# Generalized Survival Probability

David A. Zarate-Herrada1, Lea F. Santos2 and E. Jonathan Torres-Herrera1
1Instituto de Física, Universidad Autónoma de Puebla, Apt. Postal J-48,
Puebla, Puebla, 72570, Mexico  2Department of Physics, University of
Connecticut, Storrs, Connecticut 06269, USA

###### Abstract

The survival probability measures the probability that a system taken out of
equilibrium has not yet moved out from its initial state. Inspired by the
generalized entropies used to analyze nonergodic states, we introduce a
generalized version of the survival probability, and discuss how it can assist
studies of the structure of the eigenstates and ergodicity.

##  I Introduction

The square overlap between a given initial state $|\Psi(0)\rangle$ and its
time-evolved counterpart $|\Psi(t)\rangle$,

$SP(t)=|\langle\Psi(0)|\Psi(t)\rangle|^{2},$ (1)

gives the probability for finding the system still in its initial state at
time $t$. This quantity is known as survival probability, return probability,
or simply the fidelity between the initial and the evolved state. This
quantity has been extensively investigated since the early decades of quantum
mechanics, initially in the context of the uncertainty relation between time
an energy [1, 2, 3]. As stated by Fock in Ref. [3],

> [the time-energy uncertainty relation] may be viewed as a consequence of the
> general theorem of Fock and Krylov on the connection between the decay law
> and the energy distribution function.

The “connection” stated in the quote above refers to the fact that the
survival probability (decay law) is the absolute square of the Fourier
transform of the energy distribution of the initial state (energy distribution
function). That is, for a state evolving according to a Hamiltonian $H$, whose
eigenvalues and eigenstates are given by $E_{\alpha}$ and $|\alpha\rangle$,
one has that
$|\Psi(t)\rangle=\sum_{\alpha}C_{\alpha}^{(0)}e^{-iE_{\alpha}t}|\alpha\rangle$
and the survival probability can be written as

$SP(t)=\left|\sum_{\alpha}|C_{\alpha}^{(0)}|^{2}e^{-iE_{\alpha}t}\right|^{2}=\left|\int\rho(E)e^{-iEt}dE\right|^{2},$
(2)

where $C_{\alpha}^{(0)}=\langle\alpha|\Psi(0)\rangle$ and

$\rho(E)=\sum_{\alpha}|C_{\alpha}^{(0)}|^{2}\delta(E-E_{\alpha})$ (3)

is the energy distribution of the initial state. This distribution is also
known as local density of states (LDoS) or strength function, and its mean and
variance are [4]

$E^{(0)}=\sum_{\alpha}|C_{\alpha}^{(0)}|^{2}E_{\alpha}\hskip
8.5359pt\mbox{and}\hskip
8.5359pt\sigma^{2}=\sum_{\alpha}|C_{\alpha}^{(0)}|^{2}(E_{\alpha}-E^{(0)})^{2}.$
(4)

Both the survival probability and the LDoS are studied in a variety of
different fields, from quantum chaos and nuclear physics to localization and
quantum information science. These quantities received significant attention
from previous researchers of the Budker Institute in Novosibirsk, including
those to which we dedicate the present paper, namely Professor Giulio Casati
on the occasion of his 80th birthday in 2022, Professor Felix Izrailev on the
occasion of his 80th birthday in 2021, and Professor Vladimir Zelevinsky on
the occasion of his 85th birthday in 2022.

Despite the simplicity of Eq. (1), the evolution of the survival probability
in many-body quantum systems is quite rich, with different behaviors emerging
at different time scales, which reveal details about the initial state, the
spectrum and the eigenstates of the considered model. The Taylor expansion of
the phase factor in Eq. (2) shows that the survival probability at very short
times, $t\ll 1/\sigma$, presents a quadratic and universal behavior,
$SP(t)\approx 1-\sigma^{2}t^{2}$, where $\sigma$ is the width of the LDoS [see
Eq. (4)]. Beyond this point, but still at short times, $t\lesssim 1/\sigma$,
the decay is dictated by the shape of the LDoS. The shape of $\rho(E)$ was
investigated in [5, 6] in the context of banded random matrices, while in
realistic models, the transition from a Lorentzian to a Gaussian form with the
increase of the perturbation strength was discussed, for example, in [7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Depending on the initial
state and the model considered, skewed Gaussians and bimodal distributions can
also emerge [22]. Beyond the characteristic time for the initial depletion of
the initial state, $t\sim 1/\sigma$, the survival probability exhibits a
power-law decay $\propto t^{-\gamma}$ with an exponent $\gamma$ that depends
on the level of ergodicity of $|\Psi(0)\rangle$ and $|\alpha\rangle$. When the
LDoS is filled ergodically, $\gamma$ is determined by the bounds of this
energy distribution [23, 24, 25, 26, 27, 28, 29, 30, 31]. In contrast, when
$|\Psi(0)\rangle$ and $|\alpha\rangle$ are nonchaotic states, then $\gamma$
depends on the level of correlations and multifractality among the states [32,
33, 34, 35, 36, 37]. But this is not yet the end of the story. In chaotic
systems, where the energy-level statistics is similar to those of random
matrices, the survival probability does not saturate after the algebraic
decay. Instead, it reaches a value that is smaller than its infinite-time
average,

$\overline{SP}=\sum_{\alpha}|C_{\alpha}^{(0)}|^{4},$ (5)

and then grows in a ramp until it finally saturates at $\overline{SP}$. The
infinite-time average is the last term in the equation below, which is
obtained from Eq. (2),

$SP(t)=\sum_{\alpha\neq\beta}|C_{\alpha}^{(0)}|^{2}|C_{\beta}^{(0)}|^{2}e^{-i(E_{\alpha}-E_{\beta})t}+\sum_{\alpha}|C_{\alpha}^{(0)}|^{4}.$
(6)

The interval in which $SP(t)<\overline{SP}$ is known as correlation hole [38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55], and there
have been different proposals to measure it experimentally in systems out of
equilibrium (see Ref. [56] and references therein). The correlation hole is a
dynamical manifestation of spectral correlations, and as such can be used to
detect many-body quantum chaos in experiments that do not have direct access
to the spectrum, such as experiments with cold atoms and ion traps.

In this work, motivated by generalized quantities like Rényi entropies [57],
the inverse participation ratio [58, 59, 60, 61, 62, 63] and other similar
quantities [64], that have a prominent role in studies of localization and
multifractality, we introduce the generalized survival probability,
$SP_{q}(t)$, and its corresponding generalized LDoS, $\rho_{q}^{(0)}(E)$ [see
the definitions below in Eq. (10) and Eq. (12), respectively]. We discuss how
they can help with the understanding of the structure of the eigenstates.

Using the one-dimensional (1D) disordered spin-1/2 model often employed in the
analysis of many-body localization, we compare the results for the generalized
survival probability in the chaotic regime and away from it, where the
duration of the power-law decay of $SP_{q}(t)$ becomes dependent on the value
of $q$. We also compare the behavior of $SP_{q}(t)$ for the chaotic spin model
with random matrices from the Gaussian orthogonal ensemble (GOE) and, in the
latter case, provide an analytical expression for the entire evolution of the
generalized survival probability.

##  II Models

We study many-body quantum systems described by the Hamiltonian

$H=H_{0}+V,$ (7)

where a chosen eigenstate of $H_{0}$ corresponds to the initial state and $V$
is a strong perturbation that takes the system far from equilibrium. We
consider initial states that have energies
$E^{(0)}=\langle\Psi(0)|H|\Psi(0)\rangle$ close to the middle of the spectrum.
Two Hamiltonians $H$ are investigated, one is a random matrix from the GOE and
the other describes a 1D disordered Heisenberg spin-1/2 model.

###  II.1 Gaussian orthogonal ensemble

The GOE is composed by real and symmetric ${\cal{D}}\times{\cal{D}}$ matrices
completely filled with random entries from a Gaussian distribution with mean
zero and variance given by

$\left\langle
H_{jk}^{2}\right\rangle=\begin{cases}\frac{1}{2},\quad\text{for}\quad j\neq
k;\\\ 1,\quad\text{for}\quad j=k.\end{cases}$ (8)

We assume that the unperturbed Hamiltonian $H_{0}$ is the diagonal part of $H$
and $V$ is the off-diagonal part. The model is unphysical, but allows for
analytical derivations that can serve as a reference for the studies of
realistic chaotic many-body quantum systems.

###  II.2 Disordered spin-1/2 model

As a physical model, we consider the 1D Heisenberg spin-1/2 model with onsite
disorder that has been used in studies of many-body localization [65, 66, 67,
68]. The Hamiltonian is given by

$H=\sum_{k=1}^{L}h_{k}S_{k}^{z}+J\sum_{k=1}^{L}\left(S_{k}^{x}S_{k+1}^{x}+S_{k}^{y}S_{k+1}^{y}+S_{k}^{z}S_{k+1}^{z}\right),$
(9)

where $S^{x,y,z}$ are the spin-$1/2$ operators, $L$ is the system size, $J=1$
is the coupling strength, and $h_{k}$ are independent and uniformly
distributed random variables in $[-h,h]$, with $h$ being the onsite disorder
strength. We assume periodic boundary conditions. The system conserves the
total magnetization in the $z$-direction,
$\hat{S}^{z}_{\mathrm{tot}}=\sum_{k=1}^{L}\hat{S}_{k}^{z}$. Throughout our
study, we work in the largest subspace with $\hat{S}^{z}_{\mathrm{tot}}=0$
leading to ${\cal{D}}=L!/(L/2)!^{2}$. For finite sizes, $H$ shows level
statistics comparable to the GOE random matrices when $h\sim 0.5$, while level
repulsion fades away for $h>1$. We take the unperturbed Hamiltonian to consist
of the terms in the $z$-direction,
$H_{0}=\sum_{k=1}^{L}\left(h_{k}S_{k}^{z}+JS_{k}^{z}S_{k+1}^{z}\right)$, and
the perturbation to be the flip-flop term,
$V=J\sum_{k=1}^{L}\left(S_{k}^{x}S_{k+1}^{x}+S_{k}^{y}S_{k+1}^{y}\right)$.

##  III Generalized survival probability

We define the generalized survival probability as

$SP_{q}(t)=\frac{1}{{\cal{N}}_{q}^{2}}\left|\sum_{\alpha=1}^{\cal{D}}|C_{\alpha}^{(0)}|^{q}e^{-iE_{\alpha}t}\right|^{2}=\left|\int\rho_{q}(E)e^{-iE_{\alpha}t}dE\right|^{2},$
(10)

where ${\cal{N}}_{q}$ is a normalization constant given by

${\cal{N}}_{q}=\sum_{\alpha=1}^{\cal{D}}|C_{\alpha}^{(0)}|^{q},$ (11)

the parameter $q\geq 0$ is a positive real number, and

$\rho_{q}(E)=\frac{1}{{\cal{N}}_{q}}\sum_{\alpha=1}^{\cal{D}}|C_{\alpha}^{(0)}|^{q}\delta(E_{\alpha}-E)$
(12)

is the generalized LDoS (gLDoS) with mean and variance given, respectively, by

$E_{q}^{(0)}=\frac{1}{{\cal{N}}_{q}}\sum_{\alpha=1}^{\cal{D}}|C_{\alpha}^{(0)}|^{q}E_{\alpha}\quad\text{and}\quad\sigma^{2}_{q}=\frac{1}{{\cal{N}}_{q}}\sum_{\alpha=1}^{\cal{D}}|C_{\alpha}^{(0)}|^{q}(E_{\alpha}-E_{q}^{(0)})^{2}.$
(13)

The survival probability, as defined in Eq. (2), and the mean and variance
given in Eq. (4), are recovered when $q=2$. For $q=0$, Eq. (10) coincides with
the spectral form factor [69], which is a quantity used to study level
statistics in the time domain. Contrary to the (generalized) survival
probability, the spectral form factor is not a dynamical quantity, since it
does not depend on the initial state.

If one knows the generalized LDoS, we can obtain the generalized survival
probability by performing the Fourier transform in Eq. (10). We therefore
start our analysis by examining the shape of $\rho_{q}(E)$.

###  III.1 Generalized LDoS

Figure 1 depicts the generalized LDoS for a single random realization of a GOE
matrix and different values of $q$. We observe that the semicircular shape,
typical of random matrices in the limit of large $\cal{D}$, and the length of
the distribution are conserved independently of the value of $q$. This is
because all eigenstates from GOE matrices are random vectors and so is the
initial state, that is, $C_{\alpha}^{(0)}$ are random numbers from a Gaussian
distribution satisfying the constraint of normalization. Even though for
$q>1$, the larger components $C_{\alpha}^{(0)}$ get enhanced, leading to the
spikes seen in Fig. 1 (c) and Fig. 1 (d), the width of the distribution is not
affected by $q$. This means that after averages over random realizations, one
would not notice the differences between the panels. One can then state that
the robustness of the generalized LDoS for different values of $q$ is a sign
of the ergodicity of the eigenstates of the system.

Figure 1: Generalized local density of states for GOE matrices for (a)
$q=0.5$, (b) $q=1.0$, (c) $q=2.0$, and (d) $q=3.0$. Shaded areas are numerical
results and the solid curves represent the semicircle law in Eq. (14). A
single disorder realization and a single initial state are considered. Matrix
size is ${\cal{D}}=12,000$.

Since the components $|C_{\alpha}^{(0)}|^{q}$ are uncorrelated random numbers
fluctuating smoothly around the average
$\overline{|C_{\alpha}^{(0)}|^{q}}={\cal{N}}_{q}/{\cal D}$, one sees that the
gLDoS for GOE matrices coincides with the normalized density of states
$P^{\text{GOE}}(E)={\cal D}^{-1}\sum_{\alpha}\delta(E_{\alpha}-E)$. Therefore,
for GOE random matrices, we have that

$\rho_{q}^{\text{GOE}}(E)=P^{\text{GOE}}(E)=\frac{1}{\pi\sigma_{q}}\sqrt{1-\left(\frac{E}{2\sigma_{q}}\right)^{2}},$
(14)

where the standard deviation $\sigma_{q}=\sqrt{{\cal{D}}/2}$. In Fig. 3 (a),
we plot $\sigma_{q}$ as a function of $q$ and confirm that $\sigma_{q}$ is
indeed nearly constant for GOE.

For physical many-body quantum systems with two-body interactions, the density
of states is Gaussian [70, 71]. Thus, the expected shape of the LDoS for a
system perturbed far from equilibrium and an initial state in the middle of
the spectrum, as considered here, is also Gaussian, as seen in Fig. 2 (c) for
$q=2$ and $h=0.5$.

Figure 2: Generalized local density of states for the disordered spin-$1/2$
model with $h=0.5$ for (a) $q=0.5$, (b) $q=1.0$, (c) $q=2.0$ and (d) $q=3.0$.
Shaded areas are numerical results and the solid curves represent the Gaussian
expression in Eq. (15). A single disorder realization is considered. The
system size is $L=16$ with ${\cal{D}}=12,870$.

Despite the persistence of the Gaussian shape for different values of $q$,

$\rho_{q}^{\text{Spin}}(E)=\frac{1}{\sqrt{2\pi\sigma_{q}^{2}}}\exp\left[-\frac{(E_{\alpha}-E_{q}^{(0)})^{2}}{2\sigma_{q}^{2}}\right],$
(15)

Fig. 2 makes it clear that, in contrast to the GOE, the width $\sigma_{q}$
depends on $q$. As $q$ increases and the participation of the larger
$|C_{\alpha}^{(0)}|^{q}$ gets amplified, the width of
$\rho_{q}^{\text{Spin}}(E)$ gets narrower than the density of states. This
indicates that the contributions from the components at the tails of the
initial-state energy distribution, where chaotic states are nonexistent, get
erased.

The dependence of the width of the gLDoS on $q$ reveals the limited degree of
ergodicity of physical systems, even deep in the chaotic regime. Eigenstates
of physical systems are not random vectors as in random matrices and are not
random superpositions of plane waves as stated by the Berry’s conjecture [72].
How to define chaotic states in realistic systems is discussed in Refs. [73,
74, 75, 11, 76, 10, 77]. Our results add to these studies a way to quantify
the level of ergodicity in comparison to random matrices.

Figure 3: Width of the generalized LDoS normalized by the DoS for (a) GOE
matrices,
$\tilde{\sigma}^{\text{GOE}}_{q}=\sigma_{q}^{GOE}/\sigma_{DOS}^{GOE}$, and (b)
the spin model,
$\tilde{\sigma}^{\text{Spin}}_{q}=\sigma_{q}^{\text{Spin}}/\sigma_{DOS}^{\text{Spin}}$,
with $h=0.5$ (circles) and $h=2$ (squares) as a function of $q$. Each point is
an average over $10$ disorder realizations and a single initial state. Dotted
lines are guides for the eyes. ${\cal{D}}=12,000$ for GOE and
${\cal{D}}=12,870$ ($L=16$) for the spin model.

In Fig. 3, we compare the results for $\sigma_{q}$ normalized by the width of
the density of states (DoS) as a function of $q$ for the GOE model [Fig. 3
(a)] and the spin model [Fig. 3 (b)]. Each point in Fig. 3 is obtained by
performing an average over $10$ random realizations and a single initial
state. The flat curve in Fig. 3 (a) indicates the presence of fully ergodic
states throughout the spectrum, while in Fig. 3 (b)
$\sigma_{q}^{Spin}/\sigma_{DOS}^{Spin}$ clearly decays as $q$ increases. This
happens for the chaotic model with $h=0.5$ (circles), where nonchaotic states
concentrate at the edges of the spectrum, and more abruptly for the case with
$h=2$ (squares), where nonchaotic states are likely to be found also away from
the edges of the spectrum.

The reason for the abrupt decay of $\sigma_{q}^{Spin}$ with $q$ for $h=2$
becomes evident in Fig. 4, where we plot $\rho_{q}(E)$ for different values of
$q$. When $q\leq 1$ [Fig. 4 (a)-(b)], the shape of the LDoS is fragmented,
while for $q>1$ [Fig. 4 (c)-(d)], this structure is nearly erased and
$\rho_{q}(E)$ indicates a high degree of localization.

Figure 4: Generalized local density of states for the disordered spin-$1/2$
model with $h=2.0$ for (a) $q=0.5$, (b) $q=1.0$, (c) $q=2.0$, and (d) $q=3.0$.
Shaded areas are numerical results and solid curves represent the Gaussian
expression in Eq. (15). A single disorder realization and a single initial
state are considered. The system size is $L=16$ with ${\cal{D}}=12,870$.

For finite-size systems, several numerical studies supported that the
eigenstates of the disordered spin model should become multifractal in its
transition to the many-body localized phase [78, 79, 80, 35, 81, 63], although
this has not been confirmed in the thermodynamic limit [82]. The patterns
observed in Fig. 4 (a)-(b) also suggest fractality.

##  IV Evolution of the generalized survival probability under the GOE model:
Analytical expression

According to Eq. (10), the survival probability averaged over an ensemble of
initial states and random realization is written as

$\left\langle
SP_{q}(t)\right\rangle=\left\langle\frac{1}{{\cal{N}}_{q}^{2}}\sum_{\alpha\neq\beta}|C_{\alpha}^{(0)}|^{q}|C_{\beta}^{(0)}|^{q}e^{-i(E_{\alpha}-E_{\beta})t}\right\rangle+\left\langle\frac{1}{{\cal{N}}_{q}^{2}}\sum_{\alpha}|C_{\alpha}^{(0)}|^{2q}\right\rangle,$
(16)

where $\langle\dots\rangle$ denotes the average. The second term on the right-
hand-side corresponds to the infinite-time average, $\overline{SP}_{q}$, of
the generalized survival probability. For GOE random matrices, where
$C_{\alpha}^{(0)}$ are random numbers from a Gaussian distribution,

$\overline{SP}_{q}=\frac{1}{{\cal{N}}_{q}^{2}}\sum_{\alpha}|C_{\alpha}^{(0)}|^{2q}=\frac{\sqrt{\pi}\Gamma\left(q+\frac{1}{2}\right)}{{\cal{D}}\Gamma\left(\frac{q+1}{2}\right)^{2}},$
(17)

Since for random matrices the eigenvalues and the eigenstates are
statistically independent, they can be factorized (see details in [83] and the
appendix of [52]). Thus, using

$\left\langle
e^{-i(E_{\alpha}-E_{\beta})t}\right\rangle=\frac{1}{{\cal{D}}-1}\left[{\cal{D}}\frac{{\cal{J}}^{2}_{1}(2\sigma
t)}{(\sigma t)^{2}}-b_{2}\left(\frac{\sigma t}{2{\cal{D}}}\right)\right],$
(18)

and that

$\frac{1}{{\cal{N}}_{q}^{2}}\sum_{\alpha\neq\beta}|C_{\alpha}^{(0)}|^{q}|C_{\beta}^{(0)}|^{q}=1-\overline{SP}_{q},$
(19)

due to the requirement that $SP_{q}(t=0)=1$, we arrive at the analytical
expression

$\left\langle
SP_{q}(t)\right\rangle=\frac{1-\left\langle\overline{SP}_{q}\right\rangle}{{\cal{D}}-1}\left[{\cal{D}}\frac{{\cal{J}}^{2}_{1}(2\sigma
t)}{(\sigma t)^{2}}-b_{2}\left(\frac{\sigma
t}{2{\cal{D}}}\right)\right]+\left\langle\overline{SP}_{q}\right\rangle.$ (20)

Above, we wrote $\sigma_{q}=\sigma$, because $\sigma_{q}$ is constant for the
GOE. The Fourier transform of the semicircular gLDoS gives the first term on
the right-hand-side of Eq. (20), which involves the Bessel function of first
kind, ${\cal{J}}_{1}$. This first term describes the initial decay of
$\left\langle SP_{q}(t)\right\rangle$, as seen in Fig. 5. It presents
oscillations with the $nth$-zeros happening when the initial state dynamically
finds out an orthogonal state at $t_{n}\sim(\pi n+\sqrt{2}/2)/2\sigma$ with
$n=1,2,\dots$. The envelope of the oscillations decays as $t^{-3}$. The second
term on the right-hand-side of Eq. (20),
$b_{2}(t)=\left\\{t\ln\left[(2t+1)/(2t-1)\right]-1\right\\}\Theta(t-1)+[t\ln(2t+1)-2t+1]\Theta(1-t)$,
is the so-called two-level form factor that takes $\left\langle
SP_{q}(t)\right\rangle$ on a ramp to the saturation value
$\left\langle\overline{SP}_{q}\right\rangle$.

Figure 5: Generalized survival probability evolving under the GOE model for
different values of $q$. Red curves are numerical results and the black lines
correspond to the analytical expression in Eq. (20). From bottom to top,
$q=0.5$, $1.0$, $2.0$ and $3.0$. Matrix size is ${\cal{D}}=12,000$. Averages
over $10^{4}$ samples.

In Fig. 5, we compare the numerical results for $\left\langle
SP_{q}(t)\right\rangle$ with the analytical expression in Eq. (20). The
agreement is excellent. The fact that $\sigma_{q}$ for the GOE model is
independent of $q$ (see Fig. 1 and Fig. 3) becomes evident once again in Fig.
5, where the curves for different values of $q$ coincide at short times,
capturing the oscillations of the Bessel function up to the minimum value of
$\left\langle SP_{q}(t)\right\rangle$.

To derive the time scale, $t^{GOE}_{Th}$, where $\left\langle
SP_{q}(t)\right\rangle$ reaches the minimum of the correlation hole, we need
to find the point where the first and second terms in the square brackets of
Eq. (20) cross. Following Ref. [52], we get the long-time expansion of the
first term in Eq. (20) and expand the two-level form factor for short times.
Combining the two in the derivative of $\left\langle SP_{q}(t)\right\rangle$
we get

$t^{GOE}_{Th}=\left(\frac{3}{\pi}\right)^{1/4}\frac{\sqrt{\cal{D}}}{\sigma}=\left(\frac{3}{\pi}\right)^{1/4}.$
(21)

To obtain the minimum value of $\left\langle SP_{q}(t)\right\rangle$ in the
correlation hole, we evaluate Eq. (20) at $t^{GOE}_{Th}$, which results in

$\begin{split}\left.\left\langle
SP_{q}(t)\right\rangle\right|_{t=t^{GOE}_{Th}}&\approx\frac{1-\left\langle\overline{SP}_{q}\right\rangle}{{\cal{D}}-1}\left[\frac{\cal{D}}{\pi(\sigma
t^{GOE}_{Th})^{3}}-\left(1-\frac{\sigma}{\cal{D}}t^{GOE}_{Th}\right)\right]+\left\langle\overline{SP}_{q}\right\rangle\\\
&\approx\frac{1-\left\langle\overline{SP}_{q}\right\rangle}{{\cal{D}}-1}(-1)+\left\langle\overline{SP}_{q}\right\rangle.\end{split}$
(22)

Finally, using Eq. (17) for $\left\langle\overline{SP}_{q}\right\rangle$ we
arrive at

$\left.\left\langle
SP_{q}(t)\right\rangle\right|_{t=t^{GOE}_{Th}}\approx\frac{\sqrt{\pi}\Gamma\left(q+\frac{1}{2}\right)-\Gamma\left(\frac{q+1}{2}\right)^{2}}{{\cal{D}}\Gamma\left(\frac{q+1}{2}\right)^{2}}.$
(23)

For the particular case of $q=2$, Eq. (23) leads to the value $2/{\cal{D}}$
previously obtained in Ref. [52].

##  V Evolution of the generalized survival probability under the spin model

In Fig. 6, we compare the evolution of $\left\langle SP_{q}(t)\right\rangle$
under the spin model for different values of $q$. In both panels, Fig. 6 (a)
for $h=0.5$ and Fig. 6 (b) for $h=2$, the initial decay is determined by the
envelope of the gLDoS, as seen from Eq. (10). Since according to Fig. 2 and
Fig. 4, the shape of the distribution is Gaussian, one sees in Fig. 6 that

$\left\langle SP_{q}(t)\right\rangle=\exp(-\sigma_{q}^{2}t^{2})$ (24)

for $t\lesssim\sigma_{q}$. The dependence of $\sigma_{q}$ on $q$ is noticeable
in Fig. 6 (a) and evident in Fig. 6 (b).

Figure 6: Generalized survival probability evolving under the disordered
spin-$1/2$ model with (a) $h=0.5$ and (b) $h=2$ for different values of $q$.
From bottom to top, $q=0.5$, $1.0$, $1.5$, $2.0$ and $3.0$. System size is
$L=16$. Averages over $3\times 10^{4}$ samples.

Beyond the Gaussian behavior, a power-law decay emerges,

$\left\langle SP_{q}(t)\right\rangle\propto t^{-\gamma_{q}}.$ (25)

In Fig. 6 (a), where the system is chaotic, the power-law exponent should
depend on the bounds of the gLDoS. Since the gLDoS for the chaotic model in
Fig. 2 presents Gaussian tails for any $q$, we expect the same power-law
exponent for all curves in Fig. 6 (a), which is indeed what the nearly
parallel lines during the algebraic behavior suggest. In contrast, in Fig. 6
(b), it is clear that $\gamma_{q}$ decreases as $q$ increases and the minimum
of the correlation hole takes longer to be reached. In this case, the power-
law behavior reflects the correlations among the components of the initial
state, which get enhanced for larger values of $q$.

##  VI Discussion

We introduced the concepts of generalized survival probability, $SP_{q}(t)$,
and generalized local density of states, $\rho_{q}(E)$. We showed that the
width of the generalized local density of states, $\sigma_{q}$, depends on $q$
even when the many-body quantum system is deep in the chaotic regime, which
contrasts with random matrices, where the width is constant and equal to the
width of the density of states. Therefore, $\sigma_{q}$ may serve as a tool to
analyze and quantify the level of ergodicity of the states of physical systems
with respect to random matrices.

We also showed that the power-law behavior that follows the Gaussian decay of
the generalized survival probability is strongly dependent on $q$ when the
system is away from the chaotic regime. For a fixed value of the disorder
strength, the power-law decay gets stretched as $q$ increases and the power-
law exponent $\gamma_{q}$ decreases. This dependence of $\gamma_{q}$ on $q$
indicates correlations among the eigenstates. In a future work, we plan to
investigate how $\sigma_{q}$ and $\gamma_{q}$ may be used to study
multifractality.

###### Acknowledgements.

D.A.Z.-H. and E.J.T.-H. have financial support from VIEP-BUAP (Mexico),
Project No. 00270. They are also grateful to LNS-BUAP for their supercomputing
facility. D.A.Z.-H. thanks CONACyT (Mexico) for financial support. L.F.S. is
funded by the United States NSF Grant No. DMR-1936006. L.F.S. had support from
the MPS Simons Foundation Award ID: 678586. E.J.T.-H. and L.F.S. are grateful
to Felix Izrailev for his constant support and invaluable teachings on quantum
and classical chaos.

## References

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