# Effects of auto-correlated disorder on the dynamics around the many-body
localization transition

Isaías Vallejo-Fabila  E. Jonathan Torres-Herrera  Instituto de Física,
Benemérita Universidad Autónoma de Puebla, Apt. Postal J-48, Puebla, 72570,
Mexico

###### Abstract

The presence of frozen uncorrelated random on-site potential in interacting
quantum systems can induce a transition from an ergodic phase to a localized
one, the so-called many-body localization. Here we numerically study the
effects of auto-correlated disorder on the static and dynamical properties of
a one-dimensional many-body quantum system which exhibits many-body
localization. Specifically, by means of some standard measures of energy level
repulsion and localization of energy eigenstates, we show that a strong degree
of correlations between the on-site potentials in the one-dimensional spin-1/2
Heisenberg model leads to suppression of the many-body localization phase,
while level repulsion is mitigated for small disorder strengths, although
energy eigenstates remain well extended. Our findings are also remarkably
manifested in time domain, on which we put main emphasis, as shown by the time
evolution of experimentally relevant observables, like the return probability
and the spin auto-correlation function.

The celebrated, very well studied and equally understood single-particle
Anderson localization (AL) [1] is expanded in some sense to the realm of
interacting quantum systems by the still incompletely understood so-called
many-body localization (MBL), where a thermal description of the system
properties fails [2, 3, 4, 5]. Among some questions challenging MBL and some
of its features are the ones related to the role of finite-size effects [6, 7,
8], finite times [9] and self-averaging [10, 11, 12]. Deepen our knowledge
about MBL is of fundamental theoretical, experimental [13, 14, 15] and
technological [see for instance [16]] interest.

The standard scenario in the non-interacting case is that AL transition occurs
in three dimensions for strong enough disorder strength, meanwhile in one- and
two-dimensional systems at the limit of infinite system size an infinitesimal
strength of the uncorrelated disorder is enough to localize all eigenstates of
the system (see [17, 18, 19] and references therein). However, it has been
shown, theoretical and experimentally, that particular correlations between
the on-site potentials are able to produce delocalization and even the
appearance of a mobility edge in one-dimensional systems [20, 21, 22, 23, 24]
(see also [25] and references therein). Further studies of auto-correlated
disorder in low-dimensional non-interacting systems have addressed, for
instance, the onset of Bloch-like oscillations [26], localization in ultracold
atoms [27, 28] and optical lattices [29], violation of Harris criterion [30],
non-universality of AL [31] and features of the entanglement spectrum [32, 33,
34]. Recently, the effects of auto-correlated disorder have been also
investigated in the joint context of ultracold atoms and machine learning
[35]. A natural question is what could be the effects of auto-correlated
disorder in the context of MBL, however, in contrast with the non-interacting
case, effects of auto-correlated disorder in interacting many-body quantum
systems have been less explored [36, 37, 38, 39, 40, 41]. While most works are
focused on time-independent quantities, dynamics are predominantly left aside
even though they are of fundamental relevance not only by their ubiquity but
from a theoretical point of view and also for experiments dealing with many-
body quantum systems, like the ones with ultracold atoms in optical lattices
[42, 13, 43, 44], trapped ions [45] and superconducting qubit arrays [46],
where information about the system properties is usually accessed through the
dynamics. It is fair to mention that in [47] the role of speckle disorder on
the dynamics of the imbalance was part of the analysis that led to the main
conclusion that MBL transitions under uncorrelated disorder and speckle
disorder belong to the same universality class. Here, however, our aim is not
to determine the universality class of MBL or even the behavior of critical
points under our model of auto-correlated disorder, but instead to provide a
general description of its effects on the time evolution of the many-body
quantum system.

For our purposes we consider the Heisenberg model for spin-1/2 particles in a
one-dimensional lattice with pair interactions, on-site potentials and
periodic boundaries,

$H=\sum_{k=1}^{L}(S_{k}^{x}S_{k+1}^{x}+S_{k}^{y}S_{k+1}^{y}+S_{k}^{z}S_{k+1}^{z})+\sum_{k=1}^{L}h_{k}S_{k}^{z}.$
(1)

In Eq. (1) we have set $\hbar=1$ and $S_{k}^{x,y,z}=\sigma_{k}^{x,y,z}/2$ are
spin-1/2 operators in terms of Pauli matrices acting over the particle located
at site $k$. Traditional studies consider the amplitudes $h_{k}$ being
uncorrelated random numbers from an uniform distribution, that is, $h_{k}\in
U(-h,h)$, with $h$ the disorder strength. The critical point $h_{c}$ for MBL
is not exactly known but there exist some estimates for finite-size systems,
from [48, 49, 50, 51, 52] the bounds are $3<h_{c}<4$, meanwhile in [53, 54,
55, 56] an upper bound $h_{c}>4$ is given. Here we consider the amplitudes
$h_{k}$ also as random numbers from an uniform distribution with support
$[-h,h]$, but linearly correlated, that is,
$\mathbb{E}(h_{k}h_{\ell})\neq\mathbb{E}(h_{k})\mathbb{E}(h_{\ell})$ for
$k,\ell=1,2,\dots L$ and $\mathbb{E}$ stands for the statistical expectation
value. The method that we use to generate auto-correlated random numbers is
based on the cumulative distribution function $F(X)$ of the sum of a $U(0,1)$
number and a $U(0,1/c)$ number, where $c$ is any positive real number. With
$V_{k}$ an i.i.d. $U(0,1)$ we generate the sequence

$\displaystyle X_{1}$ $\displaystyle=V_{0}+V_{1}/c,$ (2) $\displaystyle X_{k}$
$\displaystyle=F(X_{k-1})+V_{k}/c,\hskip 14.22636ptk>1.$

such that the sequence of $F(X_{k})$ is auto-correlated $U(0,1)$, then the
random numbers $h_{k}$ in Hamiltonian (1) are obtained according to
$h_{k}=h[2F(X_{k})-1]$. The distribution $F(X_{k})$ depends on the value of
$c$ and it can be determined exactly as shown in [57]. By tuning $c$ any
desired degree of correlations is obtained, $c\approx 0$ means absence of
correlations, meanwhile the strength of correlations grow larger as $c$
increases.

Since Hamiltonian (1) preserves the total magnetization in $z$-direction,
${\cal{S}}^{z}=\sum_{k}S_{k}^{z}$, the Hilbert space splits in sectors with
fixed values of ${\cal{S}}^{z}$. Here we work in the subsector with
${\cal{S}}^{z}=0$ for which the dimension is ${\cal{D}}=L![(L/2)!]^{-2}$.

## Quantities.-

We describe the static quantities, as well as the time-dependent ones that
will be used in our analysis as indicators of ergodicity and localization.

We start by denoting the eigenvalues and eigenstates of Hamiltonian (1) by
$E_{\alpha}$ and $\left|\psi_{\alpha}\right>$ respectively. Next, to analyze
level repulsion we use the mean value of the _ratio between consecutive level
spacings_ $\overline{r}$ [48], defined through

$\overline{r}=\frac{1}{{\cal{D}}}\sum_{\alpha=1}^{\cal{D}}\text{min}\left(r_{\alpha},\frac{1}{r_{\alpha}}\right),\,\,\text{with}\,\,r_{\alpha}=\frac{E_{\alpha+1}-E_{\alpha}}{E_{\alpha}-E_{\alpha-1}}.$
(3)

Level repulsion is absent in the MBL phase, corresponding to spacings with
Poisson-like statistics and $\overline{r}\approx 0.386$ [48, 58]. Meanwhile,
in the ergodic phase the spacings have GOE-like statistics, that is, energy
eigenvalues present linear repulsion, for which $\overline{r}\approx 0.536$
[58]. The ratio decreases monotonically between those two limits as the
magnitude of the uncorrelated disorder increases [48].

To study the degree of localization of the eigenstates of Hamiltonian (1) we
use the _inverse participation ratio_ , IPR, given by

$\text{IPR}_{\alpha}=\sum_{n=1}^{\cal{D}}\left|C_{n}^{\alpha}\right|^{4},\,\,\text{with}\,\,C_{n}^{\alpha}=\left\langle\psi_{\alpha}|n\right\rangle.$
(4)

Where $\left\\{\left|n\right>\right\\}$ represents a suitable basis determined
by physical considerations. Since our interest is localization in real space,
we choose the basis composed by eigenstates of the $z$-part of Hamiltonian
(1), so-called site basis. A state is extended in the respective basis if
$\text{IPR}_{\alpha}\propto{\cal{D}}^{-1}$, while for localized states
$\text{IPR}_{\alpha}$ is ${\cal{O}}(1)$.

At a first glance it could appear as redundant to analyze both quantities, but
up to our knowledge no one-to-one correspondence between the behavior of
$\overline{r}$ and $IPR_{\alpha}$ for model (1) has been proved. Certainly a
proof of this last point is not the aim of this work, however, the analysis of
both features, level repulsion and degree of localization of states, is
important for our objectives and because they show up in different stages of
the time evolution of the system [59, 60, 61, 62].

Dynamics can be analyzed employing the _return probability_ , which is also
known as survival probability and non-decay probability. It measures the time
dependent probability for a given quantum system to return to its initial
state $|\left.\Psi(0)\right\rangle$, it is given by

$RP(t)=\left|\left\langle\Psi(0)|e^{-iHt}|\Psi(0)\right\rangle\right|^{2}.$
(5)

Return probability has been analyzed in several contexts, both theoretical and
experimentally. By the side of experiments in molecules [63], ultra-cold atoms
in magneto-optic traps [64], atom chip [65] and prethermalization in Floquet
systems [66]. See also an interesting proposal in [67] regarding the
experimental measuring of $RP(t)$. Theoretically in relation with the time
evolution of unstable quantum states [68] and possible implications in
cosmology of the late time behavior of false vacuum decay [69]. Non-periodic
substitution potentials [70], non-inteacting fermions [71], quantum walks and
complex networks [72, 73, 74, 75], matter-radiation interaction models like
Dicke [76] and Bose-Hubbard [77].

In addition, to further study dynamics we consider the _spin auto-correlation
function_ ,

$I(t)=\frac{4}{L}\sum_{k=1}^{L}\left<\Psi(0)\right|S_{k}^{z}(0)S_{k}^{z}(t)\left|\Psi(0)\right>.$
(6)

This quantity measures the similarity between the initial spin configuration
and the one at a late time $t$. For specific initial states, such as a Néel-
like state, it coincides with the density imbalance measured in experiments
with cold atoms [13].

Since we are dealing with a disordered system, we compute averages over
disorder realizations for the static quantities, $\overline{r}$ and
$\text{IPR}_{\alpha}$, while an additional average over initial states is done
for the dynamical quantities, $RP(t)$ and $I(t)$. Our results will be
presented using the notation for the averages as
$\left\langle\dots\right\rangle$ irrespective of their kind. The initial
states we use are part of the set $\left\\{\left|n\right>\right\\}$, that is,
$\left|\Psi(0)\right>\equiv\left|n\right>$ with energy
$E_{n}=\left<n\right|H\left|n\right>\approx 0$.

Before presenting our results, we judge as instructive and pertinent to give
an overview of the existing picture about the dynamics of many-body quantum
systems with uncorrelated disorder. Certainly, the dynamics of many-body
quantum systems could be approached by many kinds of observables, but for
decaying-in-time observables, like the return probability and the spin auto-
correlation function, a clear picture already exists [62]. For the first one,
quantum mechanics predicts a universal quadratic decay in time for very short
times, this is the region of the ancient Zeno’s paradox in the quantum realm
[78, 79]. The quadratic decay is followed by one which depends on the
interaction strength, exponential for weak strengths and Gaussian for strong
enough strengths [59]. A power-law decay is developed at larger times, it can
have two different sources, one being the inevitable bounds in the energy
spectrum [80] and the other one related to the increasing of correlations
between eigenstates components as the disorder strength increases [60]. The
former one has been used to anticipate thermalization [81, 82] in interacting
quantum systems, while the last one was investigated in relation with the
multifractality of eigenstates [60, 83]. Power-law decays have been also
observed for the imbalance, both theoretically [84, 54] and experimentally
[85]. Eigenstates correlations and their relation with the return probability
was also studied more recently in [67]. At even larger times and preceding
saturation the repulsion hole appears, a dynamical manifestation of short- and
long-range energy-level repulsion which explicitly occurs at the many-body
Thouless time and ends up at the longest time scale, the so-called Heisenberg
time. In particular, it was shown that the ratio between the Thouless and
Heisenberg times approaches unity as the system approaches the MBL phase [86],
this fact was further confirmed later in [9].

## Results.-

We now present our results, explicitly we show the effects of the correlations
between the on-site random potentials on the static and time-dependent
properties previously defined. We start with the behavior of the structure of
eigenstates [Fig. 1(a)] and repulsion between adjacent energy levels [Fig.
1(b)] with respect to the degree of correlations represented by $c$ for a
fixed system size $L=16$. In both panels the results are for three different
disorder strengths, $h=0.5$ (red solid curve), $h=3.75$ (green solid curve)
and $h=6.0$ (blue solid curve), which in a context of uncorrelated disorder
are representative of the ergodic, critical and localized regions,
respectively. We use dashed lines to represent the GOE theoretical prediction
(black) and Poisson prediction (turquoise) for both quantities,
$\text{IPR}_{\alpha}$ and $\overline{r}$. We observe two different behaviors
depending on the disorder strength as the degree of correlations increases
from a very small value of $c$ to a larger one, $c\approx 20$. For small
disorder strength, $h=0.5$, the $\text{IPR}_{\alpha}$ almost does not change
with respect to $c$, however level statistics clearly transit from GOE-like to
Poisson-like as $c$ increases. For intermediate, $h=3.75$, and strong,
$h=6.0$, disorder strengths the behaviors are similar, a crossover occurs as
$c$ increases from localized states to extended states and from Poisson-like
statistics to GOE-like statistics. It is worth to mention that results from a
single disorder realization of the static properties show big fluctuations,
this is depicted by the shaded (gray) curves in both panels of Fig. 1.

Figure 1: Inverse participation ratio (IPR) and average spacing ratio
$\overline{r}$ versus degree of correlations $c$. (a) IPR from Eq. 4 and (b)
$\overline{r}$ from Eq. 3. In each panel the shaded (gray) curves represent
raw data from a single disorder realization (see main text), while the curves
in red, green and blue are running averages for $h=0.5$, $h=3.75$ and $h=6$,
respectively. Theoretical values for GOE (black dashed line) and Poisson
(turquoise dashed line). System size is $L=16$.

Of course, those fluctuations could be reduced if an average over disorder
realizations is carried out. Also note that in Fig. 1(a) an average over $100$
energy eigenstates with energy closest to zero was performed, however this was
not enough to smooth the curves for $h=3.75$ and $h=6.0$ because
$\text{IPR}_{\alpha}$ is not self-averaging for those values [87]. To
circumvent this situation, we decided to present in a simple way running
averages over intervals of $10$ values of the static properties along $c$
which are precisely represented by the colored solid lines already explained
above.

We now move to the time-dependent quantities, in Fig. 2 we depict the behavior
of the return probability for three different disorder strengths, just like in
Fig. 1 and with the same coloring code. Four values of $c$ are selected from
large to small in order to cover different regimes of eigestates structure and
level statistics, namely $c=15$ (a), $c=8$ (b), $c=2$ (c) and $c=0.5$ (d) (see
Fig. 1).

Figure 2: Averaged return probability for different disorder strengths and
degrees of correlations $c$. (a) $c=15$, (b) $c=8$, (c) $c=2$, (d) $c=0.5$. In
each panel $h=0.5$ (red), $h=3.75$ (green), $h=6$ (blue). Dashed lines are the
corresponding saturation values. System size is $L=16$. An average over
$25\times 10^{3}$ samples ($500$ initial states and $50$ disorder
realizations). An additional running average was implemented in order to
smooth even more the curves.

The system size is also fixed to $L=16$. In general we observe similar initial
dynamics for the four values of $c$ and the three values of $h$, that is, a
typical initial non-exponential decay, explicitly a Gaussian decay [88, 89,
90, 59] which is followed by some fluctuations related to the details of the
local density of states which is not a perfect Gaussian [89, 90, 59, 60]. For
small disorder, $h=0.5$ (red solid line), the power-law decay with exponent
$2$ predicted for weak uncorrelated disorder [81, 82] is observed in the
approximated time interval $t\in[3,12]$ for all values of $c$. In the limit of
large system sizes the power-law decay with exponent 2 prevails for longer
times [91]. The survival of power-law decays with system size was also studied
for the imbalance in [54]. The repulsion hole is wiped out for $c=15$,
meanwhile the hole appears and becomes more clear as the value of $c$
decreases, as shown for $c=8$ and $c=2$ in Figs. 2(b) and (c) respectively.
The saturation values of the return probability which corresponds to the
average IPR of the initial state in the energy basis and indicated by
corresponding dashed lines in Fig. 2 remain almost the same and with a small
value, thus indicating that the average IPR of the initial states behaves at
least in a similar way as the energy eigenstates [see Fig. 1(a)]. For the two
other disorder strengths, $h=3.75$ (green solid curves) and $h=6.0$ (blue
solid curves), the dependence on $c$ of the late-time dynamics of the return
probability is quite different from the case $h=0.5$. In particular, for
$c=15$ and both values of $h$ there is practically no distinction between the
corresponding dynamics, they show a similar repulsion hole and saturation
value. The curves for the two different values of $h$ get more separated as
the value of $c$ increases. The behaviors of the repulsion hole and the
saturation value are consistent with the results of the static quantities
previously presented, while the dependence of the power-law decay on $c$ and
$h$ deserves further studies. For $c=0.5$ in Fig. 2(d) the degree of
correlations is weak enough to recover the known picture for uncorrelated
disorder [60].

Figure 3: Absolute value of the averaged spin auto-correlation function for
same parameters, coloring code and averaging procedure as used in Fig. 2.
Dashed lines are the corresponding saturation values of $|\left\langle
I(t)\right\rangle|$.

We finish our analysis by considering the time evolution of the spin auto-
correlation function, $I(t)$ [Eq. (6)]. Since $I(t)$ can have positive and
negative values even after an average over initial states and disorder
realizations, in Fig. 3 we present our results for the absolute value of the
averaged $I(t)$. The same parameters as for $RP(t)$ in Fig. 2 were used. Once
again we observe that the inclusion of correlations between the on-site random
potentials leads to rich and nontrivial dynamics, but now testified by $I(t)$.
The dependence on $c$ of this quantity is similar to that of $RP(t)$. For
large $c$ [Fig 3(a)] the evolution for $h=3.75$ and $h=6.0$ shows a repulsion
hole and small saturation values, meanwhile for $h=0.5$ there is no hole. At
this point we should note that the negative values of the averaged $I(t)$ are
present only for $h=0.5$ and $c=15,8,2$ as seen by the bumps inside the time
interval $t\in[10,100]$ of Figs. 3(a), (b) and (c). The behaviors of $I(t)$
for all disorder strengths, when the value of $c$ decreases, approaches the
ones for uncorrelated disorder [84, 62]. Although the repulsion hole is
present in $RP(t)$ and $I(t)$, it should be noted that in [92] a careful
analysis was performed to show that while for $RP(t)$ the depth of the
repulsion hole increases, for $I(t)$ and other observables the depth
apparently decreases when the system size increases.

## Conclusions.-

We revealed that the inclusion of linearly auto-correlated on-site random
potentials in a paradigmatic model for many-body localization and tuning its
strength lets to control both, static and dynamical properties of the system
in a more rich and nontrivial way that uncorrelated disorder. Strong
correlations result in thermal-like behavior of the studied dynamical probes
when the disorder strength is large. Meanwhile for small disorder strength
only energy-level indicators are significantly affected. Diminishing the
degree of correlations leads to the usual behavior when uncorrelated disorder
is considered. We hope that our results will motivate the community to
consider different observables of interest in theoretical and experimental
studies. Additional studies could also address questions related to time
scales and power-law decays. The persistence of the effects observed in this
work when approaching the thermodynamic limit is also another question that
could be addressed in further investigations.

###### Acknowledgements.

I.V.-F. and E.J.T.-H. are grateful to LNS-BUAP for their supercomputing
facility.

## References

  * Anderson [1958] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  * Nandkishore and Huse [2015] R. Nandkishore and D. Huse, Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  * Alet and Laflorencie [2018] F. Alet and N. Laflorencie, Many-body localization: An introduction and selected topics, C. R. Phys. 19, 498 (2018).
  * Altman [2018] E. Altman, Many-body localization and quantum thermalization, Nat. Phys. 14, 979 (2018).
  * Abanin  _et al._ [2019] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. of Mod. Phys. 91, 021001 (2019).
  * Potirniche  _et al._ [2019] I.-D. Potirniche, S. Banerjee, and E. Altman, Exploration of the stability of many-body localization in d$>$1, Phys. Rev. B 99, 10.1103/physrevb.99.205149 (2019).
  * Abanin  _et al._ [2021] D. Abanin, J. H. Bardarson, G. De Tomasi, S. Gopalakrishnan, V. Khemani, S. Parameswaran, F. Pollmann, A. Potter, M. Serbyn, and R. Vasseur, Distinguishing localization from chaos: Challenges in finite-size systems, Ann. Phys. 427, 168415 (2021).
  * Sierant and Zakrzewski [2022] P. Sierant and J. Zakrzewski, Challenges to observation of many-body localization, Physical Review B 105, 224203 (2022).
  * Šuntajs  _et al._ [2020] J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, Quantum chaos challenges many-body localization, Phys. Rev. E 102, 062144 (2020).
  * Schiulaz  _et al._ [2020] M. Schiulaz, E. J. Torres-Herrera, F. Pérez-Bernal, and L. F. Santos, Self-averaging in many-body quantum systems out of equilibrium: Chaotic systems, Phys. Rev. B 101, 174312 (2020).
  * Torres-Herrera  _et al._ [2020a] E. J. Torres-Herrera, G. De Tomasi, M. Schiulaz, F. Pérez-Bernal, and L. F. Santos, Self-averaging in many-body quantum systems out of equilibrium: Approach to the localized phase, Phys. Rev. B 102, 094310 (2020a).
  * Torres-Herrera  _et al._ [2020b] E. J. Torres-Herrera, I. Vallejo-Fabila, A. J. Martínez-Mendoza, and L. F. Santos, Self-averaging in many-body quantum systems out of equilibrium: Time dependence of distributions, Phys. Rev. E 102, 062126 (2020b).
  * Schreiber  _et al._ [2015] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  * Bordia  _et al._ [2016] P. Bordia, H. P. Lüschen, S. S. Hodgman, M. Schreiber, I. Bloch, and U. Schneider, Coupling identical one-dimensional many-body localized systems, Phys. Rev. Lett. 116, 140401 (2016).
  * Kohlert  _et al._ [2019] T. Kohlert, S. Scherg, X. Li, H. P. Lüschen, S. D. Sarma, I. Bloch, and M. Aidelsburger, Observation of many-body localization in a one-dimensional system with a single-particle mobility edge, Phys. Rev. Lett. 122, 170403 (2019).
  * Halpern  _et al._ [2019] N. Y. Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, Quantum engine based on many-body localization, Phys. Rev. B 99, 024203 (2019).
  * Abrahams  _et al._ [1979] E. Abrahams, P. Anderson, D. Licciardello, and T. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Lee and Ramakrishnan [1985] P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985).
  * Lagendijk  _et al._ [2009] A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, Fifty years of Anderson localization, Phys. Today 62, 24 (2009).
  * Dunlap  _et al._ [1990] D. H. Dunlap, H. Wu, and P. W. Phillips, Absence of localization in a random-dimer model, Phys. Rev. Lett. 65, 88 (1990).
  * De Moura and Lyra [1998] F. A. De Moura and M. L. Lyra, Delocalization in the 1D Anderson model with long-range correlated disorder, Phys. Rev. Lett. 81, 3735 (1998).
  * Izrailev and Krokhin [1999] F. Izrailev and A. Krokhin, Localization and the mobility edge in one-dimensional potentials with correlated disorder, Phys. Rev. Lett. 82, 4062 (1999).
  * Kuhl  _et al._ [2000] U. Kuhl, F. Izrailev, A. Krokhin, and H.-J. Stöckmann, Experimental observation of the mobility edge in a waveguide with correlated disorder, Appl. Phys. Lett. 77, 633 (2000).
  * Kuhl  _et al._ [2008] U. Kuhl, F. M. Izrailev, and A. A. Krokhin, Enhancement of localization in one-dimensional random potentials with long-range correlations, Phys. Rev. Lett. 100, 126402 (2008).
  * Izrailev  _et al._ [2012] F. M. Izrailev, A. A. Krokhin, and N. Makarov, Anomalous localization in low-dimensional systems with correlated disorder, Phys. Rep. 512, 125 (2012).
  * Domínguez-Adame  _et al._ [2003] F. Domínguez-Adame, V. Malyshev, F. De Moura, and M. Lyra, Bloch-like oscillations in a one-dimensional lattice with long-range correlated disorder, Phys. Rev. Lett. 91, 197402 (2003).
  * Sanchez-Palencia  _et al._ [2007] L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Anderson localization of expanding Bose-Einstein condensates in random potentials, Phys. Rev. Lett. 98, 210401 (2007).
  * Lugan  _et al._ [2009] P. Lugan, A. Aspect, L. Sanchez-Palencia, D. Delande, B. Grémaud, C. A. Müller, and C. Miniatura, One-dimensional Anderson localization in certain correlated random potentials, Phys. Rev. A 80, 023605 (2009).
  * Fratini and Pilati [2015] E. Fratini and S. Pilati, Anderson localization in optical lattices with correlated disorder, Phys. Rev. A 92, 063621 (2015).
  * Shima  _et al._ [2004] H. Shima, T. Nomura, and T. Nakayama, Localization-delocalization transition in one-dimensional electron systems with long-range correlated disorder, Phys. Rev. B 70, 075116 (2004).
  * Titov and Schomerus [2005] M. Titov and H. Schomerus, Nonuniversality of Anderson localization in short-range correlated disorder, Phys. Rev. Lett. 95, 126602 (2005).
  * Mondragon-Shem  _et al._ [2013] I. Mondragon-Shem, M. Khan, and T. L. Hughes, Characterizing disordered fermion systems using the momentum-space entanglement spectrum, Phys. Rev. Lett. 110, 046806 (2013).
  * Pouranvari and Yang [2014] M. Pouranvari and K. Yang, Maximally entangled mode, metal-insulator transition, and violation of entanglement area law in noninteracting fermion ground states, Phys. Rev. B 89, 115104 (2014).
  * Andrade  _et al._ [2014] E. C. Andrade, M. Steudtner, and M. Vojta, Anderson localization and momentum-space entanglement, J. Stat. Mech. Th. Exp. 2014, P07022 (2014).
  * Pilati and Pieri [2019] S. Pilati and P. Pieri, Supervised machine learning of ultracold atoms with speckle disorder, Sci. Rep. 9, 1 (2019).
  * Dukesz  _et al._ [2009] F. Dukesz, M. Zilbergerts, and L. F. Santos, Interplay between interaction and (un)correlated disorder in one-dimensional many-particle systems: delocalization and global entanglement, New J. Phys. 11, 043026 (2009).
  * Xu  _et al._ [2019] H.-Z. Xu, S.-Y. Zhang, Z.-Y. Rao, Z. Zhou, G.-C. Guo, and M. Gong, Many-body localization as a $d=\infty$ Anderson localization with correlated disorder, arXiv preprint arXiv:1910.06601 (2019).
  * Ghosh  _et al._ [2019] S. Ghosh, A. Acharya, S. Sahu, and S. Mukerjee, Many-body localization due to correlated disorder in Fock space, Phys. Rev. B 99, 165131 (2019).
  * Roy and Logan [2020] S. Roy and D. E. Logan, Localization on certain graphs with strongly correlated disorder, Phys. Rev. Lett. 125, 250402 (2020).
  * Samanta  _et al._ [2021] A. Samanta, A. Chakraborty, and R. Sensarma, Many-body localized to ergodic transitions in a system with correlated disorder, arXiv preprint arXiv:2111.11455 (2021).
  * Darius Shi  _et al._ [2022] Z. Darius Shi, V. Khemani, R. Vasseur, and S. Gopalakrishnan, Many body localization transition with correlated disorder, arXiv e-prints , arXiv:2204.06017 (2022).
  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, _et al._ , Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Ebadi  _et al._ [2021] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, _et al._ , Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021).
  * Leibfried  _et al._ [2003] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, Rev. Mod. Phys. 75, 281 (2003).
  * 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._ , Experimental characterization of the quantum many-body localization transition, Phys. Rev. Research 3, 033043 (2021).
  * Maksymov  _et al._ [2020] A. Maksymov, P. Sierant, and J. Zakrzewski, Many-body localization in a one-dimensional optical lattice with speckle disorder, Phys. Rev. B 102, 134205 (2020).
  * Oganesyan and Huse [2007] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  * Pal and Huse [2010] A. Pal and D. A. Huse, Many-body localization phase transition, Phys. Rev. B 82, 174411 (2010).
  * Berkelbach and Reichman [2010] T. C. Berkelbach and D. R. Reichman, Conductivity of disordered quantum lattice models at infinite temperature: Many-body localization, Phys. Rev. B 81, 224429 (2010).
  * Kjäll  _et al._ [2014] J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum Ising chain, Phys. Rev. Lett. 113, 107204 (2014).
  * Luitz  _et al._ [2015] D. J. Luitz, N. Laflorencie, and F. Alet, Many-body localization edge in the random-field Heisenberg chain, Phys. Rev. B 91, 081103 (2015).
  * Devakul and Singh [2015] T. Devakul and R. R. Singh, Early breakdown of area-law entanglement at the many-body delocalization transition, Phys. Rev. Lett. 115, 187201 (2015).
  * 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, Many-body localization and delocalization in large quantum chains, Phys. Rev. B 98, 174202 (2018).
  * Sierant  _et al._ [2020a] P. Sierant, M. Lewenstein, and J. Zakrzewski, Polynomially filtered exact diagonalization approach to many-body localization, Physical Review Letters 125, 156601 (2020a).
  * Sierant  _et al._ [2020b] P. Sierant, D. Delande, and J. Zakrzewski, Thouless time analysis of Anderson and many-body localization transitions, Phys. Rev. Lett. 124, 186601 (2020b).
  * Willemain and Desautels [1993] R. Willemain and A. Desautels, A method to generate autocorrelated uniform random numbers, J. Stat. Comput. Simul. 45, 23 (1993).
  * Atas  _et al._ [2013] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  * Torres-Herrera  _et al._ [2014] E. J. Torres-Herrera, M. Vyas, and L. F. Santos, General features of the relaxation dynamics of interacting quantum systems, New J. Phys. 16, 063010 (2014).
  * Torres-Herrera and Santos [2015] E. J. Torres-Herrera and L. F. Santos, Dynamics at the many-body localization transition, Phys. Rev. B 92, 014208 (2015).
  * Torres-Herrera and Santos [2017a] E. J. Torres-Herrera and L. F. Santos, Dynamical manifestations of quantum chaos: correlation hole and bulge, Philos. Trans. Royal Soc. A 375, 20160434 (2017a).
  * Torres-Herrera  _et al._ [2018] E. J. Torres-Herrera, A. M. García-García, and L. F. Santos, Generic dynamical features of quenched interacting quantum systems: Survival probability, density imbalance, and out-of-time-ordered correlator, Phys. Rev. B 97, 060303(R) (2018).
  * Rothe  _et al._ [2006] C. Rothe, S. Hintschich, and A. Monkman, Violation of the exponential-decay law at long times, Phys. Rev. Lett. 96, 163601 (2006).
  * Wilkinson  _et al._ [1997] S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram, and M. G. Raizen, Experimental evidence for non-exponential decay in quantum tunnelling, Nature 387, 575 (1997).
  * Gherardini  _et al._ [2017] S. Gherardini, C. Lovecchio, M. M. Müller, P. Lombardi, F. Caruso, and F. S. Cataliotti, Ergodicity in randomly perturbed quantum systems, Quantum Sci. Technol. 2, 015007 (2017).
  * Singh  _et al._ [2019] K. Singh, C. J. Fujiwara, Z. A. Geiger, E. Q. Simmons, M. Lipatov, A. Cao, P. Dotti, S. V. Rajagopal, R. Senaratne, T. Shimasaki, _et al._ , Quantifying and controlling prethermal nonergodicity in interacting Floquet matter, Phys. Rev. X 9, 041021 (2019).
  * Tikhonov and Mirlin [2021] K. S. Tikhonov and A. D. Mirlin, Eigenstate correlations around the many-body localization transition, Phys. Rev. B 103, 064204 (2021).
  * Chiu  _et al._ [1977] C. Chiu, E. Sudarshan, and B. Misra, Time evolution of unstable quantum states and a resolution of Zeno’s paradox, Phys. Rev. D 16, 520 (1977).
  * Krauss and Dent [2008] L. M. Krauss and J. Dent, Late time behavior of false vacuum decay: possible implications for cosmology and metastable inflating states, Phys. Rev. Lett. 100, 171301 (2008).
  * De Oliveira and Pellegrino [1999] C. R. De Oliveira and G. Q. Pellegrino, Quantum return probability for substitution potentials, J. Phys. A: Math. Gen. 32, L285 (1999).
  * Krapivsky  _et al._ [2018] P. L. Krapivsky, J.-M. Luck, and K. Mallick, Quantum return probability of a system of n non-interacting lattice fermions, J. Stat. Mech. Th. Exp. 2018, 023104 (2018).
  * Ampadu [2012] C. Ampadu, Return probability of the Fibonacci quantum walk, Commun. Theor. Phys. 58, 220 (2012).
  * Mülken and Blumen [2011] O. Mülken and A. Blumen, Continuous-time quantum walks: Models for coherent transport on complex networks, Phys. Rep. 502, 37 (2011).
  * Xu and Liu [2008] X. Xu and F. Liu, Continuous-time quantum walks on Erdös–Rényi networks, Phys. Lett. A 372, 6727 (2008).
  * Riascos and Mateos [2015] A. Riascos and J. L. Mateos, Fractional quantum mechanics on networks: Long-range dynamics and quantum transport, Phys. Rev. E 92, 052814 (2015).
  * Lerma-Hernández  _et al._ [2019] S. Lerma-Hernández, D. Villaseñor, M. Bastarrachea-Magnani, E. J. Torres-Herrera, L. F. Santos, and J. G. Hirsch, Dynamical signatures of quantum chaos and relaxation time scales in a spin-boson system, Phys. Rev. E 100, 012218 (2019).
  * de la Cruz  _et al._ [2020] J. de la Cruz, S. Lerma-Hernández, and J. G. Hirsch, Quantum chaos in a system with high degree of symmetries, Phys. Rev. E 102, 032208 (2020).
  * Misra and Sudarshan [1977] B. Misra and E. G. Sudarshan, The Zeno’s paradox in quantum theory, J. Math. Phys. 18, 756 (1977).
  * Pascazio [2014] S. Pascazio, All you ever wanted to know about the quantum Zeno effect in 70 minutes, Open Sys. & Information Dyn. 21, 1440007 (2014).
  * Khalfin [1958] L. A. Khalfin, Contribution to the decay theory of a quasi-stationary state, Sov. Phys. JETP 6, 1053 (1958).
  * Távora  _et al._ [2016] M. Távora, E. J. Torres-Herrera, and L. F. Santos, Inevitable power-law behavior of isolated many-body quantum systems and how it anticipates thermalization, Phys. Rev. A 94, 041603 (2016).
  * Távora  _et al._ [2017] M. Távora, E. J. Torres-Herrera, and L. F. Santos, Power-law decay exponents: A dynamical criterion for predicting thermalization, Phys. Rev. A 95, 013604 (2017).
  * Torres-Herrera and Santos [2017b] E. J. Torres-Herrera and L. F. Santos, Extended nonergodic states in disordered many-body quantum systems, Ann. Phys. (Berlin) 529, 1600284 (2017b).
  * Luitz  _et al._ [2016] D. J. Luitz, N. Laflorencie, and F. Alet, Extended slow dynamical regime close to the many-body localization transition, Phys. Rev. B 93, 060201 (2016).
  * Lüschen  _et al._ [2017] H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems, Phys. Rev. Lett. 119, 260401 (2017).
  * Schiulaz  _et al._ [2019] M. Schiulaz, E. J. Torres-Herrera, and L. F. Santos, Thouless and relaxation time scales in many-body quantum systems, Phys. Rev. B 99, 174313 (2019).
  * Solórzano  _et al._ [2021] A. Solórzano, L. F. Santos, and E. J. Torres-Herrera, Multifractality and self-averaging at the many-body localization transition, Phys. Rev. Research 3, L032030 (2021).
  * Izrailev and Castaneda-Mendoza [2006] F. Izrailev and A. Castaneda-Mendoza, Return probability: Exponential versus Gaussian decay, Phys. Lett. A 350, 355 (2006).
  * Torres-Herrera and Santos [2014a] E. J. Torres-Herrera and L. F. Santos, Quench dynamics of isolated many-body quantum systems, Phys. Rev. A 89, 043620 (2014a).
  * Torres-Herrera and Santos [2014b] E. J. Torres-Herrera and L. F. Santos, Nonexponential fidelity decay in isolated interacting quantum systems, Phys. Rev. A 90, 033623 (2014b).
  * Torres-Herrera and Santos [2019] E. J. Torres-Herrera and L. F. Santos, Signatures of chaos and thermalization in the dynamics of many-body quantum systems, Eur. Phys. J. Spec. Top. 227, 1897 (2019).
  * Lezama  _et al._ [2021] T. L. Lezama, E. J. Torres-Herrera, F. Pérez-Bernal, Y. B. Lev, and L. F. Santos, Equilibration time in many-body quantum systems, Phys. Rev. B 104, 085117 (2021).