 0\\
\ast & \ast & 0 & \cdots & & & \cdots & 0 \\
\ast & \ast & \ast & 0 & \cdots & & \cdots & 0\\
\vdots & & & \ddots & \ddots & & & \vdots\\
\ast & \cdots & \cdots & \cdots& \ast & 0 & \cdots & 0
\end{pmatrix}, \text{ for } i= 1,\cdots, N-1,
\\ &\hspace*{5.7cm} \text{\tiny $\uparrow N+1-i$}
\\
& \hspace*{3.6cm} \text{\tiny $\downarrow i$}
\\
&\boldsymbol{\mu}^{(i)}_+
=
\begin{pmatrix}
0 & \cdots & 0 & \ast & \cdots & \cdots & \ast \\
\vdots & & & \ddots & \ast & \cdots & \ast\\
\vdots & & & & \ddots & \ddots & \vdots \\
0 & \cdots & & & \cdots & 0 & \ast \\
0 & \cdots & & & & \cdots & 0 \\
\vdots & & & & & & \vdots \\
0 & \cdots & & & & \cdots & 0 \\
\end{pmatrix} \text{\tiny $\leftarrow N+1-i$}~,
~\text{ for } i= 1,\cdots, N,
\end{split}\end{aligned}$$ where [$i \rightarrow$]{} (or [$\downarrow i$]{}) indicates the $i$-th low (or column) of the matrices, and $\ast$ denote non-zero entries which will be constants or variables. For $N \geq 3$ we also use $$\begin{aligned}
\label{mu-0}
\begin{split}
& \hspace*{3.3cm} \text{\tiny $\downarrow i+2$}
\\
&\boldsymbol{\mu}^{(-i)}_-
=
\begin{pmatrix}
\ast & \cdots & \ast & 0 & \cdots & 0 \\
\ast & \cdots & \cdots & \ast & \ddots & \vdots \\
\vdots & & & & \ddots & 0\\
\ast & \cdots & & & \cdots & \ast \\
\ast & \cdots & & & \cdots & \ast \\
\vdots & & & & & \vdots \\
\ast & \cdots & \cdots & \cdots & \cdots & \ast \\
\end{pmatrix}
{\text{\tiny $\leftarrow N-1-i$}},
~~ \text{for $i=0,\cdots N-3$},
\\
&\boldsymbol{\mu}^{(-i)}_+
=
\text{\tiny $i+2 \rightarrow$}
\begin{pmatrix}
\ast & \cdots & \cdots & \cdots &\cdots & \ast \\[1mm]
\vdots & & & & & \vdots \\
\ast & \cdots & & & \cdots & \ast \\[1mm]
\ast & \cdots & & & \cdots & \ast \\
0 & \ddots & & & & \vdots\\
\vdots & \ddots &\ast & \cdots & \cdots & \ast \\
0 & \cdots & 0 & \ast & \cdots & \ast \\
\end{pmatrix}, ~~ \text{for $i=0,\cdots N-3$}.
\\
&\hspace{5.1cm}\text{\tiny $\uparrow N-i-1$}
\end{split}\end{aligned}$$ Using $\boldsymbol{\mu}_j, ~ j \in \mathbb{Z}_{>0}$, we denote $N$ by $N$ matrices whose entries are not identically zero. We write $(\boldsymbol{\mu}_-^{(i)} \cap \boldsymbol{\mu}_+^{(j)})$ for a matrix which has zero at $(j_1,j_2)$ if $(\boldsymbol{\mu}_-^{(i)})_{j_1,j_2}$ or $(\boldsymbol{\mu}_+^{(j)})_{j_1,j_2}$ is zero. Note that $(\boldsymbol{\mu}_-^{(i)} \cap \boldsymbol{\mu}_j)$ and $\boldsymbol{\mu}_-^{(i)}$ have the same form.
First we fix the matrices , and $\boldsymbol{\mu}_j$ for $j = 1,\cdots,m-1$ to be constant matrices in $\mathbf{M}_N(\mathbb{C})$ as $\boldsymbol{\mu}^{(i)}_- \equiv \boldsymbol{\mu}^{(i) 0}_-,
\boldsymbol{\mu}^{(i)}_+ \equiv \boldsymbol{\mu}^{(i) 0}_+$, and $\boldsymbol{\mu}_j \equiv \boldsymbol{\mu}_j^0$. Using these matrices we define a set of $N$ by $N$ polynomial matrices of the spectral parameter $z \in \mathbb{C}$; $$\label{general-T}
\boldsymbol{\mathcal{T}}_{N}(z)
=
\bigl\{ \mathbf{T}_{m;n_1,n_2}^0(z) ~|~
m \in \mathbb{Z}_{>0}, ~
n_1 \in \{1,2,\cdots, N-1\}, ~n_2 \in \{1,2,\cdots, N\} \bigr\},$$ where $\mathbf{T}_{m;n_1,n_2}^0(z)$ are defined as $$\begin{aligned}
\label{T0-general-form}
&\mathbf{T}_{m;n_1,n_2}^0(z)
=
\begin{cases}
&\boldsymbol{\mu}_-^{(n_1) 0} z^m +
\boldsymbol{\mu}_-^{(n_1-N+1) 0} z^{m-1}
+ \boldsymbol{\mu}_2^0 z^{m-2} + \cdots
+ \boldsymbol{\mu}_{m-2}^0 z^2
\\[1mm]
&\hspace{4.8cm}
+ \boldsymbol{\mu}_+^{(n_2-N) 0} z + \boldsymbol{\mu}_+^{(n_2) 0},
\text{ for $m \geq 3$},
\\[2mm]
&\boldsymbol{\mu}_-^{(n_1) 0} z^2 +
(\boldsymbol{\mu}_-^{(n_1-N+1) 0} \cap \boldsymbol{\mu}_+^{(n_2-N) 0}) \, z
+ \boldsymbol{\mu}_+^{(n_2) 0},
\text{ for $m=2$},
\\[2mm]
&(\boldsymbol{\mu}_-^{(n_1) 0} \cap \boldsymbol{\mu}_+^{(n_2-N) 0}) \, z
+ (\boldsymbol{\mu}_-^{(n_1-N+1) 0} \cap \boldsymbol{\mu}_+^{(n_2) 0}),
\text{ for $m=1$}.
\end{cases}\end{aligned}$$ When $\boldsymbol{\mu}_-^{(n_1-N+1)}$ (or $\boldsymbol{\mu}_+^{(n_2-N)}$) is not defined by , set $\boldsymbol{\mu}_-^{(n_1-N+1)} \equiv \boldsymbol{\mu}_1^0$ (or $\boldsymbol{\mu}_+^{(n_2-N)} \equiv \boldsymbol{\mu}_{m-1}^0$).
\[T-F-injection\] The map $$\begin{aligned}
\label{curve-F}
\boldsymbol{\mathcal{T}}_{N}(z) &\rightarrow \mathbb{C}[z,w]; ~
\mathbf{T}_{m;n_1,n_2}^0(z) \mapsto
F_{m;n_1,n_2}(z,w)
=
\operatorname{Det}\bigl( w \openone - \mathbf{T}_{m;n_1,n_2}^0(z) \bigr)
\end{aligned}$$ is injective.
[*Proof.*]{} Since the polynomial $F_{m;n_1,n_2}(z,w)$ has a form as , it is sufficient to check that $\mathbf{T}_{m;n_1,n_2}^0(z) \mapsto f_{N-1}(z)$ is injective. Notice $$f_{N-1}(z) = \operatorname{Det}\mathbf{T}_{m;n_1,n_2}^0(z) \cdot
\operatorname{Tr}(\mathbf{T}_{m;n_1,n_2}^0(z)^{-1}),$$ and the forms of $\boldsymbol{\mu}_+^{(n_1)}$ and $\boldsymbol{\mu}_-^{(n_2)}$ which compose $\mathbf{T}_{m;n_1,n_2}^0(z)$. Then one sees $$\deg f_{N-1}(z) = (N-1)m - n_1 + 1, ~~~
\operatorname{ord}_{z=0} f_{N-1}(z) = n_2-1.$$ Since $n_1 \in \{1, \cdots, N-1 \}$, $f_{N-1}(z)$ is classified by a triple $(m,n_1,n_2)$. In conclusion our claim is approved. $\square$\
Therefore we see that $\mathbf{T}_{m;n_1,n_2}^0(z) \in \boldsymbol{\mathcal{T}}_{N}(z)$ corresponds to $\boldsymbol{\mathcal{M}}_{F_{m;n_1,n_2}}$ injectively.
Next we set the entries of matrices , and $\boldsymbol{\mu}_j$ for $j = 1,\cdots,m-1$ to be variables, and define $N$ by $N$ monodromy matrices $\mathbf{T}_{m;n_1,n_2}(z)$ $(m \in \mathbb{Z}_{>0}, ~ n_1 \in \{1,2,\cdots, N-1\},
~n_2 \in \{1,2,\cdots, N\})$ as same as ; $$\begin{aligned}
\label{T-general-form}
&\mathbf{T}_{m;n_1,n_2}(z)
=
\begin{cases}
&\boldsymbol{\mu}_-^{(n_1)} z^m +
\boldsymbol{\mu}_-^{(n_1-N+1)} z^{m-1}
+ \boldsymbol{\mu}_2 z^{m-2} + \cdots + \boldsymbol{\mu}_{m-2} z^2
\\[1mm]
&\hspace{4.5cm}
+ \boldsymbol{\mu}_+^{(n_2-N)} z + \boldsymbol{\mu}_+^{(n_2)},
\text{ for $m \geq 3$},
\\[2mm]
&\boldsymbol{\mu}_-^{(n_1)} z^2 +
(\boldsymbol{\mu}_-^{(n_1-N+1)} \cap \boldsymbol{\mu}_+^{(n_2-N)}) \,z
+ \boldsymbol{\mu}_+^{(n_2)},
\text{ for $m=2$},
\\[2mm]
&(\boldsymbol{\mu}_-^{(n_1)} \cap \boldsymbol{\mu}_+^{(n_2-N)}) \, z
+ (\boldsymbol{\mu}_-^{(n_1-N+1)} \cap \boldsymbol{\mu}_+^{(n_2)}),
\text{ for $m=1$}.
\end{cases}\end{aligned}$$ To study $\mathbf{T}_{m;n_1,n_2}(z)$, we define a local Lax matrix as $$\begin{aligned}
\label{general-Lax}
\mathbf{L}_n(z) = \sum_{k=1}^{N-1} l_n^{(k)} \mathbf{E}_{k,k+1}
+ z l_n^{(N)} \mathbf{E}_{N,1}
+ z l_n^{(0)} \mathbf{E}_{N,2},\end{aligned}$$ where $l_n^{(k)}$ $(n \in \mathbb{Z}, k = 0,\cdots, N$) are dynamical variables.
\[lemma-Lax\] With the Lax matrix $\mathbf{L}_n(z)$ the following Poisson relation is compatible; $$\begin{aligned}
\label{L-r-poisson}
\{ \mathbf{L}_n(z) \stackrel{\otimes}{,} \mathbf{L}_m(z^\prime) \}
=
\delta_{n,m} [\, \mathbf{r}(z/z^{\prime}) ~,~
\mathbf{L}_n(z) \otimes \mathbf{L}_n(z^\prime) \,],
\end{aligned}$$ where $\mathbf{r}(z)$ is the classical $r$-matrix $$\begin{aligned}
\label{classical-r}
&{\mathbf{r}}(z)
= \frac{z + 1}{z - 1}
\sum_{k=1}^N \mathbf{E}_{k,k} \otimes \mathbf{E}_{k,k}
+
\frac{2}{z-1}
\sum_{1 \leq j < k \leq N}
\Bigl( z \, \mathbf{E}_{k,j} \otimes \mathbf{E}_{j,k}
+ \, \mathbf{E}_{j,k} \otimes \mathbf{E}_{k,j}
\Bigr).\end{aligned}$$
[*Proof.*]{} It is shown by a direct calculation. One easily sees that is consistent with the Poisson bracket algebra for $l_n^{(k)} ~(k = 0,\cdots, N)$ defined as $$\begin{aligned}
\begin{split}
&\{ l_n^{(k)} ~,~ l_m^{(j)} \} = 0,
\text{ for } 1 \leq k,j \leq N,
\\
&\{ l_n^{(0)} ~,~ l_m^{(k)} \} =
\delta_{n,m} (\delta_{k,N} - \delta_{k,1}) l_n^{(0)} l_n^{(k)}.
~~~~~ \square
\end{split}
\end{aligned}$$\
We define integers $m,m_1$ and $m_2$ by $$\begin{aligned}
\label{m-s}
m = \Bigl[\frac{L}{N(N-1)}\Bigr], ~
m_1 = \Bigl[\frac{L}{N-1}\Bigr], ~ m_2 = \Bigl[\frac{L}{N}\Bigr],\end{aligned}$$ and determine $k, k_1$ and $k_2$ using $$\begin{aligned}
\label{k-s}
L=(N-1) m_1 + k_1 = N m_2 + k_2 = N(N-1)m + k.\end{aligned}$$
\[T-Lproduct\] The monodromy matrix $\mathbf{T}_{m;n_1,n_2}(z)$ can be written as a product of $L$ Lax matrix $\mathbf{L}_n(z)$ ; $$\begin{aligned}
\label{LT-corresp}
&z^{-m_2} \prod_{n=1}^{L} \mathbf{L}_n(z)
=
\begin{cases}
\mathbf{T}_{m;1,1}(z), ~\text{ for } k_1 = k_2 = 0,
\\
\mathbf{T}_{m+1;N-k_1,k_2+1}(z), ~\text{ for } k_1, k_2 \neq 0, ~ 0 \leq k_1 - k_2 \leq N-2
\\
\mathbf{T}_{m+2;N-k_1,k_2+1}(z), ~\text{ for } k_1 - k_2 \leq -1,
\end{cases}
\end{aligned}$$ where integers $m,m_2,k_1$ and $k_2$ are defined in and .
See Appendix A for the proof. Due to Lemmas \[lemma-Lax\] and \[T-Lproduct\], it is straightforward to obtain the following proposition;
\[T-r-Poisson\] With the matrix $\mathbf{T}_{m;n_1,n_2}(z)$ the fundamental Poisson relation is compatible; $$\begin{aligned}
\label{T-Poisson}
\{\mathbf{T}_{m;n_1,n_2}(z)
\stackrel{\otimes}{,}
\mathbf{T}_{m;n_1,n_2}(z^\prime) \}
=
[\, \mathbf{r}(z/z^{\prime}) ~,~ \mathbf{T}_{m;n_1,n_2}(z) \otimes
\mathbf{T}_{m;n_1,n_2}(z^\prime) \,].\end{aligned}$$
Let $\mathcal{A}_C$ be the Poisson bracket algebra over the polynomial ring generated by the coefficients of entries in $\mathbf{T}_{m;n_1,n_2}(z)$, whose defining relation is . Then implies
[@FadTak87] \[prop:Poisson\] (i) The determinant of $\mathbf{T}_{m;n_1,n_2}(z)$ belongs to the center of $\mathcal{A}_{C}$; $$\{ \mathbf{T}_{m;n_1,n_2}(z) ~,~
\operatorname{Det}\mathbf{T}_{m;n_1,n_2}(z^\prime) \} = 0.$$ (ii) The coefficients of the characteristic polynomial of $\mathbf{T}_{m;n_1,n_2}(z)$ are Poisson commutative; $$\{ \operatorname{Det}\bigl( w \openone - \mathbf{T}_{m;n_1,n_2}(z) \bigr)~,~
\operatorname{Det}\bigl( w^\prime \openone - \mathbf{T}_{m;n_1,n_2}(z^\prime) \bigr)
\} = 0.$$
Using we define the level set of $\mathbf{T}_{m;n_1,n_2}(z)$ as $$\begin{aligned}
\{\mathbf{T}_{m;n_1,n_2}(z)\}_{F_{m;n_1,n_2}}
=
\{\mathbf{T}_{m;n_1,n_2}(z) ~|~
\operatorname{Det}( w \openone - \mathbf{T}_{m;n_1,n_2}(z))
= F_{m;n_1,n_2}(z,w)\}.\end{aligned}$$ Let $X$ be the complete algebraic curve determined by $F_{m;n_1,n_2}(z,w)=0$, and its genus be $g$. We consider the cases of $g \geq 1$. In general $\{\mathbf{T}_{m;n_1,n_2}(z)\}_{F_{m;n_1,n_2}}$ constitutes a variety whose dimension is greater than $g$. Since the isomorphism implies that $\boldsymbol{\mathcal{M}}_{F_{m;n_1,n_2}}$ is a $g$ dimensional variety, we state a problem to construct the map (a) which gives the set of representatives $\{\mathbf{M}(z)\}_{F_{m;n_1,n_2}}$ as follows;
\[problem\] For $\mathbf{T}_{m;n_1,n_2}(z)$ find a gauge matrix $\mathbf{S}$ on $\mathbf{T}_{m;n_1,n_2}(z)$, such that the set $$\begin{aligned}
\label{M-rep}
\begin{split}
&\{ \mathbf{M}(z) \}_{F_{m;n_1,n_2}}
=
\{ \mathbf{M}(z) = \mathbf{S} \, \mathbf{T}_{m;n_1,n_2}(z) \,\mathbf{S}^{-1}
~|~
\operatorname{Det}( w \openone - \mathbf{M}(z)) = F_{m;n_1,n_2}(z,w)\}
\end{split}\end{aligned}$$ constitutes a $g$ dimensional variety.
We note that the matrix $\mathbf{M}(z)$ has the same degree as $\mathbf{T}_{m;n_1,n_2}(z)$ as a polynomial matrix, and write it as $$\label{general-M}
\mathbf{M}(z) =
\boldsymbol{\eta}_0 z^m + \boldsymbol{\eta}_{1} z^{m-1} +
\cdots + \boldsymbol{\eta}_{m-1} z + \boldsymbol{\eta}_m.$$ Here the variable matrices $\boldsymbol{\eta}_i$ do not depend on $z$. Once the above problem is solved, the Poisson bracket algebra generated by the matrix elements of $\boldsymbol{\eta}_i$ is induced by $\mathcal{A}_C$, and we let $\mathcal{A}_{M}$ be this algebra. Due to Proposition \[prop:Poisson\], the coefficients of $\operatorname{Det}\bigl( w \openone - \mathbf{T}_{m;n_1,n_2}(z) \bigr)$ constitute the commuting subalgebra of $\mathcal{A}_{M}$.
In the following, without any notice we pay attention to an element of $\boldsymbol{\mathcal{T}}_N(z)$ , and abbreviate $F_{m;n_1,n_2}$ to $F$.
Eigenvector map and SoV
-----------------------
Following [@Smirnov-Zeitlin0111; @Smirnov-Zeitlin0203] we introduce the eigenvector map (b) by making use of SoV. Sklyanin refined the technique invented to solve the spectral problem of the quantum Toda lattice, and introduced the method called SoV based on the $R$-matrix structure of the monodromy matrices (See [@Sklyanin85; @Sklyanin95] and references therein). The SoV for the monodromy matrices of $SL(N)$ symmetry has been studied in detail. The cases of $N=2$ and $3$ are done by Sklyanin himself [@Sklyanin85; @Sklyanin92], and the extension to the general $N$ cases are clarified in [@Scott94; @Gekhtman95].
For classical systems this method derives the canonically conjugate variables from the poles of the eigenvector of the monodromy matrix. We review this mechanism following [@Sklyanin95]. Let $\mathbf{T}(z)$ be an $N$ by $N$ monodromy matrix which satisfies the fundamental Poisson relation as . Then the eigenvector of $\mathbf{T}(z)$ called the Baker-Akhiezer function is defined as $$\begin{aligned}
\mathbf{T}(z) \vec\phi(z) = w \vec\phi(z),
~~~ \sum_{n=1}^N a_n(z) \phi_n(z) = 1,\end{aligned}$$ where $\vec\phi(z) = (\phi_1(z), \cdots, \phi_N(z))$, and $w$ is the eigenvalue. The second equation is a normalization which uniquely determines $\vec\phi(z)$. When $\vec\phi(z)$ has a pole at $z=z_i$, the residues $\vec\phi_i = (\phi_{1,i},\cdots, \phi_{N,i}) = \mathrm{res}_{z = z_i}
\vec\phi(z)$ satisfy $$\begin{aligned}
\label{BA-function}
\mathbf{T}(z_i) \vec\phi_i = w_i \vec\phi_i,
~~~
\sum_{n=1}^N a_n(z_i) \phi_{n,i} = 0.\end{aligned}$$ Then the condition to get non-zero vector $\vec\phi_i$ becomes $$\begin{aligned}
\label{SoV-det}
\operatorname{Det}\begin{pmatrix}
a_1(z) & a_2(z) & \cdots & a_N(z) \\
T(z)_{1,1}-w & T(z)_{1,2} & \cdots & T(z)_{1,N} \\
\vdots \\
T(z)_{j-1,1} & T(z)_{j-1,2} & \cdots & T(z)_{j-1,N} \\
T(z)_{j+1,1} & T(z)_{j+1,2} & \cdots & T(z)_{j+1,N} \\
\vdots \\
T(z)_{N,1} & T(z)_{N,2} & \cdots & T(z)_{N,N}-w
\end{pmatrix}
= 0,
~~~ \text{for $j=1, \cdots N$},\end{aligned}$$ where $T(z)_{i,j} = (\mathbf{T}(z))_{i,j}$.
In our case with the monodromy matrix $\mathbf{T}_{m;n_1,n_2}(z)$ , some simple choices of the vector $\vec{a}(z) = (a_1(z), \cdots, a_N(z))$ give SoV, and reduces to two equations on $\mathbf{T}_{m;n_1,n_2}(z)$ as [@Sklyanin95] $$\begin{aligned}
\label{separation}
B(z) = 0, ~~~ w = A(z).\end{aligned}$$ Here $A(z) = A(\mathbf{T}_{m;n_1,n_2}(z))$ is a rational function of $z$ and $B(z) = B(\mathbf{T}_{m;n_1,n_2}(z))$ is a polynomial. Accordingly the zero of $B(z)$, $z_i$ uniquely determines the eigenvalue $w_i= A(z_i)$. The significant benefit of the fundamental Poisson relation is that the variables $(z_i, w_i)$ turn out to be canonically conjugate variables, namely they fulfill the canonical Poisson brackets, $$\begin{aligned}
\{ z_i ~,~ z_j \} = \{ w_i ~,~ w_j \} = 0,
~~~
\{ z_i ~,~ w_j \} = 2 \, \delta_{i,j} z_i w_i.\end{aligned}$$ These variables are nothing but the [*separated variables*]{}, and the equation $B(z) = 0$ is called the separation equation.
When we consider the level set $\{ \mathbf{T}_{m;n_1,n_2}(z) \}_F$, each pair $(z_i,w_i)$ satisfies $F(z_i, w_i) = 0$. We expect that the separation equation has a following form, $$\label{separationB}
B(z) = B_0 z^{f(n_1,n_2)}\prod_{i=1}^{g} (z - z_i),
~~~ f(n_1,n_2) \in \mathbb{Z}_{\geq 0},$$ where $g$ is the genus of the algebraic curve $X$ given by $F(z,w) = 0$. There are certainly some different choices of the separation equations depending on the vector $\vec a(z)$. To make the diagram commutative, we should choose the separation equation invariant under the gauge $\mathbf{S}$ .
To close this section, we mention the subset $D$ which appeared in the isomorphism . We assume $(z_i,w_i) \neq (z_j,w_j)$ for $i \neq j$, and a set of the $g$ separated variables $(z_i,w_i)$ determines an effective divisor $$\begin{aligned}
\label{div-P}
P = \sum_{i=1}^g [(z_i,w_i)] \in X(g). \end{aligned}$$ Then the subset $D$ should be set as [@Smirnov-Zeitlin0111] $$\label{D}
D = \{ P = \sum_{i=1}^g [(z_i, w_i)] ~|~
\operatorname{Det}\bigl( h_i(z_j,w_j) \bigr)_{1 \leq i,j \leq g} = 0 \},$$ where $h_i(z,w)$ are defined by homomorphic one-forms $\sigma_i$ on $X$ [@Griffiths-Book], $$\label{oneform}
\sigma_i(z,w) = \frac{h_i(z,w) \mathrm{d}z}{\frac{\partial}{\partial w} F(z,w)},
~~\text{ for }i = 1, \cdots, g.$$ We remark that the $g$ independent vector fields on a tangent space of $\boldsymbol{\mathcal{M}}_F$ are generated by the coefficients of $F(z,w)$ . The fundamental Poisson relation ensures that the evolution of the divisor $P$ generated by the vector fields is linearized on $J_{\text{aff}}(X)$.
Study of concrete cases
=======================
Starting with $\mathbf{T}_{m;n_1,n_2}(z)$, we study the diagram . We construct the gauge matrix $\mathbf{S}$ which gives the set of representatives $\{\mathbf{M}(z)\}_F$, and the associated separation equation which makes the map (c) well-defined. Then the isomorphic eigenvector map (II$^\prime$) is induced by (c). We explicitly discuss the cases of $N=2$ and $3$ with $g \geq 1$. Further we recall $\{\mathbf{M}(z)\}_F$ associated with $\mathbf{T}_{m;1,1}(z)$ for general $N$ [@Smirnov-Zeitlin0111; @Inoue02-JBogo].
N=2 case
--------
We have matrices $$ \boldsymbol{\mu}_-^{(1)} =
\begin{pmatrix}
0 & 0 \\
\ast & \ast
\end{pmatrix},
~~ \boldsymbol{\mu}_+^{(1)} =
\begin{pmatrix}
\ast & \ast \\
0 & \ast
\end{pmatrix},
~~ \boldsymbol{\mu}_+^{(2)} =
\begin{pmatrix}
0 & \ast \\
0 & 0
\end{pmatrix}.$$ Using them we introduce two matrices, $\mathbf{T}_{m;1,1}(z)$ and $\mathbf{T}_{m;1,2}(z)$, and derive the associated representatives;
\(i) $\mathbf{T}_{m;1,1}(z)$: We have the matrix $$\begin{aligned}
\label{T-m11}
\mathbf{T}_{m;1,1}(z) =
\boldsymbol{\mu}_-^{(1)} z^m + \boldsymbol{\mu}_1 z^{m-1} + \cdots
+ \boldsymbol{\mu}_{m-1} z + \boldsymbol{\mu}_+^{(1)},
~~ \text{for} ~ m \geq 2.\end{aligned}$$ The spectral curve $X$ is given by $$\begin{aligned}
\label{ch-2-even}
\begin{split}
&F(z,w) = \operatorname{Det}(w \openone - \mathbf{T}^0_{m;1,1}(z))
= w^2 - w f_1(z) + f_2(z) = 0, ~~
\\
& ~~\text{where} ~\text{deg}f_1(z) = m, ~~\text{deg}f_2(z) = 2m-1,
\end{split}\end{aligned}$$ and its genus is $g=m-1$. The set $\{\mathbf{M}(z)\}_F$ is obtained as the level set of $$\begin{aligned}
\label{gauge-2-even}
\mathbf{M}(z) =
{\mathbf{S}} \, \mathbf{T}_{m;1,1}(z) \,{\mathbf{S}}^{-1},
~~
{\mathbf{S}} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e}_1 \boldsymbol{\mu}_1
\end{pmatrix},\end{aligned}$$ where $\mathbf{M}(z)$ has the form as $$\mathbf{M}(z) = \boldsymbol{\eta}_0 z^m + \cdots + \boldsymbol{\eta}_m,
~~ \text{where}~
\boldsymbol{\eta}_0 =
\begin{pmatrix}
0 & 0 \\
\ast & \ast
\end{pmatrix},
~~
\boldsymbol{\eta}_1 =
\begin{pmatrix}
0 & 1 \\
\ast & \ast
\end{pmatrix},$$ and other $\boldsymbol{\eta}_i$ are the matrices without zero entries.
\(ii) $\mathbf{T}_{m;1,2}(z)$: This is the case with the matrix as $$\begin{aligned}
\mathbf{T}_{m;1,2}(z) =
\boldsymbol{\mu}_-^{(1)} z^{m} + \boldsymbol{\mu}_1 z^{m-1}
+ \cdots + \boldsymbol{\mu}_{m-1} z + \boldsymbol{\mu}_+^{(2)},
~~ \text{for} ~ m \geq 2,\end{aligned}$$ and $X$ is determined by $$\begin{aligned}
\label{ch-2-odd}
\begin{split}
&F(z,w) = \operatorname{Det}( w \openone - \mathbf{T}^0_{m;1,2}(z) )
= w^2 - w z f_1^\prime(z) + z f_2^\prime(z) = 0,
\\
&~ \text{where} ~
\text{deg} f_1^\prime(z) = m-1, ~~ \text{deg} f_2^\prime(z) = 2m-2.
\end{split}\end{aligned}$$ The genus of $X$ is $m-1$. By using the gauge matrix $$\begin{aligned}
\label{gauge-2-odd}
{\mathbf{S}} =
\begin{pmatrix}
\vec{e}_2 \boldsymbol{\mu}_-^{(1)} \\
\vec{e}_2
\end{pmatrix},\end{aligned}$$ we obtain $\mathbf{M}(z)$ with $$\boldsymbol{\eta}_0 =
\begin{pmatrix}
\ast & 0 \\
1 & 0
\end{pmatrix},
~~
\boldsymbol{\eta}_{m} =
\begin{pmatrix}
0 & \ast \\
0 & 0
\end{pmatrix},$$ and the other $\boldsymbol{\eta}_i$ are the matrices with no zero-entries.
One sees that both of $\{\mathbf{T}_{m;1,1}(z)\}_F$ and $\{\mathbf{T}_{m;1,2}(z)\}_F$ constitute the algebraic varieties of $m$ dimension which is equal to the genus of $X$. For example, by the definition one sees that $\mathbf{T}_{m;1,1}(z)$ has $(4m+1)$ variables to which the fixed characteristic equation gives $3m+1$ relations. Then we see $\{\mathbf{T}_{m;1,1}(z)\}_F$ constitutes the $m$-dimensional algebraic variety. The gauge matrix $\mathbf{S}$ reduce $\{\mathbf{T}_{m;1,1}(z)\}_F$ by one dimension, and $\{\mathbf{M}(z)\}_F$ becomes $m$ dimensional. By choosing the vector $\vec{a}(z) = (a_1(z),a_2(z))$ the separation equation is obtained as $$\begin{aligned}
\label{separation2*2}
B(z) =
\begin{cases}
{T}(z)_{1,2} = B_0 \displaystyle{\prod_{i=1}^{m-1} (z- z_i)},
~ \vec{a}(z) = (1,0) \text{ for (i)},\\
{T}(z)_{2,1} = B_0 \, z \displaystyle{\prod_{i=1}^{m-1} (z- z_i)},
~ \vec{a}(z) = (0,1) \text{ for (ii)},\\
\end{cases}\end{aligned}$$ where $T(z)_{i,j} = (\mathbf{T}_{m;n_1,n_2}(z))_{i,j}$. In both cases $B(z)$ generally has $m-1$ non-zero zeros; $z_1, \cdots, z_{m-1}$, and each of them gives an eigenvalue $$w_i =
\begin{cases}
T(z_i)_{2,2}, \text{ for (i)},\\
T(z_i)_{1,1}, \text{ for (ii)}.\\
\end{cases}$$ In the level set $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ the points $(z_i,w_i)$ on $X$ determine the effective divisor over $X$, $$P = \sum_{i=1}^{m-1} [(z_i, w_i)] \in X(g) - D.$$ We remark that this divisor is invariant under the gauge transformation induced by $\mathbf{S}$, namely the solution of the separation equation does not change after replacing each $T(z)_{i,j}$ with a matrix element of $\mathbf{M}(z)$; $M(z)_{i,j}$. In this $N=2$ case $X$ is linearly transformed to the hyperelliptic curve, and we can easily see the structure of $D$ [@Mumford-Book]. On the curve $X$, we have two infinity points $\infty_\pm$ and $m-1$ homomorphic one forms $$h_i(z,w) = z^{i-1}, ~~ \text{ for } i = 1, \cdots, m-1.$$ Then $D$ is written as [@Smirnov0001] $$D = \{ P = \sum_{i = 1}^{m-1} [(z_i, w_i)]
~|~ z_i = z_j \text{ for some } i \neq j,
\text{ or } (z_i,w_i) = \infty_{\pm} \text{ for some } i \}.$$
N=3 case
--------
The matrices and are written as $$\begin{aligned}
&\boldsymbol{\mu}_-^{(1)} =
\begin{pmatrix}
0 & 0 & 0\\
\ast & \ast & 0\\
\ast & \ast & \ast
\end{pmatrix},
~~
\boldsymbol{\mu}_-^{(2)} =
\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
\ast & \ast & 0
\end{pmatrix},
~~
\boldsymbol{\mu}_-^{(0)} =
\begin{pmatrix}
\ast & \ast & 0\\
\ast & \ast & \ast\\
\ast & \ast & \ast
\end{pmatrix},
\\[2mm]
&\boldsymbol{\mu}_+^{(1)} =
\begin{pmatrix}
\ast & \ast & \ast\\
0 & \ast & \ast\\
0 & 0 & \ast
\end{pmatrix},
~~
\boldsymbol{\mu}_+^{(2)} =
\begin{pmatrix}
0& \ast & \ast\\
0 & 0 & \ast\\
0 & 0 & 0
\end{pmatrix},
~~
\boldsymbol{\mu}_+^{(3)} =
\begin{pmatrix}
0 & 0 & \ast\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix},
~~
\boldsymbol{\mu}_+^{(0)} =
\begin{pmatrix}
\ast & \ast & \ast\\
\ast & \ast & \ast\\
0 & \ast & \ast
\end{pmatrix}.\end{aligned}$$ We study $6$ cases of $\mathbf{T}_{m;n_1,n_2}(z)$, $n_1 = 1,2$ and $n_2 = 1,2,3$. For each $\mathbf{T}_{m;n_1,n_2}(z)$ we enumerate the forms of the spectral curve $F(z,w)$ given by $\mathbf{T}_{m;n_1,n_2}^0(z)$, the gauge matrix $\mathbf{S}$ and the matrix $\mathbf{M}(z)$ . In the following, unless a concrete form is shown, $\boldsymbol{\eta}_i$ denote the matrices without zero-entries.
\(i) $\mathbf{T}_{m;1,1}(z)$: We start with the matrix $$\mathbf{T}_{m;1,1}(z) =
z^m \boldsymbol{\mu}_-^{(1)} + z^{m-1} \boldsymbol{\mu}_1
+ \cdots + z \boldsymbol{\mu}_{m-1} + \boldsymbol{\mu}_+^{(1)},$$ whose characteristic polynomial is given by $\mathbf{T}_{m;1,1}^0(z)$ as $$\label{F-3m}
F(z,w) = w^3 - f_1(z) w^2 + f_2(z) w - f_3(z),$$ where $\deg f_1(z) = m, ~\deg f_2(z) = 2m$ and $\deg f_3(z) = 3m-1$. The genus of the curve $X$ is $g = 3m-2$. The gauge matrix $$\mathbf{S} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e}_1 \boldsymbol{\mu}_1 \boldsymbol{\mu}_-^{(1)} \\
\vec{e}_1 \boldsymbol{\mu}_1
\end{pmatrix}$$ introduces of the form; $$\mathbf{M}(z) =
z^m \begin{pmatrix}
0 & 0 & 0 \\
\ast & \ast & \ast \\
0 & 1 & 0
\end{pmatrix}
+
z^{m-1} \begin{pmatrix}
0 & 0 & 1 \\
\ast & \ast & \ast \\
\ast & \ast & \ast \\
\end{pmatrix}
+ O(z^{m-2}).$$
\(ii) $\mathbf{T}_{m;2,2}(z)$: The matrix is $$\mathbf{T}_{m;2,2}(z) =
z^{m} \boldsymbol{\mu}_-^{(2)} + z^{m-1} \boldsymbol{\mu}_-^{(0)}
+ z^{m-2} \boldsymbol{\mu}_2 + \cdots + z \boldsymbol{\mu}_{m-1}
+ \boldsymbol{\mu}_+^{(2)},
~~ \text{for} ~ m \geq 2,$$ and $X$ is given by $\mathbf{T}_{m;2,2}^0(z)$, $$\label{F-3m+1}
F(z,w) = w^3 - z f_1^\prime(z) w^2 + z f_2^\prime(z) w - z f_3^\prime(z),$$ where $\deg f_1^\prime(z) = m-2, ~\deg f_2^\prime(z) = 2m-2$ and $\deg f_3^\prime(z) = 3m-3$. The genus is $g = 3m-3$. Due to the gauge matrix $$\mathbf{S} =
\begin{pmatrix}
\vec{e}_3 \boldsymbol{\mu}_-^{(2)} \\
\vec{e}_3 \boldsymbol{\mu}_-^{(2)} \boldsymbol{\mu}_+^{(2)} \\
\vec{e}_3
\end{pmatrix}$$ is obtained as $$\mathbf{M}(z) =
z^{m} \begin{pmatrix}
0 & 0 & 0 \\
\ast & 0 & 0 \\
1 & 0 & 0
\end{pmatrix}
+ \cdots +
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & \ast \\
0 & 0 & 0 \\
\end{pmatrix}.$$
\(iii) $\mathbf{T}_{m;1,3}(z)$: For the matrix $$\mathbf{T}_{m;1,3}(z) =
z^{m} \boldsymbol{\mu}_-^{(1)} + z^{m-1} \boldsymbol{\mu}_1 + \cdots
+ z \boldsymbol{\mu}_+^{(0)} + \boldsymbol{\mu}_+^{(3)},
~~ \text{for} ~ m \geq 2,$$ $X$ is given by $$\label{F-3m+2}
F(z,w) = w^3 - z f_1^\prime(z) w^2 + z^2 f_2^\prime(z) w - z^2 f_3^\prime(z),$$ where $\deg f_1^\prime(z) = m-1, ~\deg f_2^\prime(z) = 2m-2$ and $\deg f_3^\prime(z) = 3m-3$. The genus of $X$ is $g = 3m-3$. The gauge $$\mathbf{S} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e}_1 \boldsymbol{\mu}_+^{(3)} \boldsymbol{\mu}_+^{(0)} \\
\vec{e}_1 \boldsymbol{\mu}_+^{(3)}
\end{pmatrix}$$ gives , $$\mathbf{M}(z) =
z^{m} \begin{pmatrix}
0 & 0 & 0 \\
\ast & \ast & \ast \\
\ast & \ast & \ast
\end{pmatrix}
+ \cdots +
z \begin{pmatrix}
\ast & \ast & \ast \\
\ast & \ast & \ast \\
0 & 1 & 0 \\
\end{pmatrix}
+ \begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}.$$
\(iv) $\mathbf{T}_{m;2,1}(z)$: The matrix is $$\mathbf{T}_{m;2,1}(z) =
z^{m} \boldsymbol{\mu}_-^{(2)} + z^{m-1} \boldsymbol{\mu}_-^{(0)}
+ z^{m-2} \boldsymbol{\mu}_2 + \cdots + z \boldsymbol{\mu}_{m-1}
+ \boldsymbol{\mu}_+^{(1)},
~~ \text{for} ~ m \geq 2,$$ and $X$ is given by the form with $\deg f_1(z) = m-1, ~\deg f_2(z) = 2m-1$ and $\deg f_3(z) = 3m-2$. The genus of $X$ is $g = 3m-3$. The gauge matrix and the matrix are $$\mathbf{S} =
\begin{pmatrix}
\vec{e}_3 \boldsymbol{\mu}_-^{(2)} \\
\vec{e}_3 \boldsymbol{\mu}_-^{(2)} \boldsymbol{\mu}_+^{(1)} \\
\vec{e}_3
\end{pmatrix},
~~~
\mathbf{M}(z) =
z^{m} \begin{pmatrix}
0 & 0 & 0 \\
\ast & 0 & 0 \\
1 & 0 & 0
\end{pmatrix}
+ \cdots +
\begin{pmatrix}
0 & 1 & 0 \\
\ast & \ast & \ast \\
0 & 0 & \ast \\
\end{pmatrix}.$$
\(v) $\mathbf{T}_{m;1,2}(z)$: The matrix $$\mathbf{T}_{m;1,2}(z) =
z^{m} \boldsymbol{\mu}_-^{(1)} + z^{m-1} \boldsymbol{\mu}_1
+ \cdots + z \boldsymbol{\mu}_{m-1} + \boldsymbol{\mu}_+^{(2)}$$ has the spectral curve with $\deg f_1^\prime(z) = m-1, ~ \deg f_2^\prime(z) = 2m-1$ and $\deg f_3^\prime(z) = 3m-2$, whose genus is $3m-2$. The gauge matrix and the matrix are $$\mathbf{S} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e}_1 (\boldsymbol{\mu}_+^{(2)})^2 \\
\vec{e}_1 \boldsymbol{\mu}_+^{(2)}
\end{pmatrix},
~~~
\mathbf{M}(z) =
z^m \begin{pmatrix}
0 & 0 & 0 \\
\ast & \ast & \ast \\
\ast & \ast & \ast
\end{pmatrix}
+ \cdots +
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}.$$
\(vi) $\mathbf{T}_{m;2,3}(z)$: When $m \geq 3$, the matrix is defined as $$\mathbf{T}_{m;2,3}(z) =
z^{m} \boldsymbol{\mu}_-^{(2)} + z^{m-1} \boldsymbol{\mu}_-^{(0)}
+ z^{m-2} \boldsymbol{\mu}_2 + \cdots + z^2 \boldsymbol{\mu}_{m-2}
+ z \boldsymbol{\mu}_+^{(0)} + \boldsymbol{\mu}_+^{(3)}.$$ Its spectral curve is given by with $\deg f_1^\prime(z) = m-2, ~ \deg f_2^\prime(z) = 2m-3$ and $\deg f_3^\prime(z) = 3m-4$, and the genus is $3m-4$. The gauge matrix and the matrix are obtained as $$\label{3-vi}
\mathbf{S} =
\begin{pmatrix}
\vec{e}_3 \boldsymbol{\mu}_+^{(0)} \\
\vec{e}_3 \boldsymbol{\mu}_+^{(0)} \boldsymbol{\mu}_-^{(2)} \\
\vec{e}_3
\end{pmatrix},
~~
\mathbf{M}(z) =
z^{m} \begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 0 \\
0 & \ast & 0
\end{pmatrix}
+ \cdots +
z \begin{pmatrix}
\ast & \ast & \ast \\
\ast & \ast & \ast \\
1 & 0 & 0
\end{pmatrix}
+ \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & \ast \\
0 & 0 & 0
\end{pmatrix}$$ For the case of $m = 2$, we have $$\mathbf{T}_{2;2,3}(z) =
z^2 \boldsymbol{\mu}_-^{(2)}
+ z \,(\boldsymbol{\mu}_-^{(0)} \cap \boldsymbol{\mu}_+^{(0)})
+ \boldsymbol{\mu}_+^{(3)},
~~ \text{ where }
(\boldsymbol{\mu}_-^{(0)} \cap \boldsymbol{\mu}_+^{(0)}) =
\begin{pmatrix}
\ast & \ast & 0 \\
\ast & \ast & \ast \\
0 & \ast & \ast
\end{pmatrix}.$$ Following this form, the gauge becomes $$\mathbf{S} = \begin{pmatrix}
\vec{e_3}
(\boldsymbol{\mu}_-^{(0)} \cap \boldsymbol{\mu}_+^{(0)}) \\
\vec{e_3}
(\boldsymbol{\mu}_-^{(0)} \cap \boldsymbol{\mu}_+^{(0)})
\boldsymbol{\mu}_-^{(2)} \\
\vec{e_3}
\end{pmatrix}.$$ The associated characteristic polynomial and the matrix are obtained by substituting $m=2$ in and .
We construct the set of representatives $\{ \mathbf{M}(z) \}_F$ based on $F(z,w)$ and $\mathbf{M}(z)$ for each cases. One sees that in all cases the gauge matrices $\mathbf{S}$ reduce the dimension of the variety $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ by two. The separation equation differs depending on which of $\vec{e}_1$ and $\vec{e}_3$ the gauge matrix $\mathbf{S}$ includes. For the cases of (i), (iii) and (v), we have the invariant separation equation as $$B(z) = \operatorname{Det}\begin{pmatrix}
(T(z)_{1,2} , T(z)_{1,3}) \\
(T(z)_{1,2},T(z)_{1,3})
\begin{pmatrix}
T(z)_{2,2} & T(z)_{2,3} \\
T(z)_{3,2} & T(z)_{3,3}
\end{pmatrix}
\end{pmatrix} = 0,
~\text{where} ~ \vec{a} = (1,0,0),$$ and for the rest cases, $$B(z) = \operatorname{Det}\begin{pmatrix}
(T(z)_{3,1} , T(z)_{3,2}) \\
(T(z)_{3,1},T(z)_{3,2})
\begin{pmatrix}
T(z)_{1,1} & T(z)_{1,2} \\
T(z)_{2,1} & T(z)_{2,2}
\end{pmatrix}
\end{pmatrix} = 0,
~\text{where} ~ \vec{a} = (0,0,1).$$ In all cases $B(z)$ has a form as , and each of them gives the eigenvalue by $$\begin{aligned}
w_i =
\begin{cases}
\operatorname{Det}\begin{pmatrix}
T(z_i)_{1,2} & T(z_i)_{1,3} \\
T(z_i)_{3,2} & T(z_i)_{3,3}
\end{pmatrix}
/ T(z_i)_{1,2}, \text{ for (i),(iii),(v)},
\\[5mm]
\operatorname{Det}\begin{pmatrix}
T(z_i)_{1,1} & T(z_i)_{1,2} \\
T(z_i)_{3,1} & T(z_i)_{3,2}
\end{pmatrix}
/ T(z_i)_{3,2}, \text{ for (ii),(iv),(vi)}.
\end{cases}\end{aligned}$$ In conclusion, the separation equation uniquely determines the effective divisor $P \in X(g) - D$ which is invariant under the gauge $\mathbf{S}$.
General $N$ cases
-----------------
In the case of general $N_{\geq 4}$, we have $N(N-1)$ kinds of monodromy matrices $\mathbf{T}_{m;n_1,n_2}(z)$. When $n_1 = n_2 = 1$, the spectral curve $X$ is given by where $\deg f_i(z) = im$, for $i=1,\cdots,N-1$, and $f_N(z) = Nm-1$. Then the genus is $g = \frac{1}{2}(N-1)(Nm-2)$. For each $\mathbf{T}_{m;1,1}(z)$ we have a gauge matrix [@Smirnov-Zeitlin0111]; $$\begin{aligned}
\label{gauge-N}
\mathbf{S} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e_1} \boldsymbol{\mu}_1 (\boldsymbol{\mu}_-^{(1)})^{N-2} \\
\vdots \\
\vec{e_1} \boldsymbol{\mu}_1 \boldsymbol{\mu}_-^{(1)} \\
\vec{e_1} \boldsymbol{\mu}_1
\end{pmatrix},\end{aligned}$$ which reduces the variety of $\{ \mathbf{T}_{m;1,1}(z)\}_F$ by $N-1$ dimensions. Using the elements of $\mathbf{T}_{m;1,1}(z)$ given by $$\begin{aligned}
\mathbf{T}_{m;1,1}(z) =
\begin{pmatrix}
a(z) & \vec{b}(z)\\
\vec{c}(z)^T & \mathbf{d}(z)
\end{pmatrix},\end{aligned}$$ the separation equation is defined as [@Scott94; @Gekhtman95] $$\begin{aligned}
B(z)
\equiv
\operatorname{Det}\begin{pmatrix}
\vec{b}(z) \\
\vec{b}(z) \mathbf{d}(z) \\
\vec{b}(z) \mathbf{d}(z)^2 \\
\vdots \\
\vec{b}(z) \mathbf{d}(z)^{N-2} \\
\end{pmatrix}.\end{aligned}$$ Then $B(z)$ becomes a polynomial of $z$ of degree $g$, and the zeros of $B(z)$ is invariant under the gauge transformation induced by $\mathbf{S}$ [@Gekhtman95].
Instead of showing other cases, based on the above concrete studies we introduce the conjecture for $\mathbf{S}$ as follows;
\[S-conjecture\] For $\mathbf{T}_{m;n_1,n_2}(z)$ there is a gauge matrix $\mathbf{S}$ of the form $$\begin{aligned}
\label{S-matrix}
\mathbf{S} =
\begin{pmatrix}
\vec{e}_1 \\
\vec{e}_1 \boldsymbol{\mu}_a \boldsymbol{\mu}_b^{N-2} \\
\vdots \\
\vec{e}_1 \boldsymbol{\mu}_a \boldsymbol{\mu}_b \\
\vec{e}_1 \boldsymbol{\mu}_a
\end{pmatrix}, \text{ for even } L,
~~~
\begin{pmatrix}
\vec{e}_N \boldsymbol{\mu}_a \\
\vec{e}_N \boldsymbol{\mu}_a \boldsymbol{\mu}_b \\
\vdots \\
\vec{e}_N \boldsymbol{\mu}_a \boldsymbol{\mu}_b^{N-2} \\
\vec{e}_N
\end{pmatrix}, \text{ for odd } L,
\end{aligned}$$ where $\boldsymbol{\mu}_a, \boldsymbol{\mu}_b \in
\{\boldsymbol{\mu}_-^{(n_1)}, \boldsymbol{\mu}_-^{(n_1-N+1)},
\boldsymbol{\mu}_+^{(n_2)}, \boldsymbol{\mu}_+^{(n_2-N)},
\boldsymbol{\mu}_j, ~j=2,\cdots,m-2\}$, such that $\mathbf{S}$ reduces $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ to a $g$-dimensional variety $\{\mathbf{M}(z)\}_F$ and that the associated separation equation with the form as has the zeros invariant under $\mathbf{S}$.
We briefly remark on the diagram . If we get the gauge matrix $\mathbf{S}$ which solves Problem \[problem\] then (a) becomes surjective, since $\{ \mathbf{M}(z) \}_F$ is the set of representatives. Therefore the map (c) exists such that the diagram is commutative.
Integrability of LV($N,L$)
==========================
Spectral curve and Poisson structure for LV($N,L$)
--------------------------------------------------
We introduce the $N$ by $N$ Lax matrix for the extended Lotka-Volterra lattice as $$\label{Bogo-Lax}
\Tilde{\mathbf{L}}_n(z)
=
\frac{1}{z V_n^{\frac{N-1}{N}}}
\Bigl(
\sum_{k=1}^{N-1} V_n \mathbf{E}_{k,k+1}
+ z^N (-1)^{N-1} \mathbf{E}_{N,1}
+ z^N (-1)^{N-2} \mathbf{E}_{N,2}
\Bigr).$$ We have much modified the original Lax matrix [@Bogo88], and comes from $\overline{\mathbf{L}}_n(z)$ in [@Inoue02-JBogo]. Note that $\Tilde{\mathbf{L}}_n(z)$ has been normalized as $\operatorname{Det}\Tilde{\mathbf{L}}_n(z) = 1$. The monodromy matrix $\Tilde{\mathbf{T}}(z)$ of an $L$-periodic model LV($N,L$) is defined as $$\label{monodromy-LV}
\Tilde{\mathbf{T}}(z) =
\prod_{k=1}^{\stackrel{L}{\curvearrowleft}}
\Tilde{\mathbf{L}}_k(z).$$ The characteristic equation of $\Tilde{\mathbf{T}}(z)$, $$\label{curveX}
\operatorname{Det}\bigl( w \openone - \Tilde{\mathbf{T}}(z) \bigr) = 0,$$ gives an algebraic curve $\Tilde{X}$. For this equation we have the automorphism $\tau$ of order $N$, $$\tau : ~ (z,w) \mapsto (\epsilon z ,\epsilon^{-k_2} w),$$ where $\epsilon = \mathrm{e}^{\frac{2 \pi \mathrm{i}}{N}}$, and $k_2$ is defined by . We define the matrix $\mathbf{T}_{LV}(z)$, $$\begin{aligned}
\label{T-LV}
\mathbf{T}_{LV}(z) \equiv z^{\frac{k_2}{N}}
\Tilde{\mathbf{T}}(z^\frac{1}{N}),\end{aligned}$$ then its matrix elements become polynomials of $z$ and $\operatorname{Det}\mathbf{T}_{LV}(z) = z^{k_2}$. The characteristic equation of $\mathbf{T}_{LV}(z)$ gives the quotient curve $\Tilde{X}/\tau$.
On the other hand, the Hamiltonian structure of LV($N,L$) is defined by the Poisson brackets [@Bogo88] $$\label{PoissonLV}
\{ V_n, V_m \}
=
2 \,V_n V_m \sum_{k=1}^{N-1} ( \delta_{m,n+k} - \delta_{m,n-k} ),$$ and the Hamiltonian $H_1 = \sum_{n=1}^L V_n$. Using these settings, the time evolution is given by $$\frac{\partial V_n}{\partial t_1}
=
\{ V_n ~,~ H_1 \}$$ with $t = t_1$. We let $\mathcal{A}_{LV}$ be the Poisson bracket algebra for $\mathbb{C}[V_n; n \in \mathbb{Z}/L\mathbb{Z}]$ whose defining relations are given by . We have the center of $\mathcal{A}_{LV}$ denoted by $\mathcal{A}_{LV}^0$ as follows;
\[LV-center\] The center $\mathcal{A}_{LV}^0$ is generated by the variables $$\begin{aligned}
\mathcal{P}_{k}^{(i)} = \prod_{n=0}^{\frac{L}{k}-1} (V_{k n+i}),
~~ \text{for } k \in \mathcal{K},~ i \in \{1,\cdots, k\},
\end{aligned}$$ where $$\begin{aligned}
\label{k-K}
\mathcal{K} = \{ k \in \{ 1, \cdots, N \}
~|~~ k|N \text{ or } k|(N-1) \}
\sqcap \{ k ~|~~ k|L \}.\end{aligned}$$ Here $ k | L$ means that $L$ is a multiple of $k$.
See Appendix B for the proof. Since the set $\{ \mathcal{P}_{k}^{(i)} | i \in \{1,\cdots, k\} \}$ is generated by $\{ \mathcal{P}_{k^\prime}^{(j)} |
j \in \{1, \cdots, k^\prime\} \}$ for $k | k^\prime$, to generate $\mathcal{A}_{LV}^0$ it is enough to have a set $$\{ \mathcal{P}_{k}^{(i)} ~|~
k \in \mathcal{K}_0, ~i \in \{1,\cdots, k\} \},$$ where $
\mathcal{K}_0 = \{ \text{max} [ k \in \mathcal{K} \text{ for } k | N ], ~
\text{max} [k \in \mathcal{K} \text{ for } k | (N-1)] \}.
$ Then the number of independent generators of $\mathcal{A}_{LV}^0$ is $$\label{number-center}
n_0 = \sum_{k \in \mathcal{K}_0} k - ( | \mathcal{K}_0 | - 1).$$ Based on the structure of the monodromy matrix , we introduce a variable $$\mathcal{P}_0 \equiv \prod_{n=1}^L (V_n)^{- \frac{1}{N}}
= \bigl(\mathcal{P}_1^{(1)}\bigr)^{-\frac{1}{N}},$$ which is Poisson commutative with any $V_n$. Therefore $\mathcal{A}_{LV}$ is naturally extended to the Poisson bracket algebra over $\mathbb{C}(\mathcal{P}_0,V_n ; n \in \mathbb{Z}/L\mathbb{Z})$. We denote this algebra by $\mathcal{A}_{LV}^\prime$.
A family of the integrals of motion(IM) for LV($N,L$) which includes the Hamiltonian $H_1$ appears as coefficients of the characteristic equation .
[@InoueHikami98-Bogo; @HikamiInoueKomori99] \[IM-commute\] The IM compose the commuting subalgebra of $\mathcal{A}^\prime_{LV}$.
[*Proof*]{}. We show the outline of the proof. We introduce the variable transformation $$ V_n = (P_n P_{n+1} \cdots P_{n+N-1})^{-1} Q_n^{-1} Q_{n+N-1},$$ where $P_n$, $Q_n$ are canonical variables, $$\begin{aligned}
\label{canonicalPQ}
\{ P_n ~,~ Q_m \}
= \delta_{n,m} P_n \, Q_n ,
~~~~
\{ P_n ~,~ P_m \}
=
\{ Q_n ~,~ Q_m \} = 0.\end{aligned}$$ Then the matrix $\mathbf{T}_{LV}(z)$ is transformed as $$\begin{aligned}
\label{gaugeB}
\mathbf{T}_C(z) = \mathbf{B}_1 \mathbf{T}_{LV}(z) (\mathbf{B}_1)^{-1},\end{aligned}$$ using a diagonal matrix $\mathbf{B}_1 = \mathbf{B}_1(P_1,\cdots,P_{N-1},Q_1,\cdots,Q_{N-1})$. The matrix $\mathbf{T}_C(z)$ turns out to satisfy the fundamental Poisson relation $$\begin{aligned}
\label{TC-Poisson}
\{\mathbf{T}_C(z)
\stackrel{\otimes}{,}
\mathbf{T}_C(z^\prime) \}
=
[\, \mathbf{r}(z/z^{\prime}) ~,~ \mathbf{T}_C(z) \otimes
\mathbf{T}_C(z^\prime) \,],\end{aligned}$$ with the $r$-matrix . See [@HikamiInoueKomori99; @Inoue02-JBogo] for details of the gauge matrix $\mathbf{B}_1$ and $\mathbf{T}_C(z)$. Note that the characteristic equation for the matrix $\mathbf{T}_C(z)$ is obtained from by a transformation $
(z,w) \mapsto (z^{\frac{1}{N}},w z^{\frac{k_2}{N}}),
$ and that the coefficients of the characteristic polynomial belong to $\mathbb{C}[\mathcal{P}_0,V_n; n \in \mathbb{Z}]$. Then the proposition follows. $\square$
We introduce a grading on $\mathcal{A}_{LV}$ as $\deg V_n = 1$. Since the IM are obtained as homogeneous polynomials of $V_n$, we can identify each of IM based on the grading. For instance, for the Hamiltonian $H_1$ we have $\deg H_1 = 1$. Let $n_H$ be the number of the independent elements of IM in $\mathcal{A}_{LV}$. By putting the IM in the order of the grading, we obtain $$\label{IM}
H_1 , H_2 , \cdots, H_{n_H}.$$ The Proposition \[IM-commute\] yields
The family of IM generate $n_H$ independent flows for LV($N,L$) defined as $$\label{time-evol}
\frac{\partial \mathcal {O}}{\partial t_i}
\equiv \{ \mathcal{O} ~,~ H_i \},
~~~ \text{for~ } \mathcal{O} \in \mathcal{A}_{LV}^\prime,
~i = 1, \cdots, n_H.$$
We comment that in [@Suris94] the Hamiltonian structure of LV($N,L$) is studied by applying the $r$-matrix method [@ReySemenov94] to the [*big*]{} Lax matrix of $L$ by $L$, and the involution of IM is clarified by this approach. Since our aim here is to establish the eigenvector map for LV($N,L$) based on the monodromy matrix , it is important to get the [*small*]{} monodromy matrix of $N$ by $N$ with the fundamental Poisson relation .
Realization of $\mathbf{M}_F(z)$ and the integrable structure of LV($N,L$)
--------------------------------------------------------------------------
We find that the matrix $\mathbf{T}_C(z)$ gives the realization of $\mathbf{T}_{m;n_1,n_2}(z)$ , namely both of the form and the Poisson structure of $\mathbf{T}_C(z)$ coincide with those of $\mathbf{T}_{m;n_1,n_2}(z)$. We obtain a similar relation as as follows;
Under a condition $$\begin{aligned}
\label{det-condition}
\operatorname{Det}\mathbf{T}_{m;n_1,n_2}(z) =z^{n_2-1}, \end{aligned}$$ $\mathbf{T}_C(z)$ realizes $\mathbf{T}_{m;n_1,n_2}(z)$ and they are related as $$\begin{aligned}
\label{TC-Tm}
\mathbf{T}_C(z)
=
\begin{cases}
\mathbf{T}_{m;1,1}(z), ~\text{ for } k_1 = k_2 = 0,
\\
\mathbf{T}_{m+1;N-k_1,k_2+1}(z),
~\text{ for } k_1, k_2 \neq 0, ~0 \leq k_1 - k_2 \leq N-2,
\\
\mathbf{T}_{m+2;N-k_1,k_2+1}(z), ~\text{ for } k_1 - k_2 \leq -1.
\end{cases} \end{aligned}$$
[*Proof.*]{} First we check the coincidence of the form. Note that the condition comes from the normalization of $\Tilde{\mathbf{L}}_n(z)$. The Lax matrices $z^{\frac{1}{N}} \Tilde{\mathbf{L}}_n(z^{\frac{1}{N}})$ and $\mathbf{L}_n(z)$ have the same form as polynomial matrices. Then we see that $\mathbf{T}_{LV}(z)$ and $z^{-\frac{L}{N} + \frac{k_2}{N} + m_2} \mathbf{T}^{(L)}(z)$ has a same form. Since the gauge $\mathbf{B}_1$ is diagonal and does not change the form of $\mathbf{T}_{LV}(z)$, we obtain the correspondence of $\mathbf{T}_C(z) = \mathbf{B}_1 \mathbf{T}_{LV}(z) \mathbf{B}_1^{-1}$ and $\mathbf{T}^{(L)}(z)$. By using Lemma \[T-Lproduct\] and the relation $L=N m_2 + k_2$ , finally we obtain .
Next, we observe the Poisson structure. The condition and do not contradict each other, since Proposition \[prop:Poisson\] says that $\operatorname{Det}\mathbf{T}_{m;n_1,n_2}(z)$ belongs to the center of $\mathcal{A}_C$. Then from and , the monodromy matrices $\mathbf{T}_C(z)$ obviously has the same Poisson structure as that of $\mathbf{T}_{m;n_1,n_2}(z)$. $\square$\
This Proposition is the reason why we denoted the Poisson bracket algebra of $\mathbf{T}_{m;n_1,n_2}(z)$ using $\mathcal{A}_C$ in §2. Once we associate $\mathbf{T}_C(z)$ to $\mathbf{T}_{m;n_1,n_2}(z)$, $\mathbf{T}_C(z)$ realizes $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ where $\mathbf{T}_{m;n_1,n_2}^0(z)$ corresponds to the initial condition for $\mathbf{T}_C(z)$. We also see $\Tilde{X} / \tau \simeq X$.
In the following we discuss the integrability of LV($N,L$) based on the representative $\{\mathbf{M}(z)\}_F$ and the Poisson bracket algebra $\mathcal{A}_{M}$ realized by LV($N,L$). We introduce an important proposition;
\[prop:subset\] If the gauge matrix $\mathbf{S}$ which meets the conditions in Conjecture \[S-conjecture\] exists, then (1) $\mathcal{A}_{M} \subset \mathcal{A}_{LV}^\prime$, (2) the separation equation gives $g$ algebraic relations between $z_i$ $(i = 1, \cdots, g)$ and $V_n$ $(n \in \mathbb{Z}/L \mathbb{Z})$.
Remember that the matrix $\mathbf{T}_C(z)$ is no longer written in terms of the dynamical variables of LV($N,L$), but of the canonical variables . Therefore $\mathcal{A}_{M} \subset \mathcal{A}_C$ is trivial but the claim (1) in the above proposition is not. This claim was conjectured in [@Inoue02-JBogo] and now is proved in a simple way. We add the proof of Proposition \[prop:subset\] at Appendix C.
On the tangent space of $\boldsymbol{\mathcal{M}}_F$ there is the $g$ dimensional invariant vector field which induces the evolution of the divisor $P$ linearized on $J_{\text{aff}}(X)$. When $n_H$ is equal to $g$, we can identify the coordinates on $J_{\text{aff}}(X)$ with the times $t_i$ , and get $z_i$ as a functions of $t_i$; $z_i = z_i(t_1, \cdots t_g)$. Further, if $n_H = \frac{1}{2}(L - n_0)$ is satisfied, we can reduce the integrability of LV($N,L$) to $L$ independent algebraic relations between the dynamical variables of LV($N,L$) and $H_i$ , $z_i$ and $n_0$ generators of $\mathcal{A}_{LV}^0$ . We summarize the integrability of LV($N,L$) as follows;
\[def:integrability\] LV($N,L$) is algebraic completely integrable if $$\begin{aligned}
\label{IM-genus}
g = n_H = \frac{1}{2}(L - n_0)
\end{aligned}$$ and Proposition \[prop:subset\] is satisfied.
In §3, we solved Problem \[problem\] for the cases of $N=2,3$ and the special case of general $N$. We obtained the gauge matrices $\mathbf{S}$ which meet Conjecture \[S-conjecture\], then Proposition \[prop:subset\] is satisfied for these cases. The last case corresponds to LV($N,L$) with the special periodicity $L = N(N-1)m$ studied in ref. [@Inoue02-JBogo] where was proved and Proposition \[prop:subset\] was supposed. Now we have Proposition \[prop:subset\] satisfied, then we conclude that
LV($N,N(N-1)m$) is algebraic completely integrable.
In the following, we investigate Propositions \[prop:subset\] and \[def:integrability\] for the results in §3 and show Theorem \[th:N=2-3\].
LV($2,L$)
---------
Depending on the periodicity $L$ we have two cases;
\(i) $L=2m$, $\mathbf{T}_C(z) = \mathbf{T}_{m;1,1}(z)$: The IM are obtained as the coefficients of with $$f_1(z) = \mathcal{P}_0 \bigl(z^m + H_1 z^{m-1} + H_2 z^{m-2} - \cdots
+ z H_{m-1}
+ (\mathcal{P}_2^{(1)} + \mathcal{P}_2^{(2)}) \bigr).$$ Here we have $m-1$ independent IM identified by their degree, $\deg H_i = i$. The center $\mathcal{A}_{LV}^0$ is generated by two of $\mathcal{P}_1^{(1)}, \mathcal{P}_2^{(1)}$ and $\mathcal{P}_2^{(2)}$. The genus of $X$ is equal to $n_H$.
\(ii) $L=2m+1$, $\mathbf{T}_C(z) = \mathbf{T}_{m+1;1,2}(z)$: We have $m$ independent IM given by with $$f_1^\prime(z) = \mathcal{P}_0 \bigl( z^m - H_1 z^{m-1} + H_2 z^{m-2} - \cdots
+ (-)^{m} H_{m} \bigr),$$ where $\deg H_i = i$. The center $\mathcal{A}_{LV}^0$ is generated by $\mathcal{P}_1^{(1)}$ only.
In both cases is satisfied and the gauge matrices $\mathbf{S}$ and fulfill Proposition \[prop:subset\]. Therefore we conclude that LV($2,L$) is algebraic completely integrable. The correspondence of the periodicity $L$ and the genus $g$ is summarized as
$L$ $3$ $4$ $5$ $6$ $7$ $8$ $\cdots$ $2m$ $2m+1$ $\cdots$
----- ----- ----- ----- ----- ----- ----- ---------- ------- -------- ----------
$g$ $1$ $1$ $2$ $2$ $3$ $3$ $\cdots$ $m-1$ $m$ $\cdots$
.
LV($3,L$)
---------
The periodicity $L$ is classified into $6$ cases;
\(i) $L=6m$, $\mathbf{T}_C(z) = \mathbf{T}_{m;1,1}(z)$: The IM are obtained as $$\begin{aligned}
&f_1(z) = \mathcal{P}_0^2 ( f_{3m} z^m + f_{3m+1} z^{m-1} + \cdots
+ f_{4m} ),
\\
&f_2(z) = \mathcal{P}_0 ( z^{2m} + f_1 z^{2m-1} + \cdots + f_{2m} ),\end{aligned}$$ where we set $f_i$ so as to accomplish $\deg f_i = i$. The generators of $\mathcal{A}_{LV}^0$ have the ordering as $\deg \mathcal{P}_2^{(i)} = 3m, ~ \deg \mathcal{P}_3^{(i)} = 2m$, then $f_{3m}$, $f_{4m}$ and $f_{2m}$ belong to $\mathcal{A}_{LV}^0$. Actually, we have relations $$\begin{aligned}
\label{f-P-2}
&z^2 + f_{3m} z + \mathcal{P}_1^{(1)}
= (z - \mathcal{P}_2^{(1)}) (z - \mathcal{P}_2^{(2)}),
\\
\label{f-P-3}
&z^3 + z^2 f_{2m} + z f_{4m} + \mathcal{P}_1^{(1)}
= (z - \mathcal{P}_3^{(1)}) (z - \mathcal{P}_3^{(2)})
(z - \mathcal{P}_3^{(1)}).\end{aligned}$$ In conclusion we have $n_H = 3m-2$ which is equal to $g$, and $n_0 = 4$.
\(ii) $L=6m+1$, $\mathbf{T}_C(z) = \mathbf{T}_{m+1;2,2}(z)$: We have $$\begin{aligned}
&f_1^\prime(z) =
\mathcal{P}_0^2 ( f_{3m+1} z^{m-1} + f_{3m+2} z^{m-2} + \cdots
+ f_{4m} ),
\\
&f_2^\prime(z) = \mathcal{P}_0 ( z^{2m} + f_1 z^{2m-1} + \cdots + f_{2m} ).\end{aligned}$$ In this case we have only a generator of $\mathcal{A}_{LV}^0$; $\mathcal{P}_1^{(1)}$, and no $f_i$ belongs to $\mathcal{A}_{LV}^0$. Then $n_H = 3m$ and $n_0 = 1$.
\(iii) $L=6m+2$, $\mathbf{T}_C(z) = \mathbf{T}_{m+1;1,3}(z)$: $$\begin{aligned}
&f_1^\prime(z) = \mathcal{P}_0^2 ( f_{3m+1} z^{m-1} + f_{3m+2} z^{m-2} + \cdots
+ f_{4m+1} ),
\\
&f_2^\prime(z) = \mathcal{P}_0 ( z^{2m} + f_1 z^{2m-1} + \cdots + f_{2m} ).\end{aligned}$$ Since $\deg \mathcal{P}_2^{(i)} = 3m+1$, we see $f_{3m+1} \in \mathcal{A}_{LV}^0$, which satisfies a relation similar to . Then we have $n_H = 3m$ and $n_0 = 2$.
\(vi) $L=6m+3$, $\mathbf{T}_C(z) = \mathbf{T}_{m+1;2,1}(z)$: $$\begin{aligned}
&f_1(z) = \mathcal{P}_0^2 ( f_{3m+2} z^{m} + f_{3m+2} z^{m-1} + \cdots
+ f_{4m+2} ),
\\
&f_2(z) = \mathcal{P}_0 ( z^{2m+1} + f_1 z^{2m} + \cdots + f_{2m+1} ).\end{aligned}$$ Since $\deg \mathcal{P}_2^{(i)} = 2m+1$, we see $f_{4m+2}, f_{2m+1} \in \mathcal{A}_{LV}^0$, which satisfy a relation similar to . Then we have $n_H = 3m$ and $n_0 = 3$.
The remaining cases,\
(v) $L=6m+4$, $\mathbf{T}_C(z) = \mathbf{T}_{m+1;1,2}(z)$\
(iv) $L=6m+5$, $\mathbf{T}_C(z) = \mathbf{T}_{m+2;2,3}(z)$\
permit the same analysis.
For all $L$ we have $n_H$ and $n_0$ which satisfy . Remember that in §3.2 we have constructed the $\{\mathbf{M}(z)\}_F$ with the gauge matrices $\mathbf{S}$ which meet Proposition \[prop:subset\]. Herewith we prove the algebraic completely integrability of LV($3,L$). As same as the $N=2$ case, we summarize the correspondence of $L$ and $g$;
$L$ $5$ $6$ $7$ $8$ $9$ $10$ $\cdots$ $6m$ $6m+1$ $6m+2$ $6m+3$ $6m+4$ $6m+5$
----- ----- ----- ----- ----- ----- ------ ---------- -------- -------- -------- -------- -------- --------
$g$ $2$ $1$ $3$ $3$ $3$ $4$ $\cdots$ $3m-2$ $3m$ $3m$ $3m$ $3m+1$ $3m+2$
.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author thanks Prof. A. Nakayashiki for informing about [@Vanhae01; @Van1638]. She appreciates discussion with T. Takenawa and T. Yamazaki. She also thanks the referees for valuable comments which have much improved the manuscript. R. I. is a Research Fellow of the Japan Society for the Promotion of Science.
Appendix A Proof of Lemma \[T-Lproduct\] {#appendix-a-proof-of-lemma-t-lproduct .unnumbered}
------------------------------------------
We show the outline of the proof. We use the integers defined at and , and set a matrix $\mathbf{T}^{(L)}(z)$; $$\begin{aligned}
\label{T-L}
\mathbf{T}^{(L)}(z) = z^{-m_2} \prod_{n=1}^{L} \mathbf{L}_n(z).\end{aligned}$$ By definition, first we have $$\mathbf{T}^{(1)}(z)
= \mathbf{L}_1(z)
= \boldsymbol{\mu}_-^{(N-1)} z
+ (\boldsymbol{\mu}_-^{(0)} \cap \boldsymbol{\mu}_+^{(2)}).$$ Therefore we obtain the correspondence $\mathbf{T}^{(1)}(z) = \mathbf{T}_{1;N-1,2}$. Assume $\mathbf{T}^{(L)}(z) = \mathbf{T}_{m;n_1,n_2}(z)$. When we set $\mathbf{T}^{(L)}(z) = (t^{(L)}_{i,j})_{1\leq i,j\leq N}$, $\mathbf{T}^{(L+1)}(z)$ are related to $\mathbf{T}^{(L)}(z)$ as $$\begin{aligned}
\mathbf{T}^{(L+1)}(z)
=
\begin{cases}
&\displaystyle{\sum_{j=1}^{N}}
\Bigl(\displaystyle{\sum_{i=1}^{N-1}}
\mathbf{E}_{i,j} l_i^{(L+1)} t^{(L)}_{i+1,j}
+ z \mathbf{E}_{N,j}
(l_N^{(L+1)} t^{(L)}_{1,j} + l_0^{(L+1)} t^{(L)}_{2,j})
\Bigr), \text{ for } n_2 \neq N,
\\
&\displaystyle{\sum_{j=1}^{N}}
\Bigl( \frac{1}{z} \displaystyle{\sum_{i=1}^{N-1}}
\mathbf{E}_{i,j} l_i^{(L+1)} t^{(L)}_{i+1,j}
+ \mathbf{E}_{N,j}
(l_N^{(L+1)} t^{(L)}_{1,j} + l_0^{(L+1)} t^{(L)}_{2,j})
\Bigr), \text{ for } n_2 = N,
\end{cases}\end{aligned}$$ then we find the correspondence $$\begin{aligned}
\mathbf{T}^{(L+1)}(z)
=
\begin{cases}
\mathbf{T}_{m;n_1-1,n_2+1}(z), \text{ for } n_1 \neq 1, n_2 \neq N,
\\
\mathbf{T}_{m+1;N-1,n_2+1}(z), \text{ for } n_1 = 1, n_2 \neq N,
\\
\mathbf{T}_{m;N-1,1}(z), \text{ for } n_1 = 1, n_2 = N,
\\
\mathbf{T}_{m-1;n_1-1,1}(z), \text{ for } n_1 \neq 1, n_2 = N.
\end{cases}\end{aligned}$$ By induction, we obtain . $\square$
Appendix B Proof of Proposition \[LV-center\] {#appendix-b-proof-of-proposition-lv-center .unnumbered}
-----------------------------------------------
Based on the periodicity $L$ and the Poisson relations , we can set candidates for the generators of $\mathcal{A}_{LV}^0$ as $$\mathcal{P}_{k}^{(i)} = \prod_{n=0}^{\frac{L}{k}-1} (V_{k n+i}),
~~ \text{for } k \in \{1, \cdots, N\},~ k | L \text{ and }
i \in \{1,\cdots,k\}.$$ Our goal is to determine $k$. The condition for a variable $\mathcal{P}_{k}^{(i)}$ to belong to $\mathcal{A}_{LV}^0$; $$\{ V_n ~,~ \mathcal{P}_{k}^{(i)} \} = 0,
~~\text{ for } n \in \mathbb{Z}/L \mathbb{Z},$$ reduces to $$\begin{aligned}
\label{condition-0}
\sum_{m \in \mathbb{Z}/L\mathbb{Z}, ~m = i \text{ mod } k} ~
\sum_{l=1}^{N-1}
( \delta_{m,n+l} - \delta_{m,n-l}) = 0.\end{aligned}$$ Assume that we have $2j$ non-zero terms in the summation of for $j \in \{1, \cdots, N-1\}$, where $j$ of them offer $+1$ and the others offer $-1$. In the case of $j=1$ we easily obtain $k=N$ if $N|L$ is satisfied, and $k=N-1$ if $(N-1)|L$. In the case of $j=N-1$ we have $k=1$ for all $L$. In the following, we study the cases of $2 \leq k \leq N-2$.
Without limiting the generality, we consider the $n=0$ case in . Let $m = n_0$ in be the leftmost lattice point where the first $-1$ occurs for $-(N-1) \leq n_0 \leq -N+k$. In $j=2$ case, the condition for $k$ is reduced to $$\label{condition-1}
n_0 + k < 0 ~ \text{ and } ~ N-k \leq n_0 + 3k \leq N-1.$$ This situation is depicted as
(20,3)(0,-0.5) (0,0)[(1,0)[4.5]{}]{} (4.8,0)[$\dots$]{} (6,0)[(1,0)[3.5]{}]{} (9.8,0)[$\dots$]{} (11,0)[(1,0)[2.8]{}]{} (14.1,0)[$\dots$]{} (15.3,0)[(1,0)[3.2]{}]{} (0,-0.4)[(0,1)[0.8]{}]{} (-1.5,-1) (1,-0.2)[(0,1)[0.4]{}]{} (2,-0.2)[(0,1)[0.4]{}]{} (3,0) (3,1.5)[(0,-1)[1]{}]{} (2.8,2)[$n_0$]{} (4,-0.2)[(0,1)[0.4]{}]{} (6.2,-0.2)[(0,1)[0.4]{}]{} (7.2,0) (7.2,1.5)[(0,-1)[1]{}]{} (6,2)[$n_0+k$]{} (8.2,-0.2)[(0,1)[0.4]{}]{} (9.2,-0.4)[(0,1)[0.8]{}]{} (9.1,-1) (9.2,-0.2)[(0,1)[0.4]{}]{} (11.3,-0.2)[(0,1)[0.4]{}]{} (12.3,0) (12.3,1.5)[(0,-1)[1]{}]{} (11.1,2)[$n_0+2k$]{} (13.3,-0.2)[(0,1)[0.4]{}]{} (15.5,-0.2)[(0,1)[0.4]{}]{} (16.5,0) (16.5,1.5)[(0,-1)[1]{}]{} (15.3,2)[$n_0+3k$]{} (16.5,-0.2)[(0,1)[0.4]{}]{} (17.5,-0.2)[(0,1)[0.4]{}]{} (18.5,-0.4)[(0,1)[0.8]{}]{} (17.8,-1)
.
Here black circles mean where the non-zero terms are offered in . We have two critical cases for $n_0$;\
(i) when $n_0 = - (N-1)$, reduces to $$\label{1-1}
\frac{2N-1}{4} \leq k \leq \frac{2(N-1)}{3}.$$ (ii) When $n_0 = - N+k$, becomes $$\label{1-2}
\frac{2N}{5} \leq k \leq \frac{2N-1}{4}.$$ Since $\frac{2N-1}{4} \not\in \mathbb{Z}$, and are not satisfied at the same time. When $k$ satisfies (i), we should relate this $k$ to a condition\
(i’) when $n_0 = - N+k$, $n_0 + k = 0$ is imposed;
(19,3)(0,-0.5) (0,0)[(1,0)[18]{}]{} (0,-0.4)[(0,1)[0.8]{}]{} (-1.5,-1) (1,-0.2)[(0,1)[0.4]{}]{} (2,-0.2)[(0,1)[0.4]{}]{} (3,-0.2)[(0,1)[0.4]{}]{} (4,0) (4,1.5)[(0,-1)[1]{}]{} (3.8,2)[$n_0$]{} (4,-0.2)[(0,1)[0.4]{}]{} (5,-0.2)[(0,1)[0.4]{}]{} (6,-0.2)[(0,1)[0.4]{}]{} (7,-0.2)[(0,1)[0.4]{}]{} (8,-0.2)[(0,1)[0.4]{}]{} (9,-0.4)[(0,1)[0.8]{}]{} (8.9,-1) (9,0) (9,1.5)[(0,-1)[1]{}]{} (7.8,2)[$n_0+k$]{} (9,-0.2)[(0,1)[0.4]{}]{} (10,-0.2)[(0,1)[0.4]{}]{} (11,-0.2)[(0,1)[0.4]{}]{} (12,-0.2)[(0,1)[0.4]{}]{} (13,-0.2)[(0,1)[0.4]{}]{} (14,-0.2)[(0,1)[0.4]{}]{} (14,0) (15,-0.2)[(0,1)[0.4]{}]{} (16,-0.2)[(0,1)[0.4]{}]{} (17,-0.2)[(0,1)[0.4]{}]{} (18,-0.4)[(0,1)[0.8]{}]{} (17.2,-1)
.
Then we obtain $k = \frac{N}{2}$, which turns out to be the $j=1$ case.\
On the other hand, when $k$ satisfies (ii), we relate it to\
(ii’) when $n_0 = -(N-1)$, $n_0 + 2k = 0$ is required;
(17.2,3)(0,-0.5) (0,0)[(1,0)[16]{}]{} (0,-0.4)[(0,1)[0.8]{}]{} (0,0) (0,1.5)[(0,-1)[1]{}]{} (-0.2,2)[$n_0$]{} (-1.5,-1) (1,-0.2)[(0,1)[0.4]{}]{} (2,-0.2)[(0,1)[0.4]{}]{} (3,-0.2)[(0,1)[0.4]{}]{} (4,0) (4,1.5)[(0,-1)[1]{}]{} (2.8,2)[$n_0+k$]{} (5,-0.2)[(0,1)[0.4]{}]{} (6,-0.2)[(0,1)[0.4]{}]{} (7,-0.2)[(0,1)[0.4]{}]{} (8,0) (8,1.5)[(0,-1)[1]{}]{} (6.8,2)[$n_0+2k$]{} (8,-0.4)[(0,1)[0.8]{}]{} (7.9,-1) (9,-0.2)[(0,1)[0.4]{}]{} (10,-0.2)[(0,1)[0.4]{}]{} (11,-0.2)[(0,1)[0.4]{}]{} (12,0) (12,-0.2)[(0,1)[0.4]{}]{} (13,-0.2)[(0,1)[0.4]{}]{} (14,-0.2)[(0,1)[0.4]{}]{} (15,-0.2)[(0,1)[0.4]{}]{} (16,-0.4)[(0,1)[0.8]{}]{} (16,0) (15.2,-1)
.
Therefore we obtain $k = \frac{N-1}{2}$, which is a special case of $j=2$.\
The conditions (i) and (i’) do not contradict each other for $N \geq 4$, and so do not (ii) and (ii’) for $N \geq 5$. Then we obtain $k= \frac{N}{2}$ (resp. $\frac{N-1}{2}$) if $2 | N$ (resp. $2 | (N-1))$.
In general $j_{\geq 3}$ cases, reduces to $$\begin{aligned}
\label{condition-j}
n_0 +(j-1) k < 0, ~~~
\frac{N-n_0}{2 j} \leq k \leq \frac{N-1-n_0}{2 j -1}.\end{aligned}$$ Then two critical cases are written as follows;\
(i) when $n_0 = - (N-1)$, becomes $$\frac{2N-1}{2 j} \leq k \leq \frac{2(N-1)}{2 j -1}.$$ And when $n_0 = -N+k$, $n_0 + (j-1) k = 0$. Then we obtain $k = \frac{N}{j}$ for $N \geq 2 j$ and $j | N$.\
(ii) When $n_0 = - N+k$, $$\frac{2N}{2 j +1} \leq k \leq \frac{2N-1}{2 j}.$$ And when $n_0 = -(N-1)$, $n_0 + j k = 0$. Then we get $k = \frac{N-1}{j}$ for $N \geq 2 j +1$ and $j | (N-1)$.
Finally we obtain the set $\mathcal{K}$ $k$ belongs to. $\square$
Appendix C Proof of Proposition \[prop:subset\] {#appendix-c-proof-of-proposition-propsubset .unnumbered}
-------------------------------------------------
We show the first part of Proposition \[prop:subset\] in more general setting. Assume that $\mathbf{T}_{LV}(z)$ has a form as $$\mathbf{T}_{LV}(z) = \boldsymbol{\mu}_0^{LV} z^m +
\boldsymbol{\mu}_1^{LV} z^{m-1}
+ \cdots + \boldsymbol{\mu}_m^{LV}.$$ Let all matrix elements of $\boldsymbol{\mu}_i^{LV}$ belong to $\mathcal{A}_{LV}^\prime$. We relate $\mathbf{T}_{LV}(z)$ to a matrix $\mathbf{T}(z)$ by the gauge transformation $$\mathbf{T}(z) = \mathbf{B} \mathbf{T}_{LV}(z) \mathbf{B}^{-1}.$$ Here the gauge matrix $\mathbf{B}$ is a diagonal matrix independent of $z$, whose entries belong to a Poisson bracket algebra where $\mathcal{A}_{LV}^\prime$ is embedded. Then the matrix $\mathbf{T}(z)$ has a similar form to $\mathbf{T}_{LV}(z)$; $$\mathbf{T}(z) = \boldsymbol{\mu}_0 z^m + \boldsymbol{\mu}_1 z^{m-1}
+ \cdots + \boldsymbol{\mu}_m,$$ where $\boldsymbol{\mu}_i =
\mathbf{B} \boldsymbol{\mu}_i^{LV} \mathbf{B}^{-1}$. With these settings we have\
[**Proposition 4.5$\mathrm{^\prime}$**]{} [*Let $\mathcal{A}_{N}$ be a Poisson bracket algebra generated by the entries of a matrix $\mathbf{N}(z)$ related to $\mathbf{T}(z)$ by an invertible matrix $\mathbf{S}$ as $$\mathbf{N}(z) = \mathbf{S} \mathbf{T}(z) \mathbf{S}^{-1},
~~
\mathbf{S} =
\begin{pmatrix}
\vec{e}_i \boldsymbol{\mu}^{(1)} \\
\vec{e}_i \boldsymbol{\mu}^{(2)} \\
\vdots \\
\vec{e}_i \boldsymbol{\mu}^{(N)}
\end{pmatrix}.$$ Here each of $\boldsymbol{\mu}^{(i)}$ is a product of $\boldsymbol{\mu}_j$ $(j=0,\cdots, m)$. Then $\mathcal{A}_N$ is embedded in $\mathcal{A}_{LV}^\prime$.*]{}\
[*Proof*]{}. It is sufficient to show that the matrix elements of $\mathbf{N}(z)$ belong to $\mathbb{C}(\mathcal{P}_0,V_n;n \in \mathbb{Z}/L\mathbb{Z})$. Using $\mathbf{B} = \operatorname{diag}[ b_1, b_2, \cdots, b_N ]$, the matrix $\boldsymbol{\mu}^{(i)}$ is rewritten as $$\boldsymbol{\mu}^{(i)} = \mathbf{B} ~
\boldsymbol{\mu}^{(i)\,LV}~ \mathbf{B}^{-1},$$ where $\boldsymbol{\mu}^{(i)\, LV}$ is the associated product of $\boldsymbol{\mu}_j^{LV}$. Therefore the gauge matrix $\mathbf{S}$ can be written as $$\mathbf{S} = b_i \mathbf{S}_{LV} \mathbf{B}^{-1},
~~~
\mathbf{S}_{LV} =
\begin{pmatrix}
\vec{e}_i \boldsymbol{\mu}^{(1)\, LV} \\
\vec{e}_i \boldsymbol{\mu}^{(2)\, LV} \\
\vdots \\
\vec{e}_i \boldsymbol{\mu}^{(N)\, LV}
\end{pmatrix}.$$ Then $\mathbf{N}(z)$ is obtained as $$\begin{aligned}
\mathbf{N}(z) &=
b_i \mathbf{S}_{LV} \mathbf{B}^{-1} \mathbf{T}(z)
\mathbf{B} \mathbf{S}_{LV}^{-1} b_i^{-1} \\
&=
\mathbf{S}_{LV} \mathbf{T}_{LV}(z) \mathbf{S}_{LV}^{-1}.\end{aligned}$$ Since all entries of $\mathbf{S}_{LV}$ and $\mathbf{T}_{LV}(z)$ belong to $\mathcal{A}_{LV}^\prime$, the proposition follows. $\square$
When we apply this proposition to the case $\mathbf{T}(z) = \mathbf{T}_{m;n_1,n_2}(z)$, the first part (1) follows. Further, from (1) we see that the separation equation can be written in terms of entries in $\mathbf{M}_F(z)$, then we obtain the second part (2). $\square$
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|
---
abstract: |
We experimentally study the auto-oscillating spin-wave modes in NiFe/$\beta-$W constriction-based spin Hall nano-oscillators as a function of bias current, in-plane applied field strength, and azimuthal field angle, in the low-field range of 40–80 mT. We observe two different spin-wave modes: *i*) a linear-like mode confined to the minima of the internal field near the edges of the nanoconstriction, with weak frequency dependencies on the bias current and the applied field angle, and *ii*) a second, lower frequency mode that has significantly higher threshold current and stronger frequency dependencies on both bias current and the external field angle. Our micromagnetic modeling qualitatively reproduces the experimental data and reveals that the second mode is a spin-wave bullet and that the SHNO mode hops between the two modes, resulting in a substantial increase in linewidths. In contrast to the linear-like mode, the bullet is localized in the middle of the constriction and shrinks with increasing bias current. Utilizing intrinsic frequency doubling at zero field angle we can reach frequencies above 9 GHz in fields as low as 40 mT, which is important for the development of low-field spintronic oscillators with applications in microwave signal generation and neuromorphic computing.
PACS numbers
:
author:
- Hamid Mazraati
- Seyyed Ruhollah Etesami
- Seyed Amir Hossein Banuazizi
- Sunjae Chung
- Afshin Houshang
- 'Ahmad A. Awad'
- Mykola Dvornik
- Johan Åkerman
title: 'Mapping out the spin-wave modes of constriction-based spin Hall nano-oscillators in weak in-plane fields'
---
\[sec:Intro\]Introduction
=========================
Spin torque nano-oscillators—microwave signal generating devices based on spin-wave auto-oscillations—are of great interest for many kinds of nanoscale applications as they provide highly coherent and widely tunable microwave signals at room temperature. [@chen2016ieeerev] Recently, they have been succeeded by so-called spin Hall nano-oscillators (SHNOs), which utilize the spin Hall effect[@Hirsch1999; @Zhang2000; @Kato2004; @Wunderlich2005; @Saitoh2006; @Valenzuela2006a] to generate microwave signals of similar quality.[@Demidov2012b] To date, a variety of SHNO geometries and material compositions have been proposed.[@Demidov2012b; @Liu2013; @Zholud2014; @Duan2014b; @Collet2016; @Ranjbar2014; @C6NR07903B; @Pai2012; @doi:10.1063/1.4907240; @Mazraati2016; @doi:10.1063/1.5022049; @Spicer2018; @Yin2018] Most recently, a constriction-based SHNO was developed with the particular advantages of having a rather simple fabrication process and relatively low driving current.[@Demidov2014] Later, the mutual synchronization of multiple constriction-based SHNOs was experimentally demonstrated for strong oblique magnetic fields, and substantial improvements in the output power and quality factor were observed.[@Awad2016] Thanks to the unprecedented ability of constriction SHNOs to phase-lock with each other, they may also be utilized for future spintronic neuromorphic computing devices.[@Torrejon2017Nature; @Dvornik2017] However, most practical applications require these devices to operate in either zero or weak applied magnetic fields. A deeper understanding of the SHNO dynamics in such regimes is thus necessary for further developments.
Demidov *et al.*[@Demidov2014] demonstrated that, for an in-plane field of 40 mT, applied at *w.r.t* the drive current, constriction-based SHNOs exhibit a single auto-oscillating mode over a wide range of applied currents with weak negative frequency versus current tunability. Dvornik *et al.*[@Dvornik2018prappl] later showed that such auto-oscillations emerge from the linear localized mode of the nanoconstriction. Although a transition to multimode operation with substantial line broadening had also been observed in Ref. , neither was it discussed in detail nor was the origin of the additional peaks explained. Finally, optimization of the in-plane field angle is essential to achieve high output power and robust mutual synchronization of these devices in in-plane fields.[@Kendziorczyk2016prb]
In this work, we report for the first time on angular-resolved measurements of constriction based SHNO microwave signal generation under weak in-plane fields, $H_{\mathrm{IP}}$. We observe both a linear-like mode confined to the minima of the internal field near the edges of the nanoconstriction, and a lower frequency spin wave bullet mode localized in the middle of the constriction. Our micromagnetic simulations suggest that the SHNO hops rapidly between these two modes, consistent with the much larger linewidths observed in this regime. Finally, we use intrinsic frequency doubling to achieve frequencies exceeding 9 GHz in fields as low as 40 mT.
\[sec:Exp\]Experiment
=====================
\[sec:DevFabrication\]Device fabrication and measurement setup
--------------------------------------------------------------
The SHNO stack, consisting of A NiFe(5nm)/$\beta$-W(5nm) bilayer, was prepared on a c-plane sapphire substrate using dc/rf magnetron sputtering in a 2.5 mTorr Argon atmosphere, in an ultra-high vacuum chamber (base pressure below $1\times 10^{-8}$ mTorr). It was then patterned into an array of $4~\mu$m $\times$ $12~\mu$m rectangular mesas using photolithography and Argon ion milling. Nanoconstrictions with a width of 150 nm were subsequently fabricated in the center of these mesas by a combination of electron-beam lithography and dry etching. To determine the magnetic characteristics of the stack using spin-torque-induced ferromagnetic resonance (ST-FMR) measurements, 6 $\mu$m-wide bars were simultaneously fabricated next to the SHNOs. Finally, a conventional ground–signal–ground (GSG) waveguide and electrical contact pads for broad frequency range microwave measurement were fabricated by lift-off photolithography and Cu/Au sputtering on top of both the nanoconstrictions and the bars.
Fig. \[fig1\] (a) shows a schematic of the device structure, including the directions of the applied in-plane field and current: the field angles $\varphi=0\degree$ and $+90\degree$ are along the +$y$ and +$x$ axes, respectively. A negative (positive) current represents electrons flowing along the $+(-)x$ direction.
The magnetoresistance of the SHNO vs. in-plane field angle is shown in Fig. \[fig1\] (b) revealing an AMR ratio of 0.26 $\%$, similar to literature values for thin NiFi films.[@1058782] We carried out ST-FMR measurement on bars using the homodyne detection approach.[@Sankey2006a; @Sankey2007b; @Chen2009; @Cheng2013; @Collet2016; @Fazlali2016prb] A 313 Hz–pulse–modulated microwave signal was applied alongside a direct current through a bias-tee, and the modulated signal was then detected through the same bias-tee and analyzed using a lock-in amplifier. The applied field was swept from 250 mT to 0 mT while the frequency of the input microwave signal and the level of the direct current were fixed.
Microwave measurements were carried out in a custom-built setup. While a direct current was injected through the constriction area of the SHNO under an in-plane field, the auto-oscillation microwave signal was acquired by a spectrum analyzer after being amplified 35 dB using a broadband low-noise amplifier. All measurements were performed at room temperature.
\[sec:Results\]Results
----------------------
Fig. \[fig1\] (c) shows the ST-FMR spectra at different microwave frequencies from 3 to 12 GHz (dots), with each spectrum well fitted to a sum of one symmetric and one asymmetric Lorentzian (solid lines).[@Liu2011a] The extracted frequencies of the resonance peaks fit well to the Kittel formula [@Kittel1948], leading to an effective magnetization of $\mu_{\mathrm{0}}M_{\mathrm{eff}}=0.71$ T and a gyromagnetic ratio of $\gamma/2\pi=28$ GHz/T. The inset to Fig. \[fig1\] (c) shows the extracted linewidths of the corresponding peaks (black dots) and their fit to the linear model (solid red line). The obtained value of the Gilbert damping is $\alpha=0.016$.
Fig. \[fig1\] (d) shows the ST-FMR spectra for a range of bias currents from +10 mA to -10 mA, measured at a fixed microwave frequency of 5 GHz. The extracted linewidth vs. current behaviour and a fit to the linear model[@Demasius2016; @Ando2008c; @Liu2011a] are shown in the inset. The extracted value of the spin Hall efficiency, defined as the ratio of the absorbed spin and the charge density currents, is $\xi_{\mathrm{SH}}=-0.385$, which is significantly higher than in the case of NiFe/Pt stacks.[@Demidov2012b; @Liu2013; @Ulrichs2013; @Demidov2014; @Duan2014b; @Ranjbar2014; @Zholud2014; @Collet2016]
The power spectral densities (PSDs) of the SHNO vs. bias current for the fields of 40 mT, 60 mT, and 80 mT applied $\varphi=30\degree$ in-plane are shown in Fig. \[fig2\] (a)–(c). For $\mu_0H_{\mathrm{IP}}=80$ mT, there is a single dominant spin-wave mode that, according to Ref. , should originate from the linear magnonic edge mode of the constriction (hence the label “linear-like mode”). In contrast to the uniform ferromagnetic resonance of in-plane magnetized films, where the frequency decreases with the amplitude of precession, the observed auto-oscillations experience a nonmonotonic frequency vs. current behaviour. At lower currents, the frequency of the mode is almost constant while its linewidth decreases with increasing current. However, at a certain field-dependent current the mode shows a redshift (negative nonlinearity), and the linewidth starts to increase. At the same time, traces of a lower frequency and larger linewidth signal can be seen in Fig. \[fig2\](a). These are the signatures of the so-called spin-wave bullet—a nonlinear and nontopological self-localized mode nucleated in regions of negative non-linearity.[@PhysRevLett.95.237201; @Demidov2012b; @PhysRevLett.105.217204; @Boneti2012; @Spicer2018b; @Jungfleisch2016]. They become more apparent when the applied field is reduced to 60 mT (Fig. \[fig2\](b)) and eventually dominate at 40 mT (Fig. \[fig2\](c)). Our experimental data suggest that the contribution of the negative nonlinearity could increase with the applied field, shifting its onset current downwards from approximately 1.4 mA at 40 mT to 1 mA at 80 mT.
Generally, the linear-like and the bullet modes cannot coexist when overlapping in space, so the presence of both signals in the measured spectra is likely due to mode hopping.[@Boneti2012; @MuduliPrl2012; @Muduli2012prb; @Iacocca2014prb; @Heinonen2013; @Sharma2014] The reduced stability and broader linewidth of the bullet mode at higher fields indicate an increase in the nonlinearity and the suppression of the nonlinear magnetic losses[@Tiberkevich2007prb] that limit the auto-oscillation amplitude.
![\[fig3\] (Color online) PSD map of the fundamental and second harmonic of the modes versus the in-plane angle of the field for $\mu_0\mathrm{H_{IP}}=40$ mT at (a) $I_{\mathrm{dc}}=1.3$ mA, and (b) $I_{\mathrm{dc}}=1.1$ mA ](Figure3){width="3.4in"}
The fundamental and second harmonics of both modes as a function of the in-plane angle of the applied field of constant strength, $H_{\mathrm{IP}}=40$ mT, are plotted in Fig. \[fig3\] (a) and (b) at $I_{\mathrm{dc}}=1.3$ mA and 1.1 mA, respectively. The slight asymmetry of the responses with respect to the field angle is likely due to fabrication-related shape imperfections or the slight misalignment of the sample with respect to the center of the magnet. We observed no auto-oscillations for angles beyond $\lvert \varphi \rvert=45 \degree$, either because the edge modes (a) disappear due to the suppression of the spin-wave wells or (b) do not get excited as they move away from the constriction, experiencing less spin-current density. The former is unlikely since it would be accompanied by a considerable increase in frequency which we do not see experimentally. In fact, the frequency of the linear-like mode depends only weakly on the in-plane angle of the applied field, suggesting minimal changes to its localization depth.
The bullet mode was observed at higher currents (Fig. \[fig3\] (a)), and disappeared as the current dropped (Fig. \[fig3\] (b)), similar to the PSD maps shown in Fig. \[fig2\]. In contrast to the linear-like mode, the frequency of the bullet mode depends strongly on the field angle and decreases with the angle of the external field. This may be attributed to the angular dependence of the damping-like torque. At low angles, the magnetization vector points mostly antiparallel to the polarization of the spin current resulting in higher spin torque efficiency, larger bullet amplitude, and thus a higher nonlinear frequency redshift.
It is worth noting that the fundamental harmonic for any of the modes were not detected for small field angles. This is a consequence of the first derivative of the AMR curve (Fig. \[fig1\]b) approaching zero at $\varphi=0\degree$. However, both modes were instead clearly observed by their signals at twice their original frequencies, since the second derivative of the AMR curve has a maximum in the vicinity of $\varphi=0\degree$.[@Muduli2011jap] As a consequence, using this intrinsic frequency doubling, we can reach very high frequencies already at very low fields.
\[sec:Simulation\]Micromagnetic simulations
===========================================
In order to investigate the physics behind the experimentally observed spin-wave modes, we carried out micromagnetic simulations using MuMax3[@Vansteenkiste2014]. The material parameters used in the simulations—such as the NiFe saturation magnetization, the Gilbert damping, the gyromagnetic ratio, and the spin Hall efficiency of the bilayer—were obtained directly from the ST-FMR measurements. An exchange stiffness of $A_{\mathrm{ex}}=10^{-12}$ J/m was considered for NiFe. We ran the simulations for a geometry with a lateral size of 2000$\times$2000 nm$^2$ (large enough to avoid boundary effects) and a thickness of 5 nm, which is similar to the thickness of the ferromagnetic layer in the real sample. The distributions of the direct charge current and the corresponding Oersted field were obtained using COMSOL Multiphysics$\textsuperscript{\textregistered}$ software [@Dvornik2018prappl] for a NiFe/W bilayer with resistivity values of 0.90 $\mu \Omega \cdot$m and 2.12 $\mu \Omega \cdot$m for NiFe and W, respectively. The auto-oscillation spectra were obtained by applying a Fast Fourier Transform (FFT) to the net $y$ component of the total magnetization, simulated over 1000 ns. We assumed that the spin-current polarization equals $P=1$ and is independent of the angle between the directions of magnetization and of spin-current polarization.
The auto-oscillation spectra vs. current, and the splitting into two modes can be seen in Fig. \[fig4\] (a), in good agreement with our experimental observations (Fig. \[fig2\]). We additionally calculated the linear eigenmodes of the SHNOs by turning off the spin torque in our simulation and exciting the system with a magnetic field pulse (a sinc function with an amplitude of 0.5 mT and a duration of 25 ps). The FMR mode and the linear localized mode are shown in Fig. \[fig4\] (a) by green and yellow circles, respectively. Both auto-oscillation modes are far below the FMR frequency, *i.e.*, both are localized. In agreement with Ref. , the frequency of the auto-oscillation at its onset coincides with the eigenmode of the constriction. In clear contrast, the lower frequency mode that emerges at around 1.1 mA cannot be attributed to the eigenmodes of the constriction, and hence not to any deepening of the spin-wave well caused by the Oersted field.[@Demidov2014] Instead, we conclude that the lower frequency mode is a self-localized spin-wave bullet. In contrast to the in-plane magnetized extended films—where bullets typically have lower threshold currents than the quasilinear propagating spin waves—they require higher currents than the linear-like modes of the constriction. This could be attributed to the fact that self-localization in the given volume occurs only when some critical number of magnons is achieved [@PhysRevLett.95.237201], while field confinement happens even for spin waves with vanishing amplitudes.
It is worth mentioning that the bullet appears to be splitting from the linear-like mode. We, therefore, inspected the transient behaviour of the magnetization dynamics in a multimode regime ($I_{\mathrm{dc}}=1.22$) by performing time-frequency analysis using a short-time Fourier transform. Due to the small frequency gap between the linear-like and bullet modes, we did not decrease the moving window length to less than 50 ns, in order to maintain a reasonable frequency resolution (Kaiser window with $\beta=30$ and overlap of 49.95 ns). As can be seen in Fig. \[fig4\] (b), the discontinuities in the linear-like mode are followed by a sharp transition to the bullet mode—*i.e.*, mode hopping is observed. We, therefore, conclude that the bullet mode does not branch off, but instead nucleates from the linear-like mode towards a lower frequency. To this end, it could be viewed as a self-localization of the field-localized mode of the constriction.
![\[fig5\] (Color online) (*Simulation*) The mode profile of the FFT peaks at $H_{\mathrm{IP}}=60$ mT, $\varphi=30\degree$, and $I_{\mathrm{dc}}=1.26$ mA are shown in (a) for the linear-like mode confined to the edges of the nanoconstriction, and (b) for the bullet mode, which is more localized centrally in an area comparable to the constriction size. (c) FFT amplitude along the A–A’ direction on the mode profiles perpendicular to the direction of the external field. (d) Normalized volumes of the linear-like mode (black rectangles) and bullet mode (red circles) versus current.](Figure5.png){width="3.4in"}
To determine the spatial distribution of the observed modes, we performed pointwise temporal FFT over the dynamic component of the magnetization sampled in the vicinity of the constriction—that is, mode profile analysis using the <span style="font-variant:small-caps;">semargl-ng</span> package[@dvornik2011numerical; @Dvornik2013]. The profiles of the linear-like and bullet modes calculated for the applied current of $I_{\mathrm{dc}}=1.26$ mA are shown in Fig. \[fig5\] (a) and (b), respectively. While the linear-like mode is confined to the edges of the constriction where the internal field has local minima, the bullet is more localized in the center of the constriction, where the internal field instead has a local maximum. This again confirms its predominant self-localization character.
Similar to what was observed in Refs., both the linear-like and bullet modes mostly extend along the direction perpendicular to the applied field (shown by the dotted lines in Figs. \[fig5\] (a) and (b)). However, we note that the bullet mode has somewhat lower extent (a smaller “halo” is seen in Fig. \[fig5\] (b)). This is confirmed by comparing the cross-sections of the linear-like and bullet modes extracted along the major axes of their profiles (as shown by the dotted lines in Figs. \[fig5\] (a) and (b)). In fact, compared to the linear-like mode, the auto-oscillation power of the bullet mode drops faster with distance from the constriction center. The self-localization thus compresses the bullet’s profile.
To compare the degree of localization of the linear-like and bullet modes, we estimate their auto-oscillation volumes at each current by integrating the corresponding spatial profiles, as explained in Ref.. The calculated volumes are mapped against the bias current in Fig. \[fig5\] (d). First, we observe that the volume of the linear-like mode increases, starting from its onset and continuing until the bullet nucleates. Once the multimode state is achieved, the volume of the linear-like mode stays relatively constant, while its slight negative frequency vs. current slope resembles that of the eigenmode, and so could be attributed to the effect of the Oersted field. In contrast, the volume of the bullet mode drops monotonically with the applied current. According to Ref. , this non-monotonic re-localization of the auto-oscillations is due to the competition of the repulsion and attraction of magnons, caused by the reduction of the static demagnetizing field (shallowing of the spin wave wells) and enhancement of the dynamic dipolar coupling, respectively. It has been predicted that the repulsion of magnons dominates at small amplitudes, consistent with the initially positive nonlinearity of the auto-oscillations observed in Fig. \[fig4\](a). The nucleation of the bullet, thus, indicates that the attraction process becomes dominant. In the vicinity of this point, the auto-oscillations show zero nonlinearity, consistent with the minima in the generational linewidth seen in Fig. \[fig2\](d).[@kim2008prl] Although the bullet’s frequency drops well below the edge mode, its volume remains considerably higher. This again highlights the difference in the confinement mechanisms of the linear-like and bullet modes, being of a static and dynamic nature, respectively.
\[sec:Conclusion\]Conclusion
============================
We have determined the dynamic magnetic properties of nanoconstriction-based SHNOs subject to weak in-plane magnetic fields, by carrying out microwave measurements and comparing them with the micromagnetic simulation of the same structure. We found that, at high bias currents, the auto-oscillations spectra show two dominant modes: a linear-like mode that is confined to the edges of the constrictions, and a bullet mode that is confined to the center. While the former is a field-localized mode confined to the minima of the internal field, and therefore slightly depends on the strength of the current and its relative angle with regards to the external field, the frequency of the bullet rapidly decreases with its amplitude. Our simulations reveal that the two modes do not coexist at a given point in time, but instead mode hop, which also manifests itself in the experimentally observed linewidth broadening of the modes. Finally, our simulations have shown that the bullet nucleates from the linear-like mode and then experiences substantial auto-oscillation volume compression due to the nonlinear self-localization. Our findings provide a better understanding of the dynamics of nanoconstriction-based SHNOs necessary for subsequent studies of these systems, including their low-field operation and possible in-plane mutual synchronization.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and the Knut and Alice Wallenberg foundation (KAW). This work was also supported by the European Research Council (ERC) under the European Community’s Seventh Framework Programme (FP/2007-2013)/ERC Grant 307144 “MUSTANG”.
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10.1103/PhysRevB.85.174427) [ ()](https://arxiv.org/abs/1805.09212v1) [****, ()](\doibase 10.1103/PhysRevLett.116.057601) [****, ()](\doibase 10.1103/PhysRevLett.108.207203) [****, ()](\doibase 10.1103/PhysRevB.86.174408) [****, ()](\doibase 10.1103/PhysRevB.89.054402) [****, ()](\doibase 10.1109/TMAG.2013.2242866) [****, ()](\doibase 10.1063/1.4896634) [****, ()](\doibase 10.1103/PhysRevB.75.014440) [****, ()](\doibase 10.1063/1.3647759) [****, ()](\doibase 10.1063/1.4899186) **, [Ph.D. thesis](https://ore.exeter.ac.uk/repository/handle/10036/3304), () in [**](\doibase 10.1007/978-3-642-30247-3_8), Vol. , (, ) pp. [ ()](https://arxiv.org/abs/1804.01585) [****, ()](\doibase 10.1103/PhysRevLett.100.017207)
|
---
abstract: |
We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs $G$ and $W$ and an integer $k$, we are asked to find a $k$-edge weighted graph $W_k$ such that $G+W_k$ is a good spectral sparsifer of $G+W$. We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on $G$ and $W$ such that a good sparsifier exists and give a polynomial time algorithm to find the sparsifer.
As a significant application of our technique, we show that for each positive integer $k$, every $n$-vertex weighted graph has an $(n-1+k)$-edge spectral sparsifier with relative condition number at most $\frac{n}{k} \log n\, \tilde{O}(\log\log n)$ where $\tilde{O}()$ hides lower order terms. Our bound is within a factor of $\tilde{O}(\log \log n)$ from optimal. This nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems.
We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.
author:
- 'Alexandra Kolla[^1]'
- 'Yury Makarychev[^2]'
- 'Amin Saberi [^3]'
- 'Shang-Hua Teng [^4]'
title: Subgraph Sparsification and Nearly Optimal Ultrasparsifiers
---
Introduction
============
Sparsification is an important technique for designing efficient graph algorithms, especially for dense graphs. Informally, a graph $\tilde{G}$ is a sparsifer of $G$ if they are similar in a particular measure (which is important to the application that one has in mind), and that $\tilde{G}$ has linear or nearly linear number of edges. Various notions of graph approximation and sparsification have been considered in the literature. For example, Chew’s [@Che86] spanners (for shortest path planning) have the property that the distance between every pair of vertices in $\tilde{G}$ is approximately the same as in G. Benczur and Karger’s [@BK96] cut-sparsifiers (for cuts and flows) have the property that the weight of the boundary of every set of vertices is approximately the same in $G$ as in $\tilde{G}$.
In this paper, we will mainly be interested in the spectral notion of graph similarity introduced by Spielman and Teng [@ST04], [@ST08b]: we say that a weighted undirected graph $H$ is a $\kappa$-approximation of another $G$ if for all $ x\in \mathbf{R}^V$, $$\label{eq:patch:sparsif}
x^T{\mathcal{L}}_G x \leq x^T{\mathcal{L}}_{\tilde{G}} x\leq \kappa x^T{\mathcal{L}}_G x$$ where for a weighted undirected graph $G$, ${\mathcal{L}}_G$ is the [*Laplacian matrix*]{} of $G$ defined as the following: For each ${\mathcal{L}}_G(i,i)$ is equal to the sum of weights of all edges incident to vertex $i$ and for $i \neq j$, ${\mathcal{L}}_G(i,j) = -w_{i,j}$, where $w_{i,j}$ is the weight on edge $(i,j)$.
In [@ST04; @ST08b], the following spectral sparsification problem is considered. Given a weighted graph $G = (V,E,w)$, an integer $\tilde{m}\leq |E|$, and $\kappa \geq 1$, find a graph $\tilde{G} =\{V, \tilde{E}, \tilde{w}\}$ such that $|\tilde{E}| \leq \tilde{m}$ and $\tilde{G}$ is a $\kappa$-approximation of $G$. We will refer to this problem and its corresponding optimization problem as the [ Spectral Sparsification]{}. Spielman and Teng showed that every weighted graph has a nearly linear-sized spectral sparsifier and gave a nearly linear-time algorithm for computing such a sparsifier. Recently, Batson, Spielman, and Srivastava [@BSS] gave a beautiful, polynomial-time construction to produce a linear-sized spectral sparsifier.
In this paper, we introduce a variation of the spectral sparsification problem which we will refer to as the [Subgraph Sparsification.]{} In our version, we are given two weighted graphs $G$ and $W$, an integer $k$ and $\kappa\geq 1$. The goal is to find a $k$-edge weighted graph $W_k$ such that $(G+W_k)$ is a $\kappa$-approximation of $(G+W)$. The challenge in the new version of the sparsification problem is that we have to respect part of the graph, i.e., $G$, and only modify part of graph given in $W$.
As the main technical contribution of the paper, we give a nontrivial condition about $G$ and $W$ such that a good sparsifier exists. Our proof critically uses the intuition of Batson, Spielman, and Srivastava [@BSS], that uses potential functions that guide an incremental process for selecting the edges of the sparisifier. We will refer to that as as the [*BSS process*]{}. We have enhanced their approach with new understanding about subspace sparsification and spectral approximation.
Our challenge, at high level, is the following. The BSS process uses two carefully chosen barriers (see Section 2) so that at each step, all eigenvalues can be kept far enough from these barriers. They have $\Theta(n)$ edges to select. So they consider the entire $n$-dimensional space and have step size $\Theta(1/n)$ on these barriers.
On the other hand, we can only add $k$ edges, where $k$ can be arbitrarily smaller than $n$. The addition of each edge can only increase smallest eigenvalue to the second smallest eigenvalue. Therefore the addition of $k$ edges can only improve the subspace defined by the $k$ smallest eigenvalue. Now, the critical part of the argument is that to build a good sparsifier, we need to ensure that the addition of the edges does not increase the high spectra by too much. So in our incremental process, we need to keep track of two subspaces, a fixed one defined by the $k$ smallest eigenvalues and a floating one defined by the higher spectra.
We developed an analysis for performing spectral analysis in the projection of a sequence of two subspaces, which might be interesting on its own right. Our analysis also provide a nice example for using majorization.
Our ability to conduct sparsification on a subgraph enables us to obtain improved results for a few problems on spectral optimization. The first application that we consider is the problem of finding ultrasparsifiers as defined in Spielman and Teng [@ST04]. For parameters $\kappa \geq 1$ and $k\geq 1$, a weighted undirected graph $U$ is a [*$(\kappa,k)$-ultrasparsifier*]{} of another graph $G$, if $U$ has at most $n-1+k$ edges, and ${\mathcal{L}}_U \preceq {\mathcal{L}}_G \preceq
\kappa \cdot {\mathcal{L}}_U$. Ultrasarsifiers are essential in the application of the preconditioning techniques for solving linear systems [@ST04; @ST08b]. It has been shown in [@ST04] that every weighted undirected graph $G$ has a $(\frac{n}{k} \log ^{O(1)}
n,k)$ ultrasparsifiers, for any $k$. As a significant application of our subgraph sparsification technique, we show that for every positive integer $k$, every $n$-vertex weighted graph has a $(\frac{n}{k}\log
n\,\tilde{O}(\log \log n),k)$-ultrasparsifier. Our bound is within a factor of $\tilde{O}(\log \log)$ from the optimal. This new result nearly settles a question about ultrasparsifiers left open by Spielman and Teng.
At high level, our solution to ultrasparsification is quite simple, once we have our subgraph sparsification result. Given a weighted graph$G$, we first construct a low-stretch spanning tree [@AKPW; @EEST; @ABN08] $T$ of $G$. We then apply an elegant result of Spielman and Woo [@SW] which states that the sum of the relative condition numbers of ${\mathcal{L}}_G$ and ${\mathcal{L}}_T$ is equal to the total stretch to embed $G$ onto $T$. We will also use Spielman–Woo’s tail distribution bound on the number of relative eigenvalues of ${\mathcal{L}}_G$ and ${\mathcal{L}}_T$ that are larger than a given parameter.
Algorithmically, we start with the best available [@ABN08] low-stretch spanning tree $T$ of $G$ whose total stretch is $n\log n\,\tilde{O}(\log \log n)$. We then consider the subgraph sparsification problem defined by $T$ and $W = \frac{k}{n\log n \tilde{O}(\log \log n )} G$. We apply the structure theorem of Spielman and Woo [@SW] to show that $(T,W)$ satisfy our condition for subgraph sparsification and apply our result to show that there exists a $k$-edge weighted graph $W_k$ whose edges are in $W$ such that $T+W_k$ is a spectral approximation of $T+W$. It is then not hard to prove that $T+W_k$ is an a $(\frac{n}{k}\log n\,\tilde{O}(\log \log n),k)$-ultrasparsifier.
As another application of our technique on subgraph sparsification, we consider the following spectral optimization problem studied in [@BG]: Given a graph $G$ and a parameter $k$, we are asked to find $k$ edges amongst a set of candidate edges to add to $G$ so as to maximize its algebraic connectivity. Algebraic connectivity has emerged as an important parameter for measuring the robustness and stability of a network and is an essential factor in the performance of various search, routing and information diffusion algorithms.
The spectral optimization considered in this paper is known to be NP-hard [@damon] and no approximation guarantee for it was known prior to our work. We give an SDP-based approximation algorithm for the problem. Our techniques for subgraph sparsification enable us to develop a novel *rounding* scheme in order to find a combinatorial solution. Since the integrality gap of the SDP is unbounded, our analysis involves adding a separate upper bound, which is roughly the $k$-th largest eigenvalue of the Laplacian of $G$ to approximate the optimum solution.
Preliminaries
=============
**Matrix Notation and Definitions.** We denote the Laplacian of a graph $G$ by ${\mathcal{L}}_G$. For brevity, we write ${G_1} \preceq G_2$ to denote ${\mathcal{L}}_{G_1} \preceq {\mathcal{L}}_{G_2}$. For an $n\times n$ matrix $A$, let $\lambda_{\mathrm{min}}(A) \equiv \lambda_1(A) \leq \lambda_2(A) \leq \dots \leq \lambda_n(A) \equiv \lambda_{\mathrm{max}}(A)$ be the set of eigenvalues in the increasing order. Let $A^\dagger$ be the pseudoinverse of $A$. If $A$ is symmetric, $A^\dagger$ is also symmetric and $AA^\dagger = A^\dagger A = P_{\operatorname{Im}(A)}$, where $P_{\operatorname{Im}(A)}$ is the orthogonal projection on $\operatorname{Im}(A)$. Let $A{\bullet}B \equiv \operatorname{tr}A^TB$ be the Frobenius product of matrices $A$ and $B$. We define the *condition number* of a non-singular matrix $A$ as $\kappa=\|A\|\|A^{-1}\|$, which is equal to $\lambda_{\mathrm{max}}(A)/\lambda_{\mathrm{min}}(A)$ if $A$ is a (symmetric) positive definite matrix. For positive definite matrices $A$, $B$ with $\operatorname{Im}A = \operatorname{Im}B$, we define the relative condition number as $$\kappa(A,B) =\max_{x\notin\ker B} \frac{x^T A x}{x^T B x} \cdot \max_{x\notin\ker A} \frac{x^TBx}{x^TAx}.$$
#### Ultrasparsifiers.
We say that a graph is $k$–ultra-sparse if it has at most $n-1+k$ edges. We note that a spanning tree is $0$–ultra-sparse. A ($\kappa,k)$ *ultra-sparsifier* of a graph $G = (V,E,w)$ is a $k$–ultra-sparse subgraph of $G$ such that $U \preceq G \preceq \kappa \cdot U$ [@ST04].
Matrix Sparsifiers {#sec:patch:patch}
==================
In this section, we prove an analog of the sparsification theorem of Batson, Spielman, and Srivastava [@BSS].
\[def:patch\](**Graph Patch**) Let $G$ be a (weighted) graph. A graph $W$ on the vertices of $G$ is a $(k,T,\lambda^*)$-patch for $G$ if the following properties hold[^5],
1. $\lambda_{k+1}({\mathcal{L}}_G {\mathcal{L}}_{G+W}^{\dagger}) \equiv
\lambda_{k+1}(({\mathcal{L}}_{G+W}^{\dagger})^{1/2}{\mathcal{L}}_G({\mathcal{L}}_{G+W}^{\dagger})^{1/2})
\geq \lambda^*$;
2. $\operatorname{tr}({\mathcal{L}}_W {\mathcal{L}}_{G+W}^{\dagger}) \leq T$.
We prove that for every patch, there exists a “patch sparsifier” supported on $O(k)$ edges. Specifically, we prove the following theorem.
\[cl:patch\] Let $W=(V, E_W, \{w_e\}_{e\in E_W})$ be a $(k, T,\lambda^*)$-patch for $G$ with edge weights $w_e$ and $N \geq 8k$. Then there is a weighted graph $W_k = (V, E_{W_k}, \{\tilde w_e\}_{e\in E_{W_k}})$ with edge weights $\tilde w_e$ such that
1. $W_k$ has at most $N$ edges; $E_{W_k} \subseteq E_W$.
2. $c_1 \min(N/T,1) \lambda^* {\mathcal{L}}_{G+W} \preceq {\mathcal{L}}_{G+W_k} \preceq c_2 {\mathcal{L}}_{G+W}$, for some absolute constants $c_1$ and $c_2$.
3. The total weight of edges, $\sum_{e\in E_{W_k}} \tilde w_k$, is at most $\min(1, N/T) \sum_{e\in E_W} w_e$.
We say that $W_k$ is a patch sparsifier of $W$ with respect to $G$.
The claim will follow immediately from the following theorem, which is is of independent interest. We will also show another (related) application of this theorem in Section \[sec:patch:algcon\].
\[thm:mainmatrix\] Suppose we are given a positive definite $n\times n$ matrix $X$ and a sequence of matrices $Y_i = v_i v_i^T$ ($i=1,\dots, m$) with $$X + \sum_{i=1}^m Y_i = M^*,$$ and $\lambda_{\mathrm{max}}(M^*) \leq 1$. Additionaly, suppose each matrix $Y_i$ has cost $cost_i$ and $\sum_{i=1}^m cost_i = 1$. Let $\lambda^* = \lambda_{k+1}(X)$, and $T= \lceil \operatorname{tr}(M^*-X)\rceil $. Then for every $N > 8k$ there exists a set of weights $w_i$ with $|\{w_i: w_i \neq 0\}| = N$ such that the matrix $M = X + \sum_{i= 1}^m w_i Y_i$ satisfies, $$c_1\min(N/T,1) \cdot \lambda^* \cdot\lambda_{\mathrm{min}}(M^*)
\leq
\lambda_{\mathrm{min}}(M)
\leq
\lambda_{\mathrm{max}}(M)
\leq c_2,$$ where $c_1$ and $c_2$ are some absolute constants, and $\sum_{i=1}^m w_i cost_i \leq \min(1, N/T)$.
**Proof Overview.** Our proof closely follows the approach of Batson, Spielman, and Srivastava [@BSS]. We construct matrix $M$ in $N$ steps; at each step we choose an index $i$ and weight $w_i$ and add $w_i Y_i$ to the sum $X + \sum_{i=1}^m w_i Y_i$. Recall that Batson, Spielman, and Srivastava define two “barriers” $l$ and $u$ and maintain the property that all eigenvalues of $M$ lie between $l$ and $u$. At each step, they increase $l$ and $u$ and update matrix $M$ so that this property still holds. Finally, the ratio between $u$ and $l$ becomes very close to $1$, which means that $\lambda_{\mathrm{min}}(M)$ is very close to $\lambda_{\mathrm{max}}(M)$. During this process, they keep track not only of the smallest and largest eigenvalues of $M$ but of all $n$ eigenvalues to avoid accumulation of eigenvalues in neighborhoods of $l$ and $u$. To this end, they define two potential functions, the lower potential function $\Phi_l(M) = \sum_{i=1}^n \frac{1}{\lambda_i(M) - l}$ and the upper potential function $\Phi^u(M) = \sum_{i=1}^n \frac{1}{u - \lambda_i(M)}$, and then ensure that $\Phi_l(M)$ and $\Phi^u(M)$ do not increase over time. That guarantees that all eigenvalues of $M$ stay far away from $l$ and $u$.
In our proof, however, we cannot keep an eye on all eigenvalues. After each step, only one eigenvalue increases, and thus we need $\theta(n)$ steps to increase all eigenvalues participating in the definition of $\Phi_l(M)$. But our goal is to “patch” $X$ in roughly $k$ steps. So we focus our attention only on $k$ smallest and $T$ largest eigenvalues.
Let $S$ be the eigenspace of $X$ corresponding to $k$ smallest eigenvalues, and $P_S$ be the projection onto $S$. We define the lower potential function as follows, $$\Phi_l(A) = \operatorname{tr}(P_S(A-lI)P_S)^{\dagger} = \sum_{i=1}^k \frac{1}{\lambda_i( \left.A\right|_S ) - l},$$ where $\left.A\right|_S$ denotes the restriction of $A$ to the space $S$ ($\left.A\right|_S$ is a $k\times k$ matrix). Note that the space $S$ is fixed, and the eigenvector corresponding to the smallest eigenvalue will not necessarily lie in $S$ after a few steps. We want to ensure that after $N$ steps, $$\sum_{i=1}^m w_i \bigl.Y_i\bigr|_S \succeq c \min(N/T,1) \sum_{i=1}^m \bigl.Y_i\bigr|_S = c \min(N/T,1) \bigl.(M^*-X)\bigr|_S,$$ or in other words, $\lambda_{\mathrm{min}}(\left.(Z(\sum_{i=1}^m w_iY_i)Z)\right|_S) \geq c \min(N/T,1)$, where $Z = {\left((P_S(M^* - X)P_S)^{\dagger}\right)}^{1/2}$. To this end, we show how to update $M$ and $l$ so that $\Phi_l(Z(\sum_{i=1}^m w_i Y_i)Z)$ does not increase, and $l$ equals $c \min(N/T,1)$ after $N$ steps. It remains to lower bound $\lambda_{\mathrm{min}}(M)$ in the entire space. We know that all eigenvalues of $X$ (and therefore, of $M$) in $S^{\perp}$ are at least $\lambda^*$. We show that that together with an upper bound on $\lambda_{\mathrm{max}}(M)$ implies that $\lambda_{\mathrm{min}}(M) \geq c_1\min(N/T,1) \cdot \lambda^*\lambda_{\mathrm{min}}(M^*)$ (the product of the lower bounds on $\lambda_{\mathrm{min}}$ in spaces $S$ and $S^{\perp}$ divided by the upper bound on $\lambda_{\mathrm{max}}$).
Similarly, we amend the definition of the upper potential function. Since we need to bound $\lambda_{\mathrm{max}}$ in the entire space, we cannot restrict $\Phi^u(M)$ to a fixed subspace. For a matrix $A$, we consider the eigenspace of $A$ corresponding to its largest $T$ eigenvalues. Denote it by $L_A(A)$; denote the projection onto $L(A)$ by $P_{L(A)}$. Then $$\Phi^u(A) = \operatorname{tr}(P_{L(A)}(uI-A)^{-1}P_{L(A)}) = \operatorname{tr}(P_{L(A)}(uI - A)P_{L(A)})^{\dagger}
= \sum_{i=n-T + 1}^N \frac{1}{u - \lambda_i(A)}.$$ Note that both definitions of $\Phi^u(A)$ — in terms of regular inverse and in terms of pseudoinverse — are equivalent since $L(A)$ is an invariant subspace of $A$. However, $\Phi_l(A)$ is not equal to $\operatorname{tr}(P_{S}(A - lI)^{-1}P_{S})$ in general since $S$ is not necessarily an invariant subspace of $A$.
Our algorithm and analysis are similar to those of Batson, Spielman, and Srivastava [@BSS]. However, several complications arise because we are controlling eigenvalues in different subspaces and, moreover, one of these subspaces, $L(A)$, is not fixed.
Let us summarize the proof. We construct the matrix $M$ iteratively in $N$ steps. Let $A^{(q)}$ be the matrix and $w_i^{(q)}$ be the weights after $q$ steps. We define an auxiliary matrix $B^{(q)}$ as $Z(A^{(q)}-X)Z$. We have, $$A^{(q)} = X + \sum_i w_i^{(q)} Y_i;\quad
B^{(q)} = \sum_i w_i^{(q)} ZY_iZ = Z(A^{(q)}-X)Z.$$ We will ensure that the following properties hold after each step (for some values of constants $l_0$, $\delta_L$, $u_0$, $\delta_U$, $\epsilon_L$, $\epsilon_U$, which we will specify later).
1. $\Phi_{l_0}(B^{(0)}) \leq \epsilon_L$ and $\Phi^{u_0}(A^{(0)}) \leq \epsilon_U$.
2. Each matrix $A^{(q)}$ and $B^{(q)}$ is obtained by a rank-one update of the previous one: $$A^{(q+1)} =A^{(q)}+t Y_i,\quad
B^{(q+1)} =B^{(q)}+t ZY_iZ$$ for some $i$.
3. Lower and upper potentials do not increase. Namely, for every $q =0,1,\dots,N$, $$\Phi^{u_0+(q+1)\delta_U}(A^{(q+1)}) \leq \Phi^{u_0 + q\delta_U}(A^{(q)})\leq \epsilon_U
\text{ and }
\Phi_{l+(q+1)\delta_L}(B^{(q+1)}) \leq \Phi_{l_0 + q\delta_L}(B^{(q)})\leq \epsilon_L.$$
4. At each step $q$, $\lambda_{\mathrm{min}}(\bigl.B^{(q)}\bigr|_S) > l \equiv l_0 + q \delta_L$ and $\lambda_{\mathrm{max}}(A^{(q)}) < u \equiv u_0 + q \delta_U$. In particular, this condition ensures that all terms in the definitions of upper and lower potentials are positive.
5. At each step $q$, the total cost is at at most $q/\max (N,T)$: $\sum w_i^{(q)} cost_i \leq q/\max (N,T)$.
We present the complete proof in Sections \[sec:barshift\] and \[sec:thmmainproof\]. In Section \[sec:barshift\], we first find conditions under which we can update $A^{(q)}$ and $u$ (Lemma \[upper\_potential\]), and $B^{(q)}$ and $l$ (Lemma \[lower\_potential\]). Then we show that both conditions can be simultaneously satisfied (Lemma \[both\_barriers\]). In Section \[sec:basic\], we prove several theorems that we need later to deal with a non-fixed subspace $L(A)$. Finally, in Section \[sec:thmmainproof\], we combine all pieces of the proof together.
Some Basic Facts about Matrices {#sec:basic}
-------------------------------
### Sherman–Morrison Formula
We use the Sherman–Morrison Formula, which describes the behavior of the inverse of a matrix under rank-one updates. We first state the formula for regular inverse [@GV96], and then we show that a similar expression holds for the pseudoinverse.
\[lem:path:sminv\] If $A$ is a nonsingular $n\times n$ matrix and $Y = vv^T$ is a rank-one update, then $$\nonumber
(A+Y)^{-1} =A^{-1} - \frac{A^{-1} Y A^{-1}}{1+A^{-1}{\bullet}Y}$$
\[lem:path:smpseudinv\] If $A$ is a symmetric (possibly singular) $n\times n$ matrix, $Y = vv^T$ is a rank-one update, then $$(A+PYP)^{\dagger} =A^{\dagger} - \frac{A^{\dagger} Y A^{\dagger}}{1+A^{\dagger}{\bullet}Y},$$ where $P$ is the orthogonal projection on $\operatorname{Im}(A)$.
Let $\bar v = Pv$ and $\bar Y = PYP = \bar v \bar v^T$ . Note that $A^{\dagger} Y A^{\dagger} = A^{\dagger} \bar Y A^{\dagger}$, since $P A^{\dagger} = P$, and $$A^{\dagger}{\bullet}\bar Y = \operatorname{tr}A^{\dagger} \bar Y = \operatorname{tr}A^{\dagger} (PYP) =
\operatorname{tr}(PA^{\dagger}P) Y = A^{\dagger}{\bullet}Y.$$ We need to verify that $$(A+\bar Y)\left(A^{\dagger} - \frac{A^{\dagger}\bar Y A^{\dagger}}{1+A^{\dagger}{\bullet}\bar Y}\right) =\left(A^{\dagger} - \frac{A^{\dagger}\bar Y A^{\dagger}}{1+ A^{\dagger}{\bullet}\bar Y}\right)(A+\bar Y)
= P.$$ Since $A$ is a symmetric matrix, $AA^{\dagger} = A^{\dagger}A = P$. Since $P^2 = P$, $P\bar Y P = \bar Y$ and $\bar Y A^{\dagger} \bar Y = \bar v \bar v^T A \bar v \bar v^T
= \bar v (A {\bullet}\bar Y) \bar v^T = (A {\bullet}\bar Y) \bar Y$. We calculate, $$\begin{aligned}
(A+\bar Y)\left(A^{\dagger} - \frac{A^{\dagger} \bar Y A^{\dagger}}{1+ A^{\dagger} {\bullet}\bar Y}\right) &=
AA^\dagger+ \bar Y A^\dagger -\frac{\bar P}{Y A^\dagger+ \bar Y A^\dagger \bar Y A^\dagger}{1+ A^\dagger {\bullet}\bar Y}\\
&= P + \bar Y A^\dagger -\frac{(1+ A^\dagger{\bullet}\bar Y )\bar Y A^\dagger}{1
+ A^\dagger{\bullet}\bar Y}=P + \bar Y A^\dagger - \bar Y A^\dagger=P.\end{aligned}$$ Similarly, $$(A^{\dagger} - \frac{A^{\dagger} \bar Y A^{\dagger}}{1+A^{\dagger} {\bullet}\bar Y})(A+\bar Y) = P.$$
### Majorization {#sec:basic}
\[Majorization\](Majorization) For every positive semidefinite matrix $A$, every projection matrix $P$, and every $r\in\{1,\dots,n\}$ $$\label{eq:majorization}
\sum_{i= n - r + 1}^n \lambda_i(A) \geq \sum_{i= n - r + 1}^n
\lambda_i(PAP).$$ In particular, $\lambda_{\mathrm{max}}(A) \geq \lambda_{\mathrm{max}}(PAP)$.
Let $e_1, \dots, e_n$ be an orthonormal eigenbasis of $A$ so that $e_i$ has eigenvalue $\lambda_i(A)$. Similarly, let $\tilde e_1, \dots, \tilde e_n$ be an orthonormal eigenbasis of $PAP$ so that $\tilde e_i$ has eigenvalue $\lambda_i(PAP)$. Write $$\tilde e_i = \sum_{j=1}^n \langle e_j, \tilde e_i\rangle e_j.$$ Note that if $\lambda_i(PAP) \neq 0$ then $\tilde e_i \in \operatorname{Im}(P A P) \subseteq \mathrm{Im}(P)$ and $P \tilde e_i = \tilde e_i$. Then $$\lambda_i(PAP) = \tilde e_i^T PAP \tilde e_i
= \tilde e_i A \tilde e_i
= \sum_{j=1}^n \langle e_j, \tilde e_i\rangle^2 \lambda_j(A).$$ If $\lambda_i(PAP) = 0$ then trivially $$\lambda_i(PAP) = 0 \leq \sum_{j=1}^n \langle e_j, \tilde e_i\rangle^2 \lambda_j(A).$$ Therefore, $$\sum_{i=n - r + 1}^n \lambda_i(PAP) \leq
\sum_{i=n - r + 1}^n \sum_{j=1}^n \langle e_j, \tilde e_i\rangle^2 \lambda_j(A)=
\sum_{j=1}^n \left(\sum_{i=n - r + 1}^n \langle e_j, \tilde e_i\rangle^2\right) \lambda_j(A).$$ That is, $\sum_{i=n - r + 1}^n \lambda_j(PAP)$ is at most the sum of $\lambda_j(A)$ with weights $\sum_{i=n - r + 1}^n \langle e_j, \tilde
e_i\rangle^2$. The total weight of all $\lambda_1(A),\dots,\lambda_n(A)$ is $r$: $$\sum_{i=n - r + 1}^n\underbrace{\sum_{j=1}^n \langle e_j, \tilde e_i\rangle^2}_{
\|\tilde e_i\|^2} =
\sum_{i=n - r + 1}^n \|\tilde e_i\|^2 = r.$$ The weight of each eigenvalue $\lambda_j(A)$ in the sum is at most $1$: $$\sum_{i=n - r + 1}^n \langle e_j, \tilde e_i\rangle^2 \leq \sum_{i=1}^n \langle e_j,
\tilde e_i\rangle^2 = 1.$$ Therefore, the sum does not exceed the sum of the $r$ largest eigenvalues $\sum_{i=n-r+1}^n \lambda_r(A)$.
\[cor:Majorization\] For every positive semidefinite matrix $A$, every projection matrix $P$ and $u > \lambda_{\mathrm{max}}(A)$, the following inequality holds. $$\Phi^{u}(PAP) =
\sum_{i=n-T + 1}^{n}\frac{1}{u-\lambda_i(PAP)}
\leq \sum_{i=n-T+1}^{n}\frac{1}{u-\lambda_i(A)} =\Phi^u(A)$$
The statement follows from the Karamata Majorization Inequality. The inequality claims that for every two non-increasing sequences that satisfy (\[eq:majorization\]) and for every increasing convex function $f$, $$\sum_{i=n-k+1}^n f(\lambda_i(A))
\geq
\sum_{i=n-k+1}^n f(\lambda_i(PAP)).$$ Plugging in $f(x) = \frac{1}{u-x}$ (defined on $(0,u)$), we obtain the desired inequality.
\[lem:patrial\_trace\] Let $A$ be a positive semidefinite matrix such that $A \preceq I_n$. Assume $\mathrm{Tr}(A) \leq r \in \mathbb N$. Then for every positive semidefinite matrix $M$, $
A{\bullet}M \leq \sum_{i=N-r+1}^{N} \lambda_i(M)
$.
By von Neumann’s inequality [@Mirsky], $
A{\bullet}M = \operatorname{tr}(AM) \leq \sum_{i=1}^n \lambda_i(A) \lambda_i(M)
$. Since $\sum_{i=1}^n \lambda_i(A) \leq r$ and all $\lambda_i(A) \leq 1$, we can easily see that the above product achieves its maximum when the largest $r$ eigenvalues of $A$ are $1$ and the rest are $0$. In this case, we have, $
A\bullet M \leq \sum_{i=1}^n \lambda_i(A) \lambda_i(M) = \sum_{i=n-r+1}^n \lambda_i(M)
$.
As a corollary we get the following result.
\[cor:traceineq\] Let $X$, $M^*$ and $T$ be as in Theorem \[thm:mainmatrix\]. Then for any positive semidefinite matrix $U$, we have $U{\bullet}(M^* - X) \leq \sum_{i=n-T+1}^{n} \lambda_i(U)$.
Barrier Shifts {#sec:barshift}
--------------
In this section, we analyze how we can update matrices $A^{(q)}$ and $B^{(q)}$, and increment barriers $l$ and $r$ so that the upper and lower potentials do not increase. Let us think of $\Phi^u(A)$ as a function of an $n^2$ dimensional vector (consisting of entries of $A$). Then in the first approximation $\Phi^{u+\delta_U}(A + tY) \approx \Phi^{u+\delta_U}(A) + t Y{\bullet}U$, where $U$ is the gradient of $\Phi^{u+\delta_U}$ at $A$ ($U$ is an $n\times n$ matrix). Thus the potential function does not increase, $\Phi^{u+\delta_U}(A + tY) \leq \Phi^u(A)$, roughly when $t Y{\bullet}\frac{U}{\Phi^{u}(A) - \Phi^{u+\delta_U}(A)} \leq 1$. Similarly, $\Phi_{l+\delta_L}(B + tY) \leq \Phi_l(B)$, roughly when $tY{\bullet}\frac{L}{\Phi_{l+\delta_L}(B) - \Phi_{l}(B)}\geq 1$, where $L$ is the gradient of $\Phi_{l+\delta_L}$ at $B$. Following [@BSS], we make these statements precise (we need to take into account lower order terms). We define matrices $U_A$ and $L_B$, $$\begin{aligned}
U_A &= \frac{((u+\delta_U)I-A)^{-2}}{\Phi^u(A) -\Phi^{u+\delta_U}(A)}+((u+\delta_U)I-A)^{-1};\\
L_B &= \frac{(P_{S}(B-(l+\delta_L)I)P_{S})^{\dagger 2}}{\Phi_{l+\delta_L}(B) -\Phi_{l}(B)}-(P_{S}(B-(l+\delta_L)I)P_{S})^{\dagger}\end{aligned}$$
\[upper\_potential\] (Upper Barrier Shift) Suppose $\lambda_{\text{max}}(A)<u$ and $Y = v v^T$ is a rank-one update. If $
U_A {\bullet}Y \leq \frac{1}{t}
$ then $
\Phi^{u+\delta_U}(A+tY) \leq \Phi^u(A)
$ and $
\lambda_{\mathrm{max}}(A+tY)<u+\delta_U
$.
Let $u' = u+\delta_U$ and $P = P_{L(A+tY)}$. By the Sherman–Morrison formula (Lemma (\[lem:path:sminv\])), we can write the updated potential as: $$\begin{aligned}
\Phi^{u+\delta_U}(A+tY) &=
\operatorname{tr}P(u'I-A-t Y)^{-1}P
=\operatorname{tr}P\left((u'I-A)^{-1}+\frac{t(u'I-A)^{-1}Y(u'I-A)^{-1}}{1-t(u'I-A)^{-1}{\bullet}Y}\right)P\\
&=\operatorname{tr}P(u'I-A)^{-1}P+\operatorname{tr}\frac{tP(u'I-A)^{-1}Y(u'I-A)^{-1}P}{1-t(u'I-A)^{-1}{\bullet}Y}\\
&\leq \Phi^{u+\delta_U}(PAP) + \frac{t(u'I-A)^{-2}{\bullet}Y}{1-t(u'I-A)^{-1}{\bullet}Y}\\
&\leq\Phi^{u+\delta_U}(A)+\frac{t(u'I-A)^{-2}{\bullet}Y}{1-t(u'I-A)^{-1}{\bullet}Y}\\
&=\Phi^u(A) -(\Phi^u(A) -\Phi^{u+\delta_U}(A))+
\frac{(u'I-A)^{-2}{\bullet}Y}{1/t-(u'I-A)^{-1}{\bullet}Y}\end{aligned}$$ Here, we used Corollary \[cor:Majorization\] for the inequality on line 4.
Substituting $U_A {\bullet}Y \leq 1/t$ gives $\Phi^{u+\delta_U}(A+tY) \leq \Phi^u(A)$. The statement about $\lambda_{\mathrm{max}}$ follows from continuity of eigenvalues.
\[lower\_potential\] (Lower Barrier Shift) Suppose $\lambda_{\mathrm{min}}(\left.B\right|_S)>l+\delta_L$ and $Y = vv^T$ is a rank-one update. If $
L_B {\bullet}Y \geq 1/t
$ then $
\Phi_{l+\delta_L}(B+tY) \leq \Phi_l(B)
$ and $
\lambda_{\mathrm{min}}(\left.(B+tY)\right|_S)>l+\delta_L
$.
We proceed as in the proof for the upper potential. Let $l' = l + \delta_L$ and $P=P_S$. By the Sherman–Morrison formula for the pseudoinverse (Lemma \[lem:path:smpseudinv\]), we have: $$\begin{aligned}
\Phi_{l+\delta_L}(B+tY) &= \operatorname{tr}(P(B + tY-l'I)P)^{\dagger}= \operatorname{tr}(P(B-l'I)P+tPYP)^{\dagger}\\
&= \operatorname{tr}(P(B-l'I)P)^{\dagger}-\frac{t\operatorname{tr}((P(B-l'I)P)^\dagger Y (P(B-l'I)P)^\dagger)}{1+t(P(B-l'I)P)^{\dagger}{\bullet}Y}\\
&=\Phi_l(B)+(\Phi_{l+\delta_L}(B)-\Phi_l(B))-\frac{t(P(B-l'I)P)^{\dagger 2}{\bullet}Y}{1+t(P(B-l'I)P)^{\dagger}{\bullet}Y}\end{aligned}$$ Note that matrix $U_A$ is positive semidefinite. Rearranging shows that $\Phi_{l+\delta_L}(B + Y) \leq \Phi_l(B)$ when $ L_A(\pi) \geq 1/t$. It is immediate that $\lambda_{\text{min}}(P_S(A+t\pi\pi^T)P_S)>l+\delta_L$ since $\lambda_{\text{min}}(P_SAP_S)>l+\delta_L$.
Now we prove that we can choose $Y_i$ and $t$ so that conditions of both lemmas are satisfied.
\[both\_barriers\](Both Barriers) If $\Phi^{u}(A) \leq \epsilon_U$ and $\Phi_{l}(B) \leq \epsilon_L$ and $\epsilon_U, \epsilon_L,\delta_U,\delta_L$ satisfy $$0\leq \frac{1}{\delta_U}+\epsilon_U + \max(N, T) \leq \frac{1}{\delta_L} - \epsilon_L,$$ and $X$, $Y_i$, $cost_i$, $Z$, $T$ and $N$ as in Theorem \[thm:mainmatrix\], $M^*-X$ is non-singular on $S$, then there exists $i$ and positive $t$ for which $$\begin{aligned}
L_B{\bullet}(ZY_iZ) &\geq 1/t \geq U_A {\bullet}Y_i, \text{ and} \label{bothOne}\\
cost_i \cdot t &\leq 1/\max(N,T).\label{bothTwo}\end{aligned}$$
We will use the following lemma
\[boundsonUandL\] $
\sum_{i=1}^m U_A {\bullet}Y_i \leq \frac{1}{\delta_U} + \epsilon_U
$ and $
\sum_{i=1}^m L_B {\bullet}(Z Y_i Z) \geq \frac{1}{\delta_L} - \epsilon_L.
$
1\. We use Corollary \[cor:traceineq\] to bound the Frobenius product of $Y_i$ with each of the two summands in the definition of $U_A$ (note that they are positive semidefinite), we get $$\begin{aligned}
\sum_{i=1}^m U_A{\bullet}Y_i &= U_A {\bullet}\sum_{i=1}^m Y_i = U_A {\bullet}(M^*-X)\\
&=
\frac{((u+\delta_U)I-A)^{-2}}{{\Phi^u(A) -\Phi^{u+\delta_U}(A)}}{\bullet}(M^*-X)
+((u+\delta_U)I-A)^{-1}{\bullet}(M^*-X) \\
&\leq \sum_{i=n-T+1}^n \lambda_i\left(\frac{((u+\delta_U)I-A)^{-2}}{{\Phi^u(A) -\Phi^{u+\delta_U}(A)}}\right) +
\sum_{i=n-T+1}^n \lambda_i\left(((u+\delta_U)I-A)^{-1}\right)\\
&=
\frac{
\sum_{i=n-T+1}^n
\frac{1}{(u + \delta_U - \lambda_i(A))^2}}{{\Phi^u(A) -\Phi^{u+\delta_U}(A)}}
+
\sum_{i=n-T+1}^n \frac{1}{(u+\delta_U)-\lambda_i(A)}\end{aligned}$$ Note that the first term is at most $1/\delta_U$, since $$\begin{aligned}
\sum_{i=n-T+1}^n
\frac{1}{(u + \delta_U - \lambda_i(A))^2} &\leq
\sum_{i=n-T+1}^n
\frac{1}{(u - \lambda_i(A))(u + \delta_U - \lambda_i(A))}\\
&=
\frac{1}{\delta_U}\sum_{i=n-T+1}^n
\left(\frac{1}{u - \lambda_i(A)} - \frac{1}{(u + \delta_U) - \lambda_i(A)}\right)
= \frac{{\Phi^u(A) -\Phi^{u+\delta_U}(A)}}{\delta_U}\end{aligned}$$ and the second term equals $\Phi^{u+\delta_U}(A)$. Thus $\sum_{i=1}^m U_A{\bullet}Y_i
\leq \epsilon_U + 1/\delta_U$.
2\. Let $P$ be the projection on $\operatorname{Im}(M^*-X)$. Since $(M^*-X)$ is non-singular on $S$, $P P_S = P_S$. We have, $$\begin{aligned}
\sum_{i=1}^m L_B{\bullet}ZY_iZ &= L_B {\bullet}\sum_{i=1}^m ZY_iZ = L_B {\bullet}Z(M^*-X)Z = L_B {\bullet}P
\\
&=\operatorname{tr}\left(\frac{(P_S(B-(l+\delta_L)I)P_S)^{\dagger 2}}{\Phi_{l+\delta_L}(B) -\Phi_{l}(B)}-(P_{S}(B-(l+\delta_L)I)P_{S})^{\dagger}\right)\\
\\
&= \frac{\sum_{i=1}^{k}\frac{1}{(\lambda_i(\left.B\right|_S)-(l+\delta_L))^2}}{
\Phi_{l+\delta_L}(B) -\Phi_{l}(B)}-\sum_{i=1}^{k}\frac{1}{\lambda_i(\left. B \right|_S)-(l+\delta_L)}\\
&\geq 1/\delta_L - \epsilon_L,\end{aligned}$$ where the last line follows from Claim 3.6 in [@BSS].
(Of Lemma \[both\_barriers\]) For the previous lemma, we get: $\sum_{i=1}^m (U_A {\bullet}Y_i + \max(N,T)cost_i) \leq \frac{1}{\delta_U} + \epsilon_U + \max(N,T) \leq
L_B{\bullet}(ZY_iZ).$ Thus for some $i$, $U_A {\bullet}Y_i + \max(N,T) cost_i \leq L_B{\bullet}(ZY_iZ)$. Letting $t = (L_B{\bullet}(ZY_iZ))^{-1}$, we satisfy (\[bothOne\]) and (\[bothTwo\]).
Proof of Theorem \[thm:mainmatrix\] {#sec:thmmainproof}
-----------------------------------
Now we are ready to prove Theorem \[thm:mainmatrix\]. We assume that $M^*-X$ is non-singular on $S$ (which we can ensure by an arbitrary small pertrubation).
We start with $A^{(0)} = X$, $B^{(0)} = 0$ and all weights $w^{(0)}_i = 0$. We define parameters as follows, $$\begin{aligned}
\delta_L &= 1/(2\max(N,T)),&
\epsilon_L &= 1/(4\delta_L),&
l_0 &= -4k\delta_L,\\
\delta_U &=4\delta_L,&
\epsilon_U &= 1/(4\delta_L),&
u_0 &= 4T\delta_L + 1,\end{aligned}$$ so as to satisfy conditions of Lemma \[both\_barriers\], $\Phi^u(A^{(0)}) = \Phi^u(X) = \sum_{i=1}^T \frac{1}{u_0 - \lambda_{n+1-i}(X)} \leq T/(u_0-1) =\epsilon_U$, $\Phi_l (B^{(0)})= \sum_{i=1}^k \frac{1}{0 - l_0} = - k/l_0 = \epsilon_L$, $1/\delta_U + \epsilon_U + \max(N,T) = \frac{3}{2} \max(N,T)= 1/\delta_L - \epsilon_L$. Then we iteratively apply Lemma \[both\_barriers\]. At iteration $q$, we find an index $i$ and a positive $t$ such that $L_{B^{(q)}}(ZY_iZ) \geq 1/t \geq U_{A^{(q)}}(Y_i)$, $cost_i \cdot t \leq 1/\max (N,T)$, and increment the weight of matrix $Y_i$ by $t$: $w^{(q+1)}_i = w^{(q)}_i + t$; update $l = l + \delta_L$ and $u = u + \delta_U$. The total cost increases by at most $1/\max(N,T)$. Finally, after $N$ iterations we obtain matrices $A^{(N)}$ and $B^{(N)}$ with $$\begin{aligned}
\lambda_{\mathrm{max}}(A^{(N)}) &\leq u_0 + N \delta_U = 2(N + T)/\max(N,T) + 1\equiv \theta_{\mathrm{max}}\\
\lambda_{\mathrm{min}}(\bigl.B^{(N)}\bigr|_S) &\geq l_0 + N \delta_L =
(N/2 - 2k)/\max(N,T) \equiv \theta_{\mathrm{min}}.\end{aligned}$$ Now consider an arbitrary unit vector $v$. Let $v = v_S + v_{S^\perp}$, where $v_S \in S$ and $v_{S^\perp} \perp S$. Since $B^{(N)} \succeq \theta_{\mathrm{min}} P_S$ and $v_S\in S$, $$\begin{aligned}
v_S^TA^{(N)}v_S &= v_S^T(X + (P_S(M^* - X)P_S)^{1/2} B^{(N)}
(P_S(M^* - X)P_S)^{1/2}) v_S \\
&\geq v_S^T(X + (P_S(M^* - X)P_S)^{1/2} \theta_{\mathrm{min}} P_S
(P_S(M^* - X)P_S)^{1/2}) v_S \\
&=
\theta_{\mathrm{min}} v_{S}^T M^* v_{S} +
(1 - \theta_{\mathrm{min}}) v_{S}^T X v_{S} \geq \theta_{\mathrm{min}}
\lambda_{\mathrm{min}}(M^*) \|v_{S}\|^2.\end{aligned}$$ On the other hand, $v_{S^\perp}^T A^{(N)} v_{S^\perp} \leq \theta_{\mathrm{max}} \|v_{S\perp}\|$. Thus from the triangle inequality for the norm induced by $A^{(N)}$, we get $$(v^T A^{(N)} v)^{1/2} \geq \theta_{\mathrm{min}}^{1/2}
\lambda_{\mathrm{min}}(M)^{1/2}\|v_{S}\| -
\theta_{\mathrm{max}}^{1/2} \|v_{S^\perp}\| \geq \theta_{\mathrm{min}}^{1/2}\lambda_{\mathrm{min}}(M)^{1/2} -
(\theta_{\mathrm{max}}^{1/2} + \theta_{\mathrm{min}}^{1/2} \lambda_{\mathrm{min}}(M)^{1/2}) \|v_{S^\perp}\|.$$ On the other hand, since $S$ is an eigenspace of $X$ corresponding to $k$ smallest eigenvalues, $$(v^T A^{(N)} v)^{1/2}\geq (v^T X v)^{1/2} \geq (v_{S^\perp}^T X v_{S^\perp})^{1/2}
\geq {\lambda^*}^{1/2} \|v_{S^\perp}\|.$$ One of the two bounds above for $(v^T A^{(N)} v)^{1/2}$ increases and the other decreases as $\|v_{S^\perp}\|$ increases. They are equal when $\|v_{S^\perp}\| =
\frac{\theta_{\mathrm{min}}^{1/2}\lambda_{\mathrm{min}}(M^*)^{1/2}}{{\lambda^*}^{1/2}
+ \theta_{\mathrm{max}}^{1/2} + \theta_{\mathrm{min}}^{1/2}\lambda_{\mathrm{min}}(M^*)^{1/2}}$. Therefore, $(v^T A^{(N)} v)^{1/2} \geq
\frac{\theta_{\mathrm{min}}^{1/2}{\lambda^*}^{1/2}
\lambda_{\mathrm{min}}(M^*)^{1/2}}{{\lambda^*}^{1/2} + \theta_{\mathrm{max}}^{1/2} +
\theta_{\mathrm{min}}^{1/2} \lambda_{\mathrm{min}}(M^*)^{1/2}}$. We conclude that $$\lambda_{\mathrm{min}}(A^{(N)}) = \min_{v:\|v\|=1} v^T A^{(N)} v \geq
\frac{\theta_{\mathrm{min}}\lambda^*\lambda_{\mathrm{min}}(M^*)}{
\left({\lambda^*}^{1/2}
+ \theta_{\mathrm{max}}^{1/2} + \theta_{\mathrm{min}}^{1/2}
\lambda_{\mathrm{min}}(M^*)^{1/2}\right)^2}
.$$ Plugging in the values of parameters, we get the statement of the theorem for $M= A^{(N)}$. The total cost is at most $N/\max(N, T) = \min(1, N/T)$.
Finally, we prove Claim \[cl:patch\].
Let $V = \operatorname{Im}({\mathcal{L}}_{G+W}) = \ker({\mathcal{L}}_{G+W})^{\perp}$. Let ${\mathcal{L}}_e$ be the Laplacian of the edge $e$. Define $$\begin{aligned}
X &= \Bigl.\left(({\mathcal{L}}_{G+W}^{\dagger})^{1/2}{\mathcal{L}}_G ({\mathcal{L}}_{G+W}^{\dagger})^{1/2}\right)\Bigl|_V,\\
Y_e &= w_e \Bigl.\left(({\mathcal{L}}_{G+W}^{\dagger})^{1/2}{\mathcal{L}}_e ({\mathcal{L}}_{G+W}^{\dagger})^{1/2}\right)\Bigl|_V,\\
cost_e &= w_e/\left(\sum\nolimits_{d\in E_W} w_d\right).\end{aligned}$$ Since ${\mathcal{L}}_G + \sum_{e\in E_W}^m w_e {\mathcal{L}}_e = {\mathcal{L}}_{G+W}$, we have $X + \sum_{e\in E_W} Y_e = I$. By the definition of the $(k,T,\lambda^*)$-patch, $\operatorname{tr}(I-X) \leq T$ and $\lambda^* \leq \lambda^{k+1}(X) $. We apply Theorem \[thm:mainmatrix\] to matrices $X$, $Y_e$ and $M^* = I$. We obtain a set of weights $\rho_e$ — supported on at most $N$ edges — such that $$c_1\min(N/T,1) \cdot \lambda^*
\leq
\lambda_{\mathrm{min}}\left(X+ \sum\nolimits_{e\in E_W} \rho_e Y_e\right)
\leq
\lambda_{\mathrm{max}}\left(X+ \sum\nolimits_{e\in E_W} \rho_e Y_e\right)
\leq c_2,$$ Let $\tilde w_e = \rho_e w_e$. Weights $\tilde w_i$ define subgraph $W_k$ with at most $N$ edges. It follows that $$c_1 \min(N/T,1) \lambda^* {\mathcal{L}}_{G+W} \preceq {\mathcal{L}}_{G+W_k} \preceq c_2 {\mathcal{L}}_{G+W}.$$ The total weight of edges of $W_k$ is $\sum_{e\in E_W} \rho_e w_e =
(\sum_{e\in E_W} \rho_e cost_e) \sum_{d\in E_W} w_d \leq \min (1, N/T) \sum_{d\in E_W} w_d$.
Constructing Nearly-Optimal Ultrasparsifiers {#sec:patch:ultra}
============================================
We now apply our subgraph sparsification to build ultrasparsifiers. Recall that a weighted graph $U$ is a $(\kappa,k)$-ultrasparsifier of another graph $G$ if $U \preceq G \preceq \kappa \cdot U$ and $U$ has only $n-1+k$ edges, where $n$ is the number of vertices in $U$ and $G$. The main result of this section is the following theorem.
\[thm:patch:ultra\] For any integer $k>0$, every graph has an $(\frac{n}{k}\log n\, \tilde{O}(\log \log n),k)$–ultrasparsifier.
Our basic idea to build a good ultrasparsifier $U$ is quite simple. Without loss of generality, we can assume that $G$ is connected and has $O(n)$ edges. Otherwise given a graph $G$, we can first find a linear size sparsifier using [@BSS], for each of its connected components, and build a good ultrasparsifier for each component. Because $U$ is only $k$ edges aways from a tree, our construction starts with good tree $T$. As it will be much more clear below, the quality of a tree is measured by its [*stretch*]{}, as introduced by Alon, Karp, Peleg and West [@AKPW].
Suppose $T$ is a spanning tree of $G = (V,E,w)$. For any edge $e\in E$, let $e_1,\cdots, e_k \in F$ be the edges on the unique path in $T$ connecting the endpoints of $e$. The *stretch* of $e$ w.r.t. $T$ is given by $\text{st}_T(e) =w(e)(\sum_{i=1}^k \frac{1}{w(e_i)})$. The stretch of the graph $G$ with respect to $T$ is defined by $\text{st}_T(G) =\sum_{e\in E} \text{st}_T(e).$ Our construction will start with a spanning tree with the lowest possible stretch. By [@ABN08], we can in polynomial time grow a spanning tree $T$ with $$\text{st}_T(G) = O(n \log n \log \log n (\log \log \log n)^3).$$
For the sake of simplicity of the presentation, we will show the construction of ultrasparsifiers with $\Theta(k)$ edges. We note that by choosing the appropriate constants, the number of edges can be made exactly $k$.
Let $\kappa = c_1\cdot\text{st}_T(G)/k$ for a sufficiently large constant $c_1$. Our job is to choose $\Theta(k)$ more weighted edges $\tilde{W}$ and set $U = T+\tilde{W}$ such that $ c_2 \cdot U \preceq G \preceq \kappa \cdot U$, for a constant $c_2$. To this end, let $W = (1/(c_3\kappa)) \cdot G$, for some constant $c_3$. Then, $G = c_3 \kappa \cdot W \preceq c_3 \kappa \cdot (W+T) $. Also, because $T \preceq G$, we have $T+W \preceq (1+1/(c_3\kappa))G \preceq c_4 \cdot G$, for a constant $c_4$. Therefore, if we can find a $\Theta(k)$–edge subgraph $\tilde{W}$ of $W$ such that $T +\tilde{W} \preceq \Theta(1) \cdot (T+W)$, we can then build a $n-1 + \Theta(k)$ edge graph $U = T+\tilde{W}$ satisfying $ c_2 \cdot U \preceq G \preceq \kappa \cdot U$ (if we choose our constants $c_i$’s carefully).
To apply our subgraph sparsification results to construct $\tilde{W}$, we use the following structure result of Spielman and Woo ([@SW]: Theorem 2.1 and Corollary 2.2).
\[lem:patch:SW1\] (Theorem 2.1 in [@SW]) (1) $\text{Tr}({{\mathcal{L}}^\dagger_T}^{1/2}{\mathcal{L}}_G{{\mathcal{L}}^\dagger_T}^{1/2}) =\text{st}_T(G).$ (2) For every $t>0$, the number of eigenvalues of ${{\mathcal{L}}^\dagger_T}^{1/2}{\mathcal{L}}_G{{\mathcal{L}}^\dagger_T}^{1/2}$ greater than $t$ is at most $\text{st}_T(G)/t$.
We now use Lemma \[lem:patch:SW1\] to prove the following lemma, from which Theorem \[thm:patch:ultra\] follows directly.
\[lem:ultra:patch\] $W$ is a $(k,O(k),\Theta(1))$–patch for $T$.
Let $\lambda_i = \lambda_{i}(({\mathcal{L}}_{T+W}^{\dagger})^{1/2}{\mathcal{L}}_T({\mathcal{L}}_{T+W}^{\dagger})^{1/2})$ be the $i$-th eigenvalue, and $y_i$ be the corresponding eigenvector. Let $x_i = L_{T+W}^{1/2}y_i$. Then, $$\begin{aligned}
\lambda_i = \lambda_{i}(({\mathcal{L}}_{T+W}^{\dagger})^{1/2}{\mathcal{L}}_T({\mathcal{L}}_{T+W}^{\dagger})^{1/2})
= \frac{x_i^T {\mathcal{L}}_T x_i}{x_i^T {\mathcal{L}}_T x_i+ x_i^T {\mathcal{L}}_W x_i}
= \frac{x_i^T {\mathcal{L}}_T x_i}{x_i^T {\mathcal{L}}_T x_i+ x_i^T {\mathcal{L}}_G x_i/(c_3\kappa)},\end{aligned}$$ implying $$\begin{aligned}
\frac{x_i^T {\mathcal{L}}_G x_i}{x_i^T {\mathcal{L}}_T x_i} = \frac{1-\lambda_i}{\lambda_i}
c_3\kappa = \left(\frac{1-\lambda_i}{\lambda_i}\right) c_3 c_1 \frac{\text{st}_T(G)}{k}
= \frac{\text{st}_T(G)}{\frac{k}{c_1c_3}\frac{\lambda_i}{1-\lambda_i}}\end{aligned}$$ It follows from the definition of $\lambda_i$ that $0\leq \lambda_i <
1$. Hence, $(1-\lambda_{i-1})/\lambda_{i-1} \geq (1-\lambda_{i})/\lambda_{i} $. By Courant—Fischer theorem and the property 2 of Lemma \[lem:patch:SW1\], we have $k \leq
\frac{k}{c_1c_3}\frac{\lambda_{k+1}}{1-\lambda_{k+1}}.$ Therefore, $\lambda_{k+1} \geq \frac{c_1c_3}{1+c_1c_3} = \Theta(1)$. We also have,
$$\begin{aligned}
\operatorname{tr}\left(({\mathcal{L}}^{\dagger}_{T+W})^{1/2}{\mathcal{L}}_W{({\mathcal{L}}^{\dagger}_{T+W}})^{1/2}\right)
& \leq &
\operatorname{tr}\left(({\mathcal{L}}^{\dagger}_{T})^{1/2}{\mathcal{L}}_W{({\mathcal{L}}^{\dagger}_{T}})^{1/2}\right)
= \frac{1}{c_3\kappa}\operatorname{tr}\left(({\mathcal{L}}^{\dagger}_{T})^{1/2}{\mathcal{L}}_G{({\mathcal{L}}^{\dagger}_{T}})^{1/2}\right)\\
& \leq & \frac{k}{c_3c_1\text{st}_T(G)} \text{st}_T(G) =
\frac{k}{c_3c_1} = \Theta(k).\end{aligned}$$
We proved that $W$ is a $(k,O(k),\Theta(1))$–patch for $T$.
We next show that the parameters of the ultrasparsifiers we obtained are optimal, up to low order terms.
Let $G$ be a Ramanujan $d$-regular expander graph, for some constant $d$. Let $U$ a $(\kappa,N)$ ultrasparsifier for $G$. Then $
\kappa \geq \frac{n}{N} \log n.
$ \[thm:optimsparsifiers\]
Let T be a low-stretch spanning tree of $G$, as above. As mentioned in [@ABN08], $\text{st}_T(G) =\Omega( m\log n)$ where $m$ is the number of edges of the original graph. From lemma \[lem:patch:SW1\], and the conditions on the stretch of $T$ we have $
\text{Tr}({\mathcal{L}}_G{{\mathcal{L}}_T}^\dagger) =\text{st}_T(G) \geq C\cdot n \log n
$ for some constant $C$.
Since $x^T{\mathcal{L}}_G x = \Theta(1)$ for the expander, the above inequality implies that $\sum_{i =1} ^n \frac{1}{x^T {\mathcal{L}}_T x} \geq n \log n$ where $x_i$ are the eigenvectors of ${\mathcal{L}}_G({\mathcal{L}}_T)^\dagger$. It is immediate from Markov’s inequality that there exists some $k$ such that ${x_k}^T {\mathcal{L}}_T x_k \leq \frac{C_1k}{n \log n}$. Assume that for all $i \leq k$ we have ${x_i}^T {\mathcal{L}}_T x_i \leq {x_k}^T {\mathcal{L}}_T x_k \leq \frac{C_1k}{n \log n} $. (Otherwise take $k' < k$ appropriately). Then also $\lambda_k({\mathcal{L}}_T) \leq \frac{C_1k}{n \log n}$. By the minmax theorem for eigenvalues this implies that adding $N = k-2$ edges to $T$ will result to a graph $U$ with $\lambda_2({\mathcal{L}}_U) \leq \lambda_k({\mathcal{L}}_T)\leq \frac{C_1k}{n \log n}$. Thus any ultrasparsifier $U$ with $N$ edges will have $$\begin{aligned}
C_2 = \lambda_2({\mathcal{L}}_G) \leq \kappa \lambda_2({\mathcal{L}}_U) \leq \frac{C_1k}{n \log n}
\Rightarrow \kappa \geq \Omega(\frac{n\log n}{k}) =\Omega(\frac{n\log n}{N})\end{aligned}$$
Maximizing Algebraic Connectivity by Adding few edges {#sec:patch:algcon}
=====================================================
In this section, we present an approximation algorithm for the following problem: given a graph $G = (V, E_{base})$, a set of candidate edges $E_{cand}$, and a parameter $k$, add at most $k$ candidate edges to $G$ so as to maximize its algebraic connectivity, that is, find a subset $E\subset E_{cand}$ that maximizes $\lambda_2({\mathcal{L}}_{G + E})$. The problem was introduced by Ghosh and Boyd [@BG], who presented a heuristic for it. It is known that the problem is NP-hard [@damon]. But prior to this work, no approximation algorithm was known for it.
We use two upper bounds for the cost of the combinatorial solution in order to prove an approximation guarantee: one upper bound is the SDP value, $\lambda_{SDP}$, and the other is $\lambda_{k+2}({\mathcal{L}}_G)$ (see Lemma \[lem:boundOnLambdaK\]). Note that neither of these two bounds are good approximations for the value of the optimum solution by themselves (for instance, if $G$ consists of $n$ isolated vertices, $(V, E_{cand})$ is an expander, $k < n$, then the value of the combinatorial solution is $0$ but $\lambda_{SDP} \sim k/n$), but their combinations lead to a good upper bound for the optimum solution $\lambda_{OPT}$.
For clarity and simplicity of exposition, we assume here that $(V,
E_{base})$ and $(V, E_{cand})$ are bounded degree graphs with the maximum degree $\Delta$. Our algorithm uses a natural semidefinite relaxation that was also used by Ghosh and Boyd [@BG]. We introduce a variable $w_e$ (the weight of the edge $e$) for each candidate edge $e\in E_{cand}$; add constraints that all edge weights are between $0$ and $1$, and the total weight is at most $k$. Then we require that $\lambda_2({\mathcal{L}}_G +
\sum_e w_e{\mathcal{L}}_e) \geq \lambda_{SDP}$ (where ${\mathcal{L}}_e$ is the Laplacian of the edge $e$). We do that by adding an SDP constraint ${\mathcal{L}}_G +
\sum_e w_e {\mathcal{L}}_e \succeq \lambda_{SDP} P_{(1,\dots,1)^{\perp}}$, where $P_{(1,\dots,1)^{\perp}}$ is the projection on the space orthogonal to $(1,\dots,1)^{\perp}$. We get the following SDP relaxation. $$\begin{aligned}
\text{maximize: } & \lambda_{SDP},\\
\text{subject to: }& {\mathcal{L}}_G + \sum_{e\in E_{cand}} w_e {\mathcal{L}}_e \succeq \lambda_{SDP}
\cdot P_{(1,\dots,1)^{\perp}},\\
&\sum_{e\in E_{cand}} w_e \leq k,\\
&0\leq w_e \leq 1 \text{ for every } e\in E_{cand}.\end{aligned}$$
We solve the semidefinite program and obtain solution $\{w_e\}_{e\in
E_{cand}}$. The total weight of all edges is $k$, however, the number of edges involved, or the support of the solution could be significantly higher than $k$.
We use our algorithm to *sparsify* the SDP solution using Theorem \[thm:mainmatrix\]. More precisely, we apply Theorem \[thm:mainmatrix\] with $X = {\mathcal{L}}_G/(4\Delta)$ and $Y_e =
w_i {\mathcal{L}}_e/(4\Delta)$ restricted to the space $(1,\dots, 1)^{\perp}$, $N = 8k$, $T = \operatorname{tr}(\sum_e w_e {\mathcal{L}}_e)/(4\Delta) \leq k$ and $cost_i =
w_i$ (we divide ${\mathcal{L}}_G$ and ${\mathcal{L}}_e$ by $4\Delta$ to ensure that $\lambda_{\mathrm{max}}(X + \sum_e Y_i) \leq 1$). We get a set of weights $\rho_e$ supported on at most $8k$ edges s.t. $$\frac{1}{4\Delta}
\lambda_{\mathrm{2}} ({\mathcal{L}}_G + \sum_e \rho_e w_e {\mathcal{L}}_e)
= \lambda_{\mathrm{min}} (X + \sum_e \rho_e Y_e)
\geq c \lambda_{k+2}(X) \lambda_{\mathrm{min}}(X + \sum_e Y_e)
\geq c \frac{1}{(4\Delta)^2} \lambda_{k+2}({\mathcal{L}}_G) \lambda_{SDP}.$$ That is, we obtain a combinatorial weighted solution $\tilde w_e = \rho_i w_i$ whose value is at least $c\lambda_{k+2}({\mathcal{L}}_G)\lambda_{SDP}/(4\Delta)$ (if $k+2 > n$, the value is at least $c\lambda_{SDP}$). We next show that $\lambda_{SDP} \geq \lambda_{OPT}$ and $\lambda_{k+2} (G) \geq \lambda_{OPT}$. Therefore, the value of the solution is at least $c\lambda_{OPT}^2/\Delta$.
\[lem:boundOnLambdaK\] The value of the optimal solution, $\lambda_{OPT}$, is at most $\lambda_{k+2}({\mathcal{L}}_G)$.
Consider the optimal solution $E$. Let ${\mathcal{L}}_E$ be the Laplacian of the graph formed by $E$. Note that $\mathrm{rank}({\mathcal{L}}_E) \leq |E| \leq k$, therefore, $\dim \ker {\mathcal{L}}_E \geq n - k$. Let $S$ be the $k+1$-dimensional space spanned by the eigenvectors of ${\mathcal{L}}_G$ corresponding to $\lambda_2({\mathcal{L}}_G), \dots, \lambda_{k+2}({\mathcal{L}}_G)$. Since $\dim S+ \dim \ker E > n$, spaces $S$ and $\ker {\mathcal{L}}_E$ have a non-trivial intersection. Choose a unit vector $v\in \ker S\cap {\mathcal{L}}_E$. We have $v({\mathcal{L}}_{G}+{\mathcal{L}}_E)v^T \leq \lambda_{k+2}({\mathcal{L}}_G) + 0 =\lambda_{k+2}({\mathcal{L}}_G)$. Also $v$ is orthogonal to the vector $(1,\dots,1)^\perp$. Therefore, $\lambda_{OPT} = \lambda_2({\mathcal{L}}_{G}+{\mathcal{L}}_E) \leq \lambda_{k+2}({\mathcal{L}}_G)$.
The edges in the support of $\tilde w_e$, $E = \{\tilde w_e: \tilde w_e \neq 0\}$, form a non-weighted combinatorial solution. Since $\lambda_{\mathrm{max}}({\mathcal{L}}_X + \sum_e \tilde w_e {\mathcal{L}}_e) = O(\Delta)$, all weights $\tilde w_e$ are bounded by $O(\Delta)$, and thus the algebraic connectivity of $G+E$ is at least $c\lambda_{k+2}({\mathcal{L}}_G)\lambda_{SDP}/\Delta^2$.
\[thm:algconn\] There is a polynomial time approximation algorithm that finds a solution of value at least $c\lambda_{OPT}^2/\Delta$ supported on at most $8k$ edges with total weight at most $k$. If $k\geq n$ the algorithm finds a constant factor approximation.
We present two corollaries for special instances of the problem.
If it is possible to make $G$ an expander by adding $k$ edges (and thus $\lambda_{OPT} \sim \Delta$), then the algorithm finds a constant factor approximation.
Note that if the graph formed by candidate edges is an expander then the value of the following SDP solution $w_e = k/|E_{cand}|$ for each edge $e\in E_{cand}$ is $\Omega(k/n)$, thus $\lambda_{SDP} \geq ck/n$.
\[cor:algconnexpander\] If the graph formed by candidate edges is an expander, then the approximation algorithm from Theorem \[thm:algconn\] finds a solution of value at least $c \frac{k}{n\Delta}\lambda_{OPT}$.
It is possible to get rid of the dependence on $\Delta$ in Theorem \[thm:algconn\] and Corollary \[cor:algconnexpander\] and obtain approximation guarantees of $c\min(\lambda_{OPT}, \lambda_{OPT}^2)$ and $\frac{c k}{n} \lambda_{OPT}$ respectively. We omit the details in this extended abstract.
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[^1]: School of Mathematics, Institute for Advanced Study. Research supported by NSF Grant CCF-0832797.
[^2]: Toyota Technological Institute at Chicago.
[^3]: Department of Management Science and Engineering, Stanford University.
[^4]: Computer Science Department, University of Southern California. Supported by NSF grant CCF-0635102.
[^5]: we have $\lambda_{k+1}
({\mathcal{L}}_G {\mathcal{L}}_{G+W}^{\dagger}) =
\lambda_{k+1}(({\mathcal{L}}_{G+W}^{\dagger})^{1/2}{\mathcal{L}}_G({\mathcal{L}}_{G+W}^{\dagger})^{1/2})$, since $\lambda_i(AB) = \lambda_i(BA)$ for every two square matrices $A$ and $B$
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Solvable Baumslag-Solitar Groups Are Not Almost Convex
Charles F. Miller III and Michael Shapiro[^1][We wish to thank the ARC for support.]{}
Introduction
The property of [almost convexity]{} was first introduced by Cannon in [@C]. This property has very geometric in flavor, being defined in terms of the geometry of the Cayley graph. If the Cayley graph is almost convex then there are efficient algorithms for calculating in $G$, or, if you like, constructing the Cayley graph of $G$.
Given a group $G$ and a finite generating set $C \subset G$, the of $G$ with respect to $C$ is the directed, labeled graph whose vertices are the elements of $G$ and whose directed edges are the triples $(g,c,g')$ such that $g,g'\in G$, $c
\in C$ and $g'=gc$. Such an edge is directed from $g$ to $g'$ and is labeled by $c$. We denote this Cayley graph by $\Gamma=\Gamma_{C}(G)$. We will assume that $C$ is closed under inverses.
A Cayley graph has a natural base point $1 \in G$ and a natural path metric $d = d_{C}$ which results from identifying each edge with the unit interval. Each element of $G$ has a natural $\ell(G)=\ell_{C}(g)=d_{C}(1,g)$. We define the $n$ to be $$B(n) = \left\{ x \in \Gamma \mid d_{C}(1,x) \le n \right\}.$$ The group $G$ is with respect to $C$ if there is $N=N(k)$ so that if $g,g' \in B(n)$ and $d_{C}(g,g')\le k$ then there is a path $p$ from $g$ to $g'$ inside $B(n)$ whose length is at most $N$. $G$ is with respect to $C$ if it is almost convex $(k)$ with respect to $C$ for each $k$. It is a theorem of Cannon [@C] that if $G$ is almost convex $(2)$ with respect to $C$ then $G$ is almost convex with respect to $C$.
Thiel [@T] has show that almost convexity is not a group property, i.e, that there are groups which are almost convex with respect to one generating set but not another. Almost convexity is fairly well understood for the fundamental groups of closed $3$-manifold groups with uniform geometries [@SS]. The solvegeometry case was covered in a beautiful paper of Cannon, Floyd, Grayson, and Thurston [@CFGT][0 true in[^2] [Cannon has pointed out to us that there are problems with some of the details in their paper. These concern the relationship between lengths in the given group $G\subset \isom(\sol)$ and the finite index subgroup $A=G\cap\sol$. These problems are easily fixed and in our view, their paper remains quite beautiful.]{}]{}. They show that any group which acts co-compactly, discretely by isometries on cannot be almost convex with respect to any generating set. In this paper we show that their arguments apply to solvable Baumslag-Solitar groups $$G=B_{1,p}=\langle a,t \mid t^{-1}at=a^{p}\rangle$$ with $|p| > 1$.
Theorem
Let $G= B_{1,p}$ be a solvable Baumslag-Solitar group with $|p| >
1$. Then $G$ is not almost convex with respect to any generating set.
Proof of the Theorem
Let $G=B_{1,p}$ with $|p| > 1$. Then $G$ has the form $$G=\bbZ\left[{1/ p}\right]\rtimes\bbZ,$$ where the generator of $\bbZ$ acts via multiplication by $p$. Thus each element of $G$ has the form $(f,c)$ where $f$ is a fraction of the form $$f={m \over p^{n}},$$ and $m,n,c\in \bbZ$. For each element $(f,0) \in \bbZ[1/p] \subset G$, we will take $|(f,0)|=|f|$ and if $n$ is minimal such that $f=m/p^{n}$, we will say that $|p|^{n}$ is the of $(f,0)$ written $\denom(f,0)$.
We fix a generating set $$C=\{(f_{i},c_{i})\}$$ which we assume is closed under inverses. We take $$\eqalign{c&= \max\{c_{i} \mid (f_{i},c_{i}) \in C \}\cr
f^{*} &= \max\{f_{i} \mid (f_{i},c_{i}) \in C \}\cr
f^{**} &= \max\{\denom(f_{i},0) \mid (f_{i},c_{i}) \in C \}.}$$ We assume $(f_{*},c)\in C$ realizes the first of these maxima. Notice that $c>0$.
We need the following two lemmas which give information about distance with respect to the generating set $C$.
Lemma 1
There is a constant $M$ so that if $(f,0) \in B(n)$ then either $|f|
\le M |p|^{nc\over 4}$ or $\denom(f) \le M |p|^{nc\over 4}$. Further, both $|f|
\le M |p|^{nc\over 2}$ and $\denom(f) \le M |p|^{nc\over 2}$
Proof
First observe the following product formula: $$(f_{1},c_{1}) \ldots (f_{n},c_{n}) = \left( \sum_{i=1}^{n} f_{i}
p^{(0-c_{1}-\cdots - c_{i-1})}, \sum_{i=1}^{n} c_{i}\right).$$ Since $(f,0) \in B(n)$, $(f,0)$ can be written as such a product where each $(f_{i},c_{i})$ is in $C$ and $ \sum_{i=1}^{n} c_{i}=0$. For each $i$, $i=1,\ldots,n$, we set $e_{i}=0-c_{1}-\dots - c_{i-1}$. Then $e_{i}$ is positive for at most $n/ 2$ values of $i$, or $e_{i}$ is negative for at most $n/ 2$ values of $i$. Suppose that $e_{i}$ is positive for at most $n/ 2$ values of $i$. We then have $$\eqalign{|f| &= \sum_{i=1}^{n} f_{i}p^{(0-c_{1}-\cdots - c_{i-1})}\cr
&\le f^{*} \sum_{i=1}^{n} |p|^{(0-c_{1}-\cdots -
c_{i-1})}\cr
&= f^{*}\left(\sum_{e_{i}\le 0} |p|^{e_{i}} + \sum_{e_{i}> 0}
|p|^{e_{i}}\right) \cr
&\le f^{*} \left(n + \sum_{e_{i}> 0} |p|^{e_{i}}\right).}$$ Let us enumerate the $\{e_{i} \mid e_{i} >0 \}$ as $i$ increases, so that these are the $\lfloor{n/ 2}\rfloor$-tuple $(e_{i_{1}},
e_{i_{2}}, \dots, e_{i_{\lfloor {n\over 2}\rfloor}})$. If there are less than $\lfloor{n/2}\rfloor$ of these, we will consider any final entries in this list to be $0$. We now take the $m$-tuple $(e'_{i_{1}}, e'_{i_{2}}, \dots, e'_{i_{m}})$ to be $(c,2c,\dots,{nc\over 4},{nc\over 4}, \dots, 2c,c)$. Here $m$ is either $\lfloor n/2 \rfloor$ or $\lfloor n/2 \rfloor+1$. It is not hard to see that for each $j$, $e_{i_{j}}\le e'_{i_{j}}$. Consequently,
$$|f| \le f^{*} \left( n + 2|p|^{c}+ 2|p|^{2c} +\cdots+2|p|^{nc\over 4}
\right),$$ and thus for suitable choice of $M'$, $|f|\le M'|p|^{nc\over4}$.
On the other hand if more than $n/ 2$ of the $e_{i}$ are positive, then less than $n/ 2$ of them are negative, and in particular, the most negative any of these can be is ${-nc\over 4}$. It immediately follows that $$\denom(f,0) \le f^{**}|p|^{{nc\over 4}}.$$
Taking $M=\max\{M',f^{**}\}$ completes the proof of the first part of Lemma 1. After suitably enlarging $M$, a completely similar proof gives the simultaneous bound on $|f|$ and $\denom(f)$
We also need the following observation.
Lemma 2
If $h,h' \in \bbZ[1/p] \subset G$ with $d_{C}(h,h') \le r $ then $||h|-|h'||\le M |p|^{rc\over 2} $ and $|\denom(h) - \denom(h')| \le M
|p|^{rc\over 2} $.
Proof
If $d_{C}(h,h') \le r$ then (using additive notation in $\bbZ[1/p]$) $h - h' \in B(r)$. Thus $$||h|-|h'|| \le |h-h'| \le M |p|^{rc\over 2} .$$
On the other hand $h = h' +(h-h')$ and $h'= h -(h-h')$ so we have $$\denom(h) \le\max\{\denom(h'), \denom(h-h')\} \le \denom(h') +
\denom(h-h'),$$ and $$\denom(h') \le\max\{\denom(h), \denom(h-h')\} \le \denom(h) +
\denom(h-h').$$ Consequently, $$|\denom(h) - \denom(h')| \le \denom(h-h') \le M |p|^{rc\over 2} .$$
We now return to the proof of the Theorem. For each $k>0$ we take $$\eqalign{T_k &=(f_{*},c)^{-k} (1,0) (f_{*},c)^{k} = (p^{ck},0) \cr
S_k &=(f_{*},c)^{k} (1,0) (f_{*},c)^{-k} = (p^{-ck},0) }$$ We then have $T_{k}S_{k}=S_{k}T_{k}$. For some $j$ which we will fix later, we take $$\eqalign{\alpha_{k} &= S_{k}T_{k}(f_{*},c)^{-j} \cr
\beta_{k} &= T_{k}S_{k}(f_{*},c)^{j}}$$ If we take $\ell=\ell_{C}(1,0)$ and $k>j$, then $\alpha_{k}$ and $\beta_{k}$ both lie in $B(4k+2\ell-j)$ and within distance $2j$ of each other.
Suppose that, contrary to hypothesis, $G$ is almost convex. Then there is a constant $N=N(2j)$ so that $\alpha_{k}$ and $\beta_{k}$ are joined by a path of length at most $N$ lying entirely within $B(4k+2\ell-j)$. The second coordinates of points along this path vary from $-jc$ to $+jc$ changing by at most $\pm c$ along each edge. In particular, this path must pass through a point $P'_{k}$ of the form $(g_{k},i)$ with $|i|\le {c/ 2}$. We take $$\epsilon = \max\left\{\ell_{C}(0,i) \mid |i| \le {c/ 2}\right\}.$$ It follows that the point $$P_{k}=(g_{k},0)$$ lies within $B(4k+2\ell+\epsilon-j)$ and within distance ${N/2}+\epsilon+j$ of $S_{k}T_{k}$. Notice that the distance from $P_{k}$ to $S_{k}T_{k}$ is bounded by a constant independent of $k$. It is this fact that we will contradict, thus showing $G$ is not almost convex in the given generating set.
From Lemma 1 above it follows that either $$|P_{k}| \le M |p|^{c({k+{2\ell +\epsilon\over 4}-{j\over 4}})}$$ or $$\denom(P_{k}) \le M |p|^{c({k+{2\ell +\epsilon\over 4}-{j\over 4}})}.$$ Let us fix $j$ so that $$|p|^{j\over 4} > M|p|^{{2\ell +\epsilon\over 4c}+1}$$ and hence $$M |p|^{c({k+{2\ell +\epsilon\over 4}-{j\over 4}})}=
|p|^{kc}(M|p|^{{2\ell +\epsilon\over 4c}}|p|^{-{j\over 4}})
\le |p|^{kc-1}.$$ It then follows that either $$|P_{k}| \le |p|^{kc-1}$$ or $$\denom(P_{k}) \le |p|^{kc-1}.$$
Now $|S_kT_{k}| = |p^{kc} + p^{-kc}|$ and $ \denom(S_{k}T_{k})=|p|^{kc}$. Hence either $$|S_kT_{k}|- |P_{k}|
\ge |p^{kc} + p^{-kc}|-|p|^{kc-1} = |p|^{kc-1}(|p + p^{-2kc+1}|-1)>|p|^{kc-1}$$ or $$\denom(S_kT_{k}) - \denom(P_k)
\ge |p|^{kc}-|p|^{kc-1} = |p|^{kc-1}(|p|-1)\ge |p|^{kc-1}.$$ In either case by Lemma 2, as $k\to\infty$ the distance $d_C(P_k,S_kT_k)$ increases without bound. But our assumption of almost convexity implied $d_C(P_k,S_kT_k) \le N/2 +\epsilon + j$ which is a constant. This is a contradiction. Hence $G$ is not almost convex.
J. Cannon, Almost convex groups, Geom. Dedicata 22 197–210 (1987).
J. Cannon, W. Floyd, M. Grayson and W. Thurston, Solvgroups are not almost convex, Geom. Dedicata 31 no. 3 292–300 (1989).
M. Shapiro and M. Stein, Almost convexity and the eight geometries, Geom. Dedicata, 55, 125–140, (1995).
C. Theil, Zur fast-konvexität einiger nilpotenter gruppen, Doctoral dissertation, Bonn, 1991.
Department of Mathematics and Statistics University of Melbourne Parkville, VIC 3052 Australia
$$d_C(P_k,S_kT_k)\ge d_C(1,S_kT_k) - d_C(1,P_k)
\ge |p|^{kc}-|p|^{kc-1} = |p|^{kc}(|p|-1).$$
[^1]: \*
[^2]: $^{\rm\dag}$
|
---
abstract: 'Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modelled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra.'
author:
- 'N. Boulant'
- 'J. Emerson'
- 'T. F. Havel'
- 'D. G. Cory'
- 'S. Furuta'
title: Incoherent Noise and Quantum Information Processing
---
Introduction
============
One of the biggest challenges in Quantum Information Processing (QIP) is the precise control of quantum systems. Errors in the control are conveniently classified as coherent, decoherent, and incoherent [@PraviaRFI]. Coherent errors are systematic and differ from the desired operation by a unitary operation. Decoherent errors can be expressed by completely positive (CP) superoperators [@Alicki] and can be counteracted by techniques such as Quantum Error Correction (QEC) [@Shor; @NielsenChuang]. An incoherent process can also be described by a completely positive superoperator [@PraviaRFI; @BoulantEntSwap]. The apparent non-unitary behavior of the incoherent process arises due to a distribution over external experimental parameters. The incoherent process is described by a superoperator $S$ acting on Liouville space which can be written, when acting on columnized density matrices ${\vert{\rho}\rangle}$ obtained by stacking their columns on top of each other from left to right [@Havel:03], as $$\begin{aligned}
S&=& \int p(\alpha)\overline{U}(\alpha)\otimes
U(\alpha)d\alpha,\label{Sup}\end{aligned}$$ where $p(\alpha)$ is a probability density, i.e. the fraction of quantum systems within an ensemble that sees a given $U(\alpha)$ within an interval $d\alpha$, $\int p(\alpha)d\alpha = 1$, and $\overline{U}$ denotes the complex conjugate of $U$. This decomposition of a CP map into unitary Kraus operators is sometimes called a random unitary decomposition (RUD) [@LeungThesis]. A RUD exists for an incoherent process, but such a decomposition is sometimes possible even for a very general decoherent process [@Streater] when there is no correspondence between the unitary operators in the decomposition and some actual distribution of associated experimental control parameters $\alpha$. The distinction between the two therefore is practical, and depends primarily on the correlation time of the variation of experimental parameters. If the latter quantity is longer than the inverse of the typical modulation frequency, the process falls into the class of incoherent noise [@Carr; @Hahn]. The point of making this distinction is that, whilst correcting for decoherent errors requires the full power of QEC, in practice incoherent noise effects are often reduced directly through the design of the time-dependence of the control fields. This is possible since the operators underlying the incoherence $U(\alpha)$ are assumed to be time-independent over the length of the expectation value measurement. Common approaches for instance in Nuclear Magnetic Resonance (NMR) include composite and adiabatic pulses [@Tycko; @Shaka; @Levitt; @Jones; @Pines; @Silver]. Furthermore, the work done by Tycko [@Tycko] and Jones [@Jones2; @Jones3] on composite pulses finds a great application in QIP since the schemes proposed are universal and therefore work regarless of the input state.
In this paper, we demonstrate how incoherent errors introduce particular limitations to Quantum Process Tomography (QPT) [@NielsenChuang; @Childs; @BoulantQPT; @Cirac] due to the correlations they introduce with an environment in the QPT input states. Prior work has been devoted to the study of the implications of such correlations in the system’s reduced dynamics [@Buzek; @BuzekErra; @Pechukas]. However, to our knowledge they have not explicitly been studied within the context of QPT to show that the method may output non-completely positive (NCP) maps. If the existence or origin of such noise is unknown, it is shown in [@PraviaRFI] by means of explicit examples how one can eventually infer qualitative information, say its symmetry, about the probability distribution underlying the incoherent process from superoperator eigenvalue spectra. In this paper we tackle the inverse problem of extracting an effective probability distribution $p(\alpha)$ representing the incoherent superoperator, given a model for the source of the incoherent noise in the system. Such information is crucial for counteracting the incoherent errors [@PraviaRFI] which, due to their slow variation would otherwise persist during the experiment and quickly accumulate.
Incoherent Noise and Quantum Process Tomography {#QPT}
===============================================
QPT measures the experimental map associated with the implementation of a desired quantum operation by passing a complete set of input states through the gate and measuring the corresponding output states (see Fig. 1).
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This procedure is important for the experimental study of noise processes and for the design of quantum error correcting codes [@Shor]. If incoherent noise is present in the preliminary step of QPT, then the prepared (input) states will be classically correlated with the control parameter $\alpha$ characterizing the incoherence (Eq 1). Furthermore, if the correlation time of the noise in the control parameter is long compared to the coarse-grained time at which the evolution of the system is monitored, then the subsequent dynamics is non-Markovian [@Cohen-Tannoudji]. More specifically, the gate applied to the input state (i.e., the gate being characterized by QPT) will be correlated with the same (slowly-varying) control parameter, and therefore also correlated with the input state to which it is applied. In such cases, the measured dynamics are not guaranteed to be completely positive, and need not correspond to a linear map [@NielsenChuang; @Buzek; @BuzekErra; @Pechukas].
More generally, any correlations arising from non-Markovian dynamics, whether quantum or classical, can lead to incorrect interpretations of the data obtained from QPT. The “environment” which we assume from the outset is correlated with the system is in general defined by the degrees of freedom that are not part of the system. For example, the different spatial locations of individual qubits in a NMR ensemble, or the bosonic bath producing the fluctuations of the gate charge in a superconducting qubit [@Weiss].
The basic issues can be seen by exploring the QPT of a spin-$1/2$ system $A$, coupled to a second spin-$1/2$ system $B$ as its environment. Borrowing the example used in [@Buzek], the initial density matrix of the total system may be written as $$\begin{aligned}
\rho_{AB}=\frac{1}{4}(I_{AB} + \alpha_i \sigma_i\otimes I_B + \beta_j I_A \otimes
\sigma_j + \gamma_{ij} \sigma_i\otimes\sigma_j),\end{aligned}$$ where $I$ and $\sigma_i$ denote the identity and Pauli matrices respectively. The density matrix of the system $A$ is then obtained by tracing over the environment $B$: $$\begin{aligned}
\rho_{A}=\frac{1}{2}(I_{A} + \alpha_i \sigma_i).\end{aligned}$$ The dynamics of the whole system $\rho_{AB}$ is a unitary evolution $U_{AB}$, so that the resulting density matrix of $A$ is $$\begin{aligned}
\rho'_{A}&=&\sum_{\mu}{\langle{\mu}\vert}U_{AB}(\rho_A\otimes\rho_B)U_{AB}^{\dagger}{\vert{\mu}\rangle}\\
& & + \sum_{\mu}{\langle{\mu}\vert}U_{AB}\gamma^{'}_{ij}\sigma_i\otimes\sigma_j
U_{AB}^{\dagger}{\vert{\mu}\rangle},\end{aligned}$$ where $\gamma^{'}_{ij}=(\gamma_{ij}-\alpha_i\beta_j)/4$ and $\rho_B=\mathrm{Tr}_A(\rho_{AB})$. Let $\rho_B=\sum_{\nu}p_{\nu}{\vert{\nu}\rangle}{\langle{\nu}\vert}$ and $M_{\mu\nu}={\langle{\mu}\vert}U_{AB}{\vert{\nu}\rangle}$. Then the above expression can be reexpressed as $$\begin{aligned}
\rho'_{A}&=&\sum_{\mu\nu}M_{\mu\nu}\rho_AM_{\mu\nu}^{\dagger}\\
& & + \sum_{\mu}{\langle{\mu}\vert}U_{AB}\gamma^{'}_{ij}\sigma_i\otimes\sigma_j
U_{AB}^{\dagger}{\vert{\mu}\rangle}\end{aligned}$$ The first line therefore corresponds to the Kraus operator sum form [@Kraus2] of the evolution when initial correlations are not present, while the second line represents the contribution from these correlations. We can easily see that there exist $U_{AB}$, e.g., the swap gate, for which these initial correlations are not observable. In general, however, initial correlations cause the map to be non-linear or NCP.
Within the context of QPT, we now investigate an explicit example of how NCP superoperators can arise. We take a set of $4$ initial density matrices $\rho_{AB}$ such that $\rho_B$ is the same in each case, and where the input states $\rho_A$ span the Hilbert space of the system $A$ (as required by the QPT procedure) $$\begin{aligned}
\rho_{AB}^1&=&(I_{AB} + \beta I\otimes\sigma_z)/4\Rightarrow \rho^1_{A,in}=I/2\\
\rho_{AB}^2&=&(I_{AB} + \alpha \sigma_x\otimes I + \beta I\otimes \sigma_z +
\gamma \sigma_x\otimes\sigma_z)/4\Rightarrow \rho^2_{A,in}=(I+\alpha \sigma_x)/2\\
\rho_{AB}^3&=&(I_{AB} + \alpha \sigma_y\otimes I + \beta I\otimes \sigma_z +
\gamma \sigma_y\otimes\sigma_z)/4\Rightarrow \rho^3_{A,in}=(I+\alpha \sigma_y)/2\\
\rho_{AB}^4&=&(I_{AB} + \alpha \sigma_z\otimes I + \beta I\otimes \sigma_z +
\gamma \sigma_z\otimes\sigma_z)/4\Rightarrow \rho^4_{A,in}=(I+\alpha \sigma_z)/2\\\end{aligned}$$ where in each case $\rho_B=(I + \beta \sigma_z)/2$. With the example $U_{AB}=e^{-i\frac{\pi}{4}\sigma_z\otimes\sigma_z}$, the corresponding $4$ outputs are $\tilde{\rho}_{A}^1=I/2$, $\tilde{
\rho}_{A}^2=(I+\gamma
\sigma_y)/2$, $\tilde{\rho}_{A}^3=(I-\gamma
\sigma_x)/2$ and $\tilde{\rho}_{A}^4=(I+\alpha\sigma_z)/2$. We write the density matrices $\rho$ as vectors in Liouville space in the Zeeman basis [@Havel:03], which are obtained by first writing the density matrix in the Zeeman basis and then stacking their columns on top of each other from left to right. We refer to the resulting vector simply as the “columnized density matrix”, and will denote it as a ket $|\rho\rangle$. If we set $\alpha=\beta=0.5$ and $\gamma=0.6$, the map $S$ is $$\begin{aligned}[b]
\begin{smallmatrix} & \textsf{Output 1} &
\textsf{Output 2} & \textsf{Output 3} & \textsf{Output 4}
\end{smallmatrix}\\
\left[ \begin{smallmatrix}
&0.5~~~~&~0.5~~~~~ &~0.5~~~&~0.75 \\[12pt] &0~~~&~0.3i~~~~~&-0.3~~~ &~0~ \\[12pt] &0~~~ &-0.3i~~~~~ &-0.3~~~ &~0~ \\[12pt] &0.5~~ &~0.5~~~~~ &~0.5~~~
&~~0.25
\end{smallmatrix}\right]
\end{aligned}
\begin{aligned}[b]&
\begin{smallmatrix} \hspace{3em} \textsf{Input 1} &
\hspace{0.5em} \textsf{Input 2} & \hspace{0.5em} \textsf{Input 3} & \hspace{0.5em}
\textsf{Input 4}
\end{smallmatrix} \\ &
=S\cdot\left[ \begin{smallmatrix}
&0.5~~~~ &~0.5~~ &~0.5~ &~0.75 \\[12pt] &0~~~ &~0.25~~ &~0.25i~ &~0~ \\[12pt] &0~~~ &~0.25~~ &~-0.25i~ &~0~ \\[12pt] &0.5~~ &~0.5~~ &~0.5~
&~~0.25
\end{smallmatrix} \right]
\end{aligned}
~\leftrightarrow~
\begin{aligned}[b] &
\\S~=~
\left[ \begin{smallmatrix}
&~1 &0 &0 &0 \\[12pt] &0 &~1.2i &0 &0 \\[12pt] &0 &0 &~-1.2i &0 \\[12pt] &0 &0 &0
&~1
\end{smallmatrix} \right]
\end{aligned}\label{supop}$$ which is in general non-linear, since one can no longer predict the output for an arbitrary input state given the action of the gate on these four specific input states alone. However, the map can be considered to act linearly on the system $A$ Hilbert space, i.e. on the linear combinations of input states which contain the right correlations with the environment. For instance, the action of the gate on the input state $(\rho^2_{A}+\rho^3_{A})/2=(I+\frac{\alpha}{2}(\sigma_x+\sigma_y))/2$ can be computed by using the above matrix expression if the total input state is $\rho_{AB}=(I_{AB}+\frac{\alpha}{2}(\sigma_x +
\sigma_y)\otimes I +\beta I\otimes
\sigma_z + \frac{\gamma}{2}(\sigma_x + \sigma_y)\otimes \sigma_z)/4$. This result conveniently allows one to treat the map as linear. If treated as linear, the Choi matrix [@Choi] ${\mathcal C}=\sum_{i,j=0}^{N-1}(E_{ij}\otimes I)S(I\otimes E_{ij})$, where $N$ is the dimension of the system’s Hilbert space and $E_{ij}$ is the $N\times N$ elementary matrix (with a “1” in the $ij-th$ position and zeros elsewhere), corresponding to the superoperator $S$ is not positive semidefinite and consequently $S$ can not be CP [@Havel:03].
It is suggested in [@Havel:03] how the NCP part of the superoperator can be removed, namely, by removing the negative eigenvalues of the Choi matrix and then renormalizing so that the trace is equal to the dimension of the Hilbert space, $N$. We shall call this method CP-filtering. The Choi matrix corresponding to $S$ in (\[supop\]) has two non-zero eigenvalues $(2.2,-0.2)$. In this example, the CP-filtering procedure replaces the negative eigenvalue by $0$ and then renormalizes the new Choi matrix to trace $N=2$. The CP-filtering method outputs one unitary Kraus operator. On the other hand, if no initial correlations were present, the superoperator would be equal to $S=\mathrm{diag}(1,0.5i,-0.5i,1)$, and the corresponding Choi matrix would have two positive eigenvalues $(0.5,1.5)$, yielding two Kraus operators. This superoperator is therefore completely positive. Unsurprisingly, we thus observe that the superoperator obtained from the CP-filtering procedure is not equal to the superoperator obtained by removing the initial correlations. Therefore, unless the initial correlations are very small, CP-filtering is a fairly uncontrolled procedure, giving CP superoperators that may significantly misrepresent the true quantum dynamics.
We now take the case $\alpha=\beta=\gamma=0.5$ with the same $U_{AB}$ as previously. Although initial correlations are still present, the superoperator obtained by the above QPT method, without CP-filtering, is CP with one Kraus operator. In contrast, if the correlations in the initial states are removed whilst keeping all other things equal, the superoperator obtained is CP with two Kraus operators, not one. So even when initial correlations are present, the superoperator obtained via QPT may be completely positive. Therefore, one cannot rule out the presence of initial correlations merely by the existence of a valid Kraus operator sum form via QPT. Initial correlations can masquerade as CP maps, and in reality the process may not be linear with respect to arbitrary input states.
To summarize, the results of this section are: (i) incoherent errors introduce correlations between the system and the environment in the QPT input states which can persist during the implemented transformation, (ii) these correlations can yield non-completely positive superoperators or non-linear maps, (iii) the CP-filtering method suggested in [@Havel:03] is not equivalent to removing these initial correlations, and (iiii) initial correlations can masquerade as CP maps which misrepresent what is in reality a non-linear process with respect to the input states. See Table \[table\] for a summary. This simple analysis explains the apparent NCP behavior measured in experiments reported in [@BoulantQPT; @WeinsteinQFT]. This motivates the need to characterize the incoherent noise, and to provide ways to correctly interpret QPT data. If the noise can be successfully characterized, we may use this information to better counteract the noise in the first place.
CPF Corr CP Num. Kraus Op.
------ ---------- ---------- ---------- ----------------
Ex 1 $\times$ $\times$ -
1
$\times$ $\times$ 2
Ex 2 $\times$ 1
$\times$ $\times$ 2
: Summary of the two examples given in section \[QPT\]. CPF is abbreviation for CP-Filtering which is an algorithm for converting a non CP superoperator to a CP one. “Corr” is abbreviation for correlations (between subsystems $A$ and $B$). \[table\]
Extracting Probability Distributions from Superoperator Eigenvalue Spectra
==========================================================================
By applying a first order perturbation theory analysis of the eigenvalues of superoperators, we now present a method to extract the probability distribution profile $p(\alpha)$ of unitary matrices present in incoherent processes. For an incoherent noise to be refocused [@Hahn], knowledge about the noise is *a priori* required. If qualitative information about the inhomogeneity in the Hamiltonian is known, spectroscopic techniques can be used to obtain the missing quantitative information. In the following analysis, we assume that the physical origin of the incoherent noise is unknown or hidden due to the complexity of the system-apparatus interactions, but that a mathematical model is presumed.
As presented in the introduction, an incoherent process implies a random unitary distribution. Incoherent processes are thus unital, which means that the maximally mixed density matrix is left unchanged. A linear, completely positive, trace preserving and unital map is called a doubly stochastic map [@Alberti]. Although a single necessary and sufficient condition for a doubly stochastic map to possess a RUD has not been found to our knowledge, examples of doubly stochastic maps which do not possess a RUD are reported in [@LeungThesis; @Streater]. However, since many decoherent unital processes can be modelled by a stochastic Hamiltonian, i.e. semiclassically, we believe that many instances of decoherent processes can have a RUD. This belief is supported by the following two facts. Any two density operators $(\rho,\rho')$ connected by a doubly stochastic map, $\rho'
=\Lambda(\rho)$, can always be related by a transformation of the form $\rho'=\sum_i p_i U_i\rho U_i^{\dagger}$, where $\sum_i p_i=1$ and $\{U_i\}$ is a set of unitary operators [@Alberti]. Furthermore, all unital maps for a two-level quantum system always have a RUD [@Streater].
Perturbation Theory Analysis of the Eigenvalue Spectra
------------------------------------------------------
In what follows, we take examples from NMR [@Ernst] physics where the main source of incoherence comes from Radio Frequency (RF) power inhomogeneity. Due to the spatial extent of the sample, individual spins during the course of a RF field see different powers [@PraviaRFI] and evolve according to different unitary evolutions with different characteristic frequencies. Note that identical spins can have different resonance frequencies due to inhomogeneity in the static magnetic field within the ensemble, which is another source of incoherence. However, as shown in [@BoulantEntSwap], the non-unitary features arising from this static external field inhomogeneity are usually much smaller than those arising from RF inhomogeneity and will be therefore ignored in this example. Finally while the distribution of RF fields can be easily measured via a nutation experiment on a single spin, the method presented here is quite general and can for instance account for the correlation between multiple sources of incoherence (several RF fields, DC field etc...).
Let $n$ be the number of spin-1/2 particles in the ensemble, and $U_k$ denote the unitary operator generated by the RF field in the $k$th frequency interval of the RF amplitude profile. The eigenvalues of the superoperator $S=\bar U_k
\otimes U_k$ are products of the eigenvalues of $U_k$ with those of $\bar
U_k$. This yields $2^n$ eigenvalues that are equal to unity and $(2^{2n-1}
- 2^{n-1})$ pairs of eigenvalues $(\lambda,\bar{\lambda})$. In general, the eigenvalues of CP superoperators come in conjugate pairs, but only in the case of unitary superoperators do all the eigenvalues lie on the complex unit circle.
The incoherent process resulting from an inhomogeneous distribution of $U_k$ processes is given by the superoperator $S=\sum_k p_k
\overline{U}_k\otimes U_k$, where $p_k$ is the fraction of spins that sees the unitary evolution $U_k$. The more broadly the $\{p_k\}$ are distributed, the larger the degree of inhomogeneity in the evolution, and the more incoherent noise enters into the evolution. Estimates of the actual eigenvalues of $S=\sum_k p_k \overline{U}_k\otimes U_k$ will now be obtained using non-degenerate first-order perturbation theory. Because the RF pulses are not perfect, even in the absence of RF field inhomogeneity, we may assume that the unperturbed eigenvalues are generically non-degenerate. The unitary operator $U_k$ generated by the RF field acting at position $k$ may be written in exponential form as $U_k = e^{-i
H_k\, t}$ where $H_k$ represents the effective Hamiltonian of the evolution over the period $t$ for which the pulse is applied ($\hbar$ has been set equal to $1$). Defining $H_0$ to be the unperturbed (and desired) Hamiltonian, the eigenvalues $\phi_j$ and eigenstates ${\vert{\phi_j}\rangle}$ of $H_{0}$ satisfy the eigenvalue equation $$U_{0} {\vert{\phi_j}\rangle}=e^{-i\phi_j\,t}{\vert{\phi_j}\rangle},$$ where $U_0 = \exp( -i H_0\, t )$. The Hamiltonian of a particular $U_k$ is assumed to be a perturbation of the desired, homogeneous Hamiltonian $$H_k=H_{0} + K_k,$$ where $K_k$ is the perturbation. To first order, the new eigenvalues of $H_k$ are $$\tilde{\phi}_{j,k}=\phi_j+ {\langle{\phi_j}\vert} K_k {\vert{\phi_j}\rangle},$$ and the corresponding eigenvalues of $U_k$ are $$U_k{\vert{\tilde{\phi}_{j,k}}\rangle} = e^{-i\tilde{\phi}_{j,k}\,t}{\vert{\tilde{\phi}_{j,k}}\rangle}.$$ To first order, the spectral decomposition of $S$ is $$S = \sum_{k,m,j} p_k \left( e^{i\tilde{\phi}_{m,k}\,t}
\overline{{\vert{\phi_m}\rangle}}\overline{{\langle{\phi_m}\vert}} \otimes
e^{-i\tilde{\phi}_{j,k}\,t}{\vert{\phi_j}\rangle}{\langle{\phi_j}\vert}\right)$$ and the eigenvalues of $S$ are given approximately by $$\begin{aligned}
\lambda_{jm}&=&\sum_k p_k e^{-i(\tilde{\phi}_{j,k}-\,\tilde{\phi}_{m,k})\,t}\nonumber\\
\label{eigenvalues}
&=& e^{-i(\phi_j-\phi_m)t}\sum_k p_k
e^{-i\left({\langle{\phi_j}\vert}K_k{\vert{\phi_j}\rangle}-
{\langle{\phi_m}\vert}K_k{\vert{\phi_m}\rangle}\right)t}\,.\nonumber\end{aligned}$$
We now imagine the scenario where $K_k$ is given by $K_k
t=\frac{\omega_k-\omega_0}{\omega_0}K$. This result would in fact be exact for one spin on resonance. In this case, $\frac{\omega_k-\omega_0}{\omega_0}$ is the parameter $\alpha$ defined in the introduction (which parameterizes the inhomogeneity) and represents the normalized RF power deviation from the desired power $\omega_0$. Defining $\Delta\omega = \frac{\omega_k-\omega_0}{\omega_0}$ and $K_{jm}={\langle{\phi_j}\vert}K{\vert{\phi_j}\rangle}-{\langle{\phi_m}\vert}K{\vert{\phi_m}\rangle}$, the previous equation in the continuous limit becomes $$\lambda_{jm} = e^{-i(\phi_j-\phi_m)t}\int p(\Delta\omega)
e^{-iK_{jm}\Delta\omega } d\Delta\omega. \label{data}$$ We see in this case that to first order the eigenvalue $\lambda_{jm}$ is just the unperturbed eigenvalue $e^{-i(\phi_j-\phi_m)t}$ times the Fourier transform of the RF distribution profile evaluated at $K_{jm}$. This result demonstrates that the probability distribution profile $p(\Delta\omega)$ of an incoherent process can be determined, within some degree of approximation, from the eigenvalue structure $\{\lambda_{jm}\}$ of an experimental superoperator, given some model for the incoherence $K$. Knowing $K$ would indeed allow one to build the correspondence between $\lambda_{jm}$ and $K_{jm}$, and then to determine $p(\Delta\omega)$ by performing an inverse Fourier transform. Of course, this result holds for general $K$ only when the perturbation is in the first order regime, and when the unperturbed eigenvalues are non-degenerate. But if $K$ approximately commutes with $H_0$, then the first-order perturbation in the eigenvalues is close to an exact correction, and the above analysis gives a very accurate description of the incoherent process.
Recovery of the Profile
-----------------------
We now demonstrate via a numerical example how one can recover the profile $p(\Delta\omega)$ from the eigenvalue spectrum of a measured superoperator. In the theory derived in the previous subsection, $\lambda_{jm}$ is the data, i.e., the eigenvalues from the measured superoperator. A model $K$ is needed for the perturbation, while $\phi_j$ and ${\vert{\phi_j}\rangle}$ are known through the knowledge of $H_0$. Formally solving for $p(\Delta\omega)$ from (\[data\]), $$\label{formallysolve}
p(\Delta\omega) = \frac{1}{2\pi} \int \lambda_{jm} e^{i(\phi_j-\phi_m)t}\, e^{i
K_{jm}\Delta\omega} d K_{jm}.$$ The different eigenvalues $\lambda_{jm}$ multiplied by $e^{i(\phi_j-\phi_m)t}$ therefore allow us to obtain the complex function of $K_{jm}$ which we shall call $f$, corresponding to the Fourier transform of the distribution profile.
For the numerical demonstration of this technique we take a $3$-qubit system. We choose $H_0$ and $K$ such that $|{\langle{\phi_l}\vert}K{\vert{\phi_n}\rangle}/(\phi_l-\phi_n)|\simeq
0.1$ for $n\neq l$ and $[H_0,K]\approx 0$ so that first order perturbation theory can be used [@Sakurai]. We then use a measured RF inhomogeneity profile shown in Fig. 2 to construct the following superoperator acting on Liouville space: $$\begin{aligned}
S=\sum_{\Delta\omega} p(\Delta\omega) \overline{U}(\Delta\omega)\otimes
U(\Delta\omega).\end{aligned}$$
{width="7cm" height="6cm"}
Provided with this superoperator, we compute its eigenvalues and plot them on the Argand diagram (see Fig. 3). The perturbation $K$ is such that the first order limit condition is fulfilled, but that the size of its diagonal elements in the $H_0$ eigenvectors basis is large enough to generate significant dephasing and attenuation in the eigenvalues. The correspondence between $\lambda_{jm}$ and $K_{jm}$ is needed to recover the distribution profile (\[formallysolve\]).
{width="8cm" height="8cm"}
The correspondence can be established by first computing ${\langle{\overline{\phi_j},\phi_m}\vert}S{\vert{\overline{\phi_j},\phi_m}\rangle}$, then searching for the eigenvalue of $S$ closest to it. This allows us to make the correspondence between one unperturbed eigenvalue with eigenvector ${\vert{\overline{\phi_j},\phi_m}\rangle}={\vert{\overline{\phi_j}}\rangle}\otimes{\vert{\phi_m}\rangle}$ (obtained from the knowledge of $H_0$) and one eigenvalue of $S$. The function $f=\int p(\Delta\omega)e^{-i\Delta\omega K_{jm}}d\Delta\omega$ with respect to $K_{jm}$ can then be constructed. The real and imaginary parts of that function are plotted in Fig. 4.
{width="8cm" height="8cm"}
Note that we ignore the degenerate points at $f=1$ because they do not provide any information about $p(\Delta\omega)$ other than normalization. The $64$ eigenvalues of the superoperator minus the $8$ degenerate ones equal to $1$ (at $K_{jm}=0$), plus $1$ eigenvalue added at $K_{jm}=0$ to avoid a DC offset in the reciprocal domain, yield a complex function $f$ made of $57$ unequally spaced points. The function $f$ is conjugate symmetric with respect to $0$, which is consistent with the fact that we are supposed to recover a probability distribution, i.e. a real function, after computing the inverse Fourier transform. To perform the inverse Fourier transform of a function sampled at unequally spaced points, we used an algorithm prescribed in [@MathPaper]. The result is shown in Fig .5. The width of the probability distribution and its skewness are recovered to a good extent, the discrepancy being due to the lack of information about the function $f$. It is worth mentioning that with $57$ sample points, the window of $K_{jm}$ values should be large enough to allow low frequency components of the profile to be reliably extracted. If the incoherent perturbations were very small, there will not be as many large values of $K_{jm}$, and therefore less low frequency information would be available.
{width="10cm"}
However, the perturbations can be made larger without changing the mathematical model, by simply repeating the control sequence several times, provided other noise mechanisms do not play a significant role. In addition, more points could be used to get a better sampling resolution, with, for instance, a $4$-qubit superoperator yielding $241$ points. One cannot have arbitrarily many points, however, because the correspondence between the eigenvalues of $S$ and those of $S_0$ could quickly become impossible to establish unless a very good knowledge of the perturbation $K$ is available. The density of points in the Argand diagram becomes so large that eigenvalues can easily become confused. Here, the $3$-qubit superoperator was enough to recover the essential features of the probability distribution.
If $H_0$ is not exactly known, and in fact a constant offset Hamiltonian which is proportional to $K$ is present, then a different function $f$ is obtained : $$f = \int p(\Delta\omega)e^{-iK_{jm}\Delta\omega }e^{-i\beta K_{jm}}
d\Delta\omega,$$ where $\beta$ is a constant real number. Taking the inverse Fourier transform of $f$ would reveal a distribution $p(\Delta\omega)$ centered around $\beta$ rather than $0$, indicating that $H_0+\beta K$ is in fact the unperturbed Hamiltonian. Perfect knowledge about $H_0$ as a result is not required provided the offset is approximately proportional to $K$.
It is important for this method to work that the model $K$ chosen *a priori* is a reasonably faithful one, and that it approximately commutes with $H_0$. This ensures that the applied first-order perturbation theory is valid. In the extreme case, in which $K$ anticommutes with $H_0$, ${\langle{\phi_j}\vert}K{\vert{\phi_j}\rangle}=0~\forall j$ and no “data" would be available for analysis. It is also worth mentioning that this method is, needless to say, not scalable. However, in many instances, as in NMR, the inhomogeneity features are apparatus dependent, so that reasonably small physical systems can be used to probe them. The scalability of the method is not necessarily a requirement. The distribution of some control parameters, once obtained, can be valuable in designing robust control sequences [@PraviaRFI] for larger and more complex systems.
Conclusion
==========
Here we reviewed that when incoherence is present during the preparation of the input states for QPT, the resulting correlations between the system and the “environment” can play an important role on the subsequent system’s dynamics. The map obtained by right multiplying the matrix of output states by the inversion of the matrix of input states still has a meaning, but a correct interpretation of the measured data (or transformation) requires an analysis of the incoherence effects affecting the tomographic procedure. In particular, the measured map needs not be CP. If quantitative information is missing, our perturbation theory analysis of superoperator eigenvalue spectra can be used to determine an effective distribution of unitaries characterizing the process, provided a good mathematical model is available. While this requires a significant effort to measure a 3-qubit superoperator, it is certainly feasible within present experimental capabilities. Lastly, the knowledge of the distribution of control parameters should finally allow us to design more efficient control sequences aimed at counteracting these deleterious effects.
Acknowledgements
================
This work was supported by ARO, DARPA, NSF and the Cambridge-MIT institute. Correspondence and requests for materials should be addressed to D. G. Cory (e-mail: dcory@mit.edu). We thank Marcos Saraceno for valuable discussions.
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abstract: 'We study the correlators of the 2d W$_N$ minimal model in the semiclassical regime with large central charge from bulk viewpoint by utilizing open Wilson lines in $\text{sl}(N) $ Chern-Simons gauge theory. We extend previous works for the tree level of bulk theory to incorporate loop corrections in this paper. We offer a way to regularize divergences associated with loop diagrams such that three point functions with two scalars and a higher spin current agree with the values fixed by the boundary W$_N$ symmetry. With the prescription, we reproduce the conformal weight of the operator corresponding to a bulk scalar up to the two loop order for explicit examples with $N=2,3$.'
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YITP-17-92
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**Correlators in higher spin AdS$_3$ holography**
[**from Wilson lines with loop corrections** ]{}
[Yasuaki Hikida$^{a}$[^1] and Takahiro Uetoko$^b$[^2]]{}
.6 truecm
*$^a$Center for Gravitational Physics, Yukawa Institute for Theoretical Physics,*
*Kyoto University, Kyoto 606-8502, Japan*
*$^b$Department of Physical Sciences, College of Science and Engineering,*
*Ritsumeikan University, Shiga 525-8577, Japan*
0.5 truecm
0.5 truecm
Introduction
============
In [@Hikida:2017byl] we computed three point functions with two scalar operators and a higher spin current in the 2d W$_N$ minimal model with $1/N$ corrections. The main aim of this paper is to give a bulk interpretation of the conformal field theory results.[^3] The $1/N$ corrections (or $1/c$ corrections with $c$ as the central charge) in the minimal model should be interpreted as loop corrections in the bulk gravity description. However, it is notoriously difficult to deal with divergences associated with gravitational loop diagrams in general. Applying holography, it is expected that boundary theory can define bulk quantum theory of gravity generically. For our case, the minimal model would determine the way to regularize these gravitational divergences, and we would like to show that this is indeed the case in this paper.
The 2d W$_N$ minimal model has a coset description as $$\begin{aligned}
\frac{\text{su}(N)_k \oplus \text{su}(N)_1 }{ \text{su}(N)_{k+1} }
\label{coset}\end{aligned}$$ with the central charge $$\begin{aligned}
c = (N-1) \left( 1 - \frac{N(N+1)}{(k+N) (k+N+1)}\right) \, .
\label{central}\end{aligned}$$ In [@Gaberdiel:2010pz] the ’t Hooft limit with large $N$ but finite $\lambda = N/(k+N)$ of the minimal model is conjectured to be dual to the classical 3d Prokushkin-Vasiliev theory of [@Prokushkin:1998bq]. Instead of the ’t Hooft limit, we consider the semiclassical regime with large $c$ but finite $N$. The bulk description for the semiclassical regime is supposed to be given by Chern-Simons gauge theory based on $\text{sl}(N) \oplus \text{sl} (N)$ dressed by perturbative matters [@Castro:2011iw; @Gaberdiel:2012ku; @Perlmutter:2012ds]. The large $c$ regime should be realized with a negative level $k = -1 - N + \mathcal{O}(c^{-1})$, thus the conformal field theory is non-unitary in the regime.[^4] In [@Hikida:2017byl] we evaluated correlators at the ’t Hooft limit with $1/N$ corrections, but the results can be generalized for the semiclassical limit with $1/c$ corrections. We try to interpret the $1/c$ corrections in terms of $\text{sl}(N) $ Chern-Simons gauge theory.
The W$_N$ symmetry of the minimal model is generated by higher spin currents $J^{(s)} (z)$ with $s=2,3,\ldots,N$. We examine the following two and three point functions as $$\begin{aligned}
\langle \mathcal{O}_{h_+} (z_1) \bar{\mathcal{O}}_{h_+} (z_2) \rangle \, , \quad
\langle \mathcal{O}_{h_+} (z_1) \bar{\mathcal{O}}_{h_+} (z_2) J^{(s)} (z_3) \rangle
\label{2&3pt}\end{aligned}$$ including $1/c$ corrections. Here $\mathcal{O}_{h_+} $ is a scalar operator with conformal weight $h_+ = (1 - N)/2 + \mathcal{O}(c^{-1})$. The negative value of the conformal weight reflects the non-unitarity of the theory. At the leading order in $1/c$, it was claimed in [@Besken:2016ooo] that correlators or conformal blocks can be computed by the networks of open Wilson lines in $\text{sl}(N) $ Chern-Simons gauge theory.[^5] For instance, the expectation value of an open Wilson line computes the two point function $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} \rangle$. Roughly speaking, the open Wilson line corresponds to a particle running in the bulk, which is dual to the boundary two point function. Furthermore, the three point function $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} J^{(s)} \rangle$ can be evaluated with the extra insertion of the boundary current $J^{(s)}$. The main aim of this paper is to interpret the $1/c$ corrections of the correlators as loop corrections in the bulk computations with open Wilson lines. For $N=2$, the Chern-Simons theory reduces pure gravity theory as in [@Achucarro:1987vz; @Witten:1988hc], and in that case $1/c$ corrections have been examined in Virasoro conformal blocks [@Fitzpatrick:2016mtp] and the conformal weight of the scalar operator [@Besken:2017fsj]. The validity of the method with $N=2$ is formally supported by the analysis of conformal Ward identity [@Verlinde:1989ua; @Fitzpatrick:2016mtp]. See also [@Anand:2017dav] for a recent application.
During loop computations with open Wilson lines, we would meet divergences and a main issue in this paper is to propose a prescription to regularize the divergences. There are three main steps in the prescription. Firstly, we have to decide how to introduce a regulator $\epsilon$ to make integrals finite. We adopt a kind of dimensional regularization such that scaling invariance is not broken. Secondly, we have to remove the terms diverging for $\epsilon \to 0$. Here we choose to shift parameters in the open Wilson line since we cannot remove divergences in the current setup with the shift of parameters in Lagrangian as for usual quantum field theory. Finally, we have to remove ambiguities arising from $\epsilon$-independent parts in the shift of parameters. We offer a way to fix them so as to be consistent with the W$_N$ symmetry of the minimal model.
It is easy to show that the Wilson line method reproduces the leading order results for correlators in with generic $N$. For $1/c$ corrections, we mainly focus on the simplest examples with $N=2$ and $N=3$. We find that the three point functions from the Wilson line method are regularization scheme dependent at the $1/c$ order. Since the three point functions of the minimal model are fixed by the symmetry, we adopt a regularization such that the Wilson line results match the minimal model ones. For $N=2$, the authors in [@Besken:2017fsj] tried to reproduce the $1/c$ corrections in the conformal weight of the scalar operator from the bulk theory. They succeeded in doing so up to the $1/c$ order since it is regularization independent, but they failed at the $1/c^2$ order due to the regularization issue. Adopting our prescription for regularization, we succeed in reproducing the $1/c^2$ order corrections of conformal weight both for $N=2$ and $N=3$.
The organization of this paper is as follows; In the next section, we summarize the results on two and three point functions in the 2d W$_N$ minimal model of at the semiclassical limit with $1/c$ corrections. In section \[Preliminaries\], we explain our prescription to compute boundary correlators in terms of open Wilson lines in sl$(N)$ Chern-Simons gauge theory. We reproduce the minimal model results at the leading order in $1/c$ and describe our prescription to regularize divergences arising from loop diagrams. In section \[Correlators2\], we apply our method to the simplest case with $N=2$. In particular, we reproduce the result in [@Besken:2017fsj] for the two point function at the $1/c$ order and improve their argument for the next order in $1/c$ with the help of our analysis for the three point function. In section \[Correlators3\], we proceed to the $N=3$ case and show that our prescription also works for this example. In section \[conclusion\], we conclude this paper and discuss open problems.
W$_N$ minimal model in the semiclassical regime {#CFT3pt}
===============================================
In this section, we examine the two and three point functions of the coset model with large $c$ but finite $N$ in $1/c$ expansion. For this purpose we should describe the model in terms of $c,N$ instead of $k,N$ in . The parameter $k$ is related to $c,N$ as $$\begin{aligned}
k = - 1 - N + \frac{N (N^2 -1)}{c} + \frac{N (1-N^2)(1-N^3)}{c^2} + \mathcal{O} (c^{-3}) \end{aligned}$$ in $1/c$ expansion. Originally $k$ is a positive integer, but here we assume an analytic continuation of $k$ to a real value. See [@Gaberdiel:2012ku] for details on the issue. Using this relation, we can expand physical quantities in $1/c$, and terms at each order depend only on $N$.
The two point function is fixed by the symmetry as $$\begin{aligned}
\langle \mathcal{O}_{h} (z) \bar{\mathcal{O}}_{h} (0) \rangle =
\frac{1}{|z|^{4 h}} \, ,
\label{2ptcan0}\end{aligned}$$ where $h$ is the conformal weight of the scalar operator $\mathcal{O}_h$. The overall normalization can be set as $1$ by changing the definition of $\mathcal{O}_{h}$. This implies that the two point function is obtained only from knowledge of the spectrum. Throughout the paper, we only focus on the holomorphic sector, thus we may write $$\begin{aligned}
\langle \mathcal{O}_{h} (z) \bar{\mathcal{O}}_{h} (0) \rangle =
\frac{1}{z^{2 h}}
\label{2ptcan}\end{aligned}$$ instead of .
The spectrum of primary states can be obtained with finite $k,N$ by applying standard methods like coset construction as in [@Bais:1987zk]. The states are labeled as $(\Lambda_+ , \omega; \Lambda_-)$, where $\Lambda_+ , \omega, \Lambda_-$ are the highest weights of $\text{su}(N)_{k}, \text{su}(N)_{1}, \text{su}(N)_{k+1}$, respectively. The selection rule determines $\omega$ in terms of $\Lambda_+ , \Lambda_-$, so we may instead use the label $(\Lambda_+ ; \Lambda_-)$. We should take care of the field identification in [@Gepner:1989jq] as well. The conformal weight of the state can be obtained by coset construction [@Bais:1987zk] or Drinfeld-Sokolov reduction, see, e.g., [@Bershadsky:1989mf; @Bouwknegt:1992wg]. For instance, the latter gives the formula $$\begin{aligned}
h (\Lambda_+ ; \Lambda_-) = \frac{| (k+N+1) (\Lambda_+ + \hat \rho ) - (k + N) (\Lambda_- + \hat \rho)|^2 - \hat \rho^2 }{2 (k + N ) (k+N+1)} \, , \end{aligned}$$ where $\hat \rho$ is the Weyl vector of $\text{su}(N)$. According to [@Perlmutter:2012ds] (see also [@Castro:2011iw] for the original proposal), the state $(0;\Lambda_-)$ corresponds to a conical defect geometry, and the generic state $(\Lambda_+;\Lambda_-)$ is mapped to the geometry dressed by perturbative matters. In particular, the states $(0;0)$ and $(\text{f};0)$ correspond to the AdS vacuum, and a bulk scalar field on the background. Here we denote $\text{f}$ as the fundamental representation. The conformal weight of the state $(\text{f};0)$ is $$\begin{aligned}
&h_+ \equiv h(\text{f};0) = \frac{(N-1) (k + 2N + 1)}{2 N (k + N)} \, ,
\label{hexact}\end{aligned}$$ and we mainly deal with the operator $\mathcal{O}_{h_+} $ corresponding to the state in this paper.
Expanding the conformal weight $h$ in $1/c$ as $$\begin{aligned}
h = h_0 + \frac{1}{c} h_1 + \frac{1}{c^2} h_2 + \mathcal{O} (c^{-3}) \, ,
\label{hexp}\end{aligned}$$ the two point function becomes $$\begin{aligned}
\langle \mathcal{O}_{h} (z) \bar{\mathcal{O}}_{h} (0) \rangle = \frac{1}{z^{2 h_0}} \left[ 1 - \frac1c 2 h_1 \log (z) + \frac{1}{c^2} \left( 2 h_1^2 \log ^2 (z) - 2 h_2 \log (z) \right) \right] + \mathcal{O} (c^{-3}) \, . \label{2ptexp}\end{aligned}$$ For the operator $\mathcal{O}_{h_+} $ we have $$\begin{aligned}
h_0 = \frac{1-N}{2} \, , \quad
h_1 = -\frac{\left(N^2-1\right)^2}{2 } \, , \quad
h_2 = -\frac{(N+1)^2 (2 N (N+1)+1) (N-1)^3}{2} \, , \label{h012}\end{aligned}$$ which is obtained from the expression with finite $k,N$. The problem will be whether we can reproduce correct the coefficients in front of $\log (z)$ and $\log^2 (z)$ from the bulk viewpoint with open Wilson lines.
We also examine the three point functions in . In [@Hikida:2017byl] we have evaluated the three point functions by decomposing the four point function of $\mathcal{O}_{h_+} $ with Virasoro conformal blocks. As seen below, we have effectively decomposed the W$_N$ vacuum block, which is fixed by the W$_N$ symmetry in principle, and this implies that the three point functions can be fixed solely by the symmetry. Notice that the three point function with spin two current as $$\begin{aligned}
\langle \mathcal{O}_{h} (z_1) \bar{\mathcal{O}}_{h} (z_2) J^{(2)} (z_3) \rangle\end{aligned}$$ is determined by the conformal Ward identity, and our conclusion may be regarded as a higher spin generalization.
We decompose the following four point function as $$\begin{aligned}
G_{++} (z) &= \langle \mathcal{O}_{h_+} (\infty) \bar{\mathcal{O}}_{h_+} (1) \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) \rangle \, ,
\label{4pt}\end{aligned}$$ for which the expression with finite $k,N$ is given by [@Papadodimas:2011pf] $$\begin{aligned}
G_{++} (z) = |\mathcal{F}_1 (z)|^2 + \mathcal{N}_1 |\mathcal{F}_2 (z)|^2
\, .\end{aligned}$$ Here the W$_N$ conformal blocks are $$\begin{aligned}
\mathcal{F}_1 (z) = z^{- 2 h_+}(1 - z)^{ - 2 h_+ + \frac{k +2N}{k+N}}
{}_2 F_1 \left( \frac{k+N+1}{k+N} , - \frac{1}{k+N} ; - \frac{N}{k+N} ; z \right) \, ,\nonumber \\
\mathcal{F}_2 (z) = z^{- 2 h_+ + \frac{k +2N}{k+N}}(1 - x)^{ - 2 h_+ }
{}_2 F_1 \left( \frac{k+N+1}{k+N} , - \frac{1}{k+N} ; \frac{2 k + 3N}{k+N} ; z \right) \, , \end{aligned}$$ and the relative coefficient is $$\begin{aligned}
\mathcal{N}_1 = - \frac{\Gamma (\frac{k + 2N -1}{k+N}) \Gamma (\frac{-N}{k+N})^2\Gamma (\frac{2 k + 3N +1}{k+N})}{\Gamma(\frac{- k - 2N - 1}{k+N}) \Gamma (\frac{1-N}{k+N}) \Gamma (\frac{2k+3N}{k+N})^2} \, .\end{aligned}$$ From the leading terms in $z$ expansion, we can read off the conformal weights of the intermediate state. For $\mathcal{F}_1 (z)$ and $\mathcal{F}_2 (z)$, the intermediate states are found to be the identity and the state $(\text{adj};0)$, respectively. Here adj represents the adjoint representation of sl$(N)$, and the conformal weight of the state is $h (\text{adj};0) = (k +2N)/(k+N)$. This is consistent with the decomposition as $\text{f} \otimes \bar{\text{f}} = 1 \oplus \text{adj}$ with $\bar{\text{f}}$ as the anti-fundamental representation of sl$(N)$. As discussed in [@Hikida:2017byl], we only need to consider the W$_N$ vacuum block $\mathcal{F}_1 (z) $ in order to obtain the three point functions in . Therefore, we conclude that these three point functions are fixed by W$_N$ symmetry even with finite $k,N$.
We obtain the three point functions with $1/c$ corrections by slightly modifying the analysis in [@Hikida:2017byl]. We decompose the four point function as $$\begin{aligned}
|z|^{4 h_+} G_{++} (z) = \mathcal{V}_0 (z)+ \sum_{s =3}^\infty (C^{(s)})^2 \mathcal{V}_s (z)
+ \cdots \, ,
\label{cbd}\end{aligned}$$ where $\mathcal{V}_0 (z)$ is the Virasoro vacuum block and $\mathcal{V}_s (z)$ is the Virasoro block of spin $s$ current. The coefficient $C^{(s)}$ is related to the three point function in as $$\begin{aligned}
C^{(s)} = \frac{\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} J^{(s)} \rangle }{\langle J^{(s)}J^{(s)} \rangle^{1/2} } \, .\end{aligned}$$ Since $\mathcal{V}_s (z)$ start to contribute at the order of $1/c$, we expand as $$\begin{aligned}
C^{(s)} = c^{-1/2} \left[ C^{(s)}_0 + c^{-1} C^{(s)}_1 + \mathcal{O} (c^{-2}) \right] \, .\end{aligned}$$ The relevant part of the four point function can be expanded in $z$ and $1/c$ as $$\begin{aligned}
& |z|^{4 h_+} G_{++}(z) \\
& \quad \sim 1 + \frac{1}{c} \sum_{n=1}^\infty (1 - N^2)
\left( - \frac{1}{n} + \frac{N \Gamma(N) \Gamma(n)}{\Gamma(N +n)} \right) z^n + \frac{1}{c^2} \sum_{n = 2}^\infty f_c^{(n)} z^n + \cdots \, ,
\nonumber\end{aligned}$$ where we have defined $$\begin{aligned}
& \frac{f_c^{(n)}}{(1 - N^2)^2} = \frac{1}{n} \sum_{l=1}^{n-1} \frac{1}{l} + \frac{\Gamma(n) \Gamma(N) N^2}{\Gamma(N + n)}
\left( \sum_{l=0}^{n-1} \frac{N}{N + l} - \frac{1}{n} - 2 - \frac{1}{N} + \frac{1}{1+N} \right) \nonumber \\ & \qquad\qquad\qquad\qquad\qquad - \sum_{l=1}^{n-1} \frac{N\Gamma(N)\Gamma(l)}{(n-l) \Gamma(N+l)} + \left(2N + \frac{1}{1 + N}\right) \frac{1}{n} \, .\end{aligned}$$
Solving the constraint equations from , we find $$\begin{aligned}
(C^{(s)}_0)^2 = \frac{(1 - N^2) \Gamma(1+N) \Gamma(s-N)}{\Gamma(1 - N) \Gamma(s + N)} \frac{\Gamma(s)^2}{\Gamma(2s-1)}
\label{3pt0}\end{aligned}$$ for the leading order in $1/c$. The first few examples are $$\begin{aligned}
(C^{(2)}_0)^2 = \frac{1}{2} (1-N)^2 \, , \quad
(C^{(3)}_0)^2 = \frac{1}{6} \frac{(1-N)^2 (2 - N)}{(2 + N)} \, .\end{aligned}$$ The square of the three point function could be negative for $N \geq 3$, and this is related to the fact that we are working in a non-unitary theory.
Examining the equation at the next order in $1/c$, we can obtain $1/c$ corrections to the three point functions as well. At this order, the constraint equations for $s=3,4,5$ are found to be $$\begin{aligned}
&f^{(3)}_c = f^{(2)}_c + 2 C_{0}^{(3)}C_{1}^{(3)} \, , \nonumber
\\
&f^{(4)}_c = f^{(2)}_c\frac{9}{10} + \frac{(1-N)^2}{8(1+N)^2} + \frac{1-N}{10(1+N)^2} + \frac{1}{50(1+N)^2} + 2 C_{0}^{(3)}C_{1}^{(3)}\frac{3}{2} + 2 C_{0}^{(4)}C_{1}^{(4)} \, , \nonumber \\
& f^{(5)}_c = f^{(2)}_c\frac{4}{5} + \frac{(1-N)^2}{4(1+N)^2} + \frac{1-N}{5(1+N)^2} + \frac{1}{25(1+N)^2} + 2 C_{0}^{(3)}C_{1}^{(3)} \frac{12}{7} + 2 C_{0}^{(4)}C_{1}^{(4)}\cdot2 \nonumber \\
& \qquad + 2 C_{0}^{(5)}C_{1}^{(5)} + (C_{0}^{(3)})^2\left[\frac{1}{2}\frac{1-N}{1+N}+\frac{6}{7(1+N)}+\frac{18}{49(1-N^2)}\right] \, . \end{aligned}$$ From these equations, we obtain $$\begin{aligned}
&\frac{C_{1}^{(3)}}{C_{0}^{(3)}} = N^3+3 N^2-3 N-\frac{6}{N+2}+1 \, , \nonumber \\
& \frac{C_{1}^{(4)}}{C_{0}^{(4)}} = N^3+\frac{29 N^2}{4}+\frac{3 N}{2}+\frac{189}{2 (N-3)}-\frac{8}{N-2}+\frac{47}{40 (N-1)}-\frac{3}{10 (N-1)^2} \nonumber \\ & \qquad -\frac{27}{40 (N+1)}-\frac{3}{10 (N+1)^2}-\frac{6}{N+2}-\frac{36}{N+3}+\frac{161}{4} \, , \\
& \frac{C_{1}^{(5)}}{C_{0}^{(5)}} = N^3+\frac{155 N^2}{12}+\frac{29 N}{2}+\frac{800}{N-4}-\frac{180}{N-3}+\frac{25}{7 (N-1)}
-\frac{25}{7 (N+1)}-\frac{6}{N+2} \nonumber \\ & \qquad -\frac{36}{N+3}-\frac{120}{N+4}+\frac{359}{2} \, . \nonumber \end{aligned}$$ In particular, $C_{1}^{(3)}/C_{0}^{(3)} = 224/5$ for $N=3$. It is not difficult to extend the analysis for $C_{1}^{(s)}/C_{0}^{(s)}$ at least up to $s=8$ by directly applying the analysis in [@Hikida:2017ecj].
Preliminaries for bulk computations {#Preliminaries}
===================================
In this section, we explain our prescription to compute the two and three point functions from bulk theory. In the next subsection, we introduce sl$(N)$ Chern-Simons gauge theory and open Wilson lines. In subsection \[slNgen\] we explain the representation of sl$(N)$ generators in terms of $x$-derivatives. In subsection \[Tree\], we compute the two and three point functions in at the leading order in $1/c$. In subsection \[Reg\], we give a prescription to regularize divergences arising from loop diagrams, and prepare for explicit computations for $N=2,3$ in succeeding sections.
Chern-Simons gauge theory and open Wilson lines
-----------------------------------------------
In three dimensions, pure gravity with a negative cosmological constant can be described by $\text{sl}(2) \oplus \text{sl}(2)$ Chern-Simons gauge theory [@Achucarro:1987vz; @Witten:1988hc]. As a natural extension, we can construct a higher spin gauge theory using Chern-Simons theory based on a higher rank gauge algebra [@Blencowe:1988gj]. We are interested in $\text{sl}(N) \oplus \text{sl}(N)$ Chern-Simons theory, whose action is given by $$\begin{aligned}
S = S_\text{CS} [A] - S_\text{CS} [\tilde A] \, , \quad
S_\text{CS} [A] = \frac{\hat k}{4 \pi} \int \text{tr} \left( A \wedge d A + \frac{2}{3} A \wedge A \wedge A \right) \, .\end{aligned}$$ Here $\hat k$ is the level of Chern-Simons theory and $A, \tilde A$ are one forms taking values in $\text{sl}(N)$. The generators of sl$(N)$ can be decomposed in terms of the adjoint action of embedded sl$(2)$ as $$\begin{aligned}
\text{sl} (N) = \text{sl} (2) \oplus \left( \bigoplus_{s=3}^{N} g^{(s)} \right) \, .\end{aligned}$$ Here $g^{(s)}$ denotes the spin $(s-1)$ representation of sl$(2)$, and we have adopted the principal embedding of sl$(2)$. The generators in sl$(2)$ (adjoint representation) and $g^{(s)}$ are denoted as $V^2_n$ $(n=-1,0,1)$ and $V^s_n$ $(n= - s+1,-s+2,\ldots , s-1)$, respectively.
For the application to higher spin AdS$_3$ gravity, we need to assign an asymptotic AdS condition to the gauge fields. We use the metric of Euclidean AdS$_3$ as $ds^2 = d \rho^2 + e^{2\rho} dz d \bar z$, where the boundary is at $\rho \to \infty$. In a gauge choice, we can set $$\begin{aligned}
A = e^{- \rho V_0^{2}} a (z) e^{\rho V_0^{2}} dz + V_0^2 d \rho \, .\end{aligned}$$ We have a similar expression for $\tilde A$ but suppress it here and in the following. The configuration corresponding to AdS$_3$ background is given by $a(z) = V_{1}^2$. The asymptotic AdS condition restricts the form of $a(z)$ as [@Henneaux:2010xg; @Campoleoni:2010zq; @Gaberdiel:2011wb; @Campoleoni:2011hg] $$\begin{aligned}
a (z) = V_1^{2} - \frac{1}{\hat k} \sum_{s \geq 2} ^N \frac{1}{N_s} J^{(s)} (z) V^s_{-s + 1} \, , \quad N_s = \text{tr} (V_{-s +1}^s V_{s-1}^s) \, .
\label{dsgauge}\end{aligned}$$ There are residual gauge symmetries preserving the condition , and a part of them generates W$_N$ symmetry near the AdS boundary. We can define classical Poisson brackets for the reduced phase space. Moreover, we can see that $J^{(s)}(z)$ in generate the W$_N$ symmetry in terms of the Poisson brackets. At the classical level, the relation between the Chern-Simons level $\hat k$ and the central charge $c$ of the dual conformal field theory is given by the Brown-Henneaux one as [@Brown:1986nw] $$\begin{aligned}
c = 6 \hat k \, .
\label{bhclassical}\end{aligned}$$ See [@Henneaux:2010xg; @Campoleoni:2010zq; @Gaberdiel:2011wb; @Campoleoni:2011hg] for more details.
At the leading order in $1/c$, the rules for computing conformal blocks from the Chern-Simons theory with open Wilson lines were given in [@Besken:2016ooo], see also [@Bhatta:2016hpz] for $N=2$. For the two and three point functions in , we use $$\begin{aligned}
\langle \text{lw} | W (z_2 ; z_1) | \text{hw} \rangle \, , \quad
W(z_2 ; z_1) = P \exp \left(\int _{z_1}^{z_2} dz a(z) \right) \, .
\label{classicalwilson}\end{aligned}$$ Here hw and lw denote the highest and lowest weight states in finite dimensional representations of sl$(N)$, respectively, and $P$ represents the path ordering. Moreover, we remove the $\rho$-dependence in the gauge field as $A(z) = a(z)$ using a gauge transformation. We include $1/c$ corrections by extending the analysis in [@Fitzpatrick:2016mtp; @Besken:2017fsj] for $N=2$. At the leading order in $1/c$, we treat the coefficient $J^{(s)}(z)$ in as a function of $z$. At higher orders in $1/c$, we regard $J^{(s)}(z)$ as an operator, and the expectation values of open Wilson lines are evaluated by using the correlators of $J^{(s)}(z)$, which are uniquely fixed by the W$_N$ symmetry.
Generators of $\text{sl}(N)$ algebra {#slNgen}
------------------------------------
In this subsection we explain our prescription to compute the matrix elements of sl$(N)$ algebra for evaluating the expectation values of open Wilson lines as in . We start with the simplest case with $N=2$ and then extend the argument for generic $N$. For $N=2$, there are several previous works in [@Verlinde:1989ua; @Fitzpatrick:2016mtp; @Besken:2017fsj], and we start by clarifying the representation with $x$-derivatives in [@Fitzpatrick:2016mtp].
For two point functions we evaluate $$\begin{aligned}
\langle j , - j | W_{-j} (z_2 ; z_1) | j , j \rangle \, ,
\label{Gjz1z2}\end{aligned}$$ where $| j , m \rangle$ belongs to the spin $j$ representation of sl(2) with $m= -j , - j+1 , \ldots ,j$. We set the norm of these states as $$\begin{aligned}
\langle j , m| j , m' \rangle = \delta_{m,m'} \, .\end{aligned}$$ With these states, the sl(2) generators in the Wilson line are described by $(2j+1) \times (2j+1)$ matrices.
As in [@Verlinde:1989ua; @Fitzpatrick:2016mtp], it would be convenient to map the expression as $$\begin{aligned}
\langle j , - j | W_{-j} (z_2 ; z_1) | j , j \rangle =
\int dx
\langle j , - j |x \rangle W_{-j} (z_2 ; z_1) \langle x | j , j \rangle \, ,\end{aligned}$$ then the sl$(2)$ generators can be written as $$\begin{aligned}
J_+ (= V_{-1}^2)= x^2 \partial_x - 2 j x \, , \quad
J_3 (= - V_{0}^2)= - x \partial_x + j \, , \quad
J_- (= V_{+1}^2)= \partial_x \, .
\label{xsl2}\end{aligned}$$ In [@Fitzpatrick:2016mtp], they proposed that the wave functions are given by $$\begin{aligned}
\langle x | j , j \rangle = x^{2j} \, , \quad \langle j , - j |x \rangle = \delta (x) \, .
\label{basisx}\end{aligned}$$ We would like to give a derivation such that it can be extended for generic $N$. It is easy to obtain $\langle x | j , j \rangle = x^{2j}$ as a solution to the equation $J_+ | j , j \rangle = 0$. The others follow as $$\begin{aligned}
\langle x |j , m \rangle \propto (J_-)^{j-m} \langle x | j , j \rangle = \frac{\Gamma(2j+1)}{\Gamma(j+m+1)} x^{j + m} \, .\end{aligned}$$ The dual states $\langle j , m' |x \rangle $ should satisfy $$\begin{aligned}
\int dx
\langle j , m' |x \rangle \langle x | j , m \rangle = \delta_{m,m'} \, ,\end{aligned}$$ which leads to $$\begin{aligned}
\langle j , m' |x \rangle \propto \partial_x^{j + m'} \delta (x) \, .\end{aligned}$$ In particular, we have $\langle j , - j |x \rangle = \delta (x) $ as in . The normalization is set to be a convenient value.
We then apply the analysis to the case with generic $N$. A way to represent the generators of $\text{sl}(N)$ is using $N \times N$ matrices, and sl(2) generators $V^{2}_{n}$ $(n=-1,0,1)$ can be embedded as described, e.g., in appendix A of [@Castro:2011iw]. Then the other generators may be obtained as $$\begin{aligned}
V^{s}_n = (-1)^{s-1-n} \frac{(n+s-1)!}{(2s - 2)!} [V_{-1}^2 [V_{-1}^2 , ... , [ V_{-1}^2 , (V_{1}^2)^{s-1} ]]] \, ,
\label{slNgenerators}\end{aligned}$$ where $(s - n - 1)$ of $V_{-1}^2$ are inserted. The fundamental representation of sl$(N)$ can be described by an $N$ dimensional vector, which behaves as a spin $(N-1)/2$ representation under the action of the embedded sl$(2)$. Therefore, the description with $N \times N$ matrices can be given by with $j= (N-1)/2$ and open Wilson lines based on sl$(N)$ algebra. In this specific case, we can map the matrix representation to the one with $x$-derivatives using and . In the representation with $x$-derivatives, the generators of sl$(N)$ should be given by [@Bergshoeff:1991dz] $$\begin{aligned}
V^s_n = \sum_{i=0}^{s-1} (n - s + 1)_{s - 1 - i} a^{i} (s , h_0) x^{- n+i} \partial_x^{i} \, ,
\label{generators0}\end{aligned}$$ where $$\begin{aligned}
\quad a^i (s , h_0) = \binom{s-1}{i} \frac{(- 2 h_0 - s + 2)_{s - 1 - i}}{(s + i)_{s - 1 -i}}
\label{generators}\end{aligned}$$ with $h_0 = - j = (1 - N)/2$. The wave functions are precisely those in . The generators with are those of higher spin algebra hs$[\lambda]$ for $h_0 = (1 + \lambda)/2 $, and sl$(N)$ can be realized by hs$[-N]/\chi_N$ with $\chi_N$ as an ideal, which removes generators with $s > N$.
With the realization of generators, $N_s$ in are computed as $$\begin{aligned}
N_s
= \frac{3 \sqrt{\pi } \Gamma (s) (1-N)_{s-1} (N+1)_{s-1}}{ 2^{2s-2} \left(N^2-1\right) \Gamma \left(s+\frac{1}{2}\right)} \, ,\end{aligned}$$ where the first few expressions are $$\begin{aligned}
N_2 = -1 \, , \quad N_3 = \frac{1}{5} (N^2 - 4) \, ,\quad N_4 = - \frac{3}{70} (N^2 - 4) (N^2 - 9) \, . \end{aligned}$$ In particular, we have $N_3 = 1$ for $N=3$.
Correlators at the leading order in $1/c$ {#Tree}
-----------------------------------------
In order to compute the correlators in , we need to consider the expectation values of open Wilson lines with $| \text{hw} \rangle$ corresponding to the highest weight in the fundamental representation of sl$(N)$. As explained above, they can be expressed for $(z_1,z_2) = (0,z)$ as $$\begin{aligned}
W_{h_0} (z) &= \int dx \delta (x) P \exp \left[ \int^{z}_{0} d z ' \left( V_1^{2} - \frac{1}{\hat k} \sum_{s = 2}^N \frac{1}{N_s} J^{(s)} (z ') V^s_{-s + 1} \right) \right] \frac{1}{x^{2h_0}} \nonumber \\
&= \left. P \exp \left[ \int^{z}_{0} d z ' \left( V_1^{2} - \frac{1}{\hat k} \sum_{s = 2}^N \frac{1}{N_s} J^{(s)} (z ') V^s_{-s + 1} \right) \right] \frac{1}{x^{2h_0}} \right |_{x=0}
\label{Wilson}\end{aligned}$$ with $h_0= (1-N)/2$. Here the $\text{sl}(N)$ generators are written in terms of $x$-derivatives as in . We would like to treat them perturbatively in $1/\hat k$ (or $1/c$). Following the analysis in [@Besken:2017fsj], we compute $$\begin{aligned}
\frac{d}{dz} \left[e^{-z \partial_x } W_{h_0} (z) \right]
= \left( - \frac{1}{\hat k} \sum_{s = 2}^N \frac{1}{N_s} J^{(s)} (z) e^{- z\partial_x } V^s_{-s + 1} e^{ z \partial_x } \right) \left[e^{- z\partial_x } W_{h_0} (z) \right] \, .\end{aligned}$$ Integrating over $z$, we find $$\begin{aligned}
\label{Wilson1/c}
W_{h_0} (z) & = \sum_{n=0}^\infty \left( - \frac{1}{\hat k} \right )^n
\int_0^z dz_n \cdots \int_0^{z_2} dz_1 \sum_{s_j = 2}^N
\left[ \prod_{j=1}^n \frac{1 }{N_{s_j}} J^{(s_j)} (z_j) \right]
f_n^{(s_n,\ldots , s_1)} (z_n ,\ldots , z_1) \, , \end{aligned}$$ where $$\begin{aligned}
\label{fnss}
&f_n^{(s_n,\ldots , s_1)} (z_n ,\ldots , z_1) \\
& \qquad = \left.
\prod_{j=1}^n \left[
\sum_{i=0}^{s_j-1} ( - 2 s_j + 2)_{s_j - 1 - i} a^{i} (s_j , h_0) (x + z - z_j)^{s_j - 1+i} \partial_x^{i} \right] \frac{1}{(x + z)^{2h_0}} \right |_{x=0} \, , \nonumber \end{aligned}$$ see (3.3) of [@Fitzpatrick:2016mtp] for $N=2$.
According to the current prescription, the two point function of $\mathcal{O}_{h_+}$ in should be computed as $$\begin{aligned}
\langle \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) \rangle = \langle W_{h_0} (z) \rangle \, ,
\label{2ptWilson}\end{aligned}$$ where $\langle W_{h_0} (z) \rangle$ is evaluated by the correlators of $J^{(s)}$ in the W$_N$ theory. The leading order expansion in $1/\hat k$ leads to $$\begin{aligned}
\left. \langle \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) \rangle \right|_{\mathcal{O} (c^0) }= \left. \langle W_{h_0} (z) \rangle \right|_{\mathcal{O} (c^0) }=
\frac{1}{z^{2h_0}}\end{aligned}$$ as expected.
We are also interested in the three point functions in , which should be obtained as $$\begin{aligned}
\langle \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) J^{(s)} (y) \rangle =
\langle W_{h_0} (z) J^{(s)} (y) \rangle \, .
\label{3ptWilson}\end{aligned}$$ The first non-trivial contributions come from the terms of order $1/\hat k$ . At this order, we need to compute $$\begin{aligned}
\left. \langle W_{h_0} (z) J^{(s)} (y) \rangle \right|_{\mathcal{O} (c^{0}) } &= - \frac{1}{\hat k N_{s}}
\int_0^z dz_1 f^{(s)}_1 (z_1) \langle J^{(s)} (z_1) J^{(s)} (y) \rangle \\
& = - \frac{1}{\hat k N_{s}}
\int_0^z dz_1 \frac{\Gamma (2h _0+ s - 1)}{\Gamma (2 h_0)}\frac{ (z - z_1)^{s-1} z_1^{s-1}}{ z^{ s -1 + 2h_0}} \langle J^{(s)} (z_1) J^{(s)} (y) \rangle \, . \nonumber\end{aligned}$$ The normalization of higher spin currents in corresponds to (see, e.g., [@Ammon:2011ua]) $$\begin{aligned}
\left. \langle J^{(s)} (z_1) J^{(s)} (z_2) \rangle \right|_{\mathcal{O} (c) } = - (2 s -1) \hat k N_s \frac{1}{z_{12}^{2s}} \, .
\label{2ptnorm}\end{aligned}$$ Using $$\begin{aligned}
\int_0^z dz_1 \frac{(z - z_1)^{s-1} z_1^{s-1}}{(z_1 - y)^{2s} }
= \frac{z^{2s -1}}{( y- z)^s y^s} \frac{(\Gamma(s))^2}{\Gamma (2s)} \, ,\end{aligned}$$ we find $$\begin{aligned}
\left. \langle W_{h_0} (z) J^{(s)} (y) \rangle \right|_{\mathcal{O} (c^{0}) } = \frac{\Gamma (2h_0 + s - 1)}{\Gamma (2 h_0)}
\frac{(\Gamma(s))^2}{\Gamma (2s - 1)}
\left( \frac{z}{(y-z) y} \right)^s \left. \langle W_{h_0} (z) \rangle \right|_{\mathcal{O} (c^0) } \, . \label{3pttree}\end{aligned}$$ The result is consistent with in the convention of . In fact, it is the same as eq. (1.3) of [@Ammon:2011ua] up to a factor if we set $h_0 = (1 + \lambda)/2$ (or $N = - \lambda$), and this is related to the triality relation discussed in [@Gaberdiel:2012ku].
Prescription for regularization {#Reg}
-------------------------------
The $1/c$ corrections of the two and three point functions in can be evaluated from higher order contributions in using the Wilson line method. However, integrals over $z_j$ diverge when two (or more) currents $J(z_i)$ collide. Therefore, we need to decide how to deal with these divergences, and we explain our prescription in this subsection.
Let us start with the correlators of higher spin currents, which are uniquely fixed by the W$_N$ symmetry in terms of central charge $c$. In particular, we use the two point functions $$\begin{aligned}
\langle J^{(s)} (z_2) J^{(s)} (z_1) \rangle = - \frac{ (2 s -1) c N_s}{6} \frac{1}{z_{21}^{2s}} \, ,
\label{2ptnormc}\end{aligned}$$ which reduce to if we use the relation $c=6 \hat k$ in . At finite $\hat k$, the relation of should be modified, and corrections to higher spin propagators are automatically included by expanding in $1/c$ instead of $1/\hat k$, see [@Besken:2017fsj] for some arguments. Divergence would arise at the coincident point $z_2 = z_1$, and we need to decide how to regularize it. We introduce a regulator as $$\begin{aligned}
\langle J^{(s)} (z_2) J^{(s)} (z_1) \rangle = - \frac{ (2 s -1) c N_s}{6} \frac{1}{z_{21}^{2s - 2 \epsilon}}
\label{2ptnormreg}\end{aligned}$$ by shifting the conformal weight of the higher spin current as $s \to s - \epsilon$. This choice is reasonable since it does not break the scaling symmetry. Analogously, we introduce the regulator $\epsilon$ to other correlators of higher spin currents $J^{(s)}$ by shifting the conformal wights of the current.
Introducing the regulator $\epsilon$, integrals over $z_j$ become finite but have terms diverging at $\epsilon \to 0$. In the usual quantum field theory with a renormalizable Lagrangian, we can remove divergences by renormalizing the overall normalization of quantum fields and the parameters of interactions. In the current case, we offer to remove divergences in a similar manner. We first use the fact that the normalization of a two point function can be chosen arbitrarily by the redefinition of the operator. We remove a kind of divergence by changing the overall factor of the open Wilson line such that the corresponding two point function becomes the normalized one as in . We then notice that the three point interactions between two scalars and a higher spin field are governed by the coefficients in front of $J^{(s)} (z)$ in . We introduce parameters $c_s$ such that becomes $$\begin{aligned}
W_{h_0} (z) = \left. P \exp \left[ \int^{z}_{0} d z ' ( V_1^{2} - \frac{6}{c} \sum_{s =2}^N \frac{c_s}{N_s} J^{(s)} (z ') V^s_{-s + 1} ) \right] \frac{1}{x^{2h_0}} \right |_{x=0} \, .
\label{Wilsonreg}\end{aligned}$$ In terms of $1/c$ expansion, is changed as $$\begin{aligned}
\label{Wilson1/creg}
W_{h_0} (z) & = \sum_{n=0}^\infty \left( - \frac{6}{c} \right )^n
\int_0^z dz_n \cdots \int_0^{z_2} dz_1 \sum_{s_j = 2}^N
\left[ \prod_{j=1}^n \frac{c_{s_j}}{N_{s_j}} J^{(s_j)} (z_j) \right]
f_n^{(s_n,\ldots , s_1)} (z_n ,\ldots , z_1) \, , \end{aligned}$$ where $f_n^{(s_n,\ldots , s_1)} (z_n ,\ldots , z_1) $ are given by . At the leading order in $1/c$, $c = 6 \hat k$ as in and $c_s = 1$. From the next order in $1/c$, we shift the values of $c_s$ to remove divergences. Namely, we expand $c_s$ in $1/c$ as $$\begin{aligned}
c_s = 1 + \frac{1}{c} c_s^{(1)} + \frac{1}{c^2} c_s^{(2)} + \mathcal{O} (c^{-2}) \, ,
\label{csexp}\end{aligned}$$ and absorb divergences in $c_s^{(i)}$ $(i=1,2,\ldots)$ order by order. We conjecture that all divergences can be removed by these two ways of renormalization.
As explained above, we determine to remove divergences by properly choosing the “bare” values of parameters $c_s$. However, we have still freedom to choose the terms independent of $\epsilon$. Here we fix them such that the three point functions $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} J^{(s)} \rangle$ in are reproduced from the Wilson line method as in . Since the three point functions can be fixed by the W$_N$ symmetry as shown in the previous section, we would say that the regularization scheme is determined by making use of the boundary symmetry. This is expected to fix all the ambiguities left, and other physical quantities should be predictable. In the following two sections, we examine concrete examples with $N=2,3$ and show that the $1/c$ corrections in the conformal dimensions of scalar operators can be reproduced from the bulk viewpoint up to the two loop level applying the prescription described above.
Correlators for $N=2$ {#Correlators2}
=====================
In this and the next section, we explicitly evaluate the loop corrections of the correlators in terms of open Wilson lines. We start with the simpler case with $N=2$ and then move to a more involved one with $N=3$. For $N=2$, we can work with generic $h_0 = -j$, because the sl(2) generators in terms of $x$-derivatives as in are available for the generic case as argued in subsection \[slNgen\].
Two and three point functions with generic $h_0 $ are obtained from analysis of conformal field theory as follows. For $h_0 = - j$, the $1/c$ correction of conformal weight is given as with $$\begin{aligned}
h_1 = - 6 h_0 (h_0-1) \, , \quad h_2 = - 78 h_0 (h_0-1) \, ,
\label{dimcorr}\end{aligned}$$ see, e.g., [@Besken:2017fsj]. The $1/c$ expansion of the two point function is then . In the next subsection, we examine the two point function at the next leading order in $1/c$. We reproduce the order $1/c$ result as $h_1$ in , and remove a divergence by renormalizing the overall factor of the open Wilson line. The three point function is fixed by the conformal Ward identity as $$\begin{aligned}
\langle \mathcal{O}_h (z) \bar{\mathcal{O}}_h (0) J^{(2)} (y) \rangle =\left[ h_0 + \frac1c h_1 \right] \left( \frac{z}{(y - z) y} \right)^2 \langle \mathcal{O}_{h} (z) \bar{\mathcal{O}}_{h} (0) \rangle + \mathcal{O}(c^{-2})
\label{Ward}\end{aligned}$$ in the current convention of $J^{(2)}$ given by . The $c^0$ order term follows from . In subsection \[3ptN2\], we fix the parameter $c_2$ introduced in such that the $1/c$ order term is reproduced. In particular, this removes another type of divergence. With the regularization scheme, we reproduce the order $1/c^2$ term as $h_2$ in from two point function at the two loop order in subsection \[2ptN22\].
Two point function at $1/c$ order {#2ptN21}
---------------------------------
For the two point function of $\mathcal{O}_h$, we need to evaluate the expectation value of the open Wilson line $W_{h_0} (z)$ as in . With $N=2$, the $1/c$ expansion of the open Wilson line in becomes $$\begin{aligned}
W_{h_0} (z) = \frac{1}{z^{2h_0}} + \sum_{n =1} \left( \frac{6 c_2}{c} \right)^n W^{(n)}_{h_0} (z)
\label{Wexp}\end{aligned}$$ with $$\begin{aligned}
&W^{(1)}_{h_0} (z) = \int_0^z d z_1 f_1^{(2)} ( z_1) J^{(2)} (z_1) \, , \nonumber \\
&W^{(2)}_{h_0} (z) = \int_0^z d z_2 \int_0^{z_2} dz_1 f_2^{(2,2)} (z_2,z_1)
J^{(2)} (z_2) J^{(2)} (z_1) \, , \label{Wexpe} \\
&W^{(3)}_{h_0} (z) = \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3^{(2,2,2)} ( z_3,z_2 ,z_1) J^{(2)} (z_3)J^{(2)} (z_2)J^{(2)} (z_1) \, , \nonumber \\
&W^{(4)}_{h_0} (z) = \int_0^z d z_4 \cdots \int_0^{z_2} dz_1 f_4^{(2,2,2,2)} ( z_4,z_3 ,z_2,z_1) J^{(2)} (z_4)J^{(2)} (z_3) J^{(2)} (z_2) J^{(2)} (z_1) \, , \nonumber \end{aligned}$$ and so on. Here $f_n^{(2,\ldots,2)} (z_n , \ldots , z_1)$ are defined in . Since the one point function vanishes as $\langle J^{(2)} (z) \rangle = 0$, the non-trivial contribution starts from $\langle W^{(2)}_{h_0} (z) \rangle$. The contribution corresponds to the one loop correction in the two point function of $\mathcal{O}_h$ as in figure \[Wilson1\].
![Diagram contributing to the $1/c$ order correction of $\langle \mathcal{O}_h \bar{\mathcal{O}}_h \rangle$ for $N=2$. The straight line and the wavy line represent the open Wilson line and the propagator of spin two current.[]{data-label="Wilson1"}](Wilson1.pdf)
The integrals in $\langle W^{(2)}_{h_0} (z) \rangle$ over $z_1,z_2$ diverge, and we introduce a regulator $\epsilon$ as in , i.e., $$\begin{aligned}
\langle J^{(2)} (z_2) J^{(2)} (z_1) \rangle = \frac{c/2}{z_{21}^{4 - 2 \epsilon}} \end{aligned}$$ for spin two current. With the regulator, we obtain a finite result after the integration over $z_1,z_2$ as $$\begin{aligned}
& \langle W^{(2)}_{h_0} (z) \rangle = \int_0^z d z_2 \int_0^{z_2} dz_1 f_2 ^{(2,2)} (z_2,z_1)
\langle J^{(2)} (z_2) J^{(2)} (z_1) \rangle \nonumber \\
& \qquad = \frac{c}{2 z^{2h_0}} \left[ \frac{(h_0-1) h_0 }{3 \epsilon}+\frac{1}{9} h_0 \left(6 (h_0-1) \log \left(z\right)+5 h_0-2\right) \right] + \mathcal{O} (\epsilon) \, .
\label{W2}\end{aligned}$$ Using and $c_2 = 1 + \mathcal{O}(c^{-1})$, the above expression leads to $$\begin{aligned}
\langle W_{h_0} (z) \rangle = \frac{1}{z^{2h_0}} \left[ 1 + \frac{1}{c}\left(\frac{6(h_0-1) h_0 }{ \epsilon} + \left(12 h_0 (h_0 -1) \log \left(z\right)+2 h_0 (5 h_0 -2 ) \right) \right)\right]
\label{2pt1}\end{aligned}$$ up to the terms of order $\epsilon^0$ and $1/c$.
We compare the above expression in with the $1/c$ expansion of two point function in . We can see that the $\log (z)$ term correctly explains $h_1 = - 6 h_0 (h_0-1)$ in as shown in [@Besken:2017fsj]. The expression in has a term proportional to $1/\epsilon$, which diverges for $\epsilon \to 0$. We can remove the divergence by changing the overall factor of the open Wilson line as $$\begin{aligned}
\tilde W_{h_0} (z) = \left[ 1 - \frac{1}{c}\left(\frac{6(h_0-1) h_0 }{ \epsilon} + 2 h_0 (5 h_0 -2 ) \right)\right] W_{h_0} (z) \, .
\label{overall}\end{aligned}$$ With the normalization, we have $$\begin{aligned}
\langle \tilde W_{h_0} (z) \rangle = \frac{1}{z^{2 (h_0 + h_1/c)}} + \mathcal{O} (c^{-2}) \end{aligned}$$ for $\epsilon \to 0$. In other words, we choose the $\epsilon$-independent part such that the corresponding two point function has unit normalization as in .
Three point function {#3ptN2}
--------------------
We have proposed that three point functions can be computed with open Wilson lines as in and reproduced the tree level results as in . In this subsection, we examine the next leading order in $1/c$. There are two types of contribution at the order as in figure \[Wilson2\]
![Diagrams contributing to the $1/c$ order correction of $\langle \mathcal{O}_h \bar{\mathcal{O}}_h J^{(2)} \rangle$ for $N=2$.[]{data-label="Wilson2"}](Wilson2.pdf)
and we would like to examine them in turn.
The first one is from $$\begin{aligned}
\langle W_{h_0}^{(2)} (z) J^{(2)} (y) \rangle = \int_0^z d z_2 \int_0^{z_2} dz_1 f_2^{(2,2)} (z_2,z_1)
\langle J^{(2)} (z_2) J^{(2)} (z_1) J^{(2)} (y) \rangle \, ,\end{aligned}$$ which is represented as diagram (a) in figure \[Wilson2\]. Here we need to introduce the regulator $\epsilon$ to the three point function of spin two current. Our prescription is to shift the conformal weight from $2$ to $2 - \epsilon$, so we use $$\begin{aligned}
\langle J^{(2)} (z_2) J^{(2)} (z_1) J^{(2)} (y) \rangle =
\frac{c}{z_{21}^{2- \epsilon}(z_2 - y )^{2- \epsilon}(z_1 - y )^{2- \epsilon} } \, .\end{aligned}$$ The integral becomes simpler by taking $y \to - \infty$ as $$\begin{aligned}
\lim_{y \to - \infty} |y|^{4 -2 \epsilon } \langle W_{h_0}^{(2)} (z) J^{(2)} (y) \rangle
= - \frac{c h_0}{z^{2h_0 - 2}} \left[ \frac{1 }{3 \epsilon} + \frac{1}{36} (18 h_0 +12 \log (z)-13) \right]
\label{spin2first}\end{aligned}$$ up to the term of order $\epsilon^0$.
The second one is from $$\begin{aligned}
&\langle W^{(3)}_{h_0} (z) J^{(2)} (y) \rangle \nonumber
\\ & = \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3 ^{(2,2,2)} ( z_3,z_2 ,z_1) \langle J^{(2)} (z_3)J^{(2)} (z_2)J^{(2)} (z_1) J^{(2)} (y) \rangle \, .\end{aligned}$$ At the leading order in $1/c$, the four point function is given by a sum over the products of the two point function as $$\begin{aligned}
& \langle J^{(2)} (z_3)J^{(2)} (z_2)J^{(2)} (z_1) J^{(2)} (y) \rangle \nonumber \\ &= \frac{c^2/4}{z_{32}^{4 -2 \epsilon}(z_1 - y )^{4- 2 \epsilon }} +\frac{c^2/4}{z_{31} ^{4 - 2 \epsilon} (z_2 - y )^{4- 2 \epsilon } } + \frac{c^2/4}{z_{21}^{4- 2\epsilon} (z_3 - y )^{4- 2 \epsilon} }
+ \mathcal{O}(c)\, . \end{aligned}$$ Denoting $$\begin{aligned}
H^{(3)}_{ij} (z) =z^{2h_0 - 2}
\int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3 ^{(2,2,2)} ( z_3,z_2 ,z_1)
\frac{1}{z_{ji} ^{4 - 2 \epsilon}} \, ,\end{aligned}$$ we find $$\begin{aligned}
H^{(3)}_{12} (z)& =H^{(3)}_{23} (z)= \frac{(h_0-1) h_0^2 }{9 \epsilon}+\frac{2}{45} h_0 (5 (h_0-1) h_0 \log (z)+(h_0-2) (h_0+1)) \, , \\
H^{(3)}_{13} (z)& = -\frac{h_0 ((h_0-1) h_0-1) }{9 \epsilon}-\frac{1}{135} h_0 (30 ((h_0-1) h_0-1) \log (z)-h_0 (13 h_0+32)+1) \nonumber \end{aligned}$$ up to the terms of $\mathcal{O} (\epsilon^0)$. The integrals $H^{(3)}_{12}(z)$, $H^{(3)}_{13}(z)$, $H^{(3)}_{23}(z)$ correspond to the diagrams (b), (c), (d) in figure \[Wilson2\], respectively.
Combining the results so far, we find $$\begin{aligned}
&\lim_{y \to - \infty} |y|^{4 - 2 \epsilon } \langle W_{h_0} (z) J^{(2)} (y)\rangle
\nonumber \\
&= \frac{1}{z^{2h_0-2}} \left[
h_0 + \frac{h_0}{c} \left(\frac{6 (h_0(h_0-1)-1) }{\epsilon} + 10 h_0 (h_0-1) + 3 + 12 (h_0-1) h_0 \log ( z) \right) \right] + \cdots \nonumber \\
&= z^2 \left[ h_0 - \frac{6 h_0}{c} \left( \frac{1}{\epsilon} + h_0 - \frac12 \right) \right] \langle W_{h_0} (z) \rangle + \cdots \, .\end{aligned}$$ The expression diverges for $\epsilon \to 0$, and we remove the divergence by properly choosing $c_2^{(1)}$ in as $$\begin{aligned}
c_2 = 1 + \frac{6}{c} \left( \frac{1}{\epsilon} + a \right) + \mathcal{O} (c^{-2}) \, .\end{aligned}$$ Here $a$ is an arbitrary constant, which shall be fixed shortly. With this choice of the parameter $c_2$, there arises a contribution of order $1/c$ from the following term as $$\begin{aligned}
\left. \frac{6 c_2}{c} \lim_{y \to - \infty} |y|^{4 - 2 \epsilon } \langle W_{h_0}^{(1)} (z) J^{(2)} (y)\rangle
\right|_{\mathcal{O}(c^{-1})} = \frac{ h_0 }{z^{2h_0 -2}}\frac{6}{c} \left( \frac{1}{\epsilon} + a \right) \end{aligned}$$ up to the terms of order $\epsilon^0$. Here $W_{h_0}^{(1)} (z)$ is given in . With this prescription, we have $$\begin{aligned}
&\lim_{y \to - \infty} |y|^{4 - 2 \epsilon} \langle W_{h_0} (z) J^{(2)} (y)\rangle \nonumber \\
& \qquad = z^2 \left[ h_0 - \frac{6 h_0}{c} \left( h_0 - \frac12 - a \right) \right] \langle W_{h_0} (z) \rangle
+ \mathcal{O}(c^{-2})\end{aligned}$$ for $\epsilon \to 0$. Therefore, setting $a = 1/2$, we reproduce the expected result as with $h_1 = - 6 h_0 (h_0 -1)$ in . In summary, we choose the parameter $c_2$ in as $$\begin{aligned}
c_2 = 1 + \frac{1}{c} \left( \frac{6}{\epsilon} + 3 \right) + \mathcal{O} (c^{-2})
\label{level2c}\end{aligned}$$ in order to absorb a divergence from the one loop diagram and also reproduce the result from the conformal Ward identity.
Two point function at $1/c^2$ order {#2ptN22}
-----------------------------------
In the previous subsections we have regularized divergences arising up to the one loop order. Our claim is that other quantities are predictable after the renormalization. Here we would like to examine the two point function at the two loop order. Generically two loop diagrams have one loop sub-diagrams, and there would appear non-local divergences from the sub-diagrams. After all one loop divergences are removed by renormalization procedure, we should have no non-local divergences at the two loop order. There would be local divergences remaining, which can be renormalized as for the one loop computations. As discussed in [@Besken:2017fsj], two point function without proper renormalization does not reproduce the correct dependence on $\log (z)$ and $\log^2 (z)$ at the two loop order because of non-local divergences as $1/\epsilon \log (z)$. Since now it is not expected to have such divergences after the renormalization, it should be possible to reproduce the correct shift of conformal weight even at the $1/c^2$ order. We shall show that this is indeed the case in this subsection.
We first evaluate the expectation value of the open Wilson line at the $1/c^2$ order without renormalization, then we consider its effects. A contribution comes from $ \langle W_{h_0}^{(3)} (z) \rangle$ in as $$\begin{aligned}
G^{(2)}_{123} (z) = & \left( \frac{6}{c}\right)^3 \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3^{(2,2,2)} ( z_3,z_2 ,z_1) \frac{c}{z_{32}^{2- \epsilon}z_{31}^{2-\epsilon} z_{21}^{2- \epsilon} } \, ,\end{aligned}$$ which is expressed as diagram (a) in figure \[Wilson3\].
![Diagrams contributing to the $1/c^2$ order correction of $\langle \mathcal{O}_h \bar{\mathcal{O}}_h \rangle$ for $N=2$.[]{data-label="Wilson3"}](Wilson3.pdf)
The integral is computed as $$\begin{aligned}
c^2 z^{2h_0}G^{(2)}_{123} (z) = & -\frac{ 288 h_0 (h_0-1) \log (z)}{\epsilon} \nonumber \\ & +2 h_0 \left(36 \log (z) \left(-6 (h_0-1) \log (z)-2 h_0^2-9 h_0+5\right)\right) \, .\end{aligned}$$ Here we neglect the terms of $\mathcal{O}(\epsilon)$ and write down only the terms depending on $\log (z)$ or $\log ^2 (z)$. In the rest of this subsection, we include only such terms. Another type of contribution arises from $ \langle W_{h_0}^{(4)} (z) \rangle$ in . Defining $$\begin{aligned}
&G^{(2)}_{ij;kl} (z) = \left( \frac{6}{c}\right)^4 \int_0^z d z_4 \int_0^{z_4} d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_4^{(2,2,2,2)} ( z_4,z_3,z_2,z_1) \frac{c^2/4}{z_{lk} ^{4- 2 \epsilon} z_{ji}^{4- 2 \epsilon}} \, ,\end{aligned}$$ we find $$\begin{aligned}
&c^2 z^{ 2 h_0} G^{(2)}_{12;34} (z) =\frac{144 h_0^2 (h_0-1)^2 \log (z)}{\epsilon} \nonumber \\ & \quad - 96 (h_0-1) h_0 ^2 \log (z) (-3 (h_0-1) \log (z)-5 h_0+2) \, , \nonumber \\
&c^2 z^{2 h_0} G^{(2)}_{14;23} (z) = \frac{ 360 (h_0-1)^2 h_0^2 \log (z)}{5 \epsilon} \nonumber \\ & \quad +\frac{72}{5} h_0 (h_0-1) \log (z) (10 (h_0-1) h_0 \log (z)+h_0 (23 h_0-43)-16) \, , \\
&c^2 z^{2 h_0} G^{(2)}_{13;24} (z) = - \frac{ 720 h_0 \left((h_0-2) h_0^2+1\right) \log (z)}{5 \epsilon} \nonumber \\ & \quad +\frac{12}{5} h_0 \log (z) \left(-120 \left((h_0-2) h^2_0 +1\right) \log (z)+h_0 \left(-238 h_0^2+596 h_0+3\right)-241\right) \, . \nonumber \end{aligned}$$ These integrals correspond to diagrams (b), (c), (d) in figure \[Wilson3\], respectively. Summing over all contributions we find $$\begin{aligned}
\label{G2z}
&c^2 z^{2h_0} G^{(2)}_{h_0} (z) = \frac{ 72 h_0 (h_0-2) (h_0-1) (h_0+1) \log (z) }{\epsilon} \nonumber \\ & \quad +12 h_0 \log (z) \left(12 \left((h_0-2) h_0^2+1\right) \log (z) +h_0 (4 h_0 (5 h_0-7)-5)+1\right) \, . \end{aligned}$$ Therefore, a non-locally divergent term as $1/\epsilon \log (z)$ remains, and the expression cannot be compared with .
Now we include the effects of renormalization, namely, the change of overall factor as in and the shift of parameter $c_2$ as in . These effects lead to an extra contribution as $$\begin{aligned}
\langle \tilde W_{h_0} (z) \rangle
&= \left[ 1 - \frac{1}{c}\left(\frac{6(h_0-1) h_0 }{ \epsilon} + 2 h_0 (5 h_0 -2 ) \right) \right] \nonumber \\
&\times
\left[ \frac{1}{z^{2h_0}} + \left(\frac{6 }{c} \right)^2 \left( 1 + \frac{1}{c} \left( \frac{6}{\epsilon} + 3 \right) \right)^2
\langle W^{(2)}_{h_0} (z) \rangle + G^{(2)}_{h_0} (z) \right] + \cdots \nonumber \\
&= \frac{1}{z^{2h_0}} + \left(\frac{6 }{c} \right)^2\langle W^{(2)}_{h_0} (z) \rangle + G^{(2)}_{h_0} (z) + \tilde G^{(2)}_{h_0} (z) + \cdots \, , \end{aligned}$$ where $$\begin{aligned}
\tilde G^{(2)}_{h_0} (z) = \frac{1}{c}\left[ 2 \left( \frac{6}{\epsilon} + 3 \right) - \frac{6 h_0 (h_0 -1)}{\epsilon} - 2 h_0 (5h_0 -2) \right] \left( \frac{6}{c} \right)^2 \langle W^{(2)}_{h_0} (z) \rangle \, .
\label{extra}\end{aligned}$$ The extra contribution can be evaluated as $$\begin{aligned}
& c^2 z^{2h_0} \tilde G^{(2)}_{h_0} (z) =
\frac{- 72 h_0 (h_0-2) (h_0-1) (h_0+1) \log \left(z\right)}{\epsilon} \\
& \quad - 72 h_0 (h_0-2) (h_0-1) (h_0+1) \log ^2\left(z\right)
+24 h_0 (h_0 (2 h_0 (7-5 h_0)+9)-7) \log \left(z\right) \nonumber \, .\end{aligned}$$ Thus in total we arrive at $$\begin{aligned}
c^2 z^{2 h_0} \left. \langle \tilde W_{h_0} (z) \rangle \right|_{\mathcal{O}(c^{0})} =
72 h_0^2 (h_0-1)^2 \log ^2 (z)+156 h_0 (h_0-1) \log (z) \, , \end{aligned}$$ which does not have any non-local divergence. Compared with the $1/c$ expansion of two point function in , the coefficients in front of $\log (z)$ and $\log^2 (z)$ at the $1/c^2$ order are correctly reproduced with $h_1,h_2$ in .
Correlators for $N=3$ {#Correlators3}
=====================
In the previous section, we have illustrated our prescription by examining a simple example of sl$(N)$ Chern-Simons theory with $N=2$. In this section, we extend the analysis to more involved case with $N=3$. It is a rather straightforward generalization even though computations become complicated due to the existence of spin three current $J^{(3)}$. In this paper, we adopt the representation of sl$(N)$ generators with $x$-derivatives as in , which is valid for arbitrary representation with $h_0 = - j$ for $N=2$ but only for the fundamental representation with $h_0 = (1 - N)/2$ for $N \geq 3$.[^6] With $N=3$, the $1/c$ expansion of conformal weight is given by with as $$\begin{aligned}
h_0 = - 1 \, , \quad h_1 = - 32 \, , \quad h_2 = - 1600 \, .
\label{dimshift3}\end{aligned}$$ In the next subsection, we reproduce the conformal weight at the $1/c$ order as in $h_1$ above from the bulk viewpoint and renormalize open Wilson line. In subsection \[3ptN3\], we examine three point functions and fix the two parameters $c_2$ and $c_3$ in to be consistent with symmetry. In subsection \[2ptN32\], we show that our prescription correctly reproduces the conformal weight at the $1/c^2$ order as $h_2$ in .
Two point function at $1/c$ order {#2ptN31}
---------------------------------
As for $N=2$, we start by examining the two point function at the $1/c$ order. Since spin three current $J^{(3)}$ is involved along with spin two current $J^{(2)}$, there are two types of corrections as $$\begin{aligned}
\left( \frac{6}{c}\right)^2 \left[ \langle W^{(2)}_{h_0} (z) \rangle + \langle W^{(2)'}_{h_0 } (z) \rangle \right]
\label{W2h}\end{aligned}$$ at this order. The two are represents in figure \[Wilson1\] and figure \[Wilson4\], respectively.
![Diagram contributing to the $1/c$ order correction of $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} \rangle$ for $N=3$ in addition to the one in figure \[Wilson1\]. The thick wavy line represents the propagator of spin three current.[]{data-label="Wilson4"}](Wilson4.pdf)
Here $W^{(2)}_{h_0} (z) $ is defined in and $$\begin{aligned}
W^{(2) '}_{h_0} (z) = \int_0^z d z_2 \int_0^{z_2} dz_1 f^{(3,3)}_2 (z_2,z_1)
J^{(3)} (z_2) J^{(3)} (z_1) \, .
\label{W2p}\end{aligned}$$ Since we have already computed $\langle W^{(2)}_{h_0} (z) \rangle$ as in , we just need to evaluate $\langle W^{(2)'}_{h_0} (z) \rangle$. The prescription in leads us to adopt $$\begin{aligned}
\langle J^{(3)} (z_2) J^{(3)} (z_1) \rangle = - \frac{5c}{6} \frac{1}{z_{21}^{6 -2 \epsilon} }\end{aligned}$$ with the shift of conformal dimension of $J^{(3)}$ from $3$ to $3 - \epsilon$. Using this expression, we find $$\begin{aligned}
\langle W^{(2)'}_{h_0} (z) \rangle = & - \frac{5 c}{6}\left[
-\frac{h_0 (h_0 (4 (h_0-2) h_0+1)+3) z^{-2 h_0}}{15 \epsilon} \right. \\ & \left.
-\frac{1}{450} h_0 (2 h_0+1) z^{-2 h_0} (60 (h_0-1) (2 h_0-3) \log (z)+h_0 (94 h_0-115)-9) \right] \nonumber\end{aligned}$$ up to the term of order $\epsilon^0$. Inserting $h_0 = -1$, we obtain $$\begin{aligned}
z^{-2} \langle W_{-1} (z) \rangle= & 1 + \left( \frac{6}{c} \right)^2 \frac{c}{2} \left(\frac{2 }{3 \epsilon} + \frac{1}{9} \left(12 \log \left(z\right) + 7\right) \right)
\nonumber \\ & + \left( \frac{6}{c} \right)^2 \left(- \frac{5c}{6} \right) \left(-\frac{2 }{3 \epsilon}-\frac{1}{450} (600 \log (z)+200) \right) \nonumber \\
= & 1 + \frac{1}{c} \left(\frac{32 }{\epsilon }+ 64 \log (z)+\frac{82}{3} \right) \end{aligned}$$ up to the terms of orders $\epsilon^0$ and $1/c$. In particular, the $1/c$ order correction of conformal weight is read off as $h_1 = - 32$, which is consistent with . In order to remove the divergence at $\epsilon \to 0$ up to the $1/c$ order, we renormalize the Wilson line operator as $$\begin{aligned}
\tilde W_{-1} (z) = \left[ 1 - \frac{1}{c} \left(\frac{32}{ \epsilon} + \frac{82}{3} \right) \right]W_{-1} (z) \, , \label{wfren3}\end{aligned}$$ which leads to the corresponding two point function of canonical form as in .
Three point functions {#3ptN3}
---------------------
We move to three point functions with one conserved current. For $N=3$, there are two choices of currents, i.e., spin two current $J^{(2)}$ and spin three current $J^{(3)}$. We start by computing $
\langle W_{-1} (z) J^{(2)}(y) \rangle
$ up to the $1/c$ order by following the previous analysis for $N=2$. With the convention of $J^{(2)}$ in , the corresponding three point function is given by with . At the leading order in $1/c$, we have already computed as in with $s=2$ and $h_0=-1$. In the following we shall examine the next non-trivial order in $1/c$.
At the order in $1/c$, there are several types of contribution as in figure \[Wilson2\] and figure \[Wilson5\].
![Diagrams contributing to the $1/c$ order correction of $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} J^{(2)} \rangle $ for $N=3$ in addition to the ones in figure \[Wilson2\].[]{data-label="Wilson5"}](Wilson5.pdf)
One comes from $$\begin{aligned}
&\langle W_{-1}^{(2)} (z) J^{(2)}(y) \rangle = \int_0^z dz_2 \int_0^{z_2} dz_1 f^{(2,2)}_2 (z_2 , z_1) \langle J^{(2)} (z_2) J^{(2)} (z_1) J^{(2)} (y) \rangle \, , \nonumber \\
& \langle W_{-1}^{(2)'} (z) J^{(2)}(y) \rangle =
\int_0^z dz_2 \int_0^{z_2} dz_1 f^{(3,3)}_2 (z_2 , z_1) \langle J^{(3)} (z_2) J^{(3)} (z_1) J^{(2)} (y) \rangle\, , \end{aligned}$$ where $W_{-1}^{(2)} (z)$ and $W_{-1}^{(2)'} (z)$ were introduced in and , respectively. The first contribution corresponds to the diagram (a) in figure \[Wilson2\] and it has been computed as in with $h_0=-1$. For the second one corresponding to the diagram (a) in figure \[Wilson5\], we find $$\begin{aligned}
\lim_{y \to - \infty} |y|^{4 - 2 \epsilon} \langle W_{-1}^{(2)'} (z) J^{(2)}(y) \rangle = - \frac{5 c}{2} \left[ -\frac{2 z^4}{3 \epsilon} +
\frac{1}{18} z^4 (13- 12 \log (z)) \right] + \mathcal{O}(\epsilon) \, ,\end{aligned}$$ where we have used $$\begin{aligned}
\langle J^{(3)} (z_2) J^{(3)} (z_1) J^{(2)}(y) \rangle = \frac{- 5 c/2}{z_{21}^{4 - \epsilon} (z_2 - y )^{2 - \epsilon} (z_1 - y )^{2 - \epsilon} }
\label{j3j3j2}\end{aligned}$$ with the shifts of conformal weight both for $J^{(2)}$ and $J^{(3)}$.
Other types of contribution include four conserved currents. One of them involves four spin two currents as $$\begin{aligned}
\int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(2,2,2)}_3 (z_3 , z_2 , z_1 ) \langle J^{(2)} (z_3) J^{(2)} (z_2) J^{(2)} (z_1) J^{(2)} (y) \rangle \, .\end{aligned}$$ They are represented in figure \[Wilson2\] and have already been evaluated in subsection \[3ptN2\]. Others involve two spin two and two spin three currents, and the correlator of them is factorized at the leading order in $1/c$ as $$\begin{aligned}
\langle J^{(3)} (z_3) J^{(3)} (z_2)J^{(2)} (z_1) J^{(2)} (y) \rangle
= \frac{- 5c^2/12}{z_{32}^{6 -2 \epsilon} ( z_1 - y )^{4 - 2 \epsilon} } + \mathcal{O} (c) \, . \label{j2j3j3j2}\end{aligned}$$ Therefore, we need to evaluate $$\begin{aligned}
&H^{(3,2)}_{1}(z) = - \frac{5 c^2}{12} z^{-4} \int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(2,3,3)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{21}^{6 - 2\epsilon} } \, ,\nonumber \\
&H^{(3,2)}_{2} (z) = - \frac{5 c^2}{12} z^{-4}\int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(3,2,3)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{31}^{6 - 2\epsilon} } \, , \\
&H^{(3,2)}_{3} (z)= - \frac{5 c^2}{12} z^{-4} \int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(3,3,2)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{32}^{6 - 2\epsilon} } \, , \nonumber \end{aligned}$$ which correspond to diagrams (b), (c), (d) in figure \[Wilson5\], respectively. Explicitly performing the integrals, we find $$\begin{aligned}
& H^{(3,2)}_{1} (z)= H^{(3,2)}_3 (z) = - \frac{5c^2}{12} \left[\frac{2 }{9 \epsilon}+\frac{1}{9} (4 \log (z)-1) \right] \, , \nonumber \\
&H^{(3,2)}_{2} (z)= - \frac{5c^2}{12} \left[\frac{1}{9 \epsilon}+\frac{1}{54} (12 \log (z)-13) \right] \, .\end{aligned}$$
Combining all contributions so far, we have $$\begin{aligned}
&z^{-4} \lim_{y \to - \infty} |y|^{4 - 2 \epsilon} \langle W_{-1} (z) J^{(2)} (y)\rangle
= -1- \frac{1}{c} \left[ \frac{6}{\epsilon}+24 \log (z)+23 \right] \\
&+ \left(\frac{6}{c} \right)^2
\frac{-5 c }{2} \cdot \left[ -\frac{2 }{3 \epsilon} +
\frac{1}{18} (13- 12 \log (z)) \right] + \left(\frac{6}{c}\right)^3 [2 H^{(3,2)}_1 (z)+ H^{(3,2)}_2 (z)] + \cdots \, . \nonumber\end{aligned}$$ The above expression reduces to $$\begin{aligned}
- 1 + \frac{1}{c} \left( \frac{4}{\epsilon} - 64 \log (z) - \frac{139}{3} \right)
=z^{-2} \left[ -1 + \frac{1}{c} \left( \frac{36}{\epsilon} - 19 \right) \right] \langle W_{-1} (z) \rangle \, .
\label{wj2}\end{aligned}$$ As before, we choose the parameter $c_2$ in as $$\begin{aligned}
c_2 = 1 + \frac{1}{c} \left( \frac{36}{\epsilon} + 13 \right) + \mathcal{O}(c^{-2}) \, .
\label{level2c32}\end{aligned}$$ This leads to an extra contribution up to the $1/c$ order from $$\begin{aligned}
\left.
z^{-4} \lim_{y \to - \infty} |y|^{4 - 2 \epsilon} \frac{6c_2}{c} \langle W^{(1)}_{-1} (z) J^{(2)} (y) \rangle
\right|_{\mathcal{O}(c^{-1})} = - \frac{1}{c} \left( \frac{36}{\epsilon} + 13 \right) \end{aligned}$$ with $W^{(1)}_{h_0}$ in . We can see that this contribution cancels the divergence in as $$\begin{aligned}
&z^{-4} \lim_{y \to - \infty} |y|^{4 - 2 \epsilon } \langle W_{-1} (z) J^{(2)} (y)\rangle
=z^{-2} \left[ -1 - \frac{32}{c} \right] \langle W_{-1} (z) \rangle + \mathcal{O}(c^{-2})\end{aligned}$$ for $\epsilon \to 0$. The constant term in is chosen in order to reproduce with $h_1 = - 32$ as in .
We would like to compute another correlator as $
\langle W_{-1} (z) J^{(3)}(y) \rangle
$ with spin three current at the $1/c$ order. Using the leading order result in with $h_0 = -1, s=3$, and the $1/c$ correction as $C_{1}^{(3)}/C_{0}^{(3)} = 224/5$ obtained in section \[CFT3pt\], the corresponding three point function is given by $$\begin{aligned}
\langle \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) J^{(3)} (y) \rangle = \frac{1}{3}\left[ 1 + \frac1c \frac{224}{5} \right] \left( \frac{z}{(y - z) y} \right)^3 \langle \mathcal{O}_{h_+} (z) \bar{\mathcal{O}}_{h_+} (0) \rangle + \mathcal{O}(c^{-2}) \, .
\label{3pts3N3}\end{aligned}$$ Following the prescription discussed in subsection \[Reg\], we choose the parameter $c_3$ in such that the Wilson line computation reproduces this expression.
With two current insertions from an open Wilson line, we have the following type of contribution as $$\begin{aligned}
H^{(2,3)} (z)& = \lim_{y \to - \infty} |y|^{6 -2 \epsilon} z^{-5} \int_0^z dz_2 \int_0^{z_2} dz_1 \left[ f^{(3,2)}_2 (z_2 , z_1) \right . \langle J^{(3)} (z_2) J^{(2)} (z_1) J^{(3)} (y) \rangle \nonumber \\ & \qquad \qquad \qquad \qquad + \left. f^{(2,3)}_2 (z_2 , z_1) \langle J^{(2)} (z_2) J^{(3)} (z_1)J^{(3)} (y) \rangle \right] \nonumber \\
& =- \frac{5 c}{2} \left[ -\frac{4 }{15 \epsilon} + \frac{1 }{225} (107-60 \log (z)) \right] \, ,\end{aligned}$$ which come from diagrams (a), (b) in figure \[Wilson6\].
![Diagrams contributing to the $1/c$ order correction of $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} J^{(3)} \rangle $ for $N=3$.[]{data-label="Wilson6"}](Wilson6.pdf)
For the correlator of three currents, we have used . There are contributions with two spin two and two spin three currents. With the correlator in , they are given by $$\begin{aligned}
H^{(3,3)}_{1} (z)= - \frac{5 c^2 }{12} z^{-5} \int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(3,2,2)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{21}^{4 -2 \epsilon}} \, , \nonumber \\
H^{(3,3)}_{2} (z)= - \frac{5 c^2 }{12} z^{-5} \int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(2,3,2)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{31}^{4 -2 \epsilon}} \, , \\
H^{(3,3)}_{3} (z)= - \frac{5 c^2 }{12} z^{-5} \int_0^{z} dz_3 \int_0^{z_3} dz_2 \int_0^{z_2} dz_1
f^{(2,2,3)}_3 (z_3 , z_2 , z_1 ) \frac{1}{z_{32}^{4 - 2 \epsilon}} \, , \nonumber \end{aligned}$$ which correspond to diagrams (c), (d), (e) in figure \[Wilson6\]. Integrating over the variables $z_1,z_2,z_3$, we find $$\begin{aligned}
&H^{(3,3)}_{1} (z)= H^{(3,3)}_{3} (z)= - \frac{5 c^2 }{12} \cdot \left[\frac{2 }{45 \epsilon}+\frac{4}{225} (5 \log (z)-1)\right] \, , \nonumber \\
&H^{(3,3)}_{2} (z)=- \frac{5 c^2 }{12} \cdot \left[ \frac{1}{45 \epsilon}+\frac{2}{675} (15 \log (z)-26) \right] \, .\end{aligned}$$ Furthermore, we need to consider a contribution of the form as $$\begin{aligned}
\int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3 ^{(3,3,3)} ( z_3,z_2 ,z_1) \langle J^{(3)} (z_3) J^{(3)} (z_2) J^{(3)} (z_1) J^{(3)} (y) \rangle \end{aligned}$$ with $$\begin{aligned}
&\langle J^{(3)} (z_3)J^{(3)} (z_2)J^{(3)} (z_1) J^{(3)} (y) \rangle \nonumber \\ & =
\frac{(5c/6)^2}{z_{32}^{6 - 2 \epsilon} (z_1 - y )^{6 - 2 \epsilon} } +
\frac{(5c/6)^2}{ z_{31}^{6 - 2 \epsilon } ( z_2 - y )^{ 6 - 2 \epsilon }} + \frac{(5c/6)^2}{z_{21}^{6 - 2 \epsilon} (z_3 - y)^{ 6 - 2 \epsilon} } + \mathcal{O}(c)\, .\end{aligned}$$ Denoting $$\begin{aligned}
H^{(3,3)}_{ij} (z) = \left(\frac{5c}{6} \right)^2 z^{-5}
\int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3 ^{(3,3,3)} ( z_3,z_2 ,z_1)
\frac{1}{z_{ji}^{6 - 2\epsilon} } \, ,\end{aligned}$$ we find $$\begin{aligned}
H^{(3,3)}_{12} (z)& =H^{(3,3)}_{23} (z) = \left(\frac{5c}{6} \right)^2 \left[-\frac{2 }{45 \epsilon} + \frac{1}{225} (9-20 \log (z)) \right] \, ,\nonumber \\
H^{(3,3)}_{13} (z)& = \left(\frac{5c}{6} \right)^2 \left[ \frac{1}{225 \epsilon} + \frac{1}{6750}(60 \log (z)-137) \right] \, . \end{aligned}$$ Here $H^{(3,3)}_{12} (z)$, $H^{(3,3)}_{13} (z)$, $H^{(3,3)}_{23} (z)$ correspond to diagrams (f), (g), (h) in figure \[Wilson6\].
Combining the results so far as $$\begin{aligned}
& z^{ - 5} \lim_{y \to - \infty} |y|^{6 - 2 \epsilon } \langle W_{-1} (z) J^{(3)}(y) \rangle \\
& \qquad = \frac{1}{3} - \left( \frac{6}{c} \right)^2 H^{(2,3)} - \left( \frac{6}{c} \right)^3
\left[2 H^{(3,3)}_1 (z) +H^{(3,3)}_2 (z) + 2 H^{(3,3)}_{21} (z)+ H^{(3,3)}_{31} (z) \right] + \cdots \, , \nonumber\end{aligned}$$ we find $$\begin{aligned}
z^{-5 } \lim_{y \to - \infty} |y|^{6 - 2 \epsilon } \langle W_{-1} (z) J^{(3)}(y) \rangle &= \frac{1}{3} +
\frac{1}{c} \left( -\frac{4}{3 \epsilon}+\frac{64 \log (z)}{3}+\frac{1067}{45} \right) + \cdots
\nonumber \\ &= z^{-2} \left[ \frac{1}{3} +
\frac{1}{ c} \left( - \frac{12}{\epsilon} + \frac{657}{45} \right) \right] \langle W_{-1} (z) \rangle + \cdots \, .\end{aligned}$$ We remove the divergent term by properly choosing the parameter $c_3$ in as before. We propose to use $$\begin{aligned}
c_3 = 1 + \frac{1}{c} \left( \frac{36}{\epsilon} + 1 \right) + \mathcal{O} (c^{-2}) \, ,
\label{level2c33}\end{aligned}$$ which leads to an extra contribution at the $1/c$ order as $$\begin{aligned}
\left. z^{-5} \lim_{y \to - \infty} |y|^{6 - 2\epsilon } \frac{6 c_3}{c} \langle W^{(1)'}_{-1} (z) J^{(3)} (y) \rangle \right|_{\mathcal{O} (c^{-1})} = \frac{1}{c} \frac{1}{3} \left( \frac{36}{\epsilon} + 1 \right) \, .\end{aligned}$$ Here $W^{(1)'}_{-1} (z)$ is defined as $$\begin{aligned}
W^{(1)'}_{h_0} = \int_0^z d z_1 f^{(3)}_1 (z_1) J^{(3)} (z_1) \, .
\label{W1p}\end{aligned}$$ Including the effect, we obtain $$\begin{aligned}
\lim_{y \to - \infty} |y|^{6 - 2 \epsilon } \langle W_{-1} (z) J^{(3)}(y) \rangle = \frac{1}{3} z^3 \left[ 1 +
\frac{1}{ c} \frac{224}{5} \right] \langle W_{-1} (z) \rangle + \mathcal{O} (c^{-2})\end{aligned}$$ for $\epsilon \to 0$ as in .
Two point function at $1/c^2$ order {#2ptN32}
-----------------------------------
As for $N=2$, we examine the two point function up to the $1/c^2$ order and see whether we can reproduce the $1/c$ correction of conformal weight as in after adopting the regularization. As before, we first evaluate the $1/c$ correction without renormalization and then include its effects.
There are contributions involving only spin two currents, which were already evaluated in . We find $$\begin{aligned}
c^2 z^{-2} G^{(2)}_\text{spin 2} (z) = 72 \log (z) (4 \log (z)+7)\end{aligned}$$ by setting $h_0 = -1$. In this subsection, we only keep the terms involving $\log (z)$ or $\log ^2 (z)$ and not vanishing at $\epsilon \to 0$.
Furthermore, we include the effects of spin three current $J^{(3)}$. In order to make our notation simpler, we adopt the following rule. If $J^{(2)} (z_i)$ comes from the open Wilson line, then we use index $i$. If $J^{(3)} (z_i)$ enters instead of $J^{(2)} (z_i)$, then we replace the index $i$ by . We first compute those with three currents as $$\begin{aligned}
G^{(2)}_{1\text{\boldmath $23$}} (z) = & \left( \frac{6}{c}\right)^3 \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_3^{(3,3,2)} ( z_3,z_2 ,z_1) \frac{- 5 c/2}{ z_{32}^{4- \epsilon} z_{31}^{2-\epsilon} z_{21}^{2- \epsilon} } \, ,\end{aligned}$$ and $G_{\text{\boldmath $1$}2\text{\boldmath $3$}}^{(2)} (z)$, $G_{\text{\boldmath $12$}3}^{(2)} (z)$, which are represented by diagrams (a), (b), (c) in figure \[Wilson7\], respectively.
![Diagrams contributing to the $1/c^2$ order correction of $\langle \mathcal{O}_{h_+} \bar{\mathcal{O}}_{h_+} \rangle $ for $N=3$ in addition to the ones in figure \[Wilson3\].[]{data-label="Wilson7"}](Wilson7.pdf)
Integrations over $z_i$ yield $$\begin{aligned}
&c^2 z^{-2} G_{1\text{\boldmath $23$}}^{(2)} (z) = c^2 z^{-2} G_{\text{\boldmath $12$}3}^{(2)} (z) = -\frac{ 2880 \log (z) }{\epsilon} - 4320 \log (z) (\log (z)+1) \, , \nonumber \\
&c^2 z^{-2}G_{\text{\boldmath $1$}2\text{\boldmath $3$}}^{(2)} (z) =-\frac{ 2880 \log (z)}{\epsilon}- 1440 \log (z) (3 \log (z)+2) \, . \end{aligned}$$ There are also contributions with two spin two and two spin three currents such as $$\begin{aligned}
G^{(2)}_{12;\text{\boldmath $34$}} (z) = & \left( \frac{6}{c}\right)^4 \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_4^{(3,3,2,2)} ( z_4,z_3 ,z_2,z_1) \frac{- 5 c^2 /12}{z_{43}^{6- 2\epsilon} z_{21}^{4- 2\epsilon} } \, ,\end{aligned}$$ and so on. They are computed as $$\begin{aligned}
&c^2 z^{-2} G_{12;\text{\boldmath $34$}}^{(2)} (z) =c^2 z^{-2} G_{\text{\boldmath $12$;34}}^{(2)} (z) = \frac{ 960 \log (z)}{\epsilon} + 160 \log (z) (12 \log (z)+11) \, , \nonumber \\
&c^2 z^{-2} G_{14;\text{\boldmath $23$}}^{(2)} (z) = \frac{480 \log (z)}{\epsilon} + 240 \log (z) \left( 4 \log (z)+ 9\right)\, ,\nonumber \\
&c^2 z^{-2} G_{\text{\boldmath $14$};23}^{(2)} (z) = \frac{480 \log (z)}{\epsilon}+ 240 \log (z) \left( 4 \log (z)+ 5 \right) \, , \\
&c^2 z^{-2} G_{13;\text{\boldmath $24$}}^{(2)} (z) = c^2 G_{\text{\boldmath $13$;24}}^{(2)} (z) = \frac{480 \log (z)}{\epsilon} + 350 \log (z) \left( \log (z)+ 4 \right) \, , \nonumber\end{aligned}$$ which correspond to diagrams (d)-(i) in figure \[Wilson7\]. Finally, those with four spin three currents are $$\begin{aligned}
G^{(2)}_{\text{\boldmath $12$;$34$}} (z) = & \left( \frac{6}{c}\right)^4 \int_0^z d z_3 \int_0^{z_3} d z_2 \int_0^{z_2} dz_1 f_4^{(3,3,3,3)} (z_4, z_3,z_2 ,z_1) \frac{(5 c/6)^2}{z_{43}^{6- 2 \epsilon} z_{21}^{6- 2\epsilon} } \end{aligned}$$ and others with different products of the two point function. They are obtained as $$\begin{aligned}
&c^2 z^{-2} G_{\text{\boldmath $12$;$34$}}^{(2)} (z) = \frac{1600 \log (z)}{ \epsilon} + \frac{3200}{3} \left( \log (z) (3 \log (z)+2) \right) \, , \nonumber \\
&c^2 z^{-2} G_{\text{\boldmath $14$;$23$}}^{(2)} (z) = \frac{800 \log (z)}{\epsilon}+1600 \log (z) (\log (z)+1)\, , \\
&c^2 z^{-2} G_{\text{\boldmath $13$;$24$}}^{(2)} (z) = -\frac{160 \log (z) }{ \epsilon} - \frac{8}{3} \log (z) (120 \log (z)+127) \, ,\nonumber\end{aligned}$$ which are represented in diagrams (j), (k), (l) in figure \[Wilson7\], respectively. Summing up all contributions we have $$\begin{aligned}
c^2 z^{-2} G^{(2)} (z) = &
-\frac{2560 \log (z) }{\epsilon} -\frac{32}{3} \log (z) (48 \log (z)-193) \, ,\end{aligned}$$ which includes a non-locally divergent term.
Let us then examine the effects of renormalization. There are two types of $1/c$ order corrections as in before the renormalization. Multiplying the $1/c$ terms due to renormalization, some contributions at the $1/c^2$ order arise. With and , the contribution with spin two current becomes $$\begin{aligned}
&\frac{1}{c}\left[ 2 \left( \frac{36}{ \epsilon} + 13 \right) - \frac{32}{\epsilon} - \frac{82}{3} \right] \left(\frac{6}{c} \right)^2 \langle W^{(2)}_{-1} (z) \rangle \nonumber \\
& \qquad \qquad \qquad = \frac{1}{c} \left[\frac{960 z^2 \log (z)}{\epsilon} + 64 z^2 \log (z) (15 \log (z)+17) \right] \, ,\end{aligned}$$ see for the previous case with $N=2$. The contribution with spin three current is $$\begin{aligned}
&\frac{1}{c} \left[ 2 \left( \frac{36}{ \epsilon} + 1 \right) - \frac{32}{ \epsilon} - \frac{82}{3} \right] \left(\frac{6}{c} \right)^2 \langle W^{(2)'}_{-1} (z) \rangle \nonumber \\
& \qquad \qquad \qquad = \frac{1}{c} \left[ \frac{1600 z^2 \log (z)}{ \epsilon} +\frac{160}{3} z^2 \log (z) \left(30 \log (z)+ 1 \right) \right] \, ,\end{aligned}$$ where we have used and . Thus the $\log(z)$ and $\log ^2 (z)$ dependent terms in the total contribution are $$\begin{aligned}
& c^2 z^{-2} \left. \langle \tilde W_{h_0} (z) \rangle \right|_{\mathcal{O}(c^{0})} = 3200 \log (z) + 2048 \log ^2 (z) \, .\end{aligned}$$ The coefficients in front of $\log (z)$ and $\log ^2 ( z)$ are precisely those in with . We would like to emphasize again that there is cancellation among non-local divergences.
Conclusion and discussions {#conclusion}
==========================
We have examined the two and three point functions of the 2d W$_N$ minimal model in $1/c$ expansion from the bulk viewpoint. Extending a previous work of [@Besken:2016ooo] at the leading order in $1/c$, we claim that these correlators can be computed with open Wilson lines in sl$(N)$ Chern-Simons gauge theory as in and even at higher orders in $1/c$. There are divergences associated with loop diagrams in the Wilson line computations, and we have to decide how to deal with them. We offer to regularize the divergences by renormalizing the overall factor of the open Wilson line and parameters $c_s$ introduced in . The finite parts of $c_s$ are fixed such that three point functions from are consistent with the boundary W$_N$ symmetry. We confirm the validity of our prescription by reproducing the $1/c$ corrections of scalar conformal weight from including $1/c^2$ order terms.
As concrete examples, we have only examined Chern-Simons gauge theories based on sl$(N)$ with $N=2,3$. For $N \geq 4$ we see no major difference even though computations would be quite complicated. For instance, we can reproduce $h_1$ in by evaluating integrals in up to the $1/c$ order and comparing the $1/c$ expansion of the two point function in . We consider the following integral as $$\begin{aligned}
\langle W_{(1 - N)/2}^{(1,s)} (z) \rangle \equiv - \frac{ (2 s -1) N_s}{6}
\int_0^z d z_2 \int_0^{z_2} dz_1 f^{(s,s)}_2 (z_2,z_1) \frac{1}{z_{21}^{2 s- 2 \epsilon }}\end{aligned}$$ with conformal weight $(1-N)/2$. The term including $\log(z)$ at the order $\epsilon^0$ is evaluated as $$\begin{aligned}
\langle W_{(1-N)/2}^{(1,s)} (z) \rangle |_{\log , \epsilon^0}
= (2 s -1) (N^2-1) \left(- \frac{N_s}{6} \right) ^2 z^{N-1} \log(z) \end{aligned}$$ for $s=2,3, \ldots ,10$. We conjecture that the above equality also holds for $s > 10$. Then, the $1/c$ order correction of scalar conformal weight for generic $N$ can be read off as $$\begin{aligned}
- \frac{1}{2} \sum_{s = 2}^N \left(- \frac{6}{N_s} \right) ^2 (2 s -1) (N^2-1) \left(- \frac{N_s}{6} \right) ^2 = - \frac{(N^2 - 1)^2}{2} \, ,\end{aligned}$$ which matches $h_1$ in . For our purpose it is enough to work with the non-unitary duality, but other problems may require a unitary one, i.e., the ’t Hooft limit of [@Gaberdiel:2010pz], see footnote \[nonunitary\]. For the unitary duality, we should extend the analysis to the case with a higher spin algebra hs$[\lambda]$, which is a gauge algebra of 3d Prokushkin-Vasiliev theory [@Prokushkin:1998bq]. In particular, we would like to understand the precise relation between open Wilson lines and particles traveling in the bulk.
An important open problem is to confirm our proposal that correlators in the 2d W$_N$ minimal model can be computed with open Wilson lines in sl$(N)$ Chern-Simons gauge theory including $1/c$ corrections. In particular, we have to extend the checks to higher orders in $1/c$. We have conjectured that all divergences are removed by renormalizing the overall factor of the open Wilson line and the parameters $c_s$ in , but it is desirable to prove this claim. A different regulator was introduced in [@Besken:2017fsj] by shifting $1/(z_{21}^2)^a \to 1/(z_{21}^2 + \epsilon^2)^a$, but it breaks conformal symmetry. We can see that divergences from loop computations with this regulator cannot be absorbed by these changes, thus conformal symmetry in the regularization procedure should play an important role.
We have proposed our regularization prescription so as to be analogous to that for usual quantum field theory even though the precise relation is yet to be clarified. We offer to fix the interaction parameters by comparing them to “experimental data” that are obtained from dual conformal field theory in the current situation. Once they are fixed, then other quantities like the self-energy of the scalar propagator are claimed to be predictable. A particularly nice thing happens for $N=2$. In this case, the $1/c$ order of the interaction parameter $c^{(1)}_2$ was determined by using the information on $h_1$ in through . Fortunately, $h_1$ can be obtained from the expectation value of the open Wilson line as in , therefore we do not need to refer to explicit boundary data and everything is computable in terms of bulk theory. Here we have only considered to the next leading order in $1/c$, but it is natural to expect that the same is true for higher orders in $1/c$ as well. For $N=3$, we fixed the $1/c$ order of the other interaction parameter $c_3^{(1)}$ such that the equality in is satisfied. Here the number $224/5$ was borrowed from the W$_N$ minimal model. However, we believe that there should be a way to determine $c_3$ without referring to explicit boundary data, and it is an important open problem to find this out. We do not claim that our prescription is unique, and in fact a different one was adopted in [@Fitzpatrick:2016mtp] for $N=2$. It is easier to see the physical meaning in our regularization procedure, but their prescription seems to be convenient for actual computations of conformal blocks. In any case, it should be useful to understand the relation between different prescriptions.
In this paper, we have examined the duality of [@Gaberdiel:2010pz] in the semiclassical limit discussed in [@Castro:2011iw; @Gaberdiel:2012ku; @Perlmutter:2012ds] with $1/c$ corrections, but it is also possible to extend the analysis to other examples. In particular, an $\mathcal{N}=2$ supersymmetric version of duality was proposed in [@Creutzig:2011fe], and the bulk description of its semiclassical limit was argued to be given by sl$(N+1|N)$ Chern-Simons gauge theory [@Hikida:2012eu]. See [@Tan:2012xi; @Datta:2012km; @Chen:2013oxa; @Datta:2013qja; @Banados:2015tft] for conical defect or black hole solutions in higher spin supergravity. We think that supersymmetric extension is important for the following two reasons. Firstly, it is usually expected that supersymmetry suppresses quantum effects, and it would enable us to examine higher order corrections in $1/c$ systematically. Secondly, supersymmetry helps us to study relations between higher spin gauge theory and superstring theory, and concrete examples have been discussed in [@Creutzig:2013tja; @Creutzig:2014ula; @Hikida:2015nfa] with $\mathcal{N}=3$ supersymmetry and in [@Gaberdiel:2013vva; @Gaberdiel:2014cha] with $\mathcal{N}=4$ supersymmetry. We would like to report on this extension in the near future.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to Andrea Campoleoni, Pawel Caputa, Nilay Kundu, Takahiro Nishinaka, Volker Schomerus, Yuji Sugawara, Tadashi Takayanagi, and Jörg Teschner for useful discussions. YH would like to thank the organizers of the “Universität Hamburg-Kyoto University Symposium" and the workshop “New ideas on higher spin gravity and holography” at Kyung Hee University, Seoul for their hospitality. The work of YH is supported by JSPS KAKENHI Grant Number 16H02182.
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[^1]: E-mail: yhikida@yukawa.kyoto-u.ac.jp
[^2]: E-mail: rp0019fr@ed.ritsumei.ac.jp
[^3]: After completing this draft, we become aware of an interesting paper [@Giombi:2017hpr] appearing in the arXiv. The paper deals with loop corrections in two point Witten diagrams for higher spin theories on AdS$_d$. Related previous works may be found in [@Manvelyan:2008ks; @Hikida:2015nfa; @Creutzig:2015hta; @Hikida:2016wqj; @Hikida:2016cla; @Aharony:2016dwx; @Hikida:2017ecj; @Cardona:2017tsw].
[^4]: The analysis of this paper will not rely on unitarity, so we can safely work in the non-unitary regime. However, we may have to make use of unitarity for other purposes, and in that case we should come back to the ’t Hooft limit, for instance, by utilizing the analytic continuation discussed in [@Gaberdiel:2012ku]. \[nonunitary\]
[^5]: Previously, Wilson lines in $\text{sl}(N) $ Chern-Simons gauge theory were utilized to compute entanglement entropy in a holographic way [@deBoer:2013vca; @Ammon:2013hba]. For the case with $N=2$, the proposal reduces to that in [@Ryu:2006bv; @Ryu:2006ef].
[^6]: One may find the expression of sl$(3)$ generators for generic representation in terms of three parameters $x_1,x_2,x_3$, e.g., in section 15.7.4 of [@cft].
|
---
abstract: 'Superselection rules induced by the interaction with the environment are investigated with the help of exactly soluble Hamiltonian models. Starting from the examples of Araki and of Zurek more general models with scattering are presented for which the projection operators onto the induced superselection sectors do no longer commute with the Hamiltonian. The example of an environment given by a free quantum field indicates that infrared divergence plays an essential role for the emergence of induced superselection sectors. For all models the induced superselection sectors are uniquely determined by the Hamiltonian, whereas the time scale of the decoherence depends crucially on the initial state of the total system.'
---
[Exactly soluble models of decoherence[^1]]{}
[Joachim Kupsch[^2]]{}
[Fachbereich Physik, Universität Kaiserslautern\
D-67653 Kaiserslautern, Germany]{}
Introduction
============
One of the puzzles of quantum mechanics is the question, how classical objects can arise in quantum theory. Quantum mechanics is a statistical theory, but its statistics differs on a fundamental level from the statistics of classical objects. The violation of Bell’s inequalities and the context dependence of quantum mechanics (Kochen-Specker theorem) illustrate this fact, see e.g. [@Peres:1995].
It is known since a long time that the statistical results of quantum mechanics become consistent with a classical statistics of “facts”, if the superposition principle is reduced to “superselection sectors”, i.e. coherent orthogonal subspaces of the full Hilbert space. The mathematical structure of quantum mechanics and of quantum field theory provides us with only a few “superselection rules”, the most important being the charge superselection rule related to gauge invariance, see e.g. [@BLOT:1990] [@Wightman:1995] and the references given therein. But there are definitively not enough of these superselection rules to understand classical properties in quantum theory. A possible solution of this problem is the emergence of effective superselection rules due to decoherence caused by the interaction with the environment. These investigations – often related to a discussion of the process of measurement – have developed in the eighties; some references are [@Araki:1980][@Zurek:1982][@Joos/Zeh:1985], but see also the earlier publications [@Zeh:1970][@Zeh:1971] and [@Emch:1972a].
In this article decoherence and the emergence of environment induced superselection rules are investigated on the basis of exactly soluble models. After a short introduction to superselection rules and to the dynamics of subsystems in Sects. \[ssr\] and \[subsys\], several models are presented in Sect. \[models\]. For a class of simple models, which essentially go back to Araki [@Araki:1980] and Zurek [@Zurek:1982], the transition between the induced superselection sectors is suppressed uniformly in trace norm. In a more realistic example with a quantum field as environment, presented in Sect. \[free\], the infrared behaviour of the environment is of essential importance for the emergence of induced superselection rules. Here uniform estimates, which persist for arbitrary times, are only possible in the limit of infrared divergence. In Sect. \[scatt\] it is shown that additional scattering processes (by sufficiently smooth potentials) do not alter the induced superselection sectors, but the decoherence is no longer uniform with respect to the initial state of the system.
Superselection rules\[ssr\]
===========================
We start with a few mathematical notations. Let $\mathcal{H}$ be a separable Hilbert space, then the following spaces of linear operators are used.
$\mathcal{B}(\mathcal{H})$: The $\mathbf{R}$-linear space of all bounded self-adjoint operators $A$. The norm of this space is the operator norm $%
\Vert A\Vert $.
$\mathcal{T}(\mathcal{H})$: The $\mathbf{R}$-linear space of all self-adjoint nuclear operators $A$. These operators have a pure point spectrum ${\alpha _i}\in \mathbf{R},\,i=1,2,...,$ with $\sum_i|\alpha
_i|<\infty $. The natural norm of this space is the trace norm $\Vert A\Vert
_1=\mathrm{tr}\sqrt{A^{+}A}=\sum_i|\alpha _i|$. Another norm, used in the following sections, is the Hilbert-Schmidt norm $\Vert A\Vert _2=\sqrt{%
\mathrm{tr}\,A^{+}A}$. These norms satisfy the inequalities $\Vert A\Vert
\leq \Vert A\Vert _2\leq \Vert A\Vert _1$.
$\mathcal{D}(\mathcal{H})$: The set of all statistical operators, i.e. positive nuclear operators $W$ with a normalized trace, $\mathrm{tr}\,W=1 $.
$\mathcal{P}(\mathcal{H})$: The set of all rank one projection operators $%
P^1 $.
These sets satisfy the obvious inclusions $\mathcal{P}(\mathcal{H}%
)\subset \mathcal{D}(\mathcal{H})\subset \mathcal{T}(\mathcal{H})\subset
\mathcal{B}(\mathcal{H}).$
Any state of a quantum system is represented by a statistical operator $W\in
\mathcal{D}(\mathcal{H})$, the elements of $\mathcal{P}(\mathcal{H})$ thereby correspond to the pure states. Any (bounded) observable is represented by an operator $A\in \mathcal{B}(\mathcal{H})$, and the expectation of the observable $A$ in the state $W$ is the trace $\mathrm{tr}%
\,WA$. Without additional knowledge about the structure of the system we have to assume that the set of all states corresponds exactly to $\mathcal{D}%
(\mathcal{H})$, and the set of all (bounded) observables is $\mathcal{B}(%
\mathcal{H})$. The state space $\mathcal{D}(\mathcal{H})$ has an essential property: it is a convex set, i.e. $W_1,W_2\in \mathcal{D}(\mathcal{H})$ implies $\lambda _1W_1+\lambda _2W_2\in \mathcal{D}(\mathcal{H})$ if $%
\lambda _{1,2}\geq 0$ and $\lambda _1+\lambda _2=1.$ Any statistical operator $W\in \mathcal{D}(\mathcal{H})$ can be decomposed into pure states $%
W=\sum_nw_nP_n^1$ with $P_n^1\in \mathcal{P}(\mathcal{H})$ and probabilities $w_n\geq 0,\;\sum_nw_n=1$. An explicit example is the spectral decomposition of $W$. But there are many other possibilities. It is exactly this arbitrariness that does not allow a classical interpretation of quantum probability. A more detailed discussion of the state space of quantum mechanics can be found in [@Kupsch:1996a].
The arbitrariness of the decomposition of $W$ originates in the superposition principle. In quantum mechanics, especially in quantum field theory, the superposition principle can be restricted by superselection rules. Here we cannot discuss the arguments to establish such rules, for that purpose see e.g. [@BLOT:1990][@Wightman:1995] and also Chap.6 of [@GJKKSZ:1996], or to refute them, see e.g. [@Mirman:1979]. Here we only investigate the consequences for the structure of the state space. In a theory with discrete superselection rules like the charge superselection rule, the Hilbert space $\mathcal{H}$ splits into orthogonal superselection sectors $\mathcal{H}_m,\,m\in \mathbf{M,}$ such that $%
\mathcal{H=}\oplus _m\mathcal{H}_m$. Pure states with charge $m$ (in appropriate normalization) are then represented by vectors in $\mathcal{H}_m$, and superpositions of vectors with different charges have no physical interpretation. The projection operators $P_m$ onto the orthogonal subspaces $\mathcal{H}_m$ satisfy $P_mP_n=\delta _{mn}$ and $\sum_mP_m=I$. The set of states is reduced to those statistical operators which satisfy $P_mW=WP_m$ for all projection operators $P_m,m\in \mathbf{M}$. The state space of the system is then $\mathcal{D}^S=\{W\in \mathcal{D}(\mathcal{H}%
)|WP_m=P_mW,\,m\in \mathbf{M}\}$, and all statistical operators satisfy the identity $W=\sum_mP_mWP_m$. An equivalent statement is that all observables of such a system have to commute with the projection operators $P_m,\,m\in
\mathbf{M,}$ and the set of observables of the system is given by\
$%
\mathcal{B}^S=\left\{ A\in \mathcal{B}(\mathcal{H})\mid AP_m=P_mA,\,m\in
\mathbf{M}\right\} =\left\{ A\in \mathcal{B}(\mathcal{H})\mid
A=\sum_mP_mAP_m\right\} .$
The projection operators $\{P_m\mid m\in \mathbf{M}\}$ are themselves observables, which commute with all observables of the system, and they generate a nontrivial centre of the algebra of observables.
In theories with continuous superselection rules the finite or countable set of projection operators $\left\{ P_m,m\in \mathbf{M}\right\} $ is substituted by a (weakly continuous) family of projection operators $%
P(\Delta )$ indexed by measurable subsets $\Delta \subset \mathbf{R}$, see e.g. [@Piron:1969] or [@Araki:1980]. These projection operators have to satisfy $$\left\{
\begin{array}{l}
P(\Delta _1\cup \Delta _2)=P(\Delta _1)+P(\Delta _2)%
\mbox{ for all
intervalls }\,\Delta _1,\Delta _2 \\
P(\Delta _1)P(\Delta _2)=O\mbox{ \ if\ }\,\Delta _1\cap \Delta _2=\emptyset ,%
\mbox{ and }\,P(\emptyset )=O,\;P(\mathbf{R})=1.
\end{array}
\right. \label{sp}$$ $\;$The set of observables is now given by $\mathcal{B}^S=\left\{ A\in
\mathcal{B}(\mathcal{H})\mid AP(\Delta )=P(\Delta )A,\,\Delta \subset
\mathbf{R}\right\} $, but there is no formulation of the corresponding set of states within the class of nuclear statistical operators.
The importance of superselection rules for the transition from quantum probability to classical probability is obvious. But there remains an essential problem: Only very few superselection rules can be found in quantum mechanics that are compatible with the mathematical structure and with experiment. A satisfactory solution to this problem is the emergence of effective superselection rules induced by the interaction with the environment.
Dynamics of subsystems and induced superselection sectors\[subsys\]
===================================================================
In the following we consider an “open system”, i.e. a system $S$ which interacts with an “environment” $E$, such that the total system $S+E$ satisfies the usual Hamiltonian dynamics. The Hilbert space $\mathcal{H}%
_{S+E}$ of the total system $S+E$ is the tensor space $\mathcal{H}_S\otimes
\mathcal{H}_E$ of the Hilbert spaces for $S$ and for $E$. We assume that the only observables at our disposal are the operators $A\otimes I_E$ with $A\in
\mathcal{B}(\mathcal{H}_S)$. If the state of the total system is $W\in
\mathcal{D}(\mathcal{H}_{S+E})$, then all expectation values $\mathrm{tr}%
_{S+E}W(A\otimes I_E)$ can be calculated from the reduced statistical operator $\rho =\mathrm{tr}_EW$ which is an element of $\mathcal{D}(\mathcal{%
H}_S)$, such that $\mathrm{tr}_SA\rho =\mathrm{tr}_{S+E}(A\otimes I_E)W$ holds for all $A\in \mathcal{B}(\mathcal{H}_S)$. We shall refer to the statistical operator $\rho =\mathrm{tr}_EW$ as the “state” of the subsystem.
As mentioned above we assume the usual Hamiltonian dynamics for the total system, i.e. $W(t)=U(t)WU^{+}(t)$ with the unitary group $U(t)$, generated by the total Hamiltonian. Except for the trivial case that $S$ and $E$ do not interact, the dynamics of the reduced statistical operator $$\rho (t)=\mathrm{tr}_EU(t)WU^{+}(t) \label{ss.7}$$ is no longer unitary, and it is exactly this dynamics which can produce effective superselection sectors. More explicitly, the Hamiltonian of the total system can provide a family of projection operators $\left\{
P_m,\,m\in \mathbf{M}\right\} $ which are independent from the initial state, such that the statistical operator behaves like $$\rho (t)\cong \sum_mP_m\rho (t)P_m\;\mbox{ for }\,t\rightarrow \infty .
\label{ss.8}$$ An equivalent statement is that the superpositions between vectors of different sectors $P_m\mathcal{H}_S$ are strongly suppressed. Any mechanism, which leads to this effect, will be called *decoherence*.
In the case of induced continuous superselection rule the asymptotics is more appropriately described in the Heisenberg picture, as stated above. But the decoherence effect is also seen in the Schrödinger picture: $P(\Delta
_1)\rho (t)P(\Delta _2)\rightarrow 0$ for $t\rightarrow \infty $ if $\Delta
_1$ and $\Delta _2$ have a positive distance.
The statement (\[ss.8\]) is so far rather vague since it does not specify the asymptotics. A preliminary definition of a *weak* type of decoherence can be formulated as follows.
\[ss\] The subspaces $P_m\mathcal{H}_S,\,m\in \mathbf{M,}$ are denoted as induced superselection sectors, of the dynamics (\[ss.7\]), if for all observables $A\in \mathcal{B}(\mathcal{H}_S)$ which have no diagonal matrix elements, i.e. $P_mAP_m=O,\,m\in \mathbf{M,}$ the trace $$\mathrm{tr}_{S+E}(A\otimes I_E)U(t)WU^{+}(t)=\mathrm{tr}_SA\rho (t)
\label{ss.9}$$ vanishes if $t\rightarrow \infty $ for all initial states $W\in \mathcal{D}_1
$ of a dense subset $\mathcal{D}_1\subset \mathcal{D}(\mathcal{H}_{S+E})$.
It is possible to give an alternative definition with $\mathcal{D}_1$ substituted by $\mathcal{D}(\mathcal{H}_{S+E})$. These definitions are equivalent, as can be easily seen. Assume the statements of Definition \[ss\] are valid for a family of subspaces $\left\{ P_m\mathcal{H}_S,\,m\in
\mathbf{M}\right\} $, then we can find for any $W\in \mathcal{D}(\mathcal{H}%
_{S+E})$ and any $\varepsilon >0$ a statistical operator $W_1\in \mathcal{D}%
_1$ such that $\left\| W-W_1\right\| _1<\varepsilon $ and\
$\mathrm{tr}%
_{S+E}(A\otimes I_E)U(t)W_1U^{+}(t)\rightarrow 0$ if $t\rightarrow \infty $ for the specified class of observables $A$. Since\
$\left| \mathrm{tr}%
_{S+E}(A\otimes I_E)U(t)(W-W_1)U^{+}(t)\right| <\varepsilon \left\|
A\right\| $ the trace (\[ss.9\]) vanishes if $t\rightarrow \infty $ for all initial states $W\in \mathcal{D}(\mathcal{H}_{S+E})$.
The independence from the initial state justifies the terminology induced ”superselection” rules. The Definition \[ss\] has to be supplemented by statements about the time scale of the convergence. For that purpose the following models are investigated. They indicate the essential role of the initial state – especially of the components affiliated to the environment – to achieve decoherence in sufficiently short time.
Soluble models\[models\]
========================
The first class of the presented models has a discrete superselection structure such that the off-diagonal elements of the statistical operator vanish in trace norm $\left\| .\right\| _1$$$\left\| P_m\rho (t)P_n\right\| _1\rightarrow 0\mbox{ if }\,t\rightarrow
\infty \mbox{ and }\,m\neq n \label{mod.0}$$ for an arbitrary initial state $\rho (0)\in \mathcal{D}(\mathcal{H}_S)$. But the asymptotics is more complicated for the more realistic models investigated in Sects. \[free\] and \[scatt\].
The models of Sects. \[Araki\] and \[free\] have the following structure. The Hilbert space is $\mathcal{H}_{S+E}=\mathcal{H}_S\otimes
\mathcal{H}_E.$ The total Hamiltonian has the form $$H_{S+E}=H_S\otimes I_E+I_S\otimes H_E+V_S\otimes V_E \label{mod.2}$$ where $H_S$ is the Hamiltonian of S, $H_E$ is the Hamiltonian of E, $%
V_S\otimes V_E$ is the interaction term between S and E with self-adjoint operators $V_S$ on $\mathcal{H}_S$ and $V_E$ on $\mathcal{H}_E$. We make the following assumptions
1. The operators $H_S$ and $V_S$ commute, $\left[ H_S,V_S\right] =O,$ hence $\left[ H_S\otimes I_E,V_S\otimes V_E\right] =O.$
2. The operator $V_E$ has an absolutely continuous spectrum.
The assumption 1) is a rather severe restriction, which will be given up in Sect. \[scatt\], where we admit an additional scattering potential $V$, which has not to commute with any of the other operators. The assumption 2) has more technical reasons. It implies that estimates can be derived in the limit $t\rightarrow \infty $ in agreement with Definition \[ss\]. But one can also allow operators with point spectra (as done in [@Zurek:1982]), if the spacing of the eigenvalues is sufficiently small. Then the norm in (\[mod.0\]) is an almost periodic function, and the suppression of this norm takes place only during a finite time interval $0\leq t\leq T$. But $T$ can be large enough for all practical purposes.
The operator $V_S$ has the spectral representation $V_S=\int_{\mathbf{R}%
}\lambda P(d\lambda )$ with a spectral family $\left\{ P(\Delta ),\,\Delta
\subset \mathbf{R}\right\} $ which satisfies (\[sp\]). We shall see that exactly this spectral family determines the superselection sectors. If $V_S$ has a pure point spectrum, then $P(\Delta )$ is a step function with values $%
P_m$, and we can write $$V_S=\sum_m\lambda _mP_m. \label{mod.5}$$ As a consequence of assumption 1) we have $\left[ H_S,P(\Delta )\right] =O$ or $\,\left[ H_S,P_m\right] =O$ for $\Delta \subset \mathbf{R}$ or $m\in
\mathbf{M}$, respectively. The Hamiltonian (\[mod.2\]) has therefore the form (for simplicity we only write the version with the discrete spectrum (\[mod.5\])) $$\begin{aligned}
H_{S+E} &=&H_S\otimes I_E+\sum_mP_m\otimes \Gamma _m\mbox{ with}\,
\label{mod.7} \\
\Gamma _m &=&H_E+\lambda _mV_E. \label{mod.8}\end{aligned}$$ The unitary evolution $U(t):=\exp (-iH_{S+E}t)$ of the total system can be written as\
$\left( \mathrm{e}^{-\imath H_St}\otimes I_E\right)
\sum_mP_m\otimes \mathrm{e}^{-i\Gamma \lambda _mt}$. The calculation of the reduced dynamics (\[ss.7\]) then leads to $$P_m\rho (t)P_n=P_m\mathrm{e}^{-\imath H_St}\left( \mathrm{tr}_E\mathrm{e}%
^{-i\Gamma _mt}W\mathrm{e}^{i\Gamma _nt}\right) \mathrm{e}^{\imath H_St}P_n,
\label{mod.10}$$ where the operators $P_n$ are the projection operators of the spectral representation (\[mod.5\]) of $V_S$. For a factorizing initial state $%
W=\rho \otimes \omega $ with $\rho \in \mathcal{D}(\mathcal{H}_S)$ and a reference state $\omega \in \mathcal{D}(\mathcal{H}_E)$ of the environment, the operator (\[mod.10\]) simplifies to $P_m\rho (t)P_n=P_me^{-iH_St}\rho
e^{iH_St}P_n\,\chi _{m,n}(t)$ with $$\chi _{m,n}(t)=\mathrm{tr}_E\left( \mathrm{e}^{i\Gamma _nt}\mathrm{e}%
^{-i\Gamma _mt}\omega \right) \label{mod.13}$$ and the emergence of dynamically induced superselection rules depends on an estimate of this trace.
The Araki-Zurek models\[Araki\]
-------------------------------
The first soluble models for the investigation of the reduced dynamics have been given by Araki [@Araki:1980] and Zurek [@Zurek:1982], and the following construction is essentially based on these papers. In addition to the specifications made above, we demand that
1. the Hamiltonian $H_E$ and the potential $V_E$ commute, $\left[
H_E,V_E\right] =O.$
We first investigate $P_m\rho (t)P_n$ for a factorizing initial state $%
W=\rho \otimes \omega $. Under the assumption 3) the trace (\[mod.13\]) simplifies to $\chi _{m,n}(t)=\mathrm{tr}_E\left( \mathrm{e}^{-i(\lambda
_m-\lambda _n)V_Et}\omega \right) $. Let $V_E=\int_{\mathbf{R}}\lambda
P_E(d\lambda )$ be the spectral representation of the operator $V_E$. Then, as a consequence of assumption 2), for any $\omega \in \mathcal{D}(\mathcal{H%
}_E)$ the measure $d\mu (\lambda ):=\mathrm{tr}_E\left( P_E(d\lambda
)\,\omega \right) $ is absolutely continuous with respect to the Lebesgue measure, and the function $\chi (t):=\mathrm{tr}\left( \mathrm{e}^{-\imath
V_Et}\omega \right) =\int_{\mathbf{R}}\mathrm{e}^{-i\lambda t}$ $d\mu
(\lambda )$ vanishes if $t\rightarrow \infty $. But to have a decrease which is effective in sufficiently short time, we need an additional smoothness condition on $\omega $ (which does not impose restrictions on the statistical operator $\rho \in \mathcal{D}(\mathcal{H}_S)$ of the system S). If the integral operator, which represents $\omega $ in the spectral representation of $V_E$, is a sufficiently differentiable function (vanishing at the boundary points of the spectrum) we can derive estimates like $\left| \chi (t)\right| \leq C_\gamma (1+\left| t\right| )^{-\gamma }$ with arbitrarily large values of $\gamma $. Such an estimate leads to the upper bound $$|\chi _{m,n}(t)|\leq C_\gamma (1+\delta \left| t\right| )^{-\gamma }
\label{mod.17}$$ if $\left| \lambda _m-\lambda _n\right| \geq \delta >0$, and we obtain an estimate for the norm (\[mod.0\]) $$\left\| P_m\rho (t)P_n\right\| _1\leq C_\gamma (1+\delta \left| t\right|
)^{-\gamma }. \label{mod.1}$$ with arbitrary $\rho (0)\equiv \rho \in \mathcal{D}(\mathcal{H}_S)$. The constants $\gamma >0,$ $\delta >0$ and $C_\gamma >0$ do not depend on $\rho $. Moreover one can achieve large values of $\gamma $ and/or small values of the constant $C_\gamma $ if the reference state $\omega $ is sufficiently smooth.
These results depend on the reference state $\omega $ only via the decrease of $\chi (t)$. We could have chosen a more general initial state $W\in
\mathcal{D}(\mathcal{H}_{S+E})$$$W=\sum_\mu c_\mu \,\rho _\mu \otimes \omega _\mu \label{mod.18}$$ with $\rho _\mu \in \mathcal{D}(\mathcal{H}_S),\,\omega _\mu \in \mathcal{D}(%
\mathcal{H}_E)$ and numbers $c_\mu \in \mathbf{R}$ which satisfy $\sum_\mu
\left| c_\mu \right| <\infty $ and $\sum_\mu c_\mu =\mathrm{tr}\,W=1$. As a consequence of assumption 2) the space $\mathcal{H}_E$ has infinite dimension. If $\mathcal{H}_S$ is finite dimensional, the set (\[mod.18\]) of statistical operators covers the whole space $\mathcal{D}(\mathcal{H}%
_{S+E})$. If also $\mathcal{H}_S$ is infinite dimensional, this set is dense in $\mathcal{D}(\mathcal{H}_{S+E})$. With the arguments given above for factorizing initial states the statement of Definition \[ss\] can be derived for all initial states (\[mod.18\]), and the sectors $P_n\mathcal{H%
}_S$ are induced superselection sectors in the sense of this definition. Moreover, assuming that the components of the statistical operator $W$ affiliated to the environment are sufficiently smooth functions in the spectral representation of $V_E$, the sum $\sum_\mu \left| c_\mu \,\mathrm{tr%
}_E\left( \mathrm{e}^{-\imath (\lambda _m-\lambda _n)V_Et}\omega _\mu
\right) \right| $ satisfies a uniform estimate (\[mod.17\]), and (\[mod.1\]) is still valid. Hence the time scale of the decoherence can be as short as we want without restriction on $\rho (0)=\mathrm{tr}_EW=\sum_\mu
c_\mu \,\rho _\mu $.
If the potential $V_S$ has a (partially) continuous spectrum with spectral family\
$\left\{ P(\Delta ),\,\Delta \subset \mathbf{R}\right\} $, an estimate $$\left\| P(\Delta _1)\rho (t)P(\Delta _2)\right\| _2\leq C_\gamma (1+\delta
\left| t\right| )^{-\gamma } \label{mod.1c}$$ can be derived in the weaker Hilbert-Schmidt norm for arbitrary intervals $%
\Delta _1$ and $\Delta _2$ which have a non-vanishing distance, see Sect. 7.6 of [@GJKKSZ:1996].
The interaction with free fields: the role of infrared divergence for induced superselection sectors\[free\]
------------------------------------------------------------------------------------------------------------
In this section we give up the restriction 3) on the Hamiltonian. Then the estimate of the trace (\[mod.13\]) needs more involved calculations. As specific example we consider an environment given by a free Boson field. Such models can be calculated explicitly, and they have often been used as the starting point for Markov approximations.
As Hilbert space $\mathcal{H}_E$ we choose the Fock space based on the one particle space $\mathcal{H}^{(1)}=\mathcal{L}^2(\mathbf{R}_{+})$ with inner product $\left\langle f\mid g\right\rangle =\int_0^\infty \overline{f(k)}%
g(k)dk$. The one-particle Hamilton operator, denoted by $\widehat{%
\varepsilon }$, is the multiplication operator $\left( \widehat{\varepsilon }%
f\right) (k):=\varepsilon (k)f(k)$ with the energy function $\ \varepsilon
(k)=c\cdot k,\,c>0,\,k\in \mathbf{R}_{+}$, defined for all functions $f$ with $(1+\varepsilon (k))f(k)\in \mathcal{L}^2(\mathbf{R}_{+})$. The creation/annihilation operators $a_k^{+}$ and $a_k$ are normalized to $%
\left[ a_k,a_{k^{\prime }}^{+}\right] =\delta (k-k^{\prime })$. The Hamiltonian of the environment is then $$H_E=\int_0^\infty \varepsilon (k)a_k^{+}a_kdk. \label{free.1}$$ With $a^{+}(f)=\int_0^\infty f(k)a_k^{+}dk$ and $a(f)=\int_0^\infty
f(k)a_kdk $ we define field operators by $\Phi (f):=2^{-\frac 12}\left(
a^{+}(f)+a(f)\right) $ for real functions $f\in \mathcal{L}^2(\mathbf{R}%
_{+}) $. The interaction potential is chosen as $V_E=\Phi (f)$ with $$f\in \mathcal{L}^2(\mathbf{R}_{+})\mbox{ and }\,\widehat{\varepsilon }%
^{-1}f\in \mathcal{L}^2(\mathbf{R}_{+}), \label{free.2a}$$
An example for the total Hamiltonian is given by a single particle coupled to the quantum field with velocity coupling $$\begin{array}{ll}
H_{S+E} & =\frac 12P^2\otimes I_E+P\otimes \Phi (f)+I_S\otimes H_E \\
& =\frac 12\left( P\otimes I_E+I_S\otimes \Phi (f)\right) ^2+I_S\otimes
\left( H_E-\frac 12\Phi ^2(f)\right)
\end{array}
\label{free.3}$$ If the test function $f$ satisfies $\left\| \widehat{\varepsilon }^{-\frac
12}f\right\| <2^{-\frac 12}$, the Hamiltonian $H_E-\frac 12\Phi ^2(f)$ is bounded from below, and consequently $H_{S+E}$ is bounded from below. Since the particle is coupled to the free field with $V_S=P$, the reduced dynamics yields continuous superselection sectors for the momentum $P$ of the particle.
The operators (\[mod.8\]) $\Gamma _m$ are substituted by $H_\lambda
:=H_E+\lambda \Phi (f),\;\lambda \in \mathbf{R}$, which are Hamiltonians of the van Hove model [@Hove:1952]. The restrictions (\[free.2a\]) are necessary to guarantee that all operators $H_\lambda ,\,\lambda \in \mathbf{%
R,}$ are unitarily equivalent and defined on the same domain. To derive induced superselection sectors we have to estimate the time dependence of the traces $\chi _{\alpha \beta }(t):=\mathrm{tr}_EU_{\alpha \beta
}(t)\omega ,\,\alpha \neq \beta ,$ where the unitary operators $U_{\alpha
\beta }(t)$ are given by $$U_{\alpha \beta }(t):=\exp (iH_\alpha t)\exp (-iH_\beta t), \label{free.4}$$ see (\[mod.13\]). In the Appendix we prove the following results for states $\omega $ which are mixtures of coherent states.
1. Under the restrictions (\[free.2a\]) the traces $\chi _{\alpha
\beta }(t),\,\alpha \neq \beta ,$ do not vanish for $t\rightarrow \infty .$
2. If $\Phi (f)$ has contributions at arbitrarily small energies, $%
\chi _{\alpha \beta }(t)$ can nevertheless strongly decrease for $\alpha
\neq \beta $ within a very long time interval $0\leq t\leq T$. Estimates like (\[mod.1\]) or (\[mod.1c\]) are substituted by $\left\| P_m\rho
(t)P_n\right\| \leq f(t)$ or $\left\| P(\Delta )\rho (t)P(\Delta ^{\prime
})\right\| _2\leq f(t)$. But in contrast to (\[mod.1\]) or (\[mod.1c\]) the function $f(t)$ increases again for $t>T$.
3. For fixed $\alpha \neq \beta $ a limit $\chi _{\alpha \beta
}(t)\rightarrow 0$ for $t\rightarrow \infty $ is possible if $\widehat{%
\varepsilon }^{-1}f\in \mathcal{L}^2(\mathbf{R}_{+})$ is violated, i.e. in the case of infrared divergence.
A large infrared contribution is therefore essential for the emergence of induced superselection sectors. As in Sect. \[Araki\] the choice of the initial state $W$ of the total system can be extended to (\[mod.18\]) with $\rho _\mu \in \mathcal{D}(\mathcal{H}_S)$ and mixtures of coherent states $\omega _\mu \in \mathcal{D}(\mathcal{H}_E)$. This class of states is again dense in $\mathcal{D}(\mathcal{H}_{S+E})$, and, at least in the infrared divergent case, we obtain induced superselection sectors in the sense of Definition \[ss\].
Models with scattering\[scatt\]
-------------------------------
For the models presented in Sects. \[Araki\] and \[free\] the projection operators onto the effective superselection sectors $P_m\otimes I_S$ (or $%
P(\Delta )\otimes I_S$) commute with the total Hamiltonian. We now modify the Hamiltonian (\[mod.2\]) to $$H=H_{S+E}+V=H_S\otimes I_E+I_S\otimes H_E+V_S\otimes V_E+V$$ where the operator $V$ is only restricted to be a *scattering* potential. This restriction means that the wave operator $\Omega
=\lim_{t\rightarrow \infty }e^{iHt}e^{-iH_{S+E}t}$ exists as strong limit. To simplify the arguments we assume that there are no bound states such that the convergence is guaranteed on $\mathcal{H}_{S+E}$ with $\Omega
^{+}=\Omega ^{-1}$. Then the time evolution $U(t)=\exp (-iHt)$ behaves asymptotically as $U_0(t)\Omega ^{+}$ with $U_0(t)=\exp (-iH_{S+E}t).$ More precisely, we have for all $W\in \mathcal{D(H}_{S+E}\mathcal{)}$$$\lim_{t\rightarrow \infty }\,\left\| U(t)WU^{+}(t)-U_0(t)\Omega ^{+}W\Omega
U_0^{+}(t)\right\| _1=0 \label{scatt.1}$$ in trace norm. Following Sect. \[Araki\] the reduced trace $\mathrm{tr}%
_EU_0(t)\Omega ^{+}W\Omega U_0^{+}(t)$ produces the superselection sectors $%
P_m\mathcal{H}_S$ which are determined by the spectrum (\[mod.5\]) of $V_S$. The asymptotics (\[scatt.1\]) then yields (in the sense of Definition \[ss\]) the same superselection sectors for $\rho (t)=\mathrm{tr}%
_EU(t)WU^{+}(t)$. Moreover we can derive fast decoherence by additional assumptions on the initial state and on the potential. For that purpose we start with a factorizing initial state $W=\rho (0)\otimes \omega $ with smooth $\omega .$ To apply the arguments of Sect. \[Araki\] to the dynamics $U_0(t)\Omega ^{+}W\Omega U_0^{+}(t)$ the statistical operator $%
\Omega ^{+}\left( \rho \otimes \omega \right) \Omega $ has to be a sufficiently smooth operator on the tensor factor $\mathcal{H}_E$ for all $%
\rho \in \mathcal{D}(\mathcal{H}_S)$. That is guaranteed if we choose as scattering potential a smooth potential in the sense of Kato [@Kato:1966]. Then both the limits, (\[scatt.1\]) and\
$\lim_{t\rightarrow \infty
}\left\| P_m\left( \mathrm{tr}_EU_0(t)\Omega ^{+}W\Omega U_0^{+}(t)\right)
P_n\right\| _1=0,\,m\neq n,$ are reached in sufficiently short time. Hence $%
\rho (t)$ can decohere fast into the subspaces $P_m\mathcal{H}_S$ which are determined by the spectrum (\[mod.5\]) of $V_S$. But in contrast to (\[mod.1\]) one does not obtain a uniform bound with respect to the initial state $\rho (0)$, since the limit (\[scatt.1\]) is not uniform in $W\in
\mathcal{D(H}_{S+E}\mathcal{)}$.\
Remark. The restriction that $V$ is a scattering potential is essential. The dominating part $V_S\otimes V_E$ of the interaction $V_S\otimes V_E+V$ still satisfies the assumption 1). In [@Kupsch:1996a] a spin model with an interaction which violates both the constraints, the assumption 1) and the scattering condition, has been investigated. That model can produce superselection sectors only in an approximative sense, where the lower bounds on $\left\| P_m\rho
(t)P_n\right\| _1$ depend on the magnitude of the non-vanishing commutator.
Concluding remarks
------------------
The investigation of the models proves that the uniform emergence (\[mod.1\]) or (\[mod.1c\]) of effective superselection sectors is consistent with the mathematical rules of quantum mechanics. But this result depends on rather restrictive assumptions on the Hamiltonian. For the more realistic model of a quantum field presented in Sect. \[free\] the suppression persists only for a finite period of time. If the low frequency spectrum dominates, this period of time can be sufficiently large for all practical purposes. Only in the limit of infrared divergence the induced superselection sectors persist for $t\rightarrow \infty $. If there is additional scattering as considered in Sect. \[scatt\] the superselection sectors still exist. But the estimates are no longer uniform in the initial state $\rho (0)$ of the system.
For all these models the induced superselection sectors are fully determined by the Hamiltonian in the sense of Definition \[ss\]. The initial state of the total system, especially the smoothness properties of the components related to the environment, determine the time scale in which these sectors emerge.
The van Hove model\[infra\]
===========================
As Hilbert space $\mathcal{H}_E$ we take the Fock space $\mathcal{F}(%
\mathcal{H}^{(1)})$ based on the one-particle space $\mathcal{H}^{(1)}=%
\mathcal{L}^2(\mathbf{R}_{+})$. For test functions $f,g\in \mathcal{S}(%
\mathbf{R}_{+})\subset \mathcal{L}^2(\mathbf{R}_{+})$ the creation and annihilation operators $a^{+}(f)=\int_0^\infty f(k)a^{+}(k)dk$ and $%
a(g)=\int_0^\infty g(k)a(k)dk$ are normalized to $\left[
a(f),a^{+}(g)\right] =\left\langle f\mid g\right\rangle =\int_0^\infty
\overline{f(k)}g(k)dk$. The test functions $f\in \mathcal{S}(\mathbf{R}_{+})$ are rapidly decreasing $C^\infty $-functions with a support restricted to $%
\mathbf{R}_{+}=\left[ 0,\infty \right) $. The one-particle Hamiltonian of the free field is $\left( \widehat{\varepsilon }f\right) (k):=\varepsilon
(k)f(k)$ with the energy function $\ \varepsilon (k)=c\cdot k,\,c>0,$ for $%
k\geq 0$. Actually we can choose any positive monotonically increasing and polynomially bounded energy function $\varepsilon (k)$, which has excitations of arbitrarily small energy, $\varepsilon (k)/\left| k\right|
\rightarrow c>0$ if $k\rightarrow 0$. The Hamiltonian of the free field is then (\[free.1\]), and as canonical field and momentum operators we choose $\Phi (f):=\frac 1{\sqrt{2}}\left( a^{+}(f)+a(f)\right) $ and $\Pi
(f):=\frac i{\sqrt{2}}\left( a^{+}(f)-a(f)\right) .$ For real test functions we define the Weyl operators $$T(f,g):=\exp \left( -i\Pi (f)-i\Phi (g)\right) =\exp \left( -i\Pi (f)\right)
\exp \left( -i\Phi (g)\right) \mathrm{e}^{-i\left( f\mid g\right) /2}
\label{f.4}$$ These operators satisfy the Weyl relations $$T(f_1,g_1)T(f_2,g_2)=T(f_1+f_2,g_1+g_2)\mathrm{e}^{i\left( (f_1\mid
g_2)-(f_2\mid g_1)\right) /2} \label{f.5}$$ and their expectation value in the vacuum state $\Omega $ is $$\left\langle \Omega \mid T(f,g)\Omega \right\rangle =\exp \left( -\frac
14\left\| f\right\| ^2-\frac 14\left\| g\right\| ^2\right) . \label{f.5a}$$ With $U(t)=\exp (-iH_Et)$ the time evolution of the Weyl operators is $$U(-t)T(f,g)U(t)=T\left( \cos (\widehat{\varepsilon }t)\,f+\sin (\widehat{%
\varepsilon }t)\,g,\,\cos (\widehat{\varepsilon }t)\,g-\sin (\widehat{%
\varepsilon }t)\,f\right) . \label{f.6}$$ If the one-particle Hilbert space $\mathcal{L}^2(\mathbf{R}_{+})$ is restricted to the one dimensional space $\mathbf{C,}$ all these formulas become formulas of the one dimensional harmonic oscillator of frequency $%
\widehat{\varepsilon }=\ \varepsilon >0$.
The Weyl operator $T(f,0)$ is a translation operator $$\begin{aligned}
T(f,0)H_ET(-f,0) &=&\int_0^\infty \varepsilon (k)\left( a^{+}(k)+\frac 1{%
\sqrt{2}}f(k)\right) \left( a(k)+\frac 1{\sqrt{2}}f(k)\right) dk \nonumber
\\
\ &=&H_E+\Phi (\widehat{\varepsilon }f)+\frac 12\int \varepsilon (k)\left|
f(k)\right| ^2dk.\end{aligned}$$ Hence $T(\widehat{\varepsilon }^{-1}f,0)H_ET(-\widehat{\varepsilon }%
^{-1}f,0)-\frac 12\left\| \widehat{\varepsilon }^{-\frac 12}f\right\|
^2=H_E+\Phi (f)$ is the Hamiltonian of the van Hove model [@Hove:1952], see also [@Cook:1961], [@Berezin:1966] p. 166ff, and [@Emch:1972]. The operator $T(\widehat{\varepsilon }^{-1}f,0)$ is well defined if\
$%
\widehat{\varepsilon }^{-1}f\in \mathcal{L}^2(\mathbf{R}_{+}).$ The operator $\Phi (f),\,f\in \mathcal{L}^2(\mathbf{R}_{+})$, is $H_E$-bounded with relative bound smaller than one (in the sense of the Kato-Rellich theorem, see e.g. [@Reed/Simon:1975]) if $\widehat{\varepsilon }^{-\frac 12}f\in
\mathcal{L}^2(\mathbf{R}_{+})$, and $\Phi ^2(f)$ is $H_E$-bounded with relative bound smaller than one, if in addition $\left\| \widehat{%
\varepsilon }^{-\frac 12}f\right\| <2^{-1}$ holds. Hence the operators $H_E$ and $H_E+\lambda \Phi (f),\,\lambda \in \mathbf{R}$, are self-adjoint on the same domain of the Fock space if $f\in \mathcal{L}^2(\mathbf{R}_{+})$ and $%
\widehat{\varepsilon }^{-\frac 12}f\in \mathcal{L}^2(\mathbf{R}_{+})$, and moreover, the operator $H_E-\frac 12\Phi ^2(f)$ is bounded from below, if $%
\left\| \widehat{\varepsilon }^{-\frac 12}f\right\| <2^{-\frac 12}$.
The trace (\[mod.13\]) is now calculated for the model of Sect. \[free\] with the one parameter family of Hamiltonians $$H_\lambda :=T(\lambda \widehat{\varepsilon }^{-1}f,0)H_ET(-\lambda \widehat{%
\varepsilon }^{-1}f,0)-\frac{\lambda ^2}2\left\| \widehat{\varepsilon }%
^{-\frac 12}f\right\| ^2=H_E+\lambda \Phi (f),\;\lambda \in \mathbf{R.}$$ As stated above these operators are well defined if (\[free.2a\]) holds. In the following $\simeq $ indicates an identity up to a phase factor. The unitary operators (\[free.3\]) can be evaluated with the help of the Weyl relations (\[f.5\]) and the time evolution (\[f.6\]) $$\begin{array}{ll}
U_{\alpha \beta }(t) & \simeq T(\alpha \widehat{\varepsilon }%
^{-1}f,0)U(-t)T((\beta -\alpha )\widehat{\varepsilon }^{-1}f,0)U(t)T(-\beta
\widehat{\varepsilon }^{-1}f,0) \\
& \simeq T\left( (\alpha -\beta )\widehat{\varepsilon }^{-1}(1-\cos \widehat{%
\varepsilon }t)\,f,\,(\alpha -\beta )(\widehat{\varepsilon }^{-1}\sin
\widehat{\varepsilon }t)\,f\right) .
\end{array}$$ We only investigate the trace $\mathrm{tr}_EU_{\alpha \beta }(t)\omega $ for states $\omega $ which are mixtures of coherent states. Then the traces decompose into sums of matrix elements of (\[free.3\]) between coherent states $T(f_n,g_n)\Omega $ with $f_n,\,g_n\in \mathcal{L}^2(\mathbf{R}%
_{+}),\,n=1,2,...$. These matrix elements can be calculated with the help of (\[f.6\]) and the Weyl relations (\[f.5\]). Following (\[f.5a\]) the modulus of a matrix element is an exponential of the type $$\exp \left( -\frac 14\left\| a+(\alpha -\beta )(1-\cos \widehat{\varepsilon }%
t)\left( \widehat{\varepsilon }^{-1}f\right) \right\| ^2-\frac 14\left\|
b+(\alpha -\beta )\left( \sin \widehat{\varepsilon }t\right) \left( \widehat{%
\varepsilon }^{-1}f\right) \right\| ^2\right) \label{f.13}$$ where $a=f_m-f_n\in \mathcal{L}^2(\mathbf{R}_{+})$ and $b=g_m-g_n\in
\mathcal{L}^2(\mathbf{R}_{+})$ are fixed. Under the restrictions (\[free.2a\]) the norms of $(1-\cos \widehat{\varepsilon }t)\left( \widehat{%
\varepsilon }^{-1}f\right) $ and $\left( \sin \widehat{\varepsilon }t\right)
\left( \widehat{\varepsilon }^{-1}f\right) $ are uniformly bounded, and (\[f.13\]) cannot vanish for $t\rightarrow \infty $.
But nevertheless, since $\varepsilon ^{-2}(1-\cos \varepsilon t)\rightarrow
\frac 12t^2$ and $\varepsilon ^{-1}\sin \varepsilon t\rightarrow t$ if $%
\varepsilon \rightarrow 0$, the norms $\left\| (1-\cos \widehat{\varepsilon }%
t)\left( \widehat{\varepsilon }^{-1}f\right) \right\| $ and $\left\| \left(
\sin \widehat{\varepsilon }t\right) \left( \widehat{\varepsilon }%
^{-1}f\right) \right\| $ may become as large as we want at intermediate times, if $f(k)$ has large contributions at small values of $k$. The estimate (\[f.13\]) for the matrix elements is then negligible for a long period of time (for arbitrary vectors $a$ and $b$ within some bounded domain).
Only if we give up the second constraint in (\[free.2a\]), we can find test functions $f$ such that the norms of $(1-\cos \widehat{\varepsilon }%
t)\left( \widehat{\varepsilon }^{-1}f\right) $ and $\left( \sin \widehat{%
\varepsilon }t\right) \left( \widehat{\varepsilon }^{-1}f\right) $ increase indefinitely for $t\rightarrow \infty $ and (\[f.13\]) vanishes in this limit.
That behaviour can be illustrated by the coupling to a free particle. As already mentioned we can restrict the one-particle space $\mathcal{H}^{(1)}$ to the one dimensional space $\mathbf{C}$, and the free field becomes a harmonic oscillator of frequency $\widehat{\varepsilon }=\varepsilon >0$. In that case (\[f.13\]) is a periodic function of $t\in \mathbf{R}$. In the (singular) limit $\varepsilon \rightarrow 0$ we obtain functions $%
\varepsilon ^{-2}(1-\cos \varepsilon t)\rightarrow \frac 12t^2$ and $%
(\varepsilon ^{-1}\sin \varepsilon t)\rightarrow t$ which increase beyond any bound for $t\rightarrow \infty .$ This limit case corresponds to the Hamiltonian of a free particle $$H_E=\frac 12P^2\mbox{ with coupling }\,V_E=Q, \label{f.15}$$ and $\mathrm{tr}_EU_{\alpha \beta }(t)\omega $ can be calculated by standard methods, see the article [@Pfeifer:1980] of Pfeifer, who has used this model to discuss the measurement process of a spin. With (\[f.15\]) the Hamiltonian (\[mod.2\]) of the total system is unbounded from below (corresponding to infrared divergence in the field theoretic model) and we have $\mathrm{tr}_EU_{\alpha \beta }(t)\omega \rightarrow 0$ if $%
t\rightarrow \infty $ for $\alpha \neq \beta $ and for all statistical operators $\omega $ of the free particle.
[10]{}
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L. [van]{} Hove. Les difficult[é]{}s de divergences pour un mod[è]{}le particulier de champ quantifi[é]{}. , 18:145–159, 1952.
E. Joos and H. D. Zeh. The emergence of classical properties through interaction with the environment. , B59:223–243, 1985.
T. Kato. Wave operators and similarity for some non-selfadjoint operators. , 162:258–279, 1966.
J. Kupsch. The structure of the quantum mechanical state space and induced superselection rules. , 1996. quant-ph/9612033.
R. Mirman. Nonexistence of superselection rules: Definition of term [*frame of reference*]{}. , 9:283–299, 1979.
A. Peres. . Kluwer, Dordrecht, 1995.
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C. Piron. Les r[é]{}gles de supers[é]{}lection continues. , 42:330–338, 1969.
M. Reed and B. Simon. . Academic Press, New York, 1975.
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H. D. Zeh. On irreversibility of time and observation in quantum theory. In B. D’Espagnat, editor, [*[Foundations of Quantum Mechanics]{}*]{}, pages 263–273, New York, 1971. Academic Press.
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[^1]: Extended version of a talk presented at the 7th UK Conference on Mathematical and Conceptual Foundations of Modern Physics, Nottingham 7 - 11 September 1998
[^2]: e-mail: kupsch@physik.uni-kl.de
|
---
abstract: 'The Transiting Exoplanet Survey Satellite (TESS) has a goal of detecting small planets orbiting stars bright enough for mass determination via ground-based radial velocity observations. Here we present estimates of how many exoplanets the TESS mission will detect, physical properties of the detected planets, and the properties of the stars that those planets orbit. This work uses stars drawn from the TESS Input Catalog Candidate Target List and revises yields from prior studies that were based on Galactic models. We modeled the TESS observing strategy to select approximately 200,000 stars at 2-minute cadence, while the remaining stars are observed at 30-min cadence in full-frame image data. We placed zero or more planets in orbit around each star, with physical properties following measured exoplanet occurrence rates, and used the TESS noise model to predict the derived properties of the detected exoplanets. In the TESS 2-minute cadence mode we estimate that TESS will find $1250\pm70$ exoplanets (90% confidence), including 250 smaller than 2 Earth-radii. Furthermore, we predict an additional 3100 planets will be found in full-frame image data orbiting bright dwarf stars and more than 10,000 around fainter stars. We predict that TESS will find 500 planets orbiting M-dwarfs, but the majority of planets will orbit stars larger than the Sun. Our simulated sample of planets contains hundreds of small planets amenable to radial velocity follow-up, potentially more than tripling the number of planets smaller than 4 Earth-radii with mass measurements. This sample of simulated planets is available for use in planning follow-up observations and analyses.'
author:
- Thomas Barclay
- Joshua Pepper
- 'Elisa V. Quintana'
title: 'A Revised Exoplanet Yield from the Transiting Exoplanet Survey Satellite (TESS)'
---
Introduction
============
While we have known that planets orbit stars other than the Sun since the late 20th Century [@Walker1988; @Latham1989; @Wolszczan1992; @Mayor1995], it is only with the launch of the Kepler spacecraft in 2009 [@Koch2010; @Borucki2010] that we have been able to estimate the occurrence rates of terrestrial worlds. While there is not a firm consensus on the details of how common planets are as a function of size and orbital period [@Howard2010; @Gould2010; @Catanzarite2011; @Youdin2011; @Howard2012; @Traub2012; @Bonfils2013; @Swift2013; @Fressin2013; @Petigura2013a; @Petigura2013b; @Montet2014; @Kane2014; @Foreman2014; @Burke2015; @Clanton2016; @Hsu2018] it is clear that exoplanets overall are fairly commonplace, particularly orbiting the coolest of stars [@Dressing2013; @Dressing2015; @Morton2014; @Mulders2015].
Although we have a fairly large sample of planets with orbital periods of less than a few hundred days, there is still a pressing need to detect planets that are readily characterizable. The primary goal of the Transiting Exoplanet Survey Satellite (TESS), a mission led by the Massachusettes Institute of Technology, is to find small planets that are most amenable for mass measurements through precise radial velocity observations [@Ricker2015; @Ricker2016; @Collins2018]. A secondary, although unofficial, mission goal is to find targets that can be characterized through transmission spectroscopy from the James Webb Space Telescope and other future observatories.
TESS launched on April 18, 2018, and resides in an elliptical 13.7 day high-Earth orbit during a 2-year primary mission. TESS has four cameras, each with a $24^{\circ}\times24^{\circ}$ field of view. The cameras are aligned to provide continuous coverage of $96^{\circ}\times 24^{\circ}$, which is maintained for 27.4 days per pointing (known as a sector). The long axis of the observing region is aligned with a fixed ecliptic longitude, with the boresight of the fourth camera centered on the ecliptic pole, as shown in Figure \[fig:sectors\]. Every two orbits, TESS rotates $\sim$28$^{\circ}$ about the ecliptic pole. In year 1 of the mission, the spacecraft will survey 13 sectors in the southern ecliptic hemisphere, before spending year 2 in the northern ecliptic hemisphere. About 60% of the sky will be covered by a single sector of TESS observations, and a further 15% will be observed over two sectors, located in the overlap areas between two adjacent sectors. Most stars within 12 degrees of the ecliptic poles will be within the TESS continuous viewing zone (CVZ) and observable for more than 300 days (this accounts for approximately 1% of the sky per pole). Over the course of the prime mission, TESS will observe approximately 85% of the sky.
![An illustration showing the first three sectors of the TESS observing plan.[]{data-label="fig:sectors"}](tessectors.pdf){width="45.00000%"}
The TESS mission is focused on detecting small transiting planets that orbit bright stars. Although the dwell time over most of the sky is too short to permit the detection of planets in temperate orbits, that goal can be advanced by discovering planets orbiting cooler stars, especially in the TESS CVZ around the ecliptic poles.
Two observing modes will be initially implemented: the $96^{\circ} \times24^\circ$ full-frame image (FFI) will be recorded every 30-minutes, while approximately 200,000 stars will be preselected to have data recorded at 2-minute cadence. In either case, the system is integrating and reading out every 2 seconds; they differ in the number of coadds.
It is essential that a *reasonable* prediction for the scientific yield of TESS is available because (a) planning follow-up resources requires knowing the properties of the planets we might find [@Louie2018; @Crouzet2017; @Collins2018; @Kempton2018], (b) we can perform trade studies on target prioritization schemes for the 2-minute cadence targets [@Bouma2017; @Stassun2017 Pepper et al. in preparation], and when designing data analysis algorithms [@Kipping2017; @Lund2017; @pyke3; @Oelkers2018], and (c) we can manage the expectations of the scientific community and the public.
A TESS yield simulation created by @Sullivan2015 has been the standard used by both the mission team and the community. Since then, two papers have built on the work of @Sullivan2015 to refine the total mission yield and explore extended mission scenarios [@Bouma2017], and to improve estimates of the planet yield from M-dwarfs [@Ballard2018]. However, @Sullivan2015 simulations were based on a simulated stellar population rather than real stars, and used an earlier hardware configuration that provided for greater storage and downlink limits than the flight hardware being used. Therefore, now is the time to revise the TESS yield estimate using new information. Here we report on a new estimate of the exoplanet yield using the TESS Input Catalog (TIC) Candidate Target List (CTL), the same list that is used by the mission to select stars and perform photometry.
Simulating stars, planets, and detections
=========================================
The process we used to derive a population of planets detectable by TESS uses a Monte Carlo method to (1) simulate the population of stars that TESS will observe, (2) place planets in orbit around these stars, and (3) predict how many of these planets TESS will detect.
Star selection {#sec:starselction}
--------------
The first step was made relatively straightforward by the availability of the CTL - a prioritized list of target stars that the TESS Target Selection Working Group have determined represent the stars most suitable for detection of small planets by TESS. The properties of about 500 million stars were assembled in the TIC [@Stassun2017], and the CTL includes several million of those stars that are most suitable for small transit detection. We used CTL version 6.1[^1], which includes 3.8 million stars with properties such as stellar radii, masses, distances, and apparent brightness in various bandpasses. The CTL stars were then ranked using a simple metric based on stellar brightness and radius, along with the degree of blending and flux contamination (especially important given the large TESS pixels). The CTL does not include all stars. Save for stars on specially curated target lists [e.g. @Muirhead2018], stars with reduced proper motions that indicate they are red giants [@Collier2007], stars with a temperature below 5500 K and a TESS magnitude fainter than 12, or stars with temperature above 5500 K and a TESS magnitude fainter than 13, are excluded from the CTL. Such broad cuts were required in order to assemble a small enough population of stars to practically manage.
We then determined which of these stars are likely to be observed by the mission. We used [*tvguide*]{} [@Mukai2017] on each star to determine whether and for how long it is observable with TESS. We arbitrarily selected a central ecliptic longitude for the first sector of 277$^\circ$ which equates to an antisolar date of June 28 (the precise timing of the first sector is dependent on commissioning duration). Until we have on-orbit measurements of focal plane geometry, [*tvguide*]{} assumes that the cameras are uniform square detectors projected on the sky, placed precisely 24$^\circ$ apart in ecliptic latitude and with identical ecliptic longitude. Gaps between CCDs are assumed to be 0.25$^\circ$. We ended up with a total of 3.18 million individual stars on silicon.
We also needed to simulate which of these stars are likely to be observed at 2-minute cadence and ensure compliance with the TESS mission requirement that states that over the 2-year mission over 200,000 total stars should be targeted, and 10,000 stars should be observed for at least 120 days. It is somewhat less trivial than one would initially assume to simulate this requirement because we could not simply select the top 200,000 stars with the highest priority in the CTL because this would place far too many stars in the CVZ than can actually be observed there at 2-minute cadence. To ensure a realistic distribution of targets, we first divided each ecliptic hemisphere into 15 sections: a polar section with everything within 13 degrees of the pole, representing stars that primarily fall into Camera 4; an ecliptic section including everything within 6 degrees of the ecliptic to represent stars that are not observed in the prime mission; and then the remaining area was divided longitudinally into 13 northern and 13 southern adjacent sections, representing stars observable with Cameras 1–3 in Sectors 1–26. This yielded a total of 28 sections of the sky with observable stars.
A star that fell in a camera overlap region is observed in multiple sectors but only represented one unique target. We found that we could make a reasonable approximation to satisfy the requirements of 200,000 unique targets if in each polar section we selected the 6,000 stars in that region with the highest priority in the CTL, and then for each longitudinal section (representing the footprint of Cameras 1–3 in each sector) we selected the 8,200 highest priority stars in each of the regions. After removing stars that fall into CCD and camera gaps, this yielded 214,000 unique stars. We assumed that any star in an overlap region is observed in every possible sector.
![The number of CTL targets observed for a given number of 27.4-day sectors. FFI targets are shown in blue, and 2-minute cadence targets in red. In total 3.2M CTL targets are observed, of which 214,000 are observed at 2-minute cadence. Roughly three-quarters of targets are only observed for a single sector, with just 2.1% having 12 or 13 sectors of coverage. The 2-minute cadence targets are disproportionately observed for more sectors, with 4.2% of the 2-minute cadence targets receiving 12 or 13 sectors of coverage.[]{data-label="fig:stars-sectors"}](stars-sectors-both.png){width="45.00000%"}
While the CTL includes a great deal of curation, it is not infallible. A particular weakness inherent to stellar catalogs based on photometric colors is in distinguishing between dwarf stars and subgiants [@Huber2014; @Mathur2017]. CTL versions up through 6.2 use parallax information when available to determine stellar radii (and therefore luminosity class), but the vast majority of stars depend on the use of reduced proper motion (RPM) cuts to distinguish dwarfs from giants. While GAIA DR2 will shortly provide reliable parallaxes for most CTL stars [@Huber2017; @Davenport2017; @Stassun2018], the CTL will not be significantly modified until 2019. Furthermore, while the RPM method is highly reliable at distinguishing dwarfs and subgiants as a group from giant stars, it is generally not useful for distinguishing dwarfs from subgiants. Of the CTL stars that are classified as dwarfs based on the RPM cut, about 40% are actually subgiants, although roughly 35% of the CTL stars have parallax measurements confirming their spectral class. To account for this effect, we simulated a misclassified population of subgiants by increasing the stellar radius of 40% of those AFGK stars which had been selected with the RPM cut by a factor 2, with the affected stars drawn at random. That included 25% of all the AFGK stars in the CTL. This approach somewhat overestimates the radii of A-type subgiants but the effect on total planet yield is limited, because A-type stars have large radii, making detecting transiting planets challenging, and thus are already a relatively small fraction of the high-priority CTL stars.
Simulating planets
------------------
To each star in our list we assigned zero or more planets. The number of planets assigned to each star was drawn from a Poisson distribution. The mean (referred to here as $\lambda$) of the Poisson distribution we used differs between AFGK-dwarf stars and M-dwarfs because there is strong evidence that M-dwarfs host more planets on short orbital periods [@Mulders2015; @Burke2015]. For AFGK stars we used the average number of planets per star with orbital periods of up to 85 days of $\lambda=0.689$ [@Fressin2013], while for M-stars $\lambda=2.5$ planets are reported with orbital periods up to 200 days [@Dressing2015].
Each planet was then assigned six physical properties drawn at random: an orbital period ($P$), a radius ($R_p$), an eccentricity, a periastron angle, an inclination to our line of sight ($i$), and a mid-time of first transit. The orbital period and radius were selected using the exoplanet occurrence rate estimate of @Fressin2013 for AFGK stars, and @Dressing2015 for M-stars. Both @Fressin2013 and @Dressing2015 reported occurrence rates in period/radius bins. We drew at random from each of these bins with the probability to draw from a given bin weighted by the occurrence rate in that bin divided by the total occurrence rate of planets. For example, @Dressing2015 reported a 4.3% occurrence rate for planets with radii 1.25–2.0 $R_\oplus$ and orbital period 10–17 days, so in our simulation we drew planets from that bin with a frequency of 4.3 divided by the total occurrence rate in all bins. We normalized by the total occurrence rate of planets since we already took account of systems with zero or multiple planets in the Poisson draw. Once we knew which bin to select a planet from, we drew from a uniform distribution over the bin area to select an orbital period-radius pair, except for the giant planet bin where we draw from a power-law distribution in planet radius with exponent -1.7, which mirrors [@Sullivan2015]. This non-uniform giant planet size distribution reduces the number of nonphysical inflated planets, as discussed by @Mayorga2018. Occurrence rates from both @Fressin2013 and @Dressing2015 are based on Kepler data and are limited in orbital period to 0.5–85 and 0.5–200 days, respectively.
Following @Kipping2014, the orbital eccentricity was selected from a Beta distribution, with parameters $\alpha=1.03$ and $\beta=13.6$, which @Vaneylen2015 found was appropriate for transiting planets. The periastron angle was drawn from a uniform distribution between $-\pi$ and $+\pi$. The cosine of inclination was chosen to be uniform between zero and one. Planets in multiple-planet systems were assumed to be coplanar - i.e. they have the same $\cos{i}$ - which is a reasonable appumption because multiple-exoplanet systems have been found to be highly coplanar [@Xie2016]. Finally, we chose a time of first transit to be uniform between zero and the orbital period – note that this may be greater than the total observation duration, in which case no transit was recorded. We then computed the number of transits observed using the observation duration calculated previously (the number of sectors where a target is observed).
We intentionally kept planets that cross the orbit of other planets in the system because, while they are likely on unphysical orbits, to remove them would change the distribution of the number of planets per star, which is an observed property. We also assumed that none of these planets experience a significant amount of transit timing variations (Hadden et al. in preparation, address transit timing variations and period ratios in detail).
Detection model
---------------
Armed with a sample of planets and host stars, we then determined which planets are detectable. To do this we derived a transit depth modified by several factors: the flux contamination of nearby stars, the number of transits, and the transit duration. It should be noted that flux contamination is significantly more problematic for TESS than with Kepler because TESS has pixels that are 28 times larger than Kepler’s.
The raw transit depth was computed assuming a uniform disk (i.e, transit depth $T_d = (R_p / R_\star)^2$, where $R_\star$ is the stellar radius). That is, we ignored the effects of limb-darkening and grazing transits. We calculated the reduction in transit depth due to dilution from nearby stars using the value of contamination for the CTL as $T_d / (1 + d)$, where $d$ is the dilution, the fraction of light coming from stars that are not the target divided by the total star light. We then multiplied the transit depth by the square root of the transit duration ($T_{\textrm{dur}}$) in hours, with transit duration following @Winn2010 defined as, $$T_{\textrm{dur}} = \frac{P}{\pi} \arcsin\left[{ \frac{R_\star}{a} \frac{\sqrt{(1+R_p/R_\star) - b^2}}{\sqrt{1 - \cos^2{i}}}}\right],$$ where $P$ is the orbital period, $i$ is the orbital inclination relative to our line of sight, $a/R_\star$ is the semimajor axis in units of stellar radius, $b$ is the impact parameter, and $R_p / R_\star$ is the planet to star radius ratio, to derive an effective transit depth. The effective transit depth, $T_d^\prime$, is defined as $$T_d^\prime = (R_p / R_\star)^2 \times \sqrt{T_{\textrm{dur}}} \times \sqrt{N} \times \frac{1}{1 + d},$$ where $N$ is the number of transits observed.
We took the TESS photometric noise level from [@Stassun2017] who used the properties described by @Ricker2016 and tested whether the effective transit depth was greater than the TESS photometric noise at the stellar brightness of the host stars multiplied by 7.3 (i.e. SNR$\ge$7.3). A 7.3-sigma detection is the nominal value used by @Sullivan2015 and is calculated in a similar manner to the detection threshold used by Kepler [@Jenkins2010]. We also required that the impact parameter of the transit is less than 1.0 and that we observed at least 2 transits. Requiring an impact parameter of less than 1 removes a small number of grazing transits but these are difficult to distinguish from eclipsing binaries anyway [@Armstrong2017]. These detection thresholds are relatively aggressive, Section \[sec:conservative\] describes using a more conservative detection thresholds of at least 3 transits and SNR of 10.
Results {#sec:results}
=======
We performed 300 simulations using our nominal planet sample and detection criteria, this enabled us to look at the average and range from our simulations. We predict that TESS will find 4373 planets (median) orbiting stars on the CTL, with the 90% confidence interval ranging from 4260–4510 planets. Henceforth, we designate a simulation that produced the median number of planets as our fiducial simulation and the properties we show come from that simulation. All the stars in the CTL are included in Figure \[fig:skyplot-ffis\] and the detected planets are shown as red dots.
![The spatial distribution of target stars and detected planets from FFI data. The upper panel shows the southern ecliptic hemisphere and the lower panel shows the northern ecliptic hemisphere. Stars observed for 1 sector are shown in blue, two sectors in orange, 3+ sectors in green, and stars in the CVZ are shown in purple. Detected planets are shown as red dots. A total of 4373 planets are shown, of which 54% were only observed for a single sector, and 11% were observed for 12 or 13 sectors. The lower density of stars, offset from the south ecliptic pole, is centered on the south celestial pole, and is due to relatively incomplete proper-motion catalogs in the celestial south.[]{data-label="fig:skyplot-ffis"}](skyplot-ffis-v2.png){width="45.00000%"}
Our fiducial simulation has 1293 planets orbiting 2-minute cadence targets, and the 90% confidence range of planets found in 2-minute data is 1180–1310 planets. The sky distribution is shown in Figure \[fig:skyplot-2min\]. There are clear differences in features between the FFI distributions and the 2-minute cadence distributions. The FFI stars are not evenly distributed, there is a lower density of stars in the southern sky. This is caused by the use of the reduced proper motion cut to identify dwarf stars, since existing proper motion catalogs are less complete below a declination of $-30^\circ$. This low density at southern latitudes is not visible in the 2-minute cadence plots because the high quality AFGK stars chosen for 2-minute cadence observations are bright enough that the proper motion catalogs are essentially complete for them. However, M dwarfs are faint enough that the proper motion catalogs are not complete for even high priority stars below a declination of $-30^\circ$, and they are undersampled among the 2-minute targets in that region.
The Galactic plane is visibly underpopulated in the 2-minute cadence data for two related reasons. Stars near the galactic plane tend to have higher flux contamination, which depressed their calculated priority. Also, photometric catalogs have a great deal of unreliability in the galactic plane in variety of ways, including proper motions, source identification, and the effects of reddening on the stellar temperatures. Therefore the priorities of all CTL stars within 15 degrees of the galactic plane were systematically down-weighted in the CTL, except for a subset of specially identified stars.
For both the 2-minute and the FFI-observed stars, we found planets more frequently closer to the ecliptic poles, where the longer observing baseline makes transit detection easier and where it is possible to find longer-period planets.
![The spatial distribution of target stars and detected planets from 2-minute cadence data. The colors of stars and planets is the same as shown in Figure \[fig:skyplot-ffis\]. The southern hemisphere, and to a lesser extent the northern hemisphere, has a pronounced feature of the Galactic plane running through where priorities are down-weighted because the high stellar density will dilute transit signals making them harder to detect.[]{data-label="fig:skyplot-2min"}](skyplot-2min-v2.png){width="45.00000%"}
As shown in Figure \[fig:planet-radii\], our simulation predicts that TESS will find 41 Earth-sized worlds ($<$1.25 $R_\oplus$), 238 super-Earths (1.25–2.0 $R_\oplus$), 1872 sub-Neptunes (2.0–4.0 $R_\oplus$), and 2222 giant planets ($>$4.0 $R_\oplus$) orbiting stars on the CTL. In total 279 planets smaller than 2.0 $R_\oplus$ were detected in our simulation, 90% of which were orbiting targets observed at 2-minute cadence. The sub-Neptunes were split roughly evenly between those observed at 2-minute cadence and those found only in FFI data, but nearly 90% of giant planets were found in the FFI data.
![Our simulations predict that TESS will detect a total of about 4400 planets orbiting stars on the CTL, of which 1300 will be observed at 2-minute cadence. Roughly 40 Earth-sized planets will be found, almost all of which are on the 2-minute target list. One thousand super-Earths and mini-Neptunes will also be found. Many new giant planets will be discovered, primarily through FFI data. The numbers shown above the FFI bars are total planets, and include the planets found in 2-minute cadence data.[]{data-label="fig:planet-radii"}](planet-radii-both-v2.png){width="45.00000%"}
A summary of the properties of planets detected in FFIs and 2-minute cadence data is given in Table \[tab:summary\]. Full details of every planet detection in our simulation is provided in a machine readable table, with a summary shown in Table \[tab:allplanets\].
-------------------------------- -------- ------------ ------------- -------- ------------ -------------
Property Median 5th pctile 95th pctile Median 5th pctile 95th pctile
Host star radius ($R_\odot$) 1.02 0.23 2.44 1.35 0.32 3.48
Host star mass ($M_\odot$) 0.95 0.20 1.61 1.07 0.32 1.93
Host star temperature (K) 5500 3200 7200 5900 3400 8000
Host star brightness, $Ks$ 9.2 6.7 11.0 10.0 7.4 11.5
Host star brightness, TESS mag 10.4 7.5 13.5 11.0 8.2 13.1
Host star brightness, $V$ 11.3 7.9 16.3 11.7 8.8 15.4
Planet radius ($R_\oplus$) 3.1 1.4 8.9 4.2 1.9 15.1
Planet orbital period (days) 8.2 1.7 34.8 7.0 1.8 29.0
Transit duration (hours) 3.0 1.0 8.7 3.9 1.3 10.4
SNR 13.6 7.7 109 13.3 7.6 93.7
Number of transits 7 2 65 6 2 51
Distance (pc) 140 50 200 260 70 890
-------------------------------- -------- ------------ ------------- -------- ------------ -------------
Num Units Label Explanation
----- --------- --------------- ------------------------------------------------------------------------------------------
1 – TICID TESS Input Catalog ID number of star
2 deg RAdeg Right ascension 2000
3 deg DEdeg Declination 2000
4 deg ELON Ecliptic longitude
5 deg ELAT Ecliptic latitude
6 – Priority CTL v6.1 priority
7 – 2min-target Was this a 2-minute cadence target in our model? 1 = yes, 0 = no
8 – Camera TESS camera number, number between 1–4
9 d Obslen Number of days that target is observed
10 – Num-sectors Number of sectors the target is observed for
11 mag Vmag V-band magnitude
12 mag Kmag Ks-band magnitude
13 mag Jmag J-band magnitude
14 mag Tmag TESS bandpass magnitude
15 solRad Star-radius Stellar radius
16 solMass Star-mass Stellar mass
17 K Star-teff Stellar effecitve temperature
18 pc Distance Distance of the star
19 – Subgiant Was this star randomly selected to be a subgiant? 1 = yes, 0 = no
20 – Detected Was this planet detected? 1 = yes, 0 = no
21 – Detected-cons Was this planet detected using the conservative model? 1 = yes, 0 = no
22 d Planet-period Orbital period of the planet
23 Rgeo Planet-radius Radius of the planet
24 – Ntransits Number of transits the planet has, 0 if planet does not transit
25 – Ars Planet semimajor axis divided by the stellar radius
26 – Ecc Planet orbital eccentricity
27 – Rprs Planet radius divided by the stellar radius
28 – Impact Planet impact parameter
29 h Duration Planet transit duration
30 ppm Depth-obs The observed transit depth, corrected for dilution
31 – Insol Insolation flux the planet receives relative to that received by the Earth from the Sun.
32 ppm Noise-level The one-hour integrated noise level of the star
33 – SNR Combined signal-to-noise ratio of all transits, 0 if planet does not transit
About 75% of stars were observed for a single sector. Unsurprisingly, most planets (2334, 53%) were also only observed for a single sector and three-quarters of planets were observed for one or two sectors. Conversely, while just 2% of CTL stars were observed for 12 or 13 sectors, 11% of all planets detected were found around these stars. The longer observing baseline gave both higher SNR transits, and sensitivity to longer orbital period planets. The number of stars observed at 2-minute cadence for 12 or 13 sectors was fairly heavily constrained in our target selection model, therefore a relatively high fraction (60%) of planets were found in the FFI data for the high latitude fields. Overall 70% of planets were found only in the FFIs, but for stars that were observed between 4–11 sectors, just 40% of planets were found only in the FFI data.
The orbital periods of our planets ranged from 0.5–99 days, which is a somewhat artificial limitation based on the occurrence rates used. The minimum orbital period of the injected transit signals was 0.5 days. While we know of several ultra-short period planets [e.g. @Sanchis2013], they are very rare [@Winn2018] and therefore will not significantly impact the planet yield. On the long period end, we simulated M-dwarf planets with periods up to 200 days, yet no planets with periods longer than 100 days were recovered, so we are confident that few long periods planets were missing here. For hotter stars, we only simulated planets with periods up to 85 days. It is likely we were missing planets orbiting stars with periods longer than 85 days. However, we only found two planets in our M-dwarf sample with periods longer than 85 days, and in the 65–85 day period range for the AFGK sample we had just 17 planets. Since the probability of a planet to transit scales inversely with orbital distance, and the number of stars with a long enough observing baseline to detect at least two transits similarly shrinks, we do not expect more than a handful of additional long period planets. We do caution that our sample should probably not be used to estimate the yield of planets showing a single transit because the 85 day limit becomes more significant. For a study of single transiting planets we point readers to @Villanueva2018.
![The number of sectors that stars with detected planets were observed for, with a sector having an average observing window of 27.4 days. More than half of planets were observed for a single sector, with 10% being observed for 12 or 13 sectors.[]{data-label="fig:planet-sectors"}](planets-sectors-both-v2.png){width="45.00000%"}
In Figure \[fig:hitrate-sectors\] we show the ratio of stars observed to planets detected – which we define as the ‘hit rate’. Overall, the hit rate for 2-minute cadence targets was 0.60%, while for the CTL stars not on the 2-minute cadence list the hit-rate was 0.10%. Hit rate increases with observing duration, from 0.43% for 2-minute cadence targets observed for 1 sector up to 1.8% for 2-minute cadence targets with at least 12 sectors of data.
![The ratio of stars observed to planets detected as a function of the number of sectors a star is observed for. The longer a star was observed, the higher probability a planet would be detected. Targets observed at 2-minute cadence are shown in red, while blue are FFI targets. For 2-minute cadence stars the average hit-rate was 0.60%, while including all stars on the CTL drops this to 0.14%. While observing for a longer baseline increased the number of planets, the increase is not linear. For 2-minute cadence targets, an increase of 12x in observing baseline increased the hit-rate by a factor of just 4.4. There are comparatively few planets in the 12 and 13 sector bins, so we show Poisson uncertainties on these bars demonstrating that there is not a measurable difference between observing for 12 or 13 sectors. Red and blue bars are not stacked, both start at zero.[]{data-label="fig:hitrate-sectors"}](hitrate-sectors-both-v2.png){width="45.00000%"}
We found that the planet host stars range in brightness from $V$-band mag of 4.0–20, with 7 planets predicted to orbit stars brighter than 55 Cnc, currently the brightest transiting planet host [@Winn2011]. As shown in Figure \[fig:tess-mag\], in the TESS bandpass, 90% of planets orbited stars with magnitudes between 8.2–13.1, this compares with Kp=11.9–15.9 for Kepler planet candidates [@Thompson2018]. The simulated planets typically orbited stars 3 magnitudes brighter than Kepler planets. Planets around stars observed at 2-minute cadence were systematically brighter than the planets found orbiting stars observed only in FFI data, with a median TESS magnitude of 10.4 versus 11.0.
With TESS concentrating on finding planets orbiting cool stars, it is unsurprising many planets orbited stars that were bright in the infrared. The median $Ks$-band ($\sim$2.0–2.2$\mu m$) magnitude of planets in 2-minute cadence data was 9.2 and 90% of 2-minute cadence planets were brighter than Ks=10.7. None of the TESS 2-minute planets orbited stars fainter than the median infrared brightness of Kepler planet candidates of Ks=13.0.
![Brightness of the planet host stars in the TESS bandpass magnitude. The median brightness of stars with planets found in 2-minute cadence data was 10.4, with a maximum range of 3.5–15.3. For planets found only in FFI data, the median brightness was 11.3, and a maximum range of 6.1–16.4. []{data-label="fig:tess-mag"}](tess-mag-both-v2.png){width="45.00000%"}
The spectral type distribution of the detected planet host stars is shown in Figure \[fig:spectral-type\]. About a quarter of the planets found in 2-minute cadence data orbited M-dwarfs (371) with the remaining split fairly evenly between K (216), G (351), and F (299) stars. The deficit in planets orbiting K-dwarfs was caused by a deficit in K-dwarfs selected for 2-minute cadence observations. This was a result of the target prioritization strategy employed, and has been noted previously [@Stassun2017]. A few additional planets orbiting cool stars were found in FFI data (only 125 additional M-dwarfs), but 80% of FFI-only planets orbited stars larger than the Sun. In total about 10% of planets in our simulated sample orbited M-dwarfs.
![The spectral type distribution of TESS planet-hosting stars. Our simulations predict that TESS will find 496 planets orbiting M-dwarfs, of which 371 orbit stars observed at 2-minute cadence. About half the simulated planets in 2-minute cadence data orbited stars larger than the Sun, while 80% of planets found only in FFI data orbited stars larger than the Sun.[]{data-label="fig:spectral-type"}](spectral-type-both-v2.png){width="45.00000%"}
Figure \[fig:distance\] shows the distance to the simulated planets[^2]. The closest detected planet in our simulation orbited Lalande 21185, a star 2.5 pc away. We found 46 planets within 50 pc, and 234 within 100 pc, which doubles and quadruples the number of transiting planets known within 50 and 100 pc, respectively [@Akeson2013].
![The distances of planets found in our simulation in parsecs. The upper panel shows both distance and ecliptic latitude of the host stars, and the lower panel is distance plotted against planet radius. Almost all 2-minute cadence planets discovered by TESS will be within 300pc, with 77% within 200pc. FFI planets were found over 1000 pc away but 90% of planets were within 700 pc.[]{data-label="fig:distance"}](distance-both-v2.png){width="45.00000%"}
The circumstellar habitable zone concept has been popular since at least the 1950s [@Strughold1953; @Shapley1953], and refers to the spherical shell around a star where liquid water could be present on a planetary surface. @Kopparapu2013 provided models for an optimistic habitable zone with boundaries of recent Venus and early Mars, which correspond to stellar fluxes of 1.78x and 0.32x the insolation Earth receives from the Sun, respectively. Our simulation contains 69 planets in the optimistic zone, of which 9 are smaller than 2 Earth-radii. All the habitable zone planets orbit M-dwarfs.
Suitable targets for RV follow-up
---------------------------------
For the TESS mission to be successful, it must find planets smaller than 4 Earth-radii with a measurable radial velocity signal. We predict that TESS will find more than 2100 planets smaller than 4 Earth-radii, but many of these will orbit stars whose brightness makes follow-up challenging or impossible with current precision radial velocity facilities. While planets orbiting very faint stars have had their mass determined via radial velocity studies [e.g. @Koppenhoefer2013], it is typically challenging to measure masses of planets around stars fainter than V=12. We predict that TESS will find 1300 planets smaller than four Earth-radii around stars brighter than V=12. Therefore, with more than 1000 potential targets, TESS will have a plethora of targets to choose from when selecting promising RV targets. Even if just 20% are good RV targets, this will more than triple the number of planets smaller than 4 Earth-radii with measured masses.
There are 160 planets in our sample that are smaller than 2 Earth-radii and orbit stars brighter than V=12. We currently have mass and radius constraints on fewer than 60 planets smaller than 2 Earth-radii, so TESS will potentially greatly increase this number, although the precise number will depend on whether individual stars are suitable for precise radial velocity measurements.
Targets for atmospheric characterization
----------------------------------------
A second aim of the TESS mission is to find targets suitable for transmission spectroscopy using the James Webb Space Telescope (JWST). Until on-sky performance is measured, particularly the systematic noise level, there is considerable uncertainty on how JWST will perform [@Batalha2017]. However, we can identify the properties of planets that would make them good JWST targets using a few simple cuts. The host star should be bright in the infrared, and the star should be small. We identified simulated planets whose host stars have Ks$<$10, $T_{\rm eff}<3410 K$ which equates to M3V stars with a radius of approximately 0.37 solar-radii [@Pecaut2013]. In total there were 70 planets fulfilling these criteria. We show in Figure \[fig:planet-radius-insolation-cool-stars\] the simulated small planets we think make interesting candidate JWST targets in terms of insolation fluxes. There are ten planets in the boxed region in Figure \[fig:planet-radius-insolation-cool-stars\] which highlights planets that fell into the optimistic habitable zone [@Kopparapu2013], and had radii between 1.25 and 2.5 Earth-radii, implying a puffed-up atmosphere [@Lopez2014]. These planets, along with those orbiting TRAPPIST-1 [@Gillon2017] and other low mass stars [@Greene2016; @Kreidberg2016; @Morley2017; @Louie2018], will form a reference sample of temperate worlds for observation by JWST.
![Planets make good targets for transmission spectroscopy if they orbit bright, small stars. This plot shows planets that orbit stars with spectral type M3V or later, and that are brighter than Ks=10. The box is an approximate region showing planets that may have somewhat extended atmospheres (i.e. super-Earths) and are in the circumstellar habitable zone. There are 10 planets within this region, making up the prime JWST target sample from TESS.[]{data-label="fig:planet-radius-insolation-cool-stars"}](planet-radius-inxolation-cool-stars-v2.png){width="45.00000%"}
The JWST continuous viewing zone is located within 5$^\circ$ of the ecliptic poles, and is contained within the TESS CVZ, shown in Figure \[fig:sectors\]. However, because of gaps between the TESS CCDs on Camera 4 (each camera is composed of a 2x2 grid of CCDs), the central 2$^\circ$ has limited coverage. In our sample we have 74 planets with ecliptic latitude $\left|b\right|>85^\circ$, of which 29 are 2-minute cadence targets and 11 are smaller than 2 Earth-radii.
Discussion
==========
Alternative selection strategies for the 2-minute cadence targets
-----------------------------------------------------------------
In addition to the nominal 2-minute cadence target selection laid out in Section \[sec:starselction\], we also considered alternative strategies of selecting a higher or lower fraction of targets in the CVZ, which we call scenarios (a) and (b), respectively. There are justifications for both approaches. Placing more of the 2-minute cadence targets in the CVZ increases the overall number of 2-minute targets where TESS is sensitive to long-period planets, and potentially to smaller planets via increased SNR. On the other hand, placing more of the 2-minute targets outside the CVZ should increase the overall number of planets detected, since 13 stars can be observed in regions with single-sector coverage for each target in the CVZ.
To test these scenarios we selected targets in an identical manner to that described in Section \[sec:starselction\] except that in scenario (a) we included 12,000 stars in the CVZ and 2200 stars in the other cameras per sector, while in scenario (b) we select 3000 CVZ targets and 11,200 stars in the remaining cameras.
Under these two different selection strategies, we examined the number of planets found in 2-minute cadence data, compared to our nominal selection strategy. In scenario (a) we found a total of $740\pm50$ planets and in (b) we found $1380\pm60$ planets, which compares with $1250\pm70$ planets in the nominal strategy (where the reported value is the median, and uncertainties are the central 90% of the distribution, calculated by 300 Monte Carlo simulations). These results suggest that the nominal selection strategy was reasonably successful at accomplishing the goal of maximizing the number of planets with 2-minute cadence photometry, which in turn maximizes the number of planets where we can derive precise stellar parameters through asteroseismology [@Campante2016]. Scenario (b) yielded 10% more planets but the results were comparable within uncertainties, and the number of planets with orbital periods beyond 15 days was cut by about 10% in scenario (b). Scenario (a) extended the tail of the orbital period distribution – the 95th percentile shifts from 30 to 42 days – but because of the large decrease in the total number of planets, the absolute number of long period planets was unchanged.
In each scenario the total number of planets detected remained unchanged because almost all planets could be found equally well in 2-minute and FFI data, so the precise stellar selection had limited impact of the primary mission goals.
A more conservative model {#sec:conservative}
-------------------------
Our analysis so far has made two fairly optimistic assumptions, (1) that we can identify a transiting planet by observing just two transits from TESS, and (2) that we can detect all planets with a SNR$\ge$7.3. In actuality, planets with fewer than 3 observed transits are very difficult to uniquely identify using photometric survey data alone [c.f @Thompson2018; @Mullally2018]. Planets have been detected using K2 mission data [@Howell2014] with one [@Vanderburg2015] and two [@Crossfield2015] transits, but these cases occurred in systems where additional space-based follow-up assets were exploited or there were two other planets in the system, so the validity of the planets was less ambiguous [@Lissauer2012]. While with sufficient observing resources characterizing these planets is feasible to identify and confirm, they remain a challenge. Furthermore, analyses of Kepler data have shown that using a detection threshold below 8–10$\sigma$ leads to many spurious detections [@Christiansen2016; @Thompson2018; @Mullally2018]. In K2, a threshold of SNR$>$12 was typically applied [@Crossfield2016] before expending follow-up resources on a candidate planet.
With these limits in mind, we took the fiducial catalog and cut planets that either had fewer than three transits, or had a combined transit SNR$<$10. This resulted in a moderate cut in the total number of planets found to 2609 total planets, of which 820 came from the 2-minute cadence data. This was a 60% overall decrease in the total number of planets detected, but was most significant for small planets. The number of planets with radii below 2 Earth-radii decreased by a factor of two from 279 to 128 planets, with similar fractional losses in the 2–4 Earth-radii bin, but there was only a 25% decrease in detected giant planets.
![The predicted planet radius distribution using our conservative detection model where we required at least 3 transits and a combined SNR of 10. This figure is the counterpart of Figure \[fig:planet-radii\], but using our conservative detection model. The total number of planets shown is 2609, which is roughly 60% lower than our standard detection model. This change is most signification for small planets which saw a factor of two decrease. We have intentionally changed the color scheme from previous figures to differentiate between standard and conservative models.[]{data-label="fig:planet-radii-conservative"}](planet-radii-both-conservative-v2.png){width="45.00000%"}
The decrease in the number of planets amenable to radial velocity follow-up was roughly a factor of two, with planets smaller than 4 Earth-radii orbiting stars with V$<$12 dropping from 1312 to 616, and those smaller than 2 Earth-radii from 151 to 67. The number of habitable zone planets dropped from 69 to 28, and left just four smaller than 2 Earth-radii. The number of premium JWST targets sees a modest decrease. The number of planets orbiting stars cooler than 3410 K, with Ks$<$10 drops from 71 to 58, and the number in the dashed box in Figure \[fig:planet-radius-insolation-cool-stars\] dropped from 10 to 7. While these drops were significant, they are unlikely to seriously impact the primary mission goal, because there were still hundreds of small planets orbiting bright stars in the sample.
Phantom inflated planets
------------------------
This study, and other planet yield simulations [e.g. @Sullivan2015], have not paid particular attention to the physical properties of giant planets, primarily because these are not a focus for the TESS mission team. Nevertheless, we are anticipating groundbreaking scientific advances in our understanding of the atmospheres of giant planets from follow-up observations of planets found by TESS – particularly from Spitzer, HST, and JWST. As pointed out by @Mayorga2018, in the first version of this paper, there were significant numbers of giant planets that were beyond the limit of inflation for their equilibrium temperatures [@Thorngren2018]. The cause of this is that in the occurrence rate estimates of [@Fressin2013] the giant planet bin span 6–22 Earth-radii while temperate planets should rarely be larger than 12 Earth-radii. As a result of this feedback from @Mayorga2018 we changed the selection function in the giant planet bins from a log-normal function to a power law. This reduced the number of phantom planets from 8% of the total population to 1%. We caution giant planets aficionados that there are 45 over-inflated giant planets in our simulation.
The effects of Earth and Moon crossings
---------------------------------------
The nature of the TESS orbit means that a subset of observations will be obscured by the Earth or Moon passing through the field of view. Cameras that receive a significant amount of scattered light from the Earth or Moon will experience larger background flux, and photometry in any camera that receives a large portion of direct light from the Earth or Moon will likely be impossible because of saturation and bleed. However, the Earth and Moon move relatively quickly through the field of view, and Earth or Moon crossings are relatively infrequent [@Ricker2015]. @Bouma2017 estimated that the Earth and Moon will significantly affect photometric performance for 9% of all exposures, although the lost cadences will not be evenly distributed in time or focal plane location. Camera 1, and to a less degree Camera 2, are impacted, but the effect was expected to be limited for Cameras 3 and 4. Estimating how this affects the yield is non-trivial, but we can try by using the @Bouma2017 estimates that 23% of observations in Camera 1, and 12% of observations in Camera 2, will be affected. We can then assume that the SNR of transits will scale with the square root of the number of observations, so Camera 1 targets will have 11% lower SNR, and Camera 2 targets will have 6% lower SNR. This causes a 13% drop in total planets detected in our simulation, and a 9% decrease in the number of planets orbiting 2-minute cadence targets. Early commissioning results have suggested that the effect of the Moon may be more complex than anticipated, and owing to the substantial uncertainty in the impact of Earth and Moon crossings, we have not included Earth and Moon crossings in our yield statistics.
Astrophysical false positives {#sec:falsepositives}
-----------------------------
@Sullivan2015 performed a careful analysis of the sources and rates of false positives expected in the TESS 2-minute cadence data, and we have not reproduced that work here. They estimated that TESS will find over 1000 astrophysical false positives in 2-minute cadence data, but described promising mitigation strategies that utilize follow-up observations and statistical methods to reduce this by a factor of 4 or more.
The ratio of false positives to detected planets will not be uniform over all stars observed by TESS, but will vary as a function of hit-rate. In Section \[sec:results\] we showed that the hit-rate for 2-minute cadence targets is a factor of 5.5 higher than FFI-only stars. Assuming each star has the same chance of yielding a detection of an astrophysical false positive, the fraction of true planets found to false positives will be lower for the FFI-only detections than for 2-minute cadence targets. The reason is that fewer planets are found per stars observed but the same number of false positives are detected. Using the false positive rate from @Sullivan2015 of 1 false positive per 180 stars observed yields one astrophysical false positive per planet detection. However, for the FFI-only targets the ratio of false positives to planets detected increases to more than five per true planet discovered. Furthermore, stars on the CTL that are not included in our 2-minute cadence sample are, on average, 2 magnitudes fainter than the 200,000 stars observed at 2-minute cadence. This means that mitigation strategies that rely on follow-up observations will be significantly more challenging. Given essentially all small planets will be found in the 2-minute cadence data, only the most intrepid of exoplaneteers will want to commit significant resources to discovering and following-up planets in FFI data.
Planets detected around stars not in the CTL {#sec:excluded-stars}
--------------------------------------------
In Section \[sec:starselction\] we simulated planets orbiting stars that are in CTL version 6.1. This totals roughly 3.2 million stars, but includes only those stars that the TESS Target Selection Working Group considered as potential 2-minute cadence targets. The limited number of slots available for 2-minute cadence requires a careful consideration not just of the overall potential for planet detections around a given star, but also comparison of the relative planet detection potential between stars, along with the scientific value of the resulting planets. The CTL was constructed to permit a quantitative relative ranking of the best stars to select for the 2-minute cadence slots, not to identify all stars with detectable planets. While in this work we have adopted the set of several million stars in the CTL as the primary sample to investigate, stars not in the CTL might also yield some planet detections in the FFI data. The reason we adopted this approach is the same reason for the construction of the CTL in the first place – our analysis of planet yield among a population of several million stars is much more tractable than conducting the analysis for all 470 million stars in TIC-6.
Explicitly removed from the CTL are stars with a reduced proper motion that flags them as giants, stars with parallax or other information that flags them as giants or subgiants, dwarf stars that are somewhat hot and relatively faint but not as faint as some dwarf stars that are included, and faint dwarf stars. The magnitude cut used in the CTL is TESS magnitude of 12 for stars hotter than 5500 K, and TESS magnitude 13 for cooler stars, although faint cool dwarfs are explicitly included via a specially curated target list [@Muirhead2018]. The CTL therefore generally excludes hot stars, faint stars, and evolved stars, in favor of bright, cool dwarfs.
Only a handful of transiting planets have been detected around red giants [e.g. @Burrows2000; @Huber2013; @Barclay2015; @Vaneylen2016; @Grunblatt2016; @Grunblatt2017] because finding these planets is extremely challenging. Transit depth scales with the square of the stellar radius, so planets orbiting large stars are hard to find. Therefore, the frequency of planets orbiting giant stars is relatively poorly constrained. However, TESS will observe hundreds of thousands of red giants brighter than 11th magnitude in the TESS bandpass [@Huber2017a] and will certainly detect planets orbiting these stars. However, Kepler observed roughly 16,000 red giants [@Yu2018] and found only a handful of planets. With a factor 20 or so increase in the number of red giants from TESS, we might expect of order 100 new planets. This estimate is comparable to that of @Campante2016, who perform a much more careful analysis and predicted that TESS will find roughly 50 planets orbiting red giants.
The brightness cuts applied to the TIC in creating the CTL have a larger impact on our yield estimates. At 12th magnitude the TESS 1-hour integrated noise level is 600 ppm. This equates to detecting a Neptune-size planet with three transits around a solar radius star, while at 13th mag the noise is 1200 ppm which is equivalent to a 6 Earth-radii planet. So it is certainly the case that many stars not included in the CTL may have planets detectable with TESS. To detect a Jupiter with three transits around a Sun-like star would require a maximum 1-hour integrated noise of approximately 4000 ppm which corresponds to a TESS magnitude of 14.7. The TIC lists 16.0M stars with temperatures above 5500 K, log$g$ above 3.9, and TESS magnitude of 12–14.7, and 4.2M with temperature between 4000–5500 K, $\log{g}$ above 4.2, and brightness between 13–14.7 (where we cut at 4000 K because the cooler stars are included via the cool star curated list). In our fiducial sample, the frequency of detected planets larger than 4 Earth-radii was 0.069%. Assuming an equal detection rate for fainter stars in the 4+ Earth-radii bin as for brighter stars we would expect to find 14,000 additional giant planets. Even under our conservative model, the rate is 0.050%, or 10,000 additional planets.
While these planets will appear in the FFI data, they are not prime targets, hence their exclusion from the CTL, because the planets will be hard to detect and harder to follow up and confirm owing to their faintness and higher crowding. Using the logic described in Section \[sec:falsepositives\], the astrophysical false positive rate in this part of the parameter space is also very high. With a hit-rate around 0.05% and a false positive rate likely to be comparable to that found by @Sullivan2015 of 1 per 180 stars observed, we expect a factor of more than 11-to-1 false positive to true planets detected. Thus we caution that searching for planets in this regime is fraught with challenges.
The omission of these potential host stars from our analysis leads to a large underestimate in the overall planet yield of the mission, although that is almost entirely in the giant planet regime. In Figure \[fig:planet-radii-giant-planets\] we show our final distribution of planet radii and include the sample of giant planets orbiting faint stars, using the conservative yield estimate. This results in a total planet yield of 14,000 transiting planets. However, as discussed, these planets will be resource intensive both to confirm and to meaningfully analyze.
![ The predicted planet radius distribution including large planets orbiting faint stars outside of the CTL. The total number of planets that we predict TESS could find is up to 14,000. This figure is the same as Figure \[fig:planet-radii\] but includes the additional large planets orbiting faint stars. We have intentionally changed the color scheme from previous figures to differentiate from our simulated yield. []{data-label="fig:planet-radii-giant-planets"}](planet-radii-both-giant-planets-v2.png){width="45.00000%"}
One further source of additional planets is from M-dwarfs in the Southern Hemisphere. As mentioned in Section \[sec:starselction\], there is a deficit of cool stars below $-30^\circ$ declination, caused primarily by the lower completion of proper motion catalogs where northern hemisphere telescopes are unable to observe. This manifests in fewer planets detected around cool stars in the south. In the 2-minute cadence data, there are 2.6x as many planets orbiting stars cooler than 3900 K north of declination $30^\circ$ than south of declination $-30^\circ$. Including the FFI planets, this increases to 3x as many northern as southern planets (233 versus 74 planets). With GAIA data release 2 now available, it is probable that new M-dwarfs in the south will be identified. This will help to recover additional planets orbiting cool stars not identified as dwarfs in the CTL. Given that this could potentially yield new candidate planets for JWST there is a pressing need for this work.
Comparisons with earlier estimates
----------------------------------
@Sullivan2015, @Bouma2017, and @Ballard2018 have previously estimated the planet yield from TESS. These previous studies selected stars from a simulated Galactic model rather than real stars, and therefore we expect there are moderate differences between our predicted yields and previous studies. Additionally, we used different selection strategies for both 2-minute cadence targets and for FFI stars. We built a realistic 2-minute cadence star selection model that limits the stars observed at the pole cameras to just 6,000 stars per hemisphere, whereas the previous works assumed TESS can observe many more stars in the CVZ than is possible with the flight hardware configuration used. We also use a different prioritization metric than previous work, which is based on the metric used by the TESS Target Selection Working Group. For the FFI targets we primarily consider those within the CTL, whereas different cuts on brightness are made in earlier works. Therefore, we expect to see significant differences in the planet yield for giant planets.
@Sullivan2015 predicted 1700 planets in 2-minute cadence data, of which 560 are smaller than 2 Earth-radii. @Bouma2017 used the same methodology and software as @Sullivan2015, but fixed a number of software bugs and modified a number of parameters. They also predicted 1700 planets from 2-minute cadence data, of which 430 were smaller than 2 Earth-radii. The total 2-minute cadence planet yield in both these studies was about 30% larger than we have predicted, but the number of planets smaller than 2 Earth-radii in our study is lower by a factor of 1.7 and 2.3 than @Bouma2017 and @Sullivan2015, respectively. However, given the different selection strategies, it may be more reasonable to compare the combined 2-minute cadence and FFI yields. Where @Bouma2017 and @Sullivan2015 differ is in their star selection for FFI targets. @Bouma2017 limit their selection to the top ranked 3.8M stars using a similar priority metric to the one applied in CTL 6.1. This enables easy comparison with our 3.2M star sample. On the other hand, @Sullivan2015 consider all stars brighter than $Ks=15$ totalling 150M stars, which we can compare with our analysis in Section \[sec:excluded-stars\].
Our total simulated yield is remarkably similar to @Bouma2017, with 41 versus 49 Earth-sized planets, 238 versus 390 super-Earths, 1900 versus 2000 mini-Neptunes, and 2200 versus 2500 giant planets, for this work and @Bouma2017 respectively. The only area where we see a significant deviation is for super-Earths, which we attribute to differences between the Galactic model and real stars.
Compared to @Sullivan2015 [@Sullivan2017], we predict lower totals in all bins. However, as mentioned by @Bouma2017, the number of Earths and super-Earths is overestimated by around 30% owing to a bug in their calculation of the dilution from background stars. Taking this into account, our number of Earths matches both @Bouma2017 and @Sullivan2015, while the super-Earths are comparable. Our rate of giant planets predicted in Section \[sec:excluded-stars\] is consistent with @Sullivan2015 @Ballard2018 used the framework and detection rates of @Sullivan2015, but focus entirely on M1–M4 dwarfs, and made significant changes to the occurrence rates to account for covariances between planets in the same systems. In comparison, our analysis of the M-dwarf population is simplistic. @Ballard2018 predicted a 50% increase in the rate of planets orbiting these cool stars compared to the occurrence rates used by @Sullivan2015 (and this work). They predicted $990\pm350$ planets around M1–M4 stars, while we predicted 410 planets orbiting stars with temperatures of 3100–3800 K. If the @Ballard2018 occurrence rate has a similar impact to our yields as it had on @Sullivan2015, and given comparable yields between our studies, we would expect an additional 50% planets in this parameter space, which is 200 more planets orbiting cool stars. Assuming the increase is uniform in planet size, we might expect an increased yield that includes 14 additional Earths, 42 additional super-Earths, and 142 additional mini-Neptunes. The yield could be even higher if we are able to identify additional M-dwarfs in the southern sky, as discussed in Section \[sec:excluded-stars\].
Conclusions
===========
The TESS mission will find a large number of transiting planets. However, up until recently the number and physical properties of the planets that will be discovered has been estimated using simulations performed before the TESS observing strategy, 2-minute target list, and flight hardware had been finalized. Here we simulated TESS detections of transiting planets using the CTL for our star selection. We have estimated that TESS will find more than 14,000 exoplanets, of which $4400\pm110$ orbit stars in the CTL and $1250\pm70$ will be observed at 2-minute cadence. TESS will find over 2100 planets smaller than 4 Earth-radii, of which 280 will be smaller than 2 Earth-radii.
The key design feature that distinguishes TESS from Kepler is that it will observe brighter stars, emphasizing finding planets that can be followed up more readily from the ground. TESS planets range in V-band brightness from 4–20, with 80% of predicted planets orbiting stars brighter than V=13.0. Assuming V=12 as the limit for recovery of a mass via precision radial velocity observations, we predict that TESS will have a sample of 2500 planets for radial velocity observations, of which 1300 will be smaller than 4 Earth-radii, and 150 smaller than 2 Earth-radii. This will provide a plethora of planets to characterize; the TESS follow-up observers should have little problem meeting mission requirements of measuring the masses of 50 planets smaller than 4 Earth-radii. We predict that TESS will find 7 planets orbiting stars brighter than 55 Cnc, the brightest transiting planet host.
There is significant interest in finding habitable zone planets from TESS. We predict around 70 habitable zone planets will be detected and all will orbit M-dwarfs, with 9 habitable zone planets in our simulations with radii smaller than twice that of Earth’s. Our simulations predict that TESS will find 70 planets orbiting bright mid-M-dwarfs (Ks$<$10, M3V or later), 10 of which fall into the optimistic habitable zone, making them prime JWST targets.
We have shown that nearly all planets valuable for contributing to mission goals related to radial velocity and JWST targets will be found in 2-minute cadence data. This is to the great credit of the teams that worked to create the CTL. The availability of 2-minute cadence data will permit more accurate measurements of the radii and orbital configurations of the detected planets. We explored how target selection choices affect the target yield and find that the distribution of targets between the CVZ and shorter observing baseline is well balanced between collecting 2-minute cadence data for the maximum number of planets, and finding long period planets.
There are a large number of stars that are not in the CTL that might host a detectable planet. These stars were intentionally not included in the CTL, and for good reason. They are unlikely to host detectable small planets, and any planets found will be hard to follow up. While there may be as many as 10,000 additional giant planets around the faint stars in the TESS data, we have shown that the astrophysical false positive rate might be as high as 11 false positives per true planet, and there may be as few as one planet detected per 2000 stars searched. While less severe, we anticipate a high astrophysical false positive rate for stars on the CTL but not included in the 2-minute cadence sample because the ratio of detected planets to stars observed is five times lower than for stars observed at 2-minute cadence. The mission’s target of finding planets with SNR$\ge$7.3 and only two transits may be overly aggressive, based upon experience with Kepler and K2 data. We explored an alternative model that applied more conservative detection thresholds of SNR$\ge$10, and requiring three transits. This results in a decrease in the yield estimate of approximately 50% for planets smaller than 4 Earth-radii, and occurs across all parameter spaces considered. However, even if this conservative model is realized, more than enough planets will be found to ensure mission success.
This work builds upon studies by [@Sullivan2015] and [@Bouma2017], and would not be possible without their efforts. We do see a moderate decrease from previous yields estimates, although our numbers are remarkably similar to those @Bouma2017 presented, considering the different stellar selection strategies.
It will not be long before TESS planets are discovered. The real excitement will come from learning about these new worlds using data from ground and space-based facilities. The legacy of TESS will be a catalog of the planets that will be the touchstone planets for years to come. TESS will discover which of our nearest stellar neighbors have transiting planets. The brightest host star in our simulation is 70 Oph A, where we recovered a simulated Earth-sized planet. Were this simulation real, on a clear night from a dark site we could point to this star and tell our friends, “that star there has a planet.”
Planet radius as a function of distance
=======================================
Zach Berta-Thompson created a figure using data from @Sullivan2015 that has been widely shared because it is both informative of TESS’ capabilities and aesthetically pleasing. We have reproduced Berta-Thompson’s plot in Figure \[fig:zachplot\], with our revised TESS yield estimates.
![Orbital distance versus planet radii. This plot updates a widely shared figure created by Z. Berta-Thompson, to now include our new simulation results. Kepler planet candidates from @Thompson2018 are shown in blue, our simulated 2-minute cadence detections in orange, and planets detected using other telescopes in black. The size of the circle is proportional to the transit depth. A subset of nearby planets are marked. Data was extracted from the Exoplanet Archive [@Akeson2013]. Three planets in our simulation orbit stars closer than the nearest known transiting planet system HD 219134. []{data-label="fig:zachplot"}](planet-distance-simulation_V2_lowres.png){width="\textwidth"}
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[^1]: The TIC and CTL are available from the MAST archive at <http://archive.stsci.edu/tess/>.
[^2]: Only about half of the targets in our sample had distances reported in CTL version 6.1, our statistics are based on this sample. Furthermore, a small number of the CTL reported distances were unrealistically large. These issues have been fixed in CTL v6.2.
|
---
author:
- 'U. Bach'
- 'T.P. Krichbaum'
- 'A. Kraus'
- 'A. Witzel'
- 'J.A. Zensus'
date: 'Received 29 July 2005; accepted 11 November 2005'
title: 'Space-VLBI polarimetry of the BLLac object S5 0716+714: Rapid polarization variability in the VLBI core'
---
Introduction
============
Since the discovery of intra-day variability (IDV, i.e. flux density and polarization variations on time scales of less than 2 days) about 20 years ago (@1986MitAG..65..239W [@1987AJ.....94.1493H]), it has been shown that IDV is a common phenomenon among extra-galactic compact flat-spectrum radio sources. It is detected in a large fraction of this class of objects (e.g., ). The occurrence of IDV appears to be correlated with the compactness of the VLBI source structure on milliarcsecond scales: IDV is more common and more pronounced in objects dominated by a compact VLBI core than in sources that show a prominent VLBI jet (; see also @2001ApJ...554..964L). In parallel to the variability of the total flux density, variations in the linearly polarized flux density and the polarization angle have been observed in many sources (e.g., ). Both correlations (e.g. in 0716+714, @1996AJ....111.2187W) and anti-correlations (e.g. in 0917+624, @2002ChJAA...2..325Q) between the total and the polarized flux density are observed. In many cases the IDV phenomenon is explained and modelled using refractive interstellar scintillation (RISS, e.g., ). On the other hand some effects remain that cannot be easily explained by interstellar scintillation and that probably demand an explanation via source intrinsic relativistic jet physics. (e.g.). There are also sources like 0716+714 and 0954+658, where correlated intra-day variability between radio and optical wavelengths is observed, which suggests that at least part of the observed IDV has a source-intrinsic origin (e.g., ). We also note that the recent detection of IDV at millimetre wavelengths in 0716+714 () also poses a problem for the interpretation of IDV by interstellar scintillation.
Independent of the physical cause of IDV (source intrinsic, or induced by propagation effects), it is obvious that IDV sources must contain one or more ultra-compact emission regions. Using scintillation models, typical source sizes of a few ten micro-arcseconds have been derived (e.g., ). In the case of source intrinsic variability and when using the light-travel-time argument, even smaller source sizes of a few micro-arcseconds are obtained. In this case it implies apparent brightness temperatures of up to $10^{18-19}$K (in exceptional cases up to $10^{21}$K), far in excess of the inverse Compton limit of $10^{12}$K (@1969ApJ...155L..71K). These high apparent brightness temperatures can be reduced by relativistic beaming with high Doppler-factors (e.g., ). At present it is unclear if Doppler-factors larger than 50 to 100 are even possible in compact extragalactic radio sources.
One of the motivations of this VLBI monitoring, therefore, was to find out where the IDV component is located in the jet and to directly search for related rapid structural variability on milliarcsecond- to sub-milliarcsecond-scales. Previous observations with ground-based VLBI has already suggested the presence of polarization IDV in a number of objects. In the case of 0917+624 and 0954+658, the IDV could be related to a component inside, or very close to the VLBI core (@2000MNRAS.315..229G). For 0716+714, however, it has been claimed that the variable component is located at about $\sim25$mas separation from the VLBI core (@2000MNRAS.313..627G).
The BLLac object S5 0716+714 is one of the brightest BLLac objects in the sky. Only a lower limit to its redshift is known ($z >0.3$, @1996AJ....111.2187W, and references therein). The source 0716+714 is also one of the best studied IDV sources, showing rapid variability in every wavelength band where it was observed (e.g. @1996AJ....111.2187W). In the radio its linear polarization can vary up to a factor of two, often accompanied by rotation of the polarization angle ( and ref. therein). A direct conclusion drawn from such polarization angle variations is the presence of multiple sub-components, each exhibiting different compactness and polarization that interact either due to deflection in the interstellar medium or to superposition of multiple components inside the beam ().
Intraday variability is most pronounced at cm-wavelengths where the resolution of ground based radio-interferometers is limited to the mas-scale. A factor of 3 to 4 higher angular resolution now is provided with the VLBI Space Observatory Program (VSOP, @1998Sci...281.1825H [@2000PASJ...52..955H]). This offers a much closer look at the origin of the rapid variability in 0716+714. To search for structural variability on time-scales of hours to a few days, we observed 0716+714 in a multi-epoch VSOP polarization VLBI experiment in September and October 2000. In this paper we will present and discuss the detected rapid total intensity and polarization variations and compare them with similar variations seen on VLBI scales with space-VLBI.
Throughout this paper we use a flat universe with the following parameters: a Hubble constant of $H_0=71$kms$^{-1}$Mpc$^{-1}$, a pressure-less matter content of $\Omega_{\rm m}=0.3$, and a cosmological constant of $\Omega_{\rm \lambda}=0.7$ (@2003ApJS..148..175S).
The observations and the data reduction procedures will be described in Sect. \[VSOP\_sec:observations\]. This section will be followed by the presentation of the results (Sect. \[VSOP\_sec:results\]) and the discussion of our findings (Sect. \[VSOP\_sec:discussion\]). At the end we give a summary with the conclusions (Sect. \[VSOP\_sec:conclusions\]).
Observations and data reduction {#VSOP_sec:observations}
===============================
VLBI data
---------
{width="17cm"}
An array of 12 antennas consisting of the 10 stations of NRAO’s VLBA, the 100m radio telescope of the Max-Planck-Institut für Radioastronomie in Effelsberg (Germany), and the 8m HALCA antenna of the VSOP was used to follow the short-term variability of 0716+714 at 5GHz within one week. The source was observed at 5 GHz during three epochs, namely September 29 (project code: V053A2), October 4 (V053A4), and October 5 2000 (V053A5). We note that a total intensity VSOP image of 0716+714 from these experiments was used already in another publication, where we study the long-term jet kinematics of the sources (, hereafter B05). Each of the three 16h VSOP observations resulted in dense and nearly identical ($u,v$)-coverages, which are shown in Fig. \[VSOP\_fig:uvplots\]. The details of the observations are summarised in Table \[VSOP\_tab:obslog\]. The sources 0836+710 and 0615+820 were used as calibrators for the ground array to check the stability of the amplitude calibration between the epochs and to calibrate the polarization data. The calibrators were not observed by the HALCA antenna, because it is not possible to steer HALCA to different source positions during one observation.
------- ---------- ------ ------ ------ ------------------------- --------
Epoch Source
7.67 2.75 0.55 $0.58\times0.27$, $-80$ 0.492
7.67 0.00 0.58 $0.95\times0.78$, $-26$ 0.034
0836+710 1.17 0.00 2.23 $1.10\times0.88$, $-10$ 0.193
0615+820 1.17 0.00 0.74 $0.90\times0.78$, $-42$ 0.191
7.72 4.59 0.52 $0.58\times0.23$, $-79$ 0.513
7.72 0.00 0.55 $0.92\times0.77$, $-25$ 0.041
0836+710 1.20 0.00 2.28 $0.99\times0.83$, $-46$ 0.168
0615+820 1.19 0.00 0.75 $0.84\times0.82$, $-47$ 0.173
7.92 5.47 0.52 $0.58\times0.24$, $-71$ 0.580
7.92 0.00 0.56 $0.85\times0.81$, $-28$ 0.038
0836+710 1.21 0.00 2.27 $0.96\times0.81$, $-74$ 0.175
0615+820 1.23 0.00 0.76 $0.93\times0.76$, $-55$ 0.171
------- ---------- ------ ------ ------ ------------------------- --------
: Observing log. The total integration time, mutual observing time with HALCA, total flux density, the uniform weighted beam size, and the residual noise are given for each epoch and each source.[]{data-label="VSOP_tab:obslog"}
The 100m RT at Effelsberg not only participated as a VLBI antenna, but also was used to monitor the flux density and polarization variability of all programme sources. These flux density measurements were made in gaps between VLBI scans using pointing cross-scans in azimuth and elevation. From this and the flux and polarization measurements of primary calibrators (3C286, 3C48 and 3C295) before and after the VLBI experiments, the orientation of the polarization E-vector (EVPA, Electric Vector Position Angle) was also obtained (see Sect. \[sec:effred\] for details).
The data were recorded in VLBA format with two 16MHz baseband channels per circular polarization in two-bit sampling, resulting in a recording rate of 128Mbps at the ground array stations. The HALCA antenna observed only LCP and the data was recorded at several tracking stations (typically 3 per epoch). The correlation was done at the VLBA correlator in Socorro, NM.
Although HALCA is only able to observe in left circular polarization (LCP), VLBI polarimetry becomes possible, if enough other stations of the remaining VLBI array measure both circular polarizations. It is possible to cross-correlate LCP from HALCA with RCP from the ground array stations and thus obtain the cross-polarized correlations on the space baselines. Normally one needs both cross polarizations, $RL$ and $LR$, to image the linear polarization of a source; but using a complex $Q+iU$ image, it is possible to include antennas with only a single cross polarization (e.g., AIPS Memo 79, 1992, W.D. Cotton)[^1]. Further details about polarization observations using HALCA are given in, e.g., [@1999NewAR..43..691G], [@2000PASJ...52.1055K], and [@2001MNRAS.320L..49G].
The post-correlation analysis was done using NRAO’s Astronomical Image Processing System ([Aips]{}). After loading the data into [Aips]{}, the standard amplitude and phase calibrations were performed. Since HALCA does not provide pulse calibration information, a manual phase calibration was done to remove offsets between the two intermediate frequency channels (IFs).
At this point the data were exported as $(u,v)$-FITS files (task FITTP) from [Aips]{} to [Difmap]{} (@1994BAAS...26..987S), where the imaging, phase, and amplitude calibration was done, using the CLEAN () and SELFCAL procedures. Because of the small diameter of the HALCA antenna, the data from the ground array stations yield a much better signal-to-noise ratio (Fig. \[VSOP\_fig:radplot\]). Therefore, we first imaged the ground array data alone to obtain a good initial Stokes $I$ image. After this, we included the HALCA antenna in the imaging and self-calibration process, using a strong Gaussian taper (10% at 450M$\lambda$) during the first iterations and subsequently decreasing the taper. All images were obtained using uniform weighting. For the ground array images the gridding weights were scaled by the amplitude errors raised to the power $-2$, which favours high quality data with small errors, and for the VSOP images equal weights were applied to achieve highest possible angular resolution. The amplitude errors were derived by the scattering of the data when we averaged the data sampled at 4-second intervals to 60-second intervals.
The self-calibration was done in total intensity in steps of several phase-calibrations followed by careful amplitude calibration. During the iteration process, the solution interval of the amplitude self-calibration was shortened from intervals as long as the whole observational time (in the initial imaging steps) down to minutes at the end of the imaging process. Finally the self-calibrated data were reimported into [Aips]{} (using task FITLD), where the calibration of the leakage terms (D-terms) and the polarization imaging was done.
![Amplitude vs. $(u,v)$-distance of 0716+714 (Oct. 5). Clearly visible is the higher noise at the HALCA baselines at large $(u,v)$-distances. The dotted line (marginally visible in the centre of the HALCA data) represents the final clean model.[]{data-label="VSOP_fig:radplot"}](3943_fig2.eps){width="8cm"}
### Feed calibration {#VSOP_sec:dtermcal}
Calibration of the instrumental polarization and the leakage terms (D-terms) from the left to the right circular polarization feeds at each antenna was done using the task LPCAL in [Aips]{} (for the method see: @1995AJ....110.2479L). Therefore the total-intensity model produced with IMAGR was split into sub-models using the task CCEDT. The model was separated into three to four sub-models that should represent a small number of “isolated components”, with clearly defined individual polarization properties, suitable for determining the instrumental cross-polarization with LPCAL. This was done in several steps and with different sub-models, to ensure that the D-terms do not depend critically on the choice of the different source and sub-models.
The LPCAL algorithm is robust against different source structures, nevertheless one should prefer core-dominated sources for the calibration. All sources in our observations meet this criterion. The VLBA antennas and Effelsberg exhibit D-term values between 0.5% and 2%. For HALCA we derived about $4$%, similar to what was found by previous observations (e.g., @1999NewAR..43..691G). A number of feed solutions were calculated by using slightly different source models and/or different calibrator sources (0816+710 and 0615+820). The RMS differences between these D-terms are typical smaller than 0.3% for the ground array stations and reach $\sim0.5$% for the HALCA satellite. The D-terms of HALCA could only be checked with different source models of 0716+714, since the calibrators were not observed by the satellite. Plots of the real versus imaginary cross-hand polarization data indicated that a satisfactory D-term solution was obtained. This was also verified in plots of the real and imaginary cross-hand data versus $(u,v)$ parallactic angle. After applying the D-term solution, the variations in the visibility amplitudes with parallactic angle were removed. In Fig. \[VSOP\_fig:real\_imag-pa\] we illustrate this, showing some visibilities before and after application of the D-term calibration.
To estimate how the D-term variations affect the source polarization images, fractional polarization maps with different feed calibration tables were made and then subtracted. The residual amount of fractional polarization in the subtracted images can be used to measure the uncertainty introduced by the different calibrations. This yields an accuracy of $\Delta P/P = 0.1$% in fractional polarization for the VLBI core of 0716+714 and $\Delta P/P = 1.2$% for the weaker jet. The effect of different D-terms on the accuracy of the EVPA was measured in a similar way. Here we obtained an uncertainty of the orientation on the E-vector of $\Delta {\rm EVPA}= 0.3^\circ$ for the core and $\Delta {\rm EVPA}= 2.0^\circ$ for the jet.
![An example for the tests made to check the reliability of the D-term solutions: the panels show the real (top) and imaginary part (bottom) of the Stokes $U$ data on all ground array baselines to Los Alamos plotted versus the parallactic angle (Sep. 29). [**Left panels:**]{} Before application of D-term calibration. [**Right panels:**]{} After the D-term calibration was applied. The calibration successfully removes the systematic variation of the amplitude with parallactic angle.[]{data-label="VSOP_fig:real_imag-pa"}](3943_fig3456.ps){width="9cm"}
### Polarization imaging {#VSOP_sec:polimag}
The usual imaging technique for linear polarization is to form $Q$ and $U$ values from the observed $RL$ and $LR$ correlations and separately image and deconvolve the $Q$ and $U$ images. The [Aips]{} task COMB can be used to combine the $Q$ and $U$ images to obtain a linear polarization intensity map ($I_{\rm P}=\sqrt{Q^2+U^2}$) and a map that represents the electric vector polarization angle ($EVPA=\chi=0.5\,\arctan{\frac{U}{Q}}$). But this is only possible in observations where both cross-polarized correlations, $RL$ and $LR$, are available for each interferometer baseline.
Since the HALCA antenna observed only in LCP, only one of the two cross polarized correlations is available for the space-baselines. Nevertheless, it is still possible to form a linear polarization image: in this case complex imaging with a complex deconvolution of $Q+iU$ has to be applied. The [Aips]{} software package also offers these tasks. We used the procedure CXPOLN to build the complex image and beam, along with the task CXCLN, which does the complex cleaning and which provides the cleaned $Q$ and $U$ images. These images were then combined with COMB in the usual way to obtain linear polarization intensity and polarization angle images.
### EVPA calibration {#VSOP_sec:EVPA}
For the EVPA calibration we used Effelsberg measurements of the quasar 0836+710, orienting its E-vector to ${\rm P.A.}= 106.3\,^\circ$, previously determined from measurements relative to the primary calibrator 3C286. We further assumed that between the angular scales covered by the Effelsberg beam and the VLBI scale, there is no dominant polarized component that could affect the EVPA calibration of the VLBI data. The existing VLA maps of 0836+710 support this assumption, as does the comparison of the total intensity and linear polarization measurements at Effelsberg (Tables \[VSOP\_tab:ebsources\] & \[VSOP\_tab:ebpol\]) with the flux density seen with the VLBA (Table \[VSOP\_tab:obslog\] & \[VSOP\_tab:0836pol\]).
[ld[2]{}d[3]{}d[3]{}d[3]{}d[4]{}]{} Epoch & & & & &\
& & & & &\
29 Sep & 160, 1 & 106.3 , 0.3 & 131 , 13 & 125.8,1.2 &-19.5 , 1.2\
04 Oct & 160, 1 & 106.3 , 0.3 & 133 , 13 & 125.6,1.0 &-19.3 , 1.0\
05 Oct & 160, 1 & 106.3 , 0.3 & 128 , 13 & 96.0,1.1 & 10.3 , 1.1\
Effelsberg measurements {#sec:effred}
-----------------------
In gaps between VLBI scans, we measured the flux density and polarization of the program sources and calibrators. The measurements were performed using standard pointing cross-scans with slews in azimuth and elevation. In addition to the VLBI targets, we observed 0951+699 and 3C286 as additional flux density and polarization calibrators. In Table \[VSOP\_tab:ebsources\] we summarise the mean flux density (col. 2) for each source, the number of pointing scans (col. 3), the modulation index $m=100 \cdot S/\langle S\rangle $, with the mean flux density $\langle S\rangle $ (col. 4), and the reduced chi-square testing the assumption of non-variability.
\[VSOP\_tab:ebsources\]
[ld[8]{}rcc]{} & & & &\
0615+820$^a$ & 0.773 , 0.003 & 35 & 0.42 & 0.08\
0716+714$^a$ & 0.747 , 0.018 & 31 & 2.38 & 2.25\
0836+710$^a$ & 2.500 , 0.008 & 45 & 0.33 & 0.05\
0951+699 & 3.367 , 0.005 & 4 & 0.14 & 0.01\
3C286 & 7.521 , 0.074 & 11 & 0.98 & 0.42\
3C295 & 6.593 , 0.056 & 8 & 0.85 & 0.31\
3C48 & 5.537 , 0.019 & 2 & 0.34 & 0.05\
\
A detailed description of the Effelsberg data reduction procedures is given in . In this experiment we used 3C286, 3C295, and 3C48 as the main flux density calibrators ($S_{\rm 3C\,286}=7.50$Jy, $S_{\rm
3C\,295}=6.59$Jy and $S_{\rm 3C\,48}=5.63$Jy) and 0836+710 as a secondary calibrator, which was observed adjacent to each scan on 0716+714. Also 0615+820 was observed regularly as a secondary calibrator and is known to be non-variable at least on time-scales of days (@Kraus1997). The linear polarization measurements were calibrated using 0836+710 and 3C286, which both have known polarization properties, as well as 0951+699 which is unpolarized, and all only vary over longer time-scales. After calibration, the residual error of the Effelsberg EVPA was $\sim0.3^\circ$.
Results {#VSOP_sec:results}
=======
In this section the total intensity and linear polarization maps of 0716+714 from the ground array data and the VSOP data are presented and analysed. The high dynamic range of the ground array data (peak/RMS $\sim 15\,000$) enables us to follow and study the weak jet up to $\sim 12$mas core separation. The three times higher resolution in the VSOP images allows a detailed study of the core structure and its variability. The results and analysis of the total intensity and polarization measurements with the 100m RT in Effelsberg are presented in Sect. \[VSOP\_sec:Eff\]. We would like to note that during each of the three VSOP observations, 0716+714 showed only moderate variations in total intensity ($\sim 5$%, peak to peak during one epoch). A more pronounced variable source would violate the principle of stationarity in aperture synthesis and would lead to a severe degradation of the reconstructed interferometric image. Here, these effects are small and can be neglected.
Imaging simulations performed by [@Hummel1987] have shown that intra-day variability during VLBI observations mainly reduces the achievable dynamic range due to residual side lobes in the images without changing the source structure. In these simulations even larger variations ($\sim 80$%) were considered. Intensity variations of about 5%, which are present in our observations, are comparable to the usual uncertainties of the $T_{\rm sys}$ measurements ($\sim
5-10$%), which are used to calibrate the visibility amplitudes. The final images after self-calibration show a nearly constant noise level outside the source structure without any symmetric features indicating that there are no residual side lobes present. Therefore, we can neglect the effects of the IDV on our VLBI maps. The larger variability of the linear polarization (up to $\sim
40$%) probably degrades the linear polarization images to some extent, but should not severely affect the self-calibration procedure, because the phase and amplitude self-calibration of $LL$, $RR$, $LR$, and $RL$ during the imaging process was done using the total intensity data.
Ground array data {#VSOP_sec:groundarray}
-----------------
The ground array images showing the parsec-scale structure of the jet of 0716+714 are shown in Fig. \[VSOP\_fig:groundmaps\]. The panels in the left row show the total intensity contours of 0716+714 with polarization E-vectors superimposed for all three epochs. The right panels of the figure show polarization contour maps. The intensity maps reveal a one-sided core jet structure with a north-oriented jet extending up to $\sim12$mas core separation along P.A. $\approx15^\circ$. Between the core and the jet the EVPA is misaligned by $\sim 60^\circ$. In the jet and on mas-scales, the electric field vectors are aligned well with the jet axis. Provided that the jet emission is optically thin, which is supported by the high degree of fractional polarization (Fig. \[VSOP\_fig:groundprofile\]), the alignment suggests that the magnetic field is oriented perpendicular to the jet axis. The $60^\circ$ misalignment of the EVPA of the core can be either explained by Faraday rotation, the fact that the core is optically thick, or by jet bending in the inner most region near the core. High Faraday rotation has been found in many AGN cores (e.g., @2004ApJ...612..749Z [@2003ApJ...589..126Z] and references therein) with the tendency towards lower rotation measures in BLLac objects than in quasars, but we are not aware of any rotation measures for the core of 0716+714. However, recent mm-VLBI observations at 43GHz and 86GHz show that on the 0.2mas scale, the jet is strongly curved, oriented along P.A. $\approx50^\circ$ (Fig. \[VSOP\_fig:mm\_map\]). This supports the idea that the EVPA orientation follows the bent jet down to the sub-mas scale and that even in the vicinity of the core, the magnetic field is perpendicular to the jet axis.
Inspection of the three 6cm VLBI images in Fig. \[VSOP\_fig:groundmaps\] reveals no major changes in the total intensity structure between the epochs. Table \[VSOP\_tab:obslog\] shows a small and, at this stage, marginal decrease of 5% in the peak flux from the first to the last two epochs. A more detailed parametrization of the maps, however, is given by Table \[VSOP\_tab:VLBAresult\]. Here we show the integrated flux densities separately for core and jet.
The measurements were made using the tasks IMSTAT and TVSTAT in [Aips]{}, and the errors are derived from the scatter between the individual measurements on slightly different calibrated maps and with different window sizes. The task IMSTAT integrates over a rectangular region that is not necessarily in good agreement with the source structure, whereas TVSTAT integrates over a polygonal region that is specified by the user. We preferred to use IMSTAT and TVSTAT instead of model-fitting Gaussian components to the images, since they also give us more precise measurement of the extended emission.
Attempts to parameterise the total intensity and polarization images with model fitting were focused on the problem that one either needs to use a different number of components to fit $I$ and $P$ or to fix the component position in the $I$ or $P$ images. In both cases one cannot be sure that all of the extended jet emission is adequately represented by the model fit and, more important, if measurements are comparable between the epochs.
[lld[4]{}d[2]{}d[3]{}]{} Epoch & & & &\
& Core & 520.3 , 26.9 & 12.1 , 1.3 & 49.4 , 4.1\
& Jet & 56.0 ,pt 4.7 & 7.4 , 0.8 & -10.8 , 5.6\
& Core & 499.3 , 26.1 & 11.8 , 1.3 & 40.7 , 4.0\
& Jet & 54.8 ,pt 6.3 & 7.3 , 0.8 & -11.2 , 7.8\
& Core & 503.9 , 25.4 & 6.5 , 1.1 & 52.7 , 5.2\
& Jet & 54.7 ,pt 6.0 & 7.5 , 0.8 & -9.5 , 7.4\
From Table \[VSOP\_tab:VLBAresult\] one can see that the core flux density decreases from 520mJy to about 500mJy in the later epochs, whereas the jet flux density stays relatively stable at $\sim 55$mJy. Using the technique of amplitude self-calibration improves the quality of the VLBI images by applying antenna-based gain corrections to remove residual calibration errors that were not corrected by the a priory amplitude calibration. Provided that the signal-to-noise ratio is sufficiently high, after several iterations the data and the model should converge around an average value set by the a priory amplitude calibration. In these experiment time-dependent station gain corrections of a few percent (typically $\sim 5$%) were applied. Since the procedure used in all epochs was the same, the relative accuracy of the flux densities after self-calibration should be better than the absolute calibration and can be measured by using the flux of the extended jet emission seen in each experiment. Due to its parsec-scale size, the flux of the integrated jet emission should not be variable on time-scales of days. From Table \[VSOP\_tab:VLBAresult\] we obtain 56, 54.8, and 54.7mJy for the integrated jet flux for each epoch, respectively. Thus, we could attribute this 2.3% variation between the individual measurements of the extended jet emission to the remaining calibration uncertainty. While the integrated jet flux remained constant at this level, the core flux varied by $\sim 20$mJy or $\sim
4$%, a factor of two more than the jet.
In contrast to the marginal variability of the total flux density, strong variability is seen in linear polarization. A visual inspection of the polarization vectors in Fig. \[VSOP\_fig:groundmaps\] indicates that the linear polarization of the core component varies significantly. From the integrated values in Table \[VSOP\_tab:VLBAresult\], one can see that there are no changes (within the errors) in the polarized intensity between the first two epochs, but in the last epoch the polarized intensity of the core drops by $\sim$45%, from $\sim 12.0$mJy to $\sim 6.5$mJy, corresponding to a decrease in fractional polarization from $\sim2.4$% to $\sim1.3$%. As in total intensity, the polarized intensity of the jet shows no variability and stays very constant with a polarized flux of $7.4 \pm 0.1$mJy (or 1.3% accuracy) at an average degree of polarization of $\sim 13.6$%. The average EVPA of the jet is also constant at an angle of $-10.5 \pm 0.9^\circ$, whereas the core EVPA rotates by $\sim 10-13^\circ$, from $\sim50^\circ$ to $\sim40^\circ$ and back to $\sim53^\circ$ during the observations.
![Slices along ${\rm P.A.}=12.5^\circ$ through the total intensity, linear polarization intensity, fractional polarization, and EVPA images of the core and the jet of 0716+714.[]{data-label="VSOP_fig:groundprofile"}](3943_fig14.ps){width="9.3cm"}
Figure \[VSOP\_fig:groundprofile\] summarises all these results. It shows profiles of total intensity, linear polarization, degree of polarization, and the orientation of the E-vector plotted versus core separation. To avoid confusion between the different profiles, we did not display the error bars in the plots. The errors of the total intensity and the polarization are dominated by the uncertainties of the amplitude calibration, about 2%. The error of the EVPA depends on the SNR of the linear polarization and varies between $0.3\,^\circ$ for the VLBI core component and about $2\,^\circ$ in the bright parts of the jet (see also Sects. \[VSOP\_sec:dtermcal\] & \[VSOP\_sec:EVPA\]).
In summary, it seems that between the 3 observations no variation was seen in the jet, neither in intensity nor in polarization. On the other hand, the total flux density of the VLBI core component varied by about $4 \pm 2$% in total intensity and by about a factor of 2 in polarization. The polarization angle of the jet remained constant within $\sim 2^\circ$, while the polarization angle of the core varied by $\sim 10^\circ$. Therefore all variability appears in the bright and unresolved VLBI core component of 0716+714.
Noticeable are the two peaks of up to 40% fractional polarization in the jet at $r\approx 2.5$mas, $r\approx3.5$mas, and $r\approx4.5$mas distance from the core in Fig. \[VSOP\_fig:groundprofile\]. At the same positions, B05 found modelfit components in total intensity, which were used to study the kinematics in the jet. These jet components apparently move superluminally, with speeds[^2] of 6.9$c$ to 8.3$c$. This provides further evidence that these are real structures in the jet, probably shocks, where we might have a higher density that enhances the total intensity and a more ordered magnetic field that enhances the linear polarization.
Space VLBI data {#VSOP_sec:VSOP}
---------------
Combining the VLBI ground stations with the radio antenna onboard HALCA improves the resolution at 5GHz by a factor of three (see Table \[VSOP\_tab:obslog\]). Owing to the small diameter of the HALCA antenna (diameter, $\varnothing=8$m), the sensitivity on the VLBI baselines to the satellite is relatively low, leading to higher noise on the longest VLBI baselines. The nominal system equivalent flux density (SEFD) of the HALCA antenna is 15300Jy at 5GHz (@2000PASJ...52..955H) compared to 312Jy for each VLBA antenna ($\varnothing=25$m) and 18Jy for Effelsberg ($\varnothing=100$m). The uniform weighted $I$ and $P$ images are presented in Fig. \[VSOP\_fig:VSOPmaps\]. To achieve the highest possible resolution, equal weights were applied to all antennas.
In total intensity, the jet extends to core separations of up to 3.5mas. In comparison to the ground array maps, the jet appears less straight, with bends, followed by back-bends. Although the intensity is weak, the polarization vectors in the jet seem to follow these oscillations. The VLBI core itself completely dominates the weak jet. Due to the 3 times higher resolution with HALCA, it is easier to distinguish between compact emission from the core and extended emission from the mas-jet. The right panel of Fig. \[VSOP\_fig:VSOPmaps\] shows the polarization contour maps. Owing to their relatively low dynamic range of 70:1, only the VLBI core component is detected at a significant level. Most of the polarized jet emission remains only marginally visible, as indicated by the grey line, which marks the lowest contour of the emission in total intensity. In addition to the core region, a region of slightly enhanced jet polarizations is visible between 1.5mas to 2mas from the core. As for the ground array maps, we summarise the integrated flux densities of core and jet in total intensity and polarization in Table \[VSOP\_tab:VSOPresult\]. Since the marginal detection of linear polarization in the jet prevents reliable measurements, we do not give linear polarization values for the jet in Table \[VSOP\_tab:VSOPresult\].
In comparison to the ground array data (see Table \[VSOP\_tab:VLBAresult\]), we measured a core flux density that is $25-35$% lower, but a jet flux density that was comparable, although the jet is much shorter. This is most likely due to blending effects between core and jet emission, as we could not properly separate both regions using the ground array images. In all three VSOP observations, $\sim 6$% of the total flux density seen in the ground array maps is missing in the space-VLBI maps. This is reasonable, since the outer jet is partially resolved and therefore much shorter. It is possible to recover the missing flux density by using natural weighting, but this would also decrease the resolution. Since from the ground array images it is already known that the jet flux density is not variable, we can concentrate in the following on the analysis of the core-variability with the highest possible angular resolution.
[lld[4]{}d[3]{}d[3]{}]{} Epoch & & & &\
& Core & 492.5 , 25.0 & 11.8 , 2.8 & 49.6 ,pt 4.3\
& Jet & 62.4 ,pt 4.0 & &\
& Core & 465.4 , 23.6 & 10.8 , 2.6 & 40.3 , 4.1\
& Jet & 59.5 ,pt 3.7 & &\
& Core & 462.4 , 23.3 & 5.6 , 1.4 & 54.9 , 4.5\
& Jet & 60.1 ,pt 3.7 &\
The VSOP images show the same variability behaviour as the ground array images. The total intensity of the core varied at a marginal level of $\sim 6.5$%. The flux density of the jet emission remained constant at the 4.8% level. Again, the linearly polarized flux density of the core was highly variable, with an amplitude modulation of a factor 2 between October 4 and 5. The integrated polarization flux densities of the core is comparable to the ground-array images, which suggests that the linear polarized component itself is still compact on the VSOP scales ($\leq 0.25$mas). We note that 0716+714 was recently detected with VLBI at 230GHz on transatlantic baselines (@Krichbaum2004). This sets an upper limit to the source size of $\sim
20\,\mu$as, an order of magnitude smaller than the resolution obtained by VSOP.
As in the ground array maps, the EVPA in the core of 0716+714 rotates between the three epochs. Within the measurement errors, the vectors rotated by the same amount as in the ground array images, namely around $-9^\circ$ between the first and the second epochs and around $+12^\circ$ between the second and the third epochs. The source profile of the VSOP-images is shown in Fig. \[VSOP\_fig:VSOPprofile\]. For the core region, it looks very similar to the profiles from the ground-array images (Fig. \[VSOP\_fig:groundprofile\]). The VSOP profiles reveal a small but clearly visible position shift of the linear polarized peak between the epochs. Since these shifts are very small ($<50$$\mu$as), we are not confident that they are real. Small position shifts could be caused by the motion of the components, which – with regard to the short observing time interval (6 days) and the resulting extreme apparent velocities – does not appear very likely. An alternative and perhaps more realistic interpretation could invoke blending effects between two or more polarized and variable sub-components, located below the angular resolution of this data sets and inside the core region (see also Sect. \[VSOP\_sec:tb\]).
![Slices along ${\rm P.A.}=12.5^\circ$ through the VSOP images of the core of 0716+714 in total intensity, linear polarization intensity, fractional polarization and EVPA. It is a virtual zoom into the core region of Fig. \[VSOP\_fig:groundprofile\], and $r=0$ marks the same position in both plots.[]{data-label="VSOP_fig:VSOPprofile"}](3943_fig21.ps){width="9.5cm"}
Results from total flux density measurements {#VSOP_sec:Eff}
--------------------------------------------
Table \[VSOP\_tab:ebsources\] summarises the total flux density measurements, which were taken between VLBI scans with the 100m radio telescope in Effelsberg. The last two columns of the table give the modulation index $m$ and the reduced $\chi^2_{\rm r}$ for testing the variability. Comparison of the modulation indices and $\chi^2_{\rm r}$ of 0836+71, 0615+82, and 0716+714 clearly shows that 0716+714 was variable during the observations. The formal $\chi^2$-test yields a probability of significant intra-day variability of 99.984%. The same arguments apply for the polarization. In Table \[VSOP\_tab:ebpol\] we summarise the linear polarization properties of 0716+714 and the polarization calibrators. The calibrators 0615+82 and 0951+69 are not shown since they are unpolarized.
\[VSOP\_tab:ebpol\]
[ld[3]{}d[2]{}..d[3]{}.]{} & & & & & &\
& & & & & &\
0716+71 & 19.9 , 3.3 & 2.7 , 0.4 & 9.26 & 15.39 & 18.8 , 4.6 & 24.2\
0836+71 & 160.0 , 1.1 & 6.4 , 0.1 & 0.00 & 0.05 & 106.3 , 0.3 & 0.2\
3C286 & 823.3 , 12.8 & 11.0 , 0.3 & 0.78 & 0.27 & 33.0 , 0.2 & 0.2\
![Total intensity (top), linear polarization (middle), and EVPA variations of 0716+714 and the calibrator 0836+710, as measured at Effelsberg at 5GHz. The left panel shows the data from September 29 and the right panels from October 4 to 5 (note that the time axis between left and right panels differ). For intensity and polarization, we show relative variations, normalised to the mean (average taken over all three days). For the EVPA we plot the absolute difference relative to the mean EVPA. The mean values are given in Table \[VSOP\_tab:ebsources\] & \[VSOP\_tab:ebpol\].[]{data-label="VSOP_fig:eblc"}](3943_fig22.eps){width="9cm"}
Again 0716+714 is the only source that showed significant variability. The total intensity, linear polarization, and EVPA light curves for 0716+714 and 0836+710 are presented in Fig. \[VSOP\_fig:eblc\]. Since these measurements were made primarily for telescope pointing, the accuracy for flux density measurements is not as high, as it is typically the case for dedicated IDV monitoring observations. We therefore smoothed the light curves, using a three-point running mean, in order to reduce the noise. Figure \[VSOP\_fig:eblc\] clearly shows the variability of 0716+714 and the non-variability of its nearby secondary calibrator 0836+710. In 0716+714, up to 5% variations are seen in total intensity and about 40% in linear polarization. The polarization angle changed by up to 10 degrees.
Discussion {#VSOP_sec:discussion}
==========
Here we compare the intra-day variability seen in total intensity and linear polarization with the Effelsberg telescope to the variations seen in the VLBI images and discuss then their implications for the origin of the variability (Sect. \[VSOP\_sec:origIDV\] & \[VSOP\_sec:multcompos\]) and for the brightness temperature of the VLBI core of 0716+714 (Sect. \[VSOP\_sec:tb\]). We also discuss possible physical causes for our findings. Using the results of a recent VLBI study of the jet kinematics of 0716+714 (B05), we were able to derive some basic jets parameters, like speed, Doppler-factor, and inclination (Sect. \[VSOP\_sec:jetparameter\]).
Origin of variablity {#VSOP_sec:origIDV}
--------------------
A simple comparison of the average flux density measurements from Effelsberg and the integrated flux densities of the core and the jet region at the three VLBI epochs reveals that all the variablity originates in the VLBI core component (see Table \[VSOP\_tab:VLBAvsEb\]). The mean difference between the total flux density in the VLBI maps and the single-dish flux density is ($183.4\pm3.1$)mJy, corresponding to about 24.5% less flux density in the VLBI images. This is plausible, since the arcsecond-scale structure of 0716+714 shows two radio-lobes and a surrounding halo-like structure with a diameter of 10arcsec (@1986AJ.....92....1A), which is inside the Effelsberg beam but not seen on VLBI scales. Within the measurement uncertainties, the absolute values of the core variability are similar to those seen at Effelsberg. We observed a decrease of 21mJy ($\sim 3$%) with VLBI and 27.5mJy ($\sim
5$%) at Effelsberg between epochs one and two and a constant flux density level between epochs two and three. This also confirms our estimate of about 2.3% for the relative flux density error between the VLBI epochs (Sect. \[VSOP\_sec:groundarray\]).
[lld[4]{}d[2]{}d[3]{}]{} Instrument & Part & & &\
\
& Core & 520.3 , 26.9 & 12.1 , 1.3 & 49.4 , 4.1\
& Jet & 56.0 ,pt 4.7 & 7.4 , 0.8 & -10.8 , 5.6\
& Total & 576.3 , 31.6 & 17.0 , 2.6 & 27.3 , 6.0\
Eb & & 763.2 ,pt 6.9 & 21.4 , 2.6 & 23.4 , 2.1\
\
& Core & 499.3 , 26.1 & 11.8 , 1.3 & 40.7 , 4.0\
& Jet & 54.8 ,pt 6.3 & 7.3 , 0.8 & -11.2 , 7.8\
& Total & 554.1 , 32.4 & 17.3 , 2.7 & 21.3 , 4.8\
Eb & & 735.7 , 16.2 & 21.6 , 2.6 & 18.6 , 2.2\
\
& Core & 503.9 , 25.4 & 6.5 , 1.1 & 52.7 , 5.2\
& Jet & 54.7 ,pt 6.0 & 7.5 , 0.8 & -9.5 , 7.4\
& Total & 558.6 , 31.4 & 12.0 , 2.3 & 19.1 , 5.3\
Eb & & 740.2 , 14.6 & 15.7 , 1.1 & 13.3 , 2.5\
The variablity of the polarized flux density is anti-correlated with the total intensity variation. The core polarization shows nearly no variation between the first two epochs and a decrease of 5.3mJy in the VLBI images and 5.9mJy at Effelsberg between epochs two and three, which corresponds to a 40% decrease.
Only $4.2\pm0.3$mJy of linear polarization are missing in the VLBI maps in comparison to the $\sim180$mJy missing in total intensity. This yields an average fractional polarization for the large-scale structure of only 2.3%, which is small when compared to the $\sim40$% linear polarization found with the VLA at 1.4GHz in the lobe-like structure (@1987MNRAS.228..203S). However, the VLA image shows a variety of EVPA orientations, and the discrepancy is most likely due to in-beam depolarization at Effelsberg, so that polarization vectors at opposite position angles cancel. The missing polarization flux density also causes a difference between the EVPA measured from the VLBI images and at Effelsberg ranging from $2.7^\circ$ to $5.8^\circ$, but an inspection of the absolute differences between the epochs reveals that the VLBI-core EVPA varied in a similar manner to the EVPA of the Effelsberg data.
Unfortunately, the time sampling of the Effelsberg light curve was not dense enough to measure reliable intra-day variability for most of the time. However, on Oct. 4 the light curve showed a significant decrease in polarization from $(24.5\pm1.7)$mJy between $03^{00}$ UT and $07^{00}$ UT to $(20.1\pm1.1)$mJy between $14^{00}$ UT and $18^{00}$ UT. At the same time the polarization vector rotated by $(4\pm2)^\circ$. To check whether this polarization IDV is also seen in the VLBI data, the data were split into the corresponding time intervals and imaged separately. The resulting maps show a decrease in the linear polarization in the core from ($13.5\pm1.4$)mJy to ($9.5\pm1.0$)mJy (Fig. \[VSOP\_fig:oct4vlbi\] and Table \[VSOP\_tab:polIDV\]).
[ld[4]{}d[4]{}d[4]{}d[4]{}]{} UT \[h\] & & & &\
03–07 & 24.5 , 1.7 & 13.5 , 1.4 & 21.0 , 2.3 & 40.8 , 4.2\
14–18 & 20.1 , 1.1 & 9.5 , 1.0 & 17.2 , 1.8 & 43.0 , 5.1\
Although the inaccuracy of the core EVPA measurements in the VLBI images is too large to also confirm a rotation of the EVPA, the very good agreement between the intensity variation from hours to days between the VLBI-core and the single-dish measurements leaves no doubt that the IDV is coming from the VLBI core of 0716+714.
We note that similar variations were previously observed in the IDV sources 0917+624 and 0954+658, where ground-based VLBI observations revealed significant polarization IDV on mas-scales, without showing large variations in the total intensity (@2000MNRAS.315..229G).
Multiple components {#VSOP_sec:multcompos}
-------------------
The only moderate variability of the total intensity, the relatively large variation of the polarized flux density, and the relatively small variation of the polarization angle indicate that the variations are caused by the vector sum of the variability of multiple compact and differently polarized sub-components, which are located within the VLBI core region on scales that are smaller than the beam size. The slope of the polarization angle over the core region as seen with the higher resolution from the VSOP data (see Fig. \[VSOP\_fig:VSOPprofile\]) supports the idea that the VLBI core component itself is a composite of several sub-components. Such a multi-component structure is required to explain the polarization IDV by refractive interstellar scintillation (RISS) (e.g.), but is also indispensable in models using solely jet intrinsic (e.g.) or mixtures of intrinsic and extrinsic effects (@2002ChJAA...2..325Q).
If interstellar scintillation is the dominant cause of the variability of the sub-components, their size cannot be much larger than the scattering size of the interstellar medium, which at 5GHz and towards 0716+714 is believed to be of the order of several ten micro-arcseconds (e.g. ). Under these circumstances, the number of sub-components located within the VLBI-core region (smaller than about 100$\mu$as, see next section) is limited to a few. The independent and possibly partly quenched scintillation (@1986ApJ...307..564R) of the sub-components could then explain the variability of the polarization intensity and the variation in the polarization vector. The relative strength of the variability of the polarized flux density ($\sim 40$%) suggests (i) that at least one of the sub-components is considerably smaller than the scattering size (strong scintillation), and (ii) that the number of independently variable components is not very large, otherwise the time averaged vector sum of the polarization would cancel. Since the polarization angle variations are small ($\leq 15^\circ$), we conclude that either (i) the polarization vectors of the sub-components are more or less aligned or (ii) that the more variable sub-component has to be less polarized than the less variable (and most likely more extended) sub-component.
It is tempting to identify the smaller and less polarized sub-component with the optically thick jet-base and the larger and more polarized sub-component with the optically thin inner jet. The fact that the total intensity is less variable than the polarized intensity indicates that the polarized sub-components are considerably smaller than the component(s), which dominate in total intensity. Of course, this simple scenario could change, if a larger number of independently varying sub-components exist or if some sub-components vary in a coherent manner. It is also possible that the variability of the sub-components is not due to scintillation, but is source intrinsic or, more likely, a mixture of both effects.
Brightness temperature {#VSOP_sec:tb}
----------------------
Assuming that the variability has, at least partly, an intrinsic origin restricts the emitting region to being extremely compact, which then results in very high brightness temperatures exceeding the inverse-Compton limit of $10^{12}$K (@1969ApJ...155L..71K). The short time-scales observed imply linear dimensions of
$$l=\frac{c\,\delta\,\Delta t}{1+z},$$
where $\delta$ is the Doppler factor. Assuming stationarity and a redshift of 0.3, the size of the emission region responsible for the variability is around $8\cdot 10^{12}$m or $\sim 55$A.U., which is slightly larger than our solar system. The brightness temperature for a stationary source is given by
$$T_{\rm b} = 3.06\times 10^8\ S\ \left(\frac{d_{\rm L}}{\nu\ t\ (1+z)^2}\right)^2$$
where $S$ (in Jy) is the flux density, $d_{\rm L}$ (in Mpc) the luminosity distance, $\nu$ (in GHz) the observing frequency, $t$ (in yr) the variability timescale, and $z$ the redshift. The luminosity distance was calculated by adopting an analytical fit by [@1999ApJS..120...49P] resulting in $d_{\rm L}=1545.8$Mpc (see also B05). To calculate the variability time-scale we use:
$$t=\frac{\langle S\rangle }{\Delta S}\ \frac{\Delta t}{(1+z)},$$
where $\langle S\rangle $ (in Jy) is the mean flux density, $\Delta S$ (in Jy) the standard deviation, and $\Delta t$ (in yr) the duration of the variation (e.g. @1996AJ....111.2187W). Calculating the corresponding $T_{\rm b}$ of the linear polarization decrease between October 4th and 5th ($\langle S\rangle =9.2$mJy, $\Delta S=3.8$mJy, and $\Delta t=24$h) results in $T'_{\rm
b}\approx3.8\times10^{15}$K and the decrease during October 4th ($\langle S\rangle =11.5$mJy, $\Delta S=2.8$mJy, and $\Delta t=10$h) yields $T'_{\rm
b}\approx9.6\times10^{15}$K. Since the observed brightness temperature $T'_{\rm b}$ derived from variability is connected to the intrinsic brightness temperature by $T'_{\rm b}=T_{\rm b}\cdot\delta^3$, a Doppler factor of 14 to 22 is needed to reduce the brightness temperature to the inverse-Compton limit. These calculations usually require a densely sampled light curve to estimate $\langle S\rangle $, $\Delta S$, and $t$. However, considering that, due to the insufficient sampling we, have probably not observed the full amplitude or missed the most rapid variations, the calculated values of $T_{\rm b}$ are underestimates rather than overestimates of the true values.
If instead of the inverse-Compton limit, the equipartition brightness temperature of $5\times 10^{10}$K to $10^{11}$K is used (cf. @1994ApJ...426...51R), a Doppler factor of up to 60 is obtained. However, only a small departure from equipartition, e.g. by a factor of two, would bring the Doppler factor back to 30, which is well within the range of possible Doppler factors derived from our kinematic study (B05).
Gaussian modelfits to the VSOP $(u,v)$-data of the first epoch reveals a flux density of ($0.42\pm0.06$)Jy and a size of $(0.09\pm0.02$)mas for the core component. Since the core is not resolved, the size only represents an upper limit. However, this measurement already yields a brightness temperature of $(2.6\pm0.7)\times 10^{12}$K and therewith exceeds the inverse-Compton limit. To bring these temperatures down, a Doppler factor of $> 4$ is needed. We note that all these values depend on the redshift of 0716+714 which is not yet known. The used value of $z=0.3$ is based on the non-detection of an underlying host galaxy (@1996AJ....111.2187W), but newer studies already place the limit to $>0.5$ (@Sbarufatti2005) which would increase the observed $T'_{\rm b}$ by at least a factor of three. A lower limit of the Doppler factor of 2.1 comes from synchrotron self-Compton (SSC) models (@1993ApJ...407...65G).
The VSOP 5GHz AGN survey (@2000PASJ...52..997H), which contains nearly all extra-galactic flat-spectrum radio sources brighter than 1Jy and at galactic latitude $\geq 10\,^\circ$ ($\sim 300$ sources), has shown that about 50% of these types of sources have brightness temperatures of $T_{\rm
b}>10^{12}$K, and about 20% even have $T_{\rm b}>10^{13}$K (@2004ApJS..155...33S [@2004ApJ...616..110H]). The VSOP observations of a sub-sample ($\sim 30$ sources) of the Pearson-Readhead sample (@1988ApJ...328..114P) revealed a significant correlation between the IDV activity and the brightness temperature (@2001ApJ...549L..55T). The authors find that sources showing rapid IDV have higher brightness temperatures than weakly variable or non-IDV sources. Therefore, independent of the physical model for the variability, relativistic beaming seems to play a role.
Jet inclination and Lorentz factor {#VSOP_sec:jetparameter}
----------------------------------
Simulations of relativistic hydrodynamic jets show that shocks are likely to occur in highly supersonic flows. Those provide a natural way to locally enhance the magnetic field and relativistic electron density, producing knots of emission such as seen in VLBI images (e.g., @1997ApJ...482L..33G, and references therein). Assuming that the knots in the jet are shocks travelling down the jet, we can use the observed degree of polarization of such a jet component to set limits on the angle to the line of sight and on the jet Lorentz factor (@1988ApJ...332..696C). The shock model used here is based on the scenario that [@1985ApJ...298..301H] used to describe the polarization variability in BLLac. In that model, a propagating shock compresses the magnetic field of the jet so that the initially random field is ordered in a plane transverse to the jet axis. With this we can constrain the amount of compression, defined as a unit length compressed to a length $k$, and the angle between the line of sight and the shock plane in the frame of the jet ($\epsilon$) using the measured degree of polarization ($m$) in the jet. If we assume that the shock travels with the same velocity as the emitting material, we can also relate $k$ and $\epsilon$ to the apparent speed of the jet and constrain the inclination of the jet, $\theta$, and the jet Lorentz factor, $\gamma$ (@1988ApJ...332..696C).
A modelfit to the linear polarization image with JMFIT in [Aips]{} reveals that the peak of the linear polarization in the jet corresponds to component C7 (B05), which moves with a speed of $(6.9\pm0.2)\,c$ assuming $z=0.3$. Integrating the flux density in the total intensity and in the polarization image over a region of 1.5 times the beam width around the component centre ($r=
2.7\pm0.2$mas) yields a fractional polarization, $m$, of about $(24\pm4)$%. For a spectrum of $\alpha_{\rm 5/15\,GHz}=-0.7$ (B05) and therewith $s\approx2.5$, the nominal degree of polarization of synchrotron emission in a uniform magnetic field is $m_0\approx0.72$.
We would like to distinguish two cases. First, if the jet is oriented at the viewing angle that maximizes the apparent superluminal motion, then $\epsilon=0$ and $k$ gives an upper limit on the amount of compression. We find $k_{\rm min}
= 0.71\pm0.04$. Second, if we assume a maximum of compression ($k\ll 1$), then $\epsilon$ corresponds to the maximum angle between the shock plane and the line of sight, which yields $\epsilon_{\rm max} = \pm 45\pm4^\circ$.
In the first case, the measured apparent speed of $(6.9\pm0.2)\,c$ of component C7 yields a jet Lorentz factor of $\sim 7$ and an angle to the line of sight of $\sim 8^\circ$, which would not agree with the results (B05), where a minimum Lorentz factor of $11.6$ and $\theta_{\rm max}=4.9\,^\circ$ was found. The second case ($k\ll 1$ and $\epsilon =\pm 45^\circ$) yields $\gamma\approx 10$ and $\theta\approx 2.5^\circ$ or $14^\circ$ (depending on the sign of $\epsilon$), but the larger viewing angle can almost certainly be excluded from the kinematics (B05). From this the second case seems more likely, and the derived jet parameters show the same trend as the parameters ($\gamma>16$ and $\theta<2^\circ$) derived by B05. The shock scenario also favours the small-angle solution, where we have higher jet Lorentz factors and smaller viewing angles than those derived by maximization of $\beta_{\rm
app}=\frac{\beta \sin\theta}{1-\beta \cos\theta}$. The corresponding Doppler factor is about 17 for $\gamma\approx 10$ and $\theta\approx 2.5^\circ$.
Conclusions {#VSOP_sec:conclusions}
===========
In order to search for the origin of the rapid IDV in 0716+714, a multi-epoch VSOP experiment was performed in September and October 2000. The ground array and the VSOP images show a bright core and a jet oriented to the north. The linear polarization images indicate that the jet magnetic field is perpendicular to the jet axis. Compared to the jet axis, the electric vector position angle in the core is misaligned by around 60$^\circ$. This is explained either by opacity effects in the core region or by a curved jet. Jet curvature is supported by recent high resolution 3mm VLBI observations that show the inner jet structure ($r<0.1$mas) at a similar position angle as the EVPA in the core at 6cm wavelength. A misalignment between the parsec-scale and the kiloparsec-scale structure is typical of BLLacs (@1988ApJ...328..114P [@1992ApJ...391..589W]) and is also seen in 0716+714 (VLA images can be found in @1986AJ.....92....1A [@2000MNRAS.313..627G]; Bach et al., in prep.). The new VLBI observations suggest that the jet bending continues down to sub-parsec-scales. Possible explanations for such a misalignment are a helical jet, which is oriented towards us (@1993ApJ...411...89C), possibly due to Kelvin-Helmholtz instabilities (e.g., @1986CaJPh..64..484H [@1981ApJ...250L...9H]) or to being driven by precession (B05; @2005AJ....130.1466N), or due to interaction with the surrounding medium.
Simultaneous flux-density measurements with the 100m Effelsberg telescope during the VSOP observations revealed variability in total intensity ($\sim
5$%) and in linear polarization (up to $\sim 40$%) accompanied by a rotation of the polarization angle by up to $15\,^\circ$. The analysis of the VLBI data shows that in 0716+714 the intra-day variability is associated to the VLBI-core region and not to the milli-arcsecond jet. Both the ground array and the VSOP maps show a similar decrease of the flux densities in total intensity and linear polarization of the core component, which is in good agreement with the variations in the total flux density and polarization seen with the Effelsberg 100m radio-telescope. In this, 0716+714 displays a behaviour that is similar to what was previously observed in the IDV sources 0917+624 and 0954+658, where components in or near the VLBI core region were also made responsible for the IDV (@2000MNRAS.315..229G). Over the time interval of our VSOP observations, no rapid variability in the jet was observed and we cannot confirm the variability outside the core and in the jet found by [@2000MNRAS.313..627G].
The simultaneous variation of the polarization angle with the polarized intensity in the core suggests that the variations might be the result of the sum of the polarization of more than one compact sub-component on scales smaller than the beam size. Assuming that these variations are intrinsic to the source, we derived brightness temperatures of $\sim 3\times 10^{15}$K to $\sim
10^{16}$K. Doppler factors of $>20$ are needed to bring these values down to the inverse-Compton limit. These numbers agree with the observed speeds in the jet if the angle to the line of sight is very small ($\theta<2^\circ$), as already proposed by B05. Because of the unknown redshift, the derived speeds and brightness temperatures represent only lower limits.
However, interstellar scintillation effects could also explain the IDV seen in the VLBI core, if the core region consists of several compact and polarized sub-components, with sizes of a few ten micro-arcseconds. To explain the observed polarization variations, the sub-components must scintillate independently in a different manner, which means that they must have slightly different intrinsic sizes and intrinsic polarizations (i.e. ).
Independent of whether the interpretation of the IDV seen in the VLBI core is source intrinsic or extrinsic, the space-VLBI limit to the core size gives a robust lower limit to the brightness temperature of $\geq 2\times 10^{12}$K and therewith exceeds the inverse-Compton limit. This implies a lower limit to the Doppler factor of about $\geq 4$ and, independent of the model we use to explain the variability, relativistic beaming seems to play a role. Unfortunately, the possible sub-components inside the core region cannot be observed directly. Future multi-frequency polarization VLBI observations, including simultaneous observations at higher frequencies, should help distinguishing which fraction of the IDV is due to source intrinsic variations and which is caused by the interstellar medium.
To explain the enhanced degree of polarization in the superluminal jet, we applied a simple shock model developed by [@1988ApJ...332..696C] to our observations and found reasonable values for the jet Lorentz factor and the viewing angle. This supports the idea that the knots of bright emission in the jet are shocks travelling down the jet (@1985ApJ...298..114M).
We thank the anonymous referee for helpful comments and suggestions. This work made use of the VLBA, which is an instrument of the National Radio Astronomy Observatory, a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. This work is also based on observations with the 100m radio telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg. We gratefully acknowledge the VSOP Project, which is led by the Japanese Institute of Space and Astronautical Science in cooperation with many organisations and radio telescopes around the world. U.B. was partly supported by the European Community’s Human Potential Programme under contract HPRCN-CT-2002-00321 (ENIGMA Network).
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[^1]: <http://www.aoc.nrao.edu/aips/aipsmemo.html>
[^2]: $H_0=71$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$ and $\Omega_{\rm \lambda}=0.7$
|
---
abstract: 'We prove, for a finite-dimensional gentle algebra, that indecomposable $\Sigma$-pure-injective objects in the homotopy category of complexes of projectives must be shifts of string or band complexes. The key step in our proof uses a splitting result for linear relations, together with an analysis of the canonical multi-sorted language.'
author:
- 'Raphael Bennett-Tennenhaus'
bibliography:
- 'biblio.bib'
title:
- '$\Sigma$-pure-injective complexes for gentle algebras'
- '$\Sigma$-pure-injective complexes for gentle algebras'
---
Introduction.
=============
A *gentle algebra* has the form $\Lambda=kQ/\mathcal{J}$ where $k$ is a field, $Q$ is a quiver, $kQ$ is the path algebra, and $\mathcal{J}$ is the admissible ideal in $kQ$ generated by a set $\rho$ of length $2$ paths such that:
1. if $v$ is a vertex then $\vert\mathbf{A}(v\rightarrow)\vert\leq2$ and $\vert\mathbf{A}(\rightarrow v)\vert\leq2$;
2. if $y\in\mathbf{A}$ then $\vert\{ x\in \mathbf{A}(h(y)\rightarrow)\mid xy\in\mathcal{J}\}\vert\leq 1$ and $\vert \{ z\in \mathbf{A}(t(y)\rightarrow)\mid yz\in\mathcal{J}\}\vert\leq 1$;
3. if $y\in\mathbf{A}$ then $\vert \{ x\in \mathbf{A}(h(y)\rightarrow)\mid xy\notin\mathcal{J}\}\vert\leq 1$ and $\vert\{ z\in \mathbf{A}(t(y)\rightarrow)\mid yz\notin\mathcal{J}\}\vert\leq 1$.
where $\vert X\vert$ denotes the cardinality of any set $X$, and for any vertex $u$ the set $\mathbf{A}(u\rightarrow)$ (respectively $\mathbf{A}(\rightarrow u)$) consists of all arrows $a$ whose tail $t(a)$ (respectively head $h(a)$) is $u$.
A fundamental idea in model theory is to study structures via the formulas they satisfy. By the famous quantifier elimination result of Baur [@Bau1976], any such formula in the language of modules (over a fixed ring) is a Boolean combination of positive-primitive (pp) formulas. A monomorphism of modules is called pure if it remains an embedding under any tensor product functor, and a module is called pure-injective if it is injective with respect to pure monomorphisms. A module is called $\Sigma$-pure-injective if any small direct sum of copies of it is pure-injective. The direct sum over a singleton shows $\Sigma$-pure-injectives are pure-injective.
The introduction of the Ziegler spectrum (a topological space whose points are indecomposable pure-injectives) motivated the provision of examples of these modules. Various examples were given by Ringel [@Rin1995], where he conjectured a classification of pure-injective indecomposables for domestic string algebras. This conjecture was verified by Prest and Puninski [@PrePun2016] by studying coherent functors from the category of pp-pairs, defined by evaluation. More recently, in joint work [@BenCra2018] with Crawley-Boevey, we adpted the focus of [@Cra2018] to classify $\Sigma$-pure-injective modules over (possibly non-domestic) string algebras, using a classification method from representation theory known as the functorial filtrations method. As was done in [@PrePun2016], the combinatorial properties defining string algebras in [@BenCra2018] were translated into properties of pp-formulas in order to classify modules up to isomorphism.
The Ziegler spectrum of a compactly generated triangulated category was defined by Krause [@Kra2002], and Garkusha and Prest [@GarPre2005] subsequently gave a relation between the Ziegler spectrum of a (right hereditary or von Neumann regular) ring and the Ziegler spectrum of its derived category. Furthermore these authors gave a correspondence between the pp-formulas (in the canonical multi-sorted language) and coherent functors into the category of abelian groups. In this article, the combinatorial properties defining gentle algebras will be translated into properties of pp-formulas in order to classify complexes of projectives up to homotopy equivalence. We use results from [@BenCra2018] and a summary [@Ben2016] of the authors PhD thesis.
In [@Ben2016] the author considered the homotopy category $\mathcal{K}(\Lambda\text{-}\bf{Proj})$ of unbounded complexes of projective modules. Here objects were classified in the full subcategory $\mathcal{K}(\Lambda\text{-}\bf{proj})$ of complexes with finitely generated homogeneous components. Our main result in this article (Theorem \[maincor\]) provides a similar classification, in which we broaden the class of complexes we consider to *$\Sigma$-pure-injectives*. It is worth noting that the class of rings considered in [@Ben2016] strictly contains the class of gentle algebras, and examples of these rings include infinite-dimensional algebras and rings of mixed characteristic. The aforementioned broadening comes with the restriction to the smaller class of (finite-dimensional) gentle algebras.
\[maincor\] Let $\Lambda$ be a gentle algebra. Every $\Sigma$-pure-injective in $\mathcal{K}(\Lambda\text{-}\bf{Proj})$ is a direct sum of string complexes and band complexes indexed by a $\Sigma$-pure-injective $k[T,T^{-1}]$-module.
Note that any indecomposable $\Sigma$-pure-injective $k[T,T^{-1}]$-module is isomorphic to: an indecomposable finite-dimensional module; a *Prüfer* module (an injective envelope of a simple); or the function field $k(T)$. String complexes have the form $P(C)$, and are indexed by an *aperiodic homotopy word* $C$. Band complexes are indexed by a *periodic homotopy word* together with an indecomposable $k[T,T^{-1}]$-module. These words are essentially combinatorial data: see Definitions \[definition2\] and \[def.3.3\] for details. Theorem \[theorem.1.2\] characterises when two shifts of string or band complexes are isomorphic.
The article is organised as follows. In §\[4\] we recall the canonical multi-sorted language for compactly generated triangulated categories: in §\[4-1\] we recall the notion of pure monomorphisms and pure-injective objects in these categories; and in §\[4-2\] we restrict our focus to the study of $\Sigma$-pure-injective objects. In §\[2\] we recall and study string and band complexes in the context of compactly generated homotopy categories: in §\[homo\] we recall homotopy words and define associated complexes of projective modules over gentle algebras; in §\[leftbdd\] we recall results from [@Ben2018] to classify left-bounded string complexes with bounded cohomology; and in §\[cmpctgentle\] we explain why the homotopy category we are considering is a compactly generated (triangulated) category. In §\[linrels\] we study the category of linear relations; in §\[KROW\] we recall results from [@BenCra2018]; and in §\[6-1\] we consider the notion of a homotopically minimal complex, explaining how homotopy words induce linear relations on such complexes. In §\[6\] we outline the setting and definitions requried to employ the functorial filtrations method; in §\[consref\] we construct various functors involved in the proof; in §\[sec:Linear-compactness-and\] a covering property is verified for these functors evaluated on $\Sigma$-pure-injective objects; and this property is exploited in §\[Local Mapping Properties.\] to check compatibility conditions between (string and band complexes) and (linear relations given by homotopy words). In §\[sec:Proofs-of-the\] we provide a proof of Theorem \[maincor\].
The canonical multi-sorted language for compactly generated triangulated categories. {#4}
====================================================================================
There are various characterisations for the purity of a module in terms of pp-formulas. Similarly, purity in compactly generated triangulated categories may be discussed in terms of formulas in a multi-sorted language.
Suppose $\mathcal{A}$ is an additive category. Denote the hom-sets $\mathcal{A}(X,Y)$. For any set $I$ and any collection $\mathrm{B}=\{B_{i}\mid i\in I\}$ of objects in $\mathcal{A}$: if the categorical product $\prod_{i}B_{i}$ exists in $\mathcal{A}$, we write $p_{j,\mathrm{B}}:\prod_{i}B_{i}\to B_{j}$ for the natural morphisms equipping it; in which case the universal property gives unique morphisms $v_{j,\mathrm{B}}:B_{j}\to \prod_{i}B_{i}$ such that $p_{j,\mathrm{B}}v_{j,\mathrm{B}}$ is the identity on $B_{j}$ (for each $j$). Similarly $u_{j,\mathrm{B}}:B_{j}\to\bigoplus_{i}B_{i}$ will denote the morphisms equipping the coproduct $\bigoplus_{i}B_{i}$ (if it exists); in which case there are unique morphisms $q_{j,\mathrm{B}}:\bigoplus_{i}B_{i}\to B_{j}$ such that $q_{j,\mathrm{B}}u_{j,\mathrm{B}}$ is the identity.
Now fix an object $A$ in $\mathcal{A}$ and consider the covariant functor $\mathcal{A}(A,-)$. Note that both the product and coproduct of the collection $\mathcal{A}(A,\mathrm{B})=\{\mathcal{A}(A,B_{i})\mid i\in I\}$ exist in the category of abelian groups. We identify $\bigoplus_{i\in I}\mathcal{A}(A,B_{i})$ with the subgroup of $\prod_{i\in I}\mathcal{A}(A,B_{i})$ consisting of tuples $(g_{i}\mid i\in I)$ such that $g_{i}=0$ for all but finitely many $i\in I$. This means the canonical morphism $\iota_{I,\mathcal{A}(A,\mathrm{B})}$ is the inclusion of sets. If $\prod_{i}B_{i}$ exists in $\mathcal{A}$ then map $\lambda_{A,B}:\mathcal{A}(A,\prod_{i} B_{i})\to \prod_{i} \mathcal{A}(A,B_{i})$ from the universal property is given by $f\mapsto(p_{i,\mathrm{B}}f\mid i\in I)$ for each $f\in\mathcal{A}(A,\prod_{i} B_{i})$. Similarly if $\bigoplus_{i}B_{i}$ exists in $\mathcal{A}$ then then map $\gamma _{A,\mathrm{B}}:\bigoplus _{i}\mathcal{A}(A, B_{i})\to \mathcal{A}(A,\bigoplus _{i}B_{i})$ from the universal property is given by $\gamma_{A,\mathrm{B}}(g_{i}\mid i\in I)=\sum_{i}u_{i,\mathrm{B}}g_{i}$. Since the functor $\mathcal{A}(A,-)$ commutes with arbitrary limits, each of the morphisms $\lambda_{A,\mathrm{B}}$ are isomorphisms.
\[ass22\]Throughout §\[4\] fix a triangulated category $\mathcal{T}$ with suspension functor $\Sigma$. We assume that $\mathcal{T}$ is skeletally small and that $\mathcal{T}$ has arbitrary coproducts.
An object $X$ is said to be *compact* if, for any set $I$ and collection $\mathrm{Y}=\{Y_{i}\mid i\in I\}$ of objects in $\mathcal{T}$, the morphism $\gamma_{X,\mathrm{Y}}$ is an isomorphism. The category $\mathcal{T}$ is said to be *compactly generated* if there exists a set $\mathtt{G}$ of compact objects in $\mathcal{T}$, such that there are no non-zero objects $Z$ in $\mathcal{T}$ satisfying $\mathcal{T}(G,Z)=0$ for all $n\in\mathbb{Z}$ and all $G\in\mathtt{G}$. The set $\mathtt{G}$ is said to be a *generating set* if $\Sigma G\in \mathtt{G}$ for all $G\in\mathtt{G}$.
\[ass22-10\]Throughout §\[4\] we assume (in addition to Assumption \[ass22\]) that $\mathcal{T}$ is compactly generated by a generating set $\mathtt{G}$. Note that, as a consequence of the Brown representability theorem (see for example [@Kra2000 §1.3, Lemma 1.5]), under the above assumptions $\mathcal{T}$ has arbitrary products.
\[sorts\] (See for example [@DGN2016 Definition 34]). For a non-empty set $\mathtt{S}$ an $\mathtt{S}$-*sorted predicate language* $\mathfrak{L}$ is a tuple $\langle\mathrm{pred}_{\mathtt{S}},\,\mathrm{func}_{\mathtt{S}},\,\mathrm{ar}_{\mathtt{S}},\,\mathrm{sort}_{\mathtt{S}}\rangle$ where: each $s\in\mathtt{S}$ is called a *sort*; $\mathrm{pred}_{\mathtt{S}}$ is a non-empty set of *sorted predicate symbols*; $\mathrm{func}_{\mathtt{S}}$ is a set of *sorted function symbols* (considered disjoint with $\mathrm{pred}_{\mathtt{S}}$); the *arity* function $\mathrm{ar}_{\mathtt{S}}$ maps a natural number to each sorted predicate symbol and to each sorted function symbol; and the function $\mathrm{sort}_{\mathtt{S}}$ maps any $n$-ary sorted predicate (respectively function) symbol to a sequence of $n$ (respectively $n+1$) sorts.
For each sort $s$ we introduce a countable set $\mathcal{V}_{s}$ of *variables of sort* $s$. The *terms* of $\mathfrak{L}$ each have their own sort, and are defined inductively by stipulating: any variable $x$ of sort $s$ will be considered a term of sort $s$; and for any $F\in\mathrm{func}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ and any terms $t_{1},\dots,t_{n}$ of sort $s_{1},\dots,s_{n}$ (respectively) we will consider $F(t_{1},\dots,t_{n})$ as a term of sort $s$. Note that constant symbols, which are given by nullary sorted function symbols, are (therefore) also terms. The *atomic formulas* with which $\mathfrak{L}$ is equipped are built from the equality $t=_{s}t'$ between terms $t,t'$ of common sort $s$ together with the formulas $R(t_{1},\dots,t_{n})$ where $R \in\mathrm{pred}_{\mathtt{S}}$, $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ and where each $t_{i}$ is a term of sort $s_{i}$. First-order formulas $\varphi$ in $\mathfrak{L}$ are built from: the variables of each sort; the atomic formulas; Boolean connectives $\wedge$, $\vee$, $\implies$, and $\neg$; and the symbols $\forall$ and $\exists$.
By a $\mathfrak{L}$-*structure* we mean a tuple $\mathsf{Z}=\langle Z, (R(\mathsf{Z})\mid R \in\mathrm{pred}_{\mathtt{S}}), (F(\mathsf{Z})\mid F \in\mathrm{func}_{\mathtt{S}})\rangle$ where: $Z$ is a family of sets $s(\mathsf{Z})$ for each $s\in\mathtt{S}$; $R(\mathsf{Z})$ is a subset of $s_{1}(\mathsf{Z})\times\dots\times s_{n}(\mathsf{Z})$ for any $R\in\mathrm{pred}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$; and $F(\mathsf{Z})$ is a map $s_{1}(\mathsf{Z})\times\dots\times s_{n}(\mathsf{Z})\to s(\mathsf{Z})$ for any $F\in\mathrm{func}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$. If $\mathsf{Z}$ and $\mathsf{Y}$ are $\mathfrak{L}$-structures then by a $\mathfrak{L}$-*homomorphism* we mean a function $g_{s}:s(\mathsf{Z})\to s(\mathsf{Y})$ for each sort $s$ such that: for each $F\in \mathrm{func}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ we have $g_{s}(F(\mathsf{Z})(a_{1},\dots,a_{n}))=F(\mathsf{Y})(g_{s_{1}}(a_{1}),\dots,g_{s_{n}}(a_{n}))$ for all $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{Z})\times \dots \times s_{n}(\mathsf{Z})$; and for each $R\in \mathrm{pred}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ and each formula $\varphi(x_{1},\dots,x_{n})$ where ($x_{i}$ is a variable of sort $s_{i}$ and each $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{Z})\times \dots \times s_{n}(\mathsf{Z})$), if $\mathsf{Z}\models\varphi(a_{1},\dots,a_{n})$ then $\mathsf{Y}\models\varphi(g_{s_{1}}(a_{1}),\dots,g_{s_{n}}(a_{n}))$.
If $\mathsf{Z}$ and $\mathsf{Y}$ are $\mathfrak{L}$-structures then we say that $\mathsf{Z}$ is a *substructure* of $\mathsf{Y}$ provided: $s(\mathsf{Z})\subseteq s(\mathsf{Y})$ for each sort $s$; for each $F\in \mathrm{func}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(F)=(s_{1},\dots,s_{n},s)$ we have $F(\mathsf{Z})(a_{1},\dots,a_{n})=F(\mathsf{Y})(a_{1},\dots,a_{n})$ for all $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{Z})\times \dots \times s_{n}(\mathsf{Z})$; and for each $R\in \mathrm{pred}_{\mathtt{S}}$ with $\mathrm{sort}_{\mathtt{S}}(R)=(s_{1},\dots,s_{n})$ we have $R(\mathsf{Z})=R(\mathsf{Y})\cap s_{1}(\mathsf{Z})\times \dots \times s_{n}(\mathsf{Z})$. If $\mathsf{Z}$ is a substructure of $\mathsf{Y}$, then we say that $\mathsf{Z}$ is an *elementary substructure* of $\mathsf{Y}$ if, for each formula $\varphi(x_{1},\dots,x_{n})$ where $x_{i}$ is a variable of sort $s_{i}$ and each $(a_{1},\dots,a_{n})\in s_{1}(\mathsf{Z})\times \dots \times s_{n}(\mathsf{Z})$ we have ($\mathsf{Z}\models\varphi(a_{1},\dots,a_{n})$ if and only if $\mathsf{Y}\models\varphi(a_{1},\dots,a_{n})$).
For a set $\mathtt{S}$, an $\mathtt{S}$-sorted predicate language $\mathfrak{L}$ and a $\mathfrak{L}$-structure $\mathsf{Z}$ we write: $\vert\mathfrak{L}\vert$ for the largest of the cardinalities $\vert \mathbb{N}\vert$ and $\vert \mathrm{pred}_{\mathtt{S}}\sqcup\mathrm{func}_{\mathtt{S}}\vert$; and $\vert\mathsf{Z}\vert$ for the sum of the cardinalities $\vert s(\mathsf{Z})\vert$ as $s$ runs through the sorts.
We now recall the downward Löwenheim-Skolem theorem for many-sorted structures.
\[lowenskolem\]*(*See for example *[@DGN2016 Theorem 37])*. Let $\mathtt{S}$ be a set and let $\mathfrak{L}$ be an $\mathtt{S}$-sorted predicate language. Let $\mathsf{Y}$ be a $\mathfrak{L}$-structure. Fix a subset $s(R)\subseteq s(\mathsf{Y})$ for each $s\in\mathtt{S}$. Suppose that there is a cardinal $\kappa$ such that $\mathrm{max}\{\vert\mathbb{N}\vert,\vert\mathfrak{L}\vert,\vert s(R)\vert\}\leq\kappa \leq \vert\mathsf{Y}\vert$ for each $s\in\mathtt{S}$. Then there is an elementary substructure $\mathsf{Z}$ of $\mathsf{Y}$ such that $\vert\mathsf{Z}\vert=\kappa$ and for each $s\in\mathtt{S}$ we have $s(R)\subseteq s(\mathsf{Z})$.
As in Assumptions \[ass22\] and \[ass22-10\] let $\mathcal{T}$ be a triangulated category which: has arbitrary coproducts; is skeletetally small; is compactly generated by a generating set $\mathtt{G}$; and (hence) has arbitrary products. Fix a non-empty full subcategory $\mathcal{S}$ of $\mathcal{T}$.
Recall $\mathcal{S}$ is a *triangulated subcategory* if: for any object $X$ of $\mathcal{S}$ and any $n\in\mathbb{Z}$ the object $\Sigma ^{n}X$ lies in $\mathcal{S}$; and for any distinguished triangle $X\to Y\to Z\to \Sigma X$, if two of the objects $X$, $Y$, or $Z$ lies in $\mathcal{S}$, then so does the third. Note that the subcategory $\mathcal{S}$ of $\mathcal{T}$ is a triangulated subcategory if and only if: any object in $\mathcal{T}$ which is isomorphic to an object in $\mathcal{S}$ is an object in $\mathcal{S}$, and; $\mathcal{S}$ together with the restriction of $\Sigma$ defines a triangulated category, where any distinguished triangle in $\mathcal{S}$ is a distinguished triangle in $\mathcal{T}$).
We write $\mathcal{T}^{c}$ for the triangulated subcategory of $\mathcal{T}$ consisting on compact objects. We write $\mathbf{Mod}\text{-}\mathcal{T}^{c}$ for the category of additive contravariant functors $\mathcal{T}^{c}\to \mathbf{Ab}$ where $\mathbf{Ab}$ is the category of abelian groups.
\[canonlang\][@GarPre2005 §3] The *canonical language* $\mathfrak{L}^{\mathcal{T}}$ of $\mathcal{T}$ is given by a $\mathtt{G}$-sorted predicate language $\langle\mathrm{pred}_{\mathtt{G}},\,\mathrm{func}_{\mathtt{G}},\,\mathrm{ar}_{\mathtt{G}},\,\mathrm{sort}_{\mathtt{G}}\rangle$ defined as follows. The set $\mathrm{pred}_{\mathtt{G}}$ consists of a symbol $\mathsf{0}_{G}$ with $\mathrm{sort}_{\mathtt{G}}(\mathsf{0}_{G})=G$ for each $G\in\mathtt{G}$. The set $\mathrm{func}_{\mathtt{G}}$ consists of: a binary operation $+_{G}$ with $\mathrm{sort}_{\mathtt{S}}(+_{G})=(G,G,G)$ for each $G\in\mathtt{G}$; and a unary operation $-\circ \alpha$ with $\mathrm{sort}_{\mathtt{S}}(-\circ \alpha)=(H,G)$ for each $G,H\in\mathtt{G}$ and each map $\alpha\in\mathcal{T}(G,H)$. The variables of sort $G\in\mathtt{G}$ will be denoted $v_{G}$. Let $\mathrm{Ax}(\mathcal{T})$ be the set of axioms expressing the positive atomic diagram of $\mathcal{T}^{c}$, including the specification that all functions are additive. Consequently the category $\mathrm{Mod}(\mathrm{Ax}(\mathcal{T}))$ of models for $\mathrm{Ax}(\mathcal{T})$ is just the category $\mathbf{Mod}\text{-}\mathcal{T}^{c}$. The objects of $\mathcal{T}$ are regarded as structures $\mathsf{M}$ for this language via the functor which takes such an object $M$ to the functor $\mathcal{T}(-, M)$.
Purity in compactly generated triangulated categories. {#4-1}
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In what follows we discuss the notion of purity in the context of triangulated categories. Terminology about triangulated categories may be found in the book of Neeman [@Nee2001].
[@Kra2002 Definition 1.1] Let $\mathcal{T}$ be compactly generated. A morphism $L\to M$ is *pure monomorphism* if the induced map $\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ is a monomorphism for each compact object $X$, and an object $M$ of $\mathcal{T}$ is *pure*-*injective* provided every pure monomorphism $M\to N$ is a section. We say that an object $M$ of $\mathcal{T}$ is $\Sigma$-*pure*-*injective* if, for any set $I$, the coproduct $\bigoplus_{i}M$ is pure-injective.
Lemma \[elempure\] is analogous to the equivalence of (i) and (ii) in [@JenLen1989 Theorem 6.4].
\[elempure\]Let $L,M$ be objects in $\mathcal{T}$. Then the natural transformations $\mathcal{T}(-,L)\to\mathcal{T}(-,M)$ are precisely the $\mathfrak{L}^{\mathcal{T}}$-homomorphisms $\mathsf{L}\to\mathsf{M}$. Furthermore, there is a pure monomorphism $L\to M$ if and only if $\mathsf{L}$ is a pure substructure of $\mathsf{M}$.
Recall that $\mathsf{L}=\mathcal{T}(-,L)$ and $\mathsf{M}=\mathcal{T}(-,M)$ define $\mathfrak{L}^{\mathcal{T}}$-structures by setting $G(\mathsf{L})=\mathcal{T}(G,L)$ and $G(\mathsf{M})=\mathcal{T}(G,M)$ for each sort $G\in\mathtt{G}$. Since $\mathsf{L}$ and $\mathsf{M}$ are the contravariant hom-functors, any morphism $\gamma:L\to M$ defines a natural transformation $\mathcal{T}(-,L)\to\mathcal{T}(-,M)$ by postcomposition with $\gamma$. Note that any such natural transformation is, by Definition \[canonlang\], the same thing as a $\mathfrak{L}^{\mathcal{T}}$-homomorphism $\mathsf{L}\to\mathsf{M}$, since formulas and predicate and function symbols will be preserved by construction.
By [@GarPre2005 Proposition 3.1] any pp-formula $\varphi(v_{G})$ is equivalent to a divisibility formula $\exists u_{H} :v_{G}=u_{H}\alpha$ where $\alpha:G\to H$ is morphism and $G,H\in\mathtt{G}$. By definition $\mathsf{L}$ is a pure substructure of $\mathsf{M}$ if and only if (for any $(f,g)\in G(\mathsf{L})\times H(\mathsf{L})$ such that $\gamma g=\gamma f\alpha$ we must have $g=f\alpha$). This is equivalent to the condition that the morphism $\mathcal{T}(X,L)\to\mathcal{T}(X,M)$ given by $g\mapsto\gamma g$ is a monomorphism for each compact object $X$, which by definition is the same as saying $\gamma$ is a pure monomorphism. The result follows.
Let $I$ be a set and let $M$ be an object of $\mathcal{T}$. By the universal properties of the product and coproduct of the collection $\mathrm{M}=\{M\mid i\in I\}$ there exists a unique *summation morphism* $\sigma_{I,\mathrm{M}}:\bigoplus_{i} M \to M$ and a unique *canonical morphism* $\iota_{I,\mathrm{M}}:\bigoplus_{i} M\to \prod_{i} M$ satisfying $\sigma_{I,\mathrm{M}}u_{i,\mathrm{M}}=1_{M}$ and $\iota_{I,\mathrm{M}}u_{i,\mathrm{M}}=v_{i,\mathrm{M}}$ for each $i$.
There are various ways to characterise both pure-injective and $\Sigma$-pure-injective objects in a module category, see for example [@JenLen1989 Theorem 7.1] and [@JenLen1989 Theorem 8.1] respectively. It what follows we discuss analogous statements for compactly generated triangulated categories.
\[krachar\] *[@Kra2002 Theorem 1.8, (1,5)]* An object $M$ of $\mathcal{T}$ is pure-injective if and only if for each set $I$ the summation morphism $\sigma_{I,\mathrm{M}}$ factors through the canonical morphism $\iota_{I,\mathrm{M}}$.
Theorem \[krachar\] is analogous to the equivalence of (ii) and (vi) in [@JenLen1989 Theorem 7.1]. Similarly Proposition \[sigmapimapsdef\] is analogous to the equivalence of (i) and (ii) in [@JenLen1989 Theorem 8.1].
\[sigmapimapsdef\] An object $M$ of $\mathcal{T}$ is $\Sigma$-pure-injective if and only if for each set $I$ the canonical morphism $\iota_{I,\mathrm{M}}$ is a section.
Suppose that $\iota_{I,\mathrm{M}}$ is a section for each set $I$. We now show $M$ is $\Sigma$-pure-injective. Choose a set $T$. Let $N$ be the coproduct $\bigoplus_{t}M$ of the collection $\mathrm{M}=\{M\mid t\in T\}$. Let $S$ be any set and consider the collection $\mathrm{N}=\{N\mid s\in S\}$. By Theorem \[krachar\] it suffices to show find a map $\theta_{S,\mathrm{N}}:\prod_{s}N\to N$ such that $\sigma_{S,\mathrm{N}}=\theta_{S,\mathrm{N}}\iota_{S,\mathrm{N}}$. For each $(s,t)\in S\times T$ the morphisms $u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ satisfy the universal property of the coproduct $\bigoplus_{s,t}M$, and so we assume $u_{s,t,\mathrm{M}}=u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ without loss of generality. Consider the morphisms $\varphi_{s,t,\mathrm{M}}=q_{t,\mathrm{M}}p_{s,\mathrm{N}}$ for each $(s,t)\in S\times T$. Since $u_{s,t,\mathrm{M}}=u_{s,\mathrm{N}}u_{t,\mathrm{M}}$ we have $q_{s,t,\mathrm{M}}=q_{t,\mathrm{M}}q_{s,\mathrm{N}}$ by uniqueness. Consequently $\varphi_{s,t,\mathrm{M}}v_{s,\mathrm{N}}q_{s,\mathrm{N}}u_{s,t,\mathrm{M}}$ is the identity on $M$. By the universal property of the product, there is a morphism $\omega:\prod_{s}N\to \prod_{s,t}M$ such that $p_{s,t,\mathrm{M}}\omega=\varphi_{s,t,\mathrm{M}}$ for each $(s,t)$. It suffices to let $\theta_{S,\mathrm{N}}=\sigma_{S,\mathrm{N}}\pi_{S\times T,\mathrm{M}}\omega$. The proof that $\sigma_{S,\mathrm{N}}=\theta_{S,\mathrm{N}}\iota_{S,\mathrm{N}}$ is a straightforward application of the uniqueness of the involved morphisms. For the converse, suppose $M$ is $\Sigma$-pure-injective, and let $I$ be a set. Let $X$ be a compact object in $\mathcal{T}$. In general: the morphism $\lambda_{X,\mathrm{M}}$ is an isomorphism; the canonical morphism $\iota_{I,\mathcal{T}(X,\mathrm{M})}$ is injective; and $\lambda_{X,\mathrm{M}}\mathcal{T}(X,\iota_{I,\mathrm{M}})\gamma_{X,\mathrm{M}}=\iota_{I,\mathcal{T}(X,\mathrm{M})}$. Since $X$ is compact the morphism $\gamma _{X,\mathrm{M}}$ is an isomorphism, and so the morphism $\mathcal{T}(X,\iota_{I,\mathrm{M}})$ is injective. This means that the morphism $\iota_{I,\mathrm{M}}$ is a pure monomorphism. Since the domain of $\iota_{I,\mathrm{M}}$ is pure-injective by assumption, the morphism $\iota_{I,\mathrm{M}}$ is a section.
Recall the canonical language $\mathfrak{L}^{\mathcal{T}}$ from Definition \[canonlang\]. Any formula in $\mathfrak{L}^{\mathcal{T}}$ lying in the closure of the set of equations under conjunction and existential quantification is called a *pp*-*formula*, and a *pp*-*definable subgroup* is the solution set of a pp-formula. For any morphism $\alpha:G\to H$ and any object $M$ in $\mathcal{T}$ such that $G$ and $H$ are compact, let $\mathcal{T}(\alpha,M):\mathcal{T}(H, M)\to \mathcal{T}(G, M)$ be the induced map $u\mapsto u\alpha $. Following [@GarPre2005 §2] we write $M\alpha$ for the image of $\mathcal{T}(\alpha,M)$, which is a pp-definable subgroup of $M$ of sort $G$.
Parts (i) and (ii) of Lemma \[ppprops\] are analogous to [@JenLen1989 Proposition 6.6] and [@JenLen1989 Proposition 6.7(i,ii)] respectively.
\[ppprops\]Let $\varphi(v_{G})$ be a pp-formula (in one free variable of sort $G\in\mathtt{G}$) in $\mathfrak{L}^{\mathcal{T}}$.
1. Let $L,M$ be objects in $\mathcal{T}$ with $\mathfrak{L}^{\mathcal{T}}$-structures $\mathsf{L}=\mathcal{T}(-,L)$ and $\mathsf{M}=\mathcal{T}(-,M)$. If there is a pure monomorphism $L\to M$ then we have $\varphi(\mathsf{L})=\mathcal{T}(G,L)\cap \varphi(\mathsf{M})$.
2. For any set $I$ and any collection $\mathrm{M}=\{M_{i}\mid i\in I\}$ of objects in $\mathcal{T}$ with $\mathfrak{L}^{\mathcal{T}}$-structures $\mathsf{M}_{i}=\mathcal{T}(-,M_{i})$ we have $\bigoplus_{i}\varphi(\mathsf{M}_{i})\simeq \varphi(\bigoplus_{i}\mathsf{M}_{i})$ and $\varphi(\prod_{i}\mathsf{M}_{i})\simeq \prod_{i}\varphi(\mathsf{M}_{i})$.
Every subgroup $M\alpha$ of an object $M$ in $\mathcal{T}$ is the set $\varphi(M)$ of solutions $v$ to the pp-formula $\varphi(v)=\exists u:v=u\alpha $. We abuse notation by writing $\varphi(M)$ for $\varphi(\mathsf{M})$.
\(i) Any pp-formula is equivalent to a divisibility formula by [@GarPre2005 Proposition 3.1]. Hence $\varphi(M)=M\alpha$ and $\varphi(L)=L\alpha$ for some morphism $\alpha:G\to H$.
\(ii) If $t:M\to N$ and $\alpha:G\to H$ are morphisms in $\mathcal{T}$ such that $G$ and $H$ are compact, then $tv\in N\alpha$ for any $v\in M\alpha$. Hence, for any pp-formula $\varphi$ in $\mathfrak{L}^{\mathcal{T}}$, the assignment of objects $M\mapsto \varphi(M)$ from $\mathcal{T}$ to the category of abelian groups defines a functor $\varphi$.
By the existence of products and coproducts in $\mathcal{T}$ and the functorality of $\varphi$, the universal properties give morphisms $\delta:\bigoplus_{i}\varphi(M_{i})\to \varphi(\bigoplus_{i}M_{i})$ and $\mu:\varphi(\prod_{i}M_{i})\to \prod_{i}\varphi(M_{i})$. By [@GarPre2005 Lemma 4.3] the functor $\varphi$ is *coherent*, so by the equivalence of statements (1) and (3) from [@Kra2002 Theorem A] the morphisms $\delta$ and $\mu$ are isomorphisms. Note that $\varphi(M)$ is a subgroup of $\mathcal{T}(G,M)$, and $\delta$ and $\mu$ are the restrictions of $\gamma_{G,\mathrm{M}}$ and $\lambda_{G,\mathrm{M}}$ where $\mathrm{M}=\{M\mid t\in T\}$.
By [@GarPre2005 Proposition 3.2] one has quantifier elimination in $\mathfrak{L}^{\mathcal{T}}$. This is because triangulated categories have *weak kernels* and *weak cokernels*. Later we use that compactly generated categories have *weak limits* and *weak colimits* (see Remark \[homotopycolimit\]).
$\Sigma$-purity in compactly generated triangulated categories. {#4-2}
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Lemma \[sigmathendcc\], together with its subsequent proof below, is analogous to the equivalence of (ii) and (iii) in [@JenLen1989 Theorem 8.1].
\[sigmathendcc\]Let $M$ be an object $M$ in $\mathcal{T}$. Then $M$ is $\Sigma$-pure-injective if and only if, for any compact object $G$ in $\mathcal{T}$, every descending chain of (pp-definable subgroups of $M$ of sort $G$) must stabilise.
Fix a descending chain $$M\alpha_{0}\supseteq M\alpha_{1}\supseteq M\alpha_{2}\supseteq \cdots$$ of (pp-definable subgroups of $M$ of sort $G$). Hence there is a collection of compact objects $H_{n}$ in $\mathcal{T}$ such that $\alpha_{n}\in\mathcal{T}(G,H_{n})$ for each $n\in\mathbb{N}$. For a contradiction we assume this chain does not stabilise. After relabelling we can assume that $M\alpha_{n}\neq M\alpha_{n+1}$ for each $n\in\mathbb{N}$, and so we choose $f_{n}\in\mathcal{T}(H_{n},M)$ such that $f_{n}\alpha_{n}\notin M\alpha_{n+1}$. Consider the collection $\mathrm{M}=\{M\mid n\in \mathbb{N}\}$.
Since $M$ is $\Sigma$-pure-injective, by Proposition \[sigmapimapsdef\] the canonical morphism $\iota_{\mathbb{N},\mathrm{M}}:\bigoplus_{\mathbb{N}}M\to\prod_{\mathbb{N}}M$ is a section, and so there is some morphism $\pi_{\mathbb{N},\mathrm{M}}:\prod_{\mathbb{N}}M\to\bigoplus_{\mathbb{N}}M$ such that $\pi_{\mathbb{N},\mathrm{M}}\iota_{\mathbb{N},\mathrm{M}}$ is the identity on $\bigoplus_{\mathbb{N}}M$. Let $f\alpha=(f_{n}\alpha_{n}\mid n\in \mathbb{N})$, considered as an element of $\prod_{n\in\mathbb{N}}\mathcal{T}(G,M)$. Fix $n\in\mathbb{N}$ and let $\varphi_{n}(v_{G})$ be the formula $\exists u_{H_{n}} : v_{G}=u_{H_{n}}\alpha_{n}$. Let $\mathrm{M}=\{M\mid n\in\mathbb{N}\}$. Recall that the morphism $\lambda_{G,\mathrm{M}}$ is always an isomorphism, and since $G$ is compact, $\gamma_{G,\mathrm{M}}$ is also an isomorphism. Let $$\omega=(\gamma_{G,\mathrm{M}})^{-1}\mathcal{T}(G,\pi_{\mathbb{N},\mathrm{M}})(\lambda_{G,\mathrm{M}})^{-1}$$ and $\omega(f\alpha)=(w_{n}\mid n\in\mathbb{N})$. The contradiction we will find is that $w_{l}\neq 0$ for all $l\in\mathbb{N}$ (which contradicts that $\omega$ has codomain $\bigoplus_{n\in\mathbb{N}}\mathcal{T}(G,M)$). Fix $l\in\mathbb{N}$. Let $x_{n}=f_{n}\alpha_{n}$ and $y_{n}=0$ for all $n\in\mathbb{N}$ with $n\leq l$, and otherwise let $x_{n}=0$ and $y_{n}=f_{n}\alpha_{n}$. This gives $f\alpha=f\alpha_{\leq l} + f\alpha_{>l}$ where $f\alpha_{\leq l}=(x_{n}\mid n\in\mathbb{N})$ and $f\alpha_{>l}=(y_{n}\mid n\in\mathbb{N})$ and so $$f\alpha_{\leq l}=(f_{0}\alpha_{0},\dots,f_{l}\alpha_{l},0,0,\dots),\, \,f\alpha_{>l}=(0,\dots,0,f_{l+1}\alpha_{l+1},f_{l+2}\alpha_{l+2},\dots ).$$ Note that $f\alpha_{\leq l}\in \bigoplus_{n\in\mathbb{N}}\mathcal{T}(G,M)$. Furthermore, since the chain $M\alpha_{0}\supseteq M\alpha_{1}\supseteq M\alpha_{2}\supseteq \cdots$ is descending, we have $f_{n}\alpha_{n}\in\varphi_{l}(M)$ for all $n>l$ and so $f\alpha_{>l}\in\prod_{n\in\mathbb{N}}\varphi_{l+1}(M)$. As in the proof of Lemma \[ppprops\](ii), the restrictions of $(\lambda_{G,\mathrm{M}})^{-1}$ and $(\gamma_{G,\mathrm{M}})^{-1}$ respectively define isomorphisms $\prod_{n\in\mathbb{N}}\varphi_{l+1}(M)\to \varphi_{l+1}(\prod_{n\in\mathbb{N}}M)$ and $\varphi_{l+1}(\bigoplus_{n\in\mathbb{N}}M)\to \bigoplus_{n\in\mathbb{N}}\varphi_{l+1}(M)$. Similarly $\mathcal{T}(G,\pi_{\mathbb{N},\mathrm{M}})$ restricts to define a morphism $\varphi_{l+1}(\prod_{n\in\mathbb{N}}M)\to\varphi_{l+1}(\bigoplus_{n\in\mathbb{N}}M)$. Altogether we have that $\omega$ restricts to a morphism $\prod_{n\in\mathbb{N}}\varphi_{l+1}(M)\to\bigoplus_{n\in\mathbb{N}}\varphi_{l+1}(M)$. Let $\omega(f\alpha_{>l})=(z_{n}\mid n\in\mathbb{N})$, and so $z_{n}\in\varphi_{l+1}(M)$ for all $n$.
Recall it suffices to show $w_{l}\neq0$ where $\omega(f\alpha)=(w_{n}\mid n\in\mathbb{N})$. From the above we have $$\begin{array}{c}(w_{0},\dots,w_{l},w_{l+1},\dots)=\omega (f\alpha)=\omega(f\alpha_{\leq l} + f\alpha_{>l})=f\alpha_{\leq l} + \omega(f\alpha_{>l})\\=(f_{0}\alpha_{0}+z_{0},\dots,f_{l}\alpha_{l}+z_{l},f_{l+1}\alpha_{l+1}+z_{l+1},\dots),
\end{array}$$ and so $w_{l}\neq0$ as otherwise $\varphi_{l+1}(M)\ni -z_{l}=f_{l}\alpha_{l}\notin \varphi_{l+1}(M)$.
We have shown that if $M$ is $\Sigma$-pure-injective then any descending chain of (pp-definable subgroups of $M$ of sort $G$) stabilises. This was done by given an analogous to that given in the proof that (ii) implies (iii) in [@JenLen1989 Theorem 8.1]. By providing a similar analogous proof that (iii’) implies (ii’) in [@JenLen1989 Theorem 8.1], one can show that if all of the aforementioned descending chains stabilise (for each compact object $G$), then $M$ is $\Sigma$-pure-injective.
Corollary \[sigmapropsyanah\], together with its subsequent proof below, is analogous to parts (i) and (ii) of [@JenLen1989 Corollary 8.2].
\[sigmapropsyanah\] Let $M$ be a $\Sigma$-pure-injective object in $\mathcal{T}$.
1. For any set $I$ the objects $M^{(I)}$ and $M^{I}$ are $\Sigma$-pure-injective.
2. If $\lambda:L\to M$ in $\mathcal{T}$ is a pure monomorphism then $L$ is $\Sigma$-pure-injective and $\lambda$ is a section.
\(i) By Lemma \[ppprops\](ii) we have that $\varphi(M)^{(I)}\simeq \varphi(M^{(I)})$ and $\varphi(M^{I})\simeq \varphi(M)^{I}$ for any pp-formula $\varphi$. Since pp-definable subgroups are pp-definable subgroups (of a particular sort), any descending chain of such subgroups of $M^{(I)}$ (respectively $M^{I}$) gives rise to a descending chain of such subgroups in $M$. The result follows by Lemma \[sigmathendcc\].
\(ii) By Lemma \[ppprops\](i) we have that $\varphi(L)=\mathcal{T}(G,L)\cap \varphi(M)$ for any pp-formula $\varphi$ of sort $G$. By Lemma \[sigmathendcc\] this means $L$ must be $\Sigma$-pure-injective. Since this means $L$ is pure-injective, and so $\lambda$ is a section.
\[homotopycolimit\]We recall some ideas used by Beligiannis [@Bel2002] and Garkusha and Prest [@GarPre2005]. Let $\mathtt{C}$ be a set considered as a small category with arrows denoted by $ \tau_{a,b}:a\to b$ for $a,b\in\mathtt{C}$, and let $\mathcal{A}$ be an additive category. Let $H:\mathtt{C}\to \mathcal{A}$ be a functor. A *weak colimit* of $H$ is an object $N$ together with morphisms $n_{c}:H(c)\to N$ (all in $\mathcal{A}$) for each $c\in\mathtt{C}$ such that: $n_{a}=n_{b}H(\tau_{a,b})$ for any arrow $\tau_{a,b}$ in $\mathtt{C}$; and if ($\{m_{c}:H(c)\to M\mid c\in\mathtt{G}\}$ is a set of arrows in $\mathcal{A}$ such that $m_{a}=m_{b}H(\tau_{a,b})$ for any arrow $\tau_{a,b}$) then there exists a morphism $\omega:N\to M$ (which in general is not unique) such that $\omega n(c)=m(c)$ for all $c\in\mathtt{C}$.
For example, the *weak cokernel* of a morphism $f:A\to B$ in $\mathcal{A}$ is a morphism $h:B\to C$ such that $hf=0$, and for any morphism $g:B\to D$ such that $gf=0$ we have $g=ah$ for some morphism $a:C\to D$. We let $\mathrm{w.colim}(H)$ denote the weak colimit of $H$. Dually one can define the notions of a *weak kernel* of such a map $f$, and more generally the notion of a *weak limit* of such a functor $H$.
Let $\mathcal{A}$ be the compactly generated triangulated category $\mathcal{T}$. We show that any functor $H: \mathtt{C}\to\mathcal{T}$ has a weak colimit in $\mathcal{T}$. The dual argument will show that any such $H$ has a weak limit in $\mathcal{T}$. By [@Bel2002 §2.2] it suffices to show that $\mathcal{T}$ has all coproducts and all weak cokernels. We are assuming that $\mathcal{T}$ has all products, and that $\mathcal{T}$ is compactly generated. Recall that by the Brown representability theorem $\mathcal{T}$ has arbitrary coproducts.
So it suffices to show $\mathcal{T}$ has weak cokernels. Let $f:A\to B$ be an morphism in $\mathcal{T}$. Consider the morphism $h:B\to C$ given by completing $f$ to a triangle $A\to B\to C\to A[1]$. Applying the covaraint functor $\mathcal{T}(A,-):\mathcal{T}\to\mathbf{Ab}$ to this triangle yields a sequence of abelian groups. Since this sequence is exact, $h$ is the required weak cokernel of $f$.
The proof of Theorem \[sigmathenCARD\] is analogous to the respective proof that (i) implies (v) in [@JenLen1989 Theorem 8.1]. We proceed in a similar spirit to the proof of Lemma \[sigmathendcc\].
\[sigmathenCARD\] Let $M$ be a $\Sigma$-pure-injective object of $\mathcal{T}$. Then for any set $I$ the product $M^{I}$ is a direct sum of indecomposable $\Sigma$-pure-injective objects of $\mathcal{T}$.
Let $K=M^{I}$. Without loss of generality it suffices to assume $I$ is infinite. By Assumption \[ass22-10\] there are no non-zero objects $Z$ in $\mathcal{T}$ satisfying $\mathcal{T}(G,Z)=0$ for all $n\in\mathbb{Z}$ and all $G\in\mathtt{G}$. We can assume $K\neq0$, and so there is some $G\in\mathtt{G}$ for which $\mathcal{T}(G,K)\neq0$. So we choose some non-zero $f\in\mathcal{T}(G,K)$. Let $\mathsf{K}$ be the $\mathfrak{L}^{\mathcal{T}}$-structure $\mathcal{T}(-,K)$ where $\mathfrak{L}^{\mathcal{T}}$ is the canonical language for $\mathcal{T}$. Consider the formula $\psi=\neg\varphi$ where $\varphi=\exists u_{G}:u_{G}=f$. Note that the collection of $\mathfrak{L}^{\mathcal{T}}$-substructures $\mathsf{L}$ of $\mathsf{K}$ lies in the power set of the disjoint union of $\mathcal{T}(G,M)$ over all $G\in\mathtt{G}$. By Definition \[canonlang\] there is an object $L$ of $\mathcal{T}$ such that $\mathsf{L}$ is the functor $\mathcal{T}(-, L)$. Consider the set $\Psi$ consisting of all pure substructures $\mathsf{L}$ of $\mathsf{K}$ such that $\mathsf{L}\models \psi$.
Note $\mathsf{0}_{G}=\mathcal{T}(G,0)=0$ which lies in $\Psi$. Let $\{\mathsf{N}_{c}\mid c\in\mathtt{C}\}$ be a totally ordered subset in $\Psi$ such that $\mathsf{N}_{c}$ is a substructure of $\mathsf{N}_{c'}$ whenever $c<c'$ in $\mathtt{C}$. By construction there is a pure monomorphism $\gamma_{c}:N_{c}\to K$ for each $c\in\mathcal{C}$ and a morphism $t_{a,b}:N_{a}\to N_{b}$ whenever $a<b$ (in $\mathtt{C}$) such that $\gamma_{a}=\gamma_{b}t_{a,b}$. By Remark \[homotopycolimit\] there is a weak colimit $P$ of this directed system which comes equipped with morphisms $n_{c}:N_{c}\to P$ such that $n_{a}=n_{b}t_{a,b}$ whenever $a<b$. By the defining property of weak colimits there is a morphism $\omega:P \to K$ such that $\omega n_{c}=\gamma_{c}$ for each $c$. Applying the functor $\mathcal{T}(G,-)$ to these equations shows that $n_{c}$ is a pure monomorphism because $\gamma_{c}$ is a pure monomorphism. By Corollary \[sigmapropsyanah\](i) we have that $K$ is $\Sigma$-pure-injective. By Corollary \[sigmapropsyanah\](ii) this means that $n_{c}$ and $\gamma_{c}$ are sections. This shows that $\omega$ must be a pure monomorphism. So the $\mathfrak{L}^{\mathcal{T}}$-structure $\mathsf{P}=\mathcal{T}(-,P)$ is an upper bound of $\{\mathsf{N}_{c}\mid c\in\mathtt{C}\}$ in $\Psi$. Thus $\Psi$ is a non-empty partially ordered set with respect to inclusion, and every chain in $\Psi$ has an upper bound in $\Psi$. By the axiom of choice $\Psi$ contains a maximal element $\mathsf{N}$.
Since $\mathsf{N}$ is a pure substructure of $\mathsf{K}$, by Lemma \[elempure\] there is a pure monomorphism $\gamma:N\to K$. By Corollary \[sigmapropsyanah\](ii) $\gamma$ splits, and so $K\simeq N\oplus U$ for some object $U$ of $\mathcal{T}$. Note that $U\neq0$, since otherwise $K\simeq N$ which would give $\mathsf{K}\simeq\mathsf{N}$ contradicting that $\mathsf{K}\models\varphi$ and $\mathsf{N}\models\psi$. Furthermore any split monic $U\to K$ must be a pure monomorphism, and so $\mathsf{U}$ defines a pure substructure of $\mathsf{K}$. Suppose that $U\simeq V\oplus W$ in $\mathcal{T}$ and so $\mathcal{T}(G,U\simeq \mathcal{T}(G,V)\oplus \mathcal{T}(G,W)$ in the category $\mathbf{Ab}$ of abelian groups. For a contradiction suppose $V\neq 0 \neq W$. Since $f\neq0$ we cannot have $f\in\mathcal{T}(G,V)\cap\mathcal{T}(G,W)$, and so without loss of generality $f\notin\mathcal{T}(G,V)$, which means $f\notin \mathcal{T}(G,V\oplus N)$ (since otherwise $f$ would lie in either $\mathcal{T}(G,V)$ or $\mathcal{T}(G,N)$). The element $\mathsf{R}$ in $\Psi$ given by $R=N\oplus V$ contradicts the maximality of $\mathsf{N}$, and so $V=0$. This shows $U$ is indecomposable. Let $\Theta$ be the set of collections $\{\mathsf{U}_{i}\mid i\in I\}$ of pure substructures $\mathsf{U}_{i}=\mathcal{T}(-,U_{i})$ for indecomposable objects $U_{i}$ in $\mathcal{T}$ which admit a pure monomorphism $\bigoplus _{i}U_{i}\to K$. As above any chain in $\Theta$ has an upper bound in $\Theta$, and the existence of the object $U$ means $\Theta$ is non-empty. Again the choice of a maximal element of $\Theta$ shows that $\bigoplus _{i}U_{i}\simeq K$.
String and band complexes. {#3}
==========================
\[3.5\] \[2\]
Let $\mathbf{P}$ be the set of non-trivial paths $p\notin \mathcal{J}$ with head $h(p)$ and tail $t(p)$. For each $t>0$ and each vertex $v$ let $\mathbf{P}(t,v\rightarrow)$ (respectively $\mathbf{P}(t,\rightarrow v)$) be the set of paths $p\in\mathbf{P}$ of length $t$ with $t(p)=v$ (respectively $h(p)=v$). Let $\mathbf{A}$ be the set of arrows in $Q$, $\mathbf{A}(v\rightarrow)=\mathbf{P}(1,v\rightarrow)$ and $\mathbf{A}(\rightarrow v)=\mathbf{P}(1,\rightarrow v)$. The composition of $a\in\mathbf{A}(\rightarrow v)$ and $b\in\mathbf{A}(u\rightarrow)$ is $ba$ if $u=v$, and $0$ otherwise.
Homotopy words. {#homo}
---------------
We now recall the language of homotopy words developed in [@Ben2016].
\[definition2\][@Ben2016 Definition 3.1] A *homotopy letter $q$* is one of $\gamma$, $\gamma^{-1}$, $d_{\alpha}$, or $d_{\alpha}^{-1}$ for $\gamma\in\mathbf{P}$ and an arrow $\alpha$. Those of the form $\gamma$ or $d_{\alpha}$ will be called *direct*, and those of the form $\gamma^{-1}$ or $d_{\alpha}^{-1}$ will be called *inverse*. The *inverse $q^{-1}$* of a homotopy letter *$q$* is defined by setting $(\gamma)^{-1}=\gamma^{-1}$, $(\gamma^{-1})^{-1}=\gamma$, $(d_{\alpha})^{-1}=d_{\alpha}^{-1}$ and $(d_{\alpha}^{-1})^{-1}=d_{\alpha}$.
Let $I$ be one of the sets $\{0,\dots,m\}$ (for some $m\geq0$), $\mathbb{N}$, $-\mathbb{N}=\{-n\mid n\in\mathbb{N}\}$, or $\mathbb{Z}$. For $I\neq\{0\}$ a *homotopy $I$-word* is a sequence of homotopy letters $$C=\begin{cases}
l_{1}^{-1}r_{1}\dots l_{m}^{-1}r_{m} & (\mbox{if }I=\{0,\dots,m\})\\
l_{1}^{-1}r_{1}l_{2}^{-1}r_{2}\dots & (\mbox{if }I=\mathbb{N})\\
\dots l_{-1}^{-1}r_{-1}l_{0}^{-1}r_{0} & (\mbox{if }I=-\mathbb{N})\\
\dots l_{-1}^{-1}r_{-1}l_{0}^{-1}r_{0}\mid l_{1}^{-1}r_{1}l_{2}^{-1}r_{2}\dots & (\mbox{if }I=\mathbb{Z})
\end{cases}$$ (which will be written as $C=\dots l_{i}^{-1}r_{i}\dots$ to save space) such that:
1. any homotopy letter in $C$ of the form $l_{i}^{-1}$ (respectively $r_i$) is inverse (respectively direct);
2. any sequence of 2 consecutive letters in $C$, which is of the form $l_{i}^{-1}r_{i}$, is one of $\gamma^{-1}d_{\mathrm{l}(\gamma)}$ or $d_{\mathrm{l}(\gamma)}^{-1}\gamma$ for some $\gamma\in\mathbf{\mathbf{P}}$; and
3. any sequence of 4 consecutive letters in $C$ of the form $l_{i}^{-1}r_{i}l_{i+1}^{-1}r_{i+1}$ is one of
1. $\gamma^{-1}d_{\mathrm{l}(\gamma)}d_{\mathrm{l}(\lambda)}^{-1}\lambda$ where $h(\gamma)=h(\lambda)$ and $\mathrm{l}(\gamma)\neq \mathrm{l}(\lambda)$;
2. $d_{\mathrm{l}(\gamma)}^{-1}\gamma d_{\mathrm{l}(\lambda)}^{-1}\lambda$ where $t(\gamma)=h(\lambda)$ and $\mathrm{f}(\gamma)\mathrm{l}(\lambda)\in\mathcal{J}$;
3. $d_{\mathrm{l}(\gamma)}^{-1}\gamma\lambda^{-1}d_{\mathrm{l}(\lambda)}$ where $t(\gamma)=t(\lambda)$ and $\mathrm{f}(\gamma)\neq \mathrm{f}(\lambda)$;
4. $\gamma^{-1}d_{\mathrm{l}(\gamma)}\lambda^{-1}d_{\mathrm{l}(\lambda)}$ where $h(\gamma)=t(\lambda)$ and $\mathrm{f}(\lambda)\mathrm{l}(\gamma)\in\mathcal{J}$.
For $I=\{0\}$ there are *trivial homotopy words* $ 1 _{v,1}$ and $ 1 _{v,-1}$ for each vertex $v$.
The head and tail of any path $\gamma\in\mathbf{P}$ are already defined and we extend this by setting $h(d_{a}^{\pm1})=h(a)$ for any arrow $a$ and $h(q^{-1})=t(q)$ for all homotopy letters $q$. For each $i\in I$ there is an *associated vertex* $v_{C}(i)$ defined by: $v_{C}(i)=t(l_{i+1})$ for $i\leq0$ and $v_{C}(i)=t(r_{i})$ for $i>0$ provided $C=\dots l_{i}^{-1}r_{i}\dots$ is non-trivial; and $v_{ 1 _{v,\pm1}}(0)=v$ otherwise.
If $\gamma\in\mathbf{P}$ and $a=\mathrm{l}(\gamma)$ let $H(\gamma^{-1}d_{a})=-1$ and $H(d_{a}^{-1}\gamma)=1$. Let $\mu_{C}(0)=0$ and $$\mu_{C}(i)=\begin{cases}
H(l{}_{1}^{-1}r{}_{1})+\dots+H(l{}_{i}^{-1}r{}_{i}) & (\text{if } 0<i\in I)\\
-(H(l_{0}^{-1}r{}_{0})+\dots+H(l_{i+1}^{-1}r{}_{i+1}))) & (\text{if }0>i\in I)
\end{cases}$$ [@BekMer2003 Definition 2]. For $n\in\mathbb{Z}$ let $P^{n}(C)$ be the sum $\bigoplus\Lambda e_{v_{C}(i)}$ over $i\in \mu_{C}^{-1}(n)$. For each $i\in I$ let $ b _{i,C}$ denote the coset of $e_{v_{C}(i)}$ in $P(C)$ (in degree $\mu_{C}(i)$). If the dependency on $C$ is irrelevant let $b_{i,C}=b_{i}$. We define the complex $P(C)$ by extending the assignment $d_{P(C)}( b _{i})= b _{i}^{-}+ b _{i}^{+}$ linearly over $\Lambda$ for each $i\in I$, where $$\begin{array}{c}
b _{i}^{+}=\begin{Bmatrix}\alpha b _{i+1} & (\mbox{if }i+1\in I,\,l_{i+1}^{-1}r_{i+1}=d_{\mathrm{l}(\alpha)}^{-1}\alpha)\\
0 & (\mbox{otherwise})
\end{Bmatrix}\\
\\
b _{i}^{-}=\begin{Bmatrix}\beta b _{i-1} & (\mbox{if }i-1\in I,\,l_{i}^{-1}r_{i}=\beta^{-1}d_{\mathrm{l}(\beta)})\\
0 & (\mbox{otherwise})
\end{Bmatrix}
\end{array}$$ Let $[C]_{i}=[\gamma^{-1}]$ if $l_{i}^{-1}r_{i}=d_{\mathrm{l}(\gamma)}^{-1}\gamma$ and $[C]_{i}=[\gamma]$ if $l_{i}^{-1}r_{i}=\gamma^{-1}d_{\mathrm{l}(\gamma)}$. Then $[C]=\dots [C]_{i}\dots$ defines a *generalised string* or a *generalised band* as in Bekkert and Merklen [@BekMer2003 §4.1].
[@Ben2016 Definitions 3.3]\[def.3.3\] Let $C$ be a homotopy word. Write $I_{C}$ for the subset of $\mathbb{Z}$ where $C$ is a homotopy $I_{C}$-word. Let $t\in\mathbb{Z}$. If $I_{C}=\mathbb{Z}$ we let $C[t]=\dots l_{t}^{-1}r_{t}\mid l_{t+1}^{-1}r_{t+1}\dots$. That is, in the language of generalised strings and bands, if $I_{C}=\mathbb{Z}$ let $[C[t]]_{i}=[C]_{i+t}$. If instead $I_{C}\neq\mathbb{Z}$ we let $C=C[t]$.
The *inverse* $C^{-1}$ of $C$ is defined by $( 1 _{v,\delta})^{-1}= 1 _{v,-\delta}$ if $I=\{0\}$, and otherwise inverting the homotopy letters and reversing their order. Note the homotopy $\mathbb{Z}$-words are indexed so that $$\left(\dots l_{-1}^{-1}r_{-1}l_{0}^{-1}r_{0}\mid l_{1}^{-1}r_{1}l_{2}^{-1}r_{2}\dots\right)^{-1}=\dots r_{2}^{-1}l_{2}r_{1}^{-1}l_{1}\mid r_{0}^{-1}l_{0}r_{-1}^{-1}l_{-1}\dots$$ [@Ben2016 Definitions 3.5] We say $C$ is *periodic* if $I_{C}=\mathbb{Z}$, $C=C\left[p\right]$ and $\mu_{C}(p)=0$ for some $p>0$. In this case the minimal such $p$ is the *period* of $C$, and we say $C$ is $p$-*periodic*. We say $C$ is *aperiodic* if $C$ is not periodic.
If $C$ is periodic of period $p$ then by [@Ben2016 Lemma 3.4] $P^{n}(C)$ is a $\Lambda\text{-}k[T,T^{-1}]$-bimodule where $T$ acts on the right by $ b _{i}\mapsto b _{i-p}$. By translational symmetry the map $d_{P(C)}^{n}:P^{n}(C)\rightarrow P^{n+1}(C)$ is $\Lambda\otimes_{k}k[T,T^{-1}]$-linear. For a $k[T,T^{-1}]$-module $V$ we define $P(C,V)$ by $P^{n}(C,V)=P^{n}(C)\otimes_{k[T,T^{-1}]}V$ and $d_{P(C,V)}^{n}=d_{P(C)}^{n}\otimes\mathrm{id}_{V}$ for each $n\in\mathbb{Z}$.
[@Ben2016 Definition 3.12] A *string complex* has the form $P(C)$ where $C$ is aperiodic. If $V$ is a $k[T,T^{-1}]$-module we call $P(C,V)$ a *band complex* provided $C$ is a periodic homotopy $\mathbb{Z}$-word and $V$ is an indecomposable $k[T,T^{-1}]$-module.
At this point it is worth stating in full two results from [@Ben2016]. Theorem \[theorem.1.1\] classifies objects in the homotopy category of complexes with finitely generated homogeneous components. Theorem \[theorem.1.2\] characterises when two shifts of string or band complexes are isomorphic.
\[theorem.1.1\]*[@Ben2016 Theorem 1.1]* Let $\Lambda$ be a gentle algebra.
1. Every object in $\mathcal{K}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$ is isomorphic to a (possibly infinite) direct sum of shifts of string complexes $P(C)$ and shifts of band complexes $P(C,V)$.
2. Each shift of a string or band complex is an indecomposable object in $\mathcal{K}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$.
\[theorem.1.2\]*[@Ben2016 Theorem 1.2]* Let $\Lambda$ be a gentle algebra. Let $C$ and $E$ be homotopy words, let $V$ and $W$ be $k[T,T^{-1}]$-modules and let $n\in\mathbb{Z}$.
1. If $C$ and $E$ are aperiodic, then $P(C)[n]\simeq P(E)$ in $\mathcal{K}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ if and only if:
1. we have $I_{C}=\{0,\dots,m\}$ and $(I_{E},E,n)=(I_{C},C,0)\text{ or } (I_{C},C^{-1},\mu_{C}(m))$; or
2. we have $I_{C}=\pm\mathbb{N}$ and $(I_{E},E,n)=(\pm\mathbb{N},C,0)\text{ or }(\mp\mathbb{N},C^{-1},0)$; or
3. we have $I_{C}=\mathbb{Z}$ and $(I_{E},E,n)=(\mathbb{Z}, C^{\pm1}[t],\mu_{C}(\pm t))$ for some $t\in\mathbb{Z}$.
2. If $C$ and $E$ are periodic, then $P(C,V)[n]\simeq P(E,W)$ in $\mathcal{K}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ if and only if:
1. we have $E=C[t]$, $V\simeq W$ and $n=\mu_{C}(t)$ for some $t\in\mathbb{Z}$; or
2. we have $E=C^{-1}[t]$, $V\simeq \mathrm{res}_{\iota} \,W$ and $n=\mu_{C}(-t)$ for some $t\in\mathbb{Z}$.
3. If $C$ is aperiodic and $E$ is periodic, then $P(C)[n]\not\simeq P(E,V)$ in $\mathcal{K}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$.
[@Ben2016 Definition 6.8]Choose a *sign* $s(q)\in\{\pm1\}$ for each homotopy letter $q$ in the set $\mathbf{A}{}^{\pm}$ of homotopy letters of the form $\alpha$ or $\alpha^{-1}$: such that if distinct letters $q$ and $q'$ from $\mathbf{A}{}^{\pm}$ have the same head, they have the same sign if and only if $\{q,q'\}=\{\alpha^{-1},\beta\}$ with $\alpha\beta\notin\mathbf{P}$. Now let $s(\gamma)=s( \mathrm{l}( \gamma))$, $s(\gamma^{-1})=s( \mathrm{f}( \gamma)^{-1})$, and $s(d_{\alpha}^{\pm1})=-s(\alpha)$ for each $\gamma\in\mathbf{P}$ and each arrow $\alpha$. For a (non-trivial finite or $\mathbb{N}$)-homotopy word $C$ we let $h(C)$ and $s(C)$ be the head and sign of the first letter of $C$. For any vertex $v$ let $s(1_{v,\pm1})=\pm1$ and $h(1_{v,\pm1})=v$.
Let $D$ and $E$ be homotopy words where $I_{D^{-1}}\subseteq\mathbb{N}$ and $I_{E}\subseteq\mathbb{N}$. If $u=h(D^{-1})$ and $\epsilon=-s(D^{-1})$ let $D 1 _{u,\epsilon}=D$. If $v=h(E)$ and $\delta=s(E)$ we let $ 1 _{v,\delta}E=E$. The *composition* $DE$ is the concatenation of the homotopy letters in $D$ with those in $E$. The result is a homotopy word if and only if $h(D^{-1})=h(E)$ and $s(D^{-1})=-s(E)$ [@Ben2018 Proposition 2.1.13]. If $D=\dots l_{-1}^{-1}r_{-1}l_{0}^{-1}r_{0}$ is a $-\mathbb{N}$-word and $E=l_{1}^{-1}r_{1}l_{2}^{-1}r_{2}\dots$ is an $\mathbb{N}$-word, write $DE=\dots l_{0}^{-1}r_{0}\mid l_{1}^{-1}r_{1}\dots$.
\[example.2.14-1\]\[example8\](See also [@Ben2016 Example 3.2]). Consider the gentle algebra $\Lambda=kQ/\mathcal{J}$ given by $\mathcal{J}=\langle a^{2},\,gf,\,hg,\,fh,\,sr,\,ts,\,rt,\,b^{2}\rangle$ where $Q$ is the quiver $$\xymatrix@R=.2em{0\ar@(ul,ur)^{a} & 1\ar[l]_{w}\ar[dd]_{g} & & 2\ar[dl]_{t} & 3\ar@(ul,ur)^{b}\ar[l]_{x}\\
& & 4\ar[ul]_{f}\ar[dr]_{r} & & \\
5 & 6\ar[l]_{y}\ar[ur]_{h} & & 7\ar[uu]_{s} & 8\ar[l]_{z}
}$$ Let $C=d_{r}^{-1}rh d_{g}^{-1}g d_{f}^{-1}f r^{-1}d_{r} s^{-1}d_{s} d_{x}^{-1} xb^{-1}d_{b}b^{-1}d_{b}b^{-1}d_{b}$. We may depict $P(C)$ by $$\xymatrix@C=1.05em@R=1.65em{
& & & & & & & & & & \Lambda e_{3}\ar[dl]^{b}\ar@{..}[llllllllll] & & P^{-1}(C)\ar@{..}[ll]\ar[d]^{d_{P(C)}^{-1}}\\
& \Lambda e_{7}\ar[dr]^{rh}\ar@{..}[l] & & & & & & & & \Lambda e_{3}\ar[dl]^{b}\ar@{..}[llllllll] & & & P^{0}(C)\ar@{..}[lll]\ar[d]^{d_{P(C)}^{0}}\\
& & \Lambda e_{6}\ar[dr]^{g}\ar@{..}[ll] & & & & \Lambda e_{2}\ar[dl]^{s}\ar[dr]^{x}\ar@{..}[llll] & & \Lambda e_{3}\ar[dl]^{b}\ar@{..}[ll] & & & & P^{1}(C)\ar@{..}[llll]\ar[d]^{d_{P(C)}^{1}}\\
& & & \Lambda e_{1}\ar[dr]^{f}\ar@{..}[lll] & & \Lambda e_{7}\ar[dl]^{r}\ar@{..}[ll] & & \Lambda e_{3}\ar@{..}[ll] & & & & & P^{2}(C)\ar@{..}[lllll]\ar[d]^{d_{P(C)}^{2}}\\
& & & & \Lambda e_{4}\ar@{..}[llll] & & & & & & & & P^{3}(C)\ar@{..}[llllllll]
}$$ where an arrow $\Lambda e_{v}\to \Lambda e_{u}$ labelled by a path $p$ with head $v$ and tail $v$ indicates right-multiplication by $p$. The generalised string is $[C]=[(rh)^{-1}][g^{-1}][f^{-1}][r][s][x^{-1}][b][b][b]$.
Left bounded string complexes. {#leftbdd}
------------------------------
We now compute the kernel of the differential map for any string complex.
\[definition.10.5\]Let $C$ be a homotopy $I$-word. For each $i\in I$ we define the path $\kappa(i)$ by
\(a) $\kappa(i)=e_{v_{C}(i)}$ if ($i-1\notin I$ or $l_{i}^{-1}r_{i}=d_{z}^{-1}\tau$) and ($i+1\notin I$ or $l_{i+1}^{-1}r_{i+1}=\gamma^{-1}d_{y}$),
\(b) $\kappa(i)= \mathrm{f}( \tau)$ if ($i-1\in I$ and $l_{i}^{-1}r_{i}=d_{z}^{-1}\tau$) and ($i+1\in I$ and $l_{i+1}^{-1}r_{i+1}=d_{y}^{-1}\gamma$),
\(c) $\kappa(i)= \mathrm{f}( \gamma)$ if ($i-1\in I$ and $l_{i}^{-1}r_{i}=\tau^{-1}d_{z}$) and ($i+1\in I$ and $l_{i+1}^{-1}r_{i+1}=\gamma^{-1}d_{y}$),
\(d) $\kappa(i)=\beta$ if ($i-1\notin I$ and $\beta y=0$) and ($i+1\in I$ and $l_{i+1}^{-1}r_{i+1}=d_{y}^{-1}\gamma$),
\(e) $\kappa(i)=\alpha$ if ($i-1\in I$ and $l_{i}^{-1}r_{i}=\tau^{-1}d_{z}$) and ($i+1\notin I$ and $\alpha z=0$), and
\(f) $\kappa(i)=0$ (if $i-1\in I$ and $l_{i}^{-1}r_{i}=\tau^{-1}d_{z}$) and ($i+1\in I$ and $l_{i+1}^{-1}r_{i+1}=d_{y}^{-1}\gamma$).
Note that for any $i\in I$ exactly one of the ((a), (b), (c), (d), (e) and (f)) is true. We say that the $i^{\mathrm{th}}$ *kernel part* is: *full* in case (a); a *left* (resp. *right*) *arm* in case (b) (resp. (c)); a *left* (resp. *right*) *peripheral arm* in case (d) (resp. (e)); and $0$ in case (f).
\[corollarly.10.3\]*[@Ben2018 Corollary 2.7.8]* Let $C$ be a homotopy $I$-word. For any *$n\in\mathbb{Z}$* we have *$\mathrm{ker}(d_{P(C)}^{n})=\bigoplus_{i\in\mu_{C}^{-1}(n)}\Lambda\kappa(i)b_{i}$*.
To streamline proposition \[strngcmpct\] we use the following notation.
\[nesw\]For any vertex $v$ and $\delta=\pm1$, let $\mathcal{W}_{v,\delta}$ be the set of homotopy $I$-words with $I\subseteq\mathbb{N}$, head $v$ and sign $\delta$. Let $C$ be a finite homotopy word. Let $h(C)=u$, $h(C^{-1})=v$, $s(C)=\delta$ and $s(C^{-1})=\epsilon$. Note $1_{u,\delta}C=C$ and $C1_{v,-\epsilon}$ are homotopy words. Let $\mathcal{W}^{+}_{\leftarrow}(C)$ be the union of $\{1_{u,\delta}\}$ and the (potentially empty) set of homotopy $I$-words of the form $$B_{+}=
\begin{cases}
(\alpha(m))^{-1}d_{\alpha(m)}\,\dots \, (\alpha(1))^{-1}d_{\alpha(1)} & (\mbox{if }I=\{0,\dots,m\}\mbox{ for some }m>0)\\
\dots \, \alpha(3)^{-1}d_{\alpha(3)} \, (\alpha(2))^{-1} d_{\alpha(2)} \, (\alpha(1))^{-1}d_{\alpha(1)} & (\mbox{if }I=-\mathbb{N})
\end{cases}$$ where ($\alpha(i)\in\mathbf{A}$ for all $i>0$ in $I$, and) $s(\alpha(1))=\delta$.
Dually let $\mathcal{W}^{-}_{\leftarrow}(C)$ be the union of $\{1_{u,\delta}\}$ and the set of homotopy $I$-words of the form $$B_{-}=
\begin{cases}
d_{\alpha(m)}^{-1}\alpha(m)\,\dots \, d_{\alpha(1)}^{-1}\alpha(1) & (\mbox{if }I=\{0,\dots,m\}\mbox{ for some }m>0)\\
\dots \, d_{\alpha(3)}^{-1}\alpha(3) \, d_{\alpha(2)}^{-1}\alpha(2) \, d_{\alpha(1)}^{-1}\alpha(1) & (\mbox{if }I=-\mathbb{N})
\end{cases}$$ Similarly: let $\mathcal{W}^{+}_{\rightarrow}(C)$ be the union of $\{1_{v,-\epsilon}\}$ and the set of homotopy $I$-words of the form $$D_{+}=
\begin{cases}
d_{\beta(1)}^{-1}\beta(1)\,\dots \,d_{\beta(m)}^{-1}\beta(m) & (\mbox{if }I=\{0,\dots,m\}\mbox{ for some }m>0)\\
d_{\beta(1)}^{-1}\beta(1)\,d_{\beta(2)}^{-1}\beta(2)\,d_{\beta(3)}^{-1}\beta(3)\,\dots & (\mbox{if }I=\mathbb{N})
\end{cases}$$ where ($\beta(i)\in\mathbf{A}$ for all $i>0$ in $I$, and) $s(\beta(1))=\epsilon$; and let $\mathcal{W}^{-}_{\leftarrow}(C)$ be the union of $\{1_{u,-\epsilon}\}$ and the set of homotopy $I$-words of the form $$D_{-}=
\begin{cases}
(\beta(1))^{-1}d_{\beta(1)}\,\dots \,(\beta(m))^{-1}d_{\beta(m)} & (\mbox{if }I=\{0,\dots,m\}\mbox{ for some }m>0)\\
(\beta(1))^{-1}d_{\beta(1)}\,(\beta(2))^{-1}d_{\beta(2)}\,(\beta(3))^{-1}d_{\beta(3)}\,\dots & (\mbox{if }I=\mathbb{N})
\end{cases}$$ The remark above implies $B_{\pm}C$ and $CD_{\pm}$ are both homotopy words for each $B_{\pm}\in\mathcal{W}^{\pm}_{\leftarrow}(C)$ and each $D_{\pm}\in \mathcal{W}^{\pm}_{\rightarrow}(C)$. Note that, by [@Ben2018 Lemma 2.1.14], if $B_{\pm},B'_{\pm}\in\mathcal{W}^{\pm}_{\leftarrow}(C)$ and $I_{B_{\pm}}=I_{B'_{\pm}}$ then $B_{\pm}=B'_{\pm}$. This means there exists a unique homotopy $I$-word $C_{\swarrow}\in\mathcal{W}^{+}_{\leftarrow}(C)$ (respectively $C_{\nwarrow}\in\mathcal{W}^{-}_{\leftarrow}(C)$) such that $I$ is maximal with respect to inclusion. Similarly there is a unique homotopy $I$-word $C_{\searrow}\in\mathcal{W}^{+}_{\rightarrow}(C)$ (respectively $C_{\nearrow}\in\mathcal{W}^{-}_{\rightarrow}(C)$) such that $I$ is maximal with respect to inclusion. For each $\mathrm{l}\in\{\swarrow,\nwarrow\}$ and each $\mathrm{r}\in\{\searrow,\nearrow\}$: let $C(\mathrm{l})$ be the composition $C_{\mathrm{l}}\,C$ of $C_{\mathrm{l}}$ and $C$; let $C(\mathrm{r})$ be the composition $C\,C_{\mathrm{r}}$ of $C$ and $C_{\mathrm{r}}$; and let $C(\mathrm{l},\mathrm{r})$ be the composition $C_{\mathrm{l}}\,C\,C_{\mathrm{r}}$ of $C_{\mathrm{l}}$, $C$, and $C_{\mathrm{r}}$.
Consider the full subcategory $\mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$ of $\mathcal{K}(\mathbf{Proj}\text{-}\Lambda)$ (respectively $\mathcal{K}^{-,b}(\Lambda\text{-}\mathbf{proj})$ of $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$) consisting of right-bounded complexes with bounded cohomology and finitely generated homogeneous components. We now describe the objects in the category $\mathcal{K}^{+,b}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$ by adapting the proof of [@Ben2018 Lemma 2.7.5]. The proof is essentially the same, but for completeness we essentially repeat it.
\[proppss\] The following statements hold.
1. A shift of a string complex $P(A)$ lies in $\mathcal{K}^{+,b}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$ if and only if there exists a finite homotopy word $C$ such that $A=C$, $A=C(\swarrow)$, $A=C(\searrow)$ or $A=C(\swarrow,\searrow)$.
2. A shift of a band complex $P(D,V)$ lies in $\mathcal{K}^{+,b}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$ if and only if the indecomposable $k[T,T^{-1}]$-module $V$ is finitely-dimensional as a $k$-vector space.
\(i) Let $A$ be a homotopy $I$-word. Suppose firstly that there is a sequence $(i_{n}\mid n\in\mathbb{N})\in I^{\mathbb{N}}$ such that the $i_{n}^{\text{th}}$ kernel part is full for each $n$. Since $P(A)$ is bounded below $\{\mu_{A}(i_{n})\mid n\in\mathbb{N}\}$ does not have a lower bound.
This means there is a subsequence $(i_{n(r)}\mid r\in\mathbb{N})$ of $(i_{n}\mid n\in\mathbb{N})$ such that $\mu_{A}(i_{n(r)})>\mu_{A}(i_{n(r+1)})$ for all $r$. By definition, for each $r$ we have $b_{i_{n(r)}}\notin\mathrm{im}(d_{P(A)})$, and the assumption on $(i_{n})$ gives $b_{i_{n(r)}}\in\mathrm{ker}(d_{P(A)})$, which contradicts that $P(A)$ has bounded cohomology.
Hence we have shown that there are no sequences $(i_{n}\mid n\in\mathbb{N})\in I^{\mathbb{N}}$ such that the $i_{n}^{\mathrm{th}}$ kernel part is full for each $n$. So we can choose $l\in I$ such that $A_{>l}=d_{ \mathrm{l}( \gamma_{1})}^{-1}\gamma_{1}d_{ \mathrm{l}( \gamma_{2})}^{-1}\gamma_{2}\dots$ for a sequence of paths $\gamma_{j}\in\mathbf{P}$ where $ \mathrm{f}( \gamma_{j}) \mathrm{l}( \gamma_{j+1})=0$ for each $j\geq1$. Now choose $q\in\mathbb{Z}$ such that $H^{p}(P(A)=0$ for all $p<q$. Choose $t>l$ such that $\mu_{A}(i)<q$ for each $i>t$.
If there is some $j>t-l$ where $\gamma_{j}$ has length greater than $1$ then $d_{P(A)}(b_{l+j})=\gamma_{j}b_{l+j+1}$ and so $ \mathrm{f}( \gamma_{j})b_{l+j+1}\notin\mathrm{im}(d_{P(A)})$. By Corollary \[corollarly.10.3\] we have $ \mathrm{f}( \gamma_{j})b_{l+j+1}\in\mathrm{ker}(d_{P(A)})$, which contradicts that $H^{n}(P(A))=0$ where $n=\mu_{A}(l+j+1)$.
Hence $\gamma_{j}$ is an arrow for each $j>t-l$. Now let $\alpha_{h}=\gamma_{j+h}$ for each integer $h>0$. Since the quiver $Q$ is finite there is some $h>0$ such that $\alpha_{h}=\alpha_{h+n}$ for some $n>0$, which means $\alpha_{h}=\alpha_{h+n}$ for each $h>0$. Altogether we have $A_{>t}=((\alpha_{n}^{-1}d_{\alpha_{n}}\dots\alpha_{1}^{-1}d_{\alpha_{1}})^{-1})^{\infty}$, as required.
\(ii) This follows from the fact that any band complex $P(C,V)$ is a bounded complex whose homogeneous component in degree $n$ is a direct sum of $\vert\mu_{C}^{-1}(n)\vert \times \mathrm{dim}_{k}(V)$ indecomposable projective modules of the form $\Lambda e_{v}$.
Compactness in homotopy categories of gentle algebras. {#cmpctgentle}
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Let us start by noting that derived categories of modules are compactly generated.
If $A$ is a (unital) ring, then the derived category $\mathcal{D}(A\text{-}\mathbf{Mod})$ of complexes of $A$-modules is compactly generated. Here $\{A[n]\mid n\in\mathbb{Z}\}$ is a generating set, where $A[n]$ denotes the complex consisting of the $A$-module $A$ concentrated in degree $n$. Furthermore, the compact objects of $\mathcal{D}(A\text{-}\mathbf{Mod})$ are the bounded complexes of finitely generated projective $A$-modules.
We now use the above to identify which string complexes, and which band complexes, define compact objects in the triangulated category $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$. Let $(-)^{\star} = \mathrm{Hom}_{\Lambda\text{-}\mathbf{Mod}}(-,\Lambda)$, the contravariant functor from the category $\Lambda\text{-}\mathbf{Mod}$ of left $\Lambda$-modules to the category $\mathbf{Mod}\text{-}\Lambda$ of right $\Lambda$-modules. Note that for any exact sequence $0\to L\to M\to N \to 0$ in $\Lambda\text{-}\mathbf{Mod}$ the sequence $0\to N^{\star}\to M^{\star}\to L^{\star} $ in $\mathbf{Mod}\text{-}\Lambda$ is exact. Since $\Lambda$ is right noetherian, $(-)^{\star} $ restricts to a functor $(-)^{\star}|:\Lambda\text{-}\mathbf{mod}\to \mathbf{mod}\text{-}\Lambda$ between full subcategories of finitely generated modules. Similarly let ${}^{\star}(-)|$ be the restriction of the functor $\mathrm{Hom}_{\mathbf{Mod}\text{-}\Lambda}(-,\Lambda):\mathbf{Mod}\text{-}\Lambda\to\Lambda\text{-}\mathbf{Mod}$ to $\mathbf{mod}\text{-}\Lambda\to\Lambda\text{-}\mathbf{mod}$.
We write $\mathcal{K}^{c}(\Lambda\text{-}\mathbf{Proj})$ for the full subcategory of the triangulated category $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ consisting of compact objects. Proposition \[strngcmpct\] is essentially due to Jørgenson [@Jor2005]. We sketch the proof, to verify our setup is a specialisation of [@Jor2005 Setup 3.1]. We then use these details from the proof of [@ArnLakPauPre2017 Proposition 3.6].
*[@Jor2005 Theorem 3.2]*\[strngcmpct\] The following statements hold.
The restriction $(-)^{\star}|$ defines a triangle equivalence $\mathcal{K}^{c}(\Lambda\text{-}\mathbf{Proj})\rightarrow \mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$;
The triangulated category $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ is compactly generated
Any indecomposable compact object of $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ is isomorphic to:
1. a shift of a string complex of the form $P(A)$ where, for some finite homotopy word $C$, we have $A=C$, $A=C(\swarrow)$, $A=C(\searrow)$ or $A=C(\swarrow,\searrow)$;
2. or a shift of a band complex $P(D,V)$ where the indecomposable $k[T,T^{-1}]$-module $V$ is finite-dimensional over $k$.
We recall the notion of thick-ness in a triangulated category. A non-empty full triangulated subcategory $\mathcal{S}$ of $\mathcal{T}$ is called *thick* if, for any object $X$ of $\mathcal{S}$, if there is an isomorphism $X\simeq X'\oplus X''$ in $\mathcal{T}$ then $X'$ and $X''$ are objects of $\mathcal{S}$. Given a set $\mathcal{X}$ of objects in $\mathcal{S}$, the *thick subcategory* $\mathrm{thick}_{\mathcal{T}}(\mathcal{X})$ *of* $\mathcal{T}$ *generated by* $\mathcal{S}$, is defined as follows. We define, inductively, a subcategory $\mathcal{X}_{n}$ of $\mathcal{T}$ for each $n\in\mathbb{Z}$ with $n>0$. For the case $n=1$ write $\mathcal{X}_{1}$ for the full subcategory of $\mathcal{T}$ consisting of the objects in $\mathcal{X}$ together with the zero object $0$. Suppose now, for some fixed arbitrary $n\in\mathbb{Z}$ with $n>0$, subcategories $\mathcal{X}_{1}, \dots, \mathcal{X}_{n}$ of $\mathcal{T}$ have been defined. Any morphism $f: X\to Y$ in $\mathcal{T}$ defines a *mapping cone*, an object $Z$ completing $f$ to a distinguished triangle $X\to Y \to Z \to \Sigma X$. Any two mapping cones of $f$ are isomorphic, and for each morphism $f$ in $\mathcal{X}_{n}$ we choose a representative $\mathrm{c}(f)$ of the isoclass of $Z$.
Let $\mathcal{X}_{n+1}$ be the full subcategory of $\mathcal{T}$ consisting of the objects $X$ in $\mathcal{T}$ such that either: $X$ is an object in $\mathcal{X}_{n}$; or $X=\mathrm{c}(f)$ for some morphism $f$ in $\mathcal{X}_{n}$. We let $\mathrm{thick}_{\mathcal{T}}(\mathcal{X})$ be the full subcategory of $\mathcal{T}$ consisting of the objects $X$ in $\mathcal{T}$ such that $X$ lies in some $\mathcal{X}_{n}$.
Since $\Lambda$ is a finite-dimensional $k$-algebra, the jacobson radical $\mathrm{rad}(\Lambda)$ is nilpotent, and the quotient ring $\Lambda/\mathrm{rad}(\Lambda)$ is semisimple. In particular, by a well-known equivalence due to Bass [@Bas1960 Theorem P], for all integers $n\geq0$, the limit of modules of projective-dimension at most $n$ must have projective-dimension at most $n$.
By a well-known result of Lazard [@Laz1969 Théorème 1.2], over any unital ring, every flat module is a limit of free modules, which are projective. Altogether we have shown that any flat $\Lambda$-module is projective. Furthermore, $\Lambda$ is coherent because it is noetherian. So the criterion from [@Jor2005 Setup 2.1] are met. We now follow [@Jor2005 Construction 2.3], and then apply [@Jor2005 Theorems 2.4 and 3.2]. Let $M$ be an object in $\Lambda\text{-}\mathbf{mod}$, let $N=(M)^{\star}$ and let $$P_{M}=
\xymatrix{
\dots\ar[r] & P_{M}^{2}\ar[r] & P_{M}^{1}\ar[r] & P_{M}^{0}\ar[r] & 0\ar[r] & \dots}$$ be a projective resolution of $N$, where both $N$ and $P_{M}$ are considered as objects in $\mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$ such that the modules $N$ and $P_{M}^{0}$ lie in degree $0$. This means there is a quasi-isomorphism $\phi :P_{M}\to N$ in $\mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$, which corresponds to a map $\theta:P_{M}^{0}\to N$. Applying duality defines a homomorphism of left $\Lambda$-modules ${}^{\star}(\theta) :{}^{\star}(P_{M}^{0}) \to {}^{\star}(N)$, and defines a morphism ${}^{\star}(P_{M}) \to {}^{\star}(N)$ in the category $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ which we label ${}^{\star}(\phi)$, where $ {}^{\star}(N)$ is a complex concentrated in degree $0$. Note that both $P_{M}$ and ${}^{\star}(P_{M})$ depend functorially on $M$.
(i), (ii) As in [@Jor2005 Construction 2.3] note that there is a set of isomorphism classes of objects $M$ in $\Lambda\text{-}\mathbf{mod}$. Consequently there is only a set of isomorphism classes of objects in $\mathcal{K}(\mathbf{Proj}\text{-}\Lambda)$ of the form $({}^{\star}(P_{M}))[n]$ with $n\in\mathbb{Z}$, and we let $\mathtt{G}$ consist of one object from each such isomorphism class. Let $\mathtt{H}$ be the set of objects in $\mathcal{K}(\mathbf{Proj}\text{-}\Lambda)$ of the form ${}^{\star}(G)$ where $G\in\mathtt{G}$. Let $\mathcal{C}=\mathrm{thick}_{\mathcal{K}(\Lambda\text{-}\mathbf{Proj})}(\mathtt{G})$ and $\mathcal{D}=\mathrm{thick}_{\mathcal{K}(\mathbf{Proj}\text{-}\Lambda)}(\mathtt{H})$. By [@Jor2005 Theorem 2.4] the sets $\mathtt{G}$ and $\mathtt{H}$ are generating sets of compact objects. By [@Jor2005 Theorem 3.2] the restriction $(-)^{\star}|$ defines a triangle equivalence $\mathcal{C}\rightarrow \mathcal{D}$. As in the proof of [@Jor2005 Theorem 3.2] we have $\mathcal{D}=\mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$. By [@Nee1996 Theorem 2.3.1], for example, we have $\mathcal{K}^{c}(\mathbf{Proj}\text{-}\Lambda)=\mathcal{C}$.
\(ii) Let $M$ be an indecomposable object in $\mathcal{K}^{c}(\mathbf{Proj}\text{-}\Lambda)$. By part (i) we have that $(-)^{\star}$ is an object of $\mathcal{K}^{-,b}(\mathbf{proj}\text{-}\Lambda)$. This means $M$ must have been an object of $\mathcal{K}^{+,b}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$. The result follows from Proposition \[proppss\].
Lemma \[sigmacardy\] below is the analogue of the statement that (i) implies (v) in [@JenLen1989 Theorem 8.1].
\[sigmacardy\]There is a cardinal $\kappa$ such that every indecomposable pure-injective object of $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ has cardinality at most $\kappa$.
Let $\mathcal{T}$ be the triangulated category $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$. Recall the canonical language $\mathfrak{L}^{\mathcal{T}}$ defined and discussed in §\[4\]. By Proposition \[strngcmpct\] the category $\mathcal{T}$ is compactly generated by a set $\mathtt{G}$. Let $\kappa=\mathrm{max}\{\vert\mathbb{N}\vert,\vert\mathfrak{L}^{\mathcal{T}}\vert,\vert k \vert,\vert \mathtt{G} \vert\}$ (where $k$ is the ground field of $\Lambda=kQ/\mathcal{J}$). Let $U$ be a fixed indecomposable pure-injective object in $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$.
By Definition \[sorts\] we have that $\vert \mathsf{U}\vert $ is the sum of the cardinalities $\vert\mathcal{T}(G,U)\vert$ as $G$ runs through $\mathtt{G}$. This shows $\kappa\leq \vert \mathsf{U}\vert$, and so by Theorem \[lowenskolem\] there is an elementary substructure $\mathsf{Z}$ of $\mathsf{U}$ such that $\vert\mathsf{Z}\vert=\kappa$. Since $\mathsf{Z}$ is a pure substructure of $\mathsf{U}$ there is a pure monomorphism $Z\to U$ in $\mathcal{T}$. By Corollary \[sigmapropsyanah\](i) we must have that the morphism $Z\to U$ splits, and since $U$ is indecomposable, this means $U\simeq Z$ which has cardinality $\kappa$.
Linear relations. {#linrels}
=================
The proof of [@BenCra2018 Theorem 1.1] uses the *functorial filtrations* method, going back to work of Gelfand and Ponomarev [@GelPon1968], which was written in the language of *additive relations* in the sense of Mac Lane [@Mac1961]. The aforementioned method depends on a certain splitting result for finite-dimensional $k$-linear relations, see [@GelPon1968 Theorem 3.1], [@Rin1975 §2] and [@Gab1972 §7]. Given $k$-vector spaces $V$ and $W$ a *linear relation from* $V$ *to* $W$ (or *on* $V$ if $W=V$) is a $k$-subspace $C$ of the direct sum $V\oplus W$. This notion generalises the graph of a $k$-linear map $V\rightarrow W$.
Kronecker representations and relations. {#KROW}
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The category $k\text{-}\bf{Rel}$ of linear relations has as objects the pairs $(V,C)$ where $C$ is a relation on $M$, and has morphisms $(V,C)\to (W,D)$ given by $k$-linear maps $f:V\to W$ with $(f(u),f(v))\in D$ for all $(u,v)\in C$. Let $\Gamma$ be the Kronecker quiver, given by two arrows $p$ and $q$ with common tail $u$ and common head $v$, and let $k\Gamma$ be the path algebra.
Let $\alpha$ be the well-known equivalence from the category $k\Gamma\text{-}\bf{Mod}$ of left $k\Gamma$-modules to the category $k\text{-}\textbf{Rep}(\Gamma)$ of $k$-representations $(\phi_{p},\phi_{q}:L_{u}\rightarrow L_{v})$ of $\Gamma$. Any relation $C$ on $V$ defines an object $(\pi_{p},\pi_{q}:C\rightarrow V)$ of $k\text{-}\textbf{Rep}(\Gamma)$ by choosing $\pi_{p}$ (respectively $\pi_{q}$) to be the composition of the inclusion $C\subseteq V\oplus V$ with the first (respectively second) projection $V\oplus V\rightarrow V$.
In this way there is a fully-faithful additive functor $k\text{-}\textbf{Rel}\rightarrow k\text{-}\textbf{Rep}(\Gamma)$ whose essential image, denoted $k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}}$, is the full subcategory of representations $(\phi_{p},\phi_{q}:L_{u}\rightarrow L_{v})$ such that the induced map of the $k$-module product $\phi_{p}\times\phi_{q}:L_{u}\rightarrow L_{v}\oplus L_{v}$ is injective. Let $k\Gamma\text{-}\textbf{Mod}_{\mathrm{rel}}$ be the essential image of the restriction of $\alpha$ to $k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}}$. Denote the induced equivalences $$\iota:k\text{-}\textbf{Rel}\rightarrow k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}},\,\,\eta:k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}}\rightarrow k\Gamma\text{-}\textbf{Mod}_{\mathrm{rel}},$$ and the corresponding (quasi-inverses of these) equivalences $$\lambda:k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}}\rightarrow k\text{-}\textbf{Rel} ,\,\,\mu:k\Gamma\text{-}\textbf{Mod}_{\mathrm{rel}}\rightarrow k\text{-}\textbf{Rep}(\Gamma)_{\mathrm{rel}}.$$ The compositions $\eta\iota$ and $\lambda\mu$ equip $k\text{-}\bf{Rel}$ with various structural properties inherited from the category $k\Gamma\text{-}\textbf{Mod}$. We document some of the said properties below.
\[co/prodclsd\] The category $k\text{-}\bf{Rel}$ has all limits and all coproducts.
Note $k\Gamma\text{-}\textbf{Mod}_{\mathrm{rel}}$ consists of modules $X$ where $e_{u}X\to e_{v}X\oplus e_{v}X$, $x\mapsto (px,qx)$ is injective. This property is closed under taking equalisers, products and coproducts.
A sequence of relations $$0 \to (U,B) \to (V,C) \to (W,D)\to 0$$ is exact provided that the underlying sequences of $k$-vector spaces $$0\to U \to V \to W \to 0\text{
and }0\to B \to C \to D \to 0$$ are exact. For a set $I$ and an object $(V_{i},C_{i})$ of $k\text{-}\bf{Rel}$ for each $i$ the set of pairs $((v_{i}),(v'_{i}))$ with $(v_{i},v_{i}')\in C _{i}$ for each $i$ defines a the product $\prod (V_{i},C_{i})$ of the objects $(V_{i},C_{i})$. Similarly the coproduct $\bigoplus (V_{i},C_{i})$ is given by the relation on $\bigoplus V_{i}$ consisting of pairs $((v_{i}),(v'_{i}))$ as above, but where additionally $v_{i}=v_{i}'=0$ for all but finitely many $i$. If there exists an object $(V,C)$ of $k\text{-}\bf{Rel}$ with $(V_{i},C_{i})=(V,C)$ for each $i$, then the universal property of the coproduct defines a *summation* map $\sigma_{I}:\bigoplus(V,C)\to(V,C)$. By the equivalence of (ii) and (vi) in [@JenLen1989 Theorem 7.1] the object $(V,C)$ is *pure*-*injective* if, for any set $I$, $\sigma_{I}$ extends to a map $\prod (V,C)\to(V,C)$. Similarly by the equivalence of (i) and (ii) in [@JenLen1989 Theorem 8.1] the object $(V,C)$ is $\Sigma$-*pure*-*injective* if, for any set $I$, $\sigma_{I}$ is a section.
For an object $(V,C)$ of $k\text{-}\bf{Rel}$ let $Cv = \{ w\in W \colon (v,w)\in C \}$ for any $v\in V$, and for a subset $U\subseteq M$ let $CU$ be the union $ \bigcup Cu$ over $u\in U$. When $C$ is the graph of a map $f$ then $CU$ is the image of $U$ under $f$. Furthermore, let $$\begin{array}{c}
C'' = \{ m\in M : \exists \,(m_{n})\in M^{\mathbb{N}}\text{ with }(m_{n},m_{n+1})\in C\text{ and }m=m_{0}\},
\\
C' = \{ m\in M : \exists \,(m_{n})\in M^{\mathbb{N}}\text{ with }(m_{n},m_{n+1})\in C,\,m=m_{0}\text{ and }m_{n} = 0\text{ for }n\gg0\}.
\end{array}$$ In the sense of Ringel [@Rin1975 §2], $C'$ is equal to the *stable kernel* $\bigcup _{n>0}C^{n}0$, and $C''$ is a subspace of the *stable image* $\bigcap _{n>0}C^{n}V$. Furthermore if $\mathrm{dim}_{k}(V)<\infty$ then the inclusion of $C''$ in the stable image is an equality [@Cra2018 Lemma 4.2]. Define subspaces $C^{\flat}\subseteq C^{\sharp}\subseteq V$ by $$\begin{array}{c}
C^\sharp = C'' \cap (C^{-1})'',\text{ and }C^\flat = C'' \cap (C^{-1})'+(C^{-1})'' \cap C'.
\end{array}$$ By [@Cra2018 Lemma 4.5] the quotient $C^{\sharp}/C^{\flat}$ is a $k[T,T^{-1}]$-module with the action of $T$ given by $$T(v+C^{\flat})=w+C^{\flat}\text{ if and only if }w\in C^{\sharp}\cap(C^{\flat}+Cv).$$ We say $(V,D)$ is *automorphic* if both projection maps $D\to V$ are isomorphisms, and that $(V,C)$ is *split* provided that there is a subspace $W$ of $V$ such that $C^\sharp = C^\flat \oplus W$ and $(W,C|_{W})$ is automorphic [@Cra2018 §4].
In joint work with Crawley-Boevey [@BenCra2018] we considered $k$-linear relations $(V,C)$ as Kronecker modules, via the first and second projections of $D$ onto $V$, in order to prove the following.
\[meandbill\]*[@BenCra2018 Corollary 1.3]* Let $(V,C)$ be $\Sigma$-pure-injective object of $k\text{-}\bf{Rel}$. Then $(V,C)$ is split and $C^{\sharp}/C^{\flat}$ is a $\Sigma$-pure-injective $k[T,T^{-1}]$-module.
Homotopic minimality and induced relations. {#6-1}
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Let $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ and $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ be the full subcategories of $\mathcal{C}(\Lambda\text{-}\mathbf{Proj})$ and $\mathcal{K}((\Lambda\text{-}\mathbf{Proj})$ consisting of *homotopically minimal* complexes: that is, whose objects $M$ are complexes in $\mathcal{C}(\Lambda\text{-}\mathbf{Proj})$ such that $\im(d_{M}^{n})\subseteq\rad(M^{n+1})$ for all $n\in\mathbb{Z}$. Since $\Lambda$ is a finite-dimensional $k$-algebra, the jacobson radical $\mathrm{rad}(\Lambda)$ is nilpotent, and the quotient ring $\Lambda/\mathrm{rad}(\Lambda)$ is semisimple. This means $\Lambda$ is a perfect ring, and consequently every object in $\Lambda\text{-}\mathbf{Mod}$ has a projective cover.
\[corollary.9.1\]*[@Ben2016 Corollary 4.3]*. The subcategory $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ of $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ is dense.
\[isosreflect\]Let $t:M\to N$ be an isomorphism in the category $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ with inverse $s:N\to M$. Write $\tau:M\to N$ and $\sigma:N\to M$ for the corresponding morphisms in the category $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$. Consider the induced morphisms $\bar{\tau}^{n}:M^{n}/\mathrm{rad}(M^{n})\to N^{n}/\mathrm{rad}(N^{n})$ and $\bar{\sigma}^{n}:N^{n}/\mathrm{rad}(N^{n})\to M^{n}/\mathrm{rad}(M^{n})$ of $\Lambda$-modules (for each $n\in\mathbb{Z}$). By construction the morphisms $\sigma\tau-\mathrm{id}_{M}$ and $\tau\sigma-\mathrm{id}_{N}$ in $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ are null-homotopic.
Since $M$ and $N$ are homotopically minimal this means $\bar{\tau}^{n}$ is an isomorphism with inverse $(\bar{\tau}^{n})^{-1}=\bar{\sigma}^{n}$. Since $\Lambda$ is a perfect ring it must be a smeiperfect ring. By [@Ben2016 Remark 3.11] this means that each of the morphisms $\tau^{n}$ is an isomorphism, and so $\tau$ is an isomorphism. We have shown that the restriction $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})\to \mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ of the quotient functor $\mathcal{C}(\Lambda\text{-}\mathbf{Proj})\to\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$ reflects isomorphisms.
For the remainder of §\[6-1\] we let $M$ be an object of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$.
Fix an arbitrary vertex $v$. Let $ d_{v,M}$ be the $k$-linear endomorphism of $e_{v}M$ defined by the restriction of $d_{M}$. By [@BekMer2003 Lemma 5] we have $e_{v}\mathrm{rad}(M)=\bigoplus b M$ where $b$ runs through $\mathbf{A}(\rightarrow v)$. For any arrow $a\in\mathbf{A}(\rightarrow v)$ let $\pi_{a}:\bigoplus b M\to a M$ (respectively $\iota_{a}:a M\to\bigoplus b M$) be the canonical retraction (respectively section) of $k$-vector spaces.
By [@Ben2016 Lemma 6.3(i)] there is a $k$-linear endomorphism $d_{a,M}$ of $e_{h(\alpha)}M$ defined by $d_{a,M}(m)=\iota_{a}(\pi_{a}(d_{v,M}(m)))$. Furthermore, $d_{v,M}=\sum d_{c,M}$ where the sum runs over $c\in\mathbf{A}(\rightarrow v)$.
It is worth noting some properties of the maps $d_{a,M}$ defined above, which together make up [@Ben2016 Lemma 6.3(ii)]. For any $\tau\in\mathbf{P}$ and any $x\in e_{t(\tau)}M$:
1. if there exists $\sigma\in\mathbf{A}$ such that $\tau\sigma\in\mathbf{P}$ then $d_{ \mathrm{l}( \tau),M}(\tau x)=\tau d_{\sigma,M}(x)$
2. if $\tau\sigma\notin\mathbf{P}$ for all $\sigma\in\mathbf{A}$ then $d_{ \mathrm{l}( \tau),M}(\tau x)=0$;
3. if $h(\theta)=h(\tau)$ for some arrow $\theta\neq \mathrm{l}(\tau)$ then $d_{\theta,M}(\tau x)=0$;
4. if $h(\phi)=h(\tau)$ for some arrow $\phi$ then $d_{\phi,M}d_{ \mathrm{l}( \tau),M}=0$; and
5. if $\tau x\in\mathrm{im}(d_{ \mathrm{l}( \tau),M})$ then $d_{\varsigma,M}(x)=0$ for any $\varsigma\in\mathbf{A}$ where $\tau\varsigma\in\mathbf{P}$.
Let $q$ be a homotopy letter (that is, one of $\gamma$, $\gamma^{-1}$, $d_{\alpha}$ or $d_{\alpha}^{-1}$ for some path $\gamma\in\mathbf{P}$ or some arrow $\alpha$). For any subset $U$ of $e_{t(q)}M$ define the subset $qU$ of $e_{h(q)}M$ by $$\begin{array}{cc}
\gamma U=\{\gamma m\in e_{h(\gamma)}M\mid m\in U\}, & \gamma^{-1}U=\{m\in e_{t(\gamma)}M\mid\gamma m\in U\},\\
d_{\alpha}U=\{d_{\alpha,M}(m)\in e_{h(\alpha)}M\mid m\in U\}, & d_{\alpha}^{-1}U=\{m\in e_{h(\alpha)}M\mid d_{\alpha,M}(m)\in U\}.
\end{array}$$ For any vertex $v$ and any subset $U$ of $e_{v}M$ let $ 1 _{v,\pm1}U=U$. When $U=e_{t(q)}M$ we let $qM=qU$. When $U=e_{t(q)}\mathrm{rad}(M)$ we let $q\mathrm{rad}(M)=qU$. When $U=\{u\}$ we let $qu=qU$.
By [@Ben2016 Corollary 6.6] if $a$ is an arrow then $a^{-1}d_{a}\rad(M)\subseteq e_{t(a)}\rad(M)$, and furthermore given an arrow $b$ with $ab\in\mathbf{P}$ we have $(ab)^{-1}ad_{b}M=b{}^{-1}d_{b}M$. By [@Ben2016 Corollary 6.7] if $\alpha,\beta,\gamma,\sigma,\alpha\beta\in\mathbf{P}$, $h(\gamma)=h(\sigma)$ and $ \mathrm{l}( \gamma)\neq \mathrm{l}( \sigma)$ then we have the following inclusions $$\begin{array}{c}
\begin{array}{ccc}
\beta^{-1}d_{ \mathrm{l}( \beta)}M\subseteq(\alpha\beta)^{-1}d_{ \mathrm{l}( \alpha)}M, & & d_{ \mathrm{l}( \alpha)}^{-1}\alpha\beta M\subseteq d_{ \mathrm{l}( \alpha)}^{-1}\alpha M,\end{array}\\
\begin{array}{ccccc}
\alpha^{-1}d_{ \mathrm{l}( \alpha)}M\subseteq d_{ \mathrm{l}( \beta)}^{-1}\beta0, & & \gamma M\subseteq d_{ \mathrm{l}( \sigma)}^{-1}\sigma0, & & d_{ \mathrm{l}( \sigma)}M\subseteq d_{ \mathrm{l}( \sigma)}^{-1}\sigma0.\end{array}
\end{array}$$ Let $a,b\in\mathbf{A}$ and let $C$, $Ca^{-1}d_{a}$ and $Cd_{b}^{-1}b$ be homotopy words. By [@Ben2016 Corollary 6.9] we have that: if $\gamma'\in\mathbf{P}$ is longer than $\gamma\in\mathbf{P}$ and $\mathrm{f}(\gamma')=\mathrm{f}( \gamma)=a$ then $C\gamma^{-1}d_{ \mathrm{l}( \gamma)}M\subseteq C\gamma'^{-1}d_{ \mathrm{l}( \gamma')}M$; and if $\tau'\in\mathbf{P}$ is longer than $\tau\in\mathbf{P}$ and $\mathrm{l}( \tau')=\mathrm{l}( \tau)=b$ then $Cd_{ \mathrm{l}( \tau')}^{-1}\tau'M\subseteq Cd_{ \mathrm{l}( \tau)}^{-1}\tau M$.
We now define functors $C^{\pm}:\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow k\text{-}\boldsymbol{\mathrm{Mod}}$ (see [@Ben2016 Corollary 6.13]). The inclusions above are used to determine compatibility properties between these functors.
\[definition.4.14\][@Ben2016 Definition 6.12] Let $C\in\mathcal{W}_{v,\delta}$. Suppose $I_{C}$ is finite. If $a$ is an arrow and $Cd_{a}^{-1}a$ is a homotopy word let $C^{+}(M)$ be the intersection $\bigcap_{\beta} Cd_{a}^{-1}\beta\rad(M)$ over $\beta\in\mathbf{P}$ with $\mathrm{l}( \beta)=a$. By [@Ben2018 Lemma 2.1.19], if there are finitely many such $\beta$ then $C^{+}(M)= Cd_{a}^{-1}0$, and otherwise $C^{+}(M)=\bigcap_{\beta} Cd_{a}^{-1}\beta M$. If there is no such arrow $a$ we let $C^{+}(M)=CM$.
If there exists an arrow $b$ where $Cb^{-1}d_{b}$ is a homotopy word let $C^{-}(M)$ be the union $\bigcup_{\alpha} C\alpha^{-1}d_{ b}M$ over all $\alpha\in\mathbf{P}$ with $\mathrm{f}(\alpha)=b$. Otherwise let $C^{-}(M)=C(\sum d_{c(+)}M+\sum c(-)M)$ where $c(\pm)$ runs through all arrows with head $h(C^{-1})$ and sign $\pm s(C^{-1})$.
Suppose instead $I_{C}=\mathbb{N}$. In this case let $C^{+}(M)$ be the set of all $m\in e_{v}M$ with a sequence of elements $(m_{i})\in\prod_{i\in\mathbb{N}}e_{v_{C}(i)}M$ satisfying $m_{0}=m$ and $m_{i}\in l_{i+1}^{-1}r_{i+1}m_{i+1}$ for each $i\geq0$, and let $C^{-}(M)$ be the subset of $C^{+}(M)$ where each sequence $(m_{i})$ is eventually zero.
\[rem6.10\] Let $C\in\mathcal{W}_{v,\delta}$. By [@Ben2016 Corollary 6.13] we have that:
1. the assignments $M\mapsto C^{+}(M)$, $M\mapsto C^{-}(M)$ and ($M\mapsto CM$ for when $C$ is finite) respectively define subfunctors $C^{+}$, $C^{-}$ and ($C$ for when $C$ is finite) of the forgetful functor $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow k\text{-}\boldsymbol{\mathrm{Mod}}$ such that $C^{-}\leq C^{+}$;
2. if $I_{C}$ is finite then the functor $C$ preserves small coproducts and products; and
3. the functors $C^{\pm}$ preserve small coproducts.
\[Polaroid\] [@Ben2016 Remark 7.7] If $B$ and $D$ are homotopy words such that $C=B^{-1}D$ is a $p$-periodic homotopy word then there is a homotopy word $E=l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}$ with $$\begin{array}{c}
B= r_{p}^{-1}l_{p} \dots r_{1}^{-1}l_{1}r_{p}^{-1}l_{p}\dots r_{1}^{-1}l_{1}r_{p}^{-1}l_{p}\dots\text{ and } \\
D=l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}l_{1}^{-1}r_{1}\dots
\end{array}$$ In this case we write $D=E^{\,\infty}$, $B=(E^{-1})^{\,\infty}$ and $C={}^{\infty}E{}^{\,\infty}$. For each $n\in\mathbb{Z}$ let $E_{M}(n)=\{(m,m')\in e_{v}M^{n}\oplus e_{v}M^{n}\mid m'\in Em\}$. By [@Cra2018 Lemma 4.5] there is a $k$-vector space automorphism of $E_{M}(n)^{\sharp}/E_{M}(n)^{\flat}$ defined by sending $m+E_{M}(n)^{\flat}$ to $m'+E_{M}(n)^{\flat}$ if and only if $m'\in E_{M}(n)^{\sharp}\cap(E_{M}(n)^{\flat}+E_{M}(n)m)$.
\[cmpctgen\]Suppose $M$ is a $\Sigma$-pure-injective object of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$. Suppose $E=l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}$ is a homotopy word with head $v$. Let $B=(E^{-1})^{\,\infty}$ and $D=E^{\,\infty}$, so that $C=B^{-1}D$ is the $p$-periodic homotopy $\mathbb{Z}$-word ${}^{\infty}E{}^{\,\infty}$. Then for any $n\in\mathbb{Z}$ the object $(e_{v}M^n, E_{M}(n))$ of *$k\text{-}\textbf{Rel}$* is $\Sigma$-pure-injective.
Let $I$ be a set and $K=M^{I}$. By Theorem \[sigmathenCARD\] and Lemma \[sigmacardy\] there is a cardinal $\kappa$ and a set $\mathtt{S}$ such that $K\simeq \bigoplus_{s\in\mathtt{S}} U_{s}$ where each $U_{s}$ is an indecomposable $\Sigma$-pure-injective object of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ whose corresponding structure $\mathsf{U}_{s}$ (in the category of models for the canonical language of $\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$) has cardinality at most $\kappa$. By Remark \[isosreflect\] this means $K\simeq \bigoplus_{s\in\mathtt{S}} U_{s}$ in the category $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ of complexes. As in the proof of [@Ben2016 Corollary 6.13(ii)], which is precisely the statement of Remark \[rem6.10\](ii), since $\Lambda$ is semilocal and noetherian we have that $$(e_{v}\bigoplus U^{n}_{s},E_{\bigoplus U_{s}}(n))=\bigoplus(e_{v}U^{n}_{s},E_{U_{s}}(n))\text{, and }(e_{v}K^{n},E_{K}(n))=(e_{v}M^{n},E_{M}(n))
^{I}$$ where the coproducts run over $s\in\mathtt{S}$. Altogether we have shown that $(e_{v}M^{n},E_{M}(n))
^{I}$ is isomorphic to a direct sum of objects with cardinality at most $\kappa$. By considering objects in $k\text{-}\textbf{Rel}$ as modules over the Kronecker algebra, and by the equivalence of (i) and (v) in [@JenLen1989 Theorem 8.1], this shows $(e_{v}M^n, E_{M}(n))$ is $\Sigma$-pure-injective.
Functorial filtrations. {#6}
=======================
\[defintion.7.12\][@Ben2016 Definition 8.1] Let $\Sigma$ be the set of all triples $(B,D,n)$ where $B^{-1}D$ is a homotopy word (equivalently $(B,D)\in\mathcal{W}_{v,\pm 1}\times\mathcal{W}_{v,\mp 1}$) and $n$ is an integer.
\[ass9\] In §\[6\] fix $(B,D,n),(B',D',n')\in\Sigma$ and let $C=B^{-1}D$ and $C'=B'^{-1}D'$.
\[definition.7.14\][@Ben2016 Definition 8.3] We write $C\sim C'$ if and only if $C'=C^{\pm1}[t]$ for some $t\in\mathbb{Z}$. So either ($C'=C^{\pm1}$ and $I_{C}\neq\mathbb{Z}\neq I_{C'}$) or ($C'=C^{\pm1}[t]$ and $I_{C}=I_{C'}=\mathbb{Z}$) [@Ben2018 Lemma 2.2.17] (see also [@Cra2018 Lemma 2.1]). Define the *axis* $a_{B,D}\in\mathbb{Z}$ of $(B,D)$ by $C_{\leq a_{B,D}}=B^{-1}$ and $C_{>a_{B,D}}=D$. If $I_{C}=\{0,\dots,m\}$ then $a_{D,B}=m-a_{B,D}$; if $I_{C}=\pm\mathbb{N}$ then $a_{D,B}=-a_{B,D}$; and if $I_{C}=\mathbb{Z}$ then $a_{B,D}=0$ [@Ben2018 Lemma 2.2.15].
If $C\sim C'$ let $$r(B,D;B',D')=\begin{cases}
\mu_{C}(a_{B',D'})-\mu_{C}(a_{B,D}) & \mbox{(if }C'=C\mbox{ is not a homotopy }\mathbb{Z}\mbox{-word)}\\
\mu_{C}(a_{D',B'})-\mu_{C}(a_{B,D}) & \mbox{(if }C'=C^{-1}\mbox{ is not a homotopy }\mathbb{Z}\mbox{-word)}\\
\mu_{C}(\pm t) & \mbox{(if }C'=C^{\pm1}[t]\mbox{ is a homotopy }\mathbb{Z}\mbox{-word)}
\end{cases}$$ We write $(B,D,n)\sim(B',D',n')$ if and only if $$B^{-1}D\text{ and }B'^{-1}D'\text{ are equivalent and }n'-n=r(B,D;B',D').$$ By [@Ben2018 Lemma 2.2.19] we have that $r(B,D;B',D')=-r(B',D';B,D)$ and $r(B,D;B'',D'')=r(B,D;B',D')+r(B',D';B'',D'')$ for all $(B'',D'',n'')\in\Sigma$ with $B''^{-1}D''\sim B'^{-1}D'$, and so $\sim$ is an equivalence relation. Let $\Sigma(s)$ be the set of $(B,D,n)\in\Sigma$ where $B^{-1}D$ is aperiodic, and $\Sigma(b)$ the set of such $(B,D,n)$ where $B^{-1}D$ is periodic. Note that the relation $\sim$ on $\Sigma$ restricts to an equivalence relation $\sim_{s}$ (respectively $\sim_{b}$) on $\Sigma(s)$ (respectively $\Sigma(b)$). Let $\mathcal{I}(s)\subseteq\Sigma(s)$ (respectively $\mathcal{I}(b)\subseteq\Sigma(b)$) denote a chosen collection of representatives $(B,D,n)$, one for each equivalence class of $\Sigma(s)$ (respectively $\Sigma(b)$). Let $\mathcal{I}=\mathcal{I}(s)\sqcup\mathcal{I}(b)$.
Constructive and refined functors. {#consref}
----------------------------------
[@Ben2016 Definition 8.5] Let $P=P(C)[\mu_{C}(a_{B,D})-n]$ and let $V$ and $V'$ be vector spaces with bases $(v_{\lambda}\mid\lambda\in\Omega)$ and $(v'_{\lambda'}\mid\lambda'\in\Omega')$ and let $f:V\rightarrow V'$ be $k$-linear. Define $f_{\lambda',\lambda}\in k$ by $f( v_{\lambda})=\sum_{\lambda'} f_{\lambda',\lambda}v_{\lambda'}$. If $(B,D,n)\in\mathcal{I}(b)$ then $a_{B,D}=0$, $f$ is $k[T,T^{-1}]$-linear and $T$ defines automorphisms $\varphi_{V}$ of $V$ and $\varphi_{V'}$ of $V'$ with $f\varphi_{V}=\varphi_{V'}f$.
If $(B,D,n)$ lies in $\mathcal{I}(s)$ (respectively $\mathcal{I}(b)$) we use $S_{B,D,n}$ to denote a functor $k\text{-}\boldsymbol{\mathrm{Mod}}\rightarrow\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ (respectively $k[T,T^{-1}]\text{-}\boldsymbol{\mathrm{Mod}}\rightarrow\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$), defined as follows. On objects $V$ define the homogeneous component $S^{m}_{B,D,n}(V)$ of the complex $S_{B,D,n}(V)$ in degree $m\in\mathbb{Z}$ by $P^{m}\otimes_{k}V$ (respectively $P^{m}\otimes_{k[T,T^{-1}]}V$). Define the corresponding differential $d^{m}_{S_{B,D,n}(V)}$ by $d^{m}_{P}\otimes_{k}\mathrm{id}_{V}$ (respectively $d^{m}_{P}\otimes_{k[T,T^{-1}]}\mathrm{id}_{V}$) in degree $m\in\mathbb{Z}$. Similarly we can define the map $S^{m}_{B,D,n}(f)$ of the image $S_{B,D,n}(f)$ of $S_{B,D,n}$ on a morphism $f$ by $\mathrm{id}_{P}^{m}\otimes f$.
\[cor9.6\] *[@Ben2016 Corollary 8.6]* Suppose $(B,D,n)\sim (B',D',n')$ in $\Sigma$.
1. If $C$ is aperiodic then $S_{B,D,n}\simeq S_{B',D',n'}$.
2. If $C$ is periodic and $C'=C[t]$ for some $t\in\mathbb{Z}$ then $S_{B,D,n}\simeq S_{B',D',n'}$.
3. If $C$ is periodic and $C'=C^{-1}[t]$ for some $t\in\mathbb{Z}$ then $S_{B,D,n}\simeq S_{B',D',n'}\,\mathrm{res}_{\iota} $.
For the remainder of §\[6\] fix an object $M$ of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$.
[@Ben2016 Definition 7.6] For any $n\in\mathbb{Z}$ consider the $k$-subspaces of $e_{v}M^{n}$ $$\begin{array}{c}
F_{B,D,n}^{+}(M)=M^{n}\cap\left(B^{+}(M)\cap D^{+}(M)\right),\\
F_{B,D,n}^{-}(M)=M^{n}\cap\left(B^{+}(M)\cap D^{-}(M)+B^{-}(M)\cap D^{+}(M)\right),\\
G_{B,D,n}^{\pm}(M)=M^{n}\cap\left(B^{-}(M)+D^{\pm}(M)\cap B^{+}(M)\right).
\end{array}$$ Define the quotients $F_{B,D,n}(M)$ and $G_{B,D,n}(M)$ by $$\begin{array}{cc}
F_{B,D,n}(M)=F_{B,D,n}^{+}(M)/F_{B,D,n}^{-}(M), & G_{B,D,n}(M)=G_{B,D,n}^{+}(M)/G_{B,D,n}^{-}(M).
\end{array}$$ Let $$\begin{array}{cc}
\bar{F}_{B,D,n}(M)=\bar{F}_{B,D,n}^{+}(M)/\bar{F}_{B,D,n}^{-}(M), &
\bar{G}_{B,D,n}(M)=\bar{G}_{B,D,n}^{+}(M)/\bar{G}_{B,D,n}^{-}(M)
\end{array}$$ where $$\begin{array}{cc}
\bar{F}_{B,D,n}^{\pm}(M)=F_{B,D,n}^{\pm}(M)+e_{v}\rad(M^{n}), & \bar{G}_{B,D,n}^{\pm}(M)=G_{B,D,n}^{\pm}(M)+e_{v}\rad(M^{n}).
\end{array}$$ Let $ \bar{A}^{\pm}(M)=A^{\pm}(M)+e_{h(C)}\rad(M)$ for any homotopy word $A$ with $I_{A}\subseteq\mathbb{N}$.
\[remreffun\] By [@Ben2016 Lemma 7.5] we have $$\begin{array}{c}
\bar{F}_{B,D,n}^{+}(M)=e_{v}M^{n}\cap\bar{B}^{+}(M)\cap\bar{D}^{+}(M)\text{,}\\
\bar{F}_{B,D,n}^{-}(M)=e_{v}M^{n}\cap((\bar{B}^{+}(M)\cap\bar{D}^{-}(M))+(\bar{B}^{-}(M)\cap\bar{D}^{+}(M)))\text{, and}\\
\text{and }\bar{G}_{B,D,n}^{\pm}(M)=e_{v}M^{n}\cap(\bar{B}^{-}(M)+\bar{D}^{\pm}(M)\cap\bar{B}^{+}(M)).
\end{array}$$ If $C$ is aperiodic $F_{B,D,n}$, $\bar{F}_{B,D,n}$, $G_{B,D,n}$, and $\bar{G}_{B,D,n}$ all define naturally isomorphic additive functors [@Ben2018 Lemma 2.2.7]. Furthermore note that $\mathrm{im}(q)\subseteq \mathrm{rad}(N)$ for any null-homotopic morphism $q:M\to N$ between homotopically minimal complexes of projectives, and so $F_{B,D,n}$, $\bar{F}_{B,D,n}$, $G_{B,D,n}$, and $\bar{G}_{B,D,n}$ all define functors $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow k\text{-}\boldsymbol{\mathrm{Mod}}$ [@Ben2018 Corollary 2.2.8]. As above, and by [@Cra2018 Lemma 4.5], if $C$ is periodic then $F_{B,D,n}$, $\bar{F}_{B,D,n}$, $G_{B,D,n}$, and $\bar{G}_{B,D,n}$ all define naturally isomorphic functors $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow k[T,T^{-1}]\text{-}\boldsymbol{\mathrm{Mod}}$.
Recall the involution $\mathrm{res}_{\iota}$ of $k[T,T^{-1}]\text{-}\boldsymbol{\mathrm{Mod}}$ which swaps the action of $T$ and $T^{-1}$.
\[lemma.5.7\]\[corollary.6.6\] Fix $(B,D,n)$, $(B',D',n')$, $C$ and $C'$ as in Assumption \[ass9\].
1. *[@Ben2016 Lemma 8.4]* *(*see also *[@Cra2018 Lemma 7.1])*.
1. If $C$ is aperiodic then $F_{B,D,n}\simeq F_{D,B,n}$; and
2. If $C$ is periodic then $F_{B,D,n}\simeq\mathrm{res}_{\iota,k} F_{D,B,n}$.
2. \[lemma.6.7\] *[@Ben2016 Corollary 8.6]*.
1. If $C$ is aperiodic then $G_{B,D,n}\simeq G_{B',D',n'}$.
2. If $C$ is periodic and $C'=C[t]$ for some $t\in\mathbb{Z}$ then $G_{B,D,n}\simeq G_{B',D',n'}$.
3. If $C$ is periodic and $C'=C^{-1}[t]$ for some $t\in\mathbb{Z}$ then $G_{B,D,n}\simeq\mathrm{res}_{\iota,k} \,G_{B',D',n'}$.
3. *[@Ben2016 Lemma 10.5] (*see also *[@Cra2018 Lemma 8.2])*.\[lemma.31\] Let $C$ be aperiodic and let $P=P(C)$.
1. If $i\in I$ then $\bar{F}_{C(i,1),C(i,-1),n}^{+}(P[\mu_{C}(i)-n])=\bar{F}_{C(i,1),C(i,-1),n}^{-}(P[\mu_{C}(i)-n])+kb _{i}$.
2. If $C'=C$ and $n-n'=\mu_{C}(a_{B,D})-\mu_{C}(a_{B',D'})$ then $\mathrm{id}\simeq F_{B',D',n'}\,\Xi\, S_{B,D,n}$.
3. If $(B,D,n)$ is not equivalent to $(B',D',n')$ then $\bar{F}_{B',D',n'}(P[\mu_{C}(a_{B,D})-n])=0$.
4. \[lemma.7.3-1\]*[@Ben2016 Lemma 10.6] (*see also *[@Cra2018 Lemma 8.5])*. Let $C$ be aperiodic, let $V$ be an indecomposable $k[T,T^{-1}]$-module, let $P=P(C,V)$ and let $i\in\{0,\dots,p-1\}$.
1. We have $\bar{F}_{C(i,1),C(i,-1),n}^{+}(P[\mu_{C}(i)-n])=\bar{F}_{C(i,1),C(i,-1),n}^{-}(P[\mu_{C}(i)-n])+\sum_{\lambda}k b _{i,\lambda}$.
2. If $C'=C[m]$ and $n-n'=\mu_{C}(m)$ then $\mathrm{id}\simeq F_{B',D',n'}\Xi S_{B,D,n}$; and
3. If $(B,D,n)$ is not equivalent to $(B',D',n')$ then $\bar{F}_{B',D',n'}(P[-n])=0$.
\[sec:Linear-compactness-and\]Compactness and Covering.
-------------------------------------------------------
\[lemma.5.2-1\]*[@Ben2018 Lemma 2.4.1]*. Suppose $M$ is $\Sigma$-pure-injective. Let $r\in\mathbb{Z}$ and $\delta=\pm1$. Let $U$ be an $k$-subspace of $e_{v}M^{r}$ with $e_{v}\rad(M^{r})\subseteq U$.
1. *(*See also *[@Cra2018 Lemma 10.4])*. If $H$ is a linear variety in $e_{v}M^{r}$ and $m\in H\setminus U$, then there is a homotopy word $C\in\mathcal{W}_{v,\delta}$ such that $H\cap(U+m)$ meets $C^{+}(M)$ but not $C^{-}(M)$.
2. *(*See also *[@Cra2018 Lemma 10.5])*. If $m\in e_{v}M^{r}\setminus U$ then there are words $B\in\mathcal{W}_{v,\delta}$ and $D\in\mathcal{W}_{v,-\delta}$ such that $U+m$ meets $G_{B,D,r}^{+}(M)$ but not $G_{B,D,r}^{-}(M)$.
The proof of Lemma \[lemma.5.2-1\] is given at the end of §\[sec:Linear-compactness-and\].
[@Ben2018 Definition 2.1.17] If $A=\dots l_{i}^{-1}r_{i}\dots$ is a homotopy word and $i\in I_{A}$, let $A_{i}=l_{i}^{-1}r_{i}$ and $A_{\leq i}=\dots l_{i}^{-1}r_{i}$ given $i-1\in I_{A}$, and otherwise $A_{i}=A_{\leq i}= 1 _{h(A),s(A)}$. Similarly let $A_{>i}=l_{i+1}^{-1}r_{i+1}\dots$ given $i+1\in I_{A}$ and otherwise $A_{>i}= 1 _{h(A^{-1}),-s(A^{-1})}$. In the same way we can define the homotopy words $A_{<i}$ and $A_{\geq i}$ with $A_{\leq i}=A_{<i}A_{i}$ and $A_{i}A_{>i}=A_{\geq i}$.
\[lemma.5.3\]*[@Ben2018 Lemma 2.4.8] (*see also *[@Cra2018 Lemma 10.3])*. Fix an integer $r$ and some $\delta\in\{\pm1\}$. For any non-empty subset $S$ of $e_{v}M^{r}$ which does not meet *$\rad(M)$* there is a homotopy word $C\in\mathcal{W}_{v,\delta}$ such that either:
1. $C$ is finite and $S$ meets $C^{+}(M)$ but not $C^{-}(M)$; or
2. $C$ is a homotopy $\mathbb{N}$-word and $S$ meets $C_{\leq n}M$ but not $C_{\leq n}\rad(M)$ for each $n\geq0$.
In Lemma \[lemma.5.3\] we do *not* require that $M$ is an object of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{proj}})$. In Corollary \[lemma.5.3-1\] we *do* consider this setting, and show $S\cap C^{+}(M)\neq\emptyset =S\cap C^{-}(M)$ in case (ii) of Lemma \[lemma.5.3\]. Recall Definition \[nesw\].
\[noice\]*(Realisation)* If $C\in \mathcal{W}_{u,\delta}$ and $I_{C}=\{0,\dots,t\}$ then there is some $n\in I_{C(\swarrow,\searrow)}$ such that $(C(\swarrow,\searrow)_{>n})_{\leq n+t}=C$ and $$CM=\{f(b_{n,C(\swarrow,\searrow)})\mid f:P(C(\swarrow,\searrow))\to M\text{ in }\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})\}.$$
Without loss of generality assume $C$ is non-trivial, say $C=l_{1}^{-1}r_{1}\dots l_{t}^{-1}r_{t}$. Let $I= I_{C(\swarrow,\searrow)}$, the subset of $\mathbb{Z}$ such that $C(\swarrow,\searrow)$ is a homotopy $I$-word. Let $I_{-}$ and $I_{+}$ be the subsets of $\mathbb{N}$ for which $(C_{\swarrow})^{-1}$ is a homotopy $I_{-}$-word and $C_{\searrow}$ is a homotopy $I_{+}$-word. If $I_{\pm}$ is finite then we let $I_{\pm}=\{ 0,\dots, n(\pm)\}$. If $I\subseteq \mathbb{N}$ then $I_{-}$ is finite, and we let $n= n(-)$. If $I=-\mathbb{N}$ then $I_{+}$ is finite, and we let $n=-t-n(+)$. If $I=\mathbb{Z}$ let $n=0$. We firstly show $CM\subseteq X$ where we let $$X=\{f(b_{n,C(\swarrow,\searrow)})\mid f:P(C(\swarrow,\searrow))\to M\text{ in }\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})\}.$$ Let $m\in CM$, and so there exists $m_{n},\dots,m_{n+t}\in M$ such that $m=m_{n}$ and $l_{i}m_{i}=r_{i}m_{i+1}$ for all $i$ with $n\leq i<t$. To show $m\in X$ it suffices to construct a sequence $(m_{i}\mid i\in I)$ of elements $m_{i}\in M$ such that $b_{i,C(\swarrow,\searrow)}\mapsto m_{i}$ defines a morphism of complexes $f:P(C(\swarrow,\searrow))\to M$. Note that $n-h,n+t+j\in I$ for all $h\in I_{-}$ and all $j\in I_{+}$. We begin by iteratively constructing $m_{n-h}\in M$ for all $h\in I_{-}$ and $m_{n+t+j}\in M$ for all $j\in I_{+}$, noting that $m_{n}$ and $m_{n+t}$ have already been defined. Suppose that $h\in I_{-}$, $m_{n-h}$ has been defined and that $h+1\in I_{-}$. By construction $((C_{\swarrow})^{-1})_{h+1}=d_{\gamma}^{-1}\gamma$ for some arrow $\gamma$. Furthermore $\mathrm{im}(d_{\gamma,M})\subseteq \gamma M$, and we choose $m_{n-(h+1)}\in e_{t(\gamma)}M$ such that $d_{\gamma,M}(m_{n-h})=\gamma m_{n-(h+1)}$.
Similarly if ($j\in I_{+}$, $m_{n+t+j}$ has been defined and $j+1\in I_{+}$) then $(C_{\searrow})_{j+1}=d_{\beta}^{-1}\beta$ for some arrow $\beta$, and we choose $m_{n+t+(j+1)}\in e_{t(\beta)}M$ such that $d_{\beta,M}(m_{n+t+j})=\beta m_{n+t+(j+1)}$. It is straightforward to check that $f(b_{i,C(\swarrow,\searrow)})= m_{i}$ ($i\in I$) satisfies $f(b^{+}_{i,C(\swarrow,\searrow)}+b^{-}_{i,C(\swarrow,\searrow)})= d_{M}(m_{i})$ for all $i$. This is done by separting the cases $i<n$, $i=n$, $n<i<n+t$, $i=n+t$ and $i>n+t$. The cases $i<n$ and $i>n+t$ are similar. As are the cases $i=n$ and $i=n+t$. This shows $CM\subseteq X$. The proof that $X\subseteq CM$ is similar, easier, and omitted.
\[lemma.5.3-1\]If $M$ is $\Sigma$-pure-injective $I_{C}=\mathbb{N}$ then $C^{+}(M)=C_{\leq l}M$ for some $l>0$.
Recall the canonical language $\mathfrak{L}=\mathfrak{L}^{\mathcal{T}}$ of the compactly generated triangulated category $\mathcal{T}=\mathcal{K}(\Lambda\text{-}\mathbf{Proj})$. Recall that for any compact object $G$ of $\mathcal{T}$ a subgroup of finite definition of $M$ of sort $G$ has the form $Ma$, and is defined as the set of morphisms $fa\in\mathcal{T}(G,M)$ where $f$ runs through $\mathcal{T}(G,M)$, $a\in\mathcal{T}(G,H)$ is fixed and $H$ is compact.
Let $C=l_{1}^{-1}r_{1}l_{2}^{-1}r_{2}\dots$. If $l_{1}^{-1}r_{1}$ has the form $\tau^{-1}d_{\mathrm{l}(\tau)}$ then let $I=\{0\}$ and $B=1_{u,\delta}$ and $D=C_{>1}$ so that $C=B\tau^{-1}d_{\mathrm{l}(\tau)} D$. Otherwise there is a non-trivial homotopy $I$-word $B$ such that $B_{\leq s}$ has the form $d_{\mathrm{l}(\gamma(1))}^{-1}\gamma(1)\dots d_{\mathrm{l}(\gamma(s))}^{-1}\gamma(s)$ whenever $0<s\in I$, and either ($I=\mathbb{N}$ and $B=C$) or ($I=\{0,\dots,p\}$ and $C_{\leq p+1}=B\tau^{-1}d_{\mathrm{l}(\tau)}$). Let $G=P(B(\swarrow))$ and $H_{n}=P((C_{\leq n})(\swarrow,\searrow))$ for each $n$. We now define, for sufficiently large $s>0$, a morphism of complexes $a_{s}:G\to H_{s}$.
Suppose firstly $I=\{0,\dots,p\}$, so we have that $C_{\leq p}=B$ and that $l_{p+1}^{-1}r_{p+1}$ has the form $\tau^{-1}d_{\mathrm{l}(\tau)}$. So for any $s>p$ there is a morphism $a_{s}:G\to H_{n}$ given by $b_{i,B(\swarrow)}\mapsto b_{i,(C_{\leq s})(\swarrow,\searrow)}$ for any $i \in I_{B(\swarrow)}$. Suppose instead $I=\mathbb{N}$, in which case we let $p=0$. In this case $(C_{\leq s})(\swarrow,\searrow))=C(\swarrow)=B(\swarrow)$ and hence $G=H_{s}$ for all $s$, and we let $a_{s}$ be the identity $G\to G$. Consider the finite word $C_{\leq p}$. Applying Lemma \[noice\] gives some $n\in I_{C(\swarrow,\searrow)}$ such that, for any $s\geq p+1$, $C_{\leq s}M$ is the set of images $f(a_{s}(b_{n,B(\swarrow)}))$ as $f$ runs through morphisms of complexes $H_{s}\to M$.
Note that $G$ and $H_{n}$ are compact objects of $\mathcal{T}$ by Proposition \[strngcmpct\]. So the above may be rewritten as $C_{\leq n}M=\{g(b_{0,C_{\leq n}})\mid g\in Ma_{n}\}$. Let $\varphi_{n}(v_{G})$ be the pp-formula in $\mathfrak{L}$ given by $\exists u_{H_{n}}:v_{G}=fu_{H_{n}}$. Since $C_{\leq n+1}M\subseteq C_{\leq n}M$ for each $n$, we have that $
Ma_{p+1}=\varphi_{p+1}(M)\supseteq Ma_{p+2}=\varphi_{p+2}(M)\supseteq\dots$ is a descending chain of pp-definable subgroups of $M$ of sort $G$. By Lemma \[sigmathendcc\] this chain stabilises. As above this must mean the chain $C_{\leq 1}M\supseteq C_{\leq 2}M\supseteq\dots$ stabilises, as required.
\(i) Let $S=H\cap(U+m)$. Note $S\cap\rad(M)=\emptyset$ since $e_{v}\rad(M^{r})\subseteq U$ and $m\notin U$. So by Lemma \[lemma.5.3\] there is a homotopy word $C$ such that either $C$ is finite and $S\cap C^{+}(M)\neq\emptyset=S\cap C^{-}(M)$, or $C$ is a homotopy $\mathbb{N}$-word and for all $n\geq0$ we have $S\cap C_{\leq n}M\neq\emptyset=S\cap C_{\leq n}\rad(M)$. We may assume $I_{C}=\mathbb{N}$, and so $S\cap C^{+}(M)=S\cap C_{\leq l}M\neq\emptyset$ for some $l>0$ by Corollary \[lemma.5.3-1\], as required.
\(ii) The argument in the proof of [@Cra2018 Lemma 10.5] adapts with few complications.
Local and global mapping properties. {#Local Mapping Properties.}
------------------------------------
Recall: $\Sigma(s)$ (respectively $\Sigma(b)$) is the set of triples $(B,D,n)\in\Sigma$ where $B^{-1}D$ is aperiodic (respectively periodic); $\mathcal{I}(s)$ (respectively $\mathcal{I}(b)$) is a collection of representatives $(B,D,n)$; and $\mathcal{I}=\mathcal{I}(s)\sqcup\mathcal{I}(b)$ (see Definition \[definition.7.14\]).
\[lemma.7.4\]*[@Ben2016 Lemma 2.5.2] (*see also *[@Cra2018 Lemma 8.3])*. If $(B,D,n)\in\mathcal{I}(s)$ and $\mathcal{B}=(\bar{u}_{\lambda}\mid\lambda\in\Omega)$ is a $k$-basis of $F_{B,D,n}(M)$ then there is a morphism of complexes $\theta_{B,D,n,M}:\bigoplus_{\lambda}P(C)[\mu_{C}(a_{B,D})-n]\rightarrow M$ such that $F_{B,D,n}(\theta_{B,D,n,M})$ is an isomorphism.
Recall Definition \[Polaroid\]: if $E$ is a homotopy $\{0,\dots,p\}$ word such that $p>1$ and $C={}^{\infty}E^{\infty}$ is a $p$-periodic then the relation $E_{M}(n)$ on $e_{v}(M)$ is defined to be the set of pairs $(m,m')$ with $m'\in Em$. Lemma \[lemma.7.4-1\] below will be applied in the context of $\Sigma$-pure-injective complexes.
\[lemma.7.4-1\]*[@Ben2016 Lemma 12.4] (*see also *[@Cra2018 Lemma 8.6])*. Suppose $(B,D,n)\in\mathcal{I}(b)$, say where $C={}^{\infty}E{}^{\,\infty}$ is periodic of period $p>0$ and $E=l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}$. If the relation $E_{M}(n)$ on $e_{v}M^{n}$ is split then there is a morphism $\theta_{B,D,n,M}:P(C,U)[-n]\rightarrow M$ of complexes such that $F_{B,D,n}(\theta_{B,D,n,M})$ is an isomorphism.
Note that the statements of [@Ben2016 Lemma 12.4] and Lemma \[lemma.7.4-1\] are equivalent: any relation $(V,C)$ that *admits a reduction which meets in* $0$ must have been split by [@Ben2018 Corollary 1.4.33].
\[lemma.7.5\]*[@Ben2016 Lemma 12.5] (*see also *[@Cra2018 Lemma 10.5]* and *[@ButRin1987 p. 163])*. Let $\theta:P\rightarrow M$ be a morphism in *$\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$* where $M$ is $\Sigma$-pure-injective. If $F_{B,D,n}(\theta)$ is surjective for each $(B,D,n)\in\Sigma$ then $\theta^{i}$ is surjective for each $i$.
For a contradiction suppose that $\theta^{i}$ is not surjective for some $i\in\mathbb{Z}$. Since $\Lambda$ is perfect $M^{i}$ is a projective cover of $M/\mathrm{rad}(M^{i})$, and so $\mbox{rad}(M^{i})$ is a superfluous submodule of $M^{i}$. This means $e_{v}\mbox{im}(\theta^{i})+e_{v}\mbox{rad}(M^{i})$ is contained in a maximal $k$-subspace $U$ of $e_{v}M^{i}$. Since $e_{v}\mbox{rad}(M^{i})\subseteq U$ and $U\neq e_{v}M^{i}$, by Lemma \[lemma.5.2-1\](ii) for some element $m\in e_{v}M^{i}\setminus U$ there are homotopy words $B\in\mathcal{W}_{v,\delta}$ and $D\in\mathcal{W}_{v,-\delta}$ for which ($B^{-1}D$ is a homotopy word) and $U+m$ meets $G_{B,D,i}^{+}(M)$ but not $G_{B,D,i}^{-}(M)$. From here one can show $F_{B,D,i}(\theta)$ is not surjective by adapting the argument from the proof of [@Cra2018 Lemma 10.6].
\[ass12.5\] For Lemma \[lemma.7.1-1\] we fix a direct sum $N$ of shifts of string and band complexes as follows. Let $\mathcal{S}$ and $\mathcal{B}$ be sets, $\{t(\sigma),s(\beta)\mid\sigma\in\mathcal{S},\beta\in\mathcal{B}\}$ be a set of integers, $\{V^{\beta}\mid\beta\in\mathcal{B}\}$ be a set of objects from $k[T,T^{-1}]\text{-\textbf{Mod}}$ and $\{A(\sigma),E(\beta)\mid\sigma\in\mathcal{S},\beta\in\mathcal{B}\}$ be a set of homotopy words, where each $A(\sigma)$ is aperiodic and each $E(\beta)$ is $p_{\beta}$-periodic. Let $$N=\Bigl( \bigoplus_{\sigma\in\mathcal{S}}P(A(\sigma))[-t(\sigma)]\Bigr)\oplus\Bigl( \bigoplus_{\beta\in\mathcal{B}}P(E(\beta),V^{\beta})[-s(\beta)]\Bigr)$$
*[@Ben2018 Lemma 2.5.6] (*see also *[@Cra2018 Lemma 9.4])*. \[lemma.7.1-1\] Let $N$ be the direct sum of string and band complexes from Assumption \[ass12.5\]. Let $\theta:N\rightarrow M$ be a map in $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ where $\bar{F}_{B,D,n}(\theta)$ is injective for all $(B,D,n)\in\mathcal{I}$. Then each $\theta^{i}$ is injective.
\[sec:Proofs-of-the\]Completing the proof of the main theorem.
==============================================================
In §\[sec:Proofs-of-the\] we let $\mathfrak{I}$ be a set, $\Xi:\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\to\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ be the quotient and $S_{i}:\mathcal{A}_{i}\rightarrow\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ and $F_{i}:\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow\mathcal{A}_{i}$ ($i\in\mathfrak{I}$) be additive functors.
\[detect\](See [@Ben2016 Definition 4.4]). Let $\mathcal{Z}$ be a full subcategory of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$. We say that $\{(S_{i},F_{i})\mid i\in\mathfrak{I}\}$ *detects objects* in $\mathcal{Z}$ if the following statements hold.
1. For any $i\in\mathfrak{I}$:
1. the functor $F_{i}\Xi S_{i}$ is dense and reflects isomorphisms;
2. $F_{j}\Xi S_{i}\simeq0$ for each $j\in\mathfrak{I}$ with $j\neq i$;
3. $F_{i}$ preserves small coproducts; and
4. for each object $M$ in $\mathcal{Z}$ there exists an object $A_{i,M}$ in $\mathcal{A}_{i}$ and a morphism $\gamma_{i,M}:\Xi(S_{i}(A_{i,M}))\rightarrow M$ in $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ such that $F_{i}(\gamma_{i,M})$ is an isomorphism.
2. For all morphisms $\theta:N\rightarrow M$ in $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$:
1. if $M$ lies in $\mathcal{Z}$ and $F_{i}(\Xi(\theta))$ is epic for all $i\in\mathcal{I}$ then each $\theta^{n}$ is epic; and
2. if $N=\bigoplus_{i\in\mathfrak{I}}S_{i}(A_{i})$ and $F_{i}(\Xi(\theta))$ is monic for each $i\in\mathcal{I}$ then each $\theta^{n}$ is monic.
\[lemma\]*(*See *[@Ben2016 Lemma 4.5])*. Let $\mathcal{Z}$ be a full subcategory of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ where $\{(S_{i},F_{i})\mid i\in\mathfrak{I}\}$ detects objects in $\mathcal{Z}$. Any object $M$ of $\mathcal{Z}$ is isomorphic to $\bigoplus_{i\in\mathfrak{I}}\Xi(S_{i}(A_{i,M}))$.
Note that Definition \[detect\] and Lemma \[lemma\] are essentially [@Ben2016 Definition 4.4, Lemma 4.5], the difference being we have replaced the category $\mathcal{P}_{\mathcal{N}}$ of $\Lambda\text{-}\boldsymbol{\mathrm{Proj}}$ from [@Ben2016] with an arbitrary full subcategory $\mathcal{X}$ of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$. The proof of [@Ben2016 Lemma 4.5] generalises to our setting with no further complications. We now verify the hypotheses of Lemma \[lemma\].
Recall, from Definition \[definition.7.14\], the equivalence relation on the triples $(B,D,n)$ where $B^{-1}D$ is a homotopy word and $n\in\mathbb{Z}$. Recall that $\mathcal{I}=\mathcal{I}(s)\sqcup \mathcal{I}(b)$ where $\mathcal{I}(s)$ (respectively $\mathcal{I}(b)$) denotes a chosen set of representatives $(B,D,n)$ such that $B^{-1}D$ is aperiodic (respectively periodic).
Recall that if $(B,D,n)$ lies in $\mathcal{I}(s)$ (respectively $\mathcal{I}(b)$) then the functor $S_{B,D,n}$ has the form $k\text{-}\mathbf{Mod}\rightarrow \mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$ (respectively $k[T,T^{-1}]\text{-}\mathbf{Mod}\rightarrow\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})$), and the functor $F_{B,D,n}$ has the form $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})\rightarrow k\text{-}\mathbf{Mod}$ (respectively $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\mathbf{Proj})\rightarrow k[T,T^{-1}]\text{-}\mathbf{Mod}$). Recall Definition \[detect\].
\[proposition6\]*[@Ben2016 Proposition 13.1] (*see also *[@ButRin1987 p. 163, Proposition])*. Let $\mathcal{M}=\Lambda\text{-}\boldsymbol{\mathrm{Mod}}$ and $\mathfrak{I}=\mathcal{I}(s)\sqcup\mathcal{I}(b)$; and for each $i=(B,D,n)\in\mathfrak{I}$ let $$\mathfrak{A}_{i}=\left\{ \begin{array}{cc}
(k\text{-}\boldsymbol{\mathrm{Mod}}) & (\text{if }B^{-1}D\text{ is aperiodic}),\\
(k[T,T^{-1}]\text{-}\boldsymbol{\mathrm{Mod}}) & (\text{if }B^{-1}D\text{ is periodic}).
\end{array}\right.$$ The collection $\{(S_{B,D,n},F_{B,D,n})\mid(B,D,n)\in\mathcal{I}\}$ detects the objects in the full subcategory $\mathcal{Z}$ of $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ consisting of $\Sigma$-pure-injective objects.
In the proof of Proposition \[proposition6\](ii) we verify the conditions FFI, FFII, FFIII, FFIV, FFV and FFVI from Definition \[detect\]. Later we use Proposition \[proposition6\] in the context of Lemma \[lemma\].
\(i) Since $\Lambda$ is a perfect ring every $\Lambda$-module has a projective cover.
\(ii) FFI) Let $(B,D,n)\in\Sigma$ and $B^{-1}D=C$. If $C$ is aperiodic (respectively periodic) then by Lemma \[lemma.5.7\](iiib) (respectively Lemma \[lemma.5.7\](ivb)) we have $F_{B,D,n}\,\Xi\, S_{B,D,n}\simeq \mathrm{id}$.
FFII) Let $(B',D',n')\in\mathcal{I}$. If $(B',D',n')\neq (B,D,n)\in\mathcal{I}(s)$ then $\bar{F}_{B',D',n'}(P(C)[\mu_{C}(a_{B,D})-n])=0$ by Lemma \[lemma.5.7\](iiic) where $C=B^{-1}D$. This shows $F_{B',D',n'}\Xi S_{B,D,n}=0$ since $\bar{F}_{B',D',n'}\simeq F_{B',D',n'}$. If $(B,D,n)\in\mathcal{I}(b)$ then the proof is similar and uses Lemma \[lemma.5.7\](ivc).
FFIII) By Remark \[rem6.10\] each of the subfunctors $C^{\pm}$ of the forgetful functor $\mathcal{C}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})\rightarrow k\text{-}\boldsymbol{\mathrm{Mod}}$ commutes with arbitrary direct sums. It follows that $F^{\pm}_{B,D,n}$ commutes with direct sums of objects in $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$ (see [@Ben2018 Lemma 2.1.21] for details).
FFIV) Let $M$ be an object in $\mathcal{Z}$. If $(B,D,n)$ lies in $\mathcal{I}(s)$, by Lemma \[lemma.7.4\] there is a vector space $U$ and a morphism $\theta_{B,D,n}:S_{B,D,n}(U)\rightarrow M$ for which $F_{B,D,n}(\theta_{B,D,n})$ is an isomorphism. Now suppose instead $(B,D,n)$ lies in $\mathcal{I}(b)$, say where $B^{-1}D={}^{\infty}E{}^{\,\infty}$ is periodic of period $p>0$, and where $E=l_{1}^{-1}r_{1}\dots l_{p}^{-1}r_{p}$.
Note $F_{B,D,n}^{+}(M)=E(n)_{M}^{\sharp}$ and $F_{B,D,n}^{-}(M)=E(n)^{\flat}$, so $E(n)^{\sharp}/E(n)^{\flat}=F_{B,D,n}(M)$. By Lemma \[cmpctgen\] the object $(e_{v}M^n, E_{M}(n))$ of $k\text{-}\textbf{Rel}$ is $\Sigma$-pure-injective. By Corollary \[meandbill\] the relation $(e_{v}M^n, E_{M}(n))$ is split. The required morphism exists by Lemma \[lemma.7.4-1\].
FFV, FFVI) Let $\theta:N\rightarrow M$ be an morphism in the category $\mathcal{K}_{\mathrm{min}}(\Lambda\text{-}\boldsymbol{\mathrm{Proj}})$. If $M$ is an object of $\mathcal{Z}$ and $F_{B,D,n}(\theta)$ is epic for all $(B,D,n)\in\mathcal{I}$ then $\theta^{n}$ is epic for each $n\in\mathbb{N}$ by Lemma \[lemma.7.5\]. This shows FFV holds, and similarly FFVI holds by Lemma \[lemma.7.1-1\].
The statement of Theorem \[maincor\] is precisely the statement of Lemma \[lemma\] after applying Proposition \[proposition6\].
|
---
abstract: 'A procedure is proposed which accounts for final state interaction corrections for near threshold meson production in nucleon-nucleon scattering. In analogy with the Watson-Migdal approximation, it is shown that in the limit of extremely strong final state effects, the amplitude factorizes into a primary production amplitude and an elastic scattering amplitude describing a $3 \to 3$ transition. This amplitude determines the energy dependence of the reaction cross section near the reaction threshold almost solely. The approximation proposed satisfies the Fermi-Watson theorem and the coherence formalism. Application of this procedure to meson production in nucleon-nucleon scattering shows that, while corrections due to the meson-nucleon interaction are small for s-wave pion production, they are crucial for reproducing the energy dependence of the $\eta$ production cross section.'
author:
- |
A. Moalem $^{}$, L. Razdolskaja $^{}$ and E. Gedalin $^{}$\
\
date: 3 June 1994
title: ' $FSI$ Corrections For Near Threshold Meson Production In Nucleon-Nucleon Collisions'
---
3[fm$^3$]{} 3[fm$^{-3}$]{}
[BGU-94/04\
hep-ph/9505264]{}
PACS number(s) : 13.75.Cs, 14.40.Aq, 25.40.Ep
Introduction
============
The cross sections for $\pi^0$ and $\eta$ meson production via the $pp \to pp \pi^0$ and $pp \to pp \eta$ reactions, at energies near their respective thresholds, exhibit a pronounced energy dependence which deviates strongly from phase space[@meyer; @bergd]. Such deviations, which occur in other hadronic collisions as well[@wilkin], are most certainly due to final state interactions($FSI$) between the reaction products. While $FSI$ corrections due to the short range nucleon-nucleon($NN$) force between the outgoing protons seem adequate to reproduce the observed energy dependence[@meyer] and the cross section scale[@lee] for pion production, they fail to do so for the $\eta$ production[@bergd]. Particularly, several model calculations[@germo; @lagt; @vett] have considered the $NN \to NN \eta$ through a perturbative approach but none of them reproduced the energy dependence near threshold, although $FSI$ corrections were made to account for the proton-proton interaction. It is the purpose of the present work to show that while the meson-nucleon interaction influences s-wave pion production only slightly, its effects are crucial for reproducing the energy dependence of the $pp \to pp \eta$ reaction.
There exists no theory of final state effects in the presence of three strongly interacting particles and our first objective will be to develop a procedure that accounts for both the $NN$ and meson-$N$ forces. To accomplish this task we consider a production process, $ab \to cde$. A full dynamical theory of such a process would require the solution of two and three-body scattering problems. Particularly, in the Faddeev formalism[@faddeev] the full transition amplitude is decomposed as a sum of three terms depending on which of the three final particle pairs interacts last. The evaluation of these terms, the so called Faddeev amplitudes, requires the solution of a set of coupled integral equations. Such a procedure should resolve the problem at hand but, in view of the scarcity of knowledge about the $\eta N$ interaction it may turn to be unreliable and rather long and tedious to apply. In what follows we develop an approximation along exactly the lines of the Watson’s approach[@watson] for two body reactions, $\ie$, we look for a separation of the energy dependence due to $FSI$ from those of the primary production amplitude. Such an approximation may prove useful and easy to apply in analyzing near threshold meson production data from $NN$ scattering.
According to the Watson-Migdal $FSI$ theory for two body processes[@watson], the energy dependence of the full transition amplitude is governed by rescattering of final state particles. The full transition amplitude is approximated by, $$T_{if} \approx M^{(in)}_{if} \ t^{(el)}_{ff}\ \ ,
\label{eq:11}$$ where the primary production amplitude, $M^{(in)}_{if}$, is assumed to be a smooth and slowly varying function of energy and $t^{(el)}_{ff}$ is taken as the $free$ two-body elastic scattering amplitude in the exit channel. Thus the energy dependence of the full matrix element is determined mostly by $t^{(el)}_{ff}$. A decomposition as such of the amplitude into a primary production amplitude and a subsequent $FSI$ correction term is meaningful only if $t^{(el)}_{ff}$ is sufficiently large so that dominate the reaction amplitude. This requires that the distortion of the final state wave function (the deviation from a simple product of free particle wave functions) or alternatively, the $sticking\ factor$ (the probability of the particles to find each other’s vicinity) in itself be proportional to $t^{(el)}_{ff}$. Albeit, the particles spend a great deal of time close together, as should be the case in particle production processes close to threshold. Otherwise, the distortion factor of the final state (and obviously the full matrix element) must be calculated by using more precise methods[@watson].
The complete transition amplitude of a production process can be decomposed in the form shown in fig. 1, where the first term (the block $M^{(in)}$) represents the primary production amplitude which includes all possible inelastic transitions contributing to the process, while all other terms represent the contributions from rescatterings in the entrance and exit channels (the and blocks). The third term, for example, corresponds to a process in which the primary production takes place as if there were no $FSI$ and only subsequently is distorted by the Coulomb and nuclear short range interactions between the reaction products. At energies near the reaction threshold, where the relative energies of particle pairs in the final state are sufficiently small and corrections are dominating, such a term is expected to be more important than others. In analogy to the Watson-Migdal approximation we make the $ansatz$ that in the limit of weak and strong , the transition amplitude of a three-body reaction can be approximated by an expression similar to eqn. 1, with the correction term being replaced by the on mass-shell $elastic$ scattering amplitude of a $3 \to 3$ (three particles in to three particles out) transition. This amplitude is denoted by $T^{(el)}_{33}$ and, can be evaluated using the Faddeev formalism[@thomas] or others such as the Weinberg method[@weinberg]. Certainly, it is not a measurable quantity as such a $3 \to 3$ transition is not easy to realize experimentally. It is shown in sect. 2 though, that at energies near particle production threshold, $T^{(el)}_{33}$ can be estimated from two-body scattering data without having to solve the full three-body dynamics.
Within this approximation, the transition amplitude of a three-body reaction is coherent in terms of the interactions between the various final particle pairs, just as one anticipates based on general quantum mechanical considerations[@amado]. Thus the effects due to a given pair’s interaction is distributed over the entire transition amplitude and may influence strongly the energy dependence of the cross section. In sect. 3 we apply the procedure developed to production of mesons in $NN$ scattering and show that although the $\eta-N$ interaction is rather weak with respect to the $NN$ interaction it makes remarkably strong contributions. Discussion of the results and conclusions are given in sect. 4.
Theoretical Perspective
=======================
2.1 Two-body reactions {#two-body-reactions .unnumbered}
----------------------
Prior to extending the Watson-Migdal approximation to three-body reactions let us recall first the more familiar problem of corrections in two-body reactions. Assume that the full interaction between particles separates into $v\ =\ w\ +\ s$, where $w$ and $s$ stand for weak and strong terms respectively. Here $w$ is the primary interaction, such that if it were zero, the process in question would not occur. The remaining part of the interaction $s$ is responsible for rescatterings in the entrance and exit channels. We further assume that $s$ is the only strong channel energetically allowed.
We would like to find how $FSI$ affects the transition amplitude in a process in which the primary interaction is relatively weak. The full transition amplitude in an appropriate eigenchannel is defined as, $$T_{if} = \langle \Psi^{(-)}_{el,f}
|\ \hat{M}^{(in)}_{if}\ | \Psi^{(+)}_{el,i}\rangle \ \ ,
\label{eq:21}$$ where $\Psi^{(\pm)}_{el,i(f)}$ are two-particle scattering wave functions. They correspond to solutions of the equation with a Hamiltonian that contains strong $s$-interactions only and satisfy the boundary conditions that $\Psi^{(+)}_{el,i}$ ($\Psi^{(-)}_{el,f}$) tends asymptotically to free two-particle wave function $\phi_{oi}$ ($\phi_{of}$), plus outgoing (ingoing) spherical waves. These can be written in the form[@thomas], $$\Psi^{(\pm)}_{el,\lambda}(E) =
\lim_{\epsilon \to 0} \ [\ 1\ +\ G_{02} (E \pm i\epsilon)\
\hat{t}^{(el)}_{\lambda \lambda}]\ \phi_{0\lambda} \ \ , \lambda = i,\ f\ \ .
\label{eq:22}$$ Here $G_{02} (E \pm i\epsilon)$ is a free two-body Green’s functions, and $t^{(el)}_{ii}$ and $t^{(el)}_{ff}$ are two-body elastic scattering amplitude for the entrance and exit channels, respectively. Substituting eqn. 3 in 2 leads to the full transition amplitude, $$\hat{T}_{if} = \hat{M}^{(in)}_{if} + \hat{t}^{(el)}_{ii} G_{02}
\hat{M}^{(in)}_{if} +
\hat{M}^{(in)}_{if} G_{02} \hat{t}^{(el)}_{ff} +
\hat{t}^{(el)}_{ii} G_{02} \hat{M}^{(in)}_{if} G_{02}
\hat{t}^{(el)}_{ff}\ \ .
\label{eq:210}$$ Note that this expression is exact and contains both and . In the Watson-Migdal approximation, $$T_{if} \approx \langle \Psi^{(-)}_{el,f}\ |\ \phi_{0f}\rangle
\langle\ \phi_{0f}\
|\ \hat{M}^{(in)}_{if}\ | \phi_{0i}\rangle \langle\
\phi_{0i} |\Psi^{(+)}_{el,i}\rangle \ \ ,
\label{eq:23}$$ where the two overlapping integrals $\langle\phi_{0i} |\Psi^{(+)}_{el,i}\rangle$ and $\langle \Psi^{(-)}_{el,f}\ |\ \phi_{0f}\rangle$ are on mass-shell $S$ matrix elements which contain and corrections, respectively. To cast the amplitude, eqn. 5, in the form given by the Watson’s approximation, eqn. 1, one has to replace $\Psi^{(+)}_{el,i}$ by $\phi_{0i} $ so that $S_{ii}$ is equal to unity and, in the limit of extremely strong , take the on mass-shell $t^{(el)}_{ff}$ instead of the $S_{ff}$ matrix elements.
It would be instructive to repeat these arguments using a diagrammatic language. We associate the weak primary interaction $w$ with the diagram $2a$, and the strong $s$ interaction with the diagram 2b of figure 2. As usual the disconnected diagram $2c$ describes noninteracting particles. It is evident that the amplitude $T_{if}$ is the sum of all diagrams that can be constructed from the elementary diagrams $2a$ and $2b$. To single out the and blocks and obtain the full matrix element in a diagrammatic representation similar to the one of fig. 1, we first sum in the initial and final channels all diagrams that contain strong interactions only (see fig. 3). These sums yield the elastic amplitudes $t^{(el)}_{\lambda \lambda}$ ($\lambda = i,\ f$) which we identify with the appropriate and blocks, respectively. The sum of all diagrams that start or end with a weak interaction (diagram $2a$) forms the block $M^{(in)}$ which describes the $primary$ production process. When rescatterings in both of the initial and final states give sufficiently large amplitudes, the Migdal-Watson ideology tells us that the 4th term on the rhs of eqn. 4 is dominating. When $FSI$ ($ISI$) effects are small, then the second (third) term dominates. Thus the $ISI$ and $FSI$ effects can be separated from the full transition amplitude by summing diagrams in a specific order and subsequently derive the Watson-Migdal approximation by selecting an appropriate dominant term in the limit of extremely weak and extremely strong . We apply this same reasoning to obtain the Watson-Migdal approximation for three-body reactions.
2.2 Three-body reactions {#three-body-reactions .unnumbered}
------------------------
Let $T_{23}$ be the transition amplitude of a production process in which an initial two-body process goes into a final three-body state. For conciseness assume all particles to be distinguishable and spinless. This restriction simplifies the mathematics but can be easily removed to treat more general cases. Furthermore, assume that each pair in the three-body final state has a given angular momentum so that a unique elastic scattering state is assigned to each pair simultaneously. This is very often the case at energies near threshold where, only one partial wave dominates for each pair namely, $\lambda = 0$. Following the discussion given above for two-body reactions the interactions are divided into two categories: (i) elastic interactions leading to rearrangement of particle (inner) quantum numbers but not to the production of any additional hadrons, (ii) inelastic interactions which are responsible for the production (or annihilation) of particles so that particle numbers in the initial and final states are different. Likewise, we call elastic those diagrams which involve elastic interactions only and having equal number of legs to the left hand and to the right hand sides. Those which begin and end with inelastic interactions and having different number of legs to the left and right hand sides are called inelastic.
The transition amplitude $T_{23}$ of the reaction under considerations is the sum of all diagrams which start with two legs on the left and end with three legs on the right, that can be assembled using all possible elastic and inelastic diagrams. To decompose $T_{23}$ into four terms as depicted in fig. 1, these diagrams need be organized in a specific order before suming them. First note that by summing all two-particle elastic diagrams in the entrance channel and all three-particle elastic diagrams in the exit channel, one obtains the $2 \to 2$ and $3 \to 3$ $elastic$ transition amplitude of fig. 3. These elastic amplitudes which are denoted by $t^{(el)}_{22}$ and $T^{(el)}_{33}$, contain only elastic interactions and as such, they are only parts of the complete $2 \to 2$ and $3 \to 3$ amplitudes. Although at energies above particle production threshold, two and three-particle channels may well be opened in both the initial and final states, these two amplitudes are calculated by taking into account only elastic interactions. All inelastic and quasi-elastic processes are confined into $M^{(in)}$. Note also, that the $2 \to 2 $ elastic block contains connected diagrams only but, the $3 \to 3$ block contains disconnected diagrams as well (see fig. 3).
The set of all diagrams contributing to the entire transition amplitude $T_{23}$, is now separated into four partial sums each corresponding to one of the terms in the block diagram of fig. 1 : (i) the block $M^{(in)}$ is the sum of all inelastic diagrams only, (ii) the -$M^{(in)}$ term is the sum of all diagrams which start with a $2 \to 2$ elastic diagram on the left and end with an inelastic diagram on the right, (iii) the $M^{(in)}$-term is the sum of all diagrams which start with an inelastic diagram on the left and end with a $3 \to 3$ elastic diagram to the right and, (iv) the -$M^{(in)}$-term is the sum of all diagrams which start with a $2 \to 2$ elastic diagram on the left and end with a $3 \to 3 $ elastic diagram on the right. Obviously, these four partial sums exhaust the entire set of diagrams that may contribute to the complete transition amplitude.
This same presentation of $T_{23}$ as the sum of the terms of fig. 1 can as well be derived formally. By definition, the transition amplitude is, $$T_{23} = \langle \Psi^{(-)}_{el,3} | \hat{M}^{(in)} | \Psi^{(+)}_{el,2}
\rangle \ \ ,
\label{eq:221}$$ where as defined in eqn. 3, $\Psi^{(+)}_{el,2}$ stands for a two-body elastic scattering wave function and $\Psi^{(-)}_{el,3}$ is a three-particle elastic scattering wave function corresponding to the exit channel. It is a solution of a three-body Schr$\ddot{o}$dinger equation with a Hamiltonian having only two-body and three-body elastic interactions, and satisfies the boundary conditions that $\Psi^{(-)}_{el,3}$ tends asymptotically to a free three-particle wave function, $\phi_{03}$, plus outgoing spherical waves. In terms of the elastic scattering amplitudes defined above one has the expression[@thomas], $$\begin{aligned}
\Psi^{(-)}_{el,3} = \lim_{\epsilon \to 0}\
[\ 1 + G_{03}( E\ +\ i\epsilon )\ \hat{T}^{(el)}_{33}\ ]\ \phi_{03}\ \ .
\label{eq:222}\end{aligned}$$ Using this with eqn. 3 in eqn. 6 implies that, $$\hat{T}_{23} = \hat{M}^{(in)} + \hat{t}^{(el)}_{22}
\hat{G}_{02} \hat{M}^{(in)} +
\hat{M}^{(in)} \hat{G}_{03} \hat{T}^{(el)}_{33} +
\hat{t}^{(el)}_{22} \hat{G}_{02} \hat{M}^{(in)} \hat{G}_{03}
\hat{T}^{(el)}_{33}\ \ .
\label{eq:223}$$ Formally, this expression provides an exact solution of the problem and has a structure identical in form to that given in eqn. 4, for two-body reactions. A notable feature of this expression is that all inelastic transitions are confined in the block $M^{(in)}$, while all possible $elastic$ rescatterings in the entrance and exit channels are contained in $t^{(el)}_{22}$ and $T^{(el)}_{33}$.
Let us now consider corrections $a\ la\ Migdal-Watson$. In the limit of extremely weak one may replace $\Psi^{(+)}_{el,2}$ by $\phi_{02}$ in eqn. 6. This makes the second and fourth terms on the rhs of eqn. 8 disappear. Furthermore, in the limit of very strong the third term on the rhs of eqn. 8 dominates and we may approximate the transition amplitude by, $$T_{23} \approx \langle \phi_{02} | \hat{M}^{(in)} | \phi_{03}
\rangle \ T^{(el)}_{33} \ \ .
\label{eq:224}$$ We may argue that the matrix element $\langle \phi_{02} | \hat{M}^{(in)} | \phi_{03} \rangle $ is a smooth and slowly varying function of the energy and momenta and hence the energy dependence of the amplitude, eqn. 9, is determined almost solely by the elastic amplitude $T^{(el)}_{33}$. Although in form eqn. 9 resembles the Watson-Migdal approximation for two-body reactions, eqn. 1, there exists an essential difference between the two cases which makes eqn. 9 more difficult to apply. This is because the amplitude $T^{(el)}_{33}$, contrary to $t^{(el)}_{ff}$, is practically not a measurable quantity.
The two-body initial state, $\Psi_{el,2}^{(+)}$, can be calculated using distorted wave approach and the effects of slowly varying nonvanishing can be included by replacing $M^{(in)}$ in eqn. 9 with the matrix element $\langle \phi_{03} | \hat{M}^{(in)} | \Psi^{(+)}_{el,2} \rangle $. With this modification in mind eqn. 9 is more suitable to apply to data. Certainly, must be included in a full dynamical description of the process as they influence the cross section scale through the matrix element $M^{(in)}$.
2.3 The amplitude for $3 \to 3$ {#the-amplitude-for-3-to-3 .unnumbered}
--------------------------------
In the Faddeev formalism $T^{(el)}_{33}$ is written as a sum of three Faddeev amplitudes, $$T^{(el)}_{33} = \sum_{j=1}^{3} T^j\ \ .
\label{eq:8}$$ These satisfy the set of coupled integral equations[@thomas], $$T = t + F T = \sum_{ n=0 }^{\infty} \ (\ FF\ )^n\ [\ t + F t\ ]\ \ ,
\label{eq:8}$$ where we have used the notation, $$\begin{aligned}
T = \left ( \begin{array}{c}
T^1 \\
T^2 \\
T^3 \\ \end{array} \right) \ \ ,
t = \left ( \begin{array}{c}
t_1 \\
t_2 \\
t_3 \\ \end{array} \right) \ \ ,
F = \left ( \begin{array}{clcr}
0 & t_1 & t_1 \\
t_2 & 0 & t_2 \\
t_3 & t_3 & 0 \\ \end{array} \right) G_{03}\ \ ,
\label{eq:2.38}\end{aligned}$$ and final particles are labeled 1,2 and 3. Here $t_j$ are two-body scattering amplitudes in the three-body space. In momentum space and adopt the convention that ($j, l, k$) are always cyclic, $$\langle {\bf p}_j,{\bf p}_l,{\bf p}_k\ |\ t_j\ |\ {\bf p}_j',
{\bf p}_l',{\bf p}_k'\rangle
= \langle {\bf p}_l,{\bf p}_k\ |\ \hat{t}_j( E - \epsilon_j)
\ |\ {\bf p}_l',{\bf p}_k'\rangle\ \delta ({\bf p}_j - {\bf p}_j')\ \ ,
\label{eq:8}$$ with $\hat{t}_j$ being a solution of the ordinary two-body Lippmann-Schwinger equation of the energy variable $E - \epsilon_j$. E is the total three-body energy and $\epsilon_j$ the energy appropriate to momentum ${\bf p}_j$. From the definitions in eqn. 12, $$T^j
= t_j \ +\ (T^l + T^k)\ G_{03}\ t_j\ \ ,\ j\neq l\neq k \neq j\ \ ,
\ \ j = 1-3\ \ .
\label{eq:8}$$ As pointed by Amado[@amado], the contributions from the various pair interactions add coherently and the effects of a given pair’s interactions is distributed over the entire amplitude. From eqn. 14, $T^j$ is the sum of all terms contributing to $T^{(el)}_{33}$ which end with $t_j$ but, it does not include all contributions of the $l-k$ interaction. Note that the first term on the rhs represents the $l-k$ two-body scattering amplitude with the $j$th particle being a spectator and as such it corresponds to a disconnected diagram. In the center of mass of the $l-k$ pair and for a given partial wave $\lambda$, $t_j$ has the elastic scattering phase $\delta_{\lambda}(j)$. The second term in eqn. 14 represents the contributions from two or more rescatterings and involves completely connected diagrams only. These play an important role in the coherence formalism[@amado] through interference with similar terms involving the $j-l$ and $j-k$ pair interactions. Therefore, in order to account for the effects of any pair’s interaction the entire amplitude must be constructed. Adding the sum ($T^l + T^k$) to $T^j$ of eqn. 14, one obtains three equivalent forms of the entire amplitude, $$T^{(el)}_{33} =
t_j + ( T^l + T^k )\ [ 1 + G_{03} ( E )\ t_j ]\ ,\ j\neq l\neq k \neq j\ \ ,
\ \ j = 1-3\ \ .
\label{eq:8}$$ In each of the forms in eqn. 15, there appears the factor ($1 + G_{03}\ t_j$) with the $t_j$ being half on-shell. At a given $l-k$ relative momentum (${\bf q}_j$) in a given $l-k$ partial wave ($\lambda$) and following the arguments in ref.[@amado], these half-on-shell amplitudes, the factor ($1 + G_{03}\ t_j$) and, the entire amplitude have the phase $\delta_{\lambda}$(j). For this to happen, say in a region where the expression $(T^l\ + \ T^k)\ G_{03}\ t_j$ may vary rapidly with energy it must have a part which cancels the variation of $(T^l\ +\ T^k)$ so as to give to their sum the behaviour of $t_j$. This general observation on three-body amplitudes, which is true for $T_{23}$ as well as for $T_{33}$, has far reaching consequences on the energy dependence of the entire amplitude. Certainly, in order to take this cancelation into account the terms corresponding to completely connected diagrams must be included.
Another general constraint on the form of $T_{23}$ is provided by the so called Fermi-Watson theorem. This requires that the phase of the transition amplitude of the entire amplitude is determined by the phases and norms of the rescattering amplitudes in the entrance and exit channel and that in the limit of vanishing the overall phase of the entire amplitude is equal to that of the elastic scattering amplitude in the exit channel. This well established for two-body reactions[@joachim] is easily extended to the case of three-body reactions near threshold (see Appendix A). Explicitly, for the case under considerations, this theorem requires that the phase of $T_{23}$ must be identical to the overall phase of $T^{(el)}_{33}$. Both of these constraints are satisfied by the approximation, eqn. 9, in a natural way.
Turning now to estimate the rescattering amplitude, we note first that the kernel $F F$ is compact for any complex E and therefore eqns. 11 has a unique solution. By solving these equations by iteration , one can calculate the amplitude $T^{(el)}_{33}$ with the required accuracy. To first order $T^{(el)}_{33}\ \approx \ [\ t\ +\ Ft\ ]$. It is the sum of all diagrams shown in fig. 3b. The first three of these are disconnected and represent the contributions of the free $t$. As shown by Amado[@amado] and Cahill[@cahill] taking the sum of just these three diagrams would not yield a coherent amplitude and therefore would not be a satisfactory solution. The other diagrams of fig. 3b give the first order correction due to two subsequent elastic scatterings, they are all completely connected and as indicated above play an important role in the coherence mechanism[@amado; @cahill]. The next iteration introduces connected diagrams with three and four rescatterings and so on. Summing the contributions of diagrams up to the corresponding order leads, in principio, to $T^{(el)}_{33}$ with the accuracy required.
As a first step we restrict the following discussion to corrections due to one and two rescattering diagrams only (fig. 3b), , we calculate $T^{(el)}_{33}$ to first order only. Denoting contributions from double scattering diagrams by $C_{jl}$ it is shown in Appendix B that for s-wave production at energies close to threshold, $$C_{jl} = M^{(in)} (0,0)\ q_{j}\
t^{(on)}_j (0, \frac {q_{j}^2 }{ 2 \mu_{j}})\
t^{(on)}_l (0, \frac {q_{l}^2 }{ 2 \mu_{l}})\ I_{jl}\ \ .
\label{eq:2.39}$$ where $I_{jl}$ are integrals of Kowalski-Noyes[@kowalski] half-shell functions over the appropriate relative momenta of the interacting pairs. Thus the entire transition amplitude can be written as, $$T_{23} \approx M^{(in)}_{23}(0,0)\ Z_{33}\ \ ,
\label{eq:2.39}$$ where the correction factor is given by,
$$Z_{33} = \sum_{j=1}^{3} t^{(on)}_j (0, \frac {q_{j}^2 }{ 2 \mu_{j}})\ +
\sum_{j,l = 1}^{3} q_{j}\ t^{(on)}_j (0, \frac {q_{j}^2 }{ 2 \mu_{j}})\
t^{(on)}_l (0, \frac {q_{l}^2 }{ 2 \mu_{l}})\
I_{jl}\ \ .
\label{eq:2.39}$$
A further simplification can be achieved by setting all $I_{jl}$ equal unity, thus neglecting off shell effects (see Appendix B). This allows calculating the factor from two-body scattering data of the particles involved. To evaluate $T^{(el)}_{33}$ to second order we need to include $(\ FF\ )\ [\ t\ +\ Ft\ ]$ terms. The evaluation of such terms is described in Appendix C, where it is shown that the effect of the (FF) factor is to scale the first order amplitude $[\ t\ +\ Ft\ ]$ by, $$\begin{aligned}
\langle \ FF\ \rangle & \approx & q_k\
t^{(on)}_k (0\ , \frac {q^2_k} {2\ \mu_{lj}})\
q_l\ t^{(on)}_l (0\ , \frac {q^2_l} {2\ \mu_{jk}})\\ \nonumber
& \approx & \exp {i(\ \delta_k\ +\ \delta_l)}
\sin {\delta_k} \sin {\delta_l}\ \ .
\label{eq:280}\end{aligned}$$ Here we have used the standard parameterization, $$t(q, q) = \frac {1} {q(\cot {\delta } - i)}\ \ ,
\label{eq:281}$$ with the $\delta$’s being the appropriate s-wave phase shifts. These are related to the s-wave scattering length according to,
$$\begin{aligned}
|\langle \ FF\ \rangle| & \approx & \sin \delta_k \sin \delta_l =
\frac {q_k\ a_k\ q_l\ a_l}{(1\ +\ q_k\ a_k)(1\ +\ q_l\ a_l)}\ \ .
\label{eq:283}\end{aligned}$$
Thus in case of $ q_k\ a_k\ q_l\ a_l \ll 1$ , the factor $\langle \ FF\ \rangle \ll 1$ so that the evaluation of $T^{(el)}_{33}$ to first order should be sufficient.
Application
===========
In the present section we apply the procedure described above to analyze the effects of on the energy dependence of $\pi$ and $\eta$ meson production in $NN$ scattering at energies near their respective production thresholds. We demonstrate this first for $\pi$ meson production. For the $pp \to pp \pi^0$ reaction, rescatterings may include the following sequences : $pp \to pp\pi^0 \to pp\pi^0$, $pp \to pn\pi^+ \to pp\pi^0$ and $pp \to d\pi^+ \to pp\pi^0$. In considering only the first sequence is included as by definition the block includes $elastic$ interactions only. Two-nucleon mechanism contributions dominate the $pp \to pp \eta$ cross section and any reasonable calculations of the primary amplitude must include $\pi^+$ production followed by charge exchange. The effects of these other channels are assumed to be absorbed in $M^{(in)}$. The isosinglet and isotriplet pion-nucleon scattering amplitudes are calculated from eqn. 20, using the isosinglet and isotriplet pion nucleon scattering length values $a_1 = 0.245 fm$ and $a_3 = -0.143 fm$ of ref.[@slength]. The isospin average scattering length of these values is rather small so that $a\ priori$ the overall $\pi$N interaction effects on the energy dependence are expected to be very small. Similarly, the S-wave $pp$ phase shift is obtained from effective range expansion using the modified Cini-Fubini-Stanghellini formula[@noyes], $$C^2\ q\ ctn\ \delta + 2\eta_c q h(\eta_c) =
-\frac {1}{a_{pp}} + \frac {1}{2} r_{pp} q^2 -
\frac {p_1\ q^4}{ 1\ +\ p_2 k^2}\ \ ,
\label{eq:281}$$ where $a_{pp}$ and $r_{pp}$ denote the $pp$ scattering length and effective range and, $$\eta_c = \frac {e^2 E_{lab}} {P_{lab}}\ \ ,
C^2 = \frac {2\pi\ \eta_c} { (\exp{2\pi\eta_c} - 1)}\ \ ,
h(\eta_c) = \sum_{s=1}^{\infty} \frac {\eta_c^2}{s(s^2 + \eta_c^2)}
- \gamma - \ln {\eta_c}\ \ . \\
\label{eq:281}$$ The $p_i$ (i=1,2) are functions of $a_{pp}$ and $r_{pp}$. In what follows we have used the values $a_{pp} = -7.82 fm$ and $r_{pp} = 2.7 fm$ of ref.[@noyes]. To calculate the cross section, the factor $|Z_{33}|^2$, eqn. 18, is multiplied by the invariant phase space and then integrated over the appropriate momenta. The primary production amplitude is assumed to be constant and is taken outside the integral. The energy dependence of the cross section as obtained with the factor calculated to first and second orders are displayed in fig. 4 as a function of $Q_{cm}$, the energy available in the $cm$ system. The two solutions are practically identical. The $\langle \ FF\ \rangle $ factor discussed in the previous section, is indeed very small so that the sum in eqn. 18 converges and it is safe to compare data with the predictions corresponding to first order calculations of $|Z_{33}|^2$. Comparison with data is made in fig. 5. The predictions with the full $FSI$ correction of eqn. 18 are shown as the solid line. Those without the $\pi N$ interaction and with only the first disconnected diagrams of fig. 3b are given by the large-dashed and small-dashed lines, respectively. The data in the figures are taken from ref.[@meyer]. The curves are arbitrarily normalized to the lowest energy data point available. As indicated above the s-wave $\pi N$ interaction is relatively weak so that the $FSI$ correction factor is dominated by the strong interaction of the $^1 S_0$ NN system. Our predictions with the full correction factor (solid curve) agrees slightly better with data compared with those obtained by including the $NN$ force only (dashed line). Nonetheless, even in this case, where the overall meson-nucleon interaction is weak, the effects from disconnected and completely connected diagrams are comparable. It is our contention that the completely connected diagrams of fig. 3b must be included if one wishes to interpret the recent precision measurements of $\pi$ production cross section[@meyer; @lee]. As demonstrated below the importance of these completely connected diagrams becomes more apparent in applying our procedure to $\eta$ production.
For $\eta$ meson production there are no direct measurements of elastic $\eta N$ scattering and the information available are extracted indirectly using $\pi N \to \eta N$ data. Based on $\pi N$ and $\eta N$ coupled channel analysis around the $\eta N$ threshold within an isobar model, Bhalerao and Liu[@bahl] suggest a value $a_{\eta N} = (0.27 + i0.22)\ fm$ for the $\eta N$ scattering length. With this choice of $a_{\eta N}$ the results of our analysis with the factor calculated to first order are shown in fig. 6. Here as for pion production, the solution with the factor calculated to second order is nearly identical. In contrast with $\pi$ production, this process is strongly influenced by the $\eta N$ interaction giving rise to a sharp enhancement of the cross section very near threshold and thus reproduces, rather accurately, the energy dependence of the cross section up to $Q_{cm} = 20 MeV$. Note also, that in the analysis presented above, the enhancement factor $|Z_{33}|^2$ is exactly what is required to keep the primary production amplitude nearly constant over the energy range from threshold to $Q_{cm} = 20 MeV$.
It is interesting to explore how sensitive the results are for a different choice of the $a_{\eta N}$ scattering length. Since other channels such as the $\pi\pi N$ are also opened, Wilkin[@wilkin] has fixed the imaginary part to be $\Im [a_{\eta N}] = 0.30 fm$ and used the $\pi^- p \to \eta p$ data directly to determine the real part. This procedure has yielded a scattering length $a_{\eta N} = (0.55\pm0.20 + i0.30)\ fm$. Calculations with this value are given in fig. 7. The energy dependence is mostly sensitive to the real part of $a_{\eta N}$ and the results from our analysis do not support the $central$ value proposed by Wilkin[@wilkin]. It is to be noted also, that based on partial wave unitarity relations, an analysis of the $\pi^- p$ backward differential cross section data yields a value of $39^o.3 \pm 4^o.7$ for the phase of the $\eta N$ production amplitude[@ramesh] a value in agreement with Bhalerao and Liu[@bahl] but not with Wilkin[@wilkin].
Summary And Conclusions
=======================
In summary we have developed an approximate procedure which accounts for final state effects in the presence of three strongly interacting particles. In analogy with the Watson’s treatment[@watson] it was shown that, in the the limit of extremely strong the amplitude factorizes into a primary production amplitude and an elastic scattering amplitude describing a $3 \to 3$ transition. The energy dependence of the cross section in the near vicinity of the reaction threshold is determined almost entirely by $T^{(el)}_{33}$. Based on the coherence formalism of Amado[@amado] we argue that the completely connected diagrams corresponding to two rescatterings may interfere strongly and therefore, play an important role in determining the energy dependence of the cross section. In view of the scarcity of knowledge about the meson-nucleon force, the approximation is particularly useful near threshold where $T^{(el)}_{33}$ can be estimated using nucleon-nucleon and meson-nucleon scattering data. Because of the strong NN force the term dominates. Detailed analysis of $\pi$ and $\eta$ production via $NN$ collisions show that the energy dependence of the cross section can be explained in both cases, by taking into account the meson-$N$ interaction also.
In the present analysis we have not considered corrections as these vary slowly with energy and would have little influence on the energy dependence. For $NN$ scattering at bombarding energies of several 100 MeV and above, the nuclear and Coulomb distortions are expected to be very small. The Gamov factor which accounts for Coulomb distortions is equal unity within 2.5$\%$ even for the pion data analyzed above. As previously mentioned effects can be included exactly by using distorted waves for the incoming two-proton state.
Finally, the primary production amplitude is not treated explicitly so that the cross section scale was arbitrary. This deficiency can be removed by incorporating in eqn. 17 a primary production amplitude from one of the existing perturbative models[@germo; @lagt; @vett]. Such model calculations were performed for $\eta$ production, taking the $\eta$N interaction in the primary amplitude to be consistent with that occurring in the . They are found to reproduce the energy dependence as well as the scale of the cross section and will be published elsewhere[@shorer].
In view of the large effects due to the $\eta N$ force the analysis presented above could well be used to study the $\eta N$ force itself for which no direct experiments are possible.
[**Appendix A**]{}
The Fermi-Watson theorem[@joachim] can be extended to three-body reactions by applying unitarity. Let $T_{23}$ denotes the entire amplitude and let $t^{(el)}_{22'}$ and $T^{(el)}_{3',3}$ denote elastic $2 \to 2$ and $3 \to 3$ amplitudes, respectively. Then by unitarity, $$\Im \{ T_{23} \} = - \pi \sum_{2'} t^{(el)}_{22'} \delta (E - E_{2'})
T^{\dagger}_{2'3} - \pi \sum_{3'} T_{23'} \delta (E - E_{3'})
T^{(el) \dagger}_{3'3}\ \ .
\label{eq:10}$$ The $\delta$ functions force the two body and three body elastic scattering amplitudes, $t^{(el)}_{22'}$ and $T^{(el)}_{3'3}$ to be on mass-shell.
In angular momentum representation the amplitudes of eqn. 24 can be written in the form,
$$T_{ij} = | T_{ij} | \exp {i \delta _{ij}} \ \ .
\label{eq:2}$$
Note that $\delta _{2'2}$ stands for two-body phase shift but $\delta _{3'3}$ is only related to two-body phase shifts of the three interacting pairs and in itself is not a measurable quantity. Then substituting these into the unitarity condition, eqn. 24, leads to the following constraint on the $\delta$’s,
$$\begin{aligned}
\Im \{ |T_{23}| \exp {i \delta_{23}} \} & = &
- \pi \sum_{2'} |T^{(on)}_{22'}| |T^{(on)}_{2'3}|
\exp {i (\delta_{22'} - \delta_{2'3})}\nonumber \\
& &
- \pi \sum_{3'} |T^{(on)}_{23'}| |T^{(on)}_{3'3}|
\exp {i (\delta_{23'} - \delta_{3'3})}\ \ .
\label{eq:11}\end{aligned}$$
In the case of a single channel with the angular momenta of the final pairs being zero, the phase $\delta_{23}$ from eqn. 26 is,
$$\begin{aligned}
\sin {\delta_{23} } & = &
{ |t^{(on)}_{22}| \sin {\delta_{22}} -
|T^{(on)}_{33}| \sin {\delta_{33}} } / \nonumber \\
& &
[ |t^{(on)}_{22}|^2
+ |T^{(on)}_{33}|^2
- 2 |t^{(on)}_{22}| |T^{(on)}_{33}|
\cos {(\delta_{33} - \delta_{22})} ]^{1/2}\ \ .
\label{eq:12}\end{aligned}$$
Thus the phases of the rescattering amplitudes and their norms, determine the overall phase of the $2 \to 3$ transition amplitude, just as the case is for a two body reactions. In the limit of very weak (vanishing $t^{(el)}_{22}$),
$$\sin {\delta_{23} } \approx \sin { \delta_{33}} \ \ ,
\label{eq:5}$$
so that the overall phase of $T^{(el)}_{23}$ becomes identical to that of the elastic scattering amplitude $T^{(el)}_{33}$. Furthermore, the amplitudes and phases of the strong rescatterings in the entrance and exit channels are related through eqn. 26 by,
$$|t^{(el)}_{22}|\ ( \sin {\delta_{22} } + \pi \ |t^{(el)}_{22}| ) =
|T^{(el)}_{33}|\ ( \sin {\delta_{33} } + \pi \ |T^{(el)}_{33}| ) \ \ .
\label{eq:5}$$
For a real production process the amplitudes $t^{(el)}_{22}$ and $T^{(el)}_{33}$ must be independent so that the lhs and rhs of eqn. 27 must be equal to a constant. In the limit of very weak this constant must vanish so that, $$|T^{(el)}_{33}| = \sin {\delta_{33} }/ \pi\ \ .
\label{eq:7}$$ Formally, this is identical in form to the expression assumed by Watson[@watson] for two-body processes.
[**Appendix B**]{}
We evaluate the contribution of diagrams with two rescatterings. As an example consider the diagram shown in fig. 8 (crossed lines denote on mass-shell states). In the CM system, in each of the three-body states of the diagram, there are only two independent momenta which are taken to be ${\bf p}_j$ the momentum of particle $j$ and ${\bf q}_j$ the relative momentum of the remaining $l-k$ pair. Adopting the convention that $(j,l,k)$ are always cyclic, ${\bf q}_j$ is defined as $${\bf q}_j = \frac { (m_k {\bf p}_l - m_l {\bf p}_k )}
{( m_l + m_k)}\ \ .
\label{eq:2.39}$$ In terms of these momenta the total kinetic energy is , $$H_{0} = \frac {p^2_j}{2 \mu_j} + \frac {q^2_j}{2 \mu_{lk}}\ \ .
\label{eq:2.39}$$ Here $\mu_{lk}$ is the reduced mass of the $l$ and $k$ particles and $\mu_j$ is the reduced mass of the $j$ particle, , $$\begin{aligned}
\mu_{lk} = \frac {m_l m_k}{( m_l + m_k )}\ \ ;\
\mu_j = \frac {m_j (m_l + m_k) } {(m_j + m_l + m_k)} \ \ .
\label{eq:2.39}\end{aligned}$$ Now the contribution of the diagram, fig. 8, can be written in the form, $$\begin{aligned}
C_{31} & = & \int d{\bf p}_3' d{\bf q}_3'
\delta (E - \frac {p_3'^2}{2 \mu_3}- \frac {q_3'^2}{2 \mu_{12} })
M^{(in)}({\bf p}_3',{\bf q}_3')\\ \nonumber
& & \int d{\bf p}_1''d{\bf q}_1'' \delta ({\bf p}_3' - {\bf p}_3'')
t_3 ({\bf q}_3', {\bf q}_3''; E^+ - \frac {p_3''^2}{2 \mu_3})\\ \nonumber
& & [E^+ - \frac {p_1''^2}{2 \mu_1} -
\frac {q_1''^2} {2 \mu_{23}}]^{-1}
\delta ({\bf p}_1'' - {\bf p}_1) )
t_1 ({\bf q}_1'', {\bf q}_1; E^+ - \frac {p_1^2}{2 \mu_1})\ \ .
\label{eq:2.39}\end{aligned}$$ The integration over ${\bf p}_1''$ and ${\bf p}_3'$ are immediate and we obtain, $$\begin{aligned}
C_{31} & = & \int d{\bf q}_3'
\delta (E - \frac {p_3''^2}{2 \mu_3}- \frac {q_3'^2}{2 \mu_{12} })
M^{(in)}({\bf p}_3'',{\bf q}_3')\\ \nonumber
& & \int d{\bf q}_1''
t_3 ({\bf q}_3', {\bf q}_3''; E^+ - \frac {p_3''^2}{2 \mu_3})\\ \nonumber
& & [E^+ - \frac {p_1^2}{2 \mu_1} -
\frac {q_1''^2} {2 \mu_{23}}]^{-1}
t_1 ({\bf q}_1'', {\bf q}_1; E^+ - \frac {p_1^2}{2 \mu_1})\ \ .
\label{eq:2.39}\end{aligned}$$ Here $t_3$ is a half off mass-shell two-body matrix element with only the ${\bf q}_3''$ being off-shell momentum. (From a view point of invariant perturbation theory only one leg of $t_j$ is off mass-shell so that it behaves like a particle form-factor). Expanding $t_j$ in partial waves and introducing the Kowalski-Noyes half-shell function[@kowalski] $f_j(\lambda ,{\bf q}_j,{\bf q}'')$ allows writing these for a partial wave $\lambda$ as, $$t_{j}(\lambda ,{\bf q}_j,{\bf q}_j'';E^+) =
t^{(on)}_j (\lambda ,q^2_j /2 \mu_j)
\ f_j (\lambda ,{\bf q}_j,{\bf q}_j'')\ \ .
\label{eq:2.39}$$ Then for slowly varying $M^{(in)}$ and at energies near threshold one obtains, $$C_{31} = M^{(in)} (0,0)\ q_3\ t^{(on)}_3 (0, \frac {q_3^2 }{2 \mu_{12}})
\ t^{(on)}_1 (0, \frac {q_1^2 }{ 2 \mu_{23}})\ I_{31}\ \ ,
\label{eq:2.39}$$ where $I_{31}$ is an integral over the Kowalski-Noyes half-shell functions, $$I_{31} = \frac {1} {(2\pi)^2 \ \mu_{23} \ q_3} \int d{\bf q}_1''
f_3 (0,{\bf q}_3,{\bf q}_3'')
[ E - \frac {p^2_1}{2 \mu_1} - \frac {q''^2_1}{2 \mu_{23}}]^{-1}
f_1 (0,{\bf q}_1,{\bf q}_1'')\ \ ,
\label{eq:2.39}$$ where, $${\bf q}_3'' = -\frac {m_1}{m_1 + m_2 } {\bf q}_1'' -
\frac { m_2 (m_1 + m_2 + m_3) } {(m_1 + m_2) (m_1 + m_3)} {\bf p}_1\ \ .
\label{eq:2.39}$$ If we neglect off mass-shell effects the integrals $I_{jl}$ reduce to unity.
[**Appendix C**]{}
We consider here second order contributions to the transition amplitude. As indicated in the text these are obtained by multiplying first order contributions with a factor (FF). We attempt demonstrating this for the diagram shown in Fig. 9. This can be written as, $$\begin{aligned}
D_{313} & = & \int d{\bf p}_3'\ d{\bf q}_3'\ d{\bf p}_1''\ d{\bf q}_1''\
d{\bf p}_3'''\ d{\bf q}_3'''\
\delta (E - \frac {p_3'^2}{2 \mu_3}- \frac {q_3'^2}{2 \mu_{12} })\\ \nonumber
& & M^{(in)}({\bf p}_3',{\bf q}_3')\
t_3 ({\bf q}_3', {\bf q}_3''; E^+ - \frac {p_3'^2}{2 \mu_3})\
\delta ({\bf p}_3' - {\bf p}_3'')\
\\ \nonumber
& & [E^+ - \frac {p_1''^2}{2 \mu_1} -
\frac {q_1''^2} {2 \mu_{23}}]^{-1} \
\delta ({\bf p}_1'' - {\bf p}_1''') \
t_1 ({\bf q}_1'', {\bf q}_1'''; E^+ - \frac {p_1'''^2}{2 \mu_1})\\ \nonumber
& & [E^+ - \frac {p_3'''^2}{2 \mu_3} -
\frac {q_3'''^2} {2 \mu_{12}}]^{-1} \
\delta ({\bf p}_3''' - {\bf p}_3) \
t_3 ({\bf q}_3''', {\bf q}_3; E^+ - \frac {p_3^2}{2 \mu_3})\ \ .
\label{eq:2.40}\end{aligned}$$ Integrating over ${\bf p}_3'$, ${\bf p}_1''$ and ${\bf p}_3'''$ and arranging terms leads to, $$\begin{aligned}
D_{313} & = & {\Huge \{ }\ q_1\
t^{(on)}_1 (0 \ ,\ \frac {q_1'''^2}{2 \mu_{23}})\
q_3\ t^{(on)}_3 (0\ ,\ \frac {q_3^2}{2 \mu_{12}})\ \frac {1} { q_1\ q_3}
\ \int d{\bf q}_1''
\frac {1} { [E^+ - \frac {p_1'''^2}{2 \mu_1} - \frac {q_1''^2} {2 \mu_{23}}]}
\\ \nonumber
& & \int d{\bf q}_3''' \frac {f_1(0,{\bf q}_1'',{\bf q}_1''')\
f_1(0,{\bf q}_3''',{\bf q}_3)}
{[E^+ - \frac {p_3^2}{2 \mu_1} - \frac {q_3'''^2} {2 \mu_{12}}]}\
{\Large \}} \\ \nonumber
& & \int d{\bf q}_3' \ M^{(in)}({\bf p}_3',{\bf q}_3')\
\delta (E - \frac {p_3''^2}{2 \mu_3}- \frac {q_3'^2}{2 \mu_{12} })
t_3 (0, \frac {q_3'^2}{2 \mu_{12}})
f_3(0, {\bf q}_3', {\bf q}_3'')\ \ .
\label{eq:241}\end{aligned}$$ We now notice that the last integral is the first order contribution from the disconnected diagram (9.a) with a single scattering block between particles 1 and 2 while the expression in the curly bracket is just the (FF) factor. An order of magnitude of this factor can be obtained using on mass-shell values for the Kowalski-Noyes functions. This leads to, $$\langle \ FF\ \rangle \approx q_1\
t^{(on)}_1 (0\ ,\ \frac {q_1'''^2}{2 \mu_{23}})\
q_3\ t^{(on)}_3 (0\ ,\ \frac {q_3^2}{2 \mu_{12}})\ \ .
\label{eq:242}$$ Similar expressions can be calculated for the completely connected diagrams (3.b) as well.
[**Acknowledgments**]{} This work was supported in part by Israel Ministry Of Science and Technology and the Israel Ministry Of Absorption. One of us ( A. M. ) thanks C. Wilkin for suggestions and stimulating discussions concerning the results reported in this work.
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Three-Body\ Problem$ (Daniel Davey and Co., Inc., New York 1965). M. Goldberger and K. M. Watson, $Collision\ Theory$ (New York, Wiley, 1964). I. R. Afnan and A. W. Thomas, in $Modern\ Three-Hadron\
Physics$, Edited by A. W. Thomas (Springer-Verlag Berlin Heidelberg 1977). Steven Weinberg, Phys. Rev. ${\bf 133}$, B232(1964). R. D. Amado, Phys. Rev. ${\bf 158}$, 1414(1967); see also author’s contribution in ref. 10. C. J. Joachain, $ Quantum\ Collision\ Theory $ (North-Holland Publication Company, 1975). R. T. Cahill, Phys. Rev. ${\bf C9}$, 1414(1974). K. L. Kowalski, Phys. Rev. Lett. ${\bf 15}$, 798(1965); H. P. Noyes, Phys. Rev. Lett. ${\bf 15}$, 538(1965). T. Ericson and W. Weise, $Pions\ and\ Nuclei$ (Clarendon Press . Oxford, 1988). H. P. Noyes, Ann. Rev. Nucl. Sci., ${\bf 22}$, 465(1972). R. S. Bhalerao and L. C. Liu, Phys. Rev. Lett. ${\bf 54}$, 865(1985). Ramesh Bhandari and Yung-An Chao, Phys. Rev. ${\bf D15}$, 192(1977). A. Moalem, E. Gedalin, L. Razdolskaja and Z. Shorer, submitted for publication in Nucl. Phys. ${\bf A}$.
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abstract: 'We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval. The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities. Our approach closely follows and extends recent works by Short and Farrelly \[2012 New J. Phys. [**14**]{} 013063\], in particular going beyond the realm of finite-dimensional systems and large effective dimensions.'
address:
- '$^{1}$Universität Bielefeld, Fakultät für Physik, 33615 Bielefeld, Germany'
- '$^{2}$ National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa and Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa'
author:
- 'Peter Reimann$^{1}$ and Michael Kastner$^{2}$'
title: Equilibration of isolated macroscopic quantum systems
---
Introduction {#s1}
============
It is a basic everyday experience that isolated macroscopic systems, i.e. systems consisting of many particles or other microscopic degrees of freedom, approach some steady equilibrium state after a sufficiently long time evolution, no matter how far from equilibrium they started out. More precisely, for every single run of the experiment, one still may encounter certain statistical or quantum mechanical fluctuations, especially for microscopic observables, but on the average over many repetitions of the experiment, all expectation values appear to equilibrate.
The reconciliation of this irreversible behavior with quantum mechanical reversibility and revival/recurrence properties is a long standing problem [@skl93] which has recently be reconsidered from a new viewpoint, without focusing on any specific model class and without any modification/approximation of the exact quantum mechanical time evolution [@rei08; @lin09; @lin10; @rei10; @sho11; @sho12]. The key point of these works is to show that the expectation values may still exhibit everlasting small fluctuations around their equilibrium values, as well as very rare large excursions away from equilibrium (including the above-mentioned recurrences), but quantitatively these fluctuations are either unobservably small compared to any reasonably achievable resolution limit, or exceedingly rare on any realistic time scale after initial transients have died out. In this sense, the system indeed equilibrates.
Originally, these conclusions have been based on the following main assumptions [@rei08; @lin09; @lin10; @rei10]:
1. The considered observables represent experimental measurement devices with a finite instrumental range and a possibly very small but still ‘reasonable’ resolution limit.
2. The initial condition exhibits a small occupation probability of every single energy level, which is very plausible in view of the unimaginably large level density of typical macroscopic systems.
3. All energy eigenvalues are non-degenerate.
4. All energy gaps are non-degenerate, i.e. a pair of distinct energy eigenvalues never exhibits the same energy difference as some other pair.
5. Either the considered Hilbert space must be finite-dimensional [@lin09; @lin10], or some formal manipulations are not rigorously justified [@rei08; @rei10], in particular interchanging the limit of infinite dimensions with the long-time limit.
The restrictions (iii) and (iv) have recently been overcome in two very important contributions by Short [@sho11] and by Short and Farrelly [@sho12]. In our present work, we closely follow and further extend their approach by relaxing also the above conditions (ii) and (v). Namely, we only require that the second largest level population must be small, while the occupation probability of one level may be macroscopic (non-small). Such a case may e.g. be of relevance for thermal equilibrium states at extremely low temperatures. Furthermore, we will admit and treat with care (countably) infinite dimensional systems.
Our present approach also bears resemblance to recent advancements made by Goldstein and coworkers of ideas originally due to von Neumann [@gol10a; @gol10b; @gol10c; @tas10]. The main difference is that in these works an alternative notion is adopted of when a system is in or close to equilibrium, and the main emphasis is laid on macroscopic (coarse-grained) observables, exhibiting the same expectation value for most states within any quantum mechanical energy shell.
Furthermore, our approach is complementary to numerous recent investigations of equilibration for various specific systems, observables, and initial conditions, and often with a main focus on the role of (non-)integrability, see e.g. [@tas98; @sre99; @caz06; @rig07; @col07; @man07; @cra08; @bar08; @rig08; @spec; @pal10; @ban11; @pol11; @camp11; @jacobson11; @kas11; @kas12] and references therein.
Another issue closely related to equilibration is the problem of thermalization, i.e., the question whether, and to what extend, the above mentioned equilibrium states agree with any one of Gibbs’s statistical ensembles. This important issue, either for an isolated system [*per se*]{} or for an isolated system-plus-bath composite, has been recently addressed in e.g., [@rei08; @lin09; @lin10; @rei10; @gol10a; @gol10b; @gol10c; @tas10; @tas98; @sre99; @caz06; @rig07; @col07; @man07; @cra08; @bar08; @rig08; @spec; @pal10; @gog11; @ban11; @pol11; @ike11], but will not be considered here in any further detail. In other words, our notion of [*equilibrium*]{} is weaker than that of [*thermal equilibrium*]{}.
General Framework {#s2}
=================
System and Hamiltonian {#s21}
----------------------
We consider an isolated system, confined to a finite region of space and involving a finite number of particles. Later we will mainly be interested in macroscopic systems, but for the moment any finite number of degrees of freedom is admitted. In particular, we may be dealing with a compound system, consisting of a subsystem of actual interest and its environment (reservoir or thermal bath).
According to standard quantum mechanics, such a system is modeled by a time-independent Hamiltonian $H$ on a separable (i.e. at most countably infinite-dimensional) Hilbert space ${{\cal H}}$. Since we consider the system to be confined to a finite region of space, all eigenvectors of $H$ represent bound states and the spectrum of $H$ is discrete (pure point). As a consequence, the Hamiltonian can be written in the form $$H = \sum_n E_n P_n,
\label{1}$$ where the $P_n$ are projectors onto the eigenspaces of $H$ with eigenvalues $E_n$, satisfying $$\begin{aligned}
P_m P_n = \delta_{mn} P_n,
\label{2}
\\
\sum_n P_n = 1,
\label{3}
\\
E_n \neq E_m \quad \mbox{if $n\neq m$}.
\label{4}\end{aligned}$$ Here, $n,m\in\{1,\dots,d_E\}$, where the number $d_E$ of distinct energy eigenvalues may be finite or infinite. The symbol $\sum_n$ indicates a summation over all those $n$-values, $\delta_{mn}$ is the Kronecker symbol, and $1$ the identity on ${{\cal H}}$. In particular, any energy eigenvalue $E_n$ is allowed to be degenerate and its multiplicity is given by $$\mu_n:=\Tr \{ P_n \},
\label{5}$$ where $\Tr$ denotes the trace on ${{\cal H}}$. The dimension of ${{\cal H}}$ thus amounts to $\sum_n \mu_n$ and may be finite or infinite.
States and dynamics {#s22}
-------------------
The system’s state at time $t$ is captured as usual by a density operator $\rho (t)$, describing either a statistical ensemble (mixed state) or a pure state, and evolving in time according to $\rho(t)=U_t\rho(0) U_t^\dagger$ with time-evolution operator $U_t:=\exp\{-\rmi Ht\}$ and $\hbar =1$. With we can conclude that $$\rho(t)=\sum_{mn} \rho_{mn}(0) \exp[-\rmi (E_m-E_n)t],
\label{6}$$ where we have introduced the auxiliary operators $$\rho_{mn}(t):=P_m\rho(t)P_n.
\label{7}$$ While $\rho_{nn}(t)$ are thus time-independent and self-adjoint operators, the same generically does not hold for $\rho_{mn}(t)$ when $m\neq n$. In particular, $\rho_{nn}(t)$ equals $\rho_{nn}(0)$ and the time-argument will often be omitted.
Realistic observables {#s23}
---------------------
Observables are represented as usual by self-adjoint operators $A$ with expectation values $\Tr\{\rho(t)A\}$. In order to model real experimental measurements it is however not necessary to admit any arbitrary self-adjoint operator [@realobs1; @realobs2; @realobs3; @realobs4; @realobs5; @lof; @geo95; @pop06]. Rather, it is sufficient to focus on [*realistic observables*]{} in the following sense [@rei08; @rei08a; @rei10]: Any observable $A$ must represent an experimental device with a [*finite range*]{} of possible outcomes of a measurement, $${\Delta_{\! A}}:=
\sup_{\psi\in S({{\cal H}})} \langle\psi|A|\psi\rangle
- \inf_{\psi\in S({{\cal H}})} \langle\psi|A|\psi\rangle
= a_{\sup} - a_{\inf} < \infty,
\label{8}$$ where $$S({{\cal H}}):=\left\{\psi\in{{\cal H}}\,\big|\,\langle\psi|\psi\rangle=1\right\}\subset {{\cal H}}$$ denotes the set of normalized vectors in ${{\cal H}}$. Moreover, this working range ${\Delta_{\! A}}$ of the device must be limited to experimentally reasonable values compared to its [*resolution limit ${\delta \! A}$*]{}. All measurements known to the present authors yield less than 20 relevant digits, i.e. $${\Delta_{\! A}}/{\delta \! A}\leqslant 10^{20}.
\label{9}$$ Maybe some day 100 or 1000 relevant digits will become feasible, but it seems reasonable that a theory which does not go very much beyond that will do. Note that similar restrictions also apply to ‘numerical experiments’ by computer simulations.
According to , all eigenvalues of $A$ must be contained within the finite interval $[a_{\inf}, a_{\sup}]$ and the operator norm $${\|A\|}:=\sup_{\psi\in S({{\cal H}})} {\|A|\psi\rangle\|}
\label{10}$$ is finite and equal to $\max\{|a_{\inf}|, |a_{\sup}|\}$. (As usual, the vector norm on the right hand side of is the one induced by the scalar product on ${{\cal H}}$).
Level populations {#s24}
-----------------
The specific observable $A=P_n$ describes the population of the energy level $E_n$ with expectation value (occupation probability) $$p_n:=\Tr\{ P_n \rho(t) \}=\Tr \{ \rho_{nn} \}.
\label{11}$$ The last relation shows the time-independence of $p_n$ and follows from , , and the invariance of the trace under cyclic permutations.
For a system with $f$ degrees of freedom there are roughly $10^{\Or(f)}$ energy eigenstates with eigenvalues in every interval of $1$J beyond the ground state energy ([@lldiu]; for a more detailed discussion, see also section 2.1 of [@rei10]). The same estimate carries over to the number of energy eigenvalues under the assumption that their multiplicities are much smaller than $10^{\Or(f)}$. For a macroscopic system with $f=\Or(10^{23})$, the energy levels are thus unimaginably dense on any decent energy scale and even the most careful experimentalist will not be able to populate only a few of them with significant probabilities $p_n$.
We recall that $\rho(t)$ may be a pure state, but the case of foremost interest is on mixed states describing a statistical ensemble. Then, $p_n$ describes an ensemble average over many repetitions of the experiment. Hence, an ‘accidentally large’ population of a few levels in one particular experimental run is still admissible, but very unlikely to occur again when the experiment is repeated.
To obtain a rough estimate, we imagine that there are exactly $10^{(10^{23})}$ energy levels per J. Even if the experimentalist can prepare the energy of the system with a fantastically small uncertainty of $10^{-(10^{22})}$J, there still remain $N:=10^{0.9\times 10^{23}}$ energy levels which may be occupied with significant probabilities. If all of them are populated equally, we obtain $p_n=1/N$ for $N$ of the indices $n$, and $p_n=0$ for all other $n$. If not all $N$ levels are populated equally, but rather any $p_n$ may assume arbitrary values between zero and $10^{(10^{22})}$ times the average population $1/N$, we still obtain $p_n\leqslant 10^{-0.8\times 10^{23}}$. Returning to the general case, we can conclude [@rei08; @rei08a; @rei10] that even if the system’s energy is fixed up to an extremely small experimental uncertainty, and even if the energy levels are populated extremely unequally, we still expect that even the largest ensemble-averaged level population $p_n$ will be extremely small and typically satisfy the rough estimate $$\max_n p_n=10^{-\Or(f)}.
\label{12}$$
Macroscopic population of one energy level {#s25}
------------------------------------------
There is one physically significant situation in which the above arguments may become questionable. Namely, for an isolated macroscopic system which approaches a thermal equilibrium state with an extremely low temperature, it might be conceivable that the ground state energy exhibits a macroscopic population, i.e. the corresponding $p_n$ is no longer extremely small. Hence, we should omit that specific $p_n$ in the maximization , formally written as $${\max_n}' p_n=10^{-\Or(f)}.
\label{13}$$ In other words, the prime indicates that the largest $p_n$ is not included into the maximization and hence ${\max_n}' p_n$ [*represents the second largest level population*]{}.
Further situations resulting in a non-small population of one single level may be caused e.g. by certain ‘gaps’ in the energy spectrum or by one level with an extremely high multiplicity .
Note that the expected relations $p_n\geqslant 0$ and $\sum_n p_n=1$ readily follow from , , and the fact that $\rho(t)$ is non-negative and of unit trace. We thus can conclude that the maxima $\max_n p_n$ and ${\max_n'} p_n$ indeed exists—as anticipated in and —and that $0\leqslant {\sum_n'} p_n<1$, where the prime in ${\sum_n'}$ excludes the index $n$ belonging to the maximal $p_n$. It follows that $$\begin{aligned}
{\sum_n}' p_n^2 \leqslant {\max_n}' p_n
{\sum_n}' p_n \leqslant {\max_n}' p_n,
\label{14}
\\
{\max_n}' p_n = \big( {\max_n}' p_n^2 \big)^{1/2}
\leqslant \big( {\sum_n}' p_n^2 \big)^{1/2},
\label{15}\end{aligned}$$ and we can conclude that $${\max_n}' p_n \; \mbox{small}\ \Leftrightarrow\
{\sum_n}' p_n^2\; \mbox{small}.
\label{16}$$
In references [@lin09; @lin10; @sho11; @sho12], the quantity $d_{{\mathrm{eff}}}:=1/\sum_n p_n^2$, called the effective dimension of the state $\rho(t)$, is introduced. It quantifies the number of distinct energies that contribute notably to this state, and is required to be a large number. Observing that the equivalence also applies without primes, we see that the requirement of a large effective dimension is fulfilled if and only if the maximal level population $\max_n p_n$ is small. However, in the more general case including the primes in , as considered in our present work, the effective dimension $d_{{\mathrm{eff}}}$ may not be large any more.
Equilibration and equilibrium ensemble {#s26}
--------------------------------------
Generically, the statistical ensemble $\rho(t)$ is not stationary right from the beginning, in particular for an initial condition $\rho(0)$ out of equilibrium. But if the right hand side of depends on $t$ initially, it cannot approach for large $t$ any time-independent ‘equilibrium ensemble’ whatsoever. In fact, any mixed state $\rho(t)$ returns arbitrarily close (with respect to some suitable distance measure in Hilbert space) to its initial state $\rho(0)$ for certain, sufficiently large times $t$, as demonstrated for instance in appendix D of [@hob71].
We will therefore focus on the weaker notion of equilibration outlined in , requiring the existence of a time-independent ‘equilibrium state’ $\omega$ (density operator) with the property that the difference $$\sigma(t):=\Tr\{\rho(t) A\}-\Tr\{\omega A\}
\label{17}$$ between the true expectation value $\Tr\{\rho(t) A\}$ and the equilibrium reference value $\Tr\{\omega A\}$ is unresolvably small for the overwhelming majority of times $t$ contained in any sufficiently large (but finite) time interval $[0,T]$. (Note that initial transients become irrelevant if $T$ is chosen large enough.)
Heuristically, if any such equilibrium ensemble exists, then it should be given by the infinite time average of $\rho(t)$. In view of this suggests the definition $$\omega:=\sum_n\rho_{nn}.
\label{18}$$ However, from a more rigorous viewpoint, it is not so obvious that averaging over arbitrary but finite times leads to a well-defined long-time limit, which is furthermore given by . Specifically, for infinite-dimensional systems, interchanging the infinite time limit with the infinite double-sum in is problematic. We avoid all these difficulties by defining $\omega$ according to without any reference to averages over time. An alternative and entirely unproblematic viewpoint is to consider $\omega$ as the time-independent part of $\rho(t)$.
One readily sees that $\omega$ inherits from $\rho(t)$ the properties of being self-adjoint, non-negative, and of unit trace. Furthermore, $\omega$ satisfies the trivial time evolution $U_t \omega U_t^\dagger=\omega$. In other words, $\omega$ is indeed a perfectly well-defined density operator.
We finally note that, as far as the differences in are concerned, nothing changes if $A$ is replaced by $A+c\, 1$ with an arbitrary real $c$. Thus, we henceforth can assume without loss of generality that $a_{\sup}=-a_{\inf}$ in , implying with that $${\|A\|}={\Delta_{\! A}}/2.
\label{19}$$
From infinite to finite dimensions {#s3}
==================================
We focus on infinite-dimensional Hilbert spaces ${{\cal H}}$ and denote the normalized eigenvectors of the Hamiltonian $H$ by $|\nu\rangle$ with $\nu=1,2,\dots$. For any given positive integer $d$ we define the projectors $$\begin{aligned}
P := \sum_{\nu =1}^d |\nu\rangle\langle\nu|,
\label{20}
\\
Q := 1-P.
\label{21}\end{aligned}$$ For an arbitrary density operator $\rho$ and any observable $A$ it follows that $$\Tr\{\rho A\} = \Tr\{(P+Q)\rho (P+Q) A\}= R_1+R_2+R_3
\label{22}$$ with $$\begin{aligned}
R_1 := \Tr\{ P \rho P A \},
\label{23}
\\
R_2 := \Tr\{Q \rho A\},
\label{24}
\\
R_3 := \Tr\{P \rho Q A\}.
\label{25}\end{aligned}$$ Making use of $P^2=P$ and the cyclic invariance of the trace, one can rewrite $R_1$ as $\Tr\{(P \rho P) (P A P)\}$ or $$\begin{aligned}
R_1 & \phantom{:}= \Tr\{\tilde\rho\tilde A\},
\label{26}
\\
\tilde\rho & := P \rho P,
\label{27}
\\
\tilde A & := P A P.
\label{28}\end{aligned}$$
To further evaluate $R_2$ we represent the self-adjoint operator $\rho$ in terms of its eigenvalues $\rho_\nu$ and eigenvectors $|\phi_\nu\rangle$, $$\rho = \sum_{\nu=1}^\infty \rho_\nu|\phi_\nu\rangle\langle\phi_\nu|.
\label{29}$$ Since $\rho$ is non-negative, all $\rho_\nu$ are non-negative and $$\rho^{1/2} :=
\sum_{\nu=1}^\infty \sqrt{\rho_\nu}\
|\phi_\nu\rangle\langle\phi_\nu|
\label{30}$$ is a well-defined, self-adjoint operator with the property that $\rho^{1/2} \rho^{1/2}= \rho$. It follows that $$|R_2|^2 = \bigl|\Tr\{(Q \rho^{1/2}) (\rho^{1/2} A)\}\bigr|^2
\leqslant \Tr\{Q\rho Q\} \Tr\{A \rho A\},
\label{31}$$ where we exploited the Cauchy–Schwarz inequality $$\bigl|\Tr\{B^\dagger C\}\bigr|^2\leqslant \Tr\{B^\dagger B \} \Tr\{C^\dagger C \}
\label{32}$$ for the scalar product $\Tr\{B^\dagger C\}$ of arbitrary operators $B$ and $C$ (for which all traces in exist). The last term in equals $\Tr\{\rho A^2\}$, and by evaluating the trace by means of the orthonormal basis $|\phi_n\rangle$ one can infer with and that $$\Tr\{\rho A^2\}\leqslant {\|A^2\|} \Tr\{\rho\}\leqslant {\|A\|}^2,
\label{33}$$ where we exploited that $\Tr\{\rho\}=1$ and ${\|A^2\|}={\|A\|}^2$ in the last relation. Finally, we conclude from and that $$\Tr\{Q\rho Q\} = \sum_{\nu=d+1}^\infty \langle\nu|\rho|\nu\rangle
\label{34}$$ and hence $$|R_2|^2 \leqslant{\|A\|}^2 \sum_{\nu=d+1}^\infty \langle\nu|\rho|\nu\rangle.$$
Next, starting from , we rewrite $R_3$ as $\Tr\{AP\rho Q\}$. Noting that all four operators under this trace are self-adjoint and that $\Tr\{B^\dagger\}=\Tr\{B\}^\ast$ for arbitrary operators $B$, we can conclude that $|R_3|=|\Tr\{Q\rho PA\}|$. Proceeding similarly as in – and using $\Tr\{\tilde{\rho}\}\leqslant1$ one finds that $$|R_3|^2\leqslant\Tr\{Q\rho Q\}{\|A\|}^2 \Tr\{\tilde{\rho}\}\leqslant {\|A\|}^2 \sum_{\nu=d+1}^\infty \langle\nu|\rho|\nu\rangle.$$ This in turn implies via – that $$\label{35}
|\Tr\{\rho A\}-\Tr\{\tilde\rho\tilde A\}|\leqslant 2 {\|A\|} \Bigl(\sum_{\nu=d+1}^\infty \langle\nu|\rho|\nu\rangle\Bigr)^{1/2}$$ for arbitrary density operators $\rho$. Applying this relation to the specific density operators $\rho(t)$ in and $\omega$ in we obtain $$|\Tr\{\rho(t) A\}-\Tr\{\omega A\}| =
|\Tr\{\tilde\rho(t) \tilde A\}-\Tr\{\tilde \omega \tilde A\}| + R
\label{36}$$ with $$\begin{aligned}
R &\leqslant |\Tr\{\rho(t) A\}-\Tr\{\omega A\}
- \Tr\{\tilde\rho(t) \tilde A\}+\Tr\{\tilde \omega \tilde A\}|
\label{37}\\
& \leqslant 4{\|A\|} \Big(\sum_{\nu=d+1}^\infty
\langle\nu|\rho (0) |\nu\rangle\Big)^{1/2}.
\label{38}\end{aligned}$$ In we exploited the fact that $|x|-|y|\leqslant |x-y|$ for arbitrary real $x$ and $y$, while in we applied to $\rho(t)$ and $\omega$ and we took into account that, according to and , both $\langle \nu | \rho(t)|\nu\rangle$ and $\langle \nu | \omega|\nu\rangle$ are equal to $\langle \nu |\rho(0)|\nu\rangle$.
Observing that $\sum_{\nu=1}^d\langle \nu |\rho(0)|\nu\rangle$ increases with $d$ and approaches $\Tr\{\rho(0)\}=1$ for $d\to\infty$ it follows that, for any given $\epsilon>0$, there exists a finite $d(\epsilon)$ with the property that $R\leqslant {\|A\|}\epsilon$. According to , upon choosing $\epsilon=10^{-20}$ and observing , we can conclude that $$R\leqslant {\delta \! A}/2
\label{39}$$ with one common, finite $d$ for all experimentally realistic observables $A$.
As far as experimentally resolvable differences are concerned, it thus follows from and that it is sufficient to consider, instead of $\rho(t)$ and $A$, their counterparts $\tilde \rho(t)$ and $\tilde A$. According to , and these are the projections/restrictions of the original operators $\rho(t)$ and $A$ onto the finite dimensional sub-Hilbert space $\tilde{{\cal H}}\subset {{\cal H}}$, spanned by the first $d$ energy eigenvectors $\{|\nu\rangle\}_{\nu=1}^d$. Note that while $\tilde{{\cal H}}$ is independent of $A$, it does depend on $\rho(0)$ and $H$ in general.
It remains to be shown that the entire framework set out in can be consistently restricted to the finite-dimensional Hilbert space $\tilde {{\cal H}}$: Observing that the projectors $P_n$ from commute with the projector $P$ defined in implies that $$\tilde P_n:=P P_n P=P P_n=P_n P.
\label{40}$$ Setting $\tilde H:=PHP$ and keeping only indices $n$ with non-zero $\tilde P_n$, relations – remain valid, but now with finite sums $\sum_n$ and finite multiplicities $\tilde \mu_n$. Furthermore, one sees that $\tilde H$ indeed reproduces the correct time evolution of $\tilde \rho(t)$ with finite sums in . While $\tilde \rho(t)$ is still non-negative and self-adjoint, the trace now satisfies $$\Tr\{\tilde \rho(t)\}=\sum_n\tilde p_n\leqslant 1.
\label{41}$$ Likewise, with respect to the operator norm and the level population one finds that $${\|\tilde A\|} \leqslant {\|A\|},\qquad
\tilde p_n \leqslant p_n.
\label{43}$$
Finite-dimensional systems {#s4}
==========================
The main objective of this section is to establish bounds on the difference $$\tilde \sigma(t) :=\Tr\{\tilde \rho(t)\tilde A\}
-\Tr\{\tilde\omega \tilde A\},
\label{44}$$ where tildes indicate the projections/restrictions to the finite-dimensional Hilbert space $\tilde {{\cal H}}$ from the previous subsection in case the original Hilbert space ${{\cal H}}$ was infinite-dimensional (otherwise the tildes are redundant). We recall that $d<\infty$ denotes the dimension of $\tilde {{\cal H}}$ (see below ) and $\tilde d_E\leqslant d$ the number of distinct energy eigenvalues $E_n$ of $\tilde H$ (see below ).
Adopting the approach of Short and Farrelly [@sho12], we start by considering the quantity $\langle \tilde \sigma^2(t)\rangle_T$, where $\langle \cdot\rangle_T$ denotes a temporal average over the time interval $[0, T]$ with arbitrary but finite $T>0$. From and we can infer that $$\bigl\langle \tilde \sigma^2(t)\bigr\rangle_T=
\Biggl\langle \Bigl|\sum_{m\not=n} \Tr\{\tilde\rho_{mn}\tilde A\}
\exp\left[-\rmi(E_m-E_n)t\right]\Bigr|^2 \Biggr\rangle_T,
\label{45}$$ where $\tilde\rho_{mn}(0)$ is abbreviated as $\tilde\rho_{mn}$ and the sum runs over the finite set of pairs of labels $${{\cal G}}:=\bigl\{(m,n)\, |\, m ,n\in[1,\dots,\tilde d_E],\, m\not=n\bigr\}.
\label{46}$$ For any $\alpha=(m,n)\in{{\cal G}}$ we define $$G_\alpha := E_m-E_n,\qquad
v_\alpha := \Tr\{\tilde\rho_{mn}\tilde A\}.
\label{48}$$ We thus can rewrite as $$\bigl\langle \tilde \sigma^2(t)\bigr\rangle_T=
\Biggl\langle \Bigl|\sum_\alpha v_\alpha
\exp\left[-\rmi G_\alpha t\right]\Bigr|^2 \Biggr\rangle_T
=\sum_{\alpha,\beta} v^\ast_\alpha M_{\alpha\beta} v_\beta,
\label{49}$$ where we introduced the self-adjoint, non-negative, finite-dimensional matrix $M$ with matrix elements $$M_{\alpha\beta}:=\bigl\langle \exp\left[\rmi(G_\alpha-G_\beta)t\right]\bigr\rangle_T.
\label{50}$$ Denoting by ${\|M\|}$ the standard operator norm of the matrix $M$ (see ), it follows that [@sho12] $$\begin{aligned}
\bigl\langle \tilde \sigma^2(t)\bigr\rangle_T \leqslant S {\|M\|}
\label{51}
\\
S := \sum_{\alpha} |v_\alpha|^2
=\sum_{m\not = n} \bigl|\Tr\{\tilde\rho_{mn}\tilde A\}\bigr|^2.
\label{52}\end{aligned}$$
Bounds on the two factors $S$ and ${\|M\|}$ in are constructed in the following two subsections.
Bound on $S$ {#s41}
------------
We exploit , , and the cyclic invariance of the trace to conclude that $$\begin{aligned}
\Tr\{\tilde\rho_{mn}\tilde A\} & = &
\Tr\{\tilde P_m \tilde\rho \tilde P_n \tilde A \tilde P_m\}.
\label{53}\end{aligned}$$ Similarly to the derivation in –, we write $$\tilde P_m \tilde\rho \tilde P_n \tilde A\tilde P_m = (\tilde P_m \tilde\rho^{1/2})(\tilde\rho^{1/2} \tilde P_n \tilde A\tilde P_m)$$ from which it follows that $$|\Tr\{\tilde\rho_{mn}\tilde A\}|^2
\leqslant \tilde p_m
\Tr\{\tilde\rho_{nn} \tilde A \tilde P_m\tilde A\}
.
\label{54}$$
We first evaluate by means of all summands in with $n=1$, $$\begin{aligned}
S_{n=1} & \leqslant & \sum_{m\geqslant 2}
\tilde p_m \Tr\{\tilde\rho_{11} \tilde A \tilde P_m\tilde A\}
\nonumber
\\
& \leqslant &
\max_{n\geqslant 2} p_n
\Tr\Bigl\{\tilde\rho_{11} \tilde A \sum_{m\geqslant 2}\tilde P_m\tilde A\Bigr\}
\nonumber
\\
& \leqslant &
\max_{n\geqslant 2} p_n\Tr\{\tilde\rho_{11}\} \Bigl\|\tilde A \sum_{m\geqslant 2}\tilde P_m\tilde A\Bigr\|
\nonumber
\\
& \leqslant &
\max_{n\geqslant 2} p_n {\|A\|}^2.
\label{55}\end{aligned}$$ In the second line we used that $\tilde p_m\leqslant p_m \leqslant \max_{n\geqslant 2}p_n$ for all $m\geqslant 2$, see . The third line is based on a similar line of reasoning as in , exploiting that $\tilde\rho_{11}$ is a non-negative, self-adjoint operator. In the last line we used that $\Tr\{\tilde\rho_{11}\}=\tilde p_1\leqslant 1$ (see and ), that ${\|BC\|}\leqslant {\|B\|} {\|C\|}$ for arbitrary operators $B$, $C$ of finite norm, and that $\sum_{m\geqslant 2}\tilde P_m$ is a projector and hence of unit norm.
For symmetry reasons, the same estimate as in applies for the summands with $m=1$ in . The remaining summands in satisfy $m\not = n$ and $m, n\geqslant 2$. By including also those with $m=n$, the sum can only increase, resulting in $$S\leqslant 2 {\|A\|}^2 {\max_n}' p_n +
{\sum_{m,n}}' \bigl|\Tr\{\tilde\rho_{mn}\tilde A\}\bigr|^2,
\label{56}$$ where the prime indicates that indices $1$ are excluded from the maximization and the summation. Clearly, instead of this special index $1$ we could have selected any other index as well. Thus, in agreement with , the prime can and will be understood as excluding the index belonging to the maximally populated level.
The remaining sum in can be estimated by means of the two non-negative, self-adjoint operators $$\tilde\omega' := {\sum_n}' \tilde \rho_{nn},\qquad
\tilde\omega'' := {\sum_m}' \tilde p_m \tilde P_m
\label{58}$$ in the following way: $$\begin{aligned}
{\sum_{m,n}}' \bigl|\Tr\{\tilde\rho_{mn}\tilde A\}\bigr|^2
\leqslant
\Tr\{\tilde\omega' \tilde A \tilde \omega'' \tilde A\}
\leqslant
\sqrt{
\Tr\{(\tilde \omega')^2 \tilde A^2\} \Tr\{(\tilde \omega'')^2 \tilde A^2\}
}
\nonumber
\\
\leqslant
\sqrt{
\Tr\{(\tilde \omega')^2\} {\|\tilde A\|}^2
\Tr\{(\tilde \omega'')^2\} {\|\tilde A\|}^2
}
=
{\|\tilde A\|}^2 \sqrt{{\sum_n}' \Tr\{\tilde \rho_{nn}^2\}
{\sum_m}' \tilde \mu_m \tilde p_m^2}.
\label{59}\end{aligned}$$ In the first inequality, we exploited , the next two ones follow by arguments similar to the ones in and . The last equation is based on , , and .
Once again, the line of reasoning in follows very closely that of Short and Farrelly [@sho11; @sho12]. The main difference is that these authors focus, in a first step, solely on pure states and only in the end extend their result to mixed states via purification. Along this line, one actually arrives at a final result which is slightly different from , namely $${\sum_{m,n}}' |\Tr\{\tilde\rho_{mn}\tilde A\}|^2
\leqslant {\|\tilde A\|}^2 {\sum_n}' \tilde p_n^2.
\label{60}$$ Closer inspection shows that and agree if and only if all energies are non-degenerate. In any other case, one can show that the bound is sharper than . On the other hand, one readily sees that $\Tr\{\tilde\rho_{nn}^2\}\leqslant \tilde p_n^2$ and hence $$\sqrt{{\sum_n}' \Tr\{\tilde \rho_{nn}^2\}
{\sum_m}' \tilde \mu_m \tilde p_m^2}
\leqslant
{\max_m}'\sqrt{\tilde \mu_m}\,{\sum_n}' \tilde p_n^2.
\label{61}$$ The equality sign applies whenever $\rho(t)$ is a pure state contained in the maximally degenerate energy eigenspace, in any other case the inequality sign applies. In conclusion, outperforms by at most a factor of ${\max_m'}\sqrt{\tilde \mu_m}$, i.e. the square root of the maximal energy degeneracy.
The purification argument [@sho11; @sho12] behind is mathematically very appealing (somewhat reminiscent of evaluating real integrals by ‘complexification’) but its physical content remains slightly mysterious. It is reassuring that one can get at least as far as without this argument, but it is annoying that could not be fully recovered.
Working with the stronger bound by Short and Farrelly, the estimate for $S$ in combination with and takes on the simple form $$S\leqslant 3 {\|A\|}^2 {\max_n}' p_n.
\label{62}$$ While the maximization so far includes only a finite number of indices $n$, this restriction can be readily released, as the maximum can only increase in this way. As in , the last factor then represents the second largest level population of the original, possibly infinite-dimensional system.
Bound on ${\|M\|}$ in terms of energy gaps {#s42}
------------------------------------------
The main idea is that, since $M$ is a finite-dimensional matrix, its operator norm ${\|M\|}$ converges towards a well-defined limit as the averaging time $T$ (see , ) tends to infinity. Hence, ${\|M\|}$ can be readily bounded from above for all sufficiently large (but finite) $T$.
Quantitatively, by setting out from the inequality $${\|M\|}\leqslant \max_\beta\sum_\alpha |M_{\alpha \beta}|,
\label{63}$$ Short and Farrelly [@sho12] derived the estimate $${\|M\|}\leqslant N(\epsilon)
\left(1+\frac{8\log_2\! \tilde d_E}{\epsilon T}\right)
\label{64}$$ for arbitrary $\epsilon>0$ and $T>0$. Here, $\tilde d_E$ is the number of distinct energy eigenvalues $E_n$ (see below ) and $$N(\epsilon) :=
\max_{E} \bigl|\{ \alpha \in {{\cal G}}\, | \, G_\alpha \in[E,E+\epsilon)\}\bigr|.
\label{65}$$ According to and , $N(\epsilon)$ is thus the maximum number of energy gaps $G_\alpha=E_m-E_n$ in any interval of size $\epsilon$.
The main implication of this result can also be deduced directly from : since we are dealing with finite-dimensional systems, the [*finite number*]{} of all matrix elements $|M_{\alpha\beta}|$ in with $G_\alpha\not=G_\beta$ can be [*simultaneously*]{} bounded by an arbitrarily small upper limit for sufficiently large $T$. Hence, their contribution to can be made smaller than the contribution of all the remaining summands, satisfying $G_\alpha=G_\beta$ and thus $M_{\alpha\beta}=1$. It follows for all sufficiently large $T$ that $${\|M\|}\leqslant 2g,
\label{66}$$ where $$g :=
\max_\beta\left|\{ \alpha\in {{\cal G}}\, | \, G_\alpha=G_\beta\}\right|
\label{67}$$ denotes the maximal degeneracy of energy gaps. Note that only the energy eigenvalues $E_n$ of the restricted, finite-dimensional Hamiltonian $\tilde H$ contribute to ${{\cal G}}$ in and hence to the degenerate energy gaps counted in .
Problems in the infinite-dimensional limit {#s43}
------------------------------------------
So far, we have assumed a finite dimensionality $d$ of the Hilbert space $\tilde {{\cal H}}$. In this subsection we argue that it is intuitively suggestive that everything ‘should go well’ upon letting $d$ go to infinity, but that a more rigorous justification is problematic. From the latter viewpoint, the considerations in are thus indispensable for infinite-dimensional systems.
First of all, since $A$ has a finite range ${\Delta_{\! A}}$, see , it follows that $\sigma(t)$ from is contained in the finite interval $[-2{\Delta_{\! A}}, 2{\Delta_{\! A}}]$ for all times $t$. This suggest (but does not prove) that the temporal average $\langle \sigma^2(t)\rangle_T$ converges in the limit $T\to\infty$ even for infinite dimensional systems: although one can readily construct mathematical examples of bounded functions without a well-defined infinite-time average, it appears plausible that ‘reasonable’ physical models will result in functions $\sigma^2(t)$ which do not exhibit the pathologies of those examples.
On the other hand, for any given finite dimension $d$, equation has a well defined $T\to\infty$ limit. Focusing on the simplest case of non-degenerate energy gaps, $M_{\alpha\beta}$ in approaches $\delta_{\alpha\beta}$ and hence the inequality turns into an equality with ${\|M\|}=1$. Since $S$ in is positive and bounded by the $d$-independent estimate , it is once again suggestive that $S$ itself converges for $d\to\infty$. Under the further assumption that the two limits $T\to\infty$ and $d\to\infty$ commute, one then readily finds an upper bound analogous to for infinite dimensions and all sufficiently large, but finite, $T$.
Although these heuristic arguments appear plausible at first glance, some subtle open questions remain upon closer inspection: For infinite-dimensional systems one typically expects the existence of arbitrarily small, but non-vanishing energy gap differences $G_\alpha-G_\beta$. While each single matrix element then still converges for $T\to\infty$, the same is no longer clear for the entire, infinite-dimensional matrix $M$ and/or its norm ${\|M\|}$. For the same reason, $N(\epsilon)$ from is expected to diverge for $d\to\infty$ and any fixed $\epsilon>0$, so that becomes useless. Likewise, our derivation of breaks down. In other words, interchanging the limits $T\to\infty$ and $d\to\infty$ is a rather delicate, unsettled issue.
Main result and discussion {#s5}
==========================
Combining , , and , we obtain the inequality $$\bigl\langle \tilde \sigma^2(t)\bigr\rangle_T
\leqslant \case{3}{2} g{\Delta_{\! A}}^2 {\max_n}' p_n
\label{68}$$ for all sufficiently large $T$.
For any given $\epsilon>0$ and $T>0$ we define the measure of all times $t\in [0, T]$ for which $|\tilde\sigma(t)|\geqslant\epsilon$ holds true, $$\tilde T_\epsilon :=
\lambda\big( \{t\, | \, t\in [0,T] \ \mbox{and}\
|\tilde\sigma(t)|\geqslant\epsilon\}\big),
\label{69}$$ where $\lambda$ denotes the Lebesgue measure. It follows that $\tilde\sigma^2(t)\geqslant \epsilon^2$ for a set of times $t$ of measure $\tilde T_\epsilon$ and $\tilde\sigma^2(t)\geqslant 0$ for all remaining times $t$ in $[0, T]$. Hence the temporal average of $\tilde\sigma^2(t)$ over the time interval $[0, T]$ must be at least $\epsilon^2 \tilde T_\epsilon/T$, $$\bigl\langle \tilde\sigma^2(t)\bigr\rangle_T\geqslant
\frac{\epsilon^2\tilde T_\epsilon}{T}.
\label{70}$$ Choosing $\epsilon={\delta \! A}/2$, we can conclude from and that $$\frac{\tilde T_{{\delta \! A}/2}}{T}\leqslant
6g\left(\frac{{\Delta_{\! A}}}{{\delta \! A}}\right)^2 {\max_n}' p_n
\label{71}$$ for all sufficiently large $T$.
Next we infer from , and that $$|\Tr\{\rho(t)A\}-\Tr\{\omega A\}|\leqslant |\tilde\sigma(t)|+\frac{{\delta \! A}}{2}.
\label{72}$$ Analogously to , we define the measure of all times $t\in [0, T]$ with the property that $|\Tr\{\rho(t)A\}-\Tr\{\omega A\}|$ exceeds the experimental resolution limit ${\delta \! A}$, $$T_{{\delta \! A}} :=
\lambda \big( \{t\, | \, t\in [0,T] \ \mbox{and}\
|\Tr\{\rho(t)A\}-\Tr\{\omega A\}|\geqslant{\delta \! A}\}\big).
\label{73}$$ In view of with $\epsilon={\delta \! A}/2$ and , we conclude that $T_{{\delta \! A}}\leqslant\tilde T_{{\delta \! A}/2}$. With we thus arrive at the main result of our present work, $$\frac{T_{{\delta \! A}}}{T}\leqslant
6 g\left(\frac{{\Delta_{\! A}}}{{\delta \! A}}\right)^2 {\max_n}' p_n
\label{74}$$ for all sufficiently large $T$.
The left hand side of represents the fraction of all times $t\in[0, T]$ for which there is an experimentally resolvable difference between the true expectation value $\Tr\{\rho(t)A\}$ and the time-independent ‘equilibrium expectation value’ $\Tr\{\omega A\}$.
On the right hand side, $g$ is the maximal degeneracy of energy gaps from , i.e. the maximal number of (exactly) coinciding energy differences among all possible pairs of distinct energy eigenvalues of the reduced, finite-dimensional Hamiltonian $\tilde H$ from , and as such is determined by properties of both the Hamiltonian $H$ and the initial condition $\rho(0)$. Alternatively, one may also take into account all energies of the full system Hamiltonian $H$, since the maximum can only increase in this way, but this increase might possibly become prohibitively huge for infinite dimensional systems (see also ).
The next factor ${\Delta_{\! A}}/{\delta \! A}$ appearing in is the range-to-resolution ratio of $A$ from , i.e. a characteristic property of the observable $A$ only, and can be considered as bounded according to for all experimentally realistic measurements $A$. Going back to , one readily sees that one could as well replace ${\Delta_{\! A}}$ by the range of the reduced observable $\tilde A$. This range typically is somewhat smaller than the original range ${\Delta_{\! A}}$ of $A$ (see ), but this gain might often not be worth the effort.
Finally, ${\max_n}' p_n$ in stands for the second-largest, ensemble-averaged occupation probability of the (possibly degenerate) energy eigenvalues $E_n$, see sections \[s24\] and \[s25\]. Similarly as for $A$, one alternatively could maximize over the reduced level populations $\tilde p_n$, but often this will not be worthwhile. Essentially, ${\max_n}' p_n$ is thus a characteristic property of the initial condition $\rho(0)$, but obviously also the Hamiltonian $H$ itself matters. Typically, one expects that the rough upper bound applies, except if certain energy eigenvalues are so extremely highly degenerate that the multiplicities defined in severely reduce the pertinent energy level density compared to the non-degenerate case, see .
For a system with sufficiently many degrees of freedom $f$ and no exceedingly large degeneracy of the energy eigenvalues and the energy gaps, we thus can conclude from with and that the system behaves in every possible experiment exactly as if it were in the equilibrium state $\omega$ for the overwhelming majority of times within any sufficiently large (but finite) time interval $[0, T]$. In particular, $T$ must obviously be much larger than the relaxation time in case of a far-from-equilibrium initial condition $\rho(0)$. A more detailed quantitative bound on $T$ follows from the result by Short and Farrelly [@sho12].
We emphasize that even in the absence of any measurable difference between $\rho(t)$ and $\omega$, the equilibrium state $\omega$ itself is never realized in the actual system, and as such is a purely formal, theoretical construct. In particular, the difference between $\Tr\{\rho(t)A\}$ and $\Tr\{\omega A\}$ is not a quantity one can measure in a real system, not even in principle.
Further interesting physical implications of are discussed, in e.g., [@rei10].
Conclusions {#s6}
===========
To summarize, by adopting and extending recent works by Short [@sho11] and by Short and Farrelly [@sho12], we demonstrated equilibration of isolated macroscopic quantum systems in the sense that deviations from a time-independent steady state are unmeasurably small for the overwhelming majority of times within any sufficiently large time interval. This conclusion applies for arbitrary systems with (countably) infinite dimensions, initial states exhibiting a macroscopic population of at most one energy level, and Hamiltonians without exceedingly large degeneracies of energy eigenvalues and energy gaps.
As soon as a model includes at least one continuous degree of freedom (e.g. a spatial coordinate), the pertinent Hilbert space is necessarily of infinite dimension. If the system is furthermore of finite spatial extension (e.g. due to confining walls), the Hamiltonian $H$ features a discrete spectrum and can be written in the form . Both conditions are clearly satisfied in almost any model which is not based on some extreme simplifications. Moreover, it is practically impossible to prepare a real system so that only a finite number of energy eigenstates is populated. Rather, generically an infinite number of them contributes with possibly small, yet finite, amplitudes. Reducing or truncating this situation in whatever way to finite dimensions is especially problematic with respect to rigorous statements about the exact quantum mechanical evolution over arbitrarily long time intervals. More precisely, interchanging the limit of infinite dimensions with the long time limit, as discussed in , is a subtle problem, justifying the detailed treatment of the infinite-dimensional case in .
Admitting systems with degenerate energy eigenvalues and energy gaps are among the most important steps forward achieved in [@sho11; @sho12]. On the other hand, it is generally taken for granted that such degeneracies are absent in generic Hamiltonians, see e.g.[@per84; @sre99; @tas98; @rei08; @lin09; @rei10] and, in particular, section 3.2.1 of [@gol06] and references therein. Roughly speaking, Hamiltonians with such degeneracies are of measure zero compared to ‘all’ Hamiltonians. They only arise in the presence of special reasons like (perfect) symmetries, additional conserved quantities, or fine-tuning of parameters, which can be ruled out for typical real systems provided they cannot be further decomposed into non-interacting subsystems [@lin09]. It is reassuring that even some of those exceptions—namely those without exceedingly high degeneracies—are now covered in [@sho11; @sho12] and in the present work. Accidental degeneracies due to fine-tuning of parameters should not lead to high degeneracies, but the quantitative effect of symmetries on level and gap degeneracies is not clear to the present authors. In order to better understand the occurrence of degeneracies, it appears indispensable to study specific examples. The harmonic oscillator would be one which is still not admissible (as the degeneracy of energy gaps is too large), but the hope that other relevant examples will be ‘tame’ enough seems reasonable.
PR acknowledges financial support by the Deutsche Forschungsgemeinschaft under grant RE1344/7-1. MK acknowledges financial support by the Incentive Funding for Rated Researchers programme of the National Research Foundation of South Africa.
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author:
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Guangming Lang$^{1,2,3}$ [^1] Mingjie Cai$^{4}$\
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title: Incremental approaches to knowledge reduction of covering decision information systems with variations of coverings
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> [**Abstract.**]{} In practical situations, calculating approximations of concepts is the central step for knowledge reduction of dynamic covering decision information system, which has received growing interests of researchers in recent years. In this paper, the second and sixth lower and upper approximations of sets in dynamic covering information systems with variations of coverings are computed from the perspective of matrix using incremental approaches. Especially, effective algorithms are designed for calculating the second and sixth lower and upper approximations of sets in dynamic covering information systems with the immigration of coverings. Experimental results demonstrate that the designed algorithms provide an efficient and effective method for constructing the second and sixth lower and upper approximations of sets in dynamic covering information systems. Two examples are explored to illustrate the process of knowledge reduction of dynamic covering decision information systems with the covering immigration.
>
> [**Keywords:**]{} Characteristic matrix; Covering information system; Covering decision information system; Rough set\
Introduction
============
Covering-based rough set theory[@Zakowski] as a generalization of Pawlak’s rough sets is a powerful mathematical tool to deal with uncertainty and imprecise information in the field of knowledge discovery and rule acquisition. To handle with uncertainty knowledge, researchers have investigated covering-based rough set theory[@Liu7; @Chen4; @Miao; @Qian] and presented three types approximation operators summarized by Yao[@Yao2] as follows: element-based operators, granular-based operators and system-based operators for covering approximation spaces, and discussed the relationships among them. Additionally, all approximation operators are also classified into dual and non-dual operators and their inner properties are investigated.
Researchers have proposed many lower and upper approximation operators with respect to different backgrounds. Particularly, they[@Tan1; @Tan2; @Wang3; @Yang1; @Yang2; @Yao1; @Zhang1; @Zhang4; @Zhang5; @Zhang6; @Zhu1; @Zhu2; @Zhu3] have investigated approximation operators from the view of matrix. For example, Liu[@Liu1] provided a new matrix view of rough set theory for Pawlak’s lower and upper approximation operators. He also represented a fuzzy equivalence relation using a fuzzy matrix and redefined the pair of lower and upper approximation operators for fuzzy sets using the matrix representation in a fuzzy approximation space. Wang et al.[@Wang3] proposed the concepts of the type-1 and type-2 characteristic matrices of coverings and transformed the computation of the second, fifth and sixth lower and upper approximations of a set into products of the type-1 and type-2 characteristic matrices and the characteristic function of the set in covering approximation spaces. Zhang et al.[@Zhang1] proposed the matrix characterizations of the lower and upper approximations for set-valued information systems. He[@Zhang4; @Zhang5; @Zhang6] also presented efficient parallel boolean matrix based algorithms for computing rough set approximations in composite information systems and incomplete information systems. Actually, because of the dynamic characteristic of data collection, there are a lot of dynamic information systems with variations of object sets, attribute sets and attribute values, and researchers[@Chen1; @Chen2; @Chen3; @Lang1; @Lang2; @Lang3; @Lang4; @Lang5; @Li1; @Li2; @Li3; @Li4; @Liang1; @Liu2; @Liu3; @Liu4; @Liu5; @Liu6; @Luo1; @Luo2; @Luo3; @Sang; @Shu1; @Shu2; @Shu3; @Wang1; @Wang2; @Yang3; @Zhang1; @Zhang2; @Zhang3] have focused on knowledge reduction of dynamic information systems. Especially, researchers[@Zhang1; @Lang1; @Lang2] have computed approximations of sets for knowledge reduction of dynamic information systems from the view of matrix. For instance, Zhang et al.[@Zhang1] provided incremental approaches to updating the relation matrix for computing the lower and upper approximations with dynamic attribute variation in set-valued information systems. They also proposed effective algorithms of computing composite rough set approximations for dynamic data mining. Lang et al.[@Lang1; @Lang2] presented incremental algorithms for computing the second and sixth lower and upper approximations of sets from the view of matrix and investigated knowledge reduction of dynamic covering information systems with variations of objects. In practical situations, there are many dynamic covering information systems with the immigration and emigration of coverings, and computing the second and sixth lower and upper approximations of sets is time-consuming using the non-incremental algorithms in these dynamic covering information systems, it also costs more time to conduct knowledge reduction of dynamic covering information systems with variations of coverings. Therefore, it is urgent to propose effective approaches to updating the second and sixth lower and upper approximations of sets for knowledge reduction of dynamic covering decision information systems with the covering variations.
This work is to investigate knowledge reduction of dynamic covering decision information systems. First, we investigate the basic properties of dynamic covering information systems with variations of coverings. Particularly, we study the properties of the type-1 and type-2 characteristic matrices with the covering variations and the relationship between the original type-1 and type-2 characteristic matrices and the updated type-1 and type-2 characteristic matrices. We also provide incremental algorithms for updating the second and sixth lower and upper approximations of sets using the type-1 and type-2 characteristic matrices, respectively. We employ examples to illustrate how to update the second and sixth lower and upper approximations of sets with variations of coverings. Second, we generate randomly ten dynamic covering information systems with the covering variations randomly and compute the second and sixth lower and upper approximations of sets in these dynamic covering information systems. We also employ experimental results to illustrate the proposed algorithms are effective to update the second and sixth lower and upper approximations of sets in dynamic covering information systems. Third, we employ two examples to demonstrate that the designed algorithms are effective to conduct knowledge reduction of dynamic covering decision information systems with immigrations of coverings, which will enrich covering-based rough set theory from the matrix view.
The rest of this paper is organized as follows: Section 2 briefly reviews the basic concepts of covering-based rough set theory. In Section 3, we update the type-1 and type-2 characteristic matrices in dynamic covering information systems with variations of coverings. We design the incremental algorithms for computing the second and sixth lower and upper approximations of sets. We also provide examples to demonstrate how to calculate the second and sixth lower and upper approximations of sets. In Section 4, the experimental results illustrate the incremental algorithms are effective to construct the second and sixth lower and upper approximations of sets in dynamic covering information systems with the covering immigration. In Section 5, we explore two examples to illustrate how to conduct knowledge reduction of dynamic covering decision information systems with the covering immigration. Concluding remarks and further research are given in Section 6.
Preliminaries
=============
In this section, we briefly review some concepts related to covering-based rough sets.
[@Zakowski] Let $U$ be a finite universe of discourse, and $\mathscr{C}$ a family of subsets of $U$. Then $\mathscr{C}$ is called a covering of $U$ if none of elements of $\mathscr{C}$ is empty and $\bigcup\{C|C\in \mathscr{C}\}=U$. Furthermore, $(U,\mathscr{C})$ is referred to as a covering approximation space.
If $U$ is a finite universe of discourse, and $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\}$, where $\mathscr{C}_{i}(1\leq i\leq m)$ is a covering of $U$, then $(U,\mathscr{D})$ is called a covering information system, which can be viewed as a covering approximation space. Furthermore, if the coverings of $\mathscr{D}$ are classified into two categories: conditional attribute-based coverings and decision attribute-based coverings, then $(U,\mathscr{D})$ is referred to as a covering decision information system. For convenience, a covering decision information system is denoted as $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$, where $\mathscr{D}_{C}$ and $\mathscr{D}_{D}$ mean conditional attribute-based coverings and decision attribute-based coverings, respectively.
[@Wang3] Let $(U,\mathscr{C})$ be a covering approximation space, and $N(x)=\bigcap\{C_{i}|x\in C_{i}\in \mathscr{C}\}$ for $x\in U$. For any $X\subseteq U$, the second and sixth upper and lower approximations of $X$ with respect to $\mathscr{C}$ are defined as follows:
$(1)$ $SH_{\mathscr{C}}(X)=\bigcup\{C\in\mathscr{C}|C\cap X\neq
\emptyset\}$, $SL_{\mathscr{C}}(X)=[SH_{\mathscr{C}}(X^{c})]^{c}$;
$(2)$ $XH_{\mathscr{C}}(X)=\{x\in U|N(x)\cap X\neq \emptyset\}$, $XL_{\mathscr{C}}(X)=\{x\in U|N(x)\subseteq X\}$.
According to Definition 2.2, the second and sixth lower and upper approximation operators are important standards for knowledge reduction of covering information systems in covering-based rough set theory; they are also typical representatives of approximation operators for covering approximation spaces.
If $U=\{x_{1},x_{2},...,x_{n}\}$ is a finite universe of discourse, $\mathscr{C}=\{C_{1}, C_{2}, ..., C_{m}\}$ a family of subsets of $U$, and $M_{\mathscr{C}}=(a_{ij})_{n\times m}$, where $a_{ij}=\left\{
\begin{array}{ccc}
1,&{\rm}& x_{i}\in C_{j};\\
0,&{\rm}& x_{i}\notin C_{j}.
\end{array}
\right.$, then $M_{\mathscr{C}}$ is called a matrix representation of $\mathscr{C}$. Additionally, we also have the characteristic function $\mathcal {X}_{X}
=\left[\begin{array}{cccccc}
a_{1}&a_{2}&.&.&. & a_{n} \\
\end{array}
\right]^{T}$ for $X\subseteq U$, where $a_{i}=\left\{
\begin{array}{ccc}
1,&{\rm}& x_{i}\in X;\\
0,&{\rm}& x_{i}\notin X.
\end{array}
\right. $.
[@Wang3] Let $(U,\mathscr{C})$ be a covering approximation space, $A=(a_{ij})_{n\times m}$ and $B=(b_{ij})_{m\times p}$ Boolean matrices, and $A\odot B=(c_{ij})_{n\times p}$, where $c_{ij}=\bigwedge^{m}_{k=1}(b_{kj}-a_{ik}+1).$ Then
$(1)$ $\Gamma(\mathscr{C})=M_{\mathscr{C}}\bullet
M_{\mathscr{C}}^{T}=(d_{ij})_{n\times n}$ is called the type-1 characteristic matrix of $\mathscr{C}$, where $d_{ij}=\bigvee^{m}_{k=1}(a_{ik}\cdot
a_{jk})$, and $M_{\mathscr{C}}\bullet M_{\mathscr{C}}^{T}$ is the boolean product of $M_{\mathscr{C}}$ and its transpose $
M_{\mathscr{C}}^{T}$;
$(2)$ $\prod(\mathscr{C})=M_{\mathscr{C}}\odot M_{\mathscr{C}}^{T}=(e_{ij})_{n\times n}$ is called the type-2 characteristic matrix of $\mathscr{C}$.
We show the second and sixth lower and upper approximations of sets using the type-1 and type-2 characteristic matrices respectively as follows.
[@Wang3] Let $(U,\mathscr{C})$ be a covering approximation space, and $\mathcal {X}_{X}$ the characteristic function of $X$ in $U$. Then
$(1)$ $\mathcal {X}_{SH_{\mathscr{C}}(X)}=\Gamma(\mathscr{C})\bullet \mathcal
{X}_{X}$, $\mathcal {X}_{SL_{\mathscr{C}}(X)}=\Gamma(\mathscr{C})\odot \mathcal
{X}_{X}$; $(2)$ $\mathcal {X}_{XH_{\mathscr{C}}(X)}=\prod(\mathscr{C})\bullet
\mathcal {X}_{X}$, $\mathcal {X}_{XL_{\mathscr{C}}(X)}=\prod(\mathscr{C})\odot
\mathcal {X}_{X}$.
We present the concepts of the type-1 and type-2 reducts of covering decision information systems as follows.
[@Lang1] Let $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$ be a covering decision information system, where $\mathscr{D}_{C}=\{\mathscr{C}_{i}|i\in I\}$, $\mathscr{D}_{D}=\{D_{i}|i\in J\}$, I and J are indexed sets. Then $\mathscr{P}\subseteq \mathscr{D}_{C}$ is called a type-1 reduct of $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$ if it satisfies $(1)$ and $(2)$ as follows:
$(1)$ $\Gamma(\mathscr{D}_{C})\bullet
M_{\mathscr{D}_{D}}=\Gamma(\mathscr{P})\bullet M_{D_{i}},
\Gamma(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}}=\Gamma(\mathscr{P})\odot
M_{\mathscr{D}_{D}};$
$(2)$ $\Gamma(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}}\neq\Gamma(\mathscr{P^{'}})\bullet M_{\mathscr{D}_{D}},
\Gamma(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}}\neq\Gamma(\mathscr{P^{'}})\odot M_{\mathscr{D}_{D}},
\forall \mathscr{P^{'}}\subset \mathscr{P}.$
[@Lang1] Let $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$ be a covering decision information system, where $\mathscr{D}_{C}=\{\mathscr{C}_{i}|i\in I\}$, $\mathscr{D}_{D}=\{D_{i}|i\in J\}$, I and J are indexed sets. Then $\mathscr{P}\subseteq \mathscr{D}_{C}$ is called a type-2 reduct of $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$ if it satisfies $(1)$ and $(2)$ as follows:
$(1)$ $\prod(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}}=\prod(\mathscr{P})\bullet M_{\mathscr{D}_{D}},
\prod(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}}=\prod(\mathscr{P})\odot M_{\mathscr{D}_{D}};$
$(2)$ $\prod(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}}\neq\prod(\mathscr{P^{'}})\bullet M_{\mathscr{D}_{D}},
\prod(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}}\neq\prod(\mathscr{P^{'}})\odot M_{\mathscr{D}_{D}},
\forall \mathscr{P^{'}}\subset \mathscr{P}.$
Update the type-1 and type-2 characteristic matrices with variations of coverings
=================================================================================
In this section, we present incremental approaches to computing the type-1 and type-2 characteristic matrices with variations of coverings.
Let $(U,\mathscr{D})$ and $(U,\mathscr{D}^{+})$ be covering information systems, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\}$, and $\mathscr{D}^{+}=\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m},\mathscr{C}_{m+1}\}(m\geq 1)$. Then $(U,\mathscr{D}^{+})$ is called a dynamic covering information system of $(U,\mathscr{D})$.
In practical situations, the cardinalities of coverings which describes objects in covering information systems are increasing with the development of science and technology. Moreover, $(U,\mathscr{D})$ is referred to as a static covering information system of $(U,\mathscr{D}^{+})$.
Let $(U,\mathscr{D})$ be a static covering information system, where $U=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$, $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\}$, $\mathscr{C}_{1}=\{\{x_{1},x_{2},x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{C}_{2}=\{\{x_{1},x_{2}\},\{x_{3},x_{4},x_{5}\}\}$, and $\mathscr{C}_{3}=\{\{x_{1},x_{2},x_{5}\},\{x_{3},x_{4}\}\}$. By adding $\mathscr{C}_{4}=\{\{x_{1},x_{2}\},\{x_{3},x_{4}\},\{x_{5}\}\}$ into $\mathscr{D}$, we obtain a dynamic covering information system $(U,\mathscr{D}^{+})$ of $(U,\mathscr{D})$, where $\mathscr{D}^{+}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3},\mathscr{C}_{4}\}$.
In what follows, we show how to construct $\Gamma(\mathscr{D}^{+})$ based on $\Gamma(\mathscr{D})$. For convenience, we denote $M_{\mathscr{D}}=\left[\begin{array}{cccccc}
M_{\mathscr{C}_{1}}&M_{\mathscr{C}_{2}}&.&.&. & M_{\mathscr{C}_{m}}
\end{array}
\right]$, $M_{\mathscr{D}^{+}}=\left[\begin{array}{ccccccc}
M_{\mathscr{C}_{1}}&M_{\mathscr{C}_{2}}&.&.&. & M_{\mathscr{C}_{m}}&M_{\mathscr{C}_{m+1}}
\end{array}
\right]$, $M_{\mathscr{C}_{k}}=(a^{k}_{ij})_{n\times |\mathscr{C}_{k}|}$, $\Gamma(\mathscr{D})=(b_{ij})_{n\times n}$, and $\Gamma(\mathscr{D}^{+})=(c_{ij})_{n\times n}$, where $|\ast|$ denotes the cardinality of $\ast$.
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\Gamma(\mathscr{D})$ and $\Gamma(\mathscr{D}^{+})$ the type-1 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively. Then $$\begin{aligned}
\Gamma(\mathscr{D}^{+})=
\Gamma(\mathscr{D})\bigvee
\Gamma(\mathscr{C}_{m+1}),\end{aligned}$$ where $\Gamma(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\bullet M^{T}_{\mathscr{C}_{m+1}}.$
**Proof.** By Definitions 2.3 and 3.1, we get $\Gamma(\mathscr{C})$ and $\Gamma(\mathscr{C}^{+})$ as follows: $$\begin{aligned}
\Gamma(\mathscr{D})&=&M_{\mathscr{D}}\bullet
M_{\mathscr{D}}^{T}\\&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}\\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}
\end{array}
\right] \bullet \\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}\\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}
\end{array}
\right]^{T}
\\&=&\left[
\begin{array}{cccccc}
b_{11} & b_{12} & . & . & . & b_{1n} \\
b_{21} & b_{22} & . & . & . & b_{2n} \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
b_{n1} & b_{n2} & . & . & . & b_{nn}
\end{array}
\right],\\
\Gamma(\mathscr{D}^{+})&=&M_{\mathscr{D}^{+}}\bullet
M_{\mathscr{D}^{+}}^{T}\\&=&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}& a^{m+1}_{11} & a^{m+1}_{12} & . & a^{m+1}_{1|\mathscr{C}_{m+1}|} \\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}& a^{m+1}_{21} & a^{m+1}_{22} & . & a^{m+1}_{2|\mathscr{C}_{m+1}|} \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}& a^{m+1}_{n1} & a^{m+1}_{n2} & . & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right] \bullet \\ && \left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}& a^{m+1}_{11} & a^{m+1}_{12} & . & a^{m+1}_{1|\mathscr{C}_{m+1}|} \\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}& a^{m+1}_{21} & a^{m+1}_{22} & . & a^{m+1}_{2|\mathscr{C}_{m+1}|} \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}& a^{m+1}_{n1} & a^{m+1}_{n2} & . & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T} \\&=&\left[
\begin{array}{cccccc}
c_{11} & c_{12} & . & . & . & c_{1n} \\
c_{21} & c_{22} & . & . & . & c_{2n} \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
c_{n1} & c_{n2} & . & . & . & c_{nn}
\end{array}
\right].\end{aligned}$$
According to Definition 2.3, we have $$\begin{aligned}
b_{ij}&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m}_{i1} & a^{m}_{i2} & . & a^{m}_{i|\mathscr{C}_{m}|}\\
\end{array}
\right]\bullet \\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m}_{j1} & a^{m}_{j2} & . & a^{m}_{j|\mathscr{C}_{m}|}\\
\end{array}
\right]^{T}\\&=&[(a^{1}_{i1}\cdot a^{1}_{j1})\vee (a^{1}_{i2}\cdot a^{1}_{j2})\vee...\vee (a^{1}_{i|\mathscr{C}_{1}|}\cdot a^{1}_{j|\mathscr{C}_{1}|})]\vee [(a^{2}_{i1}\cdot a^{2}_{j1})\vee (a^{2}_{i2}\cdot a^{2}_{j2})\vee...\vee (a^{2}_{i|\mathscr{C}_{2}|}\cdot a^{2}_{j|\mathscr{C}_{2}|})]\vee ...\vee \\&& [(a^{m}_{i1}\cdot a^{m}_{j1})\vee (a^{m}_{i2}\cdot a^{m}_{j2})\vee...\vee (a^{m}_{i|\mathscr{C}_{m}|}\cdot a^{m}_{j|\mathscr{C}_{m}|})],\\
c_{ij}&=&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m}_{i1} & a^{m}_{i2} & . & a^{m}_{i|\mathscr{C}_{m}|}& a^{m+1}_{i1} & a^{m+1}_{i2} & . & a^{m+1}_{i|\mathscr{C}_{m+1}|} \\
\end{array}
\right]\bullet \\
&&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m}_{j1} & a^{m}_{j2} & . & a^{m}_{j|\mathscr{C}_{m}|}& a^{m+1}_{j1} & a^{m+1}_{j2} & . & a^{m+1}_{j|\mathscr{C}_{m+1}|} \\
\end{array}
\right]^{T}\\&=&[(a^{1}_{i1}\cdot a^{1}_{j1})\vee (a^{1}_{i2}\cdot a^{1}_{j2})\vee...\vee (a^{1}_{i|\mathscr{C}_{1}|}\cdot a^{1}_{j|\mathscr{C}_{1}|})]\vee [(a^{2}_{i1}\cdot a^{2}_{j1})\vee (a^{2}_{i2}\cdot a^{2}_{j2})\vee...\vee (a^{2}_{i|\mathscr{C}_{2}|}\cdot a^{2}_{j|\mathscr{C}_{2}|})]\vee ...\vee \\&&[(a^{m}_{i1}\cdot a^{m}_{j1})\vee (a^{m}_{i2}\cdot a^{m}_{j2})\vee...\vee (a^{m}_{i|\mathscr{C}_{m}|}\cdot a^{m}_{j|\mathscr{C}_{m}|})]\vee [(a^{m+1}_{i1}\cdot a^{m+1}_{j1})\vee (a^{m+1}_{i2}\cdot a^{m+1}_{j2})\vee...\vee (a^{m+1}_{i|\mathscr{C}_{m+1}|}\cdot a^{m}_{j|\mathscr{C}_{m+1}|})]\\&=&b_{ij}\vee \left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . & . & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\bullet\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T}.\end{aligned}$$
To obtain $\Gamma(\mathscr{D}^{+})$, we only need to compute $\Gamma(\mathscr{C}_{m+1})$ as follows: $$\begin{aligned}
\Gamma(\mathscr{C}_{m+1})&=&\left[\begin{array}{cccccc}
a^{m+1}_{11}&a^{m+1}_{12}&.&.&. & a^{m+1}_{1|\mathscr{C}_{m+1}|}\\
a^{m+1}_{21}&a^{m+1}_{22}&.&.&. & a^{m+1}_{2|\mathscr{C}_{m+1}|}\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
a^{m+1}_{n1}&a^{m+1}_{n2}&.&.&. & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]\bullet \left[\begin{array}{cccccc}
a^{m+1}_{11}&a^{m+1}_{12}&.&.&. & a^{m+1}_{1|\mathscr{C}_{m+1}|}\\
a^{m+1}_{21}&a^{m+1}_{22}&.&.&. & a^{m+1}_{2|\mathscr{C}_{m+1}|}\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
a^{m+1}_{n1}&a^{m+1}_{n2}&.&.&. & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, we have $$\begin{aligned}
\Gamma(\mathscr{D}^{+})=
\Gamma(\mathscr{D})\bigvee
\Gamma(\mathscr{C}_{m+1}),\end{aligned}$$ where $\Gamma(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\bullet M^{T}_{\mathscr{C}_{m+1}}.$ $\Box$
We present the non-incremental and incremental algorithms for computing $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)$ in dynamic covering information systems.
(Non-incremental algorithm of computing $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)\bf{(NIS)}$)
Step 1: Input $(U,\mathscr{D}^{+})$;
Step 2: Construct $\Gamma(\mathscr{D}^{+})=M_{\mathscr{D}^{+}}\bullet M^{T}_{\mathscr{D}^{+}};$
Step 3: Compute $\mathcal {X}_{SH_{\mathscr{D}^{+}}(X)}=\Gamma(\mathscr{D}^{+})\bullet \mathcal
{X}_{X}$ and $\mathcal {X}_{SL_{\mathscr{D}^{+}}(X)}=\Gamma(\mathscr{D}^{+})\odot \mathcal
{X}_{X}$;
Step 4: Output $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)$.
(Incremental algorithm of computing $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)\bf{(IS)}$)
Step 1: Input $(U,\mathscr{D})$ and $(U,\mathscr{D}^{+})$;
Step 2: Calculate $\Gamma(\mathscr{D})=M_{\mathscr{D}}\bullet
M^{T}_{\mathscr{D}}$;
Step 3: Construct $\Gamma(\mathscr{D}^{+})=
\Gamma(\mathscr{D})\bigvee
\Gamma(\mathscr{C}_{m+1})$, where $
\Gamma(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\bullet M^{T}_{\mathscr{C}_{m+1}};
$
Step 4: Obtain $
\mathcal {X}_{SH_{\mathscr{D}^{+}}(X)}=\Gamma(\mathscr{D}^{+})\bullet \mathcal
{X}_{X}\text{ and } \mathcal {X}_{SL_{\mathscr{D}^{+}}(X)}=\Gamma(\mathscr{D}^{+})\odot \mathcal
{X}_{X};
$
Step 5: Output $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)$.
The time complexity of computing the second lower and upper approximations of sets is $O(2n^{2}\ast\sum^{m+1}_{i=1}|\mathscr{C}_{i}|+2n^{2})$ using Algorithm 3.4. Furthermore, $O(2n^{2}\ast|\mathscr{C}_{m+1}|+3n^{2})$ is the time complexity of Algorithm 3.5. Therefore, the time complexity of the incremental algorithm is lower than that of the non-incremental algorithm.
(Continued from Example 3.2) Taking $X=\{x_{2},x_{3},x_{4}\}$. According to Definition 2.3, we first have $$\begin{aligned}
\Gamma(\mathscr{D})=M_{\mathscr{D}}\bullet
M_{\mathscr{D}}^{T}=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{4})=M_{\mathscr{C}_{4}}\bullet
M_{\mathscr{C}_{4}}^{T}=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0 \\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Theorem 3.3, we obtain $$\begin{aligned}
\Gamma(\mathscr{D}^{+})=
\Gamma(\mathscr{D}) \bigvee \Gamma(\mathscr{C}_{4})&=&\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\bigvee \left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0 \\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\\&=& \left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Third, by Definition 2.4, we get $$\begin{aligned}
\mathcal {X}_{SH_{\mathscr{D}^{+}}(X)}&=&\Gamma(\mathscr{D}^{+})\bullet \mathcal {X}_{X}
=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\bullet\left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right] =\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1
\end{array}
\right]^{T},
\\
\mathcal {X}_{SL_{\mathscr{D}^{+}}(X)}&=&\Gamma(\mathscr{D}^{+})\odot
\mathcal {X}_{X}=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right]=\left[
\begin{array}{cccccc}
0 & 0 & 0 & 0 & 0
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, $SH_{\mathscr{D}^{+}}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ and $SL_{\mathscr{D}^{+}}(X)=\emptyset$.
In Example 3.6, we only need to calculate elements of $\Gamma(\mathscr{C}_{4})$ for computing $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)$ using Algorithm 3.5. But we must construct $\Gamma(\mathscr{D}^{+})$ for computing $SH_{\mathscr{D}^{+}}(X)$ and $SL_{\mathscr{D}^{+}}(X)$ using Algorithm 3.4. Thereby, the incremental approach is more effective to compute the second lower and upper approximations of sets.
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\Gamma(\mathscr{D})=(b_{ij})_{n\times n}$ and $\Gamma(\mathscr{D}^{+})=(c_{ij})_{n\times n}$ the type-1 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively. Then $$\begin{aligned}
c_{ij}=\left\{
\begin{array}{ccc}
1,&{\rm}& b_{ij}=1;\\
\left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . & . & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\bullet\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T},&{\rm}& b_{ij}=0.
\end{array}
\right.\end{aligned}$$
**Proof.** It is straightforward by Theorem 3.3.$\Box$
(Continued from Example 3.6) According to Definition 2.3, we have $$\begin{aligned}
\Gamma(\mathscr{D})=M_{\mathscr{D}}\bullet
M_{\mathscr{D}}^{T}=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Therefore, by Theorem 3.7, we get $$\begin{aligned}
\Gamma(\mathscr{D}^{+})=\Gamma(\mathscr{D})=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\Gamma(\mathscr{D})$ and $\Gamma(\mathscr{D}^{+})$ the type-1 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively.
$(1)$ If $\Gamma(\mathscr{D})=(1)_{n\times n}$, then $\Gamma(\mathscr{D}^{+})=(1)_{n\times n}$;
$(2)$ If $\Gamma(\mathscr{D})=(0)_{n\times n}$, then $\Gamma(\mathscr{D}^{+})=\Gamma(\mathscr{C}_{m+1})$.
**Proof.** It is straightforward by Theorem 3.7.$\Box$
Subsequently, we construct $\prod(\mathscr{C}^{+})$ based on $\prod(\mathscr{C})$. For convenience, we denote $\prod(\mathscr{C})=(d_{ij})_{n\times n}$ and $\prod(\mathscr{C}^{+})=(e_{ij})_{n\times n}$.
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\prod(\mathscr{D})$ and $\prod(\mathscr{D}^{+})$ the type-2 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively. Then $$\begin{aligned}
\prod(\mathscr{D}^{+})=
\prod(\mathscr{D})\bigwedge
\prod(\mathscr{C}_{m+1}),\end{aligned}$$ where $\prod(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\odot M^{T}_{\mathscr{C}_{m+1}}.$
**Proof.** By Definitions 2.3 and 3.1, we get $\prod(\mathscr{D})$ and $\prod(\mathscr{D}^{+})$ as follows: $$\begin{aligned}
\prod(\mathscr{D})&=&M_{\mathscr{D}}\odot
M_{\mathscr{D}}^{T}\\&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}\\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}
\end{array}
\right] \odot \\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}\\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}
\end{array}
\right]^{T}
\\&=&\left[
\begin{array}{cccccc}
d_{11} & d_{12} & . & . & . & d_{1n} \\
d_{21} & d_{22} & . & . & . & d_{2n} \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
d_{n1} & d_{n2} & . & . & . & d_{nn}
\end{array}
\right],\\
\prod(\mathscr{D}^{+})&=&M_{\mathscr{D}^{+}}\odot
M_{\mathscr{D}^{+}}^{T}\\&=&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}& a^{m+1}_{11} & a^{m+1}_{12} & . & a^{m+1}_{1|\mathscr{C}_{m+1}|} \\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}& a^{m+1}_{21} & a^{m+1}_{22} & . & a^{m+1}_{2|\mathscr{C}_{m+1}|} \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}& a^{m+1}_{n1} & a^{m+1}_{n2} & . & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right] \odot\\ && \left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{11} & a^{1}_{12} & . & a^{1}_{1|\mathscr{C}_{1}|} & a^{2}_{11} & a^{2}_{12} & . & a^{2}_{1|\mathscr{C}_{2}|} & . & . & . & a^{m}_{11} & a^{m}_{12} & . & a^{m}_{1|\mathscr{C}_{m}|}& a^{m+1}_{11} & a^{m+1}_{12} & . & a^{m+1}_{1|\mathscr{C}_{m+1}|} \\
a^{1}_{21} & a^{1}_{22} & . & a^{1}_{2|\mathscr{C}_{1}|} & a^{2}_{21} & a^{2}_{22} & . & a^{2}_{2|\mathscr{C}_{2}|} & . & . & . & a^{m}_{21} & a^{m}_{22} & . & a^{m}_{2|\mathscr{C}_{m}|}& a^{m+1}_{21} & a^{m+1}_{22} & . & a^{m+1}_{2|\mathscr{C}_{m+1}|} \\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
. & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & .\\
a^{1}_{n1} & a^{1}_{n2} & . & a^{1}_{n|\mathscr{C}_{1}|} & a^{2}_{n1} & a^{2}_{n2}& . & a^{2}_{n|\mathscr{C}_{2}|}& . & . & . & a^{m}_{n1} & a^{m}_{n2} & . & a^{m}_{n|\mathscr{C}_{m}|}& a^{m+1}_{n1} & a^{m+1}_{n2} & . & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T} \\&=&\left[
\begin{array}{cccccc}
e_{11} & e_{12} & . & . & . & e_{1n} \\
e_{21} & e_{22} & . & . & . & e_{2n} \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
. & . & . & . & . & . \\
e_{n1} & e_{n2} & . & . & . & e_{nn}
\end{array}
\right].\end{aligned}$$
According to Definition 2.3, we have $$\begin{aligned}
d_{ij}&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m}_{i1} & a^{m}_{i2} & . & a^{m}_{i|\mathscr{C}_{m}|}\\
\end{array}
\right]\odot \\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m}_{j1} & a^{m}_{j2} & . & a^{m}_{j|\mathscr{C}_{m}|}\\
\end{array}
\right]^{T}\\&=&[(a^{1}_{j1}- a^{1}_{i1}+1)\wedge (a^{1}_{j2}- a^{1}_{i2}+1)\wedge...\wedge (a^{1}_{j|\mathscr{C}_{1}|}- a^{1}_{i|\mathscr{C}_{1}|}+1)]\wedge \\&& [(a^{2}_{j1}- a^{2}_{i1}+1)\wedge (a^{2}_{j2}- a^{2}_{i2}+1)\wedge...\wedge (a^{2}_{j|\mathscr{C}_{2}|}- a^{2}_{i|\mathscr{C}_{2}|}+1)]\wedge ...\wedge \\&& [(a^{m}_{j1}- a^{m}_{i1}+1)\wedge (a^{m}_{j2}- a^{m}_{i2}+1)\wedge...\wedge (a^{m}_{j|\mathscr{C}_{m}|}- a^{m}_{i|\mathscr{C}_{m}|}+1)],\\
e_{ij}&=&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m}_{i1} & a^{m}_{i2} & . & a^{m}_{i|\mathscr{C}_{m}|}& a^{m+1}_{i1} & a^{m+1}_{i2} & . & a^{m+1}_{i|\mathscr{C}_{m+1}|} \\
\end{array}
\right]\odot \\
&&\left[
\begin{array}{ccccccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m}_{j1} & a^{m}_{j2} & . & a^{m}_{j|\mathscr{C}_{m}|}& a^{m+1}_{j1} & a^{m+1}_{j2} & . & a^{m+1}_{j|\mathscr{C}_{m+1}|} \\
\end{array}
\right]^{T}\\&=&[(a^{1}_{j1}- a^{1}_{i1}+1)\wedge (a^{1}_{j2}- a^{1}_{i2}+1)\wedge...\wedge (a^{1}_{j|\mathscr{C}_{1}|}- a^{1}_{i|\mathscr{C}_{1}|}+1)]\wedge \\&& [(a^{2}_{j1}- a^{2}_{i1}+1)\wedge (a^{2}_{j2}- a^{2}_{i2}+1)\wedge...\wedge (a^{2}_{j|\mathscr{C}_{2}|}- a^{2}_{i|\mathscr{C}_{2}|}+1)]\wedge . . . \wedge \\&& [(a^{m}_{j1}- a^{m}_{i1}+1)\wedge (a^{m}_{j2}- a^{m}_{i2}+1)\wedge...\wedge (a^{m}_{j|\mathscr{C}_{m}|}- a^{m}_{i|\mathscr{C}_{m}|}+1)]\wedge\\&&[(a^{m+1}_{j1}- a^{m+1}_{i1}+1)\wedge (a^{m+1}_{j2}- a^{m+1}_{i2}+1)\wedge...\wedge (a^{m+1}_{j|\mathscr{C}_{m+1}|}- a^{m+1}_{i|\mathscr{C}_{m+1}|}+1)]
\\&=&d_{ij}\wedge \left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . &. & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\odot\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T}.\end{aligned}$$
To obtain $\prod(\mathscr{D}^{+})$, we only need to compute $\prod(\mathscr{C}_{m+1})$ as follows: $$\begin{aligned}
\prod(\mathscr{C}_{m+1})=\left[\begin{array}{cccccc}
a^{m+1}_{11}&a^{m+1}_{12}&.&.&. & a^{m+1}_{1|\mathscr{C}_{m+1}|}\\
a^{m+1}_{21}&a^{m+1}_{22}&.&.&. & a^{m+1}_{2|\mathscr{C}_{m+1}|}\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
a^{m+1}_{n1}&a^{m+1}_{n2}&.&.&. & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]\odot \left[\begin{array}{cccccc}
a^{m+1}_{11}&a^{m+1}_{12}&.&.&. & a^{m+1}_{1|\mathscr{C}_{m+1}|}\\
a^{m+1}_{21}&a^{m+1}_{22}&.&.&. & a^{m+1}_{2|\mathscr{C}_{m+1}|}\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
.&.&.&.&. & .\\
a^{m+1}_{n1}&a^{m+1}_{n2}&.&.&. & a^{m+1}_{n|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, we have $$\begin{aligned}
\prod(\mathscr{D}^{+})=
\prod(\mathscr{D})\bigwedge
\prod(\mathscr{C}_{m+1}),\end{aligned}$$ where $\prod(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\odot M^{T}_{\mathscr{C}_{m+1}}.$ $\Box$
We also provide the non-incremental and incremental algorithms for computing $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)$ in dynamic covering information systems.
(Non-incremental algorithm of computing $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)\bf{(NIX)}$)
Step 1: Input $(U,\mathscr{D}^{+})$;
Step 2: Construct $\prod(\mathscr{D}^{+})=M_{\mathscr{D}^{+}}\odot M^{T}_{\mathscr{D}^{+}};$
Step 3: Compute $\mathcal {X}_{XH_{\mathscr{D}^{+}}(X)}=\prod(\mathscr{D}^{+})\bullet \mathcal
{X}_{X}$ and $\mathcal {X}_{XL_{\mathscr{D}^{+}}(X)}=\prod(\mathscr{D}^{+})\odot \mathcal
{X}_{X}$;
Step 4: Output $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)$.
(Incremental algorithm of computing $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)\bf{(IX)}$)
Step 1: Input $(U,\mathscr{D})$ and $(U,\mathscr{D}^{+})$;
Step 2: Construct $\prod(\mathscr{D})=M_{\mathscr{D}}\odot M^{T}_{\mathscr{D}};$
Step 3: Calculate $
\prod(\mathscr{D}^{+})=
\prod(\mathscr{D})\bigwedge
\prod(\mathscr{C}_{m+1}),
$ where $\prod(\mathscr{C}_{m+1})=M_{\mathscr{C}_{m+1}}\odot M^{T}_{\mathscr{C}_{m+1}};$
Step 4: Get $
XH_{\mathscr{D}^{+}}(X)=\prod(\mathscr{D}^{+})\cdot \mathcal
{X}_{X}\text{ and }
XL_{\mathscr{D}^{+}}(X)=\prod(\mathscr{D}^{+})\odot \mathcal
{X}_{X};
$
Step 5: Output $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)$.
The time complexity of computing the sixth lower and upper approximations of sets is $O(2n^{2}\ast\sum^{m+1}_{i=1}|\mathscr{C}_{i}|+2n^{2})$ by Algorithm 3.11. Furthermore, $O(2n^{2}\ast|\mathscr{C}_{m+1}|+3n^{2})$ is the time complexity of Algorithm 3.12. Therefore, the time complexity of the incremental algorithm is lower than that of the non-incremental algorithm.
(Continued from Example 3.2) Taking $X=\{x_{2},x_{3},x_{4}\}$. By Definition 2.3, we first have $$\begin{aligned}
\prod(\mathscr{D})=M_{\mathscr{D}}\odot
M_{\mathscr{D}}^{T}=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Theorem 3.10, we get $$\begin{aligned}
\prod(\mathscr{D}^{+})=
\prod(\mathscr{D}) \bigwedge \prod(\mathscr{C}_{4})&=&\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\bigwedge \left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\\&=& \left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Third, according to Definition 2.4, we obtain $$\begin{aligned}
\mathcal {X}_{XH_{\mathscr{D}^{+}}(X)}&=&\prod(\mathscr{D}^{+})\bullet \mathcal {X}_{X}
=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\bullet\left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right] =\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 0
\end{array}
\right]^{T},
\\
\mathcal {X}_{XL_{\mathscr{D}^{+}}(X)}&=&\prod(\mathscr{D}^{+})\odot
\mathcal {X}_{X}=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right]=\left[
\begin{array}{ccccc}
0 & 0 & 1 & 1 & 0
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, $XH_{\mathscr{D}^{+}}(X)=\{x_{1},x_{2},x_{3},x_{4}\}$ and $XL_{\mathscr{D}^{+}}(X)=\{x_{3},x_{4}\}$.
In Example 3.11, we must compute $\prod(\mathscr{C}^{+})$ for constructing $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)$ using algorithm 3.11. But we only need to calculate $\prod(\mathscr{C}_{4})$ for computing $XH_{\mathscr{D}^{+}}(X)$ and $XL_{\mathscr{D}^{+}}(X)$ using Algorithm 3.12. Thereby, the incremental approach is more effective to compute the sixth lower and upper approximations of sets.
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\prod(\mathscr{C})=(d_{ij})_{n\times n}$ and $\prod(\mathscr{C}^{+})=(e_{ij})_{n\times n}$ the type-2 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively. Then $$\begin{aligned}
e_{ij}=\left\{
\begin{array}{ccc}
0,&{\rm}& d_{ij}=0;\\
\left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . & . & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\odot\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T},&{\rm}& d_{ij}=1.
\end{array}
\right.\end{aligned}$$
**Proof.** It is straightforward by Theorem 3.10.$\Box$
(Continued from Example 3.2) According to Definition 2.3, we have $$\begin{aligned}
\prod(\mathscr{D})=M_{\mathscr{D}}\odot
M_{\mathscr{D}}^{T}=\left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Therefore, by Theorem 3.14, we obtain $$\begin{aligned}
\prod(\mathscr{D}^{+})= \left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Let $(U,\mathscr{D}^{+})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\prod(\mathscr{D})$ and $\prod(\mathscr{D}^{+})$ the type-2 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively.
$(1)$ If $\prod(\mathscr{D})=(0)_{n\times n}$, then $\prod(\mathscr{D}^{+})=(0)_{n\times n}$;
$(2)$ If $\prod(\mathscr{D})=(1)_{n\times n}$, then $\prod(\mathscr{D}^{+})=\prod(\mathscr{C}_{m+1})$.
**Proof.** It is straightforward by Theorem 3.14.$\Box$
In practical situations, there are some dynamic covering information systems because of the emigration of coverings, which are presented as follows.
Let $(U,\mathscr{D})$ and $(U,\mathscr{D}^{-})$ be covering information systems, where $U=\{x_{1},x_{2},...,x_{n}\}$, $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\}$, and $\mathscr{D}^{-}=\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m-1}\}(m\geq 2 )$. Then $(U,\mathscr{D}^{-})$ is called a dynamic covering information system of $(U,\mathscr{D})$.
In other words, $(U,\mathscr{D})$ is also referred to as a static covering information system of $(U,\mathscr{D}^{-})$. Furthermore, we employ an example to illustrate dynamic covering information systems given by Definition 3.17 as follows.
Let $(U,\mathscr{D})$ be a static covering information system, where $U=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$, $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3},\mathscr{C}_{4}\}$, $\mathscr{C}_{1}=\{\{x_{1},x_{2},x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{C}_{2}=\{\{x_{1},x_{2}\},\{x_{3},x_{4},x_{5}\}\}$, $\mathscr{C}_{3}=\{\{x_{1},x_{2},x_{5}\},\{x_{3},x_{4}\}\}$, and $\mathscr{C}_{4}=\{\{x_{1},x_{2}\},\{x_{3},x_{4}\},\{x_{5}\}\}$. If we delete $\mathscr{C}_{4}$ from $\mathscr{D}$, then we obtain a dynamic covering information system $(U,\mathscr{D}^{-})$ of $(U,\mathscr{D})$, where $\mathscr{D}^{-}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\}$.
We also show how to construct $\Gamma(\mathscr{C}^{-})$ based on $\Gamma(\mathscr{C})$. For convenience, we denote $\Gamma(\mathscr{D})=(b_{ij})_{n\times n}$ and $\Gamma(\mathscr{D}^{-})=(c^{-}_{ij})_{n\times n}$.
Let $(U,\mathscr{D}^{-})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\Gamma(\mathscr{D})=(b_{ij})_{n\times n}$ and $\Gamma(\mathscr{D}^{-})=(c^{-}_{ij})_{n\times n}$ the type-1 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{+}$, respectively. Then $$\begin{aligned}
c^{-}_{ij}=\left\{
\begin{array}{ccc}
0,&{\rm}& b_{ij}=0;\\
1,&{\rm}& b_{ij}=1\wedge \triangle c_{ij}=0;\\
c^{\ast}_{ij},&{\rm}& b_{ij}=1\wedge \triangle c_{ij}=1.
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
\triangle c_{ij}&=&\left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . & . & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\bullet\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T};\\
c^{\ast}_{ij}&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m-1}_{i1} & a^{m-1}_{i2} & . & a^{m-1}_{i|\mathscr{C}_{m-1}|}\\
\end{array}
\right]\bullet\\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m-1}_{j1} & a^{m-1}_{j2} & . & a^{m-1}_{j|\mathscr{C}_{m-1}|}\\
\end{array}
\right]^{T}.\end{aligned}$$
**Proof.** It is straightforward by Theorem 3.3.$\Box$
(Continued from Example 3.14) Taking $X=\{x_{2},x_{3},x_{4}\}$. According to Definition 2.3, we first obtain $$\begin{aligned}
\Gamma(\mathscr{D})=M_{\mathscr{D}}\bullet
M_{\mathscr{D}}^{T}=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Theorem 3.19, we get $$\begin{aligned}
\Gamma(\mathscr{D}^{-})=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Third, by Definition 2.4, we have $$\begin{aligned}
\mathcal {X}_{SH_{\mathscr{D}^{-}}(X)}&=&\Gamma(\mathscr{D}^{-})\bullet \mathcal {X}_{X}
=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\bullet\left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right] =\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1
\end{array}
\right]^{T},
\\
\mathcal {X}_{SL_{\mathscr{D}^{-}}(X)}&=&\Gamma(\mathscr{D}^{-})\odot
\mathcal {X}_{X}=\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right]=\left[
\begin{array}{ccccc}
0 & 0 & 0 & 0 & 0
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, $SH_{\mathscr{D}^{-}}(X)=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ and $SL_{\mathscr{D}^{-}}(X)=\emptyset$.
We also show how to construct $\prod(\mathscr{C}^{-})$ based on $\prod(\mathscr{C})$. For convenience, we denote $\prod(\mathscr{D})=(d_{ij})_{n\times n}$ and $\prod(\mathscr{D}^{-})=(e^{-}_{ij})_{n\times n}$.
Let $(U,\mathscr{D}^{-})$ be a dynamic covering information system of $(U,\mathscr{D})$, $\prod(\mathscr{D})=(d_{ij})_{n\times n}$ and $\prod(\mathscr{D}^{-})=(e^{-}_{ij})_{n\times n}$ the type-2 characteristic matrices of $\mathscr{D}$ and $\mathscr{D}^{-}$, respectively. Then $$\begin{aligned}
e^{-}_{ij}=\left\{
\begin{array}{ccc}
1,&{\rm}& d_{ij}=1\wedge \triangle e_{ij}=1;\\
0,&{\rm}& d_{ij}=0\wedge \triangle e_{ij}=1;\\
e^{\ast}_{ij},&{\rm}& d_{ij}=0\wedge \triangle e_{ij}=0.
\end{array}
\right.\end{aligned}$$ where $$\begin{aligned}
\triangle e_{ij}&=&\left[
\begin{array}{ccccccccccc}
a^{m+1}_{i1} & a^{m+1}_{i2} & . & . & . & a^{m+1}_{i|\mathscr{C}_{m+1}|}
\end{array}
\right]\odot\left[
\begin{array}{ccccccccccc}
a^{m+1}_{j1} & a^{m+1}_{j2} & . & . & . & a^{m+1}_{j|\mathscr{C}_{m+1}|}
\end{array}
\right]^{T},\\
e^{\ast}_{ij}&=&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{i1} & a^{1}_{i2} & . & a^{1}_{i|\mathscr{C}_{1}|} & a^{2}_{i1} & a^{2}_{i2} & . & a^{2}_{i|\mathscr{C}_{2}|} & . & . & . & a^{m-1}_{i1} & a^{m-1}_{i2} & . & a^{m-1}_{i|\mathscr{C}_{m-1}|}\\
\end{array}
\right]\odot\\&&\left[
\begin{array}{ccccccccccccccc}
a^{1}_{j1} & a^{1}_{j2} & . & a^{1}_{j|\mathscr{C}_{1}|} & a^{2}_{j1} & a^{2}_{j2} & . & a^{2}_{j|\mathscr{C}_{2}|} & . & . & . & a^{m-1}_{j1} & a^{m-1}_{j2} & . & a^{m-1}_{j|\mathscr{C}_{m-1}|}\\
\end{array}
\right]^{T}.\end{aligned}$$
**Proof.** It is straightforward by Theorem 3.10.$\Box$
(Continued from Example 3.18) According to Definition 2.3, we first have $$\begin{aligned}
\prod(\mathscr{D})=M_{\mathscr{D}}\odot
M_{\mathscr{D}}^{T}=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Theorem 3.21, we get $$\begin{aligned}
\prod(\mathscr{D}^{-})=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Third, by Definition 2.4, we obtain $$\begin{aligned}
\mathcal {X}_{XH_{\mathscr{D}^{-}}(X)}&=&\prod(\mathscr{D}^{-})\bullet \mathcal {X}_{X}
=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right] \bullet \left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right] =\left[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 0
\end{array}
\right]^{T},\\
\mathcal {X}_{XL_{\mathscr{D}^{-}}(X)}&=&\prod(\mathscr{D}^{-})\odot
\mathcal {X}_{X}=\left[
\begin{array}{ccccc}
1 & 1 & 0 & 0 &0\\
1 & 1 & 0 & 0 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right] \odot \left[
\begin{array}{c}
0 \\
1 \\
1 \\
1 \\
0 \\
\end{array}
\right]=\left[
\begin{array}{ccccc}
0 & 0 & 1 & 1 & 0
\end{array}
\right]^{T}.\end{aligned}$$
Therefore, $XH_{\mathscr{D}^{-}}(X)=\{x_{1},x_{2},x_{3},x_{4}\}$ and $XL_{\mathscr{D}^{-}}(X)=\{x_{3},x_{4}\}$.
In practical situations, we compute the type-1 and type-2 characteristic matrices for dynamic covering information systems with the immigrations and emigrations of covering simultaneously using two steps as follows: (1) compute the type-1 and type-2 characteristic matrices by Theorems 3.3 and 3.10, respectively; (2) construct the type-1 and type-2 characteristic matrices by Theorems 3.19 and 3.21, respectively. Actually, there are more dynamic covering information systems given by Definition 3.1 than those defined by Definition 3.17. Therefore, the following discussion focuses on the dynamic covering information systems given by Definition 3.1.
Experimental analysis
=====================
In this section, we perform experiments to illustrate the effectiveness of Algorithms 3.5 and 3.12 for computing the second and sixth lower and upper approximations of concepts, respectively, in dynamic covering information systems with the immigration of coverings.
To test Algorithms 3.5 and 3.12, we generated randomly ten artificial covering information systems $\{(U_{i},\mathscr{D}_{i})|i=1,2,3,...,10\}$, which are outlined in Table 1, where $|U_{i}|$ means the number of objects in $U_{i}$ and $|\mathscr{D}_{i}|$ denotes the cardinality of the covering set $\mathscr{D}_{i}$. For convenience, each covering here contains five elements in each covering information system $(U_{i},\mathscr{D}_{i})$. Moreover, we conducted all computations on a PC with a Intel(R) Dual-Core(TM) i5-4590 CPU $@$ 3.30 GHZ and 8 GB memory, running 64-bit Windows 7; the software used was 64-bit Matlab R2009b.
0.6in
No. Name $|U_{i}|$ $|\mathscr{D}_{i}|$
----- ----------------------------- ----------- ---------------------
1 $(U_{1},\mathscr{D}_{1})$ 2000 1000
2 $(U_{2},\mathscr{D}_{2})$ 4000 1000
3 $(U_{3},\mathscr{D}_{3})$ 6000 1000
4 $(U_{4},\mathscr{D}_{4})$ 8000 1000
5 $(U_{5},\mathscr{D}_{5})$ 10000 1000
6 $(U_{6},\mathscr{D}_{6})$ 12000 1000
7 $(U_{7},\mathscr{D}_{7})$ 14000 1000
8 $(U_{8},\mathscr{D}_{8})$ 16000 1000
9 $(U_{9},\mathscr{D}_{9})$ 18000 1000
10 $(U_{10},\mathscr{D}_{10})$ 20000 1000
: Covering information systems for experiments.
The stability of Algorithms 3.4, 3.5, 3.11 and 3.12
---------------------------------------------------
In this section, we illustrate the stability of Algorithms 3.4, 3.5, 3.11 and 3.12 with the experimental results. First, we present the concept of sub-covering information system as follows.
Let $(U,\mathscr{D})$ be a covering information system, and $\mathscr{D}^{j}\subseteq \mathscr{D}$. Then $(U,\mathscr{D}^{j})$ is called a sub-covering information system of $(U,\mathscr{D})$.
According to Definition 4.1, we see that a sub-covering information system is a covering information system. Furthermore, the number of sub-covering covering information systems is $2^{|\mathscr{D}|}-1$ for the covering information system $(U,\mathscr{D})$.
Let $(U,\mathscr{D})$ be a covering information system, where $U=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$, $\mathscr{D}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\}$, $\mathscr{C}_{1}=\{\{x_{1},x_{2},x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{C}_{2}=\{\{x_{1},x_{2}\},\{x_{3},x_{4},x_{5}\}\}$, and $\mathscr{C}_{3}=\{\{x_{1},x_{2},x_{5}\},\{x_{3},x_{4}\}\}$. Then we obtain a sub-covering information system $(U,\mathscr{D}^{1})$ by taking $\mathscr{D}^{1}=\{\mathscr{C}_{1},\mathscr{C}_{2}\}$. Furthermore, $(U,\mathscr{D}^{2})$ is also a sub-covering information system of $(U,\mathscr{D})$, where $\mathscr{D}^{2}=\{\mathscr{C}_{1},\mathscr{C}_{3}\}$.
Second, according to Definition 4.1, we obtain ten sub-covering information systems $\{(U_{i},\mathscr{D}^{j}_{i})| j=1,2,3,...,10\}$ for covering information system $(U_{i},\mathscr{D}_{i})$ outlined in Table 1, and show these sub-covering information systems in Table 2, where $\mathscr{D}^{j}_{i}\subseteq\mathscr{D}_{i}$.
0.143in
$(U,\mathscr{D})$ $|\mathscr{D}^{1}_{i}|$ $|\mathscr{D}^{2}_{i}|$ $|\mathscr{D}^{3}_{i}|$ $|\mathscr{D}^{4}_{i}|$ $|\mathscr{D}^{5}_{i}|$ $|\mathscr{D}^{6}_{i}|$ $|\mathscr{D}^{7}_{i}|$ $|\mathscr{D}^{8}_{i}|$ $|\mathscr{D}^{9}_{i}|$ $|\mathscr{D}^{10}_{i}|$
--------------------------------- ------------------------- ------------------------- ------------------------- ------------------------- ------------------------- ------------------------- ------------------------- ------------------------- ------------------------- -------------------------- -- --
$(U_{1},\mathscr{D}^{j}_{1})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{2},\mathscr{D}^{j}_{2})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{3},\mathscr{D}^{j}_{3})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{4},\mathscr{D}^{j}_{4})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{5},\mathscr{D}^{j}_{5})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{6},\mathscr{D}^{j}_{6})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{7},\mathscr{D}^{j}_{7})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{8},\mathscr{D}^{j}_{8})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{9},\mathscr{D}^{j}_{9})$ 100 200 300 400 500 600 700 800 900 1000
$(U_{10},\mathscr{D}^{j}_{10})$ 100 200 300 400 500 600 700 800 900 1000
: Sub-covering information systems for experiments.
Third, to demonstrate the stability of Algorithms 3.4, 3.5, 3.11, and 3.12, we compute the second and sixth lower and upper approximations of sets in sub-covering information systems $\{(U_{i},\mathscr{D}^{j}_{i})|i,j=1,2,3,...,10\}$. For example, we show the process of computing the second and sixth lower and upper approximations of sets in covering information system $(U_{1},\mathscr{D}^{1}_{1})$, where $|U_{1}|=2000$ and $|\mathscr{D}^{1}_{1}|=100$ as follows:
\(1) By adding a covering into $\mathscr{D}^{1}_{1}$, we obtain the dynamic covering information system $(U_{1},\mathscr{D}^{1+}_{1})$, where $|U_{1}|=2000$ and $|\mathscr{D}^{1+}_{1}|=101$.
\(2) Taking any $X\subseteq
U_{1}$, we compute the second lower and upper approximations of $X$ in dynamic covering information system $(U_{1},\mathscr{D}^{1+}_{1})$ using Algorithms 3.4 and 3.5. Furthermore, we also compute the sixth lower and upper approximations of $X$ in dynamic covering information system $(U_{1},\mathscr{D}^{1+}_{1})$ using Algorithms 3.11 and 3.12. To confirm the accuracy of the experiment results, we conduct each experiment ten times and show the average time of ten experimental results in Table 3, where $t(s)$ denotes that the measure of time is in seconds.
0.046in \[bigtable\]
$(U,\mathscr{D})$ Algo. $\mathscr{D}^{1}_{i}$ $\mathscr{D}^{2}_{i}$ $\mathscr{D}^{3}_{i}$ $\mathscr{D}^{4}_{i}$ $\mathscr{D}^{5}_{i}$ $\mathscr{D}^{6}_{i}$ $\mathscr{D}^{7}_{i}$ $\mathscr{D}^{8}_{i}$ $\mathscr{D}^{9}_{i}$ $\mathscr{D}^{10}_{i}$
--------------------------------------- ------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ------------------------
\*[$(U_{1},\mathscr{D}^{j}_{1})$]{} NIS 0.294 0.291 0.308 0.305 0.303 0.326 0.302 0.315 0.333 0.333
NIX 0.433 0.452 0.470 0.487 0.505 0.515 0.538 0.555 0.563 0.581
IS 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019 0.019
IX 0.044 0.044 0.044 0.045 0.045 0.046 0.046 0.046 0.045 0.044
\*[$(U_{2},\mathscr{D}^{j}_{2})$]{} NIS 1.392 1.239 1.392 1.488 1.414 1.318 1.261 1.508 1.446 1.290
NIX 1.789 1.844 1.871 1.896 1.957 2.018 2.051 2.067 2.114 2.153
IS 0.116 0.116 0.116 0.115 0.115 0.114 0.114 0.115 0.113 0.114
IX 0.276 0.276 0.279 0.279 0.277 0.277 0.278 0.278 0.277 0.278
\*[$(U_{3},\mathscr{D}^{j}_{3})$]{} NIS 2.815 2.743 2.950 3.004 2.978 3.276 2.860 3.058 3.427 3.091
NIX 4.123 4.185 4.255 4.270 4.391 4.421 4.553 4.592 4.592 4.685
IS 0.281 0.280 0.282 0.282 0.280 0.281 0.280 0.281 0.280 0.280
IX 0.693 0.692 0.689 0.691 0.691 0.693 0.692 0.688 0.686 0.687
\*[$(U_{4},\mathscr{D}^{j}_{4})$]{} NIS 4.812 4.911 5.758 5.260 5.441 5.773 5.546 5.216 5.085 5.683
NIX 7.356 7.437 7.474 7.583 7.771 7.855 7.985 8.060 8.128 8.171
IS 0.536 0.534 0.534 0.535 0.535 0.535 0.534 0.533 0.534 0.534
IX 1.312 1.306 1.307 1.307 1.308 1.310 1.308 1.307 1.308 1.306
\*[$(U_{5},\mathscr{D}^{j}_{5})$]{} NIS 7.816 8.327 8.747 8.482 9.026 9.564 9.596 9.171 9.096 8.673
NIX 11.695 11.841 11.794 12.020 12.065 12.161 12.315 12.551 12.595 12.828
IS 0.899 0.900 0.901 0.901 0.901 0.900 0.899 0.898 0.901 0.901
IX 2.220 2.217 2.217 2.217 2.217 2.215 2.217 2.215 2.215 2.219
\*[$(U_{6},\mathscr{D}^{j}_{6})$]{} NIS 11.815 11.974 11.925 13.100 11.836 11.959 12.498 13.341 13.524 13.960
NIX 16.855 17.095 17.211 17.254 17.646 17.956 17.972 18.145 18.271 18.487
IS 1.311 1.310 1.310 1.311 1.312 1.314 1.314 1.312 1.313 1.313
IX 3.232 3.232 3.234 3.233 3.237 3.235 3.234 3.233 3.235 3.235
\*[$(U_{7},\mathscr{D}^{j}_{7})$]{} NIS 15.390 17.084 18.474 16.966 17.148 17.170 15.884 19.168 16.926 17.414
NIX 23.261 23.363 23.450 23.800 24.187 24.420 25.003 24.783 25.168 25.532
IS 1.816 1.812 1.816 1.815 1.816 1.814 1.814 1.816 1.813 1.813
IX 4.489 4.493 4.487 4.487 4.482 4.491 4.484 4.486 4.493 4.488
\*[$(U_{8},\mathscr{D}^{j}_{8})$]{} NIS 19.268 20.399 20.427 25.125 24.195 21.648 22.758 25.579 23.050 24.523
NIX 30.730 31.684 31.922 31.874 32.383 32.862 33.181 33.333 34.418 34.292
IS 2.329 2.348 2.334 2.339 2.335 2.333 2.331 2.335 2.336 2.329
IX 5.739 5.730 5.730 5.736 5.726 5.733 5.738 5.735 5.722 5.727
\*[$(U_{9},\mathscr{D}^{j}_{9})$]{} NIS 27.727 24.680 27.103 27.168 28.197 29.591 27.240 28.775 30.566 33.012
NIX 38.314 40.551 40.245 40.752 41.291 41.870 42.196 42.385 42.256 42.956
IS 3.081 3.079 3.077 3.084 3.080 3.077 3.082 3.082 3.077 3.076
IX 7.632 7.634 7.626 7.639 7.634 7.633 7.634 7.632 7.632 7.645
\*[$(U_{10},\mathscr{D}^{j}_{10})$]{} NIS 38.748 31.240 35.121 35.774 37.235 36.723 38.385 37.174 40.780 36.097
NIX 47.874 49.295 50.355 52.001 50.068 53.765 52.696 54.178 53.650 55.917
IS 3.728 3.730 3.725 3.727 3.719 3.723 3.724 3.721 3.722 3.721
IX 9.219 9.223 9.230 9.217 9.215 9.218 9.219 9.225 9.217 9.220
: Computational times using NIS, NIX, IS, and IX in $(U_{i},\mathscr{D}^{j}_{i})$
\(3) We compute the variance of ten experimental results for computing the approximations of sets in each dynamic covering information system and show all variance values in Table 4. According to the experimental results, we see that Algorithms 3.4, 3.5, 3.11, and 3.12 are stable to compute the second and sixth lower and upper approximations of sets in dynamic covering information systems with the immigration of coverings. Especially, Algorithms 3.5 and 3.12 are more stable to compute the second and sixth lower and upper approximations of sets than Algorithms 3.4 and 3.11, respectively, in dynamic covering information systems.
0.08in \[bigtable\]
$(U,\mathscr{D})$ Algo. $\mathscr{D}^{1}_{i}$ $\mathscr{D}^{2}_{i}$ $\mathscr{D}^{3}_{i}$ $\mathscr{D}^{4}_{i}$ $\mathscr{D}^{5}_{i}$ $\mathscr{D}^{6}_{i}$ $\mathscr{D}^{7}_{i}$ $\mathscr{D}^{8}_{i}$ $\mathscr{D}^{9}_{i}$ $\mathscr{D}^{10}_{i}$
--------------------------------------- ------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- ------------------------
\*[$(U_{1},\mathscr{D}^{j}_{1})$]{} NIS 0.005 0.004 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.001
NIX 0.004 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.001 0.001
IS 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
IX 0.000 0.001 0.001 0.000 0.001 0.001 0.000 0.000 0.000 0.001
\*[$(U_{2},\mathscr{D}^{j}_{2})$]{} NIS 0.001 0.002 0.002 0.002 0.002 0.002 0.001 0.003 0.002 0.002
NIX 0.003 0.002 0.002 0.003 0.003 0.002 0.004 0.003 0.002 0.003
IS 0.001 0.000 0.001 0.000 0.000 0.000 0.001 0.000 0.000 0.001
IX 0.001 0.001 0.001 0.001 0.002 0.001 0.001 0.001 0.001 0.002
\*[$(U_{3},\mathscr{D}^{j}_{3})$]{} NIS 0.003 0.003 0.004 0.004 0.003 0.004 0.002 0.004 0.004 0.002
NIX 0.005 0.004 0.008 0.004 0.004 0.013 0.005 0.017 0.005 0.003
IS 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.002
IX 0.006 0.003 0.002 0.001 0.003 0.003 0.002 0.004 0.003 0.002
\*[$(U_{4},\mathscr{D}^{j}_{4})$]{} NIS 0.003 0.013 0.009 0.003 0.006 0.004 0.004 0.006 0.004 0.006
NIX 0.019 0.004 0.005 0.007 0.004 0.006 0.007 0.004 0.004 0.014
IS 0.003 0.002 0.002 0.002 0.003 0.003 0.002 0.002 0.002 0.003
IX 0.004 0.004 0.002 0.004 0.003 0.007 0.004 0.003 0.004 0.002
\*[$(U_{5},\mathscr{D}^{j}_{5})$]{} NIS 0.005 0.019 0.014 0.003 0.009 0.006 0.008 0.008 0.006 0.006
NIX 0.020 0.022 0.009 0.010 0.020 0.014 0.023 0.009 0.021 0.013
IS 0.003 0.003 0.004 0.004 0.002 0.003 0.003 0.003 0.004 0.001
IX 0.008 0.007 0.007 0.007 0.005 0.005 0.005 0.003 0.004 0.005
\*[$(U_{6},\mathscr{D}^{j}_{6})$]{} NIS 0.008 0.007 0.007 0.032 0.012 0.008 0.007 0.007 0.025 0.008
NIX 0.024 0.015 0.019 0.034 0.021 0.045 0.028 0.033 0.042 0.031
IS 0.005 0.004 0.005 0.003 0.005 0.007 0.005 0.003 0.003 0.005
IX 0.004 0.004 0.004 0.007 0.006 0.007 0.005 0.008 0.005 0.005
\*[$(U_{7},\mathscr{D}^{j}_{7})$]{} NIS 0.023 0.008 0.057 0.012 0.009 0.021 0.022 0.032 0.025 0.030
NIX 0.110 0.036 0.027 0.037 0.031 0.096 0.071 0.060 0.074 0.124
IS 0.007 0.003 0.003 0.005 0.003 0.006 0.006 0.004 0.004 0.004
IX 0.008 0.011 0.007 0.009 0.003 0.006 0.005 0.005 0.014 0.005
\*[$(U_{8},\mathscr{D}^{j}_{8})$]{} NIS 0.043 0.039 0.034 0.075 0.041 0.126 0.053 0.035 0.042 0.087
NIX 0.211 0.402 0.411 0.510 0.278 0.313 0.433 0.437 0.808 0.492
IS 0.005 0.013 0.006 0.007 0.004 0.007 0.006 0.003 0.003 0.012
IX 0.011 0.011 0.010 0.007 0.010 0.011 0.010 0.012 0.006 0.007
\*[$(U_{9},\mathscr{D}^{j}_{9})$]{} NIS 0.050 0.059 0.070 0.105 0.072 0.247 0.092 0.102 0.116 0.191
NIX 0.234 0.830 0.794 0.729 0.880 0.828 0.426 0.176 0.722 0.761
IS 0.010 0.005 0.005 0.008 0.008 0.005 0.010 0.010 0.010 0.010
IX 0.006 0.009 0.008 0.013 0.006 0.015 0.011 0.007 0.007 0.019
\*[$(U_{10},\mathscr{D}^{j}_{10})$]{} NIS 0.495 0.088 0.167 0.167 0.143 0.592 0.210 0.124 0.308 0.459
NIX 0.608 1.353 1.380 1.841 0.396 1.083 0.747 1.435 1.754 1.877
IS 0.011 0.011 0.009 0.009 0.007 0.012 0.005 0.009 0.007 0.005
IX 0.008 0.008 0.017 0.013 0.011 0.011 0.015 0.016 0.016 0.018
: Variance values of computational times using NIS, NIX, IS, and IX in $(U_{i},\mathscr{D}^{j}_{i})$
The influence of the cardinality of object set
----------------------------------------------
In this section, we analyze the influence of the cardinality of object set on time of computing the second and sixth lower and upper approximations of sets using Algorithms 3.4, 3.5, 3.11, and 3.12 in dynamic covering information systems with the covering immigration.
There are ten sub-covering information systems with the same cardinality of covering sets. First, we compare the times of computing the second lower and upper approximations of sets using Algorithm 3.4 with those using Algorithm 3.5 in dynamic covering information systems with the same cardinality of covering sets. From the results in Table 3, we see that the computing times are increasing with the increasing cardinality of object sets using Algorithms 3.4 and 3.5. We also find that Algorithm 3.5 executes faster than Algorithm 3.5 in dynamic covering information systems. Second, we also compare the times of computing the sixth lower and upper approximations of sets using Algorithm 3.11 with those using Algorithm 3.12 in dynamic covering information systems with the same cardinality of covering sets. From the results in Table 3, we see that the computing times are increasing with the increasing cardinality of object sets using Algorithms 3.11 and 3.12. We also find that Algorithm 3.12 executes faster than Algorithm 3.11 in dynamic covering information systems. Third, to illustrate the effectiveness of Algorithms 3.5 and 3.12, we show these results in Figures 1-10. In each figure, $NIS,
IS, NIX,$ and $ IX$ mean Algorithms 3.4, 3.5, 3.11, and 3.12, respectively; $i$ stands for the cardinality of object set in $X$ Axis, while the y-coordinate stands for the time to construct the approximations of concepts. Therefore, Algorithms 3.5 and 3.12 are more effective to compute the second and sixth lower and upper approximations of sets, respectively, in dynamic covering information systems.
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The influence of the cardinality of covering set
------------------------------------------------
In this section, we analyze the influence of the cardinality of covering set on time of computing the second and sixth lower and upper approximations of sets using Algorithms 3.4, 3.5, 3.11, and 3.12 in dynamic covering information systems with the covering immigration.
In Table 2, there also exist ten sub-covering information systems with the same cardinality of object sets. First, we compare the times of computing the second lower and upper approximations of sets using Algorithm 3.4 with those using Algorithm 3.5 in dynamic covering information systems with the same cardinality of object sets. According to the experimental results in Table 3, we see that the computing times are almost not increasing with the increasing cardinality of covering sets using Algorithms 3.4 and 3.5. We also find that Algorithm 3.5 executes faster than Algorithm 3.4 in dynamic covering information systems. Second, we compare the times of computing the sixth lower and upper approximations of sets using Algorithm 3.11 with those using Algorithm 3.12 in dynamic covering information systems with the same cardinality of object sets. From the results in Table 3, we see that the computing times are increasing with the increasing cardinality of covering sets using Algorithms 3.11. But the computing times are almost not increasing with the increasing cardinality of covering sets using Algorithms 3.12. We also find that Algorithms 3.12 executes faster than Algorithm 3.11 in dynamic covering information systems. Third, to illustrate the effectiveness of Algorithms 3.5 and 3.12, we show these results in Figures 11-20. In each figure, $NIS,
IS, NIX,$ and $ IX$ mean Algorithms 3.4, 3.5, 3.11 and 3.12, respectively; $i$ stands for the cardinality of covering set in $X$ Axis, while the y-coordinate stands for the time to construct the approximations of concepts. Therefore, Algorithms 3.5 and 3.12 are more effective to compute the second and sixth lower and upper approximations of sets, respectively, in dynamic covering information systems with the immigration of coverings.
According to the experimental results, we see that Algorithms 3.5 and 3.12 are more effective to compute the second and sixth lower and upper approximations of sets than Algorithms 3.4 and 3.11, respectively, in dynamic covering information systems with the immigrations of objects and coverings.
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**Remark.** In the experiment, we can transform data sets downloaded from the University of California at Irvine(UCI)’s repository of machine learning databases into covering information systems. For example, we can transform the Balance Scale Weight $\&$ Distance Database with four conditional attributes into the covering information system $(U,\mathscr{D})$, where $|U|=625$ and $|\mathscr{D}|=4$. Concretely, since there are five attribute values for each conditional attribute, we can obtain a covering with five elements for each conditional attribute. Subsequently, based on Left-Weight, Left-Distance, Right-Weight, and Right-Distance, we have the covering information system $(U,\mathscr{D})$, where $|U|=625$ and $|\mathscr{D}|=4$. Furthermore, we can obtain a decision covering information system $(U,\mathscr{D}^{\ast})$ by constructing a covering based on the decision attribute, where $|U|=625$ and $|\mathscr{D}^{\ast}|=5$. Therefore, we can obtain covering information systems and decision covering information systems by transforming Irvine(UCI)’s repository of machine learning databases. Since the purpose of the experiment is to test the effectiveness of Algorithms 3.5 and 3.12 and the transformation process costs more time, we generated randomly ten artificial covering information systems $(U_{i},\mathscr{D}_{i})$ to test the designed algorithms in the experiments.
Knowledge reduction of covering decision information systems with the covering immigration
==========================================================================================
In this section, we employ examples to illustrate how to conduct knowledge reduction of covering decision information systems with the covering immigration.
Let $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$ be a covering decision information system, where $\mathscr{D}_{C}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\}$, $\mathscr{C}_{1}=\{\{x_{1},x_{2},x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{C}_{2}=\{\{x_{1},x_{2}\},\{x_{3},x_{4},x_{5}\}\}$, $\mathscr{C}_{3}=\{\{x_{1},x_{2},x_{5}\},\{x_{3},x_{4}\}\}$, $\mathscr{D}_{D}=\{D_{1},D_{2}\}$, $D_{1}=\{x_{1},x_{2}\},$ and $D_{2}=\{x_{3},x_{4},x_{5}\}$. First, according to Definition 2.3, we obtain $$\begin{aligned}
\Gamma(\mathscr{D}_{C})=M_{\mathscr{D}_{C}}\bullet
M_{\mathscr{D}_{C}}^{T}=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Definition 2.4, we have $$\begin{aligned}
\Gamma(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\bullet \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}
\right],\\
\Gamma(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right].\end{aligned}$$
Third, according to Definition 2.3, we get $$\begin{aligned}
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{1})=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{3})=\left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 &1 \\
1 & 1 & 0 & 0 &1 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right].\end{aligned}$$
Fourth, by Definition 2.4, we derive $$\begin{aligned}
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right]\bullet \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}
\right],\\
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right],\\
\Gamma(\mathscr{C}_{1})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\bullet \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
0 & 1 \\
\end{array}
\right],\\
\Gamma(\mathscr{C}_{1})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
1 & 1 & 1 & 1 &0 \\
0 & 0 & 0 & 0 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 1 \\
\end{array}
\right],\\
\Gamma(\mathscr{C}_{3})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 &1 \\
1 & 1 & 0 & 0 &1 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right]\bullet \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
0 & 1 \\
0 & 1 \\
1 & 1 \\
\end{array}
\right],\\
\Gamma(\mathscr{C}_{3})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 0 & 0 &1 \\
1 & 1 & 0 & 0 &1 \\
0 & 0 & 1 & 1 &0 \\
0 & 0 & 1 & 1 &0 \\
1 & 1 & 0 & 0 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 0 \\
\end{array}
\right].\end{aligned}$$
Therefore, according to Definition 2.5, $\{\mathscr{C}_{1},\mathscr{C}_{3}\}$ is a type-1 reduct of $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$.
In Example 5.1, we must compute $
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\bullet M_{\mathscr{D}_{D}},
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\odot M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{1})\bullet M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{1})\odot M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{3})\bullet M_{\mathscr{D}_{D}}$, and $
\Gamma(\mathscr{C}_{3})\odot M_{\mathscr{D}_{D}}
$ for constructing the type-1 reducts of covering decision information system $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$.
In what follows, we employ an example to illustrate how to compute the type-1 reducts of dynamic covering decision information systems with the immigration of coverings.
(Continued from Example 5.1) Let $(U,\mathscr{D}^{+}_{C}\cup \mathscr{D}_{D} )$ be a dynamic covering decision information system of $(U,\mathscr{D}_{C}\cup \mathscr{D}_{D} )$, where $\mathscr{D}^{+}_{C}=\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3},\mathscr{C}_{4}\}$, $\mathscr{C}_{1}=\{\{x_{1},x_{2},x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{C}_{2}=\{\{x_{1},x_{2}\},\{x_{3},x_{4},\\x_{5}\}\}$, $\mathscr{C}_{3}=\{\{x_{1},x_{2},x_{5}\},\{x_{3},x_{4}\}\}$, $\mathscr{C}_{4}=\{\{x_{1},x_{2}\},\{x_{3},x_{4}\},\{x_{5}\}\}$, $\mathscr{D}_{D}=\{D_{1},D_{2}\}$, $D_{1}=\{x_{1},x_{2}\},$ and $D_{2}=\{x_{3},x_{4},x_{5}\}$. First, by Theorem 3.3 and Example 5.1, we obtain $$\begin{aligned}
\Gamma(\mathscr{D}^{+}_{C})=\Gamma(\mathscr{D}_{C})\bigvee\Gamma(\mathscr{C}_{4})=\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right].\end{aligned}$$
Second, by Definition 2.4, we have $$\begin{aligned}
\Gamma(\mathscr{D}^{+}_{C})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\bullet \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}
\right],\\
\Gamma(\mathscr{D}^{+}_{C})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cccccc}
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
1 & 1 & 1 & 1 &1 \\
\end{array}
\right]\odot \left[
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 1 \\
\end{array}
\right]= \left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right].\end{aligned}$$
Third, by Example 5.1, we get $$\begin{aligned}
\Gamma(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}
\right],
\Gamma(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}} =\left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right],
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\bullet M_{\mathscr{D}_{D}} =\left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
\end{array}
\right],\\
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\odot M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{1})\bullet M_{\mathscr{D}_{D}} =\left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
1 & 1 \\
1 & 1 \\
0 & 1 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{1})\odot M_{\mathscr{D}_{D}} =\left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 0 \\
0 & 1 \\
\end{array}
\right],
\\
\Gamma(\mathscr{C}_{3})\bullet M_{\mathscr{D}_{D}} &=&\left[
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
0 & 1 \\
0 & 1 \\
1 & 1 \\
\end{array}
\right],
\Gamma(\mathscr{C}_{3})\odot M_{\mathscr{D}_{D}} =\left[
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
0 & 1 \\
0 & 1 \\
0 & 0 \\
\end{array}
\right].\end{aligned}$$
Therefore, according to Definition 2.5, $\{\mathscr{C}_{1},\mathscr{C}_{3}\}$ is a type-1 reduct of $(U,\mathscr{D}^{+}_{C}\cup \mathscr{D}_{D} )$.
In Example 5.2, we must compute $ \Gamma(\mathscr{D}^{+}_{C})\bullet M_{\mathscr{D}_{D}},
\Gamma(\mathscr{D}^{+}_{C})\odot M_{\mathscr{D}_{D}},
\Gamma(\mathscr{D}_{C})\bullet M_{\mathscr{D}_{D}},
\Gamma(\mathscr{D}_{C})\odot M_{\mathscr{D}_{D}},
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\bullet M_{\mathscr{D}_{D}},
\Gamma(\{\mathscr{C}_{1},\mathscr{C}_{3}\})\odot M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{1})\bullet M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{1})\odot M_{\mathscr{D}_{D}},
\Gamma(\mathscr{C}_{3})\bullet M_{\mathscr{D}_{D}}$, and $
\Gamma(\mathscr{C}_{3})\odot M_{\mathscr{D}_{D}}
$ if we construct the type-1 reducts of dynamic covering decision information system $(U,\mathscr{D}^{+}_{C}\cup \mathscr{D}_{D} )$ with the non-incremental approach. But we only need to compute $ \Gamma(\mathscr{D}^{+}_{C})\bullet M_{\mathscr{D}_{D}}$ and $
\Gamma(\mathscr{D}^{+}_{C})\odot M_{\mathscr{D}_{D}}$ for constructing the type-1 reducts of $(U,\mathscr{D}^{+}_{C}\cup \mathscr{D}_{D} )$ with the incremental approach. Therefore, the designed algorithm is effective to conduct knowledge reduction of dynamic covering decision information systems with the immigration of coverings.
Conclusions
===========
In this paper, we have updated the type-1 and type-2 characteristic matrices and designed effective algorithms for computing the second and sixth lower and upper approximations of sets in dynamic covering information systems with variations of coverings. We have employed examples to illustrate how to calculate the second and sixth lower and upper approximations of sets. We have employed experimental results to illustrate the designed algorithms are effective to calculate the second and sixth lower and upper approximations of sets in dynamic covering information systems with the immigration of coverings. We have explored two examples to demonstrate how to conduct knowledge reduction of dynamic covering decision information systems with the immigration of coverings.
In the future, we will investigate the calculation of approximations of sets in other dynamic covering information systems and propose effective algorithms for knowledge reduction of dynamic covering decision information systems. Furthermore, we will provide parallel algorithms for knowledge reduction of dynamic covering decision information systems using the type-1 and type-2 characteristic matrices.
Acknowledgments {#acknowledgments .unnumbered}
================
We would like to thank the anonymous reviewers very much for their professional comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China (NO. 61273304, 61573255,11201490,11371130,11401052,11401195), the Postdoctoral Science Foundation of China (NO.2013M542558,2015M580353), the Scientific Research Fund of Hunan Provincial Education Department(No.14C0049,15B004).
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[^1]: Corresponding author.Tel./fax: +86 021 69585800, E-mail address: langguangming1984@126.com(G.M.Lang), cmjlong@163.com (M.J.Cai).
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---
abstract: 'The standard model of cosmology $\Lambda$CDM assumes general relativity, flat space, and the presence of a positive cosmological constant. We relax these assumptions allowing spatial curvature, a time-dependent effective dark energy equation of state, as well as modifications of the Poisson equation for the lensing potential, and modifications of the growth of linear matter density perturbations in alternate combinations. Using six parameters characterizing these relations, we check $\Lambda$CDM for consistency utilizing cosmic microwave background anisotropies, cross correlations thereof with high-redshift galaxies through the integrated Sachs-Wolfe effect, the Hubble constant, supernovae, and baryon acoustic oscillation distances, as well as the relation between weak gravitational lensing and galaxy flows. In all scenarios, we find consistency of the concordance model at the 95% confidence level. However, we emphasize that constraining supplementary background parameters and parametrizations of the growth of large-scale structure separately may lead to *a priori* exclusion of viable departures from the concordance model.'
author:
- Lucas Lombriser
bibliography:
- 'lcdm\_check.bib'
title: 'Consistency check of $\Lambda$CDM phenomenology'
---
Introduction
============
The detection of the late-time acceleration of our Universe [@riess:98; @perlmutter:98] challenges the known laws of physics. General relativity and the standard model of particle physics cannot amount to the observed cosmic expansion unless we are willing to accept a seemingly random constant in the Einstein field equations or the presence of an unknown form of energy, along with large amounts of dark matter. The concordance model assumes such a cosmological constant or vacuum energy and provides a simple but successful description for the observed Universe. Given the lack of a complete understanding, it is important to repeatedly test this model against observations with the ambition of distinguishing between different explanations of the observed cosmic acceleration, e.g., a cosmological constant, dynamical dark energy, or a modification of gravity, and rule out or constrain the different models. See [@tsujikawa:10; @jain:10; @sapone:10] for recent reviews on tests of nonstandard cosmologies.
Phenomenological parametrizations for departures from the concordance model have been studied in [@chevallier:00; @linder:02; @linder:05; @koivisto:05; @caldwell:07; @zhang:07; @amendola:07; @diporto:07; @hu:07b; @linder:09]. In this paper, we shall mainly adopt the parametrization and notation of [@amendola:07; @diporto:07]. In addition to the usual cosmological parameters, including spatial curvature, we use five phenomenological parameters quantifying modifications of the Poisson equation for the lensing potential, modifications of the growth of linear matter density perturbations, as well as a time-dependent effective dark energy equation of state. The parameters are based on models of modified gravity, but may also describe properties of dark energy and provide a framework to search for phenomena which may indicate new physical effects in current and future cosmological observations.
We conduct a Markov chain Monte Carlo (MCMC) study of this parameter space using data from the cosmic microwave background (CMB) anisotropies, supernovae distances, the baryon acoustic oscillations (BAO) distances, and the Hubble constant. We also utilize information from the cross correlation between high-redshift galaxies and the CMB through the integrated Sachs-Wolfe (ISW) effect, as well as a probe of the relation between weak gravitational lensing and galaxy flows. For the predictions of the CMB anisotropies, we connect our phenomenological parameters to the parametrized post-Friedmann (PPF) framework [@hu:07b; @hu:08] and use its implementation into a standard Einstein-Boltzmann linear theory solver [@fang:08b].
In \[sec:theory\], we define the phenomenological parametrization of the modifications to standard cosmology and explain their implications on various cosmological probes in \[sec:predictions\]. Modifications to the [iswwll]{} [@ho:08; @hirata:08] code used for the galaxy-ISW (gISW) cross correlation observations are specified in \[sec:gISW\]. For the connection of our parametrization to the PPF formalism, we refer to the appendix. We present the results of our MCMC study in \[sec:results\] and discuss them in \[sec:discussion\].
Phenomenological modifications {#sec:theory}
==============================
We consider scalar linear perturbations of the Friedmann-Lemaître-Robertson-Walker background in the longitudinal gauge, hence $$ds^2 = -(1+2\Psi)dt^2 + a^2(1+2\Phi)d\vec{x}^2,$$ where $d\vec{x}^2$ is the unperturbed spatial line element with curvature $k$ and $c=1$. Motivated by modified gravity models, effects from a nonstandard cosmology may introduce the following three time- and scale-dependent phenomenological modifications on the background and at quasistatic scales of linear perturbations (see, e.g., [@caldwell:07; @zhang:07; @amendola:07; @hu:07b; @koivisto:07; @koivisto:08]; cf. [@koivisto:05]):
- A deviation from the $\Lambda$CDM expansion history, parametrized by an effective dark energy equation of state $w_{\rm eff}(a)$, $$\begin{aligned}
\left(\frac{H}{H_0}\right)^2 & = & \Omega_{\rm m} a^{-3} + \Omega_{\rm k} a^{-2} \nonumber \\
& &+ (1-\Omega_{\rm m}-\Omega_{\rm k}) a^{-3[w_{\rm eff}(a)+1]}.
\label{eq:expansionhistory}\end{aligned}$$
- An effective change of the Newton’s constant, which we can parametrize by a free function $Q(k,a)$, yielding a generalized Poisson equation, $$k^2 \Phi = 4\pi G a^2 Q \rho_{\rm m} \Delta_{\rm m}.
\label{eq:poisson_gen}$$
- A difference in the scalar linear potentials $\Psi$ and $\Phi$ parametrized by the free function $\eta(k,a)$, $$\Psi = -(1+\eta) \Phi.
\label{eq:potentials_gen}$$
Deviations from $Q=1$ and $\eta=0$, the standard values, do not necessary indicate modifications of general relativity but might, for instance, also originate from contributions of nonmatter fluids to the Poisson equation ($Q\neq1$), e.g., from clustering of dark energy or interactions between the dark components (see, e.g., [@song:10]), or nonvanishing anisotropic stress ($\eta\neq0$) (see, e.g., [@koivisto:05; @mota:07]). Dark energy models other than the cosmological constant also predict departures from $w_{\rm eff}(a)=-1$. Note that a dark energy model can always be constructed to be formally equivalent to a modification of gravity through an effective dark energy stress-energy tensor. The parameters introduced here cannot distinguish between the two descriptions. Consideration of microphysical aspects may, however, indicate which is the more reasonable picture (see, e.g., [@hu:09; @hu:07b]).
We can combine Eqs. (\[eq:poisson\_gen\]) and (\[eq:potentials\_gen\]) to obtain [@amendola:07] $$k^2 \Phi_- = \frac{3H_0^2\Omega_{\rm m}}{2a} \Sigma \Delta_{\rm m},
\label{eq:Sigma_par}$$ where $\Sigma = Q(1+\eta/2)$ and $\Phi_- = (\Phi-\Psi)/2$. This relation is in particular sensitive to weak lensing measurements and the ISW effect can be used to probe its time evolution. Modifications as in Eqs. (\[eq:poisson\_gen\]) and (\[eq:potentials\_gen\]) lead to changes in the growth of the linear matter overdensity perturbation $\Delta_{\rm m}$, which we parametrize via the growth index $\gamma$ [@linder:05; @diporto:07], $$\frac{d \ln \Delta_{\rm m}}{d \ln a} = \Omega_{\rm m}(a)^{\gamma} (1+\xi),
\label{eq:gamma_par}$$ where $\Omega_{\rm m}(a) = H_0^2\Omega_{\rm m} a^{-3} H^{-2}$. We also introduce here the parameter $\xi$ to account for growth rates beyond unity, given $\gamma>0$ and $0\leq\Omega_{\rm m}\leq1$, as can be observed in scalar-tensor theories [@diporto:07]. Hence, detection of $\xi>0$ may indicate the presence of an additional attractive force.
We decide to use the quantities defined in Eqs. (\[eq:Sigma\_par\]) and (\[eq:gamma\_par\]) to scan the cosmological data for departure from standard theory and replace the two free functions $Q$ and $\eta$ by $\Sigma$ and $(\gamma,\xi)$. Note that $\xi$ should not be interpreted as a new parameter in the gravitational dynamics like $Q$ and $\eta$, but rather as the separation $\gamma \rightarrow \gamma + \Gamma$, where we assume the specific form $\Gamma = \ln(1+\xi)/\ln\Omega_{\rm m}(a)$.
From the ordinary differential equation, which describes the correct behavior of the linear matter overdensity perturbations in the quasistatic regime, $$\Delta_{\rm m}'' + \left( 2 + \frac{H'}{H} \right) \Delta_{\rm m}' - \frac{3}{2} \frac{H_0^2 \Omega_{\rm m}}{ a^3 H^2 } \mathcal{F} \Delta_{\rm m} = 0,$$ where $\mathcal{F}= Q(1+\eta) = 2\Sigma (1+\eta)/(2+\eta)$ and primes denote derivatives with respect to $\ln a$, we derive $$\begin{aligned}
\mathcal{F} & = & \frac{2}{3} \Omega_{\rm m}(a)^{2\gamma-1} (1+\xi)^2 -
\frac{2}{3} \left\{ \gamma \left[2\frac{H'}{H}+3\right] \right. \nonumber\\
& & \left. - \left[\frac{H'}{H}+2\right] \right\} \Omega_{\rm m}(a)^{\gamma-1} (1+\xi),
\label{eq:Fcal}\end{aligned}$$ where we assumed constant $\gamma$ and $\xi$. Given $\Sigma$ and $(\gamma$, $\xi)$, we can use Eq. (\[eq:Fcal\]) to derive $Q$ and $\eta$ and thus define our modifications to $\Lambda$CDM in the framework of Eqs. (\[eq:expansionhistory\]) through (\[eq:potentials\_gen\]). Note that $$\mathcal{F} = \frac{2}{3}(1+\xi)^2 + \frac{1}{3} (1+\xi),
\label{eq:mathcalF_lim}$$ whenever $a^{w_{\rm eff}(a)} \gg 1$ and $w_{\rm eff}'(\ln a) \ll 1$ as $a\ll1$. If $|\xi|\ll1$ at $a\ll1$, standard gravity is reproduced. Otherwise, modifications to gravity persist up to high redshifts.
Parametrization {#sec:parametrization}
---------------
It is a difficult task to find general functions with a minimal set of free parameters that are flexible enough to capture the wealth of possible modifications in Eqs. (\[eq:expansionhistory\]) through (\[eq:potentials\_gen\]) or equivalently in Eqs. (\[eq:expansionhistory\]), (\[eq:Sigma\_par\]), and (\[eq:gamma\_par\]). Rather than to construct such a function for each relation, we decide to use a low number of five parameters, in addition to spatial curvature, and examine the combinational aspects of introducing modifications in each of the relations. We restrict to constant and time-dependent modifications and choose the parametrization in a way that $\Lambda$CDM is contained in the parameter space.
For the expansion history we consider the parametrization $$\begin{aligned}
H(a)^2 & = & H_0^2 \Omega_{\rm m} a^{-3} + H_0^2 \Omega_{\rm k} a^{-2} + H_0^2 (1-\Omega_{\rm m}-\Omega_{\rm k}) \nonumber\\
& & \times a^{-3(1+w_0+w_a)} \exp{\left[ 3w_a (a-1) \right]},
\label{eq:hubbleexpansion}\end{aligned}$$ where the dark energy equation of state is given by [@chevallier:00; @linder:02] $$\begin{aligned}
w_{\rm DE}(a) & = & w_0 + (1-a) w_a, \\
a^{-3[1+w_{\rm eff}(a)]} & = & \exp \left[ 3 \int_a^1 \frac{1+w_{\rm DE}(a')}{a'} da' \right].\end{aligned}$$ Note that this parametrization does not always provide a good fit to model predictions. We adopt the approach primarily due to its simplicity and wide usage. In the limit $(w_0,w_a)\rightarrow(-1,0)$, the effective dark energy term in Eq. (\[eq:hubbleexpansion\]) reduces to a cosmological constant. The parametrization is less successful, e.g., in the case of Dvali-Gabadadze-Porrati (DGP) gravity [@dvali:00]. It provides a good approach for distance measures [@linder:05] and the quasistatic regime for the self-accelerated branch, but it fails for the normal branch due to the appearance of a divergence in $w_{\rm DE}(a)$. Also note that choosing a specific form for $w_{\rm DE}$ may have a nontrivial effect on the constraints inferred for it (see, e.g., [@nesseris:10] for a model-independent approach).
Next, we need to define modifications to the rate of growth and the Poisson equation. We decide to use a constant growth index $\gamma_0$, which is a good approximation to general relativity, where $\gamma_0\approx0.55$, or the quasistatic regime of self-accelerating DGP gravity [@deffayet:00], where $\gamma_0\approx0.68$ [@lue:04; @linder:05; @linder:07]. We further use a constant $\xi=\xi_0$, which was found to provide a good fit to scalar-tensor theories where the scalar field couples to dark matter [@diporto:07]. Equation (\[eq:gamma\_par\]) relates to the parametrization of [@linder:09] as $d \ln g_{\star}/ d \ln a = \xi \Omega_m(a)^{\gamma}$, which was introduced to describe effects from early dark energy, modified gravity at high redshifts, or early acceleration. For these models, $g_{\star}$ was found to be well-described by a constant [@linder:09]. Therefore, there is only limited correspondence to a constant $\xi=\xi_0=0$. For $\Sigma$, we use a parametrization that reduces to its general relativistic value $\Sigma \rightarrow 1$ at early times. Hence, for our consistency test, we furthermore set $$\begin{aligned}
\gamma & = & \gamma_0, \\
\xi & = & \xi_0 \\
\Sigma & = & 1 + \Sigma_0 a.
$$ For reference, we denote our nonstandard cosmologies by the extra parameters we allow to be free. Hence, when, e.g., taking $\Omega_{\rm k}$ and $\gamma_0$, as well as $w_0$ and $w_a$, to be free parameters deviating from their standard values, we denote the according model by $\gamma w {\rm k}$. Note that in the limit where $\{w_0,\ w_a,\ \gamma_0,\ \xi_0,\ \Sigma_0) \rightarrow \{-1,\ 0,\ 0.55,\ 0,\ 0)$, Eqs. (\[eq:expansionhistory\]), (\[eq:Sigma\_par\]), and (\[eq:gamma\_par\]) reduce to general relativity with cold dark matter and a cosmological constant. From Eqs. (\[eq:gamma\_par\]) and (\[eq:hubbleexpansion\]), it is clear that when $w_a\approx-w_0$, there is a degeneracy between $w_a$ and $\xi_0$, which manifests itself, in particular, at high redshifts. To explore the parameter space unclosed by free $w_0$, $w_a$, and $\xi_0$, we furthermore study three models where we fix $\gamma_0=0.55$ and $\Sigma_0=0$ while allowing the following degrees of freedom:
- $\xi_0$, $w_{\rm DE} = w_0+(1-a)w_a$,
- $\xi_0$, $w_{\rm DE} = -1+(1-a)w_a$,
- $\xi_0$, $w_{\rm DE} = -1+\lambda_0(1-1.15a)$,
such that $\Lambda$CDM is a limiting case in all of the three models. The slope of $w_{\rm DE}(a)$ in model (C) is motivated by the best-fit values derived in \[sec:results\]. For numerical predictions, we connect our parametrizations to the linear PPF framework as described in Appendix \[sec:ppf\_connection\].
Note that we have ignored scale dependence of the parameters, which is a good approximation within $\Lambda$CDM. Modifications of gravity such as $f(R)$ gravity models [@carroll:03; @nojiri:03; @capozziello:03], however, may introduce a strong scale dependence in these relations. The same holds for DGP gravity, where the deficiency is, however, restricted to near-horizon and superhorizon scales [@seahra:10]. Although for these cases the parametrization is not very descriptive, it still serves as a useful tracer of nonstandard phenomenologies. Departure from the standard parameter values may indicate inconsistencies in the concordance model and point toward new physical effects, which in turn have to be addressed with more developed theories. It has been pointed out, by conducting a principal component analysis [@zhao:09], that such inconsistencies in Eqs. (\[eq:poisson\_gen\]) through (\[eq:Sigma\_par\]) are more likely to be detected in scale-dependent modifications due to weaker sensitivity in the data to time-dependent deviations [@pogosian:10; @zhao:10]. A parametrization like the one described in this section is still capable of tracing trends in the observations and disclosing new regions in the parameter space of possible modifications, especially when simultaneously allowing for time-dependent modifications in the effective dark energy equation of state.
Consistency check {#sec:constraints}
=================
We use a variety of cosmological data sets to check against nonstandard cosmology. First we use the CMB anisotropy data from the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) [@WMAP:10], the Arcminute Cosmology Bolometer Array Receiver (ACBAR) [@ACBAR:07], the Balloon Observations Of Millimetric Extragalactic Radiation and Geophysics (BOOMERanG) flight in 2003 (B03) [@B03:05], and the Cosmic Background Imager (CBI) [@CBI:04]. Next we employ data from the Supernova Cosmology Project (SCP) Union2 [@UNION2:10] compilation, the measurement of the Hubble constant from the Supernovae and $H_0$ for the Equation of State (SHOES) [@SHOES:09] program generalized by [@reid:09], and the BAO distance measurements of [@BAO:09]. Furthermore, we take gISW cross correlation observations using the [iswwll]{} code of [@ho:08; @hirata:08], and the $E_G$ measurement, probing the relation between weak gravitational lensing and galaxy flows, of [@reyes:10].
Note that we restrict to data sets amenable to linear perturbation theory. Nonlinear probes such as from the abundance of clusters and the full range of scales of weak gravitational lensing yield tight constraints on the parameters that quantify modifications to general relativity (see, e.g., [@rapetti:08; @schmidt:09; @rapetti:09; @daniel:10a; @daniel:10b; @lombriser:10]). In the case of $f(R)$ gravity models or DGP gravity, nonlinear effects have been studied in [@oyaizu:08a; @oyaizu:08b; @schmidt:08b; @schmidt:09; @schmidt:09b; @schmidt:09c; @koyama:09; @beynon:09; @ferraro:10; @zhao:10b] and spherical collapse within various dark energy models in, e.g., [@pace:10; @wintergerst:10]. However, for phenomenological parametrizations like the one we use, nonlinear behavior has not been examined in full extent. Hence, applying standard scaling relations may lead to illusive conclusions. Furthermore, note that the inclusion of gISW cross correlations, along with the ISW effect in the CMB, and $E_G$, yield competitive results to constraints from nonlinear probes (cf., e.g., [@bean:10]).
In \[sec:predictions\] we discuss the predictions for some of these observables for specific parameter values. In \[sec:results\] we present the results of a MCMC likelihood analysis, which is conducted with the publicly available [cosmomc]{} [@COSMOMC:02] package.
Cosmological observables {#sec:predictions}
------------------------
In this section we illustrate model predictions of various cosmological observables we use to derive our results. We chose the parameters of the various models that highlight results from the MCMC analysis.
As our basic set, we choose a parametrization that separates high-redshift and low-redshift constraints. Specifically we take six high-redshift parameters: the physical baryon and cold dark matter density $\Omega_bh^2$ and $\Omega_ch^2$, the ratio of sound horizon to angular diameter distance at recombination $\theta/100$, the optical depth to reionization $\tau$, the scalar tilt $n_s$, and amplitude $A_s$ at $k_* = 0.002~\textrm{Mpc}^{-1}$. We extend this parameter set with alternate combinations of the six additional parameters, describing departures from the concordance model: the spatial curvature density $\Omega_{\rm k}$, two parameters for the evolution of the effective dark energy equation of state, $w_0$ and $w_a$, the growth index $\gamma_0$ and the scaling of the growth rate $\xi_0$, as well as the first-order of a time-dependent modification of the Poisson equation for the lensing potential $\Sigma_0$.
We illustrate predictions from the maximum-likelihood $\Lambda$CDM model, as well as from the overall best-fit model (see \[sec:results\]). Hereby, we derive the parameter values using the full set of cosmological data. To demonstrate effects from the variation of a specific basic cosmological parameter on the observables, we use the corresponding 1D-marginalized 68% confidence limits from the $\Lambda$CDM model while setting the complementary parameters to their best-fit values. For a supplementary parameter, we use its 1D-marginalized 68% confidence boundary, when including it as the sole extra parameter while setting the basic parameters to their $\Lambda$CDM best-fit values. Note, however, that for the nonstandard cases, $\gamma_0$ is always simultaneously varied with any of the extra parameters. Hereby, $(w_0,w_a)$ should be counted as one parameter; i.e., $w_0$ and $w_a$ are always either fixed to $(-1,0)$ or both considered free.
### Distance measures
The comparison of the magnitudes of high-redshift to low-redshift supernovae yields a relative distance measure. Theoretical predictions for the distance modulus are related to the luminosity distance, $d_L(z) = (1+z) r(z)$, where $r(z)$ is the comoving angular diameter distance defined by $$r(z) = \left\{
\begin{array}{ll}
\sin\left[H_0 \sqrt{-\Omega_{\rm k}} \chi(z)\right]/H_0\sqrt{|\Omega_{\rm k}|}, & \Omega_{\rm k} < 0, \\
\chi(z), & \Omega_{\rm k} = 0, \\
\sinh\left[H_0 \sqrt{\Omega_{\rm k}} \chi(z)\right]/H_0\sqrt{|\Omega_{\rm k}|}, & \Omega_{\rm k} > 0,
\end{array}
\right.$$ where the comoving radial distance $\chi$ is $$\chi(z) = \int_0^z \frac{dz'}{H(z')}.$$ The supernovae magnitudes, once standardized, are related to the distance by $$m \equiv \mu + M = 5 \log_{10}{d_L}+M + 25,$$ where $d_L$ is in units of Mpc. The unknown absolute magnitude $M$ of the supernovae is a nuisance parameter in the fit and is degenerate with $H_0$. Hence supernovae measure relative distances within the set. In Fig. \[fig:UNION2\], we plot the prediction for the distance modulus from the overall best-fit model with respect to its counterpart from the best-fit $\Lambda$CDM model. The effect of varying parameters in the expansion history is illustrated in Fig. \[fig:UNION\].
The acoustic peaks in the CMB and the measurement of the local Hubble constant additionally provide absolute distance probes, which complement the relative distance measure of the supernovae. For the Hubble constant, we utilize the SHOES measurement, $H_0=74.2\pm3.6~\rm{km \: s^{-1} \: Mpc^{-1}}$, which employs Cepheid measurements to link the low-redshift supernovae to the distance scale established by the maser galaxy NGC 4258. In the analysis, we use the generalization of this measurement as a constraint on the inverse luminosity distance at $z_{\rm eff} = 0.04$, i.e., [@reid:09] $$\begin{aligned}
d_A\left( z_{\rm eff} \right)^{-1} & = & \left( 1+z_{\rm eff} \right) r\left( z_{\rm eff} \right)^{-1} \nonumber \\
& = & \left( 1+z_{\rm eff} \right)^2 d_L\left( z_{\rm eff} \right)^{-1} \nonumber \\
& \simeq & \left( 6.49 \pm 0.32 \right) \times 10^{-3}~\textrm{Mpc}^{-1},\end{aligned}$$ where $d_A$ is the angular diameter distance and the fiducial cosmology is $h=0.742$, $\Omega_{\rm m} = 0.3$, $\Omega_{\rm k} = 0$, $w_0 = -1$, $w_a = 0$. Figure \[fig:SHOES\] demonstrates predictions for $d_A(z_{\rm eff})^{-1}$ and effects on the observable from varying parameters in the expansion history. The gray band corresponds to the $1\sigma$ region of the SHOES measurement, where the solid gray line indicates the mean value.
Further, we apply the BAO distance measurement of [@BAO:09] that is obtained from analyzing the clustering of galaxies from the Sloan Digital Sky Survey (SDSS) [@SDSS:09] and the 2-degree Field Galaxy Redshift Survey (2dFGRS) [@2dFGRS:05], constraining the ratio $$d_z \equiv \frac{r_s \left(z_d\right)}{d_V(z)} \equiv \frac{ r_s \left(z_d\right) H(z)^{1/3} }{ (1+z)^{2/3} d_A(z)^{2/3} z^{1/3} }$$ at $z=0.2$ and $z=0.35$. Here $r_s \left(z_d\right)$ denotes the comoving sound horizon at the baryon drag epoch $z_d$. In Fig. \[fig:BAO\], we plot predictions for $d_z$ at $z=0.2$ and $z=0.35$ and effects on the observables from varying parameters in the expansion history. The gray bands correspond to the $1\sigma$ region of the BAO distance measurement, where the solid gray lines indicate the mean values.
These probes place tight constraints on the background parameters in Eq. (\[eq:hubbleexpansion\]) and help to identify sources for the growth of structure and break degeneracies.
### The cosmic microwave background
The CMB probes the geometry of the background history as well as the formation of large-scale structure. The latter manifests itself on the largest scales through the ISW effect from the evolution of the gravitational potential. To predict these effects we connect our parametrization to the PPF formalism and utilize the PPF modifications to CAMB [@CAMB:99] implemented in [@fang:08b]. We configure the PPF parameters as described in Appendix \[sec:ppf\_connection\]. This connection and the incorporation of the PPF formalism into a standard Einstein-Boltzmann linear theory solver yields an efficient way to obtain predictions from our parametrization for the CMB. It also prevents violations of energy-momentum conservation and avoids gauge artifacts in our results.
In Fig. \[fig:CMB\], we plot the CMB angular temperature anisotropy power spectrum for the overall best-fit model with respect to the prediction of the best-fit $\Lambda$CDM model. Effects from varying chain parameters are illustrated in Fig. \[fig:RCMB\]. Figure \[fig:CMB\_TE\] illustrates the CMB temperature-polarization (TE) cross power spectrum for $\Lambda$CDM and the overall best-fit model. The errors denote diagonal elements of the covariance matrix and include cosmic variance and instrumental noise. The overall best-fit model provides a better fit to all of the CMB data sets. This has to be attributed not only to the relative suppression of the power spectra at large scales with respect to the best-fit $\Lambda$CDM model but also to deviations at large angular multipoles.
### Weak gravitational lensing and galaxy flows
The relationship of weak gravitational lensing around galaxies to their large-scale velocities has been proposed as a smoking gun of gravity [@zhang:07]. The advantage of such a probe lies in its insensitivity to galaxy bias and initial matter fluctuations. The expectation value of the ratio of galaxy-galaxy to galaxy-velocity cross correlations of the same galaxies yields an estimator $E_G$ [@zhang:07]. We have $$E_G \equiv \left[ \frac{\nabla^2(\Psi-\Phi)}{3H_0^2a^{-1}\frac{d \ln \Delta_{\rm m}}{d \ln a}\Delta_{\rm m}} \right]_{k=\frac{\ell}{\bar{\chi}},\bar{z}} = \frac{\Omega_{\rm m}}{\Omega_{\rm m}(a)^{\gamma}} \frac{\Sigma}{1+\xi}.$$ Recently this quantity has been measured analyzing $70\,205$ luminous red galaxies (LRGs) [@SDSS:01] from the SDSS [@SDSS:00], yielding $E_G = 0.392\pm0.065$ [@reyes:10] at the redshift $z=0.32$ by averaging over scales $R=(10-50)h^{-1}~\textrm{Mpc}$.
Figure \[fig:EG\] illustrates predictions for $E_G$ for different parameter values of $q_i \in \{ \Omega_{\rm m}, \Omega_{\rm k}, \gamma_0, w_0, w_a, \Sigma_0, \xi_0 \}$, when varying only one parameter at a time and fixing the others to their overall best-fit $\Lambda$CDM values. 1D-marginalized 68% confidence limits are obtained from the MCMC analysis in \[sec:results\]. The gray shaded band corresponds to the $1\sigma$ region of the $E_G$ measurement with the gray solid line being the mean value.
### Galaxy-ISW cross correlations {#sec:gISW}
The correlation between galaxy number densities and the CMB anisotropies can be used to isolate the ISW effect in the CMB and has proven to be a useful probe for constraining modifications to standard cosmology (see, e.g., [@giannantonio:09; @lombriser:09; @bean:10; @lombriser:10]). We utilize the publicly available [iswwll]{} code [@ho:08; @hirata:08], where we turn off weak lensing contributions to the likelihood, focusing only on the gISW constraints. The galaxies in this probe are collected from the Two Micron All Sky Survey (2MASS) extended source catalog (XSC) [@jarrett:00; @skrutskie:06], the LRG samples and photometric quasars (QSO) of the SDSS [@adelman:07], and the National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) Sky Survey (NVSS) [@condon:98] and correlated with the five-year WMAP [@WMAP:08] CMB anisotropies. The resulting 42 data points of gISW cross correlations are divided into nine galaxy sample bins $j$ (2MASS0-3, LRG0-1, QSO0-1, and NVSS) based on flux (2MASS) or redshift (LRG and QSO). These data points are a selection of multipole bins from all samples, where the selection is based on the avoidance of nonlinearities and systematic effects from dust extinction, galaxy foregrounds, the thermal Sunyaev-Zel’dovich effect, and point source contamination to affect the gISW cross correlations [@ho:08].
We evaluate the gISW cross correlation in the Limber and quasistatic approximation, as it is done in the [iswwll]{} code used for the data analysis. The gISW cross correlation in this approximation reads $$\begin{aligned}
C_{\ell}^{g_jT} & \simeq & \frac{3 \Omega_{\rm m} H_0^2 T_{\rm CMB}}{(\ell+1/2)^2} \int dz \: f_j(z) H(z) \nonumber\\
& & \times \left[ D \frac{d}{dz}G \right] P \left( \frac{\ell+1/2}{\chi(z)} \right).
\label{eq:ClgT}\end{aligned}$$ Here, $P(k)$ is the matter power spectrum today. $D$ is the linear density growth rate defined by $\Delta_{\rm m}(k,z) = \Delta_{\rm m}(k,0) D/D|_{z=0}$ and $G = \Sigma D(1+z)/ \Sigma_i$ is the linear potential growth rate, where $\Sigma_i=\Sigma|_{a=a_i}$ and $a_i\ll1$.
In $\Lambda$CDM, the Limber approximation becomes accurate at the percent level for $\ell\gtrsim10$ and drops approximately as $\ell^2$ at higher $\ell$ (see, e.g., [@smith:09; @loverde:08; @afshordi:04]). This condition is satisfied by about $90\%$ of the total 42 data points that are used in the [iswwll]{} code. The approximation is also valid for, e.g., DGP and $f(R)$ gravity [@lombriser:09; @lombriser:10]. Apart from the multipole $\ell$, the error depends also on the width of the redshift distribution, which changes only little with modifications. Given the large errors of the currently available data points at low $\ell$, we conclude that it is safe to apply the Limber approximation and furthermore that it is very useful since it is numerically faster than an exact integration.
The function $f_j(z)$ relates the matter density to the observed projected galaxy overdensity with $f_j(z) = b_j(z)\Pi_j(z)$ in the absence of magnification bias. $\Pi_j(z)$ is the redshift distribution of the galaxies and the bias $b_j(z)$ is assumed independent of scale (cf. [@smith:09]) but dependent on redshift.
The term in brackets in Eq. (\[eq:ClgT\]), parametrized by Eqs. (\[eq:Sigma\_par\]) and (\[eq:gamma\_par\]), can be rewritten as $$D \frac{d}{dz}G = \left\{ \left[ 1-\Omega_{\rm m}(z)^{\gamma}(1+\xi) \right] \Sigma + (1+z) \frac{d\Sigma}{dz} \right\} D^2,
\label{eq:growthfactor}$$ where $$D = \exp\left[-\int_0^z \frac{\Omega_{\rm m}(z)^{\gamma}(1+\xi)}{z+1} dz\right].$$
For SDSS quasars, the derivation of $f_j(z)$ involves the modified linear growth factor given through Eq. (\[eq:growthfactor\]) and magnification bias, for which we use a modification of the lensing window function of [@ho:08] $$W(z,z') = \frac{3\Sigma H_0^2(1+z)}{2 H(z)} r(z)^2 \left[ \left. \frac{d\ln r(z'')}{d\chi(z'')} \right|_{z'}^{z} \right].$$
Note that the gISW analysis uses photometric LRG samples, whereas the $E_G$ measurement is based on spectroscopic LRG samples that do not overlap in redshift. Furthermore, the gISW signal is dominated by large scales and most of the error is caused by sampling variance and shot noise of galaxies. The error on $E_G$ is dominated by uncertainties in lensing and redshift space distortions and most of the signal comes from small scales around $10h^{-1}~\textrm{Mpc}$. Therefore, we can safely neglect correlations between the gISW and $E_G$ data sets.
We plot predictions from the overall best-fit model and the best-fit $\Lambda$CDM model for the gISW cross correlation in the NVSS sample in Fig. \[fig:ISWNVSS\] and illustrate effects of varying the chain parameters in Fig. \[fig:RISW\].
MCMC likelihood analysis {#sec:results}
------------------------
Parameter
----------------------- ------------------ ---------------- ------------------ ---------
$\gamma_0$ $0.57\pm0.10$ $0.53$ $0.59\pm0.10$ $0.60$
$100\:\Omega_{\rm k}$ $-0.32\pm0.53$ $-0.81$
$\gamma_0$ $0.59\pm0.11$ $0.56$ $0.62\pm0.12$ $0.60$
$w_0$ $-1.10\pm0.16$ $-1.12$ $-1.05\pm0.18$ $-1.18$
$w_a$ $(-0.19, 1.03)$ $0.44$ $(-0.79, 0.99)$ $0.58$
$100\:\Omega_{\rm k}$ $-0.54\pm0.77$ $-0.21$
$\gamma_0$ $0.57\pm0.12$ $0.62$ $0.58\pm0.11$ $0.60$
$10\:\Sigma_0$ $0.02\pm0.93$ $0.12$ $-0.03\pm0.95$ $0.24$
$100\:\Omega_{\rm k}$ $-0.30\pm0.58$ $-0.30$
$\gamma_0$ $0.58\pm0.10$ $0.57$ $0.60\pm0.10$ $0.56$
$100\:\xi_0$ $0.11\pm0.76$ $0.52$ $0.83\pm1.08$ $0.47$
$100\:\Omega_{\rm k}$ $-0.73\pm0.75$ $-0.59$
$\gamma_0$ $0.58\pm0.12$ $0.55$ $0.60\pm0.15$ $0.59$
$w_0$ $-1.11\pm0.15$ $-1.13$ $-1.05\pm0.18$ $-1.18$
$w_a$ $(-0.99, 1.05)$ $0.48$ $(-0.78, 1.13)$ $0.48$
$10\: \Sigma_0$ $0.09\pm0.92$ $-0.08$ $-0.14\pm0.94$ $-0.20$
$100\:\Omega_{\rm k}$ $-0.52\pm0.93$ $-0.25$
$\gamma_0$ $0.57\pm0.12$ $0.51$ $0.60\pm0.11$ $0.56$
$10\:\Sigma_0$ $0.07\pm0.97$ $-0.27$ $0.01\pm0.94$ $0.18$
$100\:\xi_0$ $0.15\pm0.80$ $0.33$ $0.79\pm1.07$ $0.57$
$100\:\Omega_{\rm k}$ $-0.71\pm0.77$ $-0.47$
$\gamma_0$ $0.60\pm0.11$ $0.67$ $0.63\pm0.12$ $0.60$
$w_0$ $(-1.31, -1.03)$ $-1.13$ $(-1.30, -0.93)$ $-1.29$
$w_a$ $(0.36, 1.45)$ $1.17$ $(-0.46, 1.36)$ $1.29$
$100\:\xi_0$ $(-1.46, 3.17)$ $10.06$ $(-0.58, 2.16)$ $3.72$
$100\:\Omega_{\rm k}$ $-0.64\pm0.93$ $0.23$
$\gamma_0$ $0.60\pm0.11$ $0.60$ $0.61\pm0.13$ $0.66$
$w_0$ $(-1.29, -1.01)$ $-1.15$ $(-1.30, -0.94)$ $-1.24$
$w_a$ $(0.37, 1.43)$ $1.17$ $(-0.37, 1.42)$ $1.26$
$10\:\Sigma_0$ $0.27\pm0.99$ $-0.38$ $0.06\pm0.96$ $0.50$
$100\:\xi_0$ $(-1.29, 3.14)$ $6.33$ $(-0.67, 2.35)$ $7.19$
$100\:\Omega_{\rm k}$ $-0.54\pm1.02$ $-0.41$
: Mean, standard deviations, and best-fit values for the extra degrees of freedom in the $\gamma [w] [\Sigma] [\xi] [{\rm k}]$ models from using the WMAP7, ACBAR, B03, CBI, UNION2, BAO, SHOES, gISW, and $E_G$ data sets. Values in brackets denote the 1D-marginalized 68% minimal credible intervals [@hamann:07] quoted for parameter directions in the posterior distribution with distinctive skewness.[]{data-label="tab:results_one"}
Parameter
---------------------------------- ------------------- ------------------ ------------------- ------------------ ------------------- ----------
$100\: \Omega_b h^2$ $2.229\pm0.050$ $2.244$ $2.237\pm0.056$ $2.246$ $2.231\pm0.056$ $2.196$
$\Omega_c h^2$ $0.1114\pm0.0030$ $0.1121$ $0.1066\pm0.0075$ $0.0995$ $0.1058\pm0.0077$ $0.0935$
$\theta$ $1.0397\pm0.0024$ $1.0406$ $1.0443\pm0.0063$ $1.0510$ $1.0453\pm0.0068$ $1.0560$
$\tau$ $0.085\pm0.014$ $0.087$ $0.086\pm0.015$ $0.090$ $0.086\pm0.015$ $0.091$
$n_s$ $0.960\pm0.012$ $0.962$ $(0.947, 0.991)$ $1.016$ $(0.948, 0.992)$ $1.011$
$\ln \left[ 10^{10} A_s \right]$ $3.199\pm0.036$ $3.199$ $(3.034, 3.270)$ $2.810$ $(3.009, 3.271)$ $2.744$
$\gamma_0$ $0.60\pm0.11$ $0.67$
$w_0$ $(-1.31, -1.03)$ $-1.13$ $(-1.29, -1.02)$ $-1.07$
$w_a$ $(0.36, 1.45)$ $1.17$ $(0.38, 1.47)$ $1.09$
$100\: \xi_0$ $(-1.46, 3.17)$ $10.06$ $(-1.42, 3.63)$ $9.67$
$\Omega_{\rm m}$ $0.268\pm0.014$ $0.269$ $0.258\pm0.018$ $0.233$ $0.256\pm0.019$ $0.224$
$\sigma_8$ $0.806\pm0.021$ $0.812$ $0.779\pm0.077$ $0.749$ $0.800\pm0.063$ $0.769$
$H_0$ $70.6\pm1.3$ $70.7$ $70.7\pm1.7$ $72.3$ $70.7\pm1.8$ $71.8$
Parameter
---------------------------------- ------------------- ---------------- -------------------- ---------------- ------------------- ----------
$100\: \Omega_b h^2$ $2.236\pm0.056$ $2.230$ $2.229\pm0.056$ $2.189$ $2.232\pm0.055$ $2.201$
$\Omega_c h^2$ $0.1097\pm0.0059$ $0.1150$ $0.1041\pm0.0073$ $0.0976$ $0.1023\pm0.0067$ $0.0986$
$\theta$ $1.0413\pm0.0048$ $1.0395$ $(1.0474, 1.0559)$ $1.0550$ $1.0489\pm0.0066$ $1.0544$
$\tau$ $0.085\pm0.014$ $0.086$ $0.086\pm0.015$ $0.081$ $0.087\pm0.014$ $0.085$
$n_s$ $0.963\pm0.014$ $0.959$ $(0.953, 1.002)$ $0.996$ $(0.970, 1.018)$ $0.998$
$\ln \left[ 10^{10} A_s \right]$ $3.179\pm0.059$ $3.221$ $(2.830, 2.958)$ $2.841$ $(2.775, 2.982)$ $2.864$
$w_a$ $(-0.23, 0.53)$ $-0.20$
$100\: \xi_0$ $(-0.81, 1.41)$ $-0.18$ $(5.07, 7.64)$ $7.88$ $(4.19, 8.59)$ $7.68$
$\lambda_0$ $(0.31, 1.22)$ $1.02$ $(0.71, 1.25)$ $1.02$
$\Omega_{\rm m}$ $0.267\pm0.015$ $0.274$ $0.253\pm0.021$ $0.227$ $0.248\pm0.021$ $0.232$
$\sigma_8$ $0.809\pm0.060$ $0.829$ $0.811\pm0.062$ $0.818$ $0.788\pm0.062$ $0.805$
$H_0$ $70.4\pm1.7$ $70.9$ $70.7\pm1.8$ $72.5$ $71.0\pm2.0$ $72.0$
Statistic $\Lambda$CDM $\gamma w \xi$ $\gamma w \xi {\rm k}$ $\gamma w \Sigma \xi$ $\gamma w \Sigma \xi {\rm k}$ (A) (C) *Others* (C)$^*$
----------------------------------------------------- -------------- ---------------- ------------------------ ----------------------- ------------------------------- -------- -------- ---------- ---------
$-2 \Delta \ln \mathcal{L}_{\rm max}$ $0$ $-5.2$ $-2.2$ $-3.8$ $-3.4$ $-4.9$ $-7.3$ $>-0.5$ $-6.8$
$-2 \Delta \langle \ln \mathcal{L} \rangle_{\rm s}$ $0$ $0.9$ $2.9$ $2.2$ $3.4$ $0.0$ $-0.9$ $>0$ $-2.9$
$\overline{\langle \mathcal{L}\rangle_{\rm s}}$ $1$ $2.6$ $0.6$ $1.0$ $0.6$ $2.3$ $6.5$ $<1$ $11.3$
We turn to the likelihood analysis of the cosmological parameter space. As our elementary set of parameters we use $\mathcal{C} = \left\{ \Omega_bh^2, \Omega_ch^2, \theta, \tau, n_s, \ln \left[ 10^{10} A_s \right], \gamma_0 \right\}$, where for the concordance model $\mathcal{C}_{\Lambda\textrm{CDM}} \approx \mathcal{C} \cap \{ \gamma_0 = 0.55 \}$. We implement the following flat priors: $\Omega_bh^2 \in (0.01,0.1)$, $\Omega_ch^2 \in (0.045,0.99)$, $\theta \in (0.5,10)$, $\tau \in (0.01,0.8)$, $n_s \in (0.5,1.5)$, $\ln \left[ 10^{10} A_s \right] \in (2.7,4)$ [^1], and $\gamma_0 \in (-5,5)$. In addition we use all the sets of combinations between the phenomenological parameters and spatial curvature $p_i \in \mathcal{P}$, where $\mathcal{P} = \{W, \Sigma_0, \xi_0, \Omega_{\rm k} \}$ and $W = \{w_0, w_a\}$, being free and taking on their $\Lambda$CDM values $\{w_0=-1, w_a=0, \Sigma_0=0, \xi_0=0, \Omega_{\rm k}=0 \}$. We assign the following flat priors to them: $w_0 \in (-5,5)$, $w_a \in (-10,10)$, $\Sigma_0 \in (-5,5)$, $\xi_0 \in (-0.5,0.5)$, and $\Omega_{\rm k} \in (-0.3,0.3)$. As starting centers for $p_i \in \mathcal{P}$ we use the $\Lambda$CDM values. For the (C) model, we use $\lambda_0 \in (-20,20)$ with starting center $\lambda_0=0.5$.
The [cosmomc]{} package employs the Metropolis-Hastings algorithm [@metropolis:53; @hastings:70] for the sampling and the Gelman and Rubin statistic $R$ [@gelman:92] for testing the convergence. We require $R-1 < 0.01$ for our runs with two or fewer extra degrees of freedom and $R-1 < 0.02$ for three or more extra degrees of freedom [^2].
The chain samples are used to infer marginalized probabilities and mean likelihoods of the posterior. The marginalized distribution is obtained from projecting the samples to the reduced dimensions of a subspace, ignoring information about the goodness of fit and skewness of the distribution in the marginalized directions. Averaging the likelihood for each point of a subspace produces the mean likelihood. If the two curves do not overlap, the distribution is not Gaussian or the priors are not flat. It is important to note that in this case, marginalized probabilities may be amplified due to a larger parameter volume rather than by a better fit to the data. Furthermore, for skew distributions, we quote our 1D-marginalized constraints in terms of minimal credible intervals [@hamann:07] rather than by marginalized confidence limits. The former indicate where the tails occupy equal fractions of the probability distribution, whereas the latter are constructed in such a way that any point inside of the interval has a larger posterior than a point outside of it, being the more meaningful selection in the case of skew or multimodal distributions.
We summarize our results in Table \[tab:results\_one\] through \[tab:stats\]. For comparison of the goodness of fit between the different models, we quote $$\begin{aligned}
-2 \Delta \ln \mathcal{L}_{\rm max} & = & 2 \ln \left( \mathcal{L}_{\rm max}^{\rm \Lambda CDM}/\mathcal{L}_{\rm max} \right), \label{eq:goodness_a} \\
-2 \Delta \langle \ln \mathcal{L} \rangle_{\rm s} & = & 2\left( \langle \ln \mathcal{L}_{\rm \Lambda CDM}\rangle_{\rm s} - \langle \ln \mathcal{L}\rangle_{\rm s} \right), \\
\overline{\langle \mathcal{L}\rangle_{\rm s}} & = & \langle \mathcal{L}\rangle_{\rm s}/\langle \mathcal{L}_{\rm \Lambda CDM}\rangle_{\rm s} \label{eq:goodness_b}\end{aligned}$$ in Table \[tab:stats\], where $\mathcal{L}_{\rm max}$ is the maximum likelihood of a model and $\langle \cdot \rangle_{\rm s}$ denotes the average over the Monte-Carlo samples. Note that MCMC sampling may not provide very accurate best-fit estimates (see, e.g., [@COSMOMC:02]). The best-fit values may also have a much higher likelihood than the mean, but simultaneously be confined to a very small region of the parameter space. We also give the mean likelihood of the samples $\overline{\langle \mathcal{L}\rangle_{\rm s}}$, which corresponds to taking the posterior distribution as prior in calculating the evidence and in contrary to the maximum likelihood is a quantity that penalizes fine-tuning. If $\overline{\langle \mathcal{L}\rangle_{\rm s}}$ is greater than unity, this usually suggests that, on average, the extra parameters improve the goodness of fit to the data. This, however, has to be interpreted only as a rule of thumb (see, e.g., [@COSMOMC:02]). We only give constraints on the parameters $c_i \in C \backslash \{\gamma_0\}$ for the $\gamma w \xi$, (A), (B), and (C) models in addition to $\Lambda$CDM (see Tables \[tab:results\_one\] through \[tab:results\_three\]). These models, except for the (B) model, exhibit an improved goodness of fit in terms of maximum and averaged likelihoods over the $\Lambda$CDM model. We numerically evaluate Eqs. (\[eq:goodness\_a\]) through (\[eq:goodness\_b\]) and quote the numbers for the $\gamma w [\Sigma] \xi [k]$, (A), and (C) models in Table \[tab:stats\]. For all other models, $\overline{\langle \mathcal{L}\rangle_{\rm s}}$ is smaller than unity, $-2 \Delta \langle \ln \mathcal{L} \rangle_{\rm s}$ is positive, and $-2 \Delta \ln \mathcal{L}_{\rm max}>-0.5$.
Under the assumptions made in \[sec:parametrization\], our results show that constraints on the growth index parameter $\gamma_0$ are only weakly affected by the introduction of the other extra degrees of freedom with consistency of $\Lambda$CDM, i.e., $\gamma_0\approx0.55$, at the 68% confidence level. The constraints on $\gamma_0$ are dominated by the gISW data and are competitive to existing results in the literature derived from linear and nonlinear probes (cf., e.g., [@rapetti:08; @rapetti:09; @bean:10; @dossett:10]).
In every scenario studied here, the standard values of all extra parameters, corresponding to the concordance model, lie within their 1D-marginalized 95% confidence limits. Thus, we conclude that the $\Lambda$CDM model is consistent with the joint set of WMAP7, ACBAR, B03, CBI, UNION2, BAO, SHOES, gISW, and $E_G$ data under the assumption of the existence of extra degrees of freedom of the kind described in \[sec:parametrization\] and form of priors given above. Note that constraints on a parameter direction depend a lot on the prior assumed for it or for the parameters used to derive it (see, e.g., [@COSMOMC:02]). Identifying adequate priors on extra degrees of freedom in the gravitational dynamics based on theoretical contemplations is the object of current research (see, e.g., [@ferreira:10; @song:10]).
For the maximum-likelihood $\Lambda$CDM model we obtain a fit of $-2 \ln \mathcal{L}_{\rm max}^{\rm \Lambda CDM}=8099.3$ using all of the data described above. The overall best-fit to the data was obtained in the chains of model (C) with $-2 \Delta \ln \mathcal{L}_{\rm max} = -7.3$ with respect to the best-fit $\Lambda$CDM model. Note that in principle the best-fit values of model (C) are also contained in the parameter spaces of the $\gamma w [\Sigma] \xi [{\rm k}]$ and (A) models. The fact that none of these models accessed the best-fit point can be attributed to sampling errors in the chains. However, for the maximum-likelihood values of the $\gamma w [\Sigma] \xi [{\rm k}]$ and (A) models, we also obtain increases in fit of $-2 \Delta \ln \mathcal{L}_{\rm max}\lesssim-2$ over the best-fit $\Lambda$CDM model (see Table \[tab:stats\]). In terms of $p$ values this translates to 3%, 18%, and 27% for consistency with $\Lambda$CDM in the (C), (A), and $\gamma w \xi$ scenarios, respectively, where for all other models this number is greater than $50\%$. However, several best-fit parameter values of the $\gamma w [\Sigma] \xi [{\rm k}]$, (A), and (C) models lie beyond their corresponding 1D-marginalized 68% confidence limits. In the case of, e.g., the $\gamma w \xi$ model with parameter space $\mathcal{C}\cup\{w_0,w_a,\xi_0\}$, the best fit of $\xi_0$ lies even beyond the 1D-marginalized 99% confidence level, indicating that the best-fit model occupies only a very small subspace of the parameter space. Remarkably, this increase in maximum likelihood ($\Delta\chi^2_{\rm max}\lesssim-2$) seems to appear only in cases where we allow a free, positive, $\xi_0$, phantom crossing in $w_{\rm DE}$, and $w_0\approx-w_a$, i.e., an effective dark energy equation of state that drives toward a matterlike equation of state at early times. We further observe that in these scenarios, there is no signature of $n_s\neq1$. Quite to the contrary, the overall best fit obtained in model (C), i.e., $n_s=0.996$, portrays an almost perfect scale-invariant Harrison-Zel’dovich spectrum [@harrison:69; @zeldovich:72]. Moreover, we observe a slight preference for smaller dark matter densities, smaller amplitudes for the primordial superhorizon power of the curvature perturbation, and larger ratios of sound horizon to angular diameter distance at recombination with respect to parameter values inferred for the concordance model. If we additionally assume a flat Universe and matter density perturbations that relate to lensing potentials as in $\Lambda$CDM, i.e., $\Sigma_0=0$, these scenarios induce values for $\overline{\langle \mathcal{L}\rangle_{\rm s}}$ beyond unity, suggesting an increase of the goodness of fit through the inclusion of their extra parameters.
The increase in maximum likelihood observed for models with $\xi_0>0$ and $w_a\approx-w_0>1$ is mainly attributed to slightly better fits of the CMB anisotropy data on all scales and to a smaller extent also to marginally better fits in distance measures, as can be perceived from the figures in \[sec:predictions\]. Merely the gISW data counteracts this trend in favor of $\Lambda$CDM and when removed from the joint data set for, e.g., the (C) model, here denoted by (C)$^*$, we obtain the most extreme values for our statistics, Eqs. (\[eq:goodness\_a\]) through (\[eq:goodness\_b\]) (see Table \[tab:stats\]), while $\Lambda$CDM parameter values remain consistent at the 95% confidence level (cf. Table \[tab:results\_three\]).
Figures \[fig:likes\] and \[fig:likes\_C\] show the marginalized likelihoods for the extra parameters and the parameters that exhibit a distinctive skewness or multiple modes in the case of the $\gamma w \xi$ and (C) model, respectively. In Fig. \[fig:contours\], we plot 2D-marginalized contours of the extra parameters $w_0$, $w_a$, $\gamma_0$, and $\xi_0$ within the $\gamma w \xi$ model. We also indicate the 1D-marginalized minimal credible intervals on $\lambda_0$ and $\xi_0$ obtained in model (C). Note that the best-fit points lie outside the 2D-marginalized 68% confidence level contours of $\xi_0$ and $\gamma_0$, as well as of $\xi_0$ and $w_a$, which is due to the fact that the best-fit model occupies only a small parameter subspace and a distinctive skewness of the posterior distribution in the parameter directions. Note that there is a strong degeneracy between the parameters $\xi_0$ and $w_a$ as we expected from Eq. (\[eq:gamma\_par\]). Figure \[fig:par\] illustrates the overall best-fit parameter values from model (C) and $\gamma w \xi$ for the modifications in the dark energy equation of state $w_{\rm DE}(a)$, the factor $\mathcal{F}$ that enters the ordinary differential equation of quasistatic matter overdensity perturbations, the scaling of the effective Newton’s constant $Q$, and $\eta$ that modifies the ratio of the scalar potentials. Also shown are the standard values for these quantities.
We find consistency of the $\Lambda$CDM values at the 95% confidence level for all extra parameters in all scenarios. We, therefore, do not claim detection of nonstandard cosmological effects or the necessity of extra degrees of freedom based on the increases in the maximum and mean likelihoods (see Table \[tab:stats\]). This would require a physically better-motivated model with well-understood priors and a more sophisticated analysis of the goodness of fit with, e.g., efficient exploration of the different modes and determination of the Bayes factor for model comparison (see, e.g., [@feroz:07; @feroz:08; @mukherjee:05; @shaw:07]). Note that even for a hypothetical, theoretically well-motivated model that would naturally access the best-fit region in the parameter space, there would only be a moderate preference over the concordance model, i.e., $\Delta\chi^2_{\rm max}\simeq-7$. It is not clear if such a model exists or may be developed, but we emphasize that we may *a priori* be excluding viable departures from the concordance model when constraining supplementary parameters for the background and growth of structure separately.
Discussion {#sec:discussion}
==========
We allowed departures from the concordance model by introducing six additional degrees of freedom parametrizing a time-dependent effective dark energy equation of state, modifications of the Poisson equation for the lensing potential, and modifications of the growth of linear matter density perturbations, as well as spatial curvature. We constrained alternate combinations of these parameters by performing MCMC likelihood analyses on cosmological data amenable to linear perturbation theory. In particular, we utilized all of the CMB data, including the lowest multipoles, its correlation with galaxies and the comparison of weak lensing to large-scale velocities in addition to geometrical probes from supernovae and BAO distances, as well as from the Hubble constant.
We find consistency of the concordance model at the 95% confidence level. For specific combinations of the supplementary free parameters, we, however, find an increase in the maximum likelihood of up to $\Delta\chi^2_{\rm max}\simeq-7$ over the maximum-likelihood $\Lambda$CDM model in the joint data. We observe that this increase in maximum likelihood only appears in cases where we allow a free, positive, $\xi_0$, phantom crossing in $w_{\rm DE}(a)$, and $w_0\approx-w_a$, i.e., an effective dark energy equation of state that drives toward a matterlike equation of state at early times. Moreover, we find that in these scenarios there is no preference for $n_s\neq1$. For scenarios where we adopt the assumptions of a flat Universe and a standard relation of matter density perturbations to lensing potentials, both maximum and mean likelihoods are greater than in $\Lambda$CDM, with a maximal ratio of mean likelihoods of the samples of $\overline{\langle \mathcal{L}\rangle_{\rm s}} \simeq 6$, suggesting that the introduction of their extra parameters might be justified. The increase in maximum likelihood can be attributed to better fits of the CMB anisotropy data on all scales and to a smaller extent also to better fits in distance measures. We therefore expect future CMB data from, e.g., the Planck mission [@PLANCK:10] to yield more decisive constraints on our modifications (cf., e.g., [@mortonson:08]). The gISW data sets counteract this trend in favor of the $\Lambda$CDM model and when removed, we obtain $\Delta\chi^2_{\rm max}\simeq-7$ and $\overline{\langle \mathcal{L}\rangle_{\rm s}} \simeq 11$ for the (C) model.
Given the consistency of $\Lambda$CDM parameter values with their marginalized constraints, also under removal of gISW data, and the lack of a better-motivated theory and priors, we do, however, not claim detection of nonstandard cosmological effects or the necessity of the introduction of extra degrees of freedom, but we emphasize that when constraining the background parameters and growth parameters separately, we may *a priori* be excluding viable departures from $\Lambda$CDM. We leave the search for a well-motivated model that matches the above requirements and the analysis of its performance in confrontation with new data, as well as the study of effects from scale-dependent deviations from the concordance model on our results to future work. We also point out that more sophisticated sampling methods may significantly improve our statistical analysis and offer more information through efficient exploration of the different modes and the determination of the Bayes factor for model comparison.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Uroš Seljak, Anže Slosar, Tristan Smith, Eric Linder, David Rapetti, Scott Daniel, Dragan Huterer, and Gregory Martinez for useful discussions and comments. Computational resources were provided on the Schrödinger and zBox2 supercomputers at the University of Zürich. This work was supported by the Swiss National Foundation under Contract No. 2000\_124835/1.
Connecting our parametrization to the linear PPF framework {#sec:ppf_connection}
==========================================================
Given the expansion history, the PPF framework [@hu:07b; @hu:08] is defined by three functions and one parameter. From these quantities, the dynamics are determined by conservation of energy and momentum and the Bianchi identities. The defining quantities are $g(a,k)$, which quantifies the effective anisotropic stress of the modifications and distinguishes the two gravitational potentials, $f_{\zeta}(a)$, which describes the relationship between the matter and the metric on superhorizon scales, and $f_G(a)$, which defines it in the linearized Newtonian regime. The additional parameter $c_{\Gamma}$ is the transition scale between the superhorizon and Newtonian behaviors.
From the relations defined in \[sec:theory\], we infer $g = -\eta/(\eta+2)=1 - \mathcal{F}/\Sigma$ and $f_G = (1-\Sigma)/\Sigma$ in the quasistatic regime. At superhorizon scales, we impose $f_{\zeta} = 0$ and we set the transition at the horizon, $c_{\Gamma} = 1$. Given that $g\rightarrow 0$ sufficiently fast for $a\ll1$, if $|\xi|\ll1$ and $\Sigma\rightarrow1$, we apply the relation $g=1 - \mathcal{F}/\Sigma$ to all scales.
We implement these relations and the definitions in \[sec:theory\] into the PPF modified code [@fang:08b]. This procedure produces the correct power spectra for $\Lambda$CDM and self-accelerating DGP gravity. The latter, however, only when restricting to its subhorizon effects. In connecting our parametrization to the PPF linear theory, we can take advantage of a fully consistent framework for modifications in the gravitational dynamics. In this way, we can also prevent implicit violations of the conservation of energy and momentum and avoid gauge artifacts in our parametrization.
[^1]: Parameter constraints on $\ln \left[ 10^{10} A_s \right]$ are close to its prior boundary, suggesting a non-negligible influence of the prior. We have checked that using a flat top-hat prior with the lower boundary set at 2 instead of 2.7 does not alter our results in a significant manner.
[^2]: Note that the $\gamma w \Sigma \xi [k]$ models did not acquire the desired accuracy and reached only a convergence of $R-1 \lesssim 0.1$.
|
---
abstract: 'The yields and spectra of strange hadrons have each been measured by STAR as a function of centrality in $\rm \sqrt{s_{NN}}=$ 200 GeV AuAu collisions. By comparing to pp and dAu at $\rm \sqrt{s_{NN}}=200$ GeV and in AuAu at $\rm \sqrt{s_{NN}}=62$ GeV the dependence on system size and energy is studied. Strange resonances, such as $\Sigma (1385)$ and $\Lambda (1520)$, are used to examine the dynamical evolution between production and freeze-out for these systems. Particle production is investigated by comparison to thermal models, which assume a scaling of the yield with $\rm N_{part}$. Our hyperon measurements in AuAu indicate that $\rm N_{bin}$ may be a more appropriate scale for the strangeness correlation volume. Thus canonical suppression can not be simply parameterized with the geometrical overlap volume but will depend on the individual quark content of each particle. This theory is tested by comparing the data from different collision systems and centralities.'
address: ' Physics Department, Yale University, P.O. Box 208120, New Haven, CT, 06520, USA '
author:
- 'Sevil Salur (for the STAR Collaboration)'
title: System and Energy Dependence of Strangeness Production with STAR
---
Introduction
============
RHIC has been run in various configurations of pp, CuCu, dAu, and AuAu at energies ranging from $\rm \sqrt{s_{NN}}$ =19 to 200 GeV. These rich data sets, together with STAR’s large acceptance, provide the best opportunity to study strange particles in dense systems. As the strange quark is the next-lightest after the up and down quarks and does not exist in the initial colliding system, investigation of its production and dynamics may reveal some of the properties of strongly interacting matter at high densities. A hydrodynamically inspired Blast-Wave parametrization with fit parameters kinetic temperature $\rm T_{Kin}$ at freeze-out, mean transverse flow velocity $\langle \beta_{\rm T}\rangle$, and a normalization factor, is used to fit the data [@kolb; @jeff]. The one and two $\sigma$ fit contours of the $\rm T_{Kin}$ and $\langle \beta_{\rm T}\rangle$ parameters from a Blast-Wave fit to $\pi$, K, p and strange particles are presented in Figure \[fig:BW\] for $\rm \sqrt{s_{NN}}=$ 17.3, 62.4 and 200 GeV collision energies. The variation of the fit parameters indicates that the spectral shapes are different for the different particles. The $\rm T_{Kin}$ parameter is higher (hotter source) and $\langle\beta_{\rm T}\rangle$ is lower (less flow) for baryons with higher strange quark content at the same collision energy. While the $\langle\beta_{\rm T}\rangle$ remains almost unchanged, the $\rm T_{Kin}$ parameter for the multi-strange baryons is lower at $\rm \sqrt{s_{NN}}$= 62 GeV than at 200 GeV. This behavior is different for $\pi$, K, and p where the $\langle\beta_{\rm
T}\rangle$ is larger at 200 GeV though $\rm T_{Kin}$ is the same. For SPS collisions at $\rm \sqrt{s_{NN}}$=17.3 GeV, the parameters follow similar trends
{width="20pc"}\
\[fig:BW\]
as at RHIC. The differences between the fit parameters imply variations in dynamical properties of the collision energies, while in terms of chemical properties the results from $\rm \sqrt{s_{NN}}$= 62.4 and 200 GeV collisions are equivalent [@olga].
System Size Dependence Of Strange Particles
===========================================
The energy dependence of $\Lambda$ and $\overline{\Lambda}$ yields at mid-rapidity from AuAu collisions at RHIC and PbPb collisions at SPS as a function of $\rm \sqrt{s_{NN}}$ is presented in Figure 2-a. From SPS to RHIC energies, strange baryon production is approximately constant at mid-rapidity, whereas the $\overline{\Lambda}$ rises steeply, reaching $80 \%$ of the $\Lambda$ yield at RHIC top energies. The other hyperons - $\Xi$, $\Omega$ and $\Sigma(1385)$- follow similar trends. This implies that at low energies, strange baryon production is dominated by transport from the colliding system but at RHIC it is dominated by pair production. Figure 2-b shows the $\overline{\Lambda}/\Lambda$ ratios with respect to the number of participants for the collisions at SPS and RHIC. Within errors, the $\overline{\Lambda}/\Lambda$ ratios are nearly independent of the system size in AuAu collisions of the same energy at RHIC. There is a decrease in the $\overline{\Lambda}/\Lambda$ ratio moving from pp to dAu to AuAu.
![$\rm (a)$ The $\sqrt{s_{NN}}$ dependence of $\Lambda$ and $\overline{{\Lambda}}$ yields. $\rm (b)$ The dependence of $\overline{\Lambda}/\Lambda$ on number of participants for various collision energies. $\rm (c)$ Resonance to stable particle ratios normalized to pp for pp, dAu and AuAu collisions at $\rm \sqrt{s_{\rm NN}} = $ 200 GeV [@dipak; @salur; @markert].](QM_lambdayieldvsenergy_paper.eps "fig:"){width="12.8pc"}\
\[fig:MSRatio\]
![$\rm (a)$ The $\sqrt{s_{NN}}$ dependence of $\Lambda$ and $\overline{{\Lambda}}$ yields. $\rm (b)$ The dependence of $\overline{\Lambda}/\Lambda$ on number of participants for various collision energies. $\rm (c)$ Resonance to stable particle ratios normalized to pp for pp, dAu and AuAu collisions at $\rm \sqrt{s_{\rm NN}} = $ 200 GeV [@dipak; @salur; @markert].](QM_lambdaratiovsnpart_paper.eps "fig:"){width="12.8pc"}\
\[fig:llratio\]
![$\rm (a)$ The $\sqrt{s_{NN}}$ dependence of $\Lambda$ and $\overline{{\Lambda}}$ yields. $\rm (b)$ The dependence of $\overline{\Lambda}/\Lambda$ on number of participants for various collision energies. $\rm (c)$ Resonance to stable particle ratios normalized to pp for pp, dAu and AuAu collisions at $\rm \sqrt{s_{\rm NN}} = $ 200 GeV [@dipak; @salur; @markert].](QM_resonanceratio_writeup.eps "fig:"){width="12.8pc"}\
\[fig:resratio\]
Figure 2-c shows the ratio of strange resonances to their corresponding stable particles normalized to their values in pp. While the $\Sigma(1385)/\Lambda$ ratio is independent of system size at 200 GeV and is consistent with lower energy pp values, other ratios such as $\rm K^{*}/K$ and $\Lambda(1520)/\Lambda$ show a slight suppression in AuAu collisions, independent of centrality. Due to their short lifetimes, the re-scattering of resonance decay products between chemical and thermal freeze-out is expected to cause a signal loss. While the observed suppression of $\rm K^{*}/K$ and $\Lambda(1520)/\Lambda$ corroborates the re-scattering picture, the lack of suppression of the $\Sigma(1385)/\Lambda$ ratio implies a recovery mechanism such as regeneration (e.g. $\Lambda + \pi \rightarrow \Sigma(1385)$). The total interaction cross sections with $\pi$ increases from K to p to $\pi$ respectively [@PDG]. This implies that re-scattering of $\rm K^{*}$ decaying into $\pi$ and K in the medium should be higher than that of $\Lambda(1520)$ decaying into K and p. The shorter lifetime of $\rm K^{*}$ enhances the re-scattering probability. In this scenario, assuming that the lifetime between chemical and thermal freeze-out is non-zero, the regeneration cross-section for $\rm K^{*}$ must be larger than that of $\Lambda(1520)$ due to the smaller suppression of the $\rm K^{*}$ ratios.
HBT radii show a linear dependence on $\rm
dN^{1/3}_{ch}/d\eta$, a term related to the final state geometry through the density at freeze-out [@ref:lisa]. If entropy drives the strangeness yield, results from different collision energies at the SPS and RHIC should exhibit universal scaling with entropy. Figure 3 presents the yields of $\bar{\Lambda}$, $\bar{\Xi}$ in AuAu collisions at RHIC, normalized to yields in pp and in PbPb collisions at the SPS, normalized to yields in pBe, as a function of $\rm dN_{ch}/d\eta$ ($\sim$ entropy). Strange yields in heavy ion collisions, when compared to lighter systems, seem to universally scale with $\rm dN_{ch}/d\eta$ for SPS and RHIC energies. It is also predicted that the greater the number of strange quarks in the particle, the greater the effect of phase space suppression when modeled with respect to the number of participants, $\rm N_{part}$ [@redlich]. Even though the expected ordering of the suppression is observed at RHIC, the $\bar{\Lambda}$ and $\bar{\Xi}$ measurements normalized to their pp values do not
{width="13.0pc"}\
flatten at larger $\rm N_{part}$, in variance to predictions [@helen:sqm]. This might be because strange particles scale differently from non-strange. Particles with only u and d quarks are already observed to scale with $\rm N_{part}$ while strange quarks appear to scale better with $\rm N_{bin}$. $\rm N_{part}$ and $\rm N_{bin}$ can be combined to form a correlation volume for strange particles which depends on the quark content. This combined scaling seems to represent the strange particles better than $\rm N_{part}$ alone.
\[fig:dndeta\]
Nuclear modification factors for strange particles in AuAu collisions are presented in Figure 4. At higher $\rm p_{T}$, the ratios exhibit a suppression from binary scaling, attributed to fast moving partons losing energy as they traverse a dense medium. The $\rm R_{CP}$ from $\rm \sqrt{s_{NN}}=$ 62.4 GeV shows less suppression than that at $\rm \sqrt{s_{NN}}=$ 200 GeV; however, the clear differences between baryons and mesons still exists [@rcp]. This is believed to be due to hadron production through quark coalescence at intermediate $\rm p_{T}$ [@matt]. For baryons and mesons, the suppression sets in at a different $\rm p_{T}$. Motivated by the coalescence picture, Figure 4-b shows the $\rm R_{CP}$ ratio vs $\rm p_{T}/n$ for $ \rm \sqrt {s_{NN}}=62.4 $ GeV, where n is the number of valence quarks. Thus $\rm p_{T}/n$ is the $\rm p_{T}$ of a quark. The baryon and meson sets in at the same quark $\rm p_{T}$, in agreement with the coalescence picture. This is also observed for $\rm \sqrt{s_{NN}}=200 $ GeV collisions.
{width="12.7pc"}\
\[fig:Rcp62\]
{width="12.3pc"}\
\[fig:Rcp\]
{width="12.7pc"}\
\[fig:RcpScaled\]
The measurement of $\rm R_{AA}$ with respect to $\rm p_{T}$ is shown in Figure 4-c. While the $\rm R_{AA}$ for mesons ($\rm h^{+}
+ h^{-}$, $\rm K^{0}_{S}$, $\phi$) is similar to their $\rm
R_{CP}$ values, $\rm R_{AA}$ of strange baryons shows significant differences (i.e. no suppression). Instead, there is an enhancement and ordering with strangeness content: the higher the strangeness content, the higher the $\rm R_{AA}$ measurement in the intermediate $\rm p_{T}$ region. The difference between yields in pp and peripheral AuAu may be explained by phase space (canonical) suppression in the pp data-set although this is usually attributed to low $\rm p_{T}$ particles [@redlich2].
Conclusions
===========
Dynamical properties of strange particles can be investigated by Blast-Wave fits. The $\langle \beta_{T} \rangle $ increases with collision energy, implying a higher flow and the freeze-out temperatures of multi-strange baryons are larger than those of light mesons, suggesting an earlier freeze-out (e.g. $\rm T (\Xi)
> T(\pi)$). Strange anti-baryon and baryon production are approximately equal at top RHIC energies. There is a slight decrease in the $\overline{\Lambda}/\Lambda$ ratio from pp to dAu to AuAu, indicating that baryon number transport is almost independent of system size. Resonance yields require both re-scattering and regeneration mechanisms for $\Delta t > 0$ between chemical and thermal freeze-out to describe the suppression in some of their yields. Strange quarks appear to scale better with the number of hard processes ($\rm N_{bin}$), while light quarks scale with $\rm
N_{part}$ and strange particle yields seem to universally scale with $\rm dN_{ch}/d\eta$ for SPS and RHIC. The meson/baryon separation of the nuclear modification factors also exists in $\rm
\sqrt{s_{NN}}=62$ GeV collisions which can be explained in a coalescence picture. The $\rm R_{AA}$ of strange baryons behave differently from their $\rm R_{CP}$. Canonical suppression in pp might explain the observed difference, though it is a surprise that this effect extends to intermediate $\rm p_{T}$.
[9]{} P.F. Kolb and U.Heinz, nucl-th/0305084. J. Speltz (for the STAR Collaboration), QM 2005 Poster; L. Molnar, nucl-ex/0507027.
F. Antinori et al. (NA57 Collaboration) J. Phys. G 30 (2004) 823 O. Barannikova (for the STAR Collaboration) J. Phys. G 31, (2005) S93. D. Mishra (for the STAR Collaboration), QM 2005 Poster Presentation. S. Salur (for the STAR Collaboration), Eur. Phys. J. C 40, (2005) s3.9-s3.13. C. Markert (for the STAR Collaboration), QM 2005 Poster Presentation. S. Eidelman et al., Phys. Lett. B 592, 1 (2004). M. Lisa, S. Pratt, R. Soltz, and U. Wiedemann, (2005) nucl-ex/0505014. A. Tounsi, A. Mischke and K. Redlich, Nucl. Phys. A 715, 565 (2003). H. Caines (for the STAR Collaboration), J. Phys. G 31, (2005) S101-S118. J. Adams et al. (STAR Collaboration) Phys. Rev. Lett. 92 (2004) 052302. M. Lamont (for the STAR Collaboration), J. Phys. G 30, (2004) S963-S967.
K. Redlich and A. Tounsi, Eur. Phys. J. C 24, 589-594 (2002).
|
---
author:
- 'John T. Whelan[^1]'
title: SKELETONIZATIONS OF PHASE SPACE PATHS
---
Construction of skeletonized path integrals for a particle moving on a curved spatial manifold is considered. As shown by DeWitt, Kuchař and others, while the skeletonized configuration space action can be written unambiguously as a sum of Hamilton principal functions, different choices of the measure will lead to different Schrödinger equations. On the other hand, the Liouville measure provides a unique measure for a skeletonized phase space path integral, but there is a corresponding ambiguity in the skeletonization of a path through phase space. A family of skeletonization rules described by Kuchař and referred to here as geodesic interpolation is discussed, and shown to behave poorly under the involution process, wherein intermediate points are removed by extremization of the skeletonized action. A new skeletonization rule, tangent interpolation, is defined and shown to possess the desired involution properties.
SKELETONIZED PATH INTEGRALS {#sec:skel}
===========================
The path integral method seems to provide a generic recipe for producing a quantum theory given a classical action. This recipe is based on the principle that amplitudes are given by sums, over the relevant histories, of the complex exponential of the action. For example, a wavefunction $\psi(x_i,t_i)$ is propagated to a later time $t_f>t_i$ by $$\label{formconf}
\psi(x_f,t_f)=\int\limits_{x_f\phantom{x_i}}
{{\cal{D}}}^n\! x\, e^{iS[x]/\hbar}\psi(x_i,t_i),$$ where the integral is over all paths ending at the argument $x_f$. However, in order to provide a constructive definition for an expression like [(\[formconf\])]{} one needs to give a meaning to formal concepts like an integral over paths, and it is in this step that decisions need to be made in the implementation of the quantum theory.
The system considered in this work is a free non-relativistic particle of unit mass moving on a curved $n$-dimensional *spatial* manifold, which is described by the action $$\label{freepart}
S[x]=\frac{1}{2}\int dt\,g_{ab}(x)\frac{dx^a}{dt}\frac{dx^b}{dt}.$$ This is the simplest action which exhibits the ambiguities considered here, which correspond in an operator theory to questions of operator ordering. One can also modify the action to include potential terms, but previous work [@kuchar] suggests that the results will remain essentially the same.
Configuration Space
-------------------
A formal path integral such as [(\[formconf\])]{} can be given a concrete definition via the skeletonization process, in which the interval $(t_i,t_f)$ is broken up by a series of $N$ time instants $\{t_{(I)}|I=1,\ldots N\}$. The integral over all paths $x(t)$ is realized, in the limit of an infinitely fine time slicing, as a product of integrals over the positions $\{x_{(I)}|I=0,\ldots N\}$ of the particle at the discrete times: $$\label{skelconf}
\psi(x_f,t_f)=\lim_{N\rightarrow\infty}
\left(\prod_{I=0}^N\int A(x_{(I+1)}t_{(I+1)}|x_{(I)} t_{(I)})\,d^n\! x_{(I)}\,
e^{iS(x_{(I+1)} t_{(I+1)}|x_{(I)} t_{(I)})/\hbar}\right)\psi(x_i,t_i),$$ where $t_{(0)}=t_i$ and $t_{(N+1)}=t_f$. This process replaces the action functional $S[x]$ with a function $\sum_{I=0}^N S(x_{(I+1)}
t_{(I+1)}|x_{(I)} t_{(I)})$ which is constructed by summing contributions from the intervals between time slices. Since the equations of motion are second order, there is a preferred path between a pair of endpoints $(x,t)$ and $(x',t')$, namely, the classical path between them, which for the theory described by the action [(\[freepart\])]{} is an affinely parametrized geodesic.[^2] The action functional evaluated on that piece of the path is the *Hamilton principal function* $S(x't'|x\,t)$. On the other hand, the measure $\prod_{I=0}^N\int
A(x_{(I+1)}t_{(I+1)}|x_{(I)} t_{(I)})d^n\! x_{(I)}$ for the path integral has no such natural definition in curved space. Feynman [@feyn] showed that $A(x't'|x\,t)=[2\pi i\hbar(t'-t)]^{-n/2}$ was the appropriate measure for flat space in Cartesian coördinates. The simplest generalization of the Feynman measure to curved space is [@dewitt] $$\label{feynmeas}
A_0(x't'|x\,t)=\frac{\sqrt{g}}{[2\pi i\hbar(t'-t)]^{n/2}},$$ where $g$ is the determinant of the metric $g_{ab}$ at the point $x$, which ensures that the path integral is invariant under coördinate changes. However, there are other measures which are similarly invariant and also reduce to the Feynman measure when the spatial manifold is flat. Expanding to second order in powers of the geodesic distance $\sigma'$ between $x$ and $x'$ (which turns out to be sufficient to evaluate the path integral), one family of such measures is[^3] $$\label{measlamb}
A_\lambda(x't'|x\,t)=\frac{\sqrt{g}}{[2\pi i\hbar(t'-t)]^{n/2}}
\left(1+\frac{\lambda}{3} R^{ab} y'_a y'_b+\Or(\sigma'^3)\right),$$ where $y'_a$ are the Riemann normal coördinates of the point $x'$ with respect to the origin $x$ and $R^{ab}$ is the Ricci tensor at $x$.
This ambiguity in defining the path integral measure has tangible consequences. Namely, if we use a member of the family [(\[measlamb\])]{} to define the measure in the skeletonized path integral [(\[skelconf\])]{}, the wavefunction thus propagated satisfies [@dewitt; @kuchar] a Schrödinger equation $$i\hbar\frac{\partial\psi(x,t)}{\partial t}=
\left(-\frac{\hbar^2}{2}\,g^{ab}\nabla_a\nabla_b
+\frac{1-\lambda}{6}\hbar^2 R\right)
\psi(x,t).$$ The classical Hamiltonian $g^{ab}p_a p_b$ is not quantized in the simplest way, by replacing it with an operator proportional to the covariant Laplacian, but also contains a measure-dependent term proportional to the scalar curvature. In particular, the straightforward generalization [(\[feynmeas\])]{} of the Feynman measure (in which $\lambda=0$) leads to a Schrödinger equation *with* a curvature correction term.
Phase Space
-----------
One way to resolve the measure ambiguity is to start instead with a phase space path integral [@garrod] $$\label{formphas}
\psi(x_f,t_f)=\int\limits_{x_f\phantom{x_i}}
{{\cal{D}}}^n\! p\,{{\cal{D}}}^n\! x\, e^{iS[p,x]/\hbar}\psi(x_i,t_i),$$ in which the path integral measure is expressed as a product of Liouville measures $d^n\!p\,d^n\!x/(2\pi\hbar)^n$ in the skeletonization $$\label{skelphas}
\psi(x_f,t_f)=\lim_{N\rightarrow\infty}
\left(\prod_{I=0}^N\int \frac{d^n\! p^{(I)}\,d^n\! x_{(I)}}{(2\pi\hbar)^n}
e^{iS(x_{(I+1)} t_{(I+1)}|p^{(I)} x_{(I)} t_{(I)})/\hbar}\right)\psi(x_i,t_i).$$ Unfortunately, the ambiguity has now shifted into the definition of the contributions $\{S(x_{(I+1)} t_{(I+1)}|p^{(I)} x_{(I)} t_{(I)})\}$ to the skeletonized action. This is because there is in general no classical trajectory through phase space satisfying the conditions $$\label{BCs}
x(t)=x,\qquad p(t)=p, \qquad x(t')=x',$$ since any two of these three conditions are sufficient to specify a classical phase space path.
Since there is no *classical* phase space path, we need some interpolation rule which associates a *virtual* path through phase space with the set of boundary conditions [(\[BCs\])]{}. (It should of course reproduce the classical path when the three boundary conditions are classically consistent.) A given interpolation rule defines a *phase space principal function* (PSPF) $S(x't'|p\,x\,t)$ whose value for a particular set of arguments will be the action functional evaluated along the virtual path which the rule specifies for those boundary conditions. The interpolation rule then defines a skeletonization, under which and the skeletonized canonical action appearing in [(\[skelphas\])]{} is a sum of PSPFs.
### Geodesic Interpolation.
One type of rule, introduced by Kuchař [@kuchar], prescribes the configuration space path to be the classical path from $x$ to $x'$, regardless of the value of $p$. The momentum then “goes along for the ride,” being propagated in some linear way along the geodesic (Fig. \[fig:GI\]a).
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The resulting phase space principal function is $$\label{geodpspf}
S(x't'|p\,x\,t)=p^a y'_a-\frac{t'-t}{2}G^{ab}(x'|x)p_a p_b\, ,$$ with the functional form of $G^{ab}(x'|x)$ depending on the details of the propagation rule. We call this “geodesic interpolation” (GI) because the property of the classical trajectories which is retained by all of the virtual ones is that the configuration space projection of the path is a geodesic. This motivation for selecting rules in which the second order equation of motion, involving only the path $x(t)$, is obeyed independent of the path $p(t)$, comes from the invariance of the tensor formulation under point canonical transformations \[$ x\rightarrow X(x), p\rightarrow P(p,x) $\], in which the transformation of the $x$ components of a point in phase space is independent of its $p$ components, but not under general canonical transformations \[$x\rightarrow X(p,x), p\rightarrow P(p,x)$\].
If the momentum $p$ is taken to be parallel-propagated along the geodesic, the tensor $G^{ab}(x'|x)$ in [(\[geodpspf\])]{} becomes the inverse metric $g^{ab}$ at $x$. This is one of a family of propagation rules discussed by Kuchař which, for small intervals, have the expansion $$\label{Glambda}
G_\lambda^{ab}(x'|x)=g^{ab}-\frac{\lambda}{3} R^{acbd} y'_c y'_d
+\Or(\sigma'^3),$$ where $R^{abcd}$ is the Riemann curvature tensor at $x$.
Performing the quadratic integrals over the momenta in the skeletonized path integral [(\[skelphas\])]{} induces a measure $$A(x't'|x\,t)=\frac{1}{\sqrt{[2\pi i\hbar(t'-t)]^n\det G^{ab}}}$$ on the remaining configuration space path integrals. The family of propagation rules given by [(\[Glambda\])]{} lead in this way to exactly the family of configuration space measures [(\[measlamb\])]{}. This means that they produce quantum theories whose Hamiltonians contain curvature correction terms $\frac{1-\lambda}{6}\hbar^2 R$; in particular, the parallel propagation rule ($\lambda=0$) leads to the Feynman measure [(\[feynmeas\])]{} and hence to a curvature-corrected quantum Hamiltonian. The case $\lambda=1$, which Kuchař showed arose from a propagation rule based on the equation of geodesic deviation, leads to a Hamiltonian with no curvature correction term.
This work considers a criterion for selecting a phase space skeletonization rule, and hence a principal function. This criterion, as described in the following section, is based on a property of the PSPFs and paths themselves.
THE INVOLUTION PROPERTY {#sec:invo}
=======================
Configuration space
-------------------
We will judge phase space skeletonizations by whether they obey a property analogous to the involution property satisfied by configuration space skeletonizations. The configuration space property is defined as follows: Consider the contribution to the skeletonized action from the intervals between three consecutive time instants, called $t$, $t'$ and $t''$: $$\label{compconf}
S(x''t''|x't')+S(x't'|x\,t).$$ It is a function of three positions, $x$, $x'$ and $x''$, and has associated with it a path made up of two geodesics connected at the point $x'$. If we vary the location of that point $x'$ while holding the endpoints $x$ and $x''$ fixed, the expression [(\[compconf\])]{} is minimized by the $x'$ which lies at the appropriate point on the geodesic from $x$ to $x''$. The path corresponding to that choice of $x'$ is just the geodesic connecting the two endpoints $x$ and $x''$, and its action is the Hamilton principal function $S(x''t''|x\,t)$: $$\label{invoconf}
S(x''t''|x't')+S(x't'|x\,t)\mathrel{\mathop{\Longrightarrow}^{x'}}
S(x''t''|x\,t).$$ This property is known as involution.
We say that the configuration space skeletonization procedure obeys *involution of paths* (IOP) because extremization of [(\[compconf\])]{} with respect to the intermediate position $x'$ leads to the classical path from $x$ and $x''$, and *involution of functions* (IOF) because [(\[invoconf\])]{} holds. Clearly, the latter property is implied by the former.
Phase space
-----------
Given an interpolation rule, which defines a phase space principal function $S(x't'|p\,x\,t)$, we can define the analogous property for a *phase space* skeletonization by considering two consecutive intervals in the skeletonization. The five quantities $x$, $p$, $x'$, $p'$ and $x''$ determine a phase space path between times $t$ and $t''$ according to the interpolation rule. The action for this path is $$\label{compphas}
S(x''t''|p'x't')+S(x't'|p\,x\,t).$$ Extremization with respect to the intermediate variables $x'$ and $p'$ leaves a function of $x$, $p$ and $x''$; we say that involution of *functions* holds if that is the same PSPF with which we started: $$\label{invophas}
S(x''t''|p'x't')+S(x't'|p\,x\,t)
\mathrel{\mathop{\Longrightarrow}^{p',x'}}S(x''t''|p\,x\,t).$$ The stricter condition of involution of *paths* is satisfied if the phase space path defined by the extremizing values of $x'$ and $p'$ is the same one which the interpolation rule would have constructed *a priori* from $x$, $p$ and $x''$ without making reference to the intermediate time slice $t'$.
These concepts are easily illustrated in the case where the spatial manifold is flat; in that case the preferred propagation rule associated with geodesic interpolation is simply to keep the momentum constant (as measured in a Cartesian coordinate system). This leads to a PSPF $$\label{flatpspf}
S(x't'|p\,x\,t)=p\cdot(x'-x)-\frac{t'-t}{2}p^2\, .$$ The two-step action $$\label{flatpfs}
S(x''t''|p'x't')+S(x't'|p\,x\,t)=p'\cdot x''+(p-p')\cdot x'-p\cdot x
-\frac{t''-t'}{2}p'^2-\frac{t'-t}{2}p^2$$ is minimized by $$\label{flatinvo}
p'=p\quad\hbox{and}\quad x'=x''-(t''-t')p.$$ Substituting these into [(\[flatpfs\])]{} gives $p\cdot(x''-x)-\frac{t''-t}{2}p^2=S(x''t''|p\,x\,t)$, so the GI scheme in flat-space *does* satisfy IOF. On the other hand, it is easy to see that it does *not* satisfy IOP, since the path obtained by extremization has a configuration space projection which starts at $x$ at time $t$, follows a straight path to $x''-(t''-t')p$ at $t'$, and then makes a sharp turn before heading on to $x''$ at $t''$ (Fig. \[fig:GI\]b). This is not the same as the path from $x$ to $x''$ which geodesic interpolation would dictate without the presence of the intermediate point, since that path is just a straight line from $x$ to $x''$.
If we look at the curved-space case in the limit that all of the distances are small, the zeroth-order results of course replicate the flat-space ones (including the failure of IOP). If we consider the family of PSPFs defined by [(\[Glambda\])]{}, the behavior of the first correction terms will depend on the value of $\lambda$. It turns out that for $\lambda=-1$ (and for no other value) IOF continues to hold to the lowest non-trivial order.
The reason why IOF can hold even when IOP fails is this: Since a given PSPF $S(x't'|p\,x\,t)$ corresponds to the action functional evaluated along a virtual, rather than classical, phase space path satisfying the boundary conditions [(\[BCs\])]{}, there are many different paths which yield the same function. We can use this fact to our advantage by defining a different skeletonization procedure which produces the same PSPF over small intervals as $\lambda=-1$ geodesic interpolation, and thus also obeys the IOF property, but arises from a different family of paths, which satisfies IOP.
TANGENT INTERPOLATION {#sec:TI}
=====================
We will call the skeletonization procedure which respects IOP *tangent interpolation* (TI). As opposed to the geodesic interpolation schemes, which require the configuration space part of the path to be the classical geodesic, the configuration space path in the TI scheme is not even continuous. However, the classical equation relating momentum to velocity is respected by the prescribed path, so that the momentum vector remains tangent to the configuration space trajectory. Hence the name “tangent interpolation.”
The phase space path (Fig. \[fig:TI\]a)
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determined by a given set of boundary conditions [(\[BCs\])]{} begins at $x$ with the momentum $p$ at time $t$, jumps instantaneously to a point ${\tilde{x}}$ which is prescribed (in a manner detailed below) by the boundary conditions, and then follows the classical phase space trajectory from ${\tilde{x}}$ at $t$ to $x'$ at $t'$. The point ${\tilde{x}}$ is chosen so that in a Riemann normal coördinate (RNC) system based at $x$, there is no momentum discontinuity at the initial time $t$, *i.e.*, so that the classical momentum at the beginning of the geodesic from ${\tilde{x}}$ to $x'$ has the same covariant RNC components as $p$. In flat space, this means that $p=\frac{x'-{\tilde{x}}}{t'-t}$, or $$\label{flattang}
{\tilde{x}}=x'-p(t'-t),$$ although in curved space the relation ${\tilde{x}}(x't'|p\,x\,t)$ is only implicitly defined by the matching of momenta. However, the ${\tilde{x}}$ which achieves the momentum matching condition can alternatively be found as the ${\tilde{x}}$ which minimizes $S(x't'|{\tilde{x}}\,p\,x\,t)$, the action along such a path, for a given $x$, $p$ and $x'$. This action has two contributions, one from the initial discontinuity and one from geodesic from ${\tilde{x}}$ to $x'$, giving $$\label{tildpspf}
S(x't'|{\tilde{x}}\,p\,x\,t)=p^a\tilde{y}_a+S(x't'|{\tilde{x}}t).$$ Extremization with respect to ${\tilde{x}}$ gives the curved-space counterpart of [(\[flattang\])]{}, and defines the PSPF $S(x't'|p\,x\,t)$ for the tangent interpolation recipe. Note that while GI only defines a class of PSPFs in curved space, with a recipe for propagating the momentum needed to pick out a single PSPF, the TI procedure defines one unique path and thus only one PSPF.
In flat space, it is easy to verify that the TI prescription leads to the PSPF [(\[flatpspf\])]{}, by substituting [(\[flattang\])]{} into the flat-space equivalent of [(\[tildpspf\])]{}, $$S(x't'|{\tilde{x}}\,p\,x\,t)=p\cdot({\tilde{x}}-x)+\frac{1}{2}\frac{(x'-{\tilde{x}})^2}{t'-t}.$$ The first correction term leads to the form (\[geodpspf\],\[Glambda\]) with the value $\lambda=-1$. This means, from the results of the previous section, that it satisfies IOF, at least to the first non-trivial order in the distances involved.
In fact, the TI PSPF satisfies the involution property to all orders, because the TI prescription obeys IOP, even over finite intervals. IOP can be verified for flat space as follows (Fig. \[fig:TI\]b): Since the sum of the principal functions is given by [(\[flatpfs\])]{}, the values of $x'$ and $p'$ which extremize it are again given by [(\[flatinvo\])]{}. The auxiliary point ${\tilde{x}}'$ for the second interval is given, analogously to [(\[flattang\])]{}, by ${\tilde{x}}'=x''-p'(t'-t)=x'$; since ${\tilde{x}}'$ and $x'$ coïncide, there is no discontinuity in the path from $x'$ to $x''$, which is just a classical phase space path. The point ${\tilde{x}}=x'-p(t'-t)=x''-p(t''-t)$ to which the initial jump at time $t$ is made is in the same location whether the final condition is taken to be $x''$ at $t''$ or $x'$ \[given by [(\[flatinvo\])]{}\] at $t'$.
CONCLUSIONS AND OUTLOOK
=======================
We have shown that the involution property described in Sec. \[sec:invo\] is obeyed by the phase space skeletonization procedure which we defined in Sec. \[sec:TI\] and named tangent interpolation. According to prior results, summarized in Sec. \[sec:skel\], using this skeletonization rule to construct a phase space path integral will induce a particular measure on the corresponding configuration space path integral. In this way, involution singles out a preferred realization of the skeletonized path integral. It is one which leads to a Schrödinger equation whose Hamiltonian differs from the simplest form, proportional to the covariant Laplacian, by a term $\frac{\hbar^2}{3}R$ proportional to the scalar curvature of the spatial manifold. Understanding the implications of this result rests on two as yet open questions. First, is there any physical explanarion for the modified Hamiltonian, and second, what is the significance of the involution property for the path integral prescription? Involution of the Hamilton principal function enables composition of propagators constructed via the configuration space path integral, for any choice of measure. Perhaps involution of the phase space principal function is related to composition of path integrals defining momentum space propagators. Further investigation is called for.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The author wishes to thank D. A. Craig, F. Dowker, K. V. Kuchař and S. F. Ross for helpful discussions. This work was supported by NSF grant PHY-9507719.
[9]{} Kuchař, K.V., 1983, [*J. Math. Phys.*]{}, [**24**]{}, 2122–41. Feynman, R.P., 1948 [*Rev. Mod. Phys.*]{}, [**20**]{}, 367-87. DeWitt, B.S., 1957, [*Rev. Mod. Phys.*]{}, [**29**]{}, 377–97. Parker, L., 1979, [*Phys. Rev.*]{}, [**D19**]{}, 438-41. DeWitt-Morette, C., Elworthy, K.D., Nelson, B.L. and Sammelman, G.S., 1980, [*Ann. Inst. H. Poincaré*]{}, [**32**]{}, 327–41. Garrod, C., 1966, [*Rev. Mod. Phys.*]{}, [**38**]{}, 483–94.
[^1]: e-mail: whelan@physics.utah.edu
[^2]: Throughout this paper, the points in question, even when separated by a finite distance, are taken to be close enough that there is no crossing of geodesics.
[^3]: Specific measures of this sort were discussed in [@dewitt; @meas], and placed into the family [(\[measlamb\])]{} in [@kuchar].
|
---
abstract: 'It is pointed out that an agreement of the one particle energy spectrum of the cosmic background radiation (CMBR) with Plank distribution of 2.725 \[K\] does not give a strong constraint on the coherence length of CMBR if the mean free path of CMBR is very long. The coherence length in this situation is estimated as a few times of $k_BT$. Due to this finite coherence length, the attenuation length of ultra-high energy cosmic rays (UHECR) is reduced in the $\Delta $ resonance region,i.e., around $10^{20}$ \[eV\]. The small attenuation length makes the suppression of the flux of cosmic rays in this energy region less prominent than the naive estimation.'
author:
- |
K. Ishikawa and Y. Tobita\
\
Department of Physics, Faculty of Science,\
Hokkaido University Sapporo 060-0810, Japan
title: ' Coherence length of cosmic background radiation enlarges the attenuation length of the ultra-high energy proton'
---
Introduction
============
UHECR beyond $10^{20}$ \[eV\] is expected to be suppressed by its pion production collision with CMBR by GZK bound. In this paper we assume that the proton is UHECR and study its scattering with CMBR. Recently observations of UHECR became possible by several experiments [@HaverahP; @Fly's-eye; @Lorentz; @AGASA; @Auger_pre] , and a possible signal of UHECR around this energy region has been found, although experiments are controversial. A new mechanism which modifies the pion production probability of UHECR is proposed in this letter.
CMBR has the temperature at around 2.7 \[K\] and is regarded as a wave packet in the present work. Coherence is kept within the size of wave packet, and we call it a coherence length. We show that the finite coherence length modifies the pion production probability in the ${\Delta}$ resonance region.
When the invariant mass of the initial states composed of UHECR and a CMBR exceeds the pion production threshold, the attenuation length of UHECR becomes short. GZK predicted the reduction of the flux beyond this energy. In the inelastic cross section, the $\Delta$ resonance contribution is most important. The attenuation length beyond this energy becomes the order of 20 \[Mpc\], which is less than the size of the universe. The UHECR can not propagate long distance in the universe, then. So the flux of cosmic ray should be suppressed beyond this energy, known as GZK bound. For the estimation of GZK bound, Lorentz invariance is assumed and the cross section between gamma and nucleon collisions in laboratory frame is used.
Experiments in this energy region are in progress and the source of UHECR will be identified in the future [@Auger]. Then an experimental determination of the attenuation length would become possible.
We propose a new mechanism of correction to the GZK bound in this paper. Most CMBR photons have been produced before the decoupling time in the early universe and may have finite coherence length. High energy charged particle also has a finite coherence length due to the collinear mass singularity. In the collisions of CMBR with UHECR beyond $10^{20}$ \[eV\] region, the finite coherence length makes the total photon-nucleon energy to spread. If this energy width is larger than or equivalent to the width of the $\Delta$ resonance, the total amplitude in the $\Delta$ resonance region is suppressed. The attenuation length of UHECR is modified, then.
The effect of the finite coherence length is negligible in the ordinary high energy scattering since the coherence length is much larger than the de Broglie wave length. So this effect has not been taken into account in the previous works on GZK bound. However, we show in the present work that the finite coherence length of CMBR gives a sizable effect to UHECR’s attenuation length even though the CMBR spectrum agrees with the Plank distribution.[@Stodolsky]
Wave packet and the coherence length
====================================
In ordinary scattering experiments, a position where the beam particle is produced is known and the wave packet size is determined from the mean free path of the particle in matters and its size is semi-microscopic with much larger value than de Broglie length of high energy particle. Its effect is negligible, then. Let us call this length as the first coherence length. In a dilute system we study, a position where the particle is produced is unknown and the wave packet size, i.e., the coherence length is determined by the amplitude of many particle states in the production process.
In a system where each particle has a large mean free path, particle states preserve coherence for long time and coherence length due to mean free path is negligible. In this region, particles are described by a many body wave function and coherence length due to many body effects becomes important. Final state of the scattering matrix is a linear combination of the momentum states with the weight of scattering amplitude, unless a measurement of the final state is made. Consequently a correlation for one particle state of the different momenta, which is defined from the product between the scattering amplitude and its complex conjugate, becomes finite. This correlation length due to many body wave functions becomes important in the dilute system.
We estimate this coherence length of photon from the final states of Thomson scattering of Fig. \[thomson\]. Let us focus on the final scattering of CMBR. CMBR is composed of almost free photon with the Plank distribution as far as single particle energy distribution is concerned. Thomson scattering amplitude is independent from the scattering angle at the low energy. Also the production region is not identified for CMBR. Consequently the photon in the final states of Thomson scattering is a coherent linear combination of momentum states with different orientations. This correlation of photons with different momenta is computed from these amplitudes where in the initial states the photon follows the Plank distribution with arbitrary angle and the electron follows the Fermi-Dirac distribution with arbitrary angle. We found that the correlation of final photons agrees almost perfectly with that of Gaussian wave packet[@Ishikawa-tobita-ptp]. Its width is a few times of $k_BT$. So even though photon follows the Plank distribution it has a correlation of Gaussian wave packet of the width of a few $k_BT$. We study its implications in the collision of CMBR with UHECR.
![The diagram for Thomson scattering amplitude where the initial photon follows the Planck distribution and the initial electron follows the Fermi-Dirac distribution.[]{data-label="thomson"}](thomson3.eps)
$\Delta$ resonance lives for short period and its amplitude has a finite energy width. Hence if the time scale for UHECR to overlap with CMBR is shorter than the life time of the $\Delta$ resonance, the amplitude is reduced by the finite coherence length. Then cross section due to the $\Delta$ resonance gets a sizable correction from the finite coherence length.
Scattering amplitudes for the wave packets of a finite spatial extension in relativistic field theory have been formulated in Ref. [@Ishikawa-shimomura]. It is shown that the amplitudes are consistent with the probability interpretation, despite non-orthogonality of the different states, and that the conservation of the energy and momentum is satisfied within an uncertainties given by the finite interaction area of the wave packets, since the states defined by wave packets have finite extensions. Furthermore, the asymptotic condition is satisfied with an finite initial time $T_i$ and an finite final time $T_f$, and the scattering probability has a position-dependence in addition to a momentum-dependence. From these properties, the wave packet scattering amplitude is different from a simple linear combination of the plane wave amplitude.
Gaussian wave packets, i., e., coherent states, of spherical symmetry are defined as $$\begin{aligned}
\label{eq:coherentx}
& & \langle {\vec p}|{\vec P}_0,{\vec X}_0 \rangle =
N_3{({\sigma_x^2})}^{\frac{3}{2}} e^{-i{{\vec p} \cdot {\vec X}_0} -{\sigma_x^2 \over 2}
({\vec P}_0-{\vec p})^2}
~N_3^2=(\pi {\sigma}_x^2)^{-{3 \over 2}},{{\vec P}_0 \over \hbar}={\vec k}_0.\end{aligned}$$ Generalizations to the asymmetric wave packets and to non-minimum wave packets are straightforward. The set of functions for one value of $\sigma$ satisfy the completeness condition.
The time evolution of the free wave is determined by the free Hamiltonian, and creation and annihilation operators which satisfy $[a({\vec p},t),a^{\dagger}(\vec{p}{\mspace{2mu}}^{\prime},t')]\delta(t-t') =\delta(\vec{p}-\vec{p}{\mspace{2mu}^{\prime}})\delta(t-t')$ The operator $A({\vec P}_0,{\vec X}_0 ,T_0,t)$ that annihilates the state described by the wave packet is defined by $$\begin{aligned}
\label{eq:atoA}
A({\vec P}_0,{\vec X}_0,T_0,t )=\int d{\vec p} \mspace{3mu}a({\vec p},t) \langle {\vec p
}|{\vec P}_0,{\vec X}_0,T_0\rangle
$$ and the creation operator is defined by its conjugate. The particle states expressed by the wave packets follow classical trajectories and have finite spatial extensions. Consequently overlap region is finite and the energy-momentum conservation is only approximate.
The wave packet spreads with time and the spreading velocity in the transverse direction, ${v_T}$, and the longitudinal direction, ${v_L}$,are given by $$\begin{aligned}
& &{ v}_T=\sqrt{{ 2\over \sigma_x^2}} { 1 \over E({\vec P}_0)},\
{ v}_L=\sqrt{{ 2\over \sigma_x^2}} { m^2 \over (E({\vec P}_0))^3}.\end{aligned}$$
The ${v_T}$ depends on the energy and the ${v_L}$ depends on the energy and the mass. A massive wave packet spreads in both directions but a massless wave packet spreads only in the transverse direction. After a macroscopic time, any wave packet of the massive particle spreads to huge size. These wave may be treated as a plane wave approximately. However any wave packet of the massless particle does not spread and its size is kept fixed in the longitudinal direction. Thus, the wave packet of massless particle remains for the long period and its effect is important.
Resonance in the wave packet scattering
=======================================
In the scattering of high energy proton with CMBR, the square of the center of mass energy, $S$, is defined by $$\begin{aligned}
S = (M_p^2 + 2E_pE_{\gamma} -
2|\vec{P_p}|\vec{P_{\gamma}}|\cos{\theta}) \simeq (M_p^2 +
2E_p\cdot E_{\gamma}(1 - \cos{\theta}))\end{aligned}$$ where $(E_{\gamma}, {\vec P}_{\gamma}), (E_{p},{\vec P}_{p}),
\mbox{and} \ \theta $ are four momenta of the photon, of the proton and the collision angle. The mass and width of the $\Delta$, $M_{\Delta}$ and $\Gamma $, are $ M_{\Delta} = 1232 [\mathrm{MeV}], \ \Gamma = 120 [\mathrm{MeV}] $.
Breit-Wigner partial wave amplitude and the total cross section are, $$\begin{aligned}
f_l({\theta})&=&{\sqrt{2l+1} \over p }{\Gamma/2 \over {\sqrt S}-
M_{\Delta}+i\Gamma/2 },~\sigma_l = \frac{4\pi(2l+1)}{p^2}
\frac{\left(\frac{\Gamma}{2}\right)^2} {(\sqrt{S} - M_{\Delta})^2 +
\left(\frac{\Gamma}{2}\right)^2} .\end{aligned}$$ The photon wave function of the momentum ${\vec p}_0$ which we obtained is expressed by a minimum wave packet, $$\begin{aligned}
|\psi_{\gamma}(\vec{p} - {\vec{p}}_0)| = N
\exp\left[-\frac{(\vec{p} - {\vec{p}}_0)^2}{2\sigma^2}\right] \label{gamma-wave-function}.\end{aligned}$$ We have studied also non-minimum case by multiplying a polynomial $h_m({\vec p}) $ to the last term but our conclusion of the present work is the same. For an asymmetric wave packet, an asymmetric $\sigma$ is used. Actually a high energy charged particle is combined with coherent soft photons in order to cure infra-red divergence[@infra-red_divergence; @infra-red_divergence2] caused by massless particle, photon. So a charged particle system is spread in the momentum and energy.
Cross section and attenuation length of UHECR
=============================================
The average cross section for the CMBR of finite coherence length is given by, $$\begin{aligned}
& & \sigma_{CMBR} = \frac{\int d{ \nu} U({ \nu})\Bigg|\int d^3p \ \psi_{\gamma+p}(\vec{p}-{\vec p}_0)
\frac{4\pi}{P_{CM}}
\frac{\left(\frac{\Gamma}{2}\right)}{\sqrt{S} - M_{\Delta} +
i\frac{\Gamma}{2}}\Bigg|^2}{\int d{ \nu} U({ \nu})},\label{CMBR-cross-section}\\
& & U({ \nu} , T) = \frac{8{\pi}h{{ \nu}}^3}{c^3}\frac{1}{e^{\frac{h{ \nu}}
{k_B T}}-1}, |\vec{p}_0| =h { \nu} \end{aligned}$$ where $P_{CM}$ is momentum in the center of mass frame and parameters $h$, $c$, $k_B$, $ T$, are Planck constant, speed of light, Boltzmann constant, and temperature of the CMBR. $\psi_{\gamma+p}(\vec{p})$ is the wave function of the photon-proton system. It should be noted that the integration on ${\vec p}$ is taken in the amplitude because the strict energy conservation does not hold for the wave packet scattering.
Using the cross section, we calculate the average attenuation length of UHECR in the parameter range $\sigma \leq 10 k_BT $, where the temperature is regarded as that of the decoupling time until this point. However to compare with the current observation, we use the current value of the temperature, $T= 2.725[\text{K}]$. The inelasticity of the UHECR is given by the ratio of energy loss of the UHECR, $E_{p,f}-E_{p,i}$, over the initial energy in the scattering $K_p={ E_{p,i}- E_{p,f} \over E_{p,i}}$. The attenuation length is obtained by integrating the product of the above ratio, $K_p$, with the cross section. The result is given in Fig. \[fig:1\]. The attenuation length becomes longer by factor 10 if the coherent energy spread is wide. Since the $\sigma$ includes both effects of CMBR and UHECR, larger values of the $\sigma$ is included here.
![Proton energy dependence of attenuation length is given. [**1**]{} is for plane wave, [**2**]{} is for $\sigma =
2.725[\text{K}](=T_{\text{CMBR}})$ , [**3**]{} is for $\sigma =
5.450[\text{K}]$, [**4**]{} is for $\sigma = 13.625[\text{K}]$ and [**5**]{} is for $\sigma = 27.250[\text{K}]$.[]{data-label="fig:1"}](attenuation_length_mono.eps)
Lorentz invariance
==================
By a Lorentz transformation of the frame, one value of the momentum is transformed to another value of the momentum [@Lorentz-breaking1; @Lorentz-breaking2; @Lorentz-breaking3]. Amplitude for the plane wave is covariant and the cross section is invariant under the Lorentz transformation. So the cross section for the plane wave for CMBR is computed easily from the experimental value of the photon-nucleon reactions in the laboratory frame. The amplitude for the wave packets, however, should be treated carefully, since the wave packet is a linear combination of the momentum states.
We calculated the total cross section for the wave packet explicitly and find the sizable difference compared to the plane wave. The finite size effect of the photon is negligible in the laboratory frame of the pion production process in photon-nucleon reactions but is not negligible in the present situation. In the latter system, a small coherent energy spread of CMBR leads the center of mass energy to spread finite amount. A product of the small coherent energy spread with an extremely large energy of UHECR becomes finite. The variance of center of mass energy $S$ of the laboratory frame, ${\Delta S}_l$, and of the CMBR frame, ${\Delta S}_{CMBR}$, are given by $$\begin{aligned}
{\Delta S}_l&=&2 m_p {\Delta E}_l,\\
{\Delta S}_{CMBR}&=&2 p {\Delta E}_{CMBR} \delta (1-a),\end{aligned}$$ where “a” is of order $1$. If the energy variances of the photon are the same in both systems, the ratio of two values of the center of mass energy is given by $p \over m_p$. This ratio is of order $10^{11}$ for UHECR. So in this case $\Delta S_{CMBR}$ is much larger than $\Delta S_{l}$ and the finite coherent length effect in the CMBR frame of UHECR is much larger than that of the laboratory frame. This shows the reason why a naive Lorentz invariance does not hold for the wave packet scattering.
Summary
=======
We found that the attenuation length of UHECR in the $10^{20} $ \[eV\] region varies depending on the coherence length of CMBR. It is clear from the Fig. \[fig:1\] that the attenuation length for UHECR becomes longer if the CMBR photon has a finite coherence length. The effect becomes important in the pion production threshold energy region, where a new data from Auger collaboration concluded that the UHECR in this energy region comes sources within 75 \[Mpc\] [@Auger2][@Auger3]. Our calculation at the coherence length of $3.5
k_BT$ suggests that the value becomes 150 \[Mpc\] instead of 75 \[Mpc\]. When the precise value of the flux of UHECR in wider energy region will be known, better informations will be obtained.
Our study shows the importance of the finite coherence length of CMBR in analyzing the attenuation length of UHECR.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by the special Grant-in-Aid for Promotion of Education and Science in Hokkaido University provided by the Ministry of Education, Science, Sports and Culture, a Grant-in-Aid for Scientific Research (Grant No. 19540253), and a Grant-in-Aid for Scientific Research on Priority Area ( Progress in Elementary Particle Physics of the 21st Century through Discoveries of Higgs Boson and Supersymmetry, Grant No. 16081201) provided by the Ministry of Education, Science, Sports and Culture, Japan.\
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---
abstract: 'We investigate various possible definitions of an effective temperature for a particularly simple nonequilibrium stationary system, namely a heated Brownian particle suspended in a fluid. The effective temperature based on the fluctuation dissipation ratio depends on the time scale under consideration, so that a simple Langevin description of the heated particle is impossible. The short and long time limits of this effective temperature are shown to be consistent with the temperatures estimated from the kinetic energy and Einstein relation, respectively. The fluctuation theorem provides still another definition of the temperature, which is shown to coincide with the short time value of the fluctuation dissipation ratio.'
author:
- Laurent Joly
- Samy Merabia
- 'Jean-Louis Barrat'
title: Effective temperatures of a heated Brownian particle
---
In the recent years, so called “active colloids”, *i.e.* colloidal particles that exchange with their surroundings in a non Brownian manner, have attracted considerable attention from the statistical physics community [@Schweizer2003]. These systems are of interest as possible models for simple living organisms, and the description of the corresponding nonequilibrium states using the tools of standard statistical physics raises a number of fundamental questions [@Loi2008; @Palacci2010]. The most widely studied active colloids are those that exchange momentum with the supporting solvent in a non stochastic way, resulting into self propulsion. A less studied possibility is that the colloid acts as a local heat source and is constantly surrounded by a temperature gradient. Experimentally [@Radunz2009], such a situation is achieved when colloids are selectively heated by an external source of radiation which is not absorbed by the solvent. If the heat is removed far away from the particle, or, more practically, if the particle concentration is small enough that the suspending fluid can be considered as a thermostat, a simple nonequilibrium steady state is achieved. Each colloidal particle is surrounded by a spherically symmetric halo of hot fluid, and diffuses in an *a priori* Brownian manner. The diffusion constant of such heated Brownian particles was experimentally shown to be increased compared to the one observed at equilibrium [@Radunz2009], and a semi quantitative analysis of this enhancement was presented in reference [@Rings2010], based on an analysis of the temperature dependence of the viscosity.
In this report, we use simulation to investigate in detail the statistical physics of the simple non equilibrium steady state (NESS) formed by a heated particle suspended in a fluid. The most natural way of describing such a system, in which the particles diffuse isotropically in the surrounding fluid, is to make use of a Langevin type equation for the center of mass velocity $\bm{U}$, involving in general a memory kernel $\zeta(t)$ and a random force $\bm{R}(t)$: $$M \frac{\mathrm{d}\bm{U}}{\mathrm{d}t} = - \int_{-\infty}^t \zeta(t-s) \bm{U}(s) \,\mathrm{d}s + \bm{F}_{\text{ext}} + \bm{R}(t) .
\label{langevin}$$ In a system at thermal equilibrium at temperature $T$, the correlations in the random force and the friction kernel are related by the standard fluctuation dissipation theorem, $\langle R_\alpha(t) R_\beta(t')\rangle = \delta_{\alpha \beta} \zeta(\vert t-t'\vert ) {k_{\text{B}}}T$ [@Kubo1988]. Obviously such a description is not expected to hold for a heated particle, as the system is now out of equilibrium. A generalization of Eq. \[langevin\], involving a corrected fluctuation dissipation relation with an effective temperature ${T_{\text{eff}}}$ replacing the equilibrium one, would however appear as a natural hypothesis. In fact, such an approach was shown to hold for sheared systems kept at a constant temperature by a uniform thermostat [@McPhie2001], or in the frame of the particle for a particle driven at constant average speed [@Speck2006]. The interpretation of recent experiments [@Palacci2010] also makes implicitly use of such a description in describing the sedimentation equilibrium of active particles, or in analyzing the diffusion constant for hot Brownian motion [@Rings2010].
The use of a Langevin equation with an effective temperature has several direct consequences. The kinetic energy associated with the center of mass, $\langle \frac{1}{2} M \bm{U}^2 \rangle$, is necessarily equal to the effective temperature $\frac{3}{2} {k_{\text{B}}}{T_{\text{eff}}}$. The diffusion coefficient $D$ and the mobility under the influence of an external force $\mu = U_x/F_x$ are related by an Einstein relation, $D/\mu = {k_{\text{B}}}{T_{\text{eff}}}$ [@Einstein1905]. More generally, this relation can be seen as the steady state version of the proportionality between the time dependent response function to an external force, $\chi(t) = \delta U_x(t)/\delta F_x$, and the velocity autocorrelation in the nonequilibrium steady state: $$\chi(t) = \frac{1}{{k_{\text{B}}}{T_{\text{eff}}}} \langle U_x(0) U_x(t) \rangle .
$$ This relation was explored numerically for self propelled particles in reference [@Loi2008], and shown to be consistent with the observed Einstein like relation. Independently of the use of a specific Langevin model, this relation defines an effective temperature trough a so called “fluctuation dissipation ratio”. The applicability of an effective temperature description is determined by the dependence of this fluctuation dissipation ratio on time. We show in the following that the time scale at which the fluctuation dissipation ratio of a heated particle is determined indeed matters, so that a single temperature description, even in such a seemingly simple system, is problematic.
Finally, the use of a Langevin description with an effective temperature entails the validity of several “fluctuation relations” [@Zon2003], which have been the object of numerous recent experimental and numerical tests, both in equilibrium and nonequilibrium systems. The study of the fluctuation relation for the heated particle constitutes the last part of this report.
Our work is based on a direct molecular simulation (MD) approach of a crystalline nanoparticle diffusing in a liquid. The simulation were carried out using the LAMMPS package [@LAMMPS]. Details of the model can be found in previous works [@Merabia2009; @Merabia2009a], where we used this system to investigate heat transfer from nanoparticles. The particle was made of 555 atoms with a fcc structure, tied together using FENE bonds. The liquid was made of $\sim 23000$ atoms (Fig. \[fig1\]).
![*Left*– Snapshot of the simulated system for ${T_{\text{p}}}= 3.5 {\varepsilon}/ {k_{\text{B}}}$ ($T_0 = 0.75 {\varepsilon}/ {k_{\text{B}}}$); Gray levels indicate the kinetic energy of atoms. *Right*– Steady radial temperature and density profiles for this system.[]{data-label="fig1"}](fig1){width="0.9\linewidth"}
All liquid and solid atoms interacted *via* the same Lennard-Jones (LJ) potential $v = 4 {\varepsilon}[ (\sigma/r)^{12}-(\sigma/r)^6]$, at the exclusion of solid atoms directly bonded to each others. In the following, all results will be given in LJ units, namely $\sigma$, ${\varepsilon}/{k_{\text{B}}}$ and $\tau = \sqrt{m\sigma^2/{\varepsilon}}$ for length, temperature and time, respectively. The atoms in the solid particle were held at constant temperature ${T_{\text{p}}}$ using a Nosé-Hoover thermostat, after subtracting the velocity of the center of mass. In order to mimic the bulk liquid – far from the particle – acting as a thermal bath, a rescaling thermostat was applied only to liquid atoms lying beyond $15 \sigma$ from the center of the particle (this condition being evaluated each time the thermostat was applied), to keep them at constant temperature $T_0 = 0.75 {\varepsilon}/{k_{\text{B}}}$. This amounts to an assumption that the temperature profile around the particle follows the latter instantaneously. This is a reasonable assumption, as heat diffusion is much faster than mass diffusion in our system: $D_{\text{heat}} \sim 1 \sigma^2/\tau$ [@Merabia2009a], while $D_{\text{mass}} \in [0.002;0.02] \sigma^2/\tau$ (Fig. \[fig2\]). Finally the whole system was kept at fixed pressure $p = 0.0015 {\varepsilon}/ \sigma^3$ using a Nosé-Hoover barostat. Simulations were run over typically $10^7$ timesteps in order to accumulate enough statistics.
In previous work, we have shown that nanoparticles are able to sustain extremely high heat fluxes, *via* two mechanisms: Firstly, interfacial thermal resistance at the nanoscale generates significant temperature jumps at the interface, *i.e.* the contact temperature ${T_{\text{c}}}$ of the liquid at the nanoparticle surface is much lower than the particle temperature ${T_{\text{p}}}$ (Fig. \[fig1\]). Secondly, the large curvature-induced Laplace pressure prevents the formation of a vapor layer at the interface; At the highest temperatures, only a stable depleted region is observed (Fig. \[fig1\]).
Two approaches were used to measure the effective temperature of the particle. We started by measuring the kinetic temperature ${T_{\text{K}}}$, related to the center of mass velocity of the particle. Due to the finite ratio between solid and liquid masses, care has to be taken to measure the relative velocity between the solid nanoparticle and the liquid $U_i = U_{\text{s}i} - U_{\text{l}i}$ ($i = x,y,z$), with $U_{\text{s}i}$ and $U_{\text{l}i}$ the velocities of the solid and liquid centers of mass along the $i$ direction. ${T_{\text{K}}}$ was then given by $\frac{1}{2} {k_{\text{B}}}{T_{\text{K}}}= \frac{1}{2} m_{\text{eff}} \langle U_i^2 \rangle$, where $m_{\text{eff}} = m_{\text{s}} m_{\text{l}} / (m_{\text{s}} + m_{\text{l}}$) \[$m_{\text{s}}$ and $m_{\text{l}}$ being the total mass of the solid and liquid components\]. We checked that this procedure behaved correctly for all mass ratios, even when the mass of solid atoms is increased artificially up to the point where $m_{\text{s}} = m_{\text{l}}$. All the velocity measurements presented in the following were done consistently using this procedure. ${T_{\text{K}}}$ was evaluated along the 3 degrees of freedom of the particle in order to estimate the uncertainties, which were below 1%.
We also measured the “Einstein” temperature ${T_{\text{E}}}$, defined as the ratio between the diffusion coefficient $D$ and the mobility $\mu$ of the particle [@Einstein1905]. The diffusion coefficient was computed as the plateau value of the integral of the velocity autocorrelation function (VACF) $C_{UU}(t) = \langle U_i(t)\,U_i(0) \rangle$ of the nanoparticle: $D = \lim_{t \rightarrow \infty} \mathcal{D}(t)$, with $\mathcal{D}(t) = \int_0^t C_{UU}(s) \mathrm{d}s$ (Fig. \[fig2\]a). The plateau is reached after a correlation time typically around $t_{\text{c}} \sim 30 \tau$. The mobility $\mu$ was computed by applying an external force $F = 10 \varepsilon / \sigma$ to the particle, and measuring its steady velocity $U$ in the direction of the force: $\mu = U/F$ (linear response in the applied force was carefully checked).
In Fig. \[fig2\].b, we have plotted both measures of the particle’s effective temperature as a function of ${T_{\text{p}}}$.
![(a) Integrated velocity autocorrelation functions of the particle (from bottom to top: ${k_{\text{B}}}{T_{\text{p}}}/ {\varepsilon}= 0.75, 1.5, 2, 2.5, 3, 3.5$). (b) Einstein temperature ${T_{\text{E}}}$ and kinetic temperature ${T_{\text{K}}}$ as a function of the particle temperature ${T_{\text{p}}}$; the contact temperature ${T_{\text{c}}}$ is also plotted for comparison. Lines are guides for the eye. When not indicated, uncertainties are below the symbol size.[]{data-label="fig2"}](fig2){width="1.0\linewidth"}
One can note that all temperature estimates collapse to $T_0$ at equilibrium. A striking feature of Fig. \[fig2\].b is that the two approaches to measure the effective temperature of the particle provide different results. While this is expected for active colloids with a ballistic motion at short times [@Palacci2010], it is quite surprising in the case of a simple Brownian particle, and cannot be understood in the framework of a Langevin description. As discussed before [@Rings2010], one can finally note that neither ${T_{\text{K}}}$ nor ${T_{\text{E}}}$ identify with the contact temperature ${T_{\text{c}}}$, as could be naively expected [@Mazo1974] (Fig. \[fig2\].b).
To understand the existence of two temperatures in the system, we have probed the fluctuation dissipation theorem (FDT) for the Brownian system under study. Generally speaking, considering a physical observable $A$, the response of a system driven out of equilibrium at time $t=0$ by the action of a small external field $\mathcal {F}(t)$ is characterized by the susceptibility $\chi_{AC}(t) = \frac{\langle \delta A(t) \rangle}{\delta \mathcal {F} (0)}$ where in the subscript of the susceptibility, $C$ refers to the variable conjugated to the field $\mathcal F$: $C=\frac{\delta \mathcal H} {\delta \mathcal F}$, $\mathcal H$ being the Hamiltonian of the perturbed system. The FDT states that the susceptibility $\chi_{AC}(t)$ is related to the equilibrium correlation function $C_{A C}(t) = \langle A(t) C(0) \rangle$ through: $\int_{0}^{t}\chi_{AC}(s) \mathrm{d}s = \frac{1}{{k_{\text{B}}}T} C_{AC}(t)$ where $T$ is the thermal bath temperature, and the correlation function is estimated at equilibrium. A sensitive way of probing the deviation from this relation in nonequilibrium systems, which has been extensively used for example in glassy systems [@Cugliandolo1997; @Berthier2002] consists in determining separately the integrated susceptibility function and the correlation function, and in plotting them in a parametric plot with the time as parameter. The slope of the curve is then interpreted as the inverse of an effective temperature, which may depend on the time scale [@Cugliandolo1997].
For the system under study, we obtain the integrated response to an external force $F$ by applying the force in a stationary configuration at $t=0$, and following the evolution of the particle center of mass velocity $U(t)$. The parametric plot involves then the average velocity divided by the applied force, $\mu(t) = \langle U(t) \rangle/F = \int_0^t \chi_{UX}(s) \mathrm{d}s$, versus the integrated velocity auto correlation function $C_{UX}(t) = \int_0^t C_{UU}(s) \mathrm{d}s = \mathcal{D}(t)$. To obtain the response function from the ensemble averaged particle velocity $\langle U(t)\rangle$, we have run simulations starting from $1000$ independent configurations of the system and tracked the position of the Brownian particle before a steady state is attained (corresponding to times smaller than $t_{\text{c}}$). This enabled us to obtain good statistics for the ensemble averaged velocity of the particle, in particular during the early stage of the transient $t \ll t_{\text{c}}$.
![Integrated response function as a function of the integrated VACF of the nanoparticle, for ${k_{\text{B}}}{T_{\text{p}}}/ {\varepsilon}= 0.75$ (equilibrium) and $3.5$. *Inset–* Temperatures extracted from the fit of the main graph’s curves at small and large times, as a function of the particle temperature. note that the lines are not merely guides to the eye, but correspond to the data determined independently and already reported in Fig. \[fig2\] for the kinetic and Einstein temperature.[]{data-label="fig3"}](fig3){width="0.8\linewidth"}
Figure \[fig3\] shows the resulting response/correlation parametric plot, for the different temperatures considered. When ${T_{\text{p}}}= T_0$, the nanoparticle is at equilibrium before the external force is applied, and the fluctuation dissipation theorem is obeyed. For higher values of the particle temperature ${T_{\text{p}}}$, the velocity $\langle U(t) \rangle$ depends non linearly on the integrated VACF and the fluctuation dissipation ratio is time dependent. This is particularly visible for the highest temperature considered in Fig. \[fig3\] ${T_{\text{p}}}= 3.5 {\varepsilon}/{k_{\text{B}}}$, where the two slopes $\frac{\mathrm{d} \mu}{\mathrm{d} \mathcal{D}}$ at small and large $\mathcal{D}$ differ markedly. From these two slopes, it is possible to define two temperatures ${T_{\text{K}}}'$ and ${T_{\text{E}}}'$ characterizing the response of the system respectively at short times and long times. The inset of Fig. \[fig3\] compares these two temperatures to the kinetic and Einstein temperatures defined before. Strikingly the short time effective temperature ${T_{\text{K}}}'$ is very close to the kinetic temperature of the nanoparticle ${T_{\text{K}}}$, while the long time effective temperature ${T_{\text{E}}}'$ is close to the Einstein temperature ${T_{\text{E}}}$. Therefore our system, in spite of its simplicity, exhibits a “two temperatures” behavior on the two different time scales that are separated by the typical scale set by the loss of memory in the initial velocity. The short time, fast temperature sets the kinetic energy of the particles, while the Einstein temperature which probes the steady state response is determined by the long time behavior of the integrated response.
![(a) Transient fluctuation temperature $T_t = \langle \delta W_t^2 \rangle /2\langle W_t \rangle$ as a function of the time $t$, for different temperatures ${T_{\text{p}}}$ of the nanoparticle. From bottom to top: ${k_{\text{B}}}{T_{\text{p}}}/{\varepsilon}= 0.75, 1, 1.5, 2, 2.5, 3, 3.5$. (b) Transient fluctuation temperature ${T_{\text{TFT}}}$ obtained with the long time limit of $T_t$ as a function of the particle temperature ${T_{\text{p}}}$. The lines correspond to the data for the kinetic and Einstein temperature in Fig. \[fig2\].[]{data-label="fig4"}](fig4){width="1.0\linewidth"}
For a system in contact with a thermal bath and driven out of equilibrium, the bath temperature plays also a key role in quantifying the fluctuations of the work from an external forcing [@Zon2003]. Two situations have to be distinguished depending on the time window analyzed. If we follow the evolution of a system in the transient regime before a steady state is reached, starting from a system at equilibrium, the transient fluctuation theorem (TFT) predicts: $$\label{TFT}
P(W_t)/P(-W_t)=\exp(W_t/{k_{\text{B}}}T) ,$$ where $P(W_t)$ is the density probability of the work $W_t$. In this equation $W_t$ is the work from the external force $F$, *i.e.* $W_t = \int_{0}^{t} U(s) F \mathrm{d}s$ and $T$ is the temperature of the thermal bath. On the other hand, in a a stationary situation, the steady state fluctuation theorem (SSFT) predicts $P(W_t)/P(-W_t) \rightarrow \exp(W_t/{k_{\text{B}}}T)$ when $t \gg t_{\text{c}}$ where $t_{\text{c}}$ denotes a typical equilibrium correlation time. In the SSFT, the work $W_t$ is estimated along a trajectory of length $t$: $W_t = \int_{t_i}^{t_i+t} U(s) F \mathrm{d}s$, where an average on different values of the initial $t_i$ may be performed. We have tested these fluctuation relations for the heated Brownian particles, again applying an external force $F = 10 {\varepsilon}/\sigma$ at $t=0$ and recording the statistics of the work using $1000$ independent configurations. It turned out however that the distribution of the work $W_t$ was too noisy to determine accurately the ratio $P(W_t)/P(-W_t)$ and critically assess the validity of the fluctuation theorems discussed above. To extract an effective temperature measuring the fluctuations of $W_t$, we have used the observation that the statistics of the work $W_t$ is to a good approximation Gaussian. Under these conditions, it is trivial to show that the distribution of $W_t$ obeys a law similar to Eq. \[TFT\] with an effective temperature $T_t=\langle \delta W_t^2 \rangle / 2 \langle W_t \rangle$. Note that strictly speaking the TFT implies that $T_t=T$ is independent of $t$. In Fig. \[fig4\].a we have shown the evolution of $T_t$ as a function of the time $t$ for different temperatures ${T_{\text{p}}}$ of the nanoparticle. For all the temperatures considered, the initially small values of $\langle W_t \rangle$ leads to a large uncertainty in the value of $T_t$. For longer times $t > 5 \tau$, the temperature $T_t$ is approximately independent of the time $t$. We will denote ${T_{\text{TFT}}}({T_{\text{p}}})$ the value of the effective temperature $T_t$ in this regime. Figure \[fig4\].b displays the evolution of ${T_{\text{TFT}}}$ as a function of the temperature of the nanoparticle ${T_{\text{p}}}$. It is clear that the resulting ${T_{\text{TFT}}}$ is very close to the kinetic temperature ${T_{\text{K}}}$ characterizing the particle dynamics on short time scales. While we are not aware of a theoretical analysis of this situation, we believe the reason for this proximity lies in the fact that the main contribution to fluctuations in the work function corresponds to the time regime in which the velocity is still correlated to its value at $t=0$, *i.e.* the same time regime in which the fluctuation dissipation ratio corresponds to the “fast” temperature.
Our work shows that, even in a conceptually rather simple system, in a nonequilibrium steady state, a description in terms of a Langevin model involving a single temperature is far from trivial. Further generalization and interpretation of the behavior of interacting particles in terms of Langevin models and a single noise temperature is expected to suffer similar difficulties, as can already be inferred from the results of [@Loi2008]. It would be interesting to explore, if the recent extensions of fluctuation dissipation theorems proposed in refs [@Seifert2010; @Prost2009] can be applied to the present case, *i.e.* to identify observables for which a response-correlation proportionality relation holds. Even so, the resulting observables are likely to be different from those that are naturally measured in experiments or simulations. We also note that, with the present observables, experiments using optical tweezers with a strongly absorbing particle could be used to probe the different temperatures investigated here, with the exception of the kinetic one. We expect that such experiments will be able to detect a deviation from equilibrium of the order of magnitude reported here.
We acknowledge useful exchanges with L. Bocquet, F. Cichos, K. Kroy and D. Rings, and the support of ANR project Opthermal.
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---
abstract: 'In this paper, we study the bulk motion of a classical extended charge in flat spacetime. A formalism developed by W. G. Dixon is used to determine how the details of such a particle’s internal structure influence its equations of motion. We place essentially no restrictions (other than boundedness) on the shape of the charge, and allow for inhomogeneity, internal currents, elasticity, and spin. Even if the angular momentum remains small, many such systems are found to be affected by large self-interaction effects beyond the standard Lorentz-Dirac force. These are particularly significant if the particle’s charge density fails to be much greater than its 3-current density (or vice versa) in the center-of-mass frame. Additional terms also arise in the equations of motion if the dipole moment is too large, and when the ‘center-of-electromagnetic mass’ is far from the ‘center-of-bare mass’ (roughly speaking). These conditions are often quite restrictive. General equations of motion were also derived under the assumption that the particle can only interact with the radiative component of its self-field. These are much simpler than the equations derived using the full retarded self-field; as are the conditions required to recover the Lorentz-Dirac equation.'
author:
- 'Abraham I. Harte'
date: 'January 16, 2006'
title: 'Self-forces on extended bodies in electrodynamics'
---
Introduction
============
Originally motivated by the discovery of the electron, the behavior of small (classical) electric charges has been studied in various contexts for over a century. The first results were obtained by Abraham [@Abraham] for a non-relativistic non-spinning rigid sphere. This calculation was later repeated by Schott [@Schott] and Lorentz [@Lorentz] within special relativity, and by Crowley and Nodvik [@NodvikOrig] in general (background) spacetimes. Similar results have since been obtained by a number of authors for more general charge distributions [@Schott1; @Nodvik; @Yaghjian; @Jackson; @Barut; @Crisp; @OriExtend] in flat spacetime. These results have very recently been extended to curved backgrounds as well [@PoissonExtend]. Still, these derivations did not allow for significant elasticity, charge-current coupling, and/or rotation.
Detailed reviews of different aspects of the self-force problem (in electromagnetism as well as scalar field theory and general relativity) have been given by Poisson [@SFReview1], Havas [@SFReview2], and Spohn [@SFReview3]. To summarize, though, a common theme throughout all of these works has been that the equations of motion describing sufficiently small particles were found to be independent of the details of their internal structure to a considerable degree of precision. Only a few parameters such as the rest mass and total charge entered into these equations. Ignoring the effects of spin and internal currents, the apparently universal correction to the Lorentz force law is given by the well-known Lorentz-Dirac self-force. If a particle with charge $q$ has a center-of-mass position $z^{a}(\bar{s})$ (with $\bar{s}$ being a proper time), then this self-force is equal to (using a metric with signature $-2$, and units in which $c=1$) $$\frac{2}{3} q^{2} \Big( \dddot{z}^{a}(\bar{s}) +
\dot{z}^{a}(\bar{s}) \ddot{z}^{b}(\bar{s}) \ddot{z}_{b}(\bar{s})
\Big) ~. \label{LD}$$
Although this is only an approximation to the full self-force for any realistic (extended) charge, it is natural to introduce a new concept into the theory for which it is exact – that of a point particle. The immediate problem with this is of course that the self-field diverges at the location of any point-like source, which would appear to imply that its self-force and self-energy are not well-defined. Dirac [@Dirac] removed this problem by noting that self-forces acting on a structureless particle should only arise as a reaction to emitted radiation. Letting $F^{ab}_{\mathrm{self}(+)}$ denote the advanced self-field and $F^{ab}_{\mathrm{self}(-)}$ the retarded one, we can define ‘radiative’ and ‘singular’ portions of $F^{ab}_{\mathrm{self}(-)}=F^{ab}_{\mathrm{self}(S)}+F^{ab}_{\mathrm{self}(R)}$: $$\begin{aligned}
F^{ab}_{\mathrm{self}(S)} &=& \frac{1}{2} \left(
F^{ab}_{\mathrm{self}(-)} + F^{ab}_{\mathrm{self}(+)} \right) ~,
\\
F^{ab}_{\mathrm{self}(R)} &=& \frac{1}{2} \left(
F^{ab}_{\mathrm{self}(-)} - F^{ab}_{\mathrm{self}(+)} \right) ~.
\label{radfielddefine}\end{aligned}$$ As the names imply, the singular field contains the entire divergent part of the retarded field. It is also derived from a time-symmetric Green function, so it would not be expected to contain any radiation. The self-force on a point particle should therefore be determined entirely by the radiative self-field. Combining it with the Lorentz force law immediately recovers (\[LD\]) [@Dirac].
This prescription is much simpler than any direct derivation of the equations of motion for a finite charge, and for this reason, it has been generalized to work in curved spacetime, as well as for scalar and gravitational self-forces [@detwhiting; @SFReview1] (although this is not the only way of ‘renormalizing’ point particle self-fields [@QuinnWald; @Quinn]). As before, the relevance of these extensions to realistic extended bodies has also been established in certain special cases. In linearized gravity, for example, a small nearly-Schwarzchild black hole has been found to obey the same equations of motion as a point particle [@MST; @SFReview1]. More generally, it has been shown that a nonspinning body’s internal structure is irrelevant to very high order within Post-Newtonian theory [@Damour]. Still, there remain questions of exactly how universal these results are. Will a spherical neutron star in near-equilibrium fall into a supermassive black hole in the same way as another one that is spinning rapidly and experiencing internal oscillations [@Instabilities]; or one having a mountainous (solid) surface [@NeutronMountains]? Such systems could be important sources for the upcoming Laser Interferometer Space Antenna (LISA).
Rather than addressing such questions directly, we have chosen to study the motions of charged bodies in flat spacetime (in part) as a model problem. The methods used here were specifically chosen so that very few conceptual changes would be required to consider particles (charged or not) moving in a fully dynamic spacetime. This has led to some additional complexity not strictly necessary to solve the problem at hand, although the majority of the complicating issues have been placed in the the appendix.
Our derivation is based on W. G. Dixon’s multipole formalism [@Dix67; @Dix70a; @Dix70b; @Dix74; @Dix79]. This gives a relatively simple, unified, and rigorous way of understanding the motions of arbitrarily structured bodies in both electrodynamics and general relativity. Despite the fact that Dixon’s theory decomposes the source functions into multipole moments, we never ignore any of them. By using generating functions, the entire infinite set is retained throughout all of our calculations. This only appears to be possible in this formalism. The laws of motion that are used are therefore exact (despite being ordinary differential equations). All of the approximations that we make are only used to compute the self-field.
Taking a hybrid point of view where an extended body can only interact with its radiative self-field, it is found that the Lorentz-Dirac equation follows for a wide range of nonspinning charges with small dipole moments. If the body instead interacts with its full retarded self-field, this result no longer holds. In this case, the equations of motion are drastically different if the charge and (3-) current densities have remotely similar magnitudes in the center-of-mass frame. Although it is impractical to define exactly what this means at this point, it will be shown that one of these quantities must be at least ‘second order’ compared to the other in order for these extra terms to vanish. This is because these cases usually allow the self-fields to do significant amounts of internal work (e.g. Ohmic heating). Even when the charge-current coupling in negligible, there are still extra complications in the retarded case. The conditions required to generically exclude these and other complicating effects are derived, and turn out to be surprisingly restrictive.
Sec. \[ProbMot\] reviews the various steps involved in calculating the motion of matter interacting with an electromagnetic field. Sec. \[LawsMot\] then summarizes the appropriate definitions of the center-of-mass and its laws of motion as obtained from Dixon’s formalism. It also decomposes the 4-current in a particular way that happens to be convenient in this framework. Although this reduction is not strictly required for the current problem, it is adopted throughout on the grounds that it would be essential in curved spacetime. It is derived in detail in the appendix, which also contains an in-depth review of of Dixon’s ideas.
With these basic ideas in place, Sec. \[FieldSect\] goes on to derive expansions for the advanced and retarded self-fields of an arbitrarily-structured charge using a slow motion approximation. Sec. \[ForceSect\] then combines these results with those from Sec. \[LawsMot\] to find general expressions for the self-force and self-torque. Finally, Sec. \[SpecCaseSect\] examines the equations of motion for certain simpler classes of charges, and derives some conditions under which the Lorentz-Dirac equation is applicable.
We use units in which $c = 1$ throughout. In order to facilitate a more direct comparison to Dixon’s papers, the metric is chosen to have signature $-2$ (although the rest of the notation used here frequently differs from Dixon’s). We also assume that the spacetime is flat, and adopt Minkowski coordinates for simplicity. Latin indices refer to these coordinates, while Greek ones are triad labels running from $1$ to $3$.
The Problem of Motion {#ProbMot}
=====================
In studying the dynamics of any system in a classical field theory, one has to specify ‘laws of motion’ for both the field and matter variables. In our case, the only field is of course the electromagnetic one, $F^{ab}=F^{[ab]}$. As usual, this is governed by Maxwell’s equations $$\begin{aligned}
\partial_{b}F^{ab} = J^{a} ~,
\label{maxwell}
\\
\partial^{[a} F^{bc]}=0 ~.\end{aligned}$$
We assume that the matter in our problem is completely described by its stress-energy tensor $T^{ab}$ and 4-current vector $J^{a}$. Taking the divergence of (\[maxwell\]) immediately gives our first constraint on these quantities: $$\partial_{a}J^{a}=0 ~.
\label{chargecons}$$ This equation acts as one of the laws of motion for the matter fields. The other is derived from the requirement that $\partial_{a}\left( T^{ab} + T^{ab}_{\mathrm{em}} \right) =0 $, where $T^{ab}_{\mathrm{em}}$ is the stress-energy tensor of the electromagnetic field. Combining the standard form of $T^{ab}_{\mathrm{em}}$ [@Jackson] with Maxwell’s equations then shows that the matter moves according to $$\partial_{b}T^{ab}=-F^{ab}J_{b} ~.
\label{stresscons}$$ (\[maxwell\])-(\[stresscons\]) are essentially the entire content of classical continuum mechanics in flat spacetime. Of course, different types of matter do not all move in the same way, so these equations by themselves are not sufficient to determine $T^{ab}$ and $J^{a}$ for all time (even if the $F^{ab}$ were given). One also needs to specify something analogous to a (generalized) equation of state, which can take on a rather unwieldy form.
This sort of procedure is the standard one in continuum mechanics. Unfortunately, the resulting nonlinear partial differential equations are notoriously difficult to solve. Such a detailed description of the system should not really be required, however, for problems where we are only interested in the body’s bulk motion. In these cases, a representative world line could be defined inside the (convex hull of the) spacelike-compact support of $T^{ab}$. Given that this worldline can always be parametrized by a single quantity, its tangent vector might be expected to satisfy an ordinary differential equation – at least when using certain approximations. Solving such an equation would clearly be much more straightforward than the partial differential equations that we started with.
This sort of simplification is one of the main motivations behind the many (source) multipole formalisms in the literature [@Jackson; @ThorneMult]. In these, one first fixes some particular reference frame which has, among other properties, a preferred time parameter $s$. The quantity being expanded – say $J^{a}(x)$ – can then be written in terms of an infinite set of tensors depending only on $s$: $Q^{a}(s)$, $Q^{ab}(s),\ldots$ The reverse is also true. Given $J^{a}$, there is a well-defined way to compute any moment. The set $\{Q^{\ldots} \}$ is therefore completely equivalent to $J^{a}$. This implies that the conservation equation (\[chargecons\]) may be used to find restrictions on the individual moments.
Such restrictions depend on the precise definitions that are being adopted, but can usually be divided into two general classes. The first of these consists of purely algebraic equations imposed at a fixed value of $s$. We call these the constraints. There are also a number of evolution equations which usually take the form of ordinary differential equations. Multipole expansions can therefore be used to convert (\[chargecons\]) and (\[stresscons\]) into a number of algebraic and ordinary differential equations (without any approximation).
This does not actually simplify things as much as it at first might appear. The reason is that almost all definitions for the source multipoles will lead to an infinite number of (coupled!) evolution equations. This is often dealt with in practice by assuming that all moments above a particular order are irrelevant, which leaves one with only a finite number of evolution equations. There are, however, interesting questions that require knowing the higher moments. $J^{a}$ (or $T^{ab}$) for example, cannot be reconstructed without them. This means that the self-field cannot be calculated in the near zone from only the first few moments. Although the self-force and self-torque can be found in certain cases by examining energy and momentum fluxes in the far zone [@Dirac; @DewittBrehme; @SFReview1; @MST] (which can be adequately approximated using only a finite number of moments), this is considerably less accurate than integrating the force density throughout the charge’s interior.
For these reasons and others, it is desirable to define a set of multipoles that do not require any cutoff. Remarkably, such a set exists [@Dix67], and is essentially unique [@Dix74]. Without any approximation, moments for both $J^{a}$ and $T^{ab}$ can be defined which satisfy a *finite* number of evolution equations. There remain (uncoupled) constraint equations for each moment, although these are easily solved. We adopt this formalism due to Dixon for the remainder of this paper. Its net effect is to allow us to relate the motion to the fields in a more rigorous way than is usually done (short of directly solving (\[chargecons\]) and (\[stresscons\])). It does not, however, have anything to say about the fields themselves. We therefore obtain the self-field in a standard way, and then use Dixon’s equations to find how the matter moves in response to it.
Laws of Motion {#LawsMot}
==============
As noted, we use Dixon’s method [@Dix67; @Dix70a; @Dix70b; @Dix74; @Dix79] to decompose $J^{a}$ and $T^{ab}$ into multipole moments. Each of these sets is designed to describe as simply as possible all possible forms of $J^{a}$ and $T^{ab}$ satisfying their respective conservation equations. We first assume that these matter fields are at least piecewise continuous, and have (identical) supports $W$. Any intersection of a spacelike hypersurface with this worldtube is assumed to be compact.
Now choose a timelike worldline $Z \subset W$, and a timelike unit vector field $n^{a}(s)$ defined on $Z$. It is assumed that this is always possible in any physically interesting system. $Z$ is then parameterized by the coordinate function $z^{a}(s)$, and the tangent vector to it is denoted by $v^{a}(s) := \mathrm{d} z^{a} /
\mathrm{d} s =: \dot{z}^{a}(s)$. $v^{a}$ need not be equal to $n^{a}$, although it will be convenient to normalize $s$ such that $n^{a}v_{a}=1$ (so $v^{a}v_{a} \neq 1$ in general). $v^{a}$ is called the kinematical velocity, while $n^{a}$ is the dynamical velocity. The set $\{ Z,n^{a} \}$ then defines a reference frame for the definition of the multipole moments. At this point, it should be thought of as arbitrary, although physical conditions will later be given that pick out a unique ‘center-of-mass’ frame.
A collection of spacelike hyperplanes $\{ \Sigma \}$ can easily be constructed from $Z$ and $n^{a}$. Each $\Sigma(s)$ is to pass through $z^{a}(s)$, and be (everywhere) orthogonal to $n^{a}(s)$. Assume that any point in $W$ is contained in exactly one of these planes. Unless $n^{a}$ is a constant, it is clear that this property cannot be true throughout the entire spacetime. For $x \in \Sigma(s)
\cap W$, we must therefore have that $\max|\dot{n}^{a}(s)
\big(x-z(s)\big)_{a}| < 1$ (among other conditions), which gives a weak restriction on the body’s maximum size. It is not really important, though, as any reasonable type of matter would be ripped apart long before this condition was violated.
The main results that we need from Dixon’s theory at this point are his definitions of the linear and angular momenta. These disagree with the usual ones when either $J^{a}$ or $F^{ab}$ are nonzero, although it is still convenient to label them by $p^{a}(s)$ and $S^{ab}(s)$ respectively. Unless otherwise noted, the words ‘linear and angular momenta’ will always refer to the quantities [@Dix67; @Dix70a] $$\begin{aligned}
p^{a}(s) &=& \int_{\Sigma(s)} \! \mathrm{d}\Sigma_{b} \, \left[
T^{ab} + J^{b} r_{c} \int_{0}^{1} \mathrm{d}u \; F^{ac}\big( z(s) +
ur \big) \right] ~,
\label{pdefine}
\\
S^{ab}(s) &=& 2 \int_{\Sigma(s)} \! \mathrm{d}\Sigma_{c} \, \left[
r^{[a} T^{b]c} + J^{c} r_{d} \int_{0}^{1} \mathrm{d}u \; u r^{[a}
F^{b]d} \big( z(s) + ur \big) \right] ~, \label{Sdefine}\end{aligned}$$ where $r^{a} := x^{a}-z^{a}(s)$, and $u$ is just a dummy parameter used to integrate along the line segment connecting $z^{a}(s)$ to $x^{a}=z^{a}(s)+r^{a}$.
Detailed motivations for these definitions can be found in [@Dix67; @Dix70a; @Dix74], as well as the appendix. In short, though, it can be shown that the given quantities are uniquely determined by demanding that stress-energy conservation directly affect only the first two moments of $T^{ab}$ (once the concept of a moment has been defined in a reasonable way). If $p^{a}$ and $S^{ab}$ are known in some time interval and satisfy the appropriate evolution equations, the class of all stress-energy tensors with these moments can be constructed without having to solve any differential equations. Each of these will exactly satisfy (\[stresscons\]).
This property implies that evolution equations for the quadrupole and higher moments of the stress-energy tensor are nearly unconstrained. They can be thought of as the ‘equation of state’ of the material under consideration. This type of independence of the higher moments from the conservation laws also occurs in Newtonian theory [@Dix79], and there are considerable advantages in preserving as much of that structure as possible in the relativistic regime. In particular, there is no need to discard multipole moments above a certain order. The choices (\[pdefine\]) and (\[Sdefine\]) allow us to retain many of the conveniences of a multipole formalism without its classic limitations.
The same definitions for the momenta can also be motivated by considering charged particles in curved spacetime [@Dix70a; @Dix79]. There, one can study the conserved quantities associated with Killing vectors in appropriate spacetimes (where both the metric and electromagnetic field are assumed to share the same symmetries). Fixing $Z$ and $n^{a}(s)$ allows each such quantity to be written as a linear combination of vector and antisymmetric rank 2 tensor fields on $Z$. Crucially, the definitions of these quantities do not depend on the Killing vector under consideration, so we can suppose that they are meaningful even in the absence of any symmetries. If the metric is taken to be flat, these objects reduce to (\[pdefine\]) and (\[Sdefine\]) [@Dix70a]. They are therefore the natural limits of what would generally be referred to as the linear and angular momenta of symmetric spacetimes.
It is now convenient to define a coordinate system on $W$ that is more closely adapted to the system than the Minkowski coordinates used so far. First choose an orthonormal tetrad $\{ n^{a}(s),
e^{a}_{\alpha}(s) \}$ ($\alpha =1,2,3$) along $Z$. Requiring that each of these vectors remain orthonormal to the others implies that $$\dot{e}^{a}_{\alpha} = -n^{a} \dot{n}_{b} e_{\alpha}^{b} ~,
\label{tetrad}$$ which is essentially Fermi-Walker transport. A spatial (rotation) term may also be added to this, although our calculations would then become considerably more tedious. On its own, the choice of tetrad has no physical significance, so we choose the simplest case.
Since it was assumed that $\{\Sigma\}$ foliates $W$, any point $x
\in \Sigma(s) \cap W$ may now be uniquely written in terms of $s$ and a ‘triad radius’ $r^{\alpha}$ $$x^{a}=e^{a}_{\alpha}(s) r^{\alpha} + z^{a}(s) ~. \label{XtoRS}$$ Varying $r^{\alpha}$ with a fixed value of $s$ clearly generates $\Sigma(s)$. Also, the Jacobian of the coordinate transformation $(x^{a}) \rightarrow (r^{\alpha},s)$ is equal to the lapse $N$ of the foliation, $$\begin{aligned}
N(x) &=& 1- \dot{n}_{a}(s) r^{a} ~, \nonumber
\\
&=& 1- \dot{n}_{a}(s) e^{a}_{\alpha}(s) r^{\alpha} ~. \label{lapse}\end{aligned}$$
We will be extensively transforming back and forth between these two coordinate systems, so it is convenient to abuse the notation somewhat by writing $f\left(r^{\alpha},s\right)=f\left(x^{a}\right)$ for any function $f$. The intended dependencies should always be clear from the context. It is also useful to denote quantities such as $\dot{n}_{a} e^{a}_{\alpha}$ by $\dot{n}_{\alpha}$. Note that in this notation, $\ddot{n}_{\alpha} = \ddot{n}_{a} e^{a}_{\alpha} \neq
\mathrm{d} \dot{n}_{\alpha} / \mathrm{d} s$.
Using these conventions, it is natural to split $J^{a}$ into the charge and 3-current densities seen by an observer at $r=0$ (the ‘center-of-mass observer’) $$J^{a} = \rho n^{a} + e^{a}_{\alpha} j^{\alpha} ~. \label{Jtetrad}$$
It is then shown in the appendix that there exist ‘potentials’ $\varphi$ and $H^{\alpha}$ which generate $J^{a}$ through the equations $$\begin{aligned}
\rho &=& \partial_{\alpha} \left( r^{\alpha} \varphi \right) ~,
\label{chargedensity}
\\
j^{\alpha} &=& N^{-1} \Big[ H^{\alpha} + v^{\beta}
\partial_{\beta} \left( r^{\alpha} \varphi \right) - r^{\alpha} \dot{\varphi} \Big] ~.
\label{threecurrent}\end{aligned}$$
Denoting the total charge by $q$ and letting $|r|^{2} := -r^{\alpha}
r_{\alpha} \geq 0$, $\varphi\left(r^{\alpha},s\right)$ was found to be continuous, and equal to $q/4\pi |r|^{3}$ outside $W$. Similarly, $H^{\alpha}\left(r^{\beta},s\right)$ is given by (\[HBoundary\]) outside $W$, and is piecewise continuous in $r^{\beta}$. $H^{\alpha}$ also satisfies $\partial_{\alpha} H^{\alpha} =0$. These properties guarantee that $J^{a}$ has support $W$, and is everywhere continuous. A direct calculation also shows that $$\begin{aligned}
\partial_{a} J^{a} &=& N^{-1} n_{b} \left( \frac{\partial}{\partial s} - v^{\beta} \partial_{\beta} \right) J^{b} + \partial_{\beta} j^{\beta} ~,
\\
&=& 0 ~,\end{aligned}$$ as required by (\[chargecons\]).
Any physically reasonable current vector can now be constructed by choosing potentials satisfying these rules. A physical interpretation of one’s choice is then given by substitution into (\[chargedensity\]) and (\[threecurrent\]). For example, a uniform spherical charge distribution with time-varying radius $D(s)$ is described by (assuming $n^{a}=v^{a}$) $$\begin{aligned}
\varphi(r,s) &=& \frac{q}{4\pi D^{3}(s)} \left[ \Theta \big(
D(s)-|r| \big) + \left( \frac{D(s)}{|r|} \right)^{3} \Theta \big(
|r|-D(s) \big) \right] ~, \label{phiEx}
\\
H^{\alpha}(r,s) &=& 0 ~, \label{HEx}\end{aligned}$$ where $\Theta(\cdot)$ is the Heaviside step function. (\[chargedensity\]) and (\[threecurrent\]) then show that the tetrad components of $J^{a}$ are $$\begin{aligned}
\rho(r,s) &=& \frac{3q}{4\pi D^{3}} \Theta \big( D-|r| \big) ~,
\\
j^{\alpha}(r,s) &=& \frac{3 q r^{\alpha}}{4\pi N(r,s) D^{3}(s)}
\left( \frac{\dot{D}(s)}{D(s)} \right) \Theta \big( D(s)-|r| \big)
~.\end{aligned}$$
It is also clearly possible to calculate $\varphi$ and $H^{\alpha}$ from any given $J^{a}$. This requires inverting (\[chargedensity\]), which acts as a partial differential equation for $\varphi$ on each time slice. The solution to this equation would usually have to be obtained numerically, which is clearly inconvenient. Largely for this reason, we shall consider $\varphi$ and $H^{\alpha}$ to be the given quantities for the remainder of this paper. $J^{a}$ can be derived from them using operations no more complicated than differentiation.
Another reason for this unconventional choice is that $\varphi$ and $H^{\alpha}$ contain all of the multipole moments of $J^{a}$ in a natural way. As shown in the appendix, they are closely related to the Fourier transform of a generating function for these moments. An arbitrary current moment can be obtained essentially by differentiating the inverse Fourier transforms of the potentials a suitable number of times. For example, (\[QJ\]), (\[hatJ\]), (\[Cform\]), (\[rhoA\]), and (\[Bform\]) can be used to show that the dipole moment has the general form $$Q^{ab} = \int \! \mathrm{d}^{3} r \left[ 2 n^{[a} e^{b]}_{\beta}
r^{\beta} N \left( \varphi(r,s) - \frac{q}{4 \pi |r|^{3}} \right) +
e^{a}_{\alpha} e^{b}_{\beta} \bar{H}^{\alpha \beta}(r,s) \right] ~,
\label{dipole}$$ where $\bar{H}^{\alpha \beta} = \bar{H}^{[\alpha \beta]}$ is defined by $$H^{\alpha} = \partial_{\beta} \bar{H}^{\alpha \beta} ~. \label{divH}$$ Given (\[HBoundary\]), we let $$\bar{H}^{\alpha \beta} = \frac{q v^{[\alpha} r^{\beta]} }{2 \pi
|r|^{3} }$$ outside $W$.
For the example given in (\[phiEx\]) and (\[HEx\]), the dipole moment is equal to $$Q^{ab} = \frac{1}{5} q D^{2}(s) n^{[a} \dot{n}^{b]} ~.
\label{dipoleEx}$$ This might have been expected to vanish in spherical symmetry, although it should be noted that it is only nonzero when sphericity is being defined in an accelerated reference frame.
These constructions can now be used to simplify the definitions of $p^{a}$ and $S^{ab}$. Letting $\bar{r}^{\alpha} := u r^{\alpha}$, the electromagnetic term in (\[pdefine\]) is equivalent to $$\int \mathrm{d}^{3} \bar{r} \, e_{b}^{\beta} \bar{r}_{\beta}
F^{ab}(\bar{r},s) \int_{0}^{1} \mathrm{d}u \, u^{-4}
\rho(\bar{r}/u,s)~.$$ But (\[chargedensity\]) shows that $$\begin{aligned}
\int_{0}^{1} \mathrm{d} u \, u^{-4} \rho(\bar{r}/u,s) &=&
-\int_{0}^{1} \mathrm{d} u \, \frac{\partial}{\partial u} \left[
u^{-3} \left( \varphi(\bar{r}/u,s) - \frac{q}{4\pi |\bar{r}/u|^{3}}
\right) \right] ~,
\\
&=& - \left( \varphi(\bar{r},s) - \frac{q}{4\pi |\bar{r}|^{3} }
\right) ~,\end{aligned}$$ so (\[pdefine\]) and (\[Sdefine\]) can be rewritten as $$\begin{aligned}
p^{a} &=& \int \mathrm{d}^{3} r \left[ T^{ab} n_{b} - \left( \varphi
- \frac{q}{4\pi |r|^{3}} \right) r_{\gamma} e_{c}^{\gamma} F^{ac}
\right] ~, \label{pdefine2}
\\
S^{ab} &=& 2 \int \mathrm{d}^{3} r \left[ r^{\alpha} e^{[a}_{\alpha}
T^{b]c} n_{c} - \left( \varphi - \frac{q}{4\pi |r|^{3}} \right)
r_{\gamma} e_{c}^{\gamma} r^{\alpha} e_{\alpha}^{[a} F^{b]c} \right]
~. \label{Sdefine2}\end{aligned}$$
We now need evolution equations for these quantities, which are most easily obtained by direct differentiation. For any function $I^{b}(x)$, relating the $x$-coordinates of $( r^{\alpha},
s+\mathrm{d} s )$ to those of $( r^{\alpha}, s )$ shows that $$\frac{\mathrm{d}}{\mathrm{d} s} \int_{\Sigma(s)} \mathrm{d}
\Sigma_{b} I^{b}(x) = \int \mathrm{d}^{3} r \Big[ \dot{n}_{b} I^{b}
+ \left( v^{c} - n^{c} \dot{n}_{\alpha} r^{\alpha} \right)
\partial_{c} \left( n_{b} I^{b} \right) \Big] ~.$$ If $I^{b}$ vanishes outside of some finite radius, this expression simplifies to $$\frac{\mathrm{d}}{\mathrm{d} s} \int_{\Sigma(s)} \mathrm{d}
\Sigma_{b} I^{b}(x) = \int \mathrm{d}^{3} r N \partial_{b} I^{b} ~.$$ (\[stresscons\]), (\[pdefine2\]), and (\[Sdefine2\]) can now be used to show that $$\begin{aligned}
\dot{p}^{a} &=& - \int \mathrm{d}^{3} r \, \Bigg\{ N F^{ab} J_{b} +
\frac{
\partial }{ \partial s} \left[ \left( \varphi
- \frac{q}{4\pi |r|^{3}} \right) r_{\gamma} e_{c}^{\gamma} F^{ac}
\right] \Bigg\} ~, \label{dotP1}\end{aligned}$$ and $$\begin{aligned}
\dot{S}^{ab} &=& - 2 \int \mathrm{d}^{3} r \, \Bigg\{ N r^{\alpha}
e^{[a}_{\alpha} F^{b]c} J_{c} + v^{[a} T^{b]c} n_{c} +
\frac{\partial}{ \partial s} \left[ \left( \varphi - \frac{q}{4\pi
|r|^{3}} \right) r_{\gamma} e_{c}^{\gamma} r^{\alpha}
e_{\alpha}^{[a} F^{b]c} \right] \Bigg\} ~, \nonumber
\\
&=& 2 p^{[a} v^{b]} - 2 \int \mathrm{d}^{3} r \, \Bigg\{ N
r^{\alpha} e^{[a}_{\alpha} F^{b]c} J_{c} + \frac{\partial}{ \partial
s} \left[ \left( \varphi - \frac{q}{4\pi |r|^{3}} \right) r_{\gamma}
e_{c}^{\gamma} r^{\alpha} e_{\alpha}^{[a} F^{b]c} \right] \nonumber
\\
&& ~ + \left( \varphi - \frac{q}{4\pi |r|^{3}} \right) r_{\gamma}
e_{c}^{\gamma} v^{[a} F^{b]c} \Bigg\} ~. \label{dotS1}\end{aligned}$$ If the momenta had been defined in the usual way, the $(\varphi-q/4\pi |r|^{3})$ terms would be absent from these expressions. The extra complication in the present case derives from the electromagnetic couplings in (\[pdefine\]) and (\[Sdefine\]).
Unsurprisingly, (\[dotP1\]) and (\[dotS1\]) can be directly related to the monopole and dipole moments of the force density $F^{ab} J_{b}$. Denoting such moments by $\Psi^{a}(s)$ and $\Psi^{ab}(s)$ respectively, it is possible to prove (\[fdefine\]) and (\[tdefine\]). It is then natural to refer to $-\Psi^{a}$ as the net force, and $-2 \Psi^{[ab]}$ as the net torque acting on the body. Viewing these quantities as force moments allows one to derive (\[forcedefine\]) and (\[torquedefine\]). But these results are no different than (\[dotP1\]) and (\[dotS1\]). If desired, the reader may therefore view $\Psi^{a}$ and $\Psi^{[ab]}$ to be *defined* by (\[fdefine\]) and (\[tdefine\]).
In cases where $F^{ab}$ varies slowly in the center-of-mass frame (both spatially and temporally), the expressions for the force and torque can be expanded in Taylor series involving successively higher multipole moments of the current density (denoted by $Q^{\cdots}$). Deriving such equations would be awkward using the methods introduced in this section, so we simply refer to (\[forcemult\]) and (\[torquemult\]). Despite the peculiar definitions of the current moments being used, these expansions are exactly what one would expect out of a multipole formalism, and can therefore be considered a check that Dixon’s definitions are reasonable.
In most cases where (\[forcemult\]) and (\[torquemult\]) are any simpler than the exact expressions for the force and torque, only the monopole and dipole moments will be significant. These moments can be computed from (\[chargemono\]) and (\[dipole\]) respectively. In the limit that the particle is vastly smaller than any of the field’s length scales, only the monopole moment will enter the equations of motion. In this case, the force reduces to the standard Lorentz expression and the torque vanishes, as expected.
The final ingredients required to complete this formalism are unique prescriptions for $Z$ and $n^{a}$. Simply knowing the linear and angular momenta at any point in time does not necessarily determine the body’s location in any useful way. The problem is compounded by the fact that these quantities are strongly dependent on the choice of reference frame itself. Indeed, without knowing where $W$ is, it is essentially impossible to specify $J^{a}$ in any meaningful way.
These problems can be removed by choosing $Z$ and $n^{a}$ appropriately, and then assuming that the resulting worldline provides a reasonable representation of the body’s ‘average’ position. Following [@Dix70a; @Dix74; @Dix79; @Ehl77], we first assume that for any point $z \in W$, there exists a unique future-directed timelike unit vector $n^{a}(z)$ such that $$p^{a}(z;n)=M(z;n) n^{a}(z) ~, \label{massdefine}$$ for some positive scalar $M$. Here, we have temporarily changed the dependencies of $p^{a}$ and $n^{a}$ for clarity. It is seen from (\[pdefine\]) that $p^{a}$ depends nontrivially on both the base point $z$, and on $n^{a}$, which defines the surface of integration. (\[massdefine\]) is therefore a highly implicit definition of $n^{a}$.
In any case, another condition must be also be given to fix $z$. We want this to lie on a ‘center-of-mass line’ in some sense, so it would be reasonable to expect the ‘mass dipole moment’ defined with respect to it to vanish: $$n_{b}n_{c} t^{abc} = 0 ~, \label{tempCM}$$ where $t^{abc}$ is the full dipole moment of the stress-energy tensor. This is defined by (\[stressdipole\]), so (\[tempCM\]) is equivalent to $$n_{a}\left( z \right) S^{ab}\left( z ; n \right)=0 ~.
\label{CMdefine}$$ The integral form of (\[CMdefine\]) reduces to the standard center-of-mass condition when $F^{ab}=0$ and $n^{a}=v^{a}$. It also allows us to write the angular momentum in terms of a single 3-vector $S^{a}$ $$S^{ab} = \epsilon^{abcd} n_{c} S_{d} . \label{spinang}$$
Of course, both of these properties would also have been satisfied by instead requiring $v_{a} S^{ab} = 0$. We reject this choice due to the fact that it leads to nonzero accelerations even when $F^{ab}=0$ [@Mat]. Replacing $v^{a}$ by $n^{a}$ avoids this peculiar behavior, which we consider to be an important requirement for anything deserving to be called a center-of-mass line. It also seems more natural to define the mass dipole moment in terms of $n^{a}$ rather than $v^{a}$.
For the remainder of this paper, we assume that (\[massdefine\]) and (\[CMdefine\]) are always satisfied, and call the resulting $Z$ and $n^{a}(s)$ the center of mass frame [@Dix70a; @Ehl77; @Dix79]. It is not obvious that solutions to these highly implicit equations exist, although existence and uniqueness have been proven in the gravitational case [@Schatt1]. There, it is also true that (given certain reasonable conditions) $Z$ is necessarily a timelike worldline inside the convex hull of $W$. We assume that the same results extend to electrodynamics.
The uniqueness of these definitions is actually not very critical for our purposes. The important point is that *a* solution with the given properties can presumably be chosen for a sufficiently large class of systems. While it is clearly very difficult to find the center-of-mass directly from $T^{ab}$, $J^{a}$, and $F^{ab}$, it is relatively straightforward to simply construct sets of moments which automatically incorporate (\[massdefine\]) and (\[CMdefine\]). (\[chargedensity\]) and (\[threecurrent\]) show, for example, that these definitions do not have any effect on our ability to construct arbitrary current vectors adapted to them. There is nothing preventing the moment potentials $\varphi$ and $H^{\alpha}$ from being appropriately centered around $r^{\alpha}=0$. Although it is not obvious that this can also be done for the stress-energy tensor, we conjecture that it can.
Now that (\[massdefine\]) and (\[CMdefine\]) have been assumed to hold, we need evolution equations for $M$, $n^{a}$, and $z^{a}$. These are easily found from (\[fdefine\]) and (\[tdefine\]): $$\begin{aligned}
\dot{M}&=&-n^{a}\Psi_{a} ~, \label{massevolve}
\\
M \dot{n}^{a}&=& -h^{a}_{b} \Psi^{b} ~, \label{nevolve}
\\
M(v^{a}-n^{a}) &=& S^{ab} \dot{n}_{b} - 2 \Psi^{[ab]} n_{b} ~,
\label{CMevolve}\end{aligned}$$ where $h^{a}_{b}$ is the projection operator $\delta^{a}_{b}- n^{a}
n_{b}$. Note that the last of these equations shows that $n^{a}=v^{a}$ if the spin and torque both vanish.
In general, (\[tdefine\]) and (\[massevolve\])-(\[CMevolve\]) may be used together to find the motion of the body’s center-of-mass in terms of $\Psi^{a}$ and $\Psi^{[ab]}$. Once the field is known, these quantities follow from (\[forcedefine\]) and (\[torquedefine\]). Some recipe for evolving the dipole and higher current moments in time – most conveniently expressed in terms of $\dot{\varphi}$ and $\dot{H}^{\alpha}$ – is also required. Combining all of these elements together leaves us with a well-defined initial value problem that will determine $z^{a}$, $n^{a}$, $M$, $S^{ab}$, and $J^{a}$. If the stress-energy tensor is also desired, possible forms of it could in principle be constructed from (\[stress1\]) and (\[stress2\]) in the same way that $J^{a}$ was derived from $\varphi$ and $H^{\alpha}$. Combining all of these steps would completely characterize the system, although we shall omit the last one. The result is still sufficient to answer most questions that one would be interested in asking of a nearly isolated particle.
The Field {#FieldSect}
=========
It is clear from the previous section that the equations of motion will easily follow once $\Psi^{a}$ and $\Psi^{[ab]}$ are known. These depend on the field, which we now calculate. For a reasonably isolated body, it is first convenient to split $F^{ab}$ into two parts $$F^{ab} = F^{ab}_{\mathrm{ext}} + F^{ab}_{\mathrm{self}} ~.$$ The external field is assumed to be generated by outside sources, while the self-field is entirely due to the charge itself. Since (\[forcedefine\]) and (\[torquedefine\]) are linear in $F^{ab}$, the force and torque can also be split up into ‘self’ and ‘external’ portions.
In Lorenz gauge, the vector potential sourced by the particle is given by [@Jackson] $$A^{a}_{\mathrm{self}}(x)=- \int \! \mathrm{d}^{4} x' \, J^{a}(x')
\delta \big( \sigma(x,x') \big) ~, \label{selfdefine}$$ where $$\sigma(x,x')=\frac{1}{2} (x-x')^{a}(x-x')_{a} ~, \label{sigdefine}$$ is Synge’s world function [@Synge; @SFReview1]. In realistic systems, this potential will couple to the external one via the outside matter fields. These may act as reflectors or dielectrics, or there may be an $n$-body interaction where the self-fields influence the motion of some external charged particles (obviously affecting the fields in $W$). Although it would be reasonable to group together all portions of $F^{ab}$ causally related to our particle in some way as the ‘self-field,’ this would be impossible to do with any generality. Instead, we simply define the $F^{ab}_{\mathrm{self}}$ to be the field derived from (\[selfdefine\]) in the usual way (this differs from the point of view taken in e.g. [@NewtMult]). The interactions of the self-field with the outside world will all be categorized as ‘external’ effects that we presume can be accounted for by separate methods.
If all of the external matter is sufficiently far away from $W$ (and slowly varying), $F^{ab}_{\mathrm{ext}}$ will be approximately constant within each $\Sigma(s) \cap W$ slice (and from one slice to the ‘next’). $\Psi^{a}_{\mathrm{ext}}$ and $\Psi^{[ab]}_{\mathrm{ext}}$ can therefore be approximated by (\[forcemult\]) and (\[torquemult\]) in these cases. Finding the self-force and self-torque is more complicated. For this, $F^{ab}_{\mathrm{self}}$ has to be combined with the exact integral expressions for the force and torque – (\[forcedefine\]) and (\[torquedefine\]). The detailed structure of the self-field must therefore be known throughout $W$.
This easily follows from $A_{\mathrm{self}}^{a}$: $$\begin{aligned}
F^{ab}_{\mathrm{self}}(x) &=& 2 \partial^{[a} A^{b]}_{\mathrm{self}}
~, \nonumber
\\
&=& - 2 \int \! \mathrm{d}^{4} x' \, \delta'(\sigma) (x-x')^{[a}
J^{b]} ~.\end{aligned}$$ Writing $x$ in terms of $( r^{\alpha}, s )$, and $x'$ in terms of $(
r'^{\alpha}, \tau )$, and defining $\dot{\sigma}(x,x') :=
\partial \sigma/\partial \tau$, $F^{ab}_{\mathrm{self}}$ becomes $$F^{ab}_{\mathrm{self} (\pm)}(x) = 2 \int \! \mathrm{d}^{3} r' \,
\Bigg\{ \frac{1}{|\dot{\sigma}|}
\frac{\mathrm{d}}{\mathrm{d}\tau}\left[ \frac{N}{\dot{\sigma}}
(x-x')^{[a} J^{b]} \right] \Bigg\}_{\tau= \tau_{\pm}} ~.
\label{field1}$$ $\tau_{+}$ ($>s$) represents the advanced time, and $\tau_{-}$ the retarded one. These are found by solving $\sigma(x,x')=0$ with $x$ and $r'$ held fixed.
Although we only consider the retarded field to be real, the advanced solution is also retained for now. This allows us to later construct the radiative self-field, which is considerably simpler than the full retarded field. It would be quite convenient if the self-forces generated by the two fields were identical (as Dirac assumed for point particles [@Dirac]), although we will show that this is not true in general.
Returning to the explicit form for $F^{ab}_{\mathrm{self}}$, splitting up $J^{a}$ according to (\[Jtetrad\]) shows that $$\begin{aligned}
F^{ab}_{\mathrm{self}}(x) &=& 2 \int \! \mathrm{d}^{3} r'
\frac{N}{\dot{\sigma}|\dot{\sigma}|} \, \Bigg\{ \rho \left[
(x-x')^{[a} \dot{n}^{b]} - \left( \frac{\ddot{\sigma}}{ \dot{\sigma}
} + N^{-1} \ddot{n}^{\alpha} r'_{\alpha} \right) (x-x')^{[a} n^{b]}
- v^{[a} n^{b]} \right] + \dot{\rho} (x-x')^{[a} n^{b]} \nonumber
\\
&& {} - j^{\alpha} \left[ (x-x')^{[a} e^{b]}_{\alpha} \left(
\frac{\ddot{\sigma}}{ \dot{\sigma} } + N^{-1} \ddot{n}^{\beta}
r'_{\beta} \right) + \dot{n}_{\alpha} (x-x')^{[a} n^{b]} + \left( N
n^{[a} + (h \cdot v)^{[a} \right) e^{b]}_{\alpha} \right] \nonumber
\\
&& {} + \dot{j}^{\alpha} (x-x')^{[a} e^{b]}_{\alpha} \Bigg\}~.
\label{field2}\end{aligned}$$ Here, $\dot{j}^{\alpha}(r',\tau) :=
\partial j^{\alpha}(r',\tau)/\partial \tau$, which differs from our usual convention (e.g. $\dot{n}^{\alpha}(\tau) :=
e^{\alpha}_{a}(\tau) \mathrm{d}n^{a}(\tau)/\mathrm{d}\tau$).
Moving on, (\[sigdefine\]) implies that $$\dot{\sigma}(x,x') = - \bigg( N(r',\tau) n^{a}(\tau) + (h \cdot
v)^{a}(\tau) \bigg) (x-x')_{a} ~, \label{dotsig}$$ and $$\ddot{\sigma}(x,x') = N^{2} + v^{\alpha} v_{\alpha} - \left( N
\dot{n}^{a} - n^{a} \ddot{n}^{\beta} r'_{\beta} + \frac{\mathrm{d}(h
\cdot v)^{a}}{\mathrm{d}\tau} \right) (x-x')_{a} ~.
\label{dotdotsig}$$
If we now specify how $\varphi$ and $H^{\alpha}$ vary in time (which determines $\dot{\rho}$ and $\dot{j}^{\alpha}$), we would have all of the ingredients necessary to find the body’s motion without any approximation. Unfortunately, inserting (\[field2\]) into (\[tdefine\]), (\[forcedefine\]), (\[torquedefine\]), and (\[massevolve\])-(\[CMevolve\]) leads to a set of delay integro-differential equations for the object’s motion. Such a system would be extremely difficult to solve (or even interpret) in general, although it could be a useful starting point for numerical simulations. It might also be interesting analytically if one were looking for the forces required to make a body move in some pre-determined way (e.g. a circular orbit).
Such questions will not be discussed here. We instead consider the simplest possible approximations that allow us to gain insight into a generic class of systems. In particular, it is assumed that all of the quantities in (\[field2\]) which depend on $\tau_{\pm}$ may be written in terms of quantities at $s$ (via Taylor expansion). This requires that nothing very drastic happen on timescales less than the body’s light-crossing time. This is not as trivial a condition as it might seem to be (see e.g. [@Selfosc1]), although we will not attempt to justify it.
Expressing everything in terms of $s$ rather than $\tau_{\pm}$ first requires calculating $\Delta_{\pm} := s - \tau_{\pm}$. $\tau_{\pm}$ was defined by $\sigma =0$, so $\Delta_{\pm}$ can be found by Taylor expanding this equation in $\Delta_{\pm}$. Assuming that both $n^{a}$ and $v^{a}$ are at least $C^{3}$ in time for $s \in
[\tau_{\pm}, s]$, $$\begin{aligned}
e^{a}_{\alpha}(\tau_{\pm}) &=& e^{a}_{\alpha}(s) + \Delta_{\pm}
n^{a}(s) \dot{n}_{\alpha}(s) - \frac{1}{2} \Delta_{\pm}^{2} \Big(
\dot{n}^{a}(s) \dot{n}_{\alpha}(s) + n^{a}(s) \ddot{n}_{\alpha}(s)
\Big) - \frac{1}{6} \Delta_{\pm}^{3}
\dddot{e}^{a}_{\alpha}\left(\xi_{\pm}^{(1)}\right) ~,
\label{expand1}
\\
z^{a}(\tau_{\pm}) &=& z^{a}(s) - \Delta_{\pm} v^{a}(s) + \frac{1}{2}
\Delta_{\pm}^{2} \dot{v}^{a}(s) - \frac{1}{6} \Delta_{\pm}^{3}
\ddot{v}^{a}(s) + \frac{1}{24} \Delta_{\pm}^{4}
\dddot{v}^{a}\left(\xi_{\pm}^{(2)}\right) ~, \label{expand2}\end{aligned}$$ where $\xi^{(1)}_{\pm}$ and $\xi^{(2)}_{\pm}$ are some numbers between $\tau_{\pm}$ and $s$.
Then (\[XtoRS\]) shows that $$\begin{aligned}
(x-x')^{a} &\simeq& e^{a}_{\alpha} (r-r')^{\alpha} + \Delta_{\pm}
\Big( N n^{a} + (h \cdot v)^{a} \Big) + \frac{1}{2} \Delta_{\pm}^{2}
\Big( n^{a} \ddot{n}^{\alpha} r'_{\alpha} - N \dot{n}^{a} \Big) +
\frac{1}{6} \Delta_{\pm}^{3} \ddot{n}^{a} . \label{dX1}\end{aligned}$$ Everything here has been written in terms of $s$, and all terms involving quantities such as $|\dot{n}|^{3}$, $|\ddot{n}|^{2}$, $|\dddot{n}|$, $|h \cdot v|^{2}$, and $|\dot{v}-\dot{n}|$ have been removed. Since $|\Delta_{\pm}| \sim |r-r'|$, these terms can be reasonably neglected if $|\dddot{n}| \mathcal{R}^{3} \ll |\ddot{n}|
\mathcal{R}^{2} \ll |\dot{n}| \mathcal{R} \ll 1$, and $|\dot{v}-\dot{n}| \mathcal{R} \ll |h \cdot v| \ll |\dot{n}|
\mathcal{R}$, where $\mathcal{R} := \max \big( |r|, |r'| \big)$. In a sense, we are expanding up to second order in powers of $|\dot{n}|\mathcal{R}$, and up to first order in $|v|$.
The requirement $|\dot{n}|\mathcal{R} \ll 1$ must hold for all $(r,r')$ pairs, so it is useful to define the largest possible value of $\mathcal{R}$. We call this the body’s ‘radius’ $$D(s) := \max_{\Sigma(s) \cap W} |r| ~. \label{Ddefine}$$ Using it, $|\dot{n}|\mathcal{R} \ll 1$ implies $|\dot{n}| D \ll 1$. This technically restricts the allowable size of the charge, although very few reasonable systems would actually be excluded.
Interpreting the relation satisfied by $(h \cdot v)^{a}$ ($=v^{a}-n^{a}$) isn’t quite as simple. Given (\[CMevolve\]), it is roughly equivalent to assuming that spin effects are present only to lowest nontrivial order. This is not completely accurate, though, and more precise statements will be given in the following section.
In any case, the inequalities following (\[dX1\]) will be assumed to hold from now on. Using them to expand $\sigma=0$, we find that $$\begin{aligned}
R^{2} := |r-r'|^{2} &\simeq& \Delta_{\pm} \left[ 2 v^{\alpha}
(r-r')_{\alpha} + \Delta_{\pm} N(r,s) N(r',s) + \frac{1}{3}
\Delta^{2}_{\pm} \ddot{n}^{\alpha} (r+2r')_{\alpha} + \frac{1}{12}
\Delta_{\pm}^{3} |\dot{n}|^{2} \right] ~.\end{aligned}$$ All but the second term here is already ‘small,’ but not quite negligible in our approximation. The zeroth order expression for $\Delta_{\pm}$ ($=\mp R$) may therefore be substituted into each of these terms without any overall loss of accuracy. The resulting equation is easily solved: $$\begin{aligned}
\Delta_{\pm} &\simeq& \mp R \bigg[ 1 + \frac{1}{2} \dot{n}_{\alpha}
(r+r')^{\alpha} - \frac{1}{2} \dot{n}^{\alpha} \dot{n}^{\beta}
r_{\alpha} r'_{\beta} \pm \frac{1}{6} R \ddot{n}^{\alpha}
(r+2r')_{\alpha} + \frac{1}{24} R^{2} \dot{n}^{\alpha}
\dot{n}_{\alpha} + \frac{3}{8} \Big( \dot{n}^{\alpha}
(r+r')_{\alpha} \Big)^{2} \nonumber
\\
&& {} \pm R^{-1} v^{\alpha} (r-r')_{\alpha} \bigg] ~,
\label{Deltasolve1}\end{aligned}$$ where everything is evaluated at $s$.
Although it would still be straightforward at this stage to compute the exact error in (\[Deltasolve1\]), it is not necessary. Dimensional analysis shows that the neglected terms have magnitudes $|\dddot{n}| \mathcal{R}^{4}$, $|\ddot{n}|^{2} \mathcal{R}^{5}$, $|
d(h \cdot v )/ds | \mathcal{R}^{2}$, $|d^{2}(h \cdot v)/ds^{2}|
\mathcal{R}^{3}$, $|h \cdot v|^{2} \mathcal{R}$, and so on (where each of these can be evaluated anywhere in the interval $[\tau_{\pm},s]$).
Continuing to expand quantities appearing in (\[field2\]), a long but straightforward calculation shows that (\[dotsig\]) and (\[dotdotsig\]) can be approximated by $$\begin{aligned}
\dot{\sigma} \simeq \pm R \left[ 1- \frac{1}{2} \dot{n}^{\alpha}
(r+r')_{\alpha} - \frac{1}{8} \Big( \dot{n}^{\alpha} (r-r')_{\alpha}
\Big)^{2} \mp \frac{1}{3} R \ddot{n}^{\alpha} (r+2r')_{\alpha} +
\frac{1}{8} R^{2} |\dot{n}|^{2} \right] ~,\end{aligned}$$ and $$\begin{aligned}
\ddot{\sigma} \simeq 1 - \dot{n}^{\alpha} (r+r')_{\alpha} \mp R
\ddot{n}^{\alpha} (r+2r')_{\alpha} + \dot{n}^{\alpha}
\dot{n}^{\beta} r_{\alpha} r'_{\beta} + \frac{1}{2} R^{2}
|\dot{n}|^{2} ~.\end{aligned}$$
Looking at (\[field2\]), the charge and current densities are the only quantities that have not yet been expanded away from $\tau_{\pm}$. It is instructive to temporarily leave them like this, but write out all other terms in field at $s$. The resulting expression is quite long, so we break it up into several smaller pieces by writing $F^{ab}_{\mathrm{self}(\pm)}$ in the form $$F^{ab}_{\mathrm{self}(\pm)} = 2 \int \! \mathrm{d}^{3} r'
\frac{1}{R^{3}} \, \bigg( \rho(r',\tau_{\pm}) f^{ab}_{(1)} +
\dot{\rho}(r',\tau_{\pm}) f^{ab}_{(2)} - j^{\beta} (r',\tau_{\pm})
f^{ab}_{\beta(3)} + \dot{j}^{\beta}(r',\tau_{\pm}) f^{ab}_{\beta(4)}
\bigg) ~. \label{field3}$$ The definitions of each of these coefficients is obvious from comparison with (\[field2\]).
Before computing them, we first simplify the notation by defining a quantity $T(s)>0$ such that $T^{-1}$ remains (marginally) less than $|\dot{n}|$, $|\ddot{n}|^{1/2}$, $|h \cdot v|^{1/2}/D$, etc. This is useful because a number of different objects were assumed to be negligible in this section. Some of these may be much larger than the others, so using them to writing down error estimates would become rather awkward. Introducing $T$ removes this difficulty.
Along the same lines, we also define $\epsilon := D/T \ll 1$. A simple but exceedingly tedious calculation then shows that the coefficients in (\[field3\]) are equal to $$\begin{aligned}
f^{ab}_{(1)} &\simeq& - (r-r')^{\alpha} e_{\alpha}^{[a} n^{b]}
\left[ 1+ \frac{1}{2} \dot{n}^{\beta}(r-r')_{\beta} - \frac{1}{8}
\Big( \dot{n}^{\beta} (r-r')_{\beta} \Big)^{2} + \frac{1}{2}
\dot{n}^{\beta} \dot{n}^{\gamma} r_{\beta} (r-r')_{\gamma} +
\frac{1}{8} R^{2} | \dot{n} |^{2} \right] \nonumber
\\
&& {} + \frac{1}{2} R^{2} (r-r')^{\alpha} e_{\alpha}^{[a}
\ddot{n}^{b]} - \frac{1}{2} R^{2} n^{[a} \dot{n}^{b]} \left( 1 +
\frac{1}{2} \dot{n}^{\beta} (3r-r')_{\beta} \right) \pm \frac{2}{3}
R^{3} \ddot{n}^{[a} n^{b]} + o \left( \Big[\epsilon (\mathcal{R}/
D) \Big]^{3} \mathcal{R} \right) ~, \label{f1}
\\
f^{ab}_{(2)} &\simeq& \pm R \, \Bigg\{ (r-r')^{\alpha}
e_{\alpha}^{[a} n^{b]} \left[ 1 + \dot{n}^{\beta} r_{\beta} + \Big(
\dot{n}^{\beta} r_{\beta} \Big)^{2} \pm \frac{1}{3} R
\ddot{n}^{\beta} (2r+r')_{\beta} - \frac{1}{4} R^{2} |\dot{n}|^{2}
\right] \pm \frac{1}{2} R^{2} (r-r')^{\alpha} e_{\alpha}^{[a}
\ddot{n}^{b]} \nonumber
\\
&& {} + (r-r')^{\alpha} e_{\alpha}^{[a} \dot{n}^{b]} R \left( 1 +
\frac{1}{2} \dot{n}^{\beta} (3r+r')_{\beta} \right) \mp \frac{1}{2}
R^{2} n^{[a} \dot{n}^{b]} \left( 1 + 2 \dot{n}^{\beta} r_{\beta}
\right) - \frac{1}{3} n^{[a} \ddot{n}^{b]} R^{3} - (h \cdot v)^{[a}
n^{b]} R \Bigg\} \nonumber
\\
&& {} + o \left( \Big[\epsilon (\mathcal{R}/ D) \Big]^{3}
\mathcal{R}^{2} \right) ~, \label{f2}
\\
f^{ab}_{\beta(3)} &\simeq& (r-r')^{\alpha} e_{\alpha}^{[a}
e_{\beta}^{b]} \left[ 1 + \frac{1}{2} \dot{n}^{\gamma}
(r-r')_{\gamma} - \frac{1}{8} \Big( \dot{n}^{\gamma} (r-r')_{\gamma}
\Big)^{2} + \frac{1}{2} \dot{n}^{\gamma} \dot{n}^{\lambda}
r_{\gamma} (r-r')_{\lambda} + \frac{1}{8} R^{2} |\dot{n}|^{2}
\right] \nonumber
\\
&& {} + \frac{1}{2} R^{2} \ddot{n}_{\beta} (r-r')^{\alpha}
e_{\alpha}^{[a} n^{b]} + \frac{1}{2} R^{2} \dot{n}_{\beta}
(r-r')^{\alpha} e_{\alpha}^{[a} \dot{n}^{b]} + n^{[a} e^{b]}_{\beta}
R^{2} \bigg( \frac{1}{2} \ddot{n}^{\gamma} (r-r')_{\gamma} \mp
\frac{1}{3} R |\dot{n}|^{2} \nonumber
\\
&& {} - R^{-2} v^{\gamma} (r-r')_{\gamma} \bigg) + \frac{1}{2} R^{2}
\dot{n}^{[a} e^{b]}_{\beta} \left( 1 + \frac{1}{2} \dot{n}^{\gamma}
(3r-r')_{\gamma} \right) \pm \frac{1}{3} R^{3} \ddot{n}^{[a}
e^{b]}_{\beta} + o \left( \Big[\epsilon (\mathcal{R}/ D) \Big]^{3}
\mathcal{R} \right) ~, \label{f3}
\\
f^{ab}_{\beta(4)} &\simeq& \pm R \, \Bigg\{ (r-r')^{\alpha}
e_{\alpha}^{[a} e_{\beta}^{b]} \left[ 1 + \dot{n}^{\beta} r_{\beta}
+ \Big( \dot{n}^{\beta} r_{\beta} \Big)^{2} \pm \frac{1}{3} R
\ddot{n}^{\beta} (2r+r')_{\beta} - \frac{1}{4} R^{2} |\dot{n}|^{2}
\right] \mp \frac{1}{2} R^{2} (r-r')^{\alpha} e_{\alpha}^{[a}
\nonumber
\\
&& {} \times \big( \dot{n}^{b]} \dot{n}_{\beta} -
n^{b]}\ddot{n}_{\beta} \big) - (r-r')^{\alpha} e_{\alpha}^{[a}
n^{b]} \dot{n}_{\beta} R \left( 1 + \frac{1}{2} \dot{n}^{\gamma}
(3r+r')_{\gamma} \right) - n^{[a} e^{b]}_{\beta} R \left[ 1+
\frac{1}{2} \dot{n}^{\gamma} (3r-r')_{\gamma} \right. \nonumber
\\
&& \left. {} - \frac{1}{8} \Big( \dot{n}^{\gamma} (r+r')^{\gamma}
\Big)^{2} + \dot{n}^{\gamma} \dot{n}^{\lambda} r_{\gamma} (2
r_{\lambda} + r'_{\lambda} ) \pm \frac{1}{6} R \ddot{n}^{\gamma} (5
r + r')_{\gamma} - \frac{7}{24} R^{2} |\dot{n}|^{2} \pm R^{-1}
v^{\gamma} (r-r')_{\gamma} \right] \nonumber
\\
&& {} - (h \cdot v)^{[a} e^{b]}_{\beta} R \mp \frac{1}{2}
\dot{n}^{[a} e^{b]}_{\beta} R^{2} \left( 1+2 \dot{n}^{\gamma}
r_{\gamma} \right) - \frac{1}{6} R^{3} \ddot{n}^{[a} e^{b]}_{\beta}
\Bigg\} + o\left( \Big[\epsilon (\mathcal{R}/ D) \Big]^{3}
\mathcal{R}^{2} \right) ~. \label{f4}\end{aligned}$$
Approximations {#Approximations}
--------------
At this point, it is useful to consider the physical meaning behind our approximations more carefully. Although error estimates have been given for for $f^{ab}_{(1)}$-$f^{ab}_{\beta (4)}$, the important errors are those in $F^{ab}_{\mathrm{self}}$. Unfortunately, the unbounded integrand in (\[field3\]) makes it difficult to compute these rigorously. We shall simply assume that $J^{a}$ is sufficiently homogeneous that dimensional analysis can be used to say, for example, that the absolute value of the error in $\int \! \mathrm{d}^{3} r' \, \rho(r',\tau_{\pm})
f_{(1)}^{ab}/R^{3}$ is less than about $$\epsilon^{3} \max |\rho| D ~.$$
With this established, we can also use the assumptions outlined in the previous section to provide bounds on the force and torque. Combining $|\dot{n}| < T^{-1}$ with (\[nevolve\]), $$\begin{aligned}
\left| \Psi^{\alpha} \right| < M T^{-1} ~. \label{eforcerestrict}\end{aligned}$$ Since $T \gg D$, this means that the particle cannot be accelerated up to an appreciable fraction of the speed of light within a light-crossing time – a very reasonable restriction.
Differentiating (\[nevolve\]), $$M \ddot{n}^{\alpha} = - \dot{\Psi}^{\alpha} - 2 \dot{M}
\dot{n}^{\alpha} ~.$$ Generically, this implies that $|\ddot{n}| < T^{-2}$ can be ensured by assuming $$\begin{aligned}
\left|\dot{M}\right| &<& M T^{-1} ~, \label{nforcerestrict}
\\
\left| \dot{\Psi}^{\alpha} \right| &<& M T^{-2} ~.\end{aligned}$$
Differentiating (\[nevolve\]) a second time and using (\[massevolve\]), we also have that $$\begin{aligned}
\left| \ddot{M} \right| &<& M T^{-2} ~, \label{massrestrict1}
\\
\left| \ddot{\Psi}^{\alpha} \right| &<& M T^{-3} ~,
\\
\left| n_{a} \dot{\Psi}^{a} \right| &<& M T^{-2} ~.\end{aligned}$$
Similar restrictions on the spin and torque can be found by using (\[CMevolve\]). In particular, $|v|$ will remain less than $\epsilon^{2}$ if $$\begin{aligned}
\left|S^{\alpha \beta}\right| := \left|S^{\alpha}\right| &<&
\epsilon MD ~, \label{spinrestrict}
\\
\left| \Psi^{[ab]} e_{a}^{\alpha} n_{b} \right| &<& \epsilon^{2} M
~. \label{torquerestrict2}\end{aligned}$$
Differentiating (\[CMevolve\]), $|\dot{n}-\dot{v}| D \ll
\epsilon^{2}$ is implied by $$\begin{aligned}
\left| \Psi^{[\alpha \beta]} \right| &\ll& \epsilon M ~,
\label{torquerestrict}
\\
\left| \dot{\Psi}^{[ab]} e_{a}^{\alpha} n_{b} \right| &\ll& \epsilon
MD ~.\end{aligned}$$
Although weaker restrictions than these can be adopted in special cases, we assume for simplicity that they always hold. More concisely, we are restricting ourselves to systems where the magnitude of each tetrad component of the (full self + external) force is bounded by about $M T^{-1}$, and the magnitude of each torque component is no larger than $\epsilon^{2} M$. Bounds on the $s$-derivatives of these quantities are suppressed by appropriate factors of $T$. These conditions basically mean that the body can only change significantly over timescales larger than $T$. (\[spinrestrict\]), for example, implies that this timescale sets a lower bound on the charge’s rotational period.
These interpretations should not be surprising. Unfortunately, though, they are quite not as simple as they appear. Combining them with (\[forcedefine\]) and (\[torquedefine\]) can lead to less obvious bounds on the structure of $J^{a}$ itself. One of these affects the size of the particle’s dipole moment. If the self-torque is small compared to the external one, then the magnitudes of $$Q^{[\alpha}{}_{\gamma} F^{\beta] \gamma}_{\mathrm{ext}}, \, \, n_{a}
e^{c}_{\gamma} Q^a{}_{c} F^{\beta \gamma}_{\mathrm{ext}} , \, \,
n_{a} e_{c}^{\gamma} Q^\beta{}_{\gamma} F^{ac}_{\mathrm{ext}}, \, \,
e_{a}^{[\alpha} e_{b}^{\beta]} n_{c} n_{d} Q^{ac}
F^{bd}_{\mathrm{ext}} \label{dipoleBound}$$ should all be less than about $\epsilon^{2}M$. If $q$ is negligible, these conditions ensure that both the force and torque are sufficiently small.
In most cases of interest, though, $q \neq 0$. Bounding the quantities in (\[dipoleBound\]) then guarantees only that the torque is acceptable. The (full) force will often be dominated by the monopole component of $\Psi^{a}_{\mathrm{ext}}$, in which case (applying (\[eforcerestrict\]) and (\[nforcerestrict\])) $$\begin{aligned}
M T^{-1} &>& \left| q e_{a}^{\alpha} n_{b} F^{ab}_{\mathrm{ext}}
\right| ~,
\\
&>& \left| q v_{\beta} F^{\alpha \beta}_{\mathrm{ext}} \right| ~.\end{aligned}$$ Since $v_{\beta}$ is small, this second bound allows the magnetic field to be quite large. Despite this, we choose to assume (purely for simplicity) that both the electric and magnetic fields are bounded by $M\big(|q|T\big)^{-1}$. If the fields are in fact as large as these equations allow, then the previous bounds on the dipole moment imply that $$\left|Q^{\alpha \beta}\right| \lesssim \epsilon |q| D~,
\label{dipolerestrict}$$ along with a similar restriction on $\left|Q^{ab} e_{a}^{\alpha}
n_{b}\right|$. This implies that the ‘center-of-charge’ must be very close to the center-of-mass, which considerably reduces the number of current distributions that we can allow. Note, however, that these assumptions could be relaxed somewhat if extra conditions are placed on the sizes of the field components.
To find some other implications of our approximations, we now study a particular example in detail before moving on to computing general self-forces.
Self-Forces {#ForceSect}
===========
Stationary Case
---------------
It was shown in the previous section that (\[field3\])-(\[f4\]) approximate the self-field due to a charge with an (almost) arbitrarily accelerating center-of-mass line. These equations become exact when $\dot{n}^{a} =0$ and $n^{a}=v^{a}$. Temporarily assuming that these conditions are true, $$\begin{aligned}
F^{ab}_{\mathrm{self}(\pm)}(x) &=& - 2 \int \! \mathrm{d}^{3} r'
\frac{1}{R^{3}} \, \bigg[ (r-r')^{\alpha} e_{\alpha}^{[a} n^{b]}
\Big( \rho(r',\tau_{\pm}) \mp R \dot{\rho}(r', \tau_{\pm}) \Big) -
(r-r')^{\alpha} e^{[a}_{\alpha} e^{b]}_{\beta} \Big( j^{\beta}(r',
\tau_{\pm}) \nonumber
\\
&& {} \pm R \dot{j}^{\beta} (r',\tau_{\pm}) \Big) + n^{[a}
e^{b]}_{\beta} R^{2} \dot{j}^{\beta}(r',\tau_{\pm}) \bigg] ~,
\label{statfield}\end{aligned}$$ where $\tau_{\pm} = s \pm R$.
One might at first think that the self-forces generated by this field would be identically zero, but this is not generally correct. Anything emitting a focused beam of radiation, for example, will experience some recoil. It will also lose a bit of mass over time, and can even start rotating if the the beam is offset from the source’s center-of-mass. These can all be thought of as self-force and self-torque effects, and clearly do not require $\dot{v}^{a}
\neq 0$.
Since (\[statfield\]) is exact, one could compute self-forces and self-torques for a variety of systems without any approximation. If these were nonzero, the assumptions leading to (\[statfield\]) would not be maintained unless there were also external forces and torques present that exactly balanced them. Such a situation would be rather artificial, so we instead apply the approximations of Sec. \[FieldSect\], and use the resulting self-force and self-torque to gain some intuition into more general cases.
Assuming that $| \varphi | \gg |\dot{\varphi}| D \gg
|\ddot{\varphi}| D^{2}$, except perhaps at isolated points, one can show that $$\begin{aligned}
\Psi^{\alpha}_{\mathrm{self}(\pm)} &\simeq& \int \! \mathrm{d}^{3} r
\, \mathrm{d}^{3} r' \, \frac{1}{R^{3}} \, \Bigg\{ R^{2} \rho(r,s)
\dot{H}^{\alpha}(r',s) - 2 (r-r)^{\alpha} H_{\beta}(r,s) \left(
\frac{1}{2} r'^{\beta} \dot{\varphi}(r',s) \mp R
\dot{H}^{\beta}(r',s) \right) \nonumber
\\
&& {} + \left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right) \bigg[
(r-r')^{\alpha} r_{\beta} - \delta^{\alpha}_{\beta} (r-r')^{\gamma}
r_{\gamma} \bigg] \dot{H}^{\beta}(r',s) \Bigg\} ~, \label{ePsi2}\end{aligned}$$ to first order in $\dot{\varphi}$ and $\dot{H}^{\alpha}$. This illustrates that $\Psi^{\alpha}$ is bounded by $\dot{\varphi} H
D^{5}$ (meaning $\max|\dot{\varphi}| \max|H| D^{5}$), $\varphi
\dot{H} D^{5}$, or $H \dot{H} D^{5}$ (whichever is largest). If $H^{\alpha}$ is very small compared to $\varphi$, then the lowest order contributions to the self-force can be shown to be of order $\varphi \ddot{\varphi} D^{6}$.
It is then clear that unless the system is particularly symmetric, self-forces with these magnitudes are unavoidable. They would considerably complicate the equations of motion when considering nonzero center-of-mass accelerations, so it would be convenient to ignore them. This is self-consistent with the approximations already in place if the charge’s ‘elasticity’ is bounded by $$\begin{aligned}
\dot{\varphi} D &<& \epsilon^{2} \varphi ~, \label{phisize}
\\
\ddot{\varphi} D^{2} &\ll& \epsilon^{2} \varphi ~, \label{phisize2}
\\
\dot{H} D &\ll& \epsilon^{2} H ~. \label{Hsize}\end{aligned}$$
If these relations hold, the self-force becomes negligible whenever $\dot{n}^{a}=0$ and $n^{a}=v^{a}$; i.e. a charge at rest will remain at rest unless acted on by an external field. For this reason, all of the calculations that follow will assume (\[phisize\]) and (\[phisize2\]) to be true. (\[Hsize\]) is extremely simple to relax, so it will be eventually be replaced with $\dot{H} D <
\epsilon^{2} H$. The interested reader could easily weaken these conditions even further, although we regard the extra complication to be unnecessary for the present purposes.
In writing down (\[statfield\]), both $\dot{n}^{a}=0$ and $n^{a}=v^{a}$ were required. This second condition was adopted here for simplicity, although it is by no means ‘natural.’ Given (\[CMevolve\]), it can only hold if $\Psi^{[ab]} n_{b}=0$. But combining (\[phisize\])-(\[Hsize\]) with (\[torquedefine\]) and (\[statfield\]) shows that this portion of the torque is usually nonzero even if $\dot{\varphi}=\dot{H}^{\alpha}=0$. Generically, one finds that $\Psi^{[ab]} n_{b}$ will only be small if either $\varphi \ll H$ or $H \ll \varphi$. $n^{a} \simeq v^{a}$ is not a particularly important requirement, so we choose not to impose these restrictions on the current structure. In general, then, $n^{a} \neq v^{a}$ even when $\dot{n}^{a}=0$ (and $\varphi$ can be of order $H$).
Of course, $|n-v|=|v|$ still can’t exceed $\epsilon^{2}$. As was already shown, this remains true if each tetrad component of $\Psi^{[ab]}$ is bounded by $\epsilon^{2} M$ (and $|S| < \epsilon
MD$). We now strengthen this assumption slightly, and require the self and external components of the torque to be *individually* less than $\epsilon^{2} M$. There are then terms in the self-torque which violate this unless more restrictions are placed on the particle’s current structure.
Using (\[statfield\]) and (\[torquedefine\]) again, one can see that the tetrad components of $\Psi^{[ab]}_{\mathrm{self}}$ will be of order $\varphi H D^{5}$, $\varphi \dot{\varphi} D^{6}$, etc. These types of quantities can be conveniently simplified by defining an ‘electromagnetic radius’ $D_{\mathrm{em}} \sim \left( \varphi^{2}
+ H^{2} \right) D^{6}/M \lesssim D$. $D_{\mathrm{em}}/D$ then estimates the fraction of the particle’s mass that is of (macroscopic) electromagnetic origin. This ratio will come up often, so let $$\mathcal{E} := \left( \frac{D_{\mathrm{em}}}{D} \right) \lesssim 1
~.$$
In this notation, $\varphi \dot{\varphi} D^{6} \rightarrow
\epsilon^{2} \mathcal{E} M$, and $\varphi H D^{5} \rightarrow
\mathcal{E} M$. If $\varphi$ and $H$ are comparable, the second of these expressions is clearly the one that estimates the self-torque. Enforcing (\[torquerestrict2\]) in this case therefore requires $$\mathcal{E} < \epsilon^{2} ~. \label{selfenergyrestrict}$$ Conditions weaker than this can be adopted if the particle is dominated by either $\varphi$ or $H^{\alpha}$, although it will still be true that $\mathcal{E} \ll 1$ (except in special cases). Surprisingly, our approximations place a severe restriction on the charge’s self-energy. This makes it impossible to ever take the point particle limit in a strict sense (if such a thing is even meaningful). Such a procedure is not, however, necessary to answer whether a particle’s size is important when it is much smaller than the characteristic scales of the surrounding system (the ‘physical point particle limit’).
It is interesting to note that this condition is really independent of any of Dixon’s special constructions. In almost any slow motion approximation, one would expect that $|\dot{S}| \lesssim |S| T^{-1}$ and $|S|\lesssim MD^{2} T^{-1}$. So $\left|\Psi^{[ab]}\right| \sim
|\dot{S}| \lesssim \epsilon^{2} M $ is generic. And this is exactly the condition that led to (\[selfenergyrestrict\]). The specific form of the torque needed to show this did use Dixon’s definition, but the more common one wouldn’t have changed anything. This can be verified by substituting (\[statfield\]) into the first term in (\[torquedefine\]).
Also note that when $\mathcal{E} \sim 1$, it is not clear that any slow motion assumption is even physically reasonable. It seems difficult, for example, to avoid the extremely high frequency self-sustaining oscillations discussed by Bohm and Weinstein [@Selfosc1], among others. When $\mathcal{E}$ is small, it is easy to imagine that oscillations like these would have extremely small amplitudes, or at least be highly damped in causal systems. But the degree of rigidity required to hold together a charge with very large self-energy might prevent this.
Arbitrary Motion
----------------
We now turn back to analyzing the self-forces and self-torques when $\dot{n}^{a} \neq 0$ and $n^{a} \neq v^{a}$. For simplicity, it is assumed that the internal dynamics are always ‘slow’ in the sense of (\[phisize\]) and (\[phisize2\]). It is simple enough to relax (\[Hsize\]), so we also allow the first (and only the first) derivative of $H^{\alpha}$ to appear in the self-force. Higher derivatives of both $\varphi$ and $H^{\alpha}$ could be included with relatively little extra effort, although there seems to be little reason for doing so.
### Radiative Self-Forces {#RadiativeSect}
At this point, it is useful to separately write out the retarded and radiative fields. Using (\[radfielddefine\]) and (\[field3\])-(\[f4\]), $$F^{ab}_{\mathrm{self}(R)}(x) \simeq - \frac{4}{3} q \ddot{n}^{[a}(s)
n^{b]}(s) ~, \label{radfield}$$ If $n^{a}=v^{a}$, then this is the same expression found by Dirac from the Liénard-Wiechert potential [@Dirac] (we have used the opposite sign convention for the field).
The forces and torques exerted by this field are now very simple to calculate. Inserting (\[radfield\]) in (\[forcedefine\]), $$\Psi^{\alpha}_{\mathrm{self}(R)}(s) \simeq - \frac{2}{3} q^{2}
\ddot{n}^{\alpha} + o\big( \epsilon^{2} \mathcal{E} M T^{-1} \big)
~.$$ This is just the Lorentz-Dirac force when $n^{a}=v^{a}$ (see (\[LD\])).
Substituting (\[radfield\]) into (\[torquedefine\]), the space-space components of the self-torque are $$\Psi^{[\alpha \beta]}_{\mathrm{self}(R)}(s) \simeq \frac{1}{q} \!
\int \! \mathrm{d}^{3} r \, \rho(r,s) r^{[\alpha}
\Psi^{\beta]}_{\mathrm{self}(R)} + o \big( \epsilon^{3} \mathcal{E}
M \big) ~.$$ $\int \! \mathrm{d}^{3} r \, \rho r^{\alpha}/q$ can be thought of as the separation vector between the ‘charge centroid’ and the center-of-mass ($r^{\alpha}=0$). Inverting (\[spinang\]), we can clearly convert this component of the self-torque into a 3-vector describing the rate of change of $S^{a}$. In this form, the (vector) self-torque is just the cross product of the self-force with this separation vector. Writing it in this way suggests that it arises due to the self-force not acting through the center-of-mass.
The time-space components of the self-torque are similar: $$\Psi^{[ab]}_{\mathrm{self}(R)}(s) n_{a} e_{b}^{\beta} \simeq
\frac{1}{2q} \! \int \! \mathrm{d}^{3} r \, H_{\alpha}(r,s)
\Psi^{\alpha}_{(R)\mathrm{self}} r^{\beta} + o \big( \epsilon^{3}
\mathcal{E} M \big)~. \label{ntorquerad}$$ This clearly vanishes if the ‘current centroid’ coincides with the center-of-mass.
Combining all of these results with (\[tdefine\]), (\[massevolve\]), (\[nevolve\]), and (\[CMevolve\]), one can find that the particle’s motion by simultaneously solving $$\begin{aligned}
\dot{M} &\simeq& -n_{a} \Psi^{a}_{\mathrm{ext}} + o\big(
\epsilon^{2} \mathcal{E} M T^{-1} \big) ~, \label{dotmrad1}
\\
\dot{S}^{\alpha \beta} &\simeq& - 2 \Psi^{[\alpha
\beta]}_{\mathrm{ext}} + \frac{4}{3} q \! \int \! \mathrm{d}^{3} r
\, \rho(r,s) r^{[\alpha} \ddot{n}^{\beta]} + o\big( \epsilon^{3}
\mathcal{E} M \big) ~, \label{torquerad1}
\\
M \dot{n}^{\alpha} &\simeq& -\Psi^{\alpha}_{\mathrm{ext}} +
\frac{2}{3} q^{2} \ddot{n}^{\alpha} + o\big( \epsilon^{2}
\mathcal{E} M T^{-1} \big) ~, \label{forcerad1}
\\
M \dot{v}^{\alpha} &\simeq& M \dot{n}^{\alpha} + S^{\alpha \beta}
\ddot{n}_{\beta} - \dot{M} v^{\alpha} -4 \Psi^{ [\alpha \beta]
}_{\mathrm{ext}} \dot{n}_{\beta} - 2
\dot{\Psi}^{[ab]}_{\mathrm{ext}} e_{a}^{\alpha} n_{b} + o\big(
\epsilon^{2} \mathcal{E} M T^{-1} \big) ~. \label{forcerad2}\end{aligned}$$
If the external field varies slowly over $\Sigma(s) \cap W$, (\[forcemult\]) and (\[torquemult\]) can be used to approximate $\Psi^{a}_{\mathrm{ext}}$ and $\Psi^{[ab]}_{\mathrm{ext}}$. Let the minimum characteristic length scale of the external field be denoted by $\lambda \gg D$, so that $|F^{ab}_{\mathrm{ext}}| \gtrsim \lambda
|\partial F^{ab}_{\mathrm{ext}}|$ (where the absolute value signs are meant to act on each tetrad component of the quantity inside them). It is then convenient to assume that $$\lambda \gtrsim T/\mathcal{E} ~. \label{lambdarestrict}$$ This ensures such that the dipole and higher contributions to the external force are negligible compared to the Lorentz-Dirac self-force.
Without any loss of accuracy, (\[lambdarestrict\]) allows (\[ntorquerad\])-(\[forcerad2\]) to be written as (making some weak assumptions on the magnitudes of the quadrupole and higher moments) $$\begin{aligned}
\dot{M} &\simeq& -q n_{a} e_{b}^{\beta} v_{\beta}
F^{ab}_{\mathrm{ext}} ~, \label{dotmrad2}
\\
\dot{S}^{\alpha \beta} &\simeq& -2 e_{a}^{\alpha} e_{b}^{\beta}
\left( Q^{[a}{}_{c} F^{b] c}_{\mathrm{ext}} + Q^{d[a}{}_{c}
\partial_{d} F^{b] c}_{\mathrm{ext}} \right) + \frac{4}{3} q \! \int
\! \mathrm{d}^{3} r \, \rho(r,s) r^{[\alpha} \ddot{n}^{\beta]} ~,
\label{torquerad2}
\\
M \dot{n}^{\alpha} &\simeq& - q e_{a}^{\alpha} v_{b}
F^{ab}_{\mathrm{ext}} + \frac{2}{3} q^{2} \ddot{n}^{\alpha} ~,
\label{forcerad3}
\\
M \dot{v}^{\alpha} &\simeq& M \dot{n}^{\alpha} + S^{\alpha \beta}
\ddot{n}_{\beta} - \dot{M} v^{\alpha} - 4 e_{a}^{\alpha}
e_{b}^{\beta} \dot{n}_{\beta} Q^{[a}{}_{c} F^{b]c}_{\mathrm{ext}} -
2 e^{\alpha}_{a} n_{b} \dot{Q}^{[a}{}_{c} F^{b]c}_{\mathrm{ext}} ~,
\label{forcerad4}\end{aligned}$$ where the error terms haven’t changed. Similarly, $v^{\alpha}$ can be recovered from (\[CMevolve\]): $$M v^{\alpha} \simeq S^{\alpha \beta} \dot{n}_{\beta} - 2
e_{a}^{\alpha} n_{b} \left( Q^{[a}{}_{c} F^{b] c}_{\mathrm{ext}} +
Q^{d[a}{}_{c} \partial_{d} F^{b] c}_{\mathrm{ext}} \right) -
\frac{2}{3} q \! \int \! \mathrm{d}^{3} r \, H_{\beta}(r,s)
\ddot{n}^{\beta} r^{\alpha} + o \big( \epsilon^{3} \mathcal{E} M
\big) ~. \label{vrad1}$$
The assumptions we’ve made so far – although fairly restrictive – are clearly not sufficient to recover the standard point particle result. Still, there must exist some class of charges which do behave in this way, and it is interesting to characterize it. Before doing so, it must first be mentioned that the $\bar{s}$ in (\[LD\]) is a proper time, while $s$ is not. It is close, though. Letting $s= s(\bar{s})$ and $s'(\bar{s}) := \mathrm{d}s/\mathrm{d}
\bar{s}$, $$\begin{aligned}
s' &=& \frac{1}{\sqrt{v^{a}v_{a}}} ~, \label{StoSbar}
\\
&\sim& 1 + o\left(\epsilon^{4}\right) ~.\end{aligned}$$
The fractional difference between the center-of-mass 4-velocity and $v^{a}$ is therefore of order $\epsilon^{4}$. In particular, the triad components of this 4-velocity will differ from $v^{\alpha}$ by terms no larger than $\epsilon^{6}$. Given the error estimate in (\[vrad1\]), these differences are negligible whenever $\mathcal{E} \gtrsim \epsilon^{3}$.
The proper acceleration can now be written in terms of $v^{a}$ and $\dot{v}^{a}$: $$z''^{a} = s'' v^{a} + \left( s' \right)^{2} \dot{v}^{a} ~.$$ It follows from (\[forcerad4\]) and (\[StoSbar\]) that $s'' \sim
\epsilon^{4}/T$, so $z''^{a}$ differs from $\dot{v}^{a}$ by terms of this same order. (\[forcerad2\]) then implies that these two quantities are interchangeable whenever $\mathcal{E} \gtrsim
\epsilon^{2}$.
Applying similar arguments, our results can be easily compared to the Lorentz-Dirac equation by writing it in the form $$\begin{aligned}
\big( M \dot{v}^{\alpha} \big)_{\mathrm{LD}} &\simeq& - q
e_{a}^{\alpha} v_{b} F^{ab}_{\mathrm{ext}} + \frac{2}{3} q^{2}
\ddot{n}^{\alpha} + o\Big( \epsilon^{3} M T^{-1}
\max(\epsilon,\mathcal{E}) \Big)~, \label{LDe}
\\
\big( M n_{a} \dot{v}^{a} \big)_{\mathrm{LD}} &\simeq& -q n_{a}
e_{b}^{\beta} v_{\beta} F^{ab}_{\mathrm{ext}} + o\Big( \epsilon^{3}
M T^{-1} \max(\epsilon,\mathcal{E}) \Big) ~. \label{LDn}\end{aligned}$$
In order to avoid overcomplicating the discussion, we shall say that our equations of motion reduce to the Lorentz-Dirac result if $M
\dot{v}^{\alpha}$ and $M n_{a} \dot{v}^{a}$ match (\[LDe\]) and (\[LDn\]) up to terms of order $\epsilon^{2} M T^{-1}
\max(\epsilon^{2},\mathcal{E})$. Noting that $n_{a} \dot{v}^{a} = -
\dot{n}^{\alpha} v_{\alpha}$, (\[forcerad3\]) shows that the the ‘temporal component’ of the body’s acceleration always matches (\[LDn\]) to the required accuracy.
The same is not true for the spatial acceleration. This shouldn’t be too surprising, though, as the Lorentz-Dirac equation was never intended to describe spinning particles. $S^{ab}$ should therefore be set to zero (at least instantaneously) before any reasonable comparison can be made. Even this isn’t quite sufficient, though. The situation can be remedied by assuming that $$\begin{aligned}
|S| &<& \epsilon MD \max \big( \epsilon^{2}, \mathcal{E} \big) ~,
\label{SrestrictRad}
\\
\left| Q^{[a}{}_{c} F^{b]c}_{\mathrm{ext}} e_{a}^{\alpha}
e_{b}^{\beta}\right| &<& \epsilon^{2} M \max \big( \epsilon^{2},
\mathcal{E} \big) ~, \label{dipolerestrictRad}\end{aligned}$$ along with a similar restriction on $\left| Q^{[a}{}_{c}
F^{b]c}_{\mathrm{ext}} e_{a}^{\alpha} n_{b}\right|$. Then $$\begin{aligned}
\left| v \right| &<& \epsilon^{2} \max \big( \epsilon^{2},
\mathcal{E} \big) ~, \label{vRadLD}
\\
| \dot{M} | &<& \epsilon^{2} M T^{-1} \max \big( \epsilon^{2},
\mathcal{E} \big) ~, \label{dotMRadLD}
\\
| \dot{S} | &<& \epsilon^{2} M \max \big( \epsilon^{2}, \mathcal{E}
\big) ~. \label{SboundRadLD}\end{aligned}$$ If $\mathcal{E} \gtrsim \epsilon^{2}$, $\dot{M}$ vanishes up to the maximum order that we can calculate it. The same is not necessarily true for $\dot{S}^{\alpha\beta}$ and $v^{\alpha}$, although they remain sufficiently small that (\[LDe\]) can now be recovered to the desired accuracy. There therefore exists a regime in which the equations of motion derived here reduce to the Lorentz-Dirac equation. Considering $z^{a}$ to be the only observable (as would be reasonable for an extremely small particle), this completely recovers the usual point particle result.
While this conclusion is not particularly surprising, it is interesting to note how restrictive the required assumptions are. (\[dipolerestrictRad\]) is particularly difficult to satisfy. For example, when the force is approximately given by the Lorentz (monopole) expression, external field magnitudes up to $\sim M
\big(|q|T\big)^{-1}$ are allowed. In fields this large, (\[dipolerestrictRad\]) implies that the magnitudes of the dipole moment must be less than $\sim \epsilon \mathcal{E} q D$ (when $\mathcal{E} \gtrsim \epsilon^{2}$). This is an extremely limiting condition in cases where the self-energy is small. Given (\[dipole\]), there do not appear to be any rigid ($\dot{\varphi}
= \dot{H}^{\alpha} =0$) or nearly rigid charge distributions that could satisfy it. One either needs to choose a very special class of charges, or considerably restrict the maximum allowable field strength.
Before continuing, it should first be mentioned that our definitions of the charge and three-current densities are slightly unusual. These quantities would usually be defined with respect to an orthonormal tetrad adapted to $z'^{a}\big(s(\bar{s})\big)$. We have instead defined these quantities in terms of a tetrad with temporal component $n^{a}$. Translating from one of these definitions to the other would only introduce fractional changes of order $\epsilon^{2}$, which are usually irrelevant at our level of approximation.
### Retarded Self-Forces {#RetardedSect}
It is generally accepted that detectors placed outside of $W$ will measure $F^{ab}_{\mathrm{ext}}+F^{ab}_{\mathrm{self}(-)}$ rather than $F^{ab}_{\mathrm{ext}}+F^{ab}_{\mathrm{self}(R)}$. But infinitesimal elements of an extended body cannot ‘know’ that they are part of a larger whole, so they must couple to this same field. It is therefore reasonable to consider only the retarded self-field ‘physical.’ Recalling that this is equal to the sum of the radiative and singular self-fields, any relevance of $F^{ab}_{\mathrm{self}(R)}$ itself should be derived by showing the effects of the singular self-field are irrelevant in certain cases.
To this end, we now examine the (presumably) realistic case in which the particle interacts with its full retarded self-field. Essentially all of the steps in the previous section carry over identically in this case, although most involve considerably more calculation. Starting with (\[chargedensity\]), (\[threecurrent\]), and (\[field3\])-(\[f4\]), the retarded field can be shown to be (dropping the ‘$-$’ subscript) $$\begin{aligned}
F^{ab}_{\mathrm{self}}(x) &\simeq& 2 \int \! \mathrm{d}^{3} r' \,
\frac{1}{R^{3}} \, \Bigg\{ \rho \left[ - (r-r')^{\alpha}
e_{\alpha}^{[a} n^{b]} \left(1+ \frac{1}{2} \dot{n}^{\beta}
(r-r')_{\beta} - \frac{1}{8} (\dot{n}^{\beta}(r-r')_{\beta})^{2}
+\frac{1}{8} R^{2} |\dot{n}|^{2} \right. \right.
\nonumber
\\
&& \left. {} + \frac{1}{2} \dot{n}^{\beta} \dot{n}^{\gamma}
r_{\beta} (r-r')_{\gamma} \right) + \frac{1}{2} R^{2}
(r-r')^{\alpha} e_{\alpha}^{[a} \ddot{n}^{b]} - \frac{1}{2} R^{2}
n^{[a} \dot{n}^{b]} \left( 1+ \frac{1}{2} \dot{n}^{\beta}
(3r-r')_{\beta} \right) \nonumber
\\
&& \left. {} - \frac{2}{3} R^{3} \ddot{n}^{[a} n^{b]} \right] -
\Big( H^{\beta} + v^{\sigma} \partial'_{\sigma} \left( r'^{\beta}
\varphi \right) \Big) \left[ (r-r')^{\alpha} e_{\alpha}^{[a}
e^{b]}_{\beta} \left( 1 + \frac{1}{2} \dot{n}^{\gamma}
(r+r')_{\gamma} + \frac{3}{8} \Big(\dot{n}^{\gamma}
(r+r')_{\gamma}\Big)^{2} \right. \right. \nonumber
\\
&& \left. {} + \frac{1}{8} R^{2} |\dot{n}|^{2} - \frac{1}{2}
\dot{n}^{\gamma} \dot{n}^{\lambda} r_{\gamma} r'_{\lambda} \right) +
n^{[a} e^{b]}_{\beta} R^{2} \left( \frac{1}{2} \ddot{n}^{\gamma}
(r+r')_{\gamma} + \frac{1}{3} R |\dot{n}|^{2} - R^{-2} v^{\gamma}
(r-r')_{\gamma} \right) \nonumber
\\
&& \left. {} + \frac{1}{2} R^{2} \dot{n}_{\beta} (r-r')^{\alpha}
e_{\alpha}^{[a} \dot{n}^{b]} + \frac{1}{2} R^{2} \ddot{n}_{\beta}
(r-r')^{\alpha} e^{[a}_{\alpha} n^{b]} + \frac{1}{2} R^{2}
\dot{n}^{[a} e^{b]}_{\beta} \left( 1+ \frac{1}{2}
\dot{n}^{\gamma}(3r+r')_{\gamma} \right) \right. \nonumber
\\
&& \left. {} - \frac{1}{3} R^{3} \ddot{n}^{[a} e^{b]}_{\beta}
\right] + \dot{\varphi} (r-r')^{\alpha} e_{\alpha}^{[a}
e_{\beta}^{b]} r'^{\beta} - \dot{H}^{\beta} n^{[a} e^{b]}_{\beta}
R^{2} \Bigg\} ~. \label{field4}\end{aligned}$$
The self-force is now obtained by inserting this into (\[forcedefine\]), which results in an expression of the form $$\begin{aligned}
\Psi^{a}_{\mathrm{self}}(s) &\simeq& - \int \! \mathrm{d}^{3} r \,
\mathrm{d}^{3} r' \, \mathcal{F}^{a}(r,r',s) ~,
\\
&\simeq& - \frac{1}{2} \int \! \mathrm{d}^{3} r \, \mathrm{d}^{3} r'
\, \Big( \mathcal{F}^{a}(r,r',s) + \mathcal{F}^{a}(r',r,s) \Big) ~,
\label{Newton3}\end{aligned}$$ where $\mathcal{F}^{a}(r,r',s)$ represents the force density exerted by a charge element at $r$ on a charge element at $r'$. If Newton’s third law were correct, $\mathcal{F}^{a}(r,r',s) = -
\mathcal{F}^{a}(r',r,s)$ (so $\Psi^{a}_{\mathrm{self}}$ would vanish). Of course, this is does not quite hold for the electromagnetic field (or any other fully observable field), so there is a nonzero self-force. (\[Newton3\]) therefore gives a precise form to the intuitive idea that self-forces measure the degree of failure of Newton’s third law.
After removing the components of $\mathcal{F}^{a}$ which reverse sign under interchange of $r$ and $r'$, it can be shown that $$\begin{aligned}
M \dot{n}^{\alpha} &\simeq& -\Psi^{\alpha}_{\mathrm{ext}} +
\frac{2}{3} q^{2} \ddot{n}^{\alpha} - M_{\mathrm{em}}
\dot{n}^{\alpha} - \int \! \mathrm{d}^{3}r \, \mathrm{d}^{3} r' \,
\frac{1}{R^{3}} \, \Bigg\{ \rho(r,s) H^{\beta}(r',s) R^{2} \left[
\delta^{\alpha}_{\beta} \bigg( \frac{1}{2} \ddot{n}^{\gamma}
(r+r')_{\gamma} \right. \nonumber
\\
&& \left. {} - R^{-2} v^{\gamma} (r-r')_{\gamma} \bigg) -
(r-r')^{\alpha} \ddot{n}_{\beta} \right] + \rho(r,s)
\dot{H}^{\beta}(r',s) R^{2} + v^{\gamma} \partial_{\gamma} \Big(
r_{\beta} \varphi(r,s) \Big) H^{\alpha}(r',s) (r-r')^{\beta}
\nonumber
\\
&& {} -\dot{\varphi}(r,s) H^{\gamma}(r',s) (r-r')^{\alpha}
r_{\gamma} - \left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right)
\left[ \frac{1}{2} H^{\gamma}(r',s) \bigg( R^{2} \ddot{n}^{\alpha}
r_{\gamma} - R^{2} \delta^{\alpha}_{\gamma} \ddot{n}^{\beta}
r_{\beta} \right. \nonumber
\\
&& \left. {} + \Big( (r-r')^{\alpha} r_{\gamma} -
\delta^{\alpha}_{\gamma} (r-r')^{\beta} r_{\beta} \Big)
\ddot{n}^{\sigma} (r-r')_{\sigma} \bigg) + \dot{H}^{\gamma}(r',s)
\left( (r-r')^{\alpha} r_{\gamma} - \delta^{\alpha}_{\gamma}
(r-r')^{\beta} r_{\beta} \right) \right] \Bigg\}
\nonumber
\\
&& {} + o \big( \epsilon^{2} \mathcal{E} M T^{-1} \big) ~,
\label{fullforce}\end{aligned}$$ where it was useful to define an ‘electromagnetic mass’ $$\begin{aligned}
M_{\mathrm{em}} &:=& \int \! \mathrm{d}^{3} r \, \mathrm{d}^{3} r'
\, \frac{1}{R^{3}} \, \Bigg\{ \frac{1}{2} R^{2} \bigg[ \rho(r,s)
\rho(r',s) - H_{\beta}(r,s) H^{\beta}(r',s) \Big( 1+
\dot{n}^{\gamma} (r+r')_{\gamma} \Big) \bigg] \nonumber
\\
&& {} + \left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right)
\rho(r',s) \left[ (r-r')^{\beta} r_{\beta} \left( 1 + \frac{1}{2}
\dot{n}^{\gamma} (r-r')_{\gamma} \right) - \frac{1}{2} R^{2}
\dot{n}^{\beta} r_{\beta} \right] \Bigg\} ~. \label{dM}\end{aligned}$$
The form of (\[fullforce\]) suggests an effective inertial mass $m
:= M + M_{\mathrm{em}}$. Although $M$ will very rarely remain constant, $m$ is often conserved, or at least varies slowly. To see this, substitute (\[field4\]) into (\[forcedefine\]) and use (\[massevolve\]) to show that $$\begin{aligned}
\dot{M} &\simeq& -n_{a} \Psi^{a}_{\mathrm{ext}} - \int \!
\mathrm{d}^{3} r \, \mathrm{d}^{3} r' \, \frac{1}{R^{3}} \, \Bigg\{
\left( \varphi(r,s) -\frac{q}{4\pi |r|^{3}} \right) \bigg[
\dot{\rho}(r',s) (r-r')^{\beta} r_{\beta} - \frac{1}{2} \rho(r',s)
\Big( R^{2} \ddot{n}^{\beta} r_{\beta} \nonumber
\\
&& {} -\ddot{n}^{\alpha} (r-r')_{\alpha} (r-r')^{\beta} r_{\beta}
\Big) - H^{\gamma}(r',s) \left( \frac{1}{2} R^{2} \Big(
\dot{n}_{\gamma} \dot{n}^{\beta} r_{\beta} + r_{\gamma}
|\dot{n}|^{2} \Big) - (r-r')^{\alpha} \Big( \dot{n}_{\alpha}
r_{\gamma} \right. \nonumber
\\
&& {} - r_{\alpha} \dot{n}_{\gamma} \Big) \Big(1+ \frac{1}{2}
\dot{n}^{\sigma} (r+r')_{\sigma} \Big) \bigg) \bigg] -H_{\beta}(r)
\dot{H}^{\beta}(r') R^{2} - \frac{1}{2} H_{\beta}(r) H^{\beta}(r')
R^{2} \ddot{n}^{\gamma}(r+r')_{\gamma} \nonumber
\\
&& {} - \rho(r',s) \bigg[ R^{2} H_{\beta}(r,s) \dot{n}^{\beta}
\left( 1+ \frac{1}{2} \dot{n}^{\alpha} (3 r-r')_{\alpha} \right) -
v^{\gamma} \partial_{\gamma} \Big(r_{\beta} \varphi(r,s) \Big)
(r-r')^{\beta} \bigg] \Bigg\} + o \big( \epsilon^{2} \mathcal{E} M
T^{-1} \big) ~. \label{dotM}\end{aligned}$$ Combining this with (\[dM\]) and the definition of $m$, $$\begin{aligned}
\dot{m} &\simeq & -n_{a} \Psi^{a}_{\mathrm{ext}} + \int \!
\mathrm{d}^{3} r \, \mathrm{d}^{3} r' \, \frac{1}{R^{3}} \, \Bigg\{
\left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right)
H^{\gamma}(r',s) \bigg[ \Big( \dot{n}_{\gamma} (r-r')^{\alpha}
r_{\alpha} - r_{\gamma} \dot{n}^{\alpha} (r-r')_{\alpha} \Big)
\nonumber
\\
&& {} \times \Big( 1 + \frac{1}{2} \dot{n}^{\lambda}
(r+r')_{\lambda} \Big) + \frac{1}{2} R^{2} \Big( |\dot{n}|^{2}
r_{\gamma} + \dot{n}^{\beta} r_{\beta} \dot{n}^{\gamma} \Big)
\bigg] + \rho(r',s) \bigg[ R^{2} H_{\beta}(r,s) \dot{n}^{\beta}
\left( 1 + \frac{1}{2} \dot{n}^{\alpha} (3r-r')_{\alpha} \right)
\nonumber
\\
&& {} - v^{\gamma} \partial_{\gamma} \Big( r_{\beta} \varphi(r,s)
\Big) (r-r')^{\beta} \bigg] \Bigg\} + o\big( \epsilon^{2}
\mathcal{E} M/T \big) ~, \label{dotm}\end{aligned}$$ which is considerably simpler than (\[dotM\]).
Although we now have all of the results necessary to compute each tetrad component of $\Psi^{[ab]}_{\mathrm{self}}$ to second order, the resulting expressions are extremely lengthy (and correspondingly difficult to interpret). It was also seen when examining the radiative self-fields that the second-order terms in the self-torque were not necessary to reach the point particle limit. For both of these reasons, we only compute it here to first order. Combining (\[torquedefine\]) and (\[field4\]), $$\begin{aligned}
\Psi^{[\alpha \beta]}_{\mathrm{self}} & \simeq & \int \!
\mathrm{d}^{3} r \, \mathrm{d}^{3} r' \, \frac{\dot{n}^{[\alpha}
r^{\beta]}}{R} \, \Bigg[ \frac{1}{2} \bigg( \rho(r,s) \rho(r',s) +
H_{\gamma}(r,s) H^{\gamma}(r',s) \bigg) + \left( \varphi(r,s) -
\frac{q}{4\pi |r|^{3}} \right) \rho(r',s) \Bigg] \nonumber
\\
&& {} + o \big( \epsilon^{2} \mathcal{E} M \big) ~, \label{torque}
\\
n_{a} e_{b}^{\beta} \Psi^{[ab]}_{\mathrm{self}} &\simeq& \frac{1}{2}
\int \! \mathrm{d}^{3}r \, \mathrm{d}^{3} r' \, \frac{1}{R^{3}} \,
\Bigg\{ \rho(r,s) H_{\alpha}(r',s) r'^{\beta} \left[ (r-r')^{\alpha}
\Big( 1 - \frac{1}{2} \dot{n}^{\gamma} (r-r')_{\gamma} \Big) +
\frac{1}{2} R^{2} \dot{n}^{\alpha} \right] \nonumber
\\
&& {} - \left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right) \bigg[
R^{2} r_{\gamma} \dot{n}^{[\beta} H^{\gamma]}(r',s) + 2 r_{\gamma}
(r-r')^{[\beta} H^{\gamma]}(r',s) \Big( 1 + \frac{1}{2}
\dot{n}^{\gamma} (r+r')_{\gamma} \Big) \nonumber
\\
&& {} - r_{\alpha} H^{\alpha}(r',s) \Big( r^{\beta} \dot{n}^{\gamma}
r'_{\gamma} - r'^{\beta} \dot{n}^{\gamma} r_{\gamma} \Big) +
(r-r')^{\alpha} r_{\alpha} \Big( H^{\beta}(r',s) \dot{n}^{\gamma}
r_{\gamma} - r^{\beta} \dot{n}_{\gamma} H^{\gamma}(r',s) \Big)
\bigg] \Bigg\} \nonumber
\\
&& {} + o \big( \epsilon^{2} \mathcal{E} M \big) ~. \label{ntorque}\end{aligned}$$
Special Cases {#SpecCaseSect}
=============
Current-Dominated Particles
---------------------------
The full equations of motion derived above are obviously quite complicated,, so it is useful to specialize the discussion somewhat. First assume that the particle’s charge density is sufficiently small that it can be entirely dropped without losing any accuracy. This requires that $\varphi < \epsilon^{2} H$. Then (\[torque\]) becomes equivalent to $$\dot{S}^{\alpha\beta} \simeq - 2 \Psi^{[\alpha
\beta]}_{\mathrm{ext}} + \int \! \mathrm{d}^{3}r \, \mathrm{d}^{3}r'
\, \frac{r^{[\alpha} \dot{n}^{\beta]}}{R} \, H_{\gamma}(r,s)
H^{\gamma}(r',s) + o \big( \epsilon^{2} \mathcal{E} M \big) ~.
\label{SdotCurrent1}$$ This can be quite large. In order to make sure that it satisfies (\[torquerestrict\]), let $$\mathcal{E} < \epsilon ~. \label{selfenergyrestrict2}$$
Now, $M_{\mathrm{em}} \lesssim \epsilon M$, so the remaining equations of motion can be derived from (\[fullforce\]), (\[dotm\]), and (\[ntorque\]): $$\begin{aligned}
\dot{m} &\simeq& - n_{a} \Psi^{a}_{\mathrm{ext}} + o \big(
\epsilon^{3} m T^{-1} \big) ~,
\\
m \dot{n}^{\alpha} &\simeq& - \Psi^{\alpha}_{\mathrm{ext}} + o\big(
\epsilon^{3} m T^{-1} \big) ~, \label{ndotevolveCurrent1}
\\
m v^{\alpha} &\simeq& S^{\alpha \beta} \dot{n}_{\beta} - 2
\Psi^{[ab]}_{\mathrm{ext}} e_{a}^{\alpha} n_{b} + o \big(
\epsilon^{3} m \big) ~. \label{CMevolveCurrent1}\end{aligned}$$
An expression for $\dot{v}^{\alpha}$ is obtained by differentiating (\[CMevolve\]), which gives $$\begin{aligned}
m \dot{v}^{\alpha} &\simeq& - \Psi^{\alpha}_{\mathrm{ext}} + n_{a}
\Psi^{a}_{\mathrm{ext}} v^{\alpha} + S^{\alpha \beta}
\ddot{n}_{\beta} - 4 \Psi^{[\alpha \beta]}_{\mathrm{ext}}
\dot{n}_{\beta} - 2 \dot{\Psi}^{[ab]}_{\mathrm{ext}} e_{a}^{\alpha}
n_{b} \nonumber
\\
&& {} + 2 \int \! \mathrm{d}^{3} r \, \mathrm{d}^{3} r' \,
\frac{r^{[\alpha} \dot{n}^{\beta]} \dot{n}_{\beta} }{R} \,
H_{\gamma} (r,s) H^{\gamma} (r',s) + o \big( \epsilon^{3} m T^{-1}
\big) ~.\end{aligned}$$ The presence of a (potentially) significant self-torque here is completely different than the situation that arose when considering only the radiative component of the self-field. Even adopting conditions (\[lambdarestrict\]), (\[SrestrictRad\]), and (\[dipolerestrictRad\]) would not generically recover the Lorentz-Dirac equation.
Instead, we find (assuming $\mathcal{E} \gtrsim \epsilon^{2}$) that (\[vRadLD\]) and (\[dotMRadLD\]) (with $\dot{M} \rightarrow
\dot{m}$) would remain correct, although (\[SboundRadLD\]) is weakened to $|\dot{S}| < \epsilon^{2} m$. In this case, the center-of-mass line is governed by $$m \dot{v}^{\alpha} \simeq - q e_{a}^{\alpha} v_{b}
F^{ab}_{\mathrm{ext}} + \frac{2}{3} q^{2} \ddot{n}^{\alpha} + 2
M_{\mathrm{em}} D_{\mathrm{H}}^{[\alpha} \dot{v}^{\beta]}
\dot{v}_{\beta} + o \big(\epsilon^{3} m T^{-1} \big) ~,$$ where $$D^{\alpha}_{\mathrm{H}}(s) := \frac{1}{M_{\mathrm{em}}} \int \!
\mathrm{d}^{3} r \, \mathrm{d}^{3} r' \left( \frac{r^{\alpha} }{R}
\right) H_{\gamma} (r,s) H^{\gamma} (r',s) ~.$$ Given (\[dM\]), $D^{\alpha}_{\mathrm{H}}$ appears to be related to the shift in the position of the effective center-of-mass due to the electromagnetic self-energy.
The force that it generates clearly becomes irrelevant when $M_{\mathrm{em}} D^{[\alpha}_{\mathrm{H}} \dot{v}^{\beta]}
\dot{v}_{\beta} \lesssim \epsilon^{3} m/T$, which occurs when $D^{\alpha}_{\mathrm{H}}$ (nearly) coincides with $\dot{v}^{\alpha}$, or more generically if $$\left| D_{\mathrm{H}} \right| \lesssim \epsilon D ~.
\label{Hrestrict1}$$ This happens, for example, in cases where $H^{\gamma}(r,s) \simeq
\pm H^{\gamma} (-r,s)$. (\[Hrestrict1\]) can therefore be thought of as a restriction on the allowed asymmetry in the particle’s current structure about the center-of-mass. Given (\[dipole\]), a similar intuitive interpretation can also be applied to (\[dipolerestrictRad\]), so there is some overlap between these two conditions. It does not appear that either one strictly implies the other, however.
Also note that if (\[Hrestrict1\]) holds, (\[selfenergyrestrict2\]) is no longer required. The case $\mathcal{E} \sim 1$ is then quite interesting if the spin and/or dipole moment are non-negligible. When this happens, the left-hand side of (\[CMevolveCurrent1\]) needs to be replaced by $Mv^{\alpha}$, which means that $M$ and $m$ must both be kept track of. The expression for $\dot{M}$ is not simple, so this is a considerable complication.
Charge-Dominated Particles
--------------------------
Treating the opposite case, we now consider particles with very small internal currents. In particular, let $H < \epsilon^{2}
\varphi$. Given (\[HBoundary\]), this is about as small as $H$ could possibly be without fine-tuning. (\[threecurrent\]) and (\[phisize\]) show that the internal currents which arise due to charges rearranging themselves (via elasticity, rotation, etc.) are bounded by this same amount. So we actually have that $|j| \lesssim
\epsilon^{2} \varphi$.
(\[torque\]) now reduces to $$\dot{S}^{\alpha \beta} \simeq - 2 \Psi^{[\alpha
\beta]}_{\mathrm{ext}} + \int \! \mathrm{d}^{3} r \, \mathrm{d}^{3}
r' \, \frac{ r^{[\alpha} \dot{n}^{\beta]} }{R} \, \rho(r',s) \left[
\rho(r,s) + 2 \left( \varphi(r,s) - \frac{q}{4\pi |r|^{3}} \right)
\right] + o \big( \epsilon^{2} \mathcal{E} m \big) ~.$$ Once again, this expression can easily violate (\[torquerestrict\]) unless $\mathcal{E} < \epsilon$. Assuming this, $$\begin{aligned}
\dot{m} &\simeq& -n_{a} \Psi^{a}_{\mathrm{ext}} + \int \!
\mathrm{d}^{3}r \, \mathrm{d}^{3}r' \, \frac{1}{R} v^{\alpha}
\partial_{\alpha} \rho(r,s) \rho(r',s) + o \left(
\epsilon^{3} m T^{-1} \right) ~,
\\
m \dot{n}^{\alpha} &\simeq& - \Psi^{\alpha}_{\mathrm{ext}} +
\frac{2}{3} q^{2} \ddot{n}^{\alpha} + o \left( \epsilon^{3} m T^{-1}
\right) ~,
\\
m v^{\alpha} &\simeq& S^{\alpha \beta} \dot{n}_{\beta} - 2
\Psi^{[ab]}_{\mathrm{ext}} e_{a}^{\alpha} n_{b} + o\left(
\epsilon^{3} m \right) ~.\end{aligned}$$
By analogy to the current-dominated case, we can now define $$D_{\mathrm{\varphi}}^{\alpha} := \frac{1}{M_{\mathrm{em}}} \int \!
\mathrm{d}^{3} r \, \mathrm{d}^{3} r' \left( \frac{ r^{\alpha} }{R}
\right) \rho(r',s) \left[ \rho(r,s) + 2 \left( \varphi(r,s) -
\frac{q}{4\pi |r|^{3}} \right) \right] ~.$$ Again, this looks like the center-of-electromagnetic mass, although the interpretation isn’t quite so direct as it was for $D_{\mathrm{H}}^{\alpha}$.
Now let $\mathcal{E} \gtrsim \epsilon^{2}$, and adopt (\[lambdarestrict\]), (\[SrestrictRad\]), and (\[dipolerestrictRad\]). As before, these conditions imply that $|v| < \epsilon^{3}$, so $\dot{m} < \epsilon^{3} m T^{-1}$. The Lorentz-Dirac equation is recovered when $M_{\mathrm{em}}
D_{\mathrm{\varphi}}^{[\alpha} \dot{v}^{\beta]} \dot{v}_{\beta}
\lesssim \epsilon^{3} m T^{-1}$. If the acceleration is not restricted to be in a specific direction, then it is convenient to satisfy this condition by requiring $$\left| D_{\mathrm{\varphi}} \right| \lesssim \epsilon D ~.
\label{Phirestrict1}$$ This clearly holds when (for example) $\varphi(r,s) \simeq \pm
\varphi(-r,s)$, so $|D_{\mathrm{\varphi}}|/D$ can be considered a measure of the charge’s internal symmetry.
The two examples discussed here are by the far the simplest, although other ones may also be of interest. For example, the condition $|j| \lesssim \epsilon \varphi$ could have replaced $|j|
\lesssim \epsilon^2 \varphi$ in this section. This would allow for some charge-current coupling without introducing undue complexity. Similarly, one might be interested in the case where $\varphi
\lesssim \epsilon H$, or the general scenario where $\varphi$ and $H^{\alpha}$ are symmetric (or antisymmetric) about $r=0$.
Conclusions
===========
Results
-------
Starting from the fundamental equations (\[maxwell\])-(\[stresscons\]), we have now derived equations of motion describing a very wide variety of classical extended charges in flat spacetime. One of our main motivations has been to investigate the validity of the commonly-held notion that ‘small’ charges can often be treated as though they were perfectly pointlike. Because of this, the precise implications of each approximation have been emphasized, and all of our results have been kept as general as (reasonably) possible.
To review, charge distributions were considered where all significant length scales remained of order $D(s)$ within each time slice $\Sigma(s)$. An acceleration timescale $T\gg D$ was then defined to place lower bounds on $|\dot{n}|^{-1}$ and $|\ddot{n}|^{-1/2}$. This was assumed to relate to the timescales of the body’s internal motions as well. Specifically, the $s$-derivatives of the ‘charge potential’ $\varphi$ were required to satisfy (\[phisize\]) and (\[phisize2\]). Similar restrictions were placed on the derivatives of $H^{\alpha}$ as well. The last major restriction required that the spin and torque play a relatively small but non-negligible role in the system’s behavior. This was formalized by assuming that $|n-v|=|v|$ remained less than about $\epsilon^{2}=(D/T)^{2} \ll 1$. All of these assumptions are kinematic, and cannot automatically be assumed to hold without any regard for the sizes of the external fields, spin, and so on. Self-consistency is preserved by requiring the angular momentum, force, and torque obey the bounds discussed in Sec. \[Approximations\].
The restriction given on $S^{ab}$ essentially states that the body’s rotational period can’t be less than $T$. The relations satisfied by the force and torque are not quite as simple, though. Both of these quantities depend on $J^{a}$, so any bounds placed on them must affect the class of allowable current distributions. While this can be taken into account in different ways, we have chosen to impose restrictions on the dipole moment and the ‘relative self-energy’ $D_{\mathrm{em}}/D=\mathcal{E}$. If the dominant force is just $-q
F^{ab}_{\mathrm{ext}} v_{b}$, $Q^{ab}$ must satisfy (\[dipolerestrict\]).
The bound on $\mathcal{E}$ is more complicated to summarize, as it is sensitive to the specific case under consideration. The calculation using the radiative self-field is the least restrictive; allowing all $\mathcal{E} \lesssim 1$. One could technically let the electromagnetic radius exceed the physical one in this case, although we take the point of view that this would be too unphysical. Regardless, taking into account the full retarded self-field shows that there exists a significant class of charge distributions where the self-torque will become too large whenever $\mathcal{E} \gg \epsilon^{2}$. In scenarios where the charge-current coupling is negligible, the relative self-energy can be of order $\epsilon$. Still, there are certain special cases such as spherical symmetry where the restriction on $\mathcal{E}$ can be completely relaxed.
Despite appearances, these assumptions are really no different than the standard slow-motion approximation. Using it, the radiative and retarded self-fields in a neighborhood of $W$ were shown to be approximated by (\[radfield\]) and (\[field4\]) respectively. These expressions are quite general, and although the retarded field is written using the peculiar constructions of Dixon’s theory, it can easily be translated into a more conventional notation by using (\[chargedensity\]) and (\[threecurrent\]). (\[field3\])-(\[f4\]) also provide a convenient starting point for this.
These expressions for the fields were then combined with the exact forms of Dixon’s equations of motion to yield the results contained in Sec. \[ForceSect\]. All of the approximations adopted here were therefore applied only to compute the fields. In the (unrealistic) case that the body was assumed to couple only to the radiative component of its self-field, the center-of-mass line was found to evolve according to (\[dotmrad1\])-(\[forcerad2\]). The external forces and torques that appear in these expressions are given exactly by (\[forcedefine\]) and (\[torquedefine\]), or more intuitively by (\[forcemult\]) and (\[torquemult\]).
In order to investigate the Lorentz-Dirac limit, it was first necessary to require the particle’s radius to be sufficiently small that the dipole and higher contributions to the external force could be considered negligible without having to throw away the Lorentz-Dirac component of the self-force. This idea was summarized in the condition (\[lambdarestrict\]), which transformed the equations of motion in the radiative case into (\[dotmrad2\])-(\[vrad1\]). Recovering the Lorentz-Dirac equation was then seen to require bounding the angular momentum and dipole moment by (\[SrestrictRad\]) and (\[dipolerestrictRad\]) respectively.
Since this limit is meant to describe a non-spinning particle, the bound on the angular momentum is hardly surprising. (\[dipolerestrictRad\]) is rather different, though. It shows that even when ignoring the ‘singular’ portion of the self-field, the Lorentz-Dirac equation does not apply to all small charge distributions. If a problem were to involve very large (external) field strengths and small self-energies, the class of charges which move like a point particle would in fact become extremely small. Oddly enough, this problem disappears for large self-energies. In these cases, the restriction on the dipole moment is no stronger than was required for self-consistency of the initial slow-motion assumptions.
This issue is made considerably more complex when the body is allowed to interact with its full retarded self-field. Without placing any restrictions on the spin or dipole moment, $\dot{n}^{\alpha}$ was found to be given by (\[fullforce\]). This involves an effective mass $m=M+M_{\mathrm{em}}$, which involves an electromagnetic contribution given by (\[dM\]). It does not generally remain constant, but rather evolves according to (\[dotm\]). There are also self-torques given by (\[torque\]) and (\[ntorque\]). Combining these expressions with (\[tdefine\]) and (\[CMevolve\]) recovers the full equations of motion. These are our central result. They provide a considerable generalization of the Lorentz-Dirac equation.
They are also much more complicated than the equations derived using only the radiative portion of the self-field. The majority of this complexity arises from interactions between $\varphi$ and $H^{\alpha}$, which – given (\[chargedensity\]) and (\[threecurrent\]) – can be viewed as charge-current couplings in the center-of-mass frame. Predictably then, our results are considerably simplified when only one of these quantities is significant. $\varphi$ will (generically) drop out of the equations of motion only if it is of order $\epsilon^{2} H$ or less. The reverse is also true. Both of these cases were discussed in detail in Sec. \[SpecCaseSect\]. It should be emphasized that these restrictions on the relative magnitudes of the charge and current densities are exceptionally strong. Simply saying that one of these quantities is much larger than the other without any further qualification is not sufficient to remove the coupling terms.
Even in these cases, though, the previous restrictions on $\lambda$, $S^{ab}$, and $Q^{ab}$ must be supplemented by either (\[Hrestrict1\]) or (\[Phirestrict1\]) in order to recover the Lorentz-Dirac equation. Intuitively, these relations ensure that the ‘center-of-electromagnetic mass’ is not too far from $r=0$. This is roughly what was already required by (\[dipolerestrictRad\]), although the two conditions are not quite the same.
Note that the assumptions we have given to be able to derive the Lorentz-Dirac equation are not exhaustive. They were the most obvious choices obtained by examining our general equations of motion, but are slightly more restrictive than necessary. Still, they seem to be reasonably effective for most systems that are not finely-tuned.
Discussion
----------
In summary, it was shown that the Lorentz-Dirac limit is rather delicate, and that the radiative self-field is rarely an adequate replacement for the retarded one. It is now interesting to speculate how these results might generalize in curved spacetime. One might be interested in the motion of a charged particle moving in a background spacetime, an uncharged body allowed to generate its own gravitational field, or even the general case of a massive charge. With the exception of this last possibility (which doesn’t necessarily follow from the two simpler calculations), such systems have been considered in the past by a number of authors [@NodvikOrig; @MST; @DewittBrehme; @Hobbs; @SFReview1; @QuinnWald; @PN1]. All of these calculations have either applied a point particle ansatz, or considered only very special (though usually astrophysically motivated) classes of extended bodies. Both such methods have agreed with each other, although it is clear that *some* extended bodies must exist which do not move like point particles.
Finding the scope of the existing MiSaTaQuWa equation [@SFReview1] – as well as its generalization – would likely require an analysis similar to the one given here. The methods used in this paper have in fact been chosen specifically for their ability to be easily applied to fully dynamic spacetimes (except for those used to expand the self-field). To see this, it should first be mentioned that all of the advantages of Dixon’s formalism are retained without approximation in full general relativity [@Dix74; @Dix79].
This provides two essential results. The first of these is a natural notion of a center-of-mass line. This definition has been proven to satisfy nearly all of the intuitive notions that one might expect of such an object [@Schatt1], so it can be considered a reasonable measure of a particle’s ‘average’ position. Just as importantly, it is exactly determined by a finite number of ordinary differential equations [@Dix79; @Ehl77]. The analogous equations here were (\[massevolve\])-(\[CMevolve\]), and (\[tdefine\]). These expressions are only slightly complicated by the transition to curved spacetime.
The second contribution of Dixon’s formalism is the set of stress-energy moments itself. These are all contained in the ‘stress-energy skeleton’ $\hat{T}^{ab}(r,s)$. Here $r$ is a coordinate in the tangent space of $z(s)$, which is an essential point that was not obvious in flat spacetime. Importantly, the constraints on $\hat{T}^{ab}$ implied by the generalized form of (\[stresscons\]) are exactly the same in all spacetimes. This obviously includes the flat case summarized in the appendix. Just as (\[hatJ\]) gave the general form for any current skeleton $\hat{J}^{a}$ satisfying the constraints, one can also find all possible forms of $\hat{T}^{ab}$ [@Dix74].
As with current skeleton, not all of these possibilities are physically reasonable. Some will be singular, and others will have supports extending to spatial infinity. We conjecture that such cases can be removed exactly as they were in the appendix for the current moments. This would leave a set of ‘reduced moment potentials’ that could be arbitrarily generated from some simple recipe. An automatic byproduct of this reduction process would be a relatively straightforward method of generating $T^{ab}$ from the potentials. The analogous results in this paper were that the current skeleton was determined by the freely-specifiable functions $\varphi$ and $\bar{H}^{\alpha \beta}$. These in turn generated $J^{a}$ via (\[chargedensity\]), (\[threecurrent\]), and (\[divH\]).
In the end, such reduced potentials for the stress-energy moments would serve (in combination with solutions of the ODE’s for the linear and angular momenta) as a nearly background-independent way of specifying physically reasonable conserved stress-energy tensors. Besides the intrinsic elegance of such a construction, it would also solve several problems that did not arise in the present paper. The most important of these is the location of the center-of-mass. At any $s$, this is clearly determined by $T^{ab}$ on $\Sigma(s)$. The definition is highly implicit, however, and it is almost always impractical to apply in practice. Specifying the matter via $\hat{T}^{ab}$ would avoid this problem. Constructing it in the natural way would *start* with the center-of-mass position. It would also incorporate $p^{a}$ and $S^{ab}$ automatically. These quantities obviously must solve certain differential equations, and writing down a $T^{ab}$ on each time slice with the correct momenta would be extremely difficult with any generality. These issues were the main motivations for discussing the current moments in so much detail in the appendix, as well as the use of $\varphi$ and $H^{\alpha}$ throughout this paper instead of $J^{a}$. These choices slightly obscure the results here, but would be essential in any generalization.
With this formalism in place, one would then need to compute the metric to find the body’s motion. As in the electromagnetic case, this is the most difficult step. Of course, Einstein’s equation is considerably more complicated than Maxwell’s, so it may be prohibitive to carry out the calculation by hand (except in special cases). Regardless, the above procedure could be integrated into existing numerical relativity codes. This would then allow one to rigorously study spacetimes with nonsingular matter fields without having to solve the conventional elasticity (or Navier-Stokes) equations. The most convenient such systems in this formalism may not represent the most astrophysically interesting types of matter, although there is no shortage of important problems in numerical relativity where the details of the matter distributions are not a primary concern.
Whatever the results of such inquiries, our results in electromagnetism can be used to speculate how point particle methods might break down in gravitational self-force problems. For this, it is useful to give an intuitive explanation for such failures in Maxwell’s theory. The simplest of these derives from the fact that the center-of-mass does not generally correspond to anything that could be called a ‘center-of-charge.’ In these cases, electromagnetic self-forces effectively act through a lever arm. This induces a torque, which in turn affects the particle’s overall motion. The situation is made particularly complicated when the self-force is broken up into several pieces (as is natural). Each such piece tends to act through a different point, and there’s no particular reason that any of them should coincide with the center-of-mass (although the initial restriction (\[dipolerestrict\]) on the magnitude of the dipole moment does help).
In contrast, one might expect that gravitational self-forces would always act through the center-of-mass; effectively removing this type of effect. This isn’t necessarily true, though. The center-of-mass definition only requires that $t^{abc}n_{b}n_{c}=0$, which does not necessarily mean that $t^{abc}$ completely vanishes. The portion of the self-force acting as an effective mass could also produce a significant torque. (\[stressdipole\]) shows, however, that $t^{abc}$ does vanish whenever $S^{ab} \rightarrow 0$. This is the only case in which the MiSaTaQuWa equation can be reasonably expected to hold, so imposing it at the start should remove any problems. Since the dipole moment here is entirely dependent on $S^{ab}$ (unlike in the electromagnetic case) it should be relatively easy to arrange for an initially-vanishing angular momentum to remain small. These sorts of effects might therefore be expected to be less troublesome in gravity than in electromagnetism, although they will probably still exist.
This type of mechanism did not lead to most of the complications found in this paper, however. These were instead due to couplings between the charge and 3-current densities as viewed in the center-of-mass frame. In gravity, the situation could be even worse. Similar interactions might exist between mass, 3-momentum, and stress densities. Intuitively, though, it would appear that adopting appropriate energy conditions should considerably soften these interactions.
Dixon’s Formalism {#Dixon}
=================
Current Multipoles {#CurrentMoments}
------------------
This portion of the appendix reviews Dixon’s decomposition of the electromagnetic current vector into multipole moments. It also derives natural ‘potential functions’ that can be used to generate sets of moments for all physically interesting current vectors satisfying (\[chargecons\]). These objects are shown to determine $J^{a}$ in a simple way. The main reasons for these constructions are explained in Sec. \[discussion\], although some secondary points are also mentioned in Sec. \[LawsMot\]. In short, analogs of these steps would become essential in the gravitational self-force problem, so we include them here to allow a relatively straightforward generalization.
The notation here will be that defined in Sec. \[LawsMot\]. Using it, we can define multipole moments for the current vector. Such objects are usually constructed by integrating a source function against a suitable number of radius vectors. Defining $r^{a}=x^{a}-z^{a}(s)$ for $x \in \Sigma(s)$, one might therefore expect that for $n \geq 0$, the $2n$-pole moment could be given by $$Q^{b_{1} \cdots b_{n} a}(s) := \int \! \mathrm{d}^{4}x \, r^{b_{1}}
\cdots r^{b_{n}} \hat{J}^{a}\big(x-z(s),s\big) ~, \label{QJ}$$ where we have assumed the existence of some distribution $\hat{J}^{a}(x,s)$ which is related to $J^{a}(x)$, but has compact support in $x$. This will be called the current skeleton. Note that (\[QJ\]) automatically implies that $$Q^{b_{1} \cdots b_{n} a}=Q^{(b_{1} \cdots b_{n}) a}
\label{chargesym2}$$ for all $n \geq 1$.
It is rather cumbersome to keep track of each $Q^{\cdots}$ directly, so we instead define a generating function $$G^{a}(k,s) := Q^{a}(s) + \sum_{n=1}^{\infty} \frac{(-i)^{n}}{n!}
k_{b_{1}} \cdots k_{b_{n}} Q^{b_{1} \cdots b_{n} a}(s) ~.
\label{chargegenerate}$$ An arbitrary multipole moment can now be extracted from this in the usual way: $$Q^{b_{1} \cdots b_{n} a}(s) =i^{n} \left. \Big( \partial^{b_{1}}
\cdots \partial^{b_{n}} G^{a}(k,s) \Big) \right|_{k=0} ~.
\label{momentderive}$$ $G^{a}$ is therefore completely equivalent to the set $\{ Q^{a} ,
Q^{ba} , \ldots \}$.
Although this is a useful property, the definition of $G^{a}$ is not simply a mathematical convenience. Using (\[QJ\]) and (\[chargegenerate\]), it can be shown that $$G^{a}(k,s) = \int \! \mathrm{d}^{4} r \, \hat{J}^{a}(r,s) e^{-i k
\cdot r} ~.$$ The Fourier transform of $G^{a}$, $$\widetilde{G}^{a}(r,s) := \int \! \mathrm{d}^{4}k \, G^{a}(k,s) e^{i
k \cdot r} ~,$$ is therefore proportional to $\hat{J}^{a}$: $$\widetilde{G}^{a}(r,s) = (2 \pi)^{4} \hat{J}^{a}\left(r,s\right) ~.
\label{Jhatdefine}$$ This shows that if $J^{a}$ and $\hat{J}^{a}$ equivalent in an appropriate sense, the set of moments can be used to completely reconstruct the current vector.
In order to relate the current to its skeleton, it is convenient to think of $\hat{J}^{a}$ as a linear functional on the space of all $C^{\infty}$ test functions with compact support (as is typically done in distribution theory). In particular, knowing $$\left\langle \hat{J}^{a}(r,s), \phi_{a}(x) \right\rangle := \int \!
\mathrm{d}^{4} x \, \hat{J}^{a}(r,s) \phi_{a}(x)$$ for all suitable test functions $\phi_{a}(x)$ can be used to define $\hat{J}^{a}$. An analogous statement can also be made for $J^{a}$. These two objects can then be related to each other by writing $\langle J^{a},\phi_{a} \rangle$ in terms of $\langle \hat{J}^{a},
\phi_{a} \rangle$. The latter expression depends on $s$, while the former one does not. It is therefore most straightforward to link the two by simply integrating out the $s$-dependence: $$\begin{aligned}
\Big\langle J^{a}(x), \phi_{a}(x) \Big\rangle &=& \int \!
\mathrm{d}s \, \left\langle \hat{J}^{a}(r,s), \phi_{a}(x)
\right\rangle ~, \label{JG1}
\\
&=& \frac{1}{(2\pi)^{4}} \int \! \mathrm{d}s \, \left\langle
G^{a}(k,s), \widetilde{\phi}_{a}(k) e^{-i k \cdot z(s)}
\right\rangle ~. \label{momentcurrent}\end{aligned}$$ Following [@Dix70b], we take this (along with (\[chargegenerate\])) to *define* what is meant by saying that that the $Q^{\cdots}$’s are ‘multipole moments of $J^{a}$.’
This is not unique definition, however. To remove the remaining freedom in a useful way, we simply state Dixon’s results [@Dix67; @Dix70b; @Dix74]. Let $$\begin{aligned}
n_{b_{1}} Q^{b_{1} \cdots b_{n-1} [b_{n} a]} &=& 0 ~,
\label{chargesym1}
\\
Q^{(b_{1} \cdots b_{n})} &=& 0 ~, \label{chargesym3}\end{aligned}$$ for all $n \geq 2$. Also assume that the monopole moment has the special form $$Q^{a} = q v^{a} ~, \label{chargemono}$$ where $q$ is the total charge as it is usually defined.
Now choose a test function of the form $\phi_{a}(x) = \partial_{a}
\phi(x)$, with $\phi(x)$ itself also a test function. Then (\[momentderive\]), (\[momentcurrent\]), and (\[chargesym3\]) can be used to show that $$\begin{aligned}
\big\langle \partial_{a}J^{a} , \phi \big\rangle &=& -\big\langle
J^{a} , \partial_{a} \phi \big\rangle ~, \nonumber
\\
&=& \frac{i}{(2\pi)^{4}} \int \! \mathrm{d}s \, \left\langle k_{a}
G^{a}(k,s) , \tilde{\phi}(k) e^{-i k \cdot z(s)} \right\rangle ~,
\nonumber
\\
&=& - \int \! \mathrm{d}s \, q v^{a}(s)
\partial_{a}\phi\big(z(s)\big) ~, \nonumber
\\
&=& - \int \! \mathrm{d}s \, q \frac{\mathrm{d}}{\mathrm{d}s}
\phi\big(z(s)\big)~. \label{ChargeConsCheck}\end{aligned}$$ This must vanish for all $\phi$, which can be ensured by simply requiring that $$\dot{q} = 0 ~. \label{qdot}$$ This (trivial) evolution equation is the only one implied by (\[chargecons\]). It can be shown that moments satisfying (\[chargesym2\]), (\[chargesym1\])-(\[chargemono\]), and (\[qdot\]) describe any $J^{a}$ with the given properties in a uniquely simple way [@Dix67; @Dix70b; @Dix74]. A precise statement of the theorem that was proven is contained in [@Dix74].
These conditions allow a great deal of freedom in choosing different moments. The fact that there is only one evolution equation implied by (\[chargecons\]) does not mean that the higher moments necessarily remain constant. Rather, they can be given an almost arbitrary time dependence. This is a reflection of the fact that we have not yet chosen to model any particular type of matter. Specifying how the higher moments change in time is essentially equivalent to choosing an equation of state. Although this must be given in order to have a well-defined initial value problem, self-forces and self-torques can be written down without ever having to explicitly evaluate the time derivatives of the current moments. This allows us to derive equations of motion valid for a very large class of systems.
In order to do so, we first need to pick out sets of moments (or equivalently their generating functions) which represent physically reasonable current vectors. Despite appearances, this cannot be done arbitrarily. Dixon’s theorem ensures that any nonsingular current vector with support $W$ can be described by a set of moments with the given properties, although the reverse is not necessarily true. Extra conditions need to be imposed in order to ensure that the $J^{a}$ associated with any particular $G^{a}$ (or $\hat{J}^{a}$) has the correct smoothness and support properties.
To gain some insight into this, fix some test function $\phi_{a}$. From this, construct a second test function $\phi'_{a}$ which agrees with $\phi_{a}$ everywhere except in an infinitesimal neighborhood of $Z$. Assume that the support of $\phi'_{a}$ does not include $Z$ itself. (\[chargegenerate\]) and (\[momentcurrent\]) then show that any finite number (and only finite number) of multipoles can be changed without affecting $\left\langle J^{a},\phi'_{a}
\right\rangle$. The same cannot be said for $\left\langle
J^{a},\phi_{a} \right\rangle$. But for any physical current vector, $\left\langle J^{a}, \phi'_{a} \right\rangle \simeq \left\langle
J^{a},\phi_{a} \right\rangle$. This shows that any finite subset of an admissible collection of moments is completely determined by its complement.
Because of this, it is not reasonable to impose conditions directly on the individual moments to ensure that their associated current vector is physically acceptable. Such restrictions are most easily stated in terms of $\widetilde{G}^{a}$ or $\hat{J}^{a}$, but doing so first requires finding how $\hat{J}^{a}$ is affected by (\[chargesym1\])-(\[chargemono\]), and (\[qdot\]). This is now done by finding the general form of $G^{a}$, and then taking its Fourier transform.
It is shown in [@Dix74] that the constraint equations are precisely equivalent to requiring that the generating function have the form $$G^{a}(k,s)= G_{(1)}^{a}(h \cdot k,s) + (n \cdot k) G_{(2)}^{a}(h
\cdot k,s) ~, \label{G1G2}$$ where $n \cdot k := n_{a} k^{a}$, $(h \cdot k)^{a}=h^{a}_{b} k^{b}$, and $n_{a} G^{a}_{(2)}=0$. Also, $$\begin{aligned}
k_{a}G^{a}(k,s)&=&q k_{a}v^{a}(s) ~, \label{Gtimesk}
\\
\partial^{[a} G^{b]}_{(2)}(h \cdot k,s) &=& 0 ~.
\label{dG2}\end{aligned}$$
It is now useful to take Fourier transforms of these equations to find their equivalent forms when representing the moments by $\hat{J}^{a}$ ($= \widetilde{G}^{a}/(2\pi)^{4}$). $\widetilde{G}^{a}_{(1)}$ doesn’t take on any special form, although (\[dG2\]) shows that $$\left\langle r^{c} h_{c}^{[a} \widetilde{G}^{b]}_{(2)}(r,s), \phi(r)
\right\rangle =0 ~. \label{G2constrain}$$ This equation is solved by any $\widetilde{G}^{a}_{(2)}$ of the form $$\widetilde{G}^{a}_{(2)}(r,s) = h^{a}_{b}(s) r^{b}
\widetilde{G}_{(2)}(r,s) ~, \label{G2form}$$ for all functions $\widetilde{G}_{(2)}$ (Despite the notation, it will shortly be clear that the inverse Fourier transform of $\widetilde{G}_{(2)}$ does not exist in general.). Note that this is not the most general solution of (\[G2constrain\]). A term of the form $g^{a}(n \cdot r, s) \delta^{3} (h \cdot r)$ may also be added to $\widetilde{G}^{a}_{(2)}$, although we choose to ignore this possibility.
In any case, (\[G1G2\]) and (\[G2form\]) give an explicit form for $\widetilde{G}^{a}$ $$\begin{aligned}
\widetilde{G}^{a}(k,s) &=& \widetilde{G}^{a}_{(1)} -i
h^{a}_{b}r^{b} n^{c} \partial_{c} \widetilde{G}_{(2)} ~.
\label{G2fourier}\end{aligned}$$
Since $G_{(1)}^{a}$ and $G_{(2)}^{a}$ are independent of $n \cdot
k$, their (four-dimensional) Fourier transforms must be proportional to $\delta(n \cdot r)$. It is therefore natural to define quantities $A$, $B^{a}$, and $C$ such that $$\begin{aligned}
\widetilde{G}^{a}_{(1)}(r,s) &=& (2 \pi)^{4} \delta(n \cdot r) \Big(
A(h \cdot r,s) n^{a} + B^{a}(h \cdot r,s) \Big) ~,
\label{rhojdefine}
\\
\widetilde{G}_{(2)}(r,s) &=& -i(2 \pi)^{4} \delta(n \cdot r) C(h
\cdot r,s) ~, \label{Adefine}\end{aligned}$$ where $B^{a}n_{a}=0$. Combining these expressions with (\[G2fourier\]), $\hat{J}^{a}$ is found to have the form $$\hat{J}^{a}(r,s) = \delta(n \cdot r) \Big( A(h \cdot r,s) n^{a}(s) +
B^{a}(h \cdot r,s) \Big) - \delta'(n \cdot r) C(h \cdot r,s)
h^{a}_{b}(s) r^{b} ~. \label{hatJ}$$
This is further restricted by (\[Gtimesk\]), the Fourier transform of which becomes $$\begin{aligned}
\left\langle \partial_{a} \hat{J}^{a}(x,s) , \phi(x) \right\rangle
&=& (2\pi)^{-4} i \left\langle k_{a} G^{a}(k,s) ,
\widetilde{\phi}(k) \right\rangle ~, \nonumber
\\
&=& - (2\pi)^{-4} q v^{a}(s) \left\langle \widetilde{1} ,
\partial_{a} \phi(r) \right\rangle ~, \nonumber
\\
&=& q v^{a}(s) \Big\langle \partial_{a} \delta^{4}(r) , \phi(r)
\Big\rangle ~. \label{divG}\end{aligned}$$ Comparing this to the divergence of (\[hatJ\]) shows that $$\begin{aligned}
q \delta^{3}(h \cdot r) &=& A(h \cdot r,s) - h^{a}_{b} \partial_{a}
\Big( r^{b} C(h \cdot r,s) \Big) ~, \label{Gconstrain1}
\\
q v^{a} \partial_{a} \delta^{3}(h \cdot r) &=& \partial_{a}B^{a} (h
\cdot r,s) ~. \label{Gconstrain2}\end{aligned}$$
The solution to the ‘homogeneous’ analog of (\[Gconstrain1\]), $$h^{a}_{b} \partial_{a} \Big( r^{b} C_{(0)} (h \cdot r,s) \Big) = - q
\delta^{3}(h \cdot r) ~,$$ is $$C_{(0)}(h \cdot x,s)= - \frac{q}{4 \pi |r|^{3}}~, \label{Csing}$$ where $|r|^{2}:=-h_{ab} r^{a} r^{b} \geq 0$. For future convenience, we now define new functions $\varphi(r,s)$ and $q_{0}(s)$ such that, $$C= N \left( \varphi - \frac{q_{0}(s)}{4\pi |r|^{3}} \right) ~.
\label{Cform}$$ $N$ is just the lapse, as given in (\[lapse\]).
In terms of $q_{0}$ and $\varphi$, (\[Gconstrain1\]) now becomes $$A(h \cdot r,s) = h^{a}_{b} \partial_{a} \Big( r^{b} N \varphi(h
\cdot r, s) \Big) +\Big(q-q_{0}(s)\Big) \delta^{3}(h \cdot r) +
\frac{q_{0}}{4\pi} \frac{\dot{n}^{a} r_{a}}{|r|^{3}} ~. \label{rhoA}$$ $\hat{J}^{a}$ is therefore given by (\[hatJ\]) with $A$ and $C$ having the respective forms (\[rhoA\]) and (\[Cform\]). We also have that $B^{a}$ satisfies (\[Gconstrain2\]). This effectively parameterizes all sets of moments satisfying Dixon’s constraints.
(\[JG1\]) can now be used to relate $\hat{J}^{a}$ to $J^{a}$. It is convenient to do this in the $(r^{\alpha},\tau)$ coordinates defined in Sec. \[LawsMot\]. Using them, (\[JG1\]) becomes $$\left\langle J^{a}, \phi_{a} \right\rangle = \int \! \mathrm{d}s \!
\int \! \mathrm{d}^{3}r \! \int \! \mathrm{d}\tau \, N(x)
\hat{J}^{a}\big(x-z(s),s\big) \phi_{a}(x) ~. \label{JG2}$$ Applying (\[hatJ\]), the $\tau$-integral in this equation can be carried out explicitly: $$\left\langle J^{a}, \phi_{a} \right\rangle = \int \! \mathrm{d}s \!
\int \! \mathrm{d}^{3}r \: \Bigg\{ \bigg[ A n^{a} + B^{a} + N^{-2}
e^{a}_{\alpha} r^{\alpha} \dot{n}_{\beta} v^{\beta} C + N^{-1}
e^{a}_{\alpha} v^{\beta} \partial_{\beta} \big( r^{\alpha} C \big)
\bigg] \phi_{a} + N^{-1} C e^{a}_{\alpha} r^{\alpha} \frac{\partial
\phi_{a}}{\partial s} \Bigg\} ~, \label{JG3}$$ where all quantities are evaluated at $s$.
Commuting the $s$-integral with the spatial ones and integrating the last term by parts, $$\begin{aligned}
\left\langle J^{a}, \phi_{a} \right\rangle &=& \int \!
\mathrm{d}^{3}r \! \int \! \mathrm{d}s \: \Bigg\{ n^{a} \left[A +
\frac{C}{N} \dot{n}_{\beta} r^{\beta} \right] + e^{a}_{\alpha}
\left[ B^{\alpha}+ v^{\beta} \partial_{\beta} \left( r^{\alpha}
\frac{C}{N} \right) - r^{\alpha} \frac{\partial}{\partial s} \left(
\frac{C}{N} \right) \right] \Bigg\} \phi_{a} ~. \label{JG4}\end{aligned}$$ But we also have that $\left\langle J^{a}, \phi_{a} \right\rangle =
\int \! \mathrm{d}^{3}r \! \int \! \mathrm{d}s \, N J^{a} \phi_{a}$, which gives us an obvious way to explicitly write $J^{a}$ in terms of $A$, $B^{a}$, and $C$.
The charge density with respect to the tetrad frame takes on the particularly simple form $$\begin{aligned}
\rho &:=& n_{a}J^{a} ~,
\\
&=& \partial_{\alpha}\big( r^{\alpha} \varphi \big) + N^{-1}
(q-q_{0}) \delta^{3}(r) ~. \label{chargedensity2}\end{aligned}$$ Physically, $\rho(r^{\alpha},s)$ must be nonsingular and have support $W$. Without loss of generality, we can therefore choose $q_{0}=q$. $\rho$ will then be admissible if $\varphi(r^{\alpha},s)$ is a continuous function.
$\hat{J}^{a}$ was originally introduced as something with compact support (in $r^{a}$), so $A$, $B^{a}$, and $C$ must also have compact support (in $r^{\alpha}$). Using this along with the requirement that $\rho$ vanish outside $W$ implies that $\varphi =
q/4\pi |r|^{3}$ in this region. So any choice of $\varphi$ which satisfies this ‘boundary condition’ and is continuous will generate a physically reasonable charge density.
For the 3-current $j^{\alpha} := e^{\alpha}_{a} J^{a}$, the (necessarily regular) portions contributed by $\varphi$ can be temporarily ignored to show that $$j^{\alpha} = N^{-1} \left[ B^{\alpha} + \big(\ldots\big) - v^{\beta}
\partial_{\beta} \left( \frac{q r^{\alpha}}{4\pi |r|^{3}} \right)
\right] ~.$$ The divergence of this last term is equal to $-q v^{\alpha}
\partial_{\alpha} \delta^{3}(r)$ , which exactly cancels the divergence of $B^{\alpha}$ given in (\[Gconstrain2\]). It is therefore useful to write $B^{\alpha}$ in the form $$B^{\alpha} = H^{\alpha} + v^{\beta} \partial_{\beta} \left( \frac{q
r^{\alpha}}{ 4\pi |r|^{3}} \right) ~, \label{Bform}$$ with $H^{\alpha}$ an arbitrary piecewise continuous vector field satisfying $\partial_{\alpha} H^{\alpha} =0$. This ensures that $j^{\alpha}$ is nonsingular.
Writing out the current in full, $$j^{\alpha} = N^{-1} \Big( H^{\alpha} + v^{\beta} \partial_{\beta}
\big( r^{\alpha} \varphi \big) - r^{\alpha} \dot{\varphi} \Big) ~,
\label{threecurrent2}$$ it is clear that $j^{\alpha}$ will have support $W$ if $$H^{\alpha}= -v^{\beta} \partial_{\beta} \left( \frac{q r^{\alpha}
}{4 \pi |r|^{3} } \right) \label{HBoundary}$$ outside $W$. This also guarantees that supp$(B^{\alpha})=W$, as required.
This completes our study of the current moments. Essentially all physically interesting expansions can be extracted from a $\hat{J}^{a}$ of the form (\[hatJ\]). $A$, $B^{a}$ and $C$ are all functions of $(r^{\alpha},s)$, and have support $W$. $C$ has the form (\[Cform\]), where $\varphi$ is an arbitrary continuous function equalling $q/4\pi |r|^{3}$ outside of $W$. (\[rhoA\]) shows that $A$ is also derived from $\varphi$ (with $q_{0}=q$). $B^{a}$ has the form (\[Bform\]), where $H^{\alpha}$ is a piecewise continuous (3-) vector field satisfying $\partial_{\alpha}
H^{\alpha} =0$. The relevant portions of these results are also summarized in Sec. \[LawsMot\].
Stress-Energy Moments
---------------------
Let the multipole moments of $T^{ab}$ be denoted by the set $\{
t^{bc}, t^{a b c}, \ldots, t^{a_{1} \cdots a_{n} bc}, \ldots \}$, where $t^{(a_{1} \cdots a_{n}) bc} = t^{a_{1} \cdots a_{n} (bc) } =
t^{a_{1} \cdots a_{n} bc}$. As with the current moments, it is convenient to keep track of this collection with a generating function $$G^{ab}(k,s) := \sum_{n=0}^{\infty} \frac{(-i)^{n}}{n!} k_{c_{1}}
\cdots k_{c_{n}} t^{c_{1} \cdots c_{n} ab}(s) ~.
\label{stressgenerate}$$
(\[stresscons\]) inextricably links $T^{ab}$ and $J^{a}$, so an expression for $\langle T^{ab}, \phi_{ab} \rangle$ as simple as the one for $\langle J^{a}, \phi_{a} \rangle$ is not possible while retaining simple constraint and evolution equations. Still, one might expect that $\langle T^{ab}, \phi_{ab} \rangle$ should at least be proportional to $\int \! \mathrm{d}s \, \langle
\hat{T}^{ab},\phi_{ab} \rangle$ (where $\hat{T}^{ab} := (2\pi)^{4}
\widetilde{G}^{ab}$). Define a distribution $\widetilde{\Phi}^{ab}=\widetilde{\Phi}^{(ab)}$ to make up the difference: $$\left\langle T^{ab}, \phi_{ab} \right\rangle = \int \! \mathrm{d}s
\, \left\langle \hat{T}^{ab}(r,s) + \widetilde{\Phi}^{ab}(r,s),
\phi_{ab}(x) \right\rangle ~. \label{momentstress}$$
As before, we call all sets $\{t^{\cdots}\}$ satisfying (\[stressgenerate\]) and (\[momentstress\]) ‘multipole moments of $T^{ab}$.’ This is not a unique definition, however (even if $\widetilde{\Phi}^{ab}$ were given). Following [@Dix67; @Dix74], we will now pick out a set which very simply and naturally implies (\[stresscons\]).
Using (\[ChargeConsCheck\]) as a guide, $\left\langle
\partial_{a}T^{ab}, \phi_{b} \right\rangle$ can be found by substituting a test function of the form $\phi_{ab} = \partial_{a}
\phi_{b}$ into (\[momentstress\]). It is clear that the resulting expression depends on $k_{a}G^{ab}$, which is analogous to what happened when computing $\partial_{a} J^{a}$. In that case, (\[chargesym3\]) showed that $k_{a} G^{a}$ depended only on $Q^{a}$. (\[chargecons\]) is usually interpreted as an expression of global charge conservation, so it was not unreasonable that it only restricted the monopole moment. In the case of the stress-energy tensor, we expect that (\[stresscons\]) should have something to say about both the linear and angular momenta of the body (i.e. its monopole and dipole moments). We therefore suppose that $k_{a}G^{ab}$ involves only $t^{ab}$ and $t^{abc}$. This can be accomplished by letting $$t^{(a_{1} \cdots a_{n} b)c} = 0 \label{StressCon1}$$ for all $n \geq 2$.
Using this constraint implies that $$\begin{aligned}
\left\langle \partial_{b}T^{ab}, \phi_{a} \right\rangle & \propto &
\frac{1}{(2\pi)^{4}} \int \! \mathrm{d}s \, \bigg\langle i k_{b}
G^{ab}(k,s), \widetilde{\phi}_{a}(k) e^{-i k \cdot z(s)}
\bigg\rangle ~,
\\
&=& \int \! \mathrm{d}s \, \bigg[ t^{ab} \partial_{a} \phi_{b} \big(
z(s) \big) + t^{abc} \partial_{a} \partial_{b} \phi_{c} \big( z(s)
\big) \bigg] ~.\end{aligned}$$
Dixon found that the first two moments can be given the special forms [@Dix67; @Dix74] $$\begin{aligned}
t^{ab} &=& p^{(a} v^{b)} ~,
\\
t^{abc} &=& S^{a(b} v^{c)} ~, \label{stressdipole}\end{aligned}$$ where we call $p^{a}$ the linear momentum, and $S^{ab}= S^{[ab]}$ the angular momentum. Using these expressions, $$\left\langle \partial_{b}T^{ab}, \phi_{a} \right\rangle \propto \int
\! \mathrm{d}s \, \left[ \dot{p}^{a} \phi_{a} \big(z(s)\big) +
\frac{1}{2} \left( \dot{S}^{ab} - 2 p^{[a} v^{b]} \right)
\partial_{a} \phi_{b} \big(z(s)\big) \right] . \label{hatTevolve}$$ In the absence of an electromagnetic field (or a current), the left-hand side of this equation vanishes, and the proportionality sign becomes an equality (since $\widetilde{\Phi}^{ab}$ vanishes in this case). Varying $\phi_{a}$ then recovers the standard equations of motion for a free particle: $$\begin{aligned}
\dot{p}^{a} &=& 0 ~,
\\
\dot{S}^{ab} &=& 2 p^{[a} v^{b]} ~.\end{aligned}$$
The situation is of course much more complicated when a field is present. Writing out $\left\langle \partial_{a} T^{ab},\phi_{b}
\right\rangle$ in full, (\[stresscons\]) implies that $$\int \! \mathrm{d}s \: \Bigg\{ \left[ \dot{p}^{a} \phi_{a} +
\frac{1}{2} \left( \dot{S}^{ab} - 2 p^{[a} v^{b]} \right)
\partial_{a} \phi_{b} \right] + \bigg\langle i k_{b} \Phi^{ab} ,
\widetilde{\phi}_{a} e^{-i k \cdot z(s)}\bigg\rangle\Bigg\} = -
\left\langle F^{ab}J_{b}, \phi_{a} \right\rangle ~. \label{stress1}$$ The right hand side of this equation can now be written in terms of the current moments. Using (\[momentcurrent\]), the Fourier convolution theorem, and a Taylor series, it can be shown to be [@Dix67] $$\begin{aligned}
\left\langle F^{ab} J_{b}, \phi_{a} \right\rangle &=&
\frac{1}{(2\pi)^{4}} \int \! \mathrm{d}s \, \left\langle
\sum_{n=0}^{\infty} \frac{(-i)^{n}}{n!} k_{c_{1}} \cdots k_{c_{n}}
\Psi^{c_{1} \cdots c_{n} a}, \widetilde{\phi}_{a}(k) e^{-i k \cdot
z(s)} \right\rangle ~, \label{forcedense}
\\
\Psi^{c_{1} \cdots c_{n} a}(s) &:=& \frac{1}{(2\pi)^{4}}
\left\langle \sum_{p=0}^{\infty} \frac{(-i)^{p}}{p!} l_{d_{1}}
\cdots l_{d_{p}} Q^{d_{1} \cdots d_{p} c_{1} \cdots c_{n} b} ,
\widetilde{F}^a{}_{b}(l) e^{-i l \cdot z(s)} \right\rangle ~.
\label{Psidefine}\end{aligned}$$ Here, $\widetilde{F}^{ab}$ does not quite represent the Fourier transform of $F^{ab}$, which is not well-defined. It is instead equal to the Fourier transform of some function $^{*}F^{ab}$ which coincides with the field in some neighborhood of $\Sigma(s) \cap W$, but has compact support. If $^{*}F^{ab}$ is just as smooth as $F^{ab}$, its precise form is irrelevant [@Dix67].
(\[forcedense\]) now makes it natural to interpret the $\Psi^{\cdots}$’s as multipole moments of the force density exerted on the body. We might therefore expect the net force to be proportional to $\Psi^{a}$, and the net torque to $\Psi^{[ab]}$. This identification can be made if $\Phi^{ab}$ has the form [@Dix67] $$\Phi^{ab}(k,s) = \frac{1}{(2\pi)^{4}} \left\{ \Psi^{(ab)} - i k_{c}
\left[ \Psi^{c(ab)} - \frac{1}{2} \Psi^{abc} \right] +
\sum_{n=2}^{\infty} \frac{(-i)^{n}}{n! n} k_{c_{1}} \cdots k_{c_{n}}
\left[ 2 \Psi^{c_{1} \cdots c_{n} (ab)} - \frac{n+2}{n+1}
\Psi^{(c_{1} \cdots c_{n} ab)} \right] \right\} ~. \label{stress2}$$ Combining this with (\[stress1\]) shows that $$\int \! \mathrm{d}s \, \left[ \left( \dot{p}^{a} + \Psi^{a} \right)
\phi_{a} + \frac{1}{2} \left( \dot{S}^{ab} - 2 p^{[a} v^{b]} + 2
\Psi^{[ab]} \right) \partial_{a} \phi_{b} \right] = 0 ~.$$ Given that this must hold for all possible choices of $\phi_{a}$, it follows that $$\begin{aligned}
\dot{p}^{a} &=& - \Psi^{a} ~, \label{fdefine}
\\
\dot{S}^{ab} &=& 2 \left( p^{[a} v^{b]} - \Psi^{[ab]} \right) ~.
\label{tdefine}\end{aligned}$$ By construction, these are the only evolution equations implied by (\[stresscons\]). If we impose one more constraint equation: $$n_{a_{1}} t^{a_{1} \cdots a_{n-2} [a_{n-1} [a_{n} b] c]} = 0
\label{StressCon2}$$ for $n \geq 3$, the chosen moments are unique in an appropriate sense. They are also sufficiently general to describe all physically interesting stress-energy tensors [@Dix74]. Note that $\Phi^{ab}$ only depends on $J^{a}$ and $F^{ab}$, and that the constraint equations (\[StressCon1\]) and (\[StressCon2\]) are independent of these quantities. Portions of the stress-energy tensor which depend on the current have therefore been completely isolated from those which are not.
Changes in the higher moments may once again be interpreted as ‘equation of state’ (this identification is actually more direct in this case). Their evolution is not completely arbitrary, however. Besides respecting the constraint equations, they must also be chosen so that $T^{ab}$ remains physically reasonable. These extra restrictions would be provided by analogs of (\[chargedensity2\]) and (\[threecurrent2\]). Although these will not be derived here, they can probably be constructed in a similar way. The presence of the field makes their derivation more complicated, but it should still be possible to repeat all of the steps carried out with the current moments.
It suffices to note that for our purposes, this procedure has resulted in particularly natural definitions for the stress-energy moments – most importantly the linear and angular momenta. It can be shown that the choices made here imply that these momenta are given by (\[pdefine\]) and (\[Sdefine\]) [@Dix67]. Interestingly, the net force and torque do not depend on $G^{ab}$ in any necessary way. They are apparently as independent of the details of the body’s internal structure as possible.
Before moving on, we can gain some insight into the force moments defined by (\[Psidefine\]). Only the first two of these are important here, and it is straightforward to show that $$\begin{aligned}
\Psi^{a}&=& \left\langle \hat{J}_{b}(r,s), F^{ab}(x) \right\rangle
~, \label{forceJhat}
\\
\Psi^{[ab]}&=& \left\langle \hat{J}_{c}(r,s), r^{[a} F^{b]c}(x)
\right\rangle ~. \label{torqueJhat}\end{aligned}$$ Applying (\[hatJ\]), these expressions take the more explicit forms $$\begin{aligned}
\Psi^{a} &=& \int \! \mathrm{d}^{3} r \: \Bigg\{ N J_{b} F^{ab} +
\frac{\partial}{\partial s} \left[ \left( \varphi - \frac{q}{4\pi
|r|^{3}} \right) e^{\beta}_{b} r_{\beta} F^{ab} \right] \Bigg\} ~,
\label{forcedefine}
\\
\Psi^{[ab]} &=& \int \! \mathrm{d}^{3} r \: \Bigg\{ N J_{c}
r^{\alpha} e^{[a}_{\alpha} F^{b]c} + \left( \varphi - \frac{q}{4\pi
|r|^{3}} \right) e^{\gamma}_{c} r_{\gamma} v^{[a} F^{b]c} +
\frac{\partial}{\partial s} \left[ \left( \varphi - \frac{q}{4\pi
|r|^{3}} \right) e^{\gamma}_{c} r_{\gamma} r^{\alpha}
e^{[a}_{\alpha} F^{b]c} \right] \Bigg\} ~, \label{torquedefine}\end{aligned}$$ which were derived by a different method in Sec. \[LawsMot\].
Although (\[forcedefine\]) and (\[torquedefine\]) are exact, they are rather difficult to interpret. Their meaning is made considerably more transparent if the field can be expanded in a Taylor series inside $\Sigma(s) \cap W$. Then (\[forceJhat\]) and (\[torqueJhat\]) together with (\[QJ\]) show that $$\begin{aligned}
\Psi^{a}(s) &\simeq& \sum_{\ell=0}^{L} \frac{1}{n!} \, Q^{c_{1}
\cdots c_{\ell}b} \Big( \partial_{c_{1}} \cdots
\partial_{c_{\ell}} F^a{}_{b} \Big)_{z(s)} ~, \label{forcemult}
\\
\Psi^{ab}(s) &\simeq& \sum_{\ell=0}^{L} \frac{1}{n!} \, Q^{d_{1}
\cdots d_{\ell} a c} \Big( \partial_{d_{1}} \cdots
\partial_{d_{\ell}} F^b{}_{c} \Big)_{z(s)} ~. \label{torquemult}\end{aligned}$$ When $F^{ab}$ is approximately constant throughout the charge, we recover the Lorentz force law, $\dot{p}^{a} = - \Psi^{a} \simeq - q
F^{ab} v_{b}$. Unfortunately, these series are not useful when the field varies considerably over $\Sigma(s) \cap W$. And this is exactly what the self-field does.
I would like to thank Pablo Laguna for reading over this manuscript, and for many helpful discussions leading up to the results presented here. I also thank Amos Ori for useful comments. This work was supported in part by NSF grant PHY-0244788 and the Center for Gravitational Wave Physics funded by the National Science Foundation under Cooperative Agreement PHY-0114375.
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abstract: 'Length and g-factor are fundamental parameters that characterize optical cavities. We developed a technique to measure these parameters in-situ by determining the frequency spacing between the resonances of fundamental and spatial modes of an optical cavity. Two laser beams are injected into the cavity, and their relative frequency is scanned by a phase-lock loop, while the cavity is locked to either laser. The measurement of the amplitude of their beat note in transmission reveals the resonances of the longitudinal and the transverse modes of the cavity and their spacing. This method proves particularly useful to characterize complex optical systems, including very long and/or coupled optical cavities, as in gravitational wave interferometers. This technique and the results of its application to the coupled cavities of a 40-meter-long gravitational wave interferometer prototype are here presented.'
address: |
$^1$LIGO Laboratory, MS 100-36, California Institute of Technology, Pasadena, California, 91125, USA\
$^2$Università degli Studi di Siena, Dipartimento di Fisica, Via Roma 56, 53100 Siena, Italy\
$^3$ Centre for Gravitational Physics, Department of Quantum Science, The Australian National University, Canberra, Australian Capital Territory 0200, Australia\
$^*$Corresponding author: stochino@ligo.caltech.edu
author:
- 'Alberto Stochino,$^{1,2,3,*}$ Koji Arai$^1$ and Rana X. Adhikari$^1$'
title: 'A Technique for In-situ Measurement of Free Spectral Range and Transverse Mode Spacing of Optical Cavities'
---
The absolute length and mirror curvatures are defining parameters of an optical cavity. Together these quantities uniquely determine the *Free Spectral Range* (FSR) and the *Transverse Mode Spacing* (TMS): respectively the frequency spacing between different resonances of the cavity’s longitudinal modes and the frequency spacing between the resonances of transverse and longitudinal modes.
FSR and TMS are usually accurately specified and it is thus very important to measure them with great precision. Over time, monitoring cavity length and mirror curvatures can track changes affecting the cavity geometry. For instance, variations of the cavity length may reveal drifts of the mirrors’ positions caused by ground displacement [@araya99]; changes of the mirrors’ curvatures can be evidence of deformations due to thermal effects [@rakhmanov04]. Precise measurements of FSR and TMS can also allow the modeling of the cavity reflected and transmitted power by predicting the mode distribution around the cavity working point.
Measuring these parameters proves particularly challenging in long and complex optical cavities, such as those in gravitational wave interferometers [@ligoS6; @virgostatus10; @lcgtstatus10; @geostatus10]. Precisely characterizing the optical cavities in these detectors is crucial to achieve their best sensitivity. Often three to five or more coupled optical cavities are controlled simultaneously through the sensing of RF phase modulation sidebands added to the main laser. Resonance or anti-resonance conditions of the sidebands in each part of the interferometer must be ensured at all times for optimal decoupling among the degrees of freedom [@redding2002dynamic]. When the RF modulation frequencies and the cavity lengths do not match, controlling the interferometer becomes more difficult. Also, a mismatch of more than a few centimeters in the arm cavities, or $\sim$1 mm in the recycling cavities may reduce the sensitivity of the detector. Noise from the laser and the RF modulator may then enter the interferometer’s auxiliary degrees of freedom and leak into the gravitational wave channel by cross-couplings [@ward2008dc; @wardThesis; @stochinoPhDthesis].
Ye [@Ye:04] showed that, in principle, sub-wavelength length measurement precision could be obtained by using a femtosecond laser. However this scheme is difficult to implement in situations where a dedicated ultra-short pulsed laser system may not bet readily available. Several different approaches have been tried in the past. In what was probably the simplest, Rakhmanov et al. [@rakhmanov99] measured the length of a cavity with a precision of 4 mm by an optical *vernier* obtained by swinging the end mirror. In a later experiment, Rakhmanov et al. [@rakhmanov04] measured the length of a 4 km cavity with 80 $\mu$m precision by measuring the cavity’s frequency response by frequency modulating the laser. In a similar way, Uehara and Ueda [@uehara1995accurate] measured the radius of curvature of the end mirror of a plano-concave cavity. Additionally, Araya et al. [@araya99], following DeVoe and Brewer [@devoe1984laser], estimated the length of a 300 meter cavity with a relative precision of $10^{-9}$ by simultaneously locking to the cavity the laser’s carrier and a phase-modulation sideband.
These techniques are difficult to scale and adapt to systems of very different lengths, or included in complex, coupled-cavity configurations. For example, frequency modulation techniques cannot be used in short gravitational wave interferometers since any modulation of the laser at frequencies near the cavity FSR would be suppressed by the input mode cleaner cavity. Tuning the sidebands’ frequency as in Araya’s technique, would not be possible in systems including additional input cavities.
To circumvent these limitations, we developed an alternative interferometric technique to measure the FSR and TMS of optical cavities with a larger range of lengths or in complex optical setups. We then tested this technique on the optical cavities forming the LIGO 40m gravitational wave detector prototype at the California Institute of Technology. In the following, we describe this technique and present the results obtained in our experiments.
Principles of the Technique {#sec:fp}
===========================
The FSR of an optical cavity determines the frequency spacing $\nu_{\rm FSR}$ between resonances of any given pair of cavity longitudinal modes. It is defined as[@siegman]: $$\nu_{{\rm FSR}} = \frac{c}{2L},$$ where $L$ is the cavity length and $c$ the speed of light. This definition can then be used to infer the cavity length from the direct measurement of the FSR. In a similar way, the mirrors’ curvatures are inferred from the measurement of the TMS and the estimated cavity length.
Each Hermite-Gaussian mode is characterized by a different Guoy phase determining its specific resonant frequency in the cavity [@siegman]. Because of this phase, the set of resonances of a generic TEM$_{mn}$ mode is shifted from the resonances of the fundamental TEM$_{00}$ mode by an integer multiple of the so-called *Transverse or Spatial Mode Spacing*. For a linear cavity this is defined as $$\label{eq:TMS}
\nu_{{\rm TMS}} = \nu_{{\rm FSR}}
\:\frac{m+n}{\pi}\cos^{-1}\sqrt{g_1g_2}.$$ where $g_{1}=\left( 1-L/R_{1} \right)$ and $g_{2}=\left( 1-L/R_{2} \right)$ are the *g-parameters* of the mirrors, with $R_{1}$, $R_{2}$ representing their respective radii of curvature, and $L$ the absolute length of the cavity. The product of the g-parameters $g_1g_2$ is often referred to as the cavity *g-factor*: $$\label{eq:gdef}
g = g_1 g_2$$ It follows from (\[eq:TMS\]) that if the cavity mirrors are astigmatic, the resonances of complimentary modes, TEM$_{mn}$ and TEM$_{nm}$ are split. If we assume, for simplicity, that axes of the astigmatism for the two mirrors are aligned, different g-factors $g_{\rm x}$ and $g_{\rm y}$ can be associated with each of the two transverse spatial directions $x$ and $y$, respectively[^1]. In case of astigmatic cavities, the definition of transverse mode spacing is then generalized as $$\begin{aligned}
\nu_{{\rm TMS}} &= \nu_{{\rm FSR}} \left[\frac{m}{\pi}\cos^{-1}\sqrt{g_{1x}g_{2x}} + \frac{n}{\pi}\cos^{-1}\sqrt{g_{1y}g_{2y}} \right] \\
& = m\,\nu_{{\rm TMS,x}} + n\,\nu_{{\rm TMS,y}}\,\,\,, \end{aligned}$$ where $g_{ix} = \left(1-L/R_{ix}\right)$, $g_{iy} = \left(1-L/R_{iy}\right)$, and $R_{ix}$, $R_{iy}$ represent the radius of curvature of the *i*-th mirror of the cavity in the $x$ and $y$ direction, respectively.
With this definition, the direct measurement of the TMS can then be used to estimate the cavity g-factor.
Measurement technique
---------------------
The technique presented in this article determines the FSR and the TMS of a cavity from the resonances appearing in transmission as the laser frequency is scanned. Two lasers are used for the measurement: the first, serving as a master laser, is set to resonate in the cavity in its TEM$_{00}$ mode; the second, phase-locked to the first, is held at an arbitrary offset frequency set by a local oscillator (LO) in the loop [@armor]. The phase-locked loop (PLL) ensures that the relative frequency of the two lasers remains constant.
At first, the optical cavity under test is locked to the fundamental mode of the master laser by using the Pound-Drever-Hall technique [@drever83]. The slave beam, after being appropriately mode matched, is injected into the cavity together with the main beam. At the cavity transmission, the two beams interfere producing a beat note at their differential frequency as set by the PLL’s LO frequency. Finally, the cavity resonance profile is measured by tracking the beat note’s amplitude as a function of the LO frequency.
Cavity absolute length measurement
----------------------------------
As the frequency of the PLL’s LO is swept, a resonance peak appears in transmission every time the relative frequency of the slave laser reaches a multiple of the cavity FSR. The FSR is then estimated by a least squares fit of this set of measured resonant frequencies $\nu_n$ by the linear function $\nu_n=n\times \nu_{\rm FSR} $, where $n$ is the resonance’s order.
Cavity g-factor measurement {#sec:gfactormeasurement}
---------------------------
Following the FSR measurement, the TMS is measured by coupling the laser’s fundamental mode into both the TEM$_{00}$ cavity mode and the TEM$_{01}$ or TEM$_{10}$ spatial modes. This is obtained by introducing a small misalignment between the input beam and the cavity axis [@anderson84 Sec. II]. In particular, the coupling into the cavity’s TEM$_{10}$ mode is obtained by a tilt of the cavity axis in yaw; a coupling into the cavity’s TEM$_{01}$ mode is obtained by a tilt in pitch.
{width=".97\textwidth"}
The measurement begins with the relative frequency of the two lasers set to a resonant frequency $\nu_n$ equal to an integer multiple $n$ of the cavity FSR: $\nu_n=n\times
\nu_{\rm FSR} $. Then, the frequency of the auxiliary laser is scanned, while the cavity is held locked to the main laser. As their relative frequency equals the cavity’s TMS, the auxiliary laser resonates in the cavity’s TEM$_{01}$ (or TEM$_{10}$) mode and is transmitted by the cavity. The interference between the main beam’s TEM$_{00} $ mode and the auxiliary beam’s TEM$_{01}$ (or TEM$_{10}$) mode produces a beat note on a photodetector at a frequency $\nu=\nu_n+\nu_{\rm TMS}$. The TMS is then measured by the frequency of the corresponding peak.
Both the TEM$_{01}$ and TEM$_{10}$ modes have antisymmetric phases around the beam axis. As a result, their interference with the TEM$_{00}$ mode has zero net power when integrated over the photodetector’s transverse plane. To detect the beat note it is necessary to break the symmetry of the beam spot just before the photodetector. This was done by partially clipping the beam with a razor blade in front of the photodetector but could also be done with a broadband quadrant photodetector.
Characterization of the arm cavities of a gravitational wave interferometer
===========================================================================
This technique was tested on the arm cavities of the Caltech 40m prototype, a testbed facility for the Laser Interferometer Gravitational-wave Observatory (LIGO) [@weinstein2002advanced]. For the experiment, the interferometer was set in a dual-recycled Fabry-Perot Michelson configuration. In this setup, the two ($\sim$40m long) Fabry-Perot cavities, are connected in a Michelson configuration. Similar to the Advanced LIGO interferometers [@harry2010advanced], the beam splitter’s symmetric and anti-symmetric outputs are coupled to the so-called *recycling cavities*: the Power Recycling Cavity (PRC) at the symmetric port and the Signal Recycling Cavity (SRC) at the anti-symmetric port. All the cavity mirrors are suspended and hang on wires as simple pendulums, for seismic noise isolation.
Figure \[fig:armmeas\] shows the optical layout of this experiment. The main beam illuminating the interferometer was provided by a Non-Planar-Ring-Oscillator laser (NPRO), amplified by a Master Oscillator Power Amplifier (MOPA) up to about 2W at the time of the experiment [@willke2008stabilized]. Dedicated frequency and power pre-stabilization subsystems were enabled on the laser. After these stages, phase modulation sidebands were added to the beam by electro-optic modulators (EOM) for cavity locking. Finally, prior to entering the interferometer, the laser beam passed through a 13-meter-long triangular mode cleaning cavity. The PM sideband frequencies are chosen to coincide with one of the FSRs of the mode cleaning cavity.
The auxiliary beam was injected into the anti-symmetric port of the interferometer, through the signal recycling cavity’s mirror (SRM). This solution allowed the beam to bypass the input mode cleaner cavity.
The auxiliary beam was provided by a 700 mW NPRO, located outside the interferometer’s vacuum envelope, on the same optics table hosting the dark port’s optical setup. A simple flipper mirror was then used to enable the auxiliary beam to enter the interferometer at the time of the measurement.
A phase-lock optical system was set up on the main laser table by interfering pick-off beams from the two lasers on an RF photodiode (Thorlabs PDA255 or New Focus 1611). The PLL ensured that the auxiliary laser’s frequency followed that of the main laser with an arbitrary tuneable offset, by suppressing the relative phase fluctuations between the two lasers.
Measurement results
===================
The FSR and TMS were measured separately. In both measurements, the two laser beams were resonated only in the cavity to be examined by misaligning the other parts of the interferometer. The cavity was then locked to the main laser by the Pound-Drever-Hall (PDH) technique [@drever83].
A computer controlled the PLL’s local oscillator via a GPIB I/O interface. A script swept the LO frequency and monitored a spectrum analyzer connected to the PD in transmission. The power of the PD signal at the LO frequency was recorded at each step of the frequency scan.
{width="0.47\columnwidth"} {width=".5\columnwidth"}
Arm length measurement
----------------------
The arm cavity to be measured was first locked to the main laser by controlling the end mirror with electromagnetic actuators. Then the auxiliary laser was injected into the interferometer. The PLL frequency was scanned by $\pm$ 20 MHz, a range corresponding to $\pm$5 cavity FSRs, by sweeping the auxiliary laser’s frequency first below and then above the main laser’s frequency. The data obtained recording the power of the PD signal versus the PLL frequency was plotted as in Fig. \[fig:armtransmittedpower\]. Resonance peaks were observed at multiples of the cavity FSR. The frequency $f_0$ of each peak was determined by a nonlinear least-squares fit of the PD signal amplitude $V_{\rm PD}$ with the function $$P_{\rm PD}(f)=\frac{P_0}{\sqrt{1+\left(f-f_0\right)^2/f_c^2}}+P_{\rm off}
\label{eq:resonancemodel}$$ where $P_0$, $P_{\rm off}$ and $f_{\rm c}$ are additional fit parameters.
By fitting 10 resonances of the X arm and 5 resonances of the Y arm, $f_0$ was estimated with an error of $ {\sim}6-7~{\rm Hz}$ and $ {\sim}9-11~{\rm Hz}$, respectively. The residuals from the fitting show that these errors are statistical and not systematic.
The cavity FSR was estimated by a linear least squares fit of these resonant frequencies (right plot of Fig. \[fig:armtransmittedpower\]). Table \[tab:results\] shows the result of the fitting, as well as the derived values of the cavity length. The FSRs were estimated with statistical standard errors of 0.6 Hz and 3 Hz in the X and Y arm, respectively. Accordingly, the cavity lengths were determined with a precision of $6 {\rm{\mu}m}$ and $30 {\rm{\mu}m}$, over an absolute length of $\sim$38.5m. However, the residuals in the linear fitting show deviations which are likely due to causes other than statistical fluctuations. In fact, deviations were observed at the level of $ {\sim}30-40$ Hz, equivalent to a displacement of $ {\sim}0.3-0.4$ mm (see plotted residuals in Fig. \[fig:armtransmittedpower\]). These fluctuations may be due to drift of the cavity alignment or actual length changes during the course of the measurement due to the thermal expansion of the concrete slab.
Arm cavity g-factor
-------------------
The arm cavity g-factor was measured by introducing a small misalignment in the input beam’s axis, first in pitch, then in yaw in order to detect the resonances of the TEM$_{01}$ mode and the TEM$_{10}$ mode, respectively. The measurement started by locking the cavity to the main laser’s fundamental mode. The amplitude of the PD signal was recorded and plotted against the PLL frequency.
The frequency scan revealed additional resonance peaks due to higher order modes. The first peak was produced by the TEM$_{00}$ mode resonance; the second corresponded to the spatial mode resonance.
![Measurement of transverse mode spacing in the Y-arm. The blue and red curves are the results of the frequency scans for the horizontal and vertical modes, respectively. In the upper plot, the left peak is the cavity’s fundamental resonance, while the two peaks on the right are the resonances of the spatial modes (zoomed-in in the lower plots). A fit of the data points in these peaks (magenta lines) is used to determine their frequencies.[]{data-label="fig:yarmmodes"}](abslPaperYarmMisalignmentScanKoji.pdf){width="1\columnwidth"}
The TMS was determined by measuring the frequencies of the TEM$_{10}$ and TEM$_{01}$ modes and then comparing them with the frequency of the TEM$_{00}$ mode.
The TEM$_{10}$ (or TEM$_{01}$) peak was identified by matching it with its expected location as calculated from the mirror’s nominal radius of curvature. The fitting of the TEM$_{10}$ and TEM$_{01}$ resonances allowed us to estimate $f_0$ with errors of 15 and 46 Hz. The fitting residuals indicate that statistical fluctuations are responsible for these deviations.
It should be noted that the g-factor measurements presented here and the values obtained by direct measurements of the mirrors’ radii of curvature do not agree. Phase map measurements of the mirrors obtained by a Fizeau interferometer estimated the radii of curvature of the end and input mirrors of the X and Y arms to be 57.57, 57.68, 7280, and 7210 (all in meters), respectively. These numbers, combined with the measured cavity lengths give us g-factors of $2{\sim}6~\%$ larger than the measured values. In general, the individual radii of curvature of mirrors in a two-mirror cavity cannot be directly derived from measured g-factors. However, we can still learn something about the mirror curvatures if we take into account that typically phase map measurements of flat mirrors are more accurate than those of highly curved mirrors [@Elssner:94; @Elssner:89]. For instance, phase maps measurements of the nominally-flat mirrors in use in our lab have estimated radii of curvature ranging between $-100~{\rm km}$ and $+6~{\rm km}$. Finite radii like these affect the g-factors by less than $0.6~\%$, compared to an ideally flat mirror. For this reason, most likely the observed discrepancies are due to the curvature of the end mirrors. Astigmatism in the end mirrors could explain the TEM$_{01}$/TEM$_{10}$ mode splitting and justify the difference in the g-factors by $3{\sim}4~\%$. In particular, by assuming perfectly-flat input mirrors, our measurements could be explained by an astigmatism of $\sim$1 m over end mirrors with radii of curvature of 56$\sim$57 m.
The two astigmatic modes were observed in the vertical and horizontal main axes in both arms cavities, rather than along arbitrary axes in each of them. This might occur if the astigmatism was related to the wedge angle on the cavity mirrors. Both the input and the output mirrors have wedge angles of 1 deg and 2.5 deg, respectively, and they were both installed with the angles on the horizontal axis. This non-degeneracy of the cavity’s TEM$_{10}$/TEM$_{01}$ resonances was also confirmed by an experiment in which the cavity was locked to either of these modes. By misaligning the cavity in pitch or yaw, it was possible to lock the cavity in either the TEM$_{10}$ mode or the TEM$_{01}$ but never on both at the same time. This was evidence that the modes were indeed separated by much more than the cavity half line-width of 1.6 kHz.
Power recycling cavity
----------------------
Determining the length of the other cavities, such as the recycling cavity is also very important in order to achieve a stable control of the interferometer. For this reason, we tested this technique on the Power Recycling Cavity (PRC). This cavity is formed by the Power Recycling Mirror (PRM) at one end and at the other end by the Michelson “compound mirror”, an effective mirror defined by the beam splitter and the input mirrors (ITM) of the arm cavities. The asymmetry in this short Michelson causes it to have a frequency-dependent reflectivity [@Sigg:98].
For this measurement, the auxiliary beam was introduced through a misaligned SRM. The transmitted beam was detected at the reflection port of the interferometer using the output of the Faraday isolator. The Michelson and the power recycling cavity were locked to the main laser beam, by actuating on the beam splitter and the PRM. Because the PRC length is much shorter than the arms’, the laser frequency had to be scanned by several hundreds of megahertz in order to explore one cavity FSR. This was easily accomplished thanks to the large dynamic range of the PLL, in contrast to previous length measurement techniques, which had been limited to only very long cavities with smaller FSRs.
The measured profile of the transmitted power showed a complex pattern, as expected from the frequency-dependent response of the Michelson. The data were fit with an analytical model for the cavity transmissivity (details in Appendix \[sec:appendix\]), obtaining an estimate of the cavity absolute length and of the Michelson differential length (asymmetry) with a precision of 2 mm and 3 mm, respectively (see Table \[tab:results\]).
![Frequency scan of the power recycling cavity: measured beat power (blue dots) and fitting curve (red).[]{data-label="fig:prclengthmeasresults"}](prc_AbsLpaper.pdf){width="1\columnwidth"}
--------------------------------------------------------- -- -- -- --
**[MEASURED]{} &&&\
**[X-Arm ]{} & & & &\
$\nu_{\rm FSR}$ & $3897627.5$& $\pm$ & $0.6$ &Hz\
$\nu_{\rm TMS,H}$ & $1199048$ & $\pm$ & $20$ &Hz\
$\nu_{\rm TMS,V}$ & $1213602$ & $\pm$ & $46$ & Hz\
\
**[Y-Arm]{}& & & &\
$\nu_{\rm FSR}$ & $3878678$& $\pm$ & $3$ &Hz\
$\nu_{\rm TMS,H}$ & $1207790$ & $\pm$ & $23$ &Hz\
$\nu_{\rm TMS,V}$ & $1189071$ & $\pm$ & $18$ & Hz\
\
**[DERIVED]{} & & &\
**[X-Arm]{}& & & &\
L & $38.458326$ & $\pm$ & $6 \times 10^{-6}$&m\
$g_{\rm H}$ & $0.322885$ & $\pm$ & $1.5 \times10^{-5}$\
$g_{\rm V}$ & $0.311965$ & $\pm$ & $3.5 \times10^{-5}$\
\
**[Y-Arm]{}& & & &\
L & $38.64622$ & $\pm$ & $3 \times 10^{-5}$&m\
$g_{\rm H}$ & $0.312013$ & $\pm$ & $1.7 \times10^{-5}$\
$g_{\rm V}$ & $0.326144$ & $\pm$ & $1.4 \times10^{-5}$\
\
**[PRC]{} & & &\
$l_{\rm PRC}$& $2.217$ & $\pm$ & $2 \times10^{-3} $ &m\
Asymmetry & $0.460$ & $\pm$ & $3 \times10^{-3} $&m\
**************
--------------------------------------------------------- -- -- -- --
: Summary of measurements on the arm cavities and on the power recycling cavity. $\nu_{\rm FSR}, \nu_{\rm TMS,H},$ and $\nu_{\rm TMS,V}$ are the free spectral range and the transverse mode spacing for the horizontal mode and the vertical mode, respectively. $L$ is the cavity length. $g_{\rm H}$ and $g_{\rm V}$ are the g-factors for the horizontal and vertical modes. $l_{\rm PRC}$ and Asymmetry are the length of the power recycling cavity and the difference of the Michelson arm lengths.[]{data-label="tab:results"}
Measurement Precision
---------------------
The precision of the FSR and TMS measurements listed in Table \[tab:results\] was determined by the statistical errors in the parameters of the non-linear least squares fit of the data. The difference between the two arms was due to a larger number of data points measured for the X arm.
In principle, since the cavity is locked to the main laser, the best precision of the FSR and TMS measurements achievable by this technique is set by the laser frequency and intensity noise and by the noise in the PDH loop. In reality, fluctuations of the cavity stored power due to oscillations of the cavity alignment may affect the beat note’s amplitude and thus increase the errors in the fit parameters. In our case, since the angular degrees of freedom of the cavities were uncontrolled, the mirrors’ angular motion was likely the main cause of the measured fluctuations in the beat note signal.
In the future the technique could be greatly improved by measuring the phase of the beat note rather than its amplitude. The cavity resonance could then be measured with more accuracy and precision by identifying the frequency at which the beat note’s phase flips by 180 degrees. This would render the measurement intrinsically immune to unwanted fluctuations of the beam power. For instance the phase could be detected by measuring the transfer function between the PLL’s local oscillator signal and the beat note.
Conclusions
===========
We demonstrated a new interferometric technique to measure the free spectral range and the transverse mode spacing of optical cavities of various lengths. It allowed measurements of the length and the g-factor of a Fabry-Perot cavity with a precision of 1 ppm and 10 ppm, respectively.
Compared to previous methods, this technique can be applied to optical cavities of lengths ranging from a few meters to several kilometers. With our experiment we proved that the technique is compatible with a complex optical system, comprising coupled cavities. Not requiring structural modifications to the optical system on which it is applied, it provides a convenient option in circumstances in which an in-situ and non-invasive measurement tool for cavity length and g-factor is needed.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Yoichi Aso, Robert Ward and Hiro Yamamoto for illuminating discussions on this technique. We also thank the rest of the 40m team for building and maintaining the interferometer during the course of this work.
The LIGO Observatories were constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation under cooperative agreement PHY-0757058. This paper has been assigned a LIGO Document Number of P1200048.
PRC Transmissivity {#sec:appendix}
==================
When the primary laser is resonant, the transmissivity of the Power Recycling Cavity can be written as a function of the PLL’s local oscillator frequency $\Delta\omega = \omega_{\rm psl} - \omega_{\rm aux}$ in the following way: $$\label{eq:prcmodel}
T_{\rm prc} = \left| \frac{
t_{\rm prm} r_{\rm itm} e^{-{\rm i} 2 \Delta\omega l_{\rm prc}/c} \sin \left( {\Delta\omega l_{\rm asy}}/{c} \right)
}{
1 + r_{\rm prm} r_{\rm itm} e^{-{\rm i} 2 \Delta\omega l_{\rm prc}/c} \cos \left( {\Delta\omega l_{\rm asy}}/{c} \right)
} \right|^2$$ where $t_{\rm prm}$ is the PRM transmittance; $r_{\rm itm}$ the ITM reflectance; $l_{\rm
asy}$ the asymmetry (i.e the Michelson’s differential length); and $l_{\rm prc}$ the effective length of the power recycling cavity defined as $l_{\rm prc} \equiv l_{\rm
prm-bs}+(l_{\rm x}+l_{\rm y})/2)$ where $l_{\rm prm-bs}$ is the distance between the beam splitter (BS) and the PRM, and $l_{\rm x}$ and $l_{\rm y}$ are the lengths of the Michelson arms. The amplitude of the beat note in transmission is then proportional to $\sqrt{T_{\rm prc}}$.
[^1]: For more general astigmatic cavity cases, see [@habraken07].
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