# Generalized Spectral Form Factor in Random Matrix Theory

Zhiyang Wei  MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China  Chengming Tan  Hefei National Research Center for
Physical Sciences at the Microscale and School of Physics, University of
Science and Technology of China, Hefei 230026, China  Ren Zhang
<EMAIL_ADDRESS>MOE Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Hefei National Laboratory, Hefei,
230088, China

###### Abstract

The spectral form factor (SFF) plays a crucial role in revealing the
statistical properties of energy level distributions in complex systems. It is
one of the tools to diagnose quantum chaos and unravel the universal dynamics
therein. The definition of SFF in most literature only encapsulates the two-
level correlation. In this manuscript, we extend the definition of SSF to
include the high-order correlation. Specifically, we introduce the standard
deviation of energy levels to define correlation functions, from which the
generalized spectral form factor (GSFF) can be obtained by Fourier transforms.
GSFF provides a more comprehensive knowledge of the dynamics of chaotic
systems. Using random matrices as examples, we demonstrate new dynamics
features that are encoded in GSFF. Remarkably, the GSFF is complex, and the
real and imaginary parts exhibit universal dynamics. For instance, in the two-
level correlated case, the real part of GSFF shows a dip-ramp-plateau
structure akin to the conventional counterpart, and the imaginary part for
different system sizes converges in the long time limit. For the two-level
GSFF, the analytical forms of the real part are obtained and consistent with
numerical results. The results of the imaginary part are obtained by numerical
calculation. Similar analyses are extended to three-level GSFF.

##  I introduction

The spectral form factor (SFF) is a powerful tool for characterizing and
analyzing the statistical behavior of energy levels in various physical
systems, ranging from atomic nuclei Brézin and Zee (1993); Brézin and Hikami
(1996); Papenbrock and Weidenmüller (2007); Gómez  _et al._ (2011) to quantum
chaotic systems Gómez  _et al._ (2011); Kottos and Smilansky (1997); Brézin
and Hikami (1997); Links  _et al._ (2003); Turek  _et al._ (2005); Müller  _et
al._ (2005); Gnutzmann and Smilansky (2006); Bertini  _et al._ (2018); Chan
_et al._ (2018a); Kos  _et al._ (2018); Roy  _et al._ (2022); Chen and Ludwig
(2018); del Campo  _et al._ (2018); Liu (2018); Chenu  _et al._ (2019);
Friedman  _et al._ (2019); Gaikwad and Sinha (2019); Cao  _et al._ (2022);
Winer and Swingle (2022a); Balasubramanian  _et al._ (2022); Buividovich
(2022); Stechel and Heller (1984). It measures the fluctuations of the density
of states in a complex system and reveals universal features of quantum
chaotic systems Haake (1991), such as level repulsion Bohigas  _et al._
(1984); Caurier and Grammaticos (1989), random matrix statistics Bohigas
(1991); Andreev  _et al._ (1996), and quantum ergodicity Stechel and Heller
(1984); Leitner (2015); Madhok  _et al._ (2018). One of the pioneering
developments that establish the spectrum statistical properties is known as
the Wigner-Dyson statistics Wigner (1951, 1993); Dyson (2004a, b); Wigner
(1967). It revealed that energy levels in complex systems do not merely follow
a simple random pattern but instead exhibit correlations and repulsions,
resembling the behavior of eigenvalues of random matrices Wigner (1967); Guhr
_et al._ (1998); Mehta (2004). Therefore, random matrix theory (RMT) is widely
used in the study of spectrum statistics and serves as a versatile tool for
replicating energy level distributions in complex systems Wigner (1967); Guhr
_et al._ (1998); Mehta (2004).

The SFF finds diverse applications across various disciplines. In condensed
matter physics, it has been used to diagnose the quantum phase transitions and
the level statistics of many-body systems, such as spin chains, disordered
systems, and topological insulators Rabson  _et al._ (2004); Joshi  _et al._
(2022); Kos  _et al._ (2018); Chan  _et al._ (2018b, a); Bertini  _et al._
(2018); Šuntajs  _et al._ (2020); Nivedita  _et al._ (2020); Liao  _et al._
(2020); Sierant  _et al._ (2020); Winer and Swingle (2022b); Roy  _et al._
(2022); Winer and Swingle (2023); Winer  _et al._ (2020a); Roy and Prosen
(2020); Vleeshouwers and Gritsev (2021); Sarkar  _et al._ (2023); Barney  _et
al._ (2023). For instance, the SFF is employed to analyze a hydrodynamic
system with a sound pole in a finite volume cavity, revealing an association
of the logarithm of the hydrodynamic enhancement with a quantum particle
moving in the selfsame cavity Winer and Swingle (2023). In the quadratic
Sachdev-Ye-Kitaev model, the SFF features an exponential ramp, which contrasts
with the linear slopes in chaotic models Winer  _et al._ (2020a). In Floquet
fermionic chains with long-range two-particle interactions, SSF precisely
follows the prediction of RMT in long chains and timescales surpassing the
Thouless time scales with the system size Roy and Prosen (2020). In quantum
field theory and holography, the SFF serves as a tool to probe quantum chaos
therein and information scrambling of strongly coupled systems, such as
conformal field theories, gauge theories, and black holes Buividovich (2022);
Cáceres  _et al._ (2022); Belin  _et al._ (2022); de Mello Koch  _et al._
(2019); Cotler  _et al._ (2017); Li  _et al._ (2017); Kudler-Flam  _et al._
(2020); Easther and McAllister (2006); Chen (2022); Choi  _et al._ (2023). For
example, the SFF of supersymmetric Yang-Mills theory in four dimensions
exhibits consistency with that of a Euclidean black hole in AdS5 Choi  _et
al._ (2023), which provides a geometrical interpretation of the ramp behavior.
In addition, there are many studies in the field of quantum information. A
major modification to the bipartite entanglement entropy at large sub-system
size depends only on the theoretical particle spectrum Cardy  _et al._ (2007);
The concept of SFF can be naturally extended to a subsystem via pseudo–entropy
Goto  _et al._ (2021); Also, the study of SSF have been extended to open
systems or non-Hermitian physics Li  _et al._ (2021); Xu  _et al._ (2021); Kos
_et al._ (2021); Cornelius  _et al._ (2022); Zhou  _et al._ (2023); Kawabata
_et al._ (2023); Matsoukas-Roubeas  _et al._ (2023a, b, c); Roccati  _et al._
(2023).

As of now, most of the research focuses on the SFF with a two-level
correlation, which encapsulates the energy level spacing, denoting the
intervals between two energy levels. To our best knowledge, a few works extend
SFF to the case of higher even-order correlation, such as four-point
correlation Liu (2018); Cardella (2021), and six-point correlation Gross and
Rosenhaus (2017). A universal definition for the case of arbitrary-order
correlation is still lacking. In this manuscript, we employ the standard
deviation of energy levels to redefine the correlation function and obtain the
generalized spectral form factor (GSFF) through a Fourier transform. The GSFF
elucidates the characteristics of high-order fluctuations in the density of
energy states, which manifest themselves in specific structural properties in
the time domain. By studying the GSFF in RMT, we find that it is a complex
quantity. For the case of two-level correlation, we obtain the same dip-ramp-
plateau structure as that in the conventional SFF Saad  _et al._ (2018);
Gharibyan  _et al._ (2018); Winer  _et al._ (2020b). The imaginary part of the
GSFF also presents a universal behavior akin to its real counterpart. These
findings are based on an analytical derivation and numerically validated by
adopting the Gaussian ensembles. We further find the validity of the
aforementioned results can be extended to cases with higher-level
correlations, which broadens the scope of our analysis.

The content of this manuscript is organized as follows: In Sec. II, we briefly
introduce the Gaussian ensemble and redefine the correlation function to
obtain GSFF. In Sec. III, we calculate the GSFF associated with two energy
levels in GUE and demonstrate the universal behavior of the real and imaginary
parts, respectively. In Sec. IV, we calculated the GSFF associated with three
energy levels correlation, demonstrating the universal behavior of real and
imaginary parts. Lastly, we summarize our results in Sec. V.

##  II Definition of generalized spectral form factor

###  II.1 Gaussian ensemble

For the sake of the following discussion, let us start with a brief
introduction to the Gaussian ensemble which is a pivotal concept in the realm
of RMT, owing its origins to Eugene Wigner’s groundbreaking research in the
late 1950s Wigner (1951, 1967). By investigating the distribution of random
matrices, Wigner obtained the semicircle law governing the spectrum
distribution of Gaussian ensembles. Within RMT, diverse Hamiltonians are
crafted, with their matrix elements following Gaussian distributions. These
ensembles can be categorized based on matrix element characteristics (real,
complex, or quaternion) and matrix symmetry, resulting in the Gaussian
orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian
symplectic ensemble (GSE), collectively known as GXE. These ensembles show
different Dyson’s index, denoted as $\beta$, where GOE corresponds to
$\beta=1$, GUE to $\beta=2$, and GSE to $\beta=4$ Wigner (1967); Guhr  _et
al._ (1998); Mehta (2004).

The applications of these ensembles have extended to the realm of quantum
physics, with GOE applying to systems possessing time-reversal symmetry and an
even number or no 1/2 spin particles. In contrast, GUE is associated with
systems in the absence of time-reversal symmetry, featuring random complex
symmetric Hermitian matrices. GSE finds relevance in systems characterized by
time-reversal symmetry and comprising an odd number of 1/2 spin particles,
involving random real quaternion elements symmetric Hermitian matrices. These
concepts, stemming from Wigner’s pioneering work, play an important role in
the exploration of quantum systems and statistical physics Wigner (1967); Guhr
_et al._ (1998); Mehta (2004). In this manuscript, we will use GUE as an
example to demonstrate our results.

###  II.2 Spectral correlation function and generalized spectral form factor

Taking the Gaussian ensemble as an example, we show the generalization of the
spectral correlation function and SFF. The Gaussian ensemble comprises a set
of random matrices denoted as $\left\\{M_{N\times N}\right\\}$, each having
$N$ eigenvalues labeled as $\lambda_{n}\left(n=1,2,\ldots,N\right)$. The
distribution of these eigenvalues is determined by a joint probability density
function (JPDF), denoted as
$P_{N}\left(\lambda_{1},\ldots,\lambda_{N}\right)$, which can be derived from
the Gaussian distribution. The JPDF represents the probability of finding an
energy level at each position along the energy axis defined by
$\left\\{\lambda_{1},\ldots,\lambda_{N}\right\\}$. According to RMT, for some
complex physical systems, we can employ random matrices to replicate the
energy level distribution of the system Wigner (1967); Guhr  _et al._ (1998);
Mehta (2004). The Hamiltonian is written as $H=E\times M_{N\times N}$, where
$E$ is the characteristic energy of a particular system, and serves as the
energy unit. The eigenenergies of the system are denoted by
$\lambda_{n}\left(n=1,2,\ldots,N\right)$. It assumes that each energy level is
equivalent, implying that the JPDF
$P_{N}\left(\lambda_{1},\ldots,\lambda_{N}\right)$ remains invariant under
parameter rearrangements.

Using the JPDF, we can explore the characteristics of these energy levels. In
this manuscript, we employ the standard deviation of the levels to define the
correlation among $k$ levels, which is referred to as the $k$-point
correlation function and can be expressed as

$c^{(k)}\left(\epsilon\right)=\int\frac{P[{\bm{\lambda}}]}{N^{k}}\sum_{i=1}^{N}\delta\left[\epsilon-\sigma_{m}\left(\lambda_{m_{1}},\ldots,\lambda_{m_{k}}\right)\right]D{\bm{\lambda}},$
(1)

where $k$ is a positive integer smaller than $N$,
$P[{\bm{\lambda}}]=P(\lambda_{1},\ldots,\lambda_{N})$ represents the JPDF of
the $N$ energy levels, $D{\bm{\lambda}}=d\lambda_{1}\ldots d\lambda_{N}$
denotes the $N$-dimensional integral measures, and
$\sigma_{m}\left(\lambda_{m_{1}},\ldots,\lambda_{m_{k}}\right),m_{1},\ldots,m_{k}\in\left\\{1,2,\ldots,N\right\\}$
calculates the standard deviation among any $k$ energy levels. Here, we would
like to emphasize that our definition can also be implemented in the case that
$k>2$. This is a generalization of the conventional two-point correlation
function. It is clear that the large $\epsilon$ contribution of
$c^{(k)}(\epsilon)$ comes from the energy configuration that is far-away
separated, while the small $\epsilon$ contribution comes from the
configurations that are close to each other.

The correlation function defined in Eq.(1) can be numerically computed via the
ensemble average of a random matrix ensemble. To be specific, we randomly
chose $k$ energy levels from the whole spectrum of dimension $N$. So, there
are $N^{k}$ configurations for each Hamiltonian in the ensemble. Then we
implement an ensemble average. As a result, the explicit expression of
$c^{(k)}\left(\epsilon\right)$ can be requested as

$c^{(k)}\left(\epsilon\right)=\left\langle\frac{1}{N^{k}}\sum_{m=1}^{N^{k}}\delta\left(\epsilon-\sigma_{m}\left(\lambda_{m_{1}},\ldots,\lambda_{m_{k}}\right)\right)\right\rangle_{\text{em}},$
(2)

where $\left\langle.\right\rangle_{\text{em}}$ denotes the ensemble average.
When $k=1$, $c^{(1)}\left(\epsilon\right)=1$, which is reasonable. When $k=2$,
$\sigma_{m}\left(\lambda_{m_{1}},\lambda_{m_{2}}\right)=\sigma\left(\lambda_{i},\lambda_{j}\right)=\frac{\left|\lambda_{i}-\lambda_{j}\right|}{2},i,j\in\left\\{1,2,\ldots,N\right\\}$,
and we obtain the statistical properties of the absolute energy level spacing,
which will be further discussed in subsequent sections. Importantly, this
framework can be straightforwardly extended to the higher-order correlation.

The generalization of SFF can be obtained by a Fourier transformation of the
correlation function defined in Eq.(1),

$\displaystyle g^{(k)}\left(t\right)=\int c^{(k)}\left(\epsilon\right){\rm
e}^{-i\epsilon t}d\epsilon.$ (3)

For conciseness, we take $\hbar=1$ throughout this manuscript. Employing the
GSFF, we gain more insight into the discrete nature of the energy level
distribution. The short-time behavior of $g^{(k)}\left(t\right)$ originates
from the energy configuration that is far-away separated. In contrast, the
long-time behavior is dominated by the configurations that are close to each
other.

Compared with previous research of SFF, in the case of two energy levels
correlation, GSFF is similar to SFF which is conventionally defined by the
difference between energy levels. In contrast, GSFF features an imaginary part
showing universal behavior, as we will discuss later. In literature, the high-
order energy-level correlation has been considered, but only for even order
cases, such as four order Liu (2018). Our definition of GSFF is different from
the previous ones. The GSFF in our manuscript can be extended to any order
correlation, not only even-order correlation but also odd-order correlation.

##  III Two-level Generalized spectral form factor in GUE

###  III.1 Calculation

Let us start with the two-level scenario, where the correlation between energy
levels is encapsulated in the two-point correlation function, which is defined
as:

$c^{(2)}\left(\epsilon\right)=\frac{1}{N^{2}}\int
D{\bm{\lambda}}P[{\bm{\lambda}}]\sum_{i,j=1}^{N}\delta\left[\epsilon-\sigma\left(\lambda_{i},\lambda_{j}\right)\right],$
(4)

where $\sigma(\lambda_{i},\lambda_{j})$ represents the standard deviation
between any pair of energy levels, which can be expressed as the absolute
difference between $\lambda_{i}$ and $\lambda_{j}$, i.e.,
$\sigma(\lambda_{i},\lambda_{j})=|\lambda_{i}-\lambda_{j}|/2$. The definition
of integral measure $D{\bm{\lambda}}$ is the same as that in Eq.(1), and the
integral interval for each component is $[-2,2]$. Consequently, the two-level
GSFF is formulated as follows:

$\displaystyle g^{(2)}\left(t\right)$ $\displaystyle=\int
c^{(2)}\left(\epsilon\right){\rm e}^{-i\epsilon t}d\epsilon$
$\displaystyle=\frac{1}{N^{2}}\int d\epsilon\int
D{\bm{\lambda}}P[{\bm{\lambda}}]\sum_{i,j=1}^{N}\delta\left[\epsilon-|\lambda_{i}-\lambda_{j}|/2\right]{\rm
e}^{-i\epsilon t}$ $\displaystyle=\frac{1}{N^{2}}\int
D{\bm{\lambda}}P[{\bm{\lambda}}]\sum_{i,j=1}^{N}{\rm
e}^{-it|\lambda_{i}-\lambda_{j}|/2}$
$\displaystyle=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\int
d\lambda_{i}d\lambda_{j}R^{(2)}(\lambda_{i},\lambda_{j}){\rm
e}^{-it|\lambda_{i}-\lambda_{j}|/2}$ (5)

where $R^{(2)}(\lambda_{i},\lambda_{j})=\int
P[\lambda]...d\lambda_{i-1}d\lambda_{i+1}...d\lambda_{j-1}d\lambda_{j+1}...$
is the reduced JPDF.

Considering a GUE, represented by a set of ${N\times N}$ random matrix, the
matrix elements are Gaussian random variables and follow a normal distribution
with zero mean value and standard deviation of $1/\sqrt{N}$. Therefore, the
energy levels of the system are confined to the interval $[-2,2]$. Notably,
prior researches have obtained the analytical form of reduced JPDF Mehta
(2004); Liu (2018),

$\displaystyle R^{(2)}(\lambda_{i},\lambda_{j})=\left\\{\begin{aligned}
&\frac{\sqrt{4-\lambda_{i}^{2}}}{2\pi}&i=j\\\
&\frac{N^{2}}{N(N-1)}\frac{\sqrt{\left(4-\lambda_{i}^{2}\right)\left(4-\lambda_{j}^{2}\right)}}{4\pi^{2}}\\\
&-\frac{N^{2}}{N(N-1)}\frac{\sin^{2}\left[N\left(\lambda_{i}-\lambda_{j}\right)\right]}{\left[N\pi\left(\lambda_{i}-\lambda_{j}\right)\right]^{2}}&i\neq
j\end{aligned}.\right.$ (6)

Upon the substitution of Eq. (6) into Eq. (5), we obtain the analytical
expression of the GSFF for the case of two-point correlation. It is pointed
out that the integral concerning $\lambda_{i}$ and $\lambda_{j}$ is
independent on subindex $i,j$ in Eq.(5). The summation is order of ${\cal
O}(N^{2})$ at early time. Altogether, the GSFF can be expressed as

$\displaystyle g^{(2)}\left(t\right)$
$\displaystyle=\mathcal{A}^{(2)}+\mathcal{B}^{(2)}\left(t\right)+\mathcal{C}^{(2)}\left(t\right)$
(7)

where,

$\displaystyle\mathcal{A}^{(2)}=\frac{1}{N}\int\frac{\sqrt{4-\lambda_{1}^{2}}}{2\pi}d\lambda_{1}=\frac{1}{N};$
(8)

$\displaystyle\mathcal{B}^{(2)}\left(t\right)=\int\frac{\sqrt{4-\lambda_{1}^{2}}}{2\pi}\frac{\sqrt{4-\lambda_{2}^{2}}}{2\pi}{\rm
e}^{-i\frac{\left|\lambda_{1}-\lambda_{2}\right|t}{2}}d\lambda_{1}d\lambda_{2};$
(9)

$\displaystyle\mathcal{C}^{(2)}\left(t\right)=-\int\frac{\sin^{2}\left[N\left(\lambda_{1}-\lambda_{2}\right)\right]}{\left[N\pi\left(\lambda_{1}-\lambda_{2}\right)\right]^{2}}{\rm
e}^{-i\frac{\left|\lambda_{1}-\lambda_{2}\right|t}{2}}d\lambda_{1}d\lambda_{2}.$
(10)

Now we do the integral in Eq. (9), the real part of which can be directly
obtained,

$\displaystyle\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$
$\displaystyle=\int\frac{\sqrt{4-\lambda_{1}^{2}}}{2\pi}\frac{\sqrt{4-\lambda_{2}^{2}}}{2\pi}\cos\left[\frac{\left|\lambda_{1}-\lambda_{2}\right|t}{2}\right]d\lambda_{1}d\lambda_{2}$
$\displaystyle=\frac{4J_{1}^{2}(t)}{t^{2}}.$ (11)

where $J_{1}(t)$ is the Bessel function of the first kind. In the long time
limit, the asymptotic behavior of $\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$
reads ${8\cos^{2}\left(t+\frac{\pi}{4}\right)}/{\pi t^{3}}$, which is an
oscillatory decay.

Nevertheless, a closed analytical form of the imaginary part:

$\displaystyle\text{Im}[\mathcal{B}^{(2)}\left(t\right)]=-\int$
$\displaystyle\frac{\sqrt{4-\lambda_{1}^{2}}}{2\pi}\frac{\sqrt{4-\lambda_{2}^{2}}}{2\pi}$
$\displaystyle\times\sin\left[\frac{\left|\lambda_{1}-\lambda_{2}\right|t}{2}\right]d\lambda_{1}d\lambda_{2}$
(12)

is hardly available. In the following discussion of
$\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$, we use numerics. It should be
noticed that in the long-time limit, the integrand is strongly oscillating
around zero. As such, $\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$ decays to
zero in the longtime limit, which takes the form of $-1/t$ in the long time
limit.

The integral in Eq. (10) can be done by adopting the technique detailed in
Ref.Liu (2018). Specifically, we introduce the variables
$\mu=\lambda_{1}-\lambda_{2}$ and $\nu=\lambda_{2}$, along with a cut-off
$\nu\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ based on the Wigner
semicircle law and distribution normalization, then we arrive at the following
expressions for ${\cal C}^{(2)}(t)$

$\displaystyle\text{Re}[\mathcal{C}^{(2)}\left(t\right)]$
$\displaystyle=\left\\{\begin{aligned} &\frac{t-4N}{4N^{2}}&t<4N\\\ &0&t\geq
4N\\\ \end{aligned};\right.$ (13)

and

$\displaystyle\text{Im}[\mathcal{C}^{(2)}\left(t\right)]=$
$\displaystyle\left\\{\begin{aligned}
&\frac{t\log\left(\frac{16N^{2}}{t^{2}}-1\right)+8N\coth^{-1}\left(\frac{4N}{t}\right)}{4\pi
N^{2}}&t<4N\\\ &0&t=4N\\\
&\frac{t\log\left(1-\frac{16N^{2}}{t^{2}}\right)+8N\tanh^{-1}\left(\frac{4N}{t}\right)}{4\pi
N^{2}}&t>4N\\\ \end{aligned}.\right.$ (14)

Altogether, the sum of the real parts from the three terms leads to an
analytical formulation of the real part of the two-level GSFF, expressed as:

$\displaystyle\text{Re}[g^{(2)}\left(t\right)]$
$\displaystyle=\left\\{\begin{aligned}
&\frac{1}{N}+\frac{4J_{1}^{2}(t)}{t^{2}}+\frac{t-4N}{4N^{2}}&t<4N\\\
&\frac{1}{N}+\frac{4J_{1}^{2}(t)}{t^{2}}&t\geq 4N\\\ \end{aligned}.\right.$
(15)

It can be checked that the analytical expression of
$\text{Re}[g^{(2)}\left(t\right)]$ in the above equation is consistent with
the numerics of Eq.(2). Specifically, the numerics can be done as follows:

$\displaystyle\text{Re}[g^{(2)}\left(t\right)]$
$\displaystyle=\text{Re}\left[\int\frac{d\epsilon}{N^{2}}\left\langle\sum_{i,j=1}^{N}\delta\left(\epsilon-\sigma\left(\lambda_{i},\lambda_{j}\right)\right)\right\rangle_{\text{em}}{\rm
e}^{-i\epsilon t}\right]$
$\displaystyle=\text{Re}\left[\frac{1}{N^{2}}\left\langle\sum_{i,j=1}^{N}{\rm
e}^{-i\frac{\left|\lambda_{1}-\lambda_{2}\right|}{2}t}\right\rangle_{\text{em}}\right]$
$\displaystyle=\frac{1}{N^{2}}\left\langle\sum_{i,j=1}^{N}\cos\left[\frac{\left|\lambda_{1}-\lambda_{2}\right|}{2}t\right]\right\rangle_{\text{em}}.$
(16)

where $\left\langle.\right\rangle_{\text{em}}$ denotes the ensemble average.

Figure 1: Real part of the two-level generalized spectral form factor (GSFF)
in Gaussian unitary ensembles (GUEs). Similar to SFF, the real part of the
two-level GSFF shows a universal ”dip-ramp-plateau” structure, as it is
observed over a range of Hamiltonian dimensions $N=$10 (red dashed), 100 (blue
solid),1000 (green dash-dotted), 10000 (purple dotted). $t_{\rm d}$ and
$t_{\rm p}$ denote the time of the presence of dip and plateau, respectively.

###  III.2 Discussion of Real Part

In Fig 1, we present the real part of GSFF for the random matrix of varying
dimension $N$. It is apparent that, across different $N$, the real part of the
GSFF shows the same behaviors characterized by distinct features: a dip, an
oscillation, a ramp, and a plateau. This universal pattern is referred to as
the “dip-ramp-plateau” structure, a hallmark of quantum chaotic systems Haake
(1991); Saad  _et al._ (2018); Gharibyan  _et al._ (2018); Winer  _et al._
(2020b). GSFF exhibits three distinct durations, $t<t_{\rm d}$, $t_{\rm
d}<t<t_{\rm p}$ and $t>t_{\rm p}$, where $t_{\rm d}$ denotes the time of the
presence of dip and $t_{\rm p}$ the time of the beginning of plateau. The time
duration required for the real part of the GSFF to reach the plateau phase
depends on the dimension of the system characterized by the time parameter
$t_{\rm p}=4N$. These behaviors are consistent with the conventional SFF.

To further distinguish the dominating contribution for each duration, we offer
a comparative examination of the exact results and the contribution from an
individual contribution for a specific case where $N=10000$, as shown in Fig
2. The contribution of the first term denoted as $\mathcal{A}^{(2)}$ (blue
dashed), the second term $\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$ (red
dash-dotted), and the cumulative impact of the first and third terms
$\mathcal{A}^{(2)}+\text{Re}[\mathcal{C}^{(2)}\left(t\right)]$ (green solid)
can be resolved from the curve of $\text{Re}[g^{(2)}\left(t\right)]$ as
depicted in purple. In the first duration $t<t_{\rm d}$, GSFF shows
oscillation, and our results reveal that the oscillation is mainly attributed
to $\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$. In the second duration
$t_{\rm d}<t<t_{\rm p}$, GSFF ramps to a constant value. This behavior is
attributed to $\text{Re}[\mathcal{C}^{(2)}\left(t\right)]$. In Fig. 2, we
shift $\text{Re}[\mathcal{C}^{(2)}(t)]$ by amount of ${\cal A}^{(2)}$, i.e,

$\displaystyle\text{Re}[g^{(2)}\left(t\right)]\approx\frac{t}{4N^{2}},\quad
t_{\rm d}<t<t_{\rm p}.$ (17)

Notably, the slope of this ramp is $1/(4N^{2})$. As shown in Fig. 2, the
contribution from $\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$ is much smaller
than $\mathcal{A}^{(2)}+\text{Re}[\mathcal{C}^{(2)}\left(t\right)]$. So, in
this duration, we can ignore the impact of
$\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$, as expressed in Eq. (17). In the
third duration, GSFF saturates to a constant value determined by ${\cal
A}^{(2)}$. This term is exclusively associated with Wigner’s semicircle law,
and its plateau value reflects an inverse correlation with the dimension $N$.

Figure 2: Comparative analysis of the real part of the two-level GSFF. We take
the GUE with a dimension of $N=10000$ as an example. We show
$\mathcal{A}^{(2)}$ (blue double dash),
$\text{Re}[\mathcal{B}^{(2)}\left(t\right)]$ (red dash-dotted),
$\text{Re}[\mathcal{C}^{(2)}\left(t\right)]+1/N$ (green solid), representing
the real part of the results of the first, second, and the sum of the third
and first terms of the integral defined in Eq. (7). Additionally, we also show
$\text{Re}[g^{(2)}\left(t\right)]$ (purple dotted) to encompass the entire
real part, offering a comprehensive view of the contributions of individual
integrals. Figure 3: Imaginary part of two-level GSFF. The imaginary part
presents a universal behavior at the early time and long time across various
system dimensions $N=$10 (red diamond dots line), 100 (blue square dots), 1000
(green upper triangle dots), and 10000 (purple round dots). At the early time,
the imaginary part $\text{Im}[g^{(2)}\left(t\right)]$ is primarily contributed
by $\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$(black triangle dots). At the
long time, $\text{Im}[g^{(2)}\left(t\right)]$ is primarily contributed by
$\text{Im}[\mathcal{C}^{(2)}\left(t\right)]$(black right triangle dots line).
We take the system with $N=$100 as an example. The transition time $t_{0}$
(black diamond dots) for the imaginary part to change from negative to
positive is determined by the system size, denoted as $t_{0}=2N$ (black
vertical dashed). In addition, at large $t$,
$\text{Im}[g^{(2)}\left(t\right)]$ behaves as $t^{-1}$ (yellow dashed) and
$\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$ behaves as $-t^{-1}$ (magenta
dashed).

###  III.3 Discussion of Imaginary Part

In contrast to the previous definition of SFF, a prominent new feature of GSFF
is the presence of the imaginary part, which is shown in Fig. 3. Remarkably,
our results show the imaginary part presents a universal behavior at the early
time and long time across various system dimensions ($N$). Specifically, the
imaginary part initially exhibits an identical behavior, featuring the same
dip in the region where the imaginary part is negative. Over time, each case
manifests a peak in the region where the imaginary part is positive, then
shares a power low decay of form $\sim 0.192t^{-1}$. Another notable
observation is the transition time $t_{0}$ for the imaginary part to change
from negative to positive. Our results show that this time scale is determined
by the system size, denoted as $t_{0}=2N$.

The imaginary part of the GSFF is governed by different terms in the short
time and long time limit. We take the case with $N=100$ as an example to
illustrate which term dominates in Eq.(7). In the short time limit,
$\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$ is negative and
$\text{Im}[\mathcal{C}^{(2)}\left(t\right)]$ is positive. But the latter is of
two orders of magnitude smaller than the former. So it is evident that when
$t<t_{0}$, the imaginary part $\text{Im}[g^{(2)}\left(t\right)]$ is primarily
contributed by $\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$, which is similar
to the real part. As time progresses, the magnitude of
$\text{Im}[\mathcal{B}^{(2)}\left(t\right)]$ decrease, in a form of $\sim
t^{-1}$ , and $\text{Im}[\mathcal{C}^{(2)}\left(t\right)]$ becomes the
dominated one. Additionally, based on Eq.(8) and Eq.(9), when $t=0$,
$\text{Im}[g^{(2)}\left(t\right)]=0$.

Figure 4: Comparative analysis of analytical and numerical results of the
imaginary part for two-level GSFF. At the early time, the analytical results
(blue solid) are consistent with the numerical results (red circle). In the
later stage, the analytical results decay. The numerical results exhibit an
oscillating behavior around 0 and a symmetric distribution. In numerical
calculations, we used four ensembles with different system dimensions ($N$),
in which $N=$ 25, 50, 75, and 100 respectively, and each ensemble comprises
5000 Hamiltonian matrices. The gray vertical dash lines represent the
transition time $t_{0}$ of the imaginary part in different systems.

To verify our previous analysis, a numerical statistics approach is utilized
to derive the imaginary component of GSFF. The procedure is formulated as
follows:

$\displaystyle\text{Im}[g^{(2)}\left(t\right)]$
$\displaystyle=\text{Im}\left[\int\frac{d\epsilon}{N^{2}}\left\langle\sum_{i,j=1}^{N}\delta\left(\epsilon-\sigma\left(\lambda_{i},\lambda_{j}\right)\right)\right\rangle_{\text{em}}{\rm
e}^{-i\epsilon t}\right]$
$\displaystyle=\text{Im}\left[\frac{1}{N^{2}}\left\langle\sum_{i,j=1}^{N}{\rm
e}^{-\frac{i\left|\lambda_{1}-\lambda_{2}\right|}{2}t}\right\rangle_{\text{em}}\right]$
$\displaystyle=\frac{1}{N^{2}}\left\langle-\sum_{i,j=1}^{N}\sin\left[\frac{\left|\lambda_{1}-\lambda_{2}\right|}{2}t\right]\right\rangle_{\text{em}}.$
(18)

where $\langle.\rangle_{\rm em}$ denote ensemble average. Four ensembles with
different dimensions are generated, each consisting of 5000 random matrices.
The dimensions of the random matrices for each system are $N=$25, 50, 75, and
100, respectively. The results for the imaginary part of GSFF, obtained
through numerical statistics, are presented as the red round dots in Fig. 4,
while the analytical result is shown as the blue solid line. When $t$ is
small, $\text{Im}[g^{(2)}\left(t\right)]$ is negative and forms a dip,
consistent with the analytical result. When $t$ is large,
$\text{Im}[g^{(2)}\left(t\right)]$ exhibits violent oscillation near $0$. This
means that even small changes in energy level differences can lead to drastic
changes in function value. Besides, Fig. 4 reveals a symmetrical distribution
of the oscillation around 0, indicating that the imaginary part average is 0.
This is in line with the power law decay exhibited by the analytical result
over an extended duration. Additionally, comparing systems with varying
dimensions indicates that as $N$ increases, the region for the numerical
results to be consistent with the analytical results extends, as indicated by
the gray lines. Therefore, it can be inferred that the oscillation behavior of
numerical results over an extended duration is a finite-size effect. The
imaginary part of GSFF demonstrates a decay towards zero over a long time,
which is different from the plateau of the real part in a long time.

Overall, the novel insights into the properties of these behaviors,
particularly the behavior of the imaginary part, illuminate the complex and
fascinating characteristics of GSFF. These findings, exemplified by their
intriguing time dependencies and statistical correlations, enrich our
understanding of energy level statistics and the real and imaginary parts of
GSFF. This analysis can be extended to higher-level correlation.

##  IV Numerical Statistics of Three-level Generalized Spectral Form Factor

Figure 5: Numerical results of the real part of three-level GSFF. In numerical
calculations, we used four ensembles with different system dimensions ($N$),
in which $N=$ 10 (red double dash), 15 (blue solid), 25 (green dash-dotted),
and 50 (purple dotted) respectively, and each ensemble comprises 5000
Hamiltonian matrices. The real part of three-level GSFF
$\text{Re}[g^{(3)}\left(t\right)]$ of systems of different dimensions show
similar behavior. In the long time limit, the real part shows a plateau. The
value of the plateau is related to the dimension $N$, expressed as $\text{
Re}[g^{(3)}\left(t_{\rm large}\right)]=1/N^{2}$ (gray horizontal dashed
lines).

Let us direct our focus towards the three-level GSFF, also specifically in the
context of GUE governed by a set of random matrix $\left\\{M_{N\times
N}\right\\}$. In this case, the analytical expression is hardly available. In
our following discussion, we use numerical results. Similar to the previous
choice, there are four ensembles, and each ensemble contains 5000 random
matrices. The dimensions of the matrices corresponding to each ensemble are
$N=$10,15,25,50 respectively. The representation of the three-level GSFF by
numerical method can be expressed as follows:

$\displaystyle g^{(3)}\left(t\right)$ $\displaystyle=\int
c^{(3)}\left(\epsilon\right){\rm e}^{-i\epsilon t}d\epsilon$
$\displaystyle=\int\frac{d\epsilon}{N^{3}}\left\langle\sum_{i,j,k=1}^{N}\delta\left[\epsilon-\sigma\left(\lambda_{i},\lambda_{j},\lambda_{k}\right)\right]\right\rangle_{\text{em}}{\rm
e}^{-i\epsilon t}$
$\displaystyle=\left\langle\int\frac{d\epsilon}{N^{3}}\sum_{i,j,k=1}^{N}\delta\left[\epsilon-\sigma\left(\lambda_{i},\lambda_{j},\lambda_{k}\right)\right]{\rm
e}^{-i\epsilon t}\right\rangle_{\text{em}}$
$\displaystyle=\frac{1}{N^{3}}\left\langle\sum_{i,j,k=1}^{N}{\rm
e}^{-i\sigma\left(\lambda_{i},\lambda_{j},\lambda_{k}\right)t}\right\rangle_{\text{em}},$
(19)

where $\sigma\left(\lambda_{i},\lambda_{j},\lambda_{k}\right)$ represents the
standard deviation of three energy levels. Upon numerical calculations, the
real parts $\text{Re}[g^{(3)}\left(t\right)]$ and imaginary parts
$\text{Im}[g^{(3)}\left(t\right)]$ of the GSFF can be extracted. The results
of the real part of GSFF, $\text{Re}[g^{(3)}\left(t\right)]$, are shown in
Fig. 5, which can be divided into three time durations. When $t$ is small,
$\text{Re}[g^{(3)}\left(t\right)]$ is positive, decreasing from unity, and the
behavior of the real part is completely consistent across systems of varying
sizes($N$). In the second region, a negative dip appears in
$\text{Re}[g^{(3)}\left(t\right)]$. In this region, as time increases, for
different $N$, the real parts show differences after experiencing a small
oscillation, and the behavior is no longer completely consistent. Moreover, it
can be found that the larger $N$ is, the later the real part goes from the
negative to the positive. As $t$ further increases, the real parts enter the
third region and become positive again. The behavior of the real part shows a
plateau, and the value of the plateau is related to the size of the system.
This is qualitatively similar to the two-level GSFF. Specifically,
$\text{Re}[g^{(3)}\left(t_{\rm large}\right)]=1/N^{2}$.

Figure 6: Numerical results of the imaginary part of three-level GSFF. In
numerical calculations, we used four ensembles with different system
dimensions ($N$), in which $N=$ 10 (red square dots), 15 (blue round dots), 25
(green upper triangle dots), and 50 (purple diamond dots) respectively, and
each ensemble comprises 5000 Hamiltonian matrices. The imaginary part of
three-level GSFF $\text{Im}[g^{(3)}\left(t\right)]$ of systems of different
dimensions show similar behavior. The behavior of
$\text{Im}[g^{(3)}\left(t\right)]$ has a common dip and an oscillatory decay
then oscillates around 0, which can be considered a finite-size effect.

In Fig. 6, we present the results of the imaginary part
$\text{Im}[g^{(3)}\left(t\right)]$, which behaves similarly to the counterpart
of the two-level GSFF. It is notable that, based on Eq.(19),
$\text{Im}[g^{(3)}\left(0\right)]=0$. Therefore, when $t$ is very small,
$\text{Im}[g^{(3)}\left(t\right)]$ starts to decrease from 0, and for systems
of different sizes, the behaviors of the imaginary part are the same. A dip
appears in the negative region. As $t$ increases, the behavior of
$\text{Im}[g^{(3)}\left(t\right)]$ indicates a oscillatory decay then
oscillates around 0. In addition, we observed that the oscillation amplitude
of the long-time behavior of $\text{Im}[g^{(3)}\left(t\right)]$ attenuates as
$N$ increases. This implies that the imaginary part tends to 0 in the thermal
dynamical limit. Therefore, the oscillatory behavior can also be considered a
finite-size effect.

##  V Summary

In summary, we extend the concept of SFF to incorporate the high-order level
correlation in the spectrum. The GSFF is obtained by Fourier transforming the
energy level correlation function that is defined by the standard deviation of
energy levels. In contrast to conventional approaches that only examine the
differences between two energy levels, our method enables a comprehensive
exploration of correlations across arbitrary energy levels.

In contrast to SFF, GSFF is complex. In the context of GUE, when considering
the correlation of two energy levels, the real part of the GSFF aligns with
the result of SFF. It shows the dip-ramp-plateau structure, which is
consistent with the SFF. Moreover, we extend our analysis to the imaginary
part of the GSFF, revealing structural features characterized by dip in the
negative region and universal decay in the positive region. After comparative
analysis with the results of numerical calculations, we find that the
imaginary part of GSFF decays to zero in a long time limit. Additionally, for
the GSFF in scenarios involving three energy level correlations by numerics,
the real part has a dip and a plateau structure, and the imaginary part has a
dip, an oscillatory decay, and a long-time oscillation feature. It is worth
noting that the characteristic times associated with spectral structures of
two-level and three-level correlations are intimately linked to the system
size ($N$). In short, the GSFF offers a robust computational approach for
assessing energy level correlations at higher orders, enabling a deeper
understanding of the energy level distribution of the system.

Acknowledgement. We want to thank Tian-Gang Zhou for the helpful discussion.
This work is supported by NSFC 12174300, Innovation Program for Quantum
Science and Technology (Grants No.2021ZD0302001), the Fundamental Research
Funds for the Central Universities (Grant No. 71211819000001) and Tang
Scholar.

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