11institutetext:  University of Liverpool 22institutetext: MBZUAI

# Multi-dimensional Fusion and Consistency for Semi-supervised Medical Image
Segmentation

Yixing Lu 11 Zhaoxin Fan Equal Contribution22 Min Xu Corresponding Author22

###### Abstract

In this paper, we introduce a novel semi-supervised learning framework
tailored for medical image segmentation. Central to our approach is the
innovative Multi-scale Text-aware ViT-CNN Fusion scheme. This scheme adeptly
combines the strengths of both ViTs and CNNs, capitalizing on the unique
advantages of both architectures as well as the complementary information in
vision-language modalities. Further enriching our framework, we propose the
Multi-Axis Consistency framework for generating robust pseudo labels, thereby
enhancing the semi-supervised learning process. Our extensive experiments on
several widely-used datasets unequivocally demonstrate the efficacy of our
approach.

###### Keywords:

Medical image segmentation Semi-supervise learning ViT-CNN fusion Multi-axis
consistency.

##  1 Introduction

Medical image segmentation is a pivotal and intricate process within the realm
of intelligent diagnosis, entailing the extraction of regions of interest
within medical imagery. This task is of paramount importance for enabling
precise diagnosis and tailored treatment. Over recent years, Convolutional
Neural Networks (CNNs) [15, 12, 4] and Vision Transformers (ViTs) [5, 32, 11],
both endowed with a U-shaped architecture, have witnessed significant advances
in the domain of medical image segmentation.

Medical image segmentation literature mainly employs pretrained Convolutional
Neural Networks (CNNs) [15] or Transformers [5]. The benefits of using both
CNNs and Vision Transformers (ViTs) haven’t been thoroughly explored.
Interestingly, CNNs and ViTs seem to complement each other for medical image
understanding. CNNs excel in local feature recognition [3], while ViTs are
superior in comprehending long-range dependencies [14, 7]. For medical image
segmentation, combining these strengths is crucial to understanding the organ
and its interrelations with others. This raises the question: _Can we fuse the
strengths of CNNs and ViTs into a single framework for medical image
segmentation?_ An additional noteworthy observation is that both Convolutional
Neural Networks (CNNs) and Vision Transformers (ViTs) tend to necessitate
extensive quantities of annotated data for effective training. While this may
be manageable in the realm of natural image segmentation, it poses a
formidable hurdle in the context of medical image segmentation, where the
process of annotation is both laborious and costly. As a result, a second
question emerges: _Is it feasible to reduce the model’s dependency on
annotated medical image-mask pairs without undermining the performance of the
ViT/CNN?_

In response to the aforementioned questions, we introduce a novel semi-
supervised learning framework for medical image segmentation in this study. We
first propose a simple yet efficacious Multi-scale ViT-CNN Fusion scheme. The
underlying principle of this approach is that the integration of ViT and CNN
features can equip the model with the ability to capture both intricate local
details and extensive global long-range dependency information. Additionally,
given that both the CNN and ViT are pretrained on large-scale networks, they
can retain both abstract natural features and domain-specific medical features
during fusion, thereby further enhancing the segmentation task. Moreover,
inspired by vision-language models [30, 24, 36, 39], and considering the
relative ease of obtaining text descriptions for medical images, we introduce
a text-aware language enhancement mode to further enrich the learned features.
This effectively addresses the first question. Subsequently, we incorporate a
Multi-Axis Consistency framework in our study to extend our approach to
scenarios where annotated labels are limited. Within this framework, we unify
and formulate multiple consistency regularizations under a single framework,
dubbed Multi-AXIs COnsistency (MaxiCo). This framework combines intra-model,
inter-model, and temporal consistency regularizations to generate robust
probability-aware pseudo-labels, thereby enabling the use of a large corpus of
unlabeled data for semi-supervised network training. Furthermore, we design a
voting mechanism within this module, whereby each intermediate output can
contribute to the final pseudo-label. This mechanism further enhances the
trustworthiness of pseudo-labels and bolsters the final model’s performance,
therefore providing a satisfactory answer to the second question.

To deliver a comprehensive demonstration of the efficacy of our proposed
method, we have undertaken extensive experimentation using the MoNuSeg [18]
dataset and QaTa-COV19 [9] dataset. The empirical results obtained from these
experiments substantiate that our method establishes a new benchmark in fully-
supervised settings, outperforming existing state-of-the-art methodologies.
Moreover, within semi-supervised scenarios, our strategy shows remarkable
superiority over other leading-edge techniques.

Our contribution can be summarized as: 1) We pioneer a semi-supervised
framework that harnesses the power of textual information to support fused
ViT-CNN networks for medical image segmentation, representing a unique
approach to this problem. 2) We propose a novel Multi-scale Text-aware ViT-CNN
Fusion methodology that adroitly amalgamates CNNs and ViTs to boost
segmentation accuracy. 3) We introduce a novel Multi-Axis Consistency Learning
module that capitalizes on consistency regularizations to generate reliable
pseudo-labels for semi-supervised learning, effectively addressing the issue
of data scarcity.

##  2 Related Work

Transformers in Medical Image Segmentation. The success of Vision Transformer
(ViT) [10] in various computer vision tasks has led to its integration into
medical image segmentation [41, 11, 38, 42, 22]. Some studies use transformers
for image representation [14], while others propose hybrid encoders combining
transformers and convolutional neural networks (CNNs) [7]. Cao et al. [5]
proposed a pure transformer network replacing convolutional layers with Swin
Transformer blocks [23] and the traditional Unet skip-connection with a
transformer-based channel-attention module [34]. However, these methods are
typically trained in a fully-supervised manner, which may be impractical due
to the scarcity of annotated medical data. To address this, we introduce a
semi-supervised approach that fuses ViT-CNN networks for medical image
segmentation, aiming to overcome the challenge of limited annotated data.

Semi-Supervised Medical Image Segmentation. In light of the challenge posed by
the dearth of annotated data in medical image segmentation, semi-supervised
learning has come to the fore as a promising solution [40, 33, 37, 35].
Predominant strategies for semi-supervised learning encompass pseudo labeling
[8], deep co-training [43], and entropy minimization [13]. In our work, we
adopt a consistency learning framework to generate pseudo labels for unmarked
images. There are several versions of consistency regularization methods,
including temporal consistency [19], model-level consistency [25], and pyramid
consistency [26], in existing literature. However, most of these methods only
depend on a single type of regularization, which will limit the porwer of the
model. In contrast, our approach amalgamates multiple consistency
regularizations and employs a voting mechanism to produce more robust pseudo
labels, which demonstrates better performance.

Vision-Language Fusion for Dense Predictions.In recent years, the fusion of
vision and language in large-scale pretraining has garnered significant
attention. The CLIP model [29] showcases this, demonstrating impressive
transfer learning across multiple datasets. Building on this, researchers have
explored fine-tuning the CLIP model for dense prediction tasks [30, 20, 36],
framing it as a pixel-text matching problem. Vision-language models also
enable zero-shot inference, bridging the gap between seen and unseen
categories [39]. Some research has further explored the potential of visual
prompts and their interpolation with text prompts during training [24]. In
this paper, we use textual information to enhance ViT-CNN training for medical
image segmentation, showcasing a novel application of vision-language fine-
tuning.

##  3 Methodology

###  3.1 Overview

Given an image $I\in X$, where $X$ represents the space of all possible
medical images, and a text input $T\in T$, where $T$ is the space of all
possible text inputs (e.g., medical notes or labels), the task of medical
image segmentation in our study is to learn a mapping function
$F_{\theta}:X\times T\rightarrow Y$:

$F_{\theta}:X\times T\rightarrow Y$ (1)

Where $Y$ represents the segmentation masks corresponding to the input medical
image, and $\theta$ denotes the parameters of our model. The goal is to train
the model such that the mapping function $F_{\theta}$ can accurately predict
the segmentation mask $y\in Y$ for any given input image and text $(I,T)$.

To address the first challenge, our Multi-scale Text-aware ViT-CNN Fusion
scheme integrates a pretrained ViT and CNN, incorporating text features for
increased prediction accuracy. We perform vision-language pretraining to
obtain vision and text features, aligning them to formulate ViT features.
These are then fused with CNN features at various resolutions, enabling the
efficient use of local and global features.

To facilitate semi-supervised training, we introduce a Multi-Axis Consistency
framework to generate pseudo labels, leveraging inter-model, multi-scale
intra-model, and temporal consistency. Our network makes multiple predictions
in a single pass, generating probabilistic pseudo labels via a voting
mechanism, supporting semi-supervised training.

Figure 1: The illustration of Multi-scale Text-aware ViT-CNN Fusion.

###  3.2 Multi-scale Text-aware ViT-CNN Fusion

In this section, we present a novel architectural design named Multi-scale
Text-aware ViT-CNN Fusion, as depicted in Fig. 1. This scheme is primarily
composed of three major components: _Dense Vision-Language Alignment module_ ,
_Multi-scale ViT-CNN Fusion module_ , and a _Supervised Loss function_ for
joint training. The Dense Vision-Language Alignment model is responsible for
aligning the vision and text features into a common embedding space. By
performing this alignment, we can effectively exploit the complementary
information from both modalities to enhance the feature representations. The
second component, Multi-scale ViT-CNN Fusion module, facilitates the fusion of
the features extracted by the ViT and the CNN. This fusion is carried out at
multiple scales, allowing the model to capture abstract features, domain-
specific features, local details, and global long-range dependencies at
different resolutions. Finally, the Multi-scale Supervision Loss Function. By
optimizing this loss, the network learns to predict segmentation masks in a
multi-scale and progressive manner. Next, we introduce them in detail.

Dense Vision-Language Alignment module. In our work, we incorporate both image
and text information as inputs for segmentation. The inclusion of text allows
us to capture the strengths of transformers more effectively and bolsters the
fusion of Vision Transformers (ViTs) and Convolutional Neural Networks (CNNs).
This approach leverages contextual cues from text to enhance segmentation
precision. To align the image and text features, we adopt a progressive
approach. Visual features are extracted from a sequence of layers
(${3,6,9,12}$) of a pretrained visual encoder, forming a set
$L={x_{1},x_{2},...,x_{\ell-1},x_{\ell}}$ where $x_{1}$ and $x_{\ell}$
represent the features from the shallowest and deepest layers, respectively.
We obtain text embeddings, denoted by $y$, from a pretrained clinical text
encoder.

We use transformer layers with skip connections to compactly represent visual
features, as shown in the equation below:

$\begin{split}X_{i}=\begin{cases}\mathsf{TransLayer}_{i}(W^{x}_{i}x_{i}),&i=\ell\\\
\mathsf{TransLayer}_{i}(W^{x}_{i}x_{i}+X_{i+1}),&i\in
L-\\{\ell\\}\end{cases}\end{split}$ (2)

Here, $W^{x}_{i}$ is a linear layer for dimension reduction, and
$\mathsf{TransLayer}_{i}$ denotes a Transformer layer. We reduce the dimension
of the text embeddings and transfer them to different layers using simple MLP
blocks:

$\begin{split}Y_{i}=\begin{cases}W^{y}_{i}y,&i=\ell\\\ W^{y}_{i}Y_{i+1},&i\in
L-\\{\ell\\}\end{cases}\end{split}$ (3)

Here, $W^{y}_{i}$ signifies a linear layer for dimension reduction, while
$W_{i}^{y}$ denotes an MLP block for transferring text embeddings. We perform
Vision-Text alignment using element-wise multiplication so that the alignment
is ”dense”:

$Z_{i}=W^{X}_{i}X_{i}\odot W^{Y}_{i}Y_{i},\quad i\in L$ (4)

In the equation above, $\odot$ represents element-wise multiplication, and
$W^{X}_{i}$ and $W^{Y}_{i}$ are tensor reshape operations for $X_{i}$ and
$Y_{i}$, respectively.

Multi-scale ViT-CNN Fusion module. ViTs and CNNs each have their unique
strengths in image analysis tasks. ViTs excel in capturing global
dependencies, while CNNs are particularly adept at extracting local features.
However, when dealing with complex tasks such as medical image segmentation, a
combination of these two can be beneficial, leveraging global contextual
understanding and local feature extraction. Addressing this, we propose a
dense fusion of ViT and CNN features at different resolutions. This approach
is designed to enhance local interactions and preserve global knowledge. Our
fusion method follows two guiding principles: 1) it should improve model
performance, and 2) it should maintain the robustness of each individual
feature to avoid over-dependence on either.

We begin with a non-parametric fusion method, where the fusion parameter
$\beta$ is uniformly sampled from $[0,1]$. A Unet CNN processes the input
$I\in\mathbb{R}^{H\times W\times 3}$, projecting it initially to $C_{1}$ and
then applying $(N-1)$ down/up-sampling operations to yield multi-scale
features $F_{j}^{CNN}\in\mathbb{R}^{H_{j}\times W_{j}\times C_{j}}$ at $N$
different resolutions ($N=4$ in our case).

ViT features $Z_{i}$ are projected to match the size of the corresponding CNN
features $F_{j}^{CNN}$, resulting in the ViT features $F^{ViT}_{j}$. These are
then fused with the CNN features as follows:

$F_{j}=\beta F^{CNN}_{j}+(1-\beta)F^{ViT}_{j},\quad j=1,2,...,N$ (5)

$\beta$ is sampled from $[r_{1},r_{2}](0\leq r_{1}<r_{2}\leq 1)$, and $F_{j}$
is the fused feature map at level $j$.

Beyond non-parametric fusion, we explore parametric fusion, employing a
channel attention mechanism [1] at each scale. This mechanism is defined as:

$\centering\begin{split}&\hat{X}=W\cdot\mathsf{Attention}(\hat{Q},\hat{K},\hat{V})\\\
&\mathsf{Attention}(\hat{Q},\hat{K},\hat{V})=\hat{V}\cdot\mathsf{Softmax}(\frac{\hat{K}\cdot\hat{Q}}{\alpha})\end{split}\@add@centering$
(6)

Here, $\hat{X}\in\mathbb{R}^{H\times W\times C}$ denotes the output feature
map; $\hat{Q}\in\mathbb{R}^{HW\times C},\hat{K}\in\mathbb{R}^{C\times
HW},\hat{V}\in\mathbb{R}^{HW\times C}$ are tensors reshaped from $Q,K,V$,
respectively; $W$ is a $1\times 1$ convolution for output projection; $\alpha$
is a learnable parameter to control the magnitude of $\hat{K}\cdot\hat{Q}$.
The definition of $Q,K,V$ is $Q,K,V=W^{Q}X,W^{K}X,W^{V}X$ and
$X=\mathsf{LayerNorm}([F^{CNN},F^{VIT}])$, where $[\cdot,\cdot]$ denotes
feature concatenation and $W^{(\cdot)}$ denotes point-wise $1\times 1$
convolutions.

Multi-scale Supervision Loss Function. In our proposed network architecture
that combines ViT and CNN in a multi-scale manner, multiple predictions are
generated in a single forward pass. However, relying exclusively on the final
output for training can lead to convergence issues. To address this, we
propose an end-to-end optimization of multiple predictions, which we term the
Multi-scale Supervision Loss function.

We denote the network’s final prediction as $P$ and its multi-scale
predictions as $S={Q_{1},Q_{2},...,Q_{s-1},Q_{s}}$, where $Q_{s}$ represents
the prediction at the $s$-th scale. For simplicity, we exclude the final
prediction $P$ from the set $S$. The CNN branch prediction is represented by
$R$. Then, we utilize multi-scale predictions $S+P$ and the CNN output $R$ to
compute the loss. The Multi-scale Supervision Loss function is formulated as
follow:

$\mathcal{L}_{ms}=\alpha_{1}\mathcal{L}(P,T)+\alpha_{2}\mathcal{L}(R,T)+\alpha_{3}\frac{1}{|S|}\sum_{s=1}^{|S|}\mathcal{L}(Q_{s},T)$
(7)

Here, $\mathcal{L}$ refers to the average of Dice loss and Cross Entropy loss.
$T$ denotes the ground truth label, and $|S|$ is the cardinality of set $S$.
The weights $\alpha_{1},\alpha_{2},$ and $\alpha_{3}$ are used to balance each
term in the loss function, and are set to $\alpha_{1}=\alpha_{2}=1$ and
$\alpha_{3}=0.6$ for all our experiments.

Figure 2: The illustration of Multi-Axis Consistency Framework.

###  3.3 Multi-Axis Consistency Framework

In our pursuit to accomplish semi-supervised learning, we present a novel
Multi-Axis Consistency framework as illustrated in Fig. 2. This all-
encompassing framework is made up of three main components: the _Multi-Axis
Consistency Soft-Hard Label Generation Module, the Multi-Axis Consistency
Voting Mechanism, and the Multi-scale Unsupervised Loss Function_. The Soft-
Hard Label Generation Module generates robust labels, taking into
consideration intra-model and inter-model consistency, as well as temporal
consistency. The Consistency Voting Mechanism selects the most probable
predictions across different models and scales, thereby enhancing the
robustness and accuracy of the learning process. The Multi-scale Unsupervised
Loss Function provides a metric for model optimization in scenarios where
ground truth labels are absent, promoting the extraction of valuable features
from unlabeled data. Next, we introduce them in detail.

Multi-Axis Consistency Soft-Hard Label Generation Module. This innovative
module is designed based on the Coordinate Systems we established to model
different consistency paradigms, as depicted in Fig. 2 (a). We represent the
input as $x$ and the consistency condition as $\theta=[m,s,t]^{T}$, indicating
that the output is generated by model $m$, scale $s$, and training iteration
$t$. The module’s objective is to minimize the distance between two outputs
under consistency regularization from multiple axes. The module achieves this
by applying an augmentation $\sigma$ to the input $x$ and generating a
modified input $\hat{x}$ while ensuring a small consistency relaxation
$\epsilon$. This process is expressed as follows:

$\begin{split}&\min\lVert f(x,\theta)-f(\hat{x},\theta+\epsilon)\rVert\\\
&s.t.\hat{x}=\sigma(x),\lVert\epsilon\rVert\rightarrow 0\\\ \end{split}$ (8)

Then, the module generates robust labels by predicting multiple segmentation
maps $P_{\theta}\in\mathbb{R}^{H\times W\times K},\theta\in\Theta$, where $K$
denotes the number of segmentation classes. These are the soft labels. The
module then converts these soft labels into binary hard labels using a
threshold. Next, we introduce the whole process in our ,multi-axis consistency
voting mechanism.

Multi-Axis Consistency Voting Mechanism. The Voting Mechanism is implemented
based on the Semi-Supervised Learning strategy, illustrated in Fig. 2 (b).
This mechanism samples a subset of the predicted segmentation maps within the
consistency relaxation and utilizes them to generate a probabilistic pseudo-
label. It leverages the outputs from the Vision Transformer (ViT),
Convolutional Neural Network (CNN), and multi-scale outputs to collaboratively
vote for the most probable pseudo-label. The pseudo-label includes the
probability of each pixel belonging to a specific class.

To achieve this, the mechanism first predicts multiple segmentation maps,
which can be considered as ”soft labels” indicating the probability of each
class. These soft labels are then converted into binary ”hard labels” using a
threshold of 0.5, as shown in the first part of the equation.

$\begin{split}M_{\theta}(h,w,k)&=\left\\{\begin{array}[]{rcl}1&\mbox{for}&P_{\theta}(h,w,k)\geq
0.5\\\ 0&\mbox{for}&P_{\theta}(h,w,k)<0.5\end{array}\right.\end{split}$ (9)

Here, $M_{\theta}(h,w,k)$ denotes the binary hard label of pixel $(h,w)$ for
class $k$ under condition $\theta$ and $P_{\theta}(h,w,k)$ represents the soft
label corresponding to the same. The final pseudo-label $M_{pseu}$ is then
generated by taking the average of these binary hard labels across all
conditions $\theta\in\Theta$, as expressed in the second part of the equation.

$\begin{split}M_{pseu}&=\frac{1}{|\Theta|}\sum_{\theta\in\Theta}{M_{\theta}}\end{split}$
(10)

In this equation, $M_{pseu}$ represents the final probabilistic pseudo-label
and $|\Theta|$ denotes the cardinality of set $\Theta$. In this way, the
Multi-Axis Consistency Voting Mechanism generates robust probabilistic pseudo-
labels that reflect the consensus among different models (ViT, CNN) and multi-
scale outputs, embodying the concept of multi-axis consistency.

Multi-scale Unsupervised Loss Function. The unsupervised loss function is
incorporated in our semi-supervised training via an unsupervised loss term, as
is shown in Fig. 2 (c). For unlabeled images, the function first generates
pseudo-labels according to multiple network outputs. Multiple outputs from the
current training iteration as well as previous iterations all contribute to
the generation of the pseudo-label. The function aims to minimize the distance
between contributors from the current training iteration to the pseudo-label:

$\mathcal{L}_{unsup}=\frac{1}{|\Theta|}\sum_{\theta\in\Theta}\mathcal{L}(P_{\theta},M_{pseu})$
(11)

Here, $\mathcal{L}$ represents the average of Dice loss and Cross-Entropy
loss. The final loss for semi-supervised learning $\mathcal{L}_{final}$ is
represented by the weighted sum of $L_{sup}$ and $L_{unsup}$, as shown below:

$\mathcal{L}_{final}=\mathcal{L}_{sup}+\lambda\mathcal{L}_{unsup}$ (12)

Here, $\lambda$ is a weight factor, defined by a time-dependent Gaussian
warming-up function to control the balance between the supervised loss and
unsupervised loss.

##  4 Experiments

###  4.1 Experiment Setup

We use pretrained vision and language transformers, which remain frozen during
training. We use a ViT pretrained on the ROCO dataset [28] via DINO [6] as our
vision backbone, and Clinical BERT [2] as our language backbone. We adopt
U-Net [31] as the CNN branch. We set the batch size to 4 and the initial
learning rate to $10^{-3}$, using the Adam optimizer [17] with cosine
annealing cyclic schedule. Data augmentation includes random flips and
$90^{\circ}$ rotations. All experiments are conducted on an A5000 GPU. We use
Dice and mIoU metrics as our evaluation metrics. The experiments are conducted
on MoNuSeg [18] and QaTa-COV19 [9] datasets. The MoNuSeg dataset includes
images of tissue from various patients with tumors and approximately 22,000
nuclear boundary annotations across 30 training images and 14 test images. And
the QaTa-COV19 dataset includes 9258 annotated COVID-19 chest radiographs. The
text annotations for both datasets are derived from [22].

Method | MoNuSeg | QaTa-COV19
---|---|---
Dice (%) | mIoU (%) | Dice (%) | mIoU (%)
Unet | 76.45 | 62.86 | 79.02 | 69.46
Unet++ | 77.01 | 63.04 | 79.62 | 70.25
AttUnet | 76.67 | 63.74 | 79.31 | 70.04
nnUnet | 80.06 | 66.87 | 80.42 | 70.81
MedT | 77.46 | 63.37 | 77.47 | 67.51
TransUnet | 78.53 | 65.05 | 78.63 | 69.13
GTUnet | 79.26 | 65.94 | 79.17 | 69.65
Swin-Unet | 77.69 | 63.77 | 78.07 | 68.34
UCTransNet | 79.87 | 66.68 | 79.15 | 69.60
Ours+PF | 79.91 | 66.74 | 82.29 | 72.87
Ours+NPF | 80.60 | 67.66 | 82.03 | 72.80

tableResults under Fully-supervised Learning and Comparison with the state-of-
the-arts. “PF” and “NPF” represent Parametric fusion and Non-Parametric
Fusion, respectively. Different values of $\beta$ during inference are also
included.

Setting | Labels (%) | MoNuSeg
---|---|---
Dice (%) | mIoU (%)
PF | 25 | 78.59 | 64.99
50 | 78.85 | 65.36
100 | 79.91 | 66.74
NPF | 25 | 78.47 | 64.88
50 | 79.26 | 65.94
100 | 80.16 | 67.06

tableResults under Semi-Supervised Setting. “PF” and “NPF” represent
Parametric fusion and Non-Parametric Fusion, respectively.

###  4.2 Quantitative Results

In this section, we conduct main experiments on MoNuSeg dataset on both fully-
supervised and semi-supervised settings. We also include the fully-supervised
results on QaTa-COV19 dataset.

Results on Fully-Supervised Setting. In our research, we compare our
methodology to state-of-the-art methods in a fully-supervised setting, these
methods include Unet[31], Unet++[44], AttUnet[27], nnUnet[16], MedT[32],
transUnet[7], GTUnet[21], Swin-Unet[5], and UCTransNet[34]. The results of our
approach and the other state-of-the-art methods in a fully-supervised learning
setting are presented in Table 4.1. Notably, our method demonstrates a
significant improvement over existing approaches. With the employment of
parametric ViT-CNN fusion, our method achieves results that are not only
comparable with TransUnet [7] on the MoNuSeg dataset but also surpasses it
under specific conditions. More notably, our approach exhibits superior
performance under non-parametric feature fusion, namely, random fusion with a
uniformly sampled $\beta$ during training. In this case, our method sets a new
benchmark on the MoNuSeg dataset, outperforming all other state-of-the-art
methods. This remarkable performance demonstrates the robustness of the random
fusion strategy, where both the ViT and CNN branches can learn strong
representations.

Results on Semi-Supervised Setting. In Table 4.1, we present our results under
a Semi-Supervised setting. These results are achieved by using 25% and 50%
labels for model evaluation, under our proposed Multi-Axis Consistency
framework. The performance of our method stands out in several ways: 1) Most
notably, our method delivers a comparable result to TransUnet in a fully-
supervised setting even with only 25% labels. Moreover, when we possess 50%
labels, the result improves significantly. This clearly showcases the
potential of our proposed method. It illustrates how our method can
effectively reduce the reliance on labeled data by learning from limited data
and large-scale unlabeled data, thereby alleviating the cost of labels. 2) Our
method with ViT-CNN random fusion and parametric channel attention
consistently produces strong results across all semi-supervised settings.
While the version with ViT-CNN random fusion outperforms the version with
parametric channel attention by a small margin (less than 0.5%) when using 50%
and 100% labels, the results are fairly comparable. This highlights the
advantages of our multi-scale ViT-CNN fusion, underscoring its ability to
capture both global and local interactions and retain pre-trained knowledge.
All these findings reinforce the effectiveness and efficiency of our method in
Semi-Supervised settings, demonstrating that it is a promising approach for
future research and applications.

###  4.3 Qualitative Comparisons

Fig. 3 provides a visual comparison between our method and our baseline method
[24]. Both sets of results are obtained under a fully-supervised training
setting. To facilitate a clearer distinction between the two, we have
highlighted specific areas in each sample within a red box. Upon close
examination, it’s evident that our proposed method offers superior results
with respect to the precision of boundary delineation and the accuracy of
shape representation. These improvements are most apparent within the
highlighted regions, where our method’s predictions exhibit finer detail and
higher fidelity to the original structures. A key factor contributing to this
enhanced performance is the introduction of our multi-scale text-aware ViT-CNN
fusion. This innovative approach significantly improves local feature
extraction within the target medical domain, allowing for more accurate and
detailed segmentation results. This clearly demonstrates the advantage of our
method over traditional approaches, and its potential for providing superior
outcomes in complex medical image analysis tasks.

Figure 3: Qualitative comparison. Left: Visualization results on MoNuSeg
dataset. Right: Visualization results on QaTa-COV19 dataset.

###  4.4 Ablation Study

Method | ViT-CNN Dense Fusion | MoNuSeg
---|---|---
Multi-scale Arch. | Text | ViT-CNN Fusion | Loss | Dice (%) | mIoU (%)
Baseline |  |  |  |  | 68.03 | 52.05
| ✓ |  |  |  | 76.30 | 61.92
Ours | ✓ | ✓ |  |  | 78.47 | 64.67
| ✓ | ✓ | ✓ |  | 79.14 | 65.78
| ✓ | ✓ | ✓ | ✓ | 80.16 | 67.06

(a) Ablation study on proposed modules under fully-supervised learning.

Setting | Multi-Axis Consistency | MoNuSeg
---|---|---
Intra-Model | Inter-Model | Temporal | Dice (%) | mIoU (%)
Sup. Only |  |  |  | 77.94 | 64.12
Ours | ✓ |  |  | 78.68 | 65.11
| ✓ |  | 78.52 | 64.92
|  | ✓ | 78.01 | 64.15
✓ | ✓ |  | 78.88 | 65.37
✓ |  | ✓ | 78.84 | 65.31
|  | ✓ | ✓ | 78.57 | 64.97
| ✓ | ✓ | ✓ | 79.26 | 65.94

(b) Ablation study on Multi-Axis Consistency framework under semi-supervised
learning with 50% training labels.

Table 1: Ablation studies on proposed modules.

Ablation Studies on Mutli-scale Text-aware ViT-CNN Fusion. To evaluate our
ViT-CNN Fusion, we employed the state-of-the-art vision-language transformer
dense finetuning method as our baseline, which didn’t perform well in medical
image segmentation due to over-reliance on the pretrained backbone and lack of
multi-scale dense features. To counter these issues, we propose a multi-scale
text-aware ViT-CNN fusion for optimized pretrained transformers. An ablation
study was conducted to ascertain the contribution of each component. This
analysis involved sequentially introducing multi-scale architecture, dense
vision-text alignment, ViT-CNN fusion, and a joint training loss function.
Table 1a shows the results, with significant Dice gains for each module: 8.27%
for the multi-scale architecture, 2.17% for the vision-text alignment, 0.67%
for the ViT-CNN fusion, and 1.02% for the joint training loss. The data
underscores the effectiveness of each part of our method, particularly the
substantial role of ViT-CNN Fusion in improving medical image segmentation
tasks.

Ablation Studies on Multi-Axis Consistency. Our research introduces Multi-Axis
Consistency, an innovative framework for generating robust pseudo labels in
semi-supervised learning, by integrating different consistency regularization
types. Table 1b displays our results: each consistency regularization type
improves semi-supervised performance compared to a supervised-only setting,
highlighting their importance in semi-supervised learning. Notably, peak
performance is achieved when all three types are combined, demonstrating the
effectiveness of the Multi-Axis Consistency framework. This comprehensive
approach leads to superior performance in semi-supervised learning, marking a
significant advancement in generating pseudo labels and improving model
performance.

In-depth Discussion on Multi-scale ViT-CNN Fusion. In this section, we address
two key questions experimentally: 1Why does ViT-CNN fusion work in semi-
supervised settings? Our results (Table 1a and Fig. 6) demonstrate this
module’s effectiveness in fully-supervised learning. Fig. 6 further shows that
ViT-CNN fusion is crucial in semi-supervised settings, with performance
increasing as $\beta$ decreases from 0.8 to 0.2. This suggests that both
Transformer and CNN branches can independently perform well in such settings.
2) Why is multi-scale fusion important? We conducted ablation studies on
fusion levels using different approaches: Non-Parametric Random Fusion and
Parametric Channel attention. Fig. 6 shows that increased feature fusion
levels improve model performance, underscoring the importance of multi-scale
dense features in medical image segmentation and the effectiveness of our
proposed multi-scale ViT-CNN fusion method.

Figure 4: Impact of different $\beta$ (full).

Figure 5: Impact of different $\beta$(semi).

Figure 6: Impact of fusion level numbers.

##  5 Conclusion

In this paper, we propose a novel semi-supervised learning framework for
medical image segmentation. In our work, a Text-aware ViT-CNN Fusion scheme is
proposed to take advantages of both pretrained ViTs and CNNs as well as
extracting both abstract features and medical domain specific features.
Besides, a novel Multi-Axis Consistency framework is proposed to vote for
pseudo label to encourage semi-supervised training. Experiments on serveral
widely used datasets have demonstrated the effectiveness of our method.

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# Bulk-edge correspondence of Stiefel-Whitney and Euler insulators through the
entanglement spectrum and cutting procedure

Ryo Takahashi  Advanced Institute for Materials Research, Tohoku University,
Sendai 980-8577, Japan
Tomoki Ozawa  Advanced Institute for Materials Research, Tohoku University,
Sendai 980-8577, Japan

###### Abstract

We propose an unconventional bulk-edge correspondence for two-dimensional
Stiefel-Whitney insulators and Euler insulators, which are topological
insulators protected by the $PT$ symmetry. We find that, although the energy
spectrum under the open boundary condition is generally gapped, the
entanglement spectrum is gapless when the Stiefel-Whitney or Euler class is
nonzero. The robustness of the gapless spectrum for Stiefel-Whitney insulator
can be understood through an emergent anti-unitary particle-hole symmetry. For
the Euler insulators, we propose a conjecture, which is supported by our
numerical calculation, that the Euler class is equal to the number of crossing
in the entanglement spectrum, taking into account the degree of the crossings.
We also discuss that these crossings of the entanglement spectrum are related
to the gap closing points in the cutting procedure, which is the energy
spectrum as the magnitude of the boundary hopping is varied.

##  I Introduction

Chern insulators are two dimensional energy bands whose Bloch states exhibit
non-trivial shape in momentum space, characterized by the topological
invariant called the Chern number. Mathematically, Chern number is defined as
a consequence of Bloch states being generally complex vectors in a certain
Hilbert space. For Chern insulators, an important relation known as the bulk-
edge correspondence holds, which states that the Chern number is equal to the
number of chiral edge modes around the system, which is at the heart of
various phenomena such as the integer quantum Hall effect Thouless  _et al._
(1982).

Recently, a new class of topological phases whose Bloch states are real
vectors has been proposed Fang  _et al._ (2015); Zhao and Lu (2017); Ahn  _et
al._ (2018, 2019); Ünal  _et al._ (2020); Bouhon  _et al._ (2020a); Ezawa
(2021); Zhao  _et al._ (2022). Physically such real Bloch states are relevant
for $PT$-symmetric systems, where $P$ is the space inversion and $T$ is the
time reversal. In the presence of $PT$ symmetry, one can show that the
momentum-space Hamiltonian is real, ${H}(\mathbf{k})^{*}={H}(\mathbf{k})$,
implying that the Bloch wavefunctions can be taken real. For such real Bloch
bands, topology is not characterized by the Chern number, but by other
topological invariants such as the Stiefel-Whitney (SW) classes and Euler
classes. SW classes are $\mathbb{Z}_{2}$ invariants taking either 0 or 1. The
first SW class $w_{1}$, multiplied by $\pi$, corresponds to the quantized
Berry phase in one dimension. In two spatial dimensions, insulators with bands
having nonzero second SW class $w_{2}$ is called the Stiefel-Whitney
insulator. Furthermore, if the number of occupied bands is two and $w_{1}=0$
in both directions, the bands can be characterized by the Euler class $\chi$,
which can take any integer value Bzdušek and Sigrist (2017); the Euler
insulators refer to situations with $\chi\neq 0$ Shiozaki  _et al._ (2017);
Bzdušek and Sigrist (2017); Ahn  _et al._ (2018, 2019); Ahn and Yang (2019).

Various consequences of having nonzero SW and/or Euler classes have been found
in bulk physics. In three dimensions, non-Abelian braiding of nodal lines was
found to be related to the SW classesBouhon  _et al._ (2020b); Jiang  _et al._
(2021); Peng  _et al._ (2022) In two dimensions, nonzero Euler class leads to
Wannier obstructions Zou  _et al._ (2018); Ahn  _et al._ (2019) and nonzero SW
class implies the obstructed atomic insulator Ahn and Yang (2019), and they
are relevant for physics of flat bands in the twisted bilayer graphene.
Experimental signature to detect the Euler insulator from quench dynamics has
also been proposed Ünal  _et al._ (2020). Experiments in acoustic systemsJiang
_et al._ (2021) and trapped ions Zhao  _et al._ (2022) reported detection of
nontrivial Euler number through bulk properties.

While the bulk topological properties of $PT$-symmetric insulators have been
investigated from various perspectives, little work on the bulk-edge
correspondence exists, apart from studies of higher-order topology in such
systems Ahn and Yang (2019); Lenggenhager  _et al._ (2022). One simple reason
for the absence of works on bulk-edge correspondence is that the edge spectrum
under open boundary condition is generally gapped even when $w_{2}\neq 0$ or
$\chi\neq 0$, because the $PT$ symmetry is broken on the edges, and thus the
bulk-edge correspondence in the usual sense does not hold in these systems.

In this paper, we establish an unconventional bulk-edge correspondence for
two-dimensional SW and Euler insulators. We find that, even though the edge
spectrum is generally gapped, the entanglement spectra of these systems have
gapless edge-localized eigenstates which are protected by $PT$ symmetry. We
also show that these edge-localized entanglement eigenstates are related to
spectral flow as one varies the magnitude of the hopping at the boundary,
known as the cutting procedure Teo  _et al._ (2008); Song  _et al._ (2020);
Peri  _et al._ (2020); Takahashi  _et al._ (2020); Tanaka  _et al._ (2020).

##  II Entanglement spectrum

The entanglement spectrum is defined as the eigenvalue spectrum of the reduced
density matrix, and was proposed as the tool for identifying topological
orderLi and Haldane (2008); Pollmann  _et al._ (2010). In non-interacting
systems, the entanglement spectrum can be determined from the spectrum of the
correlation functionsPeschel (2003), called single-particle entanglement
spectrum, which corresponds to the edge spectrum of spectrally flattened
Hamiltonian Fidkowski (2010); Turner  _et al._ (2010); Hughes  _et al._
(2011); Alexandradinata  _et al._ (2011); Turner  _et al._ (2012). The
spectrally flatted Hamiltonian is defined in momentum space by

$\displaystyle H_{\mathrm{flat}}(\mathbf{k})$
$\displaystyle\equiv\frac{\mathbb{I}}{2}-P_{\mathrm{occ}}(\mathbf{k}),$
$\displaystyle P_{\mathrm{occ}}(\mathbf{k})$
$\displaystyle\equiv\sum_{i:\mathrm{occ}}|u_{i}(\mathbf{k})\rangle\langle
u_{i}(\mathbf{k})|,$ (1)

where $\mathbf{k}=(k_{x},k_{y})$ is the momentum,
$P_{\mathrm{occ}}(\mathbf{k})$ is the projector to the occupied bands for a
given $\mathbf{k}$, $|u_{i}(\mathbf{k})\rangle$ is the cell-periodic part of
the $i$-th Bloch wavefunction, the sum is over the occupied bands, and
$\mathbb{I}$ is the identity matrix. From the $PT$ symmetry,
$|u_{i}(\mathbf{k})\rangle$ can be taken as a real vector, and
$P_{\mathrm{occ}}(\mathbf{k})$ is an $N$-by-$N$ real symmetric matrix, where
$N$ is the total number of bands.

The real-space representation of the spectrally flattened Hamiltonian is
obtained by Fourier transformation of $H_{\mathrm{flat}}(\mathbf{k})$. We
consider the case where $y$ direction is periodic so that the momentum $k_{y}$
is taken as a good quantum number, whereas $x$ direction can have an open
boundary condition. For a given value of $k_{y}$, the real space Hamiltonian
in $x$ direction with a periodic boundary condition is obtained by

$\displaystyle[H_{\mathrm{flat}}^{x\text{-PBC}}(k_{y})]_{x\alpha,x^{\prime}\beta}\equiv\frac{1}{L_{x}}\sum_{k_{x}}e^{ik_{x}(x-x^{\prime})}\left[H_{\mathrm{flat}}(\mathbf{k})\right]_{\alpha,\beta},$
(2)

where $L_{x}$ is the number of sites in $x$ direction, namely $1\leq x\leq
L_{x}$ and the same for $x^{\prime}$, and we take $L_{x}$ to be an even
number. The sum over $k_{x}$ is for
$k_{x}=0,2\pi\frac{1}{L_{x}},2\pi\frac{2}{L_{x}},\cdots,2\pi\frac{L_{x}-1}{L_{x}}$.
In order to impose an open boundary condition, we divide the lattice in two
parts; the first part, which we call $A$, is for $1\leq x\leq L_{x}/2$ and the
second part, which we call $B$ is for $1+L_{x}/2\leq x\leq L_{x}$ 111Although
we assume A and B to have the same length in order to derive analytical
results, results remain essentially the same even when A and B do not have the
same length, as long as both A and B are large enough.. We then write
$H_{\mathrm{flat}}^{x\text{-PBC}}(k_{y})$ in the following block-form:

$\displaystyle
H_{\mathrm{flat}}^{x\text{-PBC}}(k_{y})=\begin{pmatrix}H_{\mathrm{flat}}^{AA}(k_{y})&H_{\mathrm{flat}}^{AB}(k_{y})\\\
H_{\mathrm{flat}}^{BA}(k_{y})&H_{\mathrm{flat}}^{BB}(k_{y})\end{pmatrix},$ (3)

where, for example, $H_{\mathrm{flat}}^{AB}(k_{y})$ is a matrix obtained by
restricting
$[H_{\mathrm{flat}}^{x\text{-PBC}}(k_{y})]_{x\alpha,x^{\prime}\beta}$ to
$1\leq x\leq L_{x}/2$ and $1+L_{x}/2\leq x^{\prime}\leq L_{x}$. The flattened
Hamiltonian for an open boundary condition in $x$ direction is obtained by
only looking at $AA$-part of $H_{\mathrm{flat}}^{x\text{-PBC}}(k_{y})$, namely

$\displaystyle H^{x\text{-OBC}}_{\mathrm{flat}}(k_{y})\equiv
H_{\mathrm{flat}}^{AA}(k_{y}).$ (4)

Since the spectrum of $H^{x\text{-OBC}}_{\mathrm{flat}}(k_{y})$ is directly
related to the entanglement spectrum of the reduced density matrix of the $A$
part, we refer to the energy spectrum of
$H^{x\text{-OBC}}_{\mathrm{flat}}(k_{y})$ as the entanglement spectrum.
Interestingly, it was previously known in the context of inversion symmetric
topological insulators that the entanglement spectrum can be gapless even if
the edge spectrum of the original (not-flattened) Hamiltonian is gappedTurner
_et al._ (2010); Hughes  _et al._ (2011); Turner  _et al._ (2012).

The entanglement spectrum, which is bounded between $-1/2$ and $1/2$, is
symmetric around $0$, and this is due to the emergent antisymmetry, as we
shall explain now. For a fixed $k_{y}$, the flattened Hamiltonian with open
boundary condition $H^{x\text{-OBC}}_{\mathrm{flat}}(k_{y})$ has
$\frac{L_{x}}{2}N$ eigenvalues and eigenstates. Some eigenvalues are exactly
at $\pm 1/2$, but some others distribute between $-1/2$ and 1/2. We collect
all eigenstates with eigenvalues not equal to $\pm 1/2$ as $|\psi_{i}\rangle$:
$H^{x\text{-OBC}}_{\mathrm{flat}}(k_{y})|\psi_{i}\rangle=\epsilon_{i}|\psi_{i}\rangle$,
$\epsilon_{i}\neq\pm 1/2$.

Then, as we show in the Appendix A, the operator $\Xi$ defined by

$\displaystyle\Xi$
$\displaystyle\equiv\sum_{i}|\phi_{i}\rangle\langle\psi_{i}|,$ (5)
$\displaystyle|\phi_{i}\rangle$
$\displaystyle\equiv-\frac{H_{\mathrm{flat}}^{BA}(k_{y})}{\sqrt{1/4-\epsilon_{i}^{2}}}|\psi_{i}\rangle,$
(6)

serves as an antisymmetry for states with $\epsilon_{i}\neq\pm 1/2$. Namely,
$H_{\mathrm{flat}}(k_{y})\Xi|\psi_{i}\rangle=-\epsilon_{i}\Xi|\psi_{i}\rangle$.
We note that $|\phi_{i}\rangle=\Xi|\psi_{i}\rangle$ is normalized as shown in
the Appendix A. Generally, the operator $\Xi$ changes the side of edge states.
If the state $|\psi_{i}\rangle$ is localized on the right edge,
$\Xi|\psi_{i}\rangle$ is localized on the left edge. Existence of such an
antisymmetry $\Xi$ is a general property of flattened Hamiltonians.

In the presence of the $PT$ symmetry, since $PT$ operation switches the side
of edge states but does not change the energy, one can further construct a
state $(PT)\Xi|\psi_{i}\rangle$, which has the energy $-\epsilon_{i}$ and
localized at the same edge as $|\psi_{i}\rangle$. As shown in the Appendix A,
$(PT)\Xi$ is an antiunitary antisymmetry which squares to one in the subspace
spanned by $|\psi_{i}\rangle$. Namely, we can regard $((PT)\Xi)^{2}=+1$ as
long as we only consider states with $\epsilon_{i}\neq\pm 1/2$. This particle-
hole like symmetry will play a key role in the robustness of linear crossing
of entanglement spectrum as we see below.

##  III Stiefel-Whitney insulator

Now we study entanglement spectrum of Stiefel-Whitney insulators and show that
there is nontrivial linear crossing protected by the $PT$ symmetry. As a
concrete example, we consider the following modelAhn  _et al._ (2018):

$\displaystyle H_{\text{SW}}(\bm{k})$ $\displaystyle=\sin k_{x}\sigma_{x}+\sin
k_{y}\tau_{y}\sigma_{y}+(1-\cos k_{x}-\cos k_{y})\sigma_{z}$
$\displaystyle\quad+\frac{\tau_{z}\sigma_{z}}{2}+\frac{\tau_{x}}{4}.$ (7)

Here, $\tau_{i}$ and $\sigma_{i}$ ($i=x,y,z$) are Pauli matrices. This is a
four-band model ($N=4$), and we assume that the lower two bands are occupied.
It is known that the model has $w_{2}=1$, which can be confirmed by
calculating the Wilson loop spectrum as demonstrated in Appendix B; we note in
this case the Euler class can also be defined and $\chi=1$.

Figure 1: Numerical calculation for Stiefel-Whitney insulator model and its
perturbation. (a) Edge spectrum and (b) entanglement spectrum for
$H_{\text{SW}}$. (c,d) Entanglement spectrum with (c) $PT$-preserving mass
term [$-M_{1}\tau_{y}\sigma_{y}$] and (d) $PT$-breaking mass term
[$M_{2}(-\tau_{y}\sigma_{y}+\tau_{x}\sigma_{x}+\tau_{y}-\sigma_{y})$]. Here,
$(M_{1},M_{2})=(0.8,0.4)$. (e,f) Eigenvalue distribution at $k_{y}=0$ in (b)
and (c). Colors in (a,e,f) represent the squared amplitudes of the states in
the unit cells on either end. While the edge spectrum of the original (not
flattened) Hamiltonian [(a)] is gapped, the entanglement spectrum [(b,c)]
exhibits a gapless spectrum, and it becomes gapped under $PT$-breaking term
[(d)].

In Fig. 1(a), we show the edge spectrum of the origianl (not flattened)
Hamiltonian as a function of $k_{y}$. In this case, the spectrum is clearly
gapped; gapless edge spectrum characteristic of conventional topological
insulators is absent. On the contrary, the entanglement spectrum, plotted in
Fig. 1(b), is gapless and show linear crossing at $\epsilon=0$. Here, the
entanglement spectrum is symmetric around $\epsilon=0$ because of the
antisymmetry $\Xi$ defined in Eq. (5). Even if we add more terms to the
Hamiltonian preserving the $PT$ symmetry, the linear crossing of the
entanglement spectrum does not gap out, as exemplified in Fig. 1(c)]. If we
add $PT$-breaking terms to the Hamiltonian, $w_{2}$ and $\chi$ are no longer
well-defined, and the entanglement spectrum can become gapped, as shown in
Fig. 1(d).

This linear crossing in the entanglement spectrum is the bulk-edge
correspondence unique to the SW insulators. In fact, such a linear crossing of
the entanglement spectrum was already experimentally observed in trapped ions
Zhao  _et al._ (2022). We now show that this linear crossing is indeed robust
against $PT$-symmetric perturbations. The linear crossing of the entanglement
spectrum is the consequence of the anti-unitary particle-hole symmetry
$(PT)\Xi$. To see how this symmetry protects the linear crossing, we consider
an effective model around the crossing describing edge states on one side.
(Note that, while acting with $\Xi$ changes the side of the edge states,
$(PT)\Xi$ keeps the side.) First, we rearrange the basis appropriately so that
$(PT)\Xi=K$, where $K$ is the complex conjugation operation. (We explain in
Appendix C how one can find such a basis.) In this basis, since $(PT)\Xi=K$
must anti-commute with the effective Hamiltonian, the effective Hamiltonian
must take the following form

$\displaystyle H_{\mathrm{eff}}(k_{y})=ck_{y}\sigma_{y},$ (8)

with some nonzero constant $c$. Here, without loss of generality, we took the
crossing point to be at $k_{y}=0$. Any perturbation one can add to the system
keeping the $PT$ symmetry should also anti-commute with $(PT)\Xi=K$ and thus
the only form of the allowed mass term is $m\sigma_{y}$, and thus the
resulting effective Hamiltonian $H_{\mathrm{eff}}(k_{y})=(ck_{y}+m)\sigma_{y}$
is still gapless but the position of the gapless point is shifted to
$k_{y}=-m/c$.

We can generalize the above discussion of $2\times 2$ effective model to
include more states using the Pfaffian. Taking $(PT)\Xi=K$ as before, from the
same argument, the effective Hamiltonian $H_{\mathrm{eff}}(k_{y})$ describing
modes localized on one edge should be a skew-symmetric matrix. Then, we can
construct a real skew-symmetric matrix $T_{k_{y}}$ by

$\displaystyle T_{k_{y}}\equiv iH_{\text{eff}}(k_{y}).$ (9)

Since $T_{k_{y}}$ is a real skew-symmetric matrix, for each momentum, the
Pfaffian $\mathrm{Pf}(T_{k_{y}})$ can be defined. Taking the linear crossing
of the entanglement to be at $k_{y}=0$, we take two momenta $k_{0}<0$ and
$k_{1}>0$, and consider their respective Pfaffians $\mathrm{Pf}(T_{k_{0}})$
and $\mathrm{Pf}(T_{k_{1}})$. Even though the overall sign of a Pfaffian is
ambiguous, the sign change of the Pfaffian between $k_{0}$ and $k_{1}$ is a
valid invariant of the linear crossing, assuming that the effective
Hamiltonian is defined continuously in the region $k_{0}\leq k_{y}\leq k_{1}$
as discussed in [Carey  _et al._ , 2019]. To see that the sign change of the
Pfaffian is related to the protection of the linear crossing, first we note
that the effective Hamiltonian can be written in block-diagonal form, and then
the Pfaffian can be simply calculated as product of $\epsilon_{i}$’s:

$\displaystyle\text{Pf}(\text{diag}[i\epsilon_{1}\sigma_{y},\cdots,i\epsilon_{N_{0}}\sigma_{y}])=\epsilon_{1}\times\cdots\times\epsilon_{N_{0}}.$
(10)

Thus the sign change of the Pfaffians is the same as the sign change of
$\prod_{i}\epsilon_{i}(k_{y})$ between $k_{0}$ and $k_{1}$. Bands that do not
close the gap between $k_{0}$ and $k_{1}$ do not contribute to the sign
change, and thus the change of the sign of the Pfaffian implies that two
eigenvalues with opposite values cross somewhere in $k_{0}<k_{y}<k_{1}$.
Moreover, even if a perturbation that hybridizes different bands is added, the
sign change is preserved as long as the spectral gap at $k_{0}$ and $k_{1}$
remains open. Thus, the linear degeneracy cannot open a gap with a
perturbation which keeps the $PT$ symmetry.

Finally, we comment on the model independence of our results. The discussion
given in Ref. Bouhon  _et al._ (2020a) shows that the homotopic classification
of the $PT$-symmetric gapped Hamiltonians coincide with the classification
based on the SW and Euler classes. This implies that if two Hamiltonians have
the same SW class, they can be continuously transformed to each other, up to
addition of trivial bands, without closing the gap. Therefore, since we have
shown that a crossing in the entanglement spectrum remains robust under
continuous deformations, we can conclude that a linear crossing of
entanglement spectrum exists for any two-dimensional $PT$-symmetric models
with nontrivial SW class.

##  IV Euler insulator

Next, we consider the Euler insulator with $\chi=2$. We find that there is
either one quadratic crossing or two linear crossings in the entanglement
spectrum. We first consider the following three band model Ünal  _et al._
(2020); Ezawa (2021); Zhao  _et al._ (2022):

$\displaystyle
H_{\text{Euler}}=\begin{pmatrix}d_{1}^{2}&d_{1}d_{2}&d_{1}d_{3}\\\
d_{1}d_{2}&d_{2}^{2}&d_{2}d_{3}\\\
d_{1}d_{3}&d_{2}d_{3}&d_{3}^{2}\end{pmatrix}+c_{1}\begin{pmatrix}d_{2}^{2}&-d_{1}d_{2}&0\\\
-d_{1}d_{2}&d_{1}^{2}&0\\\ 0&0&0\end{pmatrix},$ (11)
$\displaystyle(d_{1},d_{2},d_{3})=(\sin k_{x},\sin k_{y},m-\cos k_{x}-\cos
k_{y}),$ (12)

where $m$, $c_{1}$ are real parameters, which we take to be
$(m,c_{1})=(1.2,0.1)$.

Figure 2: Entanglement spectrum for (a) $H_{\text{Euler}}$, (b)
$H_{\text{Euler}}-\frac{\sin k_{x}}{2}\bigl{(}\begin{smallmatrix}0&1&0\\\
1&0&0\\\ 0&0&0\end{smallmatrix}\bigr{)}$, and that for $H_{\text{Euler+NI}}$
with (c) $V_{0}=1.5$ and (d) $V_{0}=-0.5$. These systems have (a-c) two and
(d) three occupied bands, and have topological numbers (a-c) $\chi=2$ and (d)
$w_{2}=0$ (mod $2$), respectively.

Considering the lower two bands to be occupied, the Euler class can be
defined, and it is $\chi=2$. (We note that SW class is given by $w_{2}=\chi\
\text{mod }2=0$.) We plot the entanglement spectrum in Fig. 2(a); the
entanglement spectrum shows a quadratic degeneracy. By adding $PT$-symmetric
perturbations to the Hamiltonian, the position of the quadratic degeneracy can
shift, but we have numerically confirmed that a gap never opens unless the
bulk band gap is closed (Fig. 2(b)).

We can further understand the nature of this crossing of the entanglement
spectrum by coupling an additional trivial band to the model:

$\displaystyle
H_{\text{Euler+NI}}(\bm{k})=\begin{pmatrix}H_{\text{Euler}}&\Phi^{\dagger}\\\
\Phi&V_{0}\end{pmatrix},$ (13)
$\displaystyle\Phi=\Delta\begin{pmatrix}\sin(k_{x}+\frac{\pi}{4}),&0,&\frac{1}{2}\sin(k_{x}-\frac{\pi}{4})\end{pmatrix},$
(14)

where $\Phi$ is the coupling term and we set $\Delta=1.0$. The original
$H_{\text{Euler}}$ has a bandgap around $E=0.2$ to 0.65. By turning on
$\Delta$, one can add a band either in the occupied or unoccupied band by
adjusting $V_{0}$ to be below or above the bandgap. First, we consider adding
a trivial band to the occupied bands by setting $V_{0}=-0.5$. We consider the
lower three bands, instead of the two bands, to be occupied. Since defining
the Euler class requires the number of occupied bands to be two, the Euler
class in this new model is no longer defined. The SW class can be still
defined, and since the trivial band considered here can be added adiabatically
without closing the gap, SW class remains to be $w_{2}=0$. As a band is added,
the entanglement spectrum gaps out, as shown in Fig. 2(d), which suggests that
the quadratic band crossing of the Euler insulator is protected by the Euler
class $\chi=2$.

Next, we consider adding a trivial band to unoccupied bands by setting
$V_{0}=1.5$. Considering two lower bands to be occupied, the Euler class
remains to be $\chi=2$. In Fig. 2(c), we show the entanglement spectrum. We
see that the quadratic crossing is now split into two linear crossings. We
have numerically checked that, adding any perturbations up to next-nearest
neighbor hoppings, the entanglement spectrum is always gapless as long as the
bulk energy gap is kept open, and either one quadratic crossing or two linear
crossings exist.

From our numerical calculation, we conjecture that the Euler class equals to
the total number of band crossings in the entanglement spectrum, taking into
account the order of crossings:

$\displaystyle|\chi|=\sum_{p\in\mathbb{N}}p\cdot N_{p},$ (15)

where $N_{p}$ is the number of band crossings with degree $p$ in the
entanglement spectrum; in particular, $p=1$ corresponds to linear crossings
and $p=2$ to quadratic crossings. To further support our conjecture, we
analyzed three more models of Euler insulators in Appendix D. We have
confirmed that Eq. (15) is indeed obeyed by all the models we analyzed.

We note that the robustness of linear crossings in Euler insulators, such the
ones in Fig. 2(c) can be understood from the same reason as in the SW
insulator. However, the reasoning we used for the SW insulator does not
explain why the quadratic crossing, around which the Pfaffian (10) does not
change the sign, is robust against $PT$-symmetric perturbations; further
investigation is required to fully understand the origin of this robustness.

##  V Cutting procedure

Finally, we discuss that the crossing of the entanglement spectrum of SW and
Euler insulators is related to the degeneracy of edge spectrum upon one varies
the magnitude of the hopping at the boundary, a process known as the cutting
procedure Teo  _et al._ (2008); Song  _et al._ (2020); Peri  _et al._ (2020);
Takahashi  _et al._ (2020); Tanaka  _et al._ (2020). In the cutting procedure,
we continuously connect the original (non-flattened) Hamiltonian with periodic
boundary conditions $H_{\mathrm{PBC}}$ and the one with open boundary
conditions $H_{\mathrm{OBC}}$ by one parameter $\lambda$:

$\displaystyle H_{\lambda}=\lambda H_{\text{PBC}}+(1-\lambda)H_{\text{OBC}}.$
(16)

(We note that $H_{\mathrm{PBC}}$ and $H_{\mathrm{OBC}}$ differ just by the
boundary hopping, and in particular share the same size.) The cutting
procedure has been used to analyze the bulk-boundary correspondence of higher-
order topological insulators and fragile topological insulators, which do not
have gapless edge states in the simple open boundary conditionSong  _et al._
(2020); Peri  _et al._ (2020); Takahashi  _et al._ (2020); Tanaka  _et al._
(2020). We numerically found that SW and Euler insulators exhibit non-trivial
spectral flow with $2w_{2}$ or $2\chi$ band crossing points.

Figure 3: Spectral flow of (a) Stiefel-Whitney insulator and (b) Euler
insulator with cutting procedure. Position of the gap closing points are (a1)
$(\pm\lambda_{1},0)$ and (b1) $(\pm k_{0},\pm\lambda_{2})$, where
$\lambda_{1}=0.2$, $\lambda_{2}=0.28$ and $k_{0}=0.44$. As the bulk
Hamiltonian is flattened, these degeneracy points approach
$(\lambda,k_{y})=(0,0)$, eventually reproducing the degeneracy point of the
entanglement spectrum. (a2,b2) The Berry phase distribution of (a1) and (b1).

The spectral flow of the SW insulator $H_{\text{SW}}$ is shown in Fig. 3(a),
which have $2w_{2}=2$ gap closing points. The energy dispersion around the gap
closing points are linear in both $\lambda$ and $k_{y}$. In the presence of
the $PT$ symmetry, one can show that the Berry phase calculated along any
closed path in $\lambda$-$k_{y}$ space is quantized either to 0 or $\pi$ Chan
_et al._ (2016). Moreover, the Berry phase around each of these gap closing
points is quantized to $\pi$, and thus they are protected because one cannot
continuously change the Berry phase from $\pi$ to 0. as shown in Fig. 3(a2) As
a result, the Berry phase in the $k_{y}$ direction changes by $\pi$ across the
degeneracy point as shown in Fig. 3(a2). Here, the Berry phase in the $k_{y}$
direction is defined as follows:

$\displaystyle\theta^{\text{Berry}}_{y}(\lambda)=\text{Im}\Bigg{[}\int_{0}^{2\pi}dk_{y}\text{Tr}[A_{y}(k_{y},\lambda)]\Bigg{]},$
(17)
$\displaystyle[A_{y}(k_{y},\lambda)]_{ij}=\Big{\langle}u^{i}(k_{y},\lambda)\Big{|}\frac{\partial}{\partial
k_{y}}u^{j}(k_{y},\lambda)\Big{\rangle},$ (18)

where $|u^{i}(k_{y},\lambda)\rangle$ is the $i$-th eigenstate of
$H_{\lambda}(k_{y})$ below the gap, and $A_{y}(k_{y},\lambda)$ is the Berry
connection matrix. We note that the size of $A_{y}(k_{y},\lambda)$ is
$2N_{x}\times 2N_{x}$, where $N_{x}$ is the number of unit cells along $x$
direction, and 2 accounts for the number of occupied bulk bands.

As one flattens the Hamiltonian, the gap closing points move and merge at
$\lambda=0$ line 222Here, we need to consider a parametrized Hamiltonian of
the form $H_{\mathrm{flat},\lambda}=\lambda
H_{\mathrm{flat}}^{x\text{-PBC}}+(1-\lambda)H_{\mathrm{flat}}^{x\text{-OBC}}$.
However, our construction of the flattened Hamiltonian around Eqs. (3-4)
implies that the size of $H_{\mathrm{flat}}^{x\text{-PBC}}$ is twice larger
than that of $H_{\mathrm{flat}}^{x\text{-OBC}}$. In order to correctly define
$H_{\mathrm{flat},\lambda}$, we calculate $H_{\mathrm{flat}}^{x\text{-PBC}}$
with half the system size so that the size of
$H_{\mathrm{flat}}^{x\text{-PBC}}$ and $H_{\mathrm{flat}}^{x\text{-OBC}}$
match.. The limit of perfectly flattened Hamiltonian corresponds to the gap
closing point at $\lambda=0$; thus the edge crossing of the entanglement
Hamiltonian can be understood as a limit of the gap closing points in the
spectral flow of the cutting procedure.

We also calculate the spectral flow of the Euler insulator $H_{\text{Euler}}$
in Fig. 3(b1), which have $2\chi=4$ gap closing points, which are all linear
crossings in both $\lambda$ and $k_{y}$ directions. In this case, the Berry
phase $\theta^{\text{Berry}}_{y}(\lambda)$ changes by $2\pi\equiv 0$ (mod
$2\pi$) at $\lambda=\lambda_{2}$, because there are two degenerate points at
$\lambda=\lambda_{2}$, and for each degenerate point the Berry phase should
change by $\pi$. The same holds for $\lambda=-\lambda_{2}$. The Berry phase
thus remains unchanged as one crosses $\lambda=\pm\lambda_{2}$ as shown in
Fig. 3(b2). We numerically find that these band crossing points are robust as
long as the Euler class $\chi=2$ is unchanged. We have also checked that, by
flattening the Hamiltonian, these gap closing points merge along the line
$\lambda=0$, recovering the result of the entanglement spectrum.

##  VI Conclusion and discussion

In this paper, we found an unconventional bulk-edge correspondence for two-
dimensional lattice models with $PT$ symmetry. For the insulator characterized
by the second SW class, there is either one or zero linear edge crossing in
the entanglement spectrum depending on either $w_{2}=1$ or $w_{2}=0$. This
edge crossing is protected by the emergent anti-unitary particle-hole symmetry
in the entanglement spectrum. For the Euler insulator, we proposed a
conjecture that the Euler class equals to the sum of the number of crossings
in the entanglement spectrum taking degree into account, which is supported by
our numerics. We have also found that the crossings of entanglement spectrum
have precursors in the gap closing points of the spectral flow of the cutting
procedure.

In recent years, there is an increasing interest in experimentally detecting
the entanglement spectrum using various engineered quantum materials such as
ultracold atomic gasesPichler  _et al._ (2016); Dalmonte  _et al._ (2018);
Barfknecht  _et al._ (2021); experimental measurement of the entanglement
spectrum was already demonstrated using the IBM quantum computersChoo  _et
al._ (2018). In trapped ions, the entanglement spectrum of Euler insulators
was also experimentally reconstructed from the measurement of bulk eigenstates
Zhao  _et al._ (2022). The parameter($\lambda$)-dependent Hamiltonian we
introduced for the cutting procedure is also experimentally accessible in
metamaterials Peri  _et al._ (2020). In light of these rapid experimental
developments, the features of the bulk-edge correspondence of SW and Euler
insulators discussed in this paper should be experimentally verifiable in a
near future.

Our results show that the $PT$-symmetric topological models have
unconventional bulk-edge correspondences. In three-dimensional bulk spectra,
examples where the charge of the Weyl point and the degree of the band
crossing are related are known Zhang  _et al._ (2020), but, to our knowledge
we are not aware of any other systems where the degree of edge crossing should
be taken into account upon considering the bulk-edge correspondence. While the
SW insulator case can be clearly understood, the Euler insulator requires
further investigation; more rigorous proof of the bulk-edge correspondence of
the Euler insulator is left for future works. It is also of interest to extend
the entanglement-spectrum analysis of the bulk-edge correspondence to other
symmetry classes in the AZ+$\mathcal{I}$ classification Bzdušek and Sigrist
(2017), which is an extension of the AZ classification that includes $PT$
symmetry. We expect that similar analysis of the bulk-edge correspondence
through the entanglement spectrum and/or cutting procedure will be of use for
further analysis of fragile and delicate topological insulators.

###### Acknowledgements.

We are grateful to helpful discussion with Tomáš Bzdušek and Shin Hayashi.
This work is supported by JSPS KAKENHI Grant No. JP20H01845, JST PRESTO Grant
No. JPMJPR19L2, and JST CREST Grant No.JPMJCR19T1.

##  Appendix A Emergent particle-hole symmetry

To understand the emergent particle-hole symmetry in the flattened
Hamiltonian, it is convenient to introduce the following correlation matrix
$C_{X}$Hughes  _et al._ (2011); Turner  _et al._ (2012); Zhao  _et al._
(2022):

$\displaystyle
C_{X}(k_{y}):=\frac{\mathbb{I}}{2}-H_{\text{flat}}^{x\text{-PBC}}(k_{y}).$
(19)

Each element is defined by

$\displaystyle\left[C_{X}(k_{y})\right]_{x\alpha,x^{\prime}\beta}=\frac{1}{L_{x}}\sum_{k_{x}}e^{ik_{x}(x-x^{\prime})}\left[P_{\mathrm{occ}}(\mathbf{k})\right]_{\alpha,\beta}.$
(20)

The reason we want to introduce $C_{X}(k_{y})$ is that this is a projector,
that is, $C_{X}(k_{y})^{2}=C_{X}(k_{y})$ holds. We define four blocks of
$C_{X}(k_{y})$ by

$\displaystyle C_{X}(k_{y})=\begin{pmatrix}C_{AA}(k_{y})&C_{AB}(k_{y})\\\
C_{BA}(k_{y})&C_{BB}(k_{y})\end{pmatrix}.$ (21)

Here, as in the main text, we divided $x$ direction in two parts $A$ and $B$.
Since $C_{X}$ is a Hermitian matrix, we see that $C_{AB}=C_{BA}^{\dagger}$.
Assuming that $A$ and $B$ regions have the same number of lattice sites, one
can also generally show that $C_{AA}=C_{BB}$ and $C_{AB}=C_{BA}$. The
spectrally flattened Hamiltonian in the open boundary condition and the upper
left block of the correlation matrix are related by

$\displaystyle
H^{x\text{-OBC}}_{\text{flat}}(k_{y})=\frac{\mathbb{I}}{2}-C_{AA}(k_{y}).$
(22)

Therefore, they share the same eigenstates.

We first show that $PT$ symmetry relates two eigenstates of $C_{AA}$ with the
same eigenvalue. As in the main text, we assume that the $PT$ operation is
represented by complex conjugation $K$ in the Bloch basis. Then, the
projection to occupied states is invariant by complex conjugation:

$\displaystyle P^{*}_{\text{occ}}(\mathbf{k})=P_{\text{occ}}(\mathbf{k}).$
(23)

This leads to the following symmetry for $C_{X}$:

$\displaystyle[C_{X}(k_{y})]^{*}_{x\alpha,x^{\prime}\beta}$
$\displaystyle=\frac{1}{L_{x}}\sum_{k_{x}}e^{-ik_{x}(x-x^{\prime})}[P_{\text{occ}}(\mathbf{k})]_{\alpha,\beta}$
$\displaystyle=[C_{X}(k_{y})]_{(L_{x}+1-x)\alpha,(L_{x}+1-x^{\prime})\beta}.$
(24)

Writing this equation in a matrix form (and suppressing explicitly writing
$k_{y}$ dependence), it looks

$\displaystyle C_{X}^{*}=\begin{pmatrix}\mathbf{0}&\mathcal{P}_{0}\\\
\mathcal{P}_{0}&\mathbf{0}\end{pmatrix}C_{X}\begin{pmatrix}\mathbf{0}&\mathcal{P}_{0}\\\
\mathcal{P}_{0}&\mathbf{0}\end{pmatrix},$ (25)

where

$\displaystyle\mathcal{P}_{0}$
$\displaystyle=\begin{pmatrix}&&\mathbf{1}_{u}\\\
&\reflectbox{\text{$\ddots$}}&\\\ \mathbf{1}_{u}&&\end{pmatrix},$ (26)

where $\mathbf{1}_{u}$ is a unit matrix in the unit cell. Looking at each
block, we find the following relations

$\displaystyle C_{AA}$
$\displaystyle=\mathcal{P}_{0}C_{AA}^{*}\mathcal{P}_{0},$ (27) $\displaystyle
C_{BA}\mathcal{P}_{0}$ $\displaystyle=\mathcal{P}_{0}C_{BA}^{*}.$ (28)

Therefore, if we have an eigenstate $|{\psi_{0}}\rangle$ of $C_{AA}$,
$C_{AA}|{\psi_{0}}\rangle=\xi|{\psi_{0}}\rangle$, we can obtain another
eigenstate of $C_{AA}$, by $\mathcal{P}_{0}|{\psi_{0}}\rangle^{*}$ with the
same eigenvalue because

$\displaystyle C_{AA}\mathcal{P}_{0}|\psi_{0}\rangle^{*}$
$\displaystyle=\mathcal{P}_{0}C_{AA}^{*}\mathcal{P}_{0}\mathcal{P}_{0}|\psi_{0}\rangle^{*}$
$\displaystyle=\mathcal{P}_{0}\left(C_{AA}|\psi_{0}\rangle\right)^{*}$
$\displaystyle=\xi\mathcal{P}_{0}|\psi_{0}\rangle^{*}.$ (29)

Note that since $\mathcal{P}_{0}$ flips the position in $x$ direction, if
$|\psi_{0}\rangle$ is localized on the right edge,
$\mathcal{P}_{0}|\psi_{0}\rangle^{*}$ is localized on the left edge.

We can also define an operation $\Xi$ that change both the eigenvalues and on
which side of the edges the eigenstates are localized, by using $C_{BA}$Hughes
_et al._ (2011); Turner  _et al._ (2012). (We note that we do not need to
require $PT$ symmetry to have $\Xi$; the existence of $\Xi$ is a general
property of the correlation matrix.) Using the property
$C_{X}^{2}(k_{y})=C_{X}(k_{y})$, the following relations hold:

$\displaystyle C^{\dagger}_{BA}(k_{y})C_{BA}(k_{y})$
$\displaystyle=C_{AA}(k_{y})(1-C_{AA}(k_{y})),$ (30) $\displaystyle
C_{AA}(k_{y})C_{BA}(k_{y})$ $\displaystyle=C_{BA}(k_{y})(1-C_{AA}(k_{y})).$
(31)

For an eigenstate $|\psi_{i}\rangle$ of $C_{AA}(k_{y})$ with an eigenvalue
$\xi_{i}(k_{y})$,
$C_{AA}(k_{y})|\psi_{i}\rangle=\xi_{i}(k_{y})|\psi_{i}\rangle$, the following
relation holds from Eq.(31):

$\displaystyle C_{AA}(k_{y})C_{BA}(k_{y})|{\psi_{i}}\rangle$
$\displaystyle=C_{BA}(k_{y})(1-C_{AA}(k_{y}))|{\psi_{i}}\rangle$
$\displaystyle=(1-\xi_{i}(k_{y}))C_{BA}(k_{y})|{\psi_{i}}\rangle.$ (32)

The norm of $C_{BA}(k_{y})|{\psi_{i}}\rangle$ is calculated by using Eq.(30):

$\displaystyle||C_{BA}(k_{y})|{\psi_{i}}\rangle||^{2}=\xi_{i}(k_{y})(1-\xi_{i}(k_{y})).$
(33)

Therefore, for $\xi_{i}(k_{y})\neq 0,1$ we obtain the eigenstate
$\Xi|{\psi_{i}}\rangle$ of $C_{AA}(k_{y})$ with the eigenvalue
$(1-\xi_{i}(k_{y}))$:

$\displaystyle\Xi|{\psi_{i}}\rangle\equiv\frac{C_{BA}(k_{y})}{\sqrt{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}}|{\psi_{i}}\rangle.$
(34)

The operator $\Xi$ can be expressed as

$\displaystyle\Xi=\sum_{i}\frac{C_{BA}(k_{y})}{\sqrt{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}}|{\psi_{i}}\rangle\langle{\psi_{i}}|.$
(35)

In particular, noting that $\epsilon_{i}=1/2-\xi_{i}$ from Eq. (22), Eq. (35)
is equivalent to Eq. (5). Furthermore, for an eigenstate $|{\psi_{i}}\rangle$
with an eigenvalue $\epsilon_{i}$ of $H_{\text{flat}}^{x\text{-OBC}}(k_{y})$,
$\Xi|{\psi}\rangle$ has an eigenvalue
$\epsilon_{i}^{\prime}=(1/2-(1-\xi_{i}))=-\epsilon_{i}$.

We note that, $C_{BA}(k_{y})$ as a matrix has nonzero components typically at
lower left and upper right regions. As a result, $\Xi$ changes the side of
localization of eigenstates; if $|\psi_{i}\rangle$ is localized at the right
edge of $A$ region, $\Xi|\psi_{i}\rangle$ is localized at the left edge of $A$
region.

Combining the two operations of $PT$ and $\Xi$, we get the following anti-
linear operation $(PT)\Xi$:

$\displaystyle(PT)\Xi$
$\displaystyle\equiv\sum_{i}\frac{\mathcal{P}_{0}C_{BA}^{*}(k_{y})}{\sqrt{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}}\Big{(}|{\psi_{i}}\rangle\langle{\psi_{i}}|\Big{)}^{*}K.$
(36)

Note that this operation is defined only if the $C_{L}$ eigenvalue $\xi_{i}$
of $|{\psi_{i}}\rangle$ is neither $0$ nor $1$. The $C_{L}$ eigenvalue of
$(PT)\Xi|{\psi_{i}}\rangle$ is $(1-\xi_{i})$.

We prove that $(PT)\Xi$ is an anti-unitary operator and squares to $+1$. We
again take $|{\psi_{i}}\rangle$ to be an eigenstate of $C_{AA}$ with an
eigenvalue $\xi_{i}$, and
$|{\widetilde{\psi_{i}}}\rangle:=(PT)\Xi|{\psi_{i}}\rangle$. First, $(PT)\Xi$
preserves the norm:

$\displaystyle\langle\widetilde{\psi_{i}}|{\widetilde{\psi_{i}}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}(\langle{\psi_{i}}|)^{*}C_{BA}^{T}(k_{y})\mathcal{P}_{0}^{T}\mathcal{P}_{0}C_{BA}^{*}(k_{y})|{\psi_{i}}\rangle^{*}$
$\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}\big{[}\langle{\psi_{i}}|C_{AA}(k_{y})(1-C_{AA}(k_{y}))|{\psi_{i}}\rangle\big{]}^{*}$
$\displaystyle=$ $\displaystyle\langle\psi_{i}|{\psi_{i}}\rangle.$ (37)

Here in the second line, we used Eq. (30). Also, $(PT)\Xi$ preserves
orthogonality:

$\displaystyle\langle\widetilde{\psi_{i}}|{\widetilde{\psi_{j}}}\rangle\cdot\sqrt{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}\sqrt{\xi_{j}(k_{y})(1-\xi_{j}(k_{y}))}$
$\displaystyle=$
$\displaystyle\big{[}\langle{\psi_{i}}|C_{AA}(k_{y})(1-C_{AA}(k_{y}))|{\psi_{j}}\rangle\big{]}^{*}$
$\displaystyle=$
$\displaystyle\langle\psi_{j}|{\psi_{i}}\rangle=0\quad(\xi_{i}\neq\xi_{j}).$
(38)

Therefore, $(PT)\Xi$ is an anti-unitary operator. Finally, we show that
$(PT)\Xi$ squares to 1:

$\displaystyle(PT)\Xi|{\widetilde{\psi_{i}}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}\mathcal{P}_{0}C_{BA}^{*}(k_{y})\Big{(}\mathcal{P}_{0}C_{BA}^{*}(k_{y})|{\psi_{i}}\rangle^{*}\Big{)}^{*}$
$\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}C_{BA}^{\dagger}(k_{y})\mathcal{P}_{0}\mathcal{P}_{0}^{*}C_{BA}(k_{y})|{\psi_{i}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}C_{BA}^{\dagger}(k_{y})C_{BA}(k_{y})|{\psi_{i}}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\xi_{i}(k_{y})(1-\xi_{i}(k_{y}))}C_{AA}(k_{y})\left(1-C_{AA}(k_{y})\right)|{\psi_{i}}\rangle$
$\displaystyle=$ $\displaystyle|{\psi_{i}}\rangle.$ (39)

Since both $PT$ and $\Xi$ changes the side of localization, the combined
operation $(PT)\Xi$ thus transforms one state localized at one edge to another
state with an opposite energy localized at the same edge.

##  Appendix B Wilson loop spectra

The Euler class topology can be detected from the Wilson loop windingAhn  _et
al._ (2018); Bouhon  _et al._ (2020a, b). Specifically, Wilson loop winding
gives the absolute value of the Euler class $|\chi|$Bouhon  _et al._ (2020a).
Since the sign of the Euler class is ambiguous, unless otherwise stated, we
assume that the Euler class $\chi$ is positive by taking an appropriate
basisBouhon  _et al._ (2020b). Figure 4 shows Wilson loop spectra for several
models used in the main text. Figures 4(a-c) show that they all have non-
trivial Euler classes.

The case of Fig. 4(d) with three occupied bands is characterized by the second
Stiefel-Whitney class $w_{2}$ instead of the Euler class. From the number of
$\pi$-crossing Wilson loop spectra, $w_{2}=0$ (mod $2$) in this caseAhn  _et
al._ (2018).

Figure 4: Wilson loop spectra for (a) $H_{\text{SW}}$, (b) $H_{\text{Euler}}$
and that for $H_{\text{Euler+NI}}$ with (c) $V_{0}=1.5$ and (d) $V_{0}=-0.5$.
By counting the Wilson loop windings, we find (a) $\chi=1$, (b) $\chi=2$, and
(c) $\chi=2$, respectively. In (d), $w_{2}$ is 0 from the number of $\pi$
crossings in the Wilson loop spectra.

##  Appendix C Effective Hamiltonian

Since $(PT)\Xi$ is an antiunitary operation that squares to +1, we can take a
basis in which $(PT)\Xi$ is expressed as the complex conjugate $K$. (Note
that, in this new basis, $PT$ is no longer expressed by $K$.) Here, we briefly
explain how we can obtain a basis such that $(PT)\Xi=K$. From a pair of
eigenstates of $H_{\text{flat}}^{x\text{-OBC}}$ localized on one edge,
$|\psi_{j}\rangle$ and $(PT)\Xi|\psi_{j}\rangle$, we construct the following
two states:

$\displaystyle|{\psi_{+}^{(j)}}\rangle$
$\displaystyle:=\frac{1}{\sqrt{2}}\big{(}|{\psi_{j}}\rangle+(PT)\Xi|{\psi_{j}}\rangle\big{)},$
(40) $\displaystyle|{\psi_{-}^{(j)}}\rangle$
$\displaystyle:=\frac{1}{\sqrt{2}i}\big{(}|{\psi_{j}}\rangle-(PT)\Xi|{\psi_{j}}\rangle\big{)}.$
(41)

These states are invariant under $(PT)\Xi$, that is,
$(PT)\Xi|{\psi_{+}^{(j)}}\rangle=|{\psi_{+}^{(j)}}\rangle$ and
$(PT)\Xi|{\psi_{-}^{(j)}}\rangle=|{\psi_{-}^{(j)}}\rangle$. This invariance
combined with the fact that $(PT)\Xi$ is an anti-unitary operator implies
that, in the space spanned by $|{\psi_{+}^{(j)}}\rangle$ and
$|{\psi_{-}^{(j)}}\rangle$, $(PT)\Xi$ acts as a simple complex conjugation
$K$. Taking all the pairs of eigenstates we consider to build an effective
Hamiltonian and considering the following basis

$\displaystyle\bigg{[}|{\psi_{+}^{(1)}}\rangle,|{\psi_{-}^{(1)}}\rangle,|{\psi_{+}^{(2)}}\rangle,\cdots,|{\psi_{-}^{(N_{0})}}\rangle\bigg{]},$
(42)

where $N_{0}$ is the number of pairs of eigenstates, or half the number of all
the eigenstates, one wishes to include in the effective Hamiltonian. In this
basis, $(PT)\Xi$ acts as $K$. Since the effective Hamiltonian should anti-
commute with $(PT)\Xi=K$, the effective Hamiltonian should be a purely
imaginary matrix, and thus it should be a skew-symmetric matrix of the
following form:

$\displaystyle
H_{\text{eff}}^{x\text{-OBC}}=\begin{pmatrix}\epsilon_{1}\sigma_{y}&&\\\
&\ddots&\\\ &&\epsilon_{N_{0}}\sigma_{y}\end{pmatrix},$ (43)

which is the desired form.

##  Appendix D Analysis of more models

Here, we perform numerical calculations for more models to give further
support to our conjecture.

###  D.1 Kobayashi-Furusaki model

Figure 5: Numerical calculation for the model $H_{S1}$. (a) Edge spectrum.
Colors represent the squared amplitudes of the states in the edge unit cell.
(b) Entanglement spectrum. (c) Spectral flow with cutting procedure. Here,
$\lambda_{3}=0.18$ and $k_{1}=0.66$. (d) Wilson loop spectrum.

First, we use the model introduced by Kobayashi and Furusaki in the study of
fragile topological insulators Kobayashi and Furusaki (2021). The restriction
of their model to $k_{z}=0$ corresponds to an insulator with Euler class
$|\chi|=2$:

$\displaystyle H_{S1}(\bm{k})=$
$\displaystyle\Bigg{(}3-2\sum_{i=x,y}\cos(k_{i})\Bigg{)}\tau_{0}\sigma_{z}$
$\displaystyle+\bigg{(}2(\cos(k_{x})-\cos(k_{y}))+\mathcal{M}_{1}\bigg{)}\tau_{z}\sigma_{x}$
$\displaystyle+\Big{(}2\sin(k_{x})\sin(k_{y})+\mathcal{M}_{2}\Big{)}\tau_{x}\sigma_{x}$
$\displaystyle+\Delta\tau_{x}\sigma_{0},$ (44)

where $\mathcal{M}_{1}=0.1$, $\mathcal{M}_{2}=0.08$ and $\Delta=0.2$. The $PT$
operation is represented as complex conjugation $K$. Here, the term
$\Delta\tau_{x}\sigma_{0}$, which is absent in Kobayashi and Furusaki (2021),
is added to open a gap in the edge spectrum as shown in Fig. 5(a). The Wilson
loop spectrum (Fig. 5(d)) confirms that the model has Euler class $|\chi|=2$.
Unlike the edge spectrum, the entanglement spectrum is gapless as shown in
Fig. 5(b). There are two linear crossings at $\epsilon=0$, consistent with our
conjecture for $|\chi|=2$. Furthermore, from the spectral flow of the cutting
procedure (see Fig. 5(c)), we can see that there are four crossings as we
anticipate.

###  D.2 $C_{6}$-symmetric models

Next, we confirm that our results also hold for systems other than square
lattices. As examples, we consider two models of $C_{6}$ symmetric lattices.

The following honeycomb lattice model is an effective model for the nearly
flat bands in twisted bilayer graphene at the magic angleZou  _et al._ (2018):

$\displaystyle H_{S2}(\bm{k})=$
$\displaystyle\Bigg{(}1+2\cos\frac{\sqrt{3}k_{x}}{2}\cos\frac{k_{y}}{2}\Bigg{)}(0.4\sigma_{x}+0.6\tau_{z}\sigma_{x})$
$\displaystyle+2\sin\frac{\sqrt{3}k_{x}}{2}\cos\frac{k_{y}}{2}(0.4\sigma_{y}+0.6\tau_{z}\sigma_{y})$
$\displaystyle+\bigg{(}4\cos\frac{\sqrt{3}k_{x}}{2}\sin\frac{k_{y}}{2}-2\sin(k_{y})\bigg{)}(0.1\tau_{x}).$
(45)

Here, the $PT$ operation is represented as $\sigma_{x}K$. Figures 6(a1,a2)
show the entanglement spectrum and Wilson loop spectrum of $H_{S2}(\bf{k})$.
The Wilson loop winding in [(a2)] shows that the Euler number is $|\chi|=1$.
The entanglement spectrum [(a1)] shows one linear crossing point at
$\epsilon=0$, which is in agreement with our conjecture with $|\chi|=1$.

The final model is on a triangular lattice, originally used in the study of
triple nodal points in Lenggenhager  _et al._ (2022). Restricting the model to
$k_{z}=0$ results in the following model:

$\displaystyle\quad\ H_{S3}(\bm{k})$
$\displaystyle=-\Big{[}t_{1}+t_{2}\Big{(}\cos
k_{x}+2\cos\frac{k_{x}}{2}\cos\frac{\sqrt{3}k_{y}}{2}\Big{)}+t_{3}\Big{]}\tau_{z}\sigma_{0}$
$\displaystyle\quad\ -t_{4}(-\tau_{z}+\tau_{z}\sigma_{z})$
$\displaystyle\quad\ -t_{5}\Big{[}\Big{(}\cos
k_{x}-\cos\frac{k_{x}}{2}\cos\frac{\sqrt{3}k_{y}}{2}\Big{)}(-\tau_{x}\sigma_{z}-\tau_{x})$
$\displaystyle\quad\
+\sqrt{3}\sin\frac{k_{x}}{2}\sin\frac{\sqrt{3}k_{y}}{2}(-\tau_{z}-\tau_{z}\sigma_{z})\Big{]}$
$\displaystyle\quad\ -\sqrt{2}t_{6}\Big{[}\Big{(}\cos
k_{x}-\cos\frac{k_{x}}{2}\cos\frac{\sqrt{3}k_{y}}{2}\Big{)}(\tau_{x}\sigma_{x}-\tau_{z}\sigma_{x})$
$\displaystyle\quad\
-\sqrt{3}\sin\frac{k_{x}}{2}\sin\frac{\sqrt{3}k_{y}}{2}(\tau_{x}\sigma_{x}+\tau_{z}\sigma_{x})\Big{]},$
(46)

where $t_{1}=4$, $t_{2}=-\frac{2}{3}$, $t_{3}=-3$, $t_{4}=-\frac{1}{2}$,
$t_{5}=-\frac{1}{3}$ and $t_{6}=\frac{6}{5}$. Here, the $PT$ operation is
represented as complex conjugation $K$. Figures 6(b1,b2) show the entanglement
spectrum and Wilson loop spectrum of $H_{S3}(\bf{k})$. The Wilson loop
spectrum shows that the Euler class is $|\chi|=2$. The entanglement spectrum
[(b1)] has two linear crossing points at $\epsilon=0$, which is consistent
with our conjecture with $|\chi|=2$.

Figure 6: Numerical calculations for the models (a) $H_{S2}$ and (b) $H_{S3}$.
(a1,b1) Entanglement spectra, and (a2,b2) Wilson loop spectra.

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