# Long-range spectral statistics of the Rosenzweig-Porter model

Wouter Buijsman<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Beer-Sheva 84105, Israel  Max Planck Institute for
the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany

###### Abstract

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that
supports an ergodic, fractal, and localized phase. The names of these phases
refer to the properties of the (bulk) eigenstates. This work focuses on the
long-range spectral statistics of the recently introduced unitary equivalent
of this model. By numerically studying the Thouless time obtained from the
spectral form factor, it is argued that long-range spectral statistics can be
used to probe the transition between the ergodic and the fractal phases.
Provided that the transition between the fractal and the localized phases can
be probed through short-range level statistics such as the average ratio of
consecutive level spacings, this work establishes that spectral statistics are
sufficient to probe both transitions present in the phase diagram.

##  I Introduction

Spectral (level) statistics provide a convenient, basis-independent probe for
quantum chaos [1, 2, 3]. Starting from early studies on the spectra of heavy
atomic nuclei, spectral statistics are nowadays frequently used to
characterize phases of matter in for example studies on single-body (Anderson)
[4, 5, 6, 7] and many-body [8, 9, 10, 11, 12, 13] localization, random matrix
theory [14, 15, 16], and integrability [17, 18]. Broadly speaking, spectral
statistics come in two types: short- and long-range. Short-range spectral
statistics are commonly quantified by the level spacing distribution or the
average ratio of consecutive level spacings [19, 20], while long-range
spectral statistics are typically studied by focusing on the spectral form
factor [21, 22], next to others [3]. Often, studies on short- and long-range
spectral statistics provide complementary results.

The Rosenzweig-Porter model is a single-parameter random matrix ensemble that
supports an ergodic, fractal (also known as _delocalized yet non-ergodic_),
and localized phase [23, 24]. These names refer to the fractal properties of
the (bulk) eigenstates. In the thermodynamic limit, the phase diagram of this
model can be obtained fully by analytical methods [25, 26, 27, 28, 29]. The
Rosenzweig-Porter model serves as a natural toy model for the many-body
localization transition as there is a fractal phase in-between the ergodic and
localized phases, similar to what has been observed in physical models [30,
31]. Recently, part of the phase diagram of this model has been observed
experimentally [32]. During the last few years, various variants and
generalizations of the Rosenzweig-Porter model, for example with multifractal
eigenstates, have been proposed [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44].

Long-range spectral statistics are studied numerically most conveniently for
unitary models, for which the eigenvalues are located on the unit circle in
the complex plane. Such models typically have a uniform density of states,
meaning that no unfolding (uniformizing the density of states) is required.
The absence of spectral edges then also guarantees that results are not
affected by deviating edge statistics. A unitary equivalent of the (Hermitian)
Rosenzweig-Porter model has been proposed recently in Ref. [38]. The unitary
model can, for example, be used as a toy model for the many-body localization
transition in periodically driven (Floquet) systems [45, 46, 47, 48].
Complementing recent interest in the spectral statistics of the Rosenzweig-
Porter model and its variants [49, 40], this work focuses on long-range
spectral statistics of the unitary equivalent of the Rosenzweig-Porter model.
By numerically studying the Thouless time obtained from the spectral form
factor, it is argued that long-range spectral statistics can be used to probe
the transition between the ergodic and the fractal phases. Provided that the
transition between the fractal and the localized phases can be probed through
short-range spectral statistics, this work establishes that spectral
statistics are sufficient to probe both transitions present in the phase
diagram.

The outline of this work is as follows. Section II introduces the Rosenzweig-
Porter model and the unitary equivalent of it. Section III introduces the
probes and discusses the results for short-range spectral (level spacing)
statistics. Section IV, which contains the main results, introduces the probes
and discusses the results for long-range spectral statistics. Section V
concludes with a summary and outlook.

##  II Rosenzweig-Porter model and its unitary equivalent

The (Hermitian) Rosenzweig-Porter model with complex-valued elements consists
of matrices $H$ of the form

$H=H_{0}+\frac{1}{\sqrt{N^{\gamma}}}\,M.$ (1)

Here, $N$ is the matrix dimension, and $\gamma\geq 0$ is a tuning parameter.
The matrix $H_{0}$ is diagonal with the non-zero elements sampled
independently from the normal distribution with mean zero and unit variance.
The matrix $M$ is a sample from the Gaussian unitary random matrix ensemble.
An $N\times N$ matrix $M$ sampled from this ensemble can be constructed as

$M=\frac{1}{2}(A+A^{\dagger}),$ (2)

where $A$ is an $N\times N$ matrix with complex-valued elements
$A_{nm}=u_{nm}+iv_{nm}$ with $u_{nm}$ and $v_{nm}$ sampled independently from
the normal distribution with mean zero and variance $1/2$.

The physical properties of the Rosenzweig-Porter model are determined by the
tuning parameter $\gamma$. In the thermodynamic limit $N\to\infty$, one
distinguishes between three different phases, which can be characterized by
their type of level statistics and fractal dimension of the (bulk)
eigenstates. Here, the fractal dimension $d$ is defined in terms of the
scaling of the inverse participation ratio IPR as $\text{IPR}\sim N^{-d}$,
where

$\text{IPR}=\sum_{n}|\langle n|\psi\rangle|^{4}$ (3)

with $|\psi\rangle$ denoting the (eigen)state under consideration and the
summation running over all basis states $|n\rangle$.

For $0\leq\gamma<1$, the model is in the _ergodic_ phase. This phase is
characterized by Wigner-Dyson level spacing statistics (typically observed for
quantum-chaotic systems) and eigenstates with fractal dimension $d=1$. For
$1\leq\gamma<2$, the model is in the _fractal_ phase. This phase is
characterized by Wigner-Dyson level spacing statistics and eigenstates with
fractal dimension $d=2-\gamma$. For $\gamma\geq 2$, the model is in the
_localized_ phase, characterized by Poissonian level statistics (uncorrelated
levels, typically observed for integrable systems) and eigenstates with
fractal dimension $d=0$. A numerical investigation of finite-$N$ scalings,
which is discussed in some more detail below, can be found in Ref. [50].

This work focuses on the unitary equivalent of the Rosenzweig-Porter model,
which was introduced recently in Ref. [38] for the variant with real-valued
elements. The eigenvalues of unitary matrices are located on the unit circle
in the complex plane, and can thus be parametrized as $\exp(i\,\theta)$ for
$\theta\in[-\pi,\pi)$. The density of states for the unitary equivalent of the
Rosenzweig-Porter model is uniform, meaning that the spectra can be analyzed
without spectral unfolding or the need to take into account edge effects.
Samples of this random matrix model can be obtained through stochastic time-
evolution of a time-dependent matrix $U(t)$ which is initialized as

$U(0)=\operatorname{diag}\left(e^{i\theta_{1}},e^{i\theta_{2}},\dots,e^{i\theta_{N}}\right)$
(4)

with the phases $\theta_{n}$ ($n=1,2,\dots,N$) sampled independently from the
uniform distribution ranging over $[-\pi,\pi)$. The dynamics of this unitary
matrix are governed ny _circular Dyson Brownian motion_ [51, 52],

$U(t+dt)=U(t)\,e^{i\sqrt{dt}M},$ (5)

where $M$ is again a sample from the Gaussian unitary ensemble. This matrix is
re-sampled at each evaluation. The time step $dt$ is required to be small
enough such that $\exp(i\sqrt{dt}M)$ can be well-approximated by
$1+i\sqrt{dt}M$. For the unitary equivalent of the Rosenzweig-Porter model to
result, this stochastic process needs to be evaluated up to time
$t=N^{-\gamma}$.

Naively, numerical sampling from the unitary equivalent of the Rosenzweig-
Porter ensemble is computationally expensive as it requires many evolutions
over time intervals of infinitesimal length. As Eq. (5) gives the proper time-
evolution only up to first order, moreover the results are subject to a loss
of accuracy with progressing time. Indeed, this was the approach used in Ref.
[38]. Ref. [53] recently proposed an improved algorithm that is not subject to
these restrictions. Let

$A=U(0)+\sqrt{dt}M,$ (6)

where again $M$ is a sample from the Gaussian unitary ensemble. One can show
that a realization $U$ from the unitary equivalent of the Rosenzweig-Porter
ensemble can be obtained by from the QR-decomposition of $A$ as $U=\Lambda Q$,
where

$A=QR$ (7)

with $Q$ being unitary and $R$ being upper-triangular with real-valued
diagonal elements. The matrix $\Lambda$, making the QR-decomposition unique,
is obtained from $R$ as

$\Lambda=\operatorname{diag}\left(\frac{R_{11}}{|R_{11}|},\frac{R_{22}}{|R_{22}|},\dots,\frac{R_{NN}}{|R_{NN}|}\right).$
(8)

Within this procedure, the time step can be arbitrarily large. A sample from
the unitary equivalent of the Rosenzweig-Porter model can be obtained by
setting $dt\to N^{-\gamma}$. In what follows, numerical data is obtained using
this procedure. It can be of interest to note that this work is the first
application of this algorithm.

##  III Short-range spectral statistics

In this work, short-range level statistics are quantified through the commonly
studied average ratio of consecutive level spacings [8, 19]. Let the
eigenvalues $\lambda_{n}$ ($n=1,2,\dots,N$) of the unitary matrix that is
studied be parametrized as $\lambda_{n}=\exp(i\theta_{n})$ with
$\theta_{n}\in[-\pi,\pi)$, and sorted such that
$\theta_{1}\leq\theta_{2}\leq\dots\leq\theta_{N}$. The ratios $r_{n}$
($n=1,2,\dots,N-2$) of consecutive level spacings are then defined as

$r_{n}=\min\left(\frac{\theta_{n+2}-\theta_{n+1}}{\theta_{n+1}-\theta_{n}},\frac{\theta_{n+1}-\theta_{n}}{\theta_{n+2}-\theta_{n+1}}\right).$
(9)

By construction, $r_{n}\in[0,1]$. Poissonian and Wigner-Dyson level statistics
are characterized by an average value $\langle r\rangle$ given by $\langle
r\rangle=2\ln(2)-1\approx 0.386$ and $\langle r\rangle\approx 0.600$,
respectively.

Fig. 1 shows the average (taken over full spectra and a large number of
samples) ratio of consecutive level spacings as a function of $\gamma$ for
dimensions $N=100$, $N=1000$, and $N=10000$ (upper panel). As expected, the
curves tend towards a transition from Wigner-Dyson to Poissonian at the
transition between the fractal and the localized phases at $\gamma=2$. No
indications for the transition between the ergodic and the fractal phases at
$\gamma=1$ can be observed. Ref. [50] established for the (Hermitian)
Rosenzweig-Porter model that a collapse of finite-$N$ curves is observed when
plotting $\langle r\rangle$ as a function of
$(\gamma-\gamma_{c})\ln(N)^{1/\nu}$ with $\gamma_{c}=2$ and $\nu=1$. A similar
scaling is shown in the lower panel, where indeed a collapse can be observed.

Figure 1: The average ratio of consecutive level spacings $\langle r\rangle$
taken over full spectra for dimensions $N=100$, $N=1000$, and $N=10000$ as a
function of $\gamma$ (upper panel) and $(\gamma-\gamma_{c})\ln(N)^{1/\nu}$
with $\gamma_{c}=2$ and $\nu=1$ (lower panel).

##  IV Long-range spectral statistics

In terms of the parametrization of the eigenvalues introduced above, long-
range spectral statistics are conveniently probed by the spectral form factor

$\displaystyle K(t)$
$\displaystyle=\left\langle\frac{1}{N}\sum_{n,m}e^{i(\theta_{n}-\theta_{m})t}\right\rangle$
(10)
$\displaystyle=\left\langle\frac{1}{N}\left|\sum_{n}e^{i\theta_{n}t}\right|^{2}\right\rangle,$
(11)

where again $\langle\cdot\rangle$ denotes an ensemble average [3]. The
spectral form factor can be interpreted as the Fourier transform of the two-
point spectral correlation function, where $t$ has the interpretation of a
time. Since the phases $\theta_{n}$ can take values ranging from $-\pi$ to
$\pi$ only, the time only takes discrete values $t\in\mathbb{Z}$. For random
matrices sampled from the circular unitary ensemble, the spectral form factor
can be evaluated analytically as

$K(t)=\begin{cases}t/N&\text{if }|t|\leq N,\\\ 1&\text{if }|t|>N.\end{cases}$
(12)

For unitary matrices with Poissonian level statistics, one easily finds
$K(t)=1$.

Fig. 2 shows the spectral form factor obtained from a large number of spectra
as a function of time for values of $\gamma$ in each of the ergodic, fractal,
and localized phases for matrix dimensions $N=100$, $N=1000$, and $N=10000$.
In the ergodic phase ($\gamma<1$), the spectral form factor matches the
evaluation for the circular unitary ensemble [Eq. (12)] almost precisely. The
fractal phase ($1\leq\gamma<2$) is characterized by intermediate statistics
interpolating between the evaluations for the circular unitary ensemble and
Poissonian statistics. For $\gamma=2$, the spectral form factor appears to be
scale-invariant in the sense that it is independent of $N$ when considered in
terms of the scaled time $t/N$. In the localized phase ($\gamma\geq 2$), the
spectral form factor tends towards $K(t)=1$ as expected for localized systems
with increasing dimension.

Figure 2: The spectral form factor for dimensions $N=100$, $N=1000$, and
$N=10000$ for values of $\gamma$ in each of the ergodic, fractal, and
localized phases.

The question of interest in this work is whether (long-range) spectral
statistics can be used to probe the transition between the ergodic and the
fractal phases at $\gamma=1$. Above, it was illustrated that level spacing
(short-range) statistics are insensitive to this transition. Since the
spectral form factor can be interpreted as the Fourier transform of the two-
point spectral correlation function, one could say that the spectral form
factor evaluated at time $t$ probes two-point spectral correlations over a
separation $\sim 1/t$. This notion is commonly quantified by the _Thoulesss
time_ , after Thouless [54]. The Thouless time is typically defined as the
lowest time from which onwards the spectral form factor matches the evaluation
for the (in this case) circular unitary ensemble. The Thouless time has been
found useful to quantify the onset of quantum chaos for various physical and
random matrix models [55, 56, 57, 12, 13, 58].

The integrated value of the spectral form factor over time is fixed the
presence or absence of level repulsion [59]. Level repulsion is present
(absent) if the two-point spectral self-correlation is zero (non-zero).
Wigner-Dyson level statistics obey level repulsion, while Poissonian level
statistics do not. Specifically,

$\int_{0}^{\infty}\bigg{[}1-K\bigg{(}\frac{t}{N}\bigg{)}\bigg{]}\,dt=\begin{cases}\pi
N&\text{(level repulsion)},\\\ 0&\text{(no level repulsion)},\end{cases}$ (13)

where for convenience the sum over discrete times has been written as an
integral. Because of this constraint, one can not trivially define a time from
which onwards the spectral form factor matches the evaluation for the circular
unitary ensemble. The Thouless time is here defined as the time at which the
spectral form factor first intersects the evaluation for the circular unitary
ensemble. Arguably, this is the most precise definition of the Thouless time
that can be formulated.

Fig. 3 (upper panel) shows the Thouless time $t_{\text{Th}}$ obtained from the
spectral form factor resulting from a large number of spectra as a function of
$\gamma$ around $\gamma=1$ for matrix dimensions $N=100$, $N=1000$, and
$N=10000$. For clarity, the lower panel shows a graphical illustration of how
the Thouless time is obtained. This panel shows the spectral form factor for
$N=10000$ and $\gamma=1.3$ combined with the evaluation for the circular
unitary ensemble. As can be read off from the upper panel, the points at which
the curves first intersect (interpreted as the Thouless time) is found as
$t\approx 80.4$. Because of the constraint of Eq. (13), there is an additional
deviation between the curves at $t>t_{\text{Th}}$ to compensate for the
deviation on the interval $t\in[0,t_{\text{Th}}]$.

For $\gamma<1$, the Thouless time is of order unity, indicating agreement of
the spectral form factor with the random matrix theory expectation almost
fully. From $\gamma=1$ onwards, the Thouless time quickly increases, which
indicates a transition to a phase characterized by different long-range
spectral two-point correlations. As $\lim_{N\to\infty}t_{\text{Th}}/N\to 0$ in
the fractal phase ($1\leq\gamma<2$), short-range level statistics remain
unaffected by the increase of the Thouless time. Consistent results can be
observed in Fig. 2. These results show that the spectral form factor can be
used to probe the transition at $\gamma=1$ between the ergodic and the fractal
phases.

Figure 3: Upper panel: the Thouless time $t_{\text{Th}}$ as obtained from the
spectral form factor (see the main text for details) as a function of $\gamma$
around $\gamma=1$ for dimensions $N=100$, $N=1000$, and $N=10000$. Lower
panel: an illustration of how the Thouless time is determined. The plot shows
the spectral form factor for $N=10000$ and $\gamma=1.3$, combined with the
evaluation for the circular unitary ensemble (black dashed curve). The point
at which these curves first intersect (here, at $t\approx 80.4$) is marked as
the Thouless time.

##  V Conclusions and outlook

This work focused on the long-range spectral statistics of the recently
introduced unitary equivalent of the Rosenzweig-Porter model. Using the
algorithm of Ref. [53] to efficiently sample this model, it was argued that
the spectral form factor is able to probe the transition between the ergodic
and the fractal phases ($\gamma=1$). More precisely, it was observed that the
transition between the ergodic and the fractal phases can be marked as the
highest value of $\gamma$ for which the spectral form factor first intersects
the spectral form factor of the circular unitary ensemble at times of order
unity. It has been argued that this time can be interpreted as a Thouless
time, which is a commonly used probe for quantum ergodicity. Taking into
account that the transition between the fractal and the localized phases can
be probed through short-range spectral statistics such as the average ratio of
consecutive level spacings, this work establishes that spectral statistics are
sufficient to probe both transitions present in the phase diagram.

At an intuitive level, one might expect that the observation that ergodic and
fractal phases are characterized by different types of (long-range) spectral
statistics holds more generally. Performing a similar analysis as discussed in
this work for random matrix models related to the Rosenzweig-Porter model is
potentially a fruitful starting point to further investigate this hypothesis.
Another aspect arguably worth further investigation is the structure and
possible universality of the scale-invariant (in the sense that it is
independent of $N$ when considered in terms of the scaled time $t/N$) spectral
form factor at the transition between the fractal and the localized phases
($\gamma=2$). Finally, the constraint on the spectral form factor imposed by
the presence or absence of level repulsion [Eq. (13)] as introduced in Ref.
[59] and the way in which it is taken into account here invites for
reconsiderations on how to use the spectral form factor as a probe for long-
range spectral correlations. In particular, one could ask if the definitions
of the Thouless time used here and in the literature could be sharpened.

###### Acknowledgements.

The author acknowledges support from the Kreitman School of Advanced Graduate
Studies at Ben-Gurion University.

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