# Chaos controlled and disorder driven phase transitions by breaking
permutation symmetry

Manju C<EMAIL_ADDRESS>Department of Physics, Indian Institute
of Technology Palakkad, Kerala, 678623.  Arul Lakshminarayan
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Madras, Chennai 600036.  Uma Divakaran<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Palakkad, Kerala, 678623.

###### Abstract

Introducing disorder in a system typically breaks symmetries and can introduce
dramatic changes in its properties such as localization. At the same time, the
clean system can have distinct many-body features depending on how chaotic it
is. In this work the effect of permutation symmetry breaking by disorder is
studied in a system which has a controllable and deterministic regular to
chaotic transition. Results indicate a continuous phase transition from an
area-law to a volume-law entangled phase irrespective of whether there is
chaos or not, as the strength of the disorder is increased. The critical
disorder strength obtained by finite size scaling, indicate a strong
dependence on whether the clean system is regular or chaotic to begin with. In
the process, we also obtain the critical exponents associated with this phase
transition. Additionally, we find that a relatively small disorder is seen to
be sufficient to delocalize a chaotic system.

Deterministic chaos, originating in the lack of sufficient number of conserved
quantities, gives rise to ergodic and mixing dynamics with a positive Lyapunov
exponent in the classical case Tabor (1989); Lichtenberg and Lieberman (2013);
Ott (2002), and in the quantum case results in states with random matrix
properties, thermalization and volume law entanglement Gutzwiller (1991);
Stöckmann (1999); Mehta (2004); Wimberger (2014); D’Alessio  _et al._ (2016);
Kaufman  _et al._ (2016); Bohigas  _et al._ (1984); Srednicki (1994).
Disorder, depending on its relative strength on the other hand, can lead to
Griffiths phases Griffiths (1969), Anderson Anderson (1958) or many-body
localization Basko  _et al._ (2006); Oganesyan and Huse (2007), and can also
result in thermalization and delocalization via breaking symmetries. For
example, the Sachdev-Ye-Kitaev (SYK) model is one where randomized all-to-all
couplings results in maximal quantum chaos, maximizing a temperature dependent
quantum Lyapunov exponent. Sachdev and Ye (1993); Kitaev ; Maldacena and
Stanford (2016).

Arrays of superconducting Josephson junction based transmons, which are
nonlinear oscillators, are currently the most successful implementations of
quantum computers Koch  _et al._ (2007); Barends  _et al._ (2013); Wang  _et
al._ (2022). Disorder in the energies of the individual qubit transmons is a
necessary evil in the presence of inter-transmon couplings. A detailed study
of the consequences for these devices comes with a warning of being
“dangerously close to a phase of uncontrollable chaotic ﬂuctuations” Berke
_et al._ (2022).

On the other hand, it has long been appreciated that dynamical chaos has
serious effects on practical implementation of quantum many-body systems, as
it can increase entanglement to random state values, and small subsystems face
effective decoherence, thus questioning the reliability of quantum information
processing Zurek (2006); Hauke  _et al._ (2012). Several related aspects have
been addressed in Refs. Bandyopadhyay and Lakshminarayan (2002); Gong and
Brumer (2003); Flambaum (2000); Song and Shepelyansky (2001); Georgeot and
Shepelyansky (2000); Braun (2002); Madhok  _et al._ (2018); Shepelyansky
(2001); Piga  _et al._ (2019); Chinni  _et al._ (2021); Berke  _et al._
(2022); Lerose and Pappalardi (2020), for example it has been discussed in
Ref. Georgeot and Shepelyansky (2000) that the presence of random individual
qubit energies along with residual short range interactions can make the
system chaotic under certain limits that compromises the quantum computer. In
another work, it is shown that the error in quantum computation may increase
exponentially with system size in the chaotic regime in the presence of
imperfections Song and Shepelyansky (2001). Thus, it is important to
understand the interplay between disorder and dynamical chaos. We therefore,
in this work, bring together the clearly connected, yet disparate, fields of
chaos and disorder, and study the effect of disorder on regular or chaotic
quantum dynamics.

In particular, we investigate the dynamics of $N$ particles originally
restricted to the small $O(N)$ permutation symmetric subspace (PSS), but that
can have various dynamical regimes, from near-integrable to intermediate and
fully chaotic. The disorder breaks the permutation symmetry and results in a
phase transition into the full $O(\exp(N))$ Hilbert space. Interestingly, the
extent of chaos in the “clean” symmetric space is found to dictate the
critical strength of the disorder that ensures the dynamics has penetrated to
the full Hilbert space, in the large $N$ limit. The transition has many
possible order parameters. The block entanglement changes from area law
$\sim\ln N$, characteristic of permutation symmetric states Seshadri  _et al._
(2018) to volume law $\sim N$. Collective observables such as the total
angular momentum changes from $\sim N$ to $\sim\sqrt{N}$. This transition
shows similarities to a transition from the superradiant to the radiant phases
of collective atomic states in a cavity Scully (2015); Gegg  _et al._ (2018);
Rubies-Bigorda and Yelin (2022). A recent work studied the dynamics when the
system does not start from a permutation symmetric subspace of an otherwise
permutation symmetric Hamiltonian Qi  _et al._ (2023). They find non-generic
effects of the deep Hilbert space by studying growth of out-of-time order
correlator and entanglement. Another work studied the effect of introducing
non-uniform initial states on the permutation symmetry of the Hamiltonian
Iemini  _et al._ (2024). However, we wish to study the case when the disorder
free system starting from PSS has a controlled transition to deterministic
chaos, so that its interplay with disorder becomes transparent.

A convenient model for this study is provided by the kicked top which is a
textbook example of quantum chaos Haake  _et al._ (1987); D’Ariano  _et al._
(1992); Haake (2013), of one large spin of angular momentum $S=N/2$, that can
be considered as the collective dynamics of $N$ spin $1/2$ particles within
the permutation symmetric subspace Wang  _et al._ (2004); Madhok  _et al._
(2015). Several experiments have fruitfully implemented this to understand the
effects of chaos on entanglement Chaudhury  _et al._ (2009); Neill  _et al._
(2016); Krithika  _et al._ (2019) . A disordered version of the kicked top can
be used to investigate the effects of breaking permutation symmetry. The
specific Hamiltonian that we study is given by

$H=\frac{k}{\tau
2N}\sum_{\ell<\ell^{\prime}=1}^{N}(1+\epsilon_{\ell\ell^{\prime}})\sigma_{\ell}^{x}\sigma_{\ell^{\prime}}^{x}+\frac{p}{2}\sum_{n=-\infty}^{\infty}\sum_{\ell=1}^{N}\sigma_{\ell}^{y}\delta(t-n\tau),$
(1)

where, $\sigma_{\ell}^{\nu}$ with $\nu=x~{},y,~{}z$ are the Pauli matrices at
site $\ell$, and $k$ is the interaction strength that can be tuned such that
there is a transition from regular to deterministic chaotic dynamics in the
disorder free Hamiltonian, $\tau$ is the time period of delta kicks that is
set to unity, and $p$ is fixed to $\pi/2$ to compare with the known results.
Here, $\epsilon_{\ell\ell^{\prime}}$ are the random (quenched disorder) part
of the spin interactions taken from a normal distribution with zero mean and
standard deviation $w$, also referred to as disorder strength. The
corresponding unitary Floquet operator $U_{w}$ that evolves states
infinitesimally before consecutive kicks separated by the period $\tau$, is
given by

$\exp\left({-i\frac{k}{2N}\sum_{\ell<\ell^{\prime}=1}^{N}(1+\epsilon_{\ell\ell^{\prime}})\sigma_{\ell}^{x}\sigma_{\ell^{\prime}}^{x}}\right)\exp\left({-i\frac{\pi}{4}\sum_{\ell=1}^{N}\sigma_{\ell}^{y}}\right),\\\
$ (2)

where we have set $\hbar=1$.

The disorder free version $U_{0}$, with all $\epsilon_{\ell\ell^{\prime}}=0$,
is the quantum kicked top Haake (2013); Wimberger (2014); Haake  _et al._
(1987); Zyczkowski (1990); Sanders and Milburn (1989); Bhosale and Santhanam
(2018); Fox and Elston (1994); Wang  _et al._ (2010); Zou and Wang (2022);
Muñoz-Arias  _et al._ (2021); Fiderer and Braun (2018); Bhosale and Santhanam
(2017); Sieberer  _et al._ (2019); Lerose and Pappalardi (2020); Kumari and
Ghose (2018); Lombardi and Matzkin (2011); Anand  _et al._ (2024) where the
Hamiltonian is purely a function of the collective spin operators
$J_{\alpha}=\sum_{\ell}^{N}\sigma_{\ell}^{\alpha}/2$ where
$\alpha=x,~{}y,~{}z$. The total angular momentum
$J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$ defined in terms of these collective
spins is a constant of motion in the quantum kicked top Haake (2013); Wang
_et al._ (2004); Ghose  _et al._ (2008). The large $N$ limit of $U_{0}$ is
studied as a symplectic classical map on the unit sphere
$(X,Y,Z)\mapsto(X^{\prime},Y^{\prime},Z^{\prime})$ where $X=J_{x}/J$,
$Y=J_{y}/J$ and $Z=J_{z}/J$. This map shows regular dynamics for small values
of $k$, typically $<2$, and fully chaotic dynamics for large values of $k$ (
$k>6$), where the Lyapunov exponent may be approximated as ($\ln k-1$)
Constantoudis and Theodorakopoulos (1997); Wang and Robnik (2021). Various
aspects of long range interacting systems such as given above are discussed in
Ref. Defenu  _et al._ (2024).

If the initial state is an unentangled spin coherent state given by Glauber
and Haake (1976); Lee Loh and Kim (2015); Puri (2001)

$\ket{\psi_{0}}=\ket{\theta,\phi}=\left(\cos\frac{\theta}{2}\ket{0}+e^{i\phi}\sin\frac{\theta}{2}\ket{1}\right)^{\otimes{N}},$
(3)

then the state after $n$ Floquet periods is
$\ket{\psi_{n}}=U_{0}^{n}\ket{\psi_{0}}$. The states at all times $n$ are
restricted to the $N+1$ dimensional PSS and conserve the total angular
momentum $J^{2}=(N/2)(N/2+1)$. It has also been shown that the chaotic phase
can be described using a random matrix theory (RMT) in $N+1$ dimensions Haake
(2013). For small values of $N(\leq 4)$, the kicked top is exactly solvable
and does yield interesting insights already into the large $N$ cases Ruebeck
_et al._ (2017); Dogra  _et al._ (2019); Sreeram  _et al._ (2021).

Introducing disorder in the system will break the permutation symmetry of the
Hamiltonian, and the state $|\psi_{n}\rangle$ after $n$ kicks given by
$U_{w}^{n}\ket{\psi_{0}}$ will now span the full $2^{N}$ dimensional Hilbert
space where $J^{2}$ is no longer a constant of motion.

Figure 1: Schematic of the dynamics: Top row corresponds to disorder free
system where interactions connecting different spins have same magnitude as
depicted by connecting arrows of black color. The second term of the
Hamiltonian is the periodic kicks along $y-$direction. Bottom row corresponds
to a disordered system as shown by connecting arrows of different colors. For
times $n\tau<t<(n+1)\tau-\Delta\tau$, the interactions are dominant whereas
for $(n+1)\tau-\Delta\tau<t<(n+1)\tau$, it is the field term which is
dominant. Here, we use delta kicks which correspond to $\Delta\tau\to 0$
limit. If the radius of the sphere on the right is proportional to $N$, then
$\langle J\rangle$ for the disorder free system is restricted on the surface
of this sphere, whereas for the disordered system it is pushed inside the
sphere where $\langle J\rangle\sim\sqrt{N}$, as also shown in the figure
above.

The state $\ket{\psi_{n}}$ in presence of disorder tends to a random state on
the full Hilbert space, denoted as $\ket{\psi_{\text{RMT}}}$, in which the
expectation value of $J_{z}^{2}$ can be estimated by:

$\displaystyle\left\langle\ J_{z}^{2}\right\rangle_{{\text{RMT}}}$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{2^{N}}\lambda_{j}^{2}\overline{|\left\langle\psi_{\text{RMT}}|\phi_{j}\right\rangle|^{2}}=N/4,$
(4)

where $\ket{\phi_{j}}=\ket{j_{1}\cdots j_{N}}$, $j_{i}=\pm 1$, are
eigenvectors of $J_{z}$ with eigenvalues
$\lambda_{j}=(\sum_{\ell}j_{\ell})/2$. The ensemble average over random states
$\overline{|\left\langle\psi_{\text{RMT}}|\phi_{j}\right\rangle|^{2}}=1/2^{N}$
is used above along with the fact that
$\operatorname{Tr}J_{z}^{2}=N2^{N-1}/2$. Similarly $\left\langle
J_{x}^{2}\right\rangle_{\text{RMT}}=\left\langle
J_{y}^{2}\right\rangle_{\text{RMT}}=N/4$, and hence $\left\langle
J^{2}\right\rangle_{\text{RMT}}=3N/4$. Clearly, ${\langle J^{2}\rangle}$
exhibit a change in the scaling when the dynamics shifts from the permutation
symmetric subspace where $\langle J^{2}\rangle\sim N^{2}$ to the full Hilbert
space where the state being close to random has a $\sim N$ scaling. It is this
transition of the dynamics to the “full” Hilbert space that we shall focus in
the current work. A schematic diagram of the Hamiltonian studied highlighting
the difference in the dynamics of disorder free as well as the disordered
system is shown in Fig. 1.

One can also capture this change in the dynamics by studying von Neumann
entropy $S_{Q}(A)$ of the subsystem $A$ consisting of $Q-$ spins or qubits.
The von Neumann entropy is given by
$S_{Q}(A)=-\mbox{tr}(\rho_{A}\log_{2}\rho_{A})$, where $\rho_{A}$ is the
reduced density matrix of subsystem $A$. The expression for $S_{Q}(A)$ using
ensemble averaging over random permutation symmetric states is approximated by
Seshadri  _et al._ (2018)

$\displaystyle\left\langle
S_{Q}(A)\right\rangle_{PSS}\approx\log_{2}(Q+1)-\frac{2}{3}\left(\frac{Q+1}{N-Q+1}\right).$
(5)

We have checked numerically that $S_{Q}(A)$ even in the regular regime of
small $k$ within permutation symmetric subspace is proportional to
$~{}\log_{2}(Q+1)$ dependence. A related work calculating the von Neumann
entropy in PSS is given in Ref.Castro-Alvaredo and Doyon (2013). On the other
hand, when the dynamics traverse the full Hilbert space, $S_{Q}(A)$ is given
by the Page value Page (1993); Seshadri  _et al._ (2018); Lee  _et al._
(2017):

$\displaystyle\left\langle S_{Q}(A)\right\rangle_{2^{Q}}\approx Q-\frac{1}{\ln
2}\left(\frac{2^{Q}}{2^{N-Q+1}}\right).$ (6)

As $w$, the standard deviation of the disorder distribution or the disorder
strength is increased, various expectation values will potentially shift from
their values within the permutation symmetric subspace to the RMT values
relevant to the full Hilbert space. This is because increasing the strength of
the disorder $w$ will not only take the system out of the permutation
symmetric subspace but will also make it more chaotic since increasing $w$ is
equivalent to increasing $k$ as can be seen from Eq. 1. Thus as $w\to 0$ or
small when the dynamics is predominantly within the permutation symmetric
subspace, the scaling of long time limit of various quantities will follow the
PSS scaling, $i.e.,$ ${\langle J^{2}\rangle}\sim N^{2}$ and
${\langle{S_{Q}(A)}\rangle}\sim\log_{2}(Q+1)$. In the other extreme limit,
when $w\to\infty$ and the dynamics is described by RMT in the full
$2^{N}-$dimensions, ${\langle J^{2}\rangle}=3N/4$ and
${\langle{S_{Q}(A)}\rangle}\sim Q$.

We present the results of numerical calculations which are performed with an
initial state as given in Eq. 3 with $\theta=2.25,~{}\phi=1.1$ and is evolved
using the Floquet operator $U_{w}$ given in Eq. 2 . We use the fast Walsh
Hadamard transform to go to system sizes upto $N=16$. The evolution of
$\langle J^{2}\rangle$ and $S_{N/2}$ as a function of number of kicks, or time
$n$, is shown in Fig. 2 for $k=1$. We find that for small values of $w$,
$\langle J^{2}\rangle$ remains a constant till $n\sim N+1$ corresponding to
the Heisenberg time relevant to the PSS beyond which the system is taken out
of PSS so that its value starts decreasing. It is around $n\sim 2^{N}$, the
Heisenberg time relevant to the full Hilbert space that $\langle J^{2}\rangle$
again starts saturating, which implies that the system has realized the
presence of full Hilbert space around this time. For all further calculations,
we perform time averaging from $n=10^{5}$ to $n=3\times 10^{5}$ when a
thermalization has set in for $N=16$, the maximum system size considered in
this work. The change in scaling of $J^{2}$ and $S_{N/2}(A)$ as discussed in
the text is presented in Fig. 3(a) and (b), respectively, for different $w$.

Figure 2: (a) $\left\langle J^{2}\right\rangle$ with respect to time or number
of kicks $n$ for different disorder strengths $w$ with $k=1$, $N=12$ qubits,
averaged over 100 disorder realizations. Inset shows the entanglement
$S_{N/2}$ as a function of $n$ for the same parameters as in the main figure.

Figure 3: (a) Change in scaling of $\langle J^{2}\rangle$ from $N^{2}/4+N/2$
to $3N/4$ as $w$ is increased where the chaos parameter $k$ is set to 1.(b)
Same for $S_{N/2}(A)$ as a function of $N$ at $k=1$ as discussed in the text.
The red dashed line in Fig. (a) corresponds to $N^{2}/4+N/2$ whereas the black
dotted line corresponds to the RMT result $3N/4$. Similarly, red dashed line
in Fig. (b) corresponds to fitted line $0.41+0.54\log_{2}(N/2+1)$ and the
black dotted line is the RMT expression given by $N/2-1/2\ln 2$.

It is tempting to associate this change in the scaling of $\langle
J^{2}\rangle$ and $S_{N/2}$ with a second order phase transition. Therefore,
we now investigate the possible existence of a critical $w_{c}$ below which
the dynamics is predominantly within the permutation symmetric subspace and
above which the dynamics captures the properties of the full Hilbert space
described by RMT, and obtain relevant critical exponents of the associated
phase transition, if any. We therefore propose a finite size scaling of the
form Botet and Jullien (1983); Skinner  _et al._ (2019); Lee  _et al._ (2017);
Li  _et al._ (2018); Cardy (1988)

$\displaystyle{\langle J^{2}\rangle}=N^{\zeta/\nu}F((w-w_{c})N^{1/\nu}),$ (7)

where $\nu$ is the correlation length exponent. Fig. 4 (main) shows the
crossing of the data around $w_{c}=2.4$ when $k=1$ whereas the collapse of the
data with $\nu=0.59$ and $\zeta=0.59$ is shown in the inset of Fig. 4. In
order to confirm this phase transition further, we repeat the calculations for
other values of $k$, the results of which are shown in Figure 5. It is to be
noted that $k=4$ is a mixed phase space with considerable chaos already
present in the system.

Figure 4: $\langle J^{2}\rangle/N^{\zeta/\nu}$ as a function of disorder
strength $w$ for $k=1$ and different system sizes. These curves cross each
other at $w=w_{c}\approx 2.4$. The inset shows the collapse of the data with
$\zeta=0.59$ and $\nu=0.59$ consistent with the scaling proposed in Eq. 7.

Figure 5:  (a) The main figure shows the data corresponding to different
system sizes for $k=2$ crossing each other at $w=w_{c}\sim 0.41$ whereas the
inset shows the collapse of the data with $\nu=0.40$ and $\zeta=0.49$.(b) Same
as (a) but for $k=4$ with $w=w_{c}\sim 0.17$ and $\nu=0.3$.

Table 1 lists the critical disorder strength $w_{c}$ along with the critical
exponent $\nu$ for different values of the interaction strength or chaos
parameter $k$. Intuitively, the chaotic dynamics at larger $k$ values will
require smaller disorder to take the dynamics to the full Hilbert space, which
is also what we observe. Interestingly, we also find a weak parameter
dependence on the critical exponent $\nu$ for this phase transition. The
exponent $\nu$ lies within a range of 0.6 to 0.3 for the parameters studied.
Such a variation seems to be a characteristic of a disordered system such as
the well studied many-body-localization one Wang  _et al._ (2021). We also
find that in most cases, $\nu\sim\zeta$. The quality of the data collapse for
different values of $k$ confirm the presence of a second order phase
transition. At the same time, we do observe that identifying correct $w_{c}$
values for larger $k$ is difficult since $w_{c}$ then is very small and a
single crossing point for all system sizes is difficult to obtain. On the
other hand when $k\rightarrow 0$, the dynamics can get highly non-ergodic and
these parameters need more careful examinations. We have further confirmed
this phase transition using the entanglement entropy $S_{N/2}(A)$ with a
similar finite size scaling as in Eq. 7 giving $\nu=0.6$, $w_{c}=1.8$ when
$k=1$, as shown in Fig. 6.

It is also observed that the collapse is relatively good when the clean system
is either predominantly chaotic or predominantly regular. For a mixed phase
space, with significant fraction of both regular and chaotic regions, say at
$k=3$, the quality of the scaling seems to depend on the initial state. The
value in the table 1 for $k=3$ and $k=4$ corresponds to chaotic regions. For
regular regions in these cases, we could not observe scaling behaviour. This
may be attributed to the small values of angular quantum number $j$, that we
are forced to restrict ourselves to, as well as the complexity of mixed phase
space (see for example Madhok (2015) for cases when the initial state in a
kicked top is at the border between chaos and regular regions) and the
influence of phenomena such as chaos assisted tunneling Tomsovic and Ullmo
(1994); S.Keshavamurthy and P.Schlagheck ; Krithika  _et al._ (2023), that
warrants a detailed separate study.

$k$ |  $w_{c}$ |  $\nu$ |  $\zeta$
---|---|---|---
$0.5$ |  $3.17\pm 0.13$ |  $0.59\pm 0.04$ |  $0.63\pm 0.07$
$1.0$ |  $2.41\pm 0.06$ |  $0.59\pm 0.01$ |  $0.59\pm 0.04$
$2.0$ |  $0.41\pm 0.01$ |  $0.40\pm 0.01$ |  $0.49\pm 0.09$
$3.0$ |  $0.31\pm 0.03$ |  $0.62\pm 0.06$ |  $0.87\pm 0.05$
$4.0$ |  $0.17\pm 0.02$ |  $0.30\pm 0.05$ |  $0.32\pm 0.05$
Table 1: Critical disorder strength and exponents $\nu$, $\zeta$ for different
chaos parameter, $k$. We have used $\langle J^{2}\rangle$ with $N=12,14$ and
$16$ to identify the critical points with time averaging ranging from
$n=10^{5}$ to $3\times 10^{5}$ for 100 different realizations.  Figure 6: (a)
Plot of $\langle S_{N/2}(A)\rangle/N^{\zeta/\nu}$ vs $w$ for $k=1$
corresponding to different system sizes cross each other at $w=w_{c}\sim 1.8$.
Inset shows the collapse of the data with $\nu=0.67$ and $\zeta=0.76$.

In summary, this work has shown the existence of a critical strength of
permutation symmetry breaking, beyond which there is a second order phase
transition from area law entangled states with total angular momentum scaling
as $N$, to volume law entangled states with total angular momentum scaling as
$\sqrt{N}$. The critical disorder strength shows clear dependence on the
dynamics being regular or chaotic in the disorder-free permutation symmetric
space. The exponents on the other hand seem to show a weaker dependence.
Results not shown indicate that the criticality is robust on changing the the
initial state except when it is in the regular region of the mixed phase
space.

From an applications perspective, our results show that for disorder less than
the critical strength, permutation symmetric systems are likely to be reliably
used for their intended application, being at least not volume law entangled
in the full Hilbert space. It is possible that the proposed phase transition
be experimentally observed by extending few qubit experimental kicked top
studies Chaudhury  _et al._ (2009); Neill  _et al._ (2016); Krithika  _et al._
(2019), wherein disorder is anyway present. In fact, a disordered all-to-all
interacting system, simulating the SYK model, has been recently experimentally
studied using a combination of atomic cloud in a cavity and controllable light
shift in the large detuning limit Sauerwein  _et al._ (2023).

Among future directions, it is interesting to explore if breaking permutation
symmetry through other methods, say by introducing a power law interaction
such as in Russomanno  _et al._ (2021), will also exhibit a similar phase
transition as induced by disorder.

Acknowledgements: MC and UD acknowledege the HPC facility Chandra at IIT
Palakkad where the computations were carried out. AL thanks Sylvia Pappalardi
for discussions at an early stage of this work, at the program on “Dynamical
Foundations of Many-Body Chaos”, Institut Pascal, Université Paris-Saclay,
during March-April 2023.

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