# Non-ergodic extended states in the $\beta$-ensemble

Adway Kumar Das<EMAIL_ADDRESS>Anandamohan Ghosh
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Kolkata, Mohanpur, 741246 India
Department of Physical Sciences

###### Abstract

Matrix models showing chaotic-integrable transition in the spectral statistics
are important for understanding Many Body Localization (MBL) in physical
systems. One such example is the $\beta$-ensemble, known for its structural
simplicity. However, eigenvector properties of $\beta$-ensemble remain largely
unexplored, despite energy level correlations being thoroughly studied. In
this work we numerically study the eigenvector properties of $\beta$-ensemble
and find that the Anderson transition occurs at $\gamma=1$ and ergodicity
breaks down at $\gamma=0$ if we express the repulsion parameter as
$\beta=N^{-\gamma}$. Thus other than Rosenzweig-Porter ensemble (RPE),
$\beta$-ensemble is another example where Non-Ergodic Extended (NEE) states
are observed over a finite interval of parameter values ($0<\gamma<1$). We
find that the chaotic-integrable transition coincides with the breaking of
ergodicity in $\beta$-ensemble but with the localization transition in the RPE
or the 1-D disordered spin-1/2 Heisenberg model. As a result, the dynamical
time-scales in the NEE regime of $\beta$-ensemble behave differently than the
later models.

$\beta$-ensemble, Rosenzweig-Porter ensemble, Heisenberg model, Non-ergodic
extended states

###### pacs:

05.45.Mt

###### pacs:

02.10.Yn

###### pacs:

89.75.Da

##  1 Introduction

Canonically invariant classical ensembles including Dyson’s threefold ways [1]
and their extensions over symmetric spaces [2, 3] (e.g. Laguerre [4], Jacobi
[5], Circular [6] ensembles) are central to the paradigm of Random Matrix
Theory (RMT) [7] epitomizing completely ergodic [8] and chaotic [9] dynamics
in quantum mechanical systems. Corresponding energy levels tend to repel each
other, where the degree of repulsion is called the Dyson’s index, $\beta$
having values 1, 2 and 4 for the Gaussian Orthogonal, Unitary and Symplectic
ensembles respectively [10]. On the other hand, regular dynamics observed in
integrable systems [11, 12] is usually captured by the Poisson ensemble [13],
where energy levels are uncorrelated with inclination to be clustered (hence
can be assigned $\beta=0$). However, several physical systems (e.g. kicked top
[14], pseudo-integrable billiards [15], Harper [16], Anderson model [17] etc.)
show a spectral property intermediate between the aforementioned ideal limits.
While phenomenological models [18, 19, 20] can mimic the spectral properties
in the intermediate regions, there exist several generalizations of the
classical ensembles capturing the physics of mixed dynamics [21, 22, 23, 24,
25, 26, 27, 28, 29]. In particular, the Joint Probability Distribution
Function (JPDF) of eigenvalues for the classical ensembles can be expressed as
a Gibbs-Boltzmann weight of a 2-D system of particles, known as the Coulomb
gas model [30], where $\beta$ is no longer restrained to be quantized.
Specifically, a harmonic confining potential yields the Gaussian (also known
as the Hermite) $\beta$-ensemble characterized by the following JPDF,

$\displaystyle\begin{split}\text{P}\mathinner{\left(\vec{E}\right)}&=\frac{1}{\mathcal{Z}_{\beta}}\exp\mathinner{\left(-\sum_{i=1}^{N}\dfrac{E_{i}^{2}}{2}\right)}\prod_{i<j}\mathinner{\\!\left\lvert
E_{i}-E_{j}\right\rvert}^{\beta}\end{split}$ (1)

where $\mathcal{Z}_{\beta}$ is the normalization constant and
$\vec{E}=\\{E_{1},E_{2},\dots,E_{N}\\}$ is the set of $N$ eigenvalues [31].
Such ensembles were originally conceived as lattice gas systems [32] in
connection to the ground state wave-functions of the Calogero-Sutherland model
[33]. Following the rescaling $E_{i}\to\sqrt{\beta N}E_{i}$, we can express
the partition function $\mathcal{Z}_{\beta}$ as [34],

$\displaystyle\begin{split}\mathcal{Z}_{\beta}&\propto\int_{\mathbb{R}^{N}}\prod_{j=1}^{N}dE_{j}\exp\mathinner{\left(-\beta
N^{2}\mathcal{V}[\vec{E}]\right)},\quad\mathcal{V}[\vec{E}]=\sum_{i=1}^{N}\dfrac{E_{i}^{2}}{2N}-\sum_{i\neq
j}\dfrac{\log\mathinner{\\!\left\lvert
E_{i}-E_{j}\right\rvert}}{2N^{2}}\end{split}$ (2)

where the potential $\mathcal{V}[\vec{E}]$ has a confining term competing with
the pairwise logarithmic repulsion between $N$ fictitious particles. The
strength of such interactions is controlled by $\beta$ [35], which can be
interpreted as the inverse temperature. In the infinite temperature limit
($\beta\to 0$), the energy levels are allowed to come arbitrarily close to
each other, resulting in Poisson statistics, i.e. a signature of integrability
[13]. On the other hand, for $\beta=1$, Eq. (1) coincides with the JPDF of
Gaussian Orthogonal Ensemble (GOE), yielding Wigner-Dyson statistics
characterized by complete level repulsion, i.e. a signature of chaos [9]. Thus
tuning $\beta$, it is possible to control the degree of level repulsion in the
energy spectrum of $\beta$-ensemble with $\beta=1$ indicating the chaotic-
integrable transition. Corresponding Hamiltonians can be represented as real,
symmetric and tridiagonal $N\times N$ matrices, $H$, with following non-zero
elements [36],

$\displaystyle H_{i,i}=A_{i},\>H_{i,i+1}=H_{i+1,i}=B_{i}/\sqrt{2},\quad
A_{i}\sim\mathcal{N}(0,1),\>B_{i}\sim\chi_{(N-i)\beta}$ (3)

where $\mathcal{N}(0,1)$ is the Normal distribution and $\chi_{k}$ is the Chi-
distribution with a degree of freedom $k$. There has been extensive studies on
$\beta$-ensemble in terms of the Density of States (DOS) [37, 32, 38, 39],
associated fluctuations [40, 41, 42, 43, 44], connection to stochastic
differential operators [45, 46, 47], extreme eigenvalues [48, 49, 50, 51]. In
this work we give numerical evidence of chaotic$\rightarrow$integrable,
ergodic$\rightarrow$non-ergodic and delocalization$\rightarrow$localization
transitions by thoroughly studying the properties of eigenvalues (Sec. 2),
eigenfunctions (Sec. 3) and dynamics (Sec. 4). Therefore we identify the
critical values of $\beta$ segregating ergodic, Non-Ergodic Extended (NEE) and
localized regimes. We compare the spectral properties of the $\beta$-ensemble
with the properties of another matrix model, namely Rosenzweig-Porter ensemble
(RPE) [21, 52], where for a real symmetric matrix, $H$, all the elements are
randomly distributed with

$\displaystyle
H_{i,i}\sim\mathcal{N}(0,1),\>H_{i,j}/\sigma\sim\mathcal{N}(0,1),\quad\sigma^{2}=1/2N^{\tilde{\gamma}},\>\tilde{\gamma}\in\mathbb{R}$
(4)

Moreover, as an impetus to applications in physical systems, we compare both
$\beta$-ensemble and RPE to the widely studied 1-D disordered spin-1/2
Heisenberg model [53, 54, 55], where for simplicity we will consider the chain
to be isotropic. The Hamiltonian of such a chain of length $L$ with magnetic
field in the Z-direction is defined as,

$\displaystyle H$
$\displaystyle=-\dfrac{J}{2}\sum_{i=1}^{L}\vec{\sigma_{i}}\cdot\vec{\sigma}_{i+1}+h_{i}\hat{\sigma_{i}}^{z}$
(5)

where
$\vec{\sigma_{i}}=\set{\hat{\sigma_{i}}^{x},\hat{\sigma_{i}}^{y},\hat{\sigma_{i}}^{z}}$
are the Pauli matrices, $h_{i}$ is the random magnetic field applied in
Z-direction on the $i^{th}$ site and $J$ is the coupling constant. We assume
periodic boundary condition (i.e.
$\hat{\sigma}_{i+L}^{\alpha}=\hat{\sigma}_{i}^{\alpha}$), $J=1$ and $h_{i}$ to
be uniform random numbers sampled from $[-h,h]$, i.e. the disorder in
Heisenberg model can be controlled by tuning $h$. While the model is
integrable exactly at $h=0$, even an infinitesimal fluctuation in magnetic
fields on different sites is expected to induce chaos in the thermodynamic
limit ($L\to\infty$) [54]. Increasing $h$ further introduces more defects in
the chain leading to Many-Body Localization (MBL) where the critical disorder
strength, $h_{c}$ for ergodic to MBL transition depends on the energy density
[55]. Since the Z-component of total spin,
$S_{z}=\dfrac{1}{2}\sum_{i=1}^{L}\hat{\sigma}_{i}^{z}$ is conserved in the
Heisenberg model, we take $L$ to be even and choose the largest symmetry
sector $S_{z}=0$ having $L\choose L/2$ eigenvalues for our analysis.

One can intuitively speculate the existence of two critical points in
$\beta$-ensemble, considering that the diagonal part, $A$, is competing with
the perturbation from the off-diagonal part $B$. The overall interaction
strength can be calculated in terms of the Frobenius norm of $B$, i.e.
$||B||_{\text{F}}\approx\sqrt{\sum_{i=1}^{N-1}\overline{B_{i}^{2}}}=\sqrt{\frac{1}{2}N(N-1)\beta}\approx
N\sqrt{\beta}$, where $\overline{B_{i}^{2}}=(N-i)\beta$ is the mean value of
$B_{i}^{2}$ (as $\chi^{2}_{k}$ distribution has mean $k$). Similarly, the
strength of diagonal contribution is $||A||_{\text{F}}\approx\sqrt{N}$. Thus
for weak perturbation
($||B||_{\text{F}}<||A||_{\text{F}}\Rightarrow\beta<1/N$), one may expect the
energy states of $\beta$-ensemble to localize while they should be extended
for $||B||_{\text{F}}>||A||_{\text{F}}$. Therefore it is convenient to express
the Dyson’s index as

$\displaystyle\beta=N^{-\gamma},\quad\gamma\in\mathbb{R}.$ (6)

So the perturbation strength can be expressed as
$||B||_{\text{F}}=N^{1-\gamma/2}$ and it is reasonable to expect that
$\gamma_{\text{AT}}\equiv 1$ is the Anderson transition point such that the
energy states are localized for $\gamma>\gamma_{\text{AT}}$.

The role of the control parameter manifested in the off-diagonal terms of
$\beta$-ensemble is reminiscent of the RPE, where relative strength of
perturbation indicates that Anderson transition occurs at
$\tilde{\gamma}_{\text{AT}}=2$ [56]. Moreover, due to the random sign altering
nature of the RPE matrix elements, there exists an ergodic transition at
$\tilde{\gamma}_{\text{ET}}=1$ segregating three distinct phases: ergodic, NEE
and localized states [56]. Similarly, for $\beta$-ensemble, even though the
off-diagonal elements, $B_{i}$’s are strictly positive, the diagonal terms,
$A_{j}$’s can be positive or negative at random. If we equate the rescaled
perturbation, $||B||_{\text{F}}/\sqrt{N}$ to the total fluctuation from on-
site terms, $||A||_{\text{F}}$, we expect ergodic transition at
$\gamma_{\text{ET}}\equiv 0$ such that the energy states occupy the entire
Hilbert space in the regime $\gamma\leq\gamma_{\text{ET}}$. However, these
heuristic arguments based on the norm alone cannot account for any phase
transition. For example, eigenvectors are exponentially localized in
tridiagonal matrices with i.i.d. random elements [57]. Thus the inhomogeneity
of the off-diagonal terms evident from Eq. (3) is essential for the existence
of NEE in the $\beta$-ensemble as will be demonstrated in the following
sections.

##  2 Properties of Energy Levels

Now we would like to study the energy level properties of $\beta$-ensemble
following the Hamiltonian in Eq. (3) and identify the transition from
integrable to chaotic regimes as we vary the Dyson’s index. We also compare
the properties of $\beta$-ensemble with the results known for RPE and
Heisenberg model. Some of the results in this section are known and we list
them for completeness.

### Density of States (DOS):

As a starting point we look at the DOS, which is the marginal PDF of energy
levels. The bulk eigenvalues of $\beta$-ensemble roughly scale with system
size as $\epsilon_{\beta}=\sqrt{4+2N^{1-\gamma}}$ [58]. So we scale the
eigenvalues as $E\to E/\epsilon_{\beta}$ and obtain the DOS numerically, as
shown in Fig. 1(a) for $N=8192$ and various $\gamma$. For $\gamma>1$ we obtain
$\epsilon_{\beta}\approx 2$ such that the DOS converges to the Gaussian
distribution, $\mathcal{N}(0,1/4)$ with increasing $N$, as illustrated in Fig.
1(c) for a specfic value of $\gamma=1.1$. Exactly at $\gamma=1$, the DOS is
system size independent and follows a shape intermediate between Gaussian
distribution and Wigner semi-circle law since $\epsilon_{\beta}=\sqrt{6}$. But
for $0\leq\gamma<1$, the DOS converges to the semicircle law upon increasing
$N$, as shown in Fig. 1(b) for a specific value of $\gamma=0.6$. However, in
the limit $\beta\to\infty$ (i.e. $\gamma<0$ and $N\to\infty$), all the
eigenvalues of $\beta$-ensemble freeze and produce a picket-fence spectrum
[59]. A qualitative evolution of the shape of the DOS in $\beta$-$N$ plane is
shown in Fig. 2 of [35].

In case of the RPE, bulk eigenvalues scale as:
$\epsilon_{\text{RPE}}=\sqrt{4+2N/\sigma^{2}}=2\sqrt{1+N^{1-\tilde{\gamma}}}$.
Consequently the scaled DOS of RPE ($E\to E/\epsilon_{\text{RPE}}$) varies
from Wigner’s semicircle to Gaussian similar to $\beta$-ensemble as
$\tilde{\gamma}$ is increased from $0$. Contrarily for 1-D disordered spin-1/2
Heisenberg model, DOS always follows a Gaussian distribution spreading with
the disorder strength, which is typical of many-body systems with local
interactions [60]. However, the differences in global shapes of DOS from these
different models do not dictate the correlations in the respective local
energy scales as demonstrated below.

Figure 1: Density of States (DOS) of $\beta$-ensemble (a) averaged over 500
disordered realizations various $\gamma$ where $N=8192$. (b) $\gamma=0.6$ and
(c) $\gamma=1.1$ for various $N$. Wigner’s semicircle law and Gaussian
distribution $\mathcal{N}(0,1/4)$ shown via bold curves.

### Nearest Neighbour Spacing (NNS):

One of the most commonly investigated quantity reflecting local spectral
correlations is the distribution of NNS of the ordered and unfolded
eigenvalues (see Appx. A), which follows Wigner’s surmise and exponential
distribution for the chaotic and integrable systems respectively [7]. As the
nature of correlation present in the spectrum edge is different than that of
the bulk energy levels, we numerically evaluate the PDF of NNS choosing only
middle 25% of the spectrum. The results for $N=8192$ are shown in Fig. 2(a)
via markers along with the approximate empirical PDF of NNS (Eqs. (12) and
(13) in [35]), which is $N$-independent, provided $N\gg 1$. Such a functional
form also implies that for $s\ll 1$,
$\text{P}\mathinner{\left(\beta;s\right)}\sim s^{\beta}\;\forall\;\beta$, thus
the degree of level repulsion is indeed quantified via $\beta$, as expected
from Eq. (1). We observe a crossover from Wigner’s surmise to exponential
distribution in Fig. 2(a) as we decrease $\beta$, implying a suppression of
chaos. A similar crossover is observed in RPE [61] and 1-D disordered spin-1/2
Heisenberg model [62] as well.

### Ratio of Nearest Neighbour Spacing (RNNS):

Another notable measure of the short-range spectral correlations is the RNNS,
which is much simpler to study since unfolding of the energy spectrum is not
required [63, 64]. If we define $\tilde{r_{i}}=\min\set{r_{i},1/r_{i}}$, where
$r_{i}$ is the $i^{th}$ RNNS, then
$\text{P}\mathinner{\left(\tilde{r}\right)}=2\text{P}\mathinner{\left(r\right)}\Theta\mathinner{\left(1-r\right)}$,
with $\Theta(x)$ being the Heaviside step function. We show PDF of $\tilde{r}$
via markers in Fig. 2(b) for $N=8192$ along with the empirical PDF of RNNS
(Eq. (1) in [65]). Again a crossover w.r.t. $\beta$ is immediately apparent.
For $N\gg 1$, the density of RNNS is system size independent, as shown in Fig.
2(c) for two values of $\beta$ while varying $N$. This can inferred from Fig.
3(a) as well, where the ensemble averaged values of $\tilde{r}$ as a function
of $\beta$ collapse for different $N$ provided $N\gg 1$. In Figs. 2(d), (e)
and (f), we show the density of $\tilde{r}$ for different values of $\gamma$
while varying $N$. For any value of $\gamma<0$ as $N\to\infty$ (i.e.
$\beta\to\infty$), the energy levels are highly correlated and strongly repel
each other. The increase in level repulsion with $N$ is shown in Fig. 2(f) for
a fixed value of $\gamma=-0.3$. Exactly at $\gamma=0$ (i.e. $\beta=1$) the
density of $\tilde{r}$ is independent of $N$ and matches that of GOE (Fig.
2(e)). On the other hand for any $\gamma>0$ and $N\to\infty$ (i.e. $\beta\to
0$), the energy levels become uncorrelated and clustered as in Poisson
ensemble. In Fig. 2(d) we show that the density of $\tilde{r}$ converges
towards Poisson expression as we increase $N$ for a fixed value of
$\gamma=0.3$. The analyses above imply that the signatures of chaos in the
short-range spectral correlations are lost as we lower the repulsion parameter
$\beta$. Now we identify the exact nature of such a transition.

Figure 2: Short range spectral correlations for $\beta$-ensemble: PDF of (a)
NNS and modified RNNS (b) varying $\beta$ for $N=8192$, varying $N$ (c) for
$\beta=0.2$ (blue markers) and $\beta=2.5$ (maroon markers) (d) $\gamma=0.3$
(e) $\gamma=0$ and (f) $\gamma=-0.3$. The markers indicate numerical data
while solid lines denote empirical analytical forms (Eqs. (12) and (13) in
[35] for NNS and Eq. (1) in [65] for RNNS). The analytical expressions for
Poisson and GOE are shown via bold curves.  Figure 3: Ensemble average of
$\tilde{r}$ for $\beta$-ensemble vs. (a) $\beta$ (b) $\gamma$ (c) for RPE vs.
$\tilde{\gamma}$ and (d) Heisenberg model vs. disorder strength, $h$. For (a),
(b), (c) system size, $N$ and for (d) chain length, $L$ is varied. In (a) we
show the $\left\langle\tilde{r}\right\rangle$ for $N=3$ (Eq. (7) in [63]) via
dashed line. The insets of (b), (c) and (d) show collapsed data following the
ansatz in Eq. (7), where critical parameter and exponents are also mentioned.
In (b) and (c) we show the expected ergodic and Anderson transition point for
$\beta$-ensemble and RPE respectively.

### Criticality in chaotic-integrable transition:

For a fixed $\gamma$, the quick convergence of PDF of RNNS with system size
(Fig. 3(a)) enables us to conclude that the
$\left\langle\tilde{r}\right\rangle$ has one to one correspondence with
$\beta$ when $N\gg 1$. $\beta$ increases with $N$ for any $\gamma<0$ (as
$\beta=N^{-\gamma}$), hence $\left\langle\tilde{r}\right\rangle$ should also
increase with $N$ and vice versa. Figure 3(b) conforms to the above
expectations suggesting a scaling hypothesis for
$\left\langle\tilde{r}\right\rangle$. Let us assume that there exists a
relevant correlation length $\Xi$ showing a power law divergence around a
critical point, $\gamma_{c}$, i.e. $\Xi\sim(\gamma-\gamma_{c})^{-\nu}$, where
$\nu$ is a critical exponent. Then any quantity $A(\gamma,N)$ showing non-
analytical behaviour close to $\gamma_{c}$ should behave as

$\displaystyle A(\gamma,N)\propto f\mathinner{\left((\gamma-\gamma_{c})(\log
N)^{1/\nu}\right)}$ (7)

where $f$ is a universal function and we assume $\Xi$ to scale with $\log N$
instead of $N$. Such a scaling ansatz valid for $2^{nd}$ order phase
transition is shown to hold in case of the Kullback-Leibler divergence of RPE
[61]. We collapse the crossover curves from different system sizes based on
Eq. (7) (see Appx. A) and obtain $\gamma_{c}=0.0030$ and $\nu=1.0316$ as shown
in the inset of Fig. 3(b). Such a critical behaviour can also be inferred from
the scale invariance of $\left\langle\tilde{r}\right\rangle$ w.r.t.
$-\log\beta=\gamma\log N$ (Fig. 3(a)). Comparing this with Eq. (7), we get
$\gamma_{c}=0$ and $\nu=1$, which is consistent with our numerical analysis.

We show the $\left\langle\tilde{r}\right\rangle$ curves for different system
sizes and chain lengths in Fig. 3(c) and (d) for RPE and Heisenberg model
respectively. Again assuming a power law behaviour like Eq. (7), we are able
to collapse the data for RPE using $\tilde{\gamma}_{c}\approx
1.9750,\nu\approx 1.1359$. For Heisenberg chain we assume that
$A(\gamma,N)\propto f\mathinner{\left((h-h_{c})L^{1/\nu}\right)}$ and get
$h_{c}\approx 2.7696,\nu\approx 0.7842$. Note that the critical disorder
strength found here corresponds to the middle 25% of eigenspectrum, hence
conforms to the energy density phase diagram of MBL transition present in the
literature [55].

Thus we show that the chaotic-integrable transition in all three models is
$2^{nd}$ order in nature. The crucial difference lies in the physical
significance of these critical points. We observe that the chaotic-integrable
transition occurs at $\tilde{\gamma}_{\text{AT}}$ in the case of RPE, i.e. the
energy states localize as soon as the energy levels start to cluster.
Contrarily in case of $\beta$-ensemble, chaotic-integrable transition occurs
at $\gamma=0$ which we previously argued to be $\gamma_{\text{ET}}$, i.e.
where ergodicity breaks down. Thus in the thermodynamic limit ($N\to\infty$),
there will be extended states for which energy levels are uncorrelated, which
has a profound implication in the dynamical properties of $\beta$-ensemble
(Sec. 4). Thus our analysis shows that the eigenstate localization property is
not necessarily indicative of the degree of repulsion present in the energy
spectrum as also observed in certain structured matrix ensembles [66, 67, 68].

Figure 4: Power spectrum of noise for (a) $\beta$-ensemble and (d) RPE for
$N=8192$ as a function of dimensionless frequency, $\omega=2\pi k/N$. In (b),
power spectrum is shown for various $\gamma$ (denoted by markers similar to
(a)) and $N$ (denoted by different colors as in the legend). We see that
$\left\langle P_{\omega}^{\delta}\right\rangle$ shifts downwards with
increasing $N$ for $\gamma=-0.3$, where the dashed line $\propto 1/\omega$ is
placed as a guide to the eye. (b) $\left\langle
P_{\omega}^{\delta}\right\rangle$ for $\gamma=0.3$ while varying $N$ (data
shifted in Y-direction for clarity). Dashed and solid lines indicate
$1/\omega$ and $1/\omega^{2}$ behaviors respectively. In (a), (c) and (d),
stars denote the critical frequencies, $\omega_{c}$ separating heterogeneous
behavior. Inset of (c) shows numerically obtained $\omega_{c}$, where solid
line denotes linear fit in log-log scale. We also show the analytical
$\left\langle P^{\delta}_{\omega}\right\rangle$ for Poisson and GOE (Eq. (10)
in [69]) via bold curves.

### Power spectrum:

Short-range spectral correlations in $\beta$-ensemble exhibit criticality only
around $\gamma_{\text{ET}}=0$. We also expect a second critical point
associated with the localization transition, which can be captured by the
long-range spectral correlations, e.g. power spectrum of $\delta_{n}$
statistics defined as [69, 70],

$\displaystyle
P_{k}^{\delta}=|\hat{\delta_{k}}|^{2},\quad\hat{\delta_{k}}=\frac{1}{\sqrt{N}}\sum_{n}\delta_{n}\exp\mathinner{\left(-\frac{i2\pi
kn}{N}\right)}$ (8)

where $\delta_{n}\equiv\mathcal{E}_{n}-n$ is the fluctuation of the $n^{th}$
unfolded energy level, $\mathcal{E}_{n}$ around its mean value, $n$. Ensemble
average of $P_{k}^{\delta}$, denoted by $\left\langle
P_{k}^{\delta}\right\rangle$, is explored for 1-D disordered spin-1/2
Heisenberg model in [71, 72]. For $\gamma\geq 0$, there exists a critical
frequency $k_{c}=N^{1-\gamma}/2$ in $\beta$-ensemble [73] such that for $k\leq
k_{c},\left\langle P_{k}^{\delta}\right\rangle\propto 1/k$ identifying
completely chaotic behaviour whereas for $k>k_{c},\left\langle
P_{k}^{\delta}\right\rangle\propto 1/k^{2}$, which is a signature of Poisson
ensemble. Note that the power spectrum of some physical systems like Robnik
billiard [74], kicked top [75] exhibit a homogeneous behaviour, $\left\langle
P_{k}^{\delta}\right\rangle\propto 1/k^{\alpha}$, across all frequencies with
$1<\alpha<2$.

In Fig. 4(a), we show the power spectrum of $\beta$-ensemble as a function of
dimensionless frequency, $\omega=2\pi k/N$ for $N=8192$ and various $\gamma$
with the bold curves showing the analytical forms of $\left\langle
P^{\delta}_{\omega}\right\rangle$ for Poisson and GOE (Eq. (10) in [69]). In
Fig. 4(b), we show $\left\langle P_{\omega}^{\delta}\right\rangle$ for same
values of $\gamma$ but also by varying $N$ denoted by different colors. We see
that for finite $N$ and $\gamma<0$, $\left\langle
P_{\omega}^{\delta}\right\rangle\propto 1/\omega$ for a typical value of
$\gamma=-0.3$. However, we observe that for $N\gg 1$ and $\gamma\ll 0$ (i.e.
$\beta\to\infty$), there are fluctuations around $1/\omega$ behavior due to
the energy spectrum attaining a picket-fence structure. For $\gamma\geq 0$, we
can identify two critical points separating three distinct regimes by looking
at Fig. 4(b) or from the analytical calculations in [73]:

  * •

$\gamma=0$: $\left\langle P_{\omega}^{\delta}\right\rangle\propto 1/\omega$
for any $N\Rightarrow$ energy levels are correlated at all scales even in the
thermodynamic limit

  * •

$0<\gamma<1$: Heterogeneous spectra $\Rightarrow\omega_{c}=2\pi k_{c}/N=\pi
N^{-\gamma}$ separating Poisson and GOE like scaling. In Fig. 4(c), we show
$\left\langle P_{\omega}^{\delta}\right\rangle$ for $\gamma=0.3$ and various
$N$, which clearly reflects the heterogeneous features. In the inset we show
numerically obtained $\omega_{c}$ vs. $N$.

  * •

$\gamma\geq 1$: $\left\langle P_{\omega}^{\delta}\right\rangle\propto
1/\omega^{2}$ for any $N$ $\Rightarrow$ energy levels are uncorrelated at all
scales even in the thermodynamic limit

Note that $k_{c}\to\infty$ for $N\to\infty$ and $0<\gamma<1$, i.e. signature
of chaotic spectrum prevails over infinitely many frequencies. However, their
support set constitute a zero fraction of the set of principle frequencies as
$k_{c}/k_{\text{Nyquist}}=N^{-\gamma}\to 0$ for any $\gamma>0$
($k_{\text{Nyquist}}\approx N/2$ is the highest frequency required to fully
reconstruct the original spectrum [69]). Such a fractal behaviour suggests the
absence of ergodicity in the $\beta$-ensemble for $0<\gamma<1$. For example,
in case of RPE, eigenstates occupy zero fraction of the Hilbert space volume
despite being extended in the NEE phase ($1\leq\tilde{\gamma}<2$) [56].
Corresponding power spectrum also exhibits heterogeneous behaviour as shown in
Fig. 4(d). Thus we can attribute the heterogeneity in power spectrum of
$\delta_{n}$ statistics to the existence of NEE phase and we can conclude that
$\beta$-ensemble enters the NEE phase for $0<\gamma<1$ where ergodicity breaks
down at $\gamma_{\text{ET}}=0$ and Anderson transition occurs at
$\gamma_{\text{AT}}=1$.

With this primary evidence of the existence of NEE regime in $\beta$-ensemble,
in the next section we will study the eigenfunction properties and obtain the
fractal scaling of NEE states.

##  3 Properties of Eigenstates

Due to the canonical invariance, the eigenvectors of $N\times N$ GOE matrices
are uniformly distributed in the unit $N$-dimensional sphere, resulting in
mutually independent eigenvector components. Contrarily for $\beta$-ensemble,
all elements but the first component of the $k^{th}$ eigenvector can be
expressed in terms of the $k^{th}$ eigenvalue and different matrix elements
[36]. Hence even for typical values of $\beta$ (i.e. $\beta=1,2,4$), the
eigenvector properties of Wigner-Dyson and $\beta$-ensemble are different from
each other, although their energy level statistics are identical. This can be
readily verified from the distribution of
$\log(N\mathinner{\\!\left\lvert\Psi_{i}\right\rvert}^{2})$ ($\Psi_{i}$ is
$i^{th}$ component of the eigenstate $\ket{\Psi}$), which has a long tail for
$\beta=1$ in $\beta$-ensemble compared to GOE.

Figure 5: Eigenstate statistics: Shannon entropy, S and relative Rényi entropy
of two types ($\mathcal{R}_{1,2}$ in Eq. (9)) for $\beta$-ensemble, RPE and
1-D disordered spin-1/2 Heisenberg model, as a function of system parameters
for different matrix size, $N$ and chain length, $L$ (values given in the
legends). The critical values of parameters indicating ergodic and
localization transitions are marked by dashed line in all plots. Inset shows
collapsed data following the ansatz in Eq. (7), where numerically obtained
critical parameters and exponents are also given. Inset of (c) shows collapsed
data of $\textrm{S}^{\prime}=\textrm{S}/\log(0.48N)$.

### Localization transition:

In order to characterize the Anderson transition from the properties of
eigenstates, we begin by computing the Shannon entropy, defined as,
$\textrm{S}=-\sum_{i=1}^{N}P_{i}\log\mathinner{\left(P_{i}\right)}$ with
$P_{i}=|\Psi_{i}|^{2}$. In Fig. 5(a), we show ensemble averaged S, obtained
from the eigenstates taken from middle 25% of the spectra, exhibiting a non-
analyticity around $\gamma=1$. Assuming a power-law behaviour of the relevant
correlation length, we obtain critical parameter $\gamma_{c}=0.95$ and
exponent $\nu=0.65$ using Eq. (7), while the collapsed data is shown in the
inset of Fig. 5(a). We also observe that the Inverse Participation Ratio
(IPR), I $=\sum_{i=1}^{N}\mathinner{\\!\left\lvert\Psi_{i}\right\rvert}^{4}$
exhibits a criticality around $\gamma=1$ as shown in Fig. 6. Thus we confirm
that the Anderson transition occurs at $\gamma_{\text{AT}}\equiv 1$ for
$\beta$-ensemble, while at $\tilde{\gamma}_{\text{AT}}\simeq 2$ for RPE [Fig.
5(b)].

For 1-D disordered spin-1/2 Heisenberg model, Shannon entropy is almost
constant for a particular $L$ if disorder strength is small ($h\ll 1$) and
slowly decaying for $h\gg 1$. Similar behaviour for IPR indicates that the
energy states of the Heisenberg model in the MBL regime are extended in the
Hilbert space exhibiting a non-trivial multifractal behaviour [55]. However
according to [54], one may look at the ratio of Shannon entropies of
Heisenberg model and GOE, i.e.
$\textrm{S}^{\prime}=\textrm{S}/\textrm{S}_{\text{GOE}}\approx\textrm{S}/\log(0.48N)$,
where $N={L\choose L/2}$ is the Hilbert space dimension of the $S_{z}=0$
symmetry sector. The finite-size scaling of $\textrm{S}^{\prime}$ gives the
numerical estimate of the MBL transition point to be $h_{\text{MBL}}\approx
2.77$ for our choice of $L=8,10,\cdots,16$. This is the same critical point
beyond which energy levels start to cluster [Fig. 3(d)]. Such a conclusion is
also verified via studies of entanglement entropy, magnetization fluctuations
[55]. Thus unlike $\beta$-ensemble, eigenstates start to localize as soon as
energy levels begin to cluster for both RPE and 1-D disordered spin-1/2
Heisenberg model.

Figure 6: Inverse Participation Ratio: for $\beta$-ensemble as a function of
$\gamma$ for various system sizes, $N$. Inset shows collapsed data following
the ansatz in Eq. (7) along with critical parameter and exponent.

### Ergodic to non-ergodic transition:

We now quantify the loss of ergodicity by computing the relative Rényi
($\mathcal{R}$) entropy between a pair of eigenfunctions [76] having similar
energy densities. Let $\ket{\Psi_{i}^{j}}$ be the $i^{th}$ eigenvector of the
$j^{th}$ disordered realization of an ensemble. We define two kinds of
$\mathcal{R}$ as follows:

$\displaystyle\begin{split}\mathcal{R}_{1}&=-2\log\mathinner{\left(\sum_{k=1}^{N}\mathinner{\\!\left\lvert\Psi_{i}^{j}(k)\Psi_{i+1}^{j}(k)\right\rvert}\right)},\qquad\mathcal{R}_{2}=-2\log\mathinner{\left(\sum_{k=1}^{N}\mathinner{\\!\left\lvert\Psi_{i}^{j}(k)\Psi_{i+1}^{j^{\prime}}(k)\right\rvert}\right)}.\end{split}$
(9)

Here $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ measure similarity among
wavefunctions obtained from the same and different samples respectively. For
eigenstates of GOE,
$\mathcal{R}=-2\log\mathinner{\left(\sum_{k=1}^{N}|z_{k}|\right)},\>z_{k}=x_{k}\times
y_{k}$, where $x_{k},y_{k}$ are i.i.d. random variables and for $N\gg 1$,
$x_{k},y_{k}\sim\mathcal{N}(0,1/\sqrt{N})\;\forall\;k$ assuming wavefunctions
are normalized. Then,
$\text{P}\mathinner{\left(z\right)}=N\text{K}_{0}\mathinner{\left(N\mathinner{\\!\left\lvert
z\right\rvert}\right)}/\pi$, where $\text{K}_{0}\mathinner{\left(x\right)}$ is
the modified Bessel function of $2^{nd}$ kind. Then,

$\displaystyle\begin{split}\left\langle\mathinner{\\!\left\lvert
z\right\rvert}\right\rangle=\dfrac{2}{N\pi}\Rightarrow\left\langle\mathcal{R}\right\rangle=-2\log\mathinner{\left(\sum_{i=1}^{N}\dfrac{2}{N\pi}\right)}=2\log\mathinner{\left(\dfrac{\pi}{2}\right)}\approx
0.9032\end{split}$ (10)

Thus $\mathcal{R}\approx 0.9$ for any pair of completely extended
wavefunctions and this value is used to benchmark our numerical estimates. The
relative Rényi entropy can be viewed as a generalization of the Kullback-
Leibler divergence exhibiting critical behaviour in the case of RPE [61]. We
will investigate $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ in the similar spirit
with the premises of finding:

  * •

Ergodic regime: $\mathcal{R}$ $\sim\mathcal{O}(1)$ for any pair of
wavefunctions, as both of them are uniformly extended

  * •

Localized regime: $\mathcal{R}$ will diverge as different wavefunctions
localize at separate sites

  * •

Non-ergodic regime having two possibilities:

    * –

If energy levels of $\ket{\Psi_{i}^{j}},\ket{\Psi_{i^{\prime}}^{j}}$ repel
each other, then such energy states come from the same symmetry sector, i.e.
the same subspace of the Hilbert space. Thus
$\ket{\Psi_{i}^{j}},\ket{\Psi_{i^{\prime}}^{j}}$ are likely to hybridize if
the governing Hamiltonian is sufficiently dense [61], giving $\mathcal{R}$
$\sim\mathcal{O}(1)$.

    * –

In absence of any level repulsion, the energy states in the NEE phase are
likely to be extended over different parts of the Hilbert space, thus
$\mathcal{R}$ will diverge.

Let us now illustrate the measures for the well studied case of RPE. For two
nearby energy states $\ket{\Psi_{i}^{j}},\ket{\Psi_{i^{\prime}}^{j}}$ with
comparable energy densities, $\mathcal{R}_{1}\sim\mathcal{O}(1)$ for
$\tilde{\gamma}<\tilde{\gamma}_{\text{AT}}$ and $\mathcal{R}_{1}\gg 1$ for
$\tilde{\gamma}>\tilde{\gamma}_{\text{AT}}$ as chaotic-integrable transition
occurs at $\tilde{\gamma}_{\text{AT}}$. On the other hand, the energy states
from different samples, say
$\ket{\Psi_{i}^{j}},\ket{\Psi_{i^{\prime}}^{j^{\prime}}}$, are likely to have
different support set in the NEE phase, as different governing Hamiltonians
cannot hybridize them even if their energy densities are comparable giving
$\mathcal{R}_{2}\gg 1$ for $\tilde{\gamma}>\tilde{\gamma}_{\text{ET}}$. In
Fig. 5(e) and (h) we show $\mathcal{R}_{1},\mathcal{R}_{2}$ for RPE exhibiting
$2^{nd}$ order phase transitions clearly identifying the critical points
$\tilde{\gamma}_{\text{AT}}$ and $\tilde{\gamma}_{\text{ET}}$ respectively.

Recall that for $\beta$-ensemble chaotic-integrable transition occurs at
$\gamma_{\text{ET}}=0$, hence $\mathcal{R}_{1}$ should show non-analyticity at
the same point. Previously we argued that $\gamma_{\text{ET}}$ is also the
ergodic transition point, thus $\mathcal{R}_{2}$ should exhibit criticality
there as well. The critical behaviours of $\mathcal{R}_{1},\mathcal{R}_{2}$
are evident in Fig. 5(d) and (g), where scaling analysis gives
$\gamma_{c}\equiv\gamma_{\text{ET}}\sim 0$.

In the case of 1-D disordered spin-1/2 Heisenberg model, we find that the
scaling of $\mathcal{R}_{2}$ indicate a $h_{c}\equiv h_{\text{ET}}\approx 2$
in Fig. 5(i) while earlier we identified $h_{\text{MBL}}\approx 2.77$. This
indicates the existence of NEE in 1-D disordered spin-1/2 Heisenberg model for
an intermediate range of disorder $h\in(2,2.77)$ in agreement with the
existing study on participation entropy, survival probability [54] and
momentum distribution fluctuations [71]. Therefore one would expect
$\mathcal{R}_{1}$ to show criticality at $h_{\text{MBL}}$ since it is also the
chaotic-integrable transition point. However the Hamiltonians of 1-D
disordered spin-1/2 Heisenberg model are so sparse that they fail to
completely hybridize the NEE eigenstates even from the same subspace of the
Hilbert space. As a result $\mathcal{R}_{1}$ shows criticality at
$h_{c}\approx 2.21$ [Fig. 5(f)], a value in between $h_{\text{ET}}$ and
$h_{\text{MBL}}$. Below the critical values, $\mathcal{R}\approx 0.9$ in all
three models as expected from Eq. (10).

Figure 7: Entropic localization length for $\beta$-ensemble, RPE and 1-D
disordered spin-1/2 Heisenberg model for various system sizes and chain
lengths as a function of repulsion parameter $\beta$. We show the line
$d_{N}/N=\beta$ via dashed curve. Inset of (a) shows $\beta^{\star}$ below
which $d_{N}/N$ becomes constant.  Figure 8: (a) Phase diagram: The three
distinct phases observed in $\beta$-ensemble, RPE and 1-D disordered spin-1/2
Heisenberg model are demarcated against the critical parameter values. Markers
indicate fractal exponents $D_{1,2}$ as explained in the legend. (b) Fraction
of localized states: as a function of $\gamma$ for $\beta$-ensemble for
various system sizes. Inset shows $\alpha_{\xi}$, the system size scaling
exponent of $\xi$ vs. $\gamma$ along with a linear fit,
$\alpha_{\xi}=a\gamma+b$, where $a=0.9656\pm 0.0676$ and $b=-1.1618\pm
0.0370$.

### Localization length:

In the previous section, we noticed that the spectral properties of
$\beta$-ensemble and RPE have an important difference: chaotic-integrable
transition occurs at $\gamma_{\text{ET}}$ for $\beta$-ensemble and
$\tilde{\gamma}_{\text{AT}}$ for RPE. The degree of level repulsion, $\beta$
has been interpreted as the rescaled localization length in various systems
[77, 78, 79, 80] though there are exceptions as well [81]. Now we look at the
entropic localization length w.r.t. the Shannon entropy, $d_{N}\equiv
2.07e^{\textrm{S}}$, such that $d_{N}\approx N$ or $d_{N}\approx 1$ for a
fully ergodic or localized energy state respectively [77]. In case of RPE or
1-D disordered spin-1/2 Heisenberg model, one can numerically fit the PDF of
NNS with any phenomenological model (e.g. Brody [18], Berry-Robnik [19] etc.)
to estimate the repulsion parameter, $\beta$. However, such a numerical fit
pertains to the global shape of the PDF of NNS [77], without necessarily
reflecting the behaviour of $\text{P}\mathinner{\left(s\right)}_{s\to 0}$,
which is the true measure of level repulsion in a system. Thus we exploit the
one to one correspondence between $\beta$ and the mean value of RNNS. We
observe a sub-linear behaviour when $\beta$ is small and $d_{N}/N$ converges
to 1 when $\beta\to 1$ [Fig. 7(b) and (c)]. Hence energy states are completely
extended whenever the energy levels repel each other and this justifies the
coincidence of chaotic-integrable transition with the delocalization-
localization transition in RPE and 1-D disordered spin-1/2 Heisenberg model.

We show $d_{N}/N$ as a function of $\beta$ for various system sizes in case of
$\beta$-ensemble in Fig. 7(a). Here the relationship between the localization
length and the degree of level repulsion is super-linear throughout. Moreover,
$d_{N}/N$ becomes independent of $\beta$ for $\beta\leq\beta^{\star}\ll 1$
while $\beta^{\star}\propto 1/N$ as shown in the inset of Fig. 7(a). This
implies that the localization transition should occur roughly at $\beta=1/N$
as chaotic-integrable transition occurs at $\beta=N^{\gamma_{\text{ET}}}=1$.
This supports our earlier observation that $\gamma_{\text{AT}}=1$ does not
coincide with the chaotic-integrable transition point in the case of
$\beta$-ensemble unlike RPE or Heisenberg model.

### Scaling of eigenstate fluctuations:

It is important to analyse the eigenfunction fluctuations which can be
quantified via Rényi entropy, $S_{R}(q,N)\sim N^{D_{q}}$ where $D_{q}$’s are
fractal dimensions for different values of $q$ [82]. For $q=1$, the Rényi
entropy converges to the Shannon entropy, $\textrm{S}\sim D_{1}\log N$.
Similarly for $q=2$, one obtains the scaling in IPR $\sim N^{-D_{2}}$. In the
ergodic regime the fractal dimensions, $D_{1,2}=1$ as eigenstates occupy the
full Hilbert space volume, while $D_{1,2}=0$ in the localized regime. In the
NEE phase, $0<D_{1,2}<1$, which implies that the eigenstates are extended over
infinitely many but a zero fraction of all possible sites in the thermodynamic
limit (i.e. $N^{D_{1,2}}\to\infty$ but $N^{D_{1,2}}/N\to 0$ if $N\to\infty$).
Since distributions of Shannon entropy and IPR are quite broad and skewed
(Fig. 9(a)), median instead of mean has been used to estimate such fractal
dimensions [83]. The numerically estimated $D_{1,2}$ clearly identifies the
ergodic, NEE and localized regimes for $\beta$-ensemble and RPE as shown in
Fig. 8 ($D_{1,2}\approx 1-\gamma$ and $2-\tilde{\gamma}$ in the NEE phase for
$\beta$-ensemble and RPE respectively). However, in the case of Heisenberg
model, $D_{1,2}$ does not vanish for $h>h_{\text{MBL}}$ resulting from non-
trivial multifractality in the MBL phase [55].

To probe finer details of the eigenstructure of $\beta$-ensemble, we look at
the density of Shannon entropy. The distributions from different system sizes
collapse on top of each other at $\gamma=0,1$ upon a rescaling
$\textrm{S}\to\textrm{S}-D_{1}\log N$. However, in the NEE phase
($0<\gamma<1$), two peaks emerge in the histogram of S: i) a broad peak whose
location roughly scales as $N^{-D_{1}}$, ii) a sharp peak at $\textrm{S}=0$,
whose height decreases with $N$ (Fig. 9(a)). The existence of the $2^{nd}$
peak (which is absent for RPE, Fig. 9(b) and Heisenberg model, Fig. 9(c))
implies that a small albeit finite fraction of eigenstates are localized for
$0<\gamma<1$. In Fig. 8(b), we show $\xi$, the fraction of localized
eigenstates as a function of $\gamma$ for various $N$. Inset of the same
figure shows $\alpha_{\xi}\approx\gamma-1$, the system size scaling exponent
of $\xi$, which implies roughly $\xi\propto N^{\gamma-1}$. Thus in the
thermodynamic limit, there will be an infinite number of completely localized
states in the intermediate regime of $\beta$-ensemble (since
$N^{\gamma}\to\infty$ for $\gamma>0$), which constitutes a zero fraction of
all possible eigenstates (since $\xi\to 0$ for $\gamma<1$). By looking at the
median and mode of IPR of individual eigenstates for different system sizes,
we find that the high energy states (i.e. the ones in the middle of spectrum)
have a greater tendency to be localized compared to the ones close to the
ground state in the NEE regime. Thus contrary to RPE, $\beta$-ensemble offers
two kinds of eigenstates in the NEE phase: $N^{\gamma}$ number of completely
localized and $(N-N^{\gamma})$ number of NEE states.

Figure 9: Density of Shannon entropy: for (a) $\beta$-ensemble, $\gamma=0.6$
(b) RPE, $\tilde{\gamma}=1.5$ and (c) Heisenberg model, $h=2.4$ while varying
system sizes, $N$ and chain lengths, $L$, where $\mathcal{S}$ and S are the
scaled and original Shannon entropies respectively ($D_{1}$ is the fractal
dimension obtained from system size scaling of S.)

##  4 Properties of Dynamics

So far we have looked at the statistical picture of the energy level
correlation, eigenstate localization end ergodic properties of
$\beta$-ensemble. Now we want to look at the dynamical aspects of
$\beta$-ensemble, revealing important time and energy scales. In this regard,
one of the largest time scales is the Heisenberg time, defined as the inverse
of mean level spacing. Beyond such a time, the energy level dynamics of a
system equilibrates e.g. the spectral form factor attains a stationary state
[84]. Now we explicitly look at the time evolution of an initially localized
state and identify the relevant dynamical timescales.

### Survival probability:

An important characterization of the dynamics of a quantum mechanical system
is often done by monitoring the time evolution of a given wavefunction. We
choose a unit vector $\ket{j}$ having energy close to the spectrum centre of
$H$ as our initial state. Let $\mathinner{\left(E_{k},\ket{\phi_{k}}\right)}$
be the $k^{th}$ eigenpair of $H$ such that the time evolution of the initial
state is given by,

$\displaystyle\ket{j(t)}=e^{-iHt}\ket{j}=\sum_{k}e^{-iE_{k}t}\phi_{k}^{(j)}\ket{\phi_{k}},\quad\phi_{k}^{(j)}=\braket{\phi_{k}}{j}.$
(11)

The spread of initial state $\ket{j}$ over all other states is controlled by
the off-diagonal terms in $H$ and is quantified by the survival probability
[85]

$\displaystyle R(t)$
$\displaystyle=\mathinner{\\!\left\lvert\braket{j}{j(t)}\right\rvert}^{2}=\mathinner{\\!\left\lvert\sum_{k=1}^{N}\mathinner{\\!\left\lvert\phi_{k}^{(j)}\right\rvert}^{2}e^{-iE_{k}t}\right\rvert}^{2}$
(12)

Figure 10: Survival Probability for $\beta$-ensemble: (a) time evolution for
$\gamma=-0.3$ and various system size, $N$. We show Thouless
($t_{\textrm{Th}}$), relaxation time ($t_{\textrm{R}}$) via markers and
correlation hole ($t_{\text{hole}}$) via dashed line in each case. Inset shows
$t_{\textrm{Th}}$ and $t_{\textrm{R}}$ as a function of $N$ along with linear
fit in log-log scale via solid line, indicating a $\sqrt{N}$ dependence. (b)
$N=1024$ varying $\gamma$ (c) Correlation hole vs. $\gamma$ for various $N$.
Solid lines show smooth trend of the data using spline interpolation. Inset
shows $\gamma^{\star}$ vs. $N$ in log-log scale ($t_{\text{hole}}\approx 0$
for $\gamma\geq\gamma^{\star}$). (d) Asymptotic value of survival probability
vs. $\gamma$ for different $N$. Inset shows system size scaling
($\bar{R}\propto N^{\alpha_{\bar{R}}}$) where
$\alpha_{\bar{R}}\approx\gamma-1$ for $\gamma\leq 1$ and 0 for $\gamma>1$.

In general, the survival probability decays till $t=t_{\textrm{Th}}$, known as
the Thouless time [86]. This is the time required for $\ket{j}$ to maximally
spread over the Hilbert space. For example, in disordered (ergodic) metals, a
particle diffuses to the sample boundaries within $t_{\textrm{Th}}$. The
inverse of $t_{\textrm{Th}}$ gives the Thouless energy, $E_{\text{Th}}$ below
which the spectral correlations are similar to those of Wigner-Dyson ensemble.
Moreover, a finite-sized closed quantum system always equilibrates [87] and
the equilibrium value of survival probability is given by

$\displaystyle\bar{R}=\lim\limits_{t\to\infty}\dfrac{1}{t}\int_{0}^{t}d\tau\;R(\tau)=\sum_{k=1}^{N}\mathinner{\\!\left\lvert\phi_{k}^{(j)}\right\rvert}^{4}.$
(13)

Thus $\bar{R}$ is the IPR of initial state $\ket{j}$ in the eigenbasis
$\\{\ket{\phi_{k}}\\}$. The time required to reach $\bar{R}$ is known as the
relaxation time, $t_{\textrm{R}}$. The gap between $t_{\textrm{Th}}$ and
$t_{\textrm{R}}$ is known as the correlation hole, $t_{\text{hole}}$. A finite
$t_{\text{hole}}$ is a direct manifestation of the spectral rigidity, i.e. the
presence of long range correlation among energy levels [88, 54, 89].

The time evolution of survival probability for $\beta$-ensemble is shown in
Fig. 10(a) and (b) for various system sizes and $\gamma$ values. Tuning
$\gamma$, we observe three qualitatively different behaviours as follows:

1\. Ergodic regime $(\gamma\leq 0)\mathrel{\mathop{\ordinarycolon}}$ The
correlation hole is always present with easily identifiable Thouless and
relaxation times. $t_{\textrm{Th}}$ exhibits an approximately $\sqrt{N}$
scaling close to $\gamma_{\text{ET}}$ (inset of Fig. 10(a)), which can be
understood from sparsity of the Hamiltonian. Contrarily in the ergodic regime
of RPE, $t_{\textrm{Th}}$ is independent of system size due to the presence of
all to all coupling [56, 90]. We also observe that within $t_{\text{hole}}$,
$R(t)$ is non-monotonic unlike 1-D disordered spin-1/2 Heisenberg model [85].

2\. NEE phase $(0<\gamma<1)\mathrel{\mathop{\ordinarycolon}}$ We show
$t_{\text{hole}}$ as a function of $\gamma$ for different system sizes in Fig.
10(c) where $t_{\text{hole}}\approx 0$ for $\gamma\geq\gamma^{\star}$. The
inset shows that $\gamma^{\star}\rightarrow\gamma_{\text{ET}}=0$ as $N$
increases. Recall that chaotic-integrable transition occurs at
$\gamma_{\text{ET}}$, beyond which long range correlation among energy levels
(e.g. see power spectrum) is lost. The spectral rigidity is necessary for the
existence of $t_{\text{hole}}$ [88], which explains the absence of correlation
hole in the NEE regime in the thermodynamic limit. On the other hand, a finite
$t_{\text{hole}}$ exists in the NEE phases of RPE and 1-D disordered spin-1/2
Heisenberg model, as chaotic-integrable transition occurs at
$\tilde{\gamma}_{\text{AT}}$ and $h_{\text{MBL}}$ respectively.

3\. Localized phase $(\gamma\geq 1)\mathrel{\mathop{\ordinarycolon}}$ Exactly
at $\gamma=\gamma_{\text{AT}}$, we observe survival probability curves from
different system sizes to collapse on top of each other, showing a critical
behaviour similar to RPE. In the localized regime (i.e. $\gamma>1$),
$t_{\text{hole}}$ is completely absent while $R(t)$ converges to 1 upon
increasing either $\gamma$ or $N$.

We show the equilibrium value of survival probability, $\bar{R}$, as a
function of $\gamma$ in Fig. 10(d). We observe that $\bar{R}\approx 1$ for
$\gamma>1$ which is expected as the initial state, $\ket{j}$ is an eigenstate
in the localized regime. Inset of Fig. 10(d) shows system size scaling of
$\bar{R}$, indicating that $\bar{R}\approx N^{\gamma-1}$ in the NEE regime
denoting the extent of spread over the Hilbert space for an initially
localized state.

##  5 Conclusions

In this work we study the spectral properties of $\beta$-ensemble with a
motivation that the competition between diagonal and off-diagonal terms may
lead to a NEE phase. As customary in random matrix theory, we discuss the DOS
and short-range spectral correlations, namely, NNS, RNNS and observe a
transition from chaos to integrability at $\gamma=0$. The next pertinent
question is whether this transition can be associated with the ergodic and/or
localization transitions. A simple analysis of the power spectrum of noise in
the eigensequence identifies two critical points: ergodic transition at
$\gamma_{\text{ET}}=0$ and Anderson transition at $\gamma_{\text{AT}}=1$,
separating three distinct phases: ergodic ($\gamma\leq 0$), NEE ($0<\gamma<1$)
and localized ($\gamma\geq 1$) phase. Thus similar to RPE [56], related
ensembles [91, 92] and certain floquet systems [93, 94, 95], $\beta$-ensemble
is another matrix model where NEE phase exists over a finite interval of
system parameters.

The above observations can be consolidated from the eigenfunction properties
as both Shannon entropy and IPR show criticality at $\gamma_{\text{AT}}=1$,
confirming it to be the Anderson transition point. The system size scaling of
the above quantities gives us the fractal dimensions $D_{1,2}\approx
1-\gamma$, clearly demarcating the three phases. For Relative Rényi entropies
of type 1 and 2, criticality is seen at $\gamma_{\text{ET}}=0$, thus
confirming it to be the chaotic-integrable as well as the ergodic transition
point. Moreover, the distribution of Shannon entropy indicates that in the NEE
phase, there is a coexistence of $N^{\gamma}$ number of completely localized
and $(N-N^{\gamma})$ number of NEE states.

Finally, we identify the relevant dynamical timescales from the time evolution
of the survival probability, $R(t)$ of an initially localized state and find
that the correlation hole, $t_{\text{hole}}$, is always present in the ergodic
regime and absent in the NEE phase for $N\gg 1$ as energy levels become
uncorrelated. Moreover $R(t)\to 1$ for $N\gg 1$ and
$\gamma>\gamma_{\text{AT}}$ as expected in a localized phase. The NEE phase in
$\beta$-ensemble is quite distinct from that in RPE, where the energy levels
repel each other since integrability breaks down at the Anderson transition
point. Again the chaotic-integrable transition point is energy density
dependent and ergodicity breaks at a lower disorder strength in the case of
1-D disordered spin-1/2 Heisenberg model. We calculate the entropic
localization length to explain why chaotic-integrable transition in
$\beta$-ensemble does not coincide with the localization transition point.
These subtle differences imply that $\beta$-ensemble is not a suitable model
for spin systems like 1-D disordered spin-1/2 Heisenberg model. The
proposition that $\beta$-ensemble can model Heisenberg chain [62] has also
been contradicted in [96, 97] via analyses of higher order level spacings,
Spectral Form Factor (SFF) and in [65] by studying RNNS crossover. However, we
observe that in the ergodic regime of the $\beta$-ensemble, Thouless time
scales with the system size similar to sparse Hamiltonians [85]. Thus our
analyses suggest that the $\beta$-ensemble can imitate the spectral properties
of various Hamiltonians provided ergodicity breaks down at the chaotic-
integrable transition point in such systems.

Acknowledgment: We thank Lea F. Santos and Ivan Khaymovich for many useful
comments on the manuscript. AKD is supported by INSPIRE Fellowship, DST,
India.

##  Appendix A Numerical details

### Unfolding:

For fixed values of system parameters and size, we obtain $\mathcal{F}(E)$,
the cumulative density of eigenvalues from all disordered realizations. Next
we smooth $\mathcal{F}(E)$ using a moving average filter. Then for the
original eigenvalue $E_{i}$, unfolding implies the interpolation of
$\mathcal{F}(E_{i})$ [10].

### Scaling of crossover curves:

Let us look at an observable $y$ for a system parameter $x$. If we observe
non-analytical behavior of the corresponding crossover curves from different
system sizes, we assume $y$ to behave according to Eq. (7). So we take an
array of parameters $\vec{x}=(x(1),x(2),\dots,x(m))$ and measure $y$ for two
system sizes $N_{1}$ and $N_{2}$, giving us two arrays
$\vec{y_{i}}=(y_{i}(1),y_{i}(2),\dots,y_{i}(m)),\>i=1,2$. Following Eq. (7),
we take the function $g(x;x_{c},\nu,N)\equiv(x-x_{c})(\log N)^{1/\nu}$. Now
define the function
$\vec{x_{i}}\equiv\vec{x_{i}}(x_{c},\nu)=g(\vec{x};x_{c},\nu,N_{i})$ for
$i=1,2$. For the correct guess of $x_{c}$ and $\nu$, $\vec{y_{1}}$ vs.
$\vec{x_{1}}$ and $\vec{y_{2}}$ vs. $\vec{x_{2}}$ should behave similarly. The
observed non-analyticity of the curves $\vec{y_{1}}$ vs. $\gamma$ and
$\vec{y_{2}}$ vs. $\gamma$ gives a good idea of the initial value of $x_{c}$
and we take $\nu=1$ as the starting value. For these initial guesses of
$x_{c}$ and $\nu$, we interpolate $\vec{y_{1}}$ vs. $\vec{x_{1}}$ w.r.t.
$\vec{x_{2}}$ to create a new array $\vec{y_{1}^{\prime}}$. Next we calculate
the Residual Sum of Squares (RSS) between $\vec{y_{1}^{\prime}}$ and
$\vec{y_{2}}$, defined as
$\sum_{k}\mathinner{\left(y_{1}^{\prime}(k)-y_{2}(k)\right)}^{2}$. Since RSS
should be 0 for the correct choice of $x_{c}$ and $\nu$, we iteratively change
$x_{c}$ and $\nu$ to minimize the RSS, which gives us the desired values of
$x_{c}$ and $\nu$ for $N_{1}$ and $N_{2}$. We repeat this exercise for all
pairs of system sizes and report the average of the obtained parameters.

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