# Spectral Form Factor of a Quantum Spin Glass

Michael Winer  Joint Quantum Institute, Department of Physics, University of
Maryland, College Park, Maryland 20742, USA  Richard Barney  Joint Quantum
Institute, Department of Physics, University of Maryland, College Park,
Maryland 20742, USA  Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742, USA  Christopher L.
Baldwin  Joint Quantum Institute, Department of Physics, University of
Maryland, College Park, Maryland 20742, USA  Victor Galitski  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

It is widely expected that systems which fully thermalize are chaotic in the
sense of exhibiting random-matrix statistics of their energy level spacings,
whereas integrable systems exhibit Poissonian statistics. In this paper, we
investigate a third class: spin glasses. These systems are partially chaotic
but do not achieve full thermalization due to large free energy barriers. We
examine the level spacing statistics of a canonical infinite-range quantum
spin glass, the quantum $p$-spherical model, using an analytic path integral
approach. We find statistics consistent with a direct sum of independent
random matrices, and show that the number of such matrices is equal to the
number of distinct metastable configurations—the exponential of the spin glass
“complexity” as obtained from the quantum Thouless-Anderson-Palmer equations.
We also consider the statistical properties of the complexity itself and
identify a set of contributions to the path integral which suggest a
Poissonian distribution for the number of metastable configurations. Our
results show that level spacing statistics can probe the ergodicity-breaking
in quantum spin glasses and provide a way to generalize the notion of spin
glass complexity beyond models with a semi-classical limit.

###### Contents

  1. 1 Introduction
    1. 1.1 Review of the spectral form factor
    2. 1.2 Review of mean-field spin glasses
    3. 1.3 Summary of results and implications
  2. 2 Real-time dynamics of the quantum $p$-spherical model
    1. 2.1 The model
    2. 2.2 Schwinger-Keldysh path integral
    3. 2.3 TAP equations on the Schwinger-Keldysh contour
  3. 3 The semiclassical ramp in the ergodic phase
    1. 3.1 Effective action
    2. 3.2 Connected solutions
    3. 3.3 Contribution of connected solutions
    4. 3.4 Evaluation of the SFF
  4. 4 The semiclassical ramp in the non-ergodic phase
    1. 4.1 Effective action
    2. 4.2 Connected solutions
    3. 4.3 Contribution of connected solutions
    4. 4.4 Evaluation of the SFF
  5. 5 Higher moments of the evolution operator
    1. 5.1 Effective action
    2. 5.2 Connected solutions
    3. 5.3 Contribution of connected solutions
    4. 5.4 Evaluation of the SFF
  6. A Derivation of Schwinger-Keldysh TAP equations
  7. B Energy of a TAP state
  8. C Accounting for filter functions

##  1 Introduction

An isolated quantum many-body system which reaches an effective thermal
equilibrium state starting from an out-of-equilibrium initial state is often
called “quantum chaotic.” As commonly used, quantum chaos is a loose term
referring to a family of phenomena that typically co-occur, including the
ability of the system to serve as its own heat bath [1, 2, 3], hydrodynamic
behavior of conserved quantities [4, 5, 6, 7, 8], and random-matrix-like
energy eigenvalues [9, 10, 11, 12]. Given this variety, it is crucial to
understand the relationships between different manifestations of quantum chaos
[13, 14].

These relationships are complicated and interesting in large part because the
systems in question have structure, such as locality and symmetry. For
example, if the Hamiltonian has spatial locality, energy conservation implies
the existence of slow hydrodynamic modes and an associated long time scale,
the Thouless time, such that random-matrix behavior is only present for energy
levels closer than the inverse Thouless time [15, 16]. Similarly, if the
Hamiltonian possesses a symmetry, then it can be organized into blocks
labelled by irreducible representations of the symmetry. One finds random-
matrix statistics within each individual block, but full ergodicity is broken
because matrix elements between different blocks are forbidden [17, 18, 19,
20, 21].

It is natural to ask whether there are other ways in which ergodicity can be
lost, and if so, what the resulting spectral statistics of the Hamiltonians
are. In particular, we will better understand the relations between different
measures of quantum chaos by understanding how they are lost and what replaces
them.

Quantum spin glasses provide one well-established context to explore these
questions, since they exhibit a rich phenomenology associated with the
inability to fully thermalize [22, 23, 24, 25, 26, 27, 28] In this paper, we
determine the spectral statistics of an analytically tractable spin glass
model, the quantum $p$-spherical model. We find that up to times polynomial in
the system size, the Hamiltonian can effectively be described as approximately
block-diagonal. Each block behaves as a random matrix independent of the
others, and the number of blocks depends on the energy per particle. At high
energies, there is only one block and the system is ergodic. Below a critical
energy density, the Hamiltonian breaks into exponentially many blocks — the
average number of blocks jumps discontinuously from the high energy regime and
then decreases as the energy density decreases further. We establish these
results via a path integral computation of the spectral form factor (SFF),
which measures correlations between pairs of energy levels [29, 30, 31, 32,
33].

In the remainder of the introduction, we give some physical context by
reviewing the spectral form factor and mean-field spin glasses. We then
summarize our results in more detail and discuss implications. In Sec. 2, we
review the $p$-spherical model in detail. In Sec. 3, we calculate the SFF of
this model in the high-temperature ergodic regime, and in Sec. 4, we do so in
the non-ergodic regime. Finally, in Sec. 5, we investigate higher-moment
analogues of the SFF.

Figure 1: (Top left) Fully chaotic systems have energy levels that are
statistically similar to a Gaussian random matrix, indicated by the orange
block. (Top right) By contrast, quantum spin glasses in the non-ergodic phase
have spectral statistics that resemble a collection of many nearly-decoupled
random matrices (Bottom) Spectral statistics can be diagnosed via the spectral
form factor, denoted $\textrm{SFF}(T)$, which consists of a path integral over
a pair of real-time contours as indicated by the red lines. The universal part
of $\textrm{SFF}(T)$, which is proportional to $T$, is enhanced by the number
of effectively uncoupled sectors (other non-universal contributions are not
indicated here).

###  1.1 Review of the spectral form factor

To study the spectral correlations of a Hamiltonian $H$, a standard tool is
the spectral form factor (SFF) [34, 35], defined as

$\textrm{SFF}(T)\equiv\big{|}\textrm{Tr}e^{-iHT}\big{|}^{2}.$ (1)

In situations where the spectrum is unbounded, or when one wishes to
concentrate on a portion of the spectrum, the trace in Eq. (1) is regulated by
a filter function $f(H)$:

$\textrm{SFF}(T,f)\equiv\big{|}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}.$ (2)

One common choice is $f(H)=e^{-\beta H}$[29, 36], and another is
$f(H)=e^{-c(H-E_{0})^{2}}$. The latter allows one to study level statistics
near a specified energy $E_{0}$.

For a single Hamiltonian, the SFF is an erratic function of time [35]. Thus
one usually considers an ensemble of Hamiltonians and defines the SFF as the
average of Eq. (2) over the ensemble. Throughout this paper, we use the
notation $\mathbb{E}[\,\cdot\,]$ to denote the ensemble average.

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 $\omega$, integrated over $E$ (although the filter function allows
one to concentrate on an arbitrary subset of the spectrum).

Figure 2: The disorder-averaged SFF for the Gaussian unitary ensemble (GUE) of
matrix dimension $N=50$, computed numerically by averaging over ten thousand
realizations. The three distinct regimes — dip, ramp, plateau — are indicated.

It is conceptually useful to split the SFF 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 piece of the SFF, comes solely from the
average density of states. It is the second term, the connected piece, that
contains information on the correlation between energy levels. The assertion
of “random matrix universality” [9, 37] can be phrased as the statement that
an ensemble of quantum chaotic Hamiltonians will generically have the same
connected SFF as the canonical Gaussian ensembles of random matrix theory [11,
38] This conjectured universal behavior is illustrated in Fig. 2, which plots
the disorder-averaged SFF of the Gaussian unitary ensemble (one of the
aforementioned canonical ensembles). Note the three distinct regimes:

  * •

The “dip”, occurring at short times, comes from the disconnected piece of the
SFF (and thus its precise shape is non-universal). It 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 the canonical matrix ensembles, it is a consequence of the
result[11]

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

where $\mathfrak{b}=1$, $2$, $4$ in the orthogonal, unitary, and sympletic
ensembles respectively [11]. The right-hand side being negative is a
reflection of the well-known level repulsion in quantum chaotic systems [39].
Taking the Fourier transform with 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, results from the discreteness of the
spectrum. At times much larger than the inverse level spacing, one expects
that all off-diagonal terms in the double-trace of the SFF sum to effectively
zero, meaning that

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

As the plateau regime is both challenging to access analytically and not
particularly informative, we shall not consider it further in this work.

The bulk of our analysis in this paper is devoted to calculation of the ramp
in a well-known quantum spin glass model, the $p$-spherical model (discussed
below). The results can be understood via the elementary observation that when
a Hamiltonian is block diagonal,

$H=\begin{pmatrix}H_{1}&0&0\\\ 0&H_{2}&0&\ldots\\\ 0&0&H_{3}\\\
&\vdots&&\ddots\end{pmatrix},$ (9)

then $\textrm{Tr}e^{-iHT}=\sum_{k}\textrm{Tr}e^{-iH_{k}T}$. If the different
blocks are independent, then the variance of $\textrm{Tr}e^{-iHT}$ is the sum
of the variance of each $\textrm{Tr}e^{-iH_{k}T}$, i.e., the SFF is the sum of
the SFF for each block. In particular, the coefficient of the universal
linear-in-$T$ ramp is multiplied by the number of independent blocks. Systems
with only approximately block-diagonal Hamiltonians, for which there are small
matrix elements between blocks, have this enhancement of the ramp up to the
transition timescale between blocks. For a more detailed analysis, see Ref.
[19].

###  1.2 Review of mean-field spin glasses

Figure 3: (Left) Cartoon of the energy landscape in a 1RSB spin glass. The
y-axis is energy per spin, $E/N$, where $E$ is energy and $N$ is the number of
spins. Different points on the x-axis represent (very roughly, since the
actual configuration space is $N$-dimensional) different spin configurations
$\sigma$. The dashed line indicates the energy density $\epsilon_{d}$ below
which the system is non-ergodic. (Right) Sketch of the dynamical phase diagram
for a quantum 1RSB spin glass. The x-axis represents parameters controlling
the strength of quantum fluctuations, and the y-axis is energy density. Note
that many other types of phase transitions are also present, in particular
equilibrium transitions, but are not indicated here. See, e.g., Refs. [26, 27]
for more information.

Broadly speaking, spin glasses are systems in which the magnetic moments
$\sigma_{i}$ are frozen but disordered at low temperatures. However, this
definition (much like that of “quantum chaos”) encompasses a wide variety of
phenomena which are in many ways quite distinct, as is made clear by the
literature on the subject [22, 23, 24, 25, 26, 27, 28]. In the present paper,
we focus on what are known as “one-step replica symmetry breaking” (1RSB) spin
glass phases [27]. We are specifically interested in quantum spin glasses, but
we first review the corresponding classical case, for which configurations are
labelled by a list $\sigma\equiv\\{\sigma_{1},\cdots,\sigma_{N}\\}$ and the
Hamiltonian is simply a function of $\sigma$.

While the technical definition of 1RSB is somewhat involved, the qualitative
physics is straightforward to understand and captured by the sketch in Fig. 3.
The energy landscape, i.e., energy as a function of spin configuration, has
many deep wells and steep barriers. In particular, the number of wells is
$e^{O(N)}$ and the heights of the energy barriers separating wells are $O(N)$,
where $N$ is the number of spins. As a result, below a certain energy density
$\epsilon_{d}$, the system is extremely non-ergodic: it remains trapped within
an exponentially small fraction of the thermodynamically relevant
configuration space until exponentially long timescales. While the 1RSB
phenomenon was originally studied in the context of stochastic classical
dynamics [40, 41, 42, 43], it has recently been shown to imply exponentially
long tunneling timescales for isolated quantum dynamics as well [44, 45, 46,
47].

TAP states (named after Thouless, Anderson, and Palmer [48]) provide a more
quantitative description of such “deep wells”. Arguably the most general
definition (see Ref. [25] for others) is in terms of the Legendre transform of
the free energy with respect to local fields:

$F\big{(}\\{m_{i}\\}\big{)}=-\frac{1}{\beta}\log{\textrm{Tr}e^{-\beta
H+\beta\sum_{i}h_{i}\sigma_{i}}}+\sum_{i}h_{i}m_{i},$ (10)

where $H$ is the Hamiltonian of interest and the fields $\\{h_{i}\\}$ are
chosen so that $\langle\sigma_{i}\rangle=m_{i}$ (where
$\langle\,\cdot\,\rangle$ indicates a thermal average). TAP states are simply
the local minima of $F(\\{m_{i}\\})$. Physically, each corresponds to a
different “well” of the energy landscape, including thermal fluctuations
around the lowest point (thus TAP states do generically depend on
temperature). The partition function can be decomposed as a sum over TAP
states:

$Z\equiv\sum_{\sigma}e^{-\beta
H(\sigma)}=\sum_{\alpha}\left[\sum_{\sigma}\delta_{\sigma\in\alpha}e^{-\beta
H(\sigma)}\right]\equiv\sum_{\alpha}Z_{\alpha},$ (11)

where $\alpha$ denotes a TAP state and $\delta_{\sigma\in\alpha}$ restricts
the trace to only those states belonging to TAP state $\alpha$. Note that in
this discussion, $\sigma$ can refer to any set of degrees of freedom: Ising
spins, vector spins, continuous coordinates, etc. In all cases, Eqs. (10) and
(11) can be interpreted accordingly.

Quantum generalizations of spin glasses are usually obtained by adding non-
commuting terms to the Hamiltonian. For example, with an Ising Hamiltonian,
one often interprets $\sigma_{i}$ as the Pauli spin-$z$ operator
$\sigma_{i}^{z}$ and includes an additional transverse field
$\Gamma\sum_{i}\sigma_{i}^{x}$ [49, 50, 51, 52]. On the other hand, with
systems having continuous degrees of freedom (including the one which we study
in this paper), one can interpret $\sigma_{i}$ as a position coordinate and
include the “kinetic energy” $\sum_{i}\pi_{i}^{2}/2\mu$, where $\pi_{i}$ is
the momentum operator conjugate to $\sigma_{i}$ [53, 54]. Generically, the
resulting system has a frozen spin glass phase at low energy and small quantum
fluctuations (the latter being controlled by $\Gamma$ and $\mu^{-1}$
respectively in the examples above), and has a paramagnetic phase at either
high energy or large quantum fluctuations. A sketch of the typical phase
diagram is shown in Fig. 3, with these two phases indicated by “non-ergodic”
and “ergodic”.

It has recently been noted that quantum 1RSB spin glasses can exhibit
eigenstate phase transitions which are distinct from the above [55, 56, 57].
Qualitatively speaking, on the low energy/fluctuation side of the eigenstate
phase boundary, each eigenstate of the Hamiltonian is localized on a single
TAP state. This implies that under the system’s internal dynamics alone (i.e.,
as given by the Schrodinger equation), the system cannot tunnel between TAP
states on any timescale, even times exponential in the number of spins. On the
other side of the phase boundary, each eigenstate is delocalized over many TAP
states in accordance with random matrix behavior. As discussed in Ref. [46],
while this implies that the system does tunnel between TAP states, the
timescale for tunneling is necessarily exponential in system size, analogous
to the activation times under open-system dynamics. Only when there exists a
single TAP state can one identify the phase as genuinely thermalizing. As a
result, one finds phase diagrams like that sketched in Fig. 3, with “non-
ergodic”/“ergodic” indicating whether multiple TAP states exist and
“localized”/“delocalized” referring to the eigenstate properties.

###  1.3 Summary of results and implications

In this paper, we calculate the SFF for a particular ensemble of quantum spin
glasses, the quantum $p$-spherical model (PSM) [58, 53, 54]. We find that in
the ergodic phase, the connected part of the SFF agrees with the expectation
from random matrix theory (Eq. (62) below), while in the non-ergodic phase, it
is enhanced by a factor which is precisely the number of TAP states (Eq.
(109)). Given the discussion in Secs. 1.1 and 1.2, this makes precise and
validates the idea that each metastable state (i.e., TAP state) corresponds to
a block of the Hamiltonian that is quantum chaotic on its own but is nearly
decoupled from all others, thus making the system as a whole non-ergodic [56].
This is the main result of the present work.

Since we only calculate the SFF up to times polynomial in system size, our
results are consistent with but do not test the distinction between localized
and delocalized phases shown in Fig. 3, which is only relevant beyond the
exponentially long timescale corresponding to tunneling between TAP states. We
leave it for future work to incorporate such instanton effects into the path
integral, expecting that they will reduce the SFF to the random matrix result
precisely in the non-ergodic delocalized phase (and even then only beyond the
exponential tunneling timescale).

In addition to the SFF, we consider higher moments of the evolution operator
and identify a set of saddle points (Eq. (134)) suggesting that: i) the number
of TAP states at a given energy is Poisson-distributed, and ii) the numbers of
TAP states at different energies are independent. However, since we have not
evaluated the perturbative corrections around each saddle point, which would
generically dominate over any subleading saddle points, we cannot claim to
have an accurate calculation. It is another direction for future work to study
the distribution of TAP states more systematically.

Our results can be further understood by comparing to Refs. [18] and [19] on
one hand and Refs. [59] and [60] on the other. The first set of papers argues
that for a system which separates into weakly coupled sectors, the SFF
enhancement is the sum of return probabilities over all configurations. If the
time evolution can be considered as an effective Markov process with transfer
rates between sectors given by some matrix $M$, then the SFF enhancement
factor is $\textrm{Tr}e^{MT}$. The second set of papers argues that for a
classical spin glass undergoing Markovian stochastic dynamics with generator
$M$, the number of TAP states can be calculated — and perhaps even defined —
as $\textrm{Tr}e^{MT}$. In this sense, the present paper can be considered as
a “missing link” that extends the results of Refs. [59] and [60] to quantum
systems.

The fact that SFF enhancement is related to return probabilities suggests that
the spectral statistics of spin glasses may contain information on aging
dynamics as well. Another open question is whether the equilibrium replica-
symmetry-breaking transition has any consequences for spectral statistics.
These, as well as those already mentioned, are all promising directions for
future work.

##  2 Real-time dynamics of the quantum $p$-spherical model

###  2.1 The model

The classical $p$-spherical model (PSM) [61] is a disordered spin model with
all-to-all $p$-body interactions. It is defined by the classical Hamiltonian

$H_{\textrm{cl}}\equiv\sum_{(i_{1}\cdots i_{p})}J_{i_{1}\cdots
i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}},$ (12)

where the couplings $J_{i_{1}\dots i_{p}}$ are independent Gaussian random
variables with mean zero and variance

$\mathbb{E}{J_{i_{1}\cdots i_{p}}}^{2}=\frac{J^{2}(p-1)!}{C_{i_{1}\cdots
i_{p}}N^{p-1}}.$ (13)

Here and throughout, $\mathbb{E}$ indicates an average over couplings. The
notation $(i_{1}\cdots i_{p})$ denotes sets of $p$ indices such that $1\leq
i_{1}\leq\cdots\leq i_{p}\leq N$. The sum in Eq. (12) is over all such sets.
Our treatment differs from the standard convention by including a parameter
$J$ for the overall strength of the disorder. To recover the standard
expressions, simply set $J^{2}=p/2$. We also include the combinatorial factor
$C_{i_{1}\cdots i_{p}}=\prod_{1\leq i\leq N}n_{i}!$, where $n_{i}$ is the
number of indices set equal to $i$. This term is almost always one, but its
inclusion avoids $1/N$ corrections in the action.

The $\sigma_{i}$ are real, continuous spin variables subject to the spherical
constraint

$\sum_{i=1}^{N}\sigma_{i}^{2}=N,$ (14)

which ensures that the system has an extensive free energy. It is apparent
that this is a mean-field model without any spatial structure. This allows for
infinite free energy barriers around metastable states in the thermodynamic
limit, making the model ideal for examining the impact of metastability on the
spectral statistics of spin glasses.

In this work, we follow Refs. [58, 53, 54] in generalizing Eq. (12) to a
quantum Hamiltonian $H$. We treat the $\sigma_{i}$ as commuting position
operators, and define conjugate momentum operators $\pi_{i}$ which satisfy the
commutation relations

$[\sigma_{i},\pi_{j}]=i\delta_{ij}.$ (15)

The quantum PSM simply includes a kinetic energy term in the Hamiltonian:

$H=\sum_{i=1}^{N}\frac{\pi_{i}^{2}}{2\mu}+\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}}.$ (16)

The mass $\mu$ is an additional parameter controlling the strength of quantum
fluctuations. To incorporate the spherical constraint, we take the Hilbert
space to be the subspace in which $\sum_{i}\sigma_{i}^{2}$ has eigenvalue $N$.

The quantum PSM may be interpreted as a soft-spin version of the Ising
$p$-spin model in an external transverse field — itself the subject of much
study [62, 51, 63, 64, 65] — where $\mu^{-1}$ is analogous to the transverse
field. Alternatively, if we think of
$\sigma\equiv\\{\sigma_{1},\cdots,\sigma_{N}\\}$ as a position vector in
$N$-dimensional space, the quantum PSM has a natural interpretation as a
particle of mass $\mu$ moving on a hypersphere of radius $\sqrt{N}$. This
particle experiences the Gaussian random potential

$V(\sigma)=\sum_{(i_{1}\cdots i_{p})}J_{i_{1}\cdots
i_{p}}\sigma_{i_{1}}\cdots\sigma_{i_{p}},$ (17)

whose correlation function is

$\mathbb{E}V(\sigma)V(\sigma^{\prime})=\frac{J^{2}(p-1)!}{C_{i_{1}\cdots
i_{p}}N^{p-1}}\sum_{(i_{1}\cdots
i_{p})}\sigma_{i_{1}}\sigma^{\prime}_{i_{1}}\cdots\sigma_{i_{p}}\sigma^{\prime}_{i_{p}}=\frac{J^{2}}{pN^{p-1}}\big{(}\sigma\cdot\sigma^{\prime}\big{)}^{p}.$
(18)

Note that there is a very important difference between $p=2$ and $p>2$: the
former is a Gaussian model, essentially (but for the spherical constraint) a
system of linearly coupled harmonic oscillators. It therefore has
qualitatively different behavior than the $p>2$ models, which are genuinely
interacting and serve as reasonable toy models for rugged energy landscapes.
In this work, we exclusively consider $p>2$.

###  2.2 Schwinger-Keldysh path integral

Figure 4: Summary of the contours, order parameters, and (at least at high
temperature) equations of motion considered in this work. The left column
gives the quantities appropriate to the Schwinger-Keldysh path integral, and
the right column to the spectral form factor (SFF) path integral.
(Top row) Contours for the respective path integrals. Each of the different
branches is labelled, and directions are indicated by arrowheads. Points
connected by dashed lines are identified, making the contours periodic.
(Middle row) Relationship between order parameters of the theory and
observable quantities. $H$ and $Z$ are the $p$-spin Hamiltonian and partition
function respectively. $\mathcal{T}$ and $\widetilde{\mathcal{T}}$ denote time
ordering and anti-ordering.
(Bottom row) Equations of motion. These take the same form for both path
integrals, differing only in the contour $\mathcal{C}$ being used.

Just as other all-to-all models have a saddle-point/mean-field description at
large $N$, so too does the PSM. We start with the disorder-averaged (i.e.,
“annealed”) path integral on the Schwinger-Keldysh contour at inverse
temperature $\beta$, illustrated in the left column of Fig. 4. While it is in
general incorrect (often grossly) to disorder-average the path integral
itself, it is known that the annealed approximation is accurate in the PSM as
long as $\beta$ is less than a critical value $\beta_{s}$ [26]. We shall
assume that this is true throughout. The annealed path integral is

$\displaystyle\mathbb{E}Z_{\textrm{SK}}$
$\displaystyle=\int\mathcal{D}\sigma^{N}\exp\left[\int_{\mathcal{C}}dt\sum_{i}\left(\frac{i\mu}{2}\big{(}\partial_{t}\sigma_{i}(t)\big{)}^{2}-\frac{iz(t)}{2}\big{(}\sigma_{i}(t)^{2}-1\big{)}\right)\right]$
(19) $\displaystyle\qquad\qquad\quad\cdot\int
dP(J)\exp\left[-i\int_{\mathcal{C}}dt\sum_{(i_{1}\cdots i_{p})}J_{i_{1}\cdots
i_{p}}\sigma_{i_{1}}(t)\cdots\sigma_{i_{p}}(t)\right],$

where

$dP(J)\propto\prod_{(i_{1}\cdots i_{p})}dJ_{i_{1}\cdots
i_{p}}\exp\left[-\frac{N^{p-1}C_{i_{1}\cdots i_{p}}J_{i_{1}\cdots
i_{p}}^{2}}{2(p-1)!J^{2}}\right].$ (20)

For brevity, we use $\mathcal{C}$ to denote the entire contour. Thus
$\int_{\mathcal{C}}dt$ indicates a contour integral within the complex-$t$
plane. The Lagrange multiplier $z(t)$ is included to enforce the spherical
constraint. It can be interpreted as a time-dependent harmonic potential whose
value is chosen such that $\sum_{i}\sigma_{i}(t)^{2}=N$ at all times. Thus the
measure $\mathcal{D}\sigma^{N}$ is simply the product measure over each
$\sigma_{i}$ independently. From here, the same manipulations used to get
Schwinger-Dyson equations for the SYK model will give us equations of motion
for the PSM.

One can immediately perform the Gaussian integrals over the couplings to
obtain

$\mathbb{E}Z_{\textrm{SK}}=\int\mathcal{D}\sigma^{N}e^{-NS^{\prime}},$ (21)

where

$\displaystyle NS^{\prime}$
$\displaystyle\equiv\int_{\mathcal{C}}dt\sum_{i}\left(-\frac{i\mu}{2}\big{(}\partial_{t}\sigma_{i}(t)\big{)}^{2}+\frac{iz(t)}{2}\big{(}\sigma_{i}(t)^{2}-1\big{)}\right)$
(22) $\displaystyle\qquad\qquad+\frac{J^{2}(p-1)!}{2C_{i_{1}\cdots
i_{p}}N^{p-1}}\sum_{(i_{1}\cdots
i_{p})}\int_{\mathcal{C}}dtdt^{\prime}\sigma_{i_{1}}(t)\sigma_{i_{1}}(t^{\prime})\cdots\sigma_{i_{p}}(t)\sigma_{i_{p}}(t^{\prime})$
$\displaystyle=\int_{\mathcal{C}}dt\sum_{i}\left(-\frac{i\mu}{2}\big{(}\partial_{t}\sigma_{i}(t)\big{)}^{2}+\frac{iz(t)}{2}\big{(}\sigma_{i}(t)^{2}-1\big{)}\right)+\frac{NJ^{2}}{2p}\int_{\mathcal{C}}dtdt^{\prime}\left(\frac{1}{N}\sum_{i}\sigma_{i}(t)\sigma_{i}(t^{\prime})\right)^{p}.$

Next introduce a “fat unity”,

$\displaystyle 1=$
$\displaystyle\int\mathcal{D}\mathcal{G}\prod_{tt^{\prime}}\delta\Big{(}N\mathcal{G}(t,t^{\prime})-\sum_{i}\sigma_{i}(t)\sigma_{i}(t^{\prime})\Big{)}$
(23) $\displaystyle=$
$\displaystyle\int\mathcal{D}\mathcal{G}\mathcal{D}\mathcal{F}\exp\left[\frac{N}{2}\int_{\mathcal{C}}dtdt^{\prime}\mathcal{F}(t,t^{\prime})\left(\mathcal{G}(t,t^{\prime})-\frac{1}{N}\sum_{i}\sigma_{i}(t)\sigma_{i}(t^{\prime})\right)\right].$

The integral over the self-energy $\mathcal{F}(t,t^{\prime})$ runs along the
imaginary axis, making the second line simply the identity $\int
dpe^{ipx}=2\pi\delta(x)$ (we absorb factors of $2\pi$ into the measure
$\mathcal{D}\mathcal{F}$). However, when we ultimately evaluate the path
integral by saddle point, we shall find that the saddle point value of
$\mathcal{F}(t,t^{\prime})$ is real. Inserting Eq. (23) into the path integral
gives

$\mathbb{E}Z_{\textrm{SK}}=\int\mathcal{D}\mathcal{G}\mathcal{D}\mathcal{F}\int\mathcal{D}\sigma^{N}e^{-NS^{\prime\prime}},$
(24)

where

$\displaystyle NS^{\prime\prime}$
$\displaystyle\equiv-\frac{iN}{2}\int_{\mathcal{C}}dtz(t)+\frac{N}{2}\int_{\mathcal{C}}dtdt^{\prime}\left(\frac{J^{2}}{p}\mathcal{G}(t,t^{\prime})^{p}-\mathcal{F}(t,t^{\prime})\mathcal{G}(t,t^{\prime})\right)$
(25)
$\displaystyle\qquad\qquad+\frac{1}{2}\sum_{i}\left[\int_{\mathcal{C}}dt\left(-i\mu\big{(}\partial_{t}\sigma_{i}(t)\big{)}^{2}+iz(t)\sigma_{i}(t)^{2}\right)+\int_{\mathcal{C}}dtdt^{\prime}\sigma_{i}(t)\mathcal{F}(t,t^{\prime})\sigma_{i}(t^{\prime})\right].$

We can now perform the integral over $\sigma_{i}$, resulting in

$\mathbb{E}Z_{\textrm{SK}}=\int\mathcal{D}\mathcal{G}\mathcal{D}\mathcal{F}e^{-NS_{\textrm{eff}}},$
(26)

where

$S_{\textrm{eff}}\equiv-\frac{i}{2}\int_{\mathcal{C}}dtz(t)+\frac{1}{2}\int_{\mathcal{C}}dtdt^{\prime}\left(\frac{J^{2}}{p}\mathcal{G}(t,t^{\prime})^{p}-\mathcal{F}(t,t^{\prime})\mathcal{G}(t,t^{\prime})\right)+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(\mu\partial_{t}^{2}+z)+\mathcal{F}\Big{]}.$
(27)

At large $N$, the remaining path integral can be evaluated within the saddle
point approximation. The locations of the saddle points are determined by
setting to zero the functional derivatives of Eq. (27):

$\begin{gathered}i\big{(}\mu\partial_{t}^{2}+z(t)\big{)}\mathcal{G}(t,t^{\prime})+\int_{\mathcal{C}}dt^{\prime\prime}\mathcal{F}(t,t^{\prime\prime})\mathcal{G}(t^{\prime\prime},t^{\prime})=\delta(t-t^{\prime}),\\\
\mathcal{F}(t,t^{\prime})=J^{2}\mathcal{G}(t,t^{\prime})^{p-1},\qquad\mathcal{G}(t,t)=1.\end{gathered}$
(28)

Keep in mind that the time arguments in Eq. (28) are complex and range over
the entire Schwinger-Keldysh contour. In particular, although it is hidden in
this compact notation, the infinitesimals $dt$ acquire different phases
depending on the branch of the contour: $dt$ is a positive real infinitesimal
on the upper (“forward”) real-time branch, a negative real infinitesimal on
the lower (“backward”) real-time branch, and a negative imaginary
infinitesimal on the thermal branch.

$\mathcal{G}(t,t^{\prime})$ is the order parameter of this theory. As is clear
from the manner by which it was introduced (top line of Eq. (23)), expectation
values of $\mathcal{G}(t,t^{\prime})$ within the path integral are equivalent
to expectation values of $N^{-1}\sum_{i}\sigma_{i}(t)\sigma_{i}(t^{\prime})$.
The latter are simply time-ordered correlation functions. We shall focus on
the real-time correlation functions, for which it is more transparent to
explicitly indicate the branches by $\alpha\in\\{u,l\\}$ and have $t$ be
simply a real variable. Formally, we have that

$\displaystyle\big{<}\mathcal{G}_{uu}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}Z_{\textrm{SK}}^{-1}\textrm{Tr}e^{-\beta
H}\mathcal{T}\sigma_{i}(t)\sigma_{i}(t^{\prime})\Big{]},$
$\displaystyle\qquad\big{<}\mathcal{G}_{ul}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}Z_{\textrm{SK}}^{-1}\textrm{Tr}e^{-\beta
H}\sigma_{i}(t^{\prime})\sigma_{i}(t)\Big{]},$ (29)
$\displaystyle\big{<}\mathcal{G}_{lu}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}Z_{\textrm{SK}}^{-1}\textrm{Tr}e^{-\beta
H}\sigma_{i}(t)\sigma_{i}(t^{\prime})\Big{]},$
$\displaystyle\qquad\big{<}\mathcal{G}_{ll}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}Z_{\textrm{SK}}^{-1}\textrm{Tr}e^{-\beta
H}\widetilde{\mathcal{T}}\sigma_{i}(t)\sigma_{i}(t^{\prime})\Big{]},$

where $\mathcal{T}$ denotes time ordering and $\widetilde{\mathcal{T}}$
denotes time anti-ordering. Note that we can omit the sum over $i$ because the
different spins (upon disorder-averaging) have equivalent behavior.

A number of formal properties of
$\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})$ are evident from Eq. (29).
For one thing, $\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})$ clearly
depends only on the time difference $t-t^{\prime}$, and we shall often write
$\mathcal{G}_{\alpha\alpha^{\prime}}(t)$ with $t^{\prime}=0$. Since the four
components differ only in time ordering, we see that for any function $f(x)$,

$f\Big{(}\mathcal{G}_{uu}(t)\Big{)}+f\Big{(}\mathcal{G}_{ll}(t)\Big{)}=f\Big{(}\mathcal{G}_{ul}(t)\Big{)}+f\Big{(}\mathcal{G}_{lu}(t)\Big{)}.$
(30)

We can further express all four components in terms of a single complex-valued
function (equivalently two real-valued functions). For example, write
$\mathcal{G}_{lu}(t)$ in terms of its real and imaginary parts as
$\mathcal{G}^{R}(t)+i\mathcal{G}^{I}(t)$. Since
$\mathcal{G}_{lu}(t)^{*}=\mathcal{G}_{lu}(-t)$, $\mathcal{G}^{R}(t)$ is even
and $\mathcal{G}^{I}(t)$ is odd. One can easily confirm that

$\displaystyle\mathcal{G}_{uu}(t)$
$\displaystyle=\mathcal{G}^{R}(t)+i\textrm{sgn}[t]\mathcal{G}^{I}(t),$
$\displaystyle\qquad\mathcal{G}_{ul}(t)$
$\displaystyle=\mathcal{G}^{R}(t)-i\mathcal{G}^{I}(t),$ (31)
$\displaystyle\mathcal{G}_{lu}(t)$
$\displaystyle=\mathcal{G}^{R}(t)+i\mathcal{G}^{I}(t),$
$\displaystyle\qquad\mathcal{G}_{ll}(t)$
$\displaystyle=\mathcal{G}^{R}(t)-i\textrm{sgn}[t]\mathcal{G}^{I}(t).$

One of the most important features of
$\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})$ is the limiting behavior
at large $|t-t^{\prime}|$, as a function of the inverse temperature $\beta$.
Numerical solution of Eq. (28) demonstrates that there is a critical value
$\beta_{d}$ (which is less than $\beta_{s}$):

  * •

For $\beta<\beta_{d}$,
$\lim_{|t-t^{\prime}|\rightarrow\infty}\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})=0$.
We call this the “ergodic” phase ($\mathbb{E}\langle\sigma_{i}(t)\rangle=0$ by
symmetry regardless of temperature, and so in this phase
$\mathbb{E}\langle\sigma_{i}(t)\sigma_{i}(t^{\prime})\rangle\rightarrow\mathbb{E}\langle\sigma_{i}(t)\rangle\mathbb{E}\langle\sigma_{i}(t^{\prime})\rangle$).

  * •

For $\beta_{d}<\beta<\beta_{s}$,
$\lim_{|t-t^{\prime}|\rightarrow\infty}\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})=q_{\textrm{EA}}>0$.
We call this the “non-ergodic” phase. The quantity $q_{\textrm{EA}}$ is
referred to as the “Edwards-Anderson” order parameter.

  * •

For $\beta_{s}<\beta$, our initial annealed approximation is no longer valid.
The replica trick is required to obtain accurate results [23, 24], but (at
least for finite-time dynamical properties) the behavior is qualitatively
similar to that of the non-ergodic phase.

###  2.3 TAP equations on the Schwinger-Keldysh contour

The dynamical calculation described above only hints at the complexity of the
non-ergodic phase. A more complete picture emerges from a generalization in
the spirit of the TAP equations. Our treatment follows that of Ref. [66],
which derived TAP equations on the thermal circle for the quantum PSM. While
the extension to real-time dynamics is straightforward, we are not aware of
any explicit calculation in the literature. Thus we present a detailed
derivation of the following equations in App. A.

As discussed in Sec. 1.2, the TAP free energy (or Gibbs potential) is the
Legendre transform of the free energy with respect to local fields. It is
therefore a function of the magnetization $m_{i}$ of each spin. For the free
energy of quantum systems, the magnetization should also have an imaginary
time index $m_{i}(\tau)$. The imaginary-time correlation function
$\mathcal{G}(\tau,\tau^{\prime})$ becomes an additional order parameter.

We define the TAP action on the Schwinger-Keldysh contour analogously. It is a
function of the magnetizations $m_{i}(t)$ and the correlation function
$\mathcal{G}(t,t^{\prime})$, with $t$ again being complex-valued and ranging
over the entire contour. Specifically,

$\displaystyle iNS_{\textrm{TAP}}[m,\mathcal{G}]$
$\displaystyle\equiv\log{\int\mathcal{D}\sigma^{N}\exp\left[i\sum_{i}S_{i}^{0}-i\int_{\mathcal{C}}dt\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots i_{p}}\sigma_{i_{1}}(t)\cdots\sigma_{i_{p}}(t)\right]}$
(32)
$\displaystyle\qquad\qquad+\frac{iN}{2}\int_{\mathcal{C}}dtz(t)-i\int_{\mathcal{C}}dt\sum_{i}h_{i}(t)m_{i}(t)+\frac{iN}{2}\int_{\mathcal{C}}dtdt^{\prime}\Lambda(t,t^{\prime})\mathcal{G}(t,t^{\prime}),$

where $\mathcal{C}$ denotes the Schwinger-Keldysh contour and

$S_{i}^{0}\equiv\int_{\mathcal{C}}dt\left(\frac{\mu}{2}\big{(}\partial_{t}\sigma_{i}(t)\big{)}^{2}-\frac{z(t)}{2}\sigma_{i}(t)^{2}+h_{i}(t)\sigma_{i}(t)\right)-\frac{1}{2}\int_{\mathcal{C}}dtdt^{\prime}\Lambda(t,t^{\prime})\sigma_{i}(t)\sigma_{i}(t^{\prime}).$
(33)

The fields $h_{i}(t)$ and $\Lambda(t,t^{\prime})$ are not independent
parameters. They are instead chosen so that
$\langle\sigma_{i}(t)\rangle=m_{i}(t)$ and
$N^{-1}\sum_{i}\langle\sigma_{i}(t)\sigma_{i}(t^{\prime})\rangle=\mathcal{G}(t,t^{\prime})$,
just as $z(t)$ is again chosen to enforce
$N^{-1}\sum_{i}\langle\sigma_{i}(t)^{2}\rangle=1$, where the expectation value
is with respect to the action in Eq. (32).

Due to the Legendre-transform structure of $S_{\textrm{TAP}}$, we have that

$N\frac{\partial S_{\textrm{TAP}}}{\partial
m_{i}(t)}=-h_{i}(t),\qquad\frac{\partial
S_{\textrm{TAP}}}{\partial\mathcal{G}(t,t^{\prime})}=\frac{1}{2}\Lambda(t,t^{\prime}).$
(34)

The TAP equations are those for $m_{i}(t)$ and $\mathcal{G}(t,t^{\prime})$
which one gets by setting the right-hand sides of Eq. (34) to zero. The
solutions are therefore the values of magnetization and correlation function
which the system can consistently possess “on its own,” without any external
fields. In this sense, each solution corresponds to a distinct metastable
state. There is no reason why there cannot be many self-consistent solutions,
and indeed, spin glass models such as the PSM do have many at sufficiently low
temperature.

We calculate the TAP equations in App. A. They are simplified by the fact that
we can take $m_{i}(t)=m$ and $z(t)=z$. We also define $q_{\textrm{EA}}\equiv
N^{-1}\sum_{i}m_{i}^{2}$. The equations come out to be (together with
$\mathcal{G}(t,t)=1$)

$i\big{(}\mu\partial_{t}^{2}+z\big{)}\Big{(}\mathcal{G}(t,t^{\prime})-q_{\textrm{EA}}\Big{)}+J^{2}\int_{\mathcal{C}}dt^{\prime\prime}\Big{(}\mathcal{G}(t,t^{\prime\prime})^{p-1}-q_{\textrm{EA}}^{p-1}\Big{)}\Big{(}\mathcal{G}(t^{\prime\prime},t^{\prime})-q_{\textrm{EA}}\Big{)}=\delta(t-t^{\prime}),$
(35)
$J^{2}\int_{\mathcal{C}}dt^{\prime}\Big{(}\mathcal{G}(t,t^{\prime})^{p-1}-(p-1)q_{\textrm{EA}}^{p-2}\mathcal{G}(t,t^{\prime})+(p-2)q_{\textrm{EA}}^{p-1}\Big{)}m_{i}=-izm_{i}-i\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots i_{p}}\frac{\partial(m_{i_{1}}\cdots
m_{i_{p}})}{\partial m_{i}}.$ (36)

Note that Eq. (36) is $N$ equations, one for each spin $i$, and that it holds
equally for any value of $t$ due to time translation invariance. Defining
$\mathcal{F}(t,t^{\prime})\equiv J^{2}\mathcal{G}(t,t^{\prime})^{p-1}$, Eq.
(35) is quite similar to Eq. (28). The only difference is that Eq. (35) uses
$\Delta\mathcal{G}(t,t^{\prime})\equiv\mathcal{G}(t,t^{\prime})-q_{\textrm{EA}}$
and
$\Delta\mathcal{F}(t,t^{\prime})\equiv\mathcal{F}(t,t^{\prime})-J^{2}q_{\textrm{EA}}^{p-1}$,
which decay to zero at large $|t-t^{\prime}|$, rather than
$\mathcal{G}(t,t^{\prime})$ and $\mathcal{F}(t,t^{\prime})$ themselves.

Despite the more involved derivation, $\mathcal{G}(t,t^{\prime})$ remains a
contour-ordered expectation value. Thus, returning to the notation in which
$\alpha\in\\{u,l\\}$ labels branches and $t$ is real,
$\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$ possesses the same formal
properties as discussed in the previous subsection (Eqs. (30) and (31)). Of
particular importance will be the Fourier transform of
$\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t)$ at zero frequency, denoted
$\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(0)$, as well as its
(matrix) inverse,
$\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}^{-1}(0)$. Also define
$L\equiv\int_{-\infty}^{\infty}dt\Delta\mathcal{G}^{R}(t)$ and
$\Lambda\equiv\int_{0}^{\infty}dt\Delta\mathcal{G}^{I}(t)$. Then from Eq.
(31), we see that

$\begin{pmatrix}\Delta\widetilde{\mathcal{G}}_{uu}(0)&\Delta\widetilde{\mathcal{G}}_{ul}(0)\\\
\Delta\widetilde{\mathcal{G}}_{lu}(0)&\Delta\widetilde{\mathcal{G}}_{ll}(0)\end{pmatrix}=\begin{pmatrix}L+2i\Lambda&L\\\
L&L-2i\Lambda\end{pmatrix},$ (37)
$\begin{pmatrix}\Delta\widetilde{\mathcal{G}}_{uu}(0)&\Delta\widetilde{\mathcal{G}}_{ul}(0)\\\
\Delta\widetilde{\mathcal{G}}_{lu}(0)&\Delta\widetilde{\mathcal{G}}_{ll}(0)\end{pmatrix}^{-1}=\frac{1}{4\Lambda^{2}}\begin{pmatrix}L-2i\Lambda&-L\\\
-L&L+2i\Lambda\end{pmatrix}.$ (38)

The multiplicity of solutions to the TAP equations comes from Eq. (36). By use
of Eqs. (35), (37), and (38), it can be written (associating $u$ with 0 and
$l$ with 1)

$\displaystyle\left[(-1)^{\alpha}\sum_{\alpha^{\prime}}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}^{-1}(0)-(p-1)J^{2}q_{\textrm{EA}}^{p-2}\sum_{\alpha^{\prime}}(-1)^{\alpha^{\prime}}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(0)\right]$
$\displaystyle m_{i}$ (39)
$\displaystyle=\left[\frac{1}{2i\Lambda}-(p-1)J^{2}q_{\textrm{EA}}^{p-2}2i\Lambda\right]$
$\displaystyle m_{i}=-i\sum_{(i_{1}\cdots i_{p})}J_{i_{1}\cdots
i_{p}}\frac{\partial(m_{i_{1}}\cdots m_{i_{p}})}{\partial m_{i}}.$

Eq. (39) is identical to that which appears and has been well-studied for the
classical PSM [61, 41, 67, 26]. Thus we simply quote the following results. In
addition to the inverse temperature $\beta$, solutions to Eq. (39) are
parametrized by the quantity

$\mathcal{E}\equiv\frac{1}{NJq_{\textrm{EA}}^{p/2}}\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots i_{p}}m_{i_{1}}\cdots m_{i_{p}},$ (40)

which can be interpreted as a “normalized” potential energy density: each
magnetization has a value which is (very roughly) comparable to
$q_{\textrm{EA}}^{1/2}$, and thus the natural scale for the interaction energy
is $Jq_{\textrm{EA}}^{p/2}$. The value of $q_{\textrm{EA}}$ for a given
$\mathcal{E}<0$ is given by the largest solution to

$-2Jq_{\textrm{EA}}^{p/2-1}\Lambda=\frac{p}{2(p-1)}\left(-\mathcal{E}-\sqrt{\mathcal{E}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\right),\qquad\mathcal{E}_{\textrm{th}}\equiv-\frac{2\sqrt{p-1}}{p},$
(41)

where $\Lambda$ depends on $q_{\textrm{EA}}$ through Eq. (35). One can show
that solutions to Eq. (41) exist only for $\beta>\beta_{d}$, with $\beta_{d}$
the same as defined in Sec. 2.2. Furthermore, Eq. (41) only makes sense if
$\mathcal{E}\leq\mathcal{E}_{\textrm{th}}$. In that case, the number of
solutions $\mathcal{N}(\beta,\mathcal{E})$ to Eq. (39) — in addition to the
trivial solution $m_{i}=0$ — is exponential in system size:
$N^{-1}\log{\mathcal{N}(\beta,\mathcal{E})}\sim\Sigma(\mathcal{E})$, with111
As written, Eq. (42) is a bit sloppy. $\mathcal{N}(\mathcal{E})$ is given by
Eq. (42) when the latter is non-negative and $\beta$ is such that solutions to
Eq. (41) exist. In all other cases, $\mathcal{N}(\mathcal{E})=0$.

$\Sigma(\mathcal{E})=\frac{1}{2}\left(1+2\log{\frac{p}{2}}\right)-\frac{p\mathcal{E}^{2}}{2}+\frac{p^{2}}{8(p-1)}\Big{(}\mathcal{E}+\sqrt{\mathcal{E}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\Big{)}^{2}+\log{\Big{(}-\mathcal{E}+\sqrt{\mathcal{E}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\Big{)}}.$
(42)

The exponent $\Sigma(\mathcal{E})$ is referred to as the “complexity” in the
spin glass literature.

The connection between this TAP approach and the conventional Schwinger-
Keldysh path integral lies in the fact that: i) the inverse temperature
$\beta_{d}$ at which TAP states with non-zero magnetization appear is
identical to that at which the autocorrelation function acquires a non-zero
late-time limit; ii) the overlap determined by Eq. (41) is identical to the
late-time value of the autocorrelation function. This strongly suggests the
following picture:

  * •

For $\beta<\beta_{d}$ (the “ergodic” phase), there exists a single equilibrium
state with zero magnetization, and the correlation function decays to zero on
a finite timescale.

  * •

For $\beta_{d}<\beta$ (the “non-ergodic” phase), there exist exponentially
many metastable states having non-zero magnetization. The number of states is
given by the exponential of the complexity $\Sigma(\mathcal{E})$. Dynamically,
in the $N\rightarrow\infty$ limit, a system prepared in one metastable state
will remain in that state for all time. At finite $N$, it is only on a
timescale exponential in $N$ that the system can transition between states.

Much more can be said about these phases (in particular how the replica-
symmetry-breaking transition at $\beta_{s}$ appears within the TAP approach),
and a large body of literature is devoted to this topic. We refer in
particular to Ref. [26] as an excellent starting point.

In the present work, we determine the spectral statistics of the PSM in both
the ergodic and non-ergodic phase. Those of the former can be computed very
much along the lines of Ref. [29], which we do in Sec. 3. Those of the latter,
however, require novel calculations which we present in Sec. 4.
Unsurprisingly, the properties of TAP states shall play an essential role.

##  3 The semiclassical ramp in the ergodic phase

To reiterate, we are evaluating

$\textrm{SFF}(T,f)\equiv\mathbb{E}\big{|}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}=\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{-iHT}\textrm{Tr}f(H)e^{iHT}\Big{]},$
(43)

where $H$ is the PSM Hamiltonian (Eq. (16)) and $f$ is a filter function as
discussed in Sec. 1.1. Here we consider the ergodic phase, for which the
results are analogous to those of SYK [29]. We then consider the non-ergodic
phase in Sec. 4.

###  3.1 Effective action

The calculation begins by retracing the steps described in Sec. 2.2, only on a
modified contour. We still have upper and lower branches indicated by
$\alpha\in\\{u,l\\}$ (with $u=0$ and $l=1$), but now each is separately
periodic. Furthermore, we no longer have a thermal branch. See the right
column of Fig. 4, as compared to the left column. While some care is required
to account for the filter functions (as discussed in Appendix C), we
ultimately arrive at an expression analogous to Eq. (27):

$\textrm{SFF}(T,f)=\int\mathcal{D}G\mathcal{D}F\,f\big{(}\epsilon_{u}[G]\big{)}f\big{(}\epsilon_{l}[G]\big{)}e^{-NS_{\textrm{eff}}[G,F]},$
(44) $\displaystyle S_{\textrm{eff}}[G,F]$
$\displaystyle=-\frac{i}{2}\int_{0}^{T}dt\sum_{\alpha}(-1)^{\alpha}z_{\alpha}(t)+\frac{1}{2}\int_{0}^{T}dtdt^{\prime}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p}-F_{\alpha\alpha^{\prime}}(t,t^{\prime})G_{\alpha\alpha^{\prime}}(t,t^{\prime})\right)$
(45)
$\displaystyle\qquad\qquad+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}\mu\partial_{t}^{2}+z_{\alpha}\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{\alpha\alpha^{\prime}}\Big{]},$

where the “energy density” $\epsilon_{\alpha}[G]$ is defined as ($0^{+}$
denotes a positive infinitesimal)

$\epsilon_{\alpha}[G]\equiv-\frac{\mu}{2}\partial_{t}^{2}G_{\alpha\alpha}(0^{+},0)-\frac{iJ^{2}}{p}\int_{0}^{T}dt\sum_{\alpha^{\prime}}(-1)^{\alpha^{\prime}}G_{\alpha\alpha^{\prime}}(t,0)^{p}.$
(46)

See App. C for details. The saddle point of $S_{\textrm{eff}}$ is given by the
equations (compare to Eq. (28))

$\begin{gathered}i\big{(}\mu\partial_{t}^{2}+z_{\alpha}(t)\big{)}G_{\alpha\alpha^{\prime}}(t,t^{\prime})+\int_{0}^{T}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}F_{\alpha\alpha^{\prime\prime}}(t,t^{\prime\prime})G_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime},t^{\prime})=(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}),\\\
F_{\alpha\alpha^{\prime}}(t,t^{\prime})=J^{2}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p-1},\qquad
G_{\alpha\alpha}(t,t)=1.\end{gathered}$ (47)

Denoting averages with respect to the path integral of Eq. (44) by
$\langle\,\cdot\,\rangle$, the expectation value of $G$ is related to the
original degrees of freedom as follows (we omit the filter functions here for
brevity):

$\displaystyle\big{<}G_{uu}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}\textrm{Tr}e^{-iHT}\mathcal{T}\sigma_{i}(t)\sigma_{i}(t^{\prime})\textrm{Tr}e^{iHT}\Big{]},$
$\displaystyle\qquad\big{<}G_{ul}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}\textrm{Tr}e^{-iHT}\sigma_{i}(t)\textrm{Tr}e^{iHT}\sigma_{i}(t^{\prime})\Big{]},$
(48) $\displaystyle\big{<}G_{lu}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}\textrm{Tr}e^{-iHT}\sigma_{i}(t^{\prime})\textrm{Tr}e^{iHT}\sigma_{i}(t)\Big{]},$
$\displaystyle\qquad\big{<}G_{ll}(t,t^{\prime})\big{>}$
$\displaystyle=\mathbb{E}\Big{[}\textrm{Tr}e^{-iHT}\textrm{Tr}e^{iHT}\widetilde{\mathcal{T}}\sigma_{i}(t)\sigma_{i}(t^{\prime})\Big{]},$

where $\mathcal{T}$ denotes time ordering and $\widetilde{\mathcal{T}}$
denotes time anti-ordering. One immediately sees from Eq. (48) that:

  1. i)

all components of $\langle G_{\alpha\alpha^{\prime}}(t,t^{\prime})\rangle$ are
time-translation invariant and have period $T$;

  2. ii)

$\langle G_{uu}(t,t^{\prime})\rangle$ and $\langle
G_{ll}(t,t^{\prime})\rangle$ are even functions of $t-t^{\prime}$;

  3. iii)

$\langle G_{ul}(t,t^{\prime})\rangle$ and $\langle
G_{lu}(t,t^{\prime})\rangle$ are in fact independent of both time arguments;

  4. iv)

$\langle G_{uu}(t,t^{\prime})\rangle^{*}=\langle G_{ll}(t,t^{\prime})\rangle$;

  5. v)

$\langle G_{ul}(t,t^{\prime})\rangle^{*}=\langle G_{lu}(t,t^{\prime})\rangle$.

Solutions to Eq. (47) do not necessarily share all these properties, since
some of the symmetries may be spontaneously broken.

However, one simple solution that obeys all of the above is to take
$G_{ul}(t,t^{\prime})=G_{lu}(t,t^{\prime})=0$. The resulting action is
precisely what one would get from averaging each factor of
$\textrm{Tr}e^{-iHT}$ separately, i.e., this solution gives the disconnected
contribution to the SFF:

$\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{-iHT}\textrm{Tr}f(H)e^{iHT}\Big{]}=\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{-iHT}\Big{]}\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{iHT}\Big{]}+\cdots,$
(49)

where $\cdots$ denotes the contribution to the path integral from non-zero
$G_{ul}$ and/or $G_{lu}$. Eq. (49) holds equally well in the non-ergodic
phase, and thus the remainder of this paper will be concerned with determining
those additional contributions.

###  3.2 Connected solutions

Following Ref. [29], we construct approximate solutions to Eq. (47) which
become accurate at large $T$. We first present the solutions and justify them
afterwards. Take $\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})$ to be the
Schwinger-Keldysh correlation function at inverse temperature
$\beta_{\textrm{aux}}$, exactly as given in Sec. 2.2 (Eq. (29) in particular).
Again define $\mathcal{F}_{\alpha\alpha^{\prime}}(t,t^{\prime})\equiv
J^{2}\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p-1}$. A solution to
the SFF saddle point equations (up to terms which vanish at large $T$) is

$G_{\alpha\alpha^{\prime}}(t,t^{\prime})=\sum_{n=-\infty}^{\infty}\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime}+\delta_{\alpha\neq\alpha^{\prime}}\Delta+nT),$
(50)
$F_{\alpha\alpha^{\prime}}(t,t^{\prime})=\sum_{n=-\infty}^{\infty}\mathcal{F}_{\alpha\alpha^{\prime}}(t-t^{\prime}+\delta_{\alpha\neq\alpha^{\prime}}\Delta+nT).$
(51)

Here $\Delta$ can be any real number between 0 and $T$. Thus Eqs. (50) and
(51) constitute a two-parameter family of solutions, the parameters being
$\beta_{\textrm{aux}}$ and $\Delta$. Every such solution contributes to the
SFF.

As for the Lagrange multipliers $z_{\alpha}(t)$, they are independent of $t$
due to time translation invariance. We further have that $z_{u}=z_{l}\equiv
z$: both equal the value of the chemical potential needed to satisfy the
equilibrium spherical constraint, i.e.,
$N^{-1}\sum_{i}\textrm{Tr}Z_{\textrm{SK}}^{-1}e^{-\beta_{\textrm{aux}}H}\sigma_{i}^{2}=1$
(time translation invariance then implies that
$\mathcal{G}_{\alpha\alpha}(t,t)=1$ for all times and both branches).

To justify Eqs. (50) and (51), it is essential that
$\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$ decay exponentially to
zero as $|t-t^{\prime}|\rightarrow\infty$. Thus these solutions only apply in
the ergodic phase. With this in mind, the following comments together
establish their validity:

  * •

The sum over $n$ ensures that $G_{\alpha\alpha^{\prime}}(t-t^{\prime})$ has
period $T$, even though $\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$
does not.

  * •

Since $\mathcal{G}_{\alpha\alpha}(t-t^{\prime})$ decays exponentially,
$G_{\alpha\alpha}(0)\sim 1$ up to terms which are exponentially small in $T$.

  * •

The equation
$F_{\alpha\alpha^{\prime}}(t,t^{\prime})=J^{2}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p-1}$
is satisfied up to exponentially small terms because, when raising Eq. (50) to
the $p-1$’th power, all cross terms are exponentially small (as is the sum
over them). In other words,

$\left(\sum_{n}\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime}+nT+\delta_{\alpha\neq\alpha^{\prime}}\Delta)\right)^{p-1}\sim\sum_{n}\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime}+nT+\delta_{\alpha\neq\alpha^{\prime}}\Delta)^{p-1}.$
(52)

  * •

$\mathcal{G}_{\alpha\alpha^{\prime}}(t,t^{\prime})$ obeys Eq. (28), written
explicitly in terms of components as

$\displaystyle
i\big{(}\mu\partial_{t}^{2}+z\big{)}\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})+\int_{0}^{\infty}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\mathcal{F}_{\alpha\alpha^{\prime\prime}}(t-t^{\prime\prime})\mathcal{G}_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})$
(53)
$\displaystyle\qquad\qquad\qquad\qquad\;\;\;-i\int_{0}^{\beta_{\textrm{aux}}}d\tau^{\prime\prime}\mathcal{F}_{\alpha
v}(t+i\tau^{\prime\prime})\mathcal{G}_{v\alpha^{\prime}}(-i\tau^{\prime\prime}-t^{\prime})=(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}),$

where $v$ denotes the thermal branch of the contour. For $t,t^{\prime}\gg 1$
(which still allows $t-t^{\prime}$ to take any value), $\mathcal{G}_{\alpha
v}(t+i\tau)$ is exponentially small for all $\tau$ and the last term on the
left-hand side can be neglected. We can also take the lower limit of the
$t^{\prime\prime}$ integral to $-\infty$. Thus when checking whether Eq. (50)
satisfies Eq. (47), we have that

$\displaystyle
i\big{(}\mu\partial_{t}^{2}+z\big{)}G_{\alpha\alpha^{\prime}}(t-t^{\prime})+\int_{0}^{T}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}F_{\alpha\alpha^{\prime\prime}}(t-t^{\prime\prime})G_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})$
(54) $\displaystyle\;\;\sim
i\big{(}\mu\partial_{t}^{2}+z\big{)}\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})+\int_{-\infty}^{\infty}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\mathcal{F}_{\alpha\alpha^{\prime\prime}}(t-t^{\prime\prime})\mathcal{G}_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})$
$\displaystyle\;\;\sim(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}),$

again making use of the fact that
$\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$ is exponentially small
when $|t-t^{\prime}|$ is large. The equation is indeed satisfied.

  * •

Finally, the off-diagonal components $G_{ul}(t,t^{\prime})$ and
$G_{lu}(t,t^{\prime})$ contain the parameter $\Delta$ because they break the
separate time translation symmetries in $t$ and $t^{\prime}$ (see property iii
above). Thus if any choice of $\Delta$ solves Eq. (47), so do all choices of
$\Delta\in[0,T)$.

As noted above, we have thus identified a two-parameter family of solutions to
the SFF saddle point equations. In what follows it will be more convenient to
parametrize the solutions by the equilibrium energy density
$\epsilon(\beta_{\textrm{aux}})$ corresponding to inverse temperature
$\beta_{\textrm{aux}}$. We can express $\epsilon(\beta)$ in terms of
$\mathcal{G}$ (and thus $G$) by inserting a factor of $H$ into the Schwinger-
Keldysh contour. Since $H$ clearly commutes with the evolution operator
$e^{-\beta H}e^{iHt}e^{-iHt}$, it can be inserted at any point, in particular
at a late time for which (again because
$\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$ decays exponentially) the
thermal branch can be neglected. By following the same steps as in Appendix C,
we find that $\epsilon(\beta)$ is given precisely by Eq. (46), evaluated on
either branch:

$\displaystyle\epsilon=$
$\displaystyle-\frac{\mu}{2}\partial_{t}^{2}\mathcal{G}_{uu}(0^{+})-\frac{iJ^{2}}{p}\int_{-\infty}^{\infty}dt\Big{(}\mathcal{G}_{uu}(t)^{p}-\mathcal{G}_{ul}(t)^{p}\Big{)}$
(55) $\displaystyle=$
$\displaystyle-\frac{\mu}{2}\partial_{t}^{2}\mathcal{G}_{ll}(0^{+})+\frac{iJ^{2}}{p}\int_{-\infty}^{\infty}dt\Big{(}\mathcal{G}_{ll}(t)^{p}-\mathcal{G}_{lu}(t)^{p}\Big{)}.$

###  3.3 Contribution of connected solutions

Having demonstrated that Eqs. (50) and (51) solve the SFF saddle point
equations, it remains to calculate the action (Eq. (45)) evaluated at the
solutions. First note that, since each solution obeys Eq. (47), we can rewrite
the action as

$S_{\textrm{eff}}=-\frac{J^{2}(p-1)T}{2p}\int_{0}^{T}dt\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}G_{\alpha\alpha^{\prime}}(t)^{p}-\frac{1}{2}\sum_{\omega}\log{\textrm{Det}}\widetilde{G}_{\alpha\alpha^{\prime}}(\omega),$
(56)

where $\omega\in 2\pi\mathbb{Z}/T$ and
$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)\equiv\int_{0}^{T}dte^{i\omega
t}G_{\alpha\alpha^{\prime}}(t)$. Note that the Lagrange multiplier terms have
dropped out since $z_{u}=z_{l}$. Furthermore, since $\int
dtG_{\alpha\alpha^{\prime}}(t)^{p}\sim\int
dt\mathcal{G}_{\alpha\alpha^{\prime}}(t)^{p}$, the general relation in Eq.
(30) implies that the first term of Eq. (56) in fact vanishes.

For the second term, note that by Eq. (50),

$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)=e^{-i\delta_{\alpha\neq\alpha^{\prime}}\omega\Delta}\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega),\qquad\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)\equiv\int_{-\infty}^{\infty}dte^{i\omega
t}\mathcal{G}_{\alpha\alpha^{\prime}}(t).$ (57)

The exponential decay of $\mathcal{G}_{\alpha\alpha^{\prime}}(t)$ implies that
$\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)$ (and thus
$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)$) is an infinitely
differentiable function of $\omega$. Strictly speaking, since the path
integral is regularized by a timestep $\Delta t\rightarrow 0$,
$\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)$ is furthermore
periodic with period $2\pi/\Delta t$. The same is true of
$\log{\textrm{Det}}\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)$. Thus the
Euler-Maclaurin formula [68] gives

$\sum_{n=-\pi/\Delta t}^{\pi/\Delta
t}\log{\textrm{Det}}\widetilde{G}_{\alpha\alpha^{\prime}}\left(\frac{2\pi
n}{T}\right)\sim\frac{T}{2\pi}\int_{-\pi/\Delta t}^{\pi/\Delta
t}d\omega\log{\textrm{Det}}\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)\rightarrow\frac{T}{2\pi}\int_{-\infty}^{\infty}d\omega\log{\textrm{Det}}\widetilde{G}_{\alpha\alpha^{\prime}}(\omega),$
(58)

up to terms which vanish faster than any polynomial in $T^{-1}$. Thus
$S_{\textrm{eff}}$ is proportional to $T$, and we only need to evaluate the
proportionality constant.

Rather than calculate the integral directly, we follow Ref. [29] and evaluate
the derivative $dS_{\textrm{eff}}/dT$ starting from Eq. (45). It is convenient
to rescale time as $t\rightarrow Tt$, so that $T$ becomes simply another
parameter:

$\displaystyle S_{\textrm{eff}}$
$\displaystyle=\frac{T^{2}}{2}\int_{0}^{1}dtdt^{\prime}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p}-F_{\alpha\alpha^{\prime}}(t,t^{\prime})G_{\alpha\alpha^{\prime}}(t,t^{\prime})\right)$
(59)
$\displaystyle\qquad\qquad+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}\mu
T^{-2}\partial_{t}^{2}+z_{\alpha}\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{\alpha\alpha^{\prime}}\Big{]}.$

Note that, since $S_{\textrm{eff}}$ is evaluated at a solution of the saddle
point equations, we only need to differentiate the explicit factors of $T$:

$\displaystyle\frac{dS_{\textrm{eff}}}{dT}=$
$\displaystyle\;T\int_{0}^{1}dtdt^{\prime}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p}-F_{\alpha\alpha^{\prime}}(t,t^{\prime})G_{\alpha\alpha^{\prime}}(t,t^{\prime})\right)$
(60)
$\displaystyle\qquad\qquad-\frac{i\mu}{T^{3}}\int_{0}^{1}dt\sum_{\alpha}(-1)^{\alpha}\partial_{t}^{2}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}\mu
T^{-2}\partial_{t}^{2}+z_{\alpha}(t)\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{\alpha\alpha^{\prime}}\Big{]}^{-1}\bigg{|}_{\alpha=\alpha^{\prime},t=t^{\prime+}}.$

Returning to unscaled time and using Eq. (47), we have that

$\frac{dS_{\textrm{eff}}}{dT}=-\frac{(p-1)J^{2}}{p}\int_{0}^{T}dt\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}G_{\alpha\alpha^{\prime}}(t)^{p}-\frac{i\mu}{T}\sum_{\alpha}(-1)^{\alpha}\partial_{t}^{2}G_{\alpha\alpha}(0^{+})=0,$
(61)

again using Eqs. (30) and (55). Thus the proportionality constant is in fact
zero, i.e., $S_{\textrm{eff}}=0$.

###  3.4 Evaluation of the SFF

To finally compute the SFF, we simply need to sum over all connected
solutions, i.e., integrate over $\epsilon_{\textrm{aux}}$ and $\Delta$.
However, there are additional discrete symmetries which give further
solutions: i) we can time-reverse the off-diagonal components, i.e., take
$G_{ul}(t)=\mathcal{G}_{ul}(-t)$ and $G_{lu}(t)=\mathcal{G}_{lu}(-t)$; ii) if
$p$ is even, we can take $G_{ul}(t)=-\mathcal{G}_{ul}(t)$ and
$G_{lu}(t)=-\mathcal{G}_{lu}(t)$. These must be summed over as well, giving an
additional factor of $2(1+\delta_{p\textrm{ even}})$, where $\delta_{p\textrm{
even}}$ is the indicator function on $p$ being even (1 if true, 0 if false).
Thus our final expression is

$\displaystyle\mathbb{E}\Big{[}\textrm{Tr}f(H)e^{-iHT}\textrm{Tr}f(H)e^{iHT}\Big{]}$
$\displaystyle\sim\left|\mathbb{E}\textrm{Tr}f(H)e^{-iHT}\right|^{2}+\int\frac{d\epsilon_{\textrm{aux}}}{2\pi}f(\epsilon_{\textrm{aux}})^{2}\int_{0}^{T}d\Delta\,2\big{(}1+\delta_{p\textrm{
even}}\big{)}e^{0}$ (62)
$\displaystyle=\left|\mathbb{E}\textrm{Tr}f(H)e^{-iHT}\right|^{2}+2\big{(}1+\delta_{p\textrm{
even}}\big{)}T\int\frac{d\epsilon_{\textrm{aux}}}{2\pi}f(\epsilon_{\textrm{aux}})^{2}.$

The measure $1/2\pi$ can be derived using hydrodynamic methods [19, 29], but
its precise value is not essential for our purposes. The key feature is simply
that the linear-in-$T$ ramp has emerged.

However, keep in mind that Eq. (62) is only valid if the filter function is
such that all contributing values of $\epsilon_{\textrm{aux}}$ lie in the
ergodic phase. In the following section we modify this analysis to hold in the
non-ergodic phase as well. We shall see that it is necessary to incorporate
the structure of multiple TAP states.

##  4 The semiclassical ramp in the non-ergodic phase

As we have stressed repeatedly, the results of Sec. 3 rely heavily on having
an equilibrium correlation function which decays to zero at late times. Thus a
new approach is needed to calculate the SFF in the non-ergodic phase, where
$\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})\rightarrow
q_{\textrm{EA}}\neq 0$ as $|t-t^{\prime}|\rightarrow\infty$. More
specifically, we can no longer neglect the integral over the thermal branch in
Eq. (28), and $\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})$ no longer
solves the SFF equations of motion (Eq. (47)).

However, in the TAP equations of motion, Eq. (35), we can neglect the thermal
branch since $\mathcal{G}(t)-q_{\textrm{EA}}$ does decay to zero exponentially
quickly. This suggests that a viable strategy is to construct solutions for
the SFF using the TAP correlation function. Since TAP states are parametrized
by the quantity $\mathcal{E}$ in Eq. (40), it will be necessary to first
modify the SFF path integral so as to involve $\mathcal{E}$. We associate the
magnetizations and overlap from the TAP approach with time-averaged functions
of the spin configuration, namely

$m_{i}[\sigma]\equiv\frac{1}{T}\int_{0}^{T}dt\sigma_{iu}(t),\qquad
q[\sigma]\equiv\frac{1}{T^{2}}\int_{0}^{T}dtdt^{\prime}\frac{1}{N}\sum_{i}\sigma_{iu}(t)\sigma_{iu}(t^{\prime}).$
(63)

The choice to use only the upper contour in defining $m_{i}[\sigma]$ and
$q[\sigma]$ will become convenient in Sec. 5, but for now one could equally
well use any other combination of branches, say the average of $\sigma_{i}(t)$
over the lower branch or over both branches symmetrically. With these
definitions, we introduce $\mathcal{E}$ via Eq. (40).

###  4.1 Effective action

To begin, insert an additional fat unity into the path integral:

$\displaystyle 1$ $\displaystyle=\int
d\mathcal{E}_{\textrm{aux}}\delta\left[N\mathcal{E}_{\textrm{aux}}-\frac{1}{Jq[\sigma]^{p/2}}\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots
i_{p}}\left(\frac{1}{T}\int_{0}^{T}dt\sigma_{i_{1}u}(t)\right)\cdots\left(\frac{1}{T}\int_{0}^{T}dt\sigma_{i_{p}u}(t)\right)\right].$
(64)

With this addition, the full path integral is

SFF
$\displaystyle=\int\mathcal{D}P(J)\mathcal{D}\sigma^{N}d\mathcal{E}_{\textrm{aux}}d\lambda\exp\left[\frac{i}{2}\sum_{i}\int_{0}^{T}dt\sum_{\alpha}(-1)^{\alpha}\Big{[}\mu\big{(}\partial_{t}\sigma_{i\alpha}(t)\big{)}^{2}-z_{\alpha}(t)\big{(}\sigma_{i\alpha}(t)^{2}-1\big{)}\Big{]}\right]$
(65) $\displaystyle\qquad\cdot\exp\left[-i\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots
i_{p}}\int_{0}^{T}dt\sum_{\alpha}(-1)^{\alpha}\sigma_{i_{1}\alpha}(t)\cdots\sigma_{i_{p}\alpha}(t)\right]$
$\displaystyle\qquad\qquad\cdot\exp\left[iN\lambda\mathcal{E}_{\textrm{aux}}-\frac{i\lambda}{Jq[\sigma]^{p/2}}\sum_{(i_{1}\cdots
i_{p})}J_{i_{1}\cdots
i_{p}}\left(\frac{1}{T}\int_{0}^{T}dt\sigma_{i_{1}u}(t)\right)\cdots\left(\frac{1}{T}\int_{0}^{T}dt\sigma_{i_{p}u}(t)\right)\right].$

Proceeding as usual — averaging over disorder, introducing
$G_{\alpha\alpha^{\prime}}(t,t^{\prime})$ and
$F_{\alpha\alpha^{\prime}}(t,t^{\prime})$ as before, integrating out spins —
we arrive at

$\textrm{SFF}(T,f)=\int
d\mathcal{E}_{\textrm{aux}}d\lambda\mathcal{D}G\mathcal{D}F\,f\big{(}\epsilon_{u}[\lambda,G]\big{)}f\big{(}\epsilon_{l}[\lambda,G]\big{)}e^{-NS_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}},\lambda,G,F]},$
(66) $\displaystyle S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}},\lambda,G,F]$
$\displaystyle=-i\lambda\mathcal{E}_{\textrm{aux}}-\frac{i}{2}\int_{0}^{T}dt\sum_{\alpha}(-1)^{\alpha}z_{\alpha}(t)$
(67)
$\displaystyle\qquad+\frac{\lambda^{2}}{2p}+\frac{J\lambda}{pq[G]^{p/2}}\int_{0}^{T}dt\sum_{\alpha}(-1)^{\alpha}\left(\frac{1}{T}\int_{0}^{T}dt^{\prime}G_{\alpha
u}(t,t^{\prime})\right)^{p}$
$\displaystyle\qquad\qquad+\frac{1}{2}\int_{0}^{T}dtdt^{\prime}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{\alpha\alpha^{\prime}}(t,t^{\prime})^{p}-F_{\alpha\alpha^{\prime}}(t,t^{\prime})G_{\alpha\alpha^{\prime}}(t,t^{\prime})\right)$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}\mu\partial_{t}^{2}+z_{\alpha}\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{\alpha\alpha^{\prime}}\Big{]},$

where we are denoting $q[G]\equiv
T^{-2}\int_{0}^{T}dtdt^{\prime}G_{uu}(t,t^{\prime})$. The argument of the
filter function is modified as well; it is now

$\epsilon_{\alpha}[\lambda,G]\equiv-\frac{\mu}{2}\partial_{t}^{2}G_{\alpha\alpha}(0^{+},0)-\frac{iJ^{2}}{p}\int_{0}^{T}dt\sum_{\alpha^{\prime}}(-1)^{\alpha^{\prime}}G_{\alpha\alpha^{\prime}}(t,0)^{p}-\frac{iJ\lambda}{pq[G]^{p/2}}\left(\frac{1}{T}\int_{0}^{T}dtG_{\alpha
u}(t,0)\right)^{p}.$ (68)

Note that $\mathcal{E}_{\textrm{aux}}$ enters linearly into the action. Thus
if we were to integrate over $\mathcal{E}_{\textrm{aux}}$ at this point, we
would obtain a $\delta$-function forcing $\lambda=0$. The action would then
reduce to the ergodic-phase expression, Eq. (45). While reassuring, this would
not have accomplished anything, so we instead treat
$\mathcal{E}_{\textrm{aux}}$ as a fixed parameter for now. We obtain saddle
point equations by differentiating Eq. (67) only with respect to $\lambda$,
$z$, $G$, and $F$.

The saddle point equations, assuming time translation invariance from the
outset, are

$i\big{(}\mu\partial_{t}^{2}+z_{\alpha}\big{)}G_{\alpha\alpha^{\prime}}(t-t^{\prime})+\int_{0}^{T}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}F_{\alpha\alpha^{\prime\prime}}(t-t^{\prime\prime})G_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})=(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}),$
(69) $\displaystyle
F_{\alpha\alpha^{\prime}}(t)=J^{2}G_{\alpha\alpha^{\prime}}(t)^{p-1}$
$\displaystyle+\frac{J\lambda}{Tq[G]^{p/2}}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\left(\frac{\widetilde{G}_{\alpha\alpha^{\prime}}(0)}{T}\right)^{p-1}$
(70) $\displaystyle-\frac{J\lambda}{Tq[G]^{p/2+1}}\delta_{\alpha
u}\delta_{\alpha^{\prime}u}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\left(\frac{\widetilde{G}_{\alpha^{\prime\prime}u}(0)}{T}\right)^{p},$
$\mathcal{E}_{\textrm{aux}}=-\frac{iJT}{pq[G]^{p/2}}\sum_{\alpha}(-1)^{\alpha}\left(\frac{\widetilde{G}_{\alpha
u}(0)}{T}\right)^{p}-\frac{i\lambda}{p},$ (71)

as well as the usual requirement $G_{\alpha\alpha}(0)=1$. Here
$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)$ is the Fourier transform of
$G_{\alpha\alpha^{\prime}}(t)$.

###  4.2 Connected solutions

With $\mathcal{E}_{\textrm{aux}}$ fixed, let
$\mathcal{G}_{\alpha\alpha^{\prime}}(t)$ be the solution to the TAP equation
of motion (Eq. (35)) corresponding to inverse temperature
$\beta_{\textrm{aux}}$. Denote the Edwards-Anderson order parameter at
$\mathcal{E}_{\textrm{aux}}$ and $\beta_{\textrm{aux}}$ by $q_{\textrm{EA}}$.
Also recall the various auxiliary quantities we defined in Sec. 2.3: the self-
energy $\mathcal{F}_{\alpha\alpha^{\prime}}(t)\equiv
J^{2}\mathcal{G}_{\alpha\alpha^{\prime}}(t)^{p-1}$, the deviations
$\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t)\equiv\mathcal{G}_{\alpha\alpha^{\prime}}(t)-q_{\textrm{EA}}$
and
$\Delta\mathcal{F}_{\alpha\alpha^{\prime}}(t)\equiv\mathcal{F}_{\alpha\alpha^{\prime}}(t)-J^{2}q_{\textrm{EA}}^{p-1}$,
and the quantity $\Lambda\equiv\int_{0}^{\infty}dt\Delta\mathcal{G}^{I}(t)$.
We have that $\mathcal{G}_{\alpha\alpha^{\prime}}(t)$ and
$\mathcal{F}_{\alpha\alpha^{\prime}}(t)$ obey Eq. (35), which by taking $t$
and $t^{\prime}$ to be far from the thermal branch can be written

$i\big{(}\mu\partial_{t}^{2}+z\big{)}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t-t^{\prime})+\int_{-\infty}^{\infty}dt^{\prime\prime}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\Delta\mathcal{F}_{\alpha\alpha^{\prime\prime}}(t-t^{\prime\prime})\Delta\mathcal{G}_{\alpha^{\prime\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})=(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}).$
(72)

We also have, as a result of Eq. (39), the relationship

$\mathcal{E}_{\textrm{aux}}=\frac{2(p-1)Jq_{\textrm{EA}}^{p/2-1}\Lambda}{p}+\frac{1}{2pJq_{\textrm{EA}}^{p/2-1}\Lambda}.$
(73)

Finally, recall the expression for the complexity $\Sigma(\mathcal{E})$, the
logarithm of the number of solutions to the TAP magnetization equations at
$\mathcal{E}$:

$\Sigma(\mathcal{E})=\frac{1}{2}\left(1+2\log{\frac{p}{2}}\right)-\frac{p\mathcal{E}^{2}}{2}+\frac{p^{2}}{8(p-1)}\left(\mathcal{E}+\sqrt{\mathcal{E}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\right)^{2}+\log{\left(-\mathcal{E}+\sqrt{\mathcal{E}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\right)},$
(74)

where $\mathcal{E}_{\textrm{th}}^{2}=4(p-1)/p^{2}$. Using Eq. (73), we can
express $\Sigma(\mathcal{E}_{\textrm{aux}})$ in terms of $\Lambda$ and
$q_{\textrm{EA}}$:

$\Sigma(\mathcal{E}_{\textrm{aux}})=-\frac{p-2}{2p}-\frac{1}{8pJ^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}+\frac{2(p-1)J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}{p}-\frac{1}{2}\log{4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}.$
(75)

Our solution to the SFF saddle point equations, Eqs. (69) through (71), is
best written in the frequency domain (tildes denote Fourier transforms):

$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)=Tq_{\textrm{EA}}\delta_{\omega
0}+\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{g}_{\alpha\alpha^{\prime}}(\omega)}{T},$
(76)
$\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)=\left(TJ^{2}q_{\textrm{EA}}^{p-1}+ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\right)\delta_{\omega
0}+\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{f}_{\alpha\alpha^{\prime}}(\omega)}{T},$
(77)
$\lambda=ip\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}+\frac{\delta}{T}.$
(78)

We again take $z_{\alpha}$ to be the equilibrium value corresponding to
$\beta_{\textrm{aux}}$. The precise form of the correction terms
$\widetilde{g}_{\alpha\alpha^{\prime}}(\omega)$,
$\widetilde{f}_{\alpha\alpha^{\prime}}(\omega)$, and $\delta$ is largely
unimportant — the essential feature is simply that they are $O(1)$ and the
corrections are thus $O(T^{-1})$. Note that in the time domain, this solution
amounts to

$G_{\alpha\alpha^{\prime}}(t)=q_{\textrm{EA}}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)+\frac{g_{\alpha\alpha^{\prime}}(t)}{T},$
(79)
$F_{\alpha\alpha^{\prime}}(t)=J^{2}q_{\textrm{EA}}^{p-1}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{F}_{\alpha\alpha^{\prime}}(t+nT)+\frac{ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}}{T}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}+\frac{f_{\alpha\alpha^{\prime}}(t)}{T}.$
(80)

The sums are convergent because $\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t)$
and $\Delta\mathcal{F}_{\alpha\alpha^{\prime}}(t)$ decay rapidly to zero as
$|t|\rightarrow\infty$.

Although we have omitted it for notational simplicity, we can add a term
$\delta_{\alpha\neq\alpha^{\prime}}\Delta$ to the time arguments of
$\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)$ and
$\Delta\mathcal{F}_{\alpha\alpha^{\prime}}(t+nT)$ for any $\Delta\in[0,T)$,
exactly as in Sec. 3. Due to the separate time translation symmetry on each
branch of the SFF contour, all such solutions are equally valid and contribute
the same action. Thus we shall demonstrate the validity of Eqs. (76) through
(78) and evaluate the action only for $\Delta=0$, but then integrate over all
$\Delta\in[0,T)$ in the final expression for the SFF.

Let us first confirm that our solution satisfies the saddle point equation for
$\lambda$, Eq. (71). Referring to Eq. (37), we have that
$\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(0)=L+(-1)^{\alpha}2i\Lambda\delta_{\alpha\alpha^{\prime}}$.
Thus

$q[G]=q_{\textrm{EA}}+\frac{L+2i\Lambda}{T}+O(T^{-2}),$ (81)

and Eq. (71) becomes

$\mathcal{E}_{\textrm{aux}}=2Jq_{\textrm{EA}}^{p/2-1}\Lambda-\frac{i\lambda}{p}+O(T^{-1}).$
(82)

Solving for $\lambda$ indeed gives Eq. (78). The $O(T^{-1})$ terms determine
$\delta$ as a function of the other quantities.

Now turn to Eq. (70). In the frequency domain, the right-hand side evaluates
to222Since $\widetilde{g}_{\alpha\alpha^{\prime}}(\omega)=O(1)$ with respect
to $T$, $g_{\alpha\alpha^{\prime}}(t)$ decays to zero as
$|t|\rightarrow\infty$ (at least to leading order).

$\displaystyle\int_{0}^{T}dte^{i\omega
t}J^{2}\left(q_{\textrm{EA}}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)\right)^{p-1}+ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\delta_{\omega 0}+O(T^{-1})$ (83)
$\displaystyle\qquad\qquad\stackrel{{\scriptstyle\textrm{set}}}{{=}}\left(TJ^{2}q_{\textrm{EA}}^{p-1}+ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\right)\delta_{\omega
0}+\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{f}_{\alpha\alpha^{\prime}}(\omega)}{T}.$

Along the lines of Eq. (52), we have that

$\displaystyle
J^{2}\left(q_{\textrm{EA}}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)\right)^{p-1}$
$\displaystyle=J^{2}q_{\textrm{EA}}^{p-1}+J^{2}\sum_{r=1}^{p-1}\binom{p-1}{r}q_{\textrm{EA}}^{p-1-r}\left(\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)\right)^{r}$
(84) $\displaystyle\sim
J^{2}q_{\textrm{EA}}^{p-1}+J^{2}\sum_{r=1}^{p-1}\binom{p-1}{r}q_{\textrm{EA}}^{p-1-r}\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)^{r}$
$\displaystyle=J^{2}q_{\textrm{EA}}^{p-1}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{F}_{\alpha\alpha^{\prime}}(t+nT).$

Thus, up to $O(1)$, both sides of Eq. (83) agree.

Finally, we confirm that Eq. (69) is satisfied. At non-zero frequencies, we
have

$i\big{(}-\mu\omega^{2}+z\big{)}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)+\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime\prime}}(\omega)\Delta\widetilde{\mathcal{G}}_{\alpha^{\prime\prime}\alpha^{\prime}}(\omega)+O(T^{-1})\stackrel{{\scriptstyle\textrm{set}}}{{=}}(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}},$
(85)

which agrees at $O(1)$ due to Eq. (72). At zero frequency, we instead have

$\displaystyle
iz\left(Tq_{\textrm{EA}}+\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(0)+\frac{\widetilde{g}_{\alpha\alpha^{\prime}}(0)}{T}\right)$
(86)
$\displaystyle\quad+\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\left(TJ^{2}q_{\textrm{EA}}^{p-1}+ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime\prime}u}\big{)}+\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime\prime}}(0)+\frac{\widetilde{f}_{\alpha\alpha^{\prime\prime}}(0)}{T}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\left(Tq_{\textrm{EA}}+\Delta\widetilde{\mathcal{G}}_{\alpha^{\prime\prime}\alpha^{\prime}}(0)+\frac{\widetilde{g}_{\alpha^{\prime\prime}\alpha^{\prime}}(0)}{T}\right)\stackrel{{\scriptstyle\textrm{set}}}{{=}}(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}.$

The $O(T)$ terms come out to be

$T\left(izq_{\textrm{EA}}+\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\Big{(}J^{2}q_{\textrm{EA}}^{p-1}\Delta\widetilde{\mathcal{G}}_{\alpha^{\prime\prime}\alpha^{\prime}}(0)+q_{\textrm{EA}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime\prime}}(0)\Big{)}+ipJq_{\textrm{EA}}^{p/2}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\right)\stackrel{{\scriptstyle\textrm{set}}}{{=}}0.$
(87)

Yet from the TAP magnetization equations, Eq. (36), it follows that

$\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\Big{(}q_{\textrm{EA}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime\prime}}(0)-(p-1)J^{2}q_{\textrm{EA}}^{p-1}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime\prime}}(0)\Big{)}=-izq_{\textrm{EA}}-ipJq_{\textrm{EA}}^{p/2}\mathcal{E}_{\textrm{aux}}.$
(88)

Thus Eq. (87) evaluates to

$-ipJq_{\textrm{EA}}^{p/2}\mathcal{E}_{\textrm{aux}}+pJ^{2}q_{\textrm{EA}}^{p-1}\sum_{\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime\prime}}(0)+ipJq_{\textrm{EA}}^{p/2}\mathcal{E}_{\textrm{aux}}-2ipJ^{2}q_{\textrm{EA}}^{p-1}\Lambda=0,$
(89)

using Eq. (37). The $O(1)$ terms of Eq. (86) determine
$\widetilde{g}_{\alpha\alpha^{\prime}}(0)$ and
$\widetilde{f}_{\alpha\alpha^{\prime}}(0)$. We have therefore confirmed that
all saddle point equations are solved by Eqs. (76) through (78).

###  4.3 Contribution of connected solutions

It remains only to evaluate the action, Eq. (67), at the above solution. The
action can be written as

$\displaystyle S_{\textrm{eff}}$
$\displaystyle=-i\lambda\mathcal{E}_{\textrm{aux}}+\frac{\lambda^{2}}{2p}+\frac{JT\lambda}{pq[G]^{p/2}}\sum_{\alpha}(-1)^{\alpha}\left(\frac{\widetilde{G}_{\alpha
u}(0)}{T}\right)^{p}$ (90)
$\displaystyle\qquad+\frac{T}{2}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\int_{0}^{T}dt\left(\frac{J^{2}}{p}G_{\alpha\alpha^{\prime}}(t)^{p}-F_{\alpha\alpha^{\prime}}(t)G_{\alpha\alpha^{\prime}}(t)\right)$
$\displaystyle\qquad\qquad+\frac{1}{2}\sum_{\omega}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}-\mu\omega^{2}+z\big{)}+(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)\Big{]}.$

Interestingly, we can determine $S_{\textrm{eff}}$ up to a single additive
constant simply by noting that
$dS_{\textrm{eff}}/d\mathcal{E}_{\textrm{aux}}=-i\lambda$ (recall that
$S_{\textrm{eff}}$ is stationary with respect to variations in all quantities
other than $\mathcal{E}_{\textrm{aux}}$). With $\lambda$ given by Eq. (78) and
$q_{\textrm{EA}}^{p/2-1}\Lambda$ given by Eq. (41), we can carry out the
integral to obtain that

$S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}}]=\frac{p\mathcal{E}_{\textrm{aux}}^{2}}{2}-\frac{p^{2}}{8(p-1)}\Big{(}\mathcal{E}_{\textrm{aux}}+\sqrt{\mathcal{E}_{\textrm{aux}}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\Big{)}^{2}-\log{\Big{(}-\mathcal{E}_{\textrm{aux}}+\sqrt{\mathcal{E}_{\textrm{aux}}^{2}-\mathcal{E}_{\textrm{th}}^{2}}\Big{)}}+C,$
(91)

for some unknown constant $C$. Comparing to Eq. (42), this is highly
suggestive that $S_{\textrm{eff}}=-\Sigma(\mathcal{E}_{\textrm{aux}})$. Of
course, we do need to determine the remaining constant, and so we now turn to
a more elaborate calculation.

Rather than substitute Eqs. (76) and (77) into Eq. (90), we instead use the
simpler functions

$\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=Tq_{\textrm{EA}}\delta_{\omega
0}+\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega),$ (92)
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=\left(TJ^{2}q_{\textrm{EA}}^{p-1}+ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\right)\delta_{\omega
0}+\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{f}_{\alpha\alpha^{\prime}}(0)}{T}\delta_{\omega
0},$ (93)

and show that the error incurred in doing so vanishes at large $T$.

Let us demonstrate that the error is negligible first. At any non-zero
frequency, we have that

$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)=\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{g}_{\alpha\alpha^{\prime}}(\omega)}{T},\qquad\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)=\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)+\frac{\widetilde{f}_{\alpha\alpha^{\prime}}(\omega)}{T}.$
(94)

The partial derivatives of $S_{\textrm{eff}}$ at nonzero $\omega$ are

$\frac{\partial
S_{\textrm{eff}}}{\partial\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)}=\frac{1}{2}(-1)^{\alpha+\alpha^{\prime}}\int_{0}^{T}dte^{-i\omega
t}\Big{(}J^{2}G_{\alpha\alpha^{\prime}}(t)^{p-1}-F_{\alpha\alpha^{\prime}}(t)\Big{)},$
(95) $\frac{\partial
S_{\textrm{eff}}}{\partial\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)}=\frac{1}{2}(-1)^{\alpha+\alpha^{\prime}}\bigg{(}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}-\mu\omega^{2}+z\big{)}+(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}_{\alpha^{\prime}\alpha}(\omega)\Big{]}_{\alpha\alpha^{\prime}}^{-1}-\int_{0}^{T}dte^{-i\omega
t}G_{\alpha\alpha^{\prime}}(t)\bigg{)},$ (96)

which vanish when evaluated at
$\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)$
and
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)$.
Thus the $O(T^{-1})$ difference between
$\widetilde{G}_{\alpha\alpha^{\prime}}(\omega)$ and
$\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)$, as with
$\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)$ and
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)$, translates only to
an $O(T^{-2})$ difference in the action. Even after summing over all
$\omega\neq 0$, the total error333 Since the $G(t)^{p}$ term is not diagonal
in the frequency domain, this argument requires a bit more care. One can
easily show that
$\partial^{2}S_{\textrm{eff}}/\partial\widetilde{G}(\omega)\partial\widetilde{G}(\omega^{\prime})$
is $O(T^{-1})$ for $\omega\neq\pm\omega^{\prime}$ and $O(1)$ for
$\omega=\pm\omega^{\prime}$. Summing over all frequencies, the former case
gives a total contribution $O(T^{-3})O(T^{2})=O(T^{-1})$ and the latter gives
$O(T^{-2})O(T)=O(T^{-1})$. The total error is thus $O(T^{-1})$ as claimed.  is
only $O(T^{-1})$.

Neglecting non-zero frequencies,
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)$ is identical to
$\widetilde{F}_{\alpha\alpha^{\prime}}(\omega)$ and
$\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(\omega)$ differs only by
$\widetilde{g}_{\alpha\alpha^{\prime}}(0)\delta_{\omega 0}/T$. In the time
domain, the latter corresponds to

$G_{\alpha\alpha^{\prime}}(t)=G^{\prime}_{\alpha\alpha^{\prime}}(t)+\frac{\widetilde{g}_{\alpha\alpha^{\prime}}(0)}{T^{2}}=q_{\textrm{EA}}+\sum_{n=-\infty}^{\infty}\Delta\mathcal{G}_{\alpha\alpha^{\prime}}(t+nT)+\frac{\widetilde{g}_{\alpha\alpha^{\prime}}(0)}{T^{2}}.$
(97)

Yet

$\frac{\partial S_{\textrm{eff}}}{\partial
G_{\alpha\alpha^{\prime}}(t)}=\frac{T}{2}(-1)^{\alpha+\alpha^{\prime}}\Big{(}J^{2}G_{\alpha\alpha^{\prime}}(t)^{p-1}-F_{\alpha\alpha^{\prime}}(t)\Big{)}+O(1).$
(98)

When evaluated at $G^{\prime}_{\alpha\alpha^{\prime}}(t)$ and
$F^{\prime}_{\alpha\alpha^{\prime}}(t)$, the $O(T)$ contribution vanishes (see
Eq. (84)). Thus $\partial S_{\textrm{eff}}/\partial
G_{\alpha\alpha^{\prime}}(t)$ is $O(1)$, and an $O(T^{-2})$ change to
$G_{\alpha\alpha^{\prime}}(t)$ leads only to an $O(T^{-1})$ change in the
action even after integrating over $t$.

Since all errors are $O(T^{-1})$, we can safely evaluate $S_{\textrm{eff}}$ at
Eqs. (92) and (93) rather than the full solution (we still use Eq. (78) for
$\lambda$). The first line of Eq. (90) can be computed straightforwardly. It
comes out to be

$\frac{p\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}^{2}}{2}=\frac{1}{8pJ^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}-\frac{1}{p}+\frac{2J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}{p},$
(99)

where we used Eq. (73) to obtain the right-hand side.

Next consider the bottom line. Since
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(\omega)=\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)$
for $\omega\neq 0$, while
$\widetilde{F}^{\prime}_{\alpha\alpha^{\prime}}(0)=\widetilde{F}_{\alpha\alpha^{\prime}}(0)$,
we can write the determinant term as

$\displaystyle\frac{1}{2}\sum_{\omega}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}\big{(}-\mu\omega^{2}+z\big{)}+(-1)^{\alpha+\alpha^{\prime}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)\Big{]}$
(100)
$\displaystyle\qquad\qquad+\frac{1}{2}\log{\textrm{Det}}\Big{[}iz(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}+(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}_{\alpha\alpha^{\prime}}(0)\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad-\frac{1}{2}\log{\textrm{Det}}\Big{[}iz(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}+(-1)^{\alpha+\alpha^{\prime}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(0)\Big{]}.$

The top line vanishes by exactly the same reasoning as in Sec. 3.3: it is
proportional to $T$ by the Euler-Maclaurin formula, and then must be zero
since the derivative with respect to $T$ vanishes. Given Eq. (37), the bottom
line is simply

$\frac{1}{2}\log{\textrm{Det}}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(0)=\frac{1}{2}\log{4\Lambda^{2}}.$
(101)

For the middle line we take an indirect approach. We have that
$iz(-1)^{\alpha}\delta_{\alpha\alpha^{\prime}}+(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}_{\alpha\alpha^{\prime}}(0)$
is the matrix inverse to $\widetilde{G}_{\alpha\alpha^{\prime}}(0)$ (using the
full solution for the latter, Eq. (76)). Written out,

$\begin{pmatrix}iz+\widetilde{F}_{uu}(0)&-\widetilde{F}_{ul}(0)\\\
-\widetilde{F}_{lu}(0)&-iz+\widetilde{F}_{ll}(0)\end{pmatrix}=\begin{pmatrix}\widetilde{G}_{uu}(0)&\widetilde{G}_{ul}(0)\\\
\widetilde{G}_{lu}(0)&\widetilde{G}_{ll}(0)\end{pmatrix}^{-1}=\frac{1}{\textrm{Det}\widetilde{G}(0)}\begin{pmatrix}\widetilde{G}_{ll}(0)&-\widetilde{G}_{ul}(0)\\\
-\widetilde{G}_{lu}(0)&\widetilde{G}_{uu}(0)\end{pmatrix}.$ (102)

Rather than this $(u,l)$ basis, express Eq. (102) in the $(u+l,u-l)$ basis
(called “classical”/“quantum” in the Keldysh literature), denoted $(+,-)$:

$\begin{pmatrix}\widetilde{F}_{--}(0)&iz+\widetilde{F}_{-+}(0)\\\
iz+\widetilde{F}_{+-}(0)&\widetilde{F}_{++}(0)\end{pmatrix}=\frac{1}{\textrm{Det}\widetilde{G}(0)}\begin{pmatrix}\widetilde{G}_{--}(0)&-\widetilde{G}_{+-}(0)\\\
-\widetilde{G}_{-+}(0)&\widetilde{G}_{++}(0)\end{pmatrix}.$ (103)

We can read off that
$\textrm{Det}\widetilde{G}(0)^{-1}=\widetilde{F}_{++}(0)/\widetilde{G}_{++}(0)$.
Note that we only need $\widetilde{G}_{++}(0)$ and $\widetilde{F}_{++}(0)$ to
$O(T)$ in order to calculate the determinant to $O(1)$. Thus the middle line
of Eq. (100) evaluates to $(\log{J^{2}q_{\textrm{EA}}^{p-2}})/2$, and the
total contribution of the determinant term is

$\frac{1}{2}\log{4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}.$ (104)

Lastly consider the middle line of Eq. (90). Since
$G^{\prime}_{\alpha\alpha^{\prime}}(t)=\mathcal{G}_{\alpha\alpha^{\prime}}(t)$
(up to exponentially small corrections),
$\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}G^{\prime}_{\alpha\alpha^{\prime}}(t)^{p}=0$
by virtue of Eq. (30). We are left with

$\displaystyle-\frac{T}{2}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\int_{0}^{T}dtF^{\prime}_{\alpha\alpha^{\prime}}(t)G^{\prime}_{\alpha\alpha^{\prime}}(t)$
(105)
$\displaystyle\qquad\sim-\frac{T}{2}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\int_{0}^{T}dt\mathcal{F}_{\alpha\alpha^{\prime}}(t)\mathcal{G}_{\alpha\alpha^{\prime}}(t)$
$\displaystyle\qquad\qquad-\frac{1}{2}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}+\frac{\widetilde{f}_{\alpha\alpha^{\prime}}(0)}{T}\right)\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(0).$

The first term is again proportional to
$\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}G^{\prime}_{\alpha\alpha^{\prime}}(t)^{p}=0$.
The second term would appear to be more problematic, since
$\widetilde{f}_{\alpha\alpha^{\prime}}(0)$ (for which we have not given an
explicit expression) contributes at $O(1)$ due to
$\widetilde{G}^{\prime}_{\alpha\alpha^{\prime}}(0)$ being $O(T)$. However, we
only need the component $\widetilde{f}_{--}(0)/T=\widetilde{F}_{--}(0)$, and
from Eq. (103) we see that

$\widetilde{F}_{--}(0)=\frac{1}{\textrm{Det}\widetilde{G}(0)}\widetilde{G}_{--}(0)=\frac{1}{\textrm{Det}\widetilde{G}(0)}\frac{\textrm{Det}\widetilde{G}(0)+\widetilde{G}_{+-}(0)\widetilde{G}_{-+}(0)}{\widetilde{G}_{++}(0)}=\frac{1-4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}{2Tq_{\textrm{EA}}}+O\left(\frac{1}{T^{2}}\right).$
(106)

Eq. (105) evaluates to

$2pJq_{\textrm{EA}}^{p/2-1}\Lambda\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}-\frac{1}{2}+2J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}=\frac{1}{2}-2J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2},$
(107)

again using Eq. (73).

We finally have the large-$T$ limit of the action, given by the sum of Eqs.
(99), (104), and (107):

$S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}}]=\frac{p-2}{2p}+\frac{1}{8pJ^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}-\frac{2(p-1)J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}{p}+\frac{1}{2}\log{4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}.$
(108)

Comparing to the complexity $\Sigma(\mathcal{E})$ given in Eq. (75), we see
that $S_{\textrm{eff}}$ is precisely $-\Sigma(\mathcal{E}_{\textrm{aux}})$.

###  4.4 Evaluation of the SFF

We have shown that, at a given $\mathcal{E}_{\textrm{aux}}$ and for each value
of inverse temperature $\beta_{\textrm{aux}}$, there is a solution to the SFF
saddle point equations with
$S_{\textrm{eff}}=-\Sigma(\mathcal{E}_{\textrm{aux}})$. The full (connected)
SFF is obtained by integrating over all $\mathcal{E}_{\textrm{aux}}$ and
$\beta_{\textrm{aux}}$, as well as the symmetry-broken order parameter
$\Delta$ (which contributes an overall factor of $T$) and an additional factor
$2(1+\delta_{p\textrm{ even}})$ from the discrete symmetries. As in Sec. 3, it
is more convenient to integrate over the energy density
$\epsilon(\mathcal{E}_{\textrm{aux}},\beta_{\textrm{aux}})$. We show in App. B
that $\epsilon(\mathcal{E},\beta)$ comes out to be precisely the argument of
the filter function, Eq. (68), when evaluated at the saddle point solution.

Our final result is that444The factor $\sqrt{pN/2\pi}$ comes from the integral
over fluctuations in $\lambda$ — the variance is $p/N$ (see Eq. (90)), and the
original fat unity introducing $\mathcal{E}_{\textrm{aux}}$ comes with a
prefactor $N/2\pi$.

$\textrm{SFF}(T,f)=\big{|}\mathbb{E}\textrm{Tr}f(H)e^{-iHT}\big{|}^{2}+2\big{(}1+\delta_{p\textrm{
even}}\big{)}T\sqrt{\frac{pN}{2\pi}}\int
d\mathcal{E}_{\textrm{aux}}e^{N\Sigma(\mathcal{E}_{\textrm{aux}})}\int_{\epsilon_{-}(\mathcal{E}_{\textrm{aux}})}^{\epsilon_{+}(\mathcal{E}_{\textrm{aux}})}\frac{d\epsilon_{\textrm{aux}}}{2\pi}f(\epsilon_{\textrm{aux}})^{2},$
(109)

where the inner integral runs only over the range
$[\epsilon_{-}(\mathcal{E}_{\textrm{aux}}),\epsilon_{+}(\mathcal{E}_{\textrm{aux}})]$
in which solutions to the TAP equations exist. Furthermore, one can easily
generalize Eq. (109) by making the filter function $\mathcal{E}$-dependent,
i.e., $f(\mathcal{E},\epsilon_{\textrm{aux}})$. The resulting quantity is the
SFF for the projection of the system into certain TAP states.

Compare Eq. (109) for the non-ergodic phase to Eq. (62) for the ergodic phase,
and recall the discussion of block-diagonal Hamiltonians in Sec. 1.1. Our
result demonstrates that each metastable (i.e., TAP) state can be thought of
its own quantum chaotic subspace, one which is independent of any others. This
is the central result of our paper. While the qualitative idea has been
proposed in previous work [56], the present analysis both makes it precise and
proves it.

##  5 Higher moments of the evolution operator

In this final section we consider higher moments of $\textrm{Tr}e^{-iHT}$,
i.e., the quantities

$\textrm{SFF}^{(n)}(T,f)\equiv\mathbb{E}\Big{[}\Big{(}\textrm{Tr}f(H)e^{-iHT}\Big{)}^{n}\Big{(}\textrm{Tr}f(H)e^{iHT}\Big{)}^{n}\Big{]}.$
(110)

The saddle points of these higher moments exhibit an interesting structure
that will shed further light on the distribution of TAP states, although care
must be taken in interpreting the results. We first present the calculation
and discuss afterwards.

###  5.1 Effective action

The effective action governing the $n$’th moment is derived in exactly the
same manner as in Sec. 4. The only major difference is that now spins have a
“replica” index $a\in\\{1,\cdots,n\\}$ in addition to a contour index
$\alpha\in\\{u,l\\}$. We also include a separate fat unity defining
$\mathcal{E}_{\textrm{aux},a}$ for each replica. The result is (compare to
Eqs. (66) and (67))

$\textrm{SFF}^{(n)}(T,f)=\int
d\mathcal{E}_{\textrm{aux}}d\lambda\mathcal{D}G\mathcal{D}F\prod_{a=1}^{n}f\big{(}\epsilon_{au}[\lambda,G]\big{)}f\big{(}\epsilon_{al}[\lambda,G]\big{)}e^{-NS_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}},\lambda,G,F]},$
(111) $\displaystyle S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}},\lambda,G,F]$
$\displaystyle=-i\sum_{a}\lambda_{a}\mathcal{E}_{\textrm{aux},a}-\frac{i}{2}\int_{0}^{T}dt\sum_{a\alpha}(-1)^{\alpha}z_{a\alpha}(t)$
(112)
$\displaystyle\quad+\sum_{aa^{\prime}}\frac{\lambda_{a}\lambda_{a^{\prime}}}{2pq[G_{aa}]^{p/2}q[G_{a^{\prime}a^{\prime}}]^{p/2}}\left(\frac{1}{T^{2}}\int_{0}^{T}dtdt^{\prime}G_{au,a^{\prime}u}(t,t^{\prime})\right)^{p}$
$\displaystyle\quad\quad+\sum_{a^{\prime}}\frac{J\lambda_{a^{\prime}}}{pq[G_{a^{\prime}a^{\prime}}]^{p/2}}\int_{0}^{T}dt\sum_{a\alpha}(-1)^{\alpha}\left(\frac{1}{T}\int_{0}^{T}dt^{\prime}G_{a\alpha,a^{\prime}u}(t,t^{\prime})\right)^{p}$
$\displaystyle\quad\quad\quad+\frac{1}{2}\int_{0}^{T}dtdt^{\prime}\sum_{aa^{\prime}}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{a\alpha,a^{\prime}\alpha^{\prime}}(t,t^{\prime})^{p}-F_{a\alpha,a^{\prime}\alpha^{\prime}}(t,t^{\prime})G_{a\alpha,a^{\prime}\alpha^{\prime}}(t,t^{\prime})\right)$
$\displaystyle\quad\quad\quad\quad+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{aa^{\prime}}\delta_{\alpha\alpha^{\prime}}\big{(}\mu\partial_{t}^{2}+z_{a\alpha}\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{a\alpha,a^{\prime}\alpha^{\prime}}\Big{]},$

with energy densities

$\displaystyle\epsilon_{a\alpha}[\lambda,G]$
$\displaystyle=-\frac{\mu}{2}\partial_{t}^{2}G_{a\alpha,a\alpha}(0^{+},0)$
(113)
$\displaystyle\qquad\qquad-\frac{iJ^{2}}{p}\int_{0}^{T}dt\sum_{a^{\prime}\alpha^{\prime}}(-1)^{\alpha^{\prime}}G_{a\alpha,a^{\prime}\alpha^{\prime}}(t,0)^{p}-i\sum_{a^{\prime}}\frac{J\lambda_{a^{\prime}}}{pq[G_{a^{\prime}a^{\prime}}]^{p/2}}\left(\frac{1}{T}\int_{0}^{T}dtG_{a\alpha,a^{\prime}u}(t,0)\right)^{p}.$

The saddle point equations are therefore

$\displaystyle
i\big{(}\mu\partial_{t}^{2}+z_{a}\big{)}G_{a\alpha,a^{\prime}\alpha^{\prime}}(t-t^{\prime})+\int_{0}^{T}dt^{\prime\prime}\sum_{a^{\prime\prime}\alpha^{\prime\prime}}(-1)^{\alpha^{\prime\prime}}F_{a\alpha,a^{\prime\prime}\alpha^{\prime\prime}}(t-t^{\prime\prime})$
$\displaystyle
G_{a^{\prime\prime}\alpha^{\prime\prime},a^{\prime}\alpha^{\prime}}(t^{\prime\prime}-t^{\prime})$
(114)
$\displaystyle=(-1)^{\alpha}\delta_{aa^{\prime}}\delta_{\alpha\alpha^{\prime}}\delta(t-t^{\prime}),$
$F_{a\alpha,a^{\prime}\alpha^{\prime}}(t)=J^{2}G_{a\alpha,a^{\prime}\alpha^{\prime}}(t)^{p-1}+\frac{J}{T}\left(\frac{\lambda_{a}}{q[G_{aa}]^{p/2}}\delta_{\alpha
u}+\frac{\lambda_{a^{\prime}}}{q[G_{a^{\prime}a^{\prime}}]^{p/2}}\delta_{\alpha^{\prime}u}\right)\left(\frac{\widetilde{G}_{a\alpha,a^{\prime}\alpha^{\prime}}(0)}{T}\right)^{p-1}+O(T^{-2}),$
(115)
$\mathcal{E}_{\textrm{aux},a}=-\frac{iJT}{pq[G_{aa}]^{p/2}}\sum_{a^{\prime}\alpha^{\prime}}(-1)^{\alpha^{\prime}}\left(\frac{\widetilde{G}_{au,a^{\prime}\alpha^{\prime}}(0)}{T}\right)^{p}-\frac{i}{pq[G_{aa}]^{p/2}}\sum_{a^{\prime}}\frac{\lambda_{a^{\prime}}}{q[G_{a^{\prime}a^{\prime}}]^{p/2}}\left(\frac{\widetilde{G}_{au,a^{\prime}u}(0)}{T}\right)^{p}.$
(116)

Note that Eqs. (114) through (116) have the following permutation symmetry
with respect to replica indices. Suppose that $G$, $F$, and $\lambda$
constitute a valid solution. For any permutation $\pi$ of the set
$\\{1,\cdots,n\\}$, define $\pi_{\alpha}(a)$ to be the permuted element
$\pi(a)$ if $\alpha=l$ but simply the original element $a$ if $\alpha=u$. Then
the quantities $\overline{G}$, $\overline{F}$, and $\overline{\lambda}$
defined by

$\overline{G}_{a\alpha,a^{\prime}\alpha^{\prime}}(t,t^{\prime})\equiv
G_{\pi_{\alpha}(a)\alpha,\pi_{\alpha^{\prime}}(a^{\prime})\alpha^{\prime}}(t,t^{\prime}),\qquad\overline{F}_{a\alpha,a^{\prime}\alpha^{\prime}}(t,t^{\prime})\equiv
F_{\pi_{\alpha}(a)\alpha,\pi_{\alpha^{\prime}}(a^{\prime})\alpha^{\prime}}(t,t^{\prime}),\qquad\overline{\lambda}_{a}=\lambda_{a},$
(117)

constitute an equally valid solution. This symmetry has a nice graphical
interpretation in terms of pairings between upper and lower contours,
illustrated in Fig. 5: however contour $au$ is correlated with $a^{\prime}l$
in a given solution, there is an alternate solution in which $au$ has the same
correlation with $\pi(a^{\prime})l$.

One trivial solution to the saddle point equations is to use the solution from
Sec. 4 for $a=a^{\prime}$ while setting all cross-replica elements to zero.
The action then decomposes into a sum of single-replica actions, which we
evaluated in Sec. 4. In other words, this contribution to the $n$’th moment is
simply $\textrm{SFF}(T,f)^{n}$. However, by the permutation symmetry described
above, we actually have $n!$ such contributions:

$\textrm{SFF}^{(n)}(T,f)=n!\cdot\textrm{SFF}(T,f)^{n}+\cdots,$ (118)

where the ellipses denote additional solutions.

###  5.2 Connected solutions

Figure 5: Graphical representation of the various saddle point solutions for
the $n=2$ moment. The four contours — $1u$, $1l$, $2u$, $2l$ — are shown at
the top. Below are the four varieties of solutions: each upper contour must be
paired with a lower contour, but one is free to choose which replicas are
paired, and there is further freedom in which TAP state each pair lies within
(blue and orange lines indicate two different TAP states).

In general, for arbitrary values of $\mathcal{E}_{\textrm{aux},a}$, we have
been unable to find any further saddle points. However, when some replicas
have equal values of $\mathcal{E}_{\textrm{aux}}$, we can construct additional
solutions. Pick any set of inverse temperatures $\beta_{a}$ (not necessarily
equal), and suppose that the replicas $\\{1,\cdots,n\\}$ partition into groups
$A\equiv\\{a_{1},\cdots,a_{|A|}\\}$, such that $\mathcal{E}_{\textrm{aux},a}$
equals a common value $\mathcal{E}_{\textrm{aux},A}$ for all $a\in A$. We
again take $G_{a\alpha,a\alpha^{\prime}}(t-t^{\prime})$ to be the solution
from Sec. 4. For $a$ and $a^{\prime}$ in different groups, we still set
$G_{a\alpha,a^{\prime}\alpha^{\prime}}=0$. For $a$ and $a^{\prime}$ in the
same group $A$, however, we now set

$G_{a\alpha,a^{\prime}\alpha^{\prime}}(t-t^{\prime})=\big{(}q_{\textrm{EA},a}q_{\textrm{EA},a^{\prime}}\big{)}^{1/2},$
(119)

where $q_{\textrm{EA},a}$ is the Edwards-Anderson order parameter
corresponding to $\mathcal{E}_{\textrm{aux},A}$ and $\beta_{a}$. This
corresponds to the replicas lying within the same TAP state (see Fig. 5). We
can write this compactly as

$G_{a\alpha,a^{\prime}\alpha^{\prime}}(t)=\big{(}q_{\textrm{EA},a}q_{\textrm{EA},a^{\prime}}\big{)}^{1/2}+\delta_{aa^{\prime}}\Big{(}\Delta\mathcal{G}_{a,\alpha\alpha^{\prime}}(t)+O(T^{-1})\Big{)}.$
(120)

Inserting into Eq. (116), we have that $\lambda_{a}$ must obey

$\sum_{a^{\prime}\in
A}\lambda_{a^{\prime}}=ip\big{(}\mathcal{E}_{\textrm{aux},A}-2Jq_{\textrm{EA},a}^{p/2-1}\Lambda_{a}\big{)}+O(T^{-1}).$
(121)

Note that, by virtue of Eq. (41), $Jq_{\textrm{EA},a}^{p/2-1}\Lambda_{a}$ is a
function solely of $\mathcal{E}_{\textrm{aux},A}$. Thus Eq. (121) is
consistent among all $a\in A$. The self-energy is then given by

$F_{a\alpha,a^{\prime}\alpha^{\prime}}(t)=J^{2}\big{(}q_{\textrm{EA},a}q_{\textrm{EA},a^{\prime}}\big{)}^{\frac{p-1}{2}}+\delta_{aa^{\prime}}\Delta\mathcal{F}_{a,\alpha\alpha^{\prime}}(t)+\frac{J}{T}\left(\sqrt{\frac{q_{\textrm{EA},a^{\prime}}^{p-1}}{q_{\textrm{EA},a}}}\lambda_{a}\delta_{\alpha
u}+\sqrt{\frac{q_{\textrm{EA},a}^{p-1}}{q_{\textrm{EA},a^{\prime}}}}\lambda_{a^{\prime}}\delta_{\alpha^{\prime}u}\right)+O(T^{-1}).$
(122)

It remains only to check that Eq. (114) can be satisfied. It is automatically
solved at non-zero frequencies, since then $\widetilde{G}(\omega)$ and
$\widetilde{F}(\omega)$ reduce to
$\delta_{aa^{\prime}}\Delta\widetilde{\mathcal{G}}(\omega)$ and
$\delta_{aa^{\prime}}\Delta\widetilde{\mathcal{F}}(\omega)$ respectively. At
zero frequency we confirm that the equation is solved to $O(T)$ (the $O(1)$
terms only determine subleading corrections). Following the same steps as in
Sec. 4.2, the left-hand side of Eq. (114) simplifies to

$JT\sqrt{q_{\textrm{EA},a}^{p-1}q_{\textrm{EA},a^{\prime}}}\left(\sum_{a^{\prime\prime}\in
A}\lambda_{a^{\prime\prime}}+2i(p-1)Jq_{\textrm{EA},a}^{p/2-1}\Lambda_{a}+2iJq_{\textrm{EA},a^{\prime}}^{p/2-1}\Lambda_{a^{\prime}}-ip\mathcal{E}_{\textrm{aux},A}\right)=0,$
(123)

as desired.

Note that in this solution, only the sum $\sum_{a}\lambda_{a}$ is determined —
all orthogonal components of the vector $\lambda$ are free to take any values.
This does not imply that there are multiple such solutions, however. Returning
to the effective action in Eq. (112), the fact that the saddle point equations
determine only $G$, $F$, and $\sum_{a}\lambda_{a}$ means that, if we first
integrate over them, the resulting $\lambda$-dependent action is of the form

$S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}},\lambda]=S\left[\mathcal{E}_{\textrm{aux}},\sum_{a}\lambda_{a}\right]-i\sum_{aa^{\prime}}\sum_{b=2}^{|A|}\lambda_{a}u_{ab}u_{a^{\prime}b}\mathcal{E}_{\textrm{aux},a^{\prime}},$
(124)

for some function $S$ of the single quantity $\sum_{a}\lambda_{a}$ (as well as
all $\mathcal{E}_{\textrm{aux}}$) and for any choice of orthonormal basis
vectors $u_{ab}$ orthogonal to the all-1 vector. When we integrate over
$\sum_{a}\lambda_{a}u_{ab}$, we thus get a $\delta$-function forcing
$\sum_{a^{\prime}}u_{a^{\prime}b}\mathcal{E}_{\textrm{aux},a^{\prime}}=0$.
Together, the $\delta$-functions force all $\mathcal{E}_{\textrm{aux},a}$ to
equal a common value $\mathcal{E}_{\textrm{aux},A}$. Not only is this
consistent with our original assumption, it shows that our construction cannot
work for any other values of $\mathcal{E}_{\textrm{aux},a}$.

###  5.3 Contribution of connected solutions

To evaluate the action, note first of all that since the numbers $\beta_{a}$
define a continuous family of solutions, and since the action is by definition
stationary at these solutions, all choices of $\beta_{a}$ must give the same
value of the action. We thus take all $\beta_{a}$ to equal a common value
$\beta$ for simplicity. The action evaluated at this solution still decomposes
into a sum over groups, but now the contribution of a single group $A$ is

$\displaystyle S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}}]$
$\displaystyle=-i\mathcal{E}_{\textrm{aux}}\sum_{a\in
A}\lambda_{a}+\frac{1}{2p}\left(\sum_{a\in
A}\lambda_{a}\right)^{2}+2iJq_{\textrm{EA}}^{p/2-1}\Lambda\sum_{a\in
A}\lambda_{a}$ (125)
$\displaystyle\qquad+\frac{T}{2}\sum_{aa^{\prime}}\int_{0}^{T}dt\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\left(\frac{J^{2}}{p}G_{a\alpha,a\alpha^{\prime}}(t)^{p}-F_{a\alpha,a\alpha^{\prime}}(t)G_{a\alpha,a\alpha^{\prime}}(t)\right)$
$\displaystyle\qquad\qquad+\frac{1}{2}\log{\textrm{Det}}\Big{[}i(-1)^{\alpha}\delta_{aa^{\prime}}\delta_{\alpha\alpha^{\prime}}\big{(}\mu\partial_{t}^{2}+z\big{)}+(-1)^{\alpha+\alpha^{\prime}}F_{a\alpha,a^{\prime}\alpha^{\prime}}\Big{]}.$

Note that now $\mathcal{E}_{\textrm{aux}}$, $q_{\textrm{EA}}$, and $\Lambda$
are all independent of the replica $a$ (within a given group $A$). We are also
free to set all $\lambda_{a}=\lambda$, meaning that our saddle point solution
simplifies to (in frequency space)

$\widetilde{G}_{a\alpha,a^{\prime}\alpha^{\prime}}(\omega)=Tq_{\textrm{EA}}\delta_{\omega
0}+\delta_{aa^{\prime}}\Big{(}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)+O(T^{-1})\Big{)},$
(126)
$\widetilde{F}_{a\alpha,a^{\prime}\alpha^{\prime}}(\omega)=\left(TJ^{2}q_{\textrm{EA}}^{p-1}+\frac{ipJq_{\textrm{EA}}^{p/2-1}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}}{|A|}\big{(}\delta_{\alpha
u}+\delta_{\alpha^{\prime}u}\big{)}\right)\delta_{\omega
0}+\delta_{aa^{\prime}}\Delta\widetilde{\mathcal{F}}_{\alpha\alpha^{\prime}}(\omega)+O(T^{-1}),$
(127)
$\lambda=\frac{ip}{|A|}\big{(}\mathcal{E}_{\textrm{aux}}-2Jq_{\textrm{EA}}^{p/2-1}\Lambda\big{)}+O(T^{-1}).$
(128)

Eq. (125) can be evaluated following the same procedure as in Sec. 4.3.
Directly substituting Eqs. (126) through (128) gives

$S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}}]=\frac{p\mathcal{E}_{\textrm{aux}}^{2}}{2}-2pJ^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}-\frac{Tq_{\textrm{EA}}}{2}\sum_{aa^{\prime}}\sum_{\alpha\alpha^{\prime}}(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}_{a\alpha,a^{\prime}\alpha^{\prime}}(0)-\frac{1}{2}\sum_{\omega}\log{\textrm{Det}}\widetilde{G}_{a\alpha,a^{\prime}\alpha^{\prime}}(\omega),$
(129)

and we again must determine certain components of $\widetilde{F}(0)$ and
$\textrm{Det}\widetilde{G}(0)$. As before, it is expedient to use the $(+,-)$
basis with respect to contour indices. We also switch to the Fourier basis
with respect to factor indices: from Eq. (126),

$\displaystyle\widetilde{G}_{b\alpha,b^{\prime}\alpha^{\prime}}(\omega)$
$\displaystyle\equiv\frac{1}{|A|}\sum_{aa^{\prime}=1}^{|A|}e^{2\pi
i(ab-a^{\prime}b^{\prime})/|A|}\widetilde{G}_{a\alpha,a^{\prime}\alpha^{\prime}}(\omega)$
(130)
$\displaystyle=T|A|q_{\textrm{EA}}\delta_{b0}\delta_{b^{\prime}0}\delta_{\omega
0}+\delta_{bb^{\prime}}\Big{(}\Delta\widetilde{\mathcal{G}}_{\alpha\alpha^{\prime}}(\omega)+O(T^{-1})\Big{)}.$

Thus $\textrm{Det}\widetilde{G}(\omega)$ factors with respect to $b$, and
furthermore, $\sum_{\omega}\log{\textrm{Det}\widetilde{G}_{b}(\omega)}\sim 0$
for all $b\neq 0$ as in Secs. 3.3 and 4.3. For $b=0$, the determinant is
calculated by comparing to the $b=0$ block of
$iz(-1)^{\alpha}+(-1)^{\alpha+\alpha^{\prime}}\widetilde{F}(0)$, written in
the $(+,-)$ basis (compare to Eq. (103)):

$\begin{pmatrix}\widetilde{F}_{0-,0-}(0)&iz+\widetilde{F}_{0-,0+}(0)\\\
iz+\widetilde{F}_{0+,0-}(0)&\widetilde{F}_{0+,0+}(0)\end{pmatrix}=\frac{1}{\textrm{Det}\widetilde{G}_{0}(0)}\begin{pmatrix}\widetilde{G}_{0-,0-}(0)&-\widetilde{G}_{0+,0-}(0)\\\
-\widetilde{G}_{0-,0+}(0)&\widetilde{G}_{0+,0+}(0)\end{pmatrix}.$ (131)

We see that
$\textrm{Det}\widetilde{G}_{0}(0)=\widetilde{G}_{0+,0+}(0)/\widetilde{F}_{0+,0+}(0)\sim
1/J^{2}q_{\textrm{EA}}^{p-2}$, and $\widetilde{F}_{0-,0-}(0)$ (which is in
fact the only element of $\widetilde{F}(0)$ needed in Eq. (129)) is given by
$\widetilde{G}_{0-,0-}(0)/\textrm{Det}\widetilde{G}_{0}(0)\sim(1-4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2})/2T|A|q_{\textrm{EA}}$.
The action evaluates to

$S_{\textrm{eff}}[\mathcal{E}_{\textrm{aux}}]=\frac{p-2}{2p}+\frac{1}{8pJ^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}-\frac{2(p-1)J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}}{p}+\frac{1}{2}\log{4J^{2}q_{\textrm{EA}}^{p-2}\Lambda^{2}},$
(132)

which is again precisely $-\Sigma(\mathcal{E}_{\textrm{aux}})$.

###  5.4 Evaluation of the SFF

In the above calculation, note that we get a single contribution of complexity
for the entire group $A$. However, there is still a factor
$(2T)^{|A|}(1+\delta_{p\textrm{ even}})^{2|A|-1}$ due to the separate time
translation, time reversal, and reflection symmetries of each replica555The
contribution $(1+\delta_{p\textrm{ even}})^{2|A|-1}$, rather than
$(1+\delta_{p\textrm{ even}})^{2|A|}$, is because reflecting all spin
configurations does not change the values of any overlaps.. Finally, the sum
over all connected solutions amounts to a sum over the possible ways of
partitioning $n$ elements, in addition to the $n!$ ways of pairing upper and
lower contours. Using $P\equiv\\{A_{1},\cdots,A_{|P|}\\}$ to denote a
partition, we have that

$\displaystyle\textrm{SFF}^{(n)}(T,f)=n!\sum_{P}$ $\displaystyle\prod_{A\in
P}2^{|A|}\big{(}1+\delta_{p\textrm{
even}}\big{)}^{2|A|-1}T^{|A|}\sqrt{\frac{pN}{2\pi}}\int
d\mathcal{E}_{\textrm{aux},A}e^{N\Sigma(\mathcal{E}_{\textrm{aux},A})}$ (133)
$\displaystyle\qquad\qquad\cdot\prod_{a\in
A}\int_{\epsilon_{-}(\mathcal{E}_{\textrm{aux},A})}^{\epsilon_{+}(\mathcal{E}_{\textrm{aux},A})}\frac{\textrm{d}\epsilon_{\textrm{aux},a}}{2\pi}f(\epsilon_{\textrm{aux},a})^{2}.$

In particular, suppose the filter function is chosen so as to have a small
width $\Delta\mathcal{E}\ll 1/N$ around a certain value $\mathcal{E}$ (as in
Sec. 4.4, the above calculation can easily be modified to allow for
$\mathcal{E}$-dependent filter functions). Then the $n$’th moment simplifies
to

$\textrm{SFF}^{(n)}(T,f)=n!\sum_{P}\left(\big{(}1+\delta_{p\textrm{
even}}\big{)}^{-1}\sqrt{\frac{pN}{2\pi}}e^{N\Sigma(\mathcal{E})}\Delta\mathcal{E}\right)^{|P|}\left(2\big{(}1+\delta_{p\textrm{
even}}\big{)}^{2}T\int_{\epsilon_{-}(\mathcal{E})}^{\epsilon_{+}(\mathcal{E})}\frac{\textrm{d}\epsilon_{\textrm{aux}}}{2\pi}f(\epsilon_{\textrm{aux}})^{2}\right)^{n}.$
(134)

Eq. (134) has a nice interpretation as the $n$’th moment of a sum over a
Poisson-distributed number of Gaussians. To be precise, suppose we have an
infinite sequence of i.i.d. complex Gaussians, $\\{Z_{i}\\}_{i=1}^{\infty}$,
each with $\mathbb{E}Z_{i}=0$ and $\mathbb{E}Z_{i}Z_{i}^{*}=\sigma^{2}$.
Consider the sum $S\equiv\sum_{i=1}^{M}Z_{i}$, where $M$ is itself a Poisson-
distributed random variable with mean $\mu$. The $n$’th moment of $SS^{*}$,
averaging over both Gaussians and $M$, can be written

$\mathbb{E}\big{[}S^{n}S^{*n}\big{]}=\sum_{m=0}^{\infty}p_{\mu}(m)\mathbb{E}\left[\left(\sum_{i=1}^{m}Z_{i}\right)^{n}\left(\sum_{i=1}^{m}Z_{i}^{*}\right)^{n}\right]=n!\sum_{m=0}^{\infty}p_{\mu}(m)\big{(}m\sigma^{2}\big{)}^{n}$
(135)

where $p_{\mu}(m)$ denotes the Poisson distribution of mean $\mu$, and Wick’s
theorem is used for the latter equality. It is known that the $n$’th moment of
a Poisson distribution is $\sum_{P}\mu^{|P|}$, where the sum is again over all
partitions of $n$ elements. Thus

$\mathbb{E}\big{[}S^{n}S^{*n}\big{]}=n!\sum_{P}\mu^{|P|}\sigma^{2n}.$ (136)

If we associate $\sigma^{2}$ with the SFF of a single TAP state at
$\mathcal{E}$,

$\sigma^{2}=2\big{(}1+\delta_{p\textrm{
even}}\big{)}^{2}T\int_{\epsilon_{-}(\mathcal{E})}^{\epsilon_{+}(\mathcal{E})}\frac{\textrm{d}\epsilon_{\textrm{aux}}}{2\pi}f(\epsilon_{\textrm{aux}})^{2},$
(137)

and associate $\mu$ with the number of TAP states,

$\mu=\big{(}1+\delta_{p\textrm{
even}}\big{)}^{-1}\sqrt{\frac{pN}{2\pi}}e^{N\Sigma(\mathcal{E})}\Delta\mathcal{E},$
(138)

then Eqs. (134) and (136) are identical.

It is quite tempting to interpret this as saying that the number of TAP states
at $\mathcal{E}$ is Poisson-distributed with mean given by Eq. (138), and that
each TAP state has a Gaussian-distributed value of $\textrm{Tr}e^{-iHT}$ with
variance (i.e., SFF) given by Eq. (137). We are not aware of any results in
the literature which would contradict such a claim. However, keep in mind that
$\textrm{SFF}^{(n)}$ has perturbative corrections around each saddle point
which are suppressed by powers of $N$, whereas every connected partition in
Eq. (134) is suppressed exponentially relative to the fully disconnected one,
whose contribution is given by Eq. (118)666 At sufficiently low energies,
where the function $\Sigma(\mathcal{E})$ is negative, the situation is
reversed and the fully connected partition dominates. The issue remains,
however, that we do not calculate perturbative corrections around the dominant
saddle point. Furthermore, the relevance of these moment calculations to
individual realizations of the PSM is much more suspect when
$\Sigma(\mathcal{E})<0$. . Thus we cannot claim to have rigorously computed
the $n$’th moment to any level of accuracy beyond the disconnected piece.
Nonetheless, the structure of saddle points which we have identified is highly
suggestive and warrants further investigation.

Finally, let us briefly comment on the case $p=2$, which — being effectively a
Gaussian model — exhibits very different behavior than the $p>2$ models
considered here. One might wonder as to where our calculations break down for
$p=2$. While we leave a systematic investigation for future work, one evident
and important difference lies in the stability of the saddle point solutions:
$\partial^{2}S_{\textrm{eff}}/\partial G^{2}$ is zero when $G$ vanishes for
all $p>2$, whereas it is non-zero when $p=2$. Since our higher-moment results
in particular rely on having disconnected clusters of replicas, this
difference can very well have significant consequences

## Acknowledgements

This work was supported by:the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under award number DE-SC0001911 (V.G.); the Joint
Quantum Institute (M.W.); the Air Force Office of Scientific Research under
award numbers FA9550-17-1-0180 (M.W.) and FA9550-19-1-0360 (B.S.); the U.S.
Department of Energy, Office of Science, Office of Advanced Scientific
Computing Research, Accelerated Research for Quantum Computing program “FAR-
QC” (R.B.); the DoE ASCR Quantum Testbed Pathfinder program under award number
DE-SC0019040 (C.L.B.); the DoE ASCR Accelerated Research in Quantum Computing
program under award number DE-SC0020312 (C.L.B.); the DoE QSA, AFOSR, AFOSR
MURI, NSF PFCQC program, NSF QLCI under award number OMA-2120757 (C.L.B.); DoE
award number DE-SC0019449 (C.L.B.), ARO MURI, and DARPA SAVaNT ADVENT
(C.L.B.). This material is based upon work supported by the National Science
Foundation Graduate Research Fellowship Program under Grant No. DGE 1840340
(R.B.), and by the National Science Foundation NRC postdoctoral fellowship
program (C.L.B.).

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