# Mobility edge in long-range interacting many-body localized systems

Rozhin Yousefjani<EMAIL_ADDRESS>Institute of Fundamental and
Frontier Sciences, University of Electronic Science and Technology of China,
Chengdu 610051, China  Abolfazl Bayat<EMAIL_ADDRESS>Institute
of Fundamental and Frontier Sciences, University of Electronic Science and
Technology of China, Chengdu 610051, China

###### Abstract

As disorder strength increases in quantum many-body systems a new phase of
matter, the so-called many-body localization, emerges across the whole
spectrum. This transition is energy dependent, a phenomenon known as mobility
edge, such that the mid-spectrum eigenstates tend to localize at larger values
of disorder in comparison to eigenstates near the edges of the spectrum. Many-
body localization becomes more sophisticated in long-range interacting
systems. Here, by focusing on several quantities, we draw the phase diagram as
a function of disorder strength and energy spectrum, for a various range of
interactions. Regardless of the underlying transition type, either second-
order or Kosterlitz-Thouless, our analysis consistently determines the
mobility edge, i.e. the phase boundary across the spectrum. We show that,
long-range interaction enhances the localization effect and shifts the phase
boundary towards smaller values of disorder. In addition, we establish a
hierarchy among the studied quantities concerning their corresponding
transition boundary and critical exponents. Interestingly, we show that
deliberately discarding some information of the system can mitigate finite-
size effects and provide results in line with the analytical predictions at
the thermodynamic limit.

##  I Introduction

Ergodicity principle, as the foundation of statistical physics, is violated in
disordered systems. Many-Body Localization (MBL) is the primary example of
such phenomenon in interacting disordered systems which has attracted lots of
attention in recent years Altman (2018); Abanin and Papić (2017); Abanin et
al. (2019); Altman and Vosk (2015); Abanin et al. (2021); Polkovnikov et al.
(2011); Borgonovi et al. (2016). Several features of the MBL have been
characterized through static and dynamical analyses, including Poisson-like
level statistics Oganesyan and Huse (2007); Pal and Huse (2010); Khemani et
al. (2017a, b); Luitz et al. (2015), area-law entangled eigenstates Kjäll et
al. (2014); Bauer and Nayak (2013), logarithmic entanglement growth
Laflorencie ; Bardarson et al. (2012); Chiara et al. (2006); Žnidarič et al.
(2008); Serbyn et al. (2013); Vosk and Altman (2013); Andraschko et al.
(2014), suppression of transport Agarwal et al. (2015); Prelovšek et al.
(2017); Setiawan et al. (2017); Vasseur and Moore (2016); Žnidarič et al.
(2016), power-law decay of local correlations Ruihua Fan and Zhai (2017); Chen
et al. (2017); Huang et al. (2017); He and Lu (2017); Li et al. (2017);
Gärttner et al. (2017), emergence of memory Doggen et al. (2018); Nico-Katz et
al. (2022a); Kohlert et al. (2019); Schulz et al. (2019); Lüschen et al.
(2017); Sierant and Zakrzewski (2022); Chanda and Zakrzewski (2022); Nico-Katz
et al. (2022b), and connection to topological phases Huse et al. (2013);
Chandran et al. (2014). Unlike quantum phase transition, which is a property
of the ground state Sachdev (2011), the transition from ergodic to MBL takes
place across the whole spectrum, as disorder strength increases. This makes
the detection and characterization of the MBL transition very challenging
Potter et al. (2015); Roushan et al. (2017); Rispoli et al. (2019); Gong et
al. (2021); Sierant et al. (2020a); Sels and Polkovnikov (2021); Sels (2022a);
Solórzano et al. (2021); Schiulaz et al. (2020). In fact, each energy
eigenstate localizes at a different disorder strength, a phenomenon called
mobility edge which has been explored theoretically Luitz et al. (2015); Kjäll
et al. (2014); Chanda et al. (2020a); Mondragon-Shem et al. (2015); Xu et al.
(2020); Wei et al. (2019); Villalonga et al. (2018); Devakul and Singh (2015);
Geißler and Pupillo (2020); De Roeck et al. (2016); Wei et al. ; Maksymov et
al. (2017) and observed experimentally Guo et al. (2021a). Since MBL is a
property of the whole spectrum, its numerical investigation is mostly limited
to exact diagonalization of small systems. This makes it very challenging to
extract information about the thermodynamic limit. Thanks to the recent
development of quantum simulators, MBL experiments have been implemented in
various platforms such as superconducting devices Guo et al. (2021a); Zha et
al. (2020); Xu et al. (2018); Chiaro et al. ; Guo et al. (2021b); Mi et al.
(2022); Gong et al. ; Gong et al. (2021); Roushan et al. (2017), optical
lattices Schreiber et al. (2015); Kondov et al. (2015); Bordia et al. (2017a,
b); Choi et al. (2016); Rispoli et al. (2019); Lukin et al. (2019), ion traps
Smith et al. (2016); Morong et al. (2021); Rajabi et al. (2019), nitrogen-
vacancy centers in diamond Choi et al. (2017, 2019), and photonic systems
Nguyen et al. (2021).

Perhaps the most challenging problem in the MBL context is the
characterization of the MBL transition and its corresponding mobility edge
Abanin et al. (2021); Panda et al. (2019); Morningstar et al. (2022); Sierant
et al. (2020b); Pietracaprina et al. (2018). In a crucial analytical
contribution by Harris Harris (1974), which has been extended by others Chayes
et al. (1986); Chandran et al. , the MBL transition has been described as a
continuous second-order phase transition which is accompanied by the emergence
of a diverging length scale $\xi_{{}_{SO}}{\sim}|W-\omega|^{-\nu}$ in the
system. Here, $W$ is the strength of disorder, $\omega$ is critical disorder
strength beyond which the system is localized, and $\nu$ is a critical
exponent. Harris analysis predicts that $\nu$ has to satisfy $\nu{\geq}2/d$ in
a $d$-dimensional system. However, most of the conventional quantities that
have been studied in 1-dimensional MBL systems, such as entanglement entropy
and level statistics, violate Harris criteria and result in $\nu{\sim}1$ Luitz
et al. (2015); Nico-Katz et al. (2022a). There are some exceptions, such as
Schmidt gap Doggen et al. (2018); Gray et al. (2018) and diagonal entropy
Doggen et al. (2018); Levi et al. (2016); Sun et al. (2020), which either
satisfy the Harris criteria or at least violate it less by giving
$\nu{\sim}2$. There are two distinct explanations for the inconsistency of
numerical simulations with the Harris bound. Either the accessible system
sizes are too small to emulate the thermodynamic limit Panda et al. (2019);
Sierant et al. (2020b), or describing the MBL transition as a second-order
phase transition is not valid and one has to explain it as a Kosterlitz-
Thouless type Dumitrescu et al. (2019); Goremykina et al. (2019); Morningstar
and Huse (2019); Morningstar et al. (2020); Šuntajs et al. (2020a);
Laflorencie et al. (2020); Šuntajs et al. (2020b); Hopjan et al. (2021);
Aramthottil et al. (2021). The suggestion of describing MBL transition as a
Kosterlitz-Thouless transition type with a diverging length scale
$\xi_{{}_{KT}}{\sim}\exp{(b/\sqrt{|W-\omega|})}$, received support from real-
space renormalization group approaches based on the avalanche scenario
Dumitrescu et al. (2019); Goremykina et al. (2019); Morningstar and Huse
(2019) and has been investigated numerically Morningstar et al. (2020);
Šuntajs et al. (2020a); Laflorencie et al. (2020); Šuntajs et al. (2020b);
Hopjan et al. (2021); Aramthottil et al. (2021). Nonetheless, due to small
system sizes, the debate about the transition type is far from being settled.

Investigation of the MBL beyond the conventional nearest-neighbor interactions
may open new questions. Long-range interactions, such as Coulomb, dipole-
dipole and van der Waals, are of utmost importance as they naturally arise in
many systems. On the experimental side, some of the quantum simulators, such
as ion traps Smith et al. (2016); Morong et al. (2021); Rajabi et al. (2019)
and Rydberg atoms Choi et al. (2016); Rispoli et al. (2019); Lukin et al.
(2019), are naturally governed by long-range interactions. The existence of an
MBL phase and its principal properties in long-range interacting systems is
still under dispute Chanda and Zakrzewski (2022); Kubala et al. (2021); Roy
and Logan (2019); Nandkishore and Sondhi (2017); Yao et al. (2014); Modak and
Nag (2020a); Burin (2015a); Schiffer et al. (2019); Burin (2015b); Tikhonov
and Mirlin (2018); Maksymov and Burin (2020); Li et al. (2016); Thomson and
Schiró (2020); Deng et al. (2020); Li et al. (2021); Vu et al. (2022); De
Roeck et al. (2016); Maksymov et al. (2017); Devakul and Singh (2015). For
instance, while earlier studies of the MBL in long-range systems suggested an
algebraic growth of entanglement entropy Deng et al. (2020); Pino (2014); De
Tomasi (2019); Safavi-Naini et al. (2019); Singh et al. (2017); Nag and Garg
(2019); Modak and Nag (2020b); Roy and Sharma (2018), recent work shows non-
algebraic behavior through long-time numerical simulations Yousefjani et al. .
Interestingly, different types of long-range couplings can have different
effects on the localization issue. While long-range tunneling can increase
delocalization, long-range (Ising) interaction enhances localization Roy and
Logan (2019); Nag and Garg (2019); Deng et al. (2020); Yousefjani et al. .
Several important issues are still open in long-range MBL systems, including
(i) the main nature of the transition (ii) the phase boundary, and thus the
mobility edge, along the energy spectrum as the strength of long-range
interaction varies; and (iii) the energy dependence of the critical exponents,
associated with the transition types, along the mobility edge.

In this paper, we aim to address these issues by considering several
quantities, including level statistics ratio, entanglement entropy, diagonal
entropy, and Schmidt gap. For both of the transition types, i.e. second-order
and Kosterlitz-Thouless transition, we draw the phase diagram extracted from
the four aforementioned quantities across the whole spectrum as the strength
of long-range interaction varies. Several results have been observed. We show
that, regardless of the transition type, all the studied quantities determine
the phase boundary and the D-shape mobility edge consistently. This feature,
has already been observed, both analytically Luitz et al. (2015); Kjäll et al.
(2014); Chanda et al. (2020a); Mondragon-Shem et al. (2015); Xu et al. (2020);
Wei et al. (2019); Villalonga et al. (2018); Devakul and Singh (2015); Geißler
and Pupillo (2020); De Roeck et al. (2016); Wei et al.  and experimentally Guo
et al. (2021a), for MBL transition as a continuous second-order
transformation. However, the emergence of this characteristic for the
Kosterlitz-Thouless type has not been reported previously. Besides, we show
that long-range interaction enhances the localization and shifts the mobility
edge towards smaller values of $\omega$, consistent with previous studies Roy
and Logan (2019); Deng et al. (2020); Nandkishore and Sondhi (2017). In
addition, while in most of the literature both of the critical exponents $\nu$
(in second-order transition) and $b$ (in Kosterlitz-Thouless transition) are
only computed for the mid-spectrum, we determine them along the whole
spectrum. As the main result, our analysis establishes a hierarchy among our
quantities with respect to the phase boundary and critical exponents.
Interestingly, we show that for very wide range of long-range interactions,
Schmidt gap can give results fully consistent with the Harris criteria. This
can be explain based on this fact that Schmidt gap converges to its
thermodynamic limit faster than the other three quantities.

##  II Model.

We consider an open-boundary chain of $N$ spin-1/2 particles in the presence
of a random magnetic field. While spin tunneling is restricted between
nearest-neighbor sites, interaction between particles is taken to be long-
range which algebraically decays by exponent $\alpha{>}0$. The Hamiltonian
reads

$H=-\sum_{i=1}^{N-1}(S_{i}^{x}S_{i+1}^{x}+S_{i}^{y}S_{i+1}^{y})-\sum_{i\neq
j=1}^{N}\dfrac{1}{|i-j|^{\alpha}}S_{i}^{z}S_{j}^{z}+\sum_{i=1}^{N}h_{i}S_{i}^{z},$
(1)

here $S_{i}^{(x,y,z)}$ are the spin-$1/2$ operators for qubit $i$, and $h_{i}$
denotes a random magnetic field in the $\hat{z}$ direction acting on qubit $i$
which is drawn from a uniform distribution $[-W,W]$, with $W$ being the
strength of disorder. Note that, by varying $\alpha$ we can smoothly
interpolate between a fully connected graph (i.e. $\alpha{=}0$) and standard
nearest-neighbor 1-dimensional chain (i.e. $\alpha{\rightarrow}\infty$). Many
types of long-range models such as Coulomb and dipole-dipole interactions are
special examples of Hamiltonian $H$. Long-range interacting systems, with
tunable $\alpha$, can be realized in cold atoms in optical lattices Baier et
al. (2016), ion traps Smith et al. (2016); Morong et al. (2021); Rajabi et al.
(2019) and polar molecules Yao et al. (2014). The isotropic Hamiltonian Eq.
(1) corresponds to multipole expansion introduced in Yao et al. (2014) which
has a critical dimension $d_{c}{=}(\alpha+2)/2$ and gets localized for all
$\alpha{>}0$. The possibility of observing such consistent MBL in our model
was also corroborated by other works using resonant pairs analysis Nag and
Garg (2019); Burin  and mean-field theory Roy and Logan (2019). Indeed, the
existence of MBL in one-dimensional systems is shown through analytical
analysis Imbrie (2016a, b) as well as numerical investigations Sierant et al.
(2020b). Nonetheless, recent studies Abanin et al. (2021); Morningstar et al.
(2022); De Roeck and Huveneers (2017); Kiefer-Emmanouilidis et al. (2021);
Prasad and Garg (2021); De Tomasi et al. (2021); Sels (2022b, 2022) raised the
possibility that for some disorder ansatz formation of rare ergodic spots
(randomly formed chaotic domains of reduced disorder), may drive an avalanche,
by thermalizing nearby spins, and eventually thermalize the whole system. The
possibility of finding such rare ergodic spots increases as the system gets
larger Abanin et al. (2021); De Roeck and Huveneers (2017). The formation of
such rare ergodic spots is sensitive to the range of interaction and is mostly
endemic to models with uncorrelated disorder and systems which support the
existence of mobility edge De Roeck et al. (2016). So, our model is no
exception and avalanche effect might be observable for certain random ansatz,
though we hardly encounter such problem in our numerical analysis. This might
be due to finite sizes which are available to numerical calculations as MBL is
commonly observed in 1-dimensional systems for sufficiently large disorder $W$
Abanin et al. (2021); De Roeck and Huveneers (2017). Indeed, the numerical
limitations leave the fate of MBL with respect to the possibility of avalanche
still an open problem.

Figure 1: Upper panels: Density of states (DoS) as a function of rescaled
energy $\varepsilon$ for different disorder strengths in system with exponents
(a): $\alpha{=}0.5$ and (b): $\alpha{=}1$. Lower panels: DoS as a function of
$\varepsilon$ in system with fixed disorder strength $W{=}3$ and exponents
(c): $\alpha{=}0.5$, and (d): $\alpha{=}1$. The colored bars represent the
domains from them $M{=}50$ consecutive eigenstates in the vicinity of rescaled
energies $\varepsilon{\in}\\{0.2,\cdots,0.8\\}$ have been selected. All the
results have been obtained for a system of size $N{=}15$.

For our numerical analysis, we restrict ourselves to the subspaces of
$S^{z}_{tot}{=}0$ or $S^{z}_{tot}{=}1/2$ (with
$S^{z}_{tot}=\sum_{j}S^{z}_{j}$) for even and odd $N$’s, respectively. For
systems of length $N{=}10,11,\cdots,15$, to achieve a good statistic, we
provide $2000$ (for $N{\leqslant}13$) to $1000$ (for $N{=}14,15$) sample
realizations of the random field. For each set of random fields and with the
means of exact diagonalization, we generate $M$ eigenstates
$\\{|E_{k}\rangle\\}$ of the Hamiltonian $H$ around the rescaled energy
$\varepsilon{\in}\\{0.2,\cdots,0.8\\}$, where $\varepsilon$’s have been
calculated as $\varepsilon{=}(E-E_{min})/(E_{max}-E_{min})$ in which $E_{max}$
and $E_{min}$ are the extremal eigenvalues of $H$. In Fig. 1, we depict the
density of states (DoS) as a function of energy $\varepsilon$ in a system of
size $N{=}15$ and exponents (a): $\alpha{=}0.5$ and (b): $\alpha{=}1$. Except
for small values of disorder, the DoS is symmetric around the mid-spectrum
energy and its maximum hardly changes from $\varepsilon{=}0.5$. This is in
sharp contrast with long-range systems with random exchange couplings in which
the maximum of DoS is highly skewed towards low energy part of the spectrum Li
et al. (2016); Maksymov et al. (2017). Note that most of MBL analyzes are
focused on the energy corresponding to the peak of the DoS where the spectrum
is very dense. In our case, this is mostly around $\varepsilon{=}0.5$, namely
mid-spectrum. For the sake of completeness, in Figs. 1(c) and (d), we plot DoS
as a function of energy for the disorder strength $W{=}3$ and two different
values of $\alpha$, namely $\alpha{=}0.5$ and $\alpha{=}1$ respectively. The
colored bars represent the domain from which we select $M{=}50$ consecutive
eigenstates $\\{E_{k}\\}$ in the vicinity of the rescaled energies
$\varepsilon{\in}\\{0.2,\cdots,0.8\\}$. For every eigenstate $|E_{k}\rangle$
one can compute the desired quantity which results in $\mathcal{O}_{k}$. To
have an average behavior of eigenstates at energy $\varepsilon$, we average
over the $M{=}50$ eigenstates around that energy which results in
$\langle\mathcal{O}\rangle=1/M\sum_{k=1}^{M}\mathcal{O}_{k}$. In addition, to
observe the effect of disordered field, one has to also average over different
random samples (typically 1000-2000 samples) which is denoted by
$\overline{\langle\mathcal{O}\rangle}$.

##  III Quantities

Our first quantity is the gap ratio $r$ which is a well-established tool to
study the statistical properties of the energy levels $\\{E_{k}\\}$ of the
Hamiltonian $H$. For energy gaps $\delta_{k}{=}E_{k+1}{-}E_{k}$, the ratio
$r_{k}{=}\min(\delta_{k+1},\delta_{k}){/}\max(\delta_{k+1},\delta_{k})$ and
its average $\overline{\langle r\rangle}$, over energy levels and sample
realizations, can diagnose a global change in the spectral statistics. While
in the MBL phase the energy levels are spaced according to Poisson statistics,
yielding $\overline{\langle r\rangle}{\simeq}0.3863$, in the ergodic phase the
strong repulsion between neighboring levels causes statistics that match with
those of the Gaussian orthogonal ensembles, resulting in $\overline{\langle
r\rangle}{\simeq}0.5307$.

The second quantity of interest is the entanglement entropy $S_{{}_{EE}}$,
which is a widely used tool for quantifying entanglement between two
complementary parts of a pure many-body system. For any given eigenstate
$|E_{k}\rangle$ one can obtain the reduced density matrix
$\rho^{(k)}_{{}_{L}}$ by tracing out $\lceil L/2\rceil$ qubits on the right
side of the chain. Therefore, the entanglement entropy between left and right
cuts of the chain is defined as
$S_{{}_{EE}}(\rho^{(k)}_{{}_{L}}){=}{-}\mathrm{Tr}[\rho^{(k)}_{{}_{L}}\ln(\rho^{(k)}_{{}_{L}})]$.
In the ergodic phase, the eigenstates are expected to behave like a random
pure state and exhibit a volume law entanglement entropy determined by Page
entropy
$S^{P}_{{}_{EE}}{=}(1-\mathcal{D}_{{}_{L}})/2\mathcal{D}_{{}_{R}}+\sum_{k=\mathcal{D}_{{}_{R}}+1}^{\mathcal{D}_{{}_{L}}\mathcal{D}_{{}_{R}}}1/k$
with $\mathcal{D}_{{}_{L}}$ and $\mathcal{D}_{{}_{R}}$ as the Hilbert space
dimensions of the left and right chain cuts Page (1993). In contrast, in the
localized phase due to the emergent integrability the eigenstates exhibit an
area-law entanglement. Therefore, in deep ergodic and MBL phases, one expects
to have $\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}\sim 1$ and
$\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}\sim 0$, respectively,
for the averaged entanglement entropy.

As the third quantity, we consider the diagonal entropy of a subsystem.
Replacing $\rho^{(k)}_{{}_{L}}$ in $S_{{}_{EE}}(\rho^{(k)}_{{}_{L}})$ by the
decohered density matrix $\varrho^{(k)}_{{}_{L}}$, in which all the off-
diagonal elements in computational basis are set to zero, results in the
diagonal entropy
$S_{{}_{DE}}(\varrho^{(k)}_{{}_{L}}){=}{-}\mathrm{Tr}[\varrho^{(k)}_{{}_{L}}\ln(\varrho^{(k)}_{{}_{L}})]$.
This quantity has recently been used in the context of MBL Doggen et al.
(2018); Levi et al. (2016); Sun et al. (2020). In the ergodic phase, the
diagonal entropy provides an upper bound for entanglement entropy, namely
$S_{{}_{DE}}(\varrho^{(k)}_{{}_{L}})\geqslant
S_{{}_{EE}}(\rho^{(k)}_{{}_{L}})$. However, in the deep localized phase they
remain close to each other, namely
$S_{{}_{DE}}(\varrho^{(k)}_{{}_{L}}){\sim}S_{{}_{EE}}(\rho^{(k)})$, as
$\rho^{(k)}_{{}_{L}}$ and its decohered state $\varrho^{(k)}_{{}_{L}}$ have
high fidelity. Analogs to the entanglement entropy, by calculating the maximum
diagonal entropy for a typical random pure state as
$S^{P}_{{}_{DE}}\simeq\ln(0.48\mathcal{D}_{{}_{L}}){+}\ln(2)$ Torres-Herrera
and Santos (2017); Torres-Herrera et al. (2016), one can determine two
limiting behaviors $\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}\sim
1$ and $\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}\sim 0$ for the
ergodic and localized phases, respectively.

The fourth and final indicator is the Schmidt gap of the reduced density
matrix $\rho^{(k)}_{{}_{L}}$ which can be calculated by using its two largest
eigenvalues, denoted by $\lambda^{(k)}_{1}$ and $\lambda^{(k)}_{2}$, as
$\Delta(\rho^{(k)}_{{}_{L}}){=}\lambda^{(k)}_{1}{-}\lambda^{(k)}_{2}$. In
contrast to the entanglement entropy which considers all the spectrum of
$\rho^{(k)}_{{}_{L}}$, the Schmidt gap only focuses on two dominant
eigenvalues. In the ergodic phase $\rho^{(k)}_{{}_{L}}$ is expected to be a
thermal state with an infinite temperature (at least for mid-spectrum
eigenstates) which results in $\overline{\langle\Delta\rangle}{\sim}0$. In the
localized phase the reduced density matrix $\rho^{(k)}_{{}_{L}}$ tends to
become a pure state and thus the Schmidt gap increases such that in the deep
localized phase, one has $\overline{\langle\Delta\rangle}\rightarrow 1$.

##  IV Transition type

One of the fundamental features of quantum phase transitions is the emergence
of scale invariance in the vicinity of the transition point Sachdev (2011). In
the context of MBL, the scale invariance implies that a length scale emerges
in the system and thus all quantities are expected to scale as Gray et al.

$\overline{\langle\mathcal{O}\rangle}=f(N/\xi),$ (2)

where $N$ is the system size, $\xi$ is the emerging length scale that diverges
at the transition point, and $f(\cdot)$ is an arbitrary function which depends
on the quantity of interest. Most of the literature characterize the MBL
transition as a continuous second-order transformation which is described by a
diverging length scale Pal and Huse (2010); Chanda et al. (2020b); Gray et al.
; Khemani et al. (2017a, b); Luitz et al. (2015); Kjäll et al. (2014); Chanda
et al. (2020a); Xu et al. (2020); Wei et al. (2019); Potter et al. (2015)

$\xi_{{}_{SO}}{\sim}\dfrac{1}{|W-\omega|^{\nu}},$ (3)

where, $\omega$ is the critical disorder strength beyond which the system is
localized, and $\nu$ is a universal critical exponent. It has analytically
been shown that $\nu{\geqslant}2/d$ (known as Harris criterion), with $d$
being the dimension of the system Harris (1974); Chayes et al. (1986);
Chandran et al. . While describing MBL transition as a second-order phase
transition is supported by the phenomenological approaches based on real-space
renormalization group Potter et al. (2015), most of the numerical analyses for
one-dimensional systems, with the exception of Refs. Nico-Katz et al. (2022a);
Gray et al. (2018), give $\nu{\sim}1$ which is in contradiction with Harris
bound. The failure of capturing the right value of $\nu$, can be associated to
very finite system sizes that are accessible for numerical simulations Nico-
Katz et al. (2022a); Gray et al. (2018).

Recently, a new approach to describing the MBL transition has also been
adopted. In this approach, people argue that the very basic assumption of the
second-order phase transition might be wrong, and thus the Harris criterion is
totally irrelevant. They suggest that the transition might be of Kosterlitz-
Thouless type Dumitrescu et al. (2019); Goremykina et al. (2019); Morningstar
and Huse (2019). In such transition, while the scale invariance of Eq. (2) is
still valid, the emerging length scale is described as

$\xi_{{}_{KT}}{\sim}\exp{(\dfrac{b}{\sqrt{|W-\omega|}})},$ (4)

where $\omega$ is again the critical disorder strength beyond which the system
is localized and $b$ is a non-universal fitting parameter. Despite several
attempts Morningstar et al. (2020); Šuntajs et al. (2020a); Laflorencie et al.
(2020); Šuntajs et al. (2020b); Hopjan et al. (2021); Aramthottil et al.
(2021) for settling the type of transition, by comparing the second-order
versus Kosterlitz-Thouless transitions, the finite system sizes do not allow
to reach a conclusive answer. Therefore, in this paper, for the sake of
completeness, we perform finite-size scaling analysis for both of the
transition types.

##  V Finite-size scaling analysis

Figure 2: (a): the averaged gap ratio $\overline{\langle r\rangle}$, (b): the
normalized entanglement entropy $\overline{\langle
S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, (c): the normalized diagonal entropy
$\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and (d): the Schmidt
gap $\overline{\langle\Delta\rangle}$ as a function of disorder strength $W$
for different system sizes. All the quantities are obtained for long-range
system with interaction’s exponent $\alpha{=}0.5$ in the mid-spectrum
$\varepsilon{=}0.5$.

Figure 3: Upper panels: The finite-size scaling analysis in MBL transition as
a second-order phase transition. (a): the averaged gap ratio
$\overline{\langle r\rangle}$, (b): the normalized entanglement entropy
$\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, (c): the normalized
diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and
(d): the Schmidt gap $\overline{\langle\Delta\rangle}$ as a function of
$sN/\xi_{{}_{SO}}$ with $s=sgn{(W-\omega)}$ and
$\xi_{{}_{SO}}{\sim}|W-\omega|^{-\nu}$. For each desired quantity
$\mathcal{O}$, the extracted critical disorder
$\omega^{{}_{\mathcal{O}}}_{{}_{SO}}$, critical exponent
$\nu^{{}_{\mathcal{O}}}$, and the quality of data collapse $Q$ have been
attached on the relevant panel. Lower panels: The finite-size scaling analysis
in MBL transition as a Kosterlitz-Thouless transition type. (e): the averaged
gap ratio $\overline{\langle r\rangle}$, (f): the normalized entanglement
entropy $\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, (g): the
normalized diagonal entropy $\overline{\langle
S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and (h): the Schmidt gap
$\overline{\langle\Delta\rangle}$ as a function of $sN/\xi_{{}_{KT}}$ with
$s=sgn{(W-\omega)}$ and $\xi_{{}_{KT}}{\sim}\exp{(b|W-\omega|^{-0.5})}$. The
extracted critical disorder $\omega^{{}_{\mathcal{O}}}_{{}_{KT}}$, fitting
parameter $b^{{}_{\mathcal{O}}}$, and the quality of data collapse $Q$ have
been attached on the relevant panel. All the data are obtained in a system
with interaction’s exponent $\alpha=0.5$ in the mid-spectrum
$\varepsilon{=}0.5$.

Regardless of the transition type, at the transition point, namely
$W{=}\omega$, the length scale diverges $\xi{\rightarrow}\infty$ (see Eqs.
(3), and (4)) in the thermodynamic limit. In finite system sizes, however,
this feature is revealed by $\xi{\sim}N$. This implies that at the transition
point the ansatz in Eq. (2) becomes independent of $N$. To observe this
behavior in Figs. 2 (a)-(d) we plot the averaged gap ratio $\overline{\langle
r\rangle}$, the normalized entanglement entropy $\overline{\langle
S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, the normalized diagonal entropy
$\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$ and the Schmidt gap
$\overline{\langle\Delta\rangle}$ as a function of disorder strength $W$ for
different system sizes. As evident in the plots, all the quantities show
intersection between different lengths at around $\omega{=}2-3$. This indeed
indicates the emergence of a scale invariance behavior, of the form of Eq.
(2), in the system.

In order to have a more precise estimation of $\omega$ and other critical
exponents one has to adopt finite-size scaling analysis as a standard approach
for extracting such parameters from finite-size data. We first consider a
second-order phase transition and implement the corresponding finite-size
scaling analysis for all four aforementioned quantities. In Figs. 3(a)-(d) we
plot the averaged gap ratio $\overline{\langle r\rangle}$, the normalized
entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$,
the normalized diagonal entropy $\overline{\langle
S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and the Schmidt gap
$\overline{\langle\Delta\rangle}$ as a function of
$sN/\xi_{{}_{SO}}{=}sN|W-\omega|^{\nu}$ with $s{=}sgn(W-\omega)$. The
parameters $\omega$ and $\nu$ are optimized such that the curves for different
length collapse on each other. To achieve the best data collapse one can use
an elaborate optimization scheme and minimize a proper quality function $Q$
Houdayer and Hartmann (2004); Bhattacharjee and Seno (2001); Šuntajs et al.
(2020b); Aramthottil et al. (2021); Sorge (2015); Melchert , which is defined
and discussed in the Appendix. In our case, a perfect data collapse results in
$Q{=}0$ and any deviation from such perfect situation makes $Q$ larger. In
each panel of Figs. 3(a)-(d) we provide the optimal values of $\omega$ and
$\nu$ as well as the quality function $Q$ for the resulted data collapse.
Importantly, while the averaged gap ratio $\overline{\langle r\rangle}$ and
the normalized entanglement entropy $\overline{\langle
S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$ result in $\nu{\sim}1$, consistent with
previous studies Luitz et al. (2015); Burin (2015a); Schiffer et al. (2019);
Chanda et al. (2020b), the normalized diagonal entropy $\overline{\langle
S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$ and the Schmidt gap
$\overline{\langle\Delta\rangle}$ yield to exponent closer to $\nu{\sim}2$,
again consistent with the previous results Nico-Katz et al. (2022a); Gray et
al. (2018); Levi et al. (2016); Sun et al. (2020). This shows that the
normalized diagonal entropy and the Schmidt gap are more consistent with
Harris criterion. Interestingly, the value of $\omega$ which is obtained for
the Schmidt gap and the diagonal entropy are remarkably higher than those for
the averaged gap ratio and the entanglement entropy. These suggest faster
convergence of the Schmidt gap and the diagonal entropy towards their
thermodynamic limit in comparison with the averaged gap ratio and the
normalized entanglement entropy. In other words, while the Schmidt gap and the
diagonal entropy have almost reached their thermodynamic limit with system
sizes of $N{=}15$, the other two quantities still show significant length
dependence.

Alternatively, one can also perform finite-size scaling analysis assuming that
the underlying transition is of the Kosterlitz-Thouless type. In Figs.
3(e)-(h), we plot the averaged gap ratio $\overline{\langle r\rangle}$, the
normalized entanglement entropy $\overline{\langle
S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, the normalized diagonal entropy
$\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and the Schmidt gap
$\overline{\langle\Delta\rangle}$ as a function of
$sN/\xi_{{}_{KT}}{=}sN\exp{(-b|W-\omega|^{-0.5})}$. The parameters $\omega$
and $b$ are optimized such that the curves for different lengths collapse on
each other. Similar to the previous case the quality of the data collapse is
quantified through the same quality function $Q$. In each panel of Figs.
3(e)-(h), we provide the optimal values of $\omega$ and $b$ as well as the
quality function $Q$ for the resulted data collapse. In the case of
Kosterlitz-Thouless transition, the only relevant quantity is the transition
point $\omega$. Interestingly, similar to the previous case, the diagonal
entropy and the Schmidt gap result in higher values of $\omega$ in comparison
with the averaged gap ratio and the entanglement entropy. A remarkable
observation is that the values obtained for the transition point, namely
$\omega$, is very close for both types of transition. In addition, the quality
function $Q$ for all four quantities is very close for both transition types.
The same issue observed in previous studies Šuntajs et al. (2020b);
Aramthottil et al. (2021). The similarity of the obtained values for $\omega$
and $Q$ show that, indeed, one cannot discriminate between the two transitions
at least based on data analysis for small system sizes.

##  VI Phase diagram

Figure 4: The averaged gap ratio $\overline{\langle r\rangle}$ as function of
energy $\varepsilon$ and disorder strength $W$ for different interaction
ranges from (a): $\alpha{=}0.5$ to (f): $\alpha{\rightarrow}\infty$. The
extracted critical disorders $\omega^{{}_{\mathcal{O}}}_{{}_{SO}}$ and
$\omega^{{}_{\mathcal{O}}}_{{}_{KT}}$ are obtained by considering the
transition as second-order and Kosterlitz-Thouless type, respectively. In all
the panels, $\omega^{{}_{GR}}_{{}_{SO,KT}}$, $\omega^{{}_{EE}}_{{}_{SO,KT}}$,
$\omega^{{}_{DE}}_{{}_{SO,KT}}$, and $\omega^{{}_{SG}}_{{}_{SO,KT}}$ are
obtained from finite-size scaling analysis for the averaged gap ratio
$\overline{\langle r\rangle}$, the normalized entanglement entropy
$\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, the normalized
diagonal entropy $\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and
the Schmidt gap $\overline{\langle\Delta\rangle}$, respectively.

In this section, we extend the finite-size scaling analysis for all energy
spectrum $\varepsilon$ and various values of $\alpha$. To achieve this, at a
desired energy $\varepsilon$ we pick $M=50$ eigenstates as depicted in Figs. 1
(c)-(d). For any given $\alpha$ and energy scale $\varepsilon$, we perform
finite-size scaling analysis by considering both second-order and Kosterlitz-
Thouless transition types and extract the corresponding transition points and
critical exponents. In Figs. 4(a)-(f) we plot the averaged gap ratio
$\overline{\langle r\rangle}$ as a function of energy $\varepsilon$ and
disorder strength $W$ for $N{=}15$ and various values of $\alpha$. By
increasing the disorder strength, one can continuously move from the ergodic
phase (light region) to the localized phase (dark region). By considering a
quantity $\mathcal{O}$, which is chosen to be any of the four quantities
introduced before, at different energy scales we can use either second-order
or Kosterlitz-Thouless ansatzes to determine the transition points
$\omega^{{}_{\mathcal{O}}}_{{}_{SO}}$ and
$\omega^{{}_{\mathcal{O}}}_{{}_{KT}}$, respectively. By focusing on the
second-order transition, in Figs. 4(a)-(f) we present the transition points
$\omega^{{}_{GR}}_{{}_{SO}}$ (filled solid triangles),
$\omega^{{}_{EE}}_{{}_{SO}}$ (filled solid circles),
$\omega^{{}_{DE}}_{{}_{SO}}$ (filled solid squares), and
$\omega^{{}_{SG}}_{{}_{SO}}$ (filled solid diamonds) obtained from finite-size
scaling analysis for the averaged gap ratio $\overline{\langle r\rangle}$, the
normalized entanglement entropy $\overline{\langle
S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$, the normalized diagonal entropy
$\overline{\langle S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$, and the Schmidt gap
$\overline{\langle\Delta\rangle}$, respectively. Similarly, one can perform
finite-size scaling analysis for the four quantities using Kosterlitz-Thouless
transition ansatz. The results are also shown in Figs. 4(a)-(f) in which the
transition points $\omega^{{}_{GR}}_{{}_{KT}}$ (empty triangles),
$\omega^{{}_{EE}}_{{}_{KT}}$ (empty circles), $\omega^{{}_{DE}}_{{}_{KT}}$
(empty squares), and $\omega^{{}_{SG}}_{{}_{KT}}$ (empty diamonds) are
depicted across the whole spectrum.

Four main features can be observed from Fig 4. First, all considered
quantities, regardless of the transition type, reveal a generic D-shape phase
boundary along the spectrum, the well-known mobility edge. While this feature
has already been observed for MBL transition as a second-order phase
transition, both analytically Luitz et al. (2015); Kjäll et al. (2014); Chanda
et al. (2020a); Mondragon-Shem et al. (2015); Xu et al. (2020); Wei et al.
(2019); Villalonga et al. (2018); Devakul and Singh (2015); Geißler and
Pupillo (2020); De Roeck et al. (2016); Wei et al.  and experimentally Guo et
al. (2021a), the emergence of this feature for the Kosterlitz-Thouless
transition has not been reported previously. Second, decreasing $\alpha$ (i.e.
making the interaction more long-range) enhances the localization power such
that the localization occurs for smaller values of disorder. This can be
describe by the fact that every spin configuration of the chain induces an
effective Zeeman energy splitting at a given site. Thus, the superposition of
different spin configurations can play like an effective random field and thus
enhances the localization power which is in agreement with previous works
Yousefjani et al. . Third, apart from Schmidt gap, the other three quantities
result in very close transition points for both second-order and Kosterlitz-
Thouless transition types. This shows that the detection of the MBL phase is
very robust and hardly depends on the type of the transition. Forth, one can
clearly observe a hierarchy among the four quantities for revealing the MBL
transition point

$\omega^{{}_{SG}}_{{}_{SO,KT}}\geqslant\omega^{{}_{DE}}_{{}_{SO,KT}}\geqslant\omega^{{}_{EE}}_{{}_{SO,KT}}\geqslant\omega^{{}_{GR}}_{{}_{SO,KT}}.$
(5)

The reason that each quantity gives a distinct transition point is indeed a
finite-size effect indicating different convergence speed towards their
thermodynamic limit.

Figure 5: The critical exponent $\nu$’s (see Eq. (3)) as a function of energy
scale $\varepsilon$ for various interaction ranges from (a): $\alpha{=}0.5$ to
(f): $\alpha{\rightarrow}\infty$. These exponents are obtained from finite-
size scaling analysis for different observables including the averaged gap
ratio $\overline{\langle r\rangle}$ (dark blue triangle), the normalized
entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$
(red circle), the normalized diagonal entropy $\overline{\langle
S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$ (magenta square), and the Schmidt gap
$\overline{\langle\Delta\rangle}$ (yellow diamond).

As mentioned before, finite-size scaling analysis not only provides the
transition point $\omega$, but also reveals other exponents such as $\nu$ (for
the second-order transition) and $b$ (for the Kosterlitz-Thouless transition).
In Figs. 5(a)-(f) we plot the critical exponent $\nu$ obtained from finite-
size scaling analysis of all four quantities as a function of rescaled energy
$\varepsilon$ for different interaction range $\alpha$. For all values of
$\alpha$ and $\varepsilon$ one can observe a hierarchy among the four
quantities, namely

$\nu^{{}_{SG}}\geqslant\nu^{{}_{DE}}\geqslant\nu^{{}_{EE}}\geqslant\nu^{{}_{GR}}.$
(6)

By comparing Eqs. (5) and (6), one can see that the larger value of the
transition point $\omega$ implies a larger value for the critical exponent
$\nu$. Moreover, the critical exponent $\nu$ obtained from the averaged gap
ratio $\nu^{{}_{GR}}$ and the entanglement entropy $\nu^{{}_{EE}}$ always
remain near one, i.e. $\nu{\sim}1$, showing the usual contradiction with the
Harris bound. On the other hand, the critical exponent obtained from the
diagonal entropy ($1.5{<}\nu^{{}_{DE}}{\leqslant}2$) and the Schmidt gap
($\nu^{{}_{SG}}{\cong}2$) are far more consistent with the Harris bound. This
shows that if the underlying transition is second-order then diagonal entropy
and Schmidt gap indeed have better convergence towards their thermodynamic
limits even for system sizes as small as $N{=}15$. Schmidt gap only depends on
the two largest eigenvalues of the reduced density matrix of the half system
and ignores the rest of the spectrum. Similarly, in diagonal entropy one also
relies on partial information through setting the off-diagonal terms of the
half system reduced density matrix to zero (in computational basis). This
artificial decohering action has recently been proposed for emulating the
thermodynamic behavior in MBL context Nico-Katz et al. (2022a). In fact, in
the deep ergodic and MBL phases, the reduced density matrix is expected to be
decohered in the thermodynamic limit due to being either maximally mixed (i.e.
maximally entangled with the rest of the system) or a product pure state (i.e.
fully separable), respectively. Note that, the variation of $\nu$ at different
rescaled energy $\varepsilon$ is a consequence of several anti-crossings
between different energy eigenstates.

A side product of finite-size scaling analysis, using Kosterlitz-Thouless
ansatz, is the non-universal fitting parameter $b$, see Eq. (4). In Figs.6
(a)-(f), we present the optimal values of $b$ obtained from all the four
quantities as a function of rescaled energy $\varepsilon$ for various values
of $\alpha$. As it is evident from the figure, the quantities preserve the
same hierarchy for the values of $b$, namely

$b^{{}_{SG}}\geqslant b^{{}_{DE}}\geqslant b^{{}_{EE}}\geqslant b^{{}_{GR}}.$
(7)

Finely, we have to emphasize that our numerical analysis does not favor any of
the transition types for describing the MBL transition. This is indicated in
the quality function $Q$. While at a certain energy the second-order
transition may show slightly better $Q$, in another energy it is the opposite.
Therefore, due to the small accessible system sizes, our study cannot
conclusively settle the type of the MBL transition.

Figure 6: The fitting parameter $b$ for Kosterlitz-Thouless transition type
(see Eq. (4)) as a function of energy scale $\varepsilon$ for various
interaction ranges from (a): $\alpha{=}0.5$ to (f):
$\alpha{\rightarrow}\infty$. These parameters are obtained from finite-size
scaling analysis for different observables including the averaged gap ratio
$\overline{\langle r\rangle}$ (dark blue triangle), the normalized
entanglement entropy $\overline{\langle S_{{}_{EE}}\rangle}/S^{P}_{{}_{EE}}$
(red circle), the normalized diagonal entropy $\overline{\langle
S_{{}_{DE}}\rangle}/S^{P}_{{}_{DE}}$ (magenta square), and the Schmidt gap
$\overline{\langle\Delta\rangle}$ (yellow diamond).

##  VII Conclusion

In this paper, we have determined the phase boundary between ergodic and MBL
phases along the energy spectrum, i.e. the mobility edge, for a Heisenberg
spin-chain with long-range interaction in the presence of disordered magnetic
field. This has been done using several quantities, including the level
statistics ratio, the entanglement entropy, the diagonal entropy, and the
Schmidt gap. We showed that long-range interaction enhances the localization
effect and shifts the mobility edge in favor of MBL. Two types of transition,
namely second-order and Kosterlitz-Thouless, have been investigated for
describing the emergence of the MBL phase. Our analysis show that the
transition type does not change the description of the mobility edge and the
phase boundary is hardly affected by that. However, the choice of the quantity
may indeed shift the mobility edge significantly. This is a finite-size effect
which is related to different convergence speed of the quantities in reaching
their thermodynamic limit. We have established a hierarchy among the
quantities with respect to their corresponding transition points and critical
exponents. In the whole spectrum, larger values of the transition point
$\omega$ implies larger values for the critical exponents, $\nu$ and $b$.
Schmidt gap and diagonal entropy seem to converge faster towards their
thermodynamic limit such that in the case of second-order transition, their
critical exponent $\nu$ show significant consistency with the Harris bound,
for a wide range of the long-range interactions. The closeness of the quality
functions obtained for both second-order and Kosterlitz-Thouless transitions,
shows that one cannot conclusively determine the transition type using small
system sizes considered in our analysis.

##  VIII Acknowledgment

A.B. acknowledge support from the National Key R&D Program of China (Grant No.
2018YFA0306703), the National Science Foundation of China (Grants No.
12050410253, No. 92065115, and No. 12274059), and the Ministry of Science and
Technology of China (Grant No. QNJ2021167001L). R.Y. thanks the National
Science Foundation of China for the International Young Scientists Fund (Grant
No. 12250410242).

*

## Appendix A

In the main text, we discussed that for both second-order and Kosterlitz-
Thouless transition types, a more precise estimation of the transition point
$\omega$ and other relevant exponents, namely $\nu$ and $b$, can be obtained
by adopting a standard finite-size scaling analysis. In this method, by
rescaling data as a function of $sN/\xi_{{}_{OS}}=sN|W-\omega|^{\nu}$ for the
second-order transition or $sN/\xi_{{}_{KT}}=sN\exp{(-b|W-\omega|^{-0.5})}$
for the Kosterlitz-Thouless transition type, one can collapse all the curves
on each other. For achieving the best data collapse one can use the
optimization algorithms for minimizing a quality function $Q$. In this work,
following the Refs. Šuntajs et al. (2020b); Aramthottil et al. (2021), we
select

$Q=\dfrac{\sum_{m=1}^{\mathcal{N}-1}|X_{m+1}-X_{m}|}{\max{(X)}-\min{(X)}}-1,$
(A1)

as our quality function, which according to our experience seemed to be
stabler and more precise. Here, $X$ stands for the desired quantity with
$\mathcal{N}$ values at different disorder strength $W$ and system size $N$
which has been sorted according to non-decreasing values of
$sgn(|W-\omega|)N/\xi$. This quality function converges to zero in the case of
an ideal data collapse to a single curve and larger otherwise.

## References

  * Altman (2018) E. Altman, Nat. Phys. 14, 979 (2018).
  * Abanin and Papić (2017) D. A. Abanin and Z. Papić, Ann. Phys. 529, 1700169 (2017).
  * Abanin et al. (2019) D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  * Altman and Vosk (2015) E. Altman and R. Vosk, Annu. Rev. Condens. Matter Phys. 6, 383 (2015).
  * Abanin et al. (2021) D. A. Abanin, J. H. Bardarson, G. De Tomasi, S. Gopalakrishnan, V. Khemani, S. A. Parameswaran, F. Pollmann, A. C. Potter, M. Serbyn, and R. Vasseur, Ann. Phys. 427, 168415 (2021).
  * Polkovnikov et al. (2011) A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
  * Borgonovi et al. (2016) F. Borgonovi, F. Izrailev, L. Santos, and V. Zelevinsky, Phys. Rep. 626, 1 (2016).
  * Oganesyan and Huse (2007) V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007).
  * Pal and Huse (2010) A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010).
  * Khemani et al. (2017a) V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse, Phys. Rev. X 7, 021013 (2017a).
  * Khemani et al. (2017b) V. Khemani, D. N. Sheng, and D. A. Huse, Phys. Rev. Lett. 119, 075702 (2017b).
  * Luitz et al. (2015) D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B 91, 081103 (2015).
  * Kjäll et al. (2014) J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Phys. Rev. Lett. 113, 107204 (2014).
  * Bauer and Nayak (2013) B. Bauer and C. Nayak, J. Stat. Mech.: Theory Exp 2013, P09005 (2013).
  * (15) N. Laflorencie, eprint arXiv:2112.09102.
  * Bardarson et al. (2012) J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012).
  * Chiara et al. (2006) G. D. Chiara, S. Montangero, P. Calabrese, and R. Fazio, J. Stat. Mech. 2006, P03001 (2006).
  * Žnidarič et al. (2008) M. Žnidarič, T. c. v. Prosen, and P. Prelovšek, Phys. Rev. B 77, 064426 (2008).
  * Serbyn et al. (2013) M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. Lett. 110, 260601 (2013).
  * Vosk and Altman (2013) R. Vosk and E. Altman, Phys. Rev. Lett. 110, 067204 (2013).
  * Andraschko et al. (2014) F. Andraschko, T. Enss, and J. Sirker, Phys. Rev. Lett. 113, 217201 (2014).
  * Agarwal et al. (2015) K. Agarwal, S. Gopalakrishnan, M. Knap, M. Müller, and E. Demler, Phys. Rev. Lett. 114, 160401 (2015).
  * Prelovšek et al. (2017) P. Prelovšek, M. Mierzejewski, O. Barišić, and J. Herbrych, Ann. Phys. 529, 1600362 (2017).
  * Setiawan et al. (2017) F. Setiawan, D.-L. Deng, and J. H. Pixley, Phys. Rev. B 96, 104205 (2017).
  * Vasseur and Moore (2016) R. Vasseur and J. E. Moore, J. Stat. Mech. Theory Exp 2016, 064010 (2016).
  * Žnidarič et al. (2016) M. Žnidarič, A. Scardicchio, and V. K. Varma, Phys. Rev. Lett. 117, 040601 (2016).
  * Ruihua Fan and Zhai (2017) H. S. Ruihua Fan, Pengfei Zhang and H. Zhai, Science Bulletin 62, 707 (2017).
  * Chen et al. (2017) X. Chen, T. Zhou, D. A. Huse, and E. Fradkin, Ann. Phys. 529, 1600332 (2017).
  * Huang et al. (2017) Y. Huang, Y.-L. Zhang, and X. Chen, Ann. Phys. 529, 1600318 (2017).
  * He and Lu (2017) R.-Q. He and Z.-Y. Lu, Phys. Rev. B 95, 054201 (2017).
  * Li et al. (2017) J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Phys. Rev. X 7, 031011 (2017).
  * Gärttner et al. (2017) M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Nat. Phys. 13, 781 (2017).
  * Doggen et al. (2018) E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D. Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi, Phys. Rev. B 98, 174202 (2018).
  * Nico-Katz et al. (2022a) A. Nico-Katz, A. Bayat, and S. Bose, Phys. Rev. Research 4, 033070 (2022a).
  * Kohlert et al. (2019) T. Kohlert, S. Scherg, X. Li, H. P. Lüschen, S. Das Sarma, I. Bloch, and M. Aidelsburger, Phys. Rev. Lett. 122, 170403 (2019).
  * Schulz et al. (2019) M. Schulz, C. A. Hooley, R. Moessner, and F. Pollmann, Phys. Rev. Lett. 122, 040606 (2019).
  * Lüschen et al. (2017) H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Phys. Rev. Lett. 119, 260401 (2017).
  * Sierant and Zakrzewski (2022) P. Sierant and J. Zakrzewski, Phys. Rev. B 105, 224203 (2022).
  * Chanda and Zakrzewski (2022) T. Chanda and J. Zakrzewski, Phys. Rev. B 105, 054309 (2022).
  * Nico-Katz et al. (2022b) A. Nico-Katz, A. Bayat, and S. Bose, Phys. Rev. B 105, 205133 (2022b).
  * Huse et al. (2013) D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Phys. Rev. B 88, 014206 (2013).
  * Chandran et al. (2014) A. Chandran, V. Khemani, C. R. Laumann, and S. L. Sondhi, Phys. Rev. B 89, 144201 (2014).
  * Sachdev (2011) S. Sachdev, _Quantum Phase Transitions_ (Cambridge University Press, 2011).
  * Potter et al. (2015) A. C. Potter, R. Vasseur, and S. A. Parameswaran, Phys. Rev. X 5, 031033 (2015).
  * Roushan et al. (2017) P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, et al., Science 358, 1175 (2017).
  * Rispoli et al. (2019) M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, and M. Greiner, Nature 573, 385 (2019).
  * Gong et al. (2021) M. Gong, G. D. de Moraes Neto, C. Zha, Y. Wu, H. Rong, Y. Ye, S. Li, Q. Zhu, S. Wang, Y. Zhao, et al., Phys. Rev. Research 3, 033043 (2021).
  * Sierant et al. (2020a) P. Sierant, D. Delande, and J. Zakrzewski, Phys. Rev. Lett. 124, 186601 (2020a).
  * Sels and Polkovnikov (2021) D. Sels and A. Polkovnikov, Phys. Rev. E 104, 054105 (2021).
  * Sels (2022a) D. Sels, Phys. Rev. B 106, L020202 (2022a).
  * Solórzano et al. (2021) A. Solórzano, L. F. Santos, and E. J. Torres-Herrera, Phys. Rev. Research 3, L032030 (2021).
  * Schiulaz et al. (2020) M. Schiulaz, E. J. Torres-Herrera, F. Pérez-Bernal, and L. F. Santos, Phys. Rev. B 101, 174312 (2020).
  * Chanda et al. (2020a) T. Chanda, P. Sierant, and J. Zakrzewski, Phys. Rev. Research 2, 32045 (2020a).
  * Mondragon-Shem et al. (2015) I. Mondragon-Shem, A. Pal, T. L. Hughes, and C. R. Laumann, Phys. Rev. B 92, 064203 (2015).
  * Xu et al. (2020) Z. Xu, H. Huangfu, Y. Zhang, and S. Chen, New J. Phys. 22 (2020).
  * Wei et al. (2019) X. Wei, C. Cheng, G. Xianlong, and R. Mondaini, Phys. Rev. B 99, 165137 (2019).
  * Villalonga et al. (2018) B. Villalonga, X. Yu, D. J. Luitz, and B. K. Clark, Phys. Rev. B 97, 104406 (2018).
  * Devakul and Singh (2015) T. Devakul and R. R. P. Singh, Phys. Rev. Lett. 115, 187201 (2015).
  * Geißler and Pupillo (2020) A. Geißler and G. Pupillo, Phys. Rev. Research 2, 042037 (2020).
  * De Roeck et al. (2016) W. De Roeck, F. Huveneers, M. Müller, and M. Schiulaz, Phys. Rev. B 93, 014203 (2016).
  * (61) X. Wei, R. Mondaini, and G. Xianlong, eprint arXiv:2001.04105.
  * Maksymov et al. (2017) A. O. Maksymov, N. Rahman, E. Kapit, and A. L. Burin, Phys. Rev. A 96, 057601 (2017).
  * Guo et al. (2021a) Q. Guo, C. Cheng, Z.-H. Sun, Z. Song, H. Li, Z. Wang, W. Ren, H. Dong, D. Zheng, Y.-R. Zhang, et al., Nat. Phys. 17, 234 (2021a).
  * Zha et al. (2020) C. Zha, V. M. Bastidas, M. Gong, Y. Wu, H. Rong, R. Yang, Y. Ye, S. Li, Q. Zhu, S. Wang, et al., Phys. Rev. Lett. 125, 170503 (2020).
  * Xu et al. (2018) K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song, W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, et al., Phys. Rev. Lett. 120 (2018).
  * (66) B. Chiaro, C. Neill, A. Bohrdt, M. Filippone, F. Arute, K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends, et al., eprint arXiv:1910.06024.
  * Guo et al. (2021b) Q. Guo, C. Cheng, H. Li, S. Xu, P. Zhang, Z. Wang, C. Song, W. Liu, W. Ren, H. Dong, et al., Phys. Rev. Lett. 127 (2021b).
  * Mi et al. (2022) X. Mi, M. Ippoliti, C. Quintana, A. Greene, Z. Chen, J. Gross, F. Arute, K. Arya, J. Atalaya, R. Babbush, et al., Nature 601, 531 (2022).
  * (69) M. Gong, H.-L. Huang, S. Wang, C. Guo, S. Li, Y. Wu, Q. Zhu, Y. Zhao, S. Guo, H. Qian, et al., eprint arXiv:2201.05957.
  * 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, Science 349, 842 (2015).
  * Kondov et al. (2015) S. S. Kondov, W. R. McGehee, W. Xu, and B. DeMarco, Phys. Rev. Lett. 114, 083002 (2015).
  * Bordia et al. (2017a) P. Bordia, H. Lüschen, S. Scherg, S. Gopalakrishnan, M. Knap, U. Schneider, and I. Bloch, Phys. Rev. X 7, 041047 (2017a).
  * Bordia et al. (2017b) P. Bordia, H. Lüschen, U. Schneider, M. Knap, and I. Bloch, Nat. Phys. 13, 460 (2017b).
  * Choi et al. (2016) J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Science 352, 1547 (2016).
  * Lukin et al. (2019) A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Science 364, 256 (2019).
  * Smith et al. (2016) J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nat. Phys. 12, 907 (2016).
  * Morong et al. (2021) W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Nature 599, 393 (2021).
  * Rajabi et al. (2019) F. Rajabi, S. Motlakunta, C.-Y. Shih, N. Kotibhaskar, Q. Quraishi, A. Ajoy, and R. Islam, npj Quantum Inf. 5, 32 (2019).
  * Choi et al. (2017) S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, et al., Nature 543, 221 (2017).
  * Choi et al. (2019) J. Choi, H. Zhou, S. Choi, R. Landig, W. W. Ho, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, D. A. Abanin, et al., Phys. Rev. Lett. 122, 043603 (2019).
  * Nguyen et al. (2021) T. Nguyen, N. Andrejevic, H. C. Po, Q. Song, Y. Tsurimaki, N. C. Drucker, A. Alatas, E. E. Alp, B. M. Leu, A. Cunsolo, et al., Nano Lett. 21, 7419 (2021).
  * Panda et al. (2019) R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor, and M. Žnidarič, Epl 128, 67003 (2019).
  * Morningstar et al. (2022) A. Morningstar, L. Colmenarez, V. Khemani, D. J. Luitz, and D. A. Huse, Phys. Rev. B 105, 174205 (2022).
  * Sierant et al. (2020b) P. Sierant, M. Lewenstein, and J. Zakrzewski, Phys. Rev. Lett. 125, 156601 (2020b).
  * Pietracaprina et al. (2018) F. Pietracaprina, N. Macé, D. J. Luitz, and F. Alet, SciPost Phys. 5, 45 (2018).
  * Harris (1974) A. B. Harris, J. Phys. C: Solid State Phys. 7, 3082 (1974).
  * Chayes et al. (1986) J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer, Phys. Rev. Lett. 57, 2999 (1986).
  * (88) A. Chandran, C. R. Laumann, and V. Oganesyan, eprint arXiv:1509.04285.
  * Gray et al. (2018) J. Gray, S. Bose, and A. Bayat, Phys. Rev. B 97, 201105 (2018).
  * Levi et al. (2016) E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Phys. Rev. Lett. 116, 237203 (2016).
  * Sun et al. (2020) Z.-H. Sun, J. Cui, and H. Fan, Phys. Rev. Research 2, 013163 (2020).
  * Dumitrescu et al. (2019) P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran, M. Serbyn, and R. Vasseur, Phys. Rev. B 99, 094205 (2019).
  * Goremykina et al. (2019) A. Goremykina, R. Vasseur, and M. Serbyn, Phys. Rev. Lett. 122, 040601 (2019).
  * Morningstar and Huse (2019) A. Morningstar and D. A. Huse, Phys. Rev. B 99, 224205 (2019).
  * Morningstar et al. (2020) A. Morningstar, D. A. Huse, and J. Z. Imbrie, Phys. Rev. B 102, 125134 (2020).
  * Šuntajs et al. (2020a) J. Šuntajs, J. Bonča, T. c. v. Prosen, and L. Vidmar, Phys. Rev. E 102, 062144 (2020a).
  * Laflorencie et al. (2020) N. Laflorencie, G. Lemarié, and N. Macé, Phys. Rev. Research 2, 042033 (2020).
  * Šuntajs et al. (2020b) J. Šuntajs, J. Bonča, T. c. v. Prosen, and L. Vidmar, Phys. Rev. B 102, 064207 (2020b).
  * Hopjan et al. (2021) M. Hopjan, G. Orso, and F. Heidrich-Meisner, Phys. Rev. B 104, 235112 (2021).
  * Aramthottil et al. (2021) A. S. Aramthottil, T. Chanda, P. Sierant, and J. Zakrzewski, Phys. Rev. B 104, 214201 (2021).
  * Kubala et al. (2021) P. Kubala, P. Sierant, G. Morigi, and J. Zakrzewski, Phys. Rev. B 103, 174208 (2021).
  * Roy and Logan (2019) S. Roy and D. E. Logan, SciPost Phys. 7 (2019).
  * Nandkishore and Sondhi (2017) R. M. Nandkishore and S. L. Sondhi, Phys. Rev. X 7, 041021 (2017).
  * Yao et al. (2014) N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap, M. Müller, E. A. Demler, and M. D. Lukin, Phys. Rev. Lett. 113, 243002 (2014).
  * Modak and Nag (2020a) R. Modak and T. Nag, Phys. Rev. E 101, 052108 (2020a).
  * Burin (2015a) A. L. Burin, Phys. Rev. B 92, 104428 (2015a).
  * Schiffer et al. (2019) S. Schiffer, J. Wang, X.-J. Liu, and H. Hu, Phys. Rev. A 100, 063619 (2019).
  * Burin (2015b) A. L. Burin, Phys. Rev. B 91, 094202 (2015b).
  * Tikhonov and Mirlin (2018) K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B 97, 214205 (2018).
  * Maksymov and Burin (2020) A. O. Maksymov and A. L. Burin, Phys. Rev. B 101, 024201 (2020).
  * Li et al. (2016) H. Li, J. Wang, X.-J. Liu, and H. Hu, Phys. Rev. A 94, 063625 (2016).
  * Thomson and Schiró (2020) S. J. Thomson and M. Schiró, Phys. Rev. Research 2, 043368 (2020).
  * Deng et al. (2020) X. Deng, G. Masella, G. Pupillo, and L. Santos, Phys. Rev. Lett. 125, 010401 (2020).
  * Li et al. (2021) W.-H. Li, X. Deng, and L. Santos, Phys. Rev. Lett. 127, 260601 (2021).
  * Vu et al. (2022) D. Vu, K. Huang, X. Li, and S. Das Sarma, Phys. Rev. Lett. 128, 146601 (2022).
  * Pino (2014) M. Pino, Phys. Rev. B 90, 174204 (2014).
  * De Tomasi (2019) G. De Tomasi, Phys. Rev. B 99, 054204 (2019).
  * Safavi-Naini et al. (2019) A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Phys. Rev. A 99, 033610 (2019).
  * Singh et al. (2017) R. Singh, R. Moessner, and D. Roy, Phys. Rev. B 95, 094205 (2017).
  * Nag and Garg (2019) S. Nag and A. Garg, Phys. Rev. B 99, 224203 (2019).
  * Modak and Nag (2020b) R. Modak and T. Nag, Phys. Rev. Research 2, 012074 (2020b).
  * Roy and Sharma (2018) N. Roy and A. Sharma, Phys. Rev. B 97, 125116 (2018).
  * (123) R. Yousefjani, S. Bose, and A. Bayat, eprint arXiv:2201.03716.
  * Baier et al. (2016) S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science 352, 201 (2016).
  * (125) A. L. Burin, eprint arXiv:0611387.
  * Imbrie (2016a) J. Z. Imbrie, J. Stat. Phys. 163, 998 (2016a).
  * Imbrie (2016b) J. Z. Imbrie, Phys. Rev. Lett. 117, 027201 (2016b).
  * De Roeck and Huveneers (2017) W. De Roeck and F. m. c. Huveneers, Phys. Rev. B 95, 155129 (2017).
  * Kiefer-Emmanouilidis et al. (2021) M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Phys. Rev. B 103, 024203 (2021).
  * Prasad and Garg (2021) Y. Prasad and A. Garg, Phys. Rev. B 103, 064203 (2021).
  * De Tomasi et al. (2021) G. De Tomasi, I. M. Khaymovich, F. Pollmann, and S. Warzel, Phys. Rev. B 104, 024202 (2021).
  * Sels (2022b) D. Sels, Phys. Rev. B 106, L020202 (2022b).
  * Sels (2022) D. Sels, in _APS March Meeting Abstracts_ (2022), vol. 2022, p. Z15.004.
  * Page (1993) D. N. Page, Phys. Rev. Lett. 71, 1291 (1993).
  * Torres-Herrera and Santos (2017) E. J. Torres-Herrera and L. F. Santos, Ann. Phys. 529, 1600284 (2017).
  * Torres-Herrera et al. (2016) E. J. Torres-Herrera, J. Karp, M. Távora, and L. F. Santos, Entropy 18, 359 (2016).
  * (137) J. Gray, A. Bayat, A. Pal, and S. Bose, eprint arXiv:1908.02761.
  * Chanda et al. (2020b) T. Chanda, P. Sierant, and J. Zakrzewski, Phys. Rev. B 101, 035148 (2020b).
  * Houdayer and Hartmann (2004) J. Houdayer and A. K. Hartmann, Phys. Rev. B 70, 014418 (2004).
  * Bhattacharjee and Seno (2001) S. M. Bhattacharjee and F. Seno, J. Phys. A: Math. Gen. 34, 6375 (2001).
  * Sorge (2015) A. Sorge, pyfssa 0.7.6. Zenodo. (2015).
  * (142) O. Melchert, eprint arXiv:0910.5403.