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+1
+A Unified Visual Information Preservation
+Framework for Self-supervised Pre-training in
+Medical Image Analysis
+Hong-Yu Zhou, Student Member, IEEE, Chixiang Lu, Chaoqi Chen, Sibei Yang,
+and Yizhou Yu, Fellow, IEEE,
+Abstract—Recent advances in self-supervised learning (SSL) in computer vision are primarily comparative, whose goal is to preserve
+invariant and discriminative semantics in latent representations by comparing siamese image views. However, the preserved high-level
+semantics do not contain enough local information, which is vital in medical image analysis (e.g., image-based diagnosis and tumor
+segmentation). To mitigate the locality problem of comparative SSL, we propose to incorporate the task of pixel restoration for explicitly
+encoding more pixel-level information into high-level semantics. We also address the preservation of scale information, a powerful tool
+in aiding image understanding but has not drawn much attention in SSL. The resulting framework can be formulated as a multi-task
+optimization problem on the feature pyramid. Specifically, we conduct multi-scale pixel restoration and siamese feature comparison in
+the pyramid. In addition, we propose non-skip U-Net to build the feature pyramid and develop sub-crop to replace multi-crop in 3D
+medical imaging. The proposed unified SSL framework (PCRLv2) surpasses its self-supervised counterparts on various tasks,
+including brain tumor segmentation (BraTS 2018), chest pathology identification (ChestX-ray, CheXpert), pulmonary nodule detection
+(LUNA), and abdominal organ segmentation (LiTS), sometimes outperforming them by large margins with limited annotations. Codes
+and models are available at https://github.com/RL4M/PCRLv2.
+Index Terms—Medical image analysis, Self-supervised learning, Transfer Learning, Context restoration, Feature pyramid.
+!
+1
+INTRODUCTION
+I
+T is usual to acquire a substantial amount of manually
+labeled data before training deep neural networks. This
+condition is easy to meet in natural images, where labor
+costs and labeling difficulties are tolerable. In medical image
+analysis, however, credible annotations are mainly derived
+from domain experts’ diagnoses, which are challenging to
+obtain due to the rarity of the target disease, the need to safe-
+guard patient privacy, and the scarcity of medical resources.
+Against this background, self-supervised learning (SSL) has
+been widely accepted as a viable technique to learn medical
+image representations without specialistic annotations. We
+usually deploy SSL in the pre-training stage to obtain well-
+transferable features, which can be transferred to various
+downstream tasks for performance boosting.
+Recent advances in SSL are mostly based on compar-
+ative learning [8], [10], [15], [17]. The rationale behind is
+to learn transferable latent representations with invariant
+and discriminative semantics by maximizing the mutual
+information between a pair of siamese images. One potential
+problem of these comparative methods is that they mainly
+focus on encoding high-level global semantics in representa-
+•
+Hong-Yu Zhou, Chixiang Lu, Chaoqi Chen, and Yizhou Yu are with the
+Department of Computer Science, The University of Hong Kong, Hong
+Kong. Email: {whuzhouhongyu, luchixiang, cqchen1994}@gmail.com,
+yizhouy@acm.org.
+•
+Sibei Yang is with ShanghaiTech University and Shanghai Engineering
+Research Center of Intelligent Vision and Imaging, Shanghai, China.
+Email: yangsb@shanghaitech.edu.cn.
+•
+First two authors contributed equally.
+•
+Corresponding author: Sibei Yang and Yizhou Yu.
+tions but ignore the preservation of pixel-level information1.
+However, in medical image analysis, the latter type of
+information usually plays a vital role. For instance, in chest
+pathology detection, radiologists or clinicians are required
+to point out small lesions from a chest X-ray according to
+their textures. Sometimes, these areas of pathologies are so
+hard to identify that even medical experts have to check
+pixel-level details to tell where the lesions are. Another
+typical example lies in brain tumor segmentation, where the
+segmentation error of one voxel may cause irreparable harm
+to patients in brain surgeries, such as a permanent damage
+to the cochlear nerve when trying to remove the acoustic
+neuroma.
+An intuitive way to preserve pixel-level information in
+learned features is to restore the pixel-level content from
+latent representations directly. This methodology, known
+as context restoration [29], has already been adopted as a
+surrogate task in pretext-based SSL for natural [23], [29], [44]
+and medical images [7], [49]. Specifically, these approaches
+first apply various data augmentation strategies to a given
+image to generate a corrupted input, based on which deep
+models are trained to restore original pixels. In this way,
+we explicitly require the latent representations to preserve
+information closely related to pixels. Although pure pixel-
+based features are not as transferable as those from com-
+parative SSL [17], [48], we hypothesize it is still beneficial
+to explicitly preserve pixel-level information and global se-
+1. In 3D medical images, we often use “voxel” to denote the same
+concept as the pixel does in 2D images. For simplicity, we use “pixel”
+to denote the smallest addressable element in both 2D and 3D images
+in the rest of this paper.
+arXiv:2301.00772v1  [cs.CV]  2 Jan 2023
+
+2
+ℱ#
+ℱ$
+ℱ%
+ℱ"
+ℱ!
+Pixels
+Semantics
+Scales
+Fig. 1. Motivation illustration. We propose a unified SSL framework
+to simultaneously preserve information in visual representations from
+perspectives of pixels, semantics, and scales. {F1, F2, F3, F4, F5} de-
+note different levels in the feature pyramid, given an input image. Our
+approach restores uncorrupted inputs from the feature maps directly
+to preserve pixel-level details. In order to retain the global semantic
+information, our method compares siamese one-dimensional represen-
+tations. Last but not the least, the proposed methodology conducts pixel
+restoration and feature comparison at different scales. The rationale
+behind is to introduce multi-scale self-supervised latent representations,
+making them more transferable to various downstream tasks.
+mantics, especially in medical image analysis where details
+matter a lot.
+Besides semantics and pixels, introducing multi-scale
+representations has been proven to be quite helpful in
+aiding image understanding [12], [24], [26], [27], [32], [39].
+The common practice of these methods is to construct a
+feature pyramid during training, testing, or both stages.
+Then, various tasks, such as detection, and segmentation,
+can be conducted on the basis of multi-scale features. The
+goal of building the feature pyramid is to endow image
+representations with the ability to recognize objects at dif-
+ferent scales, which is also consistent with the law of human
+cognition [31]. However, the preservation of visual informa-
+tion at multiple scales is rarely mentioned in SSL. Thus, it
+is unclear whether introducing multi-scale self-supervised
+representations provides a stronger transfer learning ability.
+In Figure 1, we illustrate the motivation of the proposed
+unified visual information preservation framework for SSL.
+The introduced framework addresses the preservation of
+information in self-supervised visual representations from
+three aspects: pixels, semantics, and scales. Firstly, to re-
+tain pixel-level information in latent representations, our
+framework involves a reconstruction branch in the self-
+supervised model to rebuild uncorrupted images from cor-
+rupted inputs. Specifically, we ask the self-supervised model
+to restore pixels from feature maps of randomly corrupted
+inputs during training. As a result, information closely
+associated with pixels can be explicitly encoded into the
+latent representations. In practice, this type of information
+would enhance the ability of self-supervised representations
+to recognize and differentiate textures. Apart from pixel-
+level information, preserving invariant and discriminative
+semantics in visual representations is also necessary. To-
+wards this end, we adopt the existing comparative SSL
+to encode invariant semantic information by comparing
+high-level representations of siamese image patches [10].
+We empirically found the siamese SSL not only produces
+comparably (sometimes more) transferable medical image
+representations but also is much easier to implement in com-
+parison to the typical contrastive manner [17]. Last but not
+the least, the proposed unified framework introduces multi-
+scale latent representations by conducting pixel restoration
+and feature comparison in a range of scales. To achieve this
+goal, we propose a non-skip U-Net (nsUNet) that constructs a
+feature pyramid upon the U-shape architecture [32]. In prac-
+tice, nsUNet effectively avoids the production of shortcut
+solutions when performing the context restoration task. On
+the basis of nsUNet, we conduct pixel-level context restora-
+tion and siamese feature comparison in each level (i.e., scale)
+of the feature pyramid. In this way, the proposed framework
+helps improve the ability of self-supervised representations
+to recognize objects (e.g., lesions and organs in medical
+images) at different sizes and scales.
+We summarize the contributions of this paper as follows:
+•
+We present an information preservation framework
+for advancing SSL in medical image analysis. In this
+framework, we unify the preservation of visual infor-
+mation in latent representations from three aspects:
+pixels, semantics, and scales. Towards this end, pixel
+restoration and feature comparison are conducted at
+different feature scales.
+•
+We introduce non-skip U-Net (nsUNet) to construct
+the feature pyramid. Compared to the typical U-
+shape models in medical imaging [11], [32], nsUNet
+maintains more feature scales and eliminates the us-
+age of the widely adopted skip connections to avoid
+shortcut solutions to pixel restoration.
+•
+Inspired by multi-crop [5], we propose sub-crop to
+compare global volumes against local volumes. In
+order to mitigate the problem of the reduced mutual
+information between global and local views in 3D
+space, sub-crop restricts the cropping of local views
+within the 3D minimum bounding box of global
+views. Experiments on 3D medical images found that
+sub-crop is more effective than multi-crop in various
+downstream tasks.
+•
+We conduct extensive and comprehensive experi-
+ments to validate the effectiveness of the proposed
+framework. We show that the unification of pixels,
+semantics, and scales can provide impressive perfor-
+mance under the pre-training/fine-tuning protocol.
+Specifically, the proposed framework outperforms
+both self-supervised and supervised counterparts in
+chest pathology classification, pulmonary nodule de-
+tection, abdominal organ segmentation, and brain
+tumor segmentation by substantial margins.
+The conference version of this paper (PCRLv1) was pre-
+sented in [47], which demonstrates the benefits of incor-
+porating more pixel-level information besides the invariant
+and discriminative semantics obtained by contrastive learn-
+ing. In this paper, we made significant and substantial modi-
+fications to PCRLv1, and we name the improved framework
+as PCRLv2 (i.e., Preservational Comparative Representation
+Learning). The modifications and improvements in PCRLv2
+include but are not limited to (i) Besides local pixel-level
+and global semantic information, scale information is also
+
+3
+×
+×
+×
+×
+×
+×
+×
+×
+R
+R
+R
+R
+R
+𝑥
+𝑥!
+𝑥"
+t!
+t"
+R
+R
+R
+R
+R
+R Pixel restoration
+Candidate scale
+Chosen scale
+nsUNet
+Siamese nsUNet
+𝑥!
+#
+t!
+#
+𝑥"
+#
+t"
+#
+Global aug.
+Global aug. Local aug.
+Local aug.
+(a) Multi-scale pixel restoration
+×
+×
+×
+×
+×
+×
+×
+×
+C
+C
+C
+C
+C
+C Feature comparison
+Candidate scale
+Chosen scale
+nsUNet
+Siamese nsUNet
+𝑥!
+𝑥"
+𝑥!
+#
+𝑥"
+#
+𝑥
+t!
+t"
+t!
+#
+t"
+#
+Global aug.
+Global aug. Local aug.
+Local aug.
+(b) Multi-scale feature comparison
+Fig. 2. The overall structure of PCRLv2. PCRLv2 performs self-supervised visual learning on siamese feature pyramids. To achieve this goal, we
+propose non-skip U-Net (nsUNet). nsUNet consists of five feature scales and removes the skip connections to prevent network optimizers from
+finding shortcut solutions to context restoration. On the basis of nsUNet, we propose to decouple the preservation of pixel-level, semantic, and
+scale information into two tasks: (a) multi-scale pixel restoration; (b) multi-scale feature comparison. The rationale behind is to incorporate pixel
+details and semantics into features at different scales. During the training stage, we randomly choose a feature scale from the feature pyramid,
+on top of which we conduct pixel restoration and feature comparison. x denotes a batch of input images. t1 and t2 stand for two distinct global
+augmentations, while t′
+1 and t′
+2 denote the successive local augmentations.
+preserved in self-supervised visual representations. The
+motivation behind is that although multiple feature scales
+have been considered in various vision tasks, they have
+not drawn much attention in SSL. PCRLv2 shows that
+introducing multi-scale latent representations can boost the
+transfer learning performance of SSL in downstream tasks.
+(ii) PCRLv2 simplifies the attentional pixel restoration and
+hybrid feature contrast operations of PCRLv1 into a con-
+cise multi-task optimization problem. As a result, PCRLv2
+is simpler and easier to implement while achieving bet-
+ter performance, thus more practical. (iii) Compared to
+PCRLv1 that relies on the plain U-Net architecture [32],
+PCRLv2 conducts SSL on top of a new backbone, i.e., non-
+skip U-Net (nsUNet). There are two inherent advantages
+of nsUNet. First, the feature pyramid of nsUNet allows
+performing multi-scale pixel-level context restoration and
+semantic feature comparison. As a result, the unification
+of pixels, semantics, and scales produces more transferable
+visual representations. Second, nsUNet can effectively avoid
+the production of shortcut solutions, providing obvious
+performance gains over the use of the typical skip con-
+nections. (iv) We integrate the idea of multi-crop [5] in
+PCRLv2. Moreover, in 3D medical imaging, we propose sub-
+crop to produce reliable local views with increased mutual
+information by randomly cropping multiple local volumes
+within the 3D minimum bounding box of global views. In
+practice, we found that the proposed sub-crop has better
+pre-training performance than multi-crop. (v) In 5 classifica-
+tion/segmentation tasks, PCRLv2 provides more transfer-
+able pre-trained visual representations, not only surpass-
+ing previous self-supervised and supervised counterparts
+by substantial margins but also obviously outperforming
+PCRLv1 in all experiments.
+2
+RELATED WORK
+This section reviews related work in comparative SSL,
+including contrastive and non-contrastive methods, and
+lists SSL approaches that use context restoration as the
+pretext task. In the third part, we collect papers that
+emphasize the incorporation of multi-scale features in SSL.
+Comparative SSL methodologies. One of the core ideas
+behind comparative SSL is to extract and encode invari-
+ant and discriminative semantics into representations via
+feature-level comparison. Hjelm et al. [20] proposed Deep In-
+foMax to maximize the mutual information between global
+and local feature vectors of the same input image using
+InfoNCE [28]. Bachman et al. [3] augmented InfoMax by
+conducting a global-local comparison on feature vectors of
+independently-augmented versions of each input. Tian et
+al. [36] increased the number of augmented views of each
+input and extended InfoNCE to multiple views. He et al. [17]
+presented Momentum Contrast (MoCo), which comprises
+a momentum encoder to maintain the consistency among
+positive and negative feature vectors. Different from [3],
+[20], MoCo performs InfoNCE on top of global feature
+vectors only. Compared to MoCo, SimCLR removes the
+momentum architecture and defines InfoNCE on the output
+of a MLP with one hidden layer. Inspired by SimCLR,
+Chen et al. [9] proposed MoCov2, which improves MoCo
+with an additional MLP head and more augmentations.
+SwAV [5] replaces the feature vectors in InfoNCE with
+cluster assignments and introduces the multi-crop strategy
+to increase the number of views of an image with affordable
+computational overhead. Grill et al. [15] proposed BYOL
+(bootstrap your own latent), which eliminates the use of
+InfoNCE in SSL by distilling semantics from positive pairs
+only. Based on BYOL, Chen et al. [10] further removed the
+restriction of the momentum architecture and introduced
+a simple siamese learning framework named SimSiam. In
+practice, SimSiam produces comparable results to MoCov2
+in various downstream tasks. Recently, Zbontar et al. [42]
+simplified SimSiam by measuring the cross-correlation ma-
+trix between the siamese global feature vectors and trying
+to make this matrix close to the identity.
+Comparative SSL, especially InfoNCE-based method-
+ology, has also been widely adopted in medical image
+
+4
+analysis. Zhou et al. [48] proposed to integrate mixup [43]
+into MoCov2, increasing the diversity of both positive and
+negative samples in InfoNCE. Taleb et al. [34] developed 3D
+versions of existing SSL techniques and compared 2D and
+3D SSL approaches on downstream tasks. Azizi et al. [2]
+incorporated multi-instance learning into SimCLR, which
+helps utilize multiple views of each patient. Around the
+same time, Vu et al. [37] developed a method to select posi-
+tive pairs coming from views of the same patient and used
+this strategy to improve MoCov2. There are also a number
+of approaches [6], [40], [41] that tailored comparative SSL
+for semi-supervised medical image segmentation.
+However, the methodologies mentioned above fail
+to
+address
+the
+importance
+of
+integrating
+pixel-level
+information into the high-level representations with rich
+semantics, which is the primary focus of the proposed
+PCRL.
+Context restoration for preserving pixel-level information.
+Restoring original context has been treated as an important
+pretext task in SSL. Pathak et al. [29] first time conducted
+self-supervised feature learning by recovering masked input
+images. Larsson et al. [23] and Zhang et al. [44] performed
+SSL on pixels via predicting RGB color values. For medical
+images, Chen et al. [7] extended the approach in [29] with
+swapped image patches. Zhou et al. [49] showed that adding
+more augmentations to input images brings benefits to SSL.
+Tao et al. [35] presented a volume-wise context transforma-
+tion for 3D medical images. Different from the approaches
+mentioned above, Henaff [19] proposed to predict the next
+context feature vectors following an auto-regressive manner.
+We can see that context restoration is more prevalent in
+medical imaging than in natural images from the above.
+The underlying reason is that medical imaging tasks
+require more pixel-level information to make fine-grained
+yet accurate decisions. On the other hand, we observe
+that comparative SSL can produce representations with
+richer semantics. Thus, it can be beneficial to build a SSL
+framework that simultaneously integrates pixel-level and
+semantic information. As far as we are concerned, none
+of these context restoration based approaches incorporate
+such a combination.
+Multi-scale features in SSL. Although multi-scale features
+have not drawn much attention in existing SSL research,
+it has already been treated as an implicit yet effective
+regularization method for SSL in some methodologies. Deep
+InfoMax [20] contrasts high-level feature vectors with low-
+level feature maps using InfoNCE. To improve Deep Info-
+Max, Bachman et al. [3] proposed to contrast global and
+local feature vectors on multiple levels. In medical image
+analysis, preserving scale information becomes essential,
+as pathologies may show different characteristics on dif-
+ferent scales. In [6], a local contrastive loss is introduced
+to learn distinctive representations of local regions that are
+helpful to per-pixel segmentation. At the same time, global
+feature vectors are used to distill discriminative semantics
+for classification tasks. A similar idea has also been used
+in image registration [25] and one-shot segmentation [46],
+where global and local feature vectors are employed to
+provide information on semantics and position, respectively.
+However, most of these methods only perform SSL on
+two scales, i.e., one global and one local, which cannot fully
+capture multi-scale information. Besides, although these
+approaches emphasize the benefit of introducing local in-
+formation to SSL, they do not exploit pixel-level information
+that is helpful to encode locality. In contrast, this paper pro-
+poses a unified framework that can simultaneously preserve
+semantic, pixel-level, and scale information.
+3
+METHODOLOGY
+We provide an overview of PCRLv2 in Fig. 2. Suppose x
+denotes a batch of input images. We introduce cascaded
+augmentations to distort x in global and local views, respec-
+tively. To be specific, the first-stage augmentations (t1 and t2
+in Fig. 2) mainly consist of global transformations, such as
+flip and rotation, whose goal is to distort the semantics of
+input images from a global perspective. In comparison, the
+second-stage augmentations (t′
+1 and t′
+2 in Fig. 2) comprise
+local pixel-level transformations, such as random noise and
+gaussian blur, which are leveraged to perturb the local
+semantics. After two-stage augmentations, the finally aug-
+mented images x′
+1 and x′
+2 are passed to siamese networks
+to perform pixel restoration and feature comparison, while
+the results of applying t1 and t2 to x, i.e., x1 and x2, serve
+as the ground truth targets for the pixel restoration task (as
+shown in Fig. 2a).
+We perform SSL on the feature pyramid to encode multi-
+scale visual representations. Following the standard practice
+in medical image processing, we build feature pyramids
+using a U-shape model named non-skip U-Net (nsUNet).
+Compared to the typical U-Net architecture [11], [32],
+nsUNet has more feature scales and completely removes
+skip connections, both of which we empirically found help-
+ful in producing better pre-trained representations. During
+the training stage, one scale is first randomly chosen from all
+five feature scales, after which we conduct pixel restoration
+and feature comparison on the siamese feature maps at the
+chosen scale. After the pre-training stage, we fine-tune the
+encoder of nsUNet on various downstream tasks.
+3.1
+Feature pyramid in non-skip U-Net
+U-Net and its series [11], [22], [32] have been known in med-
+ical imaging for their abilities to handle image segmentation
+tasks. The most distinctive characteristic of these models is
+the skip connection that connects equal-resolution low- and
+high-level feature maps. The critical insight is to recover the
+spatial information lost in down-sampling operations of the
+encoder network, such as strided pooling or convolution.
+U-shape models use a feature pyramid to progressively
+incorporate multi-scale details brought by skip connections
+into high-level semantics, making the U-shape architecture
+an ideal choice for conducting context restoration.
+In this paper, we explore the potential of U-shape ar-
+chitecture in SSL from two perspectives: deeply fusing
+semantic and pixel-level information by removing the skip
+connections and introducing multi-scale latent representa-
+tions by conducting SSL on the feature pyramid. For the
+first perspective, we empirically found that skip connec-
+tions provide shortcuts for context restoration, as the low-
+level feature maps contain rich, high-resolution pixel-level
+
+5
+×
+×
+×
+×
+𝐻
+2 × 𝑊
+2
+𝐻×𝑊
+𝐻
+4 × 𝑊
+4
+𝐻
+8 × 𝑊
+8
+𝐻
+16 × 𝑊
+16
+𝐻
+32 × 𝑊
+32
+Skip feature maps
+Feature
+hierarchy
+Down-sampling
+Up-sampling
+×
+No skip connection
+Conv+BN
++ReLU
+×2
+Conv+BN
++ReLU
+×2
+Conv+BN
++ReLU
+×2
+Conv+BN
++ReLU
+×2
+Conv+BN
++ReLU
+×2
+Fig. 3. The architecture of non-skip U-Net (nsUNet). In comparison
+to previous U-Net series, nsUNet removes skip connections, and the
+associated skip feature maps to prevent shortcut solutions to the pixel
+restoration and feature comparison tasks. Besides, nsUNet consists of
+five levels of feature maps (denoted with different colors), where two self-
+supervised tasks are further conducted. Note that this is a 2D illustration
+of nsUNet.
+details. This characteristic does contribute to the restoration
+of context. However, it may prevent the high-level latent
+representations (with rich semantics) from incorporating
+more pixel-level information because the task of providing
+pixel-level details is assigned to low-level feature maps. To
+address this point, we remove the skip connections in U-
+shape architecture and propose non-skip U-Net (nsUNet).
+nsUNet relies on high-level representations without any
+skip connections to restore pixel-level details. In this way,
+the semantic and pixel-level information can be deeply
+fused. Meanwhile, the inherent multi-scale feature maps of
+nsUNet offer the opportunity to construct a feature pyra-
+mid, on top of which SSL can be conducted in multiple
+scales simultaneously.
+Fig. 3 presents the architecture of nsUNet. The feature
+pyramid in nsUNet comprises five levels, ranging from low
+resolution (the down-sampling rate is 32) to full resolution
+(no down-sampling). For 2D input data, we use ResNet-
+18 [18] as the encoder, while for 3D input volumes, we
+build the encoder following [11]. As illustrated in Fig. 3, the
+decoder of nsUNet maintains a shared architecture across
+all pyramid levels, which can be summarized as:
+Fi = Conv-BN-ReLU(Conv-BN-ReLU(Up(Fi−1)),
+(1)
+where i ∈ {1, 2, 3, 4, 5}. F0 denotes the output of the bottle-
+neck block, which has the lowest spatial resolution (down-
+sampling rate=32). Up represents the up-sampling opera-
+tion. Conv-BN-ReLU stands for a sequence of operations,
+including convolution (kernel size=3), batch normalization
+(BN), and ReLU activation. As a result, the bag of feature
+maps {F1, F2, F3, F4, F5} is then forwarded to following
+task-dependent heads to perform pixel restoration and fea-
+ture comparison, respectively and simultaneously.
+3.2
+Multi-scale pixel restoration
+As the name implies, multi-scale pixel restoration aims to
+preserve pixel-level and scale information in latent visual
+representations simultaneously. To achieve this goal, we ask
+the network to recover the exact pixel-level details across
+different scales, where each pair of siamese feature maps
+share one pixel restoration head. In contrast, PCRLv1 only
+restores pixel details at the full resolution, which inevitably
+loses multi-scale properties in learned representations.
+As shown in Fig. 4a, the input images x′
+1 and x′
+2 are
+intentionally corrupted via various pixel-level augmenta-
+tions, such as guassian blur and random noise. For each
+training iteration, we first randomly choose a feature scale
+Fi from {F1, F2, F3, F4, F5}. Then, we pass Fi to the pixel
+restoration head f R
+i (·) for the i-th scale, whose internal
+processing procedure can be summarized as:
+f R
+i (Fi) = Conv(Conv-BN-ReLU(Fi)),
+(2)
+where all convolution layers use a kernel size of 3 and a
+stride of 2. Similarly, we apply the shared pixel restoration
+head to the paired siamese feature map Fs
+i to acquire the
+prediction output f R
+i (Fs
+i ):
+f R
+i (Fs
+i ) = Conv(Conv-BN-ReLU(Fs
+i )),
+(3)
+Lastly, we employ the mean square error (MSE) loss to
+measure the reconstruction errors between f R
+i (Fi) and x1.
+For the siamese feature pyramid, we apply MSE loss to
+f R
+i (Fs
+i ) and x2. The cost function LR of the pixel restoration
+task in each training iteration (with mini-batch optimiza-
+tion) is as follows:
+LR =
+N
+�
+j=1,
+∀i∈H
+1[i==j] [MSE(f R
+i (Fi), x1) + MSE(f R
+i (Fs
+i ), x2)],
+(4)
+where N = 5 denotes the number of scales in each feature
+pyramid. H = {1, 2, 3, 4, 5} stands for the scale index.
+1[i==j] is an indicator function, which is equal to 1 when
+i==j is true (otherwise, 0). The explanation of LR can be
+summarized as: (i) randomly choose a feature scale Fi
+from all five scales; (ii) pass Fi and its siamese feature
+map Fs
+i to the shared task head f R
+i (·); (iii) calculate the
+MSE loss between the outputs of f R
+i (·) and uncorrupted
+images {x1, x2}. By reconstructing the same targets x1/x2
+across different feature scales, LR can encode the pixel-level
+information into multi-scale latent visual representations.
+3.3
+Multi-scale feature comparison
+PCRLv1 employs a hybrid way to conduct contrastive
+learning with the help of the momentum encoder [17]
+and mixup [43]. However, this contrastive deployment is
+complex, making PCRLv1 heavy, thus troublesome to im-
+plement and improve. To address these issues, PCRLv2
+replaces the hybrid contrastive strategies in PCRLv1 with
+the multi-scale comparison. Inspired by [10], multi-scale
+comparison conducts SSL with siamese learning, whose
+key operation is to attract the same image’s siamese views.
+Different from [10] that conducts feature comparison on one
+scale, we propose to preserve the discriminative semantics
+across different feature scales, which forces the model to
+preserve multi-scale self-supervised representations. In the
+following, we provide technical details of performing the
+multi-scale comparison.
+
+6
+Conv
+BN
+ReLU
+Conv
+Conv
+BN
+ReLU
+Conv
+𝑥!
+𝑥#
+Shared
+𝑥!
+"
+…
+𝑥#
+"
+…
+Siamese scale
+Chosen scale
+(a) Architectural details of the pixel restoration head
+Siamese scale
+Chosen scale
+GAP
+GAP
+BN
+FC
+BN
+ReLU
+FC
+BN
+FC
+BN
+ReLU
+FC
+Predictor
+Predictor
+Shared
+𝑥!
+"
+…
+𝑥#
+"
+…
+(b) Architectural details of the feature comparison head
+Fig. 4. Architectural details of the pixel restoration and feature comparison heads. Conv, BN, GAP, and FC denote the convolution, batch
+normalization, global average pooling, and fully-connected layers, respectively. The kernel size of all convolution layers is 3, and the convolution
+stride is set to 1. Note that each pair of siamese feature maps share one pixel restoration head and one feature comparison head, while different
+feature scales employ distinct task heads.
+Given the feature maps at a randomly chosen scale Fi,
+we pass them through a global average pooling layer and
+a shared batch normalization layer (as shown in Fig. 4b) to
+acquire 1D representations vi:
+vi = BN(GAP(Fi)).
+(5)
+We can get vs
+i by processing the siamese feature maps Fs
+i in
+a similar way.
+Next, we forward vi to the shared predictor fP(·), whose
+architecture is displayed in Fig. 4b and can be summarized
+as:
+fP(vi) = FC(FC-BN-ReLU(vi)).
+(6)
+where FC denotes the fully-connected layer. FC-BN-ReLU
+stands for a sequence of layers, which are the fully-
+connected layer, batch normalization layer, and ReLU ac-
+tivation. Similarly, we can acquire fP(vs
+i ) by passing vs
+i to
+the same predictor.
+We measure the similarity between siamese feature vec-
+tors with the cosine similarity:
+cos(vi, fP(vs
+i )) =
+vi
+∥vi∥2
+·
+fP(vs
+i )
+∥fP(vs
+i )∥2
+,
+(7)
+where || · ||2 denotes the L2 normalization. Symmetrically,
+we calculate cos(fP(vi), vs
+i ) as follows:
+cos(fP(vi), vs
+i )) =
+fP(vi)
+∥fP(vi)∥2
+·
+vs
+i
+∥vs
+i ∥2
+.
+(8)
+Finally, the cost function LC of multi-scale feature compari-
+son can be summarized as:
+LC =
+N
+�
+j=1,
+∀i∈H
+−1
+21[i==j] [cos(sg(vi), fP(vs
+i ))
++ cos(fP(vi), sg(vs
+i ))].
+(9)
+N
+=
+5 denotes the number of feature scales. H
+=
+{1, 2, 3, 4, 5} stands for the scale index. Following [10], we
+apply the stop-gradient operation (denoted as sg) in Eq. 9
+to prevent the network optimizer from finding shortcut
+solutions.
+Minimizing LC requires the model to maximize the
+similarity between siamese latent features across all feature
+scales. In this way, scale invariance can be implicitly incor-
+porated into the preserved latent semantics.
+1
+2
+1
+2
+3
+4
+5
+6
+7
+8
+Randomly crop two
+global patches with
+an IoU constraint
+Find the minimum 3D bounding box
+Randomly crop
+local patches
+1
+2
+3
+4
+5
+6
+7
+8
+3D Global views
+3D Local views
+Fig. 5. Illustration of sub-crop. Given a 3D local volume, we first randomly
+crop two large patches, where an intersection over union (IoU) constraint
+is applied to guarantee that two patches are partly overlapped. These
+two large patches are considered as x1 and x2 in Fig. 2 and will be
+passed to the siamese architecture to conduct the following multi-scale
+pixel restoration and feature comparison tasks. To acquire local views,
+we compute the minimum 3D bounding box of two large patches, after
+which random crop is applied to extract multiple local patches. Finally,
+we reshape these local patches to a fixed size and forward them to the
+network to extract local representations.
+3.4
+From multi-crop to sub-crop
+Multi-crop [5] has been known as a helpful strategy to im-
+prove SSL performance in natural images, which increases
+the number of input views by sampling several standard
+resolution crops and more low-resolution crops from the
+original input. One key insight behind multi-crop is to
+capture relations between parts of a scene or an object, while
+low-resolution views ensure a controllable increase in the
+computational cost.
+When applied to medical images, multi-crop works well
+in 2D X-ray data but leads to the non-convergence of the
+model in 3D volume data (such as CT and MRI). After
+careful investigation, we found the root of this problem
+lies in the contradiction between the limited input size and
+many candidate crops in three-dimensional space. Specif-
+
+7
+ically, on the one hand, we cannot afford large-sized 3D
+inputs because processing them with 3D deep models often
+costs dramatic GPU memory. On the other hand, if we
+overly reduce the size of 3D inputs, the sampled views
+would be too dispersed to guarantee the model capture the
+local-global associations.
+To mitigate the above issue, we introduce sub-crop to
+replace multi-crop in 3D medical images. The core idea of
+sub-crop is straightforward: reducing the sampling space. As
+illustrated in Fig. 5, sub-crop mainly consists of three steps:
+(i) randomly crop two extensive global views with an IoU
+constraint; (ii) find the minimum 3D bounding box over the
+cropped global patches; (iii) randomly crop multiple local
+patches within the 3D bounding box. There are two critical
+operations in sub-crop: the constraint of IoU on global views
+and the sampling of local patches within the minimum
+bounding box. In practice, the first operation guarantees the
+global-global association by ensuring the overlap between
+large patches larger than a fixed threshold. The second
+operation mitigates the disperse problem of local views and
+helps the model to discover local-global relations.
+3.5
+Overall training objective
+After applying multi-crop/sub-crop to medical images, we
+can acquire two global views {g1, g2} and ˆN local views
+{l1, l2, ..., l ˆ
+N}. For clarification, we denote the associated in-
+puts in notations of loss functions. For instance, LC(g1, g2)
+means we calculate LC on top of the extracted siamese
+representations of two global views, where g1 and g2 can
+be regarded as a pair of siamese images. At last, the overall
+training objective of PCRLv2 can be formalized as follows:
+LTotal(g1, g2, l1, ..., l ˆ
+N) =LR(g1, g2) + LC(g1, g2)
++
+�
+m∈{1,2}
+ˆ
+N
+�
+k=1
+LC(lk, gm).
+(10)
+There are three terms in LTotal: LR(g1, g2), LC(g1, g2), and
+�
+m∈{1,2}
+� ˆ
+N
+k=1 LC(lk, gm). The first term is designed to
+preserve pixel-level details in multi-scale learned repre-
+sentations. The second term addresses the importance of
+encoding multi-scale semantics into latent features. The last
+term aims to capture the multi-scale global-local semantic
+relations.
+3.6
+Short discussion: PCRLv2 vs. PCRLv1
+Simpler. PCRLv1 combines the context restoration and
+comparative SSL via transformation-conditioned attention
+and cross-model mixup. These two components make
+the framework heavy, less intuitive, and not easy to
+implement. Compared to PCRLv1, PCRLv2 exploits a
+simpler yet more intuitive design to incorporate pixel-level
+and semantic information via multi-scale learning. As
+aforementioned, PCRLv2 can be formulated as a simple
+multi-task optimization problem whose objective function
+maximizes the preservation of multi-level information in
+latent visual representations. These characteristics make it
+easier for both implementation and potential expansion.
+Faster. PCRLv1 makes heavy use of mixup (to both inputs
+and features) in its implementation, which is found to
+deliver performance gains. In PCRLv2, we eliminate mixup
+strategies and cut the training time in half. In addition,
+PCRLv2 requires less running memory in GPUs during
+the training stage, making it more practical in real-world
+scenarios.
+4
+EXPERIMENTS
+In this section, we first conduct thorough ablation studies to
+investigate the influence of different modules in PCRLv2.
+Then, we evaluate the effectiveness of PCRLv2 on both
+2D and 3D medical imaging tasks, including chest pathol-
+ogy classification, pulmonary nodule detection, abdominal
+organ segmentation, and brain tumor segmentation. For
+model evaluation, we follow the pre-training (on source
+data)→fine-tuning (on target data) protocol and employ two
+settings, which are semi-supervised learning and transfer
+learning. In the first setting, the source and target data come
+from the same dataset. Specifically, we first pre-train the
+model using all training data without labels, and then fine-
+tune the pre-trained model with limited annotations. As for
+transfer learning (the second setting), we pre-train and fine-
+tune the model on different datasets. Different from semi-
+supervised learning, we fine-tune the pre-trained model
+with both limited and full annotations in transfer learning.
+4.1
+Datasets
+NIH ChestX-ray (2D) [38] is made up of 112,120 X-
+ray
+scans
+from
+30,805
+patients.
+There
+are
+fourteen
+different chest pathologies in NIH ChestX-ray, including
+atelectasis, cardiomegaly, consolidation, edema, effusion,
+emphysema, fibrosis, hernia, infiltration, mass, nodule,
+pleural thickening, pneumonia, and pneumothorax. The
+labels of radiographs were automatically extracted from
+associated
+radiology
+reports
+using
+natural
+language
+process (NLP) techniques. We use NIH ChestX-ray in
+semi-supervised learning in our experiments and treat it as
+the target dataset in transfer learning.
+CheXpert (2D) [21] involves 224,316 chest radiographs
+from 65,240 patients for the presence of 14 common
+chest
+radiographic
+observations:
+no
+finding,
+enlarged
+cardio, cardiomegaly, lung opacity, lung Lesion, edema,
+consolidation,
+pneumonia,
+atelectasis,
+pneumothorax,
+pleural
+effusion,
+pleural
+other,
+fracture,
+and
+support
+devices. Similar to NIH ChestX-ray, an NLP labeler was
+developed to detect the presence of 14 observations in
+radiology
+reports
+automatically.
+In
+practice, CheXpert
+serves as the source data in transfer learning.
+LUNA (3D) [33] was collected for the automatic detection
+of pulmonary nodules, which involves 888 annotated
+thoracic computed tomography (CT) scans. LUNA is a
+cherry-picked subset of LIDC-IDRI [1], which excludes
+scans with a slice thickness greater than 3mm, inconsistent
+slice spacing, or missing slices. In the 888 scans, a total of
+5,855 annotations were made by the radiologists, where
+only nodules ≥ 3mm are categorized as relevant lesions,
+
+8
+and at least one radiologist checks each nodule. On LUNA,
+we perform semi-supervised learning and transfer learning
+experiments. For transfer learning, LUNA is mainly used
+for self-supervised pre-training.
+LiTS (3D) [4] releases 131 abdominal CT Volumes and
+associated annotations for training and validation. There
+are two types of labels in LiTS: the liver and tumor. In this
+paper, we only utilize the ground truth masks of the liver
+to evaluate the effectiveness of various SSL algorithms. The
+task on LiTS is abdominal organ segmentation, where LiTS
+is used for fine-tuning in transfer learning.
+BraTS (3D) has been known as a series of challenges in brain
+tumor segmentation. In this paper, we perform experiments
+on the released 351 magnetic resonance imaging (MRI) scans
+of BraTS 2018. There are three classes in BraTS: whole tumor
+(WT), tumor core (TC), and enhancing tumor (ET). Similar
+to the role of LiTS, BraTS serves as the target data in transfer
+learning.
+4.2
+Baselines
+A variety of SSL baselines are included in our extensive
+experiments, which can be roughly divided into three
+categories: 2D specific methods, 3D specific approaches,
+and generic (2D & 3D) methodologies. Details of baselines
+in each category are listed below.
+2D specific SSL methodologies consist of ImageNet-based
+pre-training (IN) [14], Comparing to Learn (C2L) [48],
+and Simple Siamese Learning (SimSiam) [10]. IN is the
+most widely adopted pre-training methodology, which
+conducts supervised pre-training on one of the biggest
+natural image datasets, i.e., ImageNet [14]. C2L is a recently
+proposed SSL approach based on momentum contrast (i.e.,
+MoCov1 [17] and MoCov2 [9]). SimSiam is a simple siamese
+SSL framework that eliminates the barrier of negative
+samples in contrastive learning and the use of a momentum
+encoder in BYOL [15]. Besides, we compare PCLRv2 against
+SimSiam to highlight the significance of the preserved
+pixel-level information and multi-scale features.
+3D specific SSL methodologies include Rubik’s cube++ [35]
+and 3D-CPC [34]. Rubik’s cube++ is the most recent SSL
+approach built on top of context restoration for 3D
+medical images. It adopts a volume-wise transformation
+for context permutation. In comparison, 3D-CPC is based
+on contrastive predictive encoding [19], a variation of
+contrastive learning, and demonstrates the most superior
+performance among different SSL approaches investigated
+in [34].
+Generic SSL methodologies involve train from scratch (TS),
+Model Genesis (MG) [49], TransVW [16], and PCRLv1 [47]
+(the conference version of our approach). MG resorts to ag-
+gressive augmentations to generate corrupted input images,
+based on which the model is asked to restore the original in-
+puts. TransVW improves MG by appending an intermediate
+classification head to encode anatomical patterns explicitly.
+PCRLv1 first proposes simultaneously preserving semantic
+and pixel-level information in SSL.
+4.3
+Implementation details
+Dataset pre-processing for pre-training. On NIH ChestX-
+ray and CheXpert, each input image is resized to 224×224
+after random crop. On LUNA, we randomly crop a
+volume from the whole CT scan with a random size
+from {64×64×32, 96×96×64, 96×96×96, 112×112×64}.
+Each cropped volume is then resized to 64×64×32. Each
+voxel’s Hounsfield Unit (HU) in the crop is truncated to
+[-1000,1000]. If a voxel’s HU is lower than -150, we regard
+it as a background voxel. In practice, if over 85% voxels
+within a crop belong to the background, we would not use
+this crop in pre-training.
+Dataset pre-processing for fine-tuning. For NIH ChestX-
+ray and CheXpert, we follow the same pre-processing
+procedures as in the pre-training stage. On LUNA, we
+randomly crop a volume for each training iteration, and the
+size of each crop is 48×48×48. On LiTS, we first localize
+the liver and expand the target volume by 30 slices on
+each axis. After random crop, the size of each crop is
+256×256×64. Unlike LUNA, we truncate the HU of each
+voxel to [-200, 200]. For BraTS, the size of each random crop
+is 112×112×112×4.
+Data augmentation and multi-crop/sub-crop. As shown in
+Fig. 2, there are two types of augmentations, i.e., global and
+local augmentations. Specifically, for 2D tasks, the global
+augmentation includes random crop, random horizontal
+flip, and random rotation. The local augmentation involves
+random grayscale, gaussian blur, and cutout. In comparison,
+for 3D tasks, the global augmentation consists of random
+flip and random affine. Local augmentation strategies are
+applied, including Gaussian blur, random noise, random
+gamma, and random swap. Note that all 3D augmentations
+are implemented following [30]. As for multi-crop in 2D
+tasks, we resort to the scale factor of random crop2 to
+generate global and local views. Specifically, we set the
+range of scale to [0.3, 1] to generate two global views. For six
+local views, the scale range is set to [0.05, 0.3]. Both global
+and local views are resized to 224×224. As for sub-crop
+in 3D tasks, we randomly sample two global views with
+a random size from {64×64×32, 96×96×64, 96×96×96,
+112×112×64}. The IoU constraint (i.e., threshold) between
+two global views is 0.3. Then, we find the minimum
+bounding box of global views, from which six local views
+are randomly cropped, each with a random size from
+{8×8×8, 16×16×16, 32×32×16, 32×32×32}. After random
+crop, all 3D global views are resized to 64×64×32, while all
+local views are resized to 16×16×16.
+Training and evaluation details. We use stochastic gradient
+descent (SGD) with momentum as the default optimizer,
+where the momentum is set to 0.9. The initial learning rate
+is 1e-2, and we employ the cosine annealing strategy for
+learning rate decay. We set the weight decay to 1e-5. The
+number of training epochs is 240. The batch sizes of 2D pre-
+training and fine-tuning (on NIH ChestX-ray or CheXpert)
+are 256 and 512, respectively. As for 3D pre-training, the
+2. https://pytorch.org/vision/main/generated/torchvision.
+transforms.RandomResizedCrop.html
+
+9
+0
+15
+Epochs
+-0.5
+-1.5
+Training MSE loss (log10)
+w/ skip
+w/o skip
+Fig. 6. Influence of skip connections in pixel restoration. We display the
+loss curve of mean square error (MSE) in the first 15 epoches.
+TABLE 1
+Impact of skip connections on chest pathology identification (NIH
+ChestX-ray), brain tumor segmentation (BraTS), and abdominal organ
+segmentation (LiTS). On NIH, We use 95% unlabeled training data for
+pre-training, while the rest 5% data with labels are used for fine-tuning.
+On BraTS and LiTS, we use 10% labeled data for fine-tuning.
+Datasets
+w/o skip
+w/ skip
+Gain
+NIH
+76.6
+75.4
+1.2
+BraTS
+73.0
+71.5
+1.5
+LiTS
+79.0
+77.6
+1.4
+batch size (on LUNA) is 32. For 3D fine-tuning tasks, the
+batch sizes on LUNA, LiTS, and BraTS are 32, 4, and
+4, respectively. The evaluation metric on NIH ChestX-ray,
+CheXpert, and LUNA is AUROC (Area Under the Receiver
+Operating Characteristics). For segmentation tasks on LiTS
+and BraTS, we use Dice similarity as the evaluation metric.
+We use 70%, 10%, and 20% of the whole dataset to build the
+training, validation, and test sets. In particular, for semi-
+supervised learning, we construct the pre-training set by
+removing a specific amount of data from the entire training
+set. At the same time, the remainder is used as the training
+set for fine-tuning. Binary cross-entropy loss is used for the
+fine-tuning of NIH ChestX-ray, CheXpert, and LUNA, while
+Dice loss is used for the fine-tuning of LiTS and BraTS.
+4.4
+Ablation studies
+Impact of skip connections on pixel restoration. In Fig. 6,
+we present the mean square error (MSE) loss (cf. Eq. 4)
+curves during the training stage. We see that the MSE
+loss, with skip connections, decreases rapidly in the first
+15 training epochs. In comparison, the proposed nsUNet
+(w/o skip) slows down the decreasing rate of MSE loss.
+These phenomena are consistent with the role of skip
+connections, which bridges the gap between low-level
+pixel details and high-level latent semantics. The existence
+of skip connections makes it easier to restore pixels
+by incorporating pixel-level details from low-level but
+high-resolution feature maps. However, nsUNet removes
+skip connections, avoiding shortcut solutions to context
+restoration. Although this design makes it harder to restore
+pixels (higher loss values in Fig. 6), it helps encode pixel-
+level information into high-level semantic representations.
+ℱ#
+ℱ$
+ℱ%
+ℱ"
+ℱ!
+ℱ#
+&
+ℱ$
+&
+ℱ%
+&
+ℱ"
+&
+ℱ"
+&
+(a) Pairwise
+ℱ#
+ℱ$
+ℱ%
+ℱ"
+ℱ!
+ℱ#
+&
+ℱ$
+&
+ℱ%
+&
+ℱ"
+&
+ℱ"
+&
+(b) Cross-scale
+Fig. 7. Two choices of how to conduct siamese feature comparison for
+multiple feature scales. Here, we primarily consider pairwise feature
+comparison and cross-scale feature comparison.
+Such advantage can be verified by the performance gains in
+Table 1, where removing skip connections brings over 1%
+improvement to chest pathology identification, brain tumor
+segmentation, and abdominal organ segmentation.
+How to conduct siamese feature comparison for multiple
+feature scales? We illustrate two intuitive choices in Fig. 7.
+Besides the adopted pairwise comparison manner (Fig. 7a),
+another obvious choice is to compare siamese features
+following a crossed way (a similar strategy was used
+in [3]). As shown in Fig. 7b, the cross-scale comparison
+aggressively compares siamese features across all feature
+scales. The motivation behind is to introduce multi-scale
+latent representations by coupling features across different
+scales. Table 2 reports the experimental results of pairwise
+and cross-scale siamese feature comparison. We find that
+cross-scale feature comparison slightly deteriorates the
+performance of semi-supervised pathology identification
+by 0.6 percents. The underlying reason might be that the
+features in each scale maintains distinct characteristics,
+and neglecting these discrepancies can lead to degenerate
+feature representations.
+Investigation of different modules in PCRLv2. In Table 3,
+we study and report the impact of different modules on
+the whole tumor (WT) and enhancing tumor (ET) classes of
+BraTS. Note that in practice, most instances of WT are much
+larger than instances from ET, making ET instances harder
+to segment. Besides, we also present the transfer learning
+results on NIH ChestX-ray.
+TABLE 2
+Results of pairwise and crossed siamese feature comparison
+(semi-supervised learning on NIH ChestX-ray). The ratio of unlabeled
+to labeled data is 9.5:0.5.
+Pairwise
+Crossed [3]
+Gain
+Mean AUROC
+76.6
+76.0
+0.6
+First of all, we investigate the influence of pixel restora-
+tion (row 0) and feature comparison (row 1), respectively.
+We directly reconstruct the full resolution uncorrupted im-
+ages for the pixel restoration task while siamese feature
+comparison is conducted on the last-layer output of the
+encoder. Comparing row 0 with row 1, we see that the
+context restoration task is more advantageous in segmenta-
+tion of small tumor regions (i.e., ET) while the comparative
+SSL is more capable of dealing with large tumor regions
+(i.e., WT) and chest pathologies. Such comparison shows
+
+10
+TABLE 3
+Impact of different modules in PCRLv2. Res. and Comp. denote the tasks of pixel restoration and feature comparison, respectively. S (N) means
+there are N scales included. MC and SC stand for the multi-crop and proposed sub-crop strategies, respectively. WT and ET denote classes of the
+whole tumor and enhancing tumor in BraTS, respectively. In most cases, instances from WT are much larger (in size) than those of ET. We
+performed these experiments by first using LUNA for self-supervised pre-training, and then we fine-tune the pre-trained model on BraTS using
+10% labeled data. NIH denotes the transfer learning on chest pathology identification, where we use CheXpert for pre-training and fine-tune the
+pre-trained model with 50% labeled data from NIH ChestX-ray.
+#
+Res.
+Comp.
+S (3)
+S (5)
+MC
+SC
+WT (BraTS)
+ET (BraTS)
+NIH
+0
+✓
+74.2
+64.9
+78.2
+1
+✓
+76.4
+63.8
+78.5
+2
+✓
+✓
+76.2
+64.6
+80.9
+3
+✓
+✓
+✓
+76.9
+66.1
+81.5
+4
+✓
+✓
+✓
+77.2
+66.8
+82.0
+5
+✓
+✓
+✓
+✓
+fail
+fail
+82.5
+6
+✓
+✓
+✓
+✓
+77.7
+67.2
+82.7
+that semantic information preservation may be more helpful
+to the detection of large objects, while segmenting small
+objects requires the incorporation of pixel-level information.
+In row 2, we can already acquire noticeable performance
+gains by directly combining pixel restoration and feature
+comparison.
+Next, we show that multi-scale representations benefit
+both pixel restoration and feature comparison tasks. By
+conducting both tasks on 3 scales, we observe a 0.7-percent
+improvement on WT, a 1.5-percent gain on ET, and a
+0.6-percent improvement on chest pathology classification.
+These results show that introducing multiple scales is more
+helpful to the segmentation of small regions. Moreover,
+by increasing the number of scales from 3 to 5, we can
+improve the accuracy of all three tasks consistently. Not
+surprisingly, ET benefits the most from the introduction of
+multiple scales, indicating the necessity of utilizing multi-
+scale representations in medical image segmentation.
+Last but not the least, we investigate the significance of
+multi-crop (row 4) and sub-crop (row 5). We empirically
+found that directly applying multi-crop to 3D medical vol-
+umes leads to the failure of model training. The underlying
+reason might be that it is difficult for cropped global and
+local views to maintain clear spatial relations in the 3D
+space as in the 2D space. In contrast, sub-crop can provide
+consistent performance gains on both types of tumor regions
+by successfully preserving the spatial relations in latent
+representations. When applying sub-crop to 2D X-rays,
+we observe a marginal improvement over multi-crop. The
+underlying reason is that sub-crop is proposed to handle
+dispersed sampled views in a 3D space to guarantee the
+model captures local-global relations. However, in a 2D
+space, the sampled views usually (partly) overlap.
+4.5
+Semi-supervised chest pathology identification
+Table 4 presents the experimental results of applying semi-
+supervised learning on NIH ChestX-ray. Specifically, we use
+a specific amount of the training set (denoted as the labeling
+ratio in Table 4) as labeled data while the remaining training
+data is used for self-supervised pre-training.
+From Table 4, we see that self-supervised pre-training
+can dramatically boost the performance compared to train
+from scratch (TS), which verify the necessity of conduct-
+ing pre-training in medical imaging. Comparing MG with
+TransVW, they show similar performance in different label-
+ing ratios. Such comparison is easy to explain as TransVW
+is built upon MG, and both are based on context restora-
+tion. TransVW performs slightly better than MG, as it
+incorporates an additional classification head to encode
+more semantics. Compared to context restoration based
+methods, comparative methodologies (C2L and SimSiam)
+display better overall and class-specific results, especially
+in small labeling ratios. The underlying reason might be
+that semantic information is more critical than pixel-level
+information in chest pathology detection. As for C2L and
+SimSiam, C2L performs better when the amount of labeled
+data is quite limited. However, SimSiam gradually produces
+better diagnosis results as the labeling ratio increases.
+After incorporating the semantic, pixel-level, and scale
+information into a unified framework, PCRLv2 outperforms
+various SSL baselines in different labeling ratios signifi-
+cantly. It surpasses the previous conference version by clear
+margins, i.e., PCRLv1. Particularly, PCRLv2 seems to have
+more advantages in small labeling ratios. For instance, when
+the labeling ratio is 5%, PCRLv2 outperforms PCRLv1 by
+2.5 percents on average, which verifies the significance of
+multi-scale latent representations.
+4.6
+Semi-supervised pulmonary nodule detection
+In Table 5, we report the experimental results of semi-
+supervised pulmonary nodule detection. Interestingly, we
+observe narrowed performance gaps between TS and SSL
+baselines than those reported in Table 4. One possible ex-
+planation is that the task of detecting pulmonary nodules
+is less sensitive to the amount of labeled data. Among
+all SSL baselines, Cube++ gives better performance when
+utilizing small amounts of labeled data, while 3D-CPC is
+more advantageous in large labeling ratios. In addition, we
+see TransVW quickly catching up with MG and Cube++ as
+the labeling ratio increases.
+PCRLv1 outperforms previous SSL approaches in dif-
+ferent labeling ratios by large margins. After incorporat-
+ing multi-scale latent representations, PCRLv2 consistently
+surpasses PCRLv1 in a range of labeling ratios. When the
+baseline SSL methods show similar performance as the
+labeling ratio increases, PCRLv2 can still provide impressive
+improvements over PCRLv1 and previous SSL approaches.
+
+11
+TABLE 4
+Semi-supervised chest pathology identification (on NIH ChestX-ray). The labeling ratio denotes the amount of data with labels in the training set
+that is used for fine-tuning while the remaining data in the training set is used for self-supervised pre-training. The best results are bolded.
+Labeling ratio
+Methodology
+Mean
+Atelectasis
+Cardiomegaly
+Effusion
+Infiltration
+Mass
+Nodule
+Pneumonia
+Pneumothorax
+Consolidation
+Edema
+Emphysema
+Fibrosis
+Pleural Thick.
+Hernia
+5%
+TS
+61.8
+58.8
+72.0
+68.8
+51.5
+63.8
+49.2
+57.4
+67.4
+61.5
+71.0
+62.7
+58.1
+60.0
+63.1
+MG [49]
+66.4
+63.4
+74.1
+72.9
+53.5
+67.2
+54.3
+59.9
+71.3
+66.5
+77.0
+65.8
+64.5
+62.8
+76.2
+TransVW [16]
+66.5
+64.2
+72.9
+72.2
+54.8
+69.4
+55.7
+59.6
+71.0
+64.8
+77.4
+66.6
+63.6
+62.8
+75.6
+C2L [48]
+71.7
+69.9
+77.9
+76.2
+59.1
+73.4
+60.0
+64.5
+76.2
+71.4
+80.3
+76.1
+69.9
+68.4
+80.4
+SimSiam [10]
+71.7
+68.9
+79.3
+77.8
+58.7
+73.0
+61.0
+65.4
+76.2
+72.1
+81.7
+75.1
+69.6
+68.1
+76.8
+PCRLv1 [47]
+74.1
+70.1
+80.3
+79.3
+61.8
+76.8
+64.6
+68.6
+77.2
+72.8
+83.7
+77.4
+71.3
+72.7
+80.8
+PCRLv2
+76.6
+75.7
+81.0
+80.3
+64.0
+76.8
+68.7
+70.7
+83.2
+77.5
+87.8
+79.2
+72.5
+73.2
+81.8
+10%
+TS
+68.1
+65.8
+77.6
+74.4
+57.1
+69.4
+54.8
+63.0
+72.9
+68.3
+78.8
+68.2
+64.3
+66.4
+72.5
+MG [49]
+70.0
+67.1
+78.1
+76.1
+57.2
+72.8
+57.5
+63.3
+75.5
+70.9
+79.5
+68.8
+67.4
+68.0
+77.6
+TransVW [16]
+70.2
+66.6
+78.9
+74.9
+58.4
+71.2
+59.5
+64.8
+72.6
+70.4
+79.4
+70.7
+67.2
+68.3
+79.5
+C2L [48]
+74.1
+72.3
+81.7
+79.9
+60.2
+74.6
+62.7
+67.6
+78.7
+73.9
+83.5
+78.2
+72.8
+69.8
+81.4
+SimSiam [10]
+74.0
+71.2
+81.4
+78.9
+60.2
+75.5
+63.2
+67.3
+78.7
+73.2
+83.5
+77.7
+72.5
+71.8
+80.8
+PCRLv1 [47]
+76.2
+73.6
+82.9
+81.2
+64.7
+77.1
+66.7
+69.7
+79.8
+74.5
+86.9
+78.8
+75.6
+74.2
+81.1
+PCRLv2
+78.2
+77.2
+84.3
+84.4
+67.4
+77.5
+68.9
+71.6
+84.4
+77.8
+89.0
+79.3
+76.1
+74.0
+82.4
+20%
+TS
+71.5
+68.9
+80.7
+77.5
+60.2
+73.6
+58.7
+66.2
+76.1
+71.7
+82.9
+72.2
+69.0
+68.7
+74.7
+MG [49]
+73.9
+71.9
+83.0
+80.0
+62.3
+75.2
+62.2
+67.5
+79.0
+73.3
+83.6
+73.4
+71.0
+70.6
+81.4
+TransVW [16]
+74.3
+71.6
+82.5
+80.1
+62.3
+76.7
+62.8
+69.2
+78.2
+73.5
+83.8
+75.4
+72.2
+71.2
+80.3
+C2L [48]
+76.4
+74.2
+83.9
+81.7
+63.8
+77.3
+64.7
+70.3
+81.5
+75.5
+86.0
+80.2
+75.2
+73.4
+81.8
+SimSiam [10]
+76.5
+73.8
+84.0
+81.4
+63.2
+78.2
+64.7
+69.6
+82.1
+76.2
+86.4
+80.7
+75.0
+73.9
+81.7
+PCRLv1 [47]
+78.8
+75.4
+86.2
+83.6
+65.1
+79.9
+69.6
+72.0
+82.3
+79.9
+88.3
+82.6
+76.5
+75.9
+81.9
+PCRLv2
+79.9
+78.1
+87.2
+85.9
+68.2
+80.5
+69.9
+72.5
+85.3
+80.4
+89.2
+83.1
+77.5
+77.0
+83.5
+30%
+TS
+73.4
+70.6
+81.9
+79.1
+61.6
+75.5
+60.7
+68.8
+78.3
+72.7
+84.3
+74.1
+70.3
+70.9
+78.9
+MG [49]
+76.1
+74.3
+84.4
+82.1
+63.6
+78.3
+64.4
+69.6
+81.2
+75.8
+85.6
+75.9
+73.6
+73.6
+82.8
+TransVW [16]
+76.7
+74.9
+84.1
+81.9
+64.9
+79.0
+65.3
+70.9
+80.3
+76.2
+86.5
+78.6
+74.5
+74.2
+82.1
+C2L [48]
+77.5
+74.3
+84.8
+82.6
+64.6
+78.3
+66.3
+71.5
+83.0
+76.8
+87.6
+81.3
+76.5
+74.4
+82.9
+SimSiam [10]
+78.0
+75.4
+85.1
+82.9
+65.0
+79.4
+67.0
+71.4
+83.4
+77.4
+87.8
+82.8
+76.1
+75.5
+82.7
+PCRLv1 [47]
+79.0
+75.5
+86.6
+83.8
+65.9
+80.7
+70.2
+72.8
+82.9
+80.4
+88.9
+83.3
+76.6
+76.5
+81.9
+PCRLv2
+81.1
+78.4
+87.6
+86.6
+69.6
+82.8
+72.0
+74.0
+86.2
+81.0
+89.9
+84.4
+79.5
+79.0
+84.6
+40%
+TS
+75.4
+72.6
+83.6
+81.5
+62.9
+77.3
+63.3
+70.1
+80.3
+74.9
+85.5
+76.4
+72.5
+73.0
+81.8
+MG [49]
+77.3
+75.4
+86.0
+83.3
+65.1
+79.0
+65.1
+70.8
+82.1
+77.0
+87.3
+76.7
+74.8
+74.9
+83.5
+TransVW [16]
+77.6
+75.0
+85.1
+82.7
+65.2
+79.7
+66.5
+72.0
+81.0
+76.7
+87.2
+79.2
+75.5
+76.5
+83.7
+C2L [48]
+79.0
+76.0
+86.1
+84.3
+66.0
+80.0
+67.9
+72.5
+84.1
+78.5
+88.5
+83.7
+77.9
+76.6
+83.8
+SimSiam [10]
+79.4
+76.7
+86.7
+84.7
+67.0
+80.9
+69.0
+73.1
+84.4
+78.9
+88.9
+83.5
+77.7
+76.6
+83.4
+PCRLv1 [47]
+79.9
+76.7
+87.1
+84.9
+67.1
+82.7
+72.2
+73.3
+83.6
+80.6
+89.2
+83.8
+77.3
+76.9
+83.2
+PCRLv2
+81.5
+78.7
+87.8
+87.0
+69.8
+83.2
+72.5
+74.7
+86.3
+81.2
+90.2
+84.9
+80.0
+79.4
+85.0
+TABLE 5
+Semi-supervised pulmonary nodule detection (on LUNA). The labeling
+ratio indicates how much data from the training set with labels is utilized
+for fine-tuning while the rest of the data is used for pre-training. Best
+results are bolded.
+Methodology
+Labeling ratio
+10%
+20%
+30%
+40%
+TS
+78.4
+83.0
+85.7
+87.5
+MG [49]
+80.2
+85.0
+87.5
+90.3
+TransVW [16]
+79.3
+84.5
+87.9
+90.5
+Cube++ [35]
+81.4
+85.2
+87.9
+90.0
+3D-CPC [34]
+80.2
+85.2
+88.3
+90.6
+PCRLv1 [47]
+84.4
+87.5
+89.8
+92.2
+PCRLv2
+85.5
+88.3
+90.3
+93.1
+4.7
+Transfer learning on chest pathology identification
+In Table 6, we validate the transferable ability of visual
+representations provided by different pre-training method-
+ologies. Specifically, we compare PCRLv2 against train from
+scratch, ImageNet-based pre-training (IN), different SSL
+baselines, and PCRLv1.
+Comparing MG/TransVW with IN, we see context
+restoration based SSL maintains the limited transferable
+ability. This phenomenon becomes more apparent when the
+target domain has quite limited annotations. The underlying
+reason is that semantic information plays a crucial role in
+transfer learning. In contrast, the significant performance
+gains brought by C2L and SimSiam again verify the effec-
+tiveness of comparative SSL. C2L and SimSiam still cannot
+outperform IN by significant margins, especially when con-
+sidering that IN is more advantageous when the labeling
+ratio is 10%.
+After integrating the benefits of context restoration
+based and comparative SSL, PCRLv1 is already capable of
+outperforming previous SSL methodologies by observable
+margins. Furthermore, by exploiting multi-scale semantic
+and pixel-level information, PCRLv2 achieves consistent
+improvements over PCRLv1 in overall and class-specific
+results in different labeling ratios.
+4.8
+Transfer learning on brain tumor segmentation
+We report the experimental results of applying transfer
+learning to brain tumor segmentation in Table 7, where
+we use LUNA dataset for self-supervised pre-training and
+fine-tune the pre-trained model with different amounts of
+labeled data.
+
+12
+TABLE 6
+Transfer learning on chest pathology identification. We pre-train the model using data from CheXpert (without labels). Then, we fine-tune the
+pre-trained model on NIH ChestX-ray with different amounts of labeled data (denotes as different labeling ratios). The best results are bolded.
+Labeling ratio
+Methodology
+Mean
+Atelectasis
+Cardiomegaly
+Effusion
+Infiltration
+Mass
+Nodule
+Pneumonia
+Pneumothorax
+Consolidation
+Edema
+Emphysema
+Fibrosis
+Pleural Thick.
+Hernia
+10%
+TS
+68.1
+67.6
+63.3
+76.8
+57.5
+71.5
+61.8
+64.2
+76.2
+69.8
+80.2
+72.4
+62.8
+68.0
+61.1
+IN [28]
+73.5
+73.3
+68.7
+81.6
+63.0
+76.6
+67.3
+70.0
+81.3
+75.6
+85.9
+78.5
+68.6
+72.5
+65.9
+MG [49]
+70.1
+69.9
+65.6
+79.2
+59.4
+72.9
+64.3
+67.0
+77.9
+72.0
+82.3
+75.8
+65.9
+69.6
+59.4
+TransVW [16]
+69.7
+69.4
+64.3
+78.2
+59.5
+72.6
+63.1
+67.2
+77.2
+70.9
+83.0
+75.3
+65.8
+68.9
+60.2
+C2L [48]
+73.1
+72.5
+68.0
+81.3
+62.4
+75.8
+67.2
+70.2
+80.6
+74.8
+85.4
+78.4
+68.3
+72.2
+66.1
+SimSiam [10]
+72.5
+71.9
+67.5
+81.2
+61.7
+75.9
+66.6
+69.6
+79.8
+74.2
+84.8
+77.6
+67.7
+71.8
+64.5
+PCRLv1 [47]
+75.8
+75.4
+70.6
+84.2
+65.5
+78.9
+69.6
+72.7
+83.5
+77.6
+88.5
+80.8
+71.3
+74.8
+67.6
+PCRLv2
+77.2
+76.8
+72.0
+85.6
+66.8
+80.2
+71.0
+74.0
+84.8
+78.9
+89.8
+82.2
+72.6
+76.2
+69.7
+20%
+TS
+71.4
+71.8
+73.1
+78.4
+59.6
+72.5
+64.5
+66.6
+77.7
+71.7
+82.0
+75.5
+69.8
+68.9
+68.2
+IN [14]
+76.2
+75.9
+78.3
+82.9
+64.2
+77.8
+68.8
+70.7
+83.0
+76.4
+87.2
+80.0
+75.3
+73.9
+73.1
+MG [49]
+73.8
+73.9
+75.4
+80.2
+61.9
+74.9
+66.5
+68.3
+80.0
+74.0
+85.1
+78.1
+72.8
+71.5
+71.3
+TransVW [16]
+73.8
+73.0
+75.5
+80.1
+62.3
+75.6
+66.7
+68.6
+80.2
+74.0
+85.2
+77.5
+72.9
+71.5
+69.4
+C2L [48]
+77.0
+76.5
+78.9
+83.4
+65.0
+78.6
+69.8
+71.8
+83.5
+77.2
+88.1
+80.8
+76.0
+74.2
+73.5
+SimSiam [10]
+76.6
+76.6
+78.7
+83.3
+64.6
+77.9
+69.2
+71.6
+83.1
+76.9
+87.8
+80.5
+75.5
+73.8
+73.6
+PCRLv1 [47]
+77.5
+77.3
+79.7
+84.3
+65.7
+78.9
+70.3
+72.8
+83.8
+77.6
+88.6
+81.1
+76.5
+74.8
+74.3
+PCRLv2
+79.4
+79.0
+81.3
+85.9
+67.3
+80.8
+72.1
+74.0
+86.0
+79.4
+90.3
+83.1
+78.4
+76.7
+76.6
+30%
+TS
+73.5
+71.7
+79.7
+79.9
+60.5
+76.5
+68.4
+66.8
+79.2
+72.8
+83.4
+76.9
+71.4
+70.5
+71.3
+IN [14]
+78.5
+77.2
+84.6
+84.3
+66.2
+80.8
+73.0
+72.3
+84.0
+78.0
+88.5
+82.0
+76.8
+75.3
+76.0
+MG [49]
+75.6
+74.1
+81.8
+81.0
+63.3
+77.9
+70.1
+69.0
+80.9
+74.8
+85.4
+79.7
+73.6
+72.6
+74.2
+TransVW [16]
+75.7
+74.8
+81.4
+81.0
+63.6
+77.7
+69.9
+69.8
+80.9
+75.4
+86.0
+79.3
+73.9
+72.3
+73.8
+C2L [48]
+78.6
+77.1
+84.5
+84.5
+66.1
+81.1
+73.0
+72.5
+84.0
+78.1
+88.3
+82.1
+76.8
+75.5
+76.8
+SimSiam [10]
+78.3
+77.0
+84.4
+84.1
+65.7
+80.7
+72.7
+72.2
+83.9
+77.9
+88.1
+82.1
+76.6
+75.2
+75.6
+PCRLv1 [47]
+79.9
+78.5
+85.8
+85.6
+67.4
+82.3
+74.2
+73.8
+85.5
+79.4
+89.7
+83.5
+78.1
+76.7
+78.1
+PCRLv2
+80.5
+79.1
+86.4
+86.2
+68.0
+82.8
+74.8
+74.3
+86.0
+80.0
+90.3
+84.1
+78.6
+77.2
+79.2
+40%
+TS
+75.4
+72.6
+80.0
+81.0
+62.5
+76.9
+69.2
+68.0
+80.7
+74.7
+85.1
+79.5
+74.0
+71.0
+79.8
+IN [14]
+79.0
+76.7
+84.2
+84.3
+66.3
+80.7
+73.6
+72.3
+84.7
+78.5
+88.6
+83.4
+77.4
+75.0
+79.7
+MG [49]
+76.5
+74.1
+81.3
+81.7
+63.9
+77.9
+71.1
+70.1
+82.5
+76.1
+85.6
+80.6
+74.5
+73.1
+77.9
+TransVW [16]
+77.3
+75.2
+82.4
+82.4
+64.4
+79.0
+71.4
+70.5
+83.2
+76.7
+86.6
+82.0
+75.8
+73.6
+78.4
+C2L [48]
+79.1
+76.9
+84.3
+84.5
+66.4
+80.8
+73.4
+72.2
+84.8
+78.3
+88.6
+83.4
+77.2
+75.4
+80.6
+SimSiam [10]
+78.9
+76.7
+83.9
+84.1
+66.6
+80.4
+73.1
+72.1
+84.7
+78.1
+88.4
+83.4
+77.2
+74.8
+80.5
+PCRLv1 [47]
+80.8
+78.5
+86.0
+86.2
+68.2
+82.4
+75.2
+74.0
+86.6
+80.2
+90.2
+85.1
+79.0
+76.9
+82.1
+PCRLv2
+81.5
+79.2
+86.6
+86.9
+68.9
+83.0
+75.8
+74.6
+87.2
+80.8
+90.9
+85.8
+79.7
+77.6
+83.4
+50%
+TS
+77.5
+75.2
+82.0
+82.0
+64.5
+79.6
+71.8
+71.3
+82.9
+75.8
+86.6
+80.9
+76.1
+75.5
+80.3
+IN
+79.5
+77.2
+84.5
+84.4
+66.6
+81.4
+73.6
+73.0
+84.6
+78.2
+89.1
+82.7
+77.9
+77.3
+82.0
+MG [49]
+77.6
+75.0
+82.8
+82.8
+64.8
+79.5
+71.8
+71.6
+82.3
+75.7
+86.7
+81.5
+76.2
+75.7
+79.5
+TransVW [16]
+77.3
+74.5
+81.9
+82.4
+64.8
+78.8
+71.5
+71.3
+82.4
+75.7
+86.8
+80.4
+75.7
+74.9
+80.6
+C2L [48]
+79.8
+77.6
+84.7
+84.5
+67.0
+81.6
+73.6
+73.4
+84.7
+78.5
+89.0
+83.1
+78.4
+78.0
+82.6
+SimSiam [10]
+80.0
+77.7
+84.9
+84.8
+67.1
+81.7
+74.0
+73.5
+84.7
+78.3
+89.5
+83.6
+78.8
+77.7
+83.2
+PCRLv1 [47]
+81.2
+78.7
+86.1
+86.3
+68.3
+82.8
+75.4
+74.5
+86.8
+80.4
+90.5
+85.3
+79.5
+78.2
+83.5
+PCRLv2
+82.5
+80.0
+87.4
+87.3
+69.6
+84.1
+76.4
+76.1
+87.4
+81.0
+91.8
+85.9
+81.0
+80.4
+86.1
+100%
+TS
+80.9
+77.7
+86.1
+85.1
+67.7
+84.2
+73.3
+73.9
+84.9
+78.7
+89.4
+85.4
+79.4
+78.5
+87.6
+IN
+80.8
+77.8
+86.3
+84.7
+67.3
+83.6
+73.0
+74.1
+84.9
+78.8
+89.5
+85.7
+79.6
+78.2
+87.0
+MG [49]
+80.8
+77.8
+86.3
+84.7
+67.3
+83.6
+73.0
+74.1
+84.9
+78.8
+89.5
+85.7
+79.6
+78.2
+87.0
+TransVW [16]
+81.2
+77.9
+86.4
+85.3
+67.6
+84.3
+73.8
+74.4
+85.1
+79.3
+89.8
+86.2
+80.0
+78.6
+88.8
+C2L [48]
+81.4
+78.2
+87.0
+85.3
+68.3
+84.8
+73.7
+74.8
+85.5
+79.6
+90.1
+86.3
+80.0
+78.6
+88.1
+SimSiam [10]
+81.6
+78.3
+87.2
+85.5
+68.3
+84.9
+74.2
+74.7
+85.7
+79.6
+90.1
+86.2
+80.2
+79.1
+89.1
+PCRLv1 [47]
+83.0
+79.8
+88.5
+87.1
+69.7
+86.1
+75.6
+76.1
+87.0
+81.2
+91.6
+87.7
+81.7
+80.4
+90.2
+PCRLv2
+84.0
+80.7
+89.3
+87.9
+70.5
+87.0
+76.4
+77.0
+87.9
+82.0
+92.5
+88.6
+82.6
+81.3
+91.6
+TABLE 7
+Transfer learning on brain tumor segmentation (on BraTS). WT, TC, and ET stand for the whole tumor, tumor core, and enhancing tumor. For all
+SSL approaches, we use LUNA for pre-training, and then fine-tune the pre-trained model on BraTS with varying amounts of labeled data. Best
+results are bolded.
+Methodology
+10%
+20%
+30%
+40%
+100%
+Mean
+WT
+TC
+ET
+Mean
+WT
+TC
+ET
+Mean
+WT
+TC
+ET
+Mean
+WT
+TC
+ET
+Mean
+WT
+TC
+ET
+TS
+66.6
+71.2
+66.7
+62.1
+72.7
+78.5
+74.3
+65.5
+76.7
+81.8
+77.9
+70.6
+77.1
+82.3
+78.3
+70.9
+81.5
+86.8
+82.8
+75.1
+MG [49]
+69.6
+72.4
+71.4
+65.1
+75.5
+80.4
+77.3
+68.9
+79.6
+84.2
+80.6
+74.1
+80.4
+85.3
+82.0
+74.0
+82.4
+87.1
+83.6
+76.6
+TransVW [16]
+70.3
+74.6
+71.7
+64.6
+75.6
+79.9
+75.4
+71.5
+79.1
+83.8
+79.9
+73.6
+80.8
+85.8
+82.1
+74.5
+82.3
+87.1
+83.3
+76.5
+Cube++ [35]
+69.0
+74.5
+70.6
+61.9
+74.9
+80.7
+75.9
+68.1
+79.3
+84.0
+79.4
+74.5
+79.7
+84.5
+80.0
+74.6
+82.2
+87.2
+82.4
+77.0
+3D-CPC [34]
+70.1
+76.7
+70.5
+63.1
+75.9
+81.6
+75.6
+70.5
+79.4
+84.6
+79.9
+73.7
+81.2
+86.5
+81.8
+75.3
+82.9
+88.0
+83.3
+77.4
+PCRLv1 [47]
+71.6
+76.9
+73.1
+65.2
+77.6
+81.4
+79.1
+72.7
+81.1
+84.9
+82.2
+76.6
+83.3
+87.5
+84.6
+78.2
+85.0
+89.0
+86.2
+80.2
+PCRLv2
+73.0
+77.7
+74.3
+67.2
+78.8
+83.2
+79.4
+74.0
+82.1
+85.1
+82.7
+78.7
+84.1
+87.9
+84.5
+80.1
+85.6
+89.4
+85.9
+81.7
+
+13
+TransVW
+PCRLv1
+PCRLv2
+GT
+TransVW
+PCRLv1
+PCRLv2
+GT
+10%
+10%
+10%
+20%
+20%
+20%
+TransVW
+PCRLv1
+PCRLv2
+GT
+10%
+20%
+30%
+TransVW
+PCRLv1
+PCRLv2
+GT
+10%
+20%
+30%
+b
+c
+Atelectasis
+TransVW
+PCRLv1
+PCRLv2
+Effusion
+Infiltration
+Mass
+Nodule
+Pneumonia
+TransVW
+PCRLv1
+PCRLv2
+a
+Fig. 8. Visual interpretation of the transfer learning on chest pathology identification (a), and segmentation results of brain tumor (b) and liver (c). We
+mainly compare PCRLv2 against PCRLv1 and TransVW. Red boxes in the top figure a denote the ground-truth (GT) annotations from radiologists.
+In figure b, we present the segmentation results of the enhancing tumor (ET) from BraTS when the labeling ratios are 10% and 20%. Similarly in
+the bottom figure, we display the liver segmentation results in three different labeling ratios (10%, 20%, and 30%).
+
+PORTABLEPORTABLEPORTABLE14
+TABLE 8
+Transfer learning on abdominal organ segmentation (on LiTS). We use
+LUNA for pre-training, and fine-tune the pre-trained model on LiTS with
+different amounts of labeled data. Best results are bolded.
+Methodology
+Labeling ratio
+10%
+20%
+30%
+40%
+100%
+TS
+71.1
+77.2
+84.1
+87.3
+90.7
+MG [49]
+73.3
+79.5
+84.3
+87.9
+91.3
+TransVW [16]
+73.8
+79.3
+85.5
+88.2
+91.4
+Cube++ [35]
+74.2
+79.3
+84.5
+88.2
+91.8
+3D-CPC [34]
+74.8
+80.2
+85.6
+88.9
+91.9
+PCRLv1 [47]
+77.3
+83.5
+87.8
+90.1
+93.7
+PCRLv2
+79.0
+86.5
+89.3
+90.9
+94.5
+Nodule
+Infiltrate
+Atelectasis
+Fig. 9. Failure case analysis on chest pathology identification. Red boxes
+stand for the lesion areas delineated by radiologists. Images are from
+NIH ChestX-ray.
+Somewhat surprisingly, we find 3D-CPC does not out-
+perform context restoration based SSL (MG, TransVW, and
+Cube++) as obviously as those in Tables 4, 5, and 7.
+This comparison is consistent with our intuition: pixel-
+level information matters a lot in medical image segmen-
+tation. Again, PCRLv1 and PCRLv2 outperform previous
+SSL methodologies in all three classes by large margins.
+Compared to PCRLv1, PCRLv2 is more advantageous in
+segmenting the enhancing tumor (ET) regions, which are
+often smaller than WT and TC, and thus harder to segment.
+The performance gains on ET again verify the effectiveness
+of multi-scale latent representations, which advances the
+segmentation of small objects.
+4.9
+Transfer learning on liver segmentation
+In Table 8, we present the results of liver segmentation.
+There exist three observable phenomena. First, we see that
+all SSL approaches provide substantial performance gains
+over train from scratch. Second, we find the comparative
+methodology, i.e., 3D-CPC, achieves comparable segmen-
+tation performance to traditional context restoration based
+SSL. This phenomenon verifies the necessity of utilizing
+pixel-level information in medical image segmentation (sim-
+ilar results also appear in Table 7). Last but not the least,
+PCRLv2 consistently outperforms PCRLv1 in all labeling
+ratios, which again validates the effectiveness of introducing
+multiple scales into SSL.
+4.10
+Visual analysis
+In Fig. 8, we visually analyze the experimental results of
+transfer learning with limited annotations on chest pathol-
+ogy identification (10%), brain tumor segmentation (10%
+and 20%), and liver segmentation (10%, 20%, and 30%).
+Here, we compare PCRLv2 against generic SSL methodolo-
+gies. Considering TransVW was developed on top of MG,
+we exclude MG and compare PCRLv2 against PCRLv1 and
+TransVW.
+Fig. 8a presents the visual interpretation of chest pathol-
+ogy diagnoses using CAM [45] on six different pathologies.
+We find that TransVW fails to capture the correct location of
+lesions on atelectasis, infiltration, nodule, and pneumonia.
+In comparison, PCRLv1 can generate more interpretable
+diagnosis results but still yields inconsistent predictions on
+infiltration and nodule. By integrating multi-scale latent rep-
+resentations, PCRLv2 can capture the small lesion areas on
+infiltration and nodule, resulting in centralized yet accurate
+diagnosis results.
+In Fig. 8b and Fig. 8c, we visualize the segmentation
+results of the enhancing tumor (ET) on BraTS and liver
+on LiTS. Compared to TransVW and PCRLv1, PCRLv2
+reduces the false positive predictions and contains richer
+fine-grained details. We believe such superiority of PCRLv2
+can be attributed to the integration of multi-scale pixel-level
+and semantic information.
+We also provide some failure examples in Fig. 9. One
+common characteristic of these detection results is that they
+include high-confidence predictions outside the lung area.
+However, in daily clinical practice, such anomalies should
+not be located outside the lung area. Similar phenomena
+have been reported in [13], where the authors summarized
+them as “shortcuts” that are common in learning systems
+based on neural networks. To mitigate this problem in self-
+supervised learning, we can add commonsense knowledge
+to pre-trained models. Besides, it is also necessary to de-
+velop more powerful machine learning tools for model
+interpretation in various downstream tasks.
+5
+CONCLUSION
+We present a unified visual information preservation frame-
+work for self-supervised learning in medical imaging. This
+framework aims to encode the pixel-level, semantic, and
+scale information into latent representations simultaneously.
+To achieve this goal, we conduct multi-scale pixel restora-
+tion and feature comparison on the feature pyramid, which
+non-skip U-Net supports. The proposed PCRLv2 outper-
+forms previous self-supervised pre-training approaches by
+large margins and yields consistent improvements over
+its conference version (PCRLv1) on four well-established
+datasets in both quantitative and qualitative validation. We
+will continue to explore how to optimally integrate different
+types of information into SSL in the future.
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+

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+arXiv:2301.01014v1  [math.DG]  3 Jan 2023
+TRICHOTOMY THEOREM FOR PRESCRIBED SCALAR AND MEAN
+CURVATURES ON COMPACT MANIFOLDS WITH BOUNDARIES
+JIE XU
+Abstract. In this article, we give results of prescribing scalar and mean curvature functions
+for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is
+equipped on a compact manifold with boundary, with dimensions at least 3. The results are clas-
+sified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a “Trichotomy
+Theorem” in terms of both scalar and mean curvature functions, which is a full extension of the
+“Trichotomy Theorem” given by Kazdan and Warner. We also discuss prescribing Gauss and geo-
+desic curvature problems on compact Riemann surfaces with boundary for metrics either pointwise
+conformal or conformally equivalent to the original metric, provided that the Euler characteristic is
+negative. The key step is a general version of monotone iteration scheme which handle the zeroth
+order nonlinear term on the boundary conditions.
+1. Introduction
+In this article, we give a “Trichotomy Theorem” on compact manifolds ( ¯
+M, g) with non-empty
+smooth boundaries ∂M, n := dim M ⩾ 3, involving both the scalar and mean curvatures. This
+is a full generalization of the “Trichotomy Theorem” on closed manifolds, given by Kazdan and
+Warner [9]. Precisely speaking, this “Trichotomy Theorem” concerns whether the given functions
+S, H can be realized as scalar and mean curvatures, respectively, of a metric ˜g either within a
+conformal class [g] or conformally equivalent to the metric g. Throughout this article, we assume
+that ¯
+M is connected since otherwise we can easily apply arguments below equally to each connected
+component. It is well-known that this problem is reduced to the existence of the positive solutions
+of the nonlinear second order elliptic PDE
+− a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2 (H ◦ φ) u
+p
+2 on ∂M.
+(1)
+Here Rg is the scalar curvature of the metric g, hg is the mean curvature. φ : ¯
+M → ¯
+M is some
+diffeomorphism on
+¯
+M.
+When φ = Id, the PDE (1) is for prescribing functions S, H within a
+conformal class [g]. The constants a, p are defined as
+a = 4(n − 1)
+n − 2 , p =
+2n
+n − 2.
+∆g is the Laplace-Beltrami operator and ν is the unique outward unit normal vector field along
+∂M. The functions S : C∞( ¯
+M) → R, and H : C∞(∂M) → R are given. We denote η1 to be the
+first eigenvalue of the conformal Laplacian □g := −a∆g + Rg with associated eigenfunction ϕ, i.e.
+ϕ is a positive, smooth function that solves the following PDE:
+−a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ
+∂ν +
+2
+p − 2hgϕ = 0 on ∂M.
+When the dimension of the manifold n = 2, we also discuss the pair of functions K, σ that can be
+realized as Gaussian and geodesic curvatures, respectively, either for a pointwise conformal metric
+1
+
+2
+J. XU
+or a conformally equivalent metric. The two dimensional case is reduced to the existence of the
+solutions of the following elliptic PDE
+− a∆gu + Kg = (K ◦ φ) e2u in M, ∂u
+∂ν + σg = (σ ◦ φ) eu on ∂M.
+(2)
+Here Kg and σg are Gaussian and geodesic curvatures of g, respectively.
+The functions K :
+C∞( ¯
+M) → R and σ : C∞(∂M) → R are given. Again when the diffeomorphism φ :
+¯
+M →
+¯
+M
+is the identity map, K, σ are prescribing Gauss and geodesic curvatures for some metric within the
+conformal class [g].
+The main results of this article are given as follows:
+Theorem 1.1. Let ( ¯
+M, g) be a connected, compact manifold with non-empty smooth boundary ∂M,
+n = dim ¯
+M ⩾ 3. Let S, H ∈ C∞( ¯
+M) be given functions.
+(i). If η1 < 0, then any function S < 0 somewhere in M can be realized as a scalar curvature
+function of some metric conformally equivalent to g, with mean curvature cH for some small
+enough constant c > 0 and any function H;
+(ii). If η1 < 0, then any function S < 0 that changes sign in M can be realized as a scalar curvature
+function of some metric conformally equivalent to g, with mean curvature cH for some small
+enough constant c > 0 and any function H;
+(iii). If η1 < 0, then any function S > 0 somewhere in M can be realized as a scalar curvature
+function of some metric pointwise conformal to g, with mean curvature cH for some small
+enough constant c > 0 and any function H.
+Case (i) is given in §3 and §4; when S < 0 everywhere on ¯
+M, we can improve the result within
+a pointwise conformal class [g] in Theorem 3.1 and Theorem 3.2; Case (ii) is given in §6; when S
+satisfies
+´
+M SdVolg < 0 in addition, we can improve the result within a pointwise conformal class
+[g], see [18, Thm .1.2]; and Case (iii) is given in §7. The significance of this is that we can choose
+arbitrary function with small enough sup-norm as our mean curvature function, provided that the
+scalar curvature function is nontrivial.
+Based on our best understanding, known results in this topic are mainly for the non-positive first
+eigenvalue cases or non-positive Euler characteristic cases. In [5], Cruz-Bl´azquez, Malchiodi and
+Ruiz discussed prescribing negative scalar functions and mean curvature functions with arbitrary
+signs by variational method, for compact manifolds with dimensions at least 2. Some of our results
+overlap their results, but with a different method and different hypotheses on prescribed functions.
+However, our results are classified by the sign of the first eigenvalue of the conformal Laplacian. For
+zero first eigenvalue case or zero Euler characteristic case, we follow the results of [18]. We point
+out that Brezis and Merle discussed the PDE −∆eu = V eu on Ω ⊂ R2 with Dirichlet boundaray
+condition in [3]. Other results for the local Yamabe equation with Dirichlet condition in higher
+dimensions could be found in [12]. For more discussions with respect to (2) in 2-dimensional case,
+we refer to [15, Ch. 13, Ch. 14]. When the first eigenvalue of conformal Laplacian is positive, a lot
+of non-existence results are given, e.g. [2] and [11], etc.
+We also give results on compact Riemann surfaces with boundary, provided that χ( ¯
+M) < 0.
+Theorem 1.2. Let ( ¯
+M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.
+Let K, σ ∈ C∞( ¯
+M) be given functions.
+(i). If K < 0 everywhere on ¯
+M, then there exists a metric pointwise conformal to g with Gauss
+curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary
+function σ;
+(ii). If K < 0 somewhere on ¯
+M, then there exists a metric conformally equivalent to g with Gauss
+curvature K and geodesic curvature cσ for some small enough constant c > 0 and arbitrary
+function σ.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+3
+Both results above are given in §5. Other than the related works on compact Riemann surfaces
+with boundary we introduced above, many work has been done on closed Riemann surface, a
+comprehensive study was given by Kazdan and Warner [8], including results for all signs of χ( ¯
+M).
+For Nirenberg problem, we refer to Chang and Yang [4] and Struwe [13], etc..
+The most common method in analyzing this type of Kazdan-Warner problem is by calculus
+of variations since we can consider the PDE as Euler-Lagrange equation with respect to some
+functional; recently Morse theory is also involved. However, a new method, inspired by Kazdan
+and Warner [10], has been developed recently. This new method applies monotone iteration scheme,
+a local version of calculus of variation to classify the existence results by sign of the first eigenvalue
+η1 of conformal Laplacian. This method has been applied to completely solve the Escobar problem
+[16], the Han-Li conjecture [17], the prescribed scalar curvature problem on compact manifolds [19],
+a trichotomy theorem in terms of prescribed scalar curvature with Dirichlet condition at boundary
+[20], and a comprehensive study of zero first eigenvalue case on compact manifolds, possibly with
+boundary, with dimensions at least 3 [18]. In this article, we apply a variation of the combination
+of monotone iteration scheme and local analysis to show the results of prescribed scalar and mean
+curvatures for the cases η1 > 0 and η1 < 0. We also develop a general monotone iteration scheme,
+which can handle nonlinear terms both in the PDE and on the boundary condition; this new
+monotone iteration scheme, see Theorem 2.3, allows us to work on 2-dimensional case without
+using the calculus of variation.
+This systematic procedure is powerful, but unfortunately this
+direct method cannot be used to the classical manifold, the unit ball with spherical boundary. We
+will explain why this direct method does not work in this case. Note that Escobar [7] has proved a
+nontrivial Kazdan-Warner type obstruction of prescribed mean curvature functions for this case.
+This paper is organized as follows:
+In §2, we introduce the essential definitions and results that will be used throughout this article.
+We assume the backgrounds of standard elliptic theory. We also introduced two versions of mono-
+tone iteration schemes. Theorem 2.2 is for the PDE (1); Theorem 2.3 is more general, works for all
+second order semi-linear elliptic PDE with Robin boundary conditions, possibly with zeroth order
+nonlinear term on the boundary condition. Theorem 2.3 works well for the PDE like (2).
+In §3, we give results for prescribing scalar curvature function S and mean curvature function
+H within a conformal class [g] on ( ¯
+M, g), n = dim ¯
+M ⩾ 3, provided that η1 < 0. When S < 0
+everywhere and arbitrary H, the results are given in Theorem 3.1 and Theorem 3.2. When S < 0
+somewhere and arbitrary H, the results are given in Theorem 3.3 and Corollary 3.1 with some
+restriction on S. The monotone iteration scheme plays a central role.
+In §4, we give results for prescribing scalar curvature function S and mean curvature function
+H for some metric conformally equivalent to g on ( ¯
+M, g), n = dim ¯
+M ⩾ 3, provided that η1 < 0.
+It follows from Corollary 3.1. We conclude in Theorem 4.1 that any S that is negative somewhere
+can be realized as a scalar curvature function of some metric conformally equivalent to g, with the
+mean curvature cH for small enough constant c > 0 and arbitrary H.
+In §5, we discuss prescribing Gauss and geodesic curvature functions K, σ on compact Riemann
+surfaces with boundary for metrics conformally equivalent to the original metric g. We show in
+Theorem 5.1 that any function K that is negative somewhere and satisfies some analytic condition
+can be realized as Gaussian curvature function for some metric conformally equivalent to g, the
+metric also has geodesic curvature cσ for some small enough constant c > 0 and arbitrary σ. The
+result in Corollary 5.1 says that when K < 0 everywhere on ¯
+M, the metric can be chosen within a
+conformal class [g].
+In §6, we give results for prescribing scalar function S and mean curvature function H for some
+metric conformally equivalent to g on ( ¯
+M, g), n = dim ¯
+M ⩾ 3, provided that η1 = 0. We show
+that any function S that changes sign can be realized as a scalar curvature function some metric
+
+4
+J. XU
+conformally equivalent to g, with the mean curvature cH for small enough constant c > 0 and
+arbitrary H in Corollary 6.2. Obviously there is a trivial case S ≡ H ≡ 0.
+In §7, we consider the prescribing scalar and mean curvature problem for η1 > 0. The results in
+Theorem 7.1 and Theorem 7.2 are for the case S > 0 somewhere and arbitrary H. We also explain
+why our method cannot work on closed Euclidean ball with some nontrivial mean curvature on the
+boundary Sn.
+2. The Preliminaries and The Monotone Iteration Scheme
+In this section, we first introduce the necessary definitions and essential results we need for the
+later sections, then introduce a general version of the monotone iteration scheme given in [18], other
+than many variations we have used in [16, 17, 19, 20, 21], with respect to the following Yamabe
+equation with Robin boundary condition
+− a∆gu + Rgu = Sup−1 in M, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2Hu
+p
+2 on ∂M.
+(3)
+for given functions S, H ∈ C∞( ¯
+M), and n = dim ¯
+M ⩾ 3. Lastly we introduce a W s,q-type regularity
+for elliptic PDE with Robin boundary conditions.
+First of all, we give definitions of Sobolev spaces, a local version and a global version.
+Let
+Ω be a connected, bounded, open subset of Rn with smooth boundary ∂Ω equipped with some
+Riemannian metric g that can be extended smoothly to ¯Ω. We call (Ω, g) a Riemannian domain.
+Throughout this article, we denote the space of smooth functions with compact support by C∞
+c ,
+smooth functions by C∞, and continuous functions by C0.
+Definition 2.1. Let (Ω, g) be a Riemannian domain.
+Let (M, g) be a closed Riemannian n-
+manifold, and ( ¯
+M, g) be a compact Riemannian n-manifold with non-empty smooth boundary, with
+volume density dVolg. Let u be a real valued function. Let ⟨v, w⟩g and |v|g = ⟨v, v⟩1/2
+g
+denote the
+inner product and norm with respect to g.
+(i) For 1 ⩽ p < ∞, we define the Lebesgue spaces on Ω and ¯
+M to be
+Lp(Ω) is the completion of
+�
+u ∈ C∞
+c (Ω) : ∥u∥p
+p :=
+ˆ
+Ω
+|u|pdx < ∞
+�
+,
+Lp(Ω, g) is the completion of
+�
+u ∈ C∞
+c (Ω) : ∥u∥p
+p,g :=
+ˆ
+Ω
+|u|p dVolg < ∞
+�
+,
+Lp(M, g) is the completion of
+�
+u ∈ C∞(M) : ∥u∥p
+p,g :=
+ˆ
+M
+|u|p dVolg < ∞
+�
+.
+(ii) For ∇u the Levi-Civita connection of g, and for u ∈ C∞(Ω) or u ∈ C∞( ¯
+M),
+|∇ku|2
+g := (∇α1 . . . ∇αku)(∇α1 . . . ∇αku).
+(4)
+In particular, |∇0u|2
+g = |u|2 and |∇1u|2
+g = |∇u|2
+g.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+5
+(iii) For s ∈ N, 1 ⩽ p < ∞, we define the (s, p)-type Sobolev spaces on Ω and ¯
+M to be
+W s,p(Ω) =
+
+
+u ∈ Lp(Ω) : ∥u∥p
+W s,p(Ω) :=
+ˆ
+Ω
+s
+�
+j=0
+��Dju
+��p dx < ∞
+
+
+ ,
+(5)
+W s,p(Ω, g) =
+
+
+u ∈ Lp(Ω, g) : ∥u∥p
+W s,p(Ω,g) =
+s
+�
+j=0
+ˆ
+Ω
+��∇ju
+��p
+g dVolg < ∞
+
+
+ ,
+W s,p(M, g) =
+
+
+u ∈ Lp(M, g) : ∥u∥p
+W s,p(M,g) =
+s
+�
+j=0
+ˆ
+M
+��∇ju
+��p
+g dVolg < ∞
+
+
+ .
+Here |Dju|p := �
+|α|=j|∂αu|p in the weak sense. Similarly, W s,p
+0 (Ω) is the completion of C∞
+c (Ω)
+with respect to the W s,p-norm.
+In particular, Hs(Ω) := W s,2(Ω) and Hs(Ω, g) := W s,2(Ω, g),
+Hs(M, g) := W s,2(M, g) are the usual Sobolev spaces. We similarly define Hs
+0(Ω), Hs
+0(Ω, g) and
+Hs
+0(M, g).
+(iv) On closed manifolds (M, g), we say that a function u ∈ Hs(M, g) if u ∈ L2(M, g) , and for
+any coordinate chart U ⊂ M, any ψ ∈ C∞
+c (U), the function ψu ∈ Hs(U, g).
+We assume the background of the standard elliptic theory, including the solvability of standard
+linear elliptic PDEs, elliptic regularity of Hs-type, trace theorem, Sobolev embedding, Schauder
+estimates, etc. We introduce a W s,q-type elliptic regularity for later use.
+Theorem 2.1. [17, Thm. 2.2] Let ( ¯
+M, g) be a compact manifold with smooth boundary ∂M. Let
+ν be the unit outward normal vector along ∂M and q > n = dim ¯
+M. Let L : C∞( ¯
+M) → C∞( ¯
+M)
+be a uniform second order elliptic operator on M with smooth coefficients up to ∂M and can be
+extended to L : W 2,q(M, g) → Lq(M, g). Let f ∈ Lq(M, g), ˜f ∈ W 1,q(M, g). Let u ∈ H1(M, g) be a
+weak solution of the following boundary value problem
+Lu = f in M, Bu = ∂u
+∂ν + c(x)u = ˜f on ∂M.
+(6)
+Here c ∈ C∞(M). Assume also that Ker(L) = {0} associated with the homogeneous Robin boundary
+condition. If, in addition, u ∈ Lq(M, g), then u ∈ W 2,q(M, g) with the following estimates
+∥u∥W 2,q(M,g) ⩽ γ′ �
+∥Lu∥Lq(M,g) + ∥Bu∥W 1,q(M,g)
+�
+(7)
+Here γ′ depends on L, q, c and the manifold ( ¯
+M, g) and is independent of u.
+We then introduce the first eigenvalue of conformal Laplacian. Note that a = 4(n−1)
+n−2
+and p =
+2n
+n−2,
+hence it only makes sense when n ⩾ 3.
+Definition 2.2. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M. We
+denote η1 be the first eigenvalue of conformal Laplacian with its corresponding eigenfunction ϕ > 0
+if and only if the following PDE holds.
+− a∆gϕ + Rgϕ = η1ϕ in M, ∂ϕ
+∂ν +
+2
+p − 2hgϕ = 0 on ∂M.
+(8)
+We now introduce a variation of the monotone iteration scheme we used in [16], [17] and [19].
+In particular, we do require hg = h > 0 to be some positive constant on ∂M, this can be done due
+to the proof of the Han-Li conjecture in [17]. We will also use other versions of monotone iteration
+schemes introduced in eariler work [16, 17, 19, 20, 21].
+Theorem 2.2. [18, Thm. 2.4] Let ( ¯
+M, g) be a compact manifold with smooth boundary ∂M. Let
+ν be the unit outward normal vector along ∂M and q > dim ¯
+M. Let S ∈ C∞( ¯
+M) and H ∈ C∞( ¯
+M)
+
+6
+J. XU
+be given functions. Let the mean curvature hg = h > 0 be some positive constant. In addition,
+we assume that sup ¯
+M|H| is small enough. Suppose that there exist u− ∈ C0( ¯
+M) ∩ H1(M, g) and
+u+ ∈ W 2,q(M, g) ∩ C0( ¯
+M), 0 ⩽ u− ⩽ u+, u− ̸≡ 0 on ¯
+M, some constants θ1 ⩽ 0, θ2 ⩾ 0 such that
+−a∆gu− + Rgu− − Sup−1
+−
+⩽ 0 in M, ∂u−
+∂ν +
+2
+p − 2hgu− ⩽ θ1u− ⩽
+2
+p − 2Hu
+p
+2
+− on ∂M
+−a∆gu+ + Rgu+ − Sup−1
++
+⩾ 0 in M, ∂u+
+∂ν +
+2
+p − 2hgu+ ⩾ θ2u+ ⩾
+2
+p − 2Hu
+p
+2
++ on ∂M
+(9)
+holds weakly. In particular, θ1 can be zero if H ⩾ 0 on ∂M, and θ1 must be negative if H < 0
+somewhere on ∂M; similarly, θ2 can be zero if H ⩽ 0 on ∂M, and θ2 must be positive if H > 0
+somewhere on ∂M. Then there exists a real, positive solution u ∈ C∞(M) ∩ C1,α( ¯
+M) of
+□gu = −a∆gu + Rgu = Sup−1 in M, Bgu = ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2Hu
+p
+2 on ∂M.
+(10)
+The following two results are necessary, which shows the existence of the solution of some local
+Yamabe-type problem. When the manifold is not locally conformally flat, we need
+Proposition 2.1. [18, Prop. 3.2] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, not locally
+conformally flat, with C∞ boundary, with Volg(Ω) and the Euclidean diameter of Ω sufficiently
+small. Let f ∈ Ω′ ⊃ Ω be a positive, smooth function in some open region Ω′. In addition, we
+assume that the first eigenvalue of Laplace-Beltrami operator −∆g on Ω with Dirichlet condition
+satisfies λ1 → ∞ as Ω shrinks. Assume Rg < 0 within the small enough closed domain ¯Ω. Then
+the Dirichlet problem
+− a∆gu + Rgu = fup−1 in Ω, u ≡ 0 on ∂Ω
+(11)
+has a real, positive, smooth solution u ∈ C∞(Ω) ∩ H1
+0(Ω, g) ∩ C0(¯Ω). The size of Ω is depending on
+the function f.
+When the manifold is locally conformally flat, we give the local solution of (11) provided that Ω
+is not topologically trivial.
+Proposition 2.2. [19, Prop. 2.5] Let (Ω, g) be a Riemannian domain in Rn, n ⩾ 3, with C∞
+boundary. Let the metric g be locally conformally flat on some open subset Ω′ ⊃ ¯Ω. For any point
+ρ ∈ Ω and any positive constant ǫ, denote the region Ωǫ to be
+Ωǫ = {x ∈ Ω||x − ρ| > ǫ}.
+Assume that Q ∈ C2(¯Ω), minx∈¯Ω Q(x) > 0 and ∇Q(ρ) ̸= 0. Then there exists some ǫ0 such that for
+every ǫ ∈ (0, ǫ0) the Dirichlet problem
+− a∆gu + Rgu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ
+(12)
+has a real, positive, smooth solution u ∈ C∞(Ωǫ) ∩ H1
+0(Ωǫ, g) ∩ C0( ¯Ωǫ).
+Remark 2.1. It is straightforward to see that under conformal change ˜g = φp−2g, we have
+˜g = φp−2g ⇒ −a∆˜g + R˜g = φ− n+2
+n−2 (−a∆g + Rg) φ ⇔ □˜g = φ1−p□gφ.
+(13)
+We call (13) the conformal invariance of the conformal Laplacian. It follows from Proposition 2.2
+and (13) that if the manifold ( ¯
+M, g) is locally conformally flat in the interior, the equation (12) is
+equivalent to
+− a∆geu = Qup−1 in Ωǫ, u = 0 on ∂Ωǫ
+(14)
+which admits a positive solution u ∈ C∞(Ωǫ) ∩ H1
+0(Ωǫ, g) ∩ C0( ¯Ωǫ).
+As a prerequisite, we also need a result in terms of the perturbation of negative first eigenvalue
+of conformal Laplacian.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+7
+Proposition 2.3. Let ( ¯
+M, g) be a compact Riemannian manifold with non-empty smooth boundary
+∂M, n = dim ¯
+M ⩾ 3. Let β > 0 be a small enough constant. If η′
+1 < 0, then the quantity
+η′
+1,β = inf
+u̸=0
+a
+´
+M|∇gu|2dVolg +
+´
+M Rgu2dVolg +
+2a
+p−2
+´
+∂M(hg + β)u2dS
+´
+M u2dVolg
+< 0.
+In particular, η′
+1,β satisfies
+− a∆gϕ + Rgϕ = η′
+1,βϕ in M, ∂ϕ
+∂ν +
+2
+p − 2(hg + β)ϕ = 0 on ∂M
+(15)
+with some positive function ϕ ∈ C∞( ¯
+M).
+Proof. Since η′
+1 < 0, the normalized first eigenfunction ϕ1, i.e.
+´
+M ϕ2
+1dVolg = 1, satisfies
+η′
+1 = a
+ˆ
+M
+|∇gϕ1|2dVolg +
+ˆ
+M
+Rgϕ2
+1dVolg +
+2a
+p − 2
+ˆ
+∂M
+hgϕ2
+1dS
+By characterization of η′
+1,β, we have
+η′
+1,β ⩽ a
+ˆ
+M
+|∇gϕ1|2dVolg +
+ˆ
+M
+Rgϕ2
+1dVolg +
+2a
+p − 2
+ˆ
+∂M
+(hg + β)ϕ2
+1dS = η′
+1 + β
+ˆ
+∂M
+ϕ2
+1dS.
+Since ϕ1 is fixed, it follows that η′
+1,β < 0 if β > 0 is small enough.
+□
+When n = 2, i.e. M or ¯
+M is a compact Riemann surface (possibly with boundary), all tools
+above are not available. We thus need a new version of the monotone iteration scheme for compact
+Riemann surfaces with non-empty smooth boundary. We point out that the monotone iteration
+scheme below works for all compact manifolds with non-empty boundary, with dimensions at least
+2.
+Theorem 2.3. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim M ⩾ 2. Let q > n be a positive integer. Let F(·, ·), G(·, ·) : ¯
+M × R → R be smooth functions.
+Let ν be the unit outward normal vector along ∂M. Let σ be some nonnegative, small enough
+constant. If
+(i) there exists two functions u+ ∈ C∞( ¯
+M) and u− ∈ C0( ¯
+M) ∩ H1(M, g) such that
+−∆gu+ ⩾ F(·, u) in M, ∂u
+∂ν + σu ⩾ G(·, u+) on ∂M;
+−∆gu− ⩽ F(·, u) in M, ∂u
+∂ν + σu ⩽ G(·, u−) on ∂M,
+(16)
+where the sub-solution may hold in the weak sense; and
+(ii) in addition, sup ¯
+M|G(·, u+)|, sup ¯
+M|∇G(·, u+)| are small enough;
+(iii) furthermore, u+ ⩾ u− pointwise on ¯
+M;
+then there exists a smooth function u ∈ C∞( ¯
+M) with u− ⩽ u ⩽ u+ such that
+− ∆gu = F(·, u) in M, ∂u
+∂ν + σu = G(·, u) on ∂M.
+(17)
+Remark 2.2. The proof of Theorem 2.3 is essentially the same as the proof of [18, Thm. 2.4],
+except some minor change, e.g. here we use general smooth functions F and G but not specific
+Yamabe equations. We therefore will give a relatively concise proof for Theorem 2.3.
+
+8
+J. XU
+Proof.
+¯
+M is compact, so extremal values of continuous functions u+, u− can be achieved. Choose
+positive constant A and nonnegative constant B such that
+A ⩾ −∂F
+∂u (x, u(x)), ∀x ∈ ¯
+M, u(x) ∈ [min
+¯
+M u−, max
+¯
+M u+];
+B ⩾ σ − ∂G
+∂u (x, u(x)), ∀x ∈ ¯
+M, u(x) ∈ [min
+¯
+M u−, max
+¯
+M u+].
+(18)
+Denote u0 = u+ ∈ C∞( ¯
+M), and consider the iteration scheme
+−∆guk + Auk = Auk−1 + F(·, uk−1) in M, k ∈ N,
+∂uk
+∂ν + Buk = Buk−1 − σuk−1 + G(·, uk−1) on ∂M, k ∈ N.
+(19)
+Since A > 0, B ⩾ 0, the operator
+�
+−∆g + A, ∂
+∂ν + B
+�
+is invertible due to the standard argument. Clearly when k = 1, the first iteration step in (19)
+gives a unique smooth solution u1 ∈ C∞( ¯
+M). The regularity argument is also standard.
+We show that u− ⩽ u ⩽ u+. For u− ⩽ u, we have to use the sub-solution in the weak sense,
+since u0 = u+ ⩾ u−, we pair (19) for k = 1 with arbitrary non-negative function v ∈ C∞( ¯
+M), and
+subtract this with the sub-solution (adding Au− and Bu− on both sides of the PDE and boundary
+conditions respectively) in the weak sense, we have
+ˆ
+M
+(A (u0 − u−) + F (x, u0) − F (x, u−)) vdVolg ⩽
+ˆ
+M
+(−∆g (u1 − u−) + A (u1 − u−)) vdVolg
+⩽
+ˆ
+∂M
+B (u1 − u−) vdS −
+ˆ
+∂M
+(B (u0 − u−) − σ (u0 − u−) + G (x, u0) − G (x, u−)) vdS
++
+ˆ
+M
+A (u1 − u−) vdVolg +
+ˆ
+M
+∇g (u1 − u−) · ∇gvdVolg.
+Taking v = w := max (u− − u1, 0), and applying the mean value theorem for F, G, due to the
+definitions of A, B in (18), we observe that
+ˆ
+M
+|∇gw|2 +
+ˆ
+∂M
+Bw2 +
+ˆ
+M
+Aw2 ⩽ 0.
+It follows that w = 0, therefore u− ⩽ u1. By a very similar argument in terms of the subtraction
+between (19) and the super-solution, we conclude that u+ ⩾ u1.
+Inductively, we may assume the existence of the solutions u1, . . . , uk with
+u− ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u1 ⩽ u0.
+By the same argument in the first iteration step, we conclude the existence of uk+1 ∈ C∞( ¯
+M); in
+addition, uk+1 satisfies
+u− ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u1 ⩽ u0.
+Therefore we show the existence of the sequence of solutions of (19) with the monotonicity
+u− ⩽ . . . ⩽ uk+1 ⩽ uk ⩽ uk−1 ⩽ . . . ⩽ u0, k ∈ N.
+(20)
+We now show the uniform boundedness of ∥uk∥C1,α( ¯
+M).
+Since q > n, showing the uniform
+boundedness of ∥uk∥C1,α( ¯
+M) is equivalent to show the uniform boundedness of ∥uk∥W 2,q(M,g). We
+have mentioned that the operator is invertible and thus the W s,q-type estimates (7) applies. We L
+and the boundary condition c to be the operators here with associated constant γ′. Mimicking the
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+9
+boundedness proof in [18, Thm. 2.4], we should require σ and sup ¯
+M|G(·, u)|, and sup ¯
+M|∇G(·, u)|
+to be small enough. Denote
+C =
+sup
+x∈ ¯
+M,u(x)∈[min ¯
+M u−,max ¯
+M u+]
+|F(x, u(x))|;
+D1 =
+sup
+x∈ ¯
+M,u(x)∈[min ¯
+M u−,max ¯
+M u+]
+|G(x, u(x))| ;
+D2 =
+sup
+x∈ ¯
+M,u(x)∈[min ¯
+M u−,max ¯
+M u+]
+|∇G(x, u(x))| ;
+(21)
+We require that G(·, u+), D1, D2 satisfies
+∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g) ⩽ 1;
+(B − σ) · γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q + 1
+�
++ D1 · Volg(M)
+1
+q
++ D2 · γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q + 1
+�
+⩽ 1.
+(22)
+By (7) and the first inequality in (22), we observe from the PDE (19) with k = 1 that
+∥u1∥W 2,q(M,g) ⩽ γ′ �
+∥Au+ + F(·, u+)∥Lq(M,g) + ∥(B − σ) u+ + G (·, u+)∥W 1,q(M,g)
+�
+⩽ γ′
+��
+A max
+¯
+M |u+| + C
+�
+· Volg(M)
+1
+q + 1
+�
+⩽ γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q + 1
+�
+.
+Inductively, assume that
+∥uk∥W 2,q(M,g) ⩽ γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q + 1
+�
+.
+(23)
+To check ∥uk+1∥W 2,q(M,g), we apply the W s,q-type elliptic estimate with the solution of (19) again,
+∥uk+1∥W 2,q(M,g) ⩽ γ′ �
+∥Auk + F(·, uk)∥Lq(M,g) + ∥(B − σ) uk + G (·, uk)∥W 1,q(M,g)
+�
+⩽ γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q
+�
++ γ′ �
+(B − σ)∥uk∥W 1,q(M,g) + ∥G(·, uk)∥Lq(M,g) + ∥∇G(·, uk)∥Lq(M,g)
+�
+⩽ γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q
+�
++
+�
+γ′�2 (B − σ)
+�
+A max
+¯
+M ((|u+|, |u−|) + C) · Volg(M)
+1
+q + 1
+�
++ γ′D1 · Volg(M)
+1
+q +
+�
+γ′�2 D2
+�
+A max
+¯
+M ((|u+|, |u−|) + C) · Volg(M)
+1
+q + 1
+�
+⩽ γ′
+��
+A max
+¯
+M (|u+|, |u−|) + C
+�
+· Volg(M)
+1
+q + 1
+�
+.
+It turns that ∥uk∥W 2,q(M,g) is uniformly bounded. The rest of the argument, in applying Arzela-
+Ascoli, the monotonicity of the sequence, and the elliptic regularity, is essentially the same as in
+[18, Thm. 2.4]. We omit the details here.
+In conclusion, the sequence uk converges classically to a smooth function u which solves (17). In
+addition, u− ⩽ u ⩽ u+ pointwise on ¯
+M.
+□
+
+10
+J. XU
+Remark 2.3. Theorem 2.2 is a special case of Theorem 2.3 by taking F(·, u) = −Rgu+Sup−1 and
+G(·, u) = −
+2
+p−2hgu +
+2
+p−2Hu
+p
+2 .
+3. Prescribed Scalar and Mean Curvature Functions under Pointwise Conformal
+Deformation When η1 < 0
+Recall the Yamabe equation with Robin condition
+− a∆gu + Rgu = Sup−1 in M, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2Hu
+p
+2 on ∂M.
+(24)
+In this section, we consider the existence of the solution of (24) for given functions S, H ∈ C∞( ¯
+M),
+provided that η1 < 0. In particular, we will discuss the following cases:
+(i). S < 0 in M, and H ⩽ 0 everywhere on ∂M, H ̸≡ 0, with η1 < 0;
+(ii). S < 0 in M, and H > 0 somewhere on ∂M, with η1 < 0;
+(iii). S changes sign in M, and H is arbitrary on ∂M, with η1 < 0.
+Note that the Case (ii) above covers the possibilities when H > 0 everywhere on ∂M, or
+´
+∂M HdS >
+0. Note also that the case S < 0 everywhere in M and H = 0 on ∂M has been discussed in [19].
+For Case (iii), obviously we have to impose some restrictions on S and H, as we shall see later;
+there is no free choice of S especially, due to Kazdan and Warner [10]. The first result concerns
+the Case (i).
+Theorem 3.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S1 < 0 be any smooth function on
+¯
+M. Let H1 ∈ C∞( ¯
+M) such that H1 < 0
+everywhere on ∂M. If η1 < 0, then there exists a small enough constant c > 0 such that (24)
+admits a positive solution u ∈ C∞( ¯
+M) with S = S1 and H = cH1. Equivalently, there exists a
+Yamabe metric ˜g = up−2g such that R˜g = S1 and h˜g = cH1
+����
+∂M
+.
+Proof. Due to the proof of Han-Li conjecture [17, Theorem], we may assume that hg = h > 0 and
+Rg < 0. Since η1 < 0, it follows that η1,β < 0 with small enough positive constant β > 0, due to
+Proposition 2.3. Any constant multiple of ϕ solves (15). Denote φ = δϕ, we choose the constant
+δ > 0 small enough so that
+η1,β inf
+¯
+M ϕ ⩾ δp−2 · inf
+¯
+M S1 · sup
+¯
+M
+ϕp−1.
+This can be done since both η1,β and S1 are negative functions. It follows that
+−a∆gφ + Rgφ = η1,βφ ⩽ S1φp−1 in M.
+Fix this δ. We check the boundary condition
+−∂φ
+∂ν +
+2
+p − 2hgφ = −β ·
+2
+p − 2φ ⩽
+2
+p − 2 · (cH1) φ
+p
+2
+for small enough positive constant c > 0. Again it works since both −β and H1 are negative. We
+set
+u− := φ.
+(25)
+The argument above shows that u− is a sub-solution of (24) with S = S1 and H = cH1 for small
+enough c. For super-solution, we set
+u+ := C ≫ 1.
+(26)
+When C large enough, we have
+−a∆gu+ + Rgu+ = RgC ⩾ S1Cp−1 in M.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+11
+Since H1 < 0, it is straightforward to check that for any c > 0, we have
+−∂u+
+∂ν +
+2
+p − 2hgu+ ⩾ 0 >
+2
+p − 2 (cH1) u
+p
+2
++.
+We can enlarge C so that C ⩾ sup ¯
+M u−. Lastly we shrink c if necessary since we require the
+smallness of the sup-norm of the prescribing mean curvature function in the proof of Theorem 2.2.
+Since 0 < u− ⩽ u+ and both u+ and u− are smooth functions, we conclude by Theorem 2.2 that
+(24) has a positive solution u ∈ C∞( ¯
+M) with S = S1 and H = cH1 for small enough c > 0.
+□
+We now consider the Case (ii) at the beginning of this section. Actually the proof is very similar
+to Theorem 3.1 above.
+Theorem 3.2. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S2 < 0 be any smooth function on
+¯
+M. Let H2 ∈ C∞( ¯
+M) such that H2 > 0
+somewhere on ∂M. If η1 < 0, then there exists a small enough constant c > 0 such that (24)
+admits a positive solution u ∈ C∞( ¯
+M) with S = S2 and H = cH2. Equivalently, there exists a
+Yamabe metric ˜g = up−2g such that R˜g = S2 and h˜g = cH2
+����
+∂M
+.
+Proof. The choice of the sub-solution is exactly the same as in Theorem 3.1. When we fix the sub-
+solution u−, we choose u+ = C ≫ 1 with C ⩾ u−, also large enough so that the same argument in
+Theorem 3.1 holds. Fix this C from now on. The only difference is that since H2 > 0 somewhere,
+we may need to shrink c, if necessary, so that
+∂C
+∂ν +
+2
+p − 2hgC ⩾
+2
+p − 2 · sup
+∂M
+(cH2)C
+p
+2
+The rest of the argument is exactly the same as in Theorem 3.1.
+□
+Remark 3.1. The method of monotone iteration scheme has its limits, as we cannot obtain the
+prescribed mean curvature to be H, due to the technical issue, see [18, Thm. 2.4].
+We now discuss the Case (iii). The following argument is inspired by Kazdan and Warner [10].
+When η1 < 0, Kazdan and Warner showed that the key is to get the super-solution of (24), if we
+are not using the variational method but instead the monotone iteration scheme. Next result shows
+that a super-solution of (24) can be converted to another relation. We point out that the following
+result is not specific for η1 < 0 case only.
+Lemma 3.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3.
+Let S, H ∈ C∞( ¯
+M) be given functions.
+Then there exists some positive function
+u ∈ C∞( ¯
+M) satisfying
+− a∆gu + Rgu ⩾ Sup−1 in M, ∂u
+∂ν +
+2
+p − 2hgu ⩾
+2
+p − 2Hu
+p
+2 on ∂M
+(27)
+if and only if there exists some positive function w ∈ C∞( ¯
+M) satisfying
+− a∆gw + (2 − p)Rgw + (p − 1)a
+p − 2
+· |∇gw|2
+w
+⩽ (2 − p)S in M, ∂w
+∂ν − 2hgw ⩽ −2Hw
+1
+2.
+(28)
+Moreover, the equality in (27) holds if and only if the equality in (28) holds.
+Proof. Assume that there is a positive function u ∈ C∞(M) that satisfies (27). Define
+w = u2−p.
+Note that 2 − p = −
+4
+n−2 < 0 since n ⩾ 3 by hypothesis. We compute that
+∇w = (2 − p)u1−p∇u ⇔ ∇u = up−1(2 − p)−1∇w,
+
+12
+J. XU
+and
+∆gw = (2 − p)u1−p∆gu + (2 − p)(1 − p)u−p|∇gu|2.
+By the inequality (27), we have
+a∆gw = (2 − p)u1−p (a∆gu) + a(2 − p)(1 − p)u−p|∇gu|2
+⩾ (p − 2)u1−p �
+−Rgu + Sup−1�
++ a(2 − p)(1 − p)(2 − p)−2u2p−2u−p|∇gv|2
+= (p − 2)S + (2 − p)Rgu1−p + a(p − 1)
+p − 2 up−2|∇gv|2
+= (p − 2)S + (2 − p)Rgw + a(p − 1)
+p − 2
+|∇gw|2
+w
+.
+Shifting (p − 2)S to the left side and a∆gw to the right side, we get the first part of the inequality
+(28). For the boundary condition, recall that u = w
+1
+2−p and p =
+2n
+n−2, it follows that
+∂u
+∂ν +
+2
+p − 2Hu
+p
+2 ⩾
+2
+p − 2hgu ⇔
+1
+2 − pw
+1
+2−p −1 ∂w
+∂ν +
+2
+p − 2hgw
+1
+2−p ⩾
+2
+p − 2Hw
+p
+2(2−p)
+⇔ − n − 2
+4
+w− n
+4 − 1
+2 ∂w
+∂ν + n − 2
+2
+hgw− n
+4 + 1
+2 ⩾ n − 2
+2
+Hw− n
+4
+⇔∂w
+∂ν − 2hgw ⩽ −2Hw
+1
+2 .
+Hence the second part of (28) holds. It is clear that the equality holds if an only if all inequalities
+above are equalities.
+If we assume (28) for some w, we just define u = w
+1
+2−p . The argument is very similar and we
+omit the details.
+□
+We now introduce the result of prescribing scalar and mean curvature functions for Case (iii),
+with a technical restriction very similar to the condition given by Kazdan and Warner.
+This
+technical condition, in principle, is to show the positivity of the function that satisfies (28). Due
+to the Han-Li conjecture [17, Theorem], we may assume that the initial metric g has Rg = λ < 0
+and hg = ζ > 0, since η1 < 0. Before we start with the special case, recall that if there exists a
+constant q > n, and some function u ∈ C∞( ¯
+M) satisfies
+∥u∥W 2,q(M,g) ⩽ γ′ �
+∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g)
+�
+for some functions F1 ∈ Lq(M, g) and F2 ∈ W 1,q(M, g), the H¨older estimates implies that
+∥u∥L∞( ¯
+M) + ∥∇u∥L∞( ¯
+M) ⩽ γ
+�
+∥F1∥Lq(M,g) + ∥F2∥W 1,q(M,g)
+�
+.
+(29)
+This inequality is due to the Sobolev embedding in [1, §2].
+Theorem 3.3. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Assume that η1 < 0, Rg = λ < 0 and hg = ζ > 0 for some constants λ and ζ. Let
+S3, H3 ∈ C∞( ¯
+M) and q > n be a positive integer. Let γ be the constant in the estimate (29). Set
+D = (p−1)a
+p−2 . If there exists a function F ∈ C∞( ¯
+M) and a positive constant A > 0, such that
+(2 − p)S3 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽
+A
+2γ (1 + (D + 1) (2 − p)λ),
+(30)
+then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯
+M)
+with S = S3 and H = cH3. Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S3
+and h˜g = cH3
+����
+∂M
+.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+13
+Proof. In this proof, we always denote Rg = λ and hg = ζ. We construct the super-solution of
+(24) first.
+Due to Lemma 3.1, it is equivalent to show the existence of some positive function
+w ∈ C∞( ¯
+M) such that (28) holds for S = S3 and H = cH3 for some constant c. Take
+δ :=
+A
+(1 + (D + 1)(2 − p)λ) > 0.
+We also choose some negative constant
+δ′ = −
+δ
+2γVolg(M)
+1
+q
+< 0.
+By standard elliptic theory, there exists a unique solution w of the following PDE
+−a∆gw + (2 − p)λw = F − δ in M, ∂w
+∂ν = δ′ on ∂M.
+The uniqueness comes from the fact that (2−p)λ > 0, which implies the invertibility of the operator
+�
+−a∆g + (2 − p)λ, ∂
+∂ν
+�
+. Clearly the constant (D + 1)δ solves the PDE
+−a∆g((D + 1)δ) + (2 − p)λ · ((D + 1)δ) = (2 − p)λ · ((D + 1)δ) in ∂M, ∂((D + 1)δ)
+∂ν
+= 0 on ∂M.
+Denote
+w0 := w − (D + 1)δ.
+The function w0 satisfies
+−a∆gw0 + (2 − p)λw0 = F − δ − (D + 1)(2 − p)λδ = F − A in M,
+∂w
+∂ν = δ′ on ∂M.
+(31)
+The first line in (31) is due to the definition of δ. Since the differential operator with the boundary
+operator is invertible, we apply W s,q-type elliptic estimates (7) as well as the estimates of (29),
+∥w0∥L∞( ¯
+M) + ∥∇w0∥L∞( ¯
+M) ⩽ γ
+�
+∥F − A∥Lq(M,g) + ∥δ′∥W 1,q(M,g)
+�
+⩽ γ
+�
+A
+2γ (1 + (D + 1)(2 − p)λ) + |δ′| · Volg(M)
+1
+q
+�
+⩽ δ.
+(32)
+The last inequality is due to the definitions of δ and δ′. By definition of w0, the inequality (32)
+implies
+∥w − (D + 1)δ∥L∞( ¯
+M) ⩽ δ, ∥∇w∥L∞(¯Ω) ⩽ δ.
+It follows that
+0 < Dδ ⩽ w ⩽ (D + 2)w on ¯
+M, sup
+¯
+M
+|∇w| ⩽ δ ⇒ (p − 1)a
+p − 2
+· |∇w|2
+w
+⩽ δ on ¯
+M.
+(33)
+With (36), (33), we have
+− a∆gw + (2 − p)λw + (p − 1)a
+p − 2
+· |∇w|2
+w
+= F − δ + (p − 1)a
+p − 2
+· |∇w|2
+w
+⩽(2 − p)S3;
+∂w
+∂ν − 2hgw = δ′ − 2hgw ⩽ −2 · (cH3) w
+1
+2 .
+The last inequality holds for small enough constant c > 0, regardless of the sign of H3 since
+δ′ − 2hgw < 0 by set-up. By (33) again, we conclude that w > 0 on
+¯
+M. By Lemma 3.1, the
+positive, smooth function
+u = w
+1
+2−p
+
+14
+J. XU
+is a super-solution of (24) with S = S3 and H = cH3. Note that u is still a super-solution if we
+make c smaller.
+For sub-solution, we apply the perturbed eigenvalue problem in Proposition 2.3 again. There
+exists a small enough constant β > 0 such that
+− a∆gϕ + λϕ = η1,βϕ in M, ∂ϕ
+∂ν +
+2
+p − 2 (ζ + β) ϕ = 0 on ∂M.
+(34)
+Any scaling of ϕ solves (35). Set the positive constant ξ ≪ 1 such that
+φ := ξϕ ⩽ u on ¯
+M
+for the fixed super-solution u defined just above. We shrink ξ further, if necessary, such that
+η1,β (ξϕ) ⩽ S3 (ξϕ)p−1 in M, −
+2
+p − 2 · β (ξϕ) ⩽
+2
+p − 2 · (cH3) · (ξϕ)
+p
+2 on ∂M.
+(35)
+Note that the boundary condition holds for every c, as long as we take ξ small enough. We point
+out that the choice of the constant c depends on the construction of the super-solution as well as
+the technical condition of the monotone iteration scheme, which only depends on the super-solution
+but not the sub-solution, see Equation (19) in [18]. Thus we can choose c first, then determine ξ.
+It follows that φ is a sub-solution of (24) with S = S3 and H = cH3. Furthermore, 0 < φ ⩽ u
+on ¯
+M. Applying Theorem 2.2, we conclude that there exists a positive function u ∈ C∞( ¯
+M) as
+desired.
+□
+The general case when η1 < 0 is a straightforward consequence of the result above.
+Corollary 3.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S4, H4 ∈ C∞( ¯
+M) and q > n be a positive integer. Let γ be the constant in the
+estimate (29) and λ be some negative constant. Set D = (p−1)a
+p−2 . Assume that η1 < 0. If there
+exists a function F ∈ C∞( ¯
+M) and a positive constant A > 0, such that
+(2 − p)S4 ⩾ F on ∂M, ∥F − A∥Lq(M,g) ⩽
+A
+2γ (1 + (D + 1) (2 − p)λ),
+(36)
+then there exists a small enough constant c > 0 such that (24) admits a positive solution u ∈ C∞( ¯
+M)
+with S = S4 and H = cH4. Equivalently, there exists a Yamabe metric ˜g = up−2g such that R˜g = S4
+and h˜g = cH4
+����
+∂M
+.
+Proof. By the result of the Han-Li conjecture [17, Theorem], there exists a conformal metric g1 =
+vp−2g such that Rg1 = λ and hg1 = ζ. We then apply Theorem 3.3 for the metric g1, i.e. there
+exists ˜g = up−2g1 with R˜g = S4 and h˜g = cH4 with small enough c > 0. The conformal change
+˜g = (uv)p−2 g
+is the desired metric.
+□
+4. Prescribed Scalar and Mean Curvature Functions for Conformal Equivalent
+Metrics When η1 < 0
+Inspired by the “Trichotomy Theorem” on closed manifolds, we would like to discuss the pre-
+scribing scalar and mean curvature problem on ( ¯
+M, g), n = dim ¯
+M ⩾ 3, but not restricted in a
+conformal class [g] only. Instead, we are interested in the conformally equivalent metrics.
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+15
+Definition 4.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, we say
+that a metric ˜g is conformally equivalent to the metric g if there exists a positive, smooth function
+u ∈ C∞( ¯
+M) and a diffeomorphism φ : ¯
+M → ¯
+M such that
+φ∗˜g = up−2g.
+Within in a conformal class, the prescribing scalar and mean curvature problem for given func-
+tions S, H ∈ C∞( ¯
+M) is reduced to the PDE (1). For conformlaly equivalent metrics, the prescribing
+scalar and mean curvature problem is reduced to the existence of a positive, smooth solution of the
+following PDE
+− a∆gu + Rgu = (S ◦ φ) up−1 in M, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2 · (H ◦ φ) · u
+p
+2 on ∂M.
+(37)
+Our next result extends the result of prescribing scalar curvature problem on closed manifolds with
+dimensions at least 3 [10, Thm. 3.3] to compact manifolds with non-empty smooth boundaries,
+provided that the first eigenvalue η1 of the conformal Laplacian with Robin boundary condition is
+negative. The method is essentially due to Kazdan and Warner [8, 10].
+Theorem 4.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S5 be any smooth function on
+¯
+M that is negative somewhere in M. Let H5 ∈
+C∞( ¯
+M) and q > n be a positive integer. If η1 < 0, then there exists a small enough constant c > 0
+and a diffeomorphism φ : ¯
+M → ¯
+M such that (37) admits a positive solution u ∈ C∞( ¯
+M) with S = S5
+and H = cH5. Equivalently, there exists a conformally equivalent metric ˜g =
+�
+φ−1�∗ �
+up−2g
+�
+such
+that R˜g = S5 and h˜g = cH5
+����
+∂M
+.
+Proof. By Han-Li conjecture [17, Theorem], we may assume that Rg = λ < 0 and hg = ζ > 0
+for some constants λ, ζ.
+Fix some constant q > n.
+Due to Theorem 3.3, it suffices to find a
+diffeomorphism φ : ¯
+M → ¯
+M, a smooth function F ∈ C∞( ¯
+M), a positive constant A > 0 and a small
+enough positive constant c such that
+(2 − p)S5 ◦ φ ⩾ F on ¯
+M, ∥F − A∥Lq(M,g) ⩽
+A
+2γ (1 + (D + 1)(2 − p)λ);
+(38)
+in addition, sup ¯
+M c (H ◦ φ) is small enough. Here γ is the constant in the estimate (29), the constant
+D is defined to be
+D = (p − 1)a
+p − 2 .
+We determine φ, F and A first. If S5 < 0 everywhere on ¯
+M, we just choose φ to be the identity
+map and set
+F = A = (2 − p) max
+¯
+M S5.
+It is straightforward to check that (38) holds.
+If S5 ⩾ 0 somewhere and changes sign, we choose A first to be any positive constant such that
+0 < A < (2 − p) min
+¯
+M S5.
+(39)
+Just note that (2 − p) < 0. We pick interior open submanifolds U, V ⊂ M such that
+V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯
+M.
+In particular, we require that
+Volg(U − V ) ⩽
+
+
+A
+2γ (1 + (D + 1)(2 − p)λ) ·
+�
+(2 − p)∥S3∥L∞( ¯
+M) − A
+�
+
+
+q
+.
+(40)
+
+16
+J. XU
+We select the diffeomorphism φ such that
+(2 − p)S3 ◦ φ > A in U.
+(41)
+We then take the function F to be
+F = A in V ;
+(2 − p) max
+¯
+M S3 ◦ φ ⩽ F ⩽ A in U − V ;
+F = (2 − p) max
+¯
+M S3 ◦ φ in ¯
+M − U.
+(42)
+Clearly F ⩽ (2 − p)S3 ◦ φ on ¯
+M by (42). The function F only differs with A in U − V , by (40), it
+is immediate to check that the second inequality in (38) holds.
+Lastly we choose c so that the condition in Theorem 2.2 holds for the function S3 ◦ φ, i.e.
+c sup ¯
+M|H5| is small enough.
+The same c applies for the smallness of c sup ¯
+M|H5 ◦ φ| since the
+diffeomorphism does not change the extremal values of a function. Therefore the function S3 ◦ φ
+and cH5 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for
+some metric φ∗˜g = up−2g where u is positive and smooth on ¯
+M. Equivalently, S5 and cH5 can
+be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g =
+�
+φ−1�∗ up−2g.
+□
+Remark 4.1. The result of Theorem 4.1 indicates that on ( ¯
+M, g) with n = dim ¯
+M ⩾ 3, any
+function that is negative somewhere can be realized as a scalar curvature function of some metric
+g, meanwhile the mean curvature function of g can be some small enough scaling of any smooth
+function, provided that the manifold admits a metric with negative first eigenvalue of the conformal
+Laplacian, or equivalently, negative Yamabe invariant [6, §1].
+5. Prescribed Gauss and Geodesic Curvature Functions When χ( ¯
+M) < 0
+In this section, we discuss the prescribing Gauss and geodesic curvatures problem within a
+conformal class [g] of compact manifolds ( ¯
+M, g) with non-empty smooth boundary ∂M, provided
+that χ( ¯
+M) < 0 and n = dim ¯
+M = 2. This is a 2-dimensional analogy of prescribing scalar and
+mean curvatures problem with η1 < 0, provided that the dimension is at least 3.
+Let K, σ ∈ C∞( ¯
+M) be given functions. This type of Kazdan-Warner problem is reduced to the
+existence of a smooth solution u of the following PDE
+− a∆gu + Kg = Ke2u in M, ∂u
+∂ν + σg = σeu on ∂M.
+(43)
+Here Kg and σg are Gaussian and geodesic curvatures of g, respectively. The solvability of this
+PDE implies that the metric ˜g = e2ug has Gauss curvature K˜g = K and geodesic curvature σ˜g = σ.
+We mainly discuss to cases:
+(i). K ⩽ 0 everywhere in ¯
+M, and arbitrary σ, with χ( ¯
+M) < 0;
+(ii). K > 0 somewhere in ¯
+M and changes sign, σ is an arbitrary function, with χ( ¯
+M) < 0.
+We would like to apply the monotone iteration scheme to solve (43), it is equivalent to construct the
+sub- and super-solutions of (43). The key is to construct the super-solution. As in §3, we convert
+the super-solution of (43) into another inequality involving derivatives.
+Lemma 5.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M = 2. Let K, σ ∈ C∞( ¯
+M) be given functions. Then there exists some function u ∈ C∞( ¯
+M)
+satisfying
+− ∆gu + Kg ⩾ Ke2u in M, ∂u
+∂ν + σg ⩾ σeu on ∂M
+(44)
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+17
+if and only if there exists some positive function w ∈ C∞( ¯
+M) satisfying
+− ∆gw − 2wKg + |∇gw|2
+w
+⩽ −2K in M, ∂w
+∂ν − 2wσg ⩽ −2σw
+1
+2 on ∂M.
+(45)
+Moreover, the equality in (44) holds if and only if the equality in (45) holds; and the inequality in
+(44) is in the reverse direction if and only if the inequality in (45) is in the reverse direction.
+Proof. Assume (44) for some function u first. Define
+w := e−2u
+We observe that
+∇gw = −2e−2u∇gu ⇒ ∇gu = −1
+2e2u∇gw,
+∆gw = −2e−2u∆gu + 4e−2u|∇gu|2 = −2e−2u∆gu + e2u|∇gw|2.
+Thus we have
+−∆gw = 2e−2u∆gu − e2u|∇gw|2 ⩽ 2e−2u �
+Kg − Ke2u�
+− |∇gw|2
+w
+= 2wKg − 2K − |∇gw|2
+w
+⇒ − ∆gw − 2wKg + |∇gw|2
+w
+⩽ −2K in M.
+For the boundary condition, we have
+∂w
+∂ν = ∂e−2u
+∂ν
+= −2e−2u ∂u
+∂ν ⩽ −2e−2u (−σg + σeu)
+= 2wσg − 2σw
+1
+2
+⇒∂w
+∂ν − 2wσg ⩽ −2σw
+1
+2 on ∂M.
+Therefore (45) holds for w = e−2u > 0 on ¯
+M. It is clear that equality holds when all inequalities
+above are equalities. It is also straightforward to see that the inequalities are in the reverse directions
+if and only if the inequalities are in the reverse directions in each step above.
+For the opposite direction, we assume (45) holds for some positive, smooth function w. Define
+u = −1
+2 log w.
+We can show that u satisfies (44).
+The argument is quite similar to above and we omit the
+details.
+□
+Due to the uniformization theorem, we may assume Kg = −1 and σg = 0 in (43) from now on, as
+our model case up to some pointwise conformal change, provided that χ( ¯
+M) < 0. In 2-dimensional
+case, we also have the W s,q-type estimates from Theorem 2.1. We choose q = 3, the estimate in
+(7) plus the Sobolev embedding into H¨older space, the inequality in (29) becomes
+∥u∥L∞( ¯
+M) + ∥∇u∥L∞( ¯
+M) ⩽ γ
+�
+∥F1∥L3(M,g) + ∥F2∥W 1,3(M,g)
+�
+.
+(46)
+Here F1, F2 and u comes from the PDE (6) with the operators L = −∆g + 2 and B =
+∂
+∂ν , so is the
+constant γ. Our main result of this section is the following, which covers both Case (i) and Case
+(ii) at the beginning of this section.
+
+18
+J. XU
+Theorem 5.1. Let ( ¯
+M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.
+Let K1, σ1 ∈ C∞( ¯
+M) be given functions. Let γ be the constant in the estimate (46). Assume that
+χ( ¯
+M) < 0. If there exists a function F ∈ C∞( ¯
+M) and a positive constant A > 0, such that
+− 2K1 ⩾ F on ∂M, ∥F − A∥L3(M,g) ⩽ A
+6γ ,
+(47)
+then there exists a small enough constant c > 0 such that (43) admits a positive solution u ∈ C∞( ¯
+M)
+with K = K1 and σ = cσ1. Equivalently, there exists a Yamabe metric ˜g = e2ug such that K˜g = K1
+and σ˜g = cσ1
+����
+∂M
+.
+Proof. The proof is essentially the same as in Theorem 4.1. By Lemma 43, the construction of the
+super-solution is equivalent to the construction of a function w that satisfies (45) for K1, σ1 and
+some small enough positive constant c. We set
+δ = A
+3 , δ′ = −
+δ
+2γVolg(M)
+1
+3
+.
+(48)
+There is a unique solution for the PDE
+−∆gw + 2w = F − δ in M, ∂w
+∂ν = δ′ on ∂M.
+Define
+w0 = w − 2δ,
+it follows that w0 satisfies the PDE
+− ∆gw0 + 2w0 = F − 3δ = F − A in M, ∂w0
+∂ν = δ′ on ∂M.
+(49)
+Apply the estimate (46) for w0 in (49), it follows that
+∥w0∥L∞( ¯
+M) + ∥∇w0∥L∞( ¯
+M) ⩽ γ
+�
+∥F − A∥L3(M,g) + ∥δ′∥W 1,3(M,g)
+�
+⩽ δ.
+It follows from the definition of w0 that
+0 < δ ⩽ w ⩽ 3δ on ¯
+M, ∥∇w∥L∞( ¯
+M) ⩽ δ.
+Therefore we conclude that
+−∆gw + 2w + |∇w|2
+w
+= F − δ + |∇w|2
+w
+⩽ F ⩽ −2K1 in M.
+In addition, we take c small enough so that
+∂w
+∂ν = δ′ ⩽ −2cσ1w
+1
+2 on ∂M.
+This can be done since δ′ < 0. It follows that the function
+u+ := −1
+2 log w
+is a super-solution of (43) with K = K1 and σ = cσ1. Clearly u+ ∈ C∞( ¯
+M).
+We construct a sub-solution now. Consider the PDE
+−∆gu0 = 1
+2 in M, ∂u0
+∂ν = C on ∂M.
+By standard elliptic PDE theory, see e.g. [14, Prop. 7.7, Ch. 4], the above PDE is solvable by some
+smooth function u0 ∈ C∞( ¯
+M) if −
+´
+M
+1
+2dVolg =
+´
+∂M CdSg. We choose the constant C < 0 so that
+the compatibility condition just mentioned holds. Clearly
+u− := u0 + C1
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+19
+solves the PDE above also for any constant C1. We just choose C1 to be very negative such that
+u− ⩽ u+ on ∂M;
+In addition,
+−∆gu− − 1 = −1
+2 ⩽ K1e2u− = K1e2u0 · e2C1 in M,
+∂u−
+∂ν = C ⩽ cσ1eu− = cσ1eu0 · eC1 on ∂M.
+These can be done since the constants on the left sides of the inequalities are both negative. Note
+that both the super-solution and sub-solution holds for smaller constant c by adjusting the constant
+C1 only.
+Note that when F(·, u) = K1e2u + 1, G(·, u) = cσ1eu, the condition (22) is independent of the
+sub-solution u− as we can see the very similar case in Theorem 2.2 for the Yamabe equation. Thus
+we take c small enough so that the hypotheses in Theorem 2.3 holds. It follows that there exists
+some smooth function u that solves (43) with K = K1 and σ = cσ1.
+□
+We can partially answer the two cases we are interested in. For Case (ii), not every function that
+changes sign can be a prescribed scalar curvature function unless it is not too positive too often. We
+show that every function that is negative everywhere can be realized as a scalar curvature function,
+meanwhile, a small enough scaling of any function can be realized as prescribed mean curvature
+function, under pointwise conformal deformation. This is Case (i).
+Corollary 5.1. Let ( ¯
+M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.
+Let K2, σ2 ∈ C∞( ¯
+M) be given functions. Assume that K2 < 0 everywhere on
+¯
+M. If χ( ¯
+M) < 0,
+then there exists a small enough constant c, a smooth function u ∈ C∞( ¯
+M) such that u solves (43)
+with K = K2 and σ = cσ2. It is equivalent to say that the metric ˜g = e2ug has Gauss curvature
+K˜g = K2 and geodesic curvature σ˜g = cσ2.
+Proof. We show that the condition (47) holds. Since K2 < 0 everywhere, we just choose
+F = A = −2 max
+¯
+M K2 ⇒ −2K2 ⩾ F, ∥F − A∥L3(M,g) = 0.
+We just need to choose a small enough c such that the hypotheses in Theorem 5.1 and Theorem
+2.3 hold.
+□
+For Case (ii), we can get a more comprehensive answer by considering the class of conformally
+equivalent metrics.
+Analogous to §4, we are looking for a metric ˜g =
+�
+φ−1�∗ e2ug with some
+diffeomorphism φ :
+¯
+M →
+¯
+M and smooth function u ∈ C∞( ¯
+M) such that the scalar and mean
+curvatures of ˜g are given functions K, σ ∈ C∞( ¯
+M), respectively. This problem is reduced to the
+PDE
+− ∆gu + Kg = (K ◦ φ) e2u in M, ∂u
+∂ν + σgu = (σ ◦ φ) eu on ∂M.
+(50)
+Similar to Theorem 4.1 for dimensions at least 3, we introduce the following result for compact
+Riemann surfaces.
+Corollary 5.2. Let ( ¯
+M, g) be a compact Riemann surface with non-empty smooth boundary ∂M.
+Let σ3 ∈ C∞( ¯
+M) be any function and K3 ∈ C∞( ¯
+M) be a function that is negative somewhere in M.
+If χ( ¯
+M) < 0, then there exists a small enough constant c, a smooth function u ∈ C∞( ¯
+M) and a
+diffeomorphism φ : ¯
+M → ¯
+M such that u solves (50) with K = K3 and σ = cσ3. It is equivalent to
+say that the metric ˜g =
+�
+φ−1�∗ e2ug has Gauss curvature K˜g = K3 and geodesic curvature σ˜g = cσ3.
+
+20
+J. XU
+Proof. The proof is essentially the same as in Theorem 4.1. We determine φ, F, A first so that (47)
+holds; then determine the constant c. We may assume that K3 is negative somewhere but not
+everywhere since otherwise it is reduced to the result of Corollary 5.1.
+We choose A first to be any positive constant such that
+0 < A < −2 min
+¯
+M K3.
+(51)
+We pick interior open submanifolds U, V ⊂ M such that
+V ⊂ ¯V ⊂ U ⊂ M ⊂ ¯
+M.
+In particular, we require that
+Volg(U − V ) ⩽
+
+
+A
+6γ ·
+�
+2∥K3∥L∞( ¯
+M) − A
+�
+
+
+3
+.
+(52)
+We select the diffeomorphism φ such that
+− 2K3 ◦ φ > A in U.
+(53)
+We then take the function F to be
+F = A in V ;
+− 2 max
+¯
+M K3 ◦ φ ⩽ F ⩽ A in U − V ;
+F = −2 max
+¯
+M K3 ◦ φ in ¯
+M − U.
+(54)
+Clearly F ⩽ −2K3 ◦ φ on ¯
+M by (54). The function F only differs with A in U − V , by (52), it is
+immediate to check that the second inequality in (47) holds.
+Lastly we choose c so that the condition in Theorem 2.3 holds for the function K3 ◦ φ, i.e.
+c sup ¯
+M|σ3| is small enough.
+The same c applies for the smallness of c sup ¯
+M|σ3 ◦ φ| since the
+diffeomorphism does not change the extremal values of a function. Therefore the function K3 ◦ φ
+and cσ3 ◦ φ can be realized as prescribed scalar and mean curvature functions, respectively, for
+some metric φ∗˜g = up−2g where u is positive and smooth on ¯
+M. Equivalently, K3 and cσ3 can
+be realized as prescribed scalar and mean curvature functions, respectively, for some metric ˜g =
+�
+φ−1�∗ up−2g.
+□
+Remark 5.1. The result of Corollary 5.2, combining Theorem 4.1 indicate that on ( ¯
+M, g) with
+n = dim ¯
+M ⩾ 2, any function that is negative somewhere can be realized as a scalar/Gauss
+curvature function of some metric g, meanwhile the mean/geodesic curvature function of g can be
+some small enough scaling of any smooth function, provided that the manifold admits a metric with
+negative first eigenvalue of the conformal Laplacian, or negative Euler characteristics, respectively,
+depending on the dimension of the manifold. This improve the result mentioned in Remark 4.1.
+6. Prescribed Scalar and Mean Curvature Functions for Conformally Equivalent
+Metrics When η1 = 0
+In this section, we discuss the prescribing scalar and mean curvatures problem for metrics con-
+formally equivalent to the metric g on compact manifolds ( ¯
+M, g) with non-empty smooth boundary
+∂M, provided that η1 = 0 and n = dim ¯
+M ⩾ 3. We gave a comprehensive study for manifolds
+with dimensions at least 3 in [18] for pointwise conformal change. Here we consider whether there
+exists some smooth function u ∈ C∞( ¯
+M) and some diffeomorphism φ :
+¯
+M →
+¯
+M such that the
+metric ˜g =
+�
+φ−1�∗ up−2g has scalar curvature S and mean curvature H for some given functions
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+21
+S, H ∈ C∞( ¯
+M). Since the model case for zero first eigenvalue case is Rg = hg = 0, the problem
+above is reduced to the existence of the solution of the following PDE
+− a∆gu = (S ◦ φ) · up−1 in M, ∂u
+∂ν =
+2
+p − 2 · (H ◦ φ) · u
+p
+2 on ∂M.
+(55)
+Recall the result of prescribing scalar and mean curvature problems for conformal metrics on ( ¯
+M, g).
+Theorem 6.1. [18, Thm. 1.4] Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary
+∂M, n = dim ¯
+M ⩾ 3. Let S, H ∈ C∞( ¯
+M) be given nonzero functions. Assume that η1 = 0. If the
+function S satisfies
+S changes sign and
+ˆ
+M
+SdVolg < 0,
+then there exists a pointwise conformal metric ˜g ∈ [g] that has scalar curvature R˜g = S and h˜g = cH
+for some small enough positive constant c.
+The conformally equivalent case follows from the result of Theorem 6.1, we show it below. Note
+that the case S = H = 0 is the trivial case.
+Theorem 6.2. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S6, H6 ∈ C∞( ¯
+M) be given nonzero functions. Assume that η1 = 0. If the function
+S satisfies
+S6 changes sign,
+then there exists a diffeomorphism φ : ¯
+M → ¯
+M and a small enough constant c > 0 such that (55)
+has a smooth solution u ∈ C∞( ¯
+M) for φ, S = S6 and H = cH6. It is equivalent to say that the
+conformally equivalent metric ˜g =
+�
+φ−1�∗ up−2g has scalar curvature R˜g = S6 and mean curvature
+h˜g = cH6.
+Proof. Due to Theorem 6.1, it suffices to show that there exist a diffeomorphism φ : ¯
+M → ¯
+M such
+that
+ˆ
+M
+(S6 ◦ φ) dVolg < 0.
+Due to the same reason in [9, 8], it is straightforward that such a diffeomorphism does exist since
+S6 changes sign. The smallness of c is then determined by S6 ◦ φ, sup ¯
+M|H6| as well as the choice
+of sub- and super-solutions in the proofs of [18, Thm. 5.1, Cor. 5.1, Cor. 5.2].
+Note that any
+diffeomorphism φ will not change the supremum of |H6| on ¯
+M.
+□
+Remark 6.1. The result of Theorem 6.2 indicates that on ( ¯
+M, g) with n = dim ¯
+M ⩾ 3, any
+function that changes sign or identically zero can be realized as a scalar curvature function of some
+metric g, meanwhile the mean curvature function of g can be some small enough scaling of any
+smooth function or zero function, respectively, provided that the manifold admits a metric with
+zero first eigenvalue of the conformal Laplacian, or equivalently, zero Yamabe invariant [6, §1].
+7. Prescribed Scalar and Mean Curvature Functions When η1 > 0
+In this section, we seek for a positive, smooth solution of the following PDE
+− a∆gu + Rgu = Sup−1 in M, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2Hu
+p
+2 on ∂M.
+(56)
+on compact manifolds ( ¯
+M, g) with non-empty smooth boundary ∂M, n = dim ¯
+M ⩾ 3, for given
+functions S, H ∈ C∞( ¯
+M), provided that η1 > 0.
+As we have shown in [16], [17] and [19], we
+need to use local analysis, gluing a super-solution, and then apply monotone iteration scheme here.
+According to the “Trichotomy Theorem” in [20], we expect few restrictions on prescribed scalar
+and mean curvature functions. We will discuss the following case:
+
+22
+J. XU
+(i). S > 0 somewhere in M, and H > 0 somewhere on ∂M, with η1 > 0;
+(ii). S > 0 somewhere in M, and H ⩽ 0 everywhere on ∂M but H ̸≡ 0, with η1 > 0.
+Note that we have discussed the case S > 0 somewhere and H ≡ 0 in [19]. Currently we do not see
+how to apply our method to the case mentioned in [7],
+− ∆eu = 0 in Bn, ∂u
+∂ν +
+2
+p − 2hgu =
+2
+p − 2Hu
+p
+2 on ∂Bn, u > 0
+(57)
+for some given function H. Escobar showed that there is an obstruction for the choice of H
+ˆ
+∂Bn X · ∇gHdS = 0.
+Here X is some conformal Killing field on ∂Bn. With standard Euclidean metric in Bn and the
+induced metric on ∂Bn, the first eigenvalue of conformal Laplacian with Robin condition is positive.
+However, since the right side is zero, we are not able to get a nontrivial local solution of the Dirichlet
+problem
+−∆eu = 0 in Ω, u = 0 on ∂Ω.
+Therefore we may need some alternative method to resolve this issue.
+However, we can get some interesting results provided that S ̸≡ 0. According to the detailed
+analysis in [19, §5], we know that there will be obstructions for the choices of prescribed scalar
+curvature functions on Sn/Γ for some Kleinian group Γ. The map Sn → Sn/Γ must be a covering
+map since otherwise Sn/Γ cannnot be a manifold. It follows that Sn/Γ has empty boundary, which
+follows that there will be no obstruction for the choice of prescribed scalar curvature functions on
+( ¯
+M, g).
+The first result concerns the Case (i) above:
+Theorem 7.1. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S7 > 0 somewhere be any smooth function on
+¯
+M. Let H7 ∈ C∞( ¯
+M) such that
+H7 > 0 somewhere on ∂M. If η1 > 0, then there exists a small enough constant c > 0 such that
+(56) admits a positive solution u ∈ C∞( ¯
+M) with S = S7 and H = cH7. Equivalently, there exists a
+Yamabe metric ˜g = up−2g such that R˜g = S7 and h˜g = cH7
+����
+∂M
+.
+Proof. Without loss of generality, we may assume that Sg > 0 and hg = h > 0 with positive
+constant h, by Theorem 2.1. According to Proposition 2.3, we fix some β < 0 small enough so that
+η1,β > 0 and satisfies
+− a∆gϕ + Rgϕ = η1,βϕ in M, ∂ϕ
+∂ν +
+2
+p − 2hgϕ = 0 on ∂M.
+(58)
+Here ϕ > 0 on ¯
+M. Any scaling of ϕ solves (58). Denote φ = δϕ for some δ > 0. We choose δ > 0
+small enough so that
+η1,β inf
+¯
+M ϕ ⩾ δp−2 sup
+¯
+M
+S7 sup
+¯
+M
+ϕp−1.
+It follows that
+−a∆gφ + Rgφ ⩾ S7φp−1 in M.
+Fix this δ. We then choose c > 0 small enough so that
+βφ ⩾ (cH7) φ
+p
+2 on ∂M.
+It follows that
+∂φ
+∂ν +
+2
+p − 2hgφ ⩾
+2
+p − 2 (cH7) φ
+p
+2 on ∂M.
+(59)
+Note that (59) still holds for any smaller c. For the sub-solution, we apply Proposition 2.1 or
+Proposition 2.2, depending on the vanishing of the Weyl tensor in the interior M, to construct local
+
+TRICHOTOMY THEOREM: PRESCRIBED SCALAR AND MEAN CURVATURES
+23
+solution u0 of the Yamabe equation with Dirichlet boundary condition on some domain Ω. Apply
+Lemma 3.2 in [19], we can construct a local super-solution f of the Yamabe equation in Ω such
+that f = φ near ∂Ω. We then define
+u− =
+�
+u0 in Ω
+0 in M\Ω
+u+ :=
+�
+f in Ω
+φ in M\Ω .
+Since u− ≡ 0 on ∂M, it follows from the same argument in Lemma 3.1 in [19] that u− is a sub-
+solution of the (56) with S = S7 and H = cH7 for any constant c. According to the construction in
+Lemma 3.2 of [19], we conclude that 0 ⩽ u− ⩽ u+, u− ̸≡ 0. In addition, u− ∈ H1(M, g) ∩ C0( ¯
+M),
+and u+ ∈ C∞( ¯
+M). According to (59), we have seen that u+ is a super-solution of the (56) with
+S = S7 and H = cH7 for small enough c. Shrinking c, if necessary, so that the hypotheses of
+smallness of c sup ¯
+M|H7| holds. A direct application of Theorem 2.2 indicates the existence of a
+positive solution u ∈ C∞( ¯
+M) with S = S7 and H = cH7.
+□
+The proof of the Case (ii) is very similar as in Theorem 7.1.
+Theorem 7.2. Let ( ¯
+M, g) be a compact manifold with non-empty smooth boundary ∂M, n =
+dim ¯
+M ⩾ 3. Let S8 > 0 somewhere be any smooth function on
+¯
+M. Let H8 ∈ C∞( ¯
+M) such that
+H8 ⩽ 0 everywhere on ∂M. If η1 > 0, then there exists a small enough constant c > 0 such that
+(56) admits a positive solution u ∈ C∞( ¯
+M) with S = S8 and H = cH8. Equivalently, there exists a
+Yamabe metric ˜g = up−2g such that R˜g = S8 and h˜g = cH8
+����
+∂M
+.
+Proof. Everything is exactly the same as in Theorem 7.1, except at (59), there is no restriction for
+the choice of the constant c. However, c should be small enough so that the hypotheses in Theorem
+2.2 holds.
+□
+Remark 7.1. The result of Theorem 7.1 and Theorem 7.2 indicate that on ( ¯
+M, g) with n =
+dim ¯
+M ⩾ 3, any function that is positive somewhere can be realized as a scalar curvature function
+of some metric g, meanwhile the mean curvature function of g can be some small enough scaling
+of any smooth function, provided that the manifold admits a metric with positive first eigenvalue
+of the conformal Laplacian, or equivalently, positive Yamabe invariant [6, §1].
+References
+[1] T. Aubin. Nonlinear Analysis on Manifolds. Monge-Amp´ere Equations. Grundlehren der mathematischen Wis-
+senschaften. Springer, Berlin, Heidelberg, New York, 1982.
+[2] S. Brendle and F. Marques. Recent progress on the Yamabe problem. arXiv:1040.4960.
+[3] H. Brezis and F. Merle. Uniform esitmates and blow-up behavior for solutions of −δu = v(x)eu in two dimensions.
+Commun. Partial. Differ., 16(8-9):1223–1253, 1991.
+[4] A. Chang and P. Yang. Prescribing Gaussian curvature on S2. Acta Math., 159:215–259, 1987.
+[5] S. Cruz-Bl´azquez, A. Malchiodi, and D. Ruiz. Conformal metrics with prscribed scalar and mean curvature.
+arXiv:2105.04185.
+[6] J. Escobar. The Yamabe problem on manifolds with boundary. J. Differential Geom., 35:21–84, 1992.
+[7] J. Escobar. Conformal metrices with prescribed mean curvature on the boundary. Calc. Var. Partial Differential
+Equations, 4:559–592, 1996.
+[8] J. Kazdan and F. Warner. Curvature functions for compact 2−manifolds. Ann. of Math., 99:14–47, 1974.
+[9] J. Kazdan and F. Warner. Existence and conformal deformations of metrices with prescribed Gaussian and scalar
+curvatures. Ann. of Math. (2), 101(2):317–331, 1975.
+[10] J. Kazdan and F. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Differential
+Geom., 10:113–134, 1975.
+
+24
+J. XU
+[11] A. Malchiodi and M. Mayer. Prescribing Morse scalar curvatures: Pinching and Morse theory. Commun. Pure
+Appl. Math, 2021.
+[12] S. Rosenberg and J. Xu. Solving the Yamabe problem by an iterative method on a small Riemannian domain.
+arXiv:2110.14543.
+[13] M. Struwe. A flow approach to Nirenberg’s problem. Duke Math. J., 128(19-64), 2005.
+[14] M. Taylor. Partial Differential Equations I. Springer-Verlag, New York, New York, 2011.
+[15] M. Taylor. Partial Differential Equations III. Springer-Verlag, New York, New York, 2011.
+[16] J. Xu. The boundary Yamabe problem, I: Minimal boundaray case. arXiv:2111:03219.
+[17] J. Xu. The boundary Yamabe problem, II: General constant mean curvature case. arXiv:2112.05674.
+[18] J. Xu. The conformal Laplacian and the Kazdan-Warner problem: Zero first eigenvalue case. arXiv:2211.15024.
+[19] J. Xu. Prescribed scalar curvature on compact manifolds under conformal deformation. arXiv:2205.15453.
+[20] J. Xu. Prescribed scalar curvature problem under conformal deformation of a Riemannian metric with Dirichlet
+boundary condition. arXiv:2208.11318.
+[21] J. Xu. Solving the Yamabe-type equations on closed manifolds by iteration schemes. arXiv: 2110.15436.
+Department of Mathematics and Statistics, Boston University, Boston, MA, U.S.A.
+Email address: xujie@bu.edu
+Institute for Theoretical Sciences, Westlake University, Hangzhou, Zhejiang Province, China
+Email address: xujie67@westlake.edu.cn
+

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+1 
+ 
+Non-centrosymmetric Sr2IrO4 obtained under High Pressure 
+ 
+Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W. Plumb4, Martha Greenblatt2, 
+Yimei Zhu3, David Walker5, Weiwei Xie1* 
+ 
+ 
+1. Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 
+2. Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 
+08854, USA 
+3. Condensed Matter Physics and Materials Science Department, Brookhaven National 
+Laboratory, Upton, New York 11973, USA 
+4. Department of Physics, Brown University, Providence, Rhode Island 02912, USA 
+5. Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA 
+ 
+ 
+‡ H.W. and M.M. contributed equally. * Email: xieweiwe@msu.edu  
+ 
+ 
+ 
+Abstract 
+ 
+Sr2IrO4 with strong spin-orbit coupling (SOC) and Hubbard repulsion (U) hosts Mott 
+insulating states. The similar crystal structure, magnetic and electronic properties, particularly the 
+d-wave gap observed in Sr2IrO4 enhanced the analogies to cuprate high-Tc superconductor, 
+La2CuO4. The incomplete analogy was due to the lack of broken inversion symmetry phases 
+observed in Sr2IrO4. Here, under high pressure and high temperature conditions, we report a non-
+centrosymmetric Sr2IrO4. The crystal structure and its noncentrosymmetric character were 
+determined by single crystal X-ray diffraction and high-resolution scanning transmission electron 
+microscopy (HR-STEM). The magnetic characterization confirms the Ir4+ with S = 1/2 at low 
+temperature in Sr2IrO4 with magnetic ordering occurred at around 86 K, where a larger moment is 
+observed than the ambient pressure Sr2IrO4. Moreover, the resistivity measurement shows three-
+dimensional Mott variable-range hopping existed in the system. This non-centrosymmetric Sr2IrO4 
+phase appears to be a unique material to offer further understanding of high-Tc superconductivity.  
+ 
+ 
+
+2 
+ 
+Introduction 
+ 
+Iridates with strong spin-orbit coupling effects can generate exotic quantum phenomena, 
+such as quantum spin liquid phases, Kitaev magnetism, and possible superconductivity.1-4 
+Different from most 3d transition metal oxides in which the spin and orbit can be distinct in the 
+energy scale, the spin and orbit interact heavily in 5d transition metal oxides. Among the Mott 
+insulating 5d transition metal oxides, Sr2IrO45-9 has attracted significant of attention due to its 
+similarity to cuprate high-temperature superconductor, La2CuO410-12. As a single-layer 
+Ruddlesden–Popper compound, Sr2IrO4 crystallizes in a tetragonal lattice with an inversion center 
+(I41/acd, #142) at ambient pressure. Sr2IrO4 contains stacked IrO2 square lattices where the unit 
+cell is doubled compared to the CuO2 square lattices in high-Tc cuprates as a result of a staggered 
+rotation of IrO6 octahedron. Although superconductivity is not yet confirmed, many phenomena 
+characteristic of the superconducting cuprates have been observed in electron-and hole-doped 
+iridates including  pseudogaps, Fermi arcs, and d-wave gaps .13-15 The Ir-d5 electrons in regular 
+IrO6 octahedron  occupy the t2g orbitals, which can be approximated as two fully filled spin-orbital 
+coupled Jeff = 3/2 bands and one half-filled Jeff = 1/2 band. The Jeff band is split into an upper and 
+lower Hubbard band by on-site Coulomb interaction. According to a previous study, as Sr2IrO4 is 
+cooled below its Néel temperature (TN, ~230 K), the spin-orbit coupled Jeff = 1/2 moments order 
+into a basal plane commensurate Néel state. Octahedral rotations in Sr2IrO4 allow for non-zero 
+Dzyaloshinskii-Moriya (DM) interactions that results in a canting of the ordered moments away 
+from the crystallographic axis and a weak ferromagnetic moment per layer.16 Such a magnetic 
+transition maintains the inversion symmetry but lowers the rotational symmetry of the system from 
+C4 to C2. However, no additional symmetry breaking has been observed by neutron or X-ray 
+diffraction, which makes the comparison of the iridate to cuprate phenomenology incomplete. To 
+date, multiple methods have been used to tune the Mott insulating states in Sr2IrO4, for example, 
+isovalent Rh doping on the Ir site.5,17-23 After partially substituting Ir with Rh, an insulator-to-metal 
+transition can be detected. However, high pressure was also used to tune the electronic states up 
+to 55 GPa without observing any metallic state in Sr2IrO4.24,25  
+ 
+In this report, we applied the high-pressure (6 GPa) high-temperature (1400 °C) method 
+for synthesizing Sr2IrO4. Under such extreme conditions, the obtained Sr2IrO4 remains in a 
+tetragonal structure but without an inversion center. The space group was determined by single-
+
+3 
+ 
+crystal X-ray diffraction (SC-XRD) as I4mm (#107). Unlike the ambient pressure phase, the high-
+pressure phase consists of the single layered IrO2 square lattice, just like CuO2 square in cuprate. 
+Magnetic susceptibility measurement on high pressure Sr2IrO4 indicate a magnetic ordering 
+temperature of approximately 86 K, which is dramatically lower than ambient pressure Sr2IrO4. 
+Interestingly, the resistivity data shows three-dimensional Mott variable-range hopping of charge 
+carriers between states localized by disorder with negligible long-range Coulomb interactions. 
+Discovering the non-centrosymmetric phase in Sr2IrO4 may accelerate the realization of 
+superconductivity and unravel the puzzle in cuprate high-Tc superconductors.   
+ 
+ 
+ 
+
+4 
+ 
+Experimental Section 
+ 
+High-Pressure Synthesis. The ambient pressure Sr2IrO4 phase was prepared accordingly by 
+thoroughly mixing and pelletizing the materials SrCO3 and IrO2 and subsequently heating them to 
+900 °C then regrinding and reannealing at 1000 °C and subsequently reannealing at 1100 °C.26 
+The ambient pressure Sr2IrO4 was pressurized to 6 GPa in 24 hours. After that, the sample was 
+heated up to 1400 °C and stayed at 1400 °C for 4 hours. Another sample was heated to 1400 °C 
+and stayed up to 28 hours to explore the optimal condition. The sample was cooled down to room 
+temperature before depressurizing to the ambient pressure. The high-pressure synthesis was 
+performed by statically compressing the sample using the Walker type multi-anvil press27 where 
+the original Sr2IrO4 was placed in a Pt capsule inside an Al2O3 crucible that was inserted into a 
+Cermacast 646 octahedra pressure medium lined on the inside with a LaCrO3 heater.  
+ 
+Phase Analysis and Chemical Composition Determinations. The phase identity and purity were 
+examined using a Bruker D2 Phaser powder X-ray diffractometer with Cu K������������ radiation (������������ = 
+1.5406 Å). Room temperature measurements were performed with a step size of 0.004° at a scan 
+speed of 0.55°/min over a Bragg angle (2������������) range of 5–90°. FullProf Suite software28,29 was 
+utilized to analyze the phase information and lattice parameters from a Rietveld refinement.  
+ 
+Structure Determination. The room temperature and low temperature (100 K) crystal structure 
+was determined using a Bruker D8 Quest Eco single crystal X-ray diffractometer, equipped with 
+Mo radiation (������������������������������������ = 0.71073 Å) with an ������������ of 2.0° per scan and an exposure time of 10 s per frame. 
+A SHELXTL package with the direct methods and full-matrix least-squares on the F2 model was 
+used to determine the crystal structure of Sr2IrO4.30,31 To confirm the crystal structure, high-
+resolution scanning transmission electron microscopy (HR-STEM) images were collected and 
+electron diffraction was conducted using a 200 kV JEOL ARM electron microscope equipped with 
+double aberration correctors. Samples for TEM analysis were crushed in an agate mortar and 
+deposited directly onto a holey carbon copper grid.   
+ 
+Physical Properties Measurement. Temperature and field-dependent magnetization, resistivity, 
+and heat capacity measurements were performed with a Quantum Design physical property 
+measurement system (PPMS) under a temperature range of 1.85–300 K and applied fields up to 9 
+
+5 
+ 
+T. Electrical resistivity measurements were accomplished with a four-probe method using 
+platinum wires on a pelletized sample of Sr2IrO4. The polycrystalline Sr2IrO4 was pressed up to 6 
+GPa and heated at a lower temperature (100 °C) to eliminate the contribution of grain boundary 
+effect but also keep the phase stable.   
+ 
+
+6 
+ 
+Results and Discussions 
+ 
+Exploring New Phase. The new Sr2IrO4 phase (I4mm, #107) was formed at 6 GPa from the 
+starting material, ambient pressure Sr2IrO4 (I41/acd, #142). The synthesis temperatures were set 
+up at 1200 °C and 1400 °C. The high pressure Sr2IrO4 phase was only produced at 1400 °C. To 
+increase the yield and grow larger crystals, the longer heating duration of 28 hours was tested. 
+However, the secondary tetragonal phase Sr3Ir2O7 simultaneously forms once the heating duration 
+was increased. As a result, only 4 hours heating process can produce the specimen consisting 
+mostly of pure phase. The resulting Le Bail fitting of the PXRD patterns for the high-pressure 
+phase Sr2IrO4 is shown in Fig. 1. An overlay of the PXRD patterns in Fig. S1 demonstrates the 
+formation of the secondary Sr3Ir2O7 phase. The pure phase synthesized at 1400 °C for 4 hours was 
+used for the physical property measurements below.  
+ 
+ 
+Fig. 1 Powder X-ray diffraction pattern of the high-pressure Sr2IrO4 phase. The experimental 
+data (red dots) was modeled with a Rietveld refinement (black line). The blue line indicates the 
+corresponding residual pattern (difference between observed and calculated patterns) along with 
+Bragg peak positions for Sr2IrO4 (green) and Al2O3 (purple) represented by the vertical tick marks.  
+ 
+ 
+ 
+
+Calc
+Diff
+Obs
+Intensity (a.u.)
+Sr2lrO4
+Al203
+10
+30
+50
+70
+90
+Sr,IrO.
+20 (degree)
+14mm (#107)7 
+ 
+Crystal Structure and Phase Determination. After 4 hours of treatment at 6 GPa and 1400 °C, 
+single crystals of Sr2IrO4 were formed, subsequently selected, and measured at both 300 K and 100 
+K using the single crystal X-ray diffractometer. High-pressure Sr2IrO4 crystallizes with good 
+agreement into the tetragonal space group I4mm, as indicated by the single crystal X-ray diffraction 
+(SCXRD) refinement information listed in Table S1. Similar to ambient pressure Sr2IrO4, the high 
+pressure Sr2IrO4 phase contains the layers of IrO6 octahedra with intercalated Sr atoms. The 
+differences between these two are half-c lattice, the disappearance of the inversion center because 
+of the nonsymmetric distortion of IrO6 octahedra, and the disappearance of IrO6 octahedral 
+rotations in the ab-plane in high-pressure Sr2IrO4 compared to the ambient pressure phase. Shown 
+in Fig. 2 are crystal structures and IrO6 octahedra stacking view of ambient pressure Sr2IrO4 
+(I41/acd), high-pressure Sr2IrO4 (I4mm), and previously reported La2CuO4 (I4/mmm), with Ir-O 
+atomic distance in the IrO6 octahedra highlighted. Atomic site vacancies and site disorder were 
+considered and refined to reveal the O3 atomic site is slightly displaced from the closer ideal 4b 
+site (1/2, 0, z) to the 8d site (x, 0, z) having a statistical occupancy of 0.5. The disordered model 
+yielded a more reasonable refinement with an R factor of 4.35 and goodness of fit (GOF) of 1.177 
+while having only one O3 atomic site resulted in an R factor of 4.62 and GOF of 1.305. As such 
+an angle ������������ can be determined from (1/2 ± ������������, 0, z) with respect to an IrO6 octahedra where the O3 
+atoms occupy the 4b site. This structural disorder has been thoroughly discussed for the ambient 
+pressure Sr2IrO4 structure.32 Additionally, the high-pressure Sr2IrO4 phase possesses a 
+nonsymmetric IrO6 octahedra elongation along the c axis, ranging in Ir-O atomic distance from 
+1.94(6)–2.27(6) Å, as indicated in Fig. 2b, which is in fact the cause of noncentrosymmetric 
+structural character. This behavior is kind of similar to the prominent feature of ambient pressure 
+Sr2IrO4 that has been speculated to originate from a Jahn Teller distortion.33-35 Previous studies 
+under high-pressure have revealed an increase in the IrO6 octahedra elongation with 
+pressurization.36,37 Compared to ambient pressure Sr2IrO4, one Ir-O along the c -axis is 
+significantly elongated, with the other almost remains the same, i.e., one oxygen atom is driven 
+away from the Ir atom, and thus the repulsion between Ir and the oxygen ligand is reduced. This 
+will lower the energy of orbitals that contains z contribution and split eg and t2g orbitals, making 
+the crystal field split of Ir d orbitals even more complicated. Together with spin-orbit coupling, 
+this may further remove orbital degeneracies. Moreover, as pressure applied for Sr2IrO4, the Ir-O-
+
+8 
+ 
+Ir angle was pushed close to 180°, which is the angle in Cu-O-Cu in La2CuO4. The structural 
+disorder was further confirmed at 100 K and the SCXRD refinement details can be found in SI.  
+ 
+Fig. 2 Crystal structure illustration. Crystal structures, octahedra stacking view along a axis, 
+and along c axis of (a) ambient pressure Sr2IrO4, (b) as-synthesized high pressure Sr2IrO4, and (c) 
+previously reported La2CuO4, with Ir(Cu)O6 octahedra and Ir(Cu)-O atomic distances presented. 
+Green, blue, dark green, dark blue, and red atoms represent Sr, Ir, La, Cu, and O atoms, 
+respectively. Single-layer square net is also highlighted.  
+ 
+ 
+
+(a)
+(b)
+1.98(1) A
+2.27(6) A
+1.95(1) A
+1.93(1) A
+1.94(1) A
+1.94(6) A
+2.11(1) A
+Sr,IrO4
+Sr,lrO4
+La,CuO
+14,/acd (#142)
+14mm (#107)
+14/mmm(#139)9 
+ 
+Transmission Electron Microscopy. The non-centrosymmetric space group and loss of IrO6 
+octahedral rotation, as well as the oxygen distortion and defects in Sr2IrO4, can at first, be 
+surprisingly interesting, thus high-pressure Sr2IrO4 was investigated by transmission electron 
+microscopy (TEM) to characterize its crystallographic nature. The High-angle annular dark-field 
+scanning transmission electron microscopy (HAADF-STEM) image was obtained along the a axis 
+shown in Fig. 3a. The TEM diffraction patterns projected down the crystalline [100] axis (Fig. 3b) 
+allowed for the determination of the orientation of the images through the d002 spacing. The c-axis 
+parameter is ~12.8 Å, agreeing with the single crystal XRD results. The electron diffraction and 
+imaging study confirmed the high quality of the nanoscale ordering in the specimen. However, the 
+fractional spots 1/2 (110)/(1-10) were observed by TEM electron diffraction in Fig. 3d. As is 
+known that IrO6 tilt/rotation along the c-axis would not introduce these fractional spots. Such 
+fractional reflection spots are related to the ordering of oxygen vacancy, which is consistent with 
+single crystal X-ray diffraction results in Fig. 3e.  
+ 
+ 
+Fig. 3 Transmission electron microscopy study of high pressure Sr2IrO4 phase. (a) HAADF-
+STEM image taken along a axis from a large area showing the high quality of the crystal Sr2IrO4. 
+(b) The zoom-in HAADF image shows the projected structure in the [100] direction, with a crystal 
+model superimposed, where Sr (green), Ir (blue), and O (orange). (c)  The diffraction pattern took 
+along the [100] direction which is consistent with the simulated pattern (Fig. 3e) based on the 
+crystal model determined by single crystal X-ray diffraction. (d) SAED pattern along the [001] 
+direction showing fractional spots of 1/2 (110)/(1-10). (e) Simulated diffraction pattern and (f) 
+projected crystal structure along the [001] direction based on the crystal structure determined by 
+
+220
+020
+X
+200
+000
+003
+220
+020
+20
+[100]
+110
+200
+10
+[001]10 
+ 
+SCXRD. The fractional spots observed in TEM were marked in red. The single crystal structure 
+of Sr2IrO4 with oxygen distortion was confirmed by both single crystal X-ray diffraction and TEM.  
+ 
+ 
+Weak Ferromagnetic Ordering. To study the magnetic properties of the high pressure Sr2IrO4 
+phase, the temperature-dependent susceptibility was measured under field cooled warming (FCW) 
+and field cooled cooling (FCC) mode at 0.1 T shown in Fig. 4a. No significant differences between 
+FCW and FCC were observed. At about 150 K, the susceptibility goes below 0, indicating a 
+diamagnetic contribution in the system, which suggests the possible breakdown of Curie-Weiss 
+behavior at high temperatures in the system. The data between 80–140 K was modeled with the 
+modified Curie-Weiss law (Eqn. 1), shown in Fig. 4b and Fig. S2b,  
+������������ = ������������0 +
+������������
+������������ − ������������cw
+(1) 
+where ������������������������������������ is the paramagnetic Curie temperature, ������������0 is the temperature independent susceptibility 
+and ������������ is the Curie constant. From the fitting, the Curie temperature, ������������������������������������, of 86(7) K was found to 
+be comparable to the magnetic ordering temperature ������������������������ ~84 K, as determined from the minimum 
+in the temperature derivative of ������������ (See Fig. S2a for details). The magnetic ordering temperature, 
+consequently, decreases when compared to ambient pressure Sr2IrO4, which has a ������������������������ ~240 K.39,40 
+On the other hand, it can be assumed that the Tc significantly decreases as the angle of Ir-O-Ir is 
+more close to 180 °, which is the one observed in Cu-O-Cu in high Tc superconductor La2CuO4. 
+The fitting also gave a negative ������������0 of -2.9(9)×10-3 emu mol-1 Oe-1, which provided a potential 
+opportunity to extrapolate our Curie-Weiss fit to higher temperature. Finally, up to 160 K was 
+included (Fig. S2c) and the fit yielded the effective moment ������������eff = 1.2(2) µB/Ir, which is more 
+agreeable with the Hund’s-rule value of 1.73 µB/Ir for S = 1/2 than the reported ������������eff = 0.33 µB/Ir 
+for ambient pressure Sr2IrO4.  
+ 
+Furthermore, the magnetization of high pressure Sr2IrO4 was measured as shown in Fig. 
+4c up to 9 T at different temperatures. It appears to saturate at ~3 T at which the magnetic saturation 
+moment (������������������������������������������������) was determined to be ~0.046 µB/Ir. This value is significantly lower than the 
+theoretical value of 1/3 µB f.u−1, however, similar to the previously reported moment for the 
+ambient pressure Sr2IrO4 phase, which originates from spin canted antiferromagnetic (AFM) 
+order.39 This could also explain why the weak ferromagnetic behavior observed in the temperature 
+
+11 
+ 
+dependence of magnetic susceptibility gives such a low value of moment. However, unlike the 
+ambient pressure Sr2IrO4 phase, the magnetization reaches a maximum at around 3 T at which 
+point the magnetization decreases. It turned out that diamagnetic transition was observed under 
+higher fields at the respective temperatures (e.g., see the 50 K and 100 K data). At 300 K, a 
+complete diamagnetic behavior was shown, consistent with ������������ < 0 shown in Fig. 4c. Subtracting 
+this by linearly fitting data from 7–9 T, the ������������������������������������������������ was modified to be 0.067 µB/Ir at 2 K and 0.014 
+µB/Ir at 100 K, as presented in Fig. 4d and 4e. Magnetic hysteresis was observed in the system 
+under 2 K from -0.6 T to 0.6 T, presented in Fig. S3, which could be interpreted as small canting 
+of the moments existed in the system.  
+
+12 
+ 
+ 
+Fig. 4 Magnetization in the dependence of temperature and field. (a) Temperature dependence 
+of magnetic susceptibility ������������ at 1000 Oe under FCW and FCC mode ranging from 2–300 K. No 
+significant difference was observed. (b) The modified inverse magnetic susceptibility data (FCW, 
+80–140 K, blue hollow circle) fitted with the modified Curie-Weiss model (orange line). (c) Field 
+dependence of magnetization up to 9 T at different temperatures. (d) Derivation of ������������sat at 2 K by 
+linearly fitting the magnetization data from 7–9 T. (e) Derivation of ������������sat at 100 K.  
+ 
+No Magnetically Induced Anomalies Observed in Specific Heat Measurement. To confirm the 
+magnetic transition, the specific heat over the temperature range of 2–200 K was measured under 
+0 T with a polycrystalline pelletized sample of Sr2IrO4, as presented in Fig. 5a. Measurements 
+
+(a)
+(b)
+1e-2
+1e3
+2
+FCW
+Cw fit
+Oe)
+FCC
+0
+FCW
+mol
+0
+8
+oo
+0
+0
+0
+1
+090
+0
+4
+8
+08
+X
+8
+0
+0
+0
+100
+200
+300
+80
+130
+180
+230
+280
+T (K)
+T (K)
+(c)
+(d)
+(e)
+1e-2
+8
+0
+10 
+20 K
+(μB per Ir ion)
+4
+0
+.4
+M
+50 K
+-8 
+100 K
+0 2K
+100K
+300 K
+"2 K"
+"100 K"
+-9
+-6-3
+3
+6
+9
+0
+36
+9
+0
+36
+9
+μoH (T)
+μoH (T)
+μoH (T)13 
+ 
+under applied fields of 0.05 T and 1 T in Fig. S3 were additionally tested to conclude no significant 
+deviation from the 0 T specific heat. No ������������ shape anomalies were observed at the whole temperature 
+regime studied, which may result from higher temperature regions being heavily dominated by the 
+phonon contribution. The specific heat data were fitted by the Debye model (Eqn. 2), and Einstein 
+model (Eqn. 3), shown in Fig. S4a and b. The Debye and Einstein temperatures could then be 
+determined as 417(2) K and 306(2) K, respectively. However, neither of these two described the 
+experimental data well.  
+������������D = 9������������������������ � ������������
+������������D
+�
+3
+�
+������������4������������������������
+(������������������������ − 1)2 ������������������������
+������������D ������������
+⁄
+0
+(2) 
+where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Debye 
+temperature.  
+������������E = 3������������������������ �������������E
+������������ �
+2
+������������
+������������E
+������������ �������������
+������������E
+������������ − 1�
+−2
+(3) 
+where ������������ is the number of atoms per formula unit, ������������ is the gas constant, and ������������������������ is the Einstein 
+temperature.  
+ 
+The specific heat data was further fitted with two Debye model (Eqn. 4) and weighted 
+Debye model (Eqn. 5), with and without the electronic contribution included, shown in Fig. 5a 
+and Fig. S4c, d, and e. The data was found to be described well with two Debye model (Fig. 5a), 
+and the Debye temperatures, ������������������������1 of 235(1) K, ������������������������2 of 708(5) K was obtained. At low temperatures, 
+the first Debye mode has a larger contribution to the specific heat. Within the temperature regime 
+studied, the expected Dulong-Petit value of 3������������������������ is not recovered, and this can be explained by the 
+high value of ������������������������2, which means that the specific heat will plateau at ������������ ≫ ������������������������2. The fitting also yields 
+������������������������1 of 3.20(3) and ������������������������2 of 4.51(2). The sum of these two seems a little larger than the expected 
+value of 7 for Sr2IrO4, which may be attributed to the impurity of Srn+1IrnO3n+1, lack of electron 
+contribution, or overestimation of photon contribution in the model. Once the electron contribution 
+term was included, ������������������������1 was slightly shifted to 238(2) K and the sum of ������������������������1 and ������������������������2 went down to 
+7.52(11).  
+������������ = 9������������D1������������ � ������������
+������������D1
+�
+3
+�
+������������4������������������������
+(������������������������ − 1)2 ������������������������
+������������D1 ������������
+⁄
+0
++ 9������������D2������������ � ������������
+������������D2
+�
+3
+�
+������������4������������������������
+(������������������������ − 1)2 ������������������������
+������������D2 ������������
+⁄
+0
+(+������������������������)
+(4) 
+
+14 
+ 
+where ������������������������1 and ������������������������2 are Debye temperatures, ������������������������1 and ������������������������2 are the oscillator strengths, and ������������������������ is the 
+electron contribution.  
+������������ = 9������������D������������ � ������������
+������������D
+�
+3
+�
+������������4������������������������
+(������������������������ − 1)2 ������������������������
+������������D ������������
+⁄
+0
++ 3������������E������������ �������������E
+������������ �
+2
+������������
+������������E
+������������ �������������
+������������E
+������������ − 1�
+−2
+(+������������������������)
+(5) 
+where ������������������������ and ������������������������ are the Debye and Einstein temperatures, ������������������������ and ������������������������ are the oscillator strengths.  
+ 
+It should be noted that the magnetic contribution cannot be quantitatively extracted from 
+the specific heat data as the phonon contribution cannot be distinguished from the magnetic 
+contribution due to the lack of a nonmagnetic analog.  
+ 
+At a low-temperature regime, of 2–20 K, the specific heat was measured, as shown in Fig. 
+S5. The data ranging from 2–3.2 K was fitted with Eqn. 6, shown in Fig. 5b.  
+������������p
+������������ = ������������ + ������������������������2
+(6) 
+From this fitting, a ������������ and ������������ value of 0.0153(2) J mol-1 K-2 and 7.1(2) × 10-4 J mol-1 K-3 
+corresponding to the electronic and phonon contributions to the specific heat, respectively, could 
+be obtained. The ������������ value recovered the Debye temperature (Eqn. 7) to be 268(2) K, which is much 
+closer to ������������������������1 rather than ������������������������2. It falls out of the temperature interval, 300–350 K, where iridates 
+most commonly exhibit Debye temperatures.41 
+������������D = �12������������4
+5������������ �������������������������
+1
+3
+(7) 
+ 
+
+15 
+ 
+ 
+Fig. 5 Specific heat data fitting of high pressure Sr2IrO4. (a) Temperature dependence of 
+specific heat over temperature (��������������p ������������
+⁄ ) for high-pressure Sr2IrO4 fitted by two Debye model in 
+orange. Green and red dotted lines refer to the 1st and 2nd Debye model. (b) ������������p ������������
+⁄  vs ������������2 between 
+2–3.2 K fitted with Eqn. 6 (orange dotted line).  
+ 
+Mott Variable-range Hopping (VRH). It is critical to investigate the electrical conductivity in 
+the high pressure Sr2IrO4 phase to compare to the Mott insulator ambient pressure Sr2IrO4. 
+Temperature-dependent resistivity measurements were performed from 2–300 K with an applied 
+field up to 9 T on a pelletized polycrystalline sample of the high pressure Sr2IrO4 phase, shown in 
+Fig. 6a. No significant field dependence was observed, which indicates the insignificance of 
+magnetoresistance for the high pressure Sr2IrO4 phase. This may be not unexpected considering 
+the small saturation moment under fields (see the discussion above). At room temperature and 0 
+T, the resistivity is relatively low, only around 4 Ω cm. However, the resistivity is increases by 6 
+orders of magnitude upon cooling, indicating the semiconducting character of the high-pressure 
+Sr2IrO4 phase.  
+ 
+To further analyze its behavior, we first tried to model the temperature dependence of ������������ with the 
+Arrhenius law (Eqn. 8),  
+������������ = ������������0������������������������������������ ������������������������
+⁄
+(8) 
+
+(a)
+(b)
+1e-1
+1e-2
+ two Debye
+OT
+... y+βT2
+OOT
+8
+Cp/T ( mol-1 K-2)
+8
+2.4
+6
+4
+2.0
+6
+2
+Debyel
+Debye2
+0
+0
+50
+100
+150
+200
+4
+8
+12
+16
+T (K)
+T2 (K2)16 
+ 
+where ������������0 is the residual resistivity, ������������������������ is the activation energy, and ������������ is the Boltzmann constant. 
+However, ������������ could not be fitted well to a ������������������������, shown in Fig. S7a, i.e., the Arrhenius law is not well 
+obeyed. Then its temperature dependence was fitted by law in the form (Eqn. 9) with ������������ of 1/2 and 
+1/4,  
+������������ = ������������0������������(������������0 ������������
+⁄ )������������
+(9) 
+where ������������0 is the residual resistivity, and ������������0 is the characteristic temperature. The fitting results were 
+presented in Fig. 6b, and Fig. S8, with parameters summarized in Table S4. The value ������������ of 1/4 is 
+favored over 1/2. While both of them indicate three-dimensional Mott variable-range hopping of 
+charge carriers between localized states, the weaker temperature dependence with ������������ of 1/4 implies 
+negligible long-range Coulomb interactions between localized electrons in the temperature regime 
+studied. This behavior is also reported in the ambient pressure Sr2IrO4.42 To explore the harboring 
+quantum states in the high-pressure Sr2IrO4 phase, further examination of its transport properties 
+is warranted.  
+ 
+ 
+ 
+Fig. 6 Details of field and temperature dependent resistivity. (a) Temperature dependence of 
+resistivity data for high-pressure phase Sr2IrO4 under fields up to 9 T. No significant derivation 
+was observed. (b) The resistivity ������������ (blue hollow circle) ranging from 80–300 K was fitted by Eqn. 
+9 with ������������ of 1/4 (orange line). A linear relationship was obtained.  
+ 
+ 
+ 
+
+(a)
+(b)
+107
+OT
+fit
+1 T
+o data
+3 T
+8
+105
+5 T
+In(p/(2 cm))
+p (Q cm)
+7 T
+9T
+103
+4 
+101
+0 
+0
+100
+200
+300
+0.2
+0.3
+0.4
+0.5
+T (K)
+T-1/4 (K-1/4)17 
+ 
+Conclusion 
+ 
+In summary, we reported the non-centrosymmetric Sr2IrO4 phase obtained under high 
+pressure and high temperature conditions. The ferromagnetic ordering temperature decreases 
+significantly to ������������c ~86 K from ~240 K in the ambient pressure Sr2IrO4, while there may be a 
+possible breakdown of the Curie-Weiss law under higher temperatures. Diamagnetism was 
+observed under room temperature and higher fields. No anomalies indicating magnetic ordering 
+were observed in the specific heat measurements, where a greater photon contribution was 
+obtained from the low-temperature regime. Temperature-dependent resistivity revealed three-
+dimensional Mott variable-range hopping of charge carriers between states localized by disorder 
+with negligible long-range Coulomb repulsions. Further transport measurements, together with 
+first-principal calculation, are expected to explore the electronic properties of the high-pressure 
+Sr2IrO4 phase. Such a system may offer a promising platform to unravel the mystery of high-Tc 
+superconductivity in cuprates.  
+ 
+ 
+ 
+Acknowledgments 
+ 
+The work at Rutgers was supported by U.S. DOE-BES under Contract DE-SC0022156. 
+The electron microscopy work at BNL was supported by U.S. DOE-BES, Materials Sciences and 
+Engineering Division under Contract No. DESC0012704. 
+ 
+Supporting Information 
+ 
+Single crystal X-ray diffraction data at room temperature and 100 K; Anisotropic 
+displacement parameters; Atomic coordinates and equivalent isotropic displacement parameters; 
+PXRD overlay of Sr2IrO4; Magnetic susceptibility and Curie-Weiss fitting; Magnetic hysteresis; 
+Field dependence of specific heat; Specific heat data fitted by Debye and Einstein model; Low 
+temperature specific heat data (2–20 K); Temperature dependence of resistivity; Resistivity data 
+fitted by Eqn. 9 with ������������ of 1/2 and 1/4; Summary of fitting parameters for resistivity data.  
+ 
+ 
+ 
+
+18 
+ 
+References  
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+Matter 2016, 28, 065601. 
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+1998, 57, R11039-R11042. 
+ 
+ 
+ 
+
+21 
+ 
+ Non-centrosymmetric Sr2IrO4 obtained under high pressure 
+ 
+ 
+Haozhe Wang1‡, Madalynn Marshall2‡, Zhen Wang3, Kemp W. Plumb4, Martha Greenblatt2, 
+Yimei Zhu3, David Walker5, Weiwei Xie1* 
+ 
+1. Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 
+2. Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 
+08854, USA 
+3. Condensed Matter Physics and Materials Science Department, Brookhaven National 
+Laboratory, Upton, New York 11973, USA 
+4. Department of Physics, Brown University, Providence, Rhode Island 02912, USA 
+5. Lamont Doherty Earth Observatory, Columbia University, Palisades, New York 10964, USA 
+ 
+ 
+‡ H.W. and M.M. contributed equally. * Email: xieweiwe@msu.edu  
+ 
+ 
+ 
+Supporting Information 
+ 
+Table S1 Single crystal X-ray diffraction data at room temperature and 100 K .......................... S2 
+Table S2 Anisotropic displacement parameters ........................................................................... S3 
+Table S3 Atomic coordinates and equivalent isotropic displacement parameters ....................... S4 
+Figure S1 PXRD overlay of Sr2IrO4 ............................................................................................ S5 
+Figure S2 Magnetic susceptibility and Curie-Weiss fitting ......................................................... S6 
+Figure S3 Magnetic hysteresis ..................................................................................................... S7 
+Figure S4 Field dependence of specific heat ............................................................................... S8 
+Figure S5 Specific heat data fitted by Debye and Einstein model ............................................... S9 
+Figure S6 Low temperature specific heat data (2–20 K) ........................................................... S10 
+Figure S7 Temperature dependence of resistivity ...................................................................... S11 
+Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4 ..................................... S12 
+Table S4 Summary of fitting parameters for resistivity data ..................................................... S13 
+ 
+
+22 
+ 
+Table S1 Single crystal X-ray diffraction data at room temperature and 100 K.  
+Temperature 
+Room Temperature 
+100 K 
+Refined formula 
+Sr2IrO4 
+Sr2IrO4 
+FW (g/mol) 
+431.44 
+431.44 
+Space group 
+I4mm 
+I4mm 
+a (Å) 
+3.8860(5) 
+3.8777(5) 
+c (Å) 
+12.826(2) 
+12.825(2) 
+V (Å3) 
+193.69(6) 
+192.85(6) 
+Extinction Coefficient 
+N/A 
+N/A 
+������������ range (°) 
+3.177–33.030 
+3.177–33.075 
+# of reflections; Rint 
+1088; 0.0627 
+1286; 0.0591 
+# of independent reflections 
+267 
+264 
+# of parameters 
+23 
+23 
+R1; ωR2 (������������ > ������������������������(������������)) 
+0.0409; 0.0651 
+0.0312; 0.0443 
+Goodness of fit (GOF) 
+1.177 
+1.125 
+Diffraction peak and hole (e-/ Å3) 
+3.658, -3.492 
+2.359, -1.96 
+ 
+ 
+ 
+
+23 
+ 
+Table S2 Anisotropic displacement parameters for Sr2IrO4 at room temperature and 100 K.  
+Sr2IrO4 at Room Temperature 
+Atom 
+U11 
+U22 
+U33 
+U23 
+U13 
+U12 
+Ir1 
+-0.0018(6) 
+-0.0018(6) 
+-0.0021(6) 
+0 
+0 
+0 
+Sr1 
+0.026(7) 
+0.026(7) 
+0.007(7) 
+0 
+0 
+0 
+Sr2 
+-0.001(4) 
+-0.001(4) 
+0.005(6) 
+0 
+0 
+0 
+O1 
+0.03(2) 
+0.03(2) 
+-0.02(2) 
+0 
+0 
+0 
+O2 
+-0.006(10) 
+-0.006(10) 
+-0.023(18) 
+0 
+0 
+0 
+O3 
+0.04(3) 
+0.003(11) 
+-0.01(3) 
+0 
+0.02(3) 
+0 
+ 
+Sr2IrO4 at 100 K 
+Atom 
+U11 
+U22 
+U33 
+U23 
+U13 
+U12 
+Ir1 
+-0.0004(3) 
+-0.0004(3) 
+0.0012(7) 
+0 
+0 
+0 
+Sr1 
+0.005(8) 
+0.005(8) 
+0.005(4) 
+0 
+0 
+0 
+Sr2 
+0.002(8) 
+0.002(8) 
+0.000(4) 
+0 
+0 
+0 
+O1 
+0.005(8) 
+0.005(8) 
+-0.033(15) 
+0 
+0 
+0 
+O2 
+0.012(10) 
+0.012(10) 
+-0.033(14) 
+0 
+0 
+0 
+O3 
+0.009(10) 
+0.005(7) 
+0.006(8) 
+0 
+0.01(3) 
+0 
+ 
+ 
+ 
+
+24 
+ 
+Table S3 Atomic coordinates and equivalent isotropic displacement parameters for Sr2IrO4 at room 
+temperature and 100 K.  (Ueq is defined as one-third of the trace of the orthogonalized Uij tensor (Å2)).  
+Sr2IrO4 at Room Temperature 
+Atom 
+Wyck. 
+x 
+y 
+z 
+Occ. 
+Ueq 
+Ir1 
+2a 
+0 
+0 
+0.1513(13) 
+1 
+-0.0019(4) 
+Sr2 
+2a 
+0 
+0 
+0.5044(4) 
+1 
+0.020(4) 
+Sr1 
+2a 
+0 
+0 
+0.79985(2) 
+1 
+0.001(3) 
+O1 
+2a 
+0 
+0 
+0.328(4) 
+1 
+0.013(18) 
+O2 
+2a 
+0 
+0 
+0.000(4) 
+1 
+-0.011(7) 
+O3 
+8d 
+0.419(9) 
+0 
+0.661(7) 
+0.5 
+0.010(15) 
+ 
+Sr2IrO4 at 100 K 
+Atom 
+Wyck. 
+x 
+y 
+z 
+Occ. 
+Ueq 
+Ir1 
+2a 
+0 
+0 
+0.1489(7) 
+1 
+0.0001(3) 
+Sr2 
+2a 
+0 
+0 
+0.5019(4) 
+1 
+0.005(5) 
+Sr1 
+2a 
+0 
+0 
+0.7978(2) 
+1 
+0.002(5) 
+O1 
+2a 
+0 
+0 
+0.321(3) 
+1 
+-0.008(6) 
+O2 
+2a 
+0 
+0 
+0.000(3) 
+1 
+-0.003(8) 
+O3 
+8d 
+0.412(4) 
+0 
+0.649(6) 
+0.5 
+0.007(4) 
+ 
+ 
+ 
+
+25 
+ 
+Figure S1 Powder X-ray diffraction pattern overlay. The experimental data of high pressure Sr2IrO4 
+phase synthesized at 1400 °C for ~4 hrs (black line) and ~28 hrs (red line) were presented. Bragg peak 
+positions are indicated as Sr2IrO4 and Sr3Ir2O7 with green and purple vertical tick marks, respectively.  
+ 
+ 
+ 
+ 
+ 
+
+TT-1412 ~4 hrs
+GG-1418 -28 hrs
+Intensity (a.u.)
+Sr2lrO4
+Sr3lr2Q7
+10
+30
+50
+70
+90
+2e (degree)26 
+ 
+Figure S2 Magnetic susceptibility and Curie-Weiss fitting. (a) Temperature derivative of magnetic 
+susceptibility ������������ at 1000 Oe under FCW mode. The minimum at around 84 K was highlighted by red circle 
+and an arrow. (b) The inverse magnetic susceptibility data (FCW, 80–140 K, blue hollow circle) fitted with 
+the modified Curie-Weiss model (orange line). (c) The Curie-Weiss fit was further extrapolated to 160 K.  
+ 
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+1e-3
+1e3
+K-1)
+Cw fit
+1.2 -
+FCW
+-1
+00
+0.6 -
+8
+-2 
+1/x
+000
+0.0 -
+000
+50
+60
+70
+80 °
+90
+100
+80
+100
+120
+140
+T (K)
+T (K)
+(c)
+1e2
+mol Oe)
+4
+CW fit
+FCW
+N
+0%
+00
+0
+80
+100
+120
+140
+160
+T (K)27 
+ 
+Figure S3 Magnetic hysteresis. Magnetic hysteresis observed in the high pressure Sr2IrO4 phase at 2 K 
+ranging from -0.6 T to 0.6 T.  
+ 
+ 
+ 
+ 
+ 
+
+1e-2
+0 2K
+1 
+M (μB per Ir ion)
+-1
+-0.6
+-0.3
+0.0
+0.3
+0.6
+μoH (T)28 
+ 
+Figure S4 Field dependence of specific heat. Temperature dependence of specific heat data over 
+temperature (������������p/������������) for high pressure Sr2IrO4 phase, under 0 T (blue), 0.05 T (orange), and 1 T (green). No 
+significant differences were observed. No ������������ shape anomalies emerged in the whole temperature regime 
+studied under either case.  
+ 
+ 
+ 
+ 
+ 
+
+le-1
+8
+6
+4 
+O T
+2
+0.05 T
+0
+1 T
+0
+0
+50
+100
+150
+200
+T (K)29 
+ 
+Figure S5 Specific heat data fitted by Debye and Einstein model. Temperature dependence of 
+specific heat data over temperature under 0 T (������������p/������������, blue hollow circle) for high pressure Sr2IrO4 phase, 
+fitted by (a) Debye model, (b) Einstein model, (c) two Debye model with the electronic contribution 
+included, and weighted Debye model (d) without and (e) with the electronic contribution included.   
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+le-1
+le-1
+8
+K-2)
+6
+6
+4
+4
+Debye
+2 
+Einstein
+0
+10
+10
+0
+0
+0
+50
+100
+150
+200
+0
+50
+100
+150
+200
+T (K)
+T (K)
+(c)
+le-1
+(d)
+le-1
+two Debye + yT
+OOT
+weightedDebye
+10
+VT
+8
+8 -
+Cp/T (I mol-1 K-2)
+K-2)
+6
+6
+4
+4
+2 
+2 
+Debyel
+Debye
+Debye2
+Einstein
+0
+0
+0
+50
+100
+150
+200
+0
+50
+100
+150
+200
+T (K)
+T (K)
+(e)
+le-1
+weighted Debye + yT o O T
+Cp/T (I mol-1 K-2)
+8
+6
+Debye
+2 
+Einstein
+yT
+0
+0
+50
+100
+150
+200
+T (K)30 
+ 
+Figure S6 Low temperature specific heat data (2–20 K). Specific heat data over temperature (������������p/������������) 
+plotted versus ������������2  under low temperature regime, 2–20 K, providing the possibility to derivate the 
+Sommerfeld parameter, ������������.  
+ 
+ 
+ 
+ 
+ 
+
+le-1
+1oo
+2 
+0
+0
+0
+0
+1 :
+0
+0
+0
+888
+0 
+.
+0
+1
+2
+3
+4
+T2 (K2)
+1e231 
+ 
+Figure S7 Temperature dependence of resistivity. Temperature dependence of the resistivity data ������������ 
+plotted as ln ������������ versus (a) ������������−1, (b)  ������������−1/2, and (c) ������������−1/4 under 0 T.  
+ 
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+16
+16
+OT
+OT
+8
+12 
+12
+In(p/(α2 cm))
+In(p/(Ω2 cm))
+8
+8 -
+4
+4 -
+0
+0.0
+0.1
+0.2
+0.0
+0.2
+0.4
+T-1 (K-1)
+T-1/2 (K-1/2)
+(c)
+16
+O T
+8
+12
+In(p/(2 cm))
+8
+4 -
+0
+0.2
+0.4
+0.6
+T-1/4 (K-1/4)32 
+ 
+Figure S8 Resistivity data fitted by Equation 9 with ������������ of 1/2 and 1/4. (a) The resistivity data ������������ 
+(blue hollow circle) ranging from 110–300 K fitted by Equation 9 with ������������ of 1/2 (orange line). (b) The 
+resistivity data ������������ in the low temperature regime ranging from 8–20 K fitted by Equation 9 with ������������ of 1/2. 
+(c) The resistivity data ������������ in the low temperature regime ranging from 10–20 K fitted by Equation 9 with ������������ 
+of 1/4. Fitting parameters were summarized in Table S4. The value ������������ of 1/4 is favored over 1/2.  
+ 
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+16
+fit
+fit
+00
+data
+data
+8 -
+In(p/(α2 cm))
+In(p/(Ω cm)
+12
+4
+8
+0
+4
+0.1
+0.2
+0.1
+0.2
+0.3
+0.4
+T-1/2 (K-1/2)
+T-1/2 (K-1/2)
+(c)
+16
+fit
+8
+data
+12
+In(p/(2 cm))
+8 -
+4 -
+0.3
+0.5
+0.7
+T-1/4 (K-1/4)33 
+ 
+ 
+Table S4. Summary of fitting parameters for resistivity data. Summary of fitting parameters for 
+the resistivity data ������������ by Equation 9. R2 is the coefficient of determination.  
+ 
+������������ = 1 2
+⁄  
+Temperature Range / K 
+������������0 / (٠cm) 
+������������0 / K 
+R2 
+110–300  
+3.01(4) × 10-5 
+6682 
+0.9996 
+8–20 
+79.6(37) 
+583 
+0.9996 
+ 
+������������ = 1 4
+⁄  
+Temperature Range / K 
+������������0 / (٠cm) 
+������������0 / K 
+R2 
+80–300  
+7.82(12) × 10-5 
+3.83 × 106 
+0.9998 
+10–20 
+-2.23(4) 
+4.10 × 105 
+0.9999 
+ 
+ 
+ 
+ 
+ 
+

49E2T4oBgHgl3EQfkAeM/content/tmp_files/2301.03974v1.pdf.txt

@@ -0,0 +1,979 @@
+Hydrogen storage in Li functionalized [2,2,2]paracyclophane 
+at cryogenic to room temperatures: A computational quest 
+Rakesh K. Sahoo, Sridhar Sahu 
+ 
+Computational Materials Research Lab, Department of Physics, Indian Institute of Technology 
+(Indian School of Mines) Dhanbad, India 
+Abstract 
+In this work, we have studied the hydrogen adsorption-desorption properties, and storage capacities of Li 
+functionalized [2,2,2]paracyclophane (PCP222) using dispersion-corrected density functional theory and 
+molecular dynamic simulation. The Li atom was found bonded strongly with the benzene ring of PCP222 
+via Dewar interaction. Subsequently, the calculation of the diffusion energy barrier revealed a significantly 
+high energy barrier of 1.38 eV, preventing the Li clustering on PCP222. The host material, PCP222-3Li 
+adsorbed up to 15H2 molecules via charge polarization mechanism with an average adsorption energy of 
+0.145 eV/5H2, suggesting physisorption type of adsorption. The PCP222 functionalized with three Li atom 
+showed maximum hydrogen uptake capacity up to 8.32 wt% which was fairly above the US-DOE criterion. 
+The practical H2storage estimation revealed that the PCP222-3Li desorbed 100% of adsorbed H2 molecules 
+at the temperature range of 260 K-300 K and pressure range of 1-10 bar. The maximum H2 desorption 
+temperature estimated by the Vant-Hoff relation was found to be 219 K and 266 K at 1 bar and 5 bar, 
+respectively. The ADMP molecular dynamics simulations assured the reversibility of adsorbed H2 and the 
+structural integrity of the host material at sufficiently above the desorption temperature (300K and 500K). 
+Therefore, the Li-functionalized PCP222 can be considered as a thermodynamically viable and potentially 
+reversible H2 storage material below room temperature. 
+Keywords: Hydrogen storage, DFT, Van’t-Hoff equation, ADMP, [2,2,2]paracyclophane, 
+PCP222, ESP 
+1 Introduction 
+The excessive consumption of traditional fossil fuels has not only led to the depletion of the energy 
+supplies but also has emerged as the prime cause of environmental pollution. The global 
+consumption of petroleum and other traditional fossil fuel is anticipated to expand up to 56% by 
+the year 2040 and the crude oil supply is expected to endure until 2060 if the current demand trend 
+continues[1]. Thus, it is essential to develop alternative energy sources that are free from the 
+drawbacks of traditional fossil fuels. To meet the world’s energy demand and reduce the pollution 
+caused by fossil fuels, hydrogen has been considered as a plausible alternative due to its natural 
+abundance, environmental friendliness, and regenerative properties. One of the distinctive quality 
+
+of hydrogen is that it produces a large amount of energy per unit mass (120 MJ/kg) without 
+releasing any pollutant by-products [2, 3]. Despite these benefits, however, the use of hydrogen in 
+practice is limited due to the obstacle of finding the most appropriate and affordable way to store 
+and deliver hydrogen under normal environmental conditions. As per the criteria proposed by the 
+United State department of energy (DOE-US) an effective hydrogen storage material should have 
+a minimum storage capacity of up to 5.5 wt% by the year 2025 at moderate thermodynamics[5, 6]. 
+In addition, as reported by many authors, the adsorption energy of hydrogen molecules of an 
+effective storage materials should be in the range of 0.1 eV/H2 to 0.6 eV/H2[4]. 
+Though, numerous varieties of materials such as; metal hydrides [7, 8], graphene [9, 10], 
+metal alloys [11, 12], metal-organic frameworks (MOF) [13, 14], covalent-organic frameworks 
+(COF) [15] and carbon nanostructures [16, 17] etc have been investigated both theoretically and 
+experimentally as potential hydrogen storage materials, but there are many drawbacks and 
+unsolved issues to handle. The metal hydrides and complex hydrides store hydrogen via 
+chemisorption process which is highly irreversible and prevents easy desorption of hydrogen [18]. 
+For example, Al(BH4)3, which yields hydrogen uptake capacity of up to 16.9 wt%, has high 
+desorption temperature (about 1000 K) that makes the material non-effective practical reversible 
+hydrogen storage applications[19]. Under ambient conditions, Mg-metal hydrides have a storage 
+capacity up to 7.6 wt%; however, it can only be used for 2-3 cycles[20]. Tavhare et al. studied the 
+hetero atom substituted Ti-benzene and reported an H2 uptake capacity up to 5.85 wt%, but at 
+relatively high desorption temperature (1193 K)[21]. Furthermore, MOF and COF applications are 
+constrained in the practical H2 storage field due to the difficulties of their heavy structure and 
+challenging step-wise production [22]. 
+The efficient use of carbonaceous materials as hydrogen storage media was initially 
+reported by Dillon et al. [23]. Carbonaceous materials are appropriate for H2 storage due to their 
+unique qualities such as, large surface area, high porosity, better stabilities, and low densities. 
+However, the early findings have shown that these pure materials are weakly interact with the 
+hydrogen molecules (with BE ~4-5 kJ/mol), thus impractical for realistic hydrogen storage at 
+ambient environment [24, 25]. Meanwhile, carbon-based pure substrates are excellent materials 
+for hydrogen storage at cryogenic temperatures. For instance, pure single wall carbon nanotube 
+
+(SWCNT) can store hydrogen molecules up to 8.25 wt%, with a substantially lower desorption 
+temperature of 80 K [26]. 
+It has been reported that the H2 interaction strength and the desorption temperature can be 
+tuned by integrating pure carbon substrates with alkali metal (AM)(Li, Na, and K), alkali earth 
+metals (Be, Mg, Ca), and transition metals (TM)(Sc, Ti, V, Y.)[27, 28, 29]. Numerous theoretical 
+investigations showed that integrating AMs and TMs with the carbon/borane substrates can bind 
+H2 molecules via charge polarization and the Kubas mechanism [30, 31]. The metallic atom 
+decorated fullerenes were first explored to investigate the impact of metal integration on pure 
+carbon substrates. According to studies by Sun et al., Li decorated fullerene could show a storage 
+capacity of 9 wt%; however, the hydrogen adsorption energy was estimated to be 0.075 eV/H2, 
+which is much lower than the DOE criterion [32]. The Li and Na-loaded C60 revealed H2 uptake 
+capacities of 4.5 wt% and 4 wt%, respectively, that were significantly below the target of DoE [33]. 
+Experimental studies of transition metals like V and Pd decorated CNT reveal 0.66 wt% and 0.69 
+wt% of hydrogen capacity respectively, while pure CNT has 0.53 wt% of storage capacity [34]. 
+Sahoo et al. reported storage of H2 on Li and Sc doped C8N8 cage via Niu-Rao-Jena and Kubas 
+interaction and estimated a desorption temperature of 286 K and 456 K, respectively [35]. The Li 
+and Na decorated on C24 fullerene could adsorb H2 molecules, with average hydrogen binding 
+energies of 0.198 eV/H2 and 0.164 eV/H2 and led to storage capacity up to 12.7 wt% and 10 wt %, 
+respectively [36]. Recently, we have investigated the H2 storage on alkali metal decorated 
+C20 fullerene and found the molecular hydrogen are physisorbed on host material via charge 
+polarization mechanism with desorption temperature of 182 -191 K [37]. Each Li and Na atom on 
+C20 could uptake up to 5H2 molecules with a total gravimetric storage capacity of 13.08 wt % and 
+10.82 wt%, respectively, and the H2 binding energies found in the range of 0.12 eV—0.13 eV/H2. 
+Other carbonaceous materials such as functionalized organometallic compounds, 
+macrocyclic compounds have also been reported recently as potential candidates for hydrogen 
+storage. For example, Mahamiya et al. revealed the H2 storage capacities of 11. 9 wt % in K and 
+Ca decorated biphenylene with an average adsorption energy of 0.24-0.33 eV [38]. Y atom doped 
+zeolite shows high capacity adsorption of H2 with binding energy 0.35 eV/H2 and the desorption 
+energy of 437K for fuel cells[39]. Lithium-doped Calixarenes show an excellent hydrogen storage 
+behaviour but at very low up to 100 K [40]. Calix[4]arene functionalized with Li metal reveals 10 
+
+wt% storage capacity via Kubas—Niu—Rao—Jena interaction, and all most all H2 desorbed at a 
+temperature of 273 K [41]. 
+Macrocyclic compounds such as, paracyclophane (PCP), a subgroup derivative of 
+cyclophanes, contains aromatic benzene rings, and their nomenclature is established on the arene 
+substitution pattern. For a [n,n]paracyclophane, the number of -CH2- moiety connecting the 
+successive benzene rings is indicated by the number in the square bracket [42]. Due to the 
+existence of aromatic benzene rings in the geometry, PCPs are easy to synthesize experimentally 
+and can be functionalized with metal atoms, making them a viable choice for hydrogen storage 
+candidates. A report on Li and Sc functionalized [4,4]paracyclophane revealed the hydrogen 
+uptake capacity up to 11.8 wt% and 13.7 wt% with an average adsorption energy of 0.08 eV/H2 and 
+0.3 eV/H2 respectively [43]. Sahoo et al. recently studied the H2 storage capacity of 
+[1,1]paracyclophane functionalized with Sc and Y metals and found an H2 gravimetric storage 
+capacity of 8.22 wt% and 6.33 wt%, respectively, with an average adsorption energy 0.36 
+eV/H2[44]. They reported the H2 desorption temperature of 439 K and 412 K for Sc and Y doped 
+PCP11, respectively, at 1 atm. The hydrogen molecules are physisorbed on Li, and Sc decorated 
+paracyclophane via Kubas-Niu-Jena interaction and show a storage capacity of 10.3 wt%, as 
+reported by Sathe et al. [45]. Many more alkali metal-doped macrocyclic compounds have also 
+been investigated for hydrogen storage candidates and found the storage capacity above the DOE 
+target; however very few reported the practical H2 capacity at various thermodynamic 
+conditions[46, 47]. Though few of PCP-based hydrogen storage systems are available in literature, 
+the [2,2,2]paracyclophane (PCP222) which is experimentally synthesized by Tabushi et al.[48] is 
+yet to be explored as hydrogen storage material. Because Li, the lightest alkali metal atom and can 
+hold H2 molecules via charge polarization mechanism, it can serve as better sorption center on 
+PCP. 
+Therefore, in the current work, we intend to investigate the hydrogen storage properties 
+and potential of Li functionalized [2,2,2]paracyclophane (PCP222). We chose the PCP222 for 
+hydrogen storage because it is already experimentally synthesized and can be decorated with metal 
+atoms to form a hydrogen storage media with a high hydrogen uptake capacity. The Li atoms are 
+functionalized as sorption centers; this is because the light-weight metal doping method is an 
+effective way to increase the capacity of H2 storage. Li being the lightest alkali metal atom, 
+
+received a lot of attention to for hydrogen sorption application. Though there are few reports 
+available based on hydrogen adsorption mechanism on metal doped macrocyclic organic 
+molecules and other Li decorated nanostructures, our work is the first to reveal the efficiency of 
+Li functionalized PCP222 using the atomistic MD simulation, practical storage capacity and 
+diffusion energy barrier estimation 
+2 Theory and Computation 
+The theoretical computations are carried out on [2.2.2] paracyclophane (PCP222) and their 
+hydrogenated derivatives within the framework of density functional theory (DFT)[49]. The 
+modern range separated hybrid functional wB97Xd is used, and molecular orbitals (MO) are 
+defined as linear combination of atom centered basis functions, with all atoms using the valence 
+diffuse and polarization function 6-311+G(d,p) basis sets. The wB97Xd, a long range separated 
+form of Becke’s 97 functional, also adds Grimme’s D2 dispersion correction[50, 51]. It is worth 
+mentioning that the wB97Xd is a reliable approach to investigate the non-covalent interaction of 
+metal doped organic molecules and their thermochemistry. The harmonic frequencies of all the 
+studied structures are calculated to confirm that they are truly in the ground state on the potential 
+surface. 
+Some of the crucial quantitative metrics, including, binding energy of metal atom on host, 
+average H2 adsorption energy and successive H2 desorption energy must be determined in order to 
+analyze the mechanism of hydrogen storage. 
+The binding strength of Li atom on the PCP222 is calculated by the following expression[44]; 
+𝐸𝑏 =
+1
+𝑚 [𝐸𝑃𝐶𝑃222 + 𝑚𝐸𝐿𝑖 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖]       
+ 
+ (1) 
+Where 𝐸𝑃𝐶𝑃222, 𝐸𝐿𝑖, and 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 are symbolize for the total energy of PCP222, energy of 
+single isolated Li atom and energy of Li-decorated PCP222 respectively. m denotes for the number 
+of Li atoms used to functionalized the PCP222. 
+The average adsorption energy of H2 molecules with Li functionalized PCP222 is estimated 
+as [52]; 
+𝐸𝑎𝑑𝑠 =
+1
+𝑛 [𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖 + 𝑛𝐸𝐻2 − 𝐸𝑃𝐶𝑃222+𝑚𝐿𝑖+𝑛𝐻2]     
+  (2) 
+
+Where, EH2, and EPCP222+mLi+nH2 represents the energy of isolated single H2 molecule and 
+hydrogen adsorbed PCP222+mLi, respectively. n is the number of H2 molecules adsorbed in each 
+Li functionalized PCP222. 
+The successive desorption energy of adsorbed H2 molecules is estimated using following 
+equation[52]. 
+𝐸𝑑𝑒𝑠 =
+1
+𝑛 [𝐸𝐻2 + 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2 − 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2]      
+ 
+ (3) 
+where 𝐸𝐻𝑜𝑠𝑡+(𝑛−1)𝐻2is the energy of previous H2 molecules adsorbed 𝐸𝐻𝑜𝑠𝑡+𝑛𝐻2. 
+The energy gap between the highest occupied molecular orbital (HOMO) and the lowest 
+unoccupied molecular orbital (LUMO) is calculated to ensure the kinetic stability of the Li 
+functionalized PCP222 and their hydrogen derivatives. The Hirshfeld charges and electrostatic 
+potential map (ESP) was used to study electronic charge transfer and interaction mechanism. 
+Further, to understand the metal and hydrogen interaction we have performed the partial density 
+of states (PDOS), and topological using the Bader’s quantum theory of atoms in molecules 
+(QTAIM). To investigate the structural integrity of the host material and H2 reversibility of the 
+system, atomistic molecular dynamic simulations were carried out using the expanded lagrangian 
+approach, atom-centered density matrix propagation (ADMP). 
+To determine the H2 adsorption capacity, gravimetric density (wt%) of hydrogen can be calculated 
+using the following expression [53]: 
+𝐻2(𝑤𝑡%) =
+𝑀𝐻2
+𝑀𝐻2+𝑀𝐻𝑜𝑠𝑡 × 100 
+ 
+ 
+ 
+   (4) 
+Here MH2 represent the mass of the total number of H2molecules adsorbed and MHost represent the 
+mass of Li functionalized PCP222. 
+3 Results and Discussion 
+3.1 Structural properties of PCP222 
+Figure 1 depicts the ground state geometrical structure of PCP222. The PCP222 comprises three 
+benzene rings, that are linked via two CH2 moiety as bridge between the adjacent rings. The lengths 
+of the nearest CH2-CH2, and the CH2 across the benzene rings are observed to be 1.5 and 5.84 Å , 
+
+respectively, that agrees with the empirically reported value by Cohen-Addad et al [54]. To 
+confirm the aromaticity of the relaxed PCP222, we calculated the Nucleus Independent Chemical 
+Shift (NICS) from center to to 3 Å above the benzene ring by increment of 1 Å. The NICS(1) is 
+found to have negative maximum (-10.1 ppm), demonstrating the aromatic character of 
+PCP222[55, 56]. This suggest that the cyclic rings of PCP222 are -electron rich and most 
+probably can bind the metal atom above (outside of PC222) the benzene rings. The Li atom then 
+functionalized above the benzene rings and on every possible site of PCP222 and allowed to relax 
+as discussed below. 
+ 
+Figure 1: (a) Optimized structure of PCP222 with adsorption site marked with red-colored 
+text, (b) Li functionalized PCP222. 
+3.2 Functionalization of Li atom on PCP222 
+To explore the hydrogen adsorption capacity in Li-functionalized PCP222, we must first carefully 
+examine the suitable adsorption site for Li atoms on the PCP222. In order to do this, we 
+investigated several PCP222 adsorption site, including the C-C bridge of benzene ring (B1), 
+CH2 moiety and benzene bridge (B2), CH2 - CH2 bridge (B3), and above the center of benzene 
+(Rc). All the possible Li adsorption sites of PCP222 are depicted in Figure 1(a). A single Li atom 
+is placed nearly 2 Å above the several probable adsorption sites of PCP222 and the structure is 
+allowed to get optimized. It is observed that functionalization of Li atom over B1 and B2 sites, it 
+migrate towards the Rc site following the optimization. On optimization of Li atom over B3 site, 
+the it moves away from the PCP222 and does not bind to the surface. We found that the Li atom 
+is stable on Rc site with binding energy of 0.32 eV that is 0.1 eV higher than that of Li on PCP44, 
+reported by Sathe et al. [43]. The Li atom is supposed to be functionalized on PCP222 via Dewar 
+mechanism, in which is due to the electronic charge transfer between the p-complex and s- orbitals 
+
+5.84
+5.858
+B1
+1.541
+1.543
+(a)
+(b)of Li atom [43, 45]. After functionalization of Li, the estimated Hirshfeld charge on benzene ring 
+of PCP222 is increased to -0.08 e.u from -0.03 e.u (in bare PCP222). These charges are transferred 
+from the metal atom, with the Hirshfeld charges on Li atom being +0.35 e.u after functionalization, 
+which make the Li atom ionic. The ionic Li atom is exposed to the guest H2 molecules and can 
+bind them via charge polarization mechanism as proposed by Niu et al. [57]. No significant change 
+in geometrical bond distances is observed after the functionalization of Li. The thermal stability 
+of the structures (host) is discussed in the molecular dynamic simulations section (section 3.5). All 
+the hydrogen adsorption/desorption simulations are performed by functionalizing the Li atom 
+above the center of benzene ring of PCP222. 
+3.2.1 Diffusion energy barrier calculation 
+ 
+Figure 2: Diffusion energy barrier plot between energy difference and diffusion coordinates of 
+Li atom on PCP222 
+ 
+The clustering of metal atoms on the substrate can reduce the hydrogen uptake capacity of the 
+system as reported earlier [17]. The barrier of metal atoms diffusion energy ultimately decides 
+whether or not the clustering will occur. With a small rise in temperature, if the Li atom migrated 
+from its adsorption location, the possibility of metal-metal clustering would increase. Since, the 
+binding energy of Li atom on the PCP222 is less than the cohesive energy of the isolated Li atom 
+(1.63eV), we calculate if there is an energy barrier for diffusion of Li atom on PCP222 that can 
+avoid the possibility of metal clustering. To calculate the energy barrier, we shift the Li atom over 
+its adsorption site (on the benzene ring) by a small distance along the path shown in the 
+Figure 2 and carried out the single point energy calculation. Then we exhibit the energy difference 
+between initial and current step energy with the diffusion coordinate as illustrated in Figure 2. The 
+
+1.4-
+AE=1.38eV
+1.2-
+1.0-
+0.8-
+ev
+0.6-
+4
+1-3)
+0.4 
+0.2 -
+0.0 -
+1
+-
+0
+1
+2
+3
+4
+5
+Diffusion coordinatesfigure shows presence of an energy barrier of 1.38 eV, that is sufficient to stop the Li atom from 
+diffusing across the PCP222 and thus prevent the metal clustering. Therefore, our calculated 
+energy barrier for diffusion of Li atom is high enough to prevent metal clustering over the studided 
+PCP222 compound. 
+3.3 Interaction of H2 with PCP222-Li 
+3.3.1 Adsorption Energy 
+ 
+Table 1: Average bond distance between carbon bridge (C-C), center of PCP222 benzene ring (Rc) 
+and Lithium atom (Rc-Li), Lithium and hydrogen molecules (Li-H2), and hydrogen Hydrogen 
+(H-H) in Å. Average adsorption energy and successive desorption energy of PCP222-Li-
+nH2 (n=1-5) 
+ 
+Name of complex 
+Bridge C-C Rc-Li 
+Li-H 
+H-H 
+Eads (eV) Edes (eV) 
+PCP222-Li 
+1.542 
+1.735 
+- 
+- 
+- 
+- 
+PCP222-Li-1H2 
+1.542 
+1.745 
+2.124 
+0.753 
+0.171 
+0.171 
+PCP222-Li-2H2 
+1.541 
+1.742 
+2.083 
+0.757 
+0.159 
+0.147 
+PCP222-Li-3H2 
+1.541 
+1.767 
+2.159 
+0.753 
+0.148 
+0.127 
+PCP222-Li-4H2 
+1.541 
+1.811 
+2.243 
+0.752 
+0.134 
+0.089 
+PCP222-Li-5H2 
+1.541 
+1.813 
+2.478 
+0.751 
+0.113 
+0.030 
+To explore the storage capacity and characteristics of Li functionalized PCP222, we introduced 
+the H2 molecules in a sequential manner to PCP222-Li. Firstly we introduced a single H2 molecule 
+at around 2Å above the Li atom on PCP222 and allowed the structure to get relaxed. It is observed 
+that, the H2 molecule is adsorbed at a distance of 2.124 Å from the Li atom with an adsorption 
+energy of 0.171 eV and the H-H bond length elongated by 0.01 Å. Sathe et al. studied the hydrogen 
+storage capacity of Li functionalized PCP11 (PCP22) and reported the adsorption energy of first 
+H2 molecule ~0.13 eV (0.11 eV) [46, 45]. Our calculated adsorption energy is slightly higher, 
+which is important in alkali metal doped H2 storage material and leads to the increase in the 
+desorption temperature. Further, we optimized the structures by adding H2 molecules sequentially 
+onto the PCP222-Li. On addition of second H2 molecule to the system, the average H2 adsorption 
+energy calculated to be 0.159 eV/H2. In this way, adsorption of 3rd, 4th and 5th H2 molecules to 
+
+PCP222-Li, the average H2 adsorption energy reduces to 0.148, 0.134 and 0.113 
+eV/H2respectively. When of more than five H2 molecules are added to the system, they fly away 
+from the Li atom and adsorption energy fall below 0.1 eV. We observed that the average adsorption 
+energy decreases with increase in number of H2 molecules in the system which is due the steric 
+hindrance between the adsorbed H2 crowed around the sorption centers and the increase in Li-
+H2 distances (Table 1). The estimated data of adsorption energy and geometrical parameters of all 
+the bare hydrogenated systems and presented in Table 1. 
+ 
+Figure 3: Optimized geometry of hydrogenated Li functionalized PCP222, (a) PCP222-Li-
+1H2, (b) PCP222-Li-2H2, (c) PCP222-Li-3H2, (d) PCP222-Li-4H2, (e) PCP222-Li-5H2. 
+3.3.2 Electrostatics potential and Hirshfeld charges 
+To get a qualitative picture of electronic charge distribution over the surface of Li functionalized 
+PCP222 and their hydrogen adsorbed systems during the hydrogen adsorbed, we generate and 
+plotted the electrostatic potential map (ESP map) on the total electron density as depicted in 
+Figure 4. The electronic charge distribution is used to identify the active adsorption site, where the 
+hydrogen molecules can be introduced. The red and blue regions in the ESP plot reflects the 
+aggregation and reduction of electronic charge density respectively. The variation in the charge 
+density is plotted with the sequence of color code as red (highest electron density)> orange > 
+yellow > green > blue (lowest electron density). The ESP map of PCP222-Li shows that the Li 
+atom has the deficiency of electronic charges as marked by the dark blue region over the Li atom, 
+this indicate that the Li atom is somewhat ionic and is prone to bind the guest H2 molecules. When 
+the first H2 molecule added to the Li atom, the colour of the region over the Li changes from dark 
+
+(a)
+b
+C
+(d)
+eblue to light blue, demonstrating the charge transfer from C atom of PCP222 and adsorbed H2 to 
+the Li atom. Further sequential adsorption of H2 molecules to PCP222-Li changes the colour of Li 
+region from blue to light blue indicating additional charge transfer. The blue region over Li almost 
+disappears on the adsorption of 5th H2 molecules suggesting the saturation of hydrogen uptake and 
+more guest H2 are unlikely to be adsorbed. The exact charge transfer is determined by calculating 
+the hirshfeld charges as discussed below. 
+ 
+ 
+Figure 4: Electrostatics potential map of (a) PCP222-Li, (b) PCP222-Li-1H2, (c) PCP222-Li-
+2H2, (d) PCP222-Li-3H2, (e) PCP222-Li-4H2, (f) PCP222-Li-5H2. 
+We have performed the hirshfeld charge analysis to quantify the charge transfer distributions on 
+the Li functionalized PCP222 and their H2 adsorbed systems. The computed average Hirshfeld 
+charges on C atoms of benzene ring (Li functionalized site), Li atom, and adsorbed H2 molecules 
+with the number of hydrogen molecules is depicted in Figure 5. The average charges on C atom 
+of benzene ring is noted to be -0.031 e which raises to -0.084 e with the functionalization of Li 
+atom. The charge on Li atom of PCP222-Li is noted to be +0.354 e, which illustrate the transfer of 
+charges from benzene ring to Li atom making the sorption center (Li) ionic and more suitable for 
+H2 adsorption. These results agree well with the aforesaid ESP analysis. On adsorption of the first 
+H2 molecule to PCP222-Li, the charge on C atom is reduced by 2.38% and at the same time the 
+charge on Li atom is increased by 16.7 %. Further addition of hydrogen molecules follows the 
+trend of decrease in charge on benzene ring and increase in charge on Li atom (Figure 5). These 
+observations suggest that, the ionic Li atom polarize the guest H2 molecules and the H2 molecules 
+are adsorbed to the sorption center via a charge polarization mechanism due to induced dipole 
+developed in H2 as suggested by the Neu-Rao-Jena [30]. It is noted that the electronic charge on 
+
+- 4.000 e-2
++ 4.000 e-2
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Sideview
+Top viewLi atom is raised by 41.36 % after the adsorption of the 5th H2 molecule. The adsorbed 
+H2 molecules are found to have an average charge of 0.027e to 0.013 e. 
+ 
+Figure 5: Hirshfeld charges before and after hydrogen adsorption on PCP222-Li 
+3.3.3 Bader’s topological analysis and PDOS 
+The nature of interaction between the Li functionalized PCP222 and the adsorbed hydrogen 
+molecules is analyzed using the topological Bader’s quantum theory of atoms in molecules 
+(QTAIM). The parameters of electron density distribution at the bond critical point (BCP), 
+including the electron density (BCP), total electron energy density (ℋBCP), and Laplacian (2BCP), 
+are computed and given in Table S1 (in Supporting Information). The electron density (r) on C-C, 
+and C-Li, of hydrogenated PCP222-Li estimated to be almost equal to that of bare host material, 
+suggesting the post-adsorption chemical stability of the material. Additionally, the 
+average BCP values on H-H in PCP222-Li-5H2 is 0.258 a.u which is same as that on isolated bare 
+H2 molecules (-0.263). This reveal that the adsorbed hydrogens are in molecular form during the 
+adsorption. According to Kumar et al., the positive value of 2BCP indicated an electron density 
+depletion in the region of bonding and implied a close-shell kind of interaction. We noticed there 
+is no BCP between the Li and H atoms which implies no chemical bond between the Li atom and 
+the adsorbed H2 molecules and the interaction is purely closed-shell type resulting from the charge 
+polarization as proposed by the Neu-Rao-Jena. 
+Figure 6 illustrate the density of state plot of Li and adsorbed H atoms of the hydrogenated 
+PCP222-Li including the first and last (5th) H2 molecules adsorbed on the system. When one 
+hydrogen molecule is bound to the sorption center (Li), the s-orbital of the H2 molecule appears 
+below the Fermi level (E = 0) and stays unaffected as in the case of bare H2in Figure S2. This 
+
+0.6
+Ring CbeforeLi decoration
+0.5
+Ring C after Li decoration
+-Liatom
+0.4-
+-Hatom
+Hirshfeld Charges (eu)
+0.3
+0.2
+0.1 -
+0.0
+0.1
+-0.2
+-0.3
+0.4
+-
+*
+*
+0
+1
+2
+3
+4
+5
+Number of H, molecules, nsignifies that there is no hybridization between the Li and adsorbed H2. This implies that the 
+adsorption of H2 molecule is owing to the induced dipole produced by charge polarization in H2. 
+With the adsorption of 5H2 molecules on PCP222-Li, the orbital of H atom splits into multiple 
+peaks ranging from -16 eV to -4 eV. This implies that the adsorption weakens as the quantity of 
+H2 molecules increases in the host. 
+ 
+Figure 6: Partial density of state on Li and H atoms of PCP222-Li-1H2 and PCP222-Li-5H2 
+3.4 Thermodynamics and storage capacity 
+3.4.1 Storage Capacity 
+ 
+Figure 7: Optimized geometry of (a) PCP222-3Li, (b) PCP222-3Li-3H2, (c) PCP222-3Li-6H2, 
+(d) PCP222-3Li-9H2, (e) PCP222-3Li-12H2, (f) PCP222-3Li-15H2. 
+ 
+
+3.0
+Li
+PCP222-Li-1H2
+2.5
+H
+2.0
+1.5
+1.0
+HOMO
+LUMO
+-7.97eV
+0.51eV
+W
+0.5
+0.0
+-18
+16
+14
+-12
+-10
+-8
+-6
+-4
+.
+-2
+0
+2
+4
+Energy (ev)
+(a)
+4.0
+Li
+PCP222-Li-5H,
+3.5
+H
+3.0
+2.5
+PDOS
+2.0
+1.5
+HOMO
+LUMO
+1.0
+-7.91eV
+0.45eV
+0.5
+0.0
+-18
+16
+14
+12
+10
+-8
+-6
+-4
+T
+-2
+0
+2
+4
+Energy (ev)
+(b)+3H2
+(a)
+(b)
+15H2
++3H2
++ 3H,To investigate the optimum hydrogen storage capacity of the studied system, we functionalized 
+the maximum possible number of Li atoms over each benzene ring of PCP222. The geometrical 
+structure of three Li functionalized PCP222 ( PCP222-3Li) is shown in Figure 7 Further, we 
+introduced H2 molecules to each Li atom of PCP222-3Li sequentially as discussed in previous 
+section (3.3.1). The computed average hydrogen adsorption energy and the geometrical parameters 
+of all the hydrogenated systems are provided in the Table S2 (in Supporting Information). It is 
+noticed that, the adsorption process of hydrogen molecules on PCP22-3Li is found similar to that 
+of on PCP222-Li. On saturation of H2 adsorption on PCP222-3Li, we found each Li atom can 
+adsorb a maximum of 5H2 molecules resulting in total gravimetric density of 8.32 wt%. The 
+estimated value of hydrogen storage capacity is fairly above the requirement of US-DOE for 
+effective hydrogen storage systems. Our results can be compared with earlier reported 
+H2 gravimetric density on metal decorated carbon-based materials for hydrogen storage, for 
+example, Li-decorated C41 allotrope (7.12 wt%) [58], Li doped MOF impregnated with Li-coated 
+fullerenes[59], Li-doped B4C3 monolayer (6.22 wt%) [4]. 
+To develop a realistically usable hydrogen storage system, a significant quantity of hydrogen 
+molecules must be adsorbed by the host material under achievable storage conditions. Further the 
+adsorbed hydrogen molecules must also be efficiently desorbed at suitable temperature (T) and 
+pressure (P). Thus, we estimated the quantity of adsorbed hydrogen that could be used at a 
+accessible range of temperature (T) and pressure (P). To calculate the number of H2 molecules 
+remain adsorbed on PCP222-3Li (Occupation number) at different T and P, we calculated the 
+empirical value of hydrogen gas chemical potential (µ). Then the occupation number (N) is 
+estimated by the following expression and plotted with various T and P in Figure 8(b)[60]. 
+𝑁 =
+∑
+𝑛𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇]
+𝑁𝑚𝑎𝑥
+𝑛=0
+∑
+𝑔𝑛𝑒[𝑛(𝜇−𝐸𝑎𝑑𝑠)/𝐾𝐵𝑇]
+𝑛𝑚𝑎𝑥
+𝑛=0
+ 
+ 
+ 
+ 
+(5) 
+Here Nmax is the maximum number of H2 molecules adsorbed at each Li atom on 
+PCP222, n and gn represents the number of H2 molecules adsorbed and configurational 
+degeneracy for a n respectively. kB is the Boltzmann constant and -Eads indicates the average 
+adsorption energy of H2 molecules to PCP222-3Li. m is the empirical value of chemical potential 
+of hydrogen gas at specific T and P, and is obtained by using the following expression [61]. 
+𝜇 = 𝐻0(𝑇) − 𝐻0(0) − 𝑇𝑆0(𝑇) + 𝐾𝐵𝑇 ln (
+𝑃
+𝑃0)      
+ 
+ (6) 
+
+Here H0(T), S0(T) are the enthalpy and entropy of H2 at pressure P0 (1 bar). 
+We can see in Figure 8(b) that the PCP222-3Li can adsorbed H2 molecules giving rise to 
+maximum hydrogen uptake capacity of ~8.32 wt% up to the temperature of 80 K and pressure of 
+30-60 bar. When the temperature rises beyond 80 K, the H2 molecules begin to desorb from the 
+PCP222-3Li and the gravimetric density closes to ~5.5 wt% (target of US-DOE by 2025) when 
+the temperature reaches 180 K under the pressure of 30-60 bar. Further rise in temperature, the 
+storage capacity of the PCP222-3Li fall below 4 wt% at 220 K and 40-bar. At a temperature range 
+of 260 K-300 K and pressure range of 1-10 bar, the studied system shows a 100% desorption of 
+hydrogen. Thus, we can propose the Li functionalized PCP222 as a low-temperature-adsorption 
+and room-temperature-desorption hydrogen storage material. Under the room temperature (300 
+K), the studied system shows up to 8.32 wt % of usable hydrogen storage capacity with 100% 
+reversibility. Thus, we believe that, our studied material Li functionalized PCP222 can be used as 
+an efficient hydrogen storage material satisfying the criteria of US-DOE. 
+ 
+Figure 8: Plot of Van’t-Hoff desorption temperature for Li functionalized PCP222 at different 
+temperature and pressure. 
+3.4.2 Desorption temperature 
+For a reversible hydrogen storage media, it is crucial to estimate the desorption temperature of 
+hydrogen molecules. We have estimated the desorption temperature (T D) of H2 for the Li 
+functionalized PCP222 using the Van’t Hoff equation [17]. 
+𝑇𝐷 = (
+𝐸𝑎𝑑𝑠
+𝐾𝐵 ) (
+∆𝑆
+𝑅 − ln 𝑝)
+−1
+ 
+ 
+ 
+ 
+ 
+(7) 
+
+280
+81
+7,488
+6.656
+7-
+260-
+5.824
+6-
+4.992
+Desorptiontemperature
+240
+219K
+5-
+4.160
+220
+G,wt%
+2.496
+200
+3
+1.664
+182K
+2-
+180
+0.000
+1
+160
+145K
+averageT,
+140-
+一minT,
+1.5
+2.0
+2.5
+3.0
+1.0
+3.5
+4.5
+5.0
+Pressure(atm)
+300
+(a)
+(b)Where, Eads represents the computed hydrogen adsorption energy, KB, and R denotes for the 
+Boltzmann constant and R the gas constant respectively. P represent s the equilibrium pressure 
+(we take a range of 1 to 5 atm with an increment of 0.5 atm) and △S is the entropy change of 
+hydrogen from its gaseous state to liquid state [62]. Using the highest and lowest adsorption energy 
+of system (with minimum and maximum H2 gravimetric density, respectively), the maximum and 
+minimum desorption temperatures (TDmax∕TDmin) are determined. While, TDmin denotes the 
+minimum temperature necessary to initiate the desorption of H2 molecules, the TDmax is the 
+temperature required for complete desorption process. The estimated desorption temperatures 
+along with the equilibrium pressure is depicted in Figure 8(a). The minimum and maximum 
+temperatures for H2 desorption are determined to be 145 K and 219 K, respectively, at 1 atm 
+pressure. The estimated average TD of Li functionalized PCP222 is 182 K at 1 atm. This result 
+reveals that, the system can adsorb its full capacity H2 at cryogenic temperature and desorb all the 
+H2 molecules bellow room temperature at 1 atm pressure. However, the desorption temperature 
+can be increases by increase in the equilibrium pressure as presented in Figure 8(a) and as 
+discussed above. 
+3.5 Molecular dynamics simulations 
+ 
+Figure 9: (a) Potential energy trajectories of hydrogenated PCP222-3Li and (b) Time 
+evolution trajectory of average bond length between the Li atom and C atoms of PCP222 at 
+300K and 500K, 
+
+968.88
+968.90
+300K
+(Hartree
+500K
+968.92
+968.94
+energy
+968.96
+968.98
+-969.00
+Potential
+-969.02
+969.04
+-969.06
+-969.08
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+1000
+Time (fs)
+(a)
+2.6
+C-Lidistance@300K
+Average C-Li distance (A)
+2.5
+C-Lidistance@50oK
+2.4
+2.3
+2.2
+2.1
+2.0
+1.9
+1.8
+1.7
+1.6
+300K,1000fs
+500K,1000fs
+1.5
+0
+100
+200
+300
+400
+500
+600
+700
+800
+006
+1000
+Time (fs)
+(b) 
+To validate the reversibility of hydrogen molecules on PCP222-3Li estimated by the DFT 
+computation, we have carried out molecular dynamics (MD) simulations using the atomistic 
+density matrix propagation (ADMP). ADMP is an extended Lagrangian approach to MD, that uses 
+the gaussian basis function and propagates the density matrix. The ADMP-MD simulations is 
+performed on system with highest storage capacity (PCP222-3Li-15H2), at two different 
+temperatures of 300K and 500 K for total time of 1 ps with the time step of 1fs. During the 
+simulations the temperature (kinetic energy thermostat) is maintained by the velocity scaling 
+approach and at every 10 fs, time step, the temperature is checked and corrected. The time 
+evolution potential energy trajectories and the snapshots are depicted in Figure 9(a) and Figure S3 
+(in supporting Information) respectively. The MD simulations at 300 K and 1ps illustrate that 
+almost all the H2 molecules fly away from the sorption centers, except 1H2 at each center. 
+Simulations at 500 K shows that all the H2 molecules are desorbed from the host material keeping 
+the host structure intact. This result suggests that the hydrogen storage in Li functionalized PCP222 
+is reversible in process. 
+For a viable reversible hydrogen storage material, it is important that the host material must not 
+distorted above the hydrogen desorption temperature. To investigate the solidity of host material 
+(PCP222-3Li), we performed the MD simulations on the bare host structure (PCP222-3Li) at room 
+temperature (300 K) and considerably above the H2 desorption temperature (500K) using ADMP. 
+The molecular dynamics simulations are performed for 1 ps with a time step of 1 fs. The time 
+evolution trajectory of average distance between Li atom and the carbon atoms of PCP222 benzene 
+rings is plotted in Figure 9(b). We noticed that the PCP222-3Li structure stays stable at 500 K and 
+almost no change in C-C and C-H bond distance is observed. The trajectory of average bond length 
+between the Li atom and C atoms of PCP222 benzene rings seem oscillate but the mean value 
+(2.25 Å) and the variation is minimal. This validates the structural integrity of the host material 
+above the H2 desorption temperature. Moreover no Li clustering is also noticed after desorption as 
+discussed earlier in Section 3.2.1. Thus, we believe that PCP222-3Li can be considered for feasible 
+reversible hydrogen storage material. 
+ 
+ 
+
+4 Summery and Conclusion 
+In this study, we investigated the thermodynamical stability and hydrogen storage capacity of Li 
+functionalized [2,2,2]paracyclophane, using the density functional theory. The Li atoms are found 
+to bind with the PCP222 via Dewar mechanism and no clustering of Li atoms over PCP222 was 
+noticed. Each Li atom on PCP222 could adsorb up to 5H2 molecules via charge polarization 
+mechanism with an average H2 adsorption energy in the range of 0.12 - 0.17 eV/H2, indicating 
+physisorption type of adsorption. Moreover, the average H-H bond distance got elongated by 0.01 
+Å, during the adsorption process, which implied that the adsorbed H2 were in molecular form and 
+this fact was also confirmed by the charge distribution analysis. When three Li atoms were 
+functionalized on PCP222, the H2 gravimetric capacity of the system was up to 8.32 wt% which 
+was fairly above the US-DOE requirements for practical hydrogen applications. During saturation 
+of H2 adsorption, the host material displayed no significant change in geometry. The 
+thermodynamic usable hydrogen capacity was found up to ~8.32 wt% at the temperature of 80 K 
+and pressure of 30-60 bar. On further increase in temperature, up to 180 K under the pressure of 
+30-60 bar, the PCP222-3Li hydrogen uptake capacity approached 5.5wt% which is the target of 
+DOE by 2025. At a temperature range of 260 K-300 K and pressure range of 1-10 bar, the PCP222-
+3Li system showed 100% desorption of H2. Molecular dynamic simulation confirmed that at 300 
+K, almost all the H2 molecules flied away except 1H2 at each center. Simulations at 500 K showed 
+that all the H2 molecules are desorbed from the host material keeping the structure of the host 
+structure intact. Since, there is no experimental works reported on Li functionalized PCP222 for 
+hydrogen storage, we hope our computational work will contribute significantly to the research of 
+hydrogen storage in macrocyclic compounds and provide supporting reference for the future 
+experiments. 
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+

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+Deep learning enhanced noise spectroscopy of a spin qubit environment
+Stefano Martina,1, 2, ∗ Santiago Hern´andez-G´omez,3, 2, † Stefano
+Gherardini,4, 2, ‡ Filippo Caruso,1, 2, § and Nicole Fabbri5, 2, ¶
+1Dipartimento di Fisica e Astronomia, Universit`a di Firenze, I-50019, Sesto Fiorentino, Italy
+2European Laboratory for Non-linear Spectroscopy (LENS),
+Universit`a di Firenze, I-50019 Sesto Fiorentino, Italy
+3Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139
+4Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO),
+Area Science Park, Basovizza, I-34149 Trieste, Italy
+5Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO), I-50019 Sesto Fiorentino, Italy
+(Dated: January 13, 2023)
+The undesired interaction of a quantum system with its environment generally leads to a coherence
+decay of superposition states in time. A precise knowledge of the spectral content of the noise induced
+by the environment is crucial to protect qubit coherence and optimize its employment in quantum
+device applications. We experimentally show that the use of neural networks can highly increase
+the accuracy of noise spectroscopy, by reconstructing the power spectral density that characterizes
+an ensemble of carbon impurities around a nitrogen-vacancy (NV) center in diamond.
+Neural
+networks are trained over spin coherence functions of the NV center subjected to different Carr-
+Purcell sequences, typically used for dynamical decoupling (DD). As a result, we determine that
+deep learning models can be more accurate than standard DD noise-spectroscopy techniques, by
+requiring at the same time a much smaller number of DD sequences.
+I.
+INTRODUCTION
+Quantum sensing combines theoretical results with ex-
+perimental and engineering techniques to carry out infer-
+ence of signals with improved accuracy and/or less com-
+putation time by making use of quantum physics [1, 2].
+A quantum sensor takes advantage of the fragility of
+its quantum properties, such as quantum coherence or
+entanglement, to improve the detection of external per-
+turbations with higher accuracy compared to any classic
+sensor.
+However, this same property implies that the
+quantum sensor is subjected to detrimental noise stem-
+ming from the coupling with its environment. For this
+reason, it is desirable to fully characterize the sensor’s
+environment, either to filter out its detrimental effect, or
+to take it into account when detecting external signals,
+for example, in algorithms using quantum optimal con-
+trol [3–6].
+Neural networks (NN) [7, 8], i.e., algorithmic models
+provided by the interconnection of a group of nodes com-
+monly called neurons, could be a powerful tool to infer
+the sensor’s environment. In this context, deep learning
+has been already proposed theoretically for the classifi-
+cation and detection of quantum noise features [9–11],
+and employed experimentally for the following tasks. (a)
+Estimating the spectra of minuscule amounts of complex
+molecules [12] for nano nuclear magnetic resonance; (b)
+the sensing of magnetic-field strength at room temper-
+∗ stefano.martina@unifi.it; Equal contribution to this work
+† shergom@mit.edu; Equal contribution to this work
+‡ stefano.gherardini@ino.cnr.it
+§ filippo.caruso@unifi.it
+¶ fabbri@lens.unifi.it
+ature with high precision [13, 14] by using nitrogen va-
+cancy (NV) centers; (c) performing error mitigation [15]
+and noise learning [16–18]; (d) the tracking of quantum
+trajectories [19]; (e) classification of many-body quantum
+states [20] in superconducting quantum circuits. How-
+ever, to our knowledge, experimental noise spectroscopy
+in single color centers in diamond via deep learning is
+still missing.
+In this paper, we demonstrate that NN can be used
+to process the data obtained by a qubit, operating as
+a quantum sensor, and then reconstruct the noise spec-
+trum that induces dephasing into the qubit itself. In par-
+ticular, we focus on a qubit under dynamical decoupling
+(DD) control sequences [21, 22] in the presence of classical
+random noise with an unknown power density spectrum,
+usually denoted as noise spectral density (NSD). Beyond
+testing numerically our machine learning models, we use
+a single NV center in diamond as a spin qubit sensor and
+we perform a spectroscopic reconstruction of the mag-
+netic noise of its local environment. The latter comprises
+13C nuclear spins randomly distributed in the diamond
+lattice [23–25] (see Fig. 1). The dephasing affecting the
+qubit sensor is analyzed by applying a set of DD con-
+trol pulses that realize filter functions [21, 22, 26, 27] in
+the frequency domain. The filter functions are designed
+to select specific noise components, without sensing all
+other system-bath interactions. A widely used DD con-
+trol pulse is the Carr-Purcell (CP) sequence [1, 28] that
+is given by N equidistant π pulses, performed between an
+initial and a final π/2 pulse. CP sequences act in the fre-
+quency domain approximately as Dirac comb filters [29];
+hence, they have been used to perform spectroscopy of in-
+tricate signals, e.g., for noise spectroscopy [30, 31]. With
+this protocol, the requirement to achieve high values of
+the noise reconstruction accuracy is to perform sequences
+arXiv:2301.05079v1  [quant-ph]  12 Jan 2023
+
+2
+with a high number of pulses meaning N ∈ [30, 120] (as in
+Ref. [32]) or higher, so that the Dirac comb filter approx-
+imation remains valid (in fact, N determines the filter
+width). This usually leads to long experiments to recon-
+struct the whole spectrum of the noise. Other techniques
+using non-equidistant or even more sophisticated DD se-
+quences [4, 33–36] have proved to be effective for noise
+sensing, but sometimes at the price of a higher computa-
+tional burden.
+For our sensing task, NN are designed to solve a re-
+gression problem, i.e., the reconstruction of the NSD.
+Here, we assume that the NSD of the bath of spins has a
+Gaussian profile [32, 37, 38]. The Gaussian NSD is thus
+parametrized as a function of key parameters, i.e., the
+mean value, variance, offset and noise power that we aim
+to reconstruct. Note that our proposal can be adapted to
+other parametrized NSD functions. The NN are trained
+over a set of synthetic data generated by simulating how
+the coherence of the qubit sensor decays over time under
+the influence of both the CP control pulses and the NSD.
+Moreover, to make the measurement statistics as close as
+possible to the ones obtained from the experiments, extra
+artificial errors are added.
+Our approach using NN entails the following advan-
+tages that we have proven experimentally. (i) NN have
+the capability to predict never-before-seen experimental
+data, and they can work with a better reconstruction
+accuracy (even up to 7 times better, as shown in the
+section Results below) than standard noise spectroscopy,
+as the ´Alvarez-Suter method [31], by making use at the
+same time of DD control sequence with a much smaller
+number of pulses. (ii) The training dataset, which can
+contain both synthetic and experimental data, is gener-
+ated just once and then it can be applied several times,
+as long as the new collected data reproduce the physical
+context under analysis. In connection with (i), we are
+going to show that the amount of data used as input to
+the NN can be smaller than the one needed to resolve the
+NSD by means of standard noise spectroscopy methods.
+From our knowledge, this work is the first experimen-
+tal proof of enhanced reconstruction performance with
+NN for carrying out noise spectroscopy in single color
+centers in diamond. We thus expect that the techniques
+discussed here could fast become a novel standard spec-
+troscopy tool both for such quantum systems and other
+quantum platforms in which regression problems have to
+be solved.
+II.
+RESULTS
+A.
+Generation of training dataset
+The training dataset is composed of synthetic data that
+are originated by simulating the coherence decay of the
+qubit sensor in a noise spectroscopy experiment based on
+DD, as the one depicted in Fig 1. This standard sensing
+procedure, which stems from Ramsey interferometry [1],
+maps information about the quantum coherence of the
+sensor into the population in |0⟩ that is then effectively
+recorded. After having initialized the qubit sensor in the
+ground state |0⟩, a π/2 pulse is applied such that the
+qubit state |ψ⟩ is the superposition (|0⟩+|1⟩)/
+√
+2. Then,
+we perform a CP control sequence consisting in a train
+of π pulses that flips repeatedly the qubit, and finally, a
+second π/2 pulse is applied in order to map the phase of
+the qubit into its population. The probability that the
+state of the quantum sensor is |0⟩, which corresponds to
+the observable population, equals to [1, 32]
+P = 1
+2 (1 + C(τ, N)) ,
+(1)
+where N is the number of π pulses and τ is the time
+between them. The coherence function C(τ, N) is sim-
+ulated numerically, for a set of different values of τ and
+N, to generate the training dataset.
+Let us now introduce the decoherence function that
+quantifies how the quantum coherence C(τ, N) is modi-
+fied under the action of both the external bath of spins
+and a set of CP control pulses.
+The control sequence
+has the effect to modulate the coherence content of the
+qubit sensor, while the interaction with the bath, asso-
+ciated to the NSD S(ω), tends on average to destroy
+such coherence. Overall, under the joint presence of con-
+trol fields and a noise source, the coherence decays as
+C(τ, N) ≡ e−χ(τ,N), where χ(τ, N) denotes the decoher-
+ence function [27, 39–41]:
+χ(τ, N) =
+�
+dω
+πω2 F(ω, τ, N)S(ω) .
+(2)
+In Eq. (2), the filter function F(ω, τ, N) ≡ |Y (ω, τ, N)|2
+is the square modulus of the Fourier transform of the
+so-called modulation function y(t, τ, N).
+The latter is
+constant piecewise, with values ±1, and switches sign at
+the times t = τ/2, 3τ/2, . . . , (N − 1/2)τ where each π
+pulse is applied [2]. Notice that we are assuming that
+the π pulses are instantaneous, a reasonable assumption
+for our experimental setup where a π pulse duration is
+∼ 0.1 µs and the time between pulses is τ ∈ [3.3, 6.1] µs.
+Let us now recall the expression, in the frequency domain,
+of the filter function for a CP sequence with even N:
+F(ω, τ, N) = 8 sin2
+�ωτN
+2
+�
+sec2 �ωτ
+2
+�
+sin4 �ωτ
+4
+�
+, (3)
+while for odd N, sin2(ωτN/2) has to be replaced with
+cos2(ωτN/2) [2, 26].
+In order to generate the training dataset, the NSD
+S(ω) is parameterized as
+S(ω) = s0 + A exp
+�
+−(ω − ωc)2
+2σ2
+�
+.
+(4)
+Thus, being a Gaussian distribution, the NSD is fully de-
+scribed by the offset s0, amplitude A, width σ and center
+ωc.
+For the training dataset in the paper, the values
+
+3
+FIG. 1: NV center and Neural Networks for noise spectroscopy. The NV center is surrounded by an
+ensemble of 13C nuclear spins (orange spheres) that collectively induce dephasing to the NV electronic spin (blue
+sphere). The NV electronic spin is controlled with a DD sequence (specifically, a Carr-Purcell (CP) sequence) with
+the aim to measure its dephasing, and therefore characterize the NSD of the nuclear spin bath, i.e., S(ω; s0, A, σ).
+The CP sequence is formed by N equidistant π pulses in between an initial and a final π/2 pulse. The time τ
+between the π pulses determines the measurement total time T = Nτ, given that the time between the first π/2 and
+the train of π pulse and the time between the last π and π/2 pulses are both equal to τ/2. Then, we measure the
+output of this experiment, which is the probability P = 1
+2(1 + C(t)) that the NV center remains in the initial state
+|0⟩. The spin coherence function C(t) – evaluated at previously-determined times in the set T ∈ {t1, t2, . . . , tn} (the
+tk’s are obtained by changing τ with N fixed) – is the input of the designed Neural Networks (NN). After being
+trained, the NN return the estimation of the NSD parameters.
+of these parameters are taken from the following inter-
+vals: s0 ∈ [4 · 10−4, 4 · 10−3] MHz; A ∈ [0.3, 0.7] MHz;
+σ ∈ [2 · 10−3, 9 · 10−3] MHz.
+Instead, ωc is kept con-
+stant.
+This is because in our experimental setup the
+NSD stems from the interaction with a large ensemble
+of unresolved 13C impurities (nuclear spin bath) around
+the NV electronic spin. Therefore, the center of the NSD
+corresponds to the Larmor frequency ωc = γB, where
+γ = 1.0705 kHz/G is the gyromagnetic ratio of the 13C
+nuclear spins, and B is the amplitude of a static mag-
+netic field aligned with the NV quantization axis, z. Such
+static magnetic field is well known during the experimen-
+tal procedure since it determines the NV electronic spin
+resonances (B = 403.2 ± 2 G).
+The training dataset is generated by uniformly sam-
+pling 104 sets of parameters within the chosen intervals.
+Hence, overall we consider 104 distinct sequences of NSD
+parameters that are used to simulate different coherence
+curves C(τ, N). These sequences are taken in the time
+intervals τ ∈ [3.3, 3.66] µs and [5.5, 6.1] µs with sampling
+time ∆τ = 1 ns (∆τ = 20 ns in the experimental case, see
+below), and for N = {1, 8, 16, 24, 32, 40, 48}. These inter-
+vals are significant for our study because they include the
+values of τ at which the coherence decay curve exhibits
+the first and second order collapses induced on the qubit
+sensor by the bath of 13C impurities [42].
+Finally, in
+order to make the synthetic data used to train the NN
+closer to the experimental setting, extra artificial errors
+sampled by a normal distribution with standard devia-
+tion equal to 0.05 (comparable with the expected error
+in our experimental measurements) are added to every
+point of the generated coherence decay curves. In this
+way, one may mitigate the over-fitting of the employed
+machine learning models that are thus expected to better
+generalize to unseen data. In general, a model trained on
+synthetic data cannot be successfully applied to real data
+without fine tuning it. But in our case, it becomes possi-
+ble, probably due to the fact that the simulated data of
+the coherence decay are quite close to the experimentally
+observed decay data induced by the environment.
+As final remark, notice that, from the 104 simulated
+curves C(τ, N), 6000 are used for the training of the NN
+and 2000 for their validation. Instead, the test step is
+performed either by using the remaining 2000 simulated
+curves, or by using experimental data as described below.
+B.
+Neural networks working principles
+Let us describe the main working features of the NN
+employed in this paper to carry out noise spectroscopy.
+Specifically, we are going to use the multi-layer percep-
+tron (MLP) that is composed of fully-connected layers,
+each of them with a variable number of artificial neurons.
+A single artificial neuron returns as output the scalar
+ˆy ≡ Σ(wT · x + b)
+(5)
+that, by definition, is provided by applying the non-linear
+function Σ : R → R to the weighted sum of the input
+vector x ∈ Rk to which the bias term b ∈ R is added.
+w ∈ Rk denotes the vector of weights. In our analysis,
+the activation function Σ is chosen equal to the rectifier
+
+4
+Σ(x) ≡ max(0, x) [43, 44]. Thus, a MLP layer composed
+of q neurons (each with k inputs) returns the vector
+ˆy ≡ Σ(W T x + b),
+(6)
+where ˆy ∈ Rq, W ∈ Rk×q is the matrix of weights (W
+collects all the weight vectors of the single neurons), and
+b ∈ Rq is the vector of the biases. Hence, a MLP with L
+layers is ruled by the recursion equation
+h[ℓ] ≡ Σ
+�
+W[ℓ]T h[ℓ − 1] + b[ℓ]
+�
+,
+(7)
+where ℓ = 1, . . . , L is the index over the number of layers
+and h[0] ≡ x. In Eq. (7), W[ℓ] and b[ℓ] are, respectively,
+the weights and the biases of the ℓ-th layer. The output
+vector of the MLP is ˆy ≡ h[L]. It is worth noting that
+the number, dimension and activation functions (they
+are usually denoted as the hyperparameters ξ) of the NN
+layers are chosen through a single optimization routine
+(cfr. Methods).
+Let us now introduce the supervised learning process.
+Ideally, the purpose of the latter is to find the parameters
+θ∗ = argminθRD(θ, ξ) that minimize the theoretical risk
+function
+RD(θ, ξ) ≡ E(x,y)∼D [L (ˆy, y)] ,
+(8)
+where θ ≡ {W[1], b[1], . . . , W[L], b[L]}, and ˆy are the
+estimated values of y.
+By definition, RD is the ex-
+pected value of the loss function L for (x, y) sampled
+from the distribution D that generates the dataset [45].
+The loss function L is a differentiable function that mea-
+sures the distance between the prediction ˆy (output of
+the MLP) and the desired output y. However, since one
+can only dispose of a finite set S = {(x, y)1, . . . , (x, y)m}
+of samples to train, validate and test the employed ML
+models, the theoretical risk function is approximated by
+the empirical risk function.
+Considering the partition
+{Str, Sva, Ste} of S in training (Str), validation (Sva) and
+test (Ste) sets, the empirical risk function is defined by:
+RStr(θ, ξ) ≡
+1
+|Str|
+�
+(x,y)∈Str
+L (ˆy, y) ,
+(9)
+where |Str| is the cardinality of the training set. In fact,
+RStr is the arithmetic mean of the loss function L eval-
+uated on the samples of the training set Str.
+In our paper, we take the loss function L equal to the
+Mean Squared Error (MSE), also called L2 loss:
+L(ˆy, y) = 1
+q
+q
+�
+i=1
+(ˆyi − yi)2
+(10)
+for the q outputs of the last layer (in our case three,
+corresponding to the noise parameters s0, A, σ). The
+MLP is trained by minimizing (step-by-step over time)
+the empirical risk function RStr(θ, ξ) with respect to θ
+by means of the mini-batch gradient descent method, so
+as to obtain the optimal value θ∗ of the NN parameters.
+Each gradient descent step is defined by
+θt+1 = θt − η∇θ
+1
+B
+B
+�
+b=1
+L(ˆyb,t, yb,t),
+(11)
+where θ0 is a randomly chosen starting point, η is the
+learning rate that defines the length of the step and
+∇θ 1
+B
+�B
+b=1 L(ˆyt,b, yt,b) is the gradient of the loss func-
+tion. The gradient is calculated for any time t on a batch
+of B elements taken from the training set, and the sub-
+script θ in ∇θ indicates that the variables of L during
+the gradient evaluation are the weights of the NN. In
+this paper, RStr is minimized by means of Adam [46]
+that is a gradient-based optimization algorithm perform-
+ing the adaptive estimation of lower-order moments. The
+minimization is stopped when the time-derivative of the
+risk function evaluated on the validation set RSva(θ∗, ξ)
+becomes positive (early stopping strategy) or after a pre-
+defined number of gradient steps using all the data of the
+training set (called epochs). Then, we use RSva(θ∗, ξ) to
+check if the MLP works also for unseen data and tune
+the hyperparameters ξ (cfr. Methods). Finally, the test
+set Ste is employed to calculate the metrics (discussed in
+detail below) used to generate the figures with the results
+that we are going to illustrate.
+C.
+Training and numerical test of neural networks
+We now show the results obtained by using the trained
+machine learning models to infer the value of the NSD
+parameters {s0, A, σ}. As already mentioned, the NN are
+tested with 2000 different NSD parameters. For each of
+these sets of parameters, the curves C(τ, N) have been
+simulated as described in the previous subsections.
+In order to determine the smallest amount of data re-
+quired to reconstruct the NSD, we perform the training,
+validation and test of the NN with sub-sets of the simu-
+lated curves. These sub-sets are defined by introducing
+the variable N that denotes the upper bound for the num-
+ber of pulses N ≤ N considered during the whole process.
+For example, for N = 16 only the curves C(τ, N) with
+N ∈ {1, 8, 16} are considered. Note that the sub-sets de-
+fined for each value of N contain the curves for all the
+different NSD parameters (6000 for training, 2000 for val-
+idation, and 2000 for testing), and for all the times τ in
+the intervals defined before.
+The results of this analysis are shown in Fig. 2 (orange
+data), where the MSE (the loss function) between the in-
+ferred parameters ( ˆs0, ˆA, ˆσ) and the original parameters
+(s0, A, σ) used to generate the dataset is plotted as a
+function of N. Remarkably, the MSE seems to achieve
+its minimum value after N = 16. This entails that the
+NN do not significantly improve their precision on the
+reconstruction of the NSD by using more data to train
+the NN beyond this point.
+
+5
+0
+10
+20
+30
+40
+50
+¯N
+0.0
+0.1
+0.2
+0.3
+MSE(s0, A, σ)
+FIG. 2: Mean-square-errors (MSE) between original
+and estimated NSD parameters for a set of 2000 test
+cases. Orange bullets with dash-dotted line are the
+mean values returned by NN. Blue squares with dotted
+line are the mean values provided by the HS method.
+Finally, shaded areas denote the standard deviation,
+taking into account all the 2000 cases.
+To establish how accurately a NN reconstructs the
+NSD, we need to compare the corresponding results with
+those of a different method. In particular, we concentrate
+on the method used in Ref. [32], which is itself based on
+Refs. [30, 31]. According to them, the decay of the coher-
+ence function C(τ, N) is analyzed as a function of N, for
+each fixed value of τi, i.e., for each fixed frequency com-
+ponent of the filter functions. In the limit of high N, the
+decay of the coherence is exponential, with a rate that is
+inversely proportional to the amplitude of the NSD [30].
+In other words, the amplitude of the NSD is directly es-
+timated for a discrete set of frequencies (each propor-
+tional to 1/τ). In contrast with the original proposals in
+Refs. [30, 31], the method in Ref [32] demonstrates that
+it is better to use the harmonics of the filter functions
+to reconstruct the NSD, in order to avoid extra broad-
+ening of the reconstructed spectrum. For this reason, we
+denote this method as Harmonics Spectroscopy (HS).
+We have analyzed the same 2000 different curves
+C(τ, N) (used to test the machine learning models) also
+with the HS method. The results are collected and shown
+in Fig. 2 (blue data), where the first point is for N = 16.
+This is due to the fact that, by definition, the HS method
+fits the decay of the coherence as a function of N. This
+is possible only for a dataset with at least three points
+(in this case N = 1, 8, 16). As one can observe in Fig. 2,
+the MSE values for the HS method (blue region) are al-
+ways above the MSE values for the NN method (orange
+region), especially for lower values of N. These results
+demonstrate that the NN method can predict the pa-
+rameters of the NSD with an improved accuracy (up to
+5 times larger) with respect to the HS method. The test
+presented in this subsection have been performed with
+simulated data. In the next subsection we are going to
+repeat the same test but with experimental data.
+D.
+Experimental test of neural networks
+By this point we know that NN can reliably predict the
+NSD from noisy simulated data. In this section, we want
+to use the NN (trained and validated with noisy simu-
+lated data) to reconstruct the NSD using experimental
+data.
+As quantum sensor we use a spin qubit encoded in the
+electronic spin of the ground state of a single nitrogen-
+vacancy (NV) center in a bulk diamond at room temper-
+ature. This system has proven as a sensitive quantum
+probe of magnetic fields, with outstanding spacial reso-
+lution and sensitivity [47, 48]. The diamond sample in
+our experiments has a natural abundance of 13C impu-
+rities (1.1%) that are randomly distributed in the dia-
+mond lattice [23–25]. The 13C nuclear spins constitute
+the external environment of the NV center.
+They act
+as a collective bath of spins that induces dephasing into
+the NV electronic spin, limiting the its coherence time
+T2 ≈ 100 µs. In the presence of strong bias magnetic
+field (≥ 150G) [32, 49], the weak coupling of the NV spin
+with these carbon impurities can be modeled as a clas-
+sical stochastic field. The latter has a power spectrum
+density (here called NSD) that follows a Gaussian dis-
+tribution centered at the Larmor frequency of the 13C
+nuclear spins. In order to measure the NV spin coher-
+ence function C(τ, N), we apply a train of π pulses (in
+our case a CP sequence) to the NV spin qubit following
+the DD protocol described in Fig 1. For more details on
+the experimental implementation and Hamiltonian of the
+system see Ref. [32]. We have performed this experiment
+for N = {1, 8, 16, 24, 32, 40, 48}, and for τ ∈ [3.3, 3.66] µs
+and [5.5, 6.1] µs with sampling time ∆t = 20 ns. The
+results are shown in Fig.3(a) (blue bullets). Then, the
+collected coherence functions have been processed and
+employed to reconstruct the NSD parameters by means
+of both the NN (trained with the generated dataset) and
+the HS method. In contrast with the test using simu-
+lated data in the previous section, in the experimental
+case we do not know the exact values of the NSD pa-
+rameters.
+Therefore, we cannot calculate the MSE to
+quantify the accuracy of the reconstructed parameters.
+In order to estimate such accuracy we have used the fol-
+lowing procedure: from the inferred NSD, the coherence
+curves C(τ, N) are simulated and then compared with
+the experimental results. An example of this comparison
+is shown in Fig.3(a), where C(τ, N) is simulated under
+the assumption that the NSD parameters are inferred ei-
+ther by the machine learning models (orange) or by the
+HS method (red), both for N = 16. Qualitatively it is
+clear that the orange curves are much closer to the ex-
+perimental data, than the red curves.
+There are several options to quantitatively compare
+the experimental data and the simulation results. Here
+we use both the reduced chi-squared χ2
+ν [50], and the
+
+6
+FIG. 3: (a) Coherence function C(τ, N). The experimental data (blue bullets) are shown together with the
+simulated ones using the NSD predicted respectively by the HS method (red lines) and machine learning models
+(orange lines), both for N = 16. (b) Reduced chi-squared χ2
+ν, obtained by comparing simulation and experimental
+data, as a function of N. As in panel (a), orange and red curves refer to the NN and HS method, respectively.
+Instead, the dashed line denotes the value of the reduced chi-squared for the HS method when we employ additional
+measurements for N = 56, 64, 72, 80 in the interval τ ∈ [5.5, 6.1] µs. Inset: Same results but quantified by the
+Mean-Absolute-Error (MAE) between the experimental data and the predicted C(τ, N).
+Mean-Absolute-Error (MAE) [51] between the exper-
+imental data and the predicted coherence functions
+C(τ, N) (see Methods for more details). The results of
+this comparison are shown in Fig. 3(b), where χ2
+ν and
+the MAE are plotted as a function of N. Remarkably,
+the NSD reconstructed by the NN for N = 16 behaves
+better that any case using the HS method. It is worth
+observing that the same experimental data used to infer
+the NSD parameters are partially used to estimate the
+χ2
+ν and MAE(C(t)). For example, for N = 16, only the
+data for N = 1, 8, 16 are used to reconstruct the NSD,
+but we employ all the data N = 1, 8, 16, . . . , 48 to ob-
+tain the χ2
+ν and MAE(C(t)). Overall, we have observed
+enhanced performance in reconstructing the NSD of the
+collective bath of spins, with a maximum improvement
+(about 7 times higher) for N = 16. In other words, for
+N = 16, once we reconstruct the NSD, the quantum sen-
+sor dynamics can be predicted with an average square
+deviation of ≃ 1.86 experimental error-bars by using the
+NN method, or with an average square deviation of ≃ 13
+error-bars if we use the HS method.
+III.
+DISCUSSION
+As shown pictorially in Fig. 1, the NN takes as input
+the spin qubit coherence functions (the coherence of the
+quantum sensor decays due to the presence of the exter-
+nal bath) obtained by using a set of different CP control
+sequences. The NN returns as output the parameters of
+the unknown NSD in the frequency domain.
+One can
+thus note that the NN, once validated, acts as a “time-
+frequency converter” (making use of a quite complicated
+deconvolution) from the measured signals living in the
+time domain – the spin coherence functions – to the NSD
+defined in the frequency domain.
+The results shown in the previous section, and sum-
+marized in Figs. 2 and 3(b), demonstrate that NN can
+be used to reconstruct the NSD affecting a quantum sen-
+sor, achieving higher precision and with considerable less
+data than the standard HS method.
+Improved values
+of the reconstruction accuracy have been obtained with
+simulated and experimental data. Both the HS and NN
+methods are comparable – in terms of NSD reconstruc-
+tion accuracy – for high values of N, but not for small
+ones, where NN gives significantly better results. More-
+over, the main result of our study is that NN trained
+with data obtained for N = 16 reconstruct the NSD
+more accurately than the best estimate provided by the
+HS method with N = 48. This improvement is remark-
+able by itself, but it becomes more significant when we
+consider that the time required to complete these exper-
+iments has a growth faster than a linear function with
+respect to N, following an arithmetic progression. As an
+example, the total time to perform all the experiments in
+the case of N = 16 and 48 is respectively ≃ 10 minutes
+and ≃ 1.2 hours [52]. This is an under-estimation of the
+time difference between methods, because we are only
+considering the bare measurement time, without taking
+into account the time delay between different experi-
+ments. Furthermore, it is worth stressing that our results
+also show that deep learning has a predictive power since
+it can be applied to never-before-seen data. This natu-
+rally provides to the employed machine learning models
+a connotation of robustness that is crucial in real appli-
+cations.
+
+7
+Let us observe that regression tasks, which are suc-
+cessfully solved by multi-layer perceptrons (one of the
+easiest form of NN), are less common with respect to the
+ones to carry out classification; a review of some exam-
+ple datasets and methods for regression is in Ref. [53].
+Hence, we expect that the synthetic data used in this
+work could be useful as a test bed also to the audience
+of machine learning researchers and developers solving
+regression problems in different contexts. With this in
+mind, we share the training dataset with synthetic data
+and our codes for their generation, as well as the code for
+machine learning experiments and NSD reconstruction
+[available on the GitHub repository (see Section “Data
+and code availability”)]. In this way, we promote the im-
+provement of machine learning models for noise sensing
+purposes and their use to solve different regression tasks
+in the quantum estimation framework.
+Conclusions & outlooks
+In this paper, we use NN to carry out noise spec-
+troscopy with a quantum sensor using dynamical decou-
+pling sequences with a much smaller number of π pulses
+and, at the same time, achieving a higher reconstruc-
+tion accuracy than standard methods (e.g., HS proto-
+col). This means that with our proposal the noise spec-
+troscopy procedure will take less time and give better
+results. More in detail, we experimentally demonstrate
+the capability of NN to reconstruct the NSD of the collec-
+tive nuclear spin bath that surrounds an electronic spin
+qubit, i.e., the ground state of a single nitrogen-vacancy
+center in bulk diamond at room temperature.
+To conclude, we outline some possible outlooks for our
+work.
+First of all, one may evaluate the performance
+of NN that are trained over input data obtained using
+DD control sequences with more degrees of freedom than
+the CP ones [54–58]. Secondly, deep learning might be
+applied to noise spectroscopy techniques beyond the HS
+methods, as for example optimal band-limited control
+protocols [34, 35] and even non-Gaussian noise charac-
+terization [59–61]. In addition, it might be worth inves-
+tigating how deep learning can be integrated to quantum
+sensing procedures that rely on the so-called stochastic
+quantum Zeno effect [62, 63], whereby the quantum probe
+is subjected to a sequence of quantum measurements that
+in the ideal case are designed to confine the dynamics of
+the probe around the initial (nominal) state [33, 64, 65].
+We are also confident that the extent of our results can
+be quite easily replicated in other experimental settings,
+as e.g., superconducting flux qubits [66, 67], trapped
+ions [68, 69], cold atoms [70, 71], quantum dots [72, 73],
+NMR experiments in molecules [31, 74], and nanoelec-
+tronic devices [75]. For such a purpose, one might slightly
+adapt the deep learning techniques used here to methods
+tailored for time series.
+IV.
+METHODS
+A.
+Technical details on the training of NN
+The NN models are developed using the PyTorch
+framework [76] on a machine with 32 CPU cores, 126Gb
+of RAM and a GeForce RTX 3090 GPU. The training
+time, including the optimization of the hyperparameters,
+is around 12 hours for each N .
+The hyperparameters optimization is implemented by
+means of the Ray Tune library [77]. The Hyperopt pack-
+age [78] uses the Tree-structured Parzen Estimators [79]
+algorithm as a Bayesian optimization to search for the
+best choice of the hyperparameters within a predefined
+search space. Hyperopt suggest the likely better configu-
+rations of the hyperparameters and the underlying model
+is updated after each trial that is run. The ASHA sched-
+uler [80] is then used to stop the run of the least promising
+trials chosen by the search algorithm, thus speeding up
+the hyperparameters optimization process.
+The optimized hyperparameters are the following. (1)
+The number of hidden layers decides the value of L −
+1 in Eq. (7). The hidden layers are between the input
+layer h[0] and the output layer h[L]. (2) The dimension
+of the hidden layers is the value of q in Eq. (6) that,
+for the sake of simplicity, is equal for all the layers in
+Eq. (7). Both the number and dimension of the hidden
+layers are chosen by sampling log-uniformly an integer
+value from the space [1, 32) and [1, 1024), respectively.
+(3) The learning rate is responsible for the length of the
+gradient descent step and it is optimized with a choice
+between 10−2, 10−3 and 10−4. (4) The batch size denotes
+the dimension of the batch on which the loss function is
+summed for the gradient calculation in a single descent
+step. The batch size is chosen between 2, 4, 8, 16, 32.
+(5) The dropout is a regularization strategy that aims
+to reduce the overfitting by randomly turn off the NN
+neurons with a predefined probability. Such probability
+is one among 0 (no dropout), 0.2 and 0.5. (6) The weight
+decay is another regularization technique that adds to the
+loss function the squared weights of the NN multiplied
+by a decay value. The latter value is optimized choosing
+between 0 (no decay), 10−6, 10−5, 10−4 and 10−3.
+B.
+Definition of quantifiers for reconstruction
+accuracy
+The accuracy NN and HS methods can be estimated by
+using the reconstructed NSD to simulate the coherence
+function C(τ, N), and ‘measuring’ the distance between
+the simulated data and the experimental values. To do
+so, we use the reduced chi-squared χ2
+ν, and the Mean-
+Absolute-Error (MAE(C)): We define Ce ± δCe (Cs) as
+the experimental (simulated) values of C(τ, N), where
+δCe is the standard deviation of the experimental data.
+
+8
+Then we can write reduced chi-squared and the MAE as
+χ2
+ν ≡ 1
+ν
+�
+n,N
+(Ce(τn, N) − Cs(τn, N))2
+δCe(τn, N)2
+(12)
+MAE(C) ≡ 1
+ν
+�
+n,N
+|Ce(τn, N) − Cs(τn, N)| ,
+(13)
+where N = {1, 8, 16, 24, . . . , N}, {τn} are the values of
+the time between pulses within the time intervals defined
+in main text, and ν is the total number of elements in the
+sum. Notice that χ2
+ν takes into account the experimental
+precision to scale the difference between experiment and
+simulation. The results showing both χ2
+ν and the MAE
+are in Fig. 3.
+DATA AND CODE AVAILABILITY
+The source codes for the generation of the train-
+ing dataset and the machine learning experiments
+are available on GitHub
+at the following address:
+https://github.com/trianam/noiseSpectroscopyNV
+ACKNOWLEDGEMENTS
+This work was supported by the European Com-
+mission’s
+Horizon
+Europe
+Framework
+Programme
+under
+the
+Research
+and
+Innovation
+Action
+GA
+n. 101070546–MUQUABIS, and by the European De-
+fence Agency under the project Q-LAMPS Contract No
+B PRJ-RT-989.
+S. H. G. acknowledges support from
+CNR-FOE-LENS-2020. F. C. and S. M. acknowledge the
+European Union’s Horizon 2020 research and innovation
+programme under FET-OPEN GA n. 828946–PATHOS.
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+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 
+Language Models sounds the Death Knell of  
+Knowledge Graphs 
+Kunal Suri  
+Optum, India 
+kunal_suri@optum.com 
+Swapna Sourav Rout 
+Optum, India 
+rout.swapnasourav@optum.com
+Atul Singh 
+Optum, India 
+atul_singh18@optum.com 
+     
+Prakhar Mishra 
+Optum, India 
+prakhar_mishra29@optum.com 
+Rajesh Sabapathy 
+Optum,India 
+rajesh_sabapathy@uhc.com 
+Abstract—Healthcare 
+domain 
+generates 
+a 
+lot 
+of 
+unstructured and semi-structured text. Natural Language 
+processing (NLP) has been used extensively to process this data. 
+Deep Learning based NLP especially Large Language Models 
+(LLMs) such as BERT have found broad acceptance and are 
+used extensively for many applications. A Language Model is a 
+probability distribution over a word sequence. Self-supervised 
+Learning on a large corpus of data automatically generates deep 
+learning-based language models. BioBERT and Med-BERT are 
+language models pre-trained for the healthcare domain. 
+Healthcare uses typical NLP tasks such as question answering, 
+information extraction, named entity recognition, and search to 
+simplify and improve processes. However, to ensure robust 
+application of the results, NLP practitioners need to normalize 
+and standardize them. One of the main ways of achieving 
+normalization and standardization is the use of Knowledge 
+Graphs. A Knowledge Graph captures concepts and their 
+relationships for a specific domain, but their creation is time-
+consuming and requires manual intervention from domain 
+experts, 
+which 
+can 
+prove 
+expensive. 
+SNOMED 
+CT 
+(Systematized Nomenclature of Medicine - Clinical Terms), 
+Unified Medical Language System (UMLS), and Gene Ontology 
+(GO) are popular ontologies from the healthcare domain. 
+SNOMED CT and UMLS capture concepts such as disease, 
+symptoms and diagnosis and GO is the world's largest source of 
+information on the functions of genes. Healthcare has been 
+dealing with an explosion in information about different types 
+of drugs, diseases, and procedures. This paper argues that using 
+Knowledge Graphs is not the best solution for solving problems 
+in this domain. We present experiments using LLMs for the 
+healthcare domain to demonstrate that language models 
+provide the same functionality as knowledge graphs, thereby 
+making knowledge graphs redundant. 
+Keywords—Medical 
+data, 
+Language 
+Models, 
+Natural 
+Language Processing, Knowledge Graphs, Deep Learning 
+I. INTRODUCTION 
+Knowledge graphs (KG) are knowledge bases that capture 
+concepts and their relationships for a specific domain using a 
+graph-structured data model. Systematized Nomenclature of 
+Medicine – Clinical Terms (SNOMED CT) (SNOMED), 
+Unified Medical Language Systems(UMLS) [Bodenreider O. 
+2004], etc., are some of the popular KG in the healthcare 
+domain. Fig. 1 shows a sample from a representative medical 
+entity, KG. On the other hand, a language model is a 
+probability distribution over a word sequence and is the 
+backbone of modern natural language processing (NLP). 
+Language models try to capture any language's linguistic 
+intuition and writing, and large language models like BERT 
+[Devlin et al., 2019] and GPT-2 [Radford et al., 2019] have 
+shown remarkable performance. The paper presents a study 
+demonstrating that language models' ability to learn 
+relationships among different entities makes knowledge 
+graphs redundant for many applications.   
+ 
+    This paper uses similar terms from SNOMED-CT KG and 
+passes them through a language model for the healthcare 
+domain BioRedditBERT to get a 768-dimensional dense 
+vector representation. The paper presents the results for 
+analyzing these embeddings. The experiments presented in 
+the paper validate that similar terms cluster together. The 
+paper uses simple heuristics to assign names to clusters. The 
+results show that the cluster names match the names in the 
+KG. Finally, the experiments demonstrate that the cosine 
+similarity of vector representation of similar terms is high and 
+vice versa. 
+ 
+    Our contributions include: (i) We propose a study to 
+demonstrate the value and application of Large Language 
+Models (LLMs) in comparison to Knowledge Graph-based 
+approaches for the task of synonym extraction. (ii) We 
+extensively evaluate our approach on a standard, widely 
+accepted dataset, and the results are encouraging. 
+ 
+ 
+                   Fig 1. Medical entity Knowledge Graph Representation 
+    The rest of the paper is organized as follows: Section II 
+presents the background required to understand the work 
+presented in this paper. Section III presents a literature survey 
+of related work on knowledge graphs and language models. 
+Section IV presents our understanding of how current days 
+language models are making knowledge graphs redundant. 
+Section V describes our proposed approach. Section VI 
+describes the experiments conducted and the results obtained. 
+Finally, section VII summarizes our work and discusses 
+possible directions for future study. 
+II. BACKGROUND 
+This section defines and describes Language Models and 
+Knowledge Graphs as used in this paper: 
+ 
+
+Medicine
+Fever
+Allergy
+Dolo
+ClaritinA. Language Models 
+ 
+A Language Model predicts the probability of a sequence of 
+words in a human language such as English. In the equation 
+below P(w1,…wm) is the probability of the word sequence 
+S, where S = (w1, w2, …, wm) and wi is the ith word in the 
+sequence. 
+ 
+ 
+ 
+    Large Language Models (LLMs) are language models 
+trained on large general corpora that learn associations and 
+relationships 
+among 
+different 
+word 
+entities 
+in 
+an 
+unsupervised manner. Large Language Models (LLMs) are 
+considered universal language learners. LLMs such as BERT 
+and GPTare deep neural networks based on transformer 
+architecture. One of many reasons for the immense popularity 
+of LLMs is that these models are pre-trained self-supervised 
+models and can be adapted or fine-tuned to cater to a wide 
+range of NLP tasks. Few-shot learning has enabled these 
+LLMs to be adapted to a given NLP task using fewer training 
+samples.  
+   
+    Another reason for the immense popularity of LLMs is that 
+a single language model is applicable for multiple 
+downstream applications such as Token classification, Text 
+classification, and Question answering. LLMs generate 
+embeddings or word vectors for words, and these embeddings 
+capture the context of the word in the corpus. This ability of 
+LLMs to generate embeddings based on the corpus makes 
+them ubiquitous in almost NLP tasks. 
+ 
+    In this paper, we use BioRedditBERT [Basaldella et al., 
+2020], a variant of BERT trained for the healthcare domain. 
+It is a domain-specific language representation model trained 
+on large-scale biomedical corpora from Reddit. 
+ 
+B. Knowledge Graphs 
+ 
+Knowledge Graphs (KGs) organize data and capture 
+relationships between different entities for a domain. Domain 
+experts create KGs to map domain-based relations between 
+various entities. 
+ 
+    Knowledge graphs are Graph data structures with nodes 
+and edges. Nodes or vertices represent entities of interest, and 
+edges represent relations between them, as shown in Fig 1. 
+KGs can map and model direct and latent relationships 
+between entities of interest. Typically, KGs are used to model 
+and map information from model sources. Once KGs are 
+designed, typically, NLP is used to populate & create the 
+knowledge base from unstructured text corpora. 
+ 
+    Knowledge graphs play a crucial role in healthcare 
+knowledge representation. There are many widely used 
+knowledge graphs like SNOMED and UMLS etc. In 
+healthcare, KGs are used for drug discovery drugs, 
+identifying tertiary symptoms for diseases and augmented 
+decision-making, etc.   
+ 
+    COMETA: A Corpus for Medical Entity Linking in social 
+media [Basaldella et al., 2020] – a corpus containing four 
+years of content in 68 health-themed subreddits and 
+annotating the most frequent with their corresponding 
+SNOMED-CT entities. In this paper, we have used COMETA 
+to obtain synonyms from SNOMED-CT. 
+III. RELATED WORK 
+In 2019, Jawahar et al. performed experiments to understand 
+the underlying language structure learned by a language 
+model like BERT [Ganesh Jawahar et al. 2019]. The authors 
+show that BERT captures the semantic information from the 
+language hierarchically through experiments. BERT captures 
+surface features in the bottom layer, syntactic elements in the 
+middle and semantic features in the top layer. The work 
+presented in this paper treats the BERT model as a black box 
+and demonstrates that BERT can learn the information in a 
+knowledge graph through experiments on real-life healthcare 
+use cases. 
+ 
+    There have been studies to generate a knowledge graph 
+directly from the output of LLMs. [Wang C et al., 2020; 
+Wang X et al. 2022] proposes a mechanism to create a KG 
+directly from LLMs. This mechanism talks about a two-step 
+mechanism to generate a KG from LLM. In the first step, 
+different candidate triplets are created from the text corpus. 
+Attention weights from a pre-trained LLM are used to get the 
+best-matched candidate triplets and then validated through a 
+beam search. In the second stage, the matched candidate 
+triplets are mapped to a pre-defined KG for validation, and 
+the unmatched candidates are used to create an open 
+knowledge graph. The work demonstrates the feasibility of 
+the idea presented in this paper that LLM can be used as a 
+substitute for knowledge graphs, especially since they 
+contain the information in the KG. 
+ 
+    There is a body of research on integrating Knowledge 
+graphs and LLMs. Structured knowledge from Knowledge 
+Graphs is effectively integrated into Language models to 
+enhance the pre-trained language models [Lei He et al., 
+2021]. However, these approaches have found limited 
+success, thereby strengthening the position in this paper that 
+LLMs contain information from KGs. 
+IV. LANGUAGE MODELS FOR KNOWLEDGE GRAPHS  
+Language Models can find associations between different 
+words based on the attention weight matrix. The 
+methodology to use attention weights as a measure of 
+relationship among the entities indicates that Knowledge 
+graphs are getting replaced by LLMs as they learn more 
+generic relationships in an unsupervised way. The proposed 
+methodology in this paper is built on this idea to demonstrate 
+that Knowledge graphs are increasingly getting redundant for 
+many NLP tasks. 
+V. PROPOSED APPROACH 
+The paper demonstrates that language models' ability to learn 
+relationships among different entities makes knowledge 
+graphs redundant for many applications. To illustrate this, we 
+have used word embeddings for all the synonyms of a set of 
+medical terms from a large language model. This work uses 
+
+m
+P(w1,..., Wm) =|[P(wi I Wi,..., Wi-1)
+i=1COMETA data to obtain synonyms for a set of medical terms. 
+In COMETA data, the work focuses on the following 
+columns: a) Example column, which contains the sentences 
+from health-themed forums on Reddit, b) Term column 
+contains the medical terms present in the Example column, c) 
+General SNOMED Label column; contains the literal 
+meaning of the Term column from the SNOMED Knowledge 
+Graphs. To obtain synonyms, we use the different values 
+from the Terms column for a specific value of the General 
+SNOMED Label column. For example, for Abdominal Wind 
+Pain General SNOMED label, we have the following three 
+synonyms that we can obtain from the Terms column: gas 
+pains, painful gas, and gas pain. 
+ 
+    To calculate the word embeddings of every synonym term, 
+we 
+use 
+the 
+word_vector 
+function 
+from 
+the 
+biobert_embeddings python module [Jitendra Jangid, 2020]. 
+Since the original code was incompatible with the current 
+version of Pytorch [Paszke, A. et al., 2019] and Huggingface 
+[Wolf et al., 2020], we modified it just enough to satisfy the 
+current version requirements – the core logic remains the 
+same. We tokenize every Term using HuggingFace 
+tokenizers 
+and 
+pass 
+the 
+tokenized 
+Term 
+through 
+BioRedditBERT model. The previous step gives us 
+embedding for the Term (or sub-terms if the model didn't see 
+the Term before). If the model has not seen the Term before, 
+then we sum up the embedding of all the subterms). We then 
+store all the embeddings for the next steps. 
+ 
+    We perform the following two experiments after 
+generating the word embeddings for the synonyms of a set of 
+medical terms. In the first experiment, we cluster the word 
+embeddings for the synonyms of a set of medical terms and 
+assign names to clusters. The word embeddings are passed 
+into UMAP to generate a 2-dimensional representation. We 
+plot the 2-dimensional representation to examine how the 
+term cluster visually. UMAP is used as the dimensionality 
+reduction technique over PCA because it is a non-linear 
+dimensionality reduction technique and does very well to 
+preserve the local and global structure of the data as 
+compared to PCA. However, unlike PCA [Karl  Pearson 
+F.R.S. , 1901], UMAP is very sensitive to hyperparameters 
+that we chose, so we visualize the embeddings for several 
+values of number of neighbours (n_neighbors) and minimum 
+distance (min_dist). This step will help us visually validate 
+that a fine-tuned LLM indeed groups together similar terms 
+while ensuring different terms are further apart.  
+ 
+    After identifying clusters from the above step, we use 
+Humans in the Loop approach to identify all terms that belong 
+together and run KMeans Clustering Algorithm [Lloyd, 
+Stuart P., 1982] on them. We identify the term closest to the 
+cluster's centroid, which becomes the Parent Node – one of 
+the core uses of Knowledge Graphs. 
+ 
+    In the second experiment, we analyze the similarity 
+between the word embeddings of the synonyms of the set of 
+medical terms. In this step, we compute the cosine similarity 
+between all the word embeddings and then we examine the 
+similarity to demonstrate that the synonyms for the same term 
+are similar with a small cosine distance between them. 
+VI. EXPERIMENTS AND RESULTS 
+We  use Term and General SNOMED Label columns from 
+COMETA dataset for our experiments. To calculate the 
+embeddings of every term, we use word_vector function from 
+biobert_embeddings package [Jitendra Jangid, 2020]. Since 
+the original code was incompatible with current version of 
+Pytorch [Paszke, A. et al., 2019] and Huggingface [Wolf et al., 
+2020], we modified it just enough to satisfy the current version 
+requirements – the core logic remains the same. 
+    To test the rich representation of language models for our 
+use case, we perform 2 experiments, (1) Cluster the word 
+embeddings for the synonyms of a set of medical terms and 
+assign names to clusters (2) Analyze the similarity between 
+the word embeddings of the synonyms of the set of medical 
+terms. 
+    For the reasons discussed in Sec. III, we use UMAP as our 
+choice of dimensionality reduction. For experiment (1), Fig. 2 
+shows that entities having similar nature are grouped together 
+and dissimilar entities are further apart which proves utility of 
+a Fine-tuned Language Models. 
+ 
+Fig 2. Clusters resulting from UMAP dimensionality reduction 
+    Next we perform KMeans clustering on mentions 
+belonging to same group using cosine similarity. The centroid 
+of each clusters were then used to identify concepts by finding 
+terms that were closest to the centers by cosine similarity. We 
+found the following terms for the concepts visible in Table. 1. 
+ 
+Concept (General SNOMED 
+Label) 
+Term (closest to the cluster) 
+Oral contraception 
+hormonal BC pills 
+Crohn's disease 
+crohns disease 
+Diabetes mellitus type 2 
+T2 diabetes 
+Analgesic 
+Pain Medication 
+Diabetes mellitus type 1 
+T1 diabetic 
+Autoimmune disease 
+autoimmune disease 
+Hypoglycemia 
+low blood sugars 
+Headache 
+head pain 
+Tachycardia 
+heart racing 
+
+10
+general_snomed_label
+Oral contraception
+Crohn's disease
+Diabetes mellitus type 2
+Analgesic
+Diabetes mellitus type 1
+Autoimmunedisease
+5
+Hypoglycemia
+E
+Headache
+Tachycardia
+Tired
+4
+Itching
+0
+2
+4
+6
+8
+10
+dim_0Tired 
+feel tired 
+Itching 
+itching 
+Table 1. Terms closest to the cluster center of each Concept 
+    While Fig. 2 illustrates global and local structure among 
+different mentions of a concept, as a part of experiment (2), 
+we also analyze distribution of similarity scores (which are 
+calculated by using cosine similarity) to visualize distribution 
+of cosine similarity among terms belonging to same concept 
+(Fig. 3 and 4) and terms belonging to different concepts (Fig. 
+5). We can see that distribution of mentions belonging to same 
+concept are closer to each other on average as compared to 
+mentions from different concepts. This point again validates 
+the utility of Language Model in finding different mentions of 
+a concept in multiple documents.  
+ 
+Fig 3. Cosine similarity between mentions from Oral Contraception 
+ 
+Fig 4. Cosine similarity between mentions from Cron’s disease 
+ 
+In addition to these plots, we also analyze similarity 
+between unrelated terms, and we see the following trend –  
+ 
+Fig 5. Cosine similarity between mentions from different concepts 
+VII. CONCLUSION AND FUTURE WORK 
+In this paper we have empirically shown how Language 
+Models fine-tuned on domain specific data can be used to 
+replace Knowledge Graphs for tasks where identifying 
+synonyms is involved.  
+     
+    Language Models do a very good job in calculating 
+embeddings which contains semantic information about 
+terms that can be used to identify if two terms are close to 
+each other or not. This information is used in this paper to 
+identify terms which are closer to each other, and which are 
+not. Once groups of similar terms have been identifying using 
+non-linear dimensionality techniques, using Humans in the 
+Loop approach we can annotate such groups.  After 
+annotating the groups, we use KMeans to identify centroids 
+of each cluster which are then used the identify terms with 
+the closest cosine distance from them. These terms can then 
+be used as parent nodes for their respective clusters. The 
+primary way in which our algorithm improves over current 
+Knowledge Graph based approaches is that unlike KGs which 
+are created by subject matter experts, our algorithm doesn’t 
+require subject matter experts for annotation.  
+ 
+    Our current algorithm handles synonym mapping quite 
+well, but it requires human intervention and for next steps, we 
+would be exploring ways in which we can extract Knowledge 
+Graphs from Language Models themselves. This would be 
+required to remove the human intervention in the current 
+process and handling cases where hypernyms are involved. 
+REFERENCES 
+ 
+[1] Bodenreider O. 2004. The Unified Medical Language System (UMLS): 
+integrating biomedical terminology. Nucleic Acids Res. 2004 Jan 
+1;32(Database issue):D267-70. 
+[2] SNOMED.   URL: http://www.snomed.org/ 
+[3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 
+2019. BERT: Pre-training of Deep Bidirectional Transformers for 
+Language Understanding. In Proceedings of the 2019 Conference of 
+the North American Chapter of the Association for Computational 
+Linguistics: Human Language Technologies, Volume 1 (Long and 
+Short Papers), pages 4171–4186, Minneapolis, Minnesota. Association 
+for Computational Linguistics. 
+[4] Tokenizer. URL:https://huggingface.co/docs/tokenizers 
+[5] Marco Basaldella, Fangyu Liu, Ehsan Shareghi, and Nigel Collier. 
+2020. COMETA: A Corpus for Medical Entity Linking in the Social 
+Media. In Proceedings of the 2020 Conference on Empirical Methods 
+in Natural Language Processing (EMNLP), pp.3122–3137, Online. 
+Association for Computational Linguistics. 
+ 
+[6] McInnes, Leland and Healy, John and Saul, Nathaniel and Grossberger, 
+Lukas. 2018. UMAP: Uniform Manifold Approximation and 
+Projection. The Journal of Open Source Software. arXiv:1802.03426v3 
+[7] Karl  Pearson  F.R.S. , 1901. LIII. On lines and planes of closest fit to 
+systems of points in space. The London, Edinburgh, and Dublin 
+Philosophical Magazine and Journal of Science, 2(11), pp.559–572. 
+[8] Lloyd, Stuart P., 1982. Least squares quantization in PCM. Information 
+Theory, IEEE Transactions on 28.2, pp.129-137 
+[9] Jitendra 
+Jangid, 
+2020. 
+https://github.com/Overfitter/biobert_embedding 
+[10] Wolf et al., 2020. Transformers: State-of-the-Art Natural Language 
+Processing. EMNLP 
+[11] Paszke, A. et al., 2019. PyTorch: An Imperative Style, High-
+Performance Deep Learning Library. In Advances in Neural 
+Information Processing Systems 32. Curran Associates, Inc., pp. 8024–
+8035. Available at: http://papers.neurips.cc/paper/9015-pytorch-an-
+imperative-style-high-performance-deep-learning-library.pdf. 
+
+Oralcontraception
+Alesse
+BCP
+Cyclen
+0.95
+Lolo
+OC, s
+0.9 
+Qlaira
+uirth control pills
+0.85
+contraceptive pills
+hormonal rirth control pills
+0.8
+honone pill
+0.75
+triphasic pills
+contro
+aceptlvepIlls
+onal
+mone
+haslc
+blrth control pWl:Crohn's disease
+CD
+Chrohns
+Crohn
+Crohn ' s
+Crohn ' s flare
+0.95
+Crohn disease
+Crohn et *$
+Crohn at' " s disease
+0.9
+Crohnie
+Crohnies
+crohns
+crohns colitis
+0.85
+crohns disease
+Crohn ' s disease
+CrohnsDisease
+0.8
+crohns flare
+3
+olitlis
+diseas eDissimilarityMatrix
+hormonal μC pills
+crohns disease
+T2 diabetes
+0.95
+Pain Medicatior
+T1 diabetic
+0.9
+autoimmune disease
+low blood sugars
+head pair
+0.85
+heart racing
+feel tired
+0.8
+itching[12] Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., & Sutskever, I. 
+(2019). Language Models are Unsupervised Multitask Learners. 
+[13] Ganesh Jawahar etal, What does BERT learn about the structure of 
+language? ;Proceedings of the 57th Annual Meeting of the Association 
+for Computational Linguistics, pages 3651–3657, 2019 
+[14] Wang, C., Liu, X., & Song, D.X. (2020). Language Models are Open 
+Knowledge Graphs. ArXiv, abs/2010.11967. 
+[15] Wang, X., He, Q., Liang, J., & Xiao, Y. (2022). Language Models as 
+Knowledge Embeddings. Proceedings of the Thirty-First International 
+Joint Conference on Artificial Intelligence (IJCAI-22) 
+[16] Lei He, Suncong Zheng, Tao Yang, and Feng Zhang. 2021. KLMo: 
+Knowledge Graph Enhanced Pretrained Language Model with Fine-
+Grained 
+Relationships. 
+In Findings 
+of 
+the 
+Association 
+for 
+Computational Linguistics: EMNLP 2021, pages 4536–4542, Punta 
+Cana, 
+Dominican 
+Republic. 
+Association 
+for 
+Computational 
+Linguistics. 
+ 
+ 
+ 
+ 
+

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+page_content='com  Atul Singh  Optum, India  atul_singh18@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='com  Prakhar Mishra  Optum, India  prakhar_mishra29@optum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='com  Rajesh Sabapathy  Optum,India  rajesh_sabapathy@uhc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='com  Abstract—Healthcare  domain  generates  a  lot  of  unstructured and semi-structured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Natural Language  processing (NLP) has been used extensively to process this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Deep Learning based NLP especially Large Language Models  (LLMs) such as BERT have found broad acceptance and are  used extensively for many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' A Language Model is a  probability distribution over a word sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Self-supervised  Learning on a large corpus of data automatically generates deep  learning-based language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' BioBERT and Med-BERT are  language models pre-trained for the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Healthcare uses typical NLP tasks such as question answering,  information extraction, named entity recognition, and search to  simplify and improve processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' However, to ensure robust  application of the results, NLP practitioners need to normalize  and standardize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' One of the main ways of achieving  normalization and standardization is the use of Knowledge  Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' A Knowledge Graph captures concepts and their  relationships for a specific domain, but their creation is time-  consuming and requires manual intervention from domain  experts,  which  can  prove  expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  SNOMED  CT  (Systematized Nomenclature of Medicine - Clinical Terms),  Unified Medical Language System (UMLS), and Gene Ontology  (GO) are popular ontologies from the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content="  SNOMED CT and UMLS capture concepts such as disease,  symptoms and diagnosis and GO is the world's largest source of  information on the functions of genes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Healthcare has been  dealing with an explosion in information about different types  of drugs, diseases, and procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This paper argues that using  Knowledge Graphs is not the best solution for solving problems  in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We present experiments using LLMs for the  healthcare domain to demonstrate that language models  provide the same functionality as knowledge graphs, thereby  making knowledge graphs redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Keywords—Medical  data,  Language  Models,  Natural  Language Processing, Knowledge Graphs, Deep Learning  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' INTRODUCTION  Knowledge graphs (KG) are knowledge bases that capture  concepts and their relationships for a specific domain using a  graph-structured data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Systematized Nomenclature of  Medicine – Clinical Terms (SNOMED CT) (SNOMED),  Unified Medical Language Systems(UMLS) [Bodenreider O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  2004], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', are some of the popular KG in the healthcare  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 1 shows a sample from a representative medical  entity, KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' On the other hand, a language model is a  probability distribution over a word sequence and is the  backbone of modern natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content="  Language models try to capture any language's linguistic  intuition and writing, and large language models like BERT  [Devlin et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2019] and GPT-2 [Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2019] have  shown remarkable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=" The paper presents a study  demonstrating that language models' ability to learn  relationships among different entities makes knowledge  graphs redundant for many applications." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  This paper uses similar terms from SNOMED-CT KG and  passes them through a language model for the healthcare  domain BioRedditBERT to get a 768-dimensional dense  vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The paper presents the results for  analyzing these embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The experiments presented in  the paper validate that similar terms cluster together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The  paper uses simple heuristics to assign names to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The  results show that the cluster names match the names in the  KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Finally, the experiments demonstrate that the cosine  similarity of vector representation of similar terms is high and  vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Our contributions include: (i) We propose a study to  demonstrate the value and application of Large Language  Models (LLMs) in comparison to Knowledge Graph-based  approaches for the task of synonym extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' (ii) We  extensively evaluate our approach on a standard, widely  accepted dataset, and the results are encouraging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Medical entity Knowledge Graph Representation  The rest of the paper is organized as follows: Section II  presents the background required to understand the work  presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Section III presents a literature survey  of related work on knowledge graphs and language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Section IV presents our understanding of how current days  language models are making knowledge graphs redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Section V describes our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Section VI  describes the experiments conducted and the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Finally, section VII summarizes our work and discusses  possible directions for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' BACKGROUND  This section defines and describes Language Models and  Knowledge Graphs as used in this paper:  Medicine  Fever  Allergy  Dolo  ClaritinA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Language Models  A Language Model predicts the probability of a sequence of  words in a human language such as English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In the equation  below P(w1,…wm) is the probability of the word sequence  S, where S = (w1, w2, …, wm) and wi is the ith word in the  sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Large Language Models (LLMs) are language models  trained on large general corpora that learn associations and  relationships  among  different  word  entities  in  an  unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Large Language Models (LLMs) are  considered universal language learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' LLMs such as BERT  and GPTare deep neural networks based on transformer  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' One of many reasons for the immense popularity  of LLMs is that these models are pre-trained self-supervised  models and can be adapted or fine-tuned to cater to a wide  range of NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Few-shot learning has enabled these  LLMs to be adapted to a given NLP task using fewer training  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Another reason for the immense popularity of LLMs is that  a single language model is applicable for multiple  downstream applications such as Token classification, Text  classification, and Question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' LLMs generate  embeddings or word vectors for words, and these embeddings  capture the context of the word in the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This ability of  LLMs to generate embeddings based on the corpus makes  them ubiquitous in almost NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  In this paper, we use BioRedditBERT [Basaldella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=',  2020], a variant of BERT trained for the healthcare domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  It is a domain-specific language representation model trained  on large-scale biomedical corpora from Reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Knowledge Graphs  Knowledge Graphs (KGs) organize data and capture  relationships between different entities for a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Domain  experts create KGs to map domain-based relations between  various entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Knowledge graphs are Graph data structures with nodes  and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Nodes or vertices represent entities of interest, and  edges represent relations between them, as shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  KGs can map and model direct and latent relationships  between entities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Typically, KGs are used to model  and map information from model sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Once KGs are  designed, typically, NLP is used to populate & create the  knowledge base from unstructured text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Knowledge graphs play a crucial role in healthcare  knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' There are many widely used  knowledge graphs like SNOMED and UMLS etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In  healthcare, KGs are used for drug discovery drugs,  identifying tertiary symptoms for diseases and augmented  decision-making, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  COMETA: A Corpus for Medical Entity Linking in social  media [Basaldella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2020] – a corpus containing four  years of content in 68 health-themed subreddits and  annotating the most frequent with their corresponding  SNOMED-CT entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In this paper, we have used COMETA  to obtain synonyms from SNOMED-CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' RELATED WORK  In 2019, Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' performed experiments to understand  the underlying language structure learned by a language  model like BERT [Ganesh Jawahar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The authors  show that BERT captures the semantic information from the  language hierarchically through experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' BERT captures  surface features in the bottom layer, syntactic elements in the  middle and semantic features in the top layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The work  presented in this paper treats the BERT model as a black box  and demonstrates that BERT can learn the information in a  knowledge graph through experiments on real-life healthcare  use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  There have been studies to generate a knowledge graph  directly from the output of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' [Wang C et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Wang X et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2022] proposes a mechanism to create a KG  directly from LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This mechanism talks about a two-step  mechanism to generate a KG from LLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In the first step,  different candidate triplets are created from the text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Attention weights from a pre-trained LLM are used to get the  best-matched candidate triplets and then validated through a  beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In the second stage, the matched candidate  triplets are mapped to a pre-defined KG for validation, and  the unmatched candidates are used to create an open  knowledge graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The work demonstrates the feasibility of  the idea presented in this paper that LLM can be used as a  substitute for knowledge graphs, especially since they  contain the information in the KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  There is a body of research on integrating Knowledge  graphs and LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Structured knowledge from Knowledge  Graphs is effectively integrated into Language models to  enhance the pre-trained language models [Lei He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=',  2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' However, these approaches have found limited  success, thereby strengthening the position in this paper that  LLMs contain information from KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' LANGUAGE MODELS FOR KNOWLEDGE GRAPHS  Language Models can find associations between different  words based on the attention weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The  methodology to use attention weights as a measure of  relationship among the entities indicates that Knowledge  graphs are getting replaced by LLMs as they learn more  generic relationships in an unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The proposed  methodology in this paper is built on this idea to demonstrate  that Knowledge graphs are increasingly getting redundant for  many NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=" PROPOSED APPROACH  The paper demonstrates that language models' ability to learn  relationships among different entities makes knowledge  graphs redundant for many applications." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' To illustrate this, we  have used word embeddings for all the synonyms of a set of  medical terms from a large language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This work uses  m  P(w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', Wm) =|[P(wi I Wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', Wi-1)  i=1COMETA data to obtain synonyms for a set of medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  In COMETA data, the work focuses on the following  columns: a) Example column, which contains the sentences  from health-themed forums on Reddit, b) Term column  contains the medical terms present in the Example column, c)  General SNOMED Label column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' contains the literal  meaning of the Term column from the SNOMED Knowledge  Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' To obtain synonyms, we use the different values  from the Terms column for a specific value of the General  SNOMED Label column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' For example, for Abdominal Wind  Pain General SNOMED label, we have the following three  synonyms that we can obtain from the Terms column: gas  pains, painful gas, and gas pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  To calculate the word embeddings of every synonym term,  we  use  the  word_vector  function  from  the  biobert_embeddings python module [Jitendra Jangid, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Since the original code was incompatible with the current  version of Pytorch [Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2019] and Huggingface  [Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2020], we modified it just enough to satisfy the  current version requirements – the core logic remains the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We tokenize every Term using HuggingFace  tokenizers  and  pass  the  tokenized  Term  through  BioRedditBERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=" The previous step gives us  embedding for the Term (or sub-terms if the model didn't see  the Term before)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' If the model has not seen the Term before,  then we sum up the embedding of all the subterms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We then  store all the embeddings for the next steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  We perform the following two experiments after  generating the word embeddings for the synonyms of a set of  medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In the first experiment, we cluster the word  embeddings for the synonyms of a set of medical terms and  assign names to clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The word embeddings are passed  into UMAP to generate a 2-dimensional representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We  plot the 2-dimensional representation to examine how the  term cluster visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' UMAP is used as the dimensionality  reduction technique over PCA because it is a non-linear  dimensionality reduction technique and does very well to  preserve the local and global structure of the data as  compared to PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' However, unlike PCA [Karl  Pearson  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' , 1901], UMAP is very sensitive to hyperparameters  that we chose, so we visualize the embeddings for several  values of number of neighbours (n_neighbors) and minimum  distance (min_dist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This step will help us visually validate  that a fine-tuned LLM indeed groups together similar terms  while ensuring different terms are further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  After identifying clusters from the above step, we use  Humans in the Loop approach to identify all terms that belong  together and run KMeans Clustering Algorithm [Lloyd,  Stuart P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 1982] on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=" We identify the term closest to the  cluster's centroid, which becomes the Parent Node – one of  the core uses of Knowledge Graphs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  In the second experiment, we analyze the similarity  between the word embeddings of the synonyms of the set of  medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In this step, we compute the cosine similarity  between all the word embeddings and then we examine the  similarity to demonstrate that the synonyms for the same term  are similar with a small cosine distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' EXPERIMENTS AND RESULTS  We  use Term and General SNOMED Label columns from  COMETA dataset for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' To calculate the  embeddings of every term, we use word_vector function from  biobert_embeddings package [Jitendra Jangid, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Since  the original code was incompatible with current version of  Pytorch [Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', 2019] and Huggingface [Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=',  2020], we modified it just enough to satisfy the current version  requirements – the core logic remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  To test the rich representation of language models for our  use case, we perform 2 experiments, (1) Cluster the word  embeddings for the synonyms of a set of medical terms and  assign names to clusters (2) Analyze the similarity between  the word embeddings of the synonyms of the set of medical  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  For the reasons discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' III, we use UMAP as our  choice of dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' For experiment (1), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2  shows that entities having similar nature are grouped together  and dissimilar entities are further apart which proves utility of  a Fine-tuned Language Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Clusters resulting from UMAP dimensionality reduction  Next we perform KMeans clustering on mentions  belonging to same group using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The centroid  of each clusters were then used to identify concepts by finding  terms that were closest to the centers by cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We  found the following terms for the concepts visible in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Concept (General SNOMED  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Label)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Term (closest to the cluster)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Oral contraception  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='hormonal BC pills  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content="Crohn's disease  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='crohns disease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Diabetes mellitus type 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='T2 diabetes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Analgesic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Pain Medication  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Diabetes mellitus type 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='T1 diabetic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Autoimmune disease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='autoimmune disease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Hypoglycemia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='low blood sugars  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Headache  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='head pain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Tachycardia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='heart racing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='general_snomed_label  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Oral contraception  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content="Crohn's disease  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Diabetes mellitus type 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Analgesic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Diabetes mellitus type 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Autoimmunedisease  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Hypoglycemia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Headache  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Tachycardia  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Tired  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Itching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='dim_0Tired  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='feel tired  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Itching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='itching  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Terms closest to the cluster center of each Concept  While Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2 illustrates global and local structure among  different mentions of a concept, as a part of experiment (2),  we also analyze distribution of similarity scores (which are  calculated by using cosine similarity) to visualize distribution  of cosine similarity among terms belonging to same concept  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 3 and 4) and terms belonging to different concepts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' We can see that distribution of mentions belonging to same  concept are closer to each other on average as compared to  mentions from different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This point again validates  the utility of Language Model in finding different mentions of  a concept in multiple documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Cosine similarity between mentions from Oral Contraception  Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Cosine similarity between mentions from Cron’s disease  In addition to these plots, we also analyze similarity  between unrelated terms, and we see the following trend –  Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Cosine similarity between mentions from different concepts  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this paper we have empirically shown how Language  Models fine-tuned on domain specific data can be used to  replace Knowledge Graphs for tasks where identifying  synonyms is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Language Models do a very good job in calculating  embeddings which contains semantic information about  terms that can be used to identify if two terms are close to  each other or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This information is used in this paper to  identify terms which are closer to each other, and which are  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Once groups of similar terms have been identifying using  non-linear dimensionality techniques, using Humans in the  Loop approach we can annotate such groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' After  annotating the groups, we use KMeans to identify centroids  of each cluster which are then used the identify terms with  the closest cosine distance from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' These terms can then  be used as parent nodes for their respective clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The  primary way in which our algorithm improves over current  Knowledge Graph based approaches is that unlike KGs which  are created by subject matter experts, our algorithm doesn’t  require subject matter experts for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Our current algorithm handles synonym mapping quite  well, but it requires human intervention and for next steps, we  would be exploring ways in which we can extract Knowledge  Graphs from Language Models themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' This would be  required to remove the human intervention in the current  process and handling cases where hypernyms are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  REFERENCES  [1] Bodenreider O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The Unified Medical Language System (UMLS):  integrating biomedical terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Nucleic Acids Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2004 Jan  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='32(Database issue):D267-70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  [2] SNOMED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='snomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='org/  [3] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' BERT: Pre-training of Deep Bidirectional Transformers for  Language Understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference of  the North American Chapter of the Association for Computational  Linguistics: Human Language Technologies, Volume 1 (Long and  Short Papers), pages 4171–4186, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Association  for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  [4] Tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' URL:https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='co/docs/tokenizers  [5] Marco Basaldella, Fangyu Liu, Ehsan Shareghi, and Nigel Collier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' COMETA: A Corpus for Medical Entity Linking in the Social  Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' In Proceedings of the 2020 Conference on Empirical Methods  in Natural Language Processing (EMNLP), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='3122–3137, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  [6] McInnes, Leland and Healy, John and Saul, Nathaniel and Grossberger,  Lukas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' UMAP: Uniform Manifold Approximation and  Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' The Journal of Open Source Software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='03426v3  [7] Karl  Pearson  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' , 1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content=' Transformers: State-of-the-Art Natural Language  Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' EMNLP  [11] Paszke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content=' PyTorch: An Imperative Style, High-  Performance Deep Learning Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content=' Available at: http://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='cc/paper/9015-pytorch-an-  imperative-style-high-performance-deep-learning-library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Oralcontraception  Alesse  BCP  Cyclen  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='95  Lolo  OC, s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='9  Qlaira  uirth control pills  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content='95  Crohn disease  Crohn et *$  Crohn at\' " s disease  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='9  Crohnie  Crohnies  crohns  crohns colitis  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content="85  crohns disease  Crohn ' s disease  CrohnsDisease  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='8  crohns flare  3  olitlis  diseas eDissimilarityMatrix  hormonal μC pills  crohns disease  T2 diabetes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='95  Pain Medicatior  T1 diabetic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='9  autoimmune disease  low blood sugars  head pair  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content='8  itching[12] Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content='  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Language Models are Unsupervised Multitask Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  [13] Ganesh Jawahar etal, What does BERT learn about the structure of  language?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='Proceedings of the 57th Annual Meeting of the Association  for Computational Linguistics, pages 3651–3657, 2019  [14] Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Language Models are Open  Knowledge Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' ArXiv, abs/2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='11967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  [15] Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=', Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Language Models as  Knowledge Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' Proceedings of the Thirty-First International  Joint Conference on Artificial Intelligence (IJCAI-22)  [16] Lei He, Suncong Zheng, Tao Yang, and Feng Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
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+page_content='  In Findings  of  the  Association  for  Computational Linguistics: EMNLP 2021, pages 4536–4542, Punta  Cana,  Dominican  Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}
+page_content='  Association  for  Computational  Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf'}

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+Topological charge quantization on localized imperfections in crystalline insulators
+and the nearsightedness principle of Kohn
+Kiryl Piasotski1, 2, ∗
+1Institut f¨ur Theorie der Kondensierten Materie,
+Karlsruher Institut f¨ur Technologie, 76131 Karlsruhe, Germany
+2Institut f¨ur QuantenMaterialien und Technologien,
+Karlsruher Institut f¨ur Technologie, 76021 Karlsruhe, Germany†
+(Dated: January 10, 2023)
+We study the quantization of the excess charge on N localized (ultra-screened) impurities in d-
+dimensional crystalline insulating systems. Solving Dyson’s equation, we demonstrate that such
+charges are topological, by expressing them as winding numbers of appropriate functionals of bulk
+position space Green’s functions. We discuss the ties of our topological invariant with the nearsight-
+edness principle of W. Kohn, stating that the electronic charge density at fixed chemical potential
+depends on the external field only locally, meaning that localized perturbations by external fields
+may only result in localized charge redistributions. We arrive at the same conclusion by demonstrat-
+ing that an adiabatic perturbation comprised of a variation of impurities’ positions and/or strengths
+may only result in the change in the occupancy of impurity-localized bound states sitting, energy-
+wise, close to the Fermi level. Finally, we conclude by discussing the relations of the nearsightedness
+principle with the topological invariants characterizing the boundary charge.
+I.
+INTRODUCTION
+With the discovery of the quantum Hall effect [1] and
+its topological origins [2, 3] the study of the topological
+structures in condensed matter systems became a style
+rather than a fashion. The possibly best-known contem-
+porary example of that is the field of topological insu-
+lators, where the boundary-localized edge states, with
+topologically granted existence and robustness, are be-
+ing studied (popular reviews are [4, 5]). Despite being a
+well-defined endeavor with its very own periodic table [6],
+this discipline leaves a number of questions open. One
+of these regards the direct experimental accessibility of
+these topological surface states. In particular, aside from
+the subspace of such states, their basis is incomplete for a
+description of the physical system accommodating them
+– a topological insulator, making it highly questionable
+whether an actual physical observable may be expanded
+into a basis of intra- and inter-surface state transition
+operators. This is, for example, not true of the excess
+charge density that these insulators accumulate at their
+boundaries as it, as such, also features the exponentially
+localized contributions of all of the occupied extended
+states. Despite that, it is clear that as the surface states
+also contribute to such an observable, the change in their
+occupancy has to have an observable effect.
+In a series of recent works [7, 8, 9, 10], the topologi-
+cal properties of the boundary-localized electronic excess
+charges (the boundary charges) in unidimensional crys-
+tals were examined. In particular, a pair of topological
+invariants characterizing the boundary charge upon two
+bulk energy spectrum-preserving transformations of crys-
+tal’s potential, translations and local inversions, were de-
+vised. Specifically, it was demonstrated that upon local
+inversion (inversion of coordinates within the unit cell),
+the boundary charge maps to its negative, up to an inte-
+gral topological quantum number known as the interface
+invariant. Likewise, upon the lattice translation by xϕ,
+the boundary charge was shown to grow linearly with
+the shift variable xϕ (with the slope being the unit cell-
+averaged average charge density in the bulk ¯ρ = ν
+L, with
+ν – filling factor and L-system’s period), whilst perform-
+ing discontinuous downward jumps by a unit of the elec-
+tron charge, as quantified by another topological quan-
+tum number – the boundary invariant. These topologi-
+cal invariants were shown to be generated by the spectral
+flow of the energies corresponding to the edge states in-
+side the energy gap that hosts the chemical potential, in
+complete analogy with the integer quantum Hall effect
+[3]. As opposed to the edge states in topological insu-
+lators, the quantization of these invariants does not rely
+on the internal symmetries of the bulk Bloch’s Hamilto-
+nian (such as particle-hole or time-reversal symmetries)
+and is instead guaranteed by a number of fundamental
+physical principles, such as charge conservation, Pauli
+principle, and the nearsightedness principle of W. Kohn
+[11, 12, 13, 14] (to be discussed further on). Moreover,
+these invariants are directly linked with the properties of
+an experimental observable, a privilege shared by both
+the quantum Hall effect and the topological defects (see
+Ref. [15] for a review), while not being entirely clear in
+the domain of the topological insulators.
+Further, in a different paper [16], rational quantization
+of boundary and interface charges was discussed. Partic-
+ularly, with the aid of the aforementioned physical princi-
+ples, a general framework for studying quantized charges
+in one dimension was laid down, allowing us to quantify
+all possible quantization patterns of the boundary charge
+in terms of the non-symmorphic symmetries of the crys-
+tal. The charges on the interfaces between pairs of in-
+sulators sharing their bulk properties were demonstrated
+to follow a lattice version of the Goldstone-Wilczek for-
+mula [17], relating the interface charge to the sum of the
+arXiv:2301.03305v1  [cond-mat.mes-hall]  9 Jan 2023
+
+2
+boundary charges right and left to their septum, mod-
+ulo an unknown integer generated by the local coupling
+between the two subsystems.
+The key feature of the method developed in Ref. [16]
+is this “modulo an unknown integer” paradigm, arising
+from the nearsightedness principle of the electronic mat-
+ter. As such, the nearsightedness principle tells us that
+(see Ref. [13, 14]), in insulators, localized perturbations
+by external fields may result in localized charge redistri-
+butions only. To be more specific, the corrections beyond
+the characteristic length scale ξg = vF
+Eg (where vF and Eg
+are the Fermi velocity and the gap opening up at the
+Fermi level, see Ref. [18] for example) are exponentially
+suppressed.
+An even further refinement of this state-
+ment would be that such perturbations may only remove
+or add an additional number of bound states whose wave
+functions are localized around the corresponding pertur-
+bations. One of the key purposes of the present paper
+is to substantiate this claim mathematically, which turns
+out to be possible in pretty general d-dimensional models.
+To be more specific, this paper concerns the topolog-
+ical properties of the electronic excess charges accumu-
+lated around point-like defects in d-dimensional insula-
+tors. Although we purposefully specify the Hamiltonian
+of the crystal under consideration to make our exposi-
+tion more transparent, the derivations presented in this
+manuscript are shown to be independent of its choice.
+What indeed matters is that the spectrum of the clean
+system consists of the energy bands occasionally sepa-
+rated by the energy gaps, that is, there exists at least
+one bulk energy gap into which we can put the chemical
+potential to promote the resulting statistical system into
+an insulator.
+Furthermore, neither we specify the internal structure
+of the impurity vertices, nor do we assume any particu-
+lar arrangement of them, making our analysis applicable
+to a wide range of experimental setups. In particular,
+quite conventionally, we may assume that a number of
+randomly located point-like impurities exerting an ultra-
+screened electrostatic force on the system’s electrons are
+scattered through the charge sampling region of a crystal
+under consideration. A slightly less familiar situation is
+inspired by the work of Nomura and Nagaosa [19] and
+may be formulated as follows. Assuming that a crystal is
+further magnetic, we know that, in an insulating regime,
+its ground state may accurately be described by a Heisen-
+berg model that, by itself, features topological defects.
+A familiar example of such a defect would be a magnetic
+skyrmion or a hedgehog texture. Assuming that the total
+spin of atoms comprising our crystal is large, these tex-
+tures may be seen as an arrangement of classical magnetic
+moments nailed down to the atomic positions. Their in-
+teraction with the electron’s spin degree of freedom may
+then be written as a sum of the Zeeman-like terms, each
+weighted with the Dirac δ-function centered at the posi-
+tion of the corresponding atom.
+Quite generically, we show that the total electronic
+excess charge accumulated around these defects is an
+integer-valued topological invariant, which we express
+as a contour integral winding number of an appropriate
+functional of bulk position space Green’s functions. Fur-
+ther analysis of this topological quantum number reveals
+that upon an adiabatic modification of positions and/or
+vertex functions of the localized scattering centers, the
+value of the invariant may only be affected by the change
+in the occupancy of the imperfection-localized bound
+states in the process of the spectral flow of their eigenen-
+ergies inside the chemical potential-accommodating en-
+ergy gap. This observation allows for an immediate in-
+terpretation in terms of the nearsightedness principle,
+as well as for a direct read-off of the central memo of
+Ref.
+[16]: “localized perturbations in insulators result
+in localized charge redistribution, leading to an addi-
+tion/removal of the corresponding perturbation-localized
+bound states to/from the occupied spectral region”. We
+conclude our analysis by commenting on the relation be-
+tween the nearsightedness principle and the topological
+invariants characterizing the boundary charge.
+In what follows, we set the reduced Plank’s constant ̵h
+and the electron charge e equal to unity ̵h = e = 1.
+II.
+ADIABATIC RESPONSE OF THE EXCESS
+CHARGE TO LOCALIZED PERTURBATIONS IN
+AN INSULATING STATE
+A.
+A translationally invariant model
+In the following, we shall specifically refer to an elec-
+tronic system governed by the following Hamiltonian
+H(0)
+x
+= p2
+2m + 1
+2m
+d
+∑
+j=1
+{ ˜Aj(x),pj} + V (x),
+(1)
+with V (x) and ˜Aj(x), j = 1, ..., d being the lattice
+periodic Nc × Nc Hermitian matrices. More specifically,
+{V (x)
+˜A(x)} = {V (x + Rm)
+˜A(x + Rm)},
+∀m ∈ Zd,
+(2)
+where Rm = ∑d
+j=1 mjaj is a lattice vector characterized
+by a d-dimensional vector of integers m = (m1 ⋯ md)
+T ,
+specifying its components in the basis of primitive vec-
+tors {aj}j spanning the unit cell of a Bravais lattice.
+Furthermore, p and x are vectorial momentum and po-
+sition operators comprised of the individual components
+pj = −i ∂
+∂xj and xj.
+This model naturally generalizes the one recently stud-
+ied in Ref. [10] in connection with the universal prop-
+erties of one-dimensional boundary charge, to higher
+dimensions.
+We remark that other models of multi-
+dimensional periodic structures [20] are expected to share
+the same physics, as the effects we are about to describe
+are rather generic to an insulating state.
+Translationally invariant systems are characterized by
+their band structure, comprised of the individual energy
+
+3
+bands dispersing as ϵα,k, α = 1, 2, ..., as a function
+of the vectorial quasimomentum variable k, confined to
+the first Brillouin zone of the reciprocal space.
+The
+eigenstates of the Hamiltonian to which ϵα,k are the
+corresponding eigenvalues are known as Bloch functions
+ψα,k(x), and may be generically expressed as
+ψα,k(x) = eik⋅xuα,k(x),
+(3)
+where uα,k(x) in the Nc-component object and is lattice
+periodic in the same sense as vector and scalar potentials
+are uα,k(x) = uα,k(x+Rm), ∀m ∈ Zd. The completeness
+and identity resolution relations may be written as
+VUC
+(2π)d ∫Rd d(d)xψ†
+α,k(x)ψα′,k′(x) = δα,α′δ(d)(k − k′),
+(4)
+VUC
+(2π)d
+∞
+∑
+α=1∫BZ d(d)kψα,k(x)ψ†
+α,k(x′) = 1Ncδ(d)(x − x′),
+(5)
+where VUC is the volume of the unit cell, defined via
+VUC = ∫UC d(d)x = det(a1∣ ⋯ ∣ad).
+(6)
+When studying charge, it is more convenient to intro-
+duce the retarded single-particle Green’s function, con-
+taining the information on both the eigenstates and the
+energy spectrum.
+In thermodynamic equilibrium, the
+Laplace image of the latter is defined as the resolvent
+of the single-particle Hamiltonian (1)
+[z − H(0)
+x ]G(0)(x,x′) = 1Ncδ(d)(x − x′),
+(7)
+where z is the complex energy variable, defined in terms
+of the physical frequency variable ω as z = ω + iη, where
+η → 0+. Owing to the identity resolution relation (5) we
+can establish the conventional Lehmann representation
+G(0)(x,x′) = VUC
+(2π)d
+∞
+∑
+α=1∫BZ d(d)k
+ψα,k(x)ψ†
+α,k(x′)
+z − ϵα,k
+. (8)
+Further, using the completeness of the basis (4), in Ap-
+pendix A, we establish the following important fusion
+rule for the bare propagators
+∫Rd d(d)x′G(0)(x,x′)G(0)(x′,x′′) = − ∂
+∂ω G(0)(x,x′′).
+(9)
+As it is shown in Appendix A, this relation holds pretty
+generally, without any reference to the Hamiltonian (1).
+B.
+Localized perturbations and Dyson’s equation
+Now we perturb the translationally invariant (on the
+scale of the unit cell) system by a finite number of point-
+like impurities
+˜V (x) =
+N
+∑
+n=1
+˜V (n)
+0
+δ(d)(x − xn),
+(10)
+where ˜V (n)
+0
+are Nc × Nc matrices describing the action
+of the nth impurity on the channel space. This action is
+further assumed to be local as prescribed by Dirac delta-
+function δ(d)(x − xn) centered at the impurity position
+xn.
+Let us remark that the problem of a Dirac delta-
+function potential is well-known to be ill-defined in spa-
+tial dimensions higher than d = 1.
+In our analysis,
+this is manifested in the ill-definiteness of the bulk po-
+sition space Green’s function at equal spatial arguments
+G(0)(x,x) due to the divergence of the defining integrals
+(8) in the ultraviolet.
+Such a divergence is not physi-
+cal and has to be circumvented by an appropriate reg-
+ularization scheme.
+In particular, in the metallic case
+˜A(x) = 0, V (x) = 0, in d > 1 the problem of the delta-
+potential has been extensively studied in both physical
+[21, 22, 23] and mathematical [24] literature and several
+meaningful regularization techniques were proposed and
+shown to produce physically sensible results. Since the
+presence of the energy gaps is of no importance in the
+deep ultraviolet regime, the same methods may be ap-
+plied in our case.
+The Dyson’s equation for the full Green’s function of
+the system is given by
+G(x,x′) =G(0)(x,x′)
++
+N
+∑
+n=1
+G(0)(x,xn) ˜V (n)
+0
+G(xn,x′).
+(11)
+First we want to consistently solve for the functions
+G(xn,x′), n = 1, ..., N. This problem is brought to
+the solution of the following matrix equation
+M(z)D(x′) = D(0)(x′),
+(12)
+where M(z) is the Nc ⋅ N × Nc ⋅ N block matrix defined
+by
+M(z) =1Nc⋅N − G(0)(z)˜V0,
+(13)
+(G(0)(z))n,n′ =G(0)(xn,xn′), (˜V0)n,n′ = δn,n′ ˜V (n)
+0
+. (14)
+Likewise, D(x′) and D(0)(x′) are the Nc ⋅ N × Nc matri-
+ces comprised of the full G(xn,x′) and bare G(0)(xn,x′)
+propagators, respectively. With these notations we ob-
+tain
+G(x,x′) =G(0)(x,x′) + D(0)†(x)˜V0D(x′)
+=G(0)(x,x′) + D(0)†(x)˜V0M−1(z)D(0)(x′),
+(15)
+where in our definition the Hermitian conjugate does not
+affect the z-variable, i.e.
+(G(0)(x,x′))† = G(0)(x′,x).
+(16)
+
+4
+C.
+Measuring the excess charge
+We define the excess charge density operator in the
+following manner:
+δ̂ρ(x) = ̂ρ(x) − ¯ρ,
+(17)
+where
+̂ρ(x) = ̂ψ†(x)̂ψ(x),
+(18)
+is the density operator, expressed in terms of the Nc-
+component fermionic field operators ̂ψ(x) and ̂ψ†(x).
+The field operators ̂ψ(x) and ̂ψ†(x) are further assumed
+to destroy/create excitations of the full Hamiltonian in-
+cluding the effect of localized scattering centers in Eq.
+(10). The constant contribution ¯ρ describes the unit cell-
+averaged average charge density in the bulk:
+¯ρ = 1
+VUC ∫VUC
+d(d)xρ(0)(x),
+(19)
+ρ(0)(x) =⟨̂ψ(0)†(x)̂ψ(0)(x)⟩
+= − 1
+π Im∫
+µ
+−∞ dωtr{G(0)(x,x)},
+(20)
+where the field operators ̂ψ(0)(x) and ̂ψ(0)†(x) describe
+the excitations of the translationally invariant system, µ
+denotes the chemical potential, and G(0)(x, x′) is the
+bare Green’s function defined by Eqs. (7) and (8).
+We measure the excess charge with the help of the clas-
+sical device, described by the envelope function f(x) (see
+Refs. [7, 8, 9, 10] and Ref. [25] for similar definitions).
+To be more specific, we define the excess charge operator
+as
+δ ̂Q = ∫Rd d(d)xf(x)δ̂ρ(x).
+(21)
+It is sensible to define the function f(x) relative to a
+certain point xp, to which the charge probe is applied,
+and further assume that the charge is sampled equiva-
+lently in all directions f(x) = f(∣x − xp∣). Additionally,
+we assume that all of the charge f(∣x − xp∣) ≈ 1 is sam-
+pled in sufficiently large vicinity of the sampling point
+xp, while the envelope function smoothly decays to zero
+f(∣x − xp∣) → 0 far away from xp. For that matter, it is
+convenient to choose
+f(∣x − xp∣) = 1 − Θlp(∣x − xp∣ − Lp),
+(22)
+where Θlp(∣x − xp∣ − Lp) is some representation of the
+Heaviside function broadened by lp. The length scales
+characteristic of the charge probe are assumed to satisfy
+Lp ≫ lp ≫ ξg,
+(23)
+where ξg ≃ vF
+Eg is the charge localization length in an insu-
+lator (also it is the charge correlation length, defining the
+exponential decay length of the density-density correla-
+tion function, see Ref. [18]), roughly defined as the ratio
+between the Fermi velocity vF and size of the energy gap
+at the Fermi level Eg.
+D.
+Topological invariant characterizing the excess
+charge
+Let us assume that N impurities, as characterized by
+the potential (10), are placed in a region of a crystal
+falling into the sampling district of the envelope function
+∣x∣ ≲ Lp. We define the total excess charge as the zero
+temperature expectation value of the excess charge op-
+erator in the grandcanonical equilibrium density matrix,
+so that
+δQ =⟨δ ̂Q⟩ = ∫Rd d(d)xf(x)(ρ(x) − ¯ρ),
+(24)
+ρ(x) = − 1
+π Im∫
+µ
+−∞ dωtr{G(x,x)}.
+(25)
+With the help of the representation (15), we obtain
+δQ = Q′ + QP ,
+(26)
+where Q′ contains the Friedel charge as well as the charge
+due to the impurity-localized bound states
+Q′ = ∫Rd d(d)xf(x)ρ′(x),
+(27)
+ρ′(x) = − 1
+π Im∫
+µ
+−∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)},
+(28)
+while QP is the so-called polarization charge given by
+QP = ∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ),
+(29)
+and, with the help of the properties of the envelope func-
+tion, is shown to be zero QP = 0 in Appendix B. It hence
+follows that
+δQ =Q′ = − 1
+π Im∫Rd d(d)xf(x)
+× ∫
+µ
+−∞ dωtr{D(0)†(x)˜V0M−1(z)D(0)(x)}.
+(30)
+Due to the branch cuts and poles of the T-matrix
+T(x,x′) = ∑n,n′[˜V0M−1(z)]n,n′δ(x − xn)δ(x′ − xn′), the
+integrand of the outer integral is exponentially sup-
+pressed ∼ e−∣x∣/ξg at large x, allowing us to set f(x) = 1.
+Interchanging the order of the integrals, we consider
+∫Rd d(d)xtr{D(0)†(x)˜V0M−1(z)D(0)(x)}
+= −
+N
+∑
+n,n′=1
+tr{[M−1(z)]n,n′ ∂
+∂ω G(0)(xn′,xn) ˜V (n)
+0
+}
+=
+N
+∑
+n,n′=1
+tr{[M−1(z)]n,n′ ∂
+∂ω [M(z)]n′,n}
+=
+N
+∑
+n=1
+tr{[M−1(z) ∂
+∂ω M(z)]
+n,n
+}
+= ∂
+∂ω tr{log M(z)} = ∂
+∂ω log det{M(z)},
+(31)
+
+5
+where, in the last line, trace and determinant of the full
+Nc ⋅N ×Nc ⋅N block matrix M(z) are understood. Using
+the result in Eq. (31), we arrive at the following compact
+formula for the total excess charge
+δQ = − 1
+π Im∫
+µ
+−∞ dω ∂
+∂ω log det{M(z)}.
+(32)
+To see why the integral in Eq. (32) may take on in-
+tegral values only, in Appendix C we find an alternative
+contour integral representation
+δQ = − ∮C
+dz
+2πi
+∂
+∂z log det{M(z)},
+(33)
+where C is an arbitrary non-self-intersecting curve that
+crosses the real axis at two points only, below the low-
+est eigenvalue of the full Hamiltonian and at the chemi-
+cal potential µ, and the direction of C is assumed to be
+clockwise.
+In the representation (33), the excess charge δQ is nec-
+essarily an integer as it is expressed as a contour integral
+winding number and the chemical potential is by def-
+inition inside one of the energy gaps (we focus on the
+insulating systems solely). In other words, the integral
+in Eq. (33) measures the degree of the mapping S1 → S1
+and is thus a member of the only non-trivial homotopy
+group of the unit circle π1(S1) = Z.
+In particular, the integral in Eq.
+(33), is a sum
+of two distinct contributions:
+the contribution of the
+branch cuts corresponding to the extended or scatter-
+ing states, and the contribution of poles corresponding
+to the imperfection-localized bound states.
+The bands in multidimensional (d > 1) and/or mul-
+tichannel (Nc > 1) systems are typically composite, i.e.
+overlapping with one another along the frequency axis.
+For that matter, it is convenient to choose the branch
+cuts to connect the bottom of the lowest sub-band with
+the top of the highest one, within every patch of the en-
+ergy bands surrounded by a pair of energy gaps.
+The bound state poles, determined as a solution of
+det{M(z)}∣z∈R = 0, are located on the complement of
+the bare Hamiltonian’s spectrum, i.e. inside the energy
+gaps and, in some cases (e.g. an attractive scalar impu-
+rity), below the bottom of the lowest energy band of the
+unperturbed Hamiltonian.
+III.
+RELATION WITH THE
+NEARSIGHTEDNESS PRINCIPLE
+A.
+Discussion
+Now we would like to discuss the topological invari-
+ant (33) in greater detail. In what follows, we specify
+the contour C as a rectangle of length µ − B in the real
+direction and width 2η in the imaginary one.
+Here B
+is by definition an energy lying below the lowest eigen-
+value of the total Hamiltonian Hx = H(0)
+x
++ ˜V (x) (i.e.
+c)
+b)
+a)
+Im{z}
+Re{z}
+C
+FIG. 1. A schematic illustration of how the spectral flow of the
+energies of the imperfection-circumscribing bound states sit-
+ting inside the gap that accommodates the Fermi level affects
+the total excess charge. The spectrum of the system is visu-
+alized through the local spectral density as looked down on
+the complex frequency plane. The occupied part of the spec-
+trum is demonstrated in blue, while the yellow color marks its
+complement (the states of the system that are unoccupied).
+Panel a) shows a rectangular contour C encircling the occu-
+pied spectral region. Panels b) and c) show the zoomed-in
+vicinity of the chemical potential before and after the per-
+turbation. As is demonstrated in panel c), the spectral flow
+results in the removal of a single bound state, carrying away
+a unity of the electron charge from the system (an inverse
+process is of course also possible).
+B ∈ (−∞,min{spec{Hx}})), and η is not necessarily an
+infinitesimal positive but is rather a finite positive num-
+ber (which is allowed as the integral is invariant under
+such contour deformations (see Appendix C)). Further-
+more, we assume that the chemical potential is located
+above the νth bulk energy band.
+Let us now consider making an adiabatic perturba-
+tion to the system that is comprised of the change in the
+positions {xn}n and/or vertex functions { ˜V (n)
+0
+}n of the
+impurities. As the span of the extended states’ energy
+bands is unaffected by such adiabatic perturbations, the
+branch cut contribution to the winding number remains
+invariant (up to the cases when the bound state merges
+with the band, as discussed below). This remark is essen-
+tially true as such deformations of the parameter space
+do not change the analytical structure of G(0)(xn, xn′),
+through the functionals of which alone our topological
+invariant is expressed.
+We hence conclude that such
+changes may only unleash themselves in the spectral flow
+of the bound state energies.
+As was anticipated in Section II D, the bound state
+energies are energy-wise located inside the energy gaps
+of the bulk system.
+This assertion also regards the
+energy gap below the bottom of the lowest band ω ∈
+(−∞, mink ϵ1,k], which can accommodate the bound
+states in the case of attractive impurities, for example.
+
+6
+The energies of the bound states ϵbs inside the energy
+gaps [maxk ϵα,k, mink ϵα+1,k] surrounded by a pair of
+bands α, α + 1, (α = 1, ..., ν − 1), are solely character-
+ized by their location within the gap. The same holds
+true for the infinite gap below the bottom of the lowest
+bulk energy band, with ϵbs now being energy-wise located
+in (−∞, mink ϵ1,k]. This implies that the spectral flow
+of these energies is constituted in the motion of ϵbs in be-
+tween the top of ϵα,k and the bottom of ϵα+1,k, or between
+the negative infinity and mink ϵ1,k shall some states be
+also found in there. When merging with one of the energy
+bands (either ϵα,k or ϵα+1,k, and ϵ1,k solely when consid-
+ering the gap preceding the entire band structure), the
+value of the contour integral winding number (33) relat-
+ing to that band gets modified by unity26. It follows that
+the motion of the bound state poles, inside such energy
+gaps below the one hosting the chemical potential, has
+absolutely no effect on the topological invariant (33) (one
+may see this result as a form of charge conservation), as
+B, by definition, resides below the lowest pole (effectively
+meaning that none of the states are allowed to escape the
+occupied spectral region from below).
+The flow of the energies of the impurity-localized
+bound states residing inside the gap separating the con-
+duction and the valence bands apart (the gap where the
+chemical potential is located), on the other hand, affects
+the winding number in Eq. (33). When a bound state
+crosses the chemical potential from above or below, the
+number of poles encompassed by the integration contour
+increases or decreases correspondingly. That means that
+the unit of the electron charge gets either pumped in or
+out of the system, modifying the topological invariant by
+±1. This discussion is summarized in Fig. 1.
+The elaboration above allows us to draw the following
+physical conclusion:
+Localized adiabatic perturbations in insulators, may only
+result in the localized charge redistributions, owing to
+the change in the occupancy of the perturbation-localized
+bound states at the Fermi level.
+This intuitive result is nothing but a direct consequence
+of the universal nearsightedness principle of W. Kohn
+[12, 13, 14] stating that, at fixed chemical potential, the
+electronic charge density depends on the external field
+(in our case being an assembly of localized scattering
+centers) only at nearby points.
+Another conclusion drawn by E. Prodan and W. Kohn
+in Ref.
+[13] (see also Ref.
+[14] for the fine details in
+d = 1) is that the adiabatic perturbations to the exter-
+nal potential, no matter how strong, have a negligible
+effect on the local charge density beyond a certain char-
+acteristic length scale, which, in the insulating regime,
+is naturally provided by the charge correlation length
+ξg. From the viewpoint of our topological invariant (33),
+this means that in the case of well-separated impurities
+∣xn −xn′∣/ξg ≫ 1, the topological invariant is expected to
+approach a sum of the individual single-impurity invari-
+ants, as distant impurities are not supposed to be able
+to “talk” with one another on such scales.
+Indeed, in
+an insulating state, it is well-known, that the two-point
+correlation functions G(0)(xn, xn′) decay exponentially
+at large distances ∼ e−∣Rmn−Rmn′ ∣/ξg (where mn labels
+the unit cell accommodating the nth scattering center),
+meaning that we can approximate
+(G(0)(z))n,n′ ≃δn,n′G(0)(xn,xn),
+(34)
+implying that
+M(z) ≃
+N
+⊕
+n=1
+(1Nc − G(0)(xn,xn)V (n)
+0
+),
+(35)
+and
+δQ ≃ −
+N
+∑
+n=1∮C
+dz
+2πi
+∂
+∂z log det{1Nc − G(0)(xn,xn)V (n)
+0
+}.
+(36)
+This result may be seen as a form of the conven-
+tional Born approximation of the linear transport theory,
+whereby, to the lowest order in the impurity density, one
+considers impurities as independent.
+B.
+An illustration: A pair of magnetic impurities
+in an illuminated quantum wire
+To illustrate some of the points highlighted in the
+above discussion, we here consider a simple model of a
+spin-orbit-interacting ballistic quantum wire, submersed
+into the background of the spatially oscillating electro-
+magnetic field. The bulk Hamiltonian assumes the form
+of the Pauli Hamiltonian with an extra Rashba-like term:
+H(0)
+x
+=
+(p + e
+cAx(x))
+2
+2m
++ kR ⋅ σ
+m
+(p + e
+cAx(x))
++ µBge
+2
+σ ⋅ B(x).
+(37)
+Above, kR = (kR,x, kR,y, kR,z) is the Rashba spin-orbit
+vector, σ = (σx, σy, σz) is the vector of the Pauli spin
+matrices, µB =
+e
+2mc is the Bohr magneton, c is the speed
+of light in vacuum, ge is the electron’s Land´e g-factor,
+and
+B(x) =∇ × A(x)∣
+x=xˆex,
+Ax(x) = ˆex ⋅ A(x)∣
+x=xˆex, (38)
+with ˆex being the ort in the x-direction, and A(x) be-
+ing the electromagnetic vector potential of the monochro-
+matic plane-wave form
+A(x) = A0 cos(q ⋅ x + ϕ),
+(39)
+in the Coulomb gauge
+∇ ⋅ A(x) = 0 ⇐⇒ q ⋅ A0 = 0.
+(40)
+
+7
+° º
+L
+° º
+2L
+0
+º
+2L
+º
+L
+k
+°2.00
+°1.75
+°1.50
+°1.25
+°1.00
+°0.75
+°0.50
+≤k
+x
+y
+z
+kR
+m(1)
+eff
+m(2)
+eff
+R
+q
+B
+E
+λ
+a)
+b)
+FIG. 2. Panel a): A schematic illustration of a ballistic quantum wire featuring Rashba-style spin-orbit coupling (defined by a
+spin-orbit vector kR) and submersed into a spatially-periodic arrangement of electric E and magnetic B fields of wavelength λ.
+The two impurity atoms, separated by distance R and carrying an effective magnetic moment of m(j)
+eff , j = 1, 2, are schematically
+shown by atomic symbols pierced with the magnetic moment-symbolizing arrows. Panel b): The bulk energy spectrum of the
+two-impurity problem. The energy bands are shown in dark blue, the chemical potential located inside the second spectral gap
+(above the fourth energy band) is depicted in orange, and the relevant spectral region is highlighted in light blue.
+The wave vector of the background electromagnetic field
+defines the fictitious lattice spacing
+L =
+2π
+ˆex ⋅ q,
+(41)
+where we have excluded the uninteresting case of the or-
+thogonally propagating wave ˆex ⋅ q = 0.
+We note that the Hamiltonian in Eq. (37) falls into
+the class of systems defined by the Hamiltonian (1), with
+d = 1 and
+V (x) =µBge
+2
+σ ⋅ B(x) + e2A2
+x(x)
+2mc2
++ ekR ⋅ σAx(x)
+mc
+, (42)
+˜Ax(x) =e
+cAx(x) + kR ⋅ σ.
+(43)
+As this demonstration is assumed to be interpretative,
+it suffices to consider the case of a pair of impurities,
+which we assume to be separated by distance R:
+˜V (x) = ˜V (1)
+0
+δ(x) + ˜V (2)
+0
+δ(x − R).
+(44)
+Note that we can place the first impurity at x = 0 with-
+out loss of generality, as its other positions inside the wire
+may be achieved by appropriate tuning of the modula-
+tion’s phase ϕ. Furthermore, we assume the impurities
+to exert both the electrostatic and the exchange “force”
+on the wire’s electrons, which we encode in the following
+form of the impurities’ vertex functions
+˜V (j)
+0
+= Ujσ0 + µBge
+2
+σ ⋅ B(j)
+eff ,
+(45)
+where B(j)
+eff is the effective (also appropriately screened to
+have a short-ranged effect only) magnetic field, produced
+by the effective magnetic moment of the impurity atom
+m(j)
+eff =
+qjgj
+2MjcS(j), with qj, gj, and Mj being the charge,
+g-factor, and mass of the jth impurity. Furthermore, Uj
+denotes the strength of the electrostatic potential, defin-
+ing the corresponding force exerted by the impurity on
+the electrons. Not going into much of the microscopic
+details, in the following, we treat Uj and B(j)
+eff as some
+constant parameters.
+The resulting setup is schemati-
+cally illustrated in panel a) of Fig. 2.
+Further, to illustrate our point, we assume that the
+associated impurity parameters {R, {Uj}j, {B(j)
+eff }j}
+evolve with a fictitious “adiabatic time” τ ∈ [0,T], in
+such a manner that their temporal derivatives remain
+much smaller than the Fermi energy ϵF times their value,
+for all τ ∈ [0,T].
+The particular form of the pumping protocol used to
+produce the numerical data and the concrete numerical
+values of the free model parameters are provided in Ap-
+pendix D. The resulting bulk energy spectrum is demon-
+strated in panel b) of Fig. 2.
+The
+numerical
+data
+for
+the
+excess
+charge-
+characterizing topological invariant,
+as well as the
+spectral flow of bound state energies inside the chem-
+ical potential-accommodating spectral gap, is shown
+in Fig.
+3.
+In particular, using the parametrization
+R(τ) = (nR − 1)L + ¯R(τ), suggested in the Appendix
+D, we present the data for five different values of
+nR ∈ {1,...,5}, as shown in five different columns of
+the corresponding figure, with upper and lower rows
+corresponding to the spectral flow and the topolog-
+ical invariant, respectively.
+Solid black and dashed
+burgundy lines mark the cases of independent and
+“interacting”
+impurities,
+correspondingly.
+By
+in-
+dependent impurities, we here understand that the
+separation between them is effectively infinite, so that
+the off-diagonal blocks of the M(z) matrix (see Eq.
+(14) for the definition) may be completely ignored.
+This means that the bound state spectrum of the
+independent impurities is provided by the solution of
+det(12 − G(0)(0,0) ˜V (1)
+0
+)det(12 − G(0)(R,R) ˜V (2)
+0
+)∣z∈R =
+
+8
+0
+T/4
+T/2
+3T/4
+T
+τ
+max ϵk,4
+min ϵk,5
+nR = 1
+0
+T/4
+T/2
+3T/4
+T
+τ
+nR = 2
+0
+T/4
+T/2
+3T/4
+T
+τ
+nR = 3
+0
+T/4
+T/2
+3T/4
+T
+τ
+nR = 4
+0
+T/4
+T/2
+3T/4
+T
+τ
+nR = 5
+0
+T/4
+T/2
+3T/4
+T
+τ
+−2
+−1
+0
+1
+2
+δQ
+0
+T/4
+T/2
+3T/4
+T
+τ
+0
+T/4
+T/2
+3T/4
+T
+τ
+0
+T/4
+T/2
+3T/4
+T
+τ
+0
+T/4
+T/2
+3T/4
+T
+τ
+FIG. 3. The figure demonstrates the adiabatic flows of both the bound state energy spectrum and the excess charge invariant
+in the toy model proposed in Section III B. Specifically, the spectral flow of the impurity-localized bound state energies is
+shown in the upper row, while the second row is dedicated to the invariant itself. As is explained in Appendix D, the position
+of the second impurity is parametrized as R(τ) = (nR − 1)L + ¯R(τ), where nR denotes the number of the unit cell hosting
+the second imperfection, and ¯R(τ) ∈ [0, L] describes its location within the unit cell. The five distinct columns in the above
+figure correspond to five choices of nR = 1, 2, 3, 4, 5. In all of the panels, red dashed lines correspond to the actual solution,
+while black solid lines relate to the case of two independent impurities (see the approximate formula (36)). As the separation
+between the impurities becomes of the order of the charge localization length ξg = O(L) (see Appendix D), both adiabatic flows
+approach the limit of two independent impurities.
+0, while the topological invariant is given by the ap-
+proximation (36).
+By the “interacting” impurities, on
+the other hand, we understand that the exact relations
+were used to produce the numerical data. The numerical
+technique for evaluation of bulk position space Green’s
+functions, as well as the topological indices of the form
+(33), is outlined in Ref.
+[10].
+In our calculations,
+the values of the contour parameters were chosen as
+η = 1, B = −30 (such a choice of B is motivated by
+the presence of the bound states below the lowest band
+ω ∈ (−∞, mink ϵk,1] in our model (37)).
+The central purpose of our demonstration is to show
+that upon the increase in the impurity’s separation be-
+yond the charge localization length ξg = O(L) (see Ap-
+pendix D), both the topological invariant and the bound
+state spectrum approach that of a pair of independent
+impurities.
+This effect is a direct consequence of the
+nearsightedness principle, telling us that a localized cause
+leads to a localized effect. Furthermore, as one may an-
+ticipate, the discontinuous jumps of the excess charge in-
+variant occur precisely at the points where bound states
+enter/leave the occupied part of the energy spectrum, as
+is explained in Section III A. Another interesting obser-
+vation is the non-zero value of the topological invariant
+at the beginning of the adiabatic evolution in τ, where
+the strengths of the electrostatic repulsion are the small-
+est 0 < Uj ≪ 1 (see Appendix D). This feature is a conse-
+quence of the presence of impurity-localized bound states
+below the bottom of the lowest energy band. Such an
+effect is well-known in the case of attractive scalar impu-
+rities, whereas here, it is generated by the non-Abelian
+structure of the model, and, to the best of our knowledge,
+was not reported previously in the literature.
+C.
+Topological invariants characterizing the
+boundary charge in unidimensional crystals
+In this section, we would like to comment on the topo-
+logical invariants characterizing boundary charges in uni-
+dimensional crystals, extensively discussed in Refs. [7,
+8, 9, 10].
+In particular, let us consider a d = 1 semi-
+infinite system described by the Hamiltonian (1), with
+the boundary placed at x = xb. An appropriate restric-
+tion of x defines the respective right and left subsystems:
+x ∈ [xb,∞),
+right sub-system,
+(46)
+x ∈ (−∞,xb],
+left sub-system.
+(47)
+In our definition, the primitive unit cell is defined as the
+one starting at the boundary of the right semi-infinite
+system UC = [xb, xb+L], with L being the lattice period.
+In this definition, the left half-system is always obtained
+from the right one by a local inversion operation, which
+acts by the inversion of local coordinates within each unit
+cell.
+Now we define the boundary charge operators corre-
+sponding to right and left semi-infinite systems as the
+envelope-weighted integrals of the expectation values of
+the appropriate excess charge density operators:
+Q(R)
+B
+=∫
+∞
+xb
+dxf(x)⟨δ̂ρR(x)⟩,
+(48)
+Q(L)
+B
+=∫
+xb
+−∞ dxf(x)⟨δ̂ρL(x)⟩,
+(49)
+where, in analogy with Eq.
+(17), δ̂ρS(x) = ̂ρS(x) − ¯ρ,
+and ̂ρS(x) is the density operator referring to the system
+S = R, L. Furthermore, the envelope function f(x) is
+chosen in accordance with Eq. (22), with xp = xb, and
+
+9
+the range of x being restricted according to Eqs. (46)
+and (47).
+Let us now consider measuring the total excess charge
+δQ accumulated around x = xb in a translationally in-
+variant system x ∈ (−∞, ∞). By the polarization charge
+neutrality condition QP = 0, demonstrated in Appendix
+B, the total excess charge also vanishes δQ = 0.
+On
+the other hand, we may consider a translationally invari-
+ant system as a sum of right and left semi-infinite sys-
+tems with a coupling corresponding to the bulk Hamilto-
+nian switched in between them. This coupling manifests
+itself as a local perturbation and, by the nearsighted-
+ness principle of Kohn, is capable of affecting the total
+charge locally by at most introducing or removing a num-
+ber of additional bound states, resulting in an integer
+contribution QI. In this connection, we conclude that
+δQ = Q(R)
+B
++ Q(L)
+B
+− QI = 0, where QI is known as the in-
+terface invariant. One of the central results of Ref. [10],
+was to demonstrate that
+QI =Q(R)
+B
++ Q(L)
+B
+= −∮C
+dz
+2πi
+∂
+∂z log det{G(0)(xb,xb)}.
+(50)
+That is, the interface invariant, characterizing the bound-
+ary charge upon local inversions, is a topological quan-
+tum number given by the winding of the determinant
+of bulk position space Green’s function evaluated at the
+location of the boundary.
+Now let us proceed with the transformations of the
+boundary charge under translations. First, we consider
+the right boundary charge of the so-called reference sys-
+tem, starting at xb = 0:
+Q(R)
+B (0) = ∫
+∞
+0
+dxf(x)(ρ(x) − ¯ρ),
+(51)
+and we would like to analyze the changes in this quantity
+upon the translation of the boundary by xϕ ∈ [0, L]. In-
+stead of shifting the boundary, we consider adding the fol-
+lowing potential ˆV (x) = ˆV0Θ(x)Θ(xϕ − x), ˆV0 → ∞. By
+the Pauli principle, the charge density becomes zero for
+x ∈ [0,xϕ] as these states sit at infinite energy above the
+chemical potential µ. From the definition of the bound-
+ary charge, we are left with the following contribution:
+δQ(R)
+B (xϕ) = ∫
+xϕ
+0
+dxf(x)(0 − ¯ρ)
+mod 1
+= −¯ρxϕ
+mod 1,
+(52)
+where
+mod 1 contribution again comes from the near-
+sightedness principle. This analysis allows us to conclude
+that:
+Q(R)
+B (xϕ) − Q(R)
+B (0) = ¯ρxϕ + I(xϕ),
+(53)
+where I(xϕ) is known as the boundary invariant. An-
+other important result of Ref. [10] was to show that
+I(xϕ) = −∮C
+dz
+2πi
+∂
+∂z lndetU(xϕ),
+(54)
+where U(xϕ) is defined via the path-ordered exponential
+U(xϕ) =Pexp{∫
+x
+0
+dx′L(x′)},
+(55)
+L(x) =[G(0)(x,x)]−1G(0)
+2 (x,x+) − iA(x),
+(56)
+and G(0)
+2 (x,x′) = ∂x′G(0)(x,x′).
+In other words, the
+boundary invariant is also a topological quantum number
+expressed as a winding of the appropriate functional of
+bulk position space Green’s functions.
+In this way, we see that the quantization of the topo-
+logical invariants characterizing the boundary charge in
+one-dimensional insulators is a direct consequence of the
+nearsightedness principle. As this intuitive physical prin-
+ciple holds beyond the single spatial dimension, one ex-
+pects the excess charges accumulated on inhomogeneities
+of various spatial co-dimensions in d-dimensional crystals
+to possess similar topological characterization schemes.
+Indeed, linear scaling of the boundary charge, along with
+its discontinuous jumps by a unit of the electron charge
+at the bound state escape/entrance spectral points, was
+recently demonstrated in a two-dimensional system [27].
+IV.
+CONCLUSIONS AND OUTLOOK
+In this paper, the quantization of the excess charges on
+localized scattering centers in d-dimensional insulators
+was discussed. Our analysis reveals that an assembly of
+such imperfections accumulates an integral excess charge,
+given by a winding number expression. We find that an
+adiabatic perturbation (no matter how strong) comprised
+of either relocation of the impurities or a modification of
+their vertex functions (or both at the same time) results
+in the change of the total charge by an integer, deter-
+mined by the saldo of the imperfection-localized bound
+states that entered or escaped the occupied spectral re-
+gion, inside the chemical potential-hosting bulk spectral
+gap. The quantization of this topological invariant was
+shown to be a direct consequence of the nearsightedness
+principle of the electronic matter, limiting the range of
+the effect of a localized cause. Additionally, this local
+behavior of the electronic matter in the insulating state
+was shown to be responsible for the quantization of the
+topological invariants characterizing the unidimensional
+boundary charge studied in [7, 8, 9, 10]. Furthermore,
+our study confirms the central paradigm of Ref.
+[16],
+namely that localized perturbations in insulators specifi-
+cally lead to the change in occupancy of the correspond-
+ing perturbation-localized bound states, modifying the
+total charge, defined as the macroscopic average on the
+scales significantly exceeding both the unit cell size L and
+the charge correlation length ξg, by at most an integer.
+As is now obvious, the present paper is of conceptual
+value only as the evaluation of the suggested topologi-
+cal invariant (33) for a specific multi-impurity (N ≫ 1)
+system poses a challenge on its own. In particular, this
+concerns questions regarding the regularization schemes
+
+10
+for the higher-dimensional equal-argument Green’s func-
+tions, as well as the basic questions regarding the numer-
+ical feasibility of the problem. Furthermore, it would be
+of future interest to study the expansion of the topolog-
+ical invariant in the interaction between the individual
+impurities, as generated by the off-diagonal blocks of the
+M(z) matrix, and analyze its ties with the conventional
+Born series for the impurity-dressed T-matrix. As it is
+suggested in the present study, in the insulating state,
+the impurity density ρI has to be always contrasted with
+the inverse charge localization length ξg, in such a man-
+ner that the condition 1 ≫ ρIξd
+g implies the validity of the
+Born approximation, treating impurities as independent.
+V.
+ACKNOWLEDGMENTS
+The author gratefully acknowledges the durable ex-
+change of ideas with M. Pletyukhov and H. Schoeller.
+Further, the author generously thanks S. Miles and M.
+Pletyukhov for their valuable comments.
+Most of the present work was done at the Institut f¨ur
+Theorie der Statistischen Physik of RWTH Aachen and
+was financially supported by the Deutsche Forschungsge-
+meinschaft via RTG 1995.
+Appendix A: Contraction of two Green’s functions
+Quite generically we may represent
+G = ⨋s
+∣s⟩⟨s∣
+z − ϵs
+,
+(A1)
+where the meta-index s labels the eigenstates ∣s⟩ and
+eigenenergies ϵs of the Hamiltonian.
+Considering the
+product of the Green’s function with itself
+GG = ⨋s ⨋s′
+∣s⟩⟨s′∣
+(z − ϵs)(z − ϵs′) ⟨s∣s′⟩
+�
+δ(s,s′)
+= ⨋s
+∣s⟩⟨s∣
+(z − ϵs)2
+= − ∂
+∂ω ⨋s
+∣s⟩⟨s∣
+z − ϵs
+= − ∂
+∂ω G.
+(A2)
+Taking the position space matrix elements
+⟨x∣GG∣x′′⟩ = − ∂
+∂ω G(x,x′′),
+and inserting
+1 = ∫Rd d(d)x∣x⟩⟨x∣,
+(A3)
+we obtain the desired identity
+∫Rd d(d)x′G(x,x′)G(x′,x′′) = − ∂
+∂ω G(x,x′′).
+(A4)
+Appendix B: Polarization charge
+We consider
+∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ)
+= ∑
+m ∫UC d(d)¯xf(¯x + Rm)(ρ(0)(¯x) − ¯ρ).
+(B1)
+Above we parametrized the position space variable x as
+x = Rm+¯x, for some vector of integers m, and ¯x is the lo-
+cal coordinate within the unit cell ¯x ∈ UC. Furthermore,
+we used the periodicity property of ρ(0)(x), implied by
+the periodicity of the equal-argument Green’s function
+G(0)(x, x) = G(0)(x + Rm, x + Rm),
+∀m ∈ Zd. (B2)
+The envelope function varies significantly only in the
+crossover region ∣Rm∣ = O(Lp), allowing us to approxi-
+mate
+f(¯x + Rm) ≈ f(Rm) + ¯x ⋅ ∇f(Rm),
+(B3)
+leading to
+∫Rd d(d)xf(x)(ρ(0)(x) − ¯ρ)
+= ∫UC d(d)¯x∑
+m
+(¯x ⋅ ∇f(Rm))(ρ(0)(¯x) − ¯ρ).
+(B4)
+Now approximating
+∑
+m
+(¯x ⋅ ∇f(Rm)) ≈
+1
+VUC ∫Rd d(d)y(¯x ⋅ ∇yf(y))
+=
+1
+VUC ∫Rd d(d)y∇y ⋅ (¯xf(y)) = 0,
+(B5)
+where in the last step we used Gauss’ divergence theorem.
+Appendix C: Contour integral representation
+First, we rewrite Eq. (32) as
+δQ = − 1
+2πi ∫
+µ
+−∞ dω ∂
+∂ω log det{M(z)}
++ 1
+2πi ∫
+µ
+−∞ dω ∂
+∂ω (log det{M(z)})∗ .
+(C1)
+Now we remind ourselves that
+(log f(z))∗ = log (f(z))∗ ≡ log f ∗(z).
+(C2)
+Furthermore, one has
+(det{M(z)})∗ = det{M†(z∗)},
+(C3)
+where, as before, the Hermitian conjugate does not affect
+the z-variable. Now we have
+det{M†(z∗)} = det{(1 − G(0)(z∗)˜V(0) )†}
+= det{1 − ˜V(0)G(0)(z∗)}
+= det{M(z∗)},
+(C4)
+
+11
+where to get from the pre-last to the last lines we em-
+ployed the Weinstein–Aronszajn identity.
+It hence follows that
+δQ = − 1
+2πi ∫
+µ
+−∞ dω ∂
+∂ω log det{M(ω + iη)}
+− 1
+2πi ∫
+−∞
+µ
+dω ∂
+∂ω log det{M(ω − iη)}
+= − ∮C
+dz
+2πi
+∂
+∂z log det{M(z)}.
+(C5)
+Above, C is the counterclockwise rectangular contour de-
+fined as a union of four segments:
+C =[B + iη,µ + iη) ∪ [µ + iη,µ − iη) ∪ [µ − iη,B − iη)
+∪ [B − iη,B + iη),
+B → −∞,
+η → 0+.
+(C6)
+We note that the integral in (C5) remains unaffected
+under continuous contour deformations, so long as the
+analytic structure of the integrand within the patch of
+the complex plane enclosed by contour C remains intact.
+In this connection, we may replace C with an arbitrary
+non-self-intersecting curve crossing the real axis at two
+points only, at any energy below the lowest eigenvalue of
+the full Hamiltonian, and at the chemical potential.
+Appendix D: Parameters and protocols
+In the numerical example provided in Section III B, the
+parameters of the model were chosen according to
+q = 2π(ex + κey)
+λ
+√
+1 + κ2
+,
+A0 = A0(κex − ey)
+√
+1 + κ2
+,
+m = 1, (D1)
+κ = 1 +
+√
+5
+2
+,
+λ = 4,
+e
+cA0 = 1.17,
+kR =
+⎛
+⎜
+⎝
+0.32
+1.39
+1.24
+⎞
+⎟
+⎠
+. (D2)
+Note that as we have set the electron’s mass m = 1 to
+unity (in addition to the electric charge e = 1 and reduced
+Plank’s constant ̵h = 1), we work in Hartree’s atomic
+units.
+In this way, the electromagnetic wave is propagating
+in the x − y plane, with the corresponding magnetic field
+being
+B(x) = 2πA0
+λ
+ez sin(q ⋅ x + ϕ).
+(D3)
+By definition, the corresponding lattice period is given
+by
+L = λ
+√
+1 + κ2 = 2
+√
+2(5 +
+√
+5).
+(D4)
+To produce the data, we used the following pumping
+protocol for the impurities’ separation
+R(τ) = (nR − 1)L + ¯R(τ),
+¯R(τ) = L
+T τ,
+L
+T ≪ vF ,
+(D5)
+where vF is the Fermi velocity and nR is an integer spec-
+ifying the number of the unit cell hosting the second
+impurity. For the impurities’ vertex functions, we fur-
+ther make an assumption of the equivalent impurities:
+U (1)(τ) = U (2)(τ) =∶ U(τ) and ∣B(1)
+eff (τ)∣ = ∣B(2)
+eff (τ)∣ =∶
+BI(τ). The direction of the magnetic moments, on the
+other hand, is allowed to be different in two scattering
+centers and is parametrized in spherical polar coordinates
+B(j)
+eff (τ)
+BI(τ) =
+⎛
+⎜
+⎝
+cosφ(j)(τ)sinθ(j)(τ)
+sinφ(j)(τ)sinθ(j)(τ)
+cosθ(j)(τ)
+⎞
+⎟
+⎠
+.
+(D6)
+In the following, we assume that, as is the case with the
+location of the second impurity within the unit cell num-
+ber nR, the impurity strength also grows linearly with
+τ
+U(τ) = U0 + δU τ
+T ,
+δU
+T ≪ ϵ2
+F .
+(D7)
+On the other hand, we assume the effective magnetic field
+of the impurity to oscillate as
+BI(τ) = B0 + δB sin(6πτ
+T ).
+(D8)
+The direction of the spins is prescribed by
+φ(1)(τ) = φ(2)(τ) = 2π sin(8πτ
+T ),
+(D9)
+θ(j)(τ) = π
+2 (1 + (−1)j τ
+T ).
+(D10)
+The rest of the parameters are chosen as
+U0 = 0,
+δU = 10,
+e
+cB0 = 3,
+e
+cδB = 1.5.
+(D11)
+Now let us estimate the charge localization length ξg
+for the second bulk spectral gap, where the chemical po-
+tential µ is assumed to be placed. According to Ref. [10],
+the Fermi velocity may roughly be estimated as vF ≈
+kF
+m ≈
+2π
+mL ≈ 0.825816. The energy gap at the Fermi level
+was numerically computed to be roughly Eg ≈ 0.271394,
+leading to the following estimate ξg ≈ 3 = O(L).
+
+12
+∗ Email: kiryl.piasotski@kit.edu
+† On the leave from Institut f¨ur Theorie der Statistischen
+Physik, RWTH Aachen, 52056 Aachen, Germany
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+matter”, PNAS 102, 11635 (2005).
+14 E. Prodan, “Nearsightedness of electronic matter in one
+dimension”, Phys. Rev. B 73, 085108 (2006).
+15 H.-R. Trebin, “The topology of non-uniform media in con-
+densed matter physics”, Adv. Phys. 31, 195 (1982).
+16 M. Pletyukhov, D. M. Kennes, K. Piasotski, J. Klinovaja,
+D. Loss, and H. Schoeller, “Rational boundary charge in
+one-dimensional systems with interaction and disorder”,
+Phys. Rev. Res. 2, 033345 (2020).
+17 J. Goldstone and F. Wilczek, “Fractional Quantum Num-
+bers on Solitons”, Phys. Rev. Lett. 47, 986 (1981).
+18 C. S. Weber, K. Piasotski, M. Pletyukhov, J. Klinovaja,
+D. Loss, H. Schoeller, and D. M. Kennes, “Universality
+of Boundary Charge Fluctuations”, Phys. Rev. Lett. 126,
+016803 (2021).
+19 K. Nomura and N. Nagaosa, “Electric Charging of Mag-
+netic Textures on the Surface of a Topological Insulator”,
+Phys. Rev. B 82, 161401(R) (2010).
+20 This includes models featuring a momentum-cubic spin-
+orbit interaction, or even tight-binding models, describing
+crystals in terms of incomplete basis sets of localized Wan-
+nier orbitals.
+21 D. K. Park, “Green’s-function approach to two- and three-
+dimensional delta-function potentials and application to
+the spin-1/2 Aharonov–Bohm problem”, J. Math. Phys.
+36, 5453 (1995).
+22 D. A. Atkinson, H. W. Crater, “An exact treatment of
+the Dirac delta function potential in the Schr¨odinger equa-
+tion”, Amer. Jour. Phys. 43, 301 (1975).
+23 R. Jackiw, “Delta-function potentials in two- and three-
+dimensional quantum mechanics” MAB B´eg memorial vol-
+ume (1991).
+24 C. N. Friedman, “Perturbations of the Schr¨odinger equa-
+tion by potentials with small support”, J. Funct. Anal. 10,
+346 (1972).
+25 S. Kivelson and J. R. Schrieffer, “Fractional charge, a sharp
+quantum observable”, Phys. Rev. B 25, 6447 (1982).
+26 In the case of one-dimensional (d = 1) single-channel
+(Nc = 1) systems, it is not possible for the bound state
+poles to coexist with the continuum (or extended) states.
+As a result, the effect of the bound state pole merging with
+the band is to increase the order of the branching pole at
+the band edge. In multichannel (Nc > 1) and/or multidi-
+mensional (d > 1) systems, on the other hand, the poles
+may, in principle, coexist with the band continuum. In ei-
+ther case, as a result, the band contribution to the winding
+number gets modified by unity.
+27 Z. Hou, C. S. Weber, D. M. Kennes, D. Loss, H. Schoeller,
+J. Klinovaja, M. Pletyukhov, “Realization of a three-
+dimensional quantum Hall effect in a Zeeman-induced sec-
+ond order topological insulator on a torus”, arXiv preprint
+arXiv:2212.09053, (2022)
+

7dAyT4oBgHgl3EQf2vnp/content/tmp_files/2301.00758v1.pdf.txt

@@ -0,0 +1,1713 @@
+> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+Analysis of a HAPS-Aided GNSS in Urban Areas 
+using a RAIM Algorithm 
+ 
+Hongzhao Zheng, Member, IEEE, Mohamed Atia, Senior Member, IEEE, and Halim Yanikomeroglu, Fellow, IEEE 
+Abstract—The global averaged civilian positioning accuracy is 
+still at meter level for all existing Global Navigation Satellite 
+Systems (GNSSs), and the performance is even worse in urban 
+areas. At lower altitudes than satellites, high altitude platform 
+stations (HAPS) offer several benefits, such as lower latency, less 
+pathloss, and likely smaller overall estimation error for the 
+parameters associated in the pseudorange equation. HAPS can 
+support GNSSs in many ways, and in this paper we treat the HAPS 
+as another type of ranging source. In so doing, we examine the 
+positioning performance of a HAPS-aided GPS system in an urban 
+area using both a simulation and physical experiment. The HAPS 
+measurements are unavailable today; therefore, they are modeled 
+in a rather simple but logical manner in both the simulation and 
+physical experiment. We show that the HAPS can improve the 
+horizontal dilution of precision (HDOP), the vertical dilution of 
+precision (VDOP), and the 3D positioning accuracy of GPS in both 
+suburban and dense urban areas. We also demonstrate the 
+applicability of a RAIM algorithm for the HAPS-aided GPS 
+system, especially in the dense urban area.  
+ 
+Index Terms—High altitude platform station (HAPS), horizontal 
+dilution of precision (HDOP), pseudorange, receiver autonomous 
+integrity monitoring (RAIM), vertical dilution of precision (VDOP). 
+I. INTRODUCTION 
+ODAY, many countries and the European union have 
+their own global navigation satellite systems (GNSSs). 
+However, 95 percent of the time, the global averaged 
+horizontal positioning accuracy of existing GNSSs is still at the 
+meter level, and it is even worse for the vertical positioning 
+accuracy [1]-[4] due to the nature of the satellite geometry. 
+Although vertical positioning performance is less important 
+than horizontal positioning performance today, it might be very 
+important in the future, for instance, for unmanned aerial 
+vehicles (UAVs) flying in the 3D aerial highways [5]. Thanks 
+to ongoing research on localization and navigation fields, there 
+are a number of techniques developed which can bring the 
+positioning accuracy of systems involving satellites to the 
+centimeter level. For example, Li et al. have shown that 
+centimeter-level positioning accuracy can be achieved using the 
+multi-constellation GNSS consisting of Beidou, Galileo, 
+GLONASS and GPS with precise point positioning (PPP) [6]. 
+Because most civilian applications use single-frequency, low-
+cost receivers for localization and navigation, many advanced 
+positioning algorithms, including PPP that delivers centimeter 
+ 
+This paper was supported in part by Huawei Canada. 
+H. Zheng, M. Atia, and H. Yanikomeroglu are with the Department of 
+Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 
+5B6, 
+Canada 
+(e-mail: 
+hongzhaozheng@cmail.carleton.ca; 
+Mohamed.Atia@carleton.ca; halim@sce.carleton.ca).  
+level positioning accuracy, cannot be implemented. Therefore, 
+the single point positioning (SPP) is the most commonly used 
+algorithm in civilian applications. But this is poised to change. 
+As increasing numbers of low-Earth-orbit (LEO) satellites are 
+launched into space, researchers are investigating the feasibility 
+of utilizing LEO satellites to aid the positioning service. For 
+instance, Li et al. have shown that a centimeter level Signal-In-
+Space Ranging Error (SISRE) in the real-time PPP application 
+can be achieved using a LEO enhanced GNSS [7]. In the event 
+that GNSS signals are unavailable in urban areas, researchers 
+are also interested in building navigation systems that 
+exclusively rely on LEO satellite signals. For example, a 
+position root mean squared error (RMSE) of 14.8 m for a UAV 
+has been proven feasible using only two Orbcomm LEO 
+satellites with the carrier phase differential algorithm [8]. 
+Compared to medium-Earth-orbit (MEO) satellites, which are 
+typically used in GNSSs, LEO satellites offer several 
+advantages, such as lower latency and less pathloss due to 
+shorter distance to ground users. LEO satellites also offer 
+greater availability due to the large number of them.  
+To further enhance high bandwidth networking coverage in 
+areas with obstacles, such as urban areas, another option is the 
+use of high altitude platform stations (HAPS1), which refer to 
+aerial platforms positioned in the stratosphere with a typical 
+altitude of about 20 km. HAPS can be utilized for many 
+technologies coming in 5G even 6G and beyond such as 
+computation offloading [9], edge computing [10], and aerial 
+base station [11]. As urban areas are where GNSS positioning 
+performance degrades severely, while also being where most 
+people live, we could improve the positioning performance of 
+GNSS by placing several HAPS above metro cities and 
+equipping them with satellite-grade atomic clocks so that HAPS 
+can be deployed as another type of ranging source. Even though 
+atomic clocks on satellites are highly accurate, they are not 
+perfect due to the time dilation postulates made in both 
+Einstein’s special theory of relativity and the general theory of 
+relativity. According to Einstein’s special theory of relativity, 
+an atomic clock on a fast-moving satellite runs slower than a 
+clock on Earth by around 7 microseconds per day. On the other 
+hand, according to the general theory of relativity, an atomic 
+clock which experiences weaker gravity on a distant satellite 
+runs faster than a clock which experiences greater gravity on 
+1 In this paper, the acronym "HAPS" is used to denote “high altitude 
+platforms station” in both singular and plural forms, in line with the convention 
+adopted in the ITU (International Telecommunications Union) documents. 
+T 
+
+> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+Earth by about 45 microseconds per day [12]. As HAPS operate 
+at an altitude of around 20 km and can be quasi-stationary, the 
+time dilation is negligible from the perspective of special 
+relativity and greatly reduced from the perspective of general 
+relativity. Therefore, the atomic clocks on HAPS will likely be 
+more accurate than that on satellites, which can make the 
+estimation error of the HAPS clock offset smaller than that of 
+the satellite clock offset. Since HAPS are positioned much 
+closer to the Earth than satellites, the pathloss of a HAPS is 
+expected to be much less, which will likely make the received 
+signal power of a HAPS stronger than that of a satellite, thereby 
+reducing the estimation error of the parameters associated in the 
+pseudorange measurement of the HAPS signal. The movement 
+of a HAPS can be confined to a cylindrical region with a radius 
+of 400 m and a height of about 700 m [13], which can reduce 
+the number of handovers during the course of navigation and 
+increase the utilization efficiency during its operation life. As 
+HAPS are positioned in the stratosphere, which is below the 
+ionosphere, their signals will likely be free of the ionospheric 
+effect, which is known to be one of the major sources of error 
+in pseudorange measurements. 
+Therefore, the overall 
+estimation error for a HAPS will likely be smaller than that of 
+a satellite. Similar to the pseudorange measurement for a 
+satellite, which incorporates the satellite position error, we 
+should also consider the position error in the pseudorange 
+measurement for HAPS. Fortunately, researchers have been 
+investigating the positioning of HAPS and have demonstrated 
+that HAPS positioning errors are comparable to or lesser than 
+satellite orbit errors. For example, Dovis et al. prove that 0.5 m 
+positioning accuracy (circular error probable [CEP] 68 percent) 
+for a HAPS is achievable using the modified RTK method [14]. 
+There are a handful of papers in the literature that have 
+investigated the HAPS-aided GNSS [15]-[18]; however, to the 
+best of our knowledge, this paper is the first to provide a 
+comprehensive study of the positioning performance of a 
+HAPS-aided GNSS in urban areas.  
+There are plenty of operational GPS satellites that could fail 
+due to the degraded signal quality for reasons such as 
+obstruction, multipath, intentional or unintentional attacks, 
+thereby impacting the positioning performance of the GNSS. In 
+this case, a signal selection algorithm like the receiver 
+autonomous integrity monitoring (RAIM) algorithm, which can 
+detect and exclude poor quality signals, can be helpful in 
+improving the positioning performance. For example, about 35 
+percent decrease in RMS positioning error of the GPS-only case 
+and 50 percent decrease in RMS positioning error of the 
+GPS/GLONASS case in a severe urban scenario have been 
+achieved on smartphone GNSS chips by using a RAIM 
+algorithm [19]. Moreover, Yang and Xu propose a robust 
+estimation-based RAIM algorithm that can detect and exclude 
+multiple faulty satellites effectively with efficiency higher than 
+the conventional least squares (LS)-based RAIM algorithm 
+[20]. In this paper, we make three postulations: 1) a HAPS 
+signal is free of the ionospheric effect; 2) the estimation error 
+of the HAPS clock offset is smaller than that of the satellite 
+clock offset; and 3) the received signal power of a HAPS is 
+higher than that of a satellite, all of which contribute toward the  
+ 
+Fig. 1. System model of the HAPS-aided GPS. 
+ 
+ 
+assumption that the overall estimation error of the parameters 
+associated in the pseudorange equation for the HAPS is smaller 
+than that for the satellite. Under this assumption, we use the SPP 
+algorithm developed in our prior work [21] to show that HAPS 
+can indeed improve the positioning performance of legacy 
+GNSSs in urban areas through both a simulation and a physical 
+experiment. We also demonstrate the applicability of the RAIM 
+algorithm to a HAPS-aided GPS system, especially in dense 
+urban areas. Since the HAPS measurements are unavailable so 
+far, they are simulated in a rather simple but logical way in both 
+the simulation and physical experiment. The contributions of 
+this paper are listed below. 
+• 
+First, using a commercial GNSS simulator, we 
+simulate the GPS pseudorange signals and generate 
+the positions of HAPS. By using the default system 
+parameters as well as a proper manipulation of the 
+number of visible satellites, we show that the 
+positioning performance of the GPS-only system in 
+both the suburban and dense urban areas are close to 
+the real scenario. Moreover, we show the positioning 
+performance of different systems where different 
+numbers of HAPS are used with or without the GPS 
+system. The issue of the ranging source geometry is 
+revealed from the simulation results. 
+• 
+Next, we apply the SPP algorithm to the real GPS data 
+collected using two commercial GNSS receivers as 
+well as the HAPS data generated using the commercial 
+GNSS software. In so doing, we show the advantage 
+of the HAPS-aided GPS system in the sense of the 
+horizontal dilution of precision (HDOP) and the 
+vertical dilution of precision (VDOP). 
+• 
+Finally, we implement a RAIM algorithm and 
+demonstrate its effectiveness in improving the 3D 
+positioning performance of the HAPS-aided GPS 
+system, especially in dense urban areas. 
+The rest of the paper is organized as follows: in Section Ⅱ, the 
+system model, the SPP algorithm, and the RAIM algorithm are 
+described. In Section Ⅲ, the simulation setup of the HAPS-
+
+Ionosphere
+HAPS
+HAPS
+20km
+HAPSfootprint
+HAPSfootprint
+15°
+cell.
+cell> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+aided GPS system and the simulation results are presented. In 
+Section Ⅳ, the physical experiment setup and results, including 
+both the DOP analysis and the 3D positioning accuracy 
+analysis, are provided. Finally, Section V offers some 
+conclusions and a discussion of future research directions. For 
+simplicity, the GNSS signal only involves the GPS C/A L1 
+signal. 
+II. SYSTEM MODEL 
+The system model of the HAPS-aided GPS system is 
+depicted in Fig. 1. There are four satellites shown in Fig. 1, this 
+is just a reminder that at least four satellites are required to 
+perform precise 3D localization using GNSS. The typical 
+choice for the elevation mask is 10 degrees. However, we use 
+15 degrees as the elevation mask for the satellites and HAPS 
+due to the following reasons: 1) the atmospheric error owing to 
+the signal refraction can be neglected if the elevation of a 
+satellite is greater than 15 degrees [22], which is likely true for 
+a HAPS as well; 2) As there is a higher chance of ensuring the 
+required number of ranging source with HAPS, we can improve 
+the positioning performance further by only using those satellite 
+signals with better quality. The pseudorange equation for 
+satellite is given by 
+ 
+ 
+𝑝𝑆𝐴𝑇 = 𝜌𝑆𝐴𝑇 + 𝑑𝑆𝐴𝑇 + 𝑐(𝑑𝑡 − 𝑑𝑇𝑆𝐴𝑇) + 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇 + 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇
++ 𝜖𝑚𝑝,𝑆𝐴𝑇 + 𝜖𝑝 
+ 
+ 
+(1) 
+where 𝑝𝑆𝐴𝑇  denotes the satellite pseudorange measurement, 
+𝜌𝑆𝐴𝑇 is the geometric range between the satellite and receiver, 
+𝑑𝑆𝐴𝑇 represents the satellite orbit error, 𝑐 is the speed of light, 
+𝑑𝑡 is the receiver clock offset from GPS time, 𝑑𝑇𝑆𝐴𝑇 is the 
+satellite clock offset from GPS time, 𝑑𝑖𝑜𝑛,𝑆𝐴𝑇  denotes the 
+ionospheric delay for satellite signals, 𝑑𝑡𝑟𝑜𝑝,𝑆𝐴𝑇  denotes the 
+tropospheric delay for satellite signals, 𝜖𝑚𝑝,𝑆𝐴𝑇  is the delay 
+caused by the multipath for satellite signals, and 𝜖𝑝 is the delay 
+caused by the receiver noise. The pseudorange equation for 
+HAPS can be expressed as follows: 
+ 
+ 
+𝑝𝐻𝐴𝑃𝑆 = 𝜌𝐻𝐴𝑃𝑆 + 𝑑𝐻𝐴𝑃𝑆 + 𝑐(𝑑𝑡 − 𝑑𝑇𝐻𝐴𝑃𝑆) + 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆
++ 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆 + 𝜖𝑝 
+ 
+(2) 
+where 𝑝𝐻𝐴𝑃𝑆  denotes the HAPS pseudorange measurement, 
+𝜌𝐻𝐴𝑃𝑆 represents the geometric range between the HAPS and 
+the receiver, 𝑑𝐻𝐴𝑃𝑆 represents the HAPS position error, 𝑑𝑇𝐻𝐴𝑃𝑆 
+is the HAPS clock offset from GPS time, 𝑑𝑡𝑟𝑜𝑝,𝐻𝐴𝑃𝑆 denotes the 
+tropospheric delay for HAPS signals, 𝜖𝑚𝑝,𝐻𝐴𝑃𝑆  is the delay 
+caused by the multipath for HAPS signals. The simulated 
+vehicle trajectory originates at Carleton University, which is in 
+a suburban area, and ends at Rideau Street, which is in a dense 
+urban part of Ottawa (see Fig. 2). There are six simulated HAPS 
+shown as transmitters on Fig. 3. As we can see, one HAPS is 
+positioned over downtown Ottawa; the other five HAPS are 
+positioned nearby, over populated areas and conservation areas. 
+HAPS is quasi-stationary, meaning that it will still be moving 
+in a variety of manners.  In this work, all the HAPS are  
+ 
+Fig. 2. Vehicle trajectory. 
+ 
+ 
+Fig. 3. Locations of the simulated HAPS.  
+ 
+ 
+simulated to be following a circular trajectory with a radius of 
+300 m. The elevation and azimuth angles of all the HAPS at the 
+beginning of the simulation are listed in Table I. The positions 
+of HAPS were chosen to provide a rich diversity in azimuth 
+angles. With one HAPS at the zenith and the others having 
+relatively low elevation angles, this constitutes a near Zenith + 
+Horizon (ZH) geometry, which can deliver a reasonably good 
+DOP [23]. To make sure the entire urban area is well covered, 
+HAPS are placed not too far away from the urban area. To better 
+understand the concept of DOP, the visual illustrations of the 
+HDOP and VDOP of the simulated HAPS constellation are 
+provided in Fig. 4 and Fig. 5, respectively. Due to various errors 
+impacting the pseudorange measurement, the estimated 
+distance between a HAPS and a receiver can be smaller or 
+larger than the geometric range. Objects with a higher elevation 
+angle will likely result in more uncertainty for the vertical  
+
+OSM + relief shading
+Tracks:
+Ottawa,On to Rideau Centre,
+Ottawa,ON
+Lyon
+Denseurban
+Areas
+yview
+Fost
+onburg
+Suburban
+Areas
+Ottawa
+Cene:45.40751.75.0087m
+Google
+Map created at CSVisualizer.com
+一净用多款Transmitter5
+deroure
+Buckinohan
+Transmitter6
+arcde
+otineau
+Transmitter3
+174
+Transmitter1
+Simulator
+417
+Embrun
+Russell
+Metcalfe
+Transmitter2
+Transmitter4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+Fig. 4. HDOP of the simulated HAPS constellation (top view). 
+ 
+ 
+Fig. 5. VDOP of the simulated HAPS constellation (front view). 
+ 
+ 
+component and less uncertainty for the horizontal component 
+from the point of view of geometry, and vice versa. The shaded 
+area is where the receiver is estimated to be. 
+ 
+A. The Single Point Positioning (SPP) Algorithm 
+The single point positioning algorithm is implemented on the 
+basis of the SPP package developed by Napat Tongkasem [24] 
+with proper modifications [21] so that HAPS can be 
+incorporated in the SPP algorithm. Fig. 6 shows the flowchart 
+of the single point positioning algorithm. We should point out 
+that the implemented single point positioning algorithm is not 
+the best positioning algorithm, and that the objective of this 
+work is to show the significance of HAPS in aiding the 
+positioning performance of a legacy GNSS. The implemented 
+SPP algorithm can be improved in many ways. For example, if 
+the knowledge of the measurement error variance is available, 
+we can apply the weighted least squares (WLS) algorithm to 
+enhance the positioning performance of the SPP algorithm by 
+lowering the weights of those observations with higher 
+variances [25]. If the knowledge of the measurement error 
+variance is unavailable, the computational complexity of the 
+SPP algorithm can be reduced by imposing the Cholesky 
+decomposition for the matrix inversion in (9) [26]. We can also  
+TABLE I 
+ELEVATION AND AZIMUTH OF THE HAPS AT THE START OF THE SIMULATION 
+ 
+HAPS index 
+Elevation angle 
+Azimuth angle 
+HAPS #1 
+81.087° 
+-14.210° 
+HAPS #2 
+24.054° 
+-128.878° 
+HAPS #3 
+27.952° 
+68.022° 
+HAPS #4 
+32.450° 
+171.477° 
+HAPS #5 
+36.554° 
+2.204° 
+HAPS #6 
+33.805° 
+-57.884° 
+ 
+ 
+use the carrier phase measurement to enhance the positioning 
+performance of the HAPS-aided GPS system, since carrier 
+performance of the HAPS-aided GPS system, since carrier 
+phase measurements come with much higher precision, which 
+usually delivers a more accurate position solution. Since the 
+HAPS clock offset in this work is not explicitly simulated, we 
+simply use 𝑑𝑇 to denote the satellite clock offset. From the data 
+collected by the GNSS receiver, we shall obtain both the 
+receiver independent exchange (RINEX) format observation 
+file and the RINEX navigation file, from which we can obtain 
+satellite information, such as the satellite pseudorange 𝒑𝑺𝑨𝑻, the 
+ionospheric parameters 𝜶 , the Keplerian parameters, the 
+pseudo-random noise (𝑷𝑹𝑵) code, which represents the unique 
+number of each satellite, the day of year ( 𝐷𝑂𝑌 ) which 
+represents the day of year at the time of measurement. We write 
+𝑷𝑹𝑵  in bold to represent a vector containing the pseudo-
+random noise code of all visible satellites at the current epoch. 
+We are able to compute the satellite positions, 𝑷𝑺𝑨𝑻 , and 
+satellite clock offset, 𝒅𝑻 , using the Keplerian parameters 
+contained in the navigation file. 𝑷𝑯𝑨𝑷𝑺  denotes a vector 
+containing the positions of all HAPS, which are generated using 
+the Skydel GNSS simulator [27], and 𝒑𝑯𝑨𝑷𝑺 denotes a vector 
+containing the HAPS pseudorange, which will be explained in 
+Section III. To compute the position solution 𝒙 , we first 
+initialize the receiver position to the center of the Earth; then 
+we initialize the receiver clock offset to zero and the change in 
+estimates 𝒅𝒙 to infinity. For each epoch of measurement, we 
+first check if the number of available ranging sources is more 
+than three, as at least four ranging sources are required to 
+perform precise 3D localization. Since the receiver position is 
+iteratively estimated, we calculate the elevation angles for both 
+satellites and HAPS with respect to the recently estimated 
+receiver position. Since both the tropospheric delay and the 
+ionospheric delay are functions of the receiver position, these 
+two atmospheric delays are estimated iteratively as well. The 
+elevation angle, satellite pseudorange, HAPS pseudorange, 
+satellite position, satellite clock offset, tropospheric delay 
+𝒅𝒕𝒓𝒐𝒑 , ionospheric delay 𝒅𝒊𝒐𝒏 , and pseudo-random noise 
+(𝑷𝑹𝑵) code are modified iteratively on the basis of the re-
+computed elevation angles for both satellites and HAPS. To 
+prepare the parameters needed for the least square method, the 
+pseudorange needs to be corrected as follows: 
+ 
+ 
+𝒑𝑺𝑨𝑻
+𝒄
+= 𝒑𝑺𝑨𝑻 + 𝑐 ∙ 𝒅𝑻 − 𝒅𝒕𝒓𝒐𝒑,𝑺𝑨𝑻 − 𝒅𝒊𝒐𝒏,𝑺𝑨𝑻 
+   (3) 
+ 
+where 𝒑𝑺𝑨𝑻
+𝒄
+ represents the corrected pseudorange for the  
+
+Transmitter 5
+Transmitter 6
+Transmitter 3
+Transmitter 2
+Transmitter 4Transmitter 5
+Transmitter 6
+Transmitter 1
+Transmitter 3
+Transmitter 2
+Transmitter 4> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+Fig. 6. Flow chart of the single point positioning algorithm. 
+ 
+ 
+satellites, and 𝒑𝑺𝑨𝑻 represents the uncorrected pseudorange for 
+the satellites. In this work, the HAPS pseudorange is modeled 
+as the sum of the geometric range and the pseudorange 
+error,which represents the overall estimation error of the 
+parameters in the HAPS pseudorange equation. Accordingly, 
+the HAPS pseudorange does not need to be corrected. Due to 
+the Earth’s rotation, the positions of satellites and HAPS at the 
+signal emission time are different from their positions at the 
+signal reception time; this is known as the Sagnac effect [28]. 
+The coordinates of satellites/HAPS can be transformed from the 
+signal emission time to the signal reception time by [28] 
+ 
+ 
+∆𝑡𝑅𝑂𝑇 = 𝑡𝑟𝑥 − 𝑡𝑡𝑥 
+(4) 
+ 
+ 
+𝑃𝑖,𝑟𝑥 = 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)𝑃𝑖,𝑡𝑥 
+(5) 
+ 
+where ∆𝑡𝑅𝑂𝑇  denotes the signal propagation time, 𝑡𝑟𝑥 
+represents the signal reception time, 𝑡𝑡𝑥 represents the signal 
+emission time, 𝑃𝑖,𝑟𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates at the 
+signal reception time, 𝑃𝑖,𝑡𝑥 is the 𝑖𝑡ℎ satellite/HAPS coordinates 
+at the signal emission time, 𝜔𝐸 denotes the Earth’s rotation rate, 
+and 𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇) is known as the rotation matrix, which 
+is described as follows: 
+ 
+ 
+𝑀𝑅𝑂𝑇(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+= [
+cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+− sin(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+cos(𝜔𝐸 × ∆𝑡𝑅𝑂𝑇)
+0
+0
+1
+] . 
+ 
+ 
+(6) 
+The line-of-sight vector 𝒗, and the true range between ranging 
+sources and receiver 𝝆, are then calculated to compute the a 
+priori range residual vector 𝒃 and the design matrix 𝑯, where 
+ 
+ 
+𝒃 = 𝒑𝒄 − 𝝆 
+(7) 
+ 
+ 
+𝑯 = [𝒗, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1] 
+(8) 
+ 
+where 𝒑𝒄 is the corrected satellite pseudorange combined with 
+the corrected HAPS pseudorange, 𝟏𝑙𝑒𝑛𝑔𝑡ℎ(𝑷𝐜)×1  denotes a 
+column vector of length being the total number of visible 
+ranging sources, and 𝑷𝐜 is a vector containing the corrected 
+positions of the visible ranging sources (i.e., satellite + HAPS). 
+Finally, the least square solution is computed as follows: 
+ 
+ 
+𝑸 = (𝑯′𝑯)−1 
+(9) 
+ 
+ 
+𝒅𝒙 = 𝑸𝑯′𝒃 
+(10) 
+ 
+ 
+𝑑𝑡 = 𝒅𝒙(4)/𝑐 
+(11) 
+ 
+where Q is known as the covariance matrix, and 𝒅𝒙(4) denotes 
+the fourth element in the vector 𝒅𝒙. To prevent the algorithm 
+from getting numerical issues, we should ensure the term being 
+inversed in (9) is non-singular; in other words, the design matrix 
+𝑯  should be non-singular. With the extra observations by 
+utilizing HAPS as additional ranging sources, the chance of 𝑯 
+being singular is likely reduced; the non-singular design matrix 
+can be ensured by avoiding the use of collinear observations, 
+which means that two or more observations have about the 
+same azimuth and elevation angle. We observe that the term 
+being inversed in (9), 𝑯′𝑯, is a Hermitian, positive definite 
+matrix, therefore the Cholesky decomposition can be imposed 
+to reduce the computational complexity [26]. The covariance 
+matrix, 𝑸, is described by 
+ 
+ 
+𝑸 =
+[
+ 
+ 
+ 
+ 𝜎𝑥
+2
+𝜎𝑥𝑦
+𝜎𝑥𝑧
+𝜎𝑥𝑡
+𝜎𝑥𝑦
+𝜎𝑦
+2
+𝜎𝑦𝑧
+𝜎𝑦𝑡
+𝜎𝑥𝑧
+𝜎𝑦𝑧
+𝜎𝑧
+2
+𝜎𝑧𝑡
+𝜎𝑥𝑡
+𝜎𝑦𝑡
+𝜎𝑧𝑡
+𝜎𝑡
+2 ]
+ 
+ 
+ 
+ 
+ 
+(12) 
+
+Initialization
+PsAT-PHAPS,PRN,DOY
+x = 04x1
+Input
+dt = x(4)/c
+PHAPS, PsAT, dT,α
+dx = x + Inf
+stop = 0
+Exit
+No
+NsAr + NHAPs ≥ 4
+IYes
+No
+dx(1:3)|> 0.01
+Yes
+Finding parameters
+For sotellites
+For HAPS
+OsAT,dtrop.dion
+HAPS
+Applying elevation mask
+For sotellites
+For HAPS
+sAT,dtrep,dion,PRN,dT,PsAr
+HAPS,PHAPS
+Pseudorange correction
+PSAT-PHAPS
+Combining the
+corrected pseudoranges
+p' = [psar.phaes]
+国
+Correcting for the Sagnaceffect
+(i.e., Earth rotation)
+PSAT,PHAPS
+Combining the corrected
+ranging source positions
+P° = [PSAT.PHAPs]
+Finding parameters
+V,p,b,H,Q
+andno
+Computing the position solution
+using the Least Square method
+3p'x
+x,dt> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+where receiver coordinates x, y, z in the Earth-centered Earth- 
+fixed (ECEF) coordinate frame and the receiver clock offset, 
+respectively. The least square solution will be found when the 
+norm of the change in receiver position 𝒅𝒙(1: 3) is sufficiently 
+small. In this work, this threshold is set as 0.01 m. We use the 
+HDOP, the VDOP and the 3D positioning accuracy as the 
+metrics to show the advantage of the proposed HAPS-aided 
+GPS system; the 3D positioning accuracy is used to show the 
+applicability of the RAIM to the HAPS-aided GPS system. To 
+compute the HDOP, we need to convert the covariance matrix 
+into the local north-east-down (NED) coordinate frame, which 
+can be done with the following equations [29]: 
+ 
+ 
+𝑸𝑵𝑬𝑫 = 𝑹′𝑸̃𝑹 = [
+𝜎𝑛
+2
+𝜎𝑛𝑒
+𝜎𝑛𝑑
+𝜎𝑛𝑒
+𝜎𝑒
+2
+𝜎𝑒𝑑
+𝜎𝑛𝑑
+𝜎𝑒𝑑
+𝜎𝑑
+2
+]  
+(13) 
+ 
+ 
+𝑸̃ = [
+𝜎𝑥
+2
+𝜎𝑥𝑦
+𝜎𝑥𝑧
+𝜎𝑥𝑦
+𝜎𝑦
+2
+𝜎𝑦𝑧
+𝜎𝑥𝑧
+𝜎𝑦𝑧
+𝜎𝑧
+2
+] 
+(14) 
+ 
+ 
+𝑹 = [
+−sin 𝜆
+cos 𝜆
+0
+− cos 𝜆 sin 𝜑
+− sin 𝜆 sin 𝜑
+cos 𝜑
+cos 𝜆 cos 𝜑
+sin 𝜆 cos 𝜑
+sin 𝜑
+] 
+(15) 
+ 
+where 𝜎𝑛, 𝜎𝑒, and 𝜎𝑑 represent the receiver position errors in 
+the local north, east, and down directions, respectively. 𝜆 and 𝜑 
+represent the longitude and latitude of the receiver, 
+respectively. Then, the HDOP is described by 
+ 
+ 
+𝐻𝐷𝑂𝑃 = √𝜎𝑛2 + 𝜎𝑒2 
+(16)   
+        
+and the VDOP is described by 
+ 
+ 
+𝑉𝐷𝑂𝑃 = √𝜎𝑑
+2. 
+(17) 
+ 
+B. The Receiver Autonomous Integrity Monitoring (RAIM) 
+Algorithm 
+The RAIM algorithm is a signal selection algorithm that can 
+detect and even exclude abnormal observations using redundant 
+measurements. It can detect an abnormal observation when the 
+number of observations is at least five; it can exclude this 
+abnormal observation when the number of observations is at 
+least six. The RAIM algorithm is typically applied to multi-
+constellation GNSSs where the number of ranging sources is 
+more than enough to perform precise 3D localization, and it is 
+typically applied to cases where there likely exists at least one 
+observation that differs from the expected value significantly. 
+Such cases include urban areas, where the pseudorange  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+measurement is highly subject to the multipath effect. With the 
+assistance from HAPS, the chance of enabling the RAIM 
+function will likely increase. Typical RAIM algorithms tend to 
+use the standard deviation of the target observable, which is the 
+pseudorange measurement in our work. As knowledge of the 
+standard deviation of the satellite pseudorange is unavailable on 
+the receivers we use, in this work the RAIM algorithm is 
+implemented on the basis of [30], which considers a 𝐶/𝑁0-
+based variance model and a computationally efficient method, 
+namely the modified Danish estimation method. 2  The 
+implemented 𝐶/𝑁0-based RAIM algorithm is given in Alg. 1, 
+where 𝑁 denotes the number of visible ranging sources. The 
+input to this algorithm consists of the position fix computed 
+using the SPP algorithm 𝒙, and the 𝑪/𝑵𝟎 of the ranging source 
+signal. Since HAPS are located at much lower altitudes than 
+2 To the best of our knowledge, RAIM is the most common algorithm used 
+for integrity monitoring. Since we only have the 𝐶/𝑁0  data which can be 
+utilized for the integrity monitoring, we could not identify in the literature any 
+other appropriate RAIM-like algorithm for comparison. However, we believe 
+that the other RAIM algorithms would also be applicable if the knowledge of 
+the standard deviation of the satellite pseudorange happens to be available. 
+Algorithm 1 The 𝐶/𝑁0-based RAIM Algorithm 
+Input: The SPP estimated position solution 𝒙 and 𝑪/𝑵𝟎; 
+Output: The SPP and RAIM jointly estimated position 
+solution 𝒙̂. 
+1: 
+Initialize the parameters 𝑠𝑡𝑜𝑝 and 𝒅𝒙; 
+2: 
+while |𝒅𝒙(1: 3)| > 0.01 do 
+3: 
+ 
+Same procedures as the SPP algorithm until 
+“Finding parameters” after correcting for the 
+Sagnac effect; 
+4: 
+ 
+for 𝑖 =  1 to 𝑁 do 
+5: 
+ 
+ 
+if 𝑠𝑡𝑜𝑝 == 1 do 
+6: 
+ 
+ 
+ 
+Find the variance of the observation 𝑖, 𝑠𝑖, 
+according to (19); 
+7: 
+ 
+ 
+end if 
+8: 
+ 
+end for 
+9: 
+ 
+Find the weight matrix 𝑾 and the design matrix 
+𝑯 , and calculate the covariance matrix 𝑸 
+according to (20); 
+10: 
+ 
+Calculate the change in estimates 𝒅𝒙 according 
+to (21), and update the position solution 𝒙; 
+11: 
+ 
+Calculate the pseudorange residual 𝒗̂ according 
+to (22), and the covariance matrix of the residuals 
+𝑪𝒗̂ according to (24); 
+12: 
+ 
+for 𝑖 =  1 to 𝑁 do 
+13: 
+ 
+ 
+Find the normalized residual of observation 𝑖 
+at the current iteration 𝑘, 𝑤̅𝑖,𝑘  according to 
+(26); 
+14: 
+ 
+ 
+if |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2) do 
+15: 
+ 
+ 
+ 
+Update the variance of the observation 𝑖 
+for the next iteration 𝑘 + 1 , 𝜎𝑖,𝑘+1
+2
+, 
+according to (25); 
+16: 
+ 
+ 
+end if 
+17: 
+ 
+end for 
+18: 
+end while 
+
+> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+satellites, in practice the 𝐶/𝑁0 value of the HAPS might be 
+higher than that for any satellite. As it is possible that a handful 
+of HAPS signals might suffer from severe multipath effects, we 
+can exclude those HAPS signals whose 𝐶/𝑁0 values are much 
+lower than the higher ones. In this work, the multipath effect is 
+not explicitly simulated for the HAPS signal; therefore, we 
+assume that the 𝐶/𝑁0 of each HAPS is equal to the maximum 
+𝐶/𝑁0 value of the available satellites at each epoch, meaning 
+that the signal quality for a HAPS will always be better than 
+that for any satellites. The variance covariance matrix (VCM) 
+𝜮 of the observations (pseudoranges) 𝒑 is defined as follows: 
+ 
+ 
+𝜮 = 𝑑𝑖𝑎𝑔(𝑠1, 𝑠2, … , 𝑠𝑛) 
+(18) 
+ 
+ 
+𝑠𝑖 = 10 + 1502 ∗ 10(−𝐶/𝑁0,𝑖)/10 
+(19) 
+ 
+where 𝑠𝑖 denotes the variance of the observation 𝑖. We assume 
+that the observations are uncorrelated, and that the errors follow 
+the normal distribution with 𝑁(𝟎, 𝜮). The weight matrix, 𝑾, is 
+defined as the inverse of the VCM, 𝜮−1 . The least square 
+equations become 
+ 
+ 
+𝑸 = (𝑯′𝑾𝑯)−1 
+(20) 
+ 
+ 
+𝒅𝒙 = 𝑸𝑯′𝑾𝑷. 
+(21) 
+ 
+The least square residuals of the pseudorange 𝒗̂ can be obtained 
+as follows: 
+ 
+ 
+𝒗̂ = 𝑯 ∙ 𝒅𝒙 − 𝑷 
+(22) 
+ 
+ 
+𝑷 = 𝒑𝒄 − 𝝆 
+(23) 
+ 
+where 𝑯  represents the design matrix, 𝒅𝒙  represents the 
+change in estimates, 𝒑𝒄 denotes the corrected pseudoranges, 
+and 𝝆 denotes the geometric range between ranging sources 
+and the receiver. The covariance matrix of the residuals, 𝑪𝒗̂, is 
+computed as 
+ 
+ 
+𝑪𝒗̂ = 𝜮 − 𝑯(𝑯𝑇𝜮−1𝑯)−1𝑯𝑇. 
+(24) 
+ 
+To detect and exclude the abnormal observations, we follow the 
+modified Danish estimation method proposed in [30].  
+ 
+ 
+𝜎𝑖,𝑘+1
+2
+= 𝜎𝑖,0
+2 ∙ {exp (
+|𝑤̅𝑖,𝑘|
+𝑇 ) ,    |𝑤̅𝑖,𝑘| > 𝑛1−(𝛼0/2)
+1,                      |𝑤̅𝑖,𝑘| ≤ 𝑛1−(𝛼0/2)
+ 
+(25) 
+ 
+with 
+ 
+ 
+𝑤̅𝑖,𝑘 =
+𝒗̂𝑖,𝑘
+√(𝑪𝒗̂𝒊,𝟏)𝑖𝑖
+ 
+(26) 
+ 
+where 𝜎𝑖,0
+2  denotes the a priori variance of the observation 𝑖 
+(i.e., s𝑖), 𝑤̅𝑖,𝑘 denotes the normalized residual of observation 𝑖 
+at iteration 𝑘,√(𝑪𝒗̂𝒊,𝟏)𝑖𝑖  represents the standard deviation of 
+observation 𝑖 from the first iteration, 𝑛1−(𝛼0/2) denotes the 𝛼0-
+quantile of the standard normal distribution, which is also called 
+the critical value, 𝛼0  is the predetermined false alarm rate 
+which is 0.5 % in this work. The modified Danish method is an 
+iteratively reweighted LS algorithm that implements a robust 
+estimator. This method detects and excludes abnormal 
+observations by comparing the absolute value of each 
+normalized pseudorange residual, |𝑤̅𝑖,𝑘|, with the critical value, 
+𝑛1−(𝛼0/2), in each iteration. The variances for the observations 
+whose normalized residuals are greater than the critical value 
+are multiplied with exponential terms, making the variances of 
+those observations larger, hence lowering the weight of those 
+observations. By iteratively multiplying the variances of the 
+abnormal observations by exponential terms, the weight of the 
+abnormal observations will likely become much smaller than 
+that of the normal observations; therefore, the abnormal 
+observations can be considered as being excluded. 
+III. SIMULATION OF THE HAPS-AIDED GPS SYSTEM 
+In this section, we will describe the simulation setup used in 
+the Skydel GNSS software [27] and present the simulation 
+results in terms of 3D positioning accuracy for several hybrid 
+systems and two standalone systems in both a suburban 
+scenario and a dense urban scenario. 
+ 
+A. Simulation Setup 
+The system model is established using the default Earth 
+orientation parameters of the Skydel GNSS simulation software 
+[27], which considers all GPS satellites orbiting around the 
+Earth and transmitting the L1 C/A code. The Saastamoinen 
+model is chosen to emulate the tropospheric effect, and the 
+Klobuchar model is chosen to emulate the ionospheric effect 
+using the default Klobuchar parameters that come with the 
+software. The output from Skydel contains the ECEF 
+coordinates of satellites at the signal emission time, the 
+ionospheric corrections, the tropospheric corrections, the 
+satellite clock offsets, the ECEF coordinates of the receiver, the 
+signal emission time, and so forth, at each time stamp from the 
+start of the simulation. The receiver clock offset in the 
+simulation is zero by default. The correction terms in the 
+pseudorange equation of satellite including the satellite orbit 
+error, the multipath error, and the receiver noise are not 
+separately considered in the simulation; instead, a pseudorange 
+error is introduced to reflect the presence of those effects. The 
+pseudorange error of satellite is featured using the built-in first 
+order Gauss-Markov process with a default time constant of 10 
+s, and the standard deviation of 6 m. The continuous model for 
+the first order Gauss-Markov process is described by [31]:  
+ 
+ 
+������̇ = −
+1
+𝑇𝑐 ������̇ + 𝑤 
+(27) 
+ 
+where ������̇  represents a random process with zero mean, 
+correlation time 𝑇𝑐, and noise 𝑤. The autocorrelation of the first  
+
+> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+TABLE Ⅱ 
+DETAILS OF THE SIMULATION SETUP 
+ 
+Item 
+Processing strategy 
+Earth orientation parameter 
+Software default Earth orientation parameters 
+Satellite signal 
+GPS L1 C/A 
+Tropospheric model 
+Saastamoinen model 
+Ionospheric model 
+Klobuchar model 
+Sampling rate 
+12.5 MS/s 
+Satellite pseudorange error 
+1st order Gauss-Markov process (time constant = 10 s; standard deviation = 6 m) 
+HAPS pseudorange error 
+Gaussian noise (mean = 0 m; std = 2 m for the suburban scenario; std = 5 m for the dense urban 
+scenario) 
+Number of GPS satellites (dense urban 
+scenario) 
+8-10 
+Number of GPS satellites (suburban 
+scenario) 
+4 
+Total number of HAPS 
+6 
+order Gauss-Markov process is described by [32]: 
+ 
+𝑅(𝛥𝑡) = 𝜎2𝑒−|𝛥𝑡|
+𝜏  (28) 
+ 
+where 𝛥𝑡 represents the sampling interval, 𝜎 and 𝜏 denote the 
+standard deviation and the time constant of the first order 
+Gauss-Markov process, respectively. The characteristics of the  
+pseudorange errors for satellites are set to be the same in both 
+the suburban scenario and the dense urban scenario. However, 
+we randomly select four satellites in the dense urban scenario 
+in order to emulate the dense urban area in a rather simple way. 
+We verify that by doing so, the standard deviation of the 3D 
+positioning accuracy for the GPS-only system in the simulation 
+is close to that in the physical experiment. The pseudorange 
+error for the HAPS is modeled using the Gaussian noise with 
+standard deviations of 2 m and 5 m, representing the suburban 
+and the dense urban scenario, respectively. Under the 
+assumption that the overall estimation error of the HAPS is less 
+than that of the satellite, the standard deviation for the HAPS 
+pseudorange error is deliberately set to be smaller than that of 
+the satellite pseudorange error in the suburban and dense urban 
+scenarios. To investigate the impact of the number of HAPS on 
+the positioning performance of the HAPS-aided GPS system, 
+we consider four hybrid systems with different numbers of 
+randomly selected HAPS at each epoch. We also consider the 
+HAPS-only system for the completeness of a research problem. 
+Under this setting, we examine the 3D positioning performance 
+of different systems in the suburban and dense urban scenarios. 
+In the suburban scenario, the number of visible satellites varies 
+between eight and ten, while in the dense urban scenario the 
+number of visible satellites is set to four. The details of the 
+simulation setup are given in Table Ⅱ. 
+ 
+B. Simulation Results 
+Fig. 7 shows the cumulative distribution function (CDF) of 
+the 3D positioning accuracy for different positioning systems in 
+the suburban scenario. With the assumption that the 
+pseudorange error for a HAPS is smaller than that of a satellite, 
+we can see from Fig. 7 that all the hybrid systems (HAPS +  
+ 
+Fig. 7. CDF of the 3D positioning accuracy for different systems (suburban 
+scenario). 
+ 
+ 
+Fig. 8. CDF of the 3D positioning accuracy for different systems (dense urban 
+scenario). 
+ 
+ 
+GPS) outperform the GPS-only system; the more HAPS, the 
+better the positioning performance of the HAPS-aided GPS  
+
+Suburbanscenario(stdofHAPSpseudorangeerror=2m)
+0.9
+0.8
+0.7
+0.6
+D
+0.5
+0.4
+0.3
+0.2
+GPS-only system
+1-HAPS with GPS system
+2-HAPS with GPS system
+3-HAPS with GPS system
+0.1
+4-HAPS with GPS system
+4-HAPS-only system
+0
+5
+10
+15
+20
+25
+30
+3D positional accuracy (m) in local NED frameDenseurbanscenario(stdofHAP
+pseudorangeerror=5m)
+0.9
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+4-GPS-only system
+1-HAPS with 4-GPS system
+2-HAPS with 4-GPS system
+0.1
+3-HAPS with 4-GPS system
+4-HAPS with 4-GPS system
+4-HAPS-only system
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+3D positionalaccuracy (m)inlocal NEDframe> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+Fig. 9. Positioning performance of M8T and M8U for horizontal and vertical 
+planes. 
+ 
+ 
+system. Nevertheless, we observe that the positioning 
+performance of the 4-HAPS-only system, where all ranging 
+sources have a much smaller pseudorange error than the 
+satellite, is not the best and can occasionally be very poor. This 
+may be due to the following reasons: 1) the 4-HAPS-only 
+system has much fewer ranging sources for computing receiver 
+positions; 2) the HAPS geometry can be poor occasionally since 
+we are randomly selecting four HAPS at each epoch. There are 
+several cases where the HAPS geometry is considered poor. For 
+example, when the four randomly selected HAPS are on the 
+same side of the receiver. The CDF of the 3D positioning 
+accuracy for different positioning systems in the dense urban 
+scenario is shown in Fig. 8, from which we can see a similar 
+trend that the more HAPS the better the positioning 
+performance of the HAPS-aided GPS system. In the dense 
+urban scenario, where only four GPS satellites are selected for 
+positioning, the 4-HAPS-only system achieves better 
+positioning performance than the 4-GPS-only system due to the 
+better signal quality for the HAPS. However, we should 
+consider using the HAPS-aided GPS system for the best 
+positioning performance. 
+IV. FIELD EXPERIMENTS 
+To verify and support the simulation results, we process the 
+raw GNSS data collected using two commercial GNSS 
+receivers. In this section, we present the experiment setup, the  
+modeling of HAPS signals, and the HAPS pseudorange error. 
+We also provide an analysis of the DOP and the 3D positioning 
+accuracy for both the GPS-only system and the HAPS-aided 
+GPS system. 
+ 
+A. Experiment Setup 
+The raw GNSS data is collected along a vehicle trajectory 
+similar to the one shown in Fig. 2, except for a slight difference 
+due to a partial road closure on the day of data collection. The 
+raw GNSS data is collected using both the Ublox EVK-M8T  
+TABLE Ⅲ 
+EVK-M8T GNSS UNIT SPECIFICATIONS [35] 
+ 
+Parameter 
+Specification 
+Serial Interfaces 
+1 USB V2.0 
+1 RS232, max.baud rate 921,6 kBd 
+DB9 +/- 12 V level 
+14 pin – 3.3 V logic 
+1 DDC (I2C compatible) max. 400 kHz 
+1 SPI-clock signal max. 5,5 MHz – SPI DATA 
+max. 1 Mbit/s 
+Timing Interfaces 
+2 Time-pulse outputs 
+1 Time-mark input 
+Dimensions 
+105 × 64 × 26 mm 
+Power Supply 
+5 V via USB or external powered via extra power 
+supply pin 14 (V5_IN) 13 (GND) 
+Normal Operating 
+Temperature 
+−40℃ to +65℃ 
+ 
+ 
+and the Ublox EVK-M8U GNSS units. The Ublox EVK-M8T 
+unit is a timing product that can provide users with precise 
+timing information for post-processing; the Ublox EVK-M8U 
+unit is a dead reckoning product equipped with inertial 
+measurement units (IMUs) such that the positioning 
+performance of this product will not be degraded much even in 
+the dense urban area. Fig. 9 shows the positioning performance 
+along both horizontal and vertical planes for both M8T and 
+M8U during the entire observation period. From Fig. 9, we can 
+see that M8U outperforms M8T for both horizontal positioning 
+accuracy and vertical positioning accuracy. As only M8T 
+provides the timing information required for post-processing, 
+the receiver positions computed by EVK-M8U are used as the 
+ground truth data for the analysis of the positioning 
+performance. The raw GNSS data is processed using the single 
+point positioning algorithm described in Section Ⅱ. Table Ⅲ 
+gives the specifications of the EVK-M8T GNSS unit. To 
+emulate realistic LOS conditions for HAPS in the urban area, 
+the LOS probability as a function of the HAPS elevation angle 
+in the urban area is implemented on the basis of [33] and [34]. 
+It is worth mentioning that the LOS probability model for the 
+HAPS provided by [33] is proposed based on the city of 
+Chicago; imposing this LOS probability model for the dense 
+urban area of Ottawa might be too harsh considering their 
+incompatible city scales. Since there is no LOS probability 
+corresponding to the suburban area in [34], the one for rural area 
+in [34] is used as the LOS probability for the HAPS in the 
+suburban area. The LOS probability for the HAPS in the rural 
+area in [34] is verified as being consistent with the LOS 
+probability for the HAPS in the suburban area in [33]. The 
+pseudorange of the HAPS in the experiment is modeled as the 
+addition of the geometric range between the satellite and 
+receiver, the receiver clock offset multiplied by the speed of 
+light, and the pseudorange error representing the sum of all 
+kinds of estimation errors. The pseudorange errors for the 
+HAPS in the suburban and dense urban areas are simulated as 
+Gaussian noise with zero mean and standard deviations of 2 m 
+and 5 m, respectively. Since the vehicle trajectory involves both  
+
+The positioning performance of M8T and M8U
+12
+Horizontal positioning accuracy (M8T)
+Horizontalpositioningaccuracy(M8U)
+Vertical positioningaccuracy(M8T)
+10
+Verticalpositioningaccuracy(M8U)
+g accuracy (m)
+8
+6
+Positioning
+2
+0
+100
+200
+300
+400
+500
+600
+700
+epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+ 
+Fig. 11. HDOP (top) and VDOP (bottom). 
+ 
+ 
+Fig. 10. HAPS availability during the entire observation period. 
+ 
+ 
+suburban and dense urban areas, the entire route is divided into 
+two segments, where the first segment is considered as the 
+suburban scenario while the second segment is considered as 
+the dense urban scenario (see Fig. 2). By observing the 
+positioning performance of the GPS-only system using the real 
+GPS data, the LOS probability for the suburban area is applied 
+to the HAPS for epochs less than 380 s, and the LOS probability 
+for the dense urban area is applied to the HAPS for epochs 
+greater than or equal to 380 s. Since the GNSS receivers we use 
+do not provide an accurate receiver clock offset with respect to 
+the GPS time, the receiver clock offset in each epoch is 
+estimated by using the ground truth receiver positions provided 
+from Ublox EVK-M8U and the precise timing information, 
+such as the receiver clock drift and the receiver clock bias, 
+provided from Ublox EVK-M8T. We should note that the 
+pseudorange of the HAPS in the experiment is modeled as a 
+function of the receiver clock offset, which is estimated with 
+best effort. Nevertheless, additional errors should be expected 
+in the pseudorange of the HAPS with the magnitude depending 
+on the quality of all visible satellite signals and the ground truth 
+receiver position. As we would expect the quality of the satellite  
+signals in the suburban area to be better than that in the dense 
+urban area, we should also expect the receiver clock offset to be 
+estimated with higher accuracy in the suburban area than in the 
+dense urban area.  
+ 
+B. Experiment Results 
+With more ranging sources, we should expect the availability 
+of the HAPS-aided GPS system to be higher than the GPS-only 
+system. The availability of HAPS and GPS satellite during the 
+entire course of observation is shown in Fig. 10. As we can see, 
+the availability of the HAPS-aided GPS system during the 
+entire observation period is 100 %, while the availability of the 
+GPS-only system is 99.71 % as there are two epochs (circled in 
+a black ellipse) where the number of GPS satellites is three. 
+While the difference between the availability of the HAPS-
+aided GPS system and the GPS-only system is not significant, 
+this is probably because Ottawa is a relatively small metro city 
+compared to the metro cities like Chicago. In the following, we 
+first present a comparison of the HDOP and VDOP between the 
+HAPS-aided GPS system and the GPS-only system. Next, we 
+analyze the 3D positioning performance for both the GPS-only 
+system and the HAPS-aided GPS system. To show the 
+improvements brought by the RAIM algorithm, we compare the 
+RAIM-enabled positioning systems, where both the SPP and 
+the RAIM algorithms are applied, with the RAIM-disabled 
+positioning systems, where only the SPP algorithm is applied. 
+ 
+1) Dilution of Precision Analysis 
+Fig. 11 shows the HDOP and VDOP of the GPS-only system 
+and the HAPS-aided GPS system. As we can see, both the 
+HDOP and VDOP of the HAPS-aided GPS system are better 
+than that of the GPS-only system. In particular, we notice that 
+there are fewer spikes on the HDOP and VDOP performance of 
+the HAPS-aided GPS system, which demonstrates that the  
+
+15
+GPS-only system
+HAPS-aided GPS system
+HDOP
+5
+0
+0
+100
+200
+300
+400
+500
+600
+700
+epoch (s)20
+GPS-only system
+HAPS-aided GPS system
+(w)
+15
+VDOI
+10
+5
+DAM
+0
+0
+100
+200
+300
+400
+500
+600
+700
+epoch(s)11
+10
+HAPSs/satellites
+9
+8
+visible
+6
+5
+of
+Number
+4
+3
+2
+HAPS
+GPSsatellite
+1
+0
+100
+200
+300
+400
+500
+600
+700
+epoch (s)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+ 
+Fig. 12. CDF of the 3D positioning accuracy in the suburban area. 
+ 
+ 
+Fig. 13. CDF of the 3D positioning accuracy in the dense urban area. 
+ 
+ 
+HDOP and VDOP performance of the HAPS-aided GPS system 
+is more stable than the GPS-only system.  
+ 
+2) 3D Positioning Accuracy Analysis 
+The CDFs of the 3D positioning accuracy for both the 
+suburban area and the dense urban area are shown in Fig. 12 
+and Fig. 13, respectively. As we can see, without enabling the 
+RAIM, the 90-percentile 3D positioning accuracy of the GPS-
+only system can be improved by 36 % in the suburban area, and 
+33.64 % in the dense urban area with the assistance from the 
+HAPS. With the RAIM turned on, we can observe that the 
+positioning performance of both the GPS-only system and the 
+HAPS-aided GPS system can be further improved. Yet we 
+notice that the improvement brought by the RAIM in the 
+suburban area for the GPS-only system is almost negligible, 
+while it is more tangible for the HAPS-aided GPS system. This 
+is because the quality of signals in the suburban area tends to be 
+great, and the HAPS-aided GPS system has a relatively higher 
+chance to enable the RAIM as there are more ranging sources 
+in the system. This observation is also applicable to the dense 
+urban scenario where the 90-percentile 3D positioning accuracy 
+of the HAPS-aided GPS system is improved by 45.2 %, which 
+is much more significant than that for the GPS-only system. For 
+the dense urban scenario where the multipath is severe, the 
+RAIM algorithm plays a more significant role, especially in the 
+HAPS-aided GPS system. The reason behind this is that the 
+number of visible satellites in the dense urban area is low, 
+making the RAIM algorithm for the GPS-only system less 
+effective. Since the implemented RAIM algorithm detects and 
+excludes an abnormal observation by multiplying its variance 
+with an exponential term if the absolute value of its normalized 
+pseudorange residual surpasses the critical value, we count the 
+number of times where the absolute value of the normalized 
+pseudorange residuals surpass the critical value for both 
+systems considered and for both the suburban scenario and the 
+dense urban scenario. For convenience, we rephrase the number 
+of times where the absolute value of the normalized 
+pseudorange residuals surpass the critical value as the number 
+of times the RAIM is enabled. With the system model 
+considered in this work, we find that the number of times the 
+RAIM is enabled for the GPS-only system is roughly 23.84 % 
+as many as the number of times the RAIM is enabled for the 
+HAPS-aided GPS system in the suburban area; and the number 
+of times the RAIM is enabled for the GPS-only system is about 
+50.72 % as many as the number of times the RAIM is enabled 
+for the HAPS-aided GPS system in the dense urban area. This 
+demonstrates the applicability of the RAIM algorithm on the 
+HAPS-aided GPS system, especially in the dense urban area. 
+V. CONCLUSION 
+HAPS have a number of advantages over satellites, including 
+(but not limited to) lower latency, lower pathloss, smaller 
+pseudorange errors, and HAPS can provide continuous 
+coverage to reduce the number of handovers for users. This 
+makes HAPS an excellent candidate to serve as another type of 
+ranging source. Since urban areas are where GNSS positioning 
+performance degrades severely, while also being where most 
+people live, deploying several HAPS as additional ranging 
+sources above metropolitan cities would improve GNSS 
+positioning performance and maximize the profit of the extra 
+payloads on HAPS. From both the simulation and physical 
+experiment results, we observed that HAPS can indeed improve 
+the HDOP, the VDOP, and the 3D positioning accuracy of a 
+legacy GNSS. With the system model considered in this work, 
+we showed that the 90-percentile 3D positioning accuracy of 
+the GPS-only system can be improved by around 35 % in both 
+suburban and dense urban areas. We demonstrated the 
+applicability of the RAIM algorithm for the HAPS-aided GPS 
+system, especially in the dense urban areas. To enhance the 
+simulation of the HAPS-aided GPS, the receiver clock offset 
+should be estimated with higher accuracy. We think the 
+effectiveness of the RAIM algorithm can be improved if the 
+standard deviation of the target observable is available. To 
+further improve the positioning performance for urban areas, 
+we can make use of terrestrial signals such as cellular network 
+signals and multipath signals. This would constitute a vertical 
+
+Suburbanarea
+0.9
+0.8
+0.7
+0.6
+DF
+0.5
+0.4
+0.3
+0.2
+GPS-only system (RAIM off)
+GPS-only system (RAIM on)
+0.1
+HAPS-aided GPS system (RAIM off)
+HAPS-aidedGPSsystem (RAIMon)
+0
+0
+5
+10
+15
+20
+25
+3Dpositioningerror(m)Dense urban area
+0.9
+0.8
+0.7
+0.6
+DF
+0.5
+0.4
+0.3
+0.2
+GPS-only system (RAIM off)
+GPS-only system (RAIM on)
+0.1
+HAPS-aided GPS system (RAIM off)
+HAPS-aided GPS system (RAIM on)
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+3D positioning error (m)> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+heterogeneous network (V-Het-Net) positioning system, which 
+we believe will yield a lower VDOP based on the DOP 
+illustration presented in this paper. 
+ACKNOWLEDGMENT 
+The Skydel software is a formal donation from Orolia to 
+Carleton University. 
+REFERENCES 
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+Canada, 2022. 
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+Simulation 
+Software,” 
+Orolia, 
+Available: 
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+87, pp. 172-177, 2016. 
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+local Cartesian coordinate systems and commutative diagrams,” in Proc. 
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+comparison between different error modeling of MEMS applied to 
+GPS/INS integrated systems,” Sens., vol. 13, no. 8, pp. 9549-9588, Jul. 
+2013.  
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+2020, pp. 481-489.  
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+Conf. Antennas Propag. (EuCAP), Krakow, Poland, 2019, pp. 1-5. 
+[34] S. Alfattani, W. Jaafar, Y. Hmamouche, H. Yanikomeroglu, and A. 
+Yongacoglu, “Link budget analysis for reconfigurable smart surfaces in 
+aerial platforms,” IEEE Open J. Commun. Soc., vol. 2, pp. 1980-1995, 
+2021. 
+[35] “EVK-M8T user guide,” Ublox, May 2018. Accessed on: Dec. 31, 2022. 
+[Online]. 
+Available: 
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+blox.com/sites/default/files/products/documents/EVK-
+M8T_UserGuide_%28UBX-14041540%29.pdf 
+ 
+ 
+
+> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 
+ 
+Hongzhao 
+Zheng 
+(Member, 
+IEEE) 
+received the B. Eng. (Hons.) degree in 
+engineering physics from the Carleton 
+University, Ottawa, ON, Canada, in 2019. 
+He is currently a PhD student at Carleton 
+University. His research interest is the 
+urban positioning using sensor-enabled 
+heterogeneous wireless infrastructure.  
+ 
+ 
+Mohamed Atia (Senior Member, IEEE) 
+received the B.S. and M.Sc. degrees in 
+computer systems from Ain Shams 
+University, Cairo, Egypt, in 2000 and 
+2006, respectively, and the Ph.D. degree 
+in electrical and computer engineering 
+from Queen’s University, Kingston, ON, 
+Canada, in 2013. He is currently an 
+Associate Professor with the Department of Systems and 
+Computer Engineering, Carleton University. He is also the 
+Founder and the Director of the Embedded and Multi-Sensory 
+Systems Laboratory (EMSLab), Carleton University. His 
+research interests include sensor fusion, navigation systems, 
+artificial intelligence, and robotics. 
+ 
+ 
+Halim Yanikomeroglu (Fellow, IEEE) 
+received the BSc degree in electrical and 
+electronics engineering from the Middle 
+East Technical University, Ankara, Turkey, 
+in 1990, and the MASc degree in electrical 
+engineering (now ECE) and the PhD degree 
+in electrical and computer engineering from 
+the University of Toronto, Canada, in 1992 
+and 1998, respectively. Since 1998 he has 
+been with the Department of Systems and Computer 
+Engineering at Carleton University, Ottawa, Canada, where he 
+is now a Full Professor. His research interests cover many 
+aspects of wireless communications and networks. He has given 
+110+ invited seminars, keynotes, panel talks, and tutorials in the 
+last five years. Dr. Yanikomeroglu’s collaborative research 
+with industry resulted in 39 granted patents. Dr. Yanikomeroglu 
+is a Fellow of the IEEE, the Engineering Institute of Canada 
+(EIC), and the Canadian Academy of Engineering (CAE). He is 
+a Distinguished Speaker for the IEEE Communications Society 
+and the IEEE Vehicular Technology Society, and an Expert 
+Panelist of the Council of Canadian Academies (CCA|CAC).  
+Dr. Yanikomeroglu is currently serving as the Chair of the 
+Steering Committee of IEEE’s flagship wireless event, 
+Wireless Communications and Networking Conference 
+(WCNC). He is also a member of the IEEE ComSoc GIMS, 
+IEEE ComSoc Conference Council, and IEEE PIMRC Steering 
+Committee. He served as the General Chair and Technical 
+Program Chair of several IEEE conferences. He has also served 
+in the editorial boards of various IEEE periodicals.  
+Dr. Yanikomeroglu received several awards for his research, 
+teaching, and service, including the IEEE ComSoc Fred W. 
+Ellersick Prize (2021), IEEE VTS Stuart Meyer Memorial 
+Award (2020), and IEEE ComSoc Wireless Communications 
+TC Recognition Award (2018). He received best paper awards 
+at IEEE Competition on Non-Terrestrial Networks for B5G and 
+6G in 2022 (grand prize), IEEE ICC 2021, IEEE WISEE 2021 
+and 2022. 
+ 
+ 
+ 
+

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+Graph Attention with Hierarchies for Multi-hop Question Answering
+Yunjie He∗
+University College London
+yunjie.he.17@ucl.ac.uk
+Ieva Stali¯unait˙e†
+Accelex Technology
+ieva.staliunaite@gmail.com
+Philip John Gorinski
+Huawei Noah’s Ark Lab, London
+philip.john.gorinski@huawei.com
+Pontus Stenetorp
+University College London
+pontus@stenetorp.se
+Abstract
+Multi-hop QA (Question Answering) is the
+task of finding the answer to a question across
+multiple documents. In recent years, a number
+of Deep Learning-based approaches have been
+proposed to tackle this complex task, as well
+as a few standard benchmarks to assess mod-
+els’ Multi-hop QA capabilities. In this paper,
+we focus on the well-established HotpotQA
+benchmark dataset, which requires models to
+perform answer span extraction as well as sup-
+port sentence prediction. We present two ex-
+tensions to the state-of-the-art Graph Neural
+Network (GNN) based model for HotpotQA,
+Hierarchical Graph Network (HGN): (i) we
+complete the original hierarchical structure by
+introducing new edges between the query and
+context sentence nodes; (ii) in the graph prop-
+agation step, we propose a novel extension
+to Hierarchical Graph Attention Network –
+GATH (Graph ATtention with Hierarchies) –
+that makes use of the graph hierarchy to up-
+date the node representations in a sequential
+fashion.
+Experiments on HotpotQA demon-
+strate the efficiency of the proposed modifica-
+tions and support our assumptions about the
+effects of model-related variables.
+1
+Introduction
+Question Answering (QA) tasks can be classified
+into single-hop and multi-hop ones, depending on
+the complexity of the underlying reasoning. Dif-
+ferent from single-hop QA (Rajpurkar et al., 2016;
+Trischler et al., 2017; Lai et al., 2017), where ques-
+tions can be answered given a single paragraph or
+single sentence in the context, multi-hop QA re-
+quires us to retrieve and reason over scattered infor-
+mation from multiple documents, as demonstrated
+in Figure 1. There are many methods proposed for
+addressing the multi-hop QA problem. One type of
+∗Work carried out as part of MSc thesis supervised by
+Huawei Noah’s Ark Lab, London
+†Work carried out while working at Huawei Noah’s Ark
+Lab, London
+Question:
+Where did the form of music played by Die
+Rhöner Säuwäntzt originate?
+Answer:
+United States
+Supports:
+Document 9
+s1:
+Die Rhöner Säuwäntzt are a Skiffle-
+Bluesband from Eichenzell-Lütter in Hessen,
+Germany.
+Document 4
+s1: Skiffle is a music genre with jazz, blues,
+folk and American folk influences [...]
+s2: Originating as a term in the United States
+in the first half of the 20th century [...]
+Figure 1: Example of a multi-hop answer and support
+prediction, as found in HotpotQA.
+these recent approaches extends well-performing
+single-hop machine reading comprehension mod-
+els to be multi-hop, such as DecompRC (Min et al.,
+2019) and QFE (Nishida et al., 2019).
+The other avenue is to develop models specifi-
+cally aimed at multi-hop QA. Among those, Graph
+Neural Networks (GNNs) have recently garnered a
+lot of attention. In GNN-based approaches, gaphs
+are employed to represent query and context con-
+tents (nodes) and the underlying relationships be-
+tween them (edges). Information between nodes is
+simultaneously propagated via the edges with the
+help of a variety of GNNs, such as Graph Convolu-
+tional Network (GCN) (Kipf and Welling, 2017),
+Graph Attention Network (GAT) (Veliˇckovi´c et al.,
+2017), or Graph Recurrent Network (GRN) (Song
+et al., 2018b). With these GNNs, node representa-
+tions are obtained conditioned on the question and
+context documents, and used for the QA task.
+In this paper, we focus on one particular GNN ap-
+proach designed for the Hotpot QA benchmark, the
+Hierarchical Graph Network (HGN) introduced in
+Fang et al. (2020). HGN constructs a hierarchical
+graph that integrates nodes from different granu-
+larity levels (question/paragraph/sentence/entity).
+The edges in the graph capture the interactions be-
+tween the information from heterogeneous levels
+of the hierarchy. This hierarchical graph structure
+arXiv:2301.11792v1  [cs.CL]  27 Jan 2023
+
+has been shown to be crucial to the model’s remark-
+able performance1 on both finding scattered pieces
+of supporting information across documents and
+the answer span prediction.
+The contribution of this work is three-fold: (i)
+we extend the edges of HGN with a new edge type
+between the query and sentences, completing its
+original structure; (ii) we introduce a novel exten-
+sion of the Graph Attention Network – Graph At-
+tention with Hierarchies (GATH). GATH allows for
+making use of the explicit hierarchical graph struc-
+ture, by propagating information through the graph
+in a sequential fashion based on the hierarchy’s
+levels, rather than updating all nodes simultane-
+ously. (iii) We perform initial experiments on the
+HotpotQA benchmark, providing evidence of the
+effectiveness of our proposed extensions.
+Code related to graph completion and GATH
+will be made publicly available at redacted.
+2
+Background
+To solve the multi-hop QA problem, two general
+research paths have been studied. The first direc-
+tion focuses on extending the successful single-
+hop machine reading comprehension method to the
+multi-hop QA. DecompRC (Min et al., 2019) de-
+composes the multi-hop reasoning problem into
+multiple single-hop sub-questions based on span
+predictions and applied traditional machine reading
+comprehension techniques on these sub-questions
+to obtain answers to the question. Query-Focused
+Extractor (QFE) (Nishida et al., 2019) reformulates
+the multi-hop QA task as a query-focused summa-
+rization task based on the extractive summarization
+model (Chen and Bansal, 2018).
+The second research direction natively addresses
+the task as a multi-hop setting, and directly tries
+to gather the information from all context doc-
+uments in order to answer the question. Many
+approaches based on the transformer architecture
+(Vaswani et al., 2017) address the multi-hop QA
+task as simply one of attention between all words
+in all available documents. In such approaches,
+the problem quickly becomes intractable due to
+the long inputs involved, and they thus typically
+focus on alleviating the problems of using a full
+attention mechanism. The Longformer (Beltagy
+et al., 2020), for example, introduces a windowed
+attention mechanism to localise the problem, allow-
+1At the time of writing, HGN achieves SOTA results on
+HotpotQA, for GNN-based approaches.
+ing for much longer input sequences to be handled
+than with standard BERT-based language models
+(Devlin et al., 2018).
+However, recently more research effort has been
+put toward approaches that employ Graph Neural
+Networks, which allow for organising information
+from various sources into a graph structure before
+addressing the core task of Question Answering,
+mitigating the need for very-long-distance attention
+functions.
+Coref-GRU (Dhingra et al., 2018) integrates mul-
+tiple evidence associated with each entity mention
+by incorporating co-reference information using
+a collection of GRU layers of a gated-attention
+reader (Dhingra et al., 2017). However, Coref-
+GRU only leverages co-references local to a sen-
+tence but ignores other useful global information.
+To address this problem, MHQA-GRN and MHQA-
+GCN (Song et al., 2018a) integrate evidence in a
+more complex entity graph, with edges that also
+connect global evidence. Similarly, De Cao et al.
+(2019) also encode different relations between en-
+tity mentions in the documents and perform the
+graph reasoning via Graph Convolutional Network
+(GCN) (Kipf and Welling, 2017).
+All of the above methods which involve Graph
+Neural Networks only consider entity nodes and the
+relations between them. The HDE-Graph (Tu et al.,
+2019) extends these works by creating a new type
+of graph with nodes corresponding to answer candi-
+dates, documents and entities. Different edges are
+included into the graph to capture the interaction
+between these nodes. DFGN (Qiu et al., 2019) con-
+structs a dynamic entity graph and performs graph
+reasoning with a fusion block. This fusion block in-
+cludes iterative interactions between the graph and
+the documents (Doc2Graph and Graph2Doc flows)
+in the graph construction process. Hierarchical
+Graph Network (Fang et al., 2020) proposes a hier-
+archical graph that incorporates nodes on different
+levels of a hierarchy, including query, paragraph,
+sentence, and entity nodes. This hierarchical graph
+allows the model to aggregate query-related data
+from many sources at various granularities.
+One limitation that all of the above conventional
+QA graph neural networks share is that their in-
+formation propagation mechanisms do not directly
+utilise the (explicit or implicit) hierarchical prop-
+erty of the graph structure. In fields outside of
+Natural Language Processing, recent studies on hi-
+erarchical graph neural networks focus on passing
+
+information on each hierarchical level to the node
+at different attention weights.
+In multi-agent reinforcement learning, HGAT
+(Ryu et al., 2020) generates hierarchical state-
+embedding of agents. This HGAT model stacks
+inter-agent and inter-group graph attention net-
+works hierarchically to capture inter-group node
+interaction. A two-level graph attention mechanism
+(Zhang et al., 2020) was developed for propagating
+information in the close neighborhood of each node
+in the constructed hierarchical graph. HATS (Kim
+et al., 2019) predicts stock trends using relational
+data on companies in the stock market. HATS
+selectively aggregates information from different
+relation types with a hierarchically designed atten-
+tion mechanism. By maintaining only important
+information at each level, HATS efficiently filters
+out relations (edges) not useful for trend prediction.
+However, all previous studies on hierarchical
+graph neural networks only exploit the possible
+hierarchical structure on the graph node itself. Dif-
+ferent from the above methods, our proposed hi-
+erarchical graph attention mechanism allows the
+graph node embeddings to be updated in the or-
+der of the hierarchical granularity level, instead of
+simultaneously.
+3
+Model
+As our proposed improvements are largely aimed at
+the established Hierarchical Graph Network (HGN)
+model (Fang et al., 2020) for HotpotQA, we briefly
+describe the original architecture. HGN builds a
+hierarchical graph with nodes from several granu-
+larity levels (question/paragraph/sentence/entity).
+This hierarchical graph structure is good at captur-
+ing the interaction between nodes from different
+granularity levels and has been shown beneficial
+to the model’s remarkable performance on both
+finding scattered pieces of supporting information
+across documents, and to answer span prediction.
+The full HGN model pipeline consists of four
+modules: (i) the Graph Construction Module se-
+lects query-related paragraphs and builds a hier-
+archical graph that contains edges between nodes
+from different granularity levels within the para-
+graphs; (ii) the Context Encoding Module gives
+an initial representation/embeddings for nodes in
+the graph via encoding layers that consist of a
+RoBERTa (Liu et al., 2019) encoder and a bi-
+attention layer; (iii) the Graph Reasoning Mod-
+ule updates the initial representation of all nodes
+via reasoning over the hierarchical graph; (iv) the
+Multi-task Prediction Module performs multiple
+sub-tasks including paragraph selection, support-
+ing facts prediction, entity prediction and answer
+span extraction, based on the representation of all
+nodes. This process is summarized in Figure 2,
+as presented by the original authors of the HGN
+model.
+We note that HGN still has limitations on its
+graph structure and the graph reasoning step, and
+in this work introduce according changes. Our
+proposed extensions aim to further improve HGN
+through a more complete graph structure, and a
+novel hierarchical graph nodes update mechanism.
+As such, our method mainly targets the Graph Con-
+struction and Graph Reasoning Modules, described
+in more detail below, while we leave the Context
+Encoding and Multi-task Prediction Modules un-
+changed.
+Graph Construction Module
+The Hierarchical Graph is built based on the given
+HotpotQA question-context pair. This construc-
+tion process consists of two steps: (i) multi-hop
+reasoning paragraph retrieval from Wikipedia, i.e.
+selecting candidate paragraphs with potential multi-
+hop relationship to the question as paragraph nodes;
+(ii) adding edges between question, sentence and
+entity nodes within the retrieved paragraphs.
+In particular, the first step consists of retriev-
+ing “first-hop” paragraphs, that is, paragraphs of
+Wikipedia entries that belong to entities mentioned
+in the question. After this, a number of “second-
+hop” paragraphs is selected, from Wikipedia arti-
+cles that are hyper-linked from these first hops.
+Our work keeps the original paragraph selection
+method, but introduces novel meaningful edges
+between graph nodes.
+Context Encoding Module
+With the hierarchical graph structure in place, rep-
+resentations of the nodes within the graph are ob-
+tained via the Context Encoding Module. In this
+encoder, query and context are concatenated and
+fed into a pretrained RoBERTa (Liu et al., 2019).
+The obtained representations are further passed into
+a bi-attention layer (Seo et al., 2018) to enhance
+the cross interactions between the question and the
+context. Through this encoding mechanism, the
+question node is finally represented as q ∈ Rd and
+the i-th paragraph/sentence/entity nodes are repre-
+sented by pi, si and ei ∈ Rd respectively.
+
+Figure 2: Model architecture of Hierarchical Graph Network (HGN). This illustration was originally introduced in
+Fang et al. (2020). We include it here for completion, to provide an overview of HGN.
+Graph Reasoning Module
+Intuitively, the initial representations of the graph
+nodes only carry the contextualized information
+contained within their local contexts. To benefit
+from the hierarchy and information across differ-
+ent contexts, the Graph Reasoning Module further
+propagates information between the graph nodes
+using a single-layered Multi-head Graph Attention
+Network (GAT) (Veliˇckovi´c et al., 2017). How-
+ever, we believe the simultaneous node-update per-
+formed by standard GAT can be improved, in the
+presence of the explicitly given hierarchical prop-
+erty of the graph. We therefore propose a novel
+hierarchical graph reasoning method that performs
+node updates sequentially, for different levels of
+the hierarchy. In this manner, nodes on certain
+granularity levels of the graph are allowed to first
+aggregate some information, before passing it on
+to their neighbours on other levels. We speculate
+that this staggered information passing paradigm
+can be beneficial to the multi-hop Question An-
+swering task, by passing on more question-specific
+contextualized information to relevant nodes.
+Multi-task Prediction Module
+The final step of the HGN model is to jointly pre-
+dict answer and supporting facts for the question
+via multi-task learning based on the updated graph
+node representations. This is decomposed into five
+sub-tasks: (i) paragraph selection determines if a
+paragraph contains the ground truth; (ii) sentence
+selection determines if a sentence from the selected
+paragraph is a supporting fact; (iii) answer span
+prediction finds the start and end indices of the
+ground-truth span; (iv) answer type prediction pre-
+dicts the type of the question; (v) entity prediction
+determines if the answer can be found among the
+selected entities. The above sub-tasks are jointly
+trained through multi-task learning with the final
+objective of the total loss from these sub-tasks:
+Ljoint =Lstart + Lend + λ1Lpara+
+λ2Lsent + λ3Lentity + λ4Ltype
+(1)
+With HGN re-introduced for completeness, we
+describe our proposed extensions to the original
+architecture in the subsequent sections.
+3.1
+Completion of the graph structure
+HGN constructs a hierarchical graph connecting
+the query node with the selected multi-hop para-
+graphs. Each selected paragraph contains sentences
+and entities which are also encoded as nodes in the
+hierarchical graph. The graph not only incorpo-
+rates the natural hierarchy existing in paragraphs,
+sentences and entities, but also includes helpful
+connections between them to faciliate the structual
+information propagation within the graph. Specif-
+ically, the graph consists of seven types of edges,
+which link the nodes in the graph. These edges
+are (i) edges between the question node and first-
+hop paragraph nodes; (ii) edges between paragraph
+nodes; (iii) edges between sentences in the same
+paragraph; (iv) edges between paragraph nodes
+and the corresponding within-paragraph sentence
+nodes; (v) edges between second-hop paragraphs
+and the hyperlinked sentences; (vi) edges between
+
+Multi-task Prediction Module
+Graph Construction Module
+Paragraph
+Supporting Facts
+Entity
+Answer Span
+Q
+Selection
+Prediction
+Prediction
+Extraction
+↑
+↑
+↑
+介
+Paragraph
+(P1
+P2
+Updated:
+Gated Attention
+Level
+hyperlink
+介
+Sentence
+S1
+S2
+S3
+S4
+S5
+Graph Reasoning Module
+Level
+Initial Representations:
+Entity
+E2
+E3
+E4
+E1
+Level
+Adriana
+个
+New York
+Virginia
+Greenwich
+Trigiani
+Village
+City
+Context Encoding Module
+↑
+Q
+P1
+P2Figure 3: Hierarchical Graph with (orange-colored)
+new question_sentence edges added.
+the question node and its matching entity nodes;
+(vii) edges between sentence nodes and their corre-
+sponding within-sentence entity nodes.
+We note that the only type of edge that seems to
+be missing from the graph are question-sentence
+edges.
+Hence, we first complete the hierarchi-
+cal graph by introducing novel question_sentence
+edges which connect the question node with all
+sentence nodes of selected paragraphs. Such new
+connections are introduced as edge (viii) in the hier-
+archical graph. The constructed hierarchical graph
+with novel edges added is illustrated in Figure 3.
+We reason that this more complete graph might
+help the model to learn more useful embedding
+because of the modification in the graph topology,
+which facilitates the information transmission be-
+tween the question and sentences.
+3.2
+Graph Attention with Hierarchies
+The Graph Reasoning Module updates the contex-
+tualized representations of graph nodes to capture
+the information aggregated from topological neigh-
+bours such that the local structures of these nodes
+can be included. In HGN, this process is realized
+by the Graph Attention Network (GAT) (Veliˇckovi´c
+et al., 2017), a well-established GNN approach.
+However, we note that in the specific setting
+of Multi-hop QA with the presence of an explicit
+hierarchical graph structure, GAT might not be
+able to make full use of the information encoded
+in the graph, as it will not directly capture the
+crucial dependencies between “levels” of the hi-
+erarchical. To address this problem, we propose a
+novel Graph Attention Network with Hierarchies
+(GATH) which updates nodes sequentially condi-
+tioned on an imposed order over the hierarchy lev-
+els. This is expected to help the model more effec-
+tively processes the local observation of each node
+into an information-condensed and contextualized
+state representation for individual nodes on specific
+levels, e.g. for paragraphs, before passing their in-
+formation on to their neighbours on other levels,
+such as to entity nodes. We expect this staggered
+flow of information might help the model aggregate
+information that is more useful and conditioned on
+the task at hand.
+The nodes in the graph are split into four cate-
+gories, and can be represented by q, P, S and E:
+P = {pi}np
+i=1
+S = {si}ns
+i=1
+E = {ei}ne
+i=1
+with each node embedded with an embedding func-
+tion as described above, into a d-dimensional vec-
+tor. These node representations are jointly repre-
+sent the graph nodes as
+H = {q, P, S, E} ∈ Rg×d, g = 1 + np + ns + ne
+GATH updates all initial node embedding H to
+H
+′ through hierarchical graph updates. Different
+from GAT, GATH updates the nodes representation
+sequentially, according to a pre-determined order
+of hierarchical levels, instead of simultaneously. It
+takes the initial node representations H as input,
+but first only updates information of node features
+of the first hierarchical level while keeping other
+node embeddings unchanged.
+For example, if the first level to be updated is
+the paragraph level, we obtain the updated graph
+representation
+Hpara = {h1, h
+′
+2, h
+′
+3, . . . h
+′
+1+np, h2+np, . . . , hg}
+Specifically,
+h
+′
+i = ∥K
+k=1LeakyRelu(
+�
+j∈Ni
+αk
+ijhjWk)
+(2)
+where ∥K
+k=1 represents concatenation of K heads,
+Wk is the weight matrix to be learned, Ni repre-
+sents the set of neighbouring nodes of node i and
+αk
+ij is the attention coefficient calculated by:
+αk
+ij =
+exp(LeakyRelu([hi; hj]wk
+eij))
+�
+t∈Ni exp(LeakyRelu([hi; ht]wkeit)) (3)
+where [hi; hj] denotes the concatenation of hi and
+hj, and wk
+eij is the weight vector corresponding to
+the edge between node i and j.
+Based on the updated embeddings on the para-
+graph level Hpara, we might next consider updating
+
+Q
+edge (vi)
+edge (i)
+Paragraph
+edge (ii)
+Level
+P1
+P2
+édge (vili)
+edge (v)
+edge (iv)
+Sentence
+Level
+S1
+S2
+S3
+S4
+S5
+edge (ii)
+edge (vii)
+Entity Level
+E1
+E2 
+E3
+E4the information on the sentence level. GATH propa-
+gates information to all nodes on the sentence level
+based on Hpara. This will output a further updated
+graph representation
+Hsent = {h1, h
+′
+2, . . . h
+′
+1+np+ns, h2+np+ns, . . . , hg}
+with all nodes in P and S updated.
+Continuing the process in this manner, we even-
+tually will have updated all node representations
+to obtain H
+′ {h
+′
+1, h
+′
+2, ..., h
+′
+g}. Algorithm 1 summa-
+rizes the above procedures in pseudo code. Ad-
+ditionally, these updating steps are combined and
+illustrated in Figure 4.
+4
+Experiments
+In this section, we present experiments comparing
+our extended HGN models with GATH with the
+original one employing GAT, and provide a detailed
+analysis of the proposed improvements and results.
+For all experiments, we use RoBERTalarge as
+the base embedding model. We train with a batch
+size of 16 and a learning rate of 1e−5 over 5 epochs,
+with λ1, λ3, λ4 = 1 and λ2 = 2, and we employ a
+dropout rate of 0.2 on the transformer outputs, and
+0.3 throughout the rest of the model.
+4.1
+Dataset
+The effects of the above proposed improvements
+are assessed based on HotpotQA (Yang et al., 2018).
+It is a dataset with 113k English Wikipedia-based
+question-answer pairs with two main features: (i) It
+requires reasoning over multiple documents with-
+out constraining itself to an existing knowledge
+base or knowledge schema; (ii) Sentence-level sup-
+porting facts are given for the answer to each ques-
+tion, which explain the information sources that the
+answer comes from. The performance of models
+on HotpotQA is mainly assessed on two metrics,
+exact match (EM) and F1 score. The model is ex-
+pected to not only provide an accurate answer to the
+question, but also to give supporting evidences for
+its solution. Thus, EM and F1 score are calculated
+for both answer spans and supporting facts.
+HotpotQA has two settings: Distractor and Full-
+wiki. In the distractor setting, context paragraphs
+consist of 2 gold truth paragraphs containing in-
+formation that is needed to solve the question, and
+8 paragraphs retrieved from Wikipedia based on
+the question, serving as related yet uninformative
+distractors for the question-answer pair. In the
+Fullwiki setting, all context paragraphs come from
+Wikipedia’s top search results, and they need to be
+pre-ranked and selected in a first step. Compared
+with the distractor setting, this setting requires us
+to propose an additional paragraph selection model
+concerned with information retrieval, before we
+address multi-hop reasoning task. As all our pro-
+posed extensions aim at the graph construction and
+reasoning steps, we only perform these initial ex-
+periments to assess the impact of our approach in
+the distractor setting, where we are independent
+from the influence of such a retrieval system.
+4.2
+Experimental Results
+Using the HotpotQA dataset, the models with our
+extensions of graph completion and GATH are com-
+pared against the baseline model of HGN with stan-
+dard GAT. Since it could reasonably be argued
+that GATH “simulates” a (partially) multi-layered
+GAT in the sense that some nodes are updated only
+after others have already been able to incorporate
+neighbouring information – which in standard GAT
+requires at least two full layers – we also include
+an HGN trained with a two-layer network rather
+than the single layer used in the original paper.
+Table 1 summarizes the results on the dev set of
+HotpotQA2.
+2Authors’ note: unfortunately, despite our best efforts we
+were not able to reproduce the numbers reported for HGN
+in Fang et al. (2020), even with their original, open-sourced
+code. We tried both the hyper-parameters as published in the
+paper, and the ones shipped with the code release however, the
+RoBERTalarge performance when training from scratch was
+consistently much lower than expected on dev (∼ 74 vs ∼ 70
+joint F1). We contacted the original authors, who were not
+able to help out with this. In light of these discrepancies, we
+
+Algorithm 1
+Graph Attention Network with Hierarchies
+(GATH)
+Input: H = hi, h2, .., hg}
+Output: H' = {hi, h2, ..,h.]
+for t in 1 : total number of levels do
+create Ht = [ ]
+foriin 1 : g do
+if node i belongs to level t then
+ht ← IIK=LeakyRelu(ZjeN; Qtit)
+k;(t)ht-1wk,(t)
+k,(t)
+EsE N, exp( LeakyRelu([ht-1;h-1]we;(t))
+else
+h ←ht-1
+end if
+Append h, to Ht
+end for
+end for
+return H' - Htotal number of levelsFigure 4: Hierarchical node representation update process. The grey-colored graph nodes are initial contextualized
+embedding given by the Context Encoding Layer. Through the paragraph level message passing layer, only the
+neighboring information of all paragraph nodes can be passed and renewed on them. Similar steps repeat for
+sentence level and entity level. For convenience of labeling indices, we set np = 2, ns = 5, ne = 4
+Completion of graph structure
+The HGN with
+new query-sentence edges improves over the base-
+line by 0.7/0.4 on Joint EM and F1 scores. This
+supports our intuition that the the missing question-
+to-sentence edges can indeed bring advantages to
+the model’s abilities of both answer span extraction
+and supporting facts prediction.
+Graph Attention with Hierarchies
+GATH al-
+lows for pre-defining the order of level updates in
+the model. Given that the order in which the hierar-
+chy levels are updated is likely to affect the model’s
+performance, we perform experiments with dif-
+ferent orders (P/S/E3,4, E/S/P, S/E/P and S/P/E)
+and compare them to the baseline models with
+one and two-layer GAT. All the GATH-based ex-
+tended models outperform the baseline model on
+the answer-span extraction by an absolute gain of
+1.6 to 2.4 points on the answer extraction metrics.
+On the other hand, the order of hierarchical lev-
+els does show an influence on the model’s evi-
+decided to focus only on dev set performance when assessing
+the impact of our extensions against re-trained vanilla HGN,
+as a fair comparison to the original model on test was not
+possible at this time.
+3P/S/E abbreviates Paragraph/Sentence/Entity
+4We exclude the query level update to make it more com-
+parable to the baseline model, which also excludes this update.
+dence collection ability. The “wrong” order leads
+to worse performance of the extended model, such
+as in the E/S/P and S/P/E cases.
+On most metrics, but specifically on joint F1
+score, the extended GATH-based model with the or-
+der S/E/P outperforms not only the baseline model,
+but also the other GATH models. It achieves a Joint
+EM/F1 score of 43.9/71.5, exceeding the baseline
+model’s performance by 1.2 each.
+Interestingly, the 2-layer GAT version of HGN
+slightly under-performs when compared to the orig-
+inal HGN setup. While gaining 0.3 points in sup-
+port prediction F1, it loses the same amount of
+performance in answer prediction and joint scores.
+We assume this is why the original HGN calls for
+only one layer, when we could intuitively have ex-
+pected multi-layered networks to perform better.
+Combined query-sentence edges and GATH
+The above experimental results demonstrate the
+individual effectiveness of these two proposed im-
+provements of graph completion and GATH. Nat-
+urally, we are also interested in the performance
+resulting from combining both. The “HGN (Com-
+bined)” row in Table 1 represents the model com-
+bining graph completion and GATH-S/E/P. This
+combined model brings slight improvement over
+
+H
+Hsent
+ Hpara
+h1
+h1
+edge (i)
+Paragraph
+edge (i)
+Paragraph
+Level
+P1
+P2
+Level
+P1
+h2
+h2
+h2
+edge (iv)
+edge (v)
+edge (iv) -
+dge (v)
+Sentence
+Sentence
+h's
+Level
+S2
+63
+h?
+h3
+Level
+S1
+edge (ii)
+edge (vii)i
+h'4
+Entity Level
+h4
+h4
+E1
+EntityLevel
+hs
+Paragraph level nodes updating
+..
+Sentence level nodes updating
+hg
+hg
+hg
+H'
+Hent
+edge(vi)
+hi
+edge (vi),
+h1
+Paragraph 
+Paragraph
+Level
+P1
+P2
+Level
+P1
+h2
+h2
+Sentence
+hs
+Level
+S2 
+S3
+ $4
+S5
+Sentence
+S1
+hs
+Level
+S3
+SA
+edge (vii) :
+h4
+h4
+edge (vi) i
+Entity Level
+E2
+E3
+E4
+EntityLevel
+hs
+E3
+hs
+1
+..
+Question level nodes updating
+hg
+hg
+Entity level nodes updatingAnswer
+Support
+Joint
+Model
+EM
+F1
+P
+R
+EM
+F1
+P
+R
+EM
+F1
+P
+R
+HGN (baseline)
+64.5
+78.3
+81.6
+79.0
+60.4
+87.4
+89.6
+87.5
+42.7
+70.3
+75.0
+70.9
+HGN (2-layer GAT)
+64.1
+78.0
+81.4
+78.9
+59.9
+87.7 89.5
+88.4
+41.6
+70.0
+74.5
+71.3
+HGN (que_sent edge)
+65.0
+79.1
+82.2
+80.1
+60.9
+87.0
+89.9
+86.9
+43.4
+70.7
+75.7
+71.5
+HGN with GATH(P/S/E)
+66.1
+80.1
+83.1
+81.1
+54.2
+80.1
+86.1
+79.5
+38.7
+66.5
+73.6
+66.9
+HGN-GATH(E/S/P)
+66.4
+80.3
+83.6
+81.2
+38.3
+74.4
+70.0 90.0
+27.2
+61.2
+59.8
+74.1
+HGN-GATH(S/E/P)
+67.0
+80.6 83.7 81.4
+60.5
+86.3 92.3 83.7
+43.9
+71.5 78.8 70.2
+HGN-GATH(S/P/E)
+66.8
+80.7 83.8
+81.6
+38.7
+74.6
+70.3 90.0
+27.3
+61.5
+60.1 74.5
+HGN (Combined)
+66.7
+80.7 83.7 81.7 61.4
+87.0
+91.2
+85.6
+43.9
+71.9 77.8
+71.8
+Table 1: Performance of the proposed HGN with completed edges (HGN que_sent), GATH, and both extensions
+combined on the development set of HotpotQA in distractor setting, against the baseline model HGN with GAT.
+the other models on most metrics. This final model
+sees further improvements, particularly in the an-
+swer span prediction task, and achieves the overall
+highest joint F1 score at 71.9, indicating that the
+contributions of graph completion and GATH are
+mutually benefitial.
+4.3
+Error Analysis
+In this section, we perform an error analysis on
+the concrete influence of the proposed HGN (com-
+bined) model based on question types. The major-
+ity of questions in HotpotQA fall under the bridge5
+and comparison reasoning categories.
+As sug-
+gested by Fang et al. (2020), we split comparison
+questions into comp-yn and comp-span. The former
+represents questions that should answer the compar-
+ison between two entities with “yes” or “no”, e.g.
+“Is Obama younger than Trump?”, while the latter
+requires an answer span, e.g. “Who is younger,
+Obama or Trump?”.
+Table 2 shows the performance of the original
+HGN model and the proposed model HGN-GATH
+(combined) on various types of reasoning questions.
+Results indicate that comp-yn questions are easiest
+for both models, and the bridge type is the hardest
+to solve. The analysis table shows that HGN (com-
+bined) is more effective than the original model
+on all of these reasoning kinds except support EM
+for comp-yn, though even here the much improved
+answer prediction leads to an overall improvement
+of 2.42 on Joint EM.
+5requiring a bridging entity between support sentences,
+needed to arrive at the answer
+HGN-GAT
+Question
+Ans EM
+Sup EM
+Joint EM
+Pct(%)
+comp-yn
+81.22
+81.44
+68.34
+6.19
+comp-span
+65.50
+71.04
+48.49
+13.90
+bridge
+63.08
+57.01
+39.73
+79.91
+HGN-GATH (Combined)
+Question
+Ans EM
+Sup EM
+Joint EM
+Pct(%)
+comp-yn
+85.81
+80.79
+70.96
+6.19
+comp-span
+68.42
+71.53
+50.34
+13.90
+bridge
+64.95
+58.08
+40.71
+79.91
+Table 2: Original HGN (top) and HGN-GATH com-
+bined (bottom) model results for various reasoning
+types. ‘Pct’ signifies percentage of all questions per
+category.
+5
+Conclusions and Future Work
+In this paper, we proposed two extensions to Hi-
+erarchical Graph Network (HGN) for the multi-
+hop Question Answering task on HotpotQA. First,
+we completed the hierarchical graph structure by
+adding new edges between the query and context
+sentence nodes. Second, we introduced GATH as
+the mechanism for neural node updates, a novel
+extension to GAT that can update node representa-
+tions sequentially, based on hierarchical levels. To
+the best of our knowledge, this is the first time the
+hierarchical graph structure is directly exploited in
+the update mechanism for information propagation.
+Experimental results indicate the validity of our
+approaches individually, as well as when used
+jointly for the multi-hop QA problem, outperform-
+ing the currently best performing graph neural net-
+work based model, HGN, on HotpotQA.
+In the future, we would particularly like to in-
+tegrate hierarchical graph attention weights into
+
+GATH, as motivated by related research in Rein-
+forcement Learning.
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+

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+Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet
+Junsen Xiang,1, ∗ Cheng Su,2, ∗ Ning Xi,3, ∗ Zhendong Fu,4 Zhuo Chen,5 Hai Jin,6 Ziyu Chen,2 Zhao-Jun Mo,7
+Yang Qi,8, 9 Jun Shen,5, 10 Long Zhang,11, 12 Wentao Jin,2, † Wei Li,3, 12, 13, ‡ Peijie Sun,1, § and Gang Su11, 12, ¶
+1Beijing National Laboratory for Condensed Matter Physics,
+Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
+2School of Physics, Beihang University, Beijing 100191, China
+3CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
+4Neutron Platform, Songshan Lake Materials Laboratory, Dongguan 523808, China
+5School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
+6Department of Astronomy, Tsinghua University, Beijing 100084, China
+7Ganjiang Innovation Academy, Chinese Academy of Sciences, Ganzhou 341119, People’s Republic of China.
+8State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
+9Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China
+10Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China
+11Kavli Institute for Theoretical Sciences, and School of Physical Sciences,
+University of Chinese Academy of Sciences, Beijng 100049, China
+12CAS Center for Excellence in Topological Quantum Computation,
+University of Chinese Academy of Sciences, Beijng 100190, China
+13Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China
+(Dated: January 10, 2023)
+By performing experimental and model studies of a triangular-lattice dipolar magnet KBaGd(BO3)2 (KBGB),
+we find the highly frustrated magnet with a planar anisotropy hosts a strongly fluctuating dipolar spin liquid
+(DSL) originating from the intriguing interplay between dipolar and Heisenberg interactions. The DSL con-
+stitutes an extended regime in the field-temperature phase diagram, which gets narrowed in temperature range
+as field increases and eventually ends with a quantum critical point (QCP) at Bc ≃ 0.75 T. Based on dipolar
+Heisenberg model calculations, we identify the DSL as a Berezinskii-Kosterlitz-Thouless (BKT) phase. Due to
+the tremendous entropy accumulation that can be related to the strong BKT and quantum fluctuations, unprece-
+dented magnetic cooling effects are observed in the DSL regime and particularly near the QCP, making KBGB
+a superior dipolar coolant over commercial Gd-based refrigerants. We establish a universal phase diagram for
+triangular-lattice dipolar quantum magnets where emergent symmetry plays an essential role, and lay down
+foundations for their applications in sub-Kelvin refrigeration.
+Introduction.— Triangular-lattice quantum antiferromag-
+nets have raised great research interest recently, due to the
+unusual quantum spin states and transitions therein [1, 2].
+One prominent example is the quantum spin liquid (QSL) [3–
+5] and its possible materialization in organic compounds [6–
+8] and rare-earth triangular magnets [9–16]. Due to the in-
+triguing spin frustration effects and two dimensionality (2D),
+Berezinskii-Kosterlitz-Thouless (BKT) physics may appear in
+the triangular quantum antiferromagnets. The Co-based quan-
+tum antiferromagnet Na2BaCo(PO4)2 hosts persistent spin
+fluctuations [17–20] till very low temperature, and is proposed
+to posses spin supersolid state with BKT fluctuations of U(1)
+phase [21]. Emergent symmetry, as a consequence of frustra-
+tion, has also been disclosed on the triangular lattice, with a
+recent example of rare-earth magnet TmMgGaO4 [22–27].
+Recently, it has been proposed that the dipolar interactions
+can give rise to QSL in triangular-lattice quantum spin sys-
+tems [29]. Lately such dipolar system has been realized in Yb-
+based triangular compounds [30–34]. However, the dipolar
+∗ These authors contributed equally to this work.
+† wtjin@buaa.edu.cn
+‡ w.li@itp.ac.cn
+§ pjsun@iphy.ac.cn
+¶ gsu@ucas.ac.cn
+interactions are rather weak and it is very challenging for con-
+ventional thermodynamic and spectroscopic measurements to
+probe the exotic spin states due to dipolar interactions. On
+the contrary, the rare-earth dipolar magnets with larger mo-
+ments, e.g., Gd-based compounds with µeff ≈ 8µB and high
+spin S = 7/2, are much less explored both in experiments
+and theories. It is expected that the dipolar frustration ef-
+fects are a priori more evident in these systems. Moreover,
+in sub-Kelvin refrigeration for space applications [35, 36] and
+quantum computations [37], high-spin frustrated magnets, es-
+pecially those with spin-liquid like behaviors [38], can have
+great entropy densities and cooling capacity, holding thus
+strong promise as superior coolants.
+In this work, we perform low-temperature thermodynam-
+ics and magnetocalorics measurements on single-crystal sam-
+ples of Gd-based triangular-lattice compound KBaGd(BO3)2
+(KBGB). The thermodynamic measurements suggest a dipo-
+lar spin liquid state with no conventional ordering but strong
+spin fluctuations, which are reflected in the algebraic specific
+heat and imaginary dynamical susceptibility (χ
+′′
+ac). We estab-
+lish a dipolar Heisenberg model with both dipole-dipole and
+Heisenberg interactions for KBGB. Monte Carlo (MC) sim-
+ulations explain excellently the experimental measurements
+and unveil the exotic spin states and transitions in the phase
+diagram. In particular, the model simulations suggest a two-
+step melting of the clock antiferromagnetic (AF) order via two
+arXiv:2301.03571v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+-0.8
+0
+0.8
+-0.8
+0
+0.8
+Yy
+Yx
+-0.8
+0
+0.8
+-0.8
+0
+0.8
+Yy
+Yx
+-0.8
+0
+0.8
+-0.8
+0
+0.8
+Yy
+Yx
+6-clock AF
+DSL
+PM
+Temperature
+(b)
+(c)
+(d)
+(e)
+(a)
+b
+a
+c
+K/Ba
+Gd
+B
+O
+b
+a
+a∗
+Si
+Sj
+eij
+FIG. 1. (a) shows the crystal structure of KBaGd(BO3)2, where (b)
+triangular-lattice layers of GdO6 octahedra are separated by the Ba/K
+layers with site mixing. The grey arrows refer to the spins on site i
+and j, and the unit vector eij is also indicated. Dipole-dipole inter-
+actions are bond-dependent and follow the ¯3m site symmetry. (c)-(e)
+are histograms of the order parameter Ψxy ≡ Ψx + iΨy for the 6-
+clock antiferromagnetic (AF) [28], dipolar spin liquid (DSL) with an
+emergent U(1) symmetry, and the paramagnetic (PM) phases.
+BKT transitions, between which a floating BKT phase emerge
+with an emergent U(1) symmetry, well accounting for the ex-
+perimentally observed spin liquid behaviors with enormous
+low-temperature entropy. Consequently, giant magnetocaloric
+effect (MCE) is observed in the quasi-adiabatic demagnetiza-
+tion measurements, where we find a clear dip in temperature
+which suggests the presence of quantum critical point (QCP)
+near Bc ≃ 0.75 T. The lowest temperature of 70 mK clearly
+surpasses that of commercial refrigerant Gd3Ga5O12 (GGG)
+under similar conditions. Overall, the triangular-lattice rare-
+earth dipolar magnets open an avenue for exploring exotic
+spin states as well as finding superior sub-Kelvin coolants.
+Crystal structure and effective model for KBaGd(BO3)2.—
+Centimeter-sized single crystals of KBGB were synthesized
+using the flux method as described in detail in Supplementary
+Materials (SM) [28], and the X-ray diffraction measurements
+suggest high quality of the single crystals. KBGB is found
+to crystallize in a trigonal structure [40, 41] with space group
+R-3m [c.f., Fig. 1(a)], and has a relatively high ionic density
+of 6.4 nm−3. As shown in Fig. 1(b), magnetic Gd3+ ions with
+4f 7 electron configuration (L = 0, S = 7/2) form perfect
+triangular lattice.
+The dipolar interaction between magnetic ions Gd3+
+has a characteristic energy Edp
+∼
+2µ0µ2
+eff/4πa3
+≈
+0.05 meV (with µeff ≈ 8µB), which determines the low-
+temperature spin states in KBGB. To simulate such Gd-
+based dipolar magnet, we consider the following Hamil-
+tonian, H
+=
+JH
+�
+⟨i,j⟩NN Si · Sj + JD
+�
+i,j[Si · Sj −
+3(Si · eij)(Sj · eij)]/r3
+ij, where eij(rij) refers to the unit vec-
+tor(distance) between site i and j. JH and JD refer to the
+nearest neighbor (NN) Heisenberg and dipole-dipole interac-
+tions, respectively. As the dipolar interactions show rapid (cu-
+bic) power-law decay and the longer range interactions can be
+washed out, we keep only NN terms as
+HDH =
+�
+⟨i,j⟩NN
+J Si · Sj − D (Si · eij)(Sj · eij),
+(1)
+where J = JH + JD/a3 is the NN isotropic coupling and
+D = 3JD/a3 refers to the dipolar anisotropic term. We find
+the NN dipolar Heisenberg (DH) model with couplings J =
+47 mK and D = 80 mK very well describe the compound
+and accurately reproduce the experimental measurements on
+KBGB. The MC simulations are performed on up to 60 × 60
+triangular lattice. Due to the high-spin state with S = 7/2,
+classical MC simulations capture well the finite-temperature
+properties of KBGB [28]. We guarantee the error bars to be
+always smaller than the symbol size in the presented data.
+Magnetic specific heat, susceptibility, and dipolar spin
+liquid.— In Fig. 2(a) we show the zero-field specific heat
+Cm measured down to 65 mK. There exists a round peak at
+T ∗ ≃ 218 mK, below which the system exhibits Cm ∼ T 2
+with algebraic scaling, resembling that of 2D Heisenberg
+or XY quantum spin model with U(1) symmetry [42, 43].
+The dipolar anisotropy in Eq. (1), like in spin-orbit magnets,
+leads to a discretized C3 rotational symmetry, and it gener-
+ically corresponds to a divergent Cm peak when transition-
+ing to low-T symmetry breaking phase.
+The presence of
+round peak and T 2 scaling in Cm is very remarkable, which
+suggests a liquid-like and strongly fluctuating spin state. In
+Fig. 2(a), when compared to the renowned Gd-based refrig-
+erant GGG [36, 39, 44, 45], KBGB has tremendous low-
+temperature specific heat, far surpassing that of GGG.
+In Fig. 2(b), we apply out-of-plane fields (B//c) to the com-
+pound, and find also round peaks in Cm curves, which move
+towards lower temperature with heights slightly reduced. This
+suggests that the spin liquid states constitute an extended
+phase that we dub as dipolar spin liquid (DSL). As field fur-
+ther increases and exceeds about 0.75 T, the DSL behaviors
+disappears [c.f., the contour plot of Cm/T in Fig. 3(b)], and
+the Cm peak moves now to high-temperature side, with the
+low-T peak and low-energy fluctuations quickly suppressed.
+In Fig. 2(c), we perform magnetization measurements
+on single-crystal sample of KBGB, and find a clear mag-
+netic anisotropy between the out-of-plane (//c axis) and in-
+plane (//a) directions.
+This anisotropy can be clearly rec-
+ognized in the different saturation magnetization moments
+and transition field values, i.e., 1 T(0.5 T) along c(a) axis.
+In Fig. 2(d), we perform low-temperature dc susceptibility
+(χdc) measurements, and find χdc also exhibits a clear easy-
+plane anisotropy. In addition, a small but sensible in-plane
+anisotropy between a and a∗ [see inset of Fig. 2(c)] is also
+observed, consistent with the intrinsic anisotropy in bond-
+dependent dipolar interaction [c.f., Eq. (1)].
+To further explore the DSL, ac magnetic susceptibilities are
+measured in Figs. 2(e,f), with χ′
+ac and χ′′
+ac for real and imag-
+inary parts, respectively. The real χ′
+ac exhibits a frequency-
+independent maximum and remains large even below the char-
+acteristic temperature scale T ∗.
+Therefore, although there
+exist K/Ba site mixing in the compound, the spin-glass sce-
+nario can be excluded in KBGB. Interestingly, the imaginary
+
+3
+1
+10
+100
+0
+2
+4
+6
+8
+10
+12
+Exp.
+Model
+χdc (emu·Oe-1·mol-1
+Gd)
+T (K)
+a
+a*
+c
+0.1 T
+-2 0 2 4 6 8 10
+0.0
+0.4
+0.8
+1.2
+χdc
+-1
+θa    -0.30 K
+θa*    -0.33 K
+   θc  
+-1.32 K
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+25
+50
+75
+T (K)
+Cm/T (J·mol-1
+Gd·K-2)
+KBGB
+GGG
+0 T
+Cm/T ~ T
+T*    218 mK
+0.1
+1
+0.0
+0.5
+1.0
+1.5
+2.0
+ 4943 Hz
+ 6253 Hz
+ 9984 Hz
+χac'' (a.u.)
+T (K)
+T*
+0
+1
+2
+3
+4
+0
+2
+4
+6
+8
+10
+B // a
+B // c
+Moment (µB/Gd)
+B (T)
+ 
+0.4 K
+       2.36
+       2.49
+Model
+ga 
+gc
+0.1
+1
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+2.4
+91 Hz
+955 Hz
+2439 Hz
+3087 Hz
+3910 Hz
+T (K)
+χac' (a.u.)
+T*
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0
+25
+50
+75
+0.25 T
+0.5 T
+0.75 T
+Cm/T (J·mol-1
+Gd·K-2)
+T (K)
+1 T
+2 T
+3 T
+4 T
+(a)
+(d)
+(e)
+(f)
+(c)
+(b)
+a*
+a
+FIG. 2. Specific heat of KBGB under (a) zero and (b) finite fields along out-of-plane direction (B//c). An algebraic Cm ∼ T 2 scaling is
+observed below the round peak temperature T ∗, and the Cm/T values far outweigh that of GGG [39] for T ≲ T ∗. In (b) we find the round
+peak in Cm/T firstly moves towards lower temperature and later for B > Bc ≃ 0.75 T the low-temperature Cm quickly gets suppressed. (c)
+shows the magnetization curves of the single-crystal KBGB sample for B//a and //c, and the results show excellent agreement with the DH
+model calculations (solid lines). The saturation moments are µsat
+a
+≃ 8.26µB and µsat
+c
+≃ 8.72µB, from which we determine the Landé factors
+ga ≃ 2.36 and gc ≃ 2.49, respectively. The as-grown KBGB single crystal is shown in the inset, with directions a and a∗ also indicated. (d)
+shows the molar dc magnetic susceptibilities (χdc) measured along the a, a∗, and c axes, respectively, where the solid lines representing the
+DH model calculations show excellent agreements. The inset shows the Curie-Weiss fittings in the paramagnetic regime 0.4 K ≤ T ≤ 10 K,
+with the fitted Curie-Weiss temperatures θa,a∗,c also indicated. (e, f) present respectively the real and imaginary ac susceptibilities measured
+with different frequencies.
+ac susceptibility χ′′
+ac(T), although being featureless for low
+frequencies ω ≲ 4 kHz, show a clear temperature-dependent
+behavior for higher frequencies in Fig. 2(f). Considering that
+χ′′(ω) can be directly related to the dynamical correlation
+S(ω) through the fluctuation-dissipation theorem, χ′′(ω) ∝
+ω
+T S(ω) (ω ≪ T), this clearly suggests the persistence of low-
+energy spin fluctuations even below T ∗ and supports the spin-
+liquid scenario.
+Magnetocaloric effect and quantum critical point.— In
+Fig. 3(a), we perform quasi-adiabatic demagnetization mea-
+surements and obtain the isentropic curves. It is found that
+KBGB clearly outperforms GGG in the minimal temperature,
+i.e., Tm ≃ 70 mK (KBGB) vs. 322 mK (GGG), when starting
+from the same initial condition of Ti = 2 K and Bi = 6 T.
+In Fig. 3(b) we provide more of the isentropic lines from dif-
+ferent initial conditions, and observe the highly asymmetric
+isentropes, which “levels off” in the bright DSL regime as in-
+dicated by large values of Cm/T.
+For KBGB, the lowest temperature Tm is achieved at the
+dip in isentropic lines and remains below 100 mK in the
+small field side. This happens also for measurements starting
+from rather low temperature Ti ≃ 95 mK, where the lowest
+Tm ≃ 33 mK. Such unprecedented MCE response strongly
+corroborates the existence of QCP at Bc ≃ 0.75 T. The mag-
+netic Grüneisen ratio ΓB =
+1
+T ( ∂T
+∂B )S has been widely used
+in the studies of heavy fermion [46–50] and low-dimensional
+quantum spin systems [51–54]. In the inset of Fig. 3 an ev-
+ident peak-dip structure with sign change is observed [55–
+58], and the peak height exceeds 4 times that of GGG. Such a
+prominent critical cooling effect provides valuable MCE evi-
+dence for QCP in the compound KBGB.
+Emergent symmetry in KBGB.— According to the magne-
+tothermal and MCE measurements above, we arrive at the
+phase diagram of KBGB in Fig. 3(b).
+The two schematic
+dashed lines, enclosing the DSL with large Cm/T, meet at a
+QCP (Bc) where the demagnetization process reaches its low-
+est temperature. Besides QCP, within the DSL regime we find
+persistent spin fluctuations and cooling effects whose origin is
+clarified by model calculations below.
+We conduct MC calculations of the DH model [Eq. (1)]
+for KBGB. As the model is highly frustrated in the out-of-
+plane direction, the order parameter lies within the ab plane.
+Note although the determined Landé factor gc ≃ 2.49 is
+slightly larger than ga ≃ 2.36, the intrinsic planar anisotropy
+of dipolar interaction leads to larger in-plane χdc (along a
+and a∗ axes) than that along the c axis. The negative Curie-
+Weiss temperatures fitted from the dc susceptibility reflect the
+AF nature, and the slightly different θa ≃ −300 mK and
+θa∗ ≃ −330 mK shows the in-plane anisotropy. In Fig. 2(d),
+we find the anisotropic susceptibility and magnetization mea-
+
+4
+0
+1
+2
+3
+4
+5
+6
+0.1
+1
+T (K)
+B (T)
+70 mK
+KBGB
+GGG
+B // c
+322 mK
+2 K
+33 mK
+0.0
+0.4
+0.8
+1.2
+0.0
+0.1
+0.2
+0.3
+B (T)
+T (K)
+0
+20
+40
+60
+Cm/T
+DSL
+QCP
+Quasi-Adiabatic
+PM
+6-clock AF
+(a)
+(b)
+ΓΒ (T-1)
+0 1 2 3 4
+-1
+0
+1
+2
+3
+4×GGG
+Bc
+FIG. 3. (a) shows the quasi-adiabatic isentropes measured in KBGB
+under out-of-plane field (see details in SM [28]). The KBGB curve
+exhibits a clear dip at the lowest temperature Tm ≃ 70 mK, much
+lower than that of GGG (Tm ≃ 322 mK). Starting from Ti ≃ 95 mK,
+KBGB is observed to cool down to remarkably low temperature
+Tm ≃ 33 mK in the dip (blue dotted line). The inset shows the mag-
+netic Grüneisen ratio ΓB deduced from the curves in (a). (b) shows
+the phase diagram of KBGB with the contour plot of Cm/T in the
+background. The bright regime with large spin fluctuations represent
+the DSL, with schematic dashed line boundaries, ending up with a
+QCP at Bc ≃ 0.75 T.
+sured along a and c axes can be well captured by the DH
+model. Besides, the model calculations of specific heat also
+obtain a round peak at about 270 mK, which again gets sup-
+pressed as field increases (see SM [28]), very much resem-
+bling the experimental data in Figs. 2(a,b). The comparisons
+confirm that the compound KBGB can indeed be accurately
+described by the DH model.
+To characterize the spin states in the phase diagram, we in-
+troduce the order parameter Ψxy ≡ meiθ = �
+j eiQrj(mx
+j +
+imy
+j ),
+where j runs over the lattice sites and Q
+=
+± 1
+2a∗, ± 1
+2b∗, ± 1
+2(a∗ − b∗) [28]. Histogram of the complex
+order parameter Ψxy at various temperature are shown in
+Figs. 1(c-e). At low temperature, the dipolar system exhibits
+a 6-clock AF order corresponding to θ = 0, ±π/3, ±2π/3,
+and π [28]. As temperature ramps up, the six points in the
+histogram prolong and merge into a circle with emergent
+U(1) symmetry, where the angle θ can choose arbitrary angle.
+As temperature further enhances, eventually the amplitude m
+vanishes and the system enters the conventional PM phase.
+Recall that the 6-state clock model with an anisotropic term
+∼ cos (6θ) undergoes two successive BKT transitions [59],
+between which the anisotropic term becomes irrelevant per-
+turbation. Based on this symmetry argument, we consider the
+intermediate DSL in the system as BKT phase with emer-
+gent U(1) symmetry and effectively described by 2D XY
+model [60–63]. The emergent symmetry extends also to the
+zero-temperature QCP as the clock term is dangerously irrele-
+vant [60], and the transition directly between the 6-fold clock
+symmetry broken and PM phases belong to the 3D XY univer-
+sality class. Therefore, the emergent symmetry constitutes a
+key for demystifying spin-liquid state and quantum criticality
+in the compound KBGB.
+0
+60
+120
+180
+0
+1
+2
+3
+4
+T (K)
+Time (min)
+4 T
+6 T
+0.1
+1
+0
+5
+10
+15
+20
+Sm (×10·J·Kg-1·K-1)
+T (K)
+∆Q
+4 T
+∆Sm
+0 T
+(a)
+0.1
+1
+10
+0
+10
+20
+30
+40
+� � m (J·Kg-1·K-1)
+T (K)
+KBGB
+GGG
+4 T
+2 T
+1 T
+(b)
+FIG. 4. (a) The quasi-adiabatic demagnetization cooling curves of
+KBGB, starting from two different initial conditions (Ti = 4 K,
+Bi = 4 T) and (Ti = 2 K, Bi = 6 T), with reached lowest tem-
+perature Tm ≃ 205 mK and 70 mK, respectively. Parasitic heat
+loads are estimated to be 0.2 µW for Ti = 4 K environment and
+0.05 µW for Ti = 2 K. The inset shows magnetic entropy under
+zero and 4 T fields, with the shaded area representing the absorbed
+heat ∆Q = 47.44 J·Kg−1 in the hold process. (b) plots the entropy
+change ∆Sm vs. T, for fields decreasing from 1 T, 2 T, and 4 T to
+zero, respectively. Comparisons to GGG are also presented [39, 44].
+Superior cooling performance.— Starting from 2 K en-
+vironment, KBGB can reach as low as 70 mK as shown
+in Fig. 4(a), such a low cooling temperature far surpasses
+other Gd-based refrigerants, e.g., GGG (322 mK) and GdLiF4
+(480 mK) [64]. Besides, KBGB also exhibits long hold time
+and large isothermal entropy change ∆Sm. In Fig. 4(a) we
+show that KBGB remains in low temperature for a long period
+after the field is exhausted. In the environment temperature of
+2 K, 0.5 g KBGB remains below 140 mK for th ≈ 2 h under
+0.05 µW heat leak, which can be ascribed to the large heat
+absorption ∆Q depicted in the inset of Fig. 4(a).
+The isothermal entropy change ∆Sm characterizes the
+cooling capacity of refrigerants. In Fig. 4(b), we compare
+∆Sm of KBGB with that of GGG, and find that in the whole
+temperature range concerned KBGB has significantly larger
+∆Sm for 1 T field. Moreover, the maximal ∆Sm of KBGB lo-
+cates below 1 K [shaded regime in Fig. 4(b)], and the entropy
+change in KBGB exceeds that of GGG in this sub-Kelvin
+regime of central interest. Overall, the low cooling temper-
+ature Tm, long hold time th, and enormous entropy change
+∆Sm in the sub-Kelvin regime lead to the conclusion that
+KBGB serves a superior quantum magnet coolant.
+Discussions and outlook.— The pursue for high entropy
+density and low ordering temperature constitutes two oppos-
+ing factors hard to fulfill simultaneously in optimizing sub-
+Kelvin refrigerants. Here the spin frustration and quantum
+criticality in the dipolar system come to the rescue. We show
+that the compound KBaGd(BO3)2 with high Gd3+ ion density
+yet form a disordered and strongly fluctuating spin liquid till
+extremely low temperature, giving rise to the superior cooling
+capacity due to the entropy accumulation near QCP. We use
+the DH model within NN interactions to describe KBGB and
+find it well reproduces the experimental results. Inclusion of
+further neighboring dipolar couplings will not change the con-
+
+5
+clusion here, as it has been shown to maintain the universality
+class of BKT transitions in planar dipolar models [63, 65].
+The scenario of DSL ending up with emergent U(1) QCP
+may also be applicable to other dipolar quantum magnets. Re-
+cent progress in experimental studies reveal several families of
+rare-earth triangular quantum dipolar antiferromagnets, e.g.,
+Ba3REB3O9/Ba3REB9O18 (with RE a rare-earth ion) [32, 33]
+and ABaRE(BO3)2 (with A an alkali ion) [66, 67]. It has been
+observed that in Ba3YbB3O9 that 80% entropy remain below
+56 mK [31], despite a dipolar energy scale of about 160 mK,
+suggesting that the DSL may also play a role in the Yb-based
+dipolar compounds. Therefore, this work opens a venue for
+hunting exotic spin states as well as superior quantum coolants
+in triangular dipolar magnets.
+Note added.— Upon finishing the present work, we are
+aware of a recent work [68] also conducting MCE study of
+KBGB with however polycrystalline samples, where they find
+strong cooling effect down to 121 mK.
+Acknowledgements.— W.L. is indebted to Yuan Wan and
+Tao Shi for helpful discussions. W.J. and C.S. acknowledge
+the support from the beamline 1W1A of the Beijing Syn-
+chrotron Radiation Facility.
+This work was supported by
+the National Natural Science Foundation of China (Grant
+Nos. 12222412, 11834014, 11974036, 12047503, 12074023,
+12074024, 12174387, and 12141002), National Key R
+& D Program of China (Grant No. 2018YFA0305800),
+Strategic
+Priority
+Research
+Program
+of
+CAS
+(Grant
+No. XDB28000000), and CAS Project for Young Scien-
+tists in Basic Research (Grant No. YSBR-057). We thank the
+HPC-ITP for the technical support and generous allocation
+of CPU time. This work was supported by the Synergetic
+Extreme Condition User Facility (SECUF).
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+magnetocaloric measurements are elaborated in Secs. 2 and 3,
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+Rev. D 36, 515 (1987).
+
+8
+Intensity (arb. units)
+Intensity (arb. units)
+(003)
+(006)
+(009)
+(0012)
+FIG. S1. (a) shows the powder XRD pattern of KBGB measured at room temperature and corresponding Rietveld refinement. The open circle
+and red solid line represent the observed and calculated intensities, respectively, while the blue solid line shows their difference. The olive
+vertical bars mark the expected reflections for KBGB. (b) Single-crystal XRD scan along the (0,0,L) direction for one representative crystal,
+revealing only peaks that are well indexed by (0,0,3n). The insets show the image of the as-grown KBGB crystals and the rocking-curve
+scan of the (0,0,12) reflection fitted by a Gaussian profile. The very narrow peak width of FWHM = 0.041◦ indicates excellent quality of the
+crystals.
+Supplementary Materials
+Dipolar Spin Liquid Ending with Quantum Critical Point in a Gd-based Triangular Magnet
+Xiang et al.
+Section 1.
+SAMPLE PREPARATION AND STRUCTURE CHARACTERIZATION
+Polycrystalline samples of KBGB were firstly prepared by standard solid-state reaction method as reported in Ref. 40. Sto-
+ichiometric mixtures of K2CO3 (99.99%), BaCO3 (99.95%), H3BO3 (99.99%) and Gd2O3 (99.99%) (with 6% excess H3BO3
+and 5% excess of K2CO3 and BaCO3) were thoroughly ground and pelletized. Then the pellet was placed into an aluminum
+crucible and sintered at 900◦C in air for 10 h. This sintering process was repeated for several times to minimize possible
+impurities.
+Single-crystal samples of KBGB were grown using the flux method as reported in Ref. 41. The pre-obtained polycrystalline
+KBGB with high purity was mixed with the H3BO3 (99.99%) and KF (99.9%) fluxes in a molar ratio of 2:3:[2-3], and thoroughly
+ground. The mixture was transferred into a Pt crucible, heated up to 980◦C in air for 24 h, and then slowly cooled to 790◦C with
+a rate of 2◦C/h. After the furnace cooling, centimeter-sized crystals were obtained on top of the fluxes.
+The phase purity of the polycrystalline KBGB sample was confirmed by powder XRD at room temperature, performed
+on a Bruker D8 ADVANCE diffractometer in Bragg-Brentano geometry with Cu-Kα radiation (λ = 1.5406 Å). As shown in
+Fig. S1(a), the powder XRD pattern can be well fitted with the previously reported trigonal phase of KBGB [40] (a = b =
+5.4676(1) Å, c = 17.9514(3) Å), without any visible impurity peaks, indicating high purity of the synthesized KBGB powders.
+The quality of the single-crystal KBGB sample was checked by high-resolution synchrotron XRD (λ = 1.54564 Å) measure-
+ments at room temperature, performed on the 1W1A beamline at the Beijing Synchrotron Radiation Facility (BSRF), China. As
+shown in Fig. S1(b), a long L scan, equivalent to a θ-2θ scan with respect to the normal direction of the plate-like KBGB crystal,
+only shows Bragg reflections well indexed by (0, 0, 3n) as expected for the R-3m space group. The peak width (full width at half
+maximum, FWHM) observed in the rocking-curve scan of the (0, 0, 12) peak is very small, 0.041(2)◦, as shown in the inset of
+Fig. S1(b), which suggests excellent crystal quality. KBGB is relatively easy to synthesize and has excellent chemical stability,
+paving its viable way for applications in advanced cryogenics.
+
+9
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+0.1
+430 mK
+280 mK
+195 mK
+150 mK
+95 mK
+T (K)
+B (T)
+0.4
+0.03
+0.06 - 0.09 T·min-1
+Bi = 3 T
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+3
+4
+KBGB
+GGG
+T (K)
+B (T)
+0.15 T·min-1
+(a)
+(c)
+(b)
+0.0
+0.6
+1.2
+0.1
+0.2
+0.3
+0.4
+0.5
+3.0
+2.5
+2.0
+0
+1
+2
+3
+-1
+0
+1
+2
+ΓB (T-1)
+Bc
+FIG. S2. (a) Illustration of the two-stage quasi-adiabatic demagnetization cooling device for the measurements of 0.5 g KBGB single crystals.
+(b) shows the measured isentropic curves of KBGB starting from various initial conditions (Ti = 2 K, Bi = 4 T), (2 K, 6 T), and (4 K, 4 T),
+respectively, where the lowest temperature are found to be significantly lower than those of GGG. The inset zooms in the small-field range
+(B ≤ 1.2 T). (c) The DR-based measurements with an initial temperature Ti ≤ 430 mK and field Bi = 3 T, where the lowest achieved
+temperature is Tm ≃ 33 mK. The inset shows the magnetic Grüneisen ratio ΓB deduced from the low-temperature isentropic T-B lines in the
+main plot, where the sign change is evident and the peak becomes more and more pronounced as the initial temperature Ti lowers.
+Section 2.
+MAGNETOTHERMAL MEASUREMENTS
+Comprehensive magnetothermal measurements were performed on single-crystal samples of KBGB. The low-temperature
+specific heat (Cp, T ≥ 50 mK) and ac susceptibility (χac, T ≥ 50 mK) measurements were conducted using the Quantum
+Design Physical Property Measurement System (PPMS) equipped with a 3He–4He dilution refrigerator (DR) insert. The specific
+heat data were measured under various out-of-plane fields (B//c) with the semi-adiabatic heat pulse method. The phonon
+contributions are negligible below 2 K as estimated via a Debye T 3 analysis of high-temperature Cp data. The ac susceptibility
+(χac), as a function of temperature, was measured in zero dc field under different ac frequencies, with the amplitude of the ac
+excitation field set as 3 Oe. The dc magnetic susceptibility χdc, as a function of temperature down to 0.4 K, was measured
+using a Quantum Design Magnetic Property Measurement System (MPMS) equipped with a 3He insert. The isothermal dc
+magnetization curves in the field up to 7 T applied along the a and c axes were measured at 0.4 K with the same setup.
+Section 3.
+MAGNETOCALORIC MEASUREMENTS
+Magnetocaloric effect (MCE) of the frustrated dipolar magnet KBGB was characterized using a homemade setup integrated
+into the PPMS, for initial temperature 2 K ≤ Ti ≤ 4 K. A DR-based setup is also exploited for MCE measurements with low
+initial temperature Ti ≤ 500 mK.
+A.
+PPMS-based setup for quasi-adiabatic demagnetization measurements
+As shown in Fig. S2, a homemade PPMS-based construction for quasi-adiabatic demagnetization process is set up, inspired
+by the Hagmann-Richards design for space applications [69]. An additional guard stage consisting of copper cylinders and
+Gd3Ga5O12 (GGG) crystals (20 g), a conventional coolant, offer thermal intercepts between the sample stage and the PPMS
+chamber. In experiments, plate-like KBGB single crystals (with a total mass of 0.5 g) are stacked along the c-axis and fixed
+on a silver foil by cryogenic glue. To improve the thermal insulation, a Vespel straw is used to support the sample pillar inside
+the copper cylinder. The guard stage is suppported by PEEK tubes to reduce the thermal exchange with the environment. The
+electrical connection of the thermometer (a field-calibrated RuO2 chip) on top of the pillar is made by two pairs of twisted
+
+bnck
+COWWGLCISI
+bbW2
+bEEK
+Ccc (To a)
+Ws12
+N6abel
+blgid2
+CCC (To a)
+KBCB 2luajG
+p 2.0-10
+manganese wires (25 µm in diameter and approximately 60 cm in length) to reduce the heat leak. A thermal shield protects
+the sample from radiant heating and reduce other parasitic heat loads from the PPMS chamber. Demagnetization cooling
+measurements are performed by gradually decreasing the fields from the initial field Bi at a rate of ˙B = 0.15 T·min−1.
+The parasitic heat load can be estimated from the temperature change rate of sample after the magnet field being exhausted,
+i.e., in the hold process with B = 0. To be specific, the parasitic heat load is estimated by ˙Q = C0 ˙T, where C0 is heat capacity
+of the sample and ˙T is the temperature change rate. For example, when starting from an initial condition of 2 K, it is found
+that ˙T ≈ 5 × 10−6 K/s. Considering C0 ≃ 0.01 J/K for 0.5 g KBGB samples, we thus figure out the parasitic heat load as
+˙Q ≈ 0.05 µW.
+In Fig. S2 we show the isentropic lines of KBGB obtained through the quasi-adiabatic demagnetization measurements, and
+make a comparison with the widely used refrigerant GGG. The results with different initial conditions lead to the same conclusion
+that KBGB clearly outperforms GGG in the lowest cooling temperature.
+B.
+The DR-based quasi-adiabatic demagnetization measurements
+To perform MCE measurements from a lower initial temperature below 500 mK, a standard DR heat capacity sample mount
+is used, which provides a quasi-adiabatic condition with high vacuum in the 3He–4He dilution insert of PPMS. The thermometer
+used is a RuO2 semiconductor. It has been carefully calibrated as functions of temperature (50 mK-4 K) and magnetic field
+(0-5 T), and also extrapolated to 30 mK according to the scaling behavior ln(R − R0) ∼ T −1/4 [67].
+The polymer strips are used to support the sample platform. A KBGB single crystal with a much smaller mass of 2.3 mg is
+used here, to avoid large magnetic torque that may break the suspended lines in the sample mount. To decrease the irreversible
+heating effect on the DR mount, the field sweep rate ˙B has been reduced to 0.06 - 0.09 T·min−1. Due to the small mass of
+the sample, the parasitic heat loads have a stronger influence in the MCE measurements. However, a prominent dip can still be
+observed in the quasi-adiabatic cooling curve that clearly signals the existence of a QCP in Fig. S2(c).
+(a) θ = 0
+(b) θ = Τ
+π 3
+(c) θ = 2 Τ
+π 3
+(d) θ = π
+(e) θ = 4 Τ
+π 3
+(f) θ = 5 Τ
+π 3
+Q1 = ±b*/2
+Q3 = ±b*/2
+Q2 = ±(b*-a*)/2
+b*
+a*
+Q1
+Q3
+Q2
+(g) First Brillouin Zone
+FIG. S3. (a)-(f) show the magnetic configurations of the stripe order with 6-fold degeneracy, i.e., 6-clock AF, which can be labeled with angle
+θ (in complex order parameter Ψxy), and also by ordering vector Q1 (blue dots), Q2 (yellow dots), and Q3 (red dots) shown in (g).
+Section 4.
+MONTE CARLO SIMULATIONS
+As the spin quantum number S = 7/2 is large in KBGB, here we use the classical Monte Carlo simulations with standard
+Metropolis algorithm and single spin update and [70, 71]. The largest system size is 60 × 60, and we calculate the snapshots of
+
+11
+the ground-state spin configurations in Figs. S3(a-f). The corresponding ordering wave vectors Q = ± 1
+2a∗, ± 1
+2b∗, ± 1
+2(a∗ −b∗),
+with a∗, b∗ are the three ordering wave vectors for the 6-clock AF order shown in Fig. S3(g). The phase angle θ of the complex
+order parameter Ψxy can only take 6 discretized values that correspond to the 6-fold degenerate ground states (corresponding to
+Q1, Q2, Q3, see below).
+In Figs. 1(c-e) of the main text, we show histograms of the complex order parameter Ψxy ≡ meiθ under magnetic field
+B = 0.68 T and at different temperature, i.e., (c) T = 0.05 K (6-clock AF), (d) T = 0.14 K (DSL), and (e) T = 0.25 K (PM),
+respectively. To count the histograms, we collect 5 × 106 MC samples on a L = 12 × 12 lattice for statistics.
+The MC simulation results of specific heat are shown in Fig. S4, where the contour plot in Fig. S4(a) resembles the experi-
+mental data in Fig. 3(b) of the main text. The round peak in Cm is located at T ∗ ≃ 270 mK, and the peak heights are converged
+with system sizes, as indicated in the inset of Fig. S4(b). As magnetic fields are applied along the out-of-plane direction, similar
+to the experiments, we also observe that the Cm peaks move towards low temperature side, with heights lowered, in Fig. S4(c).
+0.1
+0.2
+0.3
+0.4
+0.5
+1
+2
+ B= 0 T
+ B= 0.39 T
+ B= 0.68 T
+Cm
+T (K)
+L = 60
+0.2
+0.4
+1
+2
+ L=36
+ L=24
+ L=60
+ L=54
+ L=48
+Cm
+T (K)
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+B (T)
+T(K)
+0.4
+0.8
+1.2
+1.6
+2
+(a)
+(b)
+(c)
+Cm
+0.25
+0.29
+1.5
+2.5
+Cm
+T (K)
+FIG. S4. The calculated results of specific heat Cm. (a) shows the contour plot of Cm data under out-of-plane field B. The (b) zero-field Cm
+curves for different system sizes and (c) Cm curves for different fields are also presented. The inset in (b) compares the Cm data near crossover
+temperature T ∗ ≃ 270 mK. The MC simulations are performed on the HD model [Eq. (1) in the main text] with couplings J = 47 mK and
+D = 80 mK. (c) compares the specific heat curves under zero and finite magnetic fields.
+In the simulations, we use the natural unit (J = 1) in the MC calculations and thus the following process is required for
+comparing the model calculations to experimental data in SI units: (1) We replace the Si operators in Eq. (1) of the main text
+by classical vectors, Si → Sni ≡ 7/2 ni, where ni is a unit vector; (2) The value of temperature T in natural unit should
+be multiplied by a factor of J = 0.047 K; (3) Multiply the magnetic field B in natural unit (i.e., B/JS = 1) by a factor of
+JkB/(gcµB) ≃ 0.028 T.
+

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+Sensitivity analysis for transportability in multi-study,
+multi-outcome settings
+Ngoc Q. Duong, Amy J. Pitts, Soohyun Kim, Caleb H. Miles
+Department of Biostatistics, Mailman School of Public Health, Columbia University
+Abstract
+Existing work in data fusion has covered identification of causal estimands when
+integrating data from heterogeneous sources. These results typically require additional
+assumptions to make valid estimation and inference. However, there is little literature
+on transporting and generalizing causal effects in multiple-outcome setting, where the
+primary outcome is systematically missing on the study level but for which other
+outcome variables may serve as proxies. We review an identification result developed
+in ongoing work that utilizes information from these proxies to obtain more efficient
+estimators and the corresponding key identification assumption. We then introduce
+methods for assessing the sensitivity of this approach to the identification assumption.
+Keywords: Causal inference, Data fusion, External validity, Generalizability, Missing data,
+Proxy variable
+arXiv:2301.02904v1  [stat.ME]  7 Jan 2023
+
+1
+Introduction
+Research in clinical medicine and public health is often concerned with estimating the effect
+of some treatment in a specific target population. However, even in a randomized clinical
+trial, which is considered the gold-standard study design, ensuring external validity remains
+a challenge. This can be due to a variety of reasons, including non-random sampling, overly
+stringent exclusion criteria, or an ill-defined target population of interest (Tan et al., 2022;
+Kennedy-Martin et al., 2015).
+Meta-analysis of summary statistics is a commonly used
+tool to synthesize and generalize findings from published study-level summary statistics,
+but tends to rely on strong, often implausible assumptions. An alternative approach that
+allows for more control over the nuances and heterogeneity across studies is to combine
+individual-level data, when available, from multiple studies, each of which may contain
+insufficient information to address a given scientific question by itself, but which collectively
+have the power to do so. There has been a growing body of work on generalizability and
+transportability methods, which can help address the problem of external validity of the
+effect estimates from integrating individual level data across studies.
+Generalizability concerns the setting where the study population is a subset of the target
+population of interest while transportability addresses the setting where the study popula-
+tion is partially or completely external to the target population (Degtiar and Rose, 2023).
+Specifically, generalizability typically involves extending the causal effect estimate derived
+from a study as long as the covariates in the study population and the target population
+have common support (Gechter, 2015; Tipton, 2014). On the other hand, transportability
+entails extrapolating the effect estimated from a study in which some primary outcome of
+interest is observed to a population represented by a sample in which the outcome is not
+measured.
+Existing methodologies involve directly transporting some estimated causal effect, e.g.,
+the average treatmemt effect (ATE), from studies where the outcomes are observed to other
+studies with missing outcomes or across heterogeneous study designs and settings (Barein-
+1
+
+boim and Pearl, 2016; Dong et al., 2020; Pearl and Bareinboim, 2014; H¨unermund and
+Bareinboim, 2019), or to some broader target population (Dahabreh et al., 2020a,b; Lesko
+et al., 2017; Westreich et al., 2017). When considering multiple studies, it is often the case
+that one will observe different outcomes at follow up. However, existing methods do not
+take advantage of these other potentially correlated and informative outcome variables mea-
+sured at follow-up, which could potentially be leveraged to achieve large efficiency gains.
+Existing outcome proxy-blind methods typically rely on an assumption of homogeneous con-
+ditional potential outcome means for valid transportation of estimation from one population
+to another. Sensitivity analysis strategies have been proposed to study the extent to which
+the violation of these assumptions will affect the estimations and inferences drawn (Nguyen
+et al., 2017; Dahabreh and Hern´an, 2019; Dahabreh et al., 2022).
+In ongoing work, we have developed a new strategy to more efficiently estimate the
+ATE from integrated data across multi-outcome studies, with inconsistent availability of the
+primary outcome of interest at the study level. The proposed methodology takes advantage
+of the availability of follow-up measurements of potential correlates of the main outcome
+to yield more precise estimate of the causal effects.
+In this article, we consider the key
+common outcome regression (or conditional exchangeability for study selection) assumption
+for transportability while leveraging these outcome proxies, which differs slightly from the
+common outcome regression assumption that has been traditionally used for transportability.
+We discuss the resulting bias when this assumption is not met, and develop methodology for
+sensitivity analysis to the violation of this assumption.
+The remainder of the article is organized as follows. In Section 2, we discuss identification
+of the average treatment effect in the multi-study, multi-outcome setting. In Section 3, we
+discuss the bias incurred by violations of the key conditional exchangeability assumption. In
+Section 4, we compare the conditional exchangeability assumption in our setting with that
+used in settings that do not leverage outcome proxies. In Section 5, we develop methods
+for sensitivity analysis for when our assumption is violated. We demonstrate the empirical
+2
+
+performance of our proposed methods in a simulation study in Section 6, and conclude with
+a discussion in Section 7.
+2
+Data integration for studies with primary outcome
+missing systematically
+2.1
+Study and data setting
+In this setting, we let A be the treatment indicator, W be a set of covariates that are
+commonly observed across studies, Y be the primary outcome variable, the set {T1, . . . , Tk}
+be all the potential outcome proxies measured at follow-up in any study, and Js be the
+study-specific subset of {T1, . . . , Tk} that is measured in study s.
+Suppose there are S
+studies that are ordered such that for each s in the first s∗ studies, we observe the set of
+variables (Y, A, Js, W), while for each s in the remaining S − s∗ studies, only the subset
+(A, Js, W) are observed. In other words, Y is systematically missing in the latter set of
+studies. Unlike the standard setup in other works concerning effect transportability that
+only involves (Y, A, W), we introduced the use of Ts, where Ts ⊂ Js is some user-specified
+subset of Js for each study s. Ts could be chosen based on availability and subject matter
+knowledge and must be chosen such that they are observed in at least one of the studies
+{1, 2, . . . , s∗}.
+Studies can be randomized experiments or observational; however, we will not consider
+scenarios in which some studies are randomized experiments and others are observational in
+this work. Then the study-specific average treatment effect and conditional average treat-
+ment effect can be written as:
+ATE(s) = E(Y1 − Y0 | S = s)
+CATE(w, s) = E(Y1 − Y0 | W = w, S = s).
+3
+
+Accordingly, we can define the overall average treatment effect and conditional average treat-
+ment effect as:
+ATE =
+S
+�
+s=1
+ATE(s)
+CATE(w) = E(Y1 − Y0 | W = w)
+where the weights can be user-specified such that �
+s πs = 1. For instance, one can choose
+πs = P(S = s), or the marginal probability of being in each study. Alternatively, we could
+define ATE = EQW,SCATE(W, S) for a user-specified, known distribution QW,S of W and
+S.
+Since Y is not measured in s ∈ {s∗ +1, . . . , S}, we cannot directly estimate the ATE and
+CATE using data from these studies alone. Our purpose is to transport the ATE from the
+first s∗ studies where Y is observed, to the remaining S − s∗ studies while also leveraging
+the information from the outcome proxy set Ts to improve efficiency. For ease of notation,
+let σs be a subset of the first s∗ studies in which both Y and Ts are observed. We can then
+use this information from the studies that form σs to estimate the outcome regression that
+will allow us to transport the causal effects to study s. In this setting, we have shown in
+ongoing, not-yet-published work that the ATE can be nonparametrically identified as:
+ΨATE =
+s∗
+�
+s=1
+πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s}
++
+S
+�
+s=s∗+1
+πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s}
+(1)
+− E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s].
+The terms in the first sum are simply the standard identification formula for the (study-
+specific) average treatment effects when Y is observed. The second sum is identified since
+it only depends on the distribution of Y in the studies in σs, i.e., in which Y is actually
+4
+
+observed.
+Here, we introduced a modification to how transportability has traditionally been done by
+incorporating information from a set of outcomes measured at follow-up that are correlated
+with the main outcome of interest.
+2.2
+Assumptions for Identification of the ATE
+This derivation ATE can be nonparametrically identified given the assumptions that are
+standard for identification for ATE when outcomes are all observed:
+Assumption 1 (Positivity). P(A = 1 | W = w) > 0 for all w with positive probability.
+Assumption 2 (Consistency). Y = AY1 + (1 − A)Y0.
+Assumption 3 (Within-study conditional exchangeability).
+E[Y a | W, A, S = s] = E[Y a | W, S = s] for all s.
+The validity of our estimator relies on a fourth assumption that allows for the transporta-
+tion of the effect across studies:
+Assumption 4 (Common outcome regression (proxy-aware version)).
+E(Y | Ts, W, A = a, S = s) = E(Y | Ts, W, A = a, S ∈ σs) for all s.
+This is a missing at random (MAR)-type assumption, where S can in a sense be thought
+of as a missingness indicator, since missingness is systematic by study.
+We can also introduce a fifth assumption that is not necessary for identification, but
+allows for more borrowing of information across studies, which can help with efficiency:
+Assumption 5 (Common distribution of outcome proxies). Ts ⊥ S | W, A for all s.
+5
+
+This implies the distribution of Ts conditional on treatment assignment and baseline
+covariates is the same across studies. Under this additional assumption, the identification
+result simplifies to:
+ATE =
+S
+�
+s=1
+πsE[E{E(Y | Ts, W, A = 1, S ∈ σs) | W, A = 1}
+− E{E(Y | Ts, W, A = 0, S ∈ σs) | W, A = 0} | S = s].
+In ongoing work, we have developed a simple substitution estimator that involves replac-
+ing each expectation with a regression-based estimate and the outer expectation with an
+empirical mean.
+For the outcome proxy-blind approach, in addition to the first three standard internal
+validity assumptions, Assumption 4 is replaced by a slightly different mean outcome ex-
+changeability assumption: across studies assumption (exchangeability over S) (Dahabreh
+and Hern´an, 2019; Lesko et al., 2017):
+Assumption 6 (Common outcome regression (proxy-blind version)).
+E(Y | W, A = a, S = s) = E(Y | W, A = a, S ∈ σs) for all s.
+Assumption 4 differs from Assumption 6 by additionally conditioning on Ts for each study
+s. Assumptions 4 and 5 together imply Assumption 6. In this article, we will only consider
+sensitivity analysis for the violation of Assumption 4.
+When Assumption 5 is violated,
+the ATE estimator based on Assumption 4 (i.e., the substitution estimator based on the
+identification formula (1)) will remain consistent.
+6
+
+3
+Characterizing the bias resulting from violation of
+the identification assumption
+The validity of ΨATE is dependent on the key assumption 4. This assumption requires no
+heterogeneity in the conditional outcome means given treatment, covariates, and outcomes
+proxies between studies with and without missing outcome (Y ) data. This allows for trans-
+portation of the conditional outcome means, and correspondingly, the ATE and CATE,
+estimable from one study to others.
+In practice, this could be a strong assumption to make while also untestable using ob-
+served data. For instance, in previous unpublished work, we estimated the average treatment
+effect of cognitive remediation (CR) therapy on Social Behavioral Scale (SBS) score, a mea-
+sure for social functioning, using harmonized data from three trials in the NIMH Database
+of Cognitive Training and Remediation Studies (DoCTRS) database. However, the degree
+of effectiveness of CR, especially on functional and occupational outcomes, was less evident
+and has been suggested to vary depending on the setting in which the treatment was admin-
+istered (Barlati et al., 2013; Combs et al., 2008; McGurk et al., 2007; Wykes et al., 2007,
+2011). When this assumption is violated, the substitution estimators described in the pre-
+vious section will be biased. Therefore, we examine two strategies for sensitivity analysis in
+order to examine the robustness of estimates under varying degrees of assumption violation.
+To quantify the degree of violation, let the bias functions be defined as:
+u(A = 1, Ts, W) = E(Y | Ts, W, A = 1, S = s) − E(Y | Ts, W, A = 1, S ∈ σs),
+u(A = 0, Ts, W) = E(Y | Ts, W, A = 0, S = s) − E(Y | Ts, W, A = 0, S ∈ σs)
+(2)
+7
+
+Then, equation (1) when assumption 4 is violated instead becomes:
+ATE =
+s∗
+�
+s=1
+πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s)
++
+S
+�
+s=s∗+1
+πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s)}
+−E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s)} | S = s]
++
+S
+�
+s=s∗+1
+πsE[E {u (A = 1, Ts, W) | W, A = 1, S = s}
+− E {u (A = 0, TS, W) | W, A = 0, S = s} | S = s],
+where the last sum is not identified. Then, the study-specific bias for study s is:
+E [E {u (A = 1, Ts, W) | W, A = 1, S = s} − E {u (A = 0, Ts, W) | W, A = 0, S = s} | S = s]
+= E[δ∗(W)|S = s].
+(3)
+By rearranging terms, δ∗(W) can be alternatively written as:
+E [E (Y | Ts, W, A = 1, S = s) − E (Y | TS, W, A = 1, s ∈ σs) | W, A = 1, S = s]
+− E [E (Y | Ts, W, A = 0, S = s) − E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s]
+= E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s)
+− {E [E (Y | Ts, W, A = 1, s ∈ σs) | W, A = 1, S = s]
+− E [E (Y | Ts, W, A = 0, s ∈ σs) | W, A = 0, S = s]}.
+(4)
+The latter term cannot be simplified unless Assumption 5 holds.
+8
+
+4
+Comparison with bias functions in settings without
+incorporation of follow-up surrogate outcomes
+In recent work, Dahabreh and Hern´an (2019) developed sensitivity analysis for transportabil-
+ity considering a similar setting of two types of studies with and without missing outcomes.
+In the base case, there are two studies considered (missingness of the outcome variable de-
+noted by a binary indicator S). To describe this setting using our notation, we simply have
+σ0 = σ1 = {1} (i.e., study S = 1 with the observed outcome of interest is used to impute
+the conditional outcome means for study S = 0). Equivalently, for ease of interpretation in
+the base case, let S = 1 and S = 0 denote the study where the primary outcome of interest
+is observed and not observed, respectively.
+In the setting where the model used to impute conditional potential outcomes does not
+utilize information from Ts, Dahabreh and Hern´an (2019) define:
+u(A = a, W) = E[Y | A = a, W, S = 1] − E[Y | A = a, W, S = 0].
+The difference between these bias functions can then be obtained as:
+δ(W) = u(A = 1, W) − u(A = 0, W)
+= E[Y 1 − Y 0 | W, S = 1] − E[Y 1 − Y 0 | W, S = 0]
+This expression can be qualitatively expressed as the difference in the conditional average
+treatment effects between the two studies. This qualitative interpretation can aid in concep-
+tualizing and thinking about more appropriate values and range for sensitivity parameters
+when examining robustness of the results. More specifically, assuming higher levels of the
+outcome are preferred, if we believe the participants in studies with missing outcomes benefit
+less from treatment, then true δ can be assumed to be positive and vice versa (Dahabreh and
+Hern´an, 2019). Since our bias functions are conditional on the set of proxy outcomes, the
+9
+
+term δ∗(W) in (4) unfortunately cannot be reduced further to a more interpretable statistical
+entity. When we take Ts to be the empty set, the bias function δ∗(W) reduces to the same
+expression.
+5
+Accounting for violation of the common outcome re-
+gression assumption through sensitivity analyses
+We consider two scenarios in which we assume the bias terms u(A = 1, Ts, W) and u(A =
+0, Ts, W) to be 1) constants and 2) bounded functions of the outcome proxies and/or baseline
+covariates. The first scenario involves making a stronger assumption about the bias terms.
+On the other hand, the second scenario requires weaker assumptions but allow them to be
+non-constant.
+5.1
+Bias functions assumed to be some fixed values
+Although it might be more reasonable to assume that the bias functions are dependent on
+some baseline covariates, for ease of implementation of sensitivity analysis, one can also
+suppose they are constant. When u(A = 1, Ts, W) and u(A = 0, Ts, W) are independent of
+the baseline covariates W and the outcome proxy set Ts, the conditional expectations of the
+bias functions, and in turn, the term δ∗(W) in (3), reduce to:
+δ = u1 − u0, where δ, u1, and u0 ∈ R
+(5)
+The sensitivity analysis involves correcting for the above-mentioned bias term by adding it
+back to the identification formula ΨATE, which relies on the common outcome regression
+assumption.
+10
+
+ATE =
+s∗
+�
+s=1
+πsE {E (Y | W, A = 1, S ∈ σs) − E (Y | W, A = 0, S ∈ σs) | S = s}
++
+S
+�
+s=s∗+1
+πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s}
+− E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s} | S = s] +
+S
+�
+s=s∗+1
+πs (u1 − u0)
+=ΨATE +
+S
+�
+s=s∗+1
+πs (u1 − u0)
+(6)
+where u1 and u0 are scalars.
+In practice, the true bias term would be unknown. Thus, one strategy is to propose a
+grid of sensitivity parameters that covers the potential range of values in which the true bias
+term might fall. This grid of sensitivity parameters can be specified using subject-matter
+knowledge. We can then adjust for the bias term in the estimation step by adding back
+the different sensitivity parameters to the estimated ATE using our proposed method. This
+also allows for observation of the behavior of the estimated ATE as we vary the sensitivity
+parameters.
+5.2
+Bounded covariate-dependent bias functions
+One might also believe that the bias term is not constant at all levels of the baseline covariates
+and/or the outcome proxies. When the assumption of fixed-value bias terms is considered
+too strong, but the functional forms for bias terms cannot be confidently determined from
+existing knowledge of the data mechanism (as will typically be the case), one can still recover
+some information about the true ATE without having to correctly specify the bias terms. If
+we instead assume the bias terms to be some bounded functions, we can compute a bound
+around the (na¨ıve) ATE estimate that contains the true ATE by varying the bounds of these
+functions. This provides information on how far away the true ATE can be from the estimate
+11
+
+obtained constrained by the bounds of the bias term.
+Identifying the bounds for the bias term can be expressed as maximizing and minimizing
+the objective function:
+E[E[u(A = 1, Ts, W) | W, A = 1, S = s] − E[u(A = 0, Ts, W) | W, A = 0, S = s] | S = s]
+subject to the following constraints:
+|u(A = 1, Ts = ts, W = w)| ≤ γ1
+|u(A = 0, Ts = ts, W = w)| ≤ γ0
+for all ts and w, which implies |E[u(A = 1, Ts, W) | W, A = 1, S = s]| ≤ γ1 and |E[u(A =
+0, Ts, W) | W, A = 0, S = s]| ≤ γ0 where γ1, γ1 ∈ R+.
+Then we have −(γ1 + γ0) ≤ u(A = 1, Ts, W) − u(A = 0, Ts, W) ≤ γ1 + γ0. If we have no
+reason to suspect we know more about the bounds of one bias function than the other (as
+will typically be the case), we may simply choose to specify a scalar sensitivity parameter γ
+to be the maximum of γ1 and γ2, in which case we have −2γ ≤ u(A = 1, Ts, W) − u(A =
+0, Ts, W) ≤ 2γ.
+By equation (6) even though we do not know the form of the bias functions u(A =
+1, Ts, W) and u(A = 0, Ts, W), we can partially recover the true ATE using the bounds
+around the na¨ıve estimate:
+ΨATE − 2 max(γ1, γ0) ≤ ATE ≤ ΨATE + 2 max(γ1, γ0)
+ΨATE − 2γ ≤ ATE ≤ ΨATE + 2γ
+(7)
+If the bias functions are in fact bounded by some value smaller than or equal to our specified
+values for the sensitivity bounds, the true ATE would fall between [ΨATE − 2γ, ΨATE + 2γ].
+Then, the true ATE is partially identified without assumptions about the functional form
+12
+
+of u(A = 1, Ts, W) and u(A = 0, Ts, W). One can then use the bootstrap standard error for
+the substitution estimator of the identification formula (1) to determine the amount to add
+and subtract from the upper and lower bounds, respectively, in order to produce confidence
+intervals for the partial identification sets for each value of the sensitivity parameter. Since
+the sensitivity bounds are a deterministic function of the sensitivity parameter, bootstrapping
+need only be done once.
+6
+Simulations
+6.1
+Data generating mechanism
+We consider the setting of two studies, with S = 1 indicating the study where the primary
+outcome is available.
+We generate random sample draws with sample size n = 100 for
+both studies. The data generating mechanism is as follows. W, T0 come from independent
+standard normal distributions, and T1 comes from a normal distribution with mean and
+variance of 1. Then
+T = I(A = 1) × T1 + I(A = 0) × T0
+Y 0 = −4T0 + W + ϵ0
+Y 1 = 4T1 + W + ϵ1
+Y = I(A = 1) × Y1 + I(A = 0) × Y0
+where ϵ1, ϵ0 ∼ N(0, 1).
+Via these specifications, T fully mediates the relationship between A and Y (direct effect
+from A to Y is constrained to be 0). As a result, the true ATE = 4. This is also a more
+basic setting in which the vector T is observed in all studies.
+Due to the nature of the DoCTRS database, which is comprised of randomized clinical
+trials, in our base setting, we specified the marginal probability P(A = 1) = 0.5, represent-
+13
+
+ing random treatment assignment. This treatment assignment satisfies the positivity and
+exchangeability assumption.
+Specifically, to incorporate the difference in conditional outcome means between the
+two types of studies, in studies missing the outcome, we added constant bias terms to
+the counterfactual outcomes Y0 and Y1. Similar to the data generating step, we preserved
+the observed counterfactual outcome from the corresponding treatment assignment, which
+satisfies the consistency assumption. By (5), we have:
+Y 0
+S=1 = Y 0
+S=0 + u0
+Y 1
+S=1 = Y 1
+S=0 + u0 + δ
+(8)
+for u0 ∈ {−3, 0, 3}, δ ∈ {−2, 0, 2}.
+Then the bias reduces to a single parameter δ, since it is no longer a function of u0 when
+computing the ATE:
+E(Y 1 − Y 0 | S = 1) = E(Y 1 − Y 0 | S = 0) + δ
+(9)
+In the case where the bias term is a function of baseline covariates and surrogate outcome,
+we had the following specification for the true bias:
+u0 = b0 × sin (Ts + W)
+u1 = b1 ×
+exp(Ts+W)
+1+exp(Ts+W)
+for b0 ∈ {2, 3, 4} and b1 ∈ {1, 2, 3}.
+6.2
+Adjusting for sensitivity parameter in estimation step
+In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified
+such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to
+14
+
+the true ATE after compared to before. In addition, the corresponding 95% CIs are expected
+to cover the true ATE 95% of the times. Although coverage probability can be examined
+more in a more robust fashion using bootstrapped confidence intervals across all simulations,
+in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the value of the sensitivity
+parameter that reflects the degree of assumption violation all but one instance, which is in
+line with our expectations.
+Scenario 1. When the bias terms are assumed to be constants, a natural approach
+would be to specify a two-dimensional grid of sensitivity parameters for both scalars u0 and
+u1. However, by (8), it is equivalent to specifying u0 (or u1) and δ. In fact, since the u0 (or
+u1) as constant terms cancel out during adjustment, it is sufficient to specify one sensitivity
+parameter δ (9). We also note that δ being 0 does not necessarily imply assumption 4 is
+met, since the bias terms u0 and u1 could cancel exactly.
+To implement sensitivity analysis, we follow the steps:
+1. Specify a grid of sensitivity parameters δ.
+The grid should be reasonably wide to
+contain true δ.
+2. Estimate the na¨ıvely transported ATE using the identification result in (1)
+3. Sequentially add the values in the sensitivity parameter grid to the na¨ıvely estimated
+ATE, using the result in (6) to obtain the bias-corrected ATE estimates.
+We then plotted the bias-corrected estimates under different sensitivity parameters against
+the true ATE. Additionally, we bootstrapped the bias-corrected estimates to obtain the 95%
+confidence intervals and explore coverage across different values of u0 and δ.
+Scenario 2. When we want to make minimal assumptions about the functional form of
+the bias, we can still perform sensitivity analysis on the true ATE using the following steps:
+1. Specify a grid of sensitivity parameters called γ that potentially include the upper and
+lower bounds of the true bias functions
+15
+
+2. Computed the “na¨ıve” ATE estimate using the identification result in (1)
+3. Construct the upper and lower bound around the estimated ATE using (7) where γ is
+replaced with the sensitivity parameters.
+We also plot the na¨ıve ATE estimates and the bounds around these estimates at each value
+of the sensitivity parameters. In practice, the bias functions are of course unknown and
+cannot be estimated from observed data. Therefore, when specifying the grid of sensitivity
+parameters, the analyst needs to employ subject matter knowledge about the data generating
+mechanism to select values of δ and γ.
+We then explore the behavior of the adjusted estimators via simulations. In the first case,
+we focused on the general unbiasedness of the correctly-adjusted point estimate for both the
+overall ATE and ATE among studies with missing outcomes, as well as the 95% CI coverage
+across degrees of assumption violation (i.e., across values of true u0 and δ). In the second
+case, we looked for correct bounding of the true ATE.
+6.3
+Simulation Results
+6.3.1
+Bias terms as constants
+We examine the estimates produced by our method under the different degrees of violation
+of assumption 4, before and after taking into account the specified sensitivity parameter.
+Figure 1 shows the estimates (95% CI) for the true overall ATE using our method under
+varying magnitudes and directions of the bias terms from one single simulation.
+16
+
+Figure 1: Sensitivity-parameter-adjusted ATE estimate shown against the true overall ATE
+across values of the true bias and sensitivity parameter; n=100 for each study, 95% CI
+constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted
+estimate corresponds to the unadjusted estimate. Horizontal dotted line shows the true ATE
+given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ
+In the presence of non-zero bias, when the value of the sensitivity parameter δ is specified
+such that it is equal to true δ, the ATE estimate after bias adjustment tends to be closer to
+the true ATE after compared to before. In addition, the corresponding 95% CIs are expected
+to cover the true ATE 95% of the times. Although coverage probability can be examined
+more in a more robust fashion using bootstrapped confidence intervals across all simulations,
+in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the value of the sensitivity
+parameter that reflects the degree of assumption violation all but one instance, which is in
+17
+
+True  = -2
+True  = 0
+True θ = 2
+UU
+2
+Estimate
+4
+JU
+-2
+2
+Sensitivity parameterline with our expectations.
+Figure 2:
+Sensitivity-parameter-adjusted ATE estimates shown against the true study-
+specific ATE in the study in which the outcome is unobserved across values of the true
+bias and sensitivity parameter; n=100 for each study, 95% CI constructed from 1000 boot-
+strap samples. When sensitivity parameter δ = 0, the adjusted estimate corresponds to the
+unadjusted estimate. Horizontal dotted line shows the true study-specific ATE given true δ;
+vertical dotted line indicates sensitivity parameter δ equals true δ
+Figure 2 shows similar results for the study-specific ATE estimates in the study with
+missing outcomes (before and after bias adjustment) from the same simulated data. Com-
+pared to the results in Figure 1, after adjustment using the correct sensitivity parameters,
+the 95% CIs contain the true ATE more frequently than the CIs of the unadjusted estimates
+in the study with missing primary outcome. Figure 2 also shows an example where infer-
+18
+
+True  = -2
+True  = 0
+True  = 2
+10.0 -
+7.5 
+5.0 
+uo
+=
+2.5 
+-3
+0.0 
+10.0 
+7.5 
+Estimate
+rue
+5.0 
+uo
+2.5 
+II
+-
+0.0 
+7.5 
+5.0 -
+2.5 
+II
+3
+0.0 
+-2.5
+2
+0
+2
+2
+0
+2
+2
+Sensitivity parameterence is sensitive to the violation of our assumption at a magnitude of δ between -1 and -2
+(u0 = −3, bottom left panel), between which the 95% CI changes from not containing to
+zero to containing zero.
+When we increased the sample size (n=200 and n=500), we saw general reductions in the
+errors of these single estimates (Figures A.1, A.3). In most cases, even when there is error
+in the adjusted estimates, the 95% CI bootstrap confidence intervals provide good coverage
+(Figures 1, A.1, A.3). The reduction in error and improved coverage are more pronounced
+when estimating the study-specific effect in the study with missing outcomes than in the
+overall ATE combining the two studies (Figures A.2, A.4).
+We also ran 1000 simulations under the same data generating mechanism and obtained
+the unadjusted and sensitivity-parameter-adjusted estimates for each simulation. We then
+showed the mean and 2.5th and 97.5th quantiles of these estimates under each combination
+of the true bias values. We can see that when averaged across 1000 simulations, the adjusted
+estimates closely approximate the true ATE (Figures 3, 4) when the true value of δ is used
+for the sensitivity parameter.
+19
+
+Figure 3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall ATE
+across values of the true bias sensitivity parameter; mean, 2.5th and 97.5th quantiles obtained
+from 1000 simulations. When sensitivity parameter δ = 0, the adjusted estimate corresponds
+to the unadjusted estimate. Horizontal dotted line shows the true overall ATE given true δ;
+vertical dotted line indicates sensitivity parameter δ equals true δ
+20
+
+True  = -2
+True  = 0
+True  = 2
+6 
+5
+.
+True
+4
+uO = 3
+2
+6
+Estimate
+150
+True uO = 0
+4
+2
+6
+5
+.
+.
+True uO = 3
+4
+2
+2
+1
+0
+2
+-2
+-1
+0
+"
+2
+-2
+1
+0
+2
+Sensitivity parameter Figure 4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in the
+study with missing outcome across values of the true bias and sensitivity parameter; mean,
+2.5th and 97.5th quantiles obtained from 1000 simulations. When sensitivity parameter δ
+= 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal dotted line
+shows the true study-specific ATE given true δ; vertical dotted line indicates sensitivity
+parameter δ equals true δ
+When approximate sensitivity parameters δ are used (δ ∈ {−1, 1} when true δ ∈ {−2, 2}),
+the middle 95% values of adjusted estimates also cover the true ATE whereas those of
+unadjusted estimates do not (Figure 4).
+Figure 5 compares the errors in the estimates and sensitivity of associated inferences
+between the outcome proxy-blind method of Dahabreh et al. (2020b); Lesko et al. (2017)
+and our proposed method across 1000 simulations.
+21
+
+True  = -2
+True  = 0
+True  = 2
+8
+6
+True uO = -3
+4
+2
+0
+8
+6
+Estimate
+True uO = 0
+4
+2
+0
+8
+6
+True uO = 3
+4
+2
+0
+2
+U
+2
+-2
+1
+2
+0
+2
+Sensitivity parameterFigure 5: Sensitivity-parameter-adjusted ATE estimates obtained from our proposed method
+and the outcome proxy-blind method; mean, 2.5th and 97.5th quantiles obtained from 1000
+simulations. When sensitivity parameter δ = 0, the adjusted estimate corresponds to the
+unadjusted estimate. Horizontal dotted line shows the true overall ATE given true δ; vertical
+dotted line indicates sensitivity parameter δ equals true δ
+The distributions of the estimates from both methods are centered on the true parameter.
+However, the estimates tend to be more precise when we utilize the information from the
+outcome proxy (as demonstrated through the narrower 2.5th-97.5th quantile range). The
+efficiency gains have implications for the sensitivity analysis, since resulting inferences are
+not as sensitive given analogous magnitude in violation of the identification assumption 4.
+Assumption 4 implies both u0 and u1 equal 0. As a result, the true δ also equals 0.
+This suggests transportation of the conditional potential outcome means, and in turn, the
+22
+
+True  = -2
+True  = 0
+True θ = 2
+6
+5
+3
+6
+4
+3
+6
+5
+2
+2
+2
+-2
+2
+Sensitivityparameter
+Outcome regression method
+Proposed methodconditional average treatment effects, can be done without incurring bias (vertical middle
+panes, figure 3). We also observed that, when δ is 0, regardless of the values of u0 (and
+u1), there is also no bias (vertical middle panes, figure 3) in the unadjusted estimator. In
+both cases, no bias correction would be necessary, and incorporating a non-zero δ sensitivity
+parameter will actually introduce bias to the estimate.
+6.3.2
+Bias terms as bounded functions
+When the sensitivity parameter γ is greater or equal to max{γ0, γ1} for the true function
+bounds γ0 and γ1, the bounds always include the true ATE when the bias functions are
+bounded by γ0 and γ1 (Figure 6).
+23
+
+Figure 6: ATE estimates with sensitivity bounds shown against the true overall ATE across
+values of the true bias and sensitivity parameter. When sensitivity parameter γ = 0, the
+bounds collapse to a point estimate. Blue horizontal dotted line shows the true study-specific
+ATE given true bias functions
+Although this approach requires minimal assumptions about the bias functional form, it
+can also be conservative since the true bias functions are unlikely to evaluate to the bounds
+across the domain of the functions. For instance, the bottom three panels of Figure 6 show
+that when the sensitivity parameter γ is greater than or equal to max(true γ0, true γ1),
+while the bounds on the estimate contain the true ATE, they also contains the null value
+zero as well. On the other hand, these bounds do not rely on an assumption of constant bias
+functions, which we may often have no reason to believe. Here, we demonstrated through
+24
+
+True y 1 = 1
+True  1 = 2
+True y 1 = 3
+U
+Estimated ATEwith
+4
+n
+8.
+Sensitivity parameter ysimulations that sensitivity analysis with relaxed and more credible assumptions can still
+provide helpful information about the parameter of interest. However, when the bounds
+are too narrow or too wide, sensitivity analysis using bounded bias functions might not be
+accurate (i.e., not containing the true parameter) or useful (i.e., containing the null value
+when the truth is non-null), respectively.
+7
+Discussion
+In this paper, we discussed a data integrative method that utilizes information from avail-
+able proxies of the outcome of interest measured at follow-up for efficiency gains. We then
+presented two sensitivity analysis strategies specific to this approach for causal effect trans-
+portation when the identification assumption is violated. Our modification to the identifica-
+tion of the ATE in (1) allows for more efficient estimators given sufficiently strong outcome
+proxies. As a result, our bias functions also have similar, yet distinct interpretations than
+the bias functions of Dahabreh and Hern´an (2019).
+When the bias terms are assumed to be constants, we can obtain different bias-adjusted
+point estimates based on our specification of the sensitivity parameters. Additionally, via
+obtaining the 95% bootstrap confidence interval for the bias-adjusted estimates, we can
+examine the robustness of inferences made using our method under varying magnitudes of
+assumption violation. Specifically, beyond certain values of the sensitivity parameters, the
+95% CI will cross the null value 0. These are the degrees of violation that can affect inferences
+(where the 95% CI suggest a change from significant results to non-significant results).
+We also proposed sensitivity analysis using bounded bias functions as an alternative when
+one believes the assumption of a fixed-value bias term is too strong. This approach allows
+for inferences with minimal assumptions about the unobserved bias functions but can still
+provide useful information about the parameter of interest. Due to fewer assumptions being
+made, the results are more conservative and robust, hence more reasonable and credible.
+25
+
+Specifically, although we are unable to obtain a point estimate, sensitivity analysis using
+bounded bias functions can still be informative in the sense of providing information about
+the general direction of the parameter of interest (beneficial or harmful). This method is
+generally more conservative if the bounds on the functions are not close to their extreme
+values, if the bias functions are generally not close to their extreme values, or if there is a
+large difference between the extrema of the two bias functions.
+Correct specification of the bias functions would allow for more precise and informative
+estimation of the true ATE. However, since they are generally unknown and non-estimable
+from observed data, sensitivity analysis will typically be the realistic course of action.
+When conducting sensitivity analysis, the analyst can start off by specifying a wide grid
+of the sensitivity parameter and examining the behaviors of the point estimates and 95%
+CI (first approach) as well as bounds around the estimates (second approach). They can
+then search for the “critical” sensitivity parameters that still suggest rejection of the null
+hypothesis, i.e., the 95% CI (in the first case) and bounds around the estimate (in the
+second case) that do not contain 0. It can be determined if greater bias is plausible by using
+background knowledge of the data generating mechanism or further hypothesizing about
+such mechanism. If there is little or no evidence that the true bias functions exceed these
+critical sensitivity parameters, one can be more comfortable in concluding that the observed
+effect and associated inferences are robust to violation of the transportability assumption
+(Ding and VanderWeele, 2016; Cornfield et al., 1959).
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+29
+
+A
+Additional figures
+Figure A.1: Sensitivity-parameter-adjusted ATE estimates shown against the true overall
+ATE across values of the true bias and sensitivity parameter; n=200 for each study, 95% CI
+constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted
+estimate corresponds to the unadjusted estimate.
+Horizontal dotted line shows the true
+overall ATE given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ
+30
+
+True  = -2
+True  = 0
+True θ = 2
+6
+6
+stimate
+5
+rue
+uo
+4
+=
+-
+3
+6
+5
+4
+3
+3
+2
+0
+-2
+0
+2
+2
+2
+Sensitivity parameterFigure A.2: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in
+the study with missing outcome across values of the true bias and sensitivity parameter;
+n=200 for each study, 95% CI constructed from 1000 bootstrap samples. When sensitivity
+parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal
+dotted line shows the true study-specific ATE given true δ; vertical dotted line indicates
+sensitivity parameter δ equals true δ
+31
+
+True  = -2
+True  = 0
+True θ = 2
+7.5
+5.0 
+rue
+uo
+=
+2.5
+0.0 
+8
+6
+Estimate
+True
+4
+uo
+II
+-
+0 
+8
+6.
+True uO =
+4
+2.*
+3
+0
+2
+0
+2
+-2
+0
+2
+Sensitivity parameterFigure A.3: Sensitivity-parameter-adjusted ATE estimates shown against the true overall
+ATE across values of the true bias and sensitivity parameter; n=500 for each study, 95% CI
+constructed from 1000 bootstrap samples. When sensitivity parameter δ = 0, the adjusted
+estimate corresponds to the unadjusted estimate.
+Horizontal dotted line shows the true
+overall ATE given true δ; vertical dotted line indicates sensitivity parameter δ equals true δ
+32
+
+True  = -2
+True  = 0
+True θ = 2
+6
+5
+4
+3
+=3
+2
+6
+5
+Estimate
+True
+4
+uo
+=
+3
+2
+6
+True uO
+4
+II
+3
+0
+-2
+2
+Sensitivity parameter Figure A.4: Sensitivity-parameter-adjusted ATE estimates shown against the true ATE in
+the study with missing outcome across values of the true bias and sensitivity parameter;
+n=500 for each study, 95% CI constructed from 1000 bootstrap samples. When sensitivity
+parameter δ = 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal
+dotted line shows the true study-specific ATE given true δ; vertical dotted line indicates
+sensitivity parameter δ equals true δ
+33
+
+True  = -2
+True  = 0
+True θ = 2
+6
+True
+4
+u0 = -3
+2
+0.
+8
+Estimate
+6
+True uO = 0
+4
+-T
+-
+0
+8
+I
+6.
+True uO =
+4
+2
+2
+0
+-2
+0
+2
+2
+2
+Sensitivity parameterB
+Derivation of the sensitivity analysis formula when
+Assumption 4 is violated
+ATE =
+s∗
+�
+s=1
+πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s}
++
+S
+�
+s=s∗+1
+πsE [E {E (Y | Ts, W, A = 1, S = s) | W, A = 1, S = s}
+− E {E (Y | Ts, W, A = 0, S = s) | W, A = 0, S = s} | S = s]
+=
+s∗
+�
+s=1
+πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s}
++
+S
+�
+s=s∗+1
+πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) + u (A = 1, Ts, W) | W, A = 1, S = s}
+− E {E (Y | Ts, W, A = 0, S ∈ σs) + u (A = 0, Ts, W) | W, A = 0, S = s} | S = s]
+=
+s∗
+�
+s=1
+πsE{E(Y | W, A = 1, S = s) − E(Y | W, A = 0, S = s) | S = s)
++
+S
+�
+s=s∗+1
+πsE [E {E (Y | Ts, W, A = 1, S ∈ σs) | W, A = 1, S = s)}
+−E {E (Y | Ts, W, A = 0, S ∈ σs) | W, A = 0, S = s)} | S = s]
++
+S
+�
+s=s∗+1
+πsE[E {u (A = 1, Ts, W) | W, A = 1, S = s}
+− E {u (A = 0, Ts, W) | W, A = 0, S = s} | S = s]
+34
+