# Quantum many-body scars in spin-1 Kitaev chain with uniaxial single-ion
anisotropy

Wen-Yi Zhang  College of Physics, Nanjing University of Aeronautics and
Astronautics, Nanjing, 211106, China  Key Laboratory of Aerospace Information
Materials and Physics (NUAA), MIIT, Nanjing 211106, China  Ya-Nan Wang
College of Physics, Nanjing University of Aeronautics and Astronautics,
Nanjing, 211106, China  Key Laboratory of Aerospace Information Materials and
Physics (NUAA), MIIT, Nanjing 211106, China  Dongchang Liu  Mathematical
Sciences Institute, The Australian National University, Canberra ACT 2601,
Australia  Jie Ren  Department of Physics, Changshu Institute of Technology,
Changshu 215500, China  Jia Li  College of Engineering Physics, Shenzhen
Technology University, Shenzhen 518118, China  Key Laboratory of Aerospace
Information Materials and Physics (NUAA), MIIT, Nanjing 211106, China
Ning Wu<EMAIL_ADDRESS>Center for Quantum Technology Research, School of
Physics, Beijing Institute of Technology, Beijing 100081, 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 Wen-Long You<EMAIL_ADDRESS>College of Physics, Nanjing
University of Aeronautics and Astronautics, Nanjing, 211106, China  Key
Laboratory of Aerospace Information Materials and Physics (NUAA), MIIT,
Nanjing 211106, China

###### Abstract

To establish a solid-state-based framework for the coexistence of quantum
many-body scars and quantum criticality, we investigate the spin-1 Kitaev
chain with uniaxial single-ion anisotropy (SIA). In the subspace with uniform
$\mathbb{Z}_{2}$ gauge fields, this model can be exactly mapped to the
spin-1/2 effective detuned PXP Hamiltonian, where the SIA plays a role of the
static detuning term. The quench dynamics starting from the product states is
symmetric between positive and negative values of the SIA, while a quantum
phase transition from the Kitaev spin liquid to the dimer phase only occurs at
the critical point with a negative $D_{c}$, implying the spontaneous breaking
of the translational symmetry. We find that the coherent oscillations of
quantum fidelity and certain local observables are sustained against small SIA
perturbations in a quantum quench from special initial states. While the
oscillation amplitudes of these observables decay with time as the SIA
strength is increased, the system completely thermalizes upon approaching the
critical point. In contrast, the initial polarized state, which shows an
absence of revivals of quantum fidelity, will exhibit long revivals for
$D<D_{c}$. Finally, we investigate the evolution of phase boundaries of the
Kitaev spin liquid and dimer phase by introducing Heisenberg interactions,
which spoil the $\mathbb{Z}_{2}$ gauge fields. A complete phase diagram is
given by the infinite time-evolving block decimation method and the ground
state properties of each phase are accurately captured by various spin
correlations. Our work opens the door to understanding exotic connections
between many-body scars and quantum criticality in systems with higher spins.

##  I Introduction

In the past decade, there has been significant progress in understanding out-
of-equilibrium dynamics of isolated quantum systems [1]. The eigenstate
thermalization hypothesis (ETH) [2, 3, 4, 5, 6, 7] has been regarded as a
cornerstone of contemporary statistical mechanics, which states that in a
thermalizing system, the expectation value of a generic local observable in
individual eigenstates should be equivalent to its microcanonical average.
Despite the significant success of ETH in explaining thermalization of chaotic
systems, instances of ergodicity breaking are continually being discovered.
The integrable systems [8, 9, 10, 11, 12, 13] and many-body localization [14,
15, 16, 17, 18, 19, 20] are the most noteworthy exceptions. The strong
ergodicity breaking phenomena in counter-examples, where most of the
eigenstates violate the ETH, can be ascribed to the presence of conserved
quantities [21]. In an integrable system, the number of conserved quantities
is equal to the number of degrees of freedom [22]. On the other hand, many-
body localization occurring in systems where disorder and interactions prevent
the system from thermalizing can be also described by the emergence of an
extensive set of quasi-local integrals of motions [23]. Recently, a Rydberg-
atom quantum simulator [24] revealed the emergence of a new type of ETH-
violating eigenstates in certain nonintegrable quantum many-body systems,
dubbed quantum many-body scar (QMBS) states [25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35]. Some specific low-entanglement states in a many-body quantum
system are exceptional in that they violate the ETH and can retain quantum
coherence for long times, even when the system is chaotic and thermalizing
[36].

To be specific, the number of QMBSs is exponentially smaller than the Hilbert
space dimension. The discovery of QMBSs has opened up a new paradigm for
studying unusual nonequilibrium phenomena including many-body revivals and
nonthermal stationary states [37, 38]. Soon the scarred states have been
observed in a variety of physical systems, including inter alia, interacting
spin chains [39, 40], cold atom systems [41, 42, 43], superconducting qubits
[44, 45], etc. In parallel with exciting experimental advances, theoretical
studies have shown that QMBSs are not related to the usual symmetries [46].
Known systems that host QMBS states also include the Affleck-Kennedy-Lieb-
Tasaki (AKLT) model [47, 48], the spin-1 XY model [49], and the generalized
Fermi-Hubbard model [50]. The associated weak ergodicity breaking not only
challenges the validity of ETH but also poses a different scenario of
nonthermal dynamics.

Later it was pointed out theoretically that the Rydberg experiment can be
described by the one-dimensional (1D) chain of spin-1/2 degrees of freedom
[51, 52, 53, 54, 55, 56], where the spin-up state $|1\rangle$ corresponds to a
Rydberg atom occupying an excited state and the spin-down state $|0\rangle$
denotes an atom in the ground state. Such a spin-1/2 spin chain, known as the
PXP Hamiltonian and resulting from the first-order Schrieffer-Wolff
transformation applied to a tilted Ising chain, is described by

$\hat{H}_{\rm PXP}=\sum_{i=1}^{N}P_{i-1}X_{i}P_{i+1},$ (1)

where $N$ is the number of sites, $X=|0\rangle\langle 1|+|1\rangle\langle 0|$
and $P=|0\rangle\langle 0|$ is the projector onto the ground state, ensuring
that the nearby atoms are not simultaneously in the excited state.

Such Rydberg blockade induced kinetic constraint is responsible for the
atypical dynamics of QMBS states. When the system is initialized at time $t=0$
in the product state
$|\psi(0)\rangle\equiv|\mathbb{Z}_{k}\rangle\,(k=1,2,3,4)$, namely,

$\displaystyle|\mathbb{Z}_{1}\rangle$ $\displaystyle=$
$\displaystyle|0000\cdots
00\rangle,\quad~{}\,|\mathbb{Z}_{2}\rangle=|010101\cdots 01\rangle,$
$\displaystyle|\mathbb{Z}_{3}\rangle$ $\displaystyle=$
$\displaystyle|001001\cdots 001\rangle,|\mathbb{Z}_{4}\rangle=|00010001\cdots
0001\rangle,\quad\quad$ (2)

the system then follows the evolution governed by the PXP Hamiltonian,
$|\psi(t)\rangle=\exp(-i\hat{H}_{\rm PXP}t)|\psi(0)\rangle$. It was noted that
the quantum quench from either $|\mathbb{Z}_{2}\rangle$ or
$|\mathbb{Z}_{3}\rangle$ exhibits periodic revivals in the quantum fidelity

$\displaystyle F(t)=|\langle\psi(0)|\psi(t)\rangle|^{2},$ (3)

while $|\mathbb{Z}_{1}\rangle$ or $|\mathbb{Z}_{4}\rangle$ thermalize under
time evolution. The observed oscillations and apparent nonergodic dynamics are
due to the existence of equal spacing of the QMBS eigenstates [57].

Considering the experimental realization and the important role of the
emergence of QMBS in the PXP model, the intensive study of the PXP model has
been the subject of a separate thread of investigation of much current
interest [58, 59, 60, 61, 62]. In fact, this effective model has a long
history dating at least as far back as an effective Hamiltonian for the tilted
Bose-Hubbard model [63]. The PXP model has been studied in various other
contexts including Fibonacci anyon chains [64, 65, 66], Ising models on dimer
ladders [67, 68], U(1) lattice gauge theory in its quantum link [69, 70, 71,
72] dipole-conserving Hamiltonians [73], the quantum Hall effect on a thin
torus at filling $\nu$= 1/3 [74], etc. Meanwhile, the PXP model was extended
to Floquet Hamiltonians [75], higher spins [76], and higher dimensions [77].
The PXP model can be deduced from the biaxial Ising model with both transverse
and longitudinal fields at zero detuning [78] and the Bose-Hubbard model at
resonance [79]. It was also claimed that there is an intimate relation between
QMBS and quantum criticality [80] or quantum integrability [81].

Remarkably, the 1D PXP chain is shown to be embedded in the spin-1 Kitaev
model [82, 83], highlighting a solid-state-based realization of the PXP model.
The celebrated Kitaev model is renowned as a prototype model of quantum spin
liquid (QSL), which hosts massive long-range entanglement and fractional
quasiparticles from localized spins described by bosonic/fermionic spinons and
$\mathbb{Z}_{2}$ gauge fields [84]. Solid-state material realizations of the
bond-dependent Kitaev interactions with $S$=1/2 local moments have vitalized
the research in QSLs [85, 86], where strong spin-orbit coupling in a strongly
correlated Mott insulator plays an essential role. This poses $4d$ and $5d$
transition-metal compounds are proposed to be candidate materials, such as
triangular lattice $\rm YbMgGaO_{4}$ [87], $\rm 1T$-$\rm TaSe_{2}$ [88] and
$\rm NaYbS_{2}$ [89], kagome lattice $\rm ZnCu_{3}(OH)_{6}Cl_{2}$ [90] and
$\rm Na_{4}Ir_{3}O_{8}$ [91], honeycomb lattice $\rm\alpha$-$\rm RuCl_{3}$
[92], $\rm H_{3}LiIr_{2}O_{6}$ [93], $\rm Cu_{2}IrO_{3}$ [94], $\rm RuBr_{3}$
[95] and $\rm BaCo_{2}(AsO_{4})_{2}$ [96], pyrochlore lattice $\rm
Ce_{2}Zr_{2}O_{7}$ [97] and $\rm Ba_{3}Yb_{2}Zn_{5}O_{11}$ [98]. After the
groundbreaking proposal for realizing the higher-spin analogs of the Kitaev
interactions [99], a number of materials with strong Hund’s coupling among two
electrons in $e_{g}$-orbitals of transition metal ions and strong spin-orbit
coupling of anions have emerged as potential candidates for the $S=1$ Kitaev
model. Recently the importance of studying the higher-spin Kitaev physics has
attracted a lot of attention.

Both experimental and numerical analyses have been indispensably carried out
to explore the higher-spin Kitaev physics, such as $S=1$ [100, 101, 102, 103,
104, 105], $S=3/2$ [106, 107, 108], and even $S=2$ systems [109]. It is
noteworthy that non-Kitaev interactions widely exist in candidate materials,
which is a chief obstacle of keeping the system away from the pure Kitaev
limit. The ferromagnetic Heisenberg interactions are generated from
superexchange paths together with Kitaev interactions, in parallel with the
antiferromagnetic Heisenberg term from direct-exchange paths.

For Mott insulators with two or more atoms per site, the direct on-site
interactions can give rise to a nonlinear term $\propto
D\sum_{j}(S_{j}^{z})^{2}$ for $S\geq 1$, where $D$ is the so-called uniaxial
single-ion anisotropy (SIA) constant. Recently, theoretical [110, 111, 112]
and experimental [113] studies on the Kitaev model with additional SIA have
attracted increasing attention. In this work, we will show that the static
detuning in the PXP model, which describes the static frequency difference
between the ground and excited states, can be mimicked by the additional SIA
in the spin-1 Kitaev model, which normally stems from zero-field splitting due
to a crystal-field anisotropy. Upon varying the strength of the SIA, a
corresponding second-order phase transition will occur with a translational
symmetry breaking. A comprehensive study of the phase diagram has been
conducted by incorporating significant Heisenberg interactions. In the
numerical calculation, we employ the exact diagonalization (ED) method, the
time-evolving a matrix product state (MPS) with matrix product operators
(MPOs) [114] based on ITensor [115] and the infinite time-evolving block
decimation (iTEBD) algorithm [116].

The remainder of this paper is organized as follows: In Sec. II, we present
the spin-1 Kitaev model with SIA (the KD model), and deduce the effective
spin-1/2 detuned PXP model in the ground-state manifold. The quantum many-body
scars in the spin-1 KD model are studied in detail. In Sec. III, we
investigate the quantum criticality in the KD model, and find the
characteristics of the dimer phase. Under the cooperative effects of the
single-ion anisotropy and Heisenberg interactions (the KHD model), we reveal
the rich quantum phase diagram of KHD model in Sec. IV. The summary and
conclusion are given in Sec. V.

##  II Spin-1 Kitaev chain with uniaxial single-ion anisotropy

In this work, we consider a spin-1 Hamiltonian composed of the Kitaev
interaction and SIA, given by

$\displaystyle\hat{H}_{\rm
KD}\\!=\\!\sum_{j=1}^{N/2}\left(K_{2j-1}S_{2j-1}^{x}S_{2j}^{x}\\!+\\!K_{2j}S_{2j}^{y}S_{2j+1}^{y}\right)\\!+\\!\sum_{j=1}^{N}\\!D_{j}(S_{j}^{z})^{2}\\!,$

where $K_{j}$ parameterizes the strength of the bond-dependent Kitaev exchange
coupling between two neighbouring sites $\langle j,j{+}1\rangle$, and $D_{j}$
denotes the amplitude of the SIA at the $j$-th site. $S_{j}^{a}$ ($a=x,y,z$)
is the $a$-component of the spin-1 operator at the $j$-th site among total $N$
sites, obeying the SU(2) algebra, i.e.,
$[S_{i}^{a},S_{j}^{b}]=i\delta_{ij}\epsilon_{abc}S_{j}^{c}$, with the
antisymmetric tensor $\epsilon_{abc}$ and $(\bm{S}_{j})^{2}=S(S+1)=2$.

We will work with a special spin-1 representation, i.e.,

$\displaystyle|x\rangle\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{\sqrt{2}}(|-1\rangle\\!-\\!|1\rangle),\quad|y\rangle\\!=\\!\frac{i}{\sqrt{2}}(|-1\rangle\\!+\\!|1\rangle),\quad|z\rangle\\!=\\!|0\rangle,$

where $|m\rangle$ is the eigenstate of the spin operator $S^{z}$ with
eigenvalues $m$ =$-1$, $0$, $1$. In such a representation, we have
$S_{bc}^{a}=-i\epsilon_{abc}$ and $\\{S^{x},S^{y},S^{z}\\}$ are given by

$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\ 0&i&0\\\
\end{array}\right),\left(\begin{array}[]{ccc}0&0&i\\\ 0&0&0\\\ -i&0&0\\\
\end{array}\right),\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\ 0&0&0\\\
\end{array}\right).$ (15)

The corresponding site parity matrices are defined as $\Sigma_{j}^{a}\equiv
e^{i\pi S_{j}^{a}}$$=1-2(S_{j}^{a})^{2}$ and become diagonal,

$\displaystyle\\!\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\
\end{array}\right),\left(\begin{array}[]{ccc}-1&0&0\\\ 0&1&0\\\ 0&0&-1\\\
\end{array}\right),\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\\\
\end{array}\right).$ (25)

It has been revealed that different Ising interactions on odd and even bonds
in Eq. (II) can be rewritten into a similar form through a unitary
transformation on the even sites [117, 83]:

$\displaystyle U=\prod_{j}\exp(i\pi
S_{2j}^{x})\exp\left(i\frac{\pi}{2}S_{2j}^{z}\right),$ (26)

which gives $US_{2j}^{x}U^{\dagger}=S_{2j}^{y}$,
$US_{2j}^{y}U^{\dagger}=S_{2j}^{x}$, and $US_{2j}^{z}U^{\dagger}=-S_{2j}^{z}$,
as well as $U|x\rangle=|y\rangle$, $U|y\rangle=|x\rangle$,
$U|z\rangle=-|z\rangle$. Note that the order of rotations about $x$\- and
$z$\- axes in Eq. (26) is essential as they do not commute. After the unitary
transformation, the Kitaev exchange couplings in Eq. (II) take a translation-
invariant form

$\displaystyle\tilde{H}_{\rm K}=\sum_{j=1}^{N}K_{j}S_{j}^{x}S_{j+1}^{y}.$ (27)

It is easy to see that the SIA term remains in its original form and the
Hamiltonian (II) can be rewritten

$\displaystyle\tilde{H}_{\rm
KD}=\sum_{j=1}^{N}K_{j}S_{j}^{x}S_{j+1}^{y}+D_{j}(S_{j}^{z})^{2}.$ (28)

Note that the sign of the Kitaev interactions is still under debate with
conflicting results from theoretical and experimental studies [118, 119].
Hereafter the uniform couplings with $K_{j}=1$ and $D_{j}=D$ ($\forall j$) are
assumed unless otherwise specified.

Under the rotation (26), the local bond parity operators are defined by

$\displaystyle\hat{W}_{j}=\Sigma_{j}^{y}\,\Sigma_{j+1}^{x}.$ (29)

One can readily find that $\hat{W}_{j}$ is invariant by inspecting
$[\hat{W}_{j},\tilde{H}_{\rm KD}]=0$. As the eigenvalues of $\Sigma_{j}^{a}$
in Eq. (25) are $\pm 1$, the eigenvalues of $\hat{W}_{j}$ are related to
$\mathbb{Z}_{2}$-valued invariants, i.e., $w_{j}=\pm 1$. It is straightforward
to deduce from Eq. (25) that for a pair of nearest neighbor sites $\langle
j,j{+}1\rangle$, total $3{\times}3=9$ allowed states can be distinguished into
the $w_{j}=1$ sector spanned by $|xy\rangle$, $|xz\rangle$, $|yx\rangle$,
$|zy\rangle$, $|zz\rangle$ and the $w_{j}=-1$ sector spanned by $|xx\rangle$,
$|yy\rangle$, $|yz\rangle$, $|zx\rangle$. Hence, the whole Hilbert space
$\mathcal{H}$ can be decomposed into $2^{N}$ dynamically disconnected Krylov
subspaces of unequal sizes characterized by
$\vec{w}=\\{w_{1},w_{n},\cdots,w_{N}\\}$ as

$\displaystyle\mathcal{H}=\bigoplus_{n=1}^{2^{N}}\mathcal{K}_{n}.$ (30)

The Krylov subspace $\mathcal{K}_{n}$ is spanned by

$\displaystyle\\{\mathcal{K}_{n}\\}\equiv{\rm
Span}\\{|\psi_{n}\rangle,\tilde{H}_{\rm KD}|\psi_{n}\rangle,\tilde{H}_{\rm
KD}^{2}|\psi_{n}\rangle,\cdots\\},$ (31)

where $|\psi_{n}\rangle$ is the so-called root state, which is a product state
having explicit $\mathbb{Z}_{2}$ symmetries.

We have identified the ground state of spin-1 Kitaev chain lies within the
flux-free sector, i.e., $\vec{w}=\\{1,1,\cdots,1\\}$ [82]. In such a
constrained Hilbert space, there is one-to-one mapping between base
configurations $\\{\tilde{\mathcal{K}}_{S=1}\\}$ of Eq. (27) within the flux-
free sector and the configurations $\\{{\mathcal{K}}_{S=1/2}\\}$ of Eq. (1)
with nearest neighbor exclusion. The rule for constructing the mapping is
simple. The one-to-one mapping between the 5 allowed two-site configurations
for a pair of nearest neighbor sites $\langle j,j{+}1\rangle$ and spin-$1/2$
degree of freedom for the bond center $j+1/2$ is given by [120]

$\displaystyle|\cdots
zz\cdots\rangle_{j,j+1}\leftrightarrow|\cdots\downarrow\downarrow\downarrow\cdots\rangle_{j-\frac{1}{2},j+\frac{1}{2},j+\frac{3}{2}},$
$\displaystyle|\cdots
yx\cdots\rangle_{j,j+1}\leftrightarrow|\cdots\downarrow\uparrow\downarrow\cdots\rangle_{j-\frac{1}{2},j+\frac{1}{2},j+\frac{3}{2}},$
$\displaystyle|\cdots
zy\cdots\rangle_{j,j+1}\leftrightarrow|\cdots\downarrow\downarrow\uparrow\cdots\rangle_{j-\frac{1}{2},j+\frac{1}{2},j+\frac{3}{2}},$
$\displaystyle|\cdots
xz\cdots\rangle_{j,j+1}\leftrightarrow|\cdots\uparrow\downarrow\downarrow\cdots\rangle_{j-\frac{1}{2},j+\frac{1}{2},j+\frac{3}{2}},$
$\displaystyle|\cdots
xy\cdots\rangle_{j,j+1}\leftrightarrow|\cdots\uparrow\downarrow\uparrow\cdots\rangle_{j-\frac{1}{2},j+\frac{1}{2},j+\frac{3}{2}}.$
(32)

It is worthy noting that the prime lattice of the spin-1 Kitaev chain is
defined on the sites $\\{j\\}$, while the dual lattice of spin-1/2 PXP model
lives on the linking bonds at sites $\\{j+1/2\\}$. This mapping from sites to
bonds includes links to the two surrounding sites and vice verse, which
becomes subtle for open boundary conditions. As an example, the four product
states given by Eq. (I) can be mapped to the following states in
$\\{\tilde{\mathcal{K}}_{S=1}\\}$:

$\displaystyle|\widetilde{\mathbb{Z}}_{1}\rangle$ $\displaystyle=$
$\displaystyle|zzzz\cdots
zz\rangle,\quad\;\;\,|\widetilde{\mathbb{Z}}_{2}\rangle=|xyxyxy\cdots
xy\rangle,$ $\displaystyle|\widetilde{\mathbb{Z}}_{3}\rangle$ $\displaystyle=$
$\displaystyle|yxzyxz\cdots
yxz\rangle,|\widetilde{\mathbb{Z}}_{4}\rangle=|yxzzyxzz\cdots
yxzz\rangle.\quad\quad$ (33)

The simplest root configuration in $\\{\tilde{\mathcal{K}}_{S=1}\\}$ is the
product state $|\widetilde{\mathbb{Z}}_{1}\rangle$, which is the ground state
in the $D\to\infty$ limit, and the Hilbert space of this sector can be
constructed by successively applying the Hamiltonian on this root state, i.e.,

$\displaystyle\\{\tilde{\mathcal{K}}_{S=1}\\}\equiv{\rm
Span}\\{|\widetilde{\mathbb{Z}}_{1}\rangle,\tilde{H}_{\rm
K}|\widetilde{\mathbb{Z}}_{1}\rangle,\tilde{H}_{\rm
K}^{2}|\widetilde{\mathbb{Z}}_{1}\rangle,\cdots\\}.$ (34)

The corresponding dimension $d$ of the flux-free sector is proven to be a
Lucas number [83], i.e., $d=F_{N-1}+F_{N+1}$, where $F_{\ell}$ is the $\ell$th
Fibonacci number. More precisely, $d=g^{N}+g^{-N}$ with $g=(1+\sqrt{5})/2$
being the golden ratio. This exponentially large subspace belongs to the
largest Krylov subspace among the exponential number of Krylov subspaces,
implying strong fragmentation of the Hilbert space. The graphical
representation of the constrained Hilbert space in the
$\vec{w}{=}\\{1,1,{\cdots},1\\}$ subspace is schematically shown in Fig. 1 for
$N=6$. The vertices in the 18-dimensional hypercube is uniquely labeled by the
connected configurations (34), which have been arranged by the action of the
Kitaev Hamiltonian $\tilde{H}_{\rm K}$ on the product state $|\cdots
zzzz\cdots\rangle$.

Figure 1: The Hilbert space graph of the Kitaev Hamiltonian in Eq. (27) within
the $\vec{w}=\\{1,1,1,1,1,1\\}$ subspace for $N=6$ sites with periodic
boundary conditions. The nodes of the graph $|m\rangle$ ($m=0,1,2,\ldots,17)$
label the allowed product states, and the edges connect product state
configurations that differ by an excitation $|\cdots
zz\cdots\rangle\leftrightarrow|\cdots yx\cdots\rangle$ due to the action of
the Hamiltonian.

The process of bond converting $|\cdots
zz\cdots\rangle_{j,j+1}\leftrightarrow|\cdots yx\cdots\rangle_{j,j+1}$ under
the action of $\tilde{H}_{K}$ corresponds to the spin flip $|\cdots
0\cdots\rangle_{j+1/2}\leftrightarrow|\cdots 1\cdots\rangle_{j+1/2}$ in
$\\{\mathcal{K}_{S=1/2}\\}$. In this regard, the spin-1 Kitaev chain with
periodic boundary conditions can be exactly mapped to the a single qubit-flip
model represented by the effective spin-1/2 PXP model in Eq. (1). Remarkably,
we find the ground state remains in the flux-free sector even in the presence
of the SIA. The action of the SIA term on the active bases yields,

$\displaystyle D\\!\left[(S_{j}^{z})^{2}+\\!(S_{j+1}^{z})^{2}\right]\\!|\cdots
yx\cdots\rangle_{j,j+1}\\!$ $\displaystyle=$ $\displaystyle 2D|\cdots
yx\cdots\rangle_{j,j+1},$ $\displaystyle
D\\!\left[(S_{j}^{z})^{2}+\\!(S_{j+1}^{z})^{2}\right]\\!|\cdots
zz\cdots\rangle_{j,j+1}\\!$ $\displaystyle=$ $\displaystyle 0,$ (35)

which results in an effective detuning term on the spin-1/2 degrees of
freedom, such that the effective Hamiltonian can be mapped to the spin-1/2
detuned PXP model,

$\displaystyle\hat{H}_{\rm
dPXP}\\!=\sum_{i=1}^{N}P_{i-1}X_{i}P_{i+1}+2D\sum_{i=1}^{N}P_{i-1}n_{i}P_{i+1},$
(36)

where $n=1-P=|1\rangle\langle 1|$. Note that in both Eq. (1) and Eq. (36), $i$
labels the bonds between sites, while the index $j$ labels the sites in Eq.
(28). The detailed derivation of Eq. (36) can be found in Appendix A.

The detuning term is commonly prevalent in practical experiments. The static
detuning (also called chemical potential [121]) of the driving laser from the
excited state can be finely tuned in the cold-atom platforms. It has been
noted that quantum quench from initial states
$|\widetilde{\mathbb{Z}}_{2}\rangle$ or $|\widetilde{\mathbb{Z}}_{3}\rangle$
results in coherent oscillations, indicating the existence of ETH-violating
QMBSs. In our ED simulation [122], the time-evolved operator
$\exp({-i\hat{H}t})$ governed by either $\tilde{H}_{\rm KD}$ as defined in Eq.
(28) or $\hat{H}_{\rm dPXP}$ as given in Eq. (36) is discretized using time
steps of $dt=0.01$, and the time-evolved state $|\psi(t)\rangle$ is
subsequently computed using the fourth-order Runge-Kutta method within the
corresponding constraint Hilbert space. Figure 2 demonstrates these
oscillations in the dynamics of the quantum fidelity for $D=0.1$. The periodic
revivals for the spin-1 KD model (28) starting from the
$|\widetilde{\mathbb{Z}}_{2}\rangle$, $|\widetilde{\mathbb{Z}}_{3}\rangle$
initial states completely coincide with the ones observed for the spin-1/2
detuned PXP model, which starts from the corresponding
$|\mathbb{Z}_{2}\rangle$, $|\mathbb{Z}_{3}\rangle$ initial states.

Figure 2:  Characteristic quantum features of the spin-1 KD model and the
spin-1/2 detuned PXP model with $D=0.1$. Quantum fidelity $F(t)$ for
$\tilde{H}_{\rm KD}$ in Eq. (28) [$\hat{H}_{\rm dPXP}$ in Eq. (36)] starting
from the initial states: (a) $|\widetilde{\mathbb{Z}}_{2}\rangle$
($|\mathbb{Z}_{2}\rangle$) with $N=18$, and (b)
$|\widetilde{\mathbb{Z}}_{3}\rangle$ ($|\mathbb{Z}_{3}\rangle$) with $N=18$.

Figure 3:  (a) The contour map of time evolution of $\langle\hat{O}\rangle$
defined in (37) of $\tilde{H}_{\rm KD}$ (28) in a system of $N=28$ spins
prepared in $|\widetilde{\mathbb{Z}}_{2}\rangle$ obtained by ED. (b) The time
evolution of $\langle\hat{O}\rangle$ for different values of $D$. The curves
correspond to $D=0.0$, $-0.1$, $-0.2$, $-0.3$, $-0.4$, $-0.5$, $-0.655$ (from
bottom to top at $t=5$). The dashed lines are fits capturing the amplitude
decay. Inset shows inverse lifetime of $\langle\hat{O}(t)\rangle$ envelop with
increasing $D^{2}$. We extract the decay time by fitting the data to the
scaling ansatz $\langle\hat{O}(t)\rangle=Ae^{-t/\tau}\cos\omega t$ (see main
text).  Figure 4:  The dynamic evolution of the spin-1/2 detuned PXP model
starting from the initial state $|\mathbb{Z}_{1}\rangle$: (a) The quantum
fidelity $F(t)$ with respect to different $D$ for $N\\!=\\!28$; (b) Finite-
size scaling of $F(t)$ for the first peak in panel (a).

Recent studies have signified an intimate relation between QMBS and quantum
criticality [81, 80]. As $D$ is tuned to $D_{c}\approx-0.655$, the ground
state of the detuned PXP model undergoes a Ising phase transition associated
with a spontaneous breaking of $\mathbb{Z}_{2}$-symmetry [123, 124, 125, 126].
The non-thermalizing dynamics can be also captured by measuring the
expectation values of certain local observables [127], e.g.,

$\displaystyle\left\langle\hat{O}\right\rangle=\frac{1}{2}\left\langle[\left(S_{1}^{+}\right)^{2}+\left(S_{1}^{-}\right)^{2}]\right\rangle.$
(37)

Under the dual transformation (II), the correlator $\langle\hat{O}\rangle$ of
the KD model in Eq. (28) is found to be equivalent to the density imbalance,
$\langle n_{2}\rangle-\langle n_{1}\rangle$, an observable corresponding to
the staggered magnetization in the detuned PXP model in Eq. (36). Performing a
quantum quench from an initial state $|\widetilde{\mathbb{Z}}_{2}\rangle$
leads to nearly perfect coherent dynamics. The coherence oscillations persist
for long times for $D=0$, as is shown in Fig.3. Note that the values of $F(t)$
and $\langle\hat{O}\rangle$ is independent of the sign of $D$ when the system
starts from the product states, $|\widetilde{\mathbb{Z}}_{k}\rangle$ (see
details in Appendix B).

As exhibited in Fig. 3(a), these oscillations are found to be remarkably
robust to small SIA perturbations, while moderate perturbations make the
oscillations damp sharply until $D$ reaches a threshold value. One carefully
observes from Fig. 3(b) that the oscillations remain strong for deviations up
to $D\approx\pm D_{c}$, and there is barely oscillation at $D=D_{c}$, upon
which the thermalization completely sets in. Suppose that the envelope of
$\langle\hat{O}\rangle$ can be described by exponentially decaying
oscillations, $\langle\hat{O}(t)\rangle=Ae^{-t/{\tau}}\cos\omega t$ over time
$t$, with the fitting parameters $A$, $\tau$, and $\omega$. We observe that
the inverse lifetime approximately follows $\tau^{-1}\sim D^{2}$ at small $D$,
reminiscent of the Fermi’s golden rule [128]. Additionally, it is worth noting
that the decay rate of oscillations $\tau^{-1}$ at $D=0$ remains small but
finite, suggesting that the $|\widetilde{\mathbb{Z}}_{2}\rangle$ initial state
only approximates the near-perfect scar states in the standard PXP model. The
fact that the quantum critical point $D_{c}$ is negative and there is no
quantum phase transition for positive $D$, combined with the significant
difference between the ground states as $D$ tends infinity, undermine the
viewpoint that quantum many-body scars and quantum criticality are directly
bridged.

Figure 5:  Dynamics of quantum fidelity for the detuned PXP model: (a)
Starting from initial state $|\mathbb{Z}_{2}\rangle$ for $N=28$ sites; (b) The
overlaps between the product state $|\mathbb{Z}_{2}\rangle$ and the time-
evolved state starting from $|\mathbb{Z}_{2}\rangle$ (solid) and
$|\mathbb{Z}_{2}^{\prime}\rangle$ (dashed) with $D=-0.1$ for $N=28$ sites. The
inset shows the overlap of the prequench ground state with
$|\mathbb{Z}_{2}\rangle$ and $|\mathbb{Z}_{2}^{\prime}\rangle$. (c) Starting
from initial $|\mathbb{Z}_{3}\rangle$ state for $N=24$ sites.

For $D=0$, the quantum fidelity revivals do not occur for the initial state
$|\mathbb{Z}_{1}\rangle$. As $D$ decreases from zero, surprisingly, there will
be a slight revival in fidelity for the same initial state. As $D$ continues
to decrease, the oscillation becomes more clearly visible with smaller
periods, as observed in Fig. 4(a). When the value of $D$ is smaller than the
critical value $D_{c}$, the revivals become more pronounced. Finite-size
scaling in Fig. 4(b) reveals that the first peak will disappear for large $N$
before the value of $D$ exceeds $D_{c}$. When $D>D_{c}$, the intercepts of the
finite-size scaling curves become negative, which is an unphysical artifact
and indicates that the linear fit is no longer applicable. In contrast, the
first peak will always have a finite value for $D<D_{c}$ in the thermodynamic
limit, which may be related to asymptotic scars [129].

Figure 5(a) shows the quantum fidelity of $\hat{H}_{\rm dPXP}$ with different
values of $D$ for $N=28$ using the initial $|\mathbb{Z}_{2}\rangle$ state.
Remarkably, when $D$ decreases from zero, persistent oscillations first
decrease when $D>D_{c}$, then damp in the critical regime $D\approx D_{c}$,
and finally revive beyond the critical point for $D<D_{c}$. Figure 5(b) shows
the overlaps between the $|\mathbb{Z}_{2}\rangle$ state and the time evolved
state starting from $|\mathbb{Z}_{2}\rangle$ and
$|\mathbb{Z}_{2}^{\prime}\rangle$, where
$|\mathbb{Z}_{2}^{\prime}\rangle\equiv|101010\cdots 10\rangle$ is obtained by
translating one lattice spacing on $|\mathbb{Z}_{2}\rangle$.

The peaks of the oscillations of
$|\langle\mathbb{Z}_{2}|\exp(-i\hat{H}t)|\mathbb{Z}_{2}\rangle|^{2}$ and
$|\langle\mathbb{Z}_{2}|\exp(-i\hat{H}t)|\mathbb{Z}_{2}^{\prime}\rangle|^{2}$
are separated by half a period. We also show the fidelity between the
$|\mathbb{Z}_{2}\rangle$ state and the ground state $|\psi_{0}\rangle$ at
different values of $D$, as shown in the inset of Fig. 5(b). We observe that
as $D$ approaches negative infinity, the fidelity between the ground state,
and $|\mathbb{Z}_{2}\rangle$ ($|\mathbb{Z}_{2}^{\prime}\rangle$) gradually
approaches $1/2$. We remark that due to the Hilbert space constraint, at most
half of the atoms could be in the spin-up states. In fact, in the limit of
$D\to-\infty$, the ground state becomes an antiferromagnetic phase in zero-
momentum sector, i.e.,
$|\psi_{0}(D=-\infty)\rangle=(|\mathbb{Z}_{2}\rangle+|\mathbb{Z}_{2}^{\prime}\rangle)/\sqrt{2}$.
In contrast, Fig. 5(c) demonstrates a complete absence of revivals for
$D<D_{c}$ in the case of an initial state of $|\mathbb{Z}_{3}\rangle$,
featuring approximate QMBS states vanish.

Figure 6:  Evolution of the bipartite entanglement entropy in the quantum
quench from initial states: (a) $|\widetilde{\mathbb{Z}}_{2}\rangle$, (b)
$|\widetilde{\mathbb{Z}}_{3}\rangle$ for the spin-1 KD model, (c)
$|{\mathbb{Z}}_{2}\rangle$ and (d) $|{\mathbb{Z}}_{3}\rangle$ states for the
spin-1/2 detuned PXP model with $N=24$.

We next investigate the dynamics of bipartite entanglement entropies in both
the KD model and the detuned PXP model. We choose the region $A$ to be one
half of the chain, and compare the dynamics of half-chain entanglement entropy
${\cal S}$ in a quantum quench from different initial states for both the
spin-1 KD model and the spin-1/2 detuned PXP model with periodic boundary
conditions. In our numerical calculation for $S=1$ KD model [cf., Figs.
6(a-b)], we utilize the time-evolving MPS approach with MPOs [115], where the
bond dimension is set as $\chi=500$ and the time step is $dt=0.025$. The
bipartite entanglement of the evolved state starting from the initial state
$|\widetilde{\mathbb{Z}}_{2}\rangle$ is shown in Fig. 6(a). When $D$ is small
negative, ${\cal S}$ increases slowly over time while exhibits coherent
oscillations, featuring the many-body revivals. As $D$ becomes more negative,
the temporal growth rate of the bipartite entanglement increases, and the
coherent oscillation becomes weak. When $D$ approaches $D_{c}$, the
entanglement ${\cal S}$ almost increases linearly with time until saturation,
and the coherent oscillations disappear, implying that the system quickly
thermalizes.

When $D$ is smaller than $D_{c}$, the linear growth rate of entanglement
decreases with a smaller saturated value of entanglement. We find that the
growth of entanglement entropy starting from the initial state
$|\widetilde{\mathbb{Z}}_{3}\rangle$ exhibits a similar trend, as shown in
Fig. 6(b). For comparison, we show the evolution of the bipartite von Neumann
entanglement entropy of the spin-1/2 detuned PXP model when the system is
initially prepared in the state $|{\mathbb{Z}}_{2}\rangle$
($|{\mathbb{Z}}_{3}\rangle$) in Fig. 6(c) [6(d)]. One observes ${\cal S}$ is
gradually growing for $D$ being small negative, while undergoes an extremely
fast growth until saturation at $D\ \approx D_{c}$. When $D<D_{c}$, the growth
of entanglement entropy slows down again. Although the two Hamiltonians and
their corresponding initial states are unitarily equivalent under the local
transformation (II), the entanglement evolution displays noticeable
differences. Notably, the coherent oscillations for $S=1/2$ become
considerably weaker compared to those for $S=1$ when $0\geq D>D_{c}$. The
bipartite entanglement entropy heavily depends on the choice of presentations
and bipartition methods. This can be perceived by an analytical example
presented in the Appendix A of Ref. [83].

##  III Quantum phase transition of spin-1 Kitaev
chain with uniaxial single-ion anisotropy

In the previous section, we discovered a close relationship between QMBS
states and quantum criticality. Accordingly, we proceed to investigate the
quantum phase transition of the spin-1 KD model given by Eq. (II). We adopt
the iTEBD algorithm with a bond dimension of $\chi=120$ [122]. In our
calculations, we set the imaginary time as $10^{-5.5}$ to ensure a truncation
error smaller than $10^{-8}$. The advantage of using iTEBD is its capability
to treat infinite-size systems directly, providing numerical evidence for the
emergence of a symmetry-breaking phase. According to the core spirit of the
Landau-Ginzburg-Wilson paradigm, the quantum phase transition of a many-body
system can be described by a well-defined order parameter. We calculate the
two-point correlations between the $i$-th and $j$-th sites,

$\displaystyle C^{ab}(i,j)=\left\langle
S^{a}_{i}\exp\left(i\theta\sum_{l=i+1}^{j-1}S_{l}^{a}\right)S^{b}_{j}\right\rangle,a,b=x,y,z,~{}~{}$
(38)

which can detect different symmetry-breaking phases. Equation (38) reduces to
two-point correlations for $\theta=0$, while it becomes the den Nijs-Rommelse
string order parameter for $\theta=\pi$ [130, 131]. Note that there is no
phase accumulated for two nearest-neighboring sites and a general angle
$\theta$ could capture the hidden topological orders [132]. The Hamiltonian in
Eq. (II) is invariant under a joint operation that combines a $\pi/2$-rotation
about the $z$-axis and a single-site translation, which implies that on a
finite-size system

$\displaystyle C^{xx}(1,2)=C^{yy}(2,3),\quad C^{yy}(1,2)=C^{xx}(2,3),$
$\displaystyle C^{zz}(1,2)=C^{zz}(2,3).$ (39)

The joint symmetry can be expressed in the rotated Hamiltonian (28) as
$\tilde{C}^{xy}(1,2)=\tilde{C}^{xy}(2,3)$,
$\tilde{C}^{yx}(1,2)=\tilde{C}^{yx}(2,3)$, and
$\tilde{C}^{zz}(1,2)=\tilde{C}^{zz}(2,3)$.

Figure 7:  The three components of dimer order parameter $O_{D}$ as a function
of $D$. (a) From the dimer phase to the KSL phase at $J=0$. (b) From the FMz
phase to the dimer phase to the AFz phase at $D=-2$. Here we use the iTEBD
method and the bond dimension is set as $\chi=120$.

In the zero-field limit, the ground state is a gapped Kitaev spin liquid
(KSL), which is stable against nonzero perturbations [82]. Upon applying the
uniaxial single-ion anisotropy $D$, the ground state remains in the flux-free
sector, i.e., $\vec{w}{=}\\{1,1,\cdots,1\\}$. At a large positive $D$, the
spins are confined to $|z\rangle$ (i.e., $\langle S_{j}^{z}\rangle=0$), while
for a large negative $D$, the ground states are restricted to $|x\rangle$ or
$|y\rangle$ ($\langle S_{j}^{z}\rangle=\pm 1$ ). Surprisingly, unlike the KSL
phase, a notable difference between $C^{xx}(1,2)$ and $C^{yy}(2,3)$, or
equivalently, $\tilde{C}^{xy}(1,2)$ and $\tilde{C}^{xy}(2,3)$, implies the
spontaneous breaking of the translational symmetry, in the way how the system
hosts the dimer order. The dimer phase is characterized by an alternation of
nearest-neighbor spin-spin correlations, which is characterized by the
difference of $\langle\bm{S}_{i}\cdot\bm{S}_{j}\rangle$ between the odd bonds
and even bonds. A finite dimer order parameter is defined by

$\displaystyle O_{D}$ $\displaystyle=$
$\displaystyle|\langle\bm{S}_{2j-1}\cdot\bm{S}_{2j}\rangle-\langle\bm{S}_{2j}\cdot\bm{S}_{2j+1}\rangle|.$
(40)

To be more specific, we can also examine the $x$, $y$, and $z$ components of
the dimer order parameter, such as

$\displaystyle O_{D}^{x}$ $\displaystyle=$
$\displaystyle|\langle{S}_{2j-1}^{x}{S}_{2j}^{x}\rangle-\langle{S}_{2j}^{y}{S}_{2j+1}^{y}\rangle|,$
$\displaystyle O_{D}^{y}$ $\displaystyle=$
$\displaystyle|\langle{S}_{2j-1}^{y}{S}_{2j}^{y}\rangle-\langle{S}_{2j}^{x}{S}_{2j+1}^{x}\rangle|,$
$\displaystyle O_{D}^{z}$ $\displaystyle=$
$\displaystyle|\langle{S}_{2j-1}^{z}{S}_{2j}^{z}\rangle-\langle{S}_{2j}^{z}{S}_{2j+1}^{z}\rangle|.$
(41)

Note that the dimer order arises from the Kitaev interactions (II), leading to
the characterization of the $x$ and $y$ components as the differences between
distinct types of Ising interactions on odd and even bonds. Figure 7(a)
illustrates that the $x$ component of the dimer order parameter increases
smoothly from zero to a finite value as the parameter $D$ is decreased and
crosses the critical value $D_{c}=-0.655$, indicating a second-order
transition occurs at $D_{c}$. The presence of nonvanishing dimer correlations
for $D>D_{c}$ can be attributed to the limitations imposed by the finite bond
dimension. The dimer orders are associated with the spontaneous breaking of
translational symmetry of $O_{D}^{x}$ in an infinite system. The emergence of
the dimer ordering is distinct from the general mechanism for the formation of
dimerized phases, which is typically induced by inherent bond alternation and
the resulting breaking of translational symmetry. The ground state is two-fold
degenerate for $D<D_{c}$ in the thermodynamic limit, which is in contrast to
the gapped ground state for $D>D_{c}$.

##  IV Effect of Heisenberg interactions

It has been recognized the scarred states display anomalous stability in the
Kitaev phase in the vicinity of $D=0$ [83]. Quantum spin liquids are widely
believed to be crucially driven by the Kitaev interactions in spin-orbit-
coupled materials. While Kitaev interactions are highly anisotropic, the
isotropic Heisenberg interaction, ubiquitous in real materials, can also play
an essential role in the emergence of exotic phenomena in many-body systems.
The relevance of the Kitaev phase in a broader regime becomes paramount for
understanding scar stability and its potential applications in solid-state
systems. To address this point, we investigate the evolution of the phase
boundaries of the Kitaev spin liquid and the dimer phase by introducing
Heisenberg interactions that disrupt the $\mathbb{Z}_{2}$ gauge fields, as
given by

$\displaystyle\hat{H}_{\rm J}$ $\displaystyle=$ $\displaystyle
J\sum_{j=1}^{N}\bm{S}_{j}\cdot\bm{S}_{j+1}.$ (42)

When the parameters $\\{D,J\\}$ vary, the competitions of various correlations
trigger miscellaneous phase transitions. Figure 8 depicts the phase diagram
for the Kitaev-Heisenberg chain with uniaxial single-ion anisotropy (KHD
model). The phase diagram is much richer than expected. Seven distinct phases
are identified, including the KSL phase, dimer phase ($D$), the spin nematic
phase with a left-left-right-right pattern (LLRR), Haldane phase,
$x$-component ferromagnetic (FMx) phase, $z$-component ferromagnetic (FMz)
phase and $z$-component antiferromagnetic (AFz) phase.

Figure 8: Quantum phase diagram of the spin-1 Kitaev-Heisenberg model with
uniaxial single-ion anisotropy calculated by the iTEBD method with bond
dimension $\chi=120$. The quantum phase transition from the dimer phase (D) to
the KSL phase occurs at $D_{c}=-0.655$ for $J=0$ (vertical dashed line). At
$D=0$ (horizontal dashed line), the KSL is stable in the range of $|J|<0.08$.

The joint symmetry (III) is preserved in the whole KSL phase for the infinite
system. It has been reported that on the line of $D=0$ (the horizontal dashed
line in Fig. 8), the ground state of the Kitaev-Heisenberg model undergoes the
FMz phase, the LLRR phase, the KSL and the Haldane phase with increasing $J$.
The successive second-order quantum phase transitions occur at $J_{c}=-0.6$,
$-0.08$, and $0.08$, respectively [82]. For $J=0$ and $D=0$, the pure Kitaev
chain hosts only two nearest neighboring antiferromagnetic orders $\langle
S_{2j-1}^{x}S_{2j}^{x}\rangle$ and $\langle S_{2j}^{y}S_{2j+1}^{y}\rangle$
while other correlations vanish, similar to the spin-1/2 Kitaev honeycomb
model [133]. Away from the Kitav limit, the two-spin correlation functions are
found to decay exponentially and the short correlation length $\xi$ will
extend to a few sites, as shown in Fig. 9(a). Note that the ground-state
properties of integer spin chains are in stark contrast to those of half-odd
integer spin. In comparison, the ground state of spin-1/2 Kitaev chain is
$2^{N/2-1}$-fold degenerate [134], and the macroscopic degeneracy makes the
ground state vulnerable. As such, an infinitesimal Heisenberg coupling is
sufficient to lift the ground-state degeneracy and generate magnetic long-
range order [135, 136]. In contrast, the spin-1 chain supports a gapped KSL
ground state, which can sustain a finite Heisenberg coupling. It is remarkable
that the KSL phase becomes more robust against the Heisenberg interactions for
large positive $D$ . One can further observe that the size of the KSL phase
enlarges with increasing positive $D$ and becomes narrower for negative $D$.
However, it is found that the three components of the dimer order parameter
$O_{D}$ are all mismatched in a fairly small region of the parameter space
except for $J=0$, as exhibited in Fig. 7(b).

Figure 9: The correlation between site $1$ and site $j$ with $\theta=0$ for
increasing distance $r\equiv|j-1|$ for representative points in: (a) KSL phase
with $J=-0.2$, $D=2$; (b) LLRR phase with $J=-0.15$, $D=-0.8$; (c) Haldane
phase with $J=1$, $D=-0.8$; (d) FMz phase $J=-2$, $D=0.1$; (e) FMx phase
$J=-2$, $D=1.2$; and (f) AFz phase $J=1$, $D=-2$. Here we use the iTEBD method
and the bond dimension set as $\chi=120$.

A hallmark of the Haldane phase is the non-local string order parameter, which
was first introduced by den Nijs and Rommelse [130] and later refined by
Tasaki [131]. Its limiting value reveals the hidden
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry breaking

$\displaystyle O_{S}^{\alpha}(i,j)$ $\displaystyle\equiv$ $\displaystyle-{\rm
lim}_{|i-j|\rightarrow\infty}C^{aa}(i,j).$ (43)

This order parameter serves as a distinct feature of the Haldane phase. Figure
10(a) illustrates two-site correlations between sites $1$ and $50$. One
observes that $C^{z}(1,50)$ is finite in the FMz phase, while the string order
parameter $O_{S}^{x}(1,50)$ is nonvanishing in two regions, i.e.,
$-0.22\lesssim J\lesssim-0.04$ and $J\gtrsim 0.03$. To distinguish the two
phases, we plot the spin-spin correlations between site 1 and site $1+r$ for
typical parameters in Fig. 9. One can observe in Fig. 9(b) that both
$C^{x}(1,1+r)$ and $C^{y}(1,1+r)$ alternate between two successive positive
and negative values as the distance of two sites $r$ increases, indicating the
onset of the spin nematic ordering [137], while $C^{x}(1,1+r)$ and
$C^{y}(1,1+r)$ decay exponentially with respect to $r$, manifesting the
existence of the Haldane phase as demonstrated in Fig. 9(c).

Figure 10: (a) Two-point correlation $C^{z}(i,j)$ (38) with $\theta=0$ and the
string order parameter $O_{S}^{x}(i,j)$ (43) between sites $1$ and $50$ for
$D=-0.8$ and $J$ from $-1$ to $1$. (b) Two-point spin-spin correlations
$C^{\alpha}(i,j)$ (38) with $\theta=0$ at $J=-2$. Here we use the iTEBD method
and the bond dimension is set as $\chi=120$.

Furthermore, Fig. 9(d) depicts the correlations for $J=-2$, $D=0.1$, in which
the $z$-component correlations $C^{z}(1,j)$ dominates with a value close to
$1$, implying the FMz ground state, while in Fig. 9(e) the dominant
correlations $C^{x}(1,j)$ characterizes the FMx phase for $J=-2$, $D=1.2$.
Upon increasing $D$ at $J=-2$, the transition from FMz to FMx takes place at
$D_{c}=1$, see Fig. 10(b). Unlike the FMz phase, the joint symmetry (III) is
broken in the FMx phase. By further increasing the value of $D$, the ground
state evolves from the FMx state into the KSL phase, in which the joint
symmetry is restored again. In contrast, the two-site correlation functions in
Fig. 9(f) exhibit a distinct behavior. Specifically, $C^{z}(1,j)$ shows a
periodic oscillation between values close to $-1$ and 1, while $C^{x}(1,j)$
and $C^{y}(1,j)$ nearly vanish. These observations provide strong evidence
that the system is in the AFz phase.

##  V Summary and conclusions

To summarize, we have explored the physics arising from the cooperative effect
of uniaxial single-ion anisotropy (SIA) and Heisenberg interactions in the
spin-1 Kitaev chain. We studied quantum many-body scar (QMBS) states and
quantum phase transitions in spin-1 Kitaev chain with SIA. We find that the
local $\mathbb{Z}_{2}$ gauge fields, a hallmark of Kitaev model, are still
conserved in the spin-1 Kitaev chain with SIA (KD model). In this case, the
Hilbert space is fragmented into $2^{N}$ unequal subspaces characterized by
$\vec{w}=\\{w_{1},w_{n},\cdots,w_{N}\\}$. Among an exponential number of
Krylov subspaces, it has been recognized that in the uniform sector with local
$\mathbb{Z}_{2}$ gauge fields, i.e., $\vec{w}=\\{1,1,\cdots,1\\}$, is the
largest Krylov subspace, in which a local transformation maps the spin-1 KD
model onto the detuned PXP model with spin 1/2 degrees of freedom, and the SIA
acts as a static detuning. The dual transformation suggests a solid-state-
based realization of the PXP model based on Mott insulator with strong spin-
orbit and Hund’s couplings.

Considering the ground states becomes twofold degenerate
$|\mathbb{Z}_{2}\rangle$ state in the limit of $D\to-\infty$ due to the
Hilbert space constraint, a continuous transition in the detuned PXP model
occurs at $D_{c}=-0.655$. This quantum phase transition corresponds to the
emergence of dimer phase induced by the spontaneous breaking of translational
symmetry in the flux-free sector which can be described by the dimer order
parameter (40). We find the most prominent coherent oscillations of quantum
fidelity in the quantum quench from initial states $|\mathbb{Z}_{2}\rangle$
and $|\mathbb{Z}_{3}\rangle$, a characteristic of the embedded prototypical
PXP model for $D=0$. We demonstrate that these fidelity revivals are robust
against small SIA perturbation. The non-thermalizing dynamics can be also
reflected by measuring the expectation values of certain local observables,
which will vanish for $D<D_{c}$. Finally, we provide a complete phase diagram
for the spin-1 KD model by describing the interplay between Kitaev
interactions, Heisenberg interactions and SIA. In particular, we underline the
evolution of the Kitave phase in a broader regime, therefore showing the
relevance for the scar stability and possible solid-state applications. Seven
phases are identified by The numerical methods through the corresponding spin-
spin correlations, including the Kitaev spin liquid, dimer phase, LLRR phase,
Haldane phase, FMx phase, FMz phase and AFz phase. Our study on the higher-
spin Kitaev chain will likely help to identify candidate materials for Kitaev
spin liquid.

###### Acknowledgements.

The authors appreciate very insightful discussions with Hosho Katsura, Gaoyong
Sun and Zhi-Xiang Sun. We acknowledges Ming Xue for bringing Ref. [138] to our
attention. This work is supported by the National Natural Science Foundation
of China (NSFC) under Grant No. 12174194, Postgraduate Research & Practice
Innovation Program of Jiangsu Province, under Grant No. KYCX23_0347, Opening
Fund of the Key Laboratory of Aerospace Information Materials and Physics
(Nanjing University of Aeronautics and Astronautics), MIIT, Top-notch Academic
Programs Project of Jiangsu Higher Education Institutions (TAPP), and stable
supports for basic institute research under Grant No. 190101. A.M.O. kindly
acknowledges Narodowe Centrum Nauki (NCN, Poland) Project No.
2021/43/B/ST3/02166 and is grateful for support via the Alexander von Humboldt
Foundation Fellowship (Humboldt-Forschungspreis).

##  Appendix A Mapping the Kitaev model with single-ion
anisotropy within flux-free sector to the detuned PXP model

For convenience, we use the rotated Hamiltonian Eq. (28) and set $K_{j}=K$.
The local two-spin Hamiltonian is given by

$\displaystyle\tilde{H}_{j,j+1}=KS_{j}^{x}S_{j+1}^{y}+D[(S_{j}^{z})^{2}+(S_{j+1}^{z})^{2}].$
(44)

For the 5 states satisfying $w_{j}=1$, we have

$\displaystyle\tilde{H}_{j,j+1}|xy\rangle$ $\displaystyle=$ $\displaystyle
2D|xy\rangle,\quad\tilde{H}_{j,j+1}|xz\rangle=D|xz\rangle,$
$\displaystyle\tilde{H}_{j,j+1}|zy\rangle$ $\displaystyle=$ $\displaystyle
D|zy\rangle,\quad\;\;\tilde{H}_{j,j+1}|zz\rangle=K|yx\rangle,$
$\displaystyle\tilde{H}_{j,j+1}|yx\rangle$ $\displaystyle=$ $\displaystyle
K|zz\rangle+2D|yx\rangle.$ (45)

Accordingly, the Hamiltonian can be written in the matrix form as

$\displaystyle\tilde{H}_{j,j+1}=\left(\begin{array}[]{ccccc}2D&0&0&0&0\\\
0&D&0&0&0\\\ 0&0&D&0&0\\\ 0&0&0&2D&K\\\ 0&0&0&K&0\\\ \end{array}\right),$ (51)

which yields 5 energy eigenvalues $2D$,$D$,$D$,$D\pm\sqrt{D^{2}+K^{2}}$.
Hence, within the lowest-state manifold residing in the $w_{j}=1$ sector that
is spanned by $\\{|zz\rangle,|yx\rangle\\}$, Eq. (28) can be written for an
effective model of spin-1/2 degrees of freedom, which can be simplified as

$\displaystyle\hat{H}_{\rm KD,eff}=X_{i}+2Dn_{i}.$ (52)

It is noted that the Hilbert space constraint is imposed by the projector onto
the low-energy subspace spanned by configurations with no adjacent excited
states, which is written as

$\displaystyle\mathcal{P}=\prod_{j}(\mathbbm{1}-n_{j}n_{j+1}).$ (53)

The so-called detuned PXP model (36) can be derived via the Schrieffer-Wolff
transformation in the limit of strong interactions (small $\epsilon$) of the
following Hamiltonian

$\hat{H}=\hat{H}_{0}+\epsilon\hat{H}_{1}.$ (54)

The leading part of Eq. (54) $\hat{H}_{0}=\sum_{j=1}^{N}n_{j}n_{j+1}$ vanishes
in this subspace, we must consider the first non-trivial order that is given
by $H_{SW}=\epsilon\mathcal{P}H_{1}\mathcal{P}$. If $\hat{H}_{1}$ describes
the transverse field term,

$\displaystyle\hat{H}_{1}=\sum_{j=1}^{N}X_{j},$ (55)

where $X_{j}$ is defined in Eq. (1), we have

$\displaystyle\mathcal{P}H_{1}\mathcal{P}=\prod_{i=1}^{N}(1-n_{i}n_{i+1})\sum_{j=1}^{N}X_{j}\prod_{k=1}^{N}(1-n_{k}n_{k+1})$
$\displaystyle=\sum_{j}\;(1-n_{j-1}n_{j})(1-n_{j}n_{j+1})X_{j}$
$\displaystyle\quad\quad\times(1-n_{j-1}n_{j})(1-n_{j}n_{j+1})$
$\displaystyle=\sum_{j}(X_{j}-n_{j-1}X_{j}-X_{j}n_{j+1}+n_{j-1}X_{j}n_{j+1})$
$\displaystyle=\sum_{j}(1-n_{j-1})X_{j}(1-n_{j+1})$
$\displaystyle=\sum_{j}P_{j-1}X_{j}P_{j+1}.$ (56)

Then we consider the detuned term,

$\displaystyle\hat{H}_{1}=\sum_{j=1}^{N}n_{j}.$ (57)

In this case, we have

$\displaystyle\mathcal{P}H_{1}\mathcal{P}=\prod_{i}(1-n_{i}n_{i+1})\sum_{j}n_{j}\prod_{k}(1-n_{k}n_{k+1})$
$\displaystyle=\sum_{j}(1\\!-n_{j-1}n_{j})(1\\!-n_{j}n_{j+1})n_{j}(1\\!-n_{j-1}n_{j})(1\\!-n_{j}n_{j+1})$
$\displaystyle=\sum_{j}(n_{j}-n_{j-1}n_{j}-n_{j}n_{j+1}+n_{j-1}n_{j}n_{j+1})$
$\displaystyle=\sum_{j}(1-n_{j-1})n_{j}(1-n_{j+1})$
$\displaystyle=\sum_{j}P_{j-1}n_{j}P_{j+1}.$ (58)

Note that a similar form can be derived even when $\hat{H}_{1}$ is a non-
Hermitian matrix, given by

$\displaystyle\hat{H}_{1}=\sum_{j=1}^{N}iY_{j},$ (59)

we also have

$\displaystyle\mathcal{P}H_{1}\mathcal{P}=\prod_{i}(1-n_{i}n_{i+1})\sum_{j}(iY_{j})\prod_{k}(1-n_{k}n_{k+1})$
$\displaystyle=\sum_{j}(1-n_{j-1}n_{j})(1-n_{j}n_{j+1})(iY_{j})$
$\displaystyle\quad\quad\times(1-n_{j-1}n_{j})(1-n_{j}n_{j+1})$
$\displaystyle=\sum_{j}\left[(iY_{j})-n_{j-1}(iY_{j})-(iY_{j})n_{j+1}+n_{j-1}(iY_{j})n_{j+1}\right]$
$\displaystyle=\sum_{j}(1-n_{j-1})(iY_{j})(1-n_{j+1})$
$\displaystyle=\sum_{j}P_{j-1}(iY_{j})P_{j+1}.$ (60)

However, if $\hat{H}_{1}=\sum_{j=1}^{N}Z_{j}$, the $PZP$ cannot be derived. A
crucial difference is that $X_{j}P_{j}=n_{j}X_{j}$, $Y_{j}P_{j}=n_{j}Y_{j}$,
$n_{j}^{2}=n_{j}$, while $Z_{j}P_{j}\neq n_{j}Z_{j}$.

##  Appendix B Dynamical evolution in a quench
from initial product states

Here we use the similar strategy that is introduced for the dynamics of the
generalized Hubbard models [138]. Considering a Hamiltonian $\hat{H}$ that can
be separated into $\hat{H}_{0}$ and $\lambda\hat{V}$, where $\lambda$ denotes
the perturbation strength, an initial state $|\psi(0)\rangle$ evolves into
$|\psi(t)\rangle=e^{-i\hat{H}t}|\psi(0)\rangle$ at any time $t$. If we can
find an antiunitary operator $\hat{U}$ satisfies the following conditions: (i)
$\hat{U}$ anticommutes with $\hat{H}_{0}$ and commutes with $\hat{V}$, i.e.,

$\displaystyle\\{\hat{U},\hat{H}_{0}\\}=0,\quad[\hat{U},\hat{V}]=0.$ (61)

(ii) The initial state $|\psi(0)\rangle$ only acquires a global phase factor
under $\hat{U}$, i.e.,

$\displaystyle\hat{U}^{-1}|\psi(0)\rangle=e^{i\chi}|\psi(0)\rangle.$ (62)

(iii) We consider a given Hermitian operator $\hat{O}$ that is even or odd
under symmetry operation by $\hat{U}$, i.e.,

$\displaystyle\hat{U}^{-1}\hat{O}\hat{U}=\pm\hat{O},$ (63)

then we can conclude

$\displaystyle\langle\hat{O}\rangle_{+\lambda}=\pm\langle\hat{O}\rangle_{-\lambda}.$
(64)

Back to the KD model in Eq.(28), which can be rewritten as
$\tilde{H}=\tilde{H}_{\rm K}+\hat{H}_{\rm D}$. We then apply
$\hat{U}=\exp(i\pi S_{j}^{x})$, which will yield $S_{j}^{x}\rightarrow
S_{j}^{x},S_{j}^{y}\rightarrow-S_{j}^{y},S_{j}^{z}\rightarrow-S_{j}^{z}$.
Considering the condition (i), one finds

$\displaystyle\hat{U}^{-1}e^{-i(\widetilde{H}_{\rm K}+\hat{H}_{\rm
D})t}\hat{U}=e^{-i(\widetilde{H}_{\rm K}-\hat{H}_{\rm D})t}.$ (65)

To this end, in the quantum quench starting from the initial states
$|\widetilde{\mathbb{Z}}_{k}\rangle$, e.g.,
$|\widetilde{\mathbb{Z}}_{2}\rangle$, we have

$\displaystyle\langle\hat{O}\rangle_{+D}\\!$ $\displaystyle=$
$\displaystyle\langle\widetilde{\mathbb{Z}}_{2}|e^{i(\widetilde{H}_{\rm
K}+\hat{H}_{\rm D})t}\hat{O}e^{-i(\widetilde{H}_{\rm K}+\hat{H}_{\rm
D})t}|\widetilde{\mathbb{Z}}_{2}\rangle$ (66) $\displaystyle=$
$\displaystyle\langle\widetilde{\mathbb{Z}}_{2}|Ue^{i(\widetilde{H}_{\rm
K}-\hat{H}_{\rm D})t}(U^{-1}\hat{O}U)e^{-i(\widetilde{H}_{\rm K}-\hat{H}_{\rm
D})t}U^{-1}|\widetilde{\mathbb{Z}}_{2}\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{O}\rangle_{-D},$ $\displaystyle F(t)_{+D}\\!$
$\displaystyle=$
$\displaystyle\langle\widetilde{\mathbb{Z}}_{2}|e^{-i(\widetilde{H}_{\rm
K}+\hat{H}_{\rm D})t}|\widetilde{\mathbb{Z}}_{2}\rangle$ (67) $\displaystyle=$
$\displaystyle\langle\widetilde{\mathbb{Z}}_{2}|Ue^{-i(\widetilde{H}_{\rm
K}-\hat{H}_{\rm D})t}U^{-1}|\widetilde{\mathbb{Z}}_{2}\rangle=F(t)_{-D},$

where the following simple relations

$\displaystyle\hat{U}^{-1}|\widetilde{\mathbb{Z}}_{2}\rangle\\!=\\!|x(-y)\cdots
x(-y)\rangle\\!=\\!(-1)^{N/2}|xy\cdots xy\rangle,$
$\displaystyle\langle\widetilde{\mathbb{Z}}_{2}|\hat{U}\\!=\\!\langle
x(-y)\cdots x(-y)|\\!=\\!(-1)^{N/2}\langle xy\cdots xy|,$

are used. Therefore, the values of $\langle\hat{O}\rangle$ and $F(t)$ are
symmetric between positive and negative $D$ in the quantum quench starting
from the $|\widetilde{\mathbb{Z}}_{2}\rangle$ state. It is straightforward to
generalize the theorem to other initial product states
$|\widetilde{\mathbb{Z}}_{k}\rangle$ $(k\neq 2)$.

## References

  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  * Deutsch [1991] J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  * Srednicki [1994] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  * Rigol  _et al._ [2008] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008).
  * Rigol and Srednicki [2012] M. Rigol and M. Srednicki, Alternatives to eigenstate thermalization, Phys. Rev. Lett. 108, 110601 (2012).
  * Kim  _et al._ [2014] H. Kim, T. N. Ikeda, and D. A. Huse, Testing whether all eigenstates obey the eigenstate thermalization hypothesis, Phys. Rev. E 90, 052105 (2014).
  * Deutsch [2018] J. M. Deutsch, Eigenstate thermalization hypothesis, Reports on Progress in Physics 81, 082001 (2018).
  * Anderson [1958] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  * Rigol  _et al._ [2007] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons, Phys. Rev. Lett. 98, 050405 (2007).
  * Biroli  _et al._ [2010] G. Biroli, C. Kollath, and A. M. Läuchli, Effect of rare fluctuations on the thermalization of isolated quantum systems, Phys. Rev. Lett. 105, 250401 (2010).
  * Palzer  _et al._ [2009] S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Quantum Transport through a Tonks-Girardeau Gas, Phys. Rev. Lett. 103, 150601 (2009).
  * Gamayun  _et al._ [2014] O. Gamayun, O. Lychkovskiy, and V. Cheianov, Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas, Phys. Rev. E 90, 032132 (2014).
  * Barnett and Seth [2015] L. Barnett and A. K. Seth, Granger causality for state-space models, Phys. Rev. E 91, 040101 (2015).
  * Vidmar and Rigol [2016] L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, Journal of Statistical Mechanics: Theory and Experiment 2016, 064007 (2016).
  * Gornyi  _et al._ [2005] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interacting electrons in disordered wires: Anderson localization and low-$T$ transport, Phys. Rev. Lett. 95, 206603 (2005).
  * Basko  _et al._ [2006] D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Physics 321, 1126 (2006).
  * Pal and Huse [2010] A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  * Lazarides  _et al._ [2015] A. Lazarides, A. Das, and R. Moessner, Fate of many-body localization under periodic driving, Phys. Rev. Lett. 115, 030402 (2015).
  * Altman [2018] E. Altman, Many-body localization and quantum thermalization, Nature Physics 14, 979 (2018).
  * Schreiber  _et al._ [2015] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  * Pakrouski  _et al._ [2020] K. Pakrouski, P. N. Pallegar, F. K. Popov, and I. R. Klebanov, Many-Body Scars as a Group Invariant Sector of Hilbert Space, Phys. Rev. Lett. 125, 230602 (2020).
  * Rigol [2009] M. Rigol, Breakdown of thermalization in finite one-dimensional systems, Phys. Rev. Lett. 103, 100403 (2009).
  * Abanin  _et al._ [2019] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Choi  _et al._ [2019] S. Choi, C. J. Turner, H. Pichler, W. W. Ho, A. A. Michailidis, Z. Papić, M. Serbyn, M. D. Lukin, and D. A. Abanin, Emergent SU(2) Dynamics and Perfect Quantum Many-Body Scars, Phys. Rev. Lett. 122, 220603 (2019).
  * Iadecola  _et al._ [2019] T. Iadecola, M. Schecter, and S. Xu, Quantum many-body scars from magnon condensation, Phys. Rev. B 100, 184312 (2019).
  * Lin  _et al._ [2020a] C.-J. Lin, V. Calvera, and T. H. Hsieh, Quantum many-body scar states in two-dimensional Rydberg atom arrays, Phys. Rev. B 101, 220304 (2020a).
  * Iadecola and Schecter [2020] T. Iadecola and M. Schecter, Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals, Phys. Rev. B 101, 024306 (2020).
  * Bull  _et al._ [2020] K. Bull, J.-Y. Desaules, and Z. Papić, Quantum scars as embeddings of weakly broken Lie algebra representations, Phys. Rev. B 101, 165139 (2020).
  * Turner  _et al._ [2021] C. J. Turner, J.-Y. Desaules, K. Bull, and Z. Papić, Correspondence Principle for Many-Body Scars in Ultracold Rydberg Atoms, Phys. Rev. X 11, 021021 (2021).
  * Ljubotina  _et al._ [2022] M. Ljubotina, B. Roos, D. A. Abanin, and M. Serbyn, Optimal steering of matrix product states and quantum many-body scars, PRX Quantum 3, 030343 (2022).
  * Windt and Pichler [2022] B. Windt and H. Pichler, Squeezing quantum many-body scars, Phys. Rev. Lett. 128, 090606 (2022).
  * Ren  _et al._ [2022] J. Ren, C. Liang, and C. Fang, Deformed symmetry structures and quantum many-body scar subspaces, Phys. Rev. Res. 4, 013155 (2022).
  * Dooley  _et al._ [2023] S. Dooley, S. Pappalardi, and J. Goold, Entanglement enhanced metrology with quantum many-body scars, Phys. Rev. B 107, 035123 (2023).
  * Zhang  _et al._ [2023] P. Zhang, H. Dong, Y. Gao, L. Zhao, J. Hao, J.-Y. Desaules, Q. Guo, J. Chen, J. Deng, B. Liu, W. Ren, Y. Yao, X. Zhang, S. Xu, K. Wang, F. Jin, X. Zhu, B. Zhang, H. Li, C. Song, Z. Wang, F. Liu, Z. Papić, L. Ying, H. Wang, and Y.-C. Lai, Many-body Hilbert space scarring on a superconducting processor, Nature Physics 19, 120 (2023).
  * Geraedts  _et al._ [2016] S. D. Geraedts, R. Nandkishore, and N. Regnault, Many-body localization and thermalization: Insights from the entanglement spectrum, Phys. Rev. B 93, 174202 (2016).
  * Ho  _et al._ [2019] W. W. Ho, S. Choi, H. Pichler, and M. D. Lukin, Periodic orbits, entanglement, and quantum many-body scars in constrained models: Matrix product state approach, Phys. Rev. Lett. 122, 040603 (2019).
  * Moudgalya  _et al._ [2022] S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars and Hilbert space fragmentation: a review of exact results, Reports on Progress in Physics 85, 086501 (2022).
  * Gong and Duan [2013] Z.-X. Gong and L.-M. Duan, Prethermalization and dynamic phase transition in an isolated trapped ion spin chain, New Journal of Physics 15, 113051 (2013).
  * Neyenhuis  _et al._ [2017] B. Neyenhuis, J. Zhang, P. W. Hess, J. Smith, A. C. Lee, P. Richerme, Z.-X. Gong, A. V. Gorshkov, and C. Monroe, Observation of prethermalization in long-range interacting spin chains, Science Advances 3, e1700672 (2017).
  * Tang  _et al._ [2018] Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, M. Rigol, S. Gopalakrishnan, and B. L. Lev, Thermalization near Integrability in a Dipolar Quantum Newton’s Cradle, Phys. Rev. X 8, 021030 (2018).
  * Kao  _et al._ [2021] W. Kao, K.-Y. Li, K.-Y. Lin, S. Gopalakrishnan, and B. L. Lev, Topological pumping of a 1D dipolar gas into strongly correlated prethermal states, Science 371, 296 (2021).
  * Kinoshita  _et al._ [2006] T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature 440, 900 (2006).
  * Xu  _et al._ [2018a] K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, K. Huang, H. Wang, X. Zhu, D. Zheng, and H. Fan, Emulating many-body localization with a superconducting quantum processor, Phys. Rev. Lett. 120, 050507 (2018a).
  * Guo  _et al._ [2021] Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y.-R. Zhang, R. Mondaini, H. Fan, and H. Wang, Observation of energy-resolved many-body localization, Nature Physics 17, 234 (2021).
  * Ren  _et al._ [2021] J. Ren, C. Liang, and C. Fang, Quasisymmetry groups and many-body scar dynamics, Phys. Rev. Lett. 126, 120604 (2021).
  * Affleck  _et al._ [1987] I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  * Moudgalya  _et al._ [2018] S. Moudgalya, N. Regnault, and B. A. Bernevig, Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis, Phys. Rev. B 98, 235156 (2018).
  * Schecter and Iadecola [2019] M. Schecter and T. Iadecola, Weak ergodicity breaking and quantum many-body scars in spin-1 ${XY}$ magnets, Phys. Rev. Lett. 123, 147201 (2019).
  * Desaules  _et al._ [2021] J.-Y. Desaules, A. Hudomal, C. J. Turner, and Z. Papić, Proposal for Realizing Quantum Scars in the Tilted 1D Fermi-Hubbard Model, Phys. Rev. Lett. 126, 210601 (2021).
  * Jaksch  _et al._ [2000] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast Quantum Gates for Neutral Atoms, Phys. Rev. Lett. 85, 2208 (2000).
  * Turner  _et al._ [2018a] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Weak ergodicity breaking from quantum many-body scars, Nature Physics 14, 745 (2018a).
  * Khemani  _et al._ [2019] V. Khemani, C. R. Laumann, and A. Chandran, Signatures of integrability in the dynamics of Rydberg-blockaded chains, Phys. Rev. B 99, 161101 (2019).
  * Lin and Motrunich [2019] C.-J. Lin and O. I. Motrunich, Exact quantum many-body scar states in the Rydberg-blockaded atom chain, Phys. Rev. Lett. 122, 173401 (2019).
  * Mark  _et al._ [2020] D. K. Mark, C.-J. Lin, and O. I. Motrunich, Exact eigenstates in the Lesanovsky model, proximity to integrability and the PXP model, and approximate scar states, Phys. Rev. B 101, 094308 (2020).
  * Mukherjee  _et al._ [2020a] B. Mukherjee, S. Nandy, A. Sen, D. Sen, and K. Sengupta, Collapse and revival of quantum many-body scars via floquet engineering, Phys. Rev. B 101, 245107 (2020a).
  * Serbyn  _et al._ [2021] M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many-body scars and weak breaking of ergodicity, Nature Physics 17, 675 (2021).
  * Roux [2010] G. Roux, Finite-size effects in global quantum quenches: Examples from free bosons in an harmonic trap and the one-dimensional Bose-Hubbard model, Phys. Rev. A 81, 053604 (2010).
  * Sierant and Zakrzewski [2018] P. Sierant and J. Zakrzewski, Many-body localization of bosons in optical lattices, New Journal of Physics 20, 043032 (2018).
  * Zhao  _et al._ [2020] H. Zhao, J. Vovrosh, F. Mintert, and J. Knolle, Quantum many-body scars in optical lattices, Phys. Rev. Lett. 124, 160604 (2020).
  * Mukherjee  _et al._ [2020b] B. Mukherjee, A. Sen, D. Sen, and K. Sengupta, Restoring coherence via aperiodic drives in a many-body quantum system, Phys. Rev. B 102, 014301 (2020b).
  * Halimeh  _et al._ [2022] J. C. Halimeh, I. P. McCulloch, B. Yang, and P. Hauke, Tuning the topological $\theta$-angle in cold-atom quantum simulators of gauge theories, PRX Quantum 3, 040316 (2022).
  * Fendley  _et al._ [2004] P. Fendley, K. Sengupta, and S. Sachdev, Competing density-wave orders in a one-dimensional hard-boson model, Phys. Rev. B 69, 075106 (2004).
  * Trebst  _et al._ [2008] S. Trebst, M. Troyer, Z. Wang, and A. W. W. Ludwig, A Short Introduction to Fibonacci Anyon Models, Progress of Theoretical Physics Supplement 176, 384 (2008).
  * Lesanovsky and Katsura [2012] I. Lesanovsky and H. Katsura, Interacting Fibonacci anyons in a Rydberg gas, Phys. Rev. A 86, 041601 (2012).
  * Chandran  _et al._ [2020] A. Chandran, F. J. Burnell, and S. L. Sondhi, Absence of Fibonacci anyons in Rydberg chains, Phys. Rev. B 101, 075104 (2020).
  * Moessner and Sondhi [2001] R. Moessner and S. L. Sondhi, Ising models of quantum frustration, Phys. Rev. B 63, 224401 (2001).
  * Laumann  _et al._ [2012] C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi, Quantum adiabatic algorithm and scaling of gaps at first-order quantum phase transitions, Phys. Rev. Lett. 109, 030502 (2012).
  * Surace  _et al._ [2020] F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, Lattice Gauge Theories and String Dynamics in Rydberg Atom Quantum Simulators, Phys. Rev. X 10, 021041 (2020).
  * Chen and Iadecola [2021] I.-C. Chen and T. Iadecola, Emergent symmetries and slow quantum dynamics in a Rydberg-atom chain with confinement, Phys. Rev. B 103, 214304 (2021).
  * Desaules  _et al._ [2023a] J.-Y. Desaules, D. Banerjee, A. Hudomal, Z. Papić, A. Sen, and J. C. Halimeh, Weak ergodicity breaking in the schwinger model, Phys. Rev. B 107, L201105 (2023a).
  * Desaules  _et al._ [2023b] J.-Y. Desaules, A. Hudomal, D. Banerjee, A. Sen, Z. Papić, and J. C. Halimeh, Prominent quantum many-body scars in a truncated schwinger model, Phys. Rev. B 107, 205112 (2023b).
  * Sala  _et al._ [2020] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity Breaking Arising from Hilbert Space Fragmentation in Dipole-Conserving Hamiltonians, Phys. Rev. X 10, 011047 (2020).
  * Moudgalya  _et al._ [2020a] S. Moudgalya, N. Regnault, and B. A. Bernevig, $\eta$-pairing in hubbard models: From spectrum generating algebras to quantum many-body scars, Phys. Rev. B 102, 085140 (2020a).
  * Mizuta  _et al._ [2020] K. Mizuta, K. Takasan, and N. Kawakami, Exact Floquet quantum many-body scars under Rydberg blockade, Phys. Rev. Res. 2, 033284 (2020).
  * Mukherjee  _et al._ [2021] B. Mukherjee, Z. Cai, and W. V. Liu, Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models, Phys. Rev. Res. 3, 033201 (2021).
  * Michailidis  _et al._ [2020] A. A. Michailidis, C. J. Turner, Z. Papić, D. A. Abanin, and M. Serbyn, Stabilizing two-dimensional quantum scars by deformation and synchronization, Phys. Rev. Res. 2, 022065 (2020).
  * Turner  _et al._ [2018b] C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations, Phys. Rev. B 98, 155134 (2018b).
  * Su  _et al._ [2023] G.-X. Su, H. Sun, A. Hudomal, J.-Y. Desaules, Z.-Y. Zhou, B. Yang, J. C. Halimeh, Z.-S. Yuan, Z. Papić, and J.-W. Pan, Observation of many-body scarring in a Bose-Hubbard quantum simulator, Phys. Rev. Res. 5, 023010 (2023).
  * Yao  _et al._ [2022] Z. Yao, L. Pan, S. Liu, and H. Zhai, Quantum many-body scars and quantum criticality, Phys. Rev. B 105, 125123 (2022).
  * Peng and Cui [2022] C. Peng and X. Cui, Bridging quantum many-body scars and quantum integrability in Ising chains with transverse and longitudinal fields, Phys. Rev. B 106, 214311 (2022).
  * You  _et al._ [2020] W.-L. You, G. Sun, J. Ren, W. C. Yu, and A. M. Oleś, Quantum phase transitions in the spin-1 Kitaev-Heisenberg chain, Phys. Rev. B 102, 144437 (2020).
  * You  _et al._ [2022] W.-L. You, Z. Zhao, J. Ren, G. Sun, L. Li, and A. M. Oleś, Quantum many-body scars in spin-1 Kitaev chains, Phys. Rev. Res. 4, 013103 (2022).
  * Kitaev [2006] A. Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321, 2 (2006).
  * Jackeli and Khaliullin [2009] G. Jackeli and G. Khaliullin, Mott insulators in the strong spin-orbit coupling limit: From Heisenberg to a quantum compass and Kitaev models, Phys. Rev. Lett. 102, 017205 (2009).
  * Liu  _et al._ [2020] H. Liu, J. c. v. Chaloupka, and G. Khaliullin, Kitaev Spin Liquid in $3d$ Transition Metal Compounds, Phys. Rev. Lett. 125, 047201 (2020).
  * Li  _et al._ [2015] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang, X. Wang, and Q. Zhang, Rare-earth triangular lattice spin liquid: A single-crystal study of $\mathrm{Yb}\mathrm{Mg}\mathrm{Ga}\mathrm{O}_{4}$, Phys. Rev. Lett. 115, 167203 (2015).
  * Ruan  _et al._ [2021] W. Ruan, Y. Chen, S. Tang, J. Hwang, H.-Z. Tsai, R. L. Lee, M. Wu, H. Ryu, S. Kahn, F. Liou, C. Jia, A. Aikawa, C. Hwang, F. Wang, Y. Choi, S. G. Louie, P. A. Lee, Z.-X. Shen, S.-K. Mo, and M. F. Crommie, Evidence for quantum spin liquid behaviour in single-layer $\mathrm{1T}$-$\mathrm{TaSe}_{2}$ from scanning tunnelling microscopy, Nature Physics 17, 1154 (2021).
  * Wu  _et al._ [2022] J. Wu, J. Li, Z. Zhang, C. Liu, Y. H. Gao, E. Feng, G. Deng, Q. Ren, Z. Wang, R. Chen, J. Embs, F. Zhu, Q. Huang, Z. Xiang, L. Chen, Y. Wu, E. S. Choi, Z. Qu, L. Li, J. Wang, H. Zhou, Y. Su, X. Wang, G. Chen, Q. Zhang, and J. Ma, Magnetic field effects on the quantum spin liquid behaviors of $\mathrm{NaYbS_{2}}$, Quantum Frontiers 1, 13 (2022).
  * Khuntia  _et al._ [2020] P. Khuntia, M. Velazquez, Q. Barthélemy, F. Bert, E. Kermarrec, A. Legros, B. Bernu, L. Messio, A. Zorko, and P. Mendels, Gapless ground state in the archetypal quantum kagome antiferromagnet $\mathrm{ZnCu}_{3}\mathrm{(OH)}_{6}\mathrm{Cl}_{2}$, Nature Physics 16, 469 (2020).
  * Shockley  _et al._ [2015] A. C. Shockley, F. Bert, J.-C. Orain, Y. Okamoto, and P. Mendels, Frozen State and Spin Liquid Physics in $\mathrm{Na}_{4}\mathrm{Ir}_{3}\mathrm{O}_{8}$: An NMR Study, Phys. Rev. Lett. 115, 047201 (2015).
  * Banerjee  _et al._ [2017] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B. Stone, M. D. Lumsden, D. G. Mandrus, D. A. Tennant, R. Moessner, and S. E. Nagler, Neutron scattering in the proximate quantum spin liquid $\mathrm{\alpha}$-$\mathrm{Ru}\mathrm{Cl}_{3}$, Science 356, 1055 (2017).
  * Yadav  _et al._ [2018] R. Yadav, R. Ray, M. S. Eldeeb, S. Nishimoto, L. Hozoi, and J. van den Brink, Strong effect of hydrogen order on magnetic Kitaev interactions in $\mathrm{H}_{3}\mathrm{LiIr}_{2}\mathrm{O}_{6}$, Phys. Rev. Lett. 121, 197203 (2018).
  * Pal  _et al._ [2021] S. Pal, A. Seth, P. Sakrikar, A. Ali, S. Bhattacharjee, D. V. S. Muthu, Y. Singh, and A. K. Sood, Probing signatures of fractionalization in the candidate quantum spin liquid $\mathrm{Cu}_{2}\mathrm{IrO}_{3}$ via anomalous raman scattering, Phys. Rev. B 104, 184420 (2021).
  * Imai  _et al._ [2022] Y. Imai, K. Nawa, Y. Shimizu, W. Yamada, H. Fujihara, T. Aoyama, R. Takahashi, D. Okuyama, T. Ohashi, M. Hagihala, S. Torii, D. Morikawa, M. Terauchi, T. Kawamata, M. Kato, H. Gotou, M. Itoh, T. J. Sato, and K. Ohgushi, Zigzag magnetic order in the Kitaev spin-liquid candidate material $\mathrm{RuBr}_{3}$ with a honeycomb lattice, Phys. Rev. B 105, L041112 (2022).
  * Halloran  _et al._ [2023] T. Halloran, F. Desrochers, E. Z. Zhang, T. Chen, L. E. Chern, Z. Xu, B. Winn, M. Graves-Brook, M. B. Stone, A. I. Kolesnikov, Y. Qiu, R. Zhong, R. Cava, Y. B. Kim, and C. Broholm, Geometrical frustration versus Kitaev interactions in $\mathrm{BaCo_{2}(AsO_{4})_{2}}$, Proceedings of the National Academy of Sciences 120, e2215509119 (2023).
  * Gao  _et al._ [2019] B. Gao, T. Chen, D. W. Tam, C.-L. Huang, K. Sasmal, D. T. Adroja, F. Ye, H. Cao, G. Sala, M. B. Stone, C. Baines, J. A. T. Verezhak, H. Hu, J.-H. Chung, X. Xu, S.-W. Cheong, M. Nallaiyan, S. Spagna, M. B. Maple, A. H. Nevidomskyy, E. Morosan, G. Chen, and P. Dai, Experimental signatures of a three-dimensional quantum spin liquid in effective spin-1/2 $\mathrm{Ce}_{2}\mathrm{Zr}_{2}\mathrm{O}_{7}$ pyrochlore, Nature Physics 15, 1052 (2019).
  * Chern  _et al._ [2022] L. E. Chern, Y. B. Kim, and C. Castelnovo, Competing quantum spin liquids, gauge fluctuations, and anisotropic interactions in a breathing pyrochlore lattice, Phys. Rev. B 106, 134402 (2022).
  * Stavropoulos  _et al._ [2019] P. P. Stavropoulos, D. Pereira, and H.-Y. Kee, Microscopic mechanism for a higher-spin Kitaev model, Phys. Rev. Lett. 123, 037203 (2019).
  * Koga  _et al._ [2018] A. Koga, H. Tomishige, and J. Nasu, Ground-state and Thermodynamic Properties of an S = 1 Kitaev Model, Journal of the Physical Society of Japan 87, 063703 (2018).
  * Lee  _et al._ [2020] H.-Y. Lee, N. Kawashima, and Y. B. Kim, Tensor network wave function of S=1 Kitaev spin liquids, Phys. Rev. Res. 2, 033318 (2020).
  * Chen  _et al._ [2022] Y.-H. Chen, J. Genzor, Y. B. Kim, and Y.-J. Kao, Excitation spectrum of spin-1 Kitaev spin liquids, Phys. Rev. B 105, L060403 (2022).
  * Pohle  _et al._ [2023] R. Pohle, N. Shannon, and Y. Motome, Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with Kitaev interactions, Phys. Rev. B 107, L140403 (2023).
  * Taddei  _et al._ [2023] K. M. Taddei, V. O. Garlea, A. M. Samarakoon, L. D. Sanjeewa, J. Xing, T. W. Heitmann, C. dela Cruz, A. S. Sefat, and D. Parker, Zigzag magnetic order and possible Kitaev interactions in the spin-1 honeycomb lattice $\mathrm{KNiAsO}_{4}$, Phys. Rev. Res. 5, 013022 (2023).
  * Mohapatra and Balram [2023] S. Mohapatra and A. C. Balram, Pronounced quantum many-body scars in the one-dimensional spin-1 Kitaev model, Phys. Rev. B 107, 235121 (2023).
  * Xu  _et al._ [2020] C. Xu, J. Feng, M. Kawamura, Y. Yamaji, Y. Nahas, S. Prokhorenko, Y. Qi, H. Xiang, and L. Bellaiche, Possible Kitaev Quantum Spin Liquid State in 2D Materials with $S=3/2$, Phys. Rev. Lett. 124, 087205 (2020).
  * Jin  _et al._ [2022] H.-K. Jin, W. M. H. Natori, F. Pollmann, and J. Knolle, Unveiling the S=3/2 Kitaev honeycomb spin liquids, Nature Communications 13, 3813 (2022).
  * [108] W. M. H. Natori, H.-K. Jin, and J. Knolle, Quantum liquids of the S=3/2 kitaev honeycomb and related kugel-khomskii models, arXiv:2304.13378 (2023) .
  * Fukui  _et al._ [2022] K. Fukui, Y. Kato, J. Nasu, and Y. Motome, Ground-state phase diagram of spin-$S$ Kitaev-Heisenberg models, Phys. Rev. B 106, 174416 (2022).
  * Xu  _et al._ [2018b] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, Interplay between Kitaev interaction and single ion anisotropy in ferromagnetic $\mathrm{CrI}_{3}$ and $\mathrm{CrGeTe}_{3}$ monolayers, npj Computational Materials 4, 57 (2018b).
  * Bradley and Singh [2022] O. Bradley and R. R. P. Singh, Instabilities of spin-1 Kitaev spin liquid phase in presence of single-ion anisotropies, Phys. Rev. B 105, L060405 (2022).
  * Sørensen  _et al._ [2023] E. S. Sørensen, J. Riddell, and H.-Y. Kee, Islands of chiral solitons in integer-spin Kitaev chains, Phys. Rev. Res. 5, 013210 (2023).
  * Fishman [2021] R. S. Fishman, Single-ion anisotropy is necessary and appropriate to study the magnetic behavior of $\mathrm{Tb}^{3+}$ moments with ${J}_{\text{eff}}=\frac{1}{2}$ on the honeycomb lattice in $\mathrm{Tb}_{2}\mathrm{Ir}_{3}\mathrm{Ga}_{9}$, Phys. Rev. B 103, 214440 (2021).
  * Zaletel  _et al._ [2015] M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, and F. Pollmann, Time-evolving a matrix product state with long-ranged interactions, Phys. Rev. B 91, 165112 (2015).
  * Fishman  _et al._ [2022] M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calculations, SciPost Phys. Codebases , 4 (2022).
  * Vidal [2007] G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, 070201 (2007).
  * Sen  _et al._ [2010] D. Sen, R. Shankar, D. Dhar, and K. Ramola, Spin-1 Kitaev model in one dimension, Phys. Rev. B 82, 195435 (2010).
  * Sugita  _et al._ [2020] Y. Sugita, Y. Kato, and Y. Motome, Antiferromagnetic Kitaev interactions in polar spin-orbit Mott insulators, Phys. Rev. B 101, 100410 (2020).
  * Sears  _et al._ [2020] J. A. Sears, L. E. Chern, S. Kim, P. J. Bereciartua, S. Francoual, Y. B. Kim, and Y.-J. Kim, Ferromagnetic Kitaev interaction and the origin of large magnetic anisotropy in $\alpha$-$\mathrm{RuCl}_{3}$, Nature Physics 16, 837 (2020).
  * Moudgalya  _et al._ [2020b] S. Moudgalya, B. A. Bernevig, and N. Regnault, Quantum many-body scars in a Landau level on a thin torus, Phys. Rev. B 102, 195150 (2020b).
  * [121] A. Daniel, A. Hallam, J.-Y. Desaules, A. Hudomal, G.-X. Su, J. C. Halimeh, and Z. Papić, Bridging quantum criticality via many-body scarring, arXiv:2301.03631 (2023) .
  * [122] The data and the code that support the findings of this study are available from the corresponding authors upon reasonable request.
  * Byrnes  _et al._ [2002] T. M. R. Byrnes, P. Sriganesh, R. J. Bursill, and C. J. Hamer, Density matrix renormalization group approach to the massive Schwinger model, Phys. Rev. D 66, 013002 (2002).
  * Rico  _et al._ [2014] E. Rico, T. Pichler, M. Dalmonte, P. Zoller, and S. Montangero, Tensor networks for lattice gauge theories and atomic quantum simulation, Phys. Rev. Lett. 112, 201601 (2014).
  * Yang  _et al._ [2020] B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C. Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Observation of gauge invariance in a 71-site Bose–Hubbard quantum simulator, Nature 587, 392 (2020).
  * [126] M. V. Damme, J. C. Halimeh, and P. Hauke, Gauge-symmetry violation quantum phase transition in lattice gauge theories, arXiv:2010.07338 (2020) .
  * Lin  _et al._ [2020b] C.-J. Lin, A. Chandran, and O. I. Motrunich, Slow thermalization of exact quantum many-body scar states under perturbations, Phys. Rev. Res. 2, 033044 (2020b).
  * Langlett  _et al._ [2022] C. M. Langlett, Z.-C. Yang, J. Wildeboer, A. V. Gorshkov, T. Iadecola, and S. Xu, Rainbow scars: From area to volume law, Phys. Rev. B 105, L060301 (2022).
  * [129] L. Gotta, S. Moudgalya, and L. Mazza, Asymptotic quantum many-body scars, arXiv:2303.05407 (2023) .
  * den Nijs and Rommelse [1989] M. den Nijs and K. Rommelse, Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains, Phys. Rev. B 40, 4709 (1989).
  * Tasaki [1991] H. Tasaki, Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap, Phys. Rev. Lett. 66, 798 (1991).
  * Liu  _et al._ [2015] G.-H. Liu, L.-J. Kong, and W.-L. You, Quantum phase transitions in spin-1 compass chains, The European Physical Journal B 88, 284 (2015).
  * Winter  _et al._ [2017] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and materials for generalized Kitaev magnetism, Journal of Physics: Condensed Matter 29, 493002 (2017).
  * You and Tian [2008] W.-L. You and G.-S. Tian, Quantum phase transition in the one-dimensional compass model using the pseudospin approach, Phys. Rev. B 78, 184406 (2008).
  * Trousselet  _et al._ [2010] F. Trousselet, A. M. Oleś, and P. Horsch, Compass-Heisenberg model on the square lattice —spin order and elementary excitations, EPL (Europhysics Letters) 91, 40005 (2010).
  * Trousselet  _et al._ [2012] F. Trousselet, A. M. Oleś, and P. Horsch, Magnetic properties of nanoscale compass-Heisenberg planar clusters, Phys. Rev. B 86, 134412 (2012).
  * Zvyagin  _et al._ [2023] A. A. Zvyagin, V. V. Slavin, and G. A. Zvyagina, Manifestation of spin nematic ordering in the spin-1 chain system, Phys. Rev. B 107, 134421 (2023).
  * Yu  _et al._ [2017] J. Yu, N. Sun, and H. Zhai, Symmetry Protected Dynamical Symmetry in the Generalized Hubbard Models, Phys. Rev. Lett. 119, 225302 (2017).