# Cross-Element Combinatorial Selection for Multi-Element Creative in Display
Advertising

Wei Zhang, Ping Zhang, Jian Dong, Yongkang Wang  and  Pengye Zhang, Bo Zhang,
Xingxing Wang, Dong Wang  zhangwei180, zhangping18, dongjian03,
<EMAIL_ADDRESS>zhangpengye, zhangbo126, wangxingxing04,
<EMAIL_ADDRESS>MeituanBeijingChina

###### Abstract.

The effectiveness of ad creatives is greatly influenced by their visual
appearance. Advertising platforms can generate ad creatives with different
appearances by combining creative elements provided by advertisers. However,
with the increasing number of ad creative elements, it becomes challenging to
select a suitable combination from the countless possibilities. The industry’s
mainstream approach is to select individual creative elements independently,
which often overlooks the importance of interaction between creative elements
during the modeling process. In response, this paper proposes a Cross-Element
Combinatorial Selection framework for multiple creative elements, termed CECS.
In the encoder process, a cross-element interaction is adopted to dynamically
adjust the expression of a single creative element based on the current
candidate creatives. In the decoder process, the creative combination problem
is transformed into a cascade selection problem of multiple creative elements.
A pointer mechanism with a cascade design is used to model the associations
among candidates. Comprehensive experiments on real-world datasets show that
CECS achieved the SOTA score on offline metrics. Moreover, the CECS algorithm
has been deployed in our industrial application, resulting in a significant
6.02% CTR and 10.37% GMV lift, which is beneficial to the business.

Ad Creative, Creative Selection, Creative Interaction

††copyright: none††conference: Make sure to enter the correct conference title
from your rights confirmation email; 21-25 October 2023; UK††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Information systems Display
advertising; Recommender systems; Ad Creative

##  1\. INTRODUCTION

In recent years, we have witnessed a growing business in online display
advertising and its importance for Internet service providers. Image ads are
the most widely used format since they are more compact, intuitive, and
comprehensible. In the image of the advertisement, aka an $ad\>creative$,
there may be background pictures, item descriptions, promotional texts, and
other creative elements. And different combinations can quite vary in Click-
Through Rate(CTR)(Azimi et al., 2012; Mo et al., 2015). Therefore, it is of
great value to platforms to study how to improve the visual effect of
creatives.

With the increase of creative elements, it has become an emerging trend to
investigate how to combine creative elements to make a better appearance and
thus maximize the effectiveness of ads(Chen et al., 2019). However, the
difficulty lies in the exponential growth in the number of candidate
combinations. For example, a creative that contains three elements would
involve $N_{1}*N_{2}*N_{3}$ possible elements combinations. More generally,
assuming that there are $M$ types of creative elements, each with an average
of $N$ candidates, the complexity of the problem is $N^{M}$. Moreover, as new
creative elements are added, the number of possible combinations will increase
rapidly, making it difficult to apply complex models to creative selection
tasks.

Traditionally, to deal with online performance issues caused by the excessive
number of creative combinations, the creative selection is performed
offline(Choi et al., 2010), thereby reducing the number of online candidates.
However, the offline selection of creatives lacks the awareness of online
request-level data. Therefore, some online strategies are applied, such as
popularity, and preferences(Tang et al., 2013). However, this method requires
time to accumulate online feedback. Meanwhile, some studies have attempted to
classify creative elements into different types and estimate using multi-task
click-through-rate(CTR) models, and then combine the elements that are
estimated by different task towers(Richardson et al., 2007). This method
solves the problem of the combinatorial explosion. The complexity here is
reduced from $N^{M}$ to $N*M$. But for these multi-task learning routes, the
cross-category interactions are not fully captured.

To this end, we propose a Cross-Element Combinatorial Selection framework for
multiple creative elements, named CECS. To capture the association between
different types of creative elements, a cross-element interaction method is
applied to adjust the expression of every single creative element. We further
designed a cascade element selection to transform the creative combination
problem into a cascade selection problem. Experimental results show that the
method has a significant improvement compared with the baselines, indicating
that the method can select better combinations of candidate elements. Through
online real-world experiments, it is demonstrated that the method can improve
the CTR and GMV, which means the selected advertisements are appealing to the
users.

To summarize, the contribution of this paper can be described as follows:

  * •

We propose a cross-element interaction to tackle the relationship between
different creative elements. Since the modal and information may vary, we
adopt multiple attention mechanisms between each pair of creative types.

  * •

We transform the creative combination problem into a cascade selection
problem, the creative elements information is passed through the cascade
design, so that the association in ad creatives can be integrated.

  * •

We evaluate the proposed model on real-world datasets. The offline statistics
and the online feedback indicate the strong practical value of the algorithm
in the industry.

##  2\. METHODOLOGY

We first define the concepts and issues involved in this paper, then give
corresponding explanations for the details of the different modules in the
architecture.

###  2.1. Problem Setup

Definition 1. Creative Optimization. Given a user and the candidate creative
elements, the goal of creative optimization is to find the best combination of
creative elements, so as to maximize the probability of being
clicked/satisfied by the user, denoted as follows:

(1) $\displaystyle maximize\>\text{ }P(A,label=1|C,u;\Theta)$

where $A$ means an ad creative combination and $C$ is the set of candidate
elements, $label=1$ indicates the positive feedback from the user $u$, e.g.
click behavior in advertising, and $\Theta$ is the model parameter to be
optimized.

In real industrial scenarios, the creative elements in an ad may have multiple
categories, e.g., banners, description texts, shop names, etc. Every element
may contribute to the click behavior, so a design to fully explore the
relationships is necessary.

fig.2

Figure 1. An example of the creative selection problem with N types of
creatives. The CEI and CES components are applied as the encoder process and
the decoder process to capture the relationships.

###  2.2. Model Overview

In this section, we provide an overview of the model. The overall structure of
CECS is shown in Figure 1, which follows an encoder-decoder process design. In
the encoder process, the model processes candidate creative elements in a
Cross-Element Interaction (CEI), with the goal of obtaining creative elements
that contain rich interaction information. In the decoder process, creative
elements are sequentially estimated using a Cascade-Element Selection (CES),
resulting in a combined ad.

In this way, it is no longer necessary to enumerate every creative
combination($N^{M}$ solutions), but only to estimate the creative elements
sequentially through the model($N*M$ solutions). The inner relationship
between categories is expected to be captured by the model, which will be
discussed later.

###  2.3. Cross-Element Interaction

In an advertisement, every creative element contributes to the final click
behavior of a user, since different types of elements may be related to each
other at the semantic or vision level. Traditionally, the elements are modeled
separately, and the relationships that lie between different creative elements
are often overlooked, which is the key point this chapter addresses.

In order to better integrate the relationships, we apply a simple but
efficient way to combine information from all categories, so that the cross-
category relationships can be captured. Inspired by the design of the
parameter personalized network(Chang et al., 2023), we adopt a similar
interaction way, as is shown in Figure 2.

Figure 2. An illustration of procession for a single creative element in the
CEI component. The result representation integrates information from all
creative categories.

Firstly, we compute the central representation of each category of candidate
creative elements, which is shown as follows:

(2)
$\displaystyle\boldsymbol{E}_{i}=\frac{1}{N_{i}}\sum_{j=1}^{N_{i}}\boldsymbol{E}_{i,j}$

where $E_{i,j}$ represents the embedding of the $j$-th element of the $i$-th
creative category, $N_{i}$ means the number of items of the $i$-th creative
category. For short, the central representation is the mean pooling among the
specific category. Later, we adopt a $1-layer-MLP$ to transform its semantic,
i.e., $\textbf{v}_{i}=MLP(\textbf{E}_{i})$, and the dimension is made
consistent with the original representation.

Then, for each candidate’s creative element $\textbf{E}_{i,j}$, we use the
attention method to calculate the relationship with every transformed central
representation $\textbf{v}_{i}$. Here the attention mechanism is applied to
weight the importance of the different categories of creative elements, as is
shown:

(3)
$\displaystyle\textbf{E}_{i,j}^{out}=\sum_{i=1}^{k}\text{softmax}(\frac{\textbf{v}_{i}^{T}\textbf{E}_{i,j}}{\sqrt{d_{dim}}})\textbf{E}_{i,j}$

where $k$ is the total number of creative category, $d_{dim}$ is the dimension
of the embedding, $\textbf{E}_{i,j}$ is a creative element of $j$-th category,
$\textbf{v}_{i}$ is the central representation of the $i$-th creative
category, and $\textbf{E}_{i,j}^{out}$ is the output representation of the
creative.

In this way, it is possible to get the encoded creatives containing interacted
features by explicitly integrating the attention mechanism.

###  2.4. Cascade-Element Selection

The prediction process of creative optimization has two characteristics: 1)
the output is derived from the input, i.e., the predicted creative elements
are derived from the candidate ones. 2) There are relations among the
estimated creative elements, which makes it necessary to estimate in a
sequential manner. The pointer network mechanism (Vinyals et al., 2015) is a
method of selecting from existing candidate solutions, and we adopt and extend
it here.

The module Cascade-Element Selection(CES for short) estimates creative
elements step-by-step, so a cascade structure is applied here, as shown in
Figure 3. One of the key points is that in the $i$-th prediction process, the
prediction status of $(i-1)$-th needs to be considered. Therefore, The Gate
Recurrent Unit(Chung et al., 2014)(GRU for short) structure is included in the
CES. The hidden vector of GRU(also called the state vector in some scenarios)
named $h_{i}$ represents the predicted state of $(i-i)$-th elements that will
be transferred to the next CES unit, so the inter-relationship between output
elements can be integrated.

In the $i$-th step of the decoder, i.e., predicting the creative element of
the $i$-th category, the hidden vector $\textbf{h}_{i-1}$ and $\hat{y}_{i-1}$
in the previous CES are fed into the current CES, as is shown in eq(4):

(4) $\displaystyle\begin{aligned}
\textbf{h}_{i}&=GRU(\hat{y}_{i-1},\textbf{h}_{i-1})\\\
\hat{y}_{i}&=\sum_{j=1}^{N_{i}}softmax(\frac{\textbf{h}_{i}^{T}\textbf{E}_{i,j}}{\sqrt{d_{dim}}})\textbf{E}_{i,j}\end{aligned}$

where $\textbf{h}_{i}$ is the output of the GRU unit function annotated as
$GRU$, $\hat{y}_{i-1}$ and $h_{i-1}$ are the previous output and state
respectively, $N_{i}$ is the number of the $i$-th candidates,
$\textbf{E}_{i,j}$ is the interacted representation of the candidate creative
elements of the $i$-th category. For the output vector of the GRU, an
attention mechanism is applied to calculate the probability among the encoded
creative elements. Then we get the probability distribution over the
candidates.

Finally, the index of the output creative element $\hat{y}_{i}$ among
candidates is used. Pick up the corresponding index in the candidate’s
creative elements, and the solution of the $i$-th creative element can be
obtained.

fig.6

Figure 3. A single step of the cascade-element selection. The creative
information is transferred through the hidden vector along the GRU cells.

The above process repeats $k$ times to predict every creative element, and the
creative information is cascaded to the next pointer until the final creative
element is predicted. Particularly, the encoded representation of the total
candidates is used for the first CES unit, which acts as the overall upstream
for the decoder. In this way, the cascade-element selection models the
interactive relationship between different types of creative elements.

###  2.5. Model Loss and Optimization

The estimated creative elements are derived from the input, so we model the
output probability of each cascade-element selection. Therefore, we use cross-
entropy to model the loss of one type of creative element, as is shown below:

(5) $\displaystyle\begin{aligned}
Loss_{i}=-y_{j}\log(\hat{y}_{j})\end{aligned}$

where $Loss_{i}$ is the loss for the $i$-th category, $y_{i}$ and
$\hat{y}_{i}$ are the ground truth and predicted result for the $i$-th
category respectively.

The different categories of creative elements may result in different
promotional and visual effects, so the loss of different creative elements may
have a large variance. Therefore, we adopt the idea of uncertainty
modeling(Kendall et al., 2018) to provide adaptive weight coefficients for the
selection process of different types of creative elements. In this way, the
human parameter tuning process can be reduced and better-weighted loss can be
learned, shown as follows:

(6) $\displaystyle
Loss_{total}=\sum_{i=1}^{k}\left(\frac{1}{\sigma_{i}^{2}}Loss_{i}+\log{\sigma_{i}^{2}}\right)$

where $k$ is the total number of creative categories, $\sigma_{i}$ is the
corresponding learnable weight, and $Loss_{i}$ is the loss of the i-th
predicted creative element. In this way, the loss of multiple creative
elements can be adaptively adjusted.

##  3\. EXPERIMENTS

We conduct several groups of experiments to verify the model proposed, with
the purpose of answering the following research questions:

RQ1 Does our proposed method outperform the baseline methods in the creative
element selection problem?

RQ2 How do the two modules in the model work for modeling the relationship
between creative elements?

RQ3 How does the model perform when deployed online?

###  3.1. Datasets and Evaluation

####  3.1.1. Datasets.

Since there doesn’t exist an off-the-shelf benchmark dataset for such creative
selection tasks, we construct our dataset from real data online. Firstly, we
set up a small amount of random traffic online, i.e., creative elements are
randomly combined by type, so that we can obtain an unbiased copy of user
preference data.

In our scenario, an ad is composed of a banner background image, a main title,
and a sub-title, so the creative type is 3, i.e. $k=3$. Finally, we collect a
total number of 7.4 million samples, containing 5,494,110 users and 157,611
shops. We use 90% of the data for training and the rest for testing. And we
truncate each type of the creative elements sequences length of each shop to 5
uniformly, that is, up to 125($5^{3}$) potential creative combinations are
considered for each shop.

####  3.1.2. Evaluation Metrics.

For evaluation, traditional ranking evaluation metrics such as nDCG, MAP are
not suitable, because this is a combinatorial modeling problem. We focus on
the ”best combination” of a desired advertisement, and the evaluation of
creative combination needs to be considered in a holistic way.

Hit Ratio (HR). Hit ratio is a recall-based evaluation metric that measures
how much the real creative elements in an ad overlap with the predicted
elements. The formula is shown as follows:

(7) $\displaystyle HR=\frac{\sum_{i=1}^{k}w_{i}|C_{p}\cap
C_{g}|}{\sum_{i=1}^{k}w_{i}|C_{p}\cup C_{g}|}$

where $k$ is the number of creative element types, $C_{p}$ and $C_{g}$ are the
predicted result and the ground truth respectively, $w_{i}$ indicates
different weights for creative elements.

Precision (PR). The concept of precision is defined as whether the actually
clicked (positive sample) creative element is also included in the estimated
ad creative containing $k$ creative elements, formulated as follows:

(8) $\displaystyle PR=\textstyle\sum_{i=1}^{k}w_{i}I(c_{i}\in C_{g})$

where $k$ is the number of creative element types, $w_{i}$ is the weight
parameter for each type of creative element, $I(\cdot)\in\\{0,1\\}$ is the
indicator function.

In our dataset, considering the visual influence of the three types of
creatives, we set $w_{1}=0.5,w_{2}=0.3,w_{3}=0.2$, and the overall $HR$ and
$PR$ are calculated through the average metric of all testing examples.

####  3.1.3. Baselines.

We compare the proposed method(CECS) with several strategies commonly used in
online creative selection scenarios (Strategies for short). To evaluate the
ability to tackle relationships between elements, we regard the selection as a
multi-task CTR prediction task (Multi-CTR for short). Therefore, we combine
CTR models like DNN(Chen et al., 2016), DeepFM(Guo et al., 2017), AutoInt(Song
et al., 2019) with multi-task learning mechanisms like shared-bottom, MMoE(Ma
et al., 2018) and PLE(Tang et al., 2020) to fully explore the offline effects.

###  3.2. Performance Comparison (RQ1)

Table 1 shows the performance of HR and PR for the dataset for different
methods. In general, the CTR-based methods are more effective than the
Strategies cause these models are better at generalizing features. With the
improvement of the Multi-CTR model, we see a corresponding increase in the HR
and PR evaluation metrics. And the MMoE structure captures the relationships
better than the shared-bottom and slightly better than the PLE structure.
Among them, the AutoInt+MMoE model achieves the best results because the self-
attention mechanism in this model is applied to capture the correlations
between the same type of creative elements. Nevertheless, our model still
outperforms the baselines, indicating the interaction between creative
elements can be better captured.

From the table, we can conclude that the interaction between the elements in
the Multi-CTR methods are not sufficient. This is because different creative
elements are not displayed independently, but in a holistic way, and
contributes to the final visual effect of the advertisement.

Table 1. The performance for different models. Model | HR | PR
---|---|---
Strategies | E-greedy | 0.7886 | 0.6707
Popularity | 0.7912 | 0.6804
Preference | 0.8025 | 0.7141
Multi-CTR | DNN + shared-bottom | 0.8328 | 0.7669
DNN + MMoE | 0.8456 | 0.7783
DNN + PLE | 0.8402 | 0.7776
DeepFM + MMoE | 0.8491 | 0.7812
| AutoInt + MMoE | 0.8537 | 0.7873
CECS(ours.) | 0.8702 | 0.8028

###  3.3. Analysis of Model Designs (RQ2)

Here, we focus on the attention and the cascade design in CEI and CES
respectively. We conduct several ablation studies for the two designs, as is
shown in Table 2. The $\surd$ and $\times$ marks in the table indicate whether
the above designs are used or not. When not using the CEI, we use MLP instead.
Not using the CES means that no estimated element information is transmitted
through the decoder process, so we omit the hidden vector $h_{i-1}$ from the
previous CES.

It can be seen from the table that with the plug-in of different designs, the
HR and PR of the model increase. We see there’s a higher improvement in the
CES design, which mainly contributes to the performance. Because through the
design, the relationships can be transferred to other candidates to make
better selections. The improvement of the CES design means that more
information lies among different types of creative elements, and affects
users’ preference when combined together.

Table 2. Ablation experiments corresponding to different model designs. CEI | CES | HR | PR
---|---|---|---
$\times$ | $\times$ | 0.8473 | 0.7767
$\surd$ | $\times$ | 0.8501 | 0.7826
$\times$ | $\surd$ | 0.8669 | 0.7952
$\surd$ | $\surd$ | 0.8702 | 0.8028

###  3.4. Online Results(RQ3)

To evaluate how the model performs in real-world industrial scenarios, we
conduct A/B tests online. The model online is an MMoE-based Multi-CTR model,
it predicts different types of creatives at the same time. Our model focus on
the relationships between creative elements, solving them in a cascade
selection way. For the online time complexity problem, we shrink the feature
pulling time before prediction, then the inference time is within an
acceptable range. In the end, compared with the Multi-CTR method, we further
obtained an increase of CTR+6.02% and GMV+10.37%, which proved the
effectiveness of the method in the real-world industrial scene.

##  4\. CONCLUSION

In this paper, we investigate the creative element selection from the
perspective of cascade creative selection, solving the problem using a
holistic way. The cross-element interaction is applied, which captures the
associations between different types of candidate creative elements. And we
proposed the cascade element selection to model the relationship between
estimated elements. It was experimentally verified that the method outperforms
the traditional strategies and Multi-task CTR prediction method. And the
ablation study verified the importance of attention and cascading design in
the framework. The model was deployed on real-world data and significantly
improved CTR and GMV, which can be extended and applied in other advertising
scenarios.

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# Quantum Many-Body Scars in Spin-1 Kitaev Chains

Wen-Long You  College of Science, Nanjing University of Aeronautics and
Astronautics, Nanjing, 211106, China  Key Laboratory of Aerospace Information
Materials and Physics (NUAA), MIIT, Nanjing 211106, China  Zhuan Zhao  College
of Science, Nanjing University of Aeronautics and Astronautics, Nanjing,
211106, China  Key Laboratory of Aerospace Information Materials and Physics
(NUAA), MIIT, Nanjing 211106, China  Jie Ren  Department of Physics, Changshu
Institute of Technology, Changshu 215500, China  Gaoyong Sun  College of
Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106,
China  Key Laboratory of Aerospace Information Materials and Physics (NUAA),
MIIT, Nanjing 211106, China  Liangsheng Li  Science and Technology on
Electromagnetic Scattering Laboratory, Beijing 100854, China  Andrzej M. Oleś
<EMAIL_ADDRESS>Max Planck Institute for Solid State Research,
Heisenbergstrasse 1, D-70569 Stuttgart, Germany Institute of Theoretical
Physics, Jagiellonian University, Prof. Stanisława Łojasiewicza 11, PL-30348
Kraków, Poland

(20 January 2022)

###### Abstract

To provide a physical example of quantum scars, we study the many-body scars
in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly
embedded in the spectra. Regarding the conserved quantities, the Hilbert space
is fragmented into disconnected subspaces and we explore the associated
constrained dynamics. The continuous revivals of the fidelity and the
entanglement entropy when the initial state is prepared in
$|\mathbb{Z}_{k}\rangle$ ($k=2,3$) state illustrate the essential physics of
the PXP model. We study the quantum phase transitions in the one-dimensional
spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group
and Lanczos exact diagonalization methods, and determine the phase diagram. We
parametrize the two terms in the Hamiltonian by the angle $\phi$, where the
Kitaev term is $K\equiv\sin(\phi)$ and competes with the Heisenberg
$J\equiv\cos(\phi)$ term. One finds a rich ground state phase diagram as a
function of the angle $\phi$. Depending on the ratio $K/J\equiv\tan(\phi)$,
the system either breaks the symmetry to one of distinct symmetry broken
phases, or preserves the symmetry in a quantum spin liquid phase with
frustrated interactions. We find that the scarred state is stable for
perturbations which obey $\mathbb{Z}_{2}$-symmetry, while it becomes unstable
against Heisenberg-type perturbations.
Accepted for publication in Physical Review Research

##  I Introduction

With the rapid development of experimental techniques, specifically with the
advancement of stable and precisely adjustable ultrashort pulse laser
technologies, and the improvement of detection methods, the real-time
information with atomic scale and sub-picosecond resolution can be obtained.
It provides new opportunities for perceiving the phenomena in related many-
body systems. Meanwhile, quantum simulators with high isolation, long
relaxation time and experimental controllability, such as ultra-cold atoms,
ion traps, nitrogen vacancy centers in diamonds, and superconducting circuits,
have opened up a new direction in the studies of strongly correlated electron
systems in condensed matter physics. Studies of abstract quantum complex
systems is another route to provide a more complete description of quantum
phase transitions and many-body dynamics in isolated quantum systems.

Particularly in the recent years, a tremendous progress was made in describing
the non-equilibrium dynamics of isolated quantum systems. Thermalization and
localization are distinct fates of the steady quantum states. The former will
lose information of the initial state after undergoing a long-time evolution,
while the latter retains information of the initial state as much as possible.
In general, the system will tend to thermalize, as it follows the eigenstate
thermalization hypothesis (ETH), which is generally regarded as the
cornerstone of contemporary quantum statistics. A system that satisfies ETH is
considered to be ergodic. Numerical calculations show that most interacting
systems are indeed strongly ergodic [1, 2, 3, 4]. Thermal excitations enrich
the properties of spin chains and are, for instance, responsible for the
dimerization in spin-orbital systems [5].

However, currently many quantum systems were found to disobey the ETH. A
typical exception is the integrable system undergoing Anderson localization
[6], in which there are infinite local conserved quantities that can inhibit
the relaxation between eigenstates and thus retain the coherence in the
quantum system. However even a weak perturbation will suffice to break the
integrability of the system, and then the ETH would be restored [7].

Another mechanism to avoid thermalization is many-body localization—it has
been intensely investigated in the past decade [8, 9, 10, 11]. It was
demonstrated that using one-dimensional (1D) so-called PXP models for spin-1
can lead to weak ergodicity breaking [12]. The emergent integrability of these
a priori non-integrable systems prevents them from thermalization [13]. The
Anderson localization generally concerns the ground state, while the many-body
localization usually occurs at finite temperature. Recently, the phase
transitions between the thermalized and many-body localized phases,
highlighting many-body localized mobility edge, were observed in 19-qubit
superconducting systems [14].

Nowadays exact quantum many-body scars (QMBSs) and their stability in
constrained quantum chains attract attention [15]. Quantum scars are
nonthermal eigenstates characterized by low entanglement entropy, initially
detected in systems subject to nearest neighbor Rydberg blockade, the PXP
model [16, 17, 18, 19]. This model has been introduced as a simplified
description of the quantum simulator on 51 Rydberg atoms [20]. It has been
shown that the quantum quench dynamics starting from an antiferromagnetic
(AFM) state will return to the certain initial state repeatedly after reaching
equilibrium. The continuous oscillations of physical observables such as
domain-wall density show that the system exhibits a new type of non-
thermalization behavior and the peculiar behavior was dubbed as quantum many-
body scars (QMBSs). If the initial state is prepared in other generic states,
the atomic system will thermalize as expected. In addition to the Rydberg
atomic system, QMBSs were also observed in 1D dipolar Bose gas [21].

The ideas of quantum scar state were borrowed from the classic chaotic scar
phenomenon, which is characterized by a high probability density distribution
near a specific classical periodic orbit. Quantum scars were originally
proposed by Heller in 1984 based on single-particle picture [22]. QMBSs are
abnormal condensations of quantum many-body wave functions on specific
eigenstates, and these eigenstates constitute quantum scars. On one hand,
because QMBSs are different from the conventional paradigm of thermalization,
Anderson localization and many-body localization, the nature of this novel
quantum phenomenon has stimulated the interest of many physicists. On the
other hand, QMBSs may represent a new way to realize coherent quantum
mechanics, which can protect the quantum dense coding processing for a long
time. Obviously, searching for non-thermal dynamics in interacting non-
integrable quantum systems has not only fundamental theoretical significance
in condensed matter theory, but also has great practical significance in
quantum information processing.

In a nutshell, the scarred model is characterized by a subspace which is
decoupled from the rest of the energy spectrum and cannot be simply attributed
to a symmetry of the system. Taking the spin-$S$ system on a chain with length
$N$ as an example, the dimension of the full Hilbert space scales
exponentially as $(2S+1)^{N}$, and the dimension of the scar subspace scales
algebraically with the system size $N$ as ${\cal O}(N)$. These zero-measure
scar eigenstates allow the quantum system to maintain long-time coherent
dynamics. The underlying mechanism for the 51 Rydberg atoms finding the return
journey of a specific initial state in the Hilbert space of more than $4\times
10^{10}$ dimensions is fuzzy.

Ergodicity breaking in such systems can often be attributed to the presence of
symmetries (hidden, emergent, or explicit) that preclude the establishment of
a global equilibrium state [23, 24]. The existence of quantum scars may be
revelent to emergent integrability of Hamiltonian [25], the algebraic
structure of dynamic symmetry [26], while some researches rule out the
integrability [27].

A flurry of subsequent theoretical works have addressed the puzzle posed by
the experiment: the nature and origin of this new regime of ergodicity
breaking, intermediate between thermalization and strong ergodicity breaking.
Notably, an extensive research on weak ergodicity breaking has been performed
in various quantum many-body systems, such as extended Ising model [28],
integer spin XY model [29], Affleck-Kennedy-Lieb-Tasaki (AKLT) model [30, 31]
and the Hubbard model [32, 33, 34].

Kitaev models have attracted much attention currently as they provide deeper
understanding of the spin-orbital physics in transition metal oxides [35]. The
search for their phase diagrams is motivated by a topological quantum spin
liquid (QSL) ground state and Majorana excitations emerging from frustrated
Kitaev model [36]. This research has developed rapidly after a seminal paper
of Jackeli and Khaliullin [37] was published. They proposed that the QSL might
be realized in Mott insulators with strong spin-orbit coupling, opening a new
route to seek for the Kitaev QSLs.

In recent years it was recognized that $S=1$ variant could be designed by
considering strong Hund’s coupling among two electrons in $e_{g}$ orbitals and
strong spin-orbit coupling at anion sites [38], and this innovative concept
sparks an intensive theoretical studies. The 1D version of the Kitaev model
can be considered as a limiting case with vanishing interactions along $z$
bonds of hexagonal lattice. The XX- and YY-type nearest-neighbor Ising
interactions toggle sequentially between odd and even bonds in the 1D Kitaev
chain. Despite severe simplification, the Kitaev chain shares some features of
the honeycomb model.

In this paper, we study the spin-1 Kitaev chain, which is nonintegrable. We
show that this model after increasing spin to $S=1$ surprisingly harbors an
extensive set of anomalous scarred eigenstates at finite energy density that
exhibit subextensive entanglement entropy. These scarred states survive
certain Heisenberg perturbation. Our results thus firmly establish the
existence of QMBSs in the 1D spin-1 Kitaev model. The entanglement entropy of
a subregion $A$ in an eigenstate $\alpha$ is, ${\cal S}^{\alpha}_{A}=-{\rm
Tr}\left\\{\rho_{A}^{\alpha}\ln\rho_{A}^{\alpha}\right\\}$, and ETH-obeying
states have extensive volume-law entanglement entropy, ${\cal
S}_{A}^{\alpha}\propto V$. Thus, to show that the quantum states violate the
ETH, we need only to show that their entanglement entropy is subextensive.

The purpose of this paper is twofold: First, we would like to provide evidence
that scarred eigenstates indeed appear naturally in the spin-1 Kitaev model as
the PXP model is embedded in one of subspaces. Second, quantum scars in many-
body Hamiltonian relevant for electronic systems are also of interest. Thereby
we are guided by the idea that studying a wide variety of interacting systems
with exact scar states and their stability to perturbations would be
beneficial for the general understanding.

##  II Spin-1 Kitaev Chain

In the present paper, we focus on a spin-1 Kitaev chain,

$\displaystyle\hat{H}_{K}$ $\displaystyle=$ $\displaystyle
K\sum^{N/2}_{j=1}\left(S^{x}_{2j-1}S^{x}_{2j}+S^{y}_{2j}S^{y}_{2j+1}\right),$
(1)

where spin operators $\\{S_{j}^{a}\\}$ (with $a$=$\\{x,y,z\\}$) are the spin-1
operators at site $j$, for a chain of $N$ sites. The spin operators obey the
SU(2) algebra, $[S_{i}^{a},S_{j}^{b}]=i\delta_{ij}\epsilon_{abc}S_{j}^{c}$,
with the totally antisymmetric tensor $\epsilon_{abc}$ and $({\bf
S}_{j})^{2}\\!=S(S+1)\\!=2$. The spin-1 Kitaev model in Eq. (1) breaks the
global spin rotation SU(2) symmetry, while still obeys the time reversal
symmetry ${\cal T}$, ($S_{j}^{x,y,z}\to-S_{j}^{x,y,z}$) and the spatial
inversion symmetry ${\cal I}$, ($j\to N-j+1$, $S_{j}^{x,y,z}\to
S_{N+1-j}^{x,y,z}$). The sign of $K$ in Eq. (1) is irrelevant as far as a
rotation $U_{z}=\prod_{j}\Sigma_{2j}^{z}$ at each even lattice site by an
angle $\pi$ about the $z$-axis will reverse the sign.

While the sign of the Kitaev interactions is still under debate, with
conflicting results from theoretical and experimental studies, RuCl3 was
deemed as a promising candidate for the Kitaev QSL with spin $S=\frac{1}{2}$
[39]. For the present model Eq. (1) we shall investigate whether QSL is also
stable in certain regime of parameters. Previous neutron scattering studies
performed to complement density functional theory (DFT) suggest that the
Kitaev interaction may be ferromagnetic (FM) or AFM. A recent experiment on
$\alpha$-RuCl3 by measuring azimuthal dependence [40] suggests that the Kitaev
interaction is here FM.

It is convenient to use a representation for $S=1$ given by the set of
orthonormal states $\\{|-\rangle,|0\rangle,|+\rangle\\}$, where

$\displaystyle|-\rangle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}(|-1\rangle-|1\rangle),$
$\displaystyle|+\rangle\equiv$
$\displaystyle\frac{i}{\sqrt{2}}(|-1\rangle+|1\rangle),$ (2)

and $|m\rangle$ is an eigenstate of the spin operator $S^{z}_{i}$ to an
eigenvalue $m=-1,0,1$, i.e., it stands for the state $|1,m\rangle$.
Furthermore, spin-1 operators can be written as $S^{a}_{bc}=i\epsilon_{abc}$.
Explicitly, spin operators $\\{S^{x},S^{y},S^{z}\\}$ for spin $S=1$ may be
represented by matrices

$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\ 0&i&0\\\
\end{array}\right),\quad\left(\begin{array}[]{ccc}0&0&i\\\ 0&0&0\\\ -i&0&0\\\
\end{array}\right),\quad\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\ 0&0&0\\\
\end{array}\right).$ (12)

The spin components define site parity operators $\Sigma^{a}_{j}=e^{i\pi
S_{j}^{a}}$, i.e., $\\{\Sigma^{x},\Sigma^{y},\Sigma^{z}\\}$ are given by the
matrices:

$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&-1\end{array}\right),\;\left(\begin{array}[]{ccc}-1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right),\;\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\
0&0&1\end{array}\right).$ (22)

From the form of these operators we see that the eigenvalues of operators
$\Sigma^{a}_{j}$ are $\pm 1$, but $-1$ is doubly degenerate. The Hamiltonian
in Eq. (1) respects the dihedral group $D_{2}$ for a global discrete symmetry
with a rotation by an angle $\pi$ about the $\\{x,y,z\\}$ axes, i.e.,
$\prod_{j}\Sigma_{j}^{a}$. One finds that all $\Sigma^{a}_{j}$ matrices
commute with each other. In addition, $\Sigma^{a}_{j}$ commutes with
$S_{j}^{a}$ but anticommutes with $S_{j}^{b}$ ($a\\!\neq\\!b$), i.e.,
$\\{\Sigma^{a}_{j},S_{j}^{b}\\}=\\{\exp(i\pi S_{j}^{a}),S_{j}^{b}\\}=0$.

In terms of the ladder operators $S_{j}^{\pm}\equiv S_{j}^{x}\pm iS_{j}^{y}$,
one finds that

$\displaystyle[S_{j}^{+},S_{j}^{-}]$ $\displaystyle=$ $\displaystyle
2S_{j}^{z},\quad[S_{j}^{z},S_{j}^{\pm}]=\pm S_{j}^{\pm},$
$\displaystyle\\{S_{j}^{+},S_{j}^{-}\\}$ $\displaystyle=$ $\displaystyle
2\left(2-(S_{j}^{z})^{2}\right).$ (23)

The Ising terms in Eq. (1) change the total pseudospin-$z$ at both $x$-link
$(2j$-$1,2j)$ and $y$-link $(2j,2j$+$1)$ by either 0 or $\pm 2$. Thus the bond
parity operators on odd/even bonds,

$\displaystyle\hat{W}_{2j-1}=\Sigma^{y}_{2j-1}\Sigma^{y}_{2j},~{}~{}{\rm
and}~{}~{}\hat{W}_{2j}=\Sigma^{x}_{2j}\Sigma^{x}_{2j+1},$ (24)

define the invariants of the Hamiltonian in Eq. (1). The different forms on
odd and even bonds in Eq. (24) can be cured by a unitary transformation on the
even sites,

$S^{x}_{2j}\to S^{y}_{2j},~{}~{}S^{y}_{2j}\to S^{x}_{2j},~{}~{}{\rm
and}~{}~{}S^{z}_{2j}\to-S^{z}_{2j}.$ (25)

In the transformed frame, the Hamiltonian in Eq. (1) then takes a
translationally invariant form,

$\displaystyle\tilde{H}_{\rm K}=K\sum_{j=1}^{N}~{}S^{x}_{j}S^{y}_{j+1},$ (26)

and the bond parity operators on odd/even bonds then simultaneously take a
universal convenient form,

$\displaystyle\tilde{W}_{j}=\Sigma^{y}_{j}\Sigma^{x}_{j+1}.$ (27)

The eigenvalues of $\tilde{W}_{j}$ are $w_{j}=\pm 1$. One observes from Eq.
(22) that $\Sigma_{j}^{z}$= $\Sigma_{j}^{x}\Sigma_{j}^{y}$, implying two of
them are independent, e.g. $\Sigma_{j}^{x}$ and $\Sigma_{j}^{y}$. The
$\mathbb{Z}_{2}$-valued conserved quantity $\tilde{W}_{j}$ implies that the
Hilbert space can be decomposed into $2^{N}$ sectors of unequal sizes for
spin-1 chain.

The projector at a subspace with a given set of invariants,
$\\{w_{1},w_{2},\cdots,w_{N}\\}$, can be constructed as
$\hat{Q}({w_{1},w_{2},\cdots,w_{N}})=\prod_{j}(1+w_{j}\tilde{W}_{j})/2$. It
acts on a random state $|\psi_{\rm rand}\rangle$ and the dimension $\cal{D}$
can be calculated by using the technique of transfer matrix. The ground state
of $\tilde{H}_{\rm K}$ with periodic boundary conditions (PBCs) lies in the
sector with defect-free subspace [41]. Different from the spin-1/2
counterpart, where the uniform subspace, with parity being either $w_{j}=+1$
or $w_{j}=-1$ for all $j$, are all equivalent up to a unitary transformation,
but the dimensions of these sectors are nonidentical. The dimension of the
sector with all $w_{j}=-1$ is equal to ${\cal D}(\\{-1,-1,\cdots,-1\\})=2$,
which appears to be the smallest subspace, as is evidenced in Fig. 1.

The two-dimensional subspace with all $w_{j}=-1$ can be rewritten as
$|---\cdots--\rangle$ and $|+++\cdots++\rangle$ defined in the rotated bases
in Eq. (25). On the contrary, the dimension $\cal{D}$ for the subspace with
parity $w_{j}=+1$ for all $j$ is proven to be the Lucas number, i.e., ${\cal
D}(\\{1,1,\cdots,1\\})=F_{N-1}+F_{N+1}$, where $F_{n}$ is the $n$-th Fibonacci
number. More precisely, ${\cal D}(\\{1,1,\cdots,1\\})=g^{N}+g^{-N}$, where
$g=(1+\sqrt{5})/2\approx 1.618$ is the golden ratio. It is interesting to
observe that such fractal dimension-based subspace is the largest among all
the sectors, see Fig. 1.

Figure 1: The dimension of subspaces ${\cal D}$ for given parity
$(w_{1},w_{2},\cdots,w_{N})$ as a function of the defect number ${\cal
N}=N-\sum_{j=1}^{N}w_{j}$, for a chain with $N=12$ sites. The red dot denotes
the maximum (minimum) of $\cal{D}$ corresponding to the uniform subspace with
parity $w_{j}=+1$ ($w_{j}=-1$) for all $j$. The dashed line is a guide for the
eye.

The constrained Hilbert space without any adjacent spin-up atoms makes the
system identical to that of chains of spin-1/2 atoms. Hence, the Hilbert space
of a given sector can be mapped into the Hilbert space of a spin-1/2 chain
with some states excluded. Accepting the notation that the
$|{\uparrow}\rangle$ state represents the state $|-\rangle$ (head), and the
$|{\downarrow}\rangle$ state stands for either the state $|0\rangle$ (empty)
or the state $|+\rangle$ (tail), the system can be mapped to a single qubit-
flip model with nearest-neighbor exclusion represented by the effective
Hamiltonian in the sector with $w_{j}=+1$ for all $j$ [42]:

${\cal H}_{\\{1,1,\cdots,1\\}}~{}=~{}\sum_{j=1}^{N}~{}\tilde{X}_{j}.$ (28)

Here $\tilde{X}_{j}\equiv P_{j-1}X_{j}P_{j+1}$, where Pauli operators
$X_{j}=\left(\,|{\uparrow}\rangle\langle{\downarrow}|+|{\downarrow}\rangle\langle{\uparrow}|\,\right)$,
$Z_{j}=\left(\,|({\uparrow}\rangle\langle{\uparrow}|-|{\downarrow}\rangle\langle{\downarrow}|\,\right)$,
and the projectors
$P_{j}=|{\downarrow}\rangle\langle{\downarrow}|=(1-Z_{j})/{2}$, ensure that
the nearby atoms are not simultaneously in the excited state. The
corresponding dimension of the constrained Hilbert space is exactly equal to
${\cal D}(\\{1,1,\cdots,1\\})$ through an isomorphic mapping. Note that the
PXP model is an effective model of biaxial Ising model with both transverse
and longitudinal fields, where the low field $h_{x}\ll J_{z}$, and the
saturation field $h_{z}=2J_{z}$ are present. In spite of its rather friendly
form, the Hamiltonian Eq. (28) is nonintegrable due to the low-energy
constraint imposed on the Hilbert space.

Figure 2: Characteristic quantum features of the spin-1/2 PXP model: (a) The
fidelity $F(t)$ for spin-1 Kitaev model starting from the initial
$|\tilde{\mathbb{Z}}_{2}\rangle$ ($|\mathbb{Z}_{2}\rangle$) states with
$N=18$; (b) Eigenstate overlap with the Néel $\mathbb{Z}_{2}$ state for
spin-1/2 PXP model (dots) and the $w_{j}=1$ for $(j=1,2,\cdots,N)$ subspace of
spin-1 Kitaev model (squares) with $N=16$. Color scale on the right to panel
(b) indicates the density of data points, with lighter regions being more
dense.

It has been noted that PXP model specified by the Hamiltonian (28) will
exhibit QMBSs, which can be experimentally prepared and probed using a global
quench. Experiment [20] and numerical simulations on small systems [43]
revealed that the relaxation under unitary dynamics strongly depends on the
initial state of the system. It is found that special eigenstates have
anomalously high overlaps with certain product states.

The polarized state with all spins in $|{\downarrow}\rangle$ state,
$|\varnothing\rangle=\prod_{j}P_{j}|\psi_{\rm rand}\rangle$, satisfies ${\cal
H}_{\\{1,1,\cdots,1\\}}|\varnothing\rangle=0$, which allows one to identify a
dominant subset of special states in the PXP model. We initialize the system
at time $t=0$ in the state $|\psi(0)\rangle\equiv|\mathbb{Z}_{k}\rangle$,
namely,

$|\mathbb{Z}_{k}\rangle=\ldots\tilde{X}_{k}\ldots\tilde{X}_{2k}\ldots|\varnothing\rangle,$
(29)

and then follow the evolution of the initial state with the PXP Hamiltonian,
$|\psi(t)\rangle=\exp\left(-i{\cal
H}_{\\{1,1,\cdots,1\\}}t\right)|\psi(0)\rangle$. The evolution is determined
by the decomposition of $|\psi(0)\rangle$ in terms of the eigenstates of
${\cal H}_{\\{1,1,\cdots,1\\}}$. The initial states with atoms in all spin-
down state $|\mathbb{Z}_{1}\rangle\equiv|\varnothing\rangle$, or
$|\mathbb{Z}_{4}\rangle$ show fast relaxation and no revivals, characteristic
of thermalizing systems. Remarkably, the quantum quench from either period-2
state ($|\mathbb{Z}_{2}\rangle$) or the period-3 state
($|\mathbb{Z}_{3}\rangle$) will surprisingly give rise to coherent
oscillations, which can be observed in the dynamics by measuring the
expectation values of certain local observables and the quantum fidelity,

$\displaystyle F(t)=|\langle\mathbb{Z}_{k}|\exp(-iHt)|\mathbb{Z}_{k}\rangle|.$
(30)

The observed oscillations and the apparent non-ergodic dynamics are attributed
to the existence of QMBSs [44].

Intuitively, the empty state $|\varnothing\rangle$ corresponds to the state
$|0000\ldots 00\rangle$, which can be inferred from Eq. (22) that the
configuration $|0000\ldots 00\rangle$ resides in this subspace with uniform
$\mathbb{Z}_{2}$ invariants, i.e., $w_{j}=1$ for all $j$. In this context, the
akin states $|\tilde{\mathbb{Z}}_{k}\rangle$ of spin-1 Kitaev model Eq. (26)
as product states in Eq. (29), can be reconciled by acting

$|\tilde{\mathbb{Z}}_{k}\rangle=\ldots S^{x}_{k}S^{y}_{k+1}\ldots
S^{x}_{2k}S^{y}_{2k+1}\ldots|0000\cdots 00\rangle.$ (31)

Then the corresponding spin-1 states are given by:

$\displaystyle|\tilde{\mathbb{Z}}_{2}\rangle$ $\displaystyle=$
$\displaystyle|-+-+\cdots-+\rangle,$ (32)
$\displaystyle|\tilde{\mathbb{Z}}_{3}\rangle$ $\displaystyle=$
$\displaystyle|+-0+-0\cdots+-0\rangle,$ (33)
$\displaystyle|\tilde{\mathbb{Z}}_{4}\rangle$ $\displaystyle=$
$\displaystyle|+-00+-00\cdots+-00\rangle.$ (34)

It is clear that the translated states,
$|\tilde{\mathbb{Z}}^{\prime}_{k}\rangle=(T_{i\to
i+1})^{j}\,|\mathbb{Z}_{k}\rangle$ ($j=1,2\cdots,k-1)$, also belong to the
QMBSs, where $T_{i\to i+1}$ denotes the translation by one lattice site, i.e.,
$i\to i+1$. As is shown in Fig. 2(a), the revivals of the quantum fidelity for
spin-1 Kitaev model (26) [spin-1/2 PXP model (28)] starting from the initial
Néel state $|\tilde{\mathbb{Z}}_{2}\rangle$ ($|\mathbb{Z}_{2}\rangle$) are
indistinguishable, suggesting the existence of ETH-violating QMBSs.

Figure 3: Quantum features of the 24-site $S=1$ Kitaev chain: (a) the fidelity
$F$ and (b) the entanglement ${\cal S}$. The results correspond to starting
from the initial $|\tilde{\mathbb{Z}}_{k}\rangle$ ($k$=2,3,4) states. The bond
dimension is set as $\chi=300$.

Figure 3 shows that the fidelity $F$ and entanglement ${\cal S}$ with its
dynamics for different initial states are interrelated in a subtle way. One
can observe the coherent oscillations of fidelity that persist for long times
for quenches from initial product states $|\tilde{\mathbb{Z}}_{2}\rangle$ and
$|\tilde{\mathbb{Z}}_{3}\rangle$ [18], with different revival periods, and the
entanglement entropy ${\cal S}$, see Fig. 3(b), is gradually growing with
time. The weak oscillations of the entanglement are clearly visible by
eliminating the linear growth, featuring the many-body revivals [19].

By contrast, the fast damping amplitude of fidelity for quench from
$|\tilde{\mathbb{Z}}_{4}\rangle$ state indicates the negligible overlaps
between scarred eigenstates and the $|\tilde{\mathbb{Z}}_{4}\rangle$ product
state. Meanwhile, the entanglement undergoes an extremely fast growth. Note
that the bipartite entanglement approaching the maximum entanglement entropy
$\log_{2}(\chi d)$ implies a large accumulated error and thus an inevitable
breakdown of the density-matrix renormalization group (DMRG) method [45].

Furthermore, it is striking to find that the energy spectra and eigenstates
overlap with the Néel state of Eq. (28) also coincide with those of Eq. (26),
except the highly degenerate states at $E=0$, implying that a precise relation
exists between the eigenstate of spin-1 Kitaev model (26) and the celebrated
spin-1/2 PXP model (28). The zero modes of the PXP model at the center of the
spectrum are pinned by the particle-hole symmetry, i.e., $\\{{\cal C},H_{\rm
PXP}\\}=0$, with ${\cal C}=\prod_{j=1}^{N}Z_{j}$ [46]. The states with $E=0$
can be classified as eigenstates of ${\cal C}$. An exponential growth of
protected many-body zero-energy modes with system size arise as a consequence
of intertwining of spectral-reflection ${\cal C}$ symmetries and relevant
point-group symmetries [47].

Among the above macroscopic number of states, two exact scar states for PBC
have been identified in terms of matrix product states with a finite bond
dimension [46], which shows that these exact scar states have constant
entanglement. However, the bipartite entropy of the eigenstates of spin-1/2
PXP model in Fig. 4(a) and the $w_{j}=1$ $(j=1,2,\cdots,N)$ subspace of spin-1
Kitaev model in Fig. 4(b) quantitatively exhibit similar features but are not
exactly equivalent. Notably, the different choice of reference bases will
modify the values of bipartite entanglement entropy.

Figure 4: Bipartite entropy of the eigenstates for: (a) spin-1/2 PXP model and
(b) spin-1 Kitaev model, within $w_{i}=1$ $(i=1,2,\cdots,N)$ subspace. Color
scale indicates the density of data points, with lighter regions being more
dense. Here the chain has $N=16$ sites.

The explicit technical insights can be accessible by an exact solution of the
two-site model, which are presented in the Appendix A. The scarred eigenstates
are characterized by the low bipartite entanglement entropy ${\cal S}_{A}$,
and they become increasingly decoupled from the thermal bulk states as the
system size $N$ increases. The primary $\mathbb{Z}_{2}$ scar states can be
constructed from the excitations on top of the exact scar states at $E=0$. The
equally-spaced towers of QMBSs in the spectrum are closely related to perfect
revivals, but these characteristics of particular interest are currently
limited by small system sizes. Yet, they are experimentally relevant.

##  III Stability of QMBS against Heisenberg interactions

It is appealing to consider the stability of the QMBS to a nonzero
perturbation, which could possibly preserve or undermine the conservation of
local quantities characteristic of the Kitaev model. Considering atomic states
in the $t_{2g}$ manifold are immersed under the crystalline electric field
with the strong spin-orbit coupling, other interactions, such as the
Heisenberg exchange, and Dzyaloshinskii-Moriya interaction, coexist with the
Kitaev interactions [48]. In addition to the quadratic exchange interactions,
the uniaxial single-ion anisotropy (SIA) due to crystal-field effects [49] and
further-neighbor interactions are also ubiquitous [50]. In reality, these
interactions play an important role in establishing non-trivial electronic
correlations for low-dimensional models with higher spin $S>\frac{1}{2}$.

We are interested in effects induced by perturbations that leave the
intermediate symmetries intact. The SIA,

$\displaystyle\hat{H}_{\rm D}=D\sum_{j}\left(S_{j}^{z}\right)^{2},$ (35)

is one of them and leads to a quantum phase transition between a topologically
trivial phase and a nontrivial phase in spin-1 AFM Heisenberg chain as
predicted by Haldane [51, 52]. Simultaneously, the SIA leads to a low-entropy
spin Mott insulator in the FM analogy [53, 54]. It is easy to find that Eq.
(35) is invariant under the rotation (25) and commutes with local parity
operators defined in Eq. (27). We can observe in Fig. 5 that periodic
oscillations in the order parameter [55],

$O_{1}=\left\langle\left(S_{1}^{+}\right)^{2}\right\rangle,$ (36)

and the quench fidelity persist for long times when starting from
$|\tilde{\mathbb{Z}}_{2}\rangle$. As $D$ increases, the oscillation period
gets shorter.

Figure 5: Time evolution of an $18$-site Kitaev chain starting from the
initial $|\tilde{\mathbb{Z}}_{2}\rangle$ state: (a) the order parameter (36),
and (b) the quench fidelity (30). The data are presented for increasing
strength of single-ion anisotropy (35), $D=0.0$, 0.2, 0.4. see legend.

We are also interested in effects under a perturbation
$V\propto(S_{j}^{x}S_{j+1}^{z}S_{j+2}^{y}$) that obeys the same intermediate
$\mathbb{Z}_{2}$ symmetries of Eq. (26). The origin of such three-site XZY-
type interactions is explained in Appendix B. One can observe that the order
parameter $O_{1}$ and the quench fidelity $F(t)$ show quite similar
oscillations as those of SIA. The state coherence sustains a long time for a
moderate perturbation, and the revival will degrade quantitatively with
increasing amplitude of multispin perturbations, while the oscillation period
is almost unchanged.

However, it is gradually recognized that isotropic Heisenberg interactions
commonly exist in solid state materials and perturbs the systems with
frustrated interactions. One frequently considered example is the compass
model where Heisenberg interactions are responsible for the onset of possible
long range 2D order in the ground state but simultaneously excited states
preserve their unique nature which may be used for information storage. Here
the nematic order which persists in excited states may be used for correcting
the faults along the computations [56].

Another example of robust structure of excited states in presence of
perturbation is encountered in the Kitaev-Heisenberg model for $S=1/2$ spin
which has been intensely investigated and several phases with broken symmetry
have been found [39, 57, 58]. A similar situation was reported as well for
spin-1 Kitaev honeycomb model in candidate materials, such as honeycomb Ni
oxides with heavy elements of Bi and Sb, where Kitaev interaction is
accompanied by a finite FM Heisenberg interaction. In the zero-field limit,
the Kitaev QSL is destabilized when $J/K>0.08$ [59]. We thus assume that such
interactions are of Heisenberg type,

$\displaystyle\hat{H}_{\rm J}=J\sum^{N}_{i=1}{\bf S}_{i}\cdot{\bf S}_{i+1},$
(37)

where $J$ stands for the Heisenberg exchange coupling. After the rotation
(25), the perturbation by Heisenberg interaction can be rewritten as

$\displaystyle\tilde{H}_{\rm
J}=J\sum^{N}_{j=1}\left(S_{j}^{x}S_{j+1}^{y}+S_{j}^{y}S_{j+1}^{x}-S_{j}^{z}S_{j+1}^{z}\right).\quad$
(38)

The full Hamiltonian takes an anisotropic translational form:

$\displaystyle\tilde{H}_{\rm
KJ}\\!=\sum^{N}_{j=1}\\!\left\\{(K\\!+\\!J)S_{j}^{x}S_{j+1}^{y}\\!+JS_{j}^{y}S_{j+1}^{x}\\!-JS_{j}^{z}S_{j+1}^{z}\right\\}\\!.\quad$
(39)

A direct consequence is the additional Heisenberg interactions spoil the
$\mathbb{Z}_{2}$ symmetry of Eq. (1) associated with each bond. It is worth
noting that the energy levels of low-lying excited states cross at $J=0$. The
first excited state of $\tilde{H}_{\rm K}$ is $N$-fold degenerate,
corresponding to one $w_{j}=-1$ defect in the sector with all other $w_{j}=1$,
such that either $|00\\!+\\!0\ldots 00\rangle$ or $|00\\!-\\!0\ldots
00\rangle$ state occurs. A narrow gapped Kitaev phase is capable of sustaining
the perturbations of Heisenberg interactions.

The competition between spin-1 AFM Kitaev chain and isotropic Heisenberg
interactions has been analyzed for $K=1$ [60], while a complete phase diagram,
to the best of our knowledge, is elusive and deserves still a careful
investigation. To incorporate the phase diagram of spin-$1$ Kitaev-Heisenberg
model with $K=1$ [60], the ground state phase diagram with $K=-1$ is depicted
using the DMRG and exact diagonalization methods. In the DMRG simulations, we
keep up to $\chi=500$ bond states during the procedure of basis truncation and
the number of sweeps is $n=30$. These conditions guarantee that the simulation
is converged sufficiently fast and the truncation error is smaller than
$10^{-7}$.

Figure 6: The correlations between site 1 and $(r+1)$ for increasing distance
$r$ and for: (a) $J=-1.0$, (b) $J=0.03$, (c) $J=0.2$, and (d) $J=1.0$. Here we
use PBCs for the chain with $N=200$ sites.

One motivation for the study of AFM spin-1 Heisenberg chains ($K=0$, $J>0$) is
that the model hosts a symmetry-protected topological ground state with a
finite Haldane gap [61], a reminiscent of AKLT state [62]. It has been known
for decades that the indicator of the Haldane phase is beyond any local
symmetry-breaking order parameters. Instead, it can be characterized by the
exponential decay of two-spin correlation function and nonlocal string order
parameters [63, 64].

As for the coexistence the Kitaev interaction and the Heisenberg interactions,
the spatial inversion symmetry, the time-reversal symmetry, and the dihedral
$D_{2}$ symmetry are preserved, which protects the Haldane phase [65, 66].
Another specific case occurs with $J\equiv-K/2$. Then Eq. (39) can be recast
into an isotropic form,

$\displaystyle\tilde{H}_{\rm KJ}$ $\displaystyle=$
$\displaystyle\frac{1}{2}K\sum^{N}_{j=1}\left(\tilde{S}_{j}^{x}\tilde{S}_{j+1}^{x}+\tilde{S}_{j}^{y}\tilde{S}_{j+1}^{y}+\tilde{S}_{j}^{z}\tilde{S}_{j+1}^{z}\right),$
(40)

through the following spin rotation

$\displaystyle\tilde{S}_{j}^{x}$ $\displaystyle=$ $\displaystyle\cos\left(\pi
j/2\right)S_{j}^{x}-i\sin\left(\pi j/2\right)S_{j}^{y},$
$\displaystyle\tilde{S}_{j}^{y}$ $\displaystyle=$ $\displaystyle\cos\left(\pi
j/2\right)S_{j}^{y}+i\sin\left(\pi j/2\right)S_{j}^{x}.$ (41)

In this case, the ground state of Eq. (40) possesses one Goldstone mode, and
its character depends on the sign of $K$. The Goldstone mode has a quadratic
dispersion for FM Heisenberg model with $K=-1$, while it is gapped for AFM
Heisenberg model with $K=1$.

For a generic case, a well defined order parameter is a vital ingredient for
characterizing the nature of phases. In order to identify the region of the
Kitaev phase and the Haldane phase, we calculate the nonlocal correlator

$C^{a}_{i,j}=\left\langle
S^{a}_{i}\exp\left(i\theta\sum_{l=i+1}^{j-1}S_{l}^{a}\right)S^{a}_{j}\right\rangle,~{}~{}a=x,y,z.$
(42)

For $\theta=\pi$, Eq. (42) becomes the den Nijs-Rommelse string order
parameter, whose limiting value $O^{a}_{s}={\rm
lim}_{|i-j|\rightarrow\infty}\left(-C^{a}_{i,j}\right)$ reveals the hidden
symmetry breaking [67, 68]. For $\theta=0$, Eq. (42) reduces to two-point
correlations.

Figure 7: Two-point correlation $C^{a}$ (42) and string order parameter
$O^{a}$ ($a=x,z$) of 1D pristine spin-1 Kitaev-Heisenberg model for
$J\in(-1,1)$ between sites 1 and $N/2$ for $N=80$. With increasing $J$, the
order in the ground state changes from FMx through QSL and LLRR phase to the
Haldane phase.

When $J$ varies for $K=-1$, the competing correlations will trigger
miscellaneous phase transitions. For $J\simeq-1$, the $C^{x}_{i,j}$
correlations dominate, suggesting that the ground state is FMx, see Fig. 6(a).
For $J\approx 0$, the pure spin-1 Kitaev chain hosts nearest neighbor AFM
orders due to the $\mathbb{Z}_{2}$ symmetry. The especially short-range
correlations are demonstrated for $J=0.03$. The nearest neighbor spins favor
here the FM alignment as the next nearest neighbor interactions are FM, this
state is shown in Fig. 6(b). When $J$ increases further, the dominating
$y$-component correlations have a negative sign on odd bonds and a positive
sign on even bonds. These correlations indicate that spin order develops into
the left-left-right-right (LLRR) phase, as is shown in Fig. 6(c). Figure 6(d)
shows that the $C^{x}_{i,j}$ correlations dominate over $C^{y}_{i,j}$ and
$C^{z}_{i,j}$ when $J\simeq 1$.

To clarify the uniaxial order, the two-point correlation $C^{a}$ and the
string order parameter $O^{a}$ ($a=x$, $z$) between sites 1 and $N/2$ on a
$N=80$ site chain are shown for increasing $J$ in Fig. 7. One can recognize a
finite $C^{x}_{1,N/2}$ that characterizes the FMx phase for $J<-0.08$, while
all correlations vanish in the interval $-0.08<J<0.08$. Next $C^{x}_{1,N/2}<0$
for $0.08<J<0.50$, while finally $O^{z}_{1,N/2}\simeq 0.9$ is finite for
$J>0.5$ and defines the phase boundary of the Haldane phase.

##  IV Phase diagram of the Kitaev-Heisenberg chain for $S=1$

The parameters $\\{K,J\\}$ for the Kitaev and Heisenberg exchange couplings
can be set as

$K\equiv\sin(\phi),\quad\quad J\equiv\cos(\phi),$ (43)

where the angle $\phi$ $\in[0,2\pi)$ parametrizes the Hamiltonian (39). To
this end, the complete phase diagram of the 1D spin-1 Kitaev-Heisenberg model
as a function of $\phi$ is displayed in Fig. 8, where the vertical
(horizontal) axis is the Kitaev $K$ (Heisenberg $J$) exchange coupling. For
both cases with either $K>0$ or $K<0$, the phase diagram consists of four
different phases, i.e., the FM phase, the Kitaev QSL phase, the LLRR phase,
and the Haldane phase. To some extent, the symmetry between the regimes of
$K>0$ and $K<0$ is restored.

Figure 8: Ground state phase diagram of the 1D Kitaev-Heisenberg model for
$S=1$. The vertical (horizontal) axis stands for the Kitaev $K$ (Heisenberg
$J$) exchange coupling, as given in Eq. (43). Here $\phi\in[0,2\pi)$ is the
angle which determines the exchange constants, see Eq. (43).

It is interesting to note that a generalized Kennedy-Tasaki transformation,
$U_{\mathrm{KT}}=\prod_{j<k}\exp{\left(i\pi S^{y}_{j}S^{x}_{k}\right)}$ [69,
70], can be realized as the reciprocal transformation between the string order
and the local FM order. In this regard, Eqs. (1) and (37) are transformed into
equivalent ones with a minus sign standing with short-range interactions,
where the nonlocal string observable $O^{a}_{i,j}(\hat{H})$ is responsible for
the two-point correlations $C^{a}_{i,j}(\tilde{H})$, with $a=x,z$ of the
transformed Hamiltonian,

$\displaystyle\tilde{H}^{\prime}_{\rm KJ}$ $\displaystyle=$
$\displaystyle-K\sum^{N/2}_{j=1}\left(S^{x}_{2j-1}S^{x}_{2j}+S^{y}_{2j}S^{y}_{2j+1}\right)$
(44) $\displaystyle-$ $\displaystyle
J\sum_{j=1}^{N}\left(S_{j}^{x}S^{x}_{j+1}+S_{j}^{y}S^{y}_{j+1}+\tilde{W}_{j}S_{j}^{z}S^{z}_{j+1}\right).$

This suggests a likely relationship $\tilde{H}(-K,-J)\simeq\hat{H}(K,J)$.

Notably, the coupled $\mathbb{Z}_{2}$ gauge fields $\\{\tilde{W}_{j}\\}$ are
not conserved in the transformed pristine Hamiltonian. The duality becomes
exact when the strength of Heisenberg interaction $J$ is tiny
($\phi\approx\pm\pi/2$), which means that the parameters are within the
boundaries of the Kitaev QSL phases. However, Fig. 8 clearly shows that such a
relationship does not hold for moderate values of $J$ owing to quantum
fluctuations of Ising gauge fields. The system undergoes a second-order
quantum phase transition from the FM phase to the LLRR phase at
$\phi=0.8280\pi$ (i.e., for $K=1$, $J=-0.6$), while another transition from
the LLRR phase to the Haldane phase occurs at $\phi=1.8524\pi$ (i.e., for
$K=-1$, $J=0.5$). In fact, there is no phase transition at $\phi=0$ ($K=0$,
$J=1$), where the string observables $O^{a}_{i,j}$ are balanced in both $x$\-
and $z$-directions, corresponding to the restoration of hidden
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry breaking.

Figure 9: Quantum characteristics of the $S=1$ Kitaev-Heisenberg chain of
$N=36$ sites, starting from $|-+-+\cdots-+\rangle$ in Eq. (39) as the initial
state for selected values of $J$: (a) the fidelity $F$ and (b) the
entanglement entropy ${\cal S}$. The inset in (a) shows the scaling of the
first revival peak $\delta F_{1}$ versus $1/N$ for $J=0.0$ and $J=0.04$. The
bond dimension is set as $\chi=400$.

We note that the phase diagram of the $S=1$ Kitaev-Heisenberg chain resembles
to some extent the phase diagram of the Kitaev-Heisenberg chain for
$S=\frac{1}{2}$ spin [58], and one finds the following correspondence between
phases: the Haldane phase for $S=1$ corresponds to the AFM phase for
$S=\frac{1}{2}$, and the FM phase for $S=1$ corresponds again to the FM phase
when spin is smaller. But it is remarkable that the LLRR phase corresponds to
two distinct phases when $S=\frac{1}{2}$: the zigzag phase for $K>0$ and the
stripe phase for $K<0$. Thereby the regime of the LLRR phase for $K>0$ extends
over a broader range of angle $\phi$, similar to the broader range of $\phi$
for the zigzag phase at $S=\frac{1}{2}$ [58]. In addition, both the FM and the
Haldane phase consist here of the two distinct phases, depending on whether
$K>0$ or $K<0$. This demonstrates that the quantum components of $S=1$ spins
are nonequivalent and the SU(2) symmetry of the Heisenberg term does not hold
when Kitaev interactions are finite.

Finally, we investigate the scar stability in a broader regime, which is
relevant for possible solid state applications. Then we consider scar
stability of the Kitaev model with Heisenberg perturbations, and show that
they display anomalous stability in the Kitaev phase. Figure 9 shows the
fidelity with periodic slowly-decaying revivals in the dynamics for $J=0$. In
contrast, it completely collapses for $J=0.2$ with $N=36$. The coherent
dynamics is retained in the Kitaev phase while it is lost quickly when the
phase enters other phases. Away from the Kitaev point, one can observe how the
entanglement grows rapidly with time. These results suggest that exact scars
are a generic property of Kitaev models.

##  V Summary and conclusions

In this paper, we studied the eigenstate properties of Hamiltonian relevant
for spin-orbit-coupled electronic system, where the PXP model was embedded.
Thus, the many-body quantum scars and the associated constrained dynamics can
be unveiled. This phenomenology is characterized by the fact that the dynamics
is anomalously slow provided the initial state has a non-negligible overlap
with scarred states. The ETH violations have been explicitly detected in the
uniform $w_{j}=1$ ($\forall j$) sector, where the spin-1 Kitaev model can be
rigorously mapped into a spin-$\frac{1}{2}$ PXP model. The PXP model acts here
as an archetypal model to possess quantum scars, which are introduced to
explain the slow dynamics observed by evolving a charge-density wave initial
state in the Rydberg atom chain realized in Ref. [20]. As a consequence of
nearest-neighbor Rydberg blockaded, the dimension of constrained Hilbert space
grows as $g^{N}$, where $g=(1+\sqrt{5})/2$ is the golden ratio. A few scarred
eigenstates spread throughout the spectrum, and their presence is evidenced by
sub-extensive entanglement entropy and persistent oscillation of local
observables. We have shown that there is coexistence of volume-law and area-
law entangled eigenstates throughout the spectrum.

These ETH violating eigenstates survive despite hybridization with an
exponential number of thermal eigenstates and motivated the study of their
stability against perturbation. We consider the perturbations by Heisenberg
interactions, see Eq. (44). In order to characterize the ground state
properties, we adopt both the local and nonlocal correlations, which identify
distinct phases when the Kitaev coupling $K$ and the Heisenberg coupling $J$
vary, see Fig. 8. For large negative $J$, the FM order along $z$ ($x$) axis is
favored for $K=1$ ($K=-1$). Increasing the value of $J$, the ground state
evolves from the FM state into the stripy phase, in which the spin structure
has a left-left-right-right pattern. The magnetic order vanishes when $J$
approaches the Kitaev limit. In stark contrast to the gapless ground state of
$S=\frac{1}{2}$ Kitaev chain, the gapped ground state supports a narrow Kitaev
phase in the vicinity of $J=0$, which is characterized by the extremely short-
range correlations.

As a universal feature, the system enters into the Haldane phase upon
increasing $J$. The Haldane phase maintains its topological character and
cannot evolve adiabatically to other phases, since it is protected by the
combination of the spatial inversion symmetry, the time-reversal symmetry, and
the dihedral $D_{2}$ symmetry. The continuous revivals for starting from
particular initial states remains in the disordered Kitaev phase, while the
state coherence breaks down swiftly in symmetry-breaking phases. Nevertheless,
with the help of examples of single-ion term and multispin interactions, we
demonstrate the scarred states appear to be robust against
$\mathbb{Z}_{2}$-symmetry preserving perturbations, at least for currently
experimentally accessible system-sizes. Our findings highlight that spin-1
Kitaev systems exhibit a rich variety of phenomena and thus provide a generic
non-integrable constrained quantum many-body system to study non-ergodic
behaviors, the stability under generic perturbations and quantum analogues of
the Kolmogorov-Arnold-Moser theorem.

###### Acknowledgements.

The authors appreciate very insightful discussions with Wojciech Brzezicki,
Hosho Katsura, and Zhi-Xiang Sun. This work is supported by the National
Natural Science Foundation of China (NSFC) under Grants No. 12174194, the
startup fund of Nanjing University of Aeronautics and Astronautics Grant No.
1008-YAH20006, Top-notch Academic Programs Project of Jiangsu Higher Education
Institutions (TAPP) and stable supports for basic institute research, Grant
No. 190101. A. M. Oleś kindly acknowledges support by Narodowe Centrum Nauki
(NCN, Poland) under Project No. 2016/23/B/ST3/00839 and is grateful for
support by the Alexander von Humboldt Foundation (Humboldt-Forschungspreis).

##  Appendix A Bipartite entropy of two-site spin-1/2 PXP model and spin-1
Kitaev model

Here we consider $N=2$. The Hamiltonian of spin-1/2 PXP model within the bases
in the constrained Hilbert space,

$\displaystyle\\{|{\downarrow\downarrow}\rangle,|{\downarrow\uparrow}\rangle,|{\uparrow\downarrow}\rangle\\},$
(45) $\displaystyle H=\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&0\\\ 1&0&0\\\
\end{array}\right),$ (49)

and the ones for the spin-1 Kitaev model within $(w_{i}=1$, $i=1,2,\cdots,N)$
subspace

$\displaystyle\\{|-+\rangle,|+-\rangle,|00\rangle\\},$ (50) $\displaystyle
H=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&1\\\ 1&1&0\\\ \end{array}\right).$
(54)

There is a one-to-one correspondence between Eqs. (45) and (50):

$\displaystyle|-+\rangle\Rightarrow|{\downarrow\uparrow}\rangle,|+-\rangle\Rightarrow|{\uparrow\downarrow}\rangle,|00\rangle\Rightarrow|{\downarrow\downarrow}\rangle.$
(55)

The eigenvectors of Eq. (49)

$\displaystyle|\psi_{1}\rangle$ $\displaystyle=$
$\displaystyle-0.7071\,|{\downarrow\downarrow}\rangle+0.5000\,|{\downarrow\uparrow}\rangle+0.5000\,|{\uparrow\downarrow}\rangle,\quad$
$\displaystyle|\psi_{2}\rangle$ $\displaystyle=$
$\displaystyle\;\;\,0.0000\,|{\downarrow\downarrow}\rangle-0.7071\,|{\downarrow\uparrow}\rangle+0.7071\,|{\uparrow\downarrow}\rangle,\quad$
$\displaystyle|\psi_{3}\rangle$ $\displaystyle=$
$\displaystyle\;\;\,0.7071\,|{\downarrow\downarrow}\rangle+0.5000\,|{\downarrow\uparrow}\rangle+0.5000\,|{\uparrow\downarrow}\rangle.\quad$
(56)

The entanglement spectra and von Neumann entropy

$\displaystyle\lambda_{1,3}^{(1,2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\pm\frac{\sqrt{3}}{4}\,,\quad{\rm with}\quad
S_{A}=0.3546,$ (57) $\displaystyle\lambda_{2}^{(1,2)}$ $\displaystyle=$
$\displaystyle\frac{1}{2},\frac{1}{2},\quad\quad\quad{\rm with}\quad
S_{A}=1.0.$ (58)

Similarly, the eigenvectors of Eq. (54)

$\displaystyle|\phi_{1}\rangle$ $\displaystyle=$ $\displaystyle
0.5000\,|{-+}\rangle+0.5000\,|{+-}\rangle-0.7071\,|00\rangle,$
$\displaystyle|\phi_{2}\rangle$ $\displaystyle=$ $\displaystyle
0.7071\,|{-+}\rangle-0.7071\,|{+-}\rangle+0.0000\,|00\rangle,$
$\displaystyle|\phi_{3}\rangle$ $\displaystyle=$ $\displaystyle
0.5000\,|{-+}\rangle+0.5000\,|{+-}\rangle+0.7071\,|00\rangle.$ (59)

The entanglement spectra and von Neumann entropy are:

$\displaystyle\lambda_{1,3}^{(1,2,3)}$ $\displaystyle=$
$\displaystyle\frac{1}{4},\frac{1}{4},\frac{1}{2},\quad{\rm with}\quad
S_{A}=1.5,$ (60) $\displaystyle\lambda_{2}^{(1,2,3)}$ $\displaystyle=$
$\displaystyle\frac{1}{2},\frac{1}{2},0,\,\quad{\rm with}\quad S_{A}=1.0.$
(61)

To this end, the diverse choice of bases in Eqs. (45) and (50) will lead to
multiple values of bipartite entanglement entropy.

Figure 10: Quantum properties of the Kitaev chain with $N=18$ sites starting
from the initial $|\tilde{\mathbb{Z}}_{2}\rangle$ state with XZY interactions
and for $\lambda=0.0$, 0.2, 0.4: (a) the order parameter (36), and (b) the
quench fidelity $F(t)$.

##  Appendix B The Kitaev chain under XZY-type perturbations

The energy current follows from the continuity equation [71],

$\hat{J}_{\rm
E}=\sum_{j=1}^{N}i[h_{j},h_{j+1}]=K^{2}\sum_{j=1}^{N}S_{j}^{x}S_{j+1}^{z}S_{j+2}^{y}\,.$
(62)

This operator acts on three adjacent sites and contains also the $z$ component
of spin-$1$ operators, which commutes with $\tilde{W}_{j}$. The presence of an
effective energy flow will manifest itself in the effective Hamiltonian
followed by a Lagrange multiplier $\lambda$:

$\displaystyle\tilde{H}_{\rm
K}=K\sum_{j=1}^{N}~{}S^{x}_{j}S^{y}_{j+1}-\lambda\hat{J}_{\rm E}.$ (63)

It can be verified that $[\tilde{W}_{j},\tilde{W}_{k}]=0$,
$[\tilde{W}_{j},\tilde{H}_{\rm K}]=0$. In this respect, the three-site XZY-
type interactions belong to $\mathbb{Z}_{2}$-symmetry preserved perturbations.
The numerical results for different values of $\lambda$ are shown in Fig. 10.
One finds that the oscilations of the order parameter (36) are more damped and
the fidelity $F(t)$ has a richer structure at larger time $t$ when $\lambda$
increases.

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