# Meson Instability of Quantum Many-body Scars in a 1D Lattice Gauge Theory

Zi-Yong Ge<EMAIL_ADDRESS>Theoretical Quantum Physics Laboratory, Cluster
for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Yu-Ran
Zhang<EMAIL_ADDRESS>Theoretical Quantum Physics Laboratory, Cluster
for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan  Quantum
Information Physics Theory Research Team, Center for Quantum Computing, RIKEN,
Wako-shi, Saitama 351-0198, Japan  Franco Nori<EMAIL_ADDRESS>Theoretical
Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi,
Saitama 351-0198, Japan  Quantum Information Physics Theory Research Team,
Center for Quantum Computing, RIKEN, Wako-shi, Saitama 351-0198, Japan
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040,
USA

###### Abstract

We investigate the stability of meson excitations (particle-antiparticle bound
states) in quantum many-body scars of a 1D $\mathbb{Z}_{2}$ lattice gauge
theory coupled to a dynamical spin-$\frac{1}{2}$ chain as a matter field. By
introducing a string representation of the physical Hilbert space, we express
a scar state $\ket{\Psi_{n,l}}$ as a superposition of all string bases with an
identical string number $n$ and a total length $l$. For the small-$l$ scar
state $\ket{\Psi_{n,l}}$, the gauge-invariant spin exchange correlation
function of the matter field hosts an exponential decay as the distance
increases, indicating the existence of stable mesons. However, for large $l$,
the correlation function exhibits a power-law decay, signaling the emergence
of a meson instability. Furthermore, we show that this mesonic-nonmesonic
crossover can be detected by the quench dynamics, starting from two low-
entangled initial states, respectively, which are experimentally feasible in
quantum simulators. Our results expand the physics of quantum many-body scars
in lattice gauge theories and reveal that the nonmesonic state can also
manifest ergodicity breaking.

Introduction.—Due to the development of quantum simulations [1, 2, 3, 4, 5, 6,
7, 8], out-of-equilibrium quantum many-body physics has been attracting
growing interests [9]. The Eigenstate Thermalization Hypothesis (ETH)
postulates that generic isolated nonintegrable quantum many-body systems
exhibit ergodicity [10, 11, 12, 13, 14], and thus, the unitary quantum
evolution of the systems can result in an equilibrium state described by
statistical mechanics. Though ETH was thought to be general, there are several
counter-examples, e.g., quantum integrable systems [15, 16] and many-body
localizations [17, 18, 19, 20, 21, 22, 23]. These two examples are called
strong ergodicity breaking, since most of the eigenstates violate the ETH.
Recent experimental and theoretical works demonstrate that there exist a new
type of ETH-violating eigenstates in some specific nonintegrable quantum many-
body systems, dubbed quantum many-body scar (QMBS) states [24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Generally,
the number of QMBS states is exponentially smaller than the Hilbert space
dimension, so they can be considered as weak ergodicity breaking. One typical
class of QMBS states are constructed by spectrum-generated algebras [45, 38,
37, 43], whose eigenenergies are equally spaced, dubbed towers of QMBSs [43].
Thus, if the initial state is a superposition of these scar states, there will
exist a perfect revival dynamics indicating the ETH violation.

Empirically, kinetically constrained systems are thought more likely to host
QMBSs. Thus, as a typical instance, lattice gauge theories (LGTs) [46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] have attracted
considerable interest to study QMBSs [63, 64, 65, 66, 67, 68, 69, 70].
Meanwhile, particles can be pairwise confined into mesons in LGTs, which is
closely related to non-thermal states [63, 64, 65, 66, 67, 71, 72]. For
instance, the scar dynamics in the PXP model [65] can also be understood as
the string inversion of a $U(1)$ LGT, where a particle and an antiparticle are
always bounded into a stable meson. Therefore, one natural question is whether
stable mesons are a necessary condition of ETH violation in LGTs. A recent
work [36] reports a special type of QMBSs in a spin chain with conserved
domain wall (equivalent to a $\mathbb{Z}_{2}$ LGT), which are generated by
nonlocal operators. Based on these QMBSs, we address the above questions, and
further reveal more nontrivial physics when investigating QMBSs in LGTs.

In this Letter, we study this type of QMBSs in a $\mathbb{Z}_{2}$ LGT and
demonstrate that it can manifest both mesonic and nonmesonic features. First,
we introduce a string representation to describe the physical Hilbert space in
a specific gauge sector. In this representation, the exact wave function of
the QMBS state $\ket{\Psi_{n,l}}$ is written as an equal superposition of all
string bases with an identical string number $n$ and a total string length
$l$. We identify the instability of mesons in $\ket{\Psi_{n,l}}$ by
calculating the gauge-invariant spin exchange correlation function of the
matter field. Our results show that mesons are stable for small $l$, while
these become unstable for large $l$. Moreover, we propose a feasible approach
to detect the crossover between stable and unstable mesons in quantum
simulators, by observing the quench dynamics of the system initially at two
different low-entangled states, respectively.

Model.—Here we consider a 1D $\mathbb{Z}_{2}$ LGT minimally coupled to a
dynamical spin-$\frac{1}{2}$ chain as a matter field. The Hamiltonian has a
form $\hat{H}=\hat{H}_{K}+H_{E}+\hat{H}_{\mu}$, with

$\displaystyle\hat{H}_{K}=-J\sum_{j=1}^{L}\big{(}\hat{\sigma}^{+}_{j}\hat{\tau}^{z}_{j+\frac{1}{2}}\hat{\sigma}^{-}_{j+1,}+\text{H.c.}\big{)},$
$\displaystyle\hat{H}_{E}=-h\sum_{j=1}^{L}\hat{\tau}_{j+\frac{1}{2}}^{x},\ \ \
\ \hat{H}_{\mu}=\mu\sum_{j=1}^{L}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j},$
(1)

where $\hat{\sigma}^{\alpha}_{j}$ and $\hat{\tau}_{j+\frac{1}{2}}^{\alpha}$
are both Pauli matrices describing the matter and gauge fields, respectively,
and $L$ is the system size. The matter field lives on the site, while the
gauge field is on the link. The kinetic term $\hat{H}_{K}$ describes the
minimal gauge-matter coupling with strength $J$, the second term $\hat{H}_{E}$
describes an electric field with strength $h$, and the last term
$\hat{H}_{\mu}$ denotes the potential of the matter field. We consider
periodic boundary conditions, i.e.,
$\hat{\sigma}^{\alpha}_{1}=\hat{\sigma}^{\alpha}_{L+1}$ and
$\hat{\tau}_{1+\frac{1}{2}}^{\alpha}=\hat{\tau}_{L+1+\frac{1}{2}}^{\alpha}$.
The Hamiltonian $\hat{H}$ is $\mathbb{Z}_{2}$ gauge invariant with a generator
$\hat{G}_{j}=\hat{\tau}_{j-\frac{1}{2}}^{x}\hat{\sigma}^{z}_{j}\hat{\tau}_{j+\frac{1}{2}}^{x}$.
In addition to the gauge structure, there also exists the spin $U(1)$ symmetry
in $\hat{H}$, where the total spin charge
$\sum_{j=1}^{L}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j}$ is conserved. Note
that this $\mathbb{Z}_{2}$ LGT can be experimentally addressed in various
quantum simulators [73, 74, 75, 76, 77, 78, 79, 80].

Without loss of generality, we fix the system to the gauge sector
$\hat{G}_{j}=1$, with an even number of total spin charges for each physical
basis. With a dual transformation, the original Hamiltonian $\hat{H}$ can be
mapped to a local spin chain [81]

$\displaystyle\hat{H}_{K}=-\frac{J}{2}\sum_{j}(\hat{Z}_{j+\frac{1}{2}}-\hat{X}_{j-\frac{1}{2}}\hat{Z}_{j+\frac{1}{2}}\hat{X}_{j+\frac{3}{2}}),$
$\displaystyle\hat{H}_{E}=-h\sum_{j}\hat{X}_{j+\frac{1}{2}},\ \ \ \
\hat{H}_{\mu}=\frac{\mu}{2}\sum_{j}(1-\hat{X}_{j-\frac{1}{2}}\hat{X}_{j+\frac{1}{2}}),$
(2)

where $\hat{X}_{j+\frac{1}{2}}=\hat{\tau}^{x}_{j+\frac{1}{2}}$ and
$\hat{Z}_{j+\frac{1}{2}}=\hat{\sigma}^{x}_{j}\hat{\tau}^{z}_{j+\frac{1}{2}}\hat{\sigma}^{x}_{j+1}$
are also Pauli matrices.

Figure S1(a) plots the half-chain von Neumann entropies,
$S=\text{Tr}\hat{\rho}_{L/2}\ln\hat{\rho}_{L/2}$, of the whole eigenstates in
the half-filling case
($\sum_{j=1}^{L}\hat{\sigma}^{+}_{j}\hat{\sigma}^{-}_{j}=L/2$), where
$\hat{\rho}_{L/2}$ is the half-chain density matrix. The entanglement
entropies of states near the middle of the spectrum approach the value for a
random state $S^{\text{ran}}=(L\ln 2-1)/2$ [82], which demonstrates that most
of the eigenstates obey ETH in this gauge sector.

As shown in Ref. [36], a pyramid-like structure of scar states exists for the
dual Hamiltonian (Meson Instability of Quantum Many-body Scars in a 1D Lattice
Gauge Theory), see Figs. S1(a,b). These scar states cannot be generated by
local operators and are very distinct from the conventional towers of QMBSs.
However, they have not been fully investigated. Specifically, it is still
unclear whether stable meson excitations dominate the scar dynamics, like most
of conventional QMBSs in LGTs. In addition, an experimental proposal to detect
these QMBSs from the nontrivial quench dynamics in quantum simulators is also
a relevant issue. Hereafter, we investigate the QMBSs of $\hat{H}$ from the
viewpoint of LGTs, and uncover whether these can be described by mesonic
physics. Note that while analytical discussions are based on the original
Hamiltonian Eq. (Meson Instability of Quantum Many-body Scars in a 1D Lattice
Gauge Theory), the numerical results are obtained from the Hamiltonian (Meson
Instability of Quantum Many-body Scars in a 1D Lattice Gauge Theory) by exact
diagonalization.

Figure 1: Half-chain von Neumann entropy $S$. (a) The distribution of $S$ for
all eigenstates of Hamiltonian (Meson Instability of Quantum Many-body Scars
in a 1D Lattice Gauge Theory) in the half-filling sector (i.e., string number
$n=L/4$) with $L=16$, $J=1$, $h=0.5$. The color codes the density of states
(warmer colors imply higher density). The blue dashed line represents the
entanglement entropy of the random state $S^{\text{ran}}=(L\ln 2-1)/2\approx
5.05$. The data points in red circles correspond to the eigenstates in Eq.
(6). (b) Pyramid-like structure of $\ket{\Psi_{n,l}}$ for $L=20$. Different
colors represent different numbers of strings $n$. (c) The size scaling of the
entanglement entropy for the QMBS state $\ket{\Psi_{L/4,L/2}}$. The dashed
line shows linear fitting: $S\sim\ln L$.

String representation.—Before discussing the QMBSs, we introduce string bases
in the $\hat{G}_{j}=1$ sector, which are convenient for discussing meson
excitations. Due to gauge invariance, the physical Hilbert space in the fixed
gauge sector can be represented by open strings. The vacuum state of string
excitations in the $\hat{G}_{j}=1$ sector can be defined as

$\displaystyle\ket{\Omega}:=\ket{\downarrow\downarrow...\downarrow\downarrow}\otimes\ket{++...++},$
(3)

where $\ket{\downarrow\downarrow...\downarrow\downarrow}$ is a ferromagnetic
state of matter fields, and $\ket{\tau}=\ket{++...++}$ is the state of gauge
fields with all links being polarized at $\hat{\tau}^{x}=1$. A state with one
string excitation can be written as

$\displaystyle\ket{\mathcal{S}_{k,\ell}}:=\hat{\mathcal{S}}_{k,\ell}^{\dagger}\ket{\Omega}=\hat{\sigma}_{k}^{+}\big{(}\prod_{k\leq
j<k+\ell}\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}\big{)}\hat{\sigma}_{k+\ell}^{+}\ket{\Omega},$
(4)

where $\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}=\ket{-}\bra{+}$, and $k$ denotes
the string position, and $\ell$ denotes the string length. We can also define
the parity as $P_{\mathcal{S}_{k,\ell}}:=\exp{(-i\pi k)}$, which is determined
by the string position. Note that, if the operator
$\hat{\mathcal{S}}_{k,\ell}^{\dagger}$ is local [i.e., $\ell\sim O(1)$], the
corresponding string excitation can be regarded as a meson.

An arbitrary gauge invariant basis of the Hamiltonian (Meson Instability of
Quantum Many-body Scars in a 1D Lattice Gauge Theory) can be written as

$\displaystyle\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}=\hat{\mathcal{S}}_{k_{1},\ell_{1}}^{\dagger}\hat{\mathcal{S}}_{k_{2},\ell_{2}}^{\dagger}...\hat{\mathcal{S}}_{k_{n-1},\ell_{n-1}}^{\dagger}\hat{\mathcal{S}}_{k_{n},\ell_{n}}^{\dagger}\ket{\Omega},$
(5)

where $k_{j}>k_{i}+\ell_{i}$, for $j>i$, $n$ is the number of strings,
$l:=\sum_{j=1}^{n}\ell_{j}$ is the total string length, and the parity is
$P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}=\exp{(-i\pi\sum_{j=1}^{n}k_{j})}$.
Here, each string excitation contains two spin charges, so the string number
$n$ equals half of total spin charges. Due to the $U(1)$ symmetry of
$\hat{H}$, the quantum number $n$ is conserved. However, the total string
length and the parity are not invariant under the action of $\hat{H}_{K}$.
While $l$ determines the energy of the electric-field term
$\hat{H}_{E}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}=h(2l-L)\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}$,
$n$ determines the energy of the potential term
$\hat{H}_{\mu}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}=2\mu
n\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}$.

Exact quantum many-body scars.—We introduce the pyramid-like QMBS states in
this $\mathbb{Z}_{2}$ LGT [36], whose wave functions in the string
representation are written as

$\displaystyle\ket{\Psi_{n,l}}=\mathcal{N}_{n,l}\sum_{\\{k_{j},\ell_{j}\\}}P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}},$
(6)

where $\mathcal{N}_{n,l}$ is a normalization factor. That is,
$\ket{\Psi_{n,l}}$ is an equal superposition of all string bases with both the
same string number $n$ and total length $l$, and the phase is determined by
the parity of each basis. Since $0<\ell_{j}<L$, the quantum numbers $n$ and
$l$ satisfy $n\leq l\leq L-n$. Thus, there are $(L-2n+1)$ of eigenstates in
the sector with $\sum_{j=1}^{L}\hat{\sigma}_{j}^{+}\hat{\sigma}_{j}^{-}=2n$
total spin charge. It can be proved that $\hat{H}_{K}\ket{\Psi_{n,l}}=0$ [see
more details in Supplemental Material (SM) [83]], and the eigenenergy of
$\ket{\Psi_{n,l}}$ is ${\varepsilon}_{n,l}=2hl+2\mu n-hL$, which can be away
from edges of the spectrum corresponding to a high-energy eigenstate, see Fig.
S1(a). In addition, the scar states $\ket{\Psi_{n,l}}$ host the sub-volume-law
entanglement entropy, i.e., $S\sim\ln L$, demonstrating the ETH violation, see
Fig. S1(c).

The scar state $\ket{\Psi_{n,l}}$ can also be expressed in terms of generating
operators. First, we consider a simple case $\ket{\Psi_{n,n}}$, which only
contains $n$ length-$1$ string excitation. We can construct a ladder operator
[66]
$\hat{S}^{\dagger}:=\sum_{j}P_{\mathcal{S}_{j,1}}\hat{\mathcal{S}}_{j,1}^{\dagger}$,
and the eigenstate $\ket{\Psi_{n,n}}$ can be obtained as

$\displaystyle\ket{\Psi_{n,n}}=\mathcal{A}_{n}(\hat{S}^{\dagger})^{n}\ket{\Omega},$
(7)

where $\mathcal{A}_{n}$ is a normalization factor. Then, we introduce another
operator [36] $\hat{L}^{\dagger}_{m}=\sum_{j}\big{(}\sum_{k\leq
m}\prod_{\ell\leq
k}\hat{\mathcal{P}}^{-}_{j+\frac{1}{2}-\ell}\big{)}\hat{\sigma}^{-}_{j}\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}\hat{\sigma}^{+}_{j+1}$,
where $\hat{\mathcal{P}}^{-}:=\ket{-}\bra{-}$. The action of
$\hat{L}^{\dagger}_{m}$ is to enlarge the total string length by 1 without
changing the parity. Using $\hat{L}_{m}^{\dagger}$, we obtain the eigenstate
$\ket{\Psi_{n,n+m}}$ as [36, 83]

$\displaystyle\ket{\Psi_{n,n+m}}=\mathcal{D}_{n,m}\hat{L}_{m}^{\dagger}\ket{\Psi_{n,n+m-1}}$
(8)

where $\mathcal{D}_{n,m}$ is a normalization factor. Equations (7, 8) indicate
that the scar state $\ket{\Psi_{n,n}}$, like the conventional tower of QMBSs,
is generated by local operators, while $\ket{\Psi_{n,n+m}}$ is generated by
nonlocal operators. Thus, intuitively, as the total string length $l$
increases, the stability of mesons for the scar state $\ket{\Psi_{n,l}}$ is
expected to be significantly broken.

Figure 2:  Stability of mesons. (a) Expectation values of the electric field
for all eigenstates at half filling with $L=16$ and $h=0.5$. The red scatters
correspond to scar states. (b,c) Gauge-invariant spin exchange correlation
function of the matter field $C_{\sigma}(r)$ defined in Eq. (9) for $l=L/4$
and $l=L/2$, respectively. The total string number is $n=L/4$. The black
dashed line is for the fit: $|C_{\sigma}(r)|\sim 1/r^{\alpha}$, where
$\alpha\approx 0.35$. (d) The half-chain correlation function
$C_{\sigma}(L/2)$ versus the ratio $l/L$ for different system sizes with
$n=L/4$.

Emergent instability of mesons.—Mesons, as a type of particle-antiparticle
bound states, play an important role in the dynamics of LGTs. If the system is
in a confined phase, the low-energy excitation is described by mesons. In
addition, in a high-energy regime, meson dynamics also closely relate to the
ETH. Previous works have shown that almost all of the nonthermal dynamics in
LGTs originate from stable meson excitations [63, 64, 65, 66, 67, 71, 72],
including the revival dynamics in the PXP model [65]. Thus, one natural
question is whether a stable meson excitation is a necessary condition to
induce an ETH violation in LGTs. For the Hamiltonian (Meson Instability of
Quantum Many-body Scars in a 1D Lattice Gauge Theory), the $\sigma$-spin is
confined in the ground state with an arbitrary finite $h$ [81], where the low-
energy excitation is a meson. However, for the high-energy dynamics,
especially the scar dynamics, whether mesons are still stable is an open
question.

According to Eq. (5), the $\sigma$-spin charges appear pairwise forming string
excitations. For small $l$, e.g., $\ket{\Psi_{n,n}}$, two $\sigma$-spin
charges are always bonded together on two nearest-neighbor sites, i.e., there
only exist local string excitations (mesons). This suggests that these scar
states should be described by stable mesons. However, as $l$ increases, the
distance between two $\sigma$-spin charges of a string excitation becomes
large, and nonlocal string excitations can emerge. Hence, intuitively,
isolated $\sigma$-spin charges are expected to exist in this case, leading to
the instability of mesons. In Fig. 2(a), we present the expectation value of
the electric field, defined as
$\hat{M}_{x}:=\sum_{j=1}^{L}\hat{\tau}_{j+\frac{1}{2}}^{x}$. It shows that
$\braket{\hat{M}_{x}}$ of small-$l$ scar states are located at the edge of the
spectrum, which is similar to the confinement-induced non-thermal states in
Refs. [63, 64]. This suggests that mesons in these scar states would be
stable. However, for large-$l$ scar states, $\braket{\hat{M}_{x}}$ is located
in the main spectrum, implying that mesons may be unstable.

To further verify the above picture, we perform numerical simulations by
calculating the gauge-invariant spin exchange correlation function versus the
distance $r$ [81]

$\displaystyle
C_{\sigma}(r):=\bra{\Psi_{n,l}}\big{[}\hat{\sigma}_{j}^{+}(\prod_{j\leq
k<j+r}\hat{\tau}^{z}_{k+\frac{1}{2}})\hat{\sigma}^{-}_{j+r}+h.c.\big{]}\ket{\Psi_{n,l}},$
(9)

which identifies elementary excitations. As shown in Fig. 2, the QMBS states
$\ket{\Psi_{L/4,L/4}}$ and $\ket{\Psi_{L/4,L/2}}$ are studied with $L=32$. For
$\ket{\Psi_{L/4,L/4}}$, we can find that $C_{\sigma}(r)$ approximately
exhibits an exponential decay with increasing $r$, see Fig. 2(b). It indicates
that $\sigma$-spin charges tend to form stable bound states (i.e., mesons) for
small-$l$ scar states [81]. However, $C_{\sigma}(r)$ can exhibit a power-law
decay for the state $\ket{\Psi_{L/4,L/2}}$, showing that the isolated
$\sigma$-spin charges emerge for large $l$, see Fig. 2(c). Thus, the
instability of mesons in QMBSs $\ket{\Psi_{n,l}}$ arises, when increasing $l$.
Therefore, there exists a mesonic-nonmesonic crossover for these QMBSs, see
Fig. 2(d). Figure 2 also reveals that the stable meson excitation is not a
necessary condition to violate the ETH in LGTs. Note that the meson
instability is a collective effect, which only emerges in many-particle
systems, i.e., the filling factor $n/L$ is finite.

Figure 3: Time evolution of the fidelity $\mathcal{F}(t)$ after a quantum
quench for the initial states (a) $\ket{\psi_{1}}$ and (b) $\ket{\psi_{2}}$ in
Eq. (10). Here, we choose $J=1$, $h=0.5$, and $L=16$.

Quench dynamics.—Another problem is whether the QMBS states $\ket{\Psi_{n,l}}$
can lead to nontrivial quench dynamics, which can be experimentally observed
in quantum simulators. Here, we introduce two initial states

$\displaystyle\ket{\psi_{1}}=\mathcal{B}\prod_{j}\big{[}1+(-1)^{j}\hat{\sigma}^{+}_{j}\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}\hat{\sigma}^{+}_{j+1}\big{]}\ket{\Omega},$
(10a)
$\displaystyle\ket{\psi_{2}}=\frac{1}{2^{L/2}}\sum_{n,l}\sum_{\\{k_{j},\ell_{j}\\}}P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}},$
(10b)

where $\mathcal{B}$ is the normalization factor. Here, $\ket{\psi_{1}}$ is a
superposition of scar states $\ket{\Psi_{n,n}}$, i.e.,
$\ket{\psi_{1}}=\sum_{n}\alpha_{n}\ket{\Psi_{n,n}}$, and $\ket{\psi_{2}}$ is a
superposition of all scar states $\ket{\Psi_{n,l}}$, i.e.,
$\ket{\psi_{2}}=\sum_{n,l}\beta_{n,l}\ket{\Psi_{n,l}}$. It is obvious that
both initial states host low entanglement entropies. Specifically, with the
dual transformation in Eq. (Meson Instability of Quantum Many-body Scars in a
1D Lattice Gauge Theory), $\ket{\psi_{1}}$ is related to the ground state of
the PXP model [66], and
$\ket{\psi_{2}}=\bigotimes_{j}\ket{V_{2j+\frac{1}{2},2j+\frac{3}{2}}}$, with
$\ket{V_{2j+\frac{1}{2},2j+\frac{3}{2}}}=(\ket{++}-\ket{+-}+\ket{-+}+\ket{--})/2$.

In Fig. 3, we present the fidelity
$\mathcal{F}(t):=|\braket{\psi_{1,2}}{e^{-i\hat{H}t}}{\psi_{1,2}}|^{2}$, where
the initial state evolves with $\hat{H}$. It shows that $\mathcal{F}(t)$ can
exhibit perfect revival dynamics for both initial states in Eq. (10). For the
initial state $\ket{\psi_{1}}$, the oscillation period is $T=\pi/(h+\mu)$, see
Fig. 3(a). For the initial state $\ket{\psi_{2}}$, if $h/\mu=p/q$, with $p$
and $q$ being relatively prime, then the time for a perfect revival is
$T=p\pi/h=q\pi/\mu$, see Fig. 3(b). The oscillation period is consistent with
the eigenenergies of $\ket{\Psi_{n,l}}$. Moreover, the revival dynamics
signals the ETH violation for QMBSs $\ket{\Psi_{n,l}}$.

We also probe the stability of mesons during the quench dynamics. Since
$\ket{\psi_{1}}$ is a superposition of small-$l$ scar states, we expect a
mesonic quench dynamics, i.e., mesons are always stable during the dynamics,
like the scar dynamics in the PXP model [65]. For the initial state
$\ket{\psi_{2}}$, although it is a superposition of all scar states, the
large-$l$ scar states should be dominant [83], e.g.,
$|\beta_{L/4,L/2}|\gg|\beta_{L/4,L/4}|$. Thus, it leads to nonmesonic
dynamics. We calculate the gauge invariant correlation function
$C_{\sigma}(r)$ in Eq. (9) to identify whether meson excitations are stable
during the quench dynamics. For $\ket{\psi_{1}}$, we can find that
$\sigma$-spins are always short-range correlated, and $C_{\sigma}(r)$ exhibits
an exponential decay as $r$ increases during the quench dynamics, see Figs.
4(a,b). This indicates that mesons are very stable and cannot be decomposed
into isolated spin charges. However, the situation becomes different for
$\ket{\psi_{2}}$, where $\sigma$-spins can be long-range correlated
[$C_{\sigma}(r)\sim\text{const}$ for $r\rightarrow\infty$] at the specific
time, see Figs. 4(c,d). Mesons in this case are unstable and can be decomposed
into isolated spin charges. Therefore, the initial states in Eq. (10) can be
used to detect the mesonic-nonmesonic crossover for QMBSs $\ket{\Psi_{n,l}}$
during their quench dynamics.

Figure 4: Time evolution of the correlation function $C_{\sigma}(r)$ for two
different initial states (a,b) $\ket{\psi_{1}}$ and (c,d) $\ket{\psi_{2}}$.
Here, we choose $J=1$, $h=0.5$, $\mu=0.4$, and $L=24$.

Experimental proposal.—The preparation of the initial states in Eq. (10) in
dual systems is convenient in quantum simulators. In addition, the dual
Hamiltonian (Meson Instability of Quantum Many-body Scars in a 1D Lattice
Gauge Theory), with three-body interaction terms, has also been realized by
quantum gates, e.g, in superconducting circuits [78, 84]. Therefore, the
mesonic-nonmesonic crossover for QMBSs in this $\mathbb{Z}_{2}$ LGT can be
experimentally detected with digital quantum simulations.

Summary.—We have investigated the instability of mesons in QMBSs of a
$\mathbb{Z}_{2}$ LGT, minimally coupled to a dynamical spin-$\frac{1}{2}$
chain as a matter field. By introducing the string representation, we express
the wave function of each QMBS as an equal superposition of all string bases
with an identical string number and total string length. We demonstrate that
scar states with a small total string length is described by stable mesons,
like conventional nonthermal states in LGTs, while we find the meson
instability in the scar states with large total string length. Furthermore,
this instability of mesons in QMBSs can be observed from the quench dynamics
with two experimentally accessible initial states. Our results bring new
insights into QMBSs in LGTs and reveal that the nonmesonic states can also
host ergodicity breaking in LGTs, which can also be experimentally verified in
quantum simulators.

Here, the mesonic-nonmesonic crossover is reminiscent of the asymptotic
freedom of quarks [85, 86]. The eigenenergy of the scar state
$\ket{\Psi_{n,l}}$ becomes large when increasing the total string length $l$.
Thus, the meson instability in large-$l$ QMBSs is in analogy with the
asymptotic freedom of quarks in the high-energy regime of quantum
chromodynamics. This is an interesting correspondence that deserves further
study. The conventional QMBSs are generated by local operators, which
correspond to non-fractionalized excitations [87], e.g., magnons in PXP [88]
and spin-$1$ XY models [33], and $\eta$-pairs in Hubbard-like models [37, 38].
However, for nonmesonic scar states in this work, isolated spin charges emerge
due to nonlocal generating operators. The corresponding excitations have some
analogies with spinons [51], which are fractionalized excitations and
different from conventional QMBSs. Therefore, our results also reveal that
nonlocal generating operators may lead to fractionalized excitations in QMBSs,
providing an inspiration for studying the nontrivial excitation in QMBSs.
Another interesting issue is whether the above physics can be generalized to
other gauge groups or high-dimensional LGTs [89, 56].

###### Acknowledgements.

Acknowledgements.—We thank M. Dalmonte, Z. Gong, and R.-Z Huang for insightful
discussions and useful suggestions. This work is supported in part by: Nippon
Telegraph and Telephone Corporation (NTT) Research, the Japan Science and
Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP), and
the Moonshot R&D Grant Number JPMJMS2061], the Asian Office of Aerospace
Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the
Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.

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##  I Supplemental Material:
Meson Instability of Quantum Many-body Scars in a 1D Lattice Gauge Theory

##  II Quantum Many-Body Scars

###  II.1 Proof of $\hat{H}_{K}\ket{\Psi_{n,l}}=0$

In the main text, we show that the wave function

$\displaystyle\ket{\Psi_{n,l}}=\mathcal{N}_{n,l}\sum
P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}},$
(S1)

is an exact eigenstate of $\hat{H}$. Here we present details for proving this
result. It is not difficult to find that
$\hat{H}_{E}\ket{\Psi_{n,l}}=h(2l-L)\ket{\Psi_{n,l}}$, so we only need to
prove $\hat{H}_{K}\ket{\Psi_{n,l}}=0$. Since the action of $\hat{H}_{K}$ is
increasing or reducing the total string length by one, while keeping $n$
invariant, we have

$\displaystyle\hat{H}_{K}\ket{\Psi_{n,l}}=\sum
c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l-1}}\ket{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l-1}}+\sum
c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l+1}}\ket{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l+1}}.$
(S2)

Here, the factors have forms

$\displaystyle
c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l-1}}=\mathcal{N}_{n,l}\sum[$
$\displaystyle(1-\delta_{k^{\prime}_{1}+\ell^{\prime}_{1}+1,k^{\prime}_{2}})(P_{\mathcal{S}_{k^{\prime}_{1},\ell^{\prime}_{1}+1},\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}}...}+P_{\mathcal{S}_{k^{\prime}_{1},\ell^{\prime}_{1}},\mathcal{S}_{k^{\prime}_{2}-1,\ell^{\prime}_{2}+1}...})$
$\displaystyle+$
$\displaystyle(1-\delta_{k^{\prime}_{2}+\ell^{\prime}_{2}+1,k^{\prime}_{3}})(P_{...,\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}+1},\mathcal{S}_{k^{\prime}_{3},\ell^{\prime}_{3}}...}+P_{...,\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}},\mathcal{S}_{k^{\prime}_{3}-1,\ell^{\prime}_{3}+1}...})+...$
$\displaystyle
c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l+1}}=\mathcal{N}_{n,l}\sum[$
$\displaystyle(1-\delta_{\ell^{\prime}_{1},1})(P_{\mathcal{S}_{k^{\prime}_{1}-1,\ell^{\prime}_{1}-1},\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}}...}+P_{\mathcal{S}_{k^{\prime}_{1},\ell^{\prime}_{1}-1},\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}}...})$
$\displaystyle+$
$\displaystyle(1-\delta_{\ell^{\prime}_{2},1})(P_{...,\mathcal{S}_{k^{\prime}_{2}-1,\ell^{\prime}_{2}-1},\mathcal{S}_{k^{\prime}_{3},\ell^{\prime}_{3}}...}+P_{...,\mathcal{S}_{k^{\prime}_{2},\ell^{\prime}_{2}-1},\mathcal{S}_{k^{\prime}_{3},\ell^{\prime}_{3}}...})+....$
(S3)

Since the parity satisfies
$P_{\mathcal{S}_{k^{\prime}_{1},\ell^{\prime}_{1}},...,\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{1}},...,\mathcal{S}_{k^{\prime}_{n},\ell^{\prime}_{n}}}=\exp{(i\pi\sum_{j}k^{\prime}_{j})}$,
we have

$\displaystyle
P_{...\mathcal{S}_{k^{\prime}_{j-1},\ell^{\prime}_{j-1}},\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}+1},\mathcal{S}_{k^{\prime}_{j+1},\ell^{\prime}_{j+1}}...}=-P_{...\mathcal{S}_{k^{\prime}_{j-1},\ell^{\prime}_{j-1}},\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}},\mathcal{S}_{k^{\prime}_{j+1}-1,\ell^{\prime}_{j+1}+1}...}$
$\displaystyle
P_{...\mathcal{S}_{k^{\prime}_{j-1},\ell^{\prime}_{j-1}},\mathcal{S}_{k^{\prime}_{j}-1,\ell^{\prime}_{j}-1},\mathcal{S}_{k^{\prime}_{j+1},\ell^{\prime}_{j+1}}...}=-P_{...\mathcal{S}_{k^{\prime}_{j-1},\ell^{\prime}_{j-1}},\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}-1},\mathcal{S}_{k^{\prime}_{j+1},\ell^{\prime}_{j+1}}...}.$
(S4)

Therefore,
$c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l-1}}=c_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{l+1}}=0$,
i.e., $\hat{H}_{K}\ket{\Psi_{n,l}}=0$.

###  II.2 Proof of
$\ket{\Psi_{n,n+m}}=\mathcal{D}_{n,m}\hat{L}_{m}^{\dagger}\ket{\Psi_{n,n+m-1}}$

Next we show the detail of proving Eq. (8) in the main text, i.e,

$\displaystyle\ket{\Psi_{n,n+m}}=\mathcal{D}_{n,m}\hat{L}_{m}^{\dagger}\ket{\Psi_{n,n+m-1}},$
(S5)

where $\mathcal{D}_{n,m}$ is a normalization factor, and

$\displaystyle\hat{L}^{\dagger}_{m}=\sum_{j}\big{(}\sum_{k\leq
m}\prod_{\ell\leq
k}\hat{\mathcal{P}}^{-}_{j+\frac{1}{2}-\ell}\big{)}\hat{\sigma}^{-}_{j}\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}\hat{\sigma}^{+}_{j+1}$
(S6)

It is not difficult to demonstrate

$\displaystyle\big{(}\sum_{k\leq m}\prod_{\ell\leq
k}\hat{\mathcal{P}}^{-}_{j+\frac{1}{2}-\ell}\big{)}\hat{\sigma}^{-}_{j}\hat{\tau}^{\text{M}}_{j+\frac{1}{2}}\hat{\sigma}^{+}_{j+1}\ket{\mathcal{S}_{k,\ell}}=\begin{cases}\ell\delta_{j,k+\ell}\ket{\mathcal{S}_{k,\ell+1}}\quad\ell\leq
m\\\ m\delta_{j,k+\ell}\ket{\mathcal{S}_{k,\ell+1}}\quad\ell>m.\end{cases}$
(S7)

Thus, the action of $\hat{L}^{\dagger}_{m}$ is increasing the total string
length of a basis without changing the parity and string number. Therefore,

$\displaystyle\hat{L}_{m}^{\dagger}\ket{\Psi_{n,n+m-1}}=\sum\alpha_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}\ket{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}.$
(S8)

For the wave function $\ket{\Psi_{n,n+m-1}}=\mathcal{N}_{n,n+m-1}\sum
P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{n+m-1}}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{n+m-1}}$,
the length of each string satisfies $\ell_{j}\leq m$. Hence, the factor has
the form

$\displaystyle\alpha_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}=\mathcal{N}_{n,n+m-1}P_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}[(1-\delta_{\ell^{\prime}_{1},1})(\ell^{\prime}_{1}-1)+(1-\delta_{\ell^{\prime}_{2},1})(\ell^{\prime}_{2}-1)+...+(1-\delta_{\ell^{\prime}_{n},1})(\ell^{\prime}_{n}-1)].$
(S9)

If $\ell^{\prime}_{j}=1$, then
$(1-\delta_{\ell^{\prime}_{j},1})(\ell^{\prime}_{j}-1)=(\ell^{\prime}_{j}-1)=0$,
and if $\ell^{\prime}_{j}\neq 1$, then
$(1-\delta_{\ell^{\prime}_{j},1})(\ell^{\prime}_{j}-1)=(\ell^{\prime}_{j}-1)$.
Thus

$\displaystyle\alpha_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}=\mathcal{N}_{n,n+m-1}P_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}\sum_{j=1}^{n}(\ell^{\prime}_{j}-1)=\mathcal{N}_{n,n+m-1}P_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}(m-1).$
(S10)

Therefore, we have

$\displaystyle\hat{L}_{m}^{\dagger}\ket{\Psi_{n,n+m-1}}=(m-1)\mathcal{N}_{n,n+m-1}\sum
P_{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}\ket{\\{\mathcal{S}_{k^{\prime}_{j},\ell^{\prime}_{j}}\\}_{n}^{n+m}}=\frac{(m-1)\mathcal{N}_{n,n+m-1}}{\mathcal{N}_{n,n+m}}\ket{\Psi_{n,n+m}}.$
(S11)

That is, Eq. (8) is proved, and the normalization factor satisfies

$\displaystyle\mathcal{D}_{n,m}=\frac{\mathcal{N}_{n,n+m}}{(m-1)\mathcal{N}_{n,n+m-1}}.$
(S12)

Figure S1: Distribution of $\mathcal{N}_{n,l}^{-2}$ for $L=32$ and $n=8$ (half
filling). The orange dashed curve is a Gaussian fit.

##  III Initial state

Here we discuss the initial state $\ket{\psi_{2}}$ in Eq. (10b) of the main
text, where it reads

$\displaystyle\ket{\psi_{2}}$
$\displaystyle=\frac{1}{2^{L/2}}\sum_{n,l}\sum_{\\{k_{j},\ell_{j}\\}}P_{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}\ket{\\{\mathcal{S}_{k_{j},\ell_{j}}\\}_{n}^{l}}=\sum_{n,l}\beta_{n,l}\ket{\Psi_{n,l}}.$
(S13)

The amplitude $\beta_{n,l}$ satisfies
$\beta_{n,l}=1/\mathcal{N}_{n,l}2^{L/2}$, where $\mathcal{N}_{n,l}$ is the
normalization factor defined in Eq. (6) of the main text. In addition,
$\mathcal{N}_{n,l}^{-2}$ is the number of string bases for the scar state
$\ket{\Psi_{n,l}}$, and it can be obtained as

$\displaystyle\mathcal{N}_{n,l}^{-2}=\binom{l-1}{n-1}\bigg{[}\binom{L-l-1}{n}+2\binom{L-l-1}{n-1}\bigg{]}+\binom{L-l-1}{n-1}\binom{l-1}{n},$
(S14)

where $\binom{\cdot}{\cdot}$ is the combinatorial number. In Fig. S1, we show
the result of $\mathcal{N}_{n,l}^{-2}$ versus $l$ for $L=32$ and $n=8$ (half
filling). We can find that $\mathcal{N}_{n,l}^{-2}$ nearly satisfies a
Gaussian distribution with the symmetric point at $l=L/2$. Therefore, for the
initial state $\ket{\psi_{2}}$ the nonmesonic scar states dominate.