# Emergent Spectral Form Factors in Sonic Systems

Michael Winer  Joint Quantum Institute, Department of Physics, University of
Maryland, College Park, Maryland 20742, USA  Brian Swingle  Department of
Physics, Brandeis University, Waltham, Massachusetts 02453, USA

###### Abstract

We study the spectral form factor (SFF) for hydrodynamic systems with a sound
pole, a large class including any fluid with momentum conservation and energy
conservation, or any extended system with spontaneously broken continuous
symmetry. We study such systems in a finite volume cavity and find that the
logarithm of the hydrodynamic enhancement to the SFF is closely related to the
spectral form factor of a quantum particle moving in the selfsame cavity.
Depending upon the dimensionality and nature of the effective single-particle
physics, these systems exhibit a range of behaviors including an intricate
resonance phenomenon, emergent integrability in the SFF, and anomalously large
fluctuations of the SFF.

##  1 Introduction

The statistics of the energy spectrum is one of the most important diagnostics
of quantum chaos [1, 2, 3]. There is strong evidence that ensembles of chaotic
systems have the same Hamiltonian spectral statistics as ensembles of Gaussian
random matrices, with examples including nuclear systems [4, 5], mechanical
systems [6, 7, 8], condensed matter systems [9, 10, 11, 12, 13, 14, 15, 16,
17, 18], holographic models [19, 20, 21, 22, 23, 24] and (generalizing to
time-dependent Hamiltonians) circuit models [25, 26, 27, 28, 29]. A central
quantity in spectral statistics in the Spectral Form Factor (SFF). Among other
interesting properties [30, 31], this quantity diagnoses whether energy levels
repel as they do in random matrices [9, 32, 33], have independent Poissonian
statistics [6], or have some more exotic behavior [34, 35, 36, 37]. The SFF
has spawned relatives including the partial SFF [38], the entanglement SFF
[39, 40, 41], and the Loschmidt SFF [42, 43, 44].

The SFF can be written as

$\text{SFF}(T,f)=\overline{|\text{Tr}[U(T)f(H)]|^{2}},$ (1)

where $U(T)=e^{-iHT}$ is the time evolution operator, $f$ is a filter function
(perhaps a Gaussian centered on a specific energy band), and overline denotes
a disorder average over an ensemble of Hamiltonians. These Hamiltonians can
differ either through microscopic disorder or, as we will see, through the
overall shape of the region containing the system. For a discussion of how
different the Hamiltonians must be to constitute “sufficient coverage”, see
[42, 43, 44, 45, 46]. This disorder average is necessary because while the
disorder-averaged SFF has smooth behavior as seen below, the SFF of a single
Hamiltonian is a highly erratic function of time. A central problem in the
field of quantum chaos is thus to calculate the SFF for a variety of physical
systems in order to understand the general conditions under which random
matrix behavior emerges.

In [47, 48], the authors set forth a hydrodynamic theory of the spectral form
factor. This effective theory predicts pure random matrix behavior at late
time and computes corrections at earlier times due to slow modes. Intuitively,
the presence of slow modes is related to atypically small matrix elements in
the Hamiltonian, and atypically small matrix elements suppress the level
repulsion which is characteristic of a random matrix. These corrections in the
SFF persist until the system’s “Thouless time”, after which the random matrix
behavior is recovered. In particular, approximate symmetries enhance the ramp
by an amount exponential in system size, consistent with [26, 49, 50, 51] .
This enhancement factor was calculated for simple models of a system with
conserved modes and with spontaneous symmetry breaking. These calculations
included tree-level effects and diagrammatic corrections not seen in
traditional hydrodynamics. [47] dealt exclusively with theories first order in
time, and [48] dealt with systems with no spatial extent. However, the problem
of hydrodynamic systems with sound poles was left unstudied. In this work, we
fill in this gap, pointing out that sound pole effects can lead to new
phenomena such as exponentially increasing SFFs and spectral form factors with
sensitive qualitative dependence on the precise shape of the system.

###  1.1 Summary of Results and Sketch of Paper

The theory in [47, 48] expressed the enhancement of the SFF in linear
hydrodynamics as a product over all modes. Here hydrodynamics (hydro) refers
to a effective theory of a system’s long-wavelength and long-time dynamics
which incorporates the effects of conservation laws and other general
constraints while coarse-graining over physics at more microscopic scales.
[47] considered only cases where the hydro equations were first-order in time
while [48] only studied systems with no spatial extent. As such, this is the
first work to use the methods of hydrodynamics to study SFFs in systems with
sound poles. This framework is meant to apply to systems for which the
underlying microscopic theory is quantum chaotic and the long-wavelength
physics is described by oscillatory sound modes. We treat the sound modes at
the quadratic level within hydrodynamics (except for an appendix discussing
some effects of hydrodynamic interactions), but this quadratic approximation
still allows for the the modes to decay and is consistent with an underlying
microscopic chaotic dynamics.

For the SFF enhancement, our hydrodynamic theory predicts that an oscillatory
mode with angular frequency $\omega$ decaying at rate $\lambda$ contributes to
the enhancement like $\left|\frac{1}{1-e^{i\omega T-\lambda T}}\right|^{2}$.
This expression, which is obtained in Section 2, rapidly oscillates with
angular frequency $\omega$, and the product of many such terms can generate
much richer time-dependence than is possible with purely decaying modes (such
as diffusive modes). The work of Sections 3, 4, and 5 is dedicated entirely to
the study of these products. For this purpose, the spectrum of allowed
$\omega$s is very important, and we find a variety of different behaviors
depending on the statistical properties of the $\omega$s. This spectrum is
determined by Laplacian of the region, or “cavity”, in which the system
resides. In particular, it is crucial to understand whether the spectrum of
sound modes admits constructive or destructive interference in the product
over $\omega$s.

In the case of 1d linear hydrodynamics, we find that patterns of interference
impart an intricate fractal structure on the graph of the spectral form
factor, as seen in figure 4. There is an exponentially tall peak at every
rational time in units of the system length. Unfortunately, this result is
delicate; at the very least, hydrodynamic interactions destroy the sound pole
in 1d [52], with the dynamics flowing to to the KPZ universality class [53,
54, 55]. However, we still include this discussion as a particularly simple
example of the dramatic effects of oscillations and because the sound pole can
survive in some special situations, such as when the local Hilbert space
dimension is large.

More generally, for cavities in higher dimensions, one sees an enhancement
factor that is qualitatively exponential in the single-particle SFF for a
billiard of the same shape as the cavity. This surprising statement means that
even though the fundamental system is made out of many microscopic degrees of
freedom undergoing aperiodic motion, the full system SFF depends very strongly
on the motion of wavepackets of sound. We distinguish between “integrable”
cavities, in which the spectrum of the Laplacian is Poissonian, and “chaotic”
cavities in which the spectrum of the Laplacian is random matrix like
(although we stress that this integrable or chaotic adjective refers to the
structure of the cavity modes, not to the many-body levels, which are always
ultimately random-matrix-like in our models).

For example, local disordered systems in integrable cavities such as tori or
ellipsoids see an exponential (in volume) enhancement to the many-body SFF
corresponding to the Poissonian statistics of sound wavepackets, whereas
disordered systems in a chaotic cavity, such a two-dimensional region in the
shape a Bunimovich stadium billiard, see an exponentially growing enhancement
to their SFF corresponding to the ramp in the SFF of a sonic wavepacket.

The rest of the paper is organized as follows. In the remainder of the
introduction, we briefly review the SFF. In Section 2, we review the
hydrodynamic theory of [47] and discuss its application to systems with a
sound pole. In Section 3, we treat a one-dimensional system where the SFF
exhibits an intricate fractal structure. In Section 4, we move to higher
dimensions and discuss the case of a sound pole when the system is confined to
an integrable cavity. In Section 5, we discuss the analogous problem in a
chaotic cavity. In Section 6, we discuss interaction effects, with the
preceeding discussion all at the quadratic level. This is followed by a brief
discussion of the results; appendix A also contains a comparison to our prior
work on SYK2.

###  1.2 Review of the form factor

Before proceeding to the derivations of these results, we first review the SFF
in more detail. The starting point for any understanding of the spectral form
factor is level repulsion. If we take the well-known joint probability density
function for the $N$ eigenvalues of an $N$ by $N$ random Hermitian (GUE)
matrix, we have

$\begin{split}P(\lambda_{1},\lambda_{2},\lambda_{3}...\lambda_{N})=\frac{1}{Z_{N}}\exp(\frac{N}{2}\sum_{i}\lambda_{i}^{2})\prod_{ij}(\lambda_{i}-\lambda_{j})^{2}=\\\
\frac{1}{Z_{N}}\exp\left(\int_{\lambda_{1}>\lambda_{2}}2\log(\lambda_{1}-\lambda_{2})\rho(\lambda_{1})\rho(\lambda_{2})d\lambda_{1}d\lambda_{2}-\int\frac{N}{2}\lambda^{2}\rho(\lambda)d\lambda\right).\end{split}$
(2)

It is the last term in the product on the first line that is responsible for
level repulsion. Intuitively, if we zoom in on a near-degeneracy involving two
levels, then the $2$ by $2$ submatrix involving those two levels only has an
exact degeneracy if the coefficients of each basis matrix (Pauli $\sigma_{x}$,
$\sigma_{y}$, and $\sigma_{z}$) are all zero, so near-degeneracy requires a
special degree of fine-tuning.

The SFF is closely related to the correlation function of the density of
states. Formally, the (filtered) density of states is given by

$\rho(E,f)\equiv\sum_{n}f(E_{n})\delta(E-E_{n})=\textrm{Tr}f(H)\delta(E-H),$
(3)

where $n$ labels the eigenstate of $H$ with eigenvalue $E_{n}$, and its
correlation function is

$C(E,\omega,f)\equiv\mathbb{E}\left[\rho\left(E+\frac{\omega}{2},f\right)\rho\left(E-\frac{\omega}{2},f\right)\right].$
(4)

We have that

$\displaystyle\textrm{SFF}(T,f)$
$\displaystyle=\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{-iHT}\textrm{Tr}f(H)e^{iHT}\Big{]}$
(5) $\displaystyle=\int dEd\omega\,e^{-i\omega
T}\mathbb{E}\left[\textrm{Tr}f(H)\delta\left(E+\frac{\omega}{2}-H\right)\textrm{Tr}f(H)\delta\left(E-\frac{\omega}{2}-H\right)\right]$
$\displaystyle=\int d\omega\,e^{-i\omega T}\int dE\,C(E,\omega,f).$

The SFF is simply the Fourier transform of the correlation function with
respect to $\omega$.

The SFF can be split into two contributions:

$\textrm{SFF}(T,f)=\big{|}\mathbb{E}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}+\bigg{(}\mathbb{E}\left[\big{|}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}\right]-\big{|}\mathbb{E}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}\bigg{)}.$
(6)

The first term, the disconnected part of the SFF, comes solely from the
average density of states. It is the absolute value square of the density’s
Fourier Transform.

It is the second term, the connected part of the SFF, that contains
interesting information on the correlation between energy densities. “Random
matrix universality” [2, 56] is the principle that an ensemble of quantum
chaotic Hamiltonians will generically have the same connected SFF as the
canonical Gaussian ensembles of random matrix theory [3, 57]. This behavior is
illustrated in Fig. 1, which plots the disorder-averaged SFF of the Gaussian
unitary ensemble (one of the aforementioned canonical ensembles). The graph
shows the three regimes of the random matrix theory SFF:

Figure 1: The disorder-averaged SFF for the Gaussian unitary ensemble (GUE).
The matrices in this ensemble have dimension $N=50$. The SFF was computed
numerically by averaging over ten thousand realizations. The three regimes —
dip, ramp, plateau — are labeled.

  * •

The “dip”, occurring at early times, comes from the disconnected piece of the
SFF (and thus its precise shape is non-universal and depends on the details of
$f$ and the thermodynamics of the system). Its downward nature reflects a loss
of constructive interference — the different terms of $\textrm{Tr}e^{-iHT}$
acquire different phase factors as $T$ increases.

  * •

The “ramp”, occurring at intermediate times, is arguably the most interesting
regime. In canonical random matrix ensembles, the ramp follows from the result
[3]

$\mathbb{E}\left[\rho\left(E+\frac{\omega}{2}\right)\rho\left(E-\frac{\omega}{2}\right)\right]-\mathbb{E}\left[\rho\left(E+\frac{\omega}{2}\right)\right]\mathbb{E}\left[\rho\left(E-\frac{\omega}{2}\right)\right]\sim-\frac{1}{\mathfrak{b}\pi^{2}\omega^{2}},$
(7)

where $\mathfrak{b}=1$, $2$, $4$ for the orthogonal, unitary, and symplectic
ensembles respectively [3]. The fact that the right hand side is negative is
known as level repulsion in quantum chaotic systems [5]. Since the energy
levels of a system repel each other, fluctuations in the density of states are
suppressed. Long-wavelength (short time) fluctuations are suppressed the most,
short-wavelength (long time) fluctuations are suppressed the least. Taking the
Fourier transform of equation 7 respect to $\omega$ gives a term proportional
to $T$ for the connected SFF. Such a linear-in-$T$ ramp is often taken as a
defining signature of quantum chaos.

  * •

The “plateau”, occurring at late times, is fundamentally a consequence of the
discreteness of the spectrum. At times much larger than the inverse level
spacing or “Heisenberg time”, all off-diagonal terms in the double-trace of
the SFF average to zero, meaning that

$\textrm{SFF}(T,f)\approx\sum_{mn}e^{-i(E_{m}-E_{n})T}f(E_{m})f(E_{n})\sim\sum_{n}f(E_{n})^{2}.$
(8)

For integrable systems, the plateau is reached very quickly with little to no
ramp regime [6, 35, 36]. For chaotic systems it isn’t reached until a time
exponential in system size. Nonetheless, it is the long-term fate of any
system without a degenerate spectrum.

This paper will focus on the connected SFF at time scales towards the
beginning of the ramp, scaling polynomially in system size instead of
exponentially. At this time scale, physical systems do not behave exactly like
random matrices, and exciting new phenomena can be observed. The timescale
separating these non-universal phenomena from the linear ramp of RMT is known
as the Thouless time, see e.g. [58, 26] for recent discussions in the many-
body context.

Before we specialize to the case of hydrodynamics, a word on filter functions
$f$. One common choice of $f$ is $f(E)=e^{-\beta E}$. It is a matter of
Fourier analysis to show that this would give a ramp looking like
$\int_{E_{\min}}^{E_{\max}}dEe^{-2\beta E}\frac{T}{2\pi}$. This integral would
be dominated by the lowest-energy region of the spectrum. Importantly, the
region of the spectrum which in thermodynamics corresponds to inverse
temperature $\beta$ is often very heavily suppressed. The physical reason why
the coefficient of the ramp doesn’t depend on the density of states (and hence
why the densest region of the spectrum cannot dominate) can be seen from the
equation (2). The distribution of the eigenvalues can be interpreted as a
Boltzmann distribution where the eigenvalues are particles in some confining
potential $NV(x)$ ($V=\frac{x^{2}}{2}$ for a Gaussian random matrix) that
repel each other with potential $2\log|\lambda_{i}-\lambda_{j}|$. If we
approximate $\rho$ as a continuous density instead of a train of $\delta$
functions, then there is some saddle point density which can be solved from an
integro-differential equation involving $V$. The connected SFF, however can be
seen as the fluctations in $\rho$. At short times and large energies, where
this approximation is valid, we see that the energy function we are trying to
minimize is purely quadratic in the $\rho$s, so fluctuations shouldn’t depend
on the background $\rho$. A longer discussion on this point can be found in
[1].

Because canonical SFFs select only dynamics near the ground state, we need a
different approach. If we are interested in spectral statistics far from the
ground state, we must instead choose an $f$ like
$f(E)=e^{-(E-E_{0})^{2}/4\sigma^{2}}$, which samples around the energy window
of interest. This well-known but unusual property puts spectral statistics in
contrast with thermodynamics, where canonical and microcanonical ensembles are
equivalent.

In this paper, we discuss the connected SFF. We restrict ourselves to
timescales long enough for hydrodynamic effects to kick in. We pay attention
only to times much less than the many-body Heisenberg time, after which
generic systems have no interesting spectral properties. Our results are non-
trivial only at times less than the decay-time for the slowest sound mode,
which is quadratic in system length. For a more careful discussion of
different timescales in this work, see the end of section 5.

##  2 Overview of the Hydrodynamic Spectral Form Factor

Our approach to the problem is to formulate an effective theory of the SFF
contour. We do this by comparing two different but related sets of contours
and arguing for a relationship between their effective descriptions. We will
suppress the time $T$ and filter function $f$ in many of the expressions
below.

Case 1: The Schwinger-Keldysh (S-K or Kel) contour (Figure 2, left), which
computes

$Z_{\text{Kel}}=\text{tr}(U[A_{1}]\rho U[A_{2}]^{\dagger}),$ (9)

where $\rho$ is the initial state and the $A_{i}$ are backgrounds fields along
the forward ($A_{1}$) and backward ($A_{2}$) legs of the contour. In this
case, we have a single trace and an initial state which sets the background
value of the energy and other conserved charges.

Figure 2: The Schwinger-Keldysh (S-K) contour (left) has a thermal circle of
inverse temperature $\beta$ and forward and backwards time evolution legs. By
integrating out local microscopic degrees of freedom (bottom), one arrives at
an effective theory at the same temperature. The integration couples the left
and right legs of the contour. The SFF contour (right) consists of two totally
disconnected legs each with periodic boundary condition. By integrating out
local microscopic degrees of freedom (bottom), one arrives at the same
effective theory as in the Schwinger-Keldysh case (at least up to loop
corrections). Remarkably, this integration results in a coupling between the
two otherwise decoupled legs. This figure first appeared in [47]

Case 2: The form factor contour (Figure 2, right), which computes

$\text{SFF}=\tr(f(H)U[A_{1}])\tr(f(H)U[A_{2}]^{\dagger}),$ (10)

where again we allow for different background fields on the forward and
backward contours. In this case, we have two traces and no initial state,
although we can consider the addition of a filter function $f$ which can be
used to select part of the energy spectrum of interest.

It is important to note that the SFF will be an erratic function of time; we
must include some implicit averaging, for example, over time or over an
ensemble of Hamiltonians, to give a smooth function of time. We can also
compute the same ensemble average of the Schwinger-Keldysh contour, but we
will here assume that the SK contour computes self-averaging observables that
are only weakly affected by the ensemble average. Henceforth, we compare
ensemble averaged versions of Case 1 and Case 2.

The Schwinger-Keldysh contour (Case 1) computes a generating function that
gives access, via derivatives with respect to the background fields, to
correlators and response functions. It is typically used as a tool to compute
dynamical properties out of equilibrium. For our purposes here, we have in
mind the Schwinger-Keldysh contour as a tool to compute the long wavelength
hydrodynamic response of the system. As we review below, one can formulate an
effective theory using the degrees of freedom on the SK contour which can be
used to describe the hydrodynamic response of the system.

The SFF contour (Case 2) computes a generalization of the spectral form
factor, with the usual form factor being recovered when the background fields
are set to zero on each contour. At late times, our expectation is that the
SFF will approach that of an appropriate random matrix, but at early times,
especially in a system with slow hydrodynamic modes, one can expect systematic
deviations from the random matrix result. We are interested in formulating an
effective theory for the SFF contour, analogous to the hydrodynamic effective
theory of the SK contour.

In a prior work, we gave a proposal for an effective theory of the SFF
contour. We now review that proposal and the arguments in favor of it. The
basic observation is that the Schwinger-Keldysh and SFF contours can both be
viewed as different ways summing matrix elements of the same unitary operator,
$U[A_{1}]\otimes U[A_{2}]^{*}$, defined on two copies of the system. For the
Schwinger-Keldysh contour (Case 1), we connect the two copies to each other
via a link at the initial time which encodes the initial state and a link at
the final time which encodes the single trace. For the SFF contour (Case 2),
we connect each copy to itself via a link from the initial time to the final
time which encodes the two traces.

Based on this clue, we proposed that an effective theory of the Schwinger-
Keldysh contour should also give an effective theory of the SFF contour
provided the boundary conditions are modified appropriately. As a reminder,
this proposal should be viewed as a statement about the ensemble averaged
Schwinger-Keldysh contour and the ensemble averaged SFF contour. For the
Schwinger-Keldysh contour, the boundary conditions set the value of the total
energy (initial state) and introduce a fine-grained correlation between the
final states of the two legs (final trace). For the SFF contour, there is no
built in correlation between the fine-grained states of the two legs, but the
ensemble average produces such a correlation. The SFF contour also does not
have a set value of the total energy (or other conserved quantities), so we
must integrate over them.

The final proposal is then as follows. Suppose, for a given total energy
$E_{0}$, we have some effective fields $\phi_{1,2}$ which compute
$Z_{\text{Kel}}$ as

$Z_{\text{Kel}}[A]=\int\mathcal{D}\phi e^{i\int
dtd^{d}xL_{\text{Kel}}[\phi,A;E]},$ (11)

where the effective action $W_{SK}$ depends on the background energy $E_{0}$.
Then, we can write a similar effective theory to compute the SFF,

$\text{SFF}=\int dE_{0}\int\mathcal{D}^{\prime}\phi e^{i\int
dtd^{d}xL_{\text{SFF}}[\phi,A;E_{0}]},$ (12)

where

  * •

we now integrate over the total energy $E_{0}$ (and other conserved charges),

  * •

we modify the boundary conditions of $\phi$ to be periodic in time on each
contour,

  * •

and we have $L_{\text{SFF}}\approx L_{\text{Kel}}$.

The argument for this final point is that the fast decaying degrees of freedom
which have been integrated out to yield $L_{\text{Kel}}$ are not significantly
affected by the change in boundary conditions provided the time duration $T$
is much larger than the lifetime $\tau_{\text{fast}}$ of any integrated out
mode. Performing the same integrating out procedure on the SFF contour thus
gives

$L_{\text{SFF}}=L_{\text{Kel}}+\mathcal{O}(e^{-T/\tau_{\text{fast}}}).$ (13)

In subsection 2.1 we quickly review the closed time path (CTP) formalism which
deals with the Schwinger-Keldysh contour. In subsection 2.2 we discuss the
doubled periodic time (DPT) effective theory, a cousin of CTP which deals with
the SFF contour. Finally in subsection 2.3 we discuss a special modification
of the CTP and DPT for systems with oscillatory modes.

###  2.1 Review of Closed Time Path Formalism

Hydrodynamics is the program of creating effective field theories (EFTs) for
systems based on the principle that long-time and long-range physics is driven
primarily by conservation laws and other protected slow modes. One
particularly useful formulation is the CTP formalism explained concisely in
[59] and in more detail in [60, 61, 62]. Other approaches to fluctuating
hydrodyamics can be found in [63, 64, 65, 66].

The CTP formalism lives on the Schwinger-Keldysh contour, pictured in figure
2. Its central object of study is the partition function

$Z_{\text{Kel}}[A^{\mu}_{1}(t,x),A^{\mu}_{2}(t,x)]=\tr\left(e^{-\beta
H}\mathcal{P}e^{i\int dtd^{d}xA^{\mu}_{1}j_{1\mu}}\mathcal{P}e^{-i\int
dtd^{d}xA^{\mu}_{2}j_{2\mu}}\right),$ (14)

where $\mathcal{P}$ is a path ordering on the Schwinger-Keldish contour. The
$j$ operators are local conserved currents.

For $A_{1}=A_{2}=0$, $Z_{\text{Kel}}$ reduces to thermal partition function at
inverse temperature $\beta$. Differentiating $Z_{\text{Kel}}$ with respect to
the $A$s generates insertions of the conserved current density $j_{\mu}$ along
either leg of the Schwinger-Keldysh contour. Thus $Z_{\text{Kel}}$ is the
generating function of all possible contour-ordered correlation functions of
current operators. In particular, for systems with a conserved energy, the
energy density operator can always be extracted from the hydrodynamic action.

One can write $Z_{\text{Kel}}$ as

$Z_{\text{Kel}}[A^{\mu}_{1},A^{\mu}_{2}]=\int\mathcal{D}\phi^{i}_{1}\mathcal{D}\phi^{i}_{2}\exp\left(i\int
dtd^{d}xL_{\text{Kel}}[A_{1\mu},A_{2\mu},\phi^{i}_{1},\phi^{i}_{2}]\right),$
(15)

for some collection of local fields $\phi$s. The fundamental insight of
hydrodynamics is that at long times and distances, any massive $\phi$s can be
integrated out. All that’s left over is one $\phi$ per contour to enforce the
conservation law $\partial^{\mu}j_{i\mu}=0$. Our partition function can be
written

$\begin{split}Z_{\text{Kel}}[A^{\mu}_{1},A^{\mu}_{2}]=\int\mathcal{D}\phi_{1}\mathcal{D}\phi_{2}\exp\left(i\int
dtd^{d}xL_{\text{Kel}}[B_{1\mu},B_{2\mu}]\right),\\\
B_{i\mu}(t,x)=\partial_{\mu}\phi_{i}(t,x)+A_{i\mu}(t,x).\end{split}$ (16)

Insertions of the currents are obtained by differentiating $Z_{\text{Kel}}$
with respect to the background gauge fields $A_{i\mu}$. A single such
functional derivative gives a single insertion of the current, and so one
presentation of current conservation is the identity
$\partial_{\mu}\frac{\delta Z_{\text{Kel}}}{\delta A_{i\mu}}=0$.

The effective Lagrangian $L_{\text{Kel}}$ satisfies a number of constraints.
The most important is locality. There are no slow modes besides $\phi$, and at
long enough distance and time scales integrating out fast modes should yield a
local Lagrangian depending on $B_{1,2}$ and their derivatives. There are
several additional constraints following from unitarity. They are best
expressed in terms of

$\begin{split}B_{a}=B_{1}-B_{2},\\\ B_{r}=\frac{B_{1}+B_{2}}{2}.\end{split}$
(17)

The key constraints, which will not be proven here but are proven in, say,
[60], are:

  * •

All terms in $L_{\text{Kel}}$ have at least one factor of $B_{a}$, that is
$L_{\text{Kel}}=0$ when $B_{a}=0$.

  * •

Terms odd (even) in $B_{a}$ make a real (imaginary) contribution to the
action.

  * •

All imaginary contributions to the action are positive imaginary (or zero).

  * •

Any correlator in which the chronologically last variable has $a$-type will
evaluate to 0 (known as the last time theorem or LTT).

  * •

A KMS constraint imposing fluctuation-dissipation relations.

  * •

Unless the symmetry is spontaneously broken, all factors of $B_{r}$ have at
least one time derivative. This condition will be lifted in this paper, as we
will be considering systems with a spontaneously broken symmetry.

For many applications, including calculating SFFs, one typically sets the
external sources $A$ to zero, so the action can be written purely in terms of
the derivatives of the $\phi$s.

The $\phi$s often have a physical interpretation depending on the precise
symmetry in question. In the case of time translation, the $\phi$s are the
physical time corresponding to a given fluid time (and are often denoted
$\sigma$). In the case of a U(1) internal symmetry, they are local fluid
phases. One simple quadratic action which is consistent with the above rules
and which describes an energy-conserving system exhibiting diffusive energy
transport is (with $\phi_{a,r}\rightarrow\sigma_{a,r}$)

$L_{\text{Kel}}^{\text{(diffusion)}}=\sigma_{a}\left(\kappa\beta^{-1}\partial_{t}^{2}\sigma_{r}-D\kappa\beta^{-1}\nabla^{2}\partial_{t}\sigma_{r}\right)+i\beta^{-2}\kappa(\nabla\sigma_{a})^{2}.$
(18)

Here $\kappa$ and $D$ can all be viewed as functions of the background energy
$E_{0}$ which is set by the temperature $\beta$. Later, it will be convenient
to also view the $\beta$ appearing in (18) as a function of the background
energy $E_{0}$.

###  2.2 Review of the Doubled Periodic Time Formalism

We now explain in more detail how our DPT theory [47], which is built from the
CTP formalism, can be used to compute SFFs. We focus for concreteness on the
diffusive action (18) as an example. The hydro approach to the SFF predicts
that the SFF can be obtained by evaluating the following path integral (still
with $\phi_{a,r}\rightarrow\sigma_{a,r}$),

$\textrm{SFF}(T,f)=\int_{\mathcal{C}}\mathcal{D}\sigma_{1}\mathcal{D}\sigma_{2}f(E_{1})f(E_{2})e^{i\int
dtd^{d}xL_{\text{Kel}}(\sigma_{1},\sigma_{2})}.$ (19)

Here $\mathcal{C}$ represents the SFF contour seen on the right of figure 2.
This contour has two disconnected legs each with real-time periodicity $T$.
$\sigma_{1,2}$ represent time reparameterization modes on the two legs of the
contour.

The contour in equation (19) should be contrasted with the Schwinger-Keldysh
contour. These contours both have two long legs, but very different boundary
conditions. The fact that an action initially written to calculate two-point
functions in fluctuating hydrodynamics on the SK contour can—when evaluated
with different boundary conditions—calculate the spectral form factor is the
surprising result of [47].

The key change is in the boundary conditions of the fields. To see this, we
first define

$\begin{split}\rho=\frac{\partial
L_{\text{Kel}}}{\partial(\partial_{t}\sigma_{a})}=-\kappa\beta^{-1}\partial_{t}\sigma_{r},\end{split}$
(20)

which can be thought of as the average energy density on the two contours.
Consider the spatial- and time-zero-modes of $\rho$ and $\sigma_{a}$,
$\rho_{0}$ and $\sigma_{a,0}$. $\rho_{0}$ is nothing but the total energy
$E_{0}$ while $\sigma_{a,0}$ is the total relative time shift between the two
contours. In the SK contour, $\rho_{0}$ is set by the initial state, which
gives a strongly peaked probability distribution for $\rho_{0}$. Similarly, in
the SK contour, $\sigma_{a,0}$ is fixed to be zero since the times on the two
contours are fixed to agree in the far future. By contrast, on the SFF contour
the overall energy is not constrained by an initial state. The filter
functions can select an energy, and they contribute to the path integral as
$f(\rho_{0})^{2}$. (Contributions in which the energies on the two contours
are substantially different are suppressed.) Similarly, on the SFF contour the
overall time shift is not fixed to be zero, so $\sigma_{a,0}$ should now be
integrated over. The domain of integration is periodic since time is
identified. In this way, the zero modes produce the expected random matrix
ramp,

$\text{SFF}\sim\int d\sigma_{a,0}\int dE_{0}=T\int dE_{0}.$ (21)

Now we review the calculation in more detail, including both zero- and non-
zero-modes. Using the definition of $\rho$, we can rewrite equation (18)

$L_{\text{Kel}}^{\text{(diffusion)}}=-\sigma_{a}\left(\partial_{t}\rho-D\nabla^{2}\rho\right)+i\beta^{-2}\kappa(\nabla\sigma_{a})^{2}.$
(22)

Since the action is entirely Gaussian, we can evaluate the path integral
exactly. We first break into Fourier modes in the spatial directions. The
integral becomes

$\prod_{k}\int\mathcal{D}\epsilon_{k}\mathcal{D}\sigma_{ak}f(E_{1})f(E_{2})\exp\left(-i\int
dt\sigma_{ak}\partial_{t}\rho_{k}+Dk^{2}\sigma_{ak}\rho_{k}-\beta^{-2}\kappa
k^{2}\sigma_{ak}^{2}\right)$ (23)

For $k\neq 0$, breaking the path integral into time modes gives an infinite
product which evalutes [47] to $\frac{1}{1-e^{-Dk^{2}T}}$ (and there is indeed
no explicit $\kappa$ dependence). For $k=0$, we just integrate over the full
manifold of possible $\sigma_{a}$s and $\rho$s to get $\frac{T}{2\pi}\int
f^{2}(E_{0})dE_{0}$, up to an overall factor that depends on the measure. So
the full connected SFF for systems with a single diffusive mode is

$\textrm{SFF}=\frac{T}{2\pi}\int
f^{2}(E_{0})dE_{0}\prod_{k}\frac{1}{1-e^{-D(E)k^{2}T}}.$ (24)

We emphasize again that $D(E_{0})$ depends on the background energy $E_{0}$
and we are integrating over $E_{0}$.

Let us now suppose we work in the thermodynamic limit and use a Gaussian
filter $f$ to select a particular background energy $E_{0}$ (up to
subextensive fluctuations). In this large-volume limit, the product over non-
zero modes can be evaluated by taking the log and then Taylor expanding
$\log(1-e^{-Dk^{2}T})$. The result is

$\log\frac{\textrm{SFF}}{\text{SFF}_{\text{zero-modes}}}\approx
V\left(\frac{1}{4\pi DT}\right)^{d/2}\zeta(1+d/2),$ (25)

where $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ is the famous Riemann zeta
function. This approximation breaks down at times near the Thouless time of
the system $T=1/(DL^{2})$, where $L$ is the characteristic length. At this
point the product over modes in equation (24) can be better approximated as
just 1, and the ramp takes on the value one would expect from conventional
random matrix theory. The form of (25) agrees with the expression derived in
[49] in the limit of large local Hilbert space dimension; the hydro theory
predicts (25) just given the diffusive dynamics and it can be used to show
that that the leading perturbative correction due to hydrodynamic interactions
(non-quadratic terms in $L_{\text{Kel}}$) is small when the volume is large.

The for a huge array of systems (including many with sound modes), the product
$\prod_{k}\frac{1}{1-e^{-D(E)k^{2}T}}$ actually has a physical interpretation
as the Total Return Probability (TRP) [47]. If one partitions the
configuration space into sectors labeled by $i,j$, then one can define
$p_{i\to j}(T)$ as the probability that a system starting in sector $i$ is in
sector $j$ at time $T$. The total return probability is

$\textrm{TRP}(T)=\sum_{i}p_{i\to i}(T).$ (26)

Remarkably, if the sectors are small enough that one can’t tell where in a
sector the system started after time $T$, then this quantity is very resilient
to how exactly sectors are chosen. For instance cutting a sector $i$ into
$i^{\prime},i^{\prime\prime}$ will replace $p_{i\to i}$ with $p_{i^{\prime}\to
i^{\prime}}+p_{i^{\prime}\to i^{\prime}}=p_{i\to i^{\prime}}+p_{i\to
i^{\prime\prime}}=p_{i\to i}$. If one chooses to have each sector be a single
configuration $\psi$ then the TRP can be written

$\textrm{TRP}=\sum_{\psi}\left|\expectationvalue{e^{-iHT}}{\psi}\right|^{2}.$
(27)

The TRP has an interpretation as measuring how much a system still remembers
after time $T$. If a system has not spread through configuration space, the
TRP will be large, whereas if the system has forgotten its initial
configuration the TRP will be one. In purely dissipative systems, the TRP will
always be greater than one, and the connected SFF will always be larger than
the RMT result. But for systems with some oscillatory character then TRP’s
behavior can be much more complicated.

That completes our review of the DPT formalism in the context of diffusive
dynamics. For a general quadratic hydro theory, the frequencies
$\omega=iDk^{2}$ will be replaced by a more general set of modes
$\omega_{j}(k)$. The parameters specifying this dispersion may also depend on
the background energy and the background values of other conserved quantities.
The general SFF predicted by the DPT formalism is then

$\textrm{SFF}=\frac{T}{2\pi}\int f^{2}(E_{0})dE_{0}\prod_{\textrm{Slow Modes
}\phi_{j}}\prod_{k}\frac{1}{1-e^{i\omega_{j}(k)T}}.$ (28)

We again assume for simplicity that $f$ is chosen to select a particular
background energy for which the parameters of $\omega_{j}(k)$ take some
particular value. Then in terms of the random matrix form factor,

$\textrm{SFF}_{\textrm{GUE}}=\frac{T}{2\pi}\int f^{2}(E)dE,$ (29)

the ramp is enhanced by a factor of

$Z_{\textrm{enh}}=\frac{\textrm{SFF}}{\textrm{SFF}_{\textrm{GUE}}}=\prod_{\textrm{Slow
Modes }\phi_{j}}\prod_{k}\frac{1}{1-e^{i\omega_{j}(k)T}}.$ (30)

The rest of this paper will focus on the computation the product in (30) for
various systems with sound poles. We find our results depend in detail on the
geometry of the system, in stark contrast with the diffusive result in
equation 25, which depends on on volume.

###  2.3 Doubled Periodic Time Formalism with a Sound Pole

We are finally ready to tackle the problem of sound poles SFF hydrodynamics.
To do so, we simple need to specify the frequencies $\omega_{j}$ entering into
(30). We use a simple model of sound poles described by the hydro Lagrangian

$L_{\text{Kel}}^{\text{(sonic)}}=\frac{1}{2}\phi_{a}\left(\partial_{t}^{2}+\frac{2\Gamma}{c^{2}}\partial_{t}^{3}-c^{2}\partial_{\mu}^{2}\right)\phi_{r}+\frac{2i\Gamma}{\beta
c^{2}}\phi_{a}\partial_{t}^{2}\phi_{a}+\textrm{higher derivative terms}.$ (31)

This sort of Lagrangian might arise in a superfluid, where $\phi$ plays the
role of an order parameter. Alternatively this can be taken as a minimal
schematic for a system with a more conventional sound pole, such as familiar
fluid systems with energy and momentum conservation.

This system has a characteristic length scale $\ell=\Gamma/c$. Physically,
this is the scale below which hydrodynamics breaks down, and corresponds
roughly to the mean free path of the constituent molecules.

In general this Lagrangian is cubic in frequency, and the equations of motion
gives us three solutions for $\omega$. For small $k$ such that $k\ell\ll 1$,
these solutions are $\omega\approx\pm ck+i\Gamma k^{2}$, and $\omega\approx
i\frac{2\Gamma}{c^{2}}$. This last mode is fast, and can be ignored in the
infrared. Note that the solutions for $i\omega$ are either real or come in
complex-conjugate pairs, so the enhancement in (30) is real as it must be.

Ignoring the fast decaying mode, we finally have our formula for the SFF
enhancement in the presence of a sound pole,

$\log
Z_{\textrm{enh}}=-\sum_{k,\mathfrak{s}=\pm}\log(1-\exp\\{(i\mathfrak{s}ck-\Gamma
k^{2})T\\}).$ (32)

where sums over $k$ are always over the numbers $k>0$ such that $k^{2}$ is an
eigenvalue of the Laplacian in the system holding our fluid.

By Taylor expanding the log on the right, we also have the alternate formula

$\log
Z_{\textrm{enh}}=\sum_{j=1}^{\infty}\sum_{k^{2},\mathfrak{s}}\frac{\exp(j(i\mathfrak{s}ck-\Gamma
k^{2})T)}{j}.$ (33)

The presence of the complex exponentials in equation (33) leads to
qualitatively new behavior not seen in purely diffusive hydro. While systems
without sound poles see $Z_{\textrm{enh}}$ decay monotonicly as $T$ increases,
in sonic systems we see an intricate interplay between the positive and
negative terms in (33).

We will now compute the enhancement in a variety of sonic scenarios.
Corrections to the formula (32) arise from higher-derivative terms and non-
Gaussian terms not included in $L_{\text{Kel}}^{\text{(sonic)}}$. Corrections
can also arise from the finite width of the filter function.

##  3 Babylon: The 1D Sound Pole

In this section we will concern ourselves with a very specific problem: the
SFF enhancement for a 1D system with periodic boundary conditions. In this
case, the $k$s are just $2\pi n/L$. In order to get our bearings, let’s first
evaluate equation (32) numerically. For a particular parameter choice, the
results are shown in figure 3.

Figure 3: The first image shows the log of the SFF enhancement factor for a
system of length $L=10^{5}$ with $c=1$, $\Gamma=1$, as calculated in equation
35. The graph cuts off at a lower bound of $T=9\times 10^{4}$, otherwise the
$T=0$ divergence would overwhelm the rest of the image. The second graph shows
a more sedate choice of $L=50$ $c=1$, $\Gamma=1$. At this smaller scale, only
fifty times the mean free path, fewer terms in product 30 and the enhancement
is small enough that the sonic SFF and the (connected) GUE SFF can be plotted
on the same axes. Note the oscillatory behavior, and the fact that at certain
times the sonic SFF is actually smaller than the pure random matrix system.

Before we can understand this fascinating picture, we need to discuss the
function

$f(x)=\begin{cases}\frac{1}{n}&\text{if }x=\tfrac{m}{n}\quad(x\text{ is
rational), with }m\in\mathbb{Z}\text{ and }n\in\mathbb{N}\text{ coprime}\\\
0&\text{if }x\text{ is irrational.}\end{cases}$ (34)

This function is known by many fanciful names including the popcorn function,
the raindrop function, the countable cloud function, and, best of all, the
Stars over Babylon.

Figure 4: The Stars over Babylon. The Stars over Babylon function in this
range corresponds to one period in figure 3.

What does this fairytale function have to do the sums in equations (32)? The
answer is that there is a huge enhancement whenever the system is in
resonance, that is whenever $cT/L$ is rational. Setting $k=2\pi q/L$ and
$cT/L=m/n$ the sum in equation (32) becomes

$\begin{split}\log
Z_{\textrm{enh}}=-2\sum_{q>0,\mathfrak{s}=\pm}\log(1-\exp\left\\{\left(2\pi
i\mathfrak{s}\frac{cq}{L}-\Gamma\frac{4\pi^{2}q^{2}}{L^{2}}\right)T\right\\})\\\
=-2\sum_{q>0,\mathfrak{s}=\pm}\log(1-\exp\left\\{\left(2\pi
i\mathfrak{s}\frac{mq}{n}-\Gamma\frac{4\pi^{2}q^{2}}{L^{2}}\right)T\right\\}).\end{split}$
(35)

so every $n$th term in the sum over $q$ contributes a term
$-\log(1-\exp(-\Gamma k^{2}T))$, which is very close to $-\log 0$ when $k$ is
small. Since only a $1/n$ fraction of the modes contribute, and contributions
like this swamp out any others, we get a structure like equation (34). A more
careful accounting in the next subsection will reveal that the Babylon formula
is actually modified to $n^{-3/2}$ from $1/n$. Here $n$, as the denominator in
a Babylon-like function, plays the role of a sort of order. Smaller $n$ spikes
are more resilient to dissipation while larger $n$ spikes get wiped out more
quickly.

###  3.1 Details on the 1D Spectral Form Factor

Not content to observe the Stars over Babylon pattern, let’s work out some
quantitative details. For instance, how tall should one expect the peaks at
full resonance to be? These are the times $T$ satisfying $cT/L=m$ for some
integer $m$. So we have

$\log
Z_{\textrm{enh}}(T=mL/c)=-2\sum_{q\in\mathbb{N}/\\{0\\}}\log(1-\exp(-\Gamma\left(\frac{2\pi
q}{L}\right)^{2}T))$ (36)

If we replace $\Gamma$ with $D$, we see that at these special resonant times,
this is also the enhancement factor for normal diffusion. So for times $T\ll
L^{2}/\Gamma$ we can thus use the same methods leading to (25) (namely Taylor
expanding the log and approximating the sum over $q$s with an integral) to get

$\log Z_{\textrm{enh}}\left(T=\frac{mL}{c}\right)=2L\left(\frac{1}{4\pi\Gamma
T}\right)^{1/2}\zeta(3/2).$ (37)

For a picture of this envelope versus the actual function, see figure 5.

Figure 5: Equation (37) does a good job estimating the envelope for
$L=1,000,000$, $c=1$, $\Gamma=1$. The source of the discrepancy comes from
approximating the sum in equation (36) as an integral. This approximation is
valid in the limit $cL\gg\Gamma$ where many terms contribute to the sum.

We can also evaluate an envelope for the shorter peaks for $n>1$. In these
cases it is best to break the sum into sets of $n$ consecutive terms. We can
write this exactly as

$\log
Z_{\textrm{enh}}\left(T=\frac{mL}{nc}\right)=-\sum_{q\in\mathbb{N}/\\{0\\},\mathfrak{s}=\pm}\sum_{q_{2}=0}^{n-1}\log(1-\exp(-\Gamma\left(\frac{2\pi(nq-
q_{2})}{L}\right)^{2}T+\mathfrak{s}2\pi iq_{2}/n)).$ (38)

In the thermodynamic limit $cL\gg\Gamma$ we can treat the Gaussian as constant
within each inner sum. We can then use the identity

$\sum_{q=0}^{n-1}\log(1-a\exp(2\pi iq/n))=\log(1-a^{n})$ (39)

to perform the inner sum. We have

$\log
Z_{\textrm{enh}}\left(T=\frac{mL}{nc}\right)=-\sum_{q\in\mathbb{N}/\\{0\\}}\sum_{q_{2}=0}^{n-1}\log(1-\exp(-n\Gamma\left(\frac{2\pi
nq}{L}\right)^{2}T)).$ (40)

This is the same sum again but with $\Gamma$ replaced with $n^{3}$ $\Gamma$.
This means that we have

$\log
Z_{\textrm{enh}}\left(T=\frac{mL}{nc}\right)=L\frac{1}{n^{3/2}}\left(\frac{1}{4\pi\Gamma
T}\right)^{1/2}\zeta(3/2).$ (41)

This $n^{-3/2}$ is interesting and surprising. Again, in the original Babylon
function, there is just an $1/n$. We justified this by saying that $1/n$ of
the terms contribute exploding positive contributions to the sum. However, the
other $n-1$ terms can be shown to on average contribute slightly negative
suppressing terms. This provides the intuition for the $n^{-3/2}$ in equation
(41).

A final word about when these approximations become valid. Based on
numerically evaluating equation (32), it seems that the spikes at integers
don’t become clearly visible until this system size is a thousand relaxation
lengths, and the full fractal structure isn’t visible until ten thousand.
Figures (3) and (5) use a system size of a million. Needless to say, this is
quite beyond the realm of any foreseeable exact diagonalization calculation or
experimental technology.

###  3.2 A Fourier Perspective and the Width of the Peaks

The analysis in the above subsection tells us the height of the peaks, but
doesn’t tell us anything about their breadth or shape. We can extract this
information by taking the Taylor expansion of
$\log(1-x)=\sum_{j=1}^{\infty}-\frac{1}{j}x^{j}$

$\log
Z_{\textrm{enh}}=2\sum_{q>0,\mathfrak{s}=\pm}\sum_{j=1}^{\infty}\frac{1}{j}\exp\left(2\pi
iq\mathfrak{s}\frac{cT}{L}j\right)\exp\left(-j\frac{4\pi^{2}\Gamma
q^{2}}{L^{2}}T\right),$ (42)

For each $j$ this is the Fourier series representation of a chain of tight
peaks (missing a $q=0$ term). The peaks have Gaussian shape, and recur after
time $\frac{L}{cj}$. Thus we see that the $j$th term in this sum corresponds
only to the peaks at $T=\frac{mL}{jc}$, or in other words we can identify $j$
with $n$ in the previous section.

The total area under the Gaussian from the $j$th sum is given by the amplitude
of the missing $q=0$ term times the period. This is just an amplitude of
$\frac{1}{j}$ times a period of $\frac{L}{cj}$ for a total area of
$\frac{L}{cj^{2}}$.

Combined with the results of the previous subsection, we know that the width
of the Gaussian is something like $\\#\frac{1}{c}\sqrt{\frac{\Gamma T}{n}}$.
This implies that at sufficiently large $n$, the width of the peaks does
become greater than the area between the peaks, which means that the lovely
Stars over Babylon structure does not have infinite resolution.

###  3.3 Instability to Interactions

It is also worth investigating how higher derivative terms and interactions
would affect the qualitative result of the Babylon-shaped enhancement factor.
In the thermodynamic limit, the higher derivative terms don’t substantially
affect the pattern. The higher derivative terms only affect larger values of
$k$, which are already suppressed since they have factors of $e^{-\Gamma
k^{2}T}$.

Interactions, however, can have a more noticeable impact. A more detailed
discussion will need to wait until appendix C; we will quote the results of
that appendix here. An important point is that while in the CTP formalism the
velocity would merely be renormalized by a smooth amount, in DPT the velocity
can be renormalized by an amount depending sensitively on frequency. We
hypothesize that this irregularity would substantially derange the intricate
pattern, likely in favor of a more erratic pattern of spikes.

If there were some large number of local degrees of freedom, in the tradition
of [26, 49, 50, 67], then these interactions would be heavily suppressed. But
a large local number of degrees of freedom combined with a large system size
means a truly large Hilbert space dimensions, making the empirical observation
of these patterns a remote possibility for the foreseeable future.

Moreover, in the Schwinger-Keldysh case, 1d sonic hydrodynamics is known to be
unstable [52], and flows to the KPZ universality class [53, 54, 55]. For SFF
hydrodynamics, these concerns are modified. The periodic time changes the
significance of diagrammatic corrections in complicated ways [47], and it is
unclear in what dimensions weakly coupled hydrodynamics is stable in the IR
limit. This will be discussed more in the outlook.

That said, it is far from obvious that the KPZ scaling completely destroys the
Stars. This is because the ballistic propagation of sound, which is
responsible for the basic resonance structure, is still present in the KPZ
case. Said differently, one can view the flow to the KPZ universality class as
modifying the dispersion from $\omega=ck-i\Gamma k^{2}+\cdots$ to
$\omega=ck-i\tilde{\Gamma}k^{3/2}+\cdots$. This suggests that the KPZ physics
most strongly modifies the envelope function and may have a weaker effect on
the resonances. We emphasize again that this requires further study and is a
non-trivial problem in the DPT formalism, one that combines relevant
interactions with periodic time.

However, we can sketch out one starting point for the analysis. One way to
interpret $Z_{\text{enh}}$ for the unstable sonic fixed point is via a pair of
biased diffusion equations, one for the left movers and one for the right
movers. These equations are

$\partial_{t}\rho_{\pm}=\pm
c\partial_{x}\rho_{\pm}+\Gamma\partial_{x}^{2}\rho_{\pm}+\xi_{\pm},$ (43)

where $\pm$ refers to the left and right movers and we included a stochastic
force $\xi_{\pm}$ to describe hydrodynamic fluctuations. The $Z_{\text{enh}}$
is obtained as the probability that a given initial condition
$\rho_{\pm}(x,0)$ is recovered at later time at later time $T$. Focusing on
the $+$ component, we have

$Z_{\text{enh}}(T)=\int
D\rho_{+}(0)D\xi_{+}P(\xi_{+})\delta[\rho_{+}^{\text{diff}}(T;\xi_{+})-\rho_{+}(0)],$
(44)

where $D\rho_{+}(0)D\xi$ denotes a functional integral over the noise and the
initial condition and $\rho_{+}^{\text{diff}}(T;\xi)$ solves the biased
diffusion equation with noise $\xi$.

The resonance condition, $mL=ncT$, arises as follows. If we ignore the
$\Gamma$ term and the stochastic term, then a given profile $\rho_{+}(x,0)$ is
merely translated by the dynamics to $\rho_{+}(x,T)=\rho_{+}(x+cT,0)$. Hence,
for a localized wavepacket the probability to return is zero unless

$mL=cT$ (45)

for some integer $m$. The full set of resonances arises by considering
perturbations of specific wavelenths, e.g. the $n=2$ case corresponds to
perturbations of wavelength $\lambda=L/2,L/4,L/6,\cdots$ that need only be
translated by half the length of the system to return to themselves. The
effect of the noise and $\Gamma$ term is to broaden these sharp features as
discussed above.

Now consider the KPZ case. Because of the peculiarities of 1+1d kinematics,
the decomposition into left and right movers still provides an approximate
starting point for the analysis. Focusing again on $\rho_{+}$, the simplest
equation which captures the relevant effects is

$\partial_{t}\rho_{+}=c\partial_{x}\rho_{+}+\frac{c^{\prime}}{2}\partial_{x}\rho_{+}^{2}+D\partial_{x}^{2}\rho_{+}+\xi_{+}.$
(46)

The new term is the $c^{\prime}$ term, which turns out to be relevant in the
scaling sense. Before proceeding, we emphasize again that it is not clear how
the time periodicity modifies the standard analysis and whether the displayed
terms are sufficient to capture all the physics of interest in our case.

With that caveat, the enhancement takes the same form,

$Z_{\textrm{enh}}(T)=\int
D\rho_{+}(0)D\xi_{+}P(\xi_{+})\delta[\rho_{+}^{\text{KPZ}}(T;\xi_{+})-\rho_{+}(0)],$
(47)

where now $\rho_{+}^{\text{KPZ}}(T;\xi_{+})$ solves (46) instead of the biased
diffusion equation. We see immediately that the translating effect of the
$c\partial_{x}\rho_{+}$ term is still present, so a localized wavepacket will
still have vanishing return probability unless $mL=cT$. Of course, the other
terms are crucial, but as in the diffusive case, their effect is plausibly to
broaden these primary resonances rather than to destroy them. On the other
hand, the subleading peaks, which arose from profiles with special wavelengths
that enjoyed an enhanced translation symmetry, e.g., by $L/2$, are harder to
analyze in the non-linear theory given by (46). Indeed, we cannot analyze the
physics wavelength-by-wavelength since the non-linear term couples different
wavelengths. We think it is plausible that there could still be enhancements
in $Z_{\textrm{enh}}$ corresponding to the subleading resonances, but we
cannot say for sure without a more complete analysis of the KPZ return
probability.

##  4 Sound In An Integrable Cavity

We will now make the jump from quadratic hydrodynamics in 1D to higher
dimensions. In this case we are now faced with the choice of what shape our
system should take: spherical, toroidal, or something more exotic. While in
most cases—including the diffusive hydro SFF—the answers to hydrodynamic
questions do not depend on this sort of choice, the sonic hydrodynamic SFF
will care about the precise shape we choose.

We see that equation (32) depends sensitively on the detailed eigenvalues of
the Laplacian in our system. In this section we will examine the case where
the Laplacian operator in our cavity has Poissonian spectral statistics. This
might occur, for instance, if we are doing hydrodynamics in a torus, a
rectangular prism, or an ellipsoid. More generally, Poissonian spectral
statistics for the Laplacian is thought to describe the case where the cavity,
when viewed as a stadium/billiard table, gives rise to integrable dynamics for
a particle moving inside the stadium [6]. For this reason, we will call this
case an integrable cavity. But it is important to remember that while the
emergent dynamics of an individual sound mode may be integrable, our
hydrodynamic assumption requires that the full many-body dynamics be chaotic
in terms of microscopic degrees of freedom.

For a general hydrodynamic system, equation (32) can be rewritten as

$Z_{\textrm{enh}}=\prod_{k}\frac{1}{(1-\exp\\{(ick-\Gamma
k^{2})T\\})(1-\exp\\{(-ick-\Gamma k^{2})T\\})}.$ (48)

Using the Poissonian statistics, we can calculate the expected value of this
enhancement factor. In any Poissonian system, the density of Laplacian
“energy” eigenvalues in the interval $[E_{1},E_{1}+dE_{1}]$ is independent of
that in region $[E_{2},E_{2}+dE_{2}]$ for $E_{1}\neq E_{2}$. This independence
implies a similar independence for $k\sim\sqrt{E}$. So we can evaluate the
expected value of equation (48) by partitioning the spectrum of $k$ into non-
overlapping regions $[k_{i},k_{i}+\delta k]$, finding the expected value of
the product over eigenvalues in that region, and then multiplying the results
together.

For concreteness, we will treat the values of $k$ as a Poisson process with
intensity $\bar{\rho}(k)$. This means that in any given region $[k,k+dk]$ the
product is $[(1-\exp\\{(ick-\Gamma k^{2})T\\})(1-\exp\\{(-ick-\Gamma
k^{2})T\\})]^{-1}$ with probability $\bar{\rho}dk$ (eigenvalue present) and
$1$ with probability $1-\bar{\rho}dk$ (eigenvalue absent). The expected value
is thus

$1+\left(\frac{1}{(1-\exp\\{(ick-\Gamma k^{2})T\\})(1-\exp\\{(-ick-\Gamma
k^{2})T\\})}-1\right)\bar{\rho}dk.$ (49)

This means that in expectation we can write

$\langle{Z_{\textrm{enh}}}\rangle=\exp\left(\int_{0}^{\infty}dk\left(\frac{1}{(1-\exp\\{(ick-\Gamma
k^{2})T\\})(1-\exp\\{(-ick-\Gamma k^{2})T\\})}-1\right)\bar{\rho}(k)\right),$
(50)

where the angle brackets represent an average over different cavity
configurations.

This formula isn’t the result of an expansion, it assumes only a purely
Poissonian density for $k$. Figure 6 shows numerics backing up this
prediction. To create figure 6, we used the most Poissonian process possible:
independent random numbers. It does not correspond to a Laplacian on any
particular cavity shape.

Figure 6: A graph illustrating the agreement of numerical (blue) vs
theoretical (orange) predictions for equation (50). We chose $c=1,\Gamma=0.1$,
$\bar{\rho}(k)=1$. Because equation (50) has a divergence at low $k$ we had a
cutoff of $k=0.1$. This cutoff is responsible for the interesting steplike
behavior seen in both functions. The blue line is an average of 1971459
samples.

It is worth noting that equation (48) is a product over many terms. As such,
depending on context, the mean value might not be a good representation of
typical values, for the same reason that log-normal distributions are not
well-clustered around their mean. The expected value of the log of the
coefficient can be calculated straightforwardly by taking the log of equation
(48).

Another caveat is that real billiard systems, even integrable ones, do not
have exactly Poissonian spectral statistics. The clearest exhibition of this
is in the case of periodic orbits. To explain how this affects the result,
consider a torus of dimensions $2\pi\times 4\pi$. The eigenfunctions of the
Laplacian are parameterized by a wavevector of the form $(n_{x},n_{y})$ where
$n_{x}$ is integer valued and $n_{y}$ can also take half-integer values. For
example, one possible Laplacian eigenstate has wavevector $(1,\frac{1}{2})$,
which gives $k^{2}=\frac{5}{4},k=\frac{\sqrt{5}}{2}$. There is another
eigenstate with wavevector $(2,1)$, which corresponds to $k=\sqrt{5}$. In
general, there will be states with $k=n\frac{\sqrt{5}}{2}$ for all positive
integers $n$. When we multiply the enhancement factors for all of these
states, we get the same intricate pattern seen in figure (3).

We think that this same effect can exist more generally in any cavity in which
a classical particle can take a closed periodic path. To show this fact about
the eigenvalues of the Laplacian, we imagine that we start a quantum
wavepacket moving under a fictitious Hamiltonian
$H_{\text{fict}}=-\frac{1}{2m_{\text{fict}}}\gradient^{2}$ at velocity $v$
around a periodic path of length $L$. The wavepacket does not have a definite
energy, but instead a spread of energies well-centered on
$E_{\text{fict}}=\frac{k^{2}}{2m_{\text{fict}}}=\frac{m_{\text{fict}}v^{2}}{2}$.
If we choose a $k$ very large compared to the inverse system size (well into
the semiclassical limit) then all of the energy eigenvalues contributing to
this wavepacket are close to $E_{\text{fict}}$.

Because the classical motion is periodic, the wavefunction is going to be
approximately periodic in time with period $L/v$. This means that the
wavepacket has overlap with $H_{\text{fict}}$ eigenstates with energies that
differ by integer multiples of $\frac{2\pi v}{L}$. So there will be values of
$k=\sqrt{2m_{\text{fict}}E_{\text{fict}}}$ spaced out, separated by integer
multiples of $\frac{2\pi}{L}$. These evenly spaced out modes violate our
assumption of Poissonness, and will lead to contributions like in figure (3).
These effects rely only on the existence of periodic orbits in the cavity,
which are are present in both integrable or chaotic cavities.

It is likely, however that these effects will drown out after a comparatively
short time once the wavefunction has time to spread out. For chaotic systems
this time is known as the Ehrenfest time, and is on the order of
$\frac{\log\frac{\textrm{Phase Space Volume}}{h^{d}}}{\lambda}$, where
$\lambda$ is the Lyapunov exponent. For integrable systems the time is given
by the same star destroying effects as in subsection (3.3). For a generic
integrable system, $\omega$ will depend on $d$ commuting quantum numbers. The
dependence will be well approximated as linear, but any higher-derivative
terms or interactions in the hydro theory will break that perfect
interference.

##  5 Sound in a Chaotic Cavity

In this section we will turn our attention to the case of quadratic
hydrodynamic enhancements in a chaotic cavity. As a reminder, the expression
for the enhancement from sound poles in a generic cavity is

$\begin{split}\log
Z_{\textrm{enh}}=-\sum_{k^{2},\mathfrak{s}=\pm}\log(1-\exp\\{(i\mathfrak{s}ck-\Gamma
k^{2})T\\})\\\
=\sum_{j=1}^{\infty}\sum_{k,\mathfrak{s}=\pm}\frac{1}{j}\exp((i\mathfrak{s}ck-\Gamma
k^{2})jT)\end{split}$ (51)

Let’s imagine the shape of the cavity is a Bunimovich stadium or a Sinai
billiard (figure 7). Then the $k^{2}$s become the eigenvalues of a level-
repelling chaotic Hamiltonian of the Gaussian Orthogonal Ensemble (GOE)
universality class. The $k$s are stretched eigenvalues, but still exhibit GOE-
type level repulsion.

Figure 7: The eigenvalues of the Laplacian are known to have Wigner-Dyson/RMT-
like statistics in both Bunimovich Stadium (left) and the Sinai Billiard
(right). In this section we explore the spectral statistics of fluids filling
up the gray regions in these two shapes. Figure 8: RMT Result (Blue) versus
sound-enhanced results in 3 different chaotic cavities with area 40 (Orange,
Green, and Red). Notice how the system with one very slow mode (Orange) seems
to have fairly different behavior than the faster decaying Green and Reds. As
system sizes become larger compared to the mean free path, and more modes can
oscillate without dying, interference between various oscillations matters
more than the magnitude of any one slow mode.

We can view the right hand side of equation (51) as a sum over partition
functions with imaginary temperature $x$ associated with the $k$ spectrum,

$Z(x,f)=\int dk\rho(k)e^{ikx}f(k)$ (52)

where $\rho$ is the exact density of states and $f$ is a filter function. In
particular, we have

$\log
Z_{\textrm{enh}}=\sum_{j=1,\,\mathfrak{s}=\pm}^{\infty}\frac{1}{j}Z\left[j\mathfrak{s}cT,f_{j}=\exp(-j\Gamma
Tk^{2})\right].$ (53)

If we fix particular stadium, then we expect that the enhancement factor will
be an erratic function of time. To get a smooth result which is calculable, we
average over a number of configurations for our stadium (being careful to
remain in the GOE universality class) instead of calculating the logarithm of
$Z_{\textrm{enh}}$ for a given realization of the stadium. Denoting by $W$ the
quantity in equation (53), we have $\log\langle
Z_{\textrm{enh}}\rangle=\log\langle e^{W}\rangle$, which is by definition the
sum of the cumulants of $W$.

Doing a cumulant expansion, we get

$\log\langle
Z_{\textrm{enh}}\rangle=\sum_{\ell}\frac{1}{\ell!}c_{\ell}=\mathbb{E}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]+\frac{1}{2}\textrm{var}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]+\dots$
(54)

For times smaller than the Heisenberg time of the single-particle system, we
can safely approximate the cumulant expansion with just the first two terms,
knowing all subsequent terms will suppressed by factors of the cavity volume.
We will consider these two terms in turn.

The first cumulant is obtained from the average density of states. Our system
has some density of states $\rho(k)$ which fluctuates depending on the precise
shape of the cavity. Averaging over cavity shapes we get that $\rho(k)$
fluctuates about some $\bar{\rho}(k)$. $\bar{\rho}$ can be calculated in a
semiclassical approximation. If the cavity has $d$-dimensional volume $V$, we
can expect a density of states

$\bar{\rho}(k)=\frac{V}{(2\pi)^{d}}S_{d-1}k^{d-1},$ (55)

where $S_{d-1}=\frac{2\pi^{\frac{d-1}{2}}}{\Gamma(\frac{d-1}{2})}$ is the
surface area of a $d-1$ sphere.

So the expected value of the imaginary-time partition function is

$\mathbb{E}\left[\frac{1}{j}Z(j\mathfrak{s}cT,f_{j}=\exp(-j\Gamma
Tk^{2}))\right]=\frac{VS_{d-1}}{(2\pi)^{d}j}\int_{0}^{\infty}dkk^{d-1}\exp(ij\mathfrak{s}cTk)\exp(-j\Gamma
Tk^{2}).$ (56)

After including the sum over $\mathfrak{s}$ and relabeling $k\rightarrow-k$ in
the $\mathfrak{s}=-$ term, we have

$\sum_{\mathfrak{s}}\mathbb{E}\left[\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]=\frac{VS_{d-1}}{(2\pi)^{d}j}\int_{-\infty}^{\infty}dk|k|^{d-1}\exp(ijcTk)\exp(-j\Gamma
Tk^{2})$ (57)

Right away, we notice that the behavior is qualitatively different for even
versus odd $d$. For odd $d$, we have the Fourier transform of an analytic
function, while for even $d$, there is a non-analyticity at $k=0$.

If we evaluate the expression, we get

$\sum_{\mathfrak{s}}\mathbb{E}\left[\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]=\begin{cases}\frac{VS_{d-1}}{(2\pi)^{d}j}\sqrt{\frac{2\pi}{j\Gamma
T}}(jT)^{1-d}\partial_{c}^{d-1}\exp(-j\frac{c^{2}}{4\Gamma}T)&\text{if
}d\textrm{ odd},\\\
\frac{2VS_{d-1}}{(2\pi)^{d}j}\frac{(d-1)!}{(ijcT)^{d}}+O(T^{-d-1})&\text{if
}d\textrm{ even}.\end{cases}$ (58)

As we see, in odd dimensions this is an exponential decay, while in even
dimensions it is a power-law decay. This is related to the fact that sound
waves have sharp edges in odd dimensions and soft edges in even dimensions.
The terms in (58) can plugged into (54) and the sum over $j$ computed to get

$\log\langle{Z_{\textrm{enh}}}\rangle\supset\begin{cases}\frac{VS_{d-1}}{(2\pi)^{d}}\sqrt{\frac{2\pi}{\Gamma
T}}(T)^{1-d}\partial_{c}^{d-1}\exp(-\frac{c^{2}}{4\Gamma}T)&\text{if
}d\textrm{ odd},\\\
\zeta(d+1)\frac{2VS_{d-1}}{(2\pi)^{d}}\frac{(d-1)!}{(icT)^{d}}+O(T^{-d-1})&\text{if
}d\textrm{ even}.\end{cases}$ (59)

The odd dimension result is zero at times greater much than
$\frac{\Gamma}{c^{2}}$. Unfortunately, before this time hydrodynamics is
dominated by higher derivative corrections, so in odd dimensions equation 59
must be approached with caution. This fast decay of the exponential-in-volume
enhancement factor can be contrasted with the diffusive result quoted in
equation 25, which decays slowly in all dimensions. In contrast the rapid
decay in odd dimensions, the even-dimensional part of equation 59 is large
until times extensive in the system length, and could, in principle, be
observed. But it still decays far faster than equation 25.

If we stopped here, at the first term in the cumulant expansion, we would be
making the approximation $\log\langle{Z_{\textrm{enh}}}\rangle=\langle{\log
Z_{\textrm{enh}}}\rangle$. By analogy with the glass literature, we refer to
$\log\langle{Z_{\textrm{enh}}}\rangle$ as the annealed average and
$\langle{\log Z_{\textrm{enh}}}\rangle$ as the quenched average. The above
approximation is thus analogous to the approximation that quenched equals
annealed. Typically, the quenched value of a partition function is the value
for a typical realization of disorder, whereas the unquenched or annealed
value is dominated by more extreme terms. If one were to take a small number
of cavity shapes and evaluate the SFF enhancement coefficients, most of the
coefficients would look like equation (59). This is because we have $\log
Z_{\textrm{enh}}\sim\rho$ sample by sample, so the quenched average contains
only a linear-in-$\rho$ term whereas the annealed average has contributions
from higher cumulants. Equation (59) also gives the quenched answer for an
integrable system.

Now onto the second cumulant, which is

$\textrm{var}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]=\sum_{j,j^{\prime},\mathfrak{s},\mathfrak{s}^{\prime}}\frac{1}{jj^{\prime}}\int
dkdk^{\prime}\mathbb{E}(\rho(k)\rho(k^{\prime}))_{\text{conn}}e^{ijk\mathfrak{s}cT+ij^{\prime}k^{\prime}\mathfrak{s}^{\prime}cT}f_{j}(k)f_{j^{\prime}}(k^{\prime}),$
(60)

where $\mathbb{E}(\rho(k)\rho(k^{\prime}))_{\text{conn}}$ is the connected
pair correlation function. Further progress can be made under the assumption
that the connected correlator has only weak dependence on $k+k^{\prime}$. In
this case, only terms with $j=j^{\prime}$ and
$\mathfrak{s}=-\mathfrak{s}^{\prime}$ contribute, giving

$\textrm{var}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]\approx\frac{1}{j^{2}}\sum_{j,\mathfrak{s}}\int
dkdk^{\prime}\mathbb{E}(\rho(k)\rho(k^{\prime}))_{\text{conn}}e^{ij\mathfrak{s}(k-k^{\prime})cT}f_{j}(k)f_{j}(k^{\prime}).$
(61)

Now, this expression is essentially a sum over single-particle spectral form
factors evaluated at the times $jcT$ and with filter function $f_{j}$. For
each of these we may use the standard GOE result (assuming the single particle
SFF is always in the ramp phase) to obtain

$\textrm{var}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]\approx\sum_{j}\frac{2}{j^{2}}\int_{0}^{\infty}d\bar{k}\left(\frac{jcT}{\pi}\right)\exp(-2j\Gamma
T\bar{k}^{2}).$ (62)

The integral is straightforward, and the total variance is thus

$\textrm{var}\left[\sum_{j,\mathfrak{s}}\frac{1}{j}Z(j\mathfrak{s}cT,f_{j})\right]\approx\sum_{j}\frac{cT}{j\pi}\sqrt{\frac{\pi}{2j\Gamma
T}}=\zeta(3/2)\sqrt{\frac{c^{2}T}{2\pi\Gamma}}.$ (63)

This answer is eerily reminiscent of equation (37). But most interestingly, it
is a highly universal contribution to the ramp of a hydrodynamic system with a
sound pole. It doesn’t depend on any details of the shape of the system, or
even its overall size. Just that the sound waves propagating around experience
chaotic dynamics.

Equation (63) increases indefinitely, and it is worth knowing when exactly it
starts to fail. The answer is that at times on the order of the single-
particle Heisenberg time (inverse level spacing of the single-particle system)
the second cumulant reaches a plateau, and cumulants after the second stop
being negligible. After a few Heisenberg times, there is no longer any residue
of the single-particle level repulsion and the enhancement for chaotic
billiards should resemble that for integrable billiards. This, in turn, falls
to 1 as the various sound modes die down, long before the full many-body
Heisenberg time.

In summary, here is a list of important time scales for the connected sound
pole SFF in a chaotic cavity:

  * •

Hydrodynamic scattering time of order $\Gamma/c^{2}$. This is the time-scale
after which hydrodynamics becomes a valid approximation.

  * •

Single particle Thouless time, on the order of $L/c$. This is the time scale
at which equation (63) becomes a good approximation.

  * •

The single particle Heisenberg scale on the order of
$\left(\frac{1}{c}V\Gamma^{-\frac{d-1}{2}}\right)^{\frac{2}{d+1}}$. Equation
(63) is dominated by modes with $k$ on the order of $k\sim(\Gamma T)^{-1/2}$.
The single particle Heisenberg scale is the time at which $T$ exceeds the
density of states in this region. Above this time scale a chaotic billiard
should behave like an integrable billiard.

  * •

The lifetime of the slowest sound modes $L^{2}/\Gamma$. This can also be
thought of as a many-body Thouless time. After this time the ramp looks like
the pure random matrix theory result.

  * •

The many-body Heisenberg time at order $e^{S}$. This is the time scale at
which the SFF of the full many-body system plateaus.

##  6 Discussion and Outlook

In this paper, we expanded the theory of the hydrodynamic spectral form factor
to the more realistic case of hydrodynamics with a second-time derivative.
This more sophisticated theory and its accompanying sound pole structure
allowed a new phenomenon: interference in the emergent hydrodynamic SFF
formalism. We discovered a variety of different interference patterns, all
connecting intimately with the SFF of a single particle problem in various
cavities. We started with the simplest case, the (necessarily integrable)
dynamics of a sound mode in a single dimension, and got the remarkable
function in figure 3 in the absence of dispersion. In the case of higher
dimensional cavities, our results mostly concerned disorder-averages over
cavities. Using Wigner-Dyson versus Poissonian statistics for the eigenvalues
of the Laplacian of the cavity, we derived the results of sections 4 and 5. We
derive equations (50) and (54) respectively, and graph the predicted
enhancements for toy examples of the laplacian spectral statistics.

Throughout this work, we find that the expected spectral form factor depends
sensitively on both the dimensionality of the system as well as whether the
cavity supports chaotic or integrable billiard dynamics. For example, equation
(59) take on entirely different forms in odd versus even dimensions, for the
same reason that sound has a non-analytic shockwave in odd but not even
dimensions. We also note that, if momentum is exactly conserved, then there
will be distinct blocks in the many-body Hamiltonian labelled by the many-body
momentum (similar to the discussion of conserved quantities in [48]).

This work opens the gates for calculations in a wide array of settings,
including CFTs (which necessarily have both momentum and energy conservation)
and systems with spontaneously broken symmetry. In particular, CFTs on spheres
have been shown [68] to have eternal non-decaying hydrodynamic modes. This
follows from the conformal symmetry, where certain ladder operators $L^{+}$
satisfy the commutation relation $[H,L^{+}]=\frac{1}{R}L^{+}$. Just like in a
simple harmonic oscillator, this leads to two-point functions which oscillate
exactly, forever, even at times much longer than the Heisenberg time of the
system. Each primary operator with energy $\frac{\Delta}{R}$ sits at the
bottom of a tower of states which contribute $\frac{e^{i\Delta
T/R}}{(1-e^{iT/R})^{D}}$ to the SFF. These $\frac{1}{(1-e^{iT/R})^{D}}$
factors would lead to strong enhancements in the SFF out to arbitrary times,
even deep into the plateau region. It would be interesting to look for a
hydrodynamic explanation for these effects, and relate them to the above
formalism.

As our ability to numerically calculate the SFFs of quantum fields theories
improves [69], one might look for at least the leading peaks in a numerical
spectral form factor. One highly optimistic possibility is that signatures of
these results could be found in the original home of Wigner-Dyson statistics:
atomic nuclei. These nuclei are hydrodynamic systems with vibrational modes,
these modes might have a signature in the spectral form factor.

An important direction for future work is seeing whether this emergent
spectral effect can give rise to enhanced 2-point functions in the CTP
hydrodynamic theory. Figure (9) shows a diagram which depends on an internal
integral over the frequencies of the two legs. If the two-point function of
these frequencies is not analytic (as is the case for both integrable and
Wigner-Dyson spectral statistics), that leads to a long-time tail (albeit
suppressed by a factor of volume).

Figure 9: The loop integral in this diagram depends on the second moment of
the spectral density.

Finally, the fate of DPT hydro in the long-range limit is still an open
question. It is known that in conventional hydrodynamics, the 1D theory is
always strongly coupled. In the DPT theory, each leg in each Feynman diagram
is modified due to each mode wrapping around the periodic time (see equation
80). One could imagine these modes destructively interfering at times $T$
incommensurate with the period of the sound modes, thus rescuing the diffusive
theory. On the other hand, it is known [47] that for purely diffusive dynamics
the additional wrapping makes periodic-time hydro strongly coupled even up to
two dimensions. We leave this question for future work.

This work was supported by the Joint Quantum Institute (M.W.) and by the AFOSR
under FA9550-19-1-0360 (B.G.S.).

##  Appendix A More On The Total Return Probability

Figure 10: A simple example of chaotic dynamics with two slowly thermalizing
sectors. The ball will quickly spread out within the Hilbert Space of a given
chamber, but only much more slowly will it transfer between the two sectors.
During this time the connected SFF will be larger than the RMT prediction.

The Total Return Probability (TRP) is a measure of how much a system has
thermalized after time $T$. Remarkably, it has also been shown [47] to be the
enhancement factor for the connected SFF at time $T$. One simple example of a
system with slow thermalization is a single quantum particle bouncing around a
partitioned Bunimovich stadium in figure 10 (not to be confused with the much
more complicated system discussed in section 5, a many-body system filling a
Bunimovich stadium). The two chambers of this stadium are our two sectors.
This particle will bounce around and quickly (after the Ehrenfest time) be
spread out within a given sector. At this point $p_{1\to 1}$ and $p_{2\to 2}$
will both be 1 and the TRP will be 2. As more time passes, both $p_{1\to 1}$
and $p_{2\to 2}$ will decrease towards the equilibrium probabilities of being
in chamber 1 or 2 respectively. Eventually the TRP will be one. More RMT-
focused readers might prefer the example where $H$ is a matrix with

$H=\begin{pmatrix}H_{1}&H_{\textrm{small}}\\\
H_{\textrm{small}}^{\dagger}&H_{2}\end{pmatrix}$ (64)

where some weak coupling $H_{\textrm{small}}$ induces jumping between two RMT-
like sectors. In this case the TRP can be calculated using Fermi’s golden
rule. We get

$\textrm{TRP}(T)=1+e^{-\lambda T}$ (65)

for a specific value of $\lambda$ characterizing the strength of the off-
block-diagonal matrix elements.

Figure 11: (Left) the RMT vs nearly block diagonal connected SFFs during the
ramp regime. (Right) A comparison of the numerical ratio (blue) versus the TRP
in equation 65 (orange).

A more complicated case would be a single particle diffusing through a
complicated medium. In this case the sectors would be regions of space and
$p_{i\to i}$ would be calculated by solving the diffusion equation. This
procedure would agree with previous results obtained through sigma model
methods[70, 11]. An even more complicated case would be a many-body system
with a conserved charge, where the sectors would be coarse-grained charge
distributions $\rho_{i}(x)$. $p_{i\to i}$ is calculated using fluctating
hydrodynamics as discussed in subsection 2.2 and in more detail in [47]. The
results agree with a large-$q$ Floquet calculation done in [49].

##  Appendix B SYK2 At Finite Temperature

In [35, 36] the authors investigated the many-body SFF in a specific system of
free-fermions with random two-body interactions. By specifically studying the
many-body SFF of these They find an exponential ramp in the infinite-
temperature SFF, in particular with a ramp of the form $SFF\propto N^{\\#T}$,
where the number depended on the specific model under consideration. This
exponential ramp behavior can be argued to come from the same cumulant
expansion as equation (63).

In the case of the SYK2 model with no charge/number conservation, we have

$H=\frac{1}{2}\psi_{i}M^{ij}\psi_{j},$ (66)

where $M$ is an anti-symmetric real matrix. One can show that the eigenvalues
of $M$ follow a semicircle law, and obey GUE level repulsion (the reason for
this is that even though the matrix is real, the eigenvalues come in pairs
$\pm\lambda$, and there is repulsion for both the positive and negative
eigenvalues). We can write the full-system SFF at inverse temperature $\beta$
as

$\log
SFF=\sum_{\lambda>0}2\textrm{Re}\log(1+\exp(-\beta\lambda+iT\lambda))=\sum_{j\neq
0}\frac{(-1)^{j}}{j}\sum_{\lambda>0}\exp(-\beta
j\lambda+ijT\lambda))=\sum_{j\neq 0}\frac{(-1)^{j}}{j}Z(jT,j\beta)$ (67)

Note the similarity between this form and equation (51). And while at infinite
temperature all terms of the cumulant expansion are needed, we will show that
at finite temperature only the first two are important.

For most random matrix ensembles, we can write the probability density as

$dP(M)=\exp(-N\tr V(M))dM.$ (68)

In terms of the eigenvalues density
$\rho(\lambda)=\frac{1}{N}\sum_{i}\delta(\lambda-\lambda_{1})$ and the
eigenvalues distribution density as

$dP(\rho)\propto\exp(-\int N^{2}V(\lambda)\rho(\lambda)d\lambda+\int
N^{2}\rho(\lambda_{1})\rho(\lambda_{2})\log|\lambda_{1}-\lambda_{2}|d\lambda_{1}d\lambda_{2}+\int
N\rho(\lambda)\log\rho(\lambda)d\lambda)\mathcal{D}\rho$ (69)

At large $N$, this is the quadratic functional

$dP(\rho)=\exp(-\int N^{2}V(\lambda)\rho(\lambda)d\lambda+\int
N^{2}\rho(\lambda_{1})\rho(\lambda_{2})\log|\lambda_{1}-\lambda_{2}|d\lambda_{1}d\lambda_{2})\mathcal{D}\rho.$
(70)

This means that we can analyze quantities related to $\rho$ by linear
transformations (including $Z(T,f)$, or the resolvent $R$) using only the
first two cumulants. This breaks down only when $\rho$ gets extremely close to
zero. Let us start with $\rho=\bar{\rho}$, the saddle point, and see what
temperatures cause trouble.

For instance, the SFF of the SYK2 model can be written

$SFF=\int
dP(\rho)\exp(N\int\rho\log(1+\exp(iT\lambda-\beta\lambda))d\lambda+cc)$ (71)

We can make use of

$\log(1+e^{ix})=\sum_{n}\log\frac{x-(2n+1)\pi}{(2n+1)\pi}$ (72)

This means we can write the SFF as

$SFF=\int
dP(\rho)\exp(N\int\rho\sum_{n}\log\frac{x-(2n+1)\pi/(T+i\beta)}{(2n+1)\pi/(T+i\beta)}+\log\frac{x-(2n+1)\pi/(T-i\beta)}{(2n+1)\pi/(T-i\beta)}d\lambda)$
(73)

Combining with the quadratic large $N$ measure in equation (70) we can solve
for the new saddle-point value of $\rho$. We have

$\begin{split}\rho=\rho_{0}+\delta\rho\\\
\delta\rho=\frac{1}{N}\sum_{n}(2n+1)\frac{\beta}{T^{2}+\beta^{2}}\frac{1}{(x-\frac{(2n+1)\pi
T}{T^{2}+\beta^{2}})^{2}+(2n+1)^{2}\pi^{2}\frac{\beta^{2}}{(T^{2}+\beta^{2})^{2}}}\end{split}$
(74)

In the limit of small $\beta$, this is just a train of delta functions of mass
$\frac{1}{N}$. The deepest well will be of depth
$\frac{T^{2}+\beta^{2}}{N\pi^{2}\beta}$. If this becomes $O(1)$, comparable to
$\rho_{0}$, the cumulant expansion breaks down. For any fixed nonzero $\beta$
this isn’t a concern, except at very long times. But if $\beta\sim N^{-1}$, we
get the more complicated $N^{\\#T}$ behavior.

In the regime where the cumulant expansion works, and where $T\gg\beta$, we
can evaluate the first two terms $c_{1},c_{2}$ in the cumulant expansion. The
first is non-universal and dependent on the density of states:

$c_{1}=\textrm{E}\sum_{j\neq
0}\frac{(-1)^{j}}{j}Z(ijT-j\beta)=\sum_{j}{(-1)^{j}}j\int
d\lambda\rho(\lambda)\exp(ijT\lambda-j\beta\lambda)$ (75)

In the case of the semicircle law, this becomes a sum of Bessel functions.

The next term is

$c_{2}=\textrm{var}\sum_{j\neq
0}{(-1)^{j}}jZ(ijT-j\beta)=\frac{1}{\mathfrak{b}\pi}\sum_{j\neq
0}\frac{1}{j^{2}}\int d\lambda\exp(-2j\beta\lambda)jT$ (76)

The sum can be evaluated for any values of the parameters. In the case where
$\beta E_{\text{max}}\gg 1$, that sum becomes simple:

$\textrm{var}\sum_{j\neq
0}{(-1)^{j}}jZ(ijT-j\beta)=\frac{1}{\mathfrak{b}\pi}\sum_{j\neq
0}\frac{T}{2\beta j^{2}}=\frac{\pi T}{12\beta}$ (77)

We can graph $\log\left(\text{SFF}(T,\beta)\right)-c_{1}$ and $c_{2}$ as
functions of time. We see good alignment in figure 12.

Figure 12: The quantity in blue is a numerical estimate for the SFF divided by
the theoretical value from just the first cumulant. The value in orange is the
second cumulant term, which seems to explain all the remaining discrepancy. At
later times the two curves seem to diverge, suggesting that at sufficiently
long times later terms in teh cumulant expansion might be necessary.

These findings can be interestingly compared with the so-called Sigma model,
one of the state-of-the-art tools in spectral statistics. This technique
consists of studying a many-body version of a one-body chaotic system [71, 72,
73]. By calculating the partition function of a Sigma Model, one can extract
the spectral determinant, and thus the SFF, of the system of interest. The
work in this paper as well as [35, 36] shows that not just the partition
function but also the SFF of the many-body system contains signatures of the
single-body SFF.

##  Appendix C Nonlinear Hydrodynamics and Interacting Sound Waves

In this appendix, we sketch the inclusion of hydrodynamic interactions using a
diagrammatic perturbation theory along the lines discussed in [47]. We
consider the basic diagramatic setup and, as an example, evaluate a single
diagram. We do not give a comprehensive analysis of interaction effects and we
discuss some key open issues at the end of the section.

Let’s modify our action in equation (18) to include interactions among the
various modes. We now have a Lagrangian that includes a huge number of
possible cubic and higher-order interactions. This leads to perturbative
corrections to $\log Z_{\textrm{enh}}$. For concreteness, we will choose the
Lagrangian

$L_{\text{Kel}}=\frac{1}{2}\phi_{a}\left(\partial_{t}^{2}+\frac{2\Gamma}{c^{2}}\partial_{t}^{3}-c^{2}\partial_{\mu}^{2}\right)\phi_{r}+\frac{2i\Gamma}{\beta
c^{2}}\phi_{a}\partial_{t}^{2}\phi_{a}+\lambda(\partial_{\mu}\phi_{r})^{2}\partial_{t}\phi_{r}\partial_{t}\phi_{a},$
(78)

though there are other quadratic terms with comparable effects, and the action
we wrote isn’t even KMS invariant.

We evaluate corrections to $Z_{\textrm{enh}}$ diagrammatically, according to
the rules in [47]. The propagators in the CTP formalism are

$\begin{split}G^{CTP}_{ra}(t,k)=\textrm{Re}\frac{e^{-\Gamma
k^{2}t}}{ic|k|-\Gamma k^{2}}\exp\left(ic|k|t\right)\theta(t),\\\
G^{CTP}_{rr}(t,k)=\textrm{Re}\frac{e^{-\Gamma k^{2}|t|}}{ic|k|-\Gamma
k^{2}}\exp\left(ic|k||t|\right).\end{split}$ (79)

In the DPT formalism, propagators are wrapped around the time circle according
to

$G^{DPT}(t,k)=\sum_{n=-\infty}^{\infty}G^{CTP}(t+nT,k).$ (80)

Note that the IR divergence for a sound-pole system is much less pronounced
than in the case of diffusive hydrodynamics.

With our propagators in hand, it is time to evaluate Feynman diagrams. Since
the SFF is a partition function on an unorthodox manifold, the correction to
the log of the SFF is given by a sum of bubble diagrams. The leading diagram
is given in figure 13.

Figure 13: The dashed line represents $\phi_{a}$, the solid line represents
$\phi_{r}$.

This can be factored to

$\Delta L=-\lambda\left(\int
d^{d}k_{1}k_{1}^{2}G^{DPT}_{rr}(t=0,k_{1})\right)\left(\int
d^{d}k_{2}\partial_{t}^{2}G^{DPT}_{ra}(t=0,k_{2})\right)$ (81)

The first of the integrals is UV divergent. We will impose a hard cutoff at
$k=\Lambda$. In keeping with realistic hydrodynamics, we will chose
$\Lambda\ll\frac{c}{\Gamma}$. With this cutoff, the first integral works out
to $\frac{S_{d-1}}{d+2}\frac{\Gamma}{c^{2}}\Lambda^{d+2}$. The second integral
can be evaluated as

$\begin{split}\int
d^{d}k_{2}\partial_{t}^{2}G^{DPT}_{ra}(t=0,k)=\sum_{j=1}^{\infty}\int
d^{d}k\partial_{t}^{2}G^{CTP}_{ra}(jT,k)=\textrm{Re}\sum_{j=1}^{\infty}\int
d^{d}k(ic|k|-\Gamma k^{2}){e^{-j\Gamma k^{2}T}}\exp\left(ijc|k|T\right)=\\\
S_{d-1}\textrm{Re}\sum_{j=1}^{\infty}\int_{0}^{\infty}dk(ick-\Gamma
k^{2})|k|^{d-1}{e^{-j\Gamma k^{2}T}}\exp\left(ijckT\right)=\\\
\frac{S_{d-1}}{2}\sum_{j=1}^{\infty}\int_{-\infty}^{\infty}dk(ick-\Gamma
k^{2})|k|^{d-1}{e^{-j\Gamma k^{2}T}}\exp\left(ijckT\right)\end{split}$ (82)

Much like equation (58), the integral in equation (82) has long-time behavior
if $d$ is even, and a quick decay if $d$ is odd. The integral works out to

$\int_{-\infty}^{\infty}dk(ick-\Gamma k^{2})|k|^{d-1}{e^{-j\Gamma
k^{2}T}}\exp\left(ijckT\right)=\begin{cases}\sqrt{\frac{2\pi}{j\Gamma
T}}\left(c\frac{1}{(jT)^{d}}\partial_{c}^{d}-\Gamma\frac{1}{(jT)^{d+1}}\partial_{c}^{d+1}\right)\exp(-j\frac{c^{2}}{4\Gamma}T)&\text{if
}d\textrm{ odd}\\\ c\frac{(d)!}{(ijcT)^{d+1}}+O(T^{-d-2})&\text{if }d\textrm{
even}\end{cases}$ (83)

For odd $d$, the sum in equation (82) is dominated by the $j=1$ term, while
for even $d$ we extract a value of
$c\zeta(d+1)\frac{(d)!}{(icT)^{d+1}}+O(T^{-d-2})$.

Multiplying everything together, we get

$\Delta\text{(action)}/(VT)\approx\begin{cases}-\lambda\frac{S_{d-1}^{2}}{2(d+2)}\frac{\Gamma}{c^{2}}\Lambda^{d+2}\sqrt{\frac{2\pi}{\Gamma
T}}\left(c\frac{1}{(T)^{d}}\partial_{c}^{d}-\Gamma\frac{1}{(T)^{d+1}}\partial_{c}^{d+1}\right)\exp(-\frac{c^{2}}{4\Gamma}T)&\text{if
}d\textrm{ odd},\\\
-\lambda\frac{S_{d-1}^{2}}{2(d+2)}\frac{\Gamma}{c}\Lambda^{d+2}\zeta(d+1)\frac{(d)!}{(icT)^{d+1}}&\text{if
}d\textrm{ even}.\\\ \end{cases}$ (84)

This can be interpreted as a renormalization of $c$ in equation (58). More
complicated diagrams have less straightforward interpretations, but any
individual diagram can be evaluated using these Feynman rules.

There are also more complicated effects with no obvious diagrammatic
interpretation. All of the effects in this section are renormalizations to the
first cumulant in the exponent. But presumably higher cumulants of the single-
particle density of states can also be renormalized by interactions. We do not
know how to systematically investigate these effects. For example, in writing
equation (81) we are assuming a density of states where the number of
Laplacian eigenvalues between $k^{2}$ and $(k+\delta k)^{2}$ is
$\frac{V}{(2\pi)^{d}}S_{d-1}k^{d-1}\delta k$. More realistically, this density
would be a fluctuating variable. Would the overlaps between different
eigenstates depend on the relative energies and on the density of states? We
don’t know. The higher-cumulant analogues of the calculation in this appendix
are an open question.

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