# A two-way approach to out-of-time-order correlators

Yingfei Gu, Alexei Kitaev, Pengfei Zhang
California Institute of Technology, Pasadena, CA 91125, U.S.A.

(March 3, 2022)

###### Abstract

Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos.
Of the four operators involved, one pair may be regarded as a source and the
other as a probe. A usual approach, applicable to large-$N$ systems such as
the SYK model, is to replace the actual source with some mean-field
perturbation and solve for the probe correlation function on the double
Keldysh contour. We show how to obtain the OTOC by combining two such
solutions for perturbations propagating forward and backward in time. These
dynamical perturbations, or scrambling modes, are considered on the
thermofield double background and decomposed into a coherent and an incoherent
part. For the large-$q$ SYK, we obtain the OTOC in a closed form. We also
prove a previously conjectured relation between the Lyapunov exponent and
high-frequency behavior of the spectral function.

###### Contents

  1. 1 Introduction
  2. 2 The big picture
  3. 3 Derivation of the main equation
  4. 4 Small perturbations to the thermofield double
    1. 4.1 The double system
    2. 4.2 Retarded vertex function as a deformation of the TFD
    3. 4.3 Size operator
  5. 5 Nonlinear theory
    1. 5.1 Matrix element interpretation of $F^{\mathrm{R}}$, $F^{\mathrm{A}}$, $h^{\mathrm{R}}$, $h^{\mathrm{A}}$
    2. 5.2 Kinetic equation on the double Keldysh contour
  6. 6 Example: the large-$q$ SYK model
    1. 6.1 Preliminaries
    2. 6.2 Differential form and solution of the kinetic equation
    3. 6.3 Vertex functions and late-time OTOC
  7. 7 Discussion
  8. A Past and future horizon representations
  9. B Brownian OTOC
  10. C Numerical study of the SYK model at finite $q$
    1. C.1 Solution of the kinetic equation
    2. C.2 High-frequency noise vs. Lyapunov exponent

##  1 Introduction

Out-of-time-order correlators (OTOCs) are interesting objects for several
reasons. First, they characterize quantum chaos in a way that is comparable to
the classical picture of divergent trajectories [1]. For systems in which
connected correlators are suppressed by a large factor $C$, OTOCs behave as
$1-C^{-1}e^{\varkappa t}$ as long as $C^{-1}e^{\varkappa t}\ll 1$; at later
times, they decay to zero. The number $\varkappa$, called the Lyapunov
exponent, is bounded by $\frac{2\pi}{\beta}$ [2]. OTOCs give access to the
near-horizon region of black holes [3, 4, 5], in which case
$\varkappa=\frac{2\pi}{\beta}$. Another reason to study OTOCs is their
relevance to quantum scrambling and certain information-theoretic tasks [6, 7,
8].

In this paper, we give a conceptual picture and a general expression for OTOCs
in many-body systems with all-to-all interactions such as the SYK model [9,
10, 11, 12]. For such systems, the previously mentioned parameter $C$ is
proportional to the number of elementary degrees of freedom $N$. The early-
time behavior is already well-studied. For the SYK model at low temperature,
the Lyapunov exponent $\varkappa$ matches that of a black hole [10]. Maldacena
and Stanford [13] found a finite-temperature correction to $\varkappa$ and
calculated the Lyapunov exponent at an arbitrary temperature in the large-$q$
limit. A general ansatz for the early-time OTOC was introduced in [12], and a
relation between the ratio $C/N$ and the Lyapunov exponent was established in
[14].

In principle, the calculation of a correlator like
$\langle{(A(t)B(0)C(t)D(0)}\rangle$ can be carried out by propagating $B(0)$,
$D(0)$ forward or $A(t)$, $C(t)$ backward in time using the Heisenberg
equation. This leads to the operator growth picture [15]. At infinite
temperature, the decomposition of the Heisenberg-evolved operator
$\mathcal{O}$ into basis operators (i.e. products of $\chi_{j}$ in the SYK
case) may be interpreted as a quantum wave function [16]. Qi and Streicher
[17] have extended this method to finite temperatures by constructing the
quantum state $|\Psi\rangle=\big{|}\mathcal{O}\rho^{1/2}\bigr{\rangle}$, where
$\rho$ is the thermal density matrix and $\big{|}\rho^{1/2}\bigr{\rangle}$ is
the thermofield double state (TFD). However, they still define the “size” of
$|\Psi\rangle$ relative to the infinite-temperature TFD. We will adjust their
definition as follows. Consider the free-fermion state $\varrho$ (referred to
as the “naive model” in [12]) reproducing the Wightman function of the SYK
model at a given temperature. The corresponding state
$\big{|}\varrho^{1/2}\bigr{\rangle}$ of the double system may be regarded as a
fermionic vacuum. Then the size of a perturbed TFD is defined as the average
number of quasiparticles with some energy-dependent weights.

At low temperatures, essential properties of the SYK model are governed by the
Schwarzian action [11, 13, 12], which describes a black hole in $1+1$
dimensions [18]. The relevant part of black hole physics goes back to the work
of Dray and ’t Hooft [19], who discovered that an infalling object generates a
gravitational “shock wave” at the past horizon. A classical shock affects
quantum correlation functions between the two sides of an eternal black hole
[3]. In fact, a two-sided black hole perturbed by a shock is analogous to the
state $|\Psi\rangle$ we have previously discussed. ’T Hooft also proposed a
fully quantum model of gravitational shocks, which mediate the scattering
between incoming and outgoing particles [20]. The scattering amplitude is
$a(p^{\mathrm{out}},p^{\mathrm{in}})=\exp(ifp^{\mathrm{out}}p^{\mathrm{in}})$,
where $p^{\mathrm{out}}$, $p^{\mathrm{in}}$ are the particles’ null momenta
and $f$ depends on their angular coordinates. Since the states of particles
inside a black hole are inaccessible, the physical meaning of this theory is
rather elusive and was understood long after ’t Hooft’s work. If one wants to
operate with quantum observables only on one side, the relevant quantity is an
out-of-time-order correlator. Such correlators were calculated by Shenker and
Stanford for black holes [5] and by Maldacena, Stanford, and Yang for the SYK
model at low temperature [18]. This includes the nonlinear, late-time regime.
Stanford, Yang, and Yao [21] conjectured that the general form of these
solutions is valid for large-$N$ models at any temperature. We will show it is
indeed the case.

A surprising connection exists between OTOCs and two-point functions. In a
nutshell, chaotic dynamics contributes to equilibrium noise. Parker, Cao,
Avdoshkin, Scaffidi, and Altman [22] conjectured a lower bound on high-
frequency tails of the spectral function in terms of the Lyapunov exponent.
They proved it at infinite temperature, and Avdoshkin and Dymarsky extended
the derivation to finite temperatures [23]. We will give a different proof in
section 7.

##  2 The big picture

| $3$$4$$2$$1$
---|---
(a) | (b)
Figure 1: (a) The complex times $\theta_{1},\theta_{2},\theta_{3},\theta_{4}$
in terms of the variable $e^{i\theta}=e^{-t}e^{i\tau}$. (b) A symmetric case,
where the points 1,2,3,4 are placed on a cylinder with the axial coordinate
$t$ (going left to right) and connected by a contour such that the contour
ordering coincides with the $\tau$ ordering.

First, let us fix the notation and define the exact problem. We set
$\beta=2\pi$ and consider complex times $\theta=\tau+it$ with $\tau$ defined
modulo $2\pi$. In these units, the Lyapunov exponent satisfies the inequality
$0<\varkappa<1$; the limits $\varkappa\to 0$ and $\varkappa\to 1$ can also be
considered. To ensure the finiteness of correlation functions, operators
should be ordered by $\tau$. The ordering involves fermionic parity, which
will be indicated by a variable $\zeta=0,1$. Let $X_{1},X_{2}$ have parity
$\zeta_{1}=\zeta_{2}$ and let $X_{3},X_{4}$ have parity $\zeta_{3}=\zeta_{4}$.
The main function to study is this one:

$\displaystyle\operatorname{OTOC}_{X_{1},X_{2},X_{3},X_{4}}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$
$\displaystyle=\bigl{\langle}{\operatorname{\mathbf{T}}X_{1}(\theta_{1})X_{2}(\theta_{2})X_{3}(\theta_{3})X_{4}(\theta_{4})}\bigr{\rangle}$
(2.1)
$\displaystyle=(-1)^{\zeta_{2}\zeta_{3}}\bigl{\langle}{X_{1}(\theta_{1})X_{3}(\theta_{3})X_{2}(\theta_{2})X_{4}(\theta_{4})}\bigr{\rangle},$

where (see also figure 1)

$\theta_{j}=\tau_{j}+it_{j},\qquad\tau_{1}\geqslant\tau_{3}\geqslant\tau_{2}\geqslant\tau_{4}\geqslant\tau_{1}-2\pi,\qquad
t_{1}\approx t_{2}>t_{3}\approx t_{4}.$ (2.2)

Using the time translation symmetry, we can arrange that $t_{1},t_{2}$ are
positive, $t_{3},t_{4}$ are negative, and all four times are approximately
equal in magnitude. This allows for a symmetric interpretation of the OTOC,
whereby $X_{3}(\theta_{3})$ and $X_{4}(\theta_{4})$ create a perturbation to
the thermofield double that propagates forward in time, while
$X_{1}(\theta_{1})$ and $X_{2}(\theta_{2})$ create a backward-propagating
perturbation. These counter-propagating “waves” interact at times around $0$,
where they are both relatively weak. More exactly, let us assume that

$|t_{1}-t_{2}|,\,|t_{3}-t_{4}|\lesssim
1,\qquad\frac{t_{1}+t_{2}}{2},\,-\frac{t_{3}+t_{4}}{2}\gg 1,\qquad
C^{-1}e^{(t_{1}+t_{2})/2},\,C^{-1}e^{-(t_{3}+t_{4})/2}\ll 1.$ (2.3)

The quantity relevant to the OTOC, $C^{-1}e^{(t_{1}+t_{2}-t_{3}-t_{4})/2}$, is
generally of the order of one.

We will show that the OTOC has the same general form as in the black hole
problem [5]:

$\displaystyle\begin{aligned}
&\operatorname{OTOC}_{X_{1},X_{2},X_{3},X_{4}}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})\\\
&\hskip
42.67912pt=\int_{0}^{\infty}\\!dy_{\mathrm{A}}\int_{0}^{\infty}\\!dy_{\mathrm{R}}\,e^{-\lambda
y_{\mathrm{A}}y_{\mathrm{R}}}\,h^{\mathrm{R}}_{X_{1},X_{2}}\bigl{(}y_{\mathrm{A}};\theta_{1}-\theta_{2}\bigr{)}\,h^{\mathrm{A}}_{X_{3},X_{4}}\bigl{(}y_{\mathrm{R}};\theta_{3}-\theta_{4}\bigr{)},\end{aligned}$
(2.4)
$\displaystyle\text{where}\qquad\lambda=C^{-1}e^{i\varkappa(\pi-\theta_{1}-\theta_{2}+\theta_{3}+\theta_{4})/2}.$
(2.5)

Let us explain this result before deriving it. The integration variables
$y_{\mathrm{A}}$ and $y_{\mathrm{R}}$ are analogous to the null momenta on the
past and future horizons ($p^{\mathrm{out}}$, $p^{\mathrm{in}}$ in [20] or
$p^{u}_{1}$, $p^{v}_{2}$ in [5]), which are due to particles moving and
generating gravitational shocks along the opposite horizons. More exactly,
$y_{\mathrm{A}}$ (representing a perturbation source in the future) and
$y_{\mathrm{R}}$ (representing a similar source in the past) determine the
magnitudes $z_{\mathrm{A}}$, $z_{\mathrm{R}}$ of the backward-propagating
(advanced) and forward-propagating (retarded) modes, respectively:

$z_{\mathrm{A}}=C^{-1}e^{-i\varkappa(\theta_{1}+\theta_{2})/2}y_{\mathrm{A}},\qquad
z_{\mathrm{R}}=C^{-1}e^{i\varkappa(\theta_{3}+\theta_{4})/2}y_{\mathrm{R}}.$
(2.6)

The exponential factor in (2.4),

$a(y_{\mathrm{A}},y_{\mathrm{R}}):=e^{-\lambda
y_{\mathrm{A}}y_{\mathrm{R}}}=\exp\bigl{(}-Ce^{i\varkappa\pi/2}z_{\mathrm{A}}z_{\mathrm{R}}\bigr{)},$
(2.7)

describes interaction between the counter-propagating modes and may be
understood as a scattering amplitude.

For a sharper interpretation, let us assume that

$\tau_{1}=\tau_{3}=\frac{\tau}{2},\qquad\tau_{2}=\tau_{4}=-\frac{\tau}{2},\qquad
0\leqslant\tau\leqslant 2\pi$ (2.8)

so that $z_{\mathrm{A}}=C^{-1}e^{\varkappa(t_{1}+t_{2})}y_{\mathrm{A}}$ and
$z_{\mathrm{R}}=C^{-1}e^{-\varkappa(t_{3}+t_{4})}y_{\mathrm{R}}$ are real and
positive.111As a generalization, it is sufficient to assume that
$\pi\geqslant\tau_{1}\geqslant\tau_{3}\geqslant
0\geqslant\tau_{2}\geqslant\tau_{4}\geqslant-\pi$. In this case, one can
arrange for $z_{\mathrm{A}}$, $z_{\mathrm{R}}$ given by Eq. (2.6) to be real
and positive by taking the integrals over $y_{\mathrm{A}},y_{\mathrm{R}}$ in
(2.4) along certain paths in the complex plane. In section 4.3, we will
interpret the quantities $Cz_{\mathrm{A}}$, $Cz_{\mathrm{R}}$ as eigenvalues
of certain operators $Q_{\mathrm{A}}$, $Q_{\mathrm{R}}$ acting on the double
system. Condition (2.8) also allows for the definition of in- and out-states
and an $S$-matrix [12, 14]. However, let us take the idea of scattering
informally and posit that the operators $X_{2}(\theta_{2})$ and
$X_{1}^{\dagger}(\theta_{1})$ create some local excitations (around the time
$t_{1}\approx t_{2}$) dressed with a backward-propagating mode characterized
by $y_{\mathrm{A}}$, and similarly, $X_{4}(\theta_{4})$ and
$X_{3}^{\dagger}(\theta_{3})$ create local excitations dressed with a forward-
propagating mode characterized by $y_{\mathrm{R}}$. The complex numbers
$h^{\mathrm{R}}_{X_{1},X_{2}}(y_{\mathrm{A}};\theta_{1}-\theta_{2})$ and
$h^{\mathrm{A}}_{X_{3},X_{4}}(y_{\mathrm{R}};\theta_{3}-\theta_{4})$ represent
the inner products between the local states projected onto the given values of
$y_{\mathrm{A}}$ and $y_{\mathrm{R}}$. In particular, if
$X_{1}^{\dagger}=X_{2}$, $t_{1}=t_{2}$, $X_{3}^{\dagger}=X_{4}$,
$t_{3}=t_{4}$, then the functions

$w^{\mathrm{R}}(y_{\mathrm{A}})=\frac{h^{\mathrm{R}}_{X_{1},X_{2}}(y_{\mathrm{A}};\tau)}{\langle{X_{1}(\tau)X_{2}(0)}\rangle},\qquad
w^{\mathrm{A}}(y_{\mathrm{R}})=\frac{h^{\mathrm{A}}_{X_{3},X_{4}}(y_{\mathrm{R}};\tau)}{\langle{X_{3}(\tau)X_{4}(0)}\rangle}$
(2.9)

are real, nonnegative, and properly normalized so that they may be interpreted
as the probability distributions of $y_{\mathrm{A}}$, $y_{\mathrm{R}}$.

Perhaps the most important observation is concerned with the scattering
amplitude (2.7). In the situation we have described, where
$z_{\mathrm{A}}\geqslant 0$ and $z_{\mathrm{R}}\geqslant 0$, one finds that
$|a(y_{\mathrm{A}},y_{\mathrm{R}})|=1$ if $\varkappa=1$. If $\varkappa<1$, the
amplitude $a(y_{\mathrm{A}},y_{\mathrm{R}})$ has absolute value less than $1$.
This is a signature of inelastic scattering due to the production of strings
[5] or similar but simpler objects [14].

##  3 Derivation of the main equation

We now derive equation (2.4), postponing one step (namely, a matrix element
interpretation of $h^{\mathrm{R}}$, $h^{\mathrm{A}}$, and related functions)
until section 5.1. First, let us consider the early-time OTOC. We assume that
there is one fastest-growing scrambling mode (which comes in the retarded and
advanced variants). This assumption amounts to the following ansatz
[12]:222Here, $C^{-1}$, $\Upsilon^{\mathrm{R}}$, $\Upsilon^{\mathrm{A}}$ are
defined up to constant factors; only their product is fixed. By requiring
positivity in certain cases (e.g. when $\theta_{4}=-\theta_{3}^{*}$ and
$X_{4}=X_{3}^{\dagger}$; see section 4.2), the ambiguity is reduced down to
positive factors. We also assume that
$\Upsilon^{\mathrm{R}}_{X_{1},X_{2}}(\theta)$ and
$\Upsilon^{\mathrm{A}}_{X_{3},X_{4}}(\theta)$ are of the order of $1$ if
$\theta\sim 1$, whereas $C^{-1}$ is small.

$\displaystyle\operatorname{OTOC}_{X_{1},X_{2},X_{3},X_{4}}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})$
$\displaystyle\approx\langle{X_{1}(\theta_{1})X_{2}(\theta_{2})}\rangle\langle{X_{3}(\theta_{3})X_{4}(\theta_{4})}\rangle$
(3.1)
$\displaystyle-\lambda\,\Upsilon^{\mathrm{R}}_{X_{1},X_{2}}(\theta_{1}-\theta_{2})\,\Upsilon^{\mathrm{A}}_{X_{3},X_{4}}(\theta_{3}-\theta_{4}),$

where
$\lambda=C^{-1}e^{i\varkappa(\pi-\theta_{1}-\theta_{2}+\theta_{3}+\theta_{4})/2}$.
The approximation is valid if $t:=\frac{t_{1}+t_{2}-t_{3}-t_{4}}{2}$ is
sufficiently large such that any decaying or more slowly growing modes may be
neglected. We also suppose that $|\lambda|=C^{-1}e^{\varkappa t}$ is much less
than $1$.

For the SYK model with Majorana operators $\chi_{1},\dots,\chi_{N}$, equation
(3.1) is specialized as follows:

$\frac{1}{N^{2}}\sum_{j,k}\bigl{\langle}{\operatorname{\mathbf{T}}\chi_{j}(\theta_{1})\chi_{j}(\theta_{2})\chi_{k}(\theta_{3})\chi_{k}(\theta_{4})}\bigr{\rangle}\approx\langle{\chi(\theta_{1})\chi(\theta_{2})}\rangle\langle{\chi(\theta_{3})\chi(\theta_{4})}\rangle-\lambda\,\Upsilon^{\mathrm{R}}(\theta_{1}-\theta_{2})\,\Upsilon^{\mathrm{A}}(\theta_{3}-\theta_{4}).$
(3.2)

The coefficient $C$ in this case is of the order of $N$. (At low temperatures,
$C\sim\frac{N}{\beta J}$ [13], but we assume that $\beta$ and $J$ are fixed,
whereas $N\to\infty$.) The second term in (3.2) is given by a sum of ladder
diagrams. Let us replace it with a schematic drawing:

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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{20.6pt}{0.0pt}\pgfsys@curveto{19.975pt}{0.0pt}{19.6625pt}{-2.0pt}{19.0375pt}{-2.0pt}\pgfsys@curveto{18.585pt}{-2.0pt}{18.19502pt}{-1.02399pt}{17.7875pt}{0.0pt}\pgfsys@curveto{17.38pt}{1.02399pt}{16.99pt}{2.0pt}{16.5375pt}{2.0pt}\pgfsys@curveto{16.085pt}{2.0pt}{15.69502pt}{1.02399pt}{15.2875pt}{0.0pt}\pgfsys@curveto{14.88pt}{-1.02399pt}{14.49pt}{-2.0pt}{14.0375pt}{-2.0pt}\pgfsys@curveto{13.585pt}{-2.0pt}{13.19502pt}{-1.02399pt}{12.7875pt}{0.0pt}\pgfsys@curveto{12.38pt}{1.02399pt}{11.99pt}{2.0pt}{11.5375pt}{2.0pt}\pgfsys@curveto{11.085pt}{2.0pt}{10.69502pt}{1.02399pt}{10.2875pt}{0.0pt}\pgfsys@curveto{9.88pt}{-1.02399pt}{9.49pt}{-2.0pt}{9.0375pt}{-2.0pt}\pgfsys@curveto{8.585pt}{-2.0pt}{8.19502pt}{-1.02399pt}{7.7875pt}{0.0pt}\pgfsys@curveto{7.38pt}{1.02399pt}{6.99pt}{2.0pt}{6.5375pt}{2.0pt}\pgfsys@curveto{6.085pt}{2.0pt}{5.69502pt}{1.02399pt}{5.2875pt}{0.0pt}\pgfsys@curveto{4.88pt}{-1.02399pt}{4.49pt}{-2.0pt}{4.0375pt}{-2.0pt}\pgfsys@curveto{3.585pt}{-2.0pt}{3.19502pt}{-1.02399pt}{2.7875pt}{0.0pt}\pgfsys@curveto{2.38pt}{1.02399pt}{1.99pt}{2.0pt}{1.5375pt}{2.0pt}\pgfsys@curveto{1.085pt}{2.0pt}{0.69502pt}{1.02399pt}{0.2875pt}{0.0pt}\pgfsys@curveto{-0.12pt}{-1.02399pt}{-0.51pt}{-2.0pt}{-0.9625pt}{-2.0pt}\pgfsys@curveto{-1.415pt}{-2.0pt}{-1.80498pt}{-1.02399pt}{-2.2125pt}{0.0pt}\pgfsys@curveto{-2.62pt}{1.02399pt}{-3.01pt}{2.0pt}{-3.4625pt}{2.0pt}\pgfsys@curveto{-3.915pt}{2.0pt}{-4.30498pt}{1.02399pt}{-4.7125pt}{0.0pt}\pgfsys@curveto{-5.12pt}{-1.02399pt}{-5.51pt}{-2.0pt}{-5.9625pt}{-2.0pt}\pgfsys@curveto{-6.415pt}{-2.0pt}{-6.80498pt}{-1.02399pt}{-7.2125pt}{0.0pt}\pgfsys@curveto{-7.62pt}{1.02399pt}{-8.01pt}{2.0pt}{-8.4625pt}{2.0pt}\pgfsys@curveto{-8.915pt}{2.0pt}{-9.30498pt}{1.02399pt}{-9.7125pt}{0.0pt}\pgfsys@curveto{-10.12pt}{-1.02399pt}{-10.51pt}{-2.0pt}{-10.9625pt}{-2.0pt}\pgfsys@curveto{-11.415pt}{-2.0pt}{-11.80498pt}{-1.02399pt}{-12.2125pt}{0.0pt}\pgfsys@curveto{-12.62pt}{1.02399pt}{-13.01pt}{2.0pt}{-13.4625pt}{2.0pt}\pgfsys@curveto{-13.915pt}{2.0pt}{-14.30498pt}{1.02399pt}{-14.7125pt}{0.0pt}\pgfsys@curveto{-15.12pt}{-1.02399pt}{-15.51pt}{-2.0pt}{-15.9625pt}{-2.0pt}\pgfsys@curveto{-16.415pt}{-2.0pt}{-16.80498pt}{-1.02399pt}{-17.2125pt}{0.0pt}\pgfsys@curveto{-17.62pt}{1.02399pt}{-18.01pt}{2.0pt}{-18.4625pt}{2.0pt}\pgfsys@lineto{-20.6pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=\lambda\,\Upsilon^{\mathrm{R}}(\theta_{1}-\theta_{2})\,\Upsilon^{\mathrm{A}}(\theta_{3}-\theta_{4}),$
(3.3)

where the functions $\Upsilon^{\mathrm{R}}$, $\Upsilon^{\mathrm{A}}$ are
associated with the two vertices and $\lambda$ (the scrambling mode
propagator) with the wavy line. It is sometimes convenient to cut the wavy
line in half:

$\leavevmode\hbox to44.45pt{\vbox
to28.15pt{\pgfpicture\makeatletter\hbox{\hskip 44.0513pt\lower-14.07385pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{}\pgfsys@moveto{-21.0pt}{0.0pt}\pgfsys@curveto{-21.0pt}{2.20917pt}{-22.79083pt}{4.0pt}{-25.0pt}{4.0pt}\pgfsys@curveto{-27.20917pt}{4.0pt}{-29.0pt}{2.20917pt}{-29.0pt}{0.0pt}\pgfsys@curveto{-29.0pt}{-2.20917pt}{-27.20917pt}{-4.0pt}{-25.0pt}{-4.0pt}\pgfsys@curveto{-22.79083pt}{-4.0pt}{-21.0pt}{-2.20917pt}{-21.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-25.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.11127pt}{3.11127pt}\pgfsys@lineto{-33.48529pt}{8.48529pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.71829pt}{6.22974pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.11127pt}{-3.11127pt}\pgfsys@lineto{-33.48529pt}{-8.48529pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.71829pt}{-10.74084pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-20.6pt}{0.0pt}\pgfsys@curveto{-19.975pt}{0.0pt}{-19.6625pt}{2.0pt}{-19.0375pt}{2.0pt}\pgfsys@curveto{-18.585pt}{2.0pt}{-18.19502pt}{1.02399pt}{-17.7875pt}{0.0pt}\pgfsys@curveto{-17.38pt}{-1.02399pt}{-16.99pt}{-2.0pt}{-16.5375pt}{-2.0pt}\pgfsys@curveto{-16.085pt}{-2.0pt}{-15.69502pt}{-1.02399pt}{-15.2875pt}{0.0pt}\pgfsys@curveto{-14.88pt}{1.02399pt}{-14.49pt}{2.0pt}{-14.0375pt}{2.0pt}\pgfsys@curveto{-13.585pt}{2.0pt}{-13.19502pt}{1.02399pt}{-12.7875pt}{0.0pt}\pgfsys@curveto{-12.38pt}{-1.02399pt}{-11.99pt}{-2.0pt}{-11.5375pt}{-2.0pt}\pgfsys@curveto{-11.085pt}{-2.0pt}{-10.69502pt}{-1.02399pt}{-10.2875pt}{0.0pt}\pgfsys@curveto{-9.88pt}{1.02399pt}{-9.49pt}{2.0pt}{-9.0375pt}{2.0pt}\pgfsys@curveto{-8.585pt}{2.0pt}{-8.19502pt}{1.02399pt}{-7.7875pt}{0.0pt}\pgfsys@curveto{-7.38pt}{-1.02399pt}{-6.99pt}{-2.0pt}{-6.5375pt}{-2.0pt}\pgfsys@curveto{-6.085pt}{-2.0pt}{-5.69502pt}{-1.02399pt}{-5.2875pt}{0.0pt}\pgfsys@curveto{-4.88pt}{1.02399pt}{-4.49pt}{2.0pt}{-4.0375pt}{2.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=C^{-1}e^{-i\varkappa(\theta_{1}+\theta_{2})/2}\,\Upsilon^{\mathrm{R}}(\theta_{1}-\theta_{2}),\qquad\leavevmode\hbox
to44.45pt{\vbox to28.15pt{\pgfpicture\makeatletter\hbox{\hskip
0.4pt\lower-14.07385pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{}\pgfsys@moveto{29.0pt}{0.0pt}\pgfsys@curveto{29.0pt}{2.20917pt}{27.20917pt}{4.0pt}{25.0pt}{4.0pt}\pgfsys@curveto{22.79083pt}{4.0pt}{21.0pt}{2.20917pt}{21.0pt}{0.0pt}\pgfsys@curveto{21.0pt}{-2.20917pt}{22.79083pt}{-4.0pt}{25.0pt}{-4.0pt}\pgfsys@curveto{27.20917pt}{-4.0pt}{29.0pt}{-2.20917pt}{29.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.11127pt}{3.11127pt}\pgfsys@lineto{33.48529pt}{8.48529pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.21829pt}{6.22974pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{{}{}}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{28.11127pt}{-3.11127pt}\pgfsys@lineto{33.48529pt}{-8.48529pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.21829pt}{-10.74084pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}
{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}}
}}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{20.6pt}{0.0pt}\pgfsys@curveto{19.975pt}{0.0pt}{19.6625pt}{-2.0pt}{19.0375pt}{-2.0pt}\pgfsys@curveto{18.4125pt}{-2.0pt}{18.1pt}{0.0pt}{17.475pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=C^{-1}e^{i\varkappa(\theta_{3}+\theta_{4})/2}\,\Upsilon^{\mathrm{A}}(\theta_{3}-\theta_{4}).$
(3.4)

To glue these pieces back together, one needs to insert the factor
$Ce^{i\varkappa\pi/2}$. (The cutting and gluing of ladder diagrams was used in
[14] to derive an expression for the coefficient $C$ in terms of the retarded
kernel, the elementary unit of such ladders.)

Figure 2: An SYK diagram contributing to the OTOC at late times, such that
$\lambda\sim 1$.

When $\lambda\sim 1$, simple ladders are not sufficient, and we have to
include more complex diagrams such as shown in figure 2. This particular
diagram has an additional small factor $N^{-1}\sim C^{-1}$ due to branching,
and the overall value can be estimated as follows:

$C^{-2}e^{\varkappa
t_{L}}\bigl{(}e^{\varkappa(t-t_{L}-t_{R})}\bigr{)}^{2}e^{\varkappa
t_{R}}=|\lambda|^{2}e^{-\varkappa(t_{L}+t_{R})}.$ (3.5)

Therefore, the diagram is significant if $\varkappa(t_{L}+t_{R})\lesssim 1$.
In general, significant diagrams consist of parallel ladders that join near
the initial and final times, with no branching in the middle. They may be
depicted like this:

$\leavevmode\hbox to109.42pt{\vbox
to39.46pt{\pgfpicture\makeatletter\hbox{\hskip
54.70816pt\lower-19.73071pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{}\pgfsys@moveto{-22.0pt}{0.0pt}\pgfsys@curveto{-22.0pt}{4.41833pt}{-25.58167pt}{8.0pt}{-30.0pt}{8.0pt}\pgfsys@curveto{-34.41833pt}{8.0pt}{-38.0pt}{4.41833pt}{-38.0pt}{0.0pt}\pgfsys@curveto{-38.0pt}{-4.41833pt}{-34.41833pt}{-8.0pt}{-30.0pt}{-8.0pt}\pgfsys@curveto{-25.58167pt}{-8.0pt}{-22.0pt}{-4.41833pt}{-22.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-30.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-30.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@fill{.75}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (3.6)

Thus, we arrive at a nonlinear generalization [14] of the single-mode ansatz:

$\operatorname{OTOC}_{X_{1},X_{2},X_{3},X_{4}}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})=\sum_{m=0}^{\infty}\frac{(-\lambda)^{m}}{m!}\,\Upsilon_{X_{1},X_{2}}^{\mathrm{R},m}(\theta_{1}-\theta_{2})\,\Upsilon_{X_{3},X_{4}}^{\mathrm{A},m}(\theta_{3}-\theta_{4}).$
(3.7)

In particular,
$\Upsilon_{X_{1},X_{2}}^{\mathrm{R},0}(\theta_{1}-\theta_{2})=\langle{X_{1}(\theta_{1})X_{2}(\theta_{2})}\rangle$
and
$\Upsilon_{X_{1},X_{2}}^{\mathrm{R},1}=\Upsilon_{X_{1},X_{2}}^{\mathrm{R}}$.
The expansion (3.7) is only asymptotic; see section 6.3 for an illustration.

The next step is to cut all wavy lines in half, perform an independent
summation on each side, and figure how to combine the results. For example, on
the left side, we get:

$F^{\mathrm{R}}_{X_{1},X_{2}}(z;\theta_{1},\theta_{2})=\,\leavevmode\hbox
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;+\;\cdots$ (3.8)

More exactly, the two one-sided sums are defined as follows:

$\displaystyle F_{X_{1},X_{2}}^{\mathrm{R}}(z;\theta_{1},\theta_{2})$
$\displaystyle=\sum_{m=0}^{\infty}\frac{\bigl{(}-e^{-i\varkappa(\theta_{1}+\theta_{2})/2}z\bigr{)}^{m}}{m!}\,\Upsilon_{X_{1},X_{2}}^{\mathrm{R},m}(\theta_{1}-\theta_{2}),$
(3.9) $\displaystyle F_{X_{3},X_{4}}^{\mathrm{A}}(z;\theta_{3},\theta_{4})$
$\displaystyle=\sum_{m=0}^{\infty}\frac{\bigl{(}-e^{i\varkappa(\theta_{3}+\theta_{4})/2}z\bigr{)}^{m}}{m!}\,\Upsilon_{X_{3},X_{4}}^{\mathrm{A},m}(\theta_{3}-\theta_{4}).$

Here, $z$ is an abstract parameter that represents the strength of some mean-
field perturbation to the thermofield double. (In the big picture we are
trying to justify, the argument of $F^{\mathrm{R}}$ is
$e^{i\varkappa\pi/2}z_{\mathrm{R}}$ and the argument of $F^{\mathrm{A}}$ is
$e^{i\varkappa\pi/2}z_{\mathrm{A}}$.) Thus, $F_{X_{1},X_{2}}^{\mathrm{R}}$,
$F_{X_{3},X_{4}}^{\mathrm{A}}$ are correlation functions on the perturbed
background; they can be found by solving mean-field equations on the double
Keldysh contour. The linear version of this problem has been studied
extensively, see e.g. [10, 13, 24, 14]. The nonlinear equations have been
solved for various weakly coupled models where they can be reduced to an
analogue of the Boltzmann equation [25]. For the SYK model, these equations
have been written explicitly (but not solved) in [26]. We will obtain an
analytic solution in the large-$q$ limit and numerical solutions for $q=4,6$.

As will be seen in concrete examples, the Taylor series (3.9) have finite
convergence radius. However, the path to our goal, equation (2.4), lies
through inverse Laplace transform, which requires analytic continuation. To
this end, we will show in section 5.1 that the function values
$F_{X_{1},X_{2}}^{\mathrm{R}}(z;\theta_{1},\theta_{2})$ and
$F_{X_{3},X_{4}}^{\mathrm{A}}(z;\theta_{3},\theta_{4})$ can be represented as
matrix elements of $\exp(-zQ_{\mathrm{A}})$ and $\exp(-zQ_{\mathrm{R}})$,
respectively, where $Q_{\mathrm{A}}$ and $Q_{\mathrm{R}}$ are some positive-
semidefinite operators acting on the double system.333Quantum states of the
double system, such as $|A\rangle$, correspond to operators acting on the
single system (in this case, $A$). Furthermore, $\langle
A|B\rangle=\operatorname{Tr}(A^{{\dagger}}B)$; see section 4.1 for exact
rules. For example,

$F^{\mathrm{R}}_{X_{1},X_{2}}(z;\theta_{1},\theta_{2})=\bigl{\langle}X_{1}(\theta_{1})^{\dagger}\rho^{1/2}\big{|}e^{-zQ_{\mathrm{A}}}\big{|}X_{2}(\theta_{2})\rho^{1/2}\bigr{\rangle}\quad\;\text{for
}\,\pi\geqslant\operatorname{Re}\theta_{1}\geqslant
0\geqslant\operatorname{Re}\theta_{2}\geqslant-\pi.$ (3.10)

The conditions on $\theta_{1},\theta_{2}$ guarantee that the states
$|X_{1}(\theta_{1})^{\dagger}\rho^{1/2}\rangle$ and
$|X_{2}(\theta_{2})\rho^{1/2}\rangle$ have bounded norm even in the
$N\to\infty$ limit. (In section 7, these conditions will be relaxed, leading
to a bound on the high-frequency decay of the spectral function.) Furthermore,
$\|e^{-zQ_{\mathrm{A}}}\|\leqslant 1$ for $\operatorname{Re}z\geqslant 0$
because $Q_{\mathrm{A}}$ is positive-semidefinite. It follows that for given
$\theta_{1}$ and $\theta_{2}$, the matrix element (3.10) is analytic in the
half-plane $\operatorname{Re}z>0$, tending to $0$ as
$\operatorname{Re}z\to+\infty$. The function
$F_{X_{3},X_{4}}^{\mathrm{A}}(z;\theta_{3},\theta_{4})$ has similar analytic
properties.

For convenience, let us absorb the common factors in (3.9) into the variable
$z$:

$\displaystyle F_{X_{1},X_{2}}^{\mathrm{R}}(z;\theta_{1},\theta_{2})$
$\displaystyle=f_{X_{1},X_{2}}^{\mathrm{R}}\bigl{(}e^{-i\varkappa(\theta_{1}+\theta_{2})/2}z;\,\theta_{1}-\theta_{2}\bigr{)},\quad\>$
$\displaystyle f_{X_{1},X_{2}}^{\mathrm{R}}(x;\theta)$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(-x)^{m}}{m!}\,\Upsilon_{X_{1},X_{2}}^{\mathrm{R},m}(\theta),$
(3.11) $\displaystyle F_{X_{3},X_{4}}^{\mathrm{A}}(z;\theta_{3},\theta_{4})$
$\displaystyle=f_{X_{3},X_{4}}^{\mathrm{A}}\bigl{(}e^{i\varkappa(\theta_{3}+\theta_{4})/2}z;\,\theta_{3}-\theta_{4}\bigr{)},\quad\>$
$\displaystyle f_{X_{3},X_{4}}^{\mathrm{A}}(x;\theta)$
$\displaystyle=\sum_{m=0}^{\infty}\frac{(-x)^{m}}{m!}\,\Upsilon_{X_{3},X_{4}}^{\mathrm{A},m}(\theta).$

The new functions $f_{X_{1},X_{2}}^{\mathrm{R}}(x;\theta)$ and
$f_{X_{3},X_{4}}^{\mathrm{A}}(x;\theta)$ are analytic in $x$ for $|\arg
x|<\frac{\pi}{2}+\phi$, where

$\phi=\frac{\varkappa}{2}\min\\{\tau,2\pi-\tau\\},\quad\tau=\operatorname{Re}\theta,$
(3.12)

and tend to $0$ as $|x|$ goes to infinity while $\arg x$ remaining constant.
Thus, they can be expressed using the Laplace transform,

$f_{X_{1},X_{2}}^{\mathrm{R}}(x;\theta)=\int_{0}^{\infty}e^{-xy}\,h_{X_{1},X_{2}}^{\mathrm{R}}(y;\theta)\,dy,\qquad
f_{X_{3},X_{4}}^{\mathrm{A}}(x;\theta)=\int_{0}^{\infty}e^{-xy}\,h_{X_{3},X_{4}}^{\mathrm{A}}(y;\theta)\,dy,$
(3.13)

where $h_{X_{1},X_{2}}^{\mathrm{R}}(y;\theta)$ and
$h_{X_{1},X_{2}}^{\mathrm{A}}(y;\theta)$ are analytic in $y$ in the domain
$|\arg y|<\phi$.

Now, using the definition of $f_{X_{1},X_{2}}^{\mathrm{R}}$,
$f_{X_{3},X_{4}}^{\mathrm{A}}$ and equation (3.13), we find that

$\Upsilon_{X_{1},X_{2}}^{\mathrm{R},m}(\theta)=\int_{0}^{\infty}y^{m}\,h_{X_{1},X_{2}}^{\mathrm{R}}(y;\theta)\,dy,\qquad\Upsilon_{X_{3},X_{4}}^{\mathrm{A},m}(\theta)=\int_{0}^{\infty}y^{m}\,h_{X_{3},X_{4}}^{\mathrm{A}}(y;\theta)\,dy.$
(3.14)

Plugging these expressions into (3.7), we obtain equation (2.4).

##  4 Small perturbations to the thermofield double

In the linear (early-time) regime, scrambling modes are treated as small
perturbations to the thermofield double state. A mode is generally in a
superposition of mean-field states of different amplitudes, but in the linear
approximation, only the average amplitude $c$ and the corresponding change in
the two-point function
$\delta\langle{\chi_{\alpha}(\tau_{1})\chi_{\beta}(\tau_{2})}\rangle\propto c$
matter. (Here, $\alpha,\beta,\ldots$ refer to Majorana modes in an SYK-like
setting.)

We find it convenient to use a Hilbert space formulation of mean-field theory,
known as the “naive model” [12]. Formally, it is just a representation of the
two-point function by a Gaussian fermionic state. To give it more meaning,
consider the operators $\chi_{\alpha}(it)$ with $t$ in an interval much
shorter than the scrambling time. Their thermal expectation values obey Wick’s
theorem with $1/N$ accuracy, and so it is tempting to say that the quantum
state is almost Gaussian. This is actually not correct because $1/N$ errors
add up; for example, one cannot calculate the expectation value of the SYK
Hamiltonian $\sum_{j<k<l<m}J_{jklm}\chi_{j}\chi_{k}\chi_{l}\chi_{m}$ using
Wick’s theorem. (The accumulation of errors can be illustrated by constructing
$N$ linear combinations of basic Majorana operators that almost anticommute
pairwise but not in the bigger set. Indeed, there exist $N$ unit vectors with
$O(1/N)$ mutual inner products that are nonetheless linearly dependent.) To
avoid this problem, we assume that $\alpha,\beta,\ldots$ belong to a small
subset of modes. Then the operators $\chi_{\alpha}(it)$ may be regarded as
free-fermion, i.e. $\\{\chi_{\alpha}(it),\chi_{\beta}(it^{\prime})\\}$ is just
a number with acceptable accuracy. Furthermore, the restriction of the thermal
state to the subalgebra generated by these operators is Gaussian. For the
double system and similarly doubled subalgebra, the TFD is reduced to a pure
Gaussian state, which may be called a free-fermion vacuum.

In this section, we mathematically define the double system and consider the
perturbed TFD. Although it is still a pure state, its restriction $R$ to the
aforementioned subalgebra is not. We examine the structure of $R$ and
decompose it into a coherent and an incoherent part. (This is the only task
that requires the use of the naive model, and it is tangential to our main
goal.) Then we introduce “size operators” $Q_{\mathrm{R}}$, $Q_{\mathrm{A}}$
that measure the magnitudes of forward-growing (retarded) and backward-growing
(advanced) scrambling modes, respectively. These operators are positive-
semidefinite, vanish on the TFD, and are sensitive only to the incoherent
perturbation parts, which are related to the commutator OTOC.

###  4.1 The double system

Let $\mathcal{H}$ be the Hilbert space of a physical system, and let us
introduce its almost identical copy with the Hilbert space $\mathcal{H}^{*}$,
that is, the space of linear functional on $\mathcal{H}$ (or equivalently,
bra-vectors). The algebra $\mathbb{L}(\mathcal{H})$ of operators acting on
$\mathcal{H}$ is canonically isomorphic to
$\mathcal{H}\otimes\mathcal{H}^{*}$. Thus, any operator
$A\in\mathbb{L}(\mathcal{H})$ may be interpreted as a vector
$|A\rangle\in\mathcal{H}\otimes\mathcal{H}^{*}$. In particular, $|{\rm
TFD}\rangle=|\rho^{\frac{1}{2}}\rangle$ corresponds to the square root of the
thermal density matrix $\rho=Z^{-1}e^{-\beta H}$. The Hermitian inner product
on $\mathcal{H}\otimes\mathcal{H}^{*}$ has the following expression in the
operator language:

$\langle A|B\rangle=\operatorname{Tr}(A^{{\dagger}}B).$ (4.1)

Now, we consider operators acting on the double system. In the bosonic case,
the formalism is quite simple. Note that the operator algebra
$\mathbb{L}(\mathcal{H}^{*})$ is isomorphic to $\mathbb{L}(\mathcal{H})$ with
the multiplication order reversed; this new algebra is denoted by
$\mathbb{L}(\mathcal{H})^{\rm op}$. Thus,
$\mathbb{L}(\mathcal{H}\otimes\mathcal{H}^{*})\cong\mathbb{L}(\mathcal{H})\otimes\mathbb{L}(\mathcal{H})^{\rm
op}$. An element
$X\boldsymbol{\cdot}Y\in\mathbb{L}(\mathcal{H})\otimes\mathbb{L}(\mathcal{H})^{\rm
op}$ acts as follows:

$(X\boldsymbol{\cdot}Y)|A\rangle=|XAY\rangle.$ (4.2)

In the $\mathbb{Z}_{2}$-graded case, we define the left and right actions of
$\mathbb{L}(\mathcal{H})$ on itself so as to produce a representation of the
$\mathbb{Z}_{2}$-graded algebra product $\mathbb{L}(\mathcal{H})\otimes_{\rm
gr}\mathbb{L}(\mathcal{H})^{\rm op}$:

$\displaystyle(X\boldsymbol{\cdot}I)|A\rangle=|XA\rangle\qquad\text{(left
action)},$ (4.3)
$\displaystyle(I\boldsymbol{\cdot}Y)|A\rangle=i^{\deg(AY)-\deg(A)}|AY\rangle\qquad\text{(right
action)}.$

Here, the phase factor $i^{\deg(AY)-\deg(A)}$ is designed to ensure that the
left and right actions commute up to $(-1)^{\deg X\cdot\deg Y}$. For a general
element $X\boldsymbol{\cdot}Y=(X\boldsymbol{\cdot}I)(I\boldsymbol{\cdot}Y)$ of
$\mathbb{L}(\mathcal{H})\otimes_{\mathrm{gr}}\mathbb{L}(\mathcal{H})^{\rm
op}$, we have

$\displaystyle(X_{1}\boldsymbol{\cdot}Y_{1})(X_{2}\boldsymbol{\cdot}Y_{2})$
$\displaystyle=(-1)^{\deg Y_{1}\cdot\deg
X_{2}}(X_{1}X_{2}\boldsymbol{\cdot}Y_{2}Y_{1}),$ (4.4)
$\displaystyle(X\boldsymbol{\cdot}Y)^{\dagger}$ $\displaystyle=(-1)^{\deg
X\cdot\deg Y}(X^{\dagger}\boldsymbol{\cdot}Y^{\dagger}).$

For an arbitrary $X\in\mathbb{L}(\mathcal{H})$, we define an “annihilation
operator” $a(X)$ such that $a(X)|\rho^{\frac{1}{2}}\rangle=0$:

$a(X)=\rho^{\frac{1}{4}}X\rho^{-\frac{1}{4}}\boldsymbol{\cdot}I-i^{-\deg
X}I\boldsymbol{\cdot}\rho^{-\frac{1}{4}}X\rho^{\frac{1}{4}},$ (4.5)

where the phase factor $i^{-\deg X}$ is consistent with the choice in (4.3).
Note that

$a(X)\,|\rho^{\frac{1}{4}}Y\rho^{\frac{1}{4}}\rangle=\big{|}\rho^{\frac{1}{4}}[X,Y]_{\mathrm{gr}}\rho^{\frac{1}{4}}\bigr{\rangle},$
(4.6)

where

$[X,Y]_{\mathrm{gr}}=XY-(-1)^{\deg X\cdot\deg Y}YX.\vskip 6.0pt plus 2.0pt
minus 2.0pt$ (4.7)

Now, let $\chi_{\alpha}(it)$ be some Majorana fields satisfying Wick’s
theorem. (The example of interest is $\chi(it)=\chi_{j}(it)$ for a fixed site
$j$ of the SYK model, considered at the time scale of the order of $1$ so that
$1/N$ corrections are not exponentially amplified.) We work in the operator
algebra generated by $\chi_{\alpha}(it)$, or equivalently, by
$\tilde{\chi}_{\alpha}(\omega)=\int_{-\infty}^{+\infty}\chi_{\alpha}(it)e^{i\omega
t}dt$ with the commutation relations
$\\{\tilde{\chi}_{\alpha}(\omega),\tilde{\chi}_{\beta}(\omega^{\prime})\\}=A_{\alpha\beta}(\omega)\,2\pi\delta(\omega+\omega^{\prime})$,
where the spectral function $A$ also determines the correlation function:

$\langle\tilde{\chi}_{\alpha}(\omega)\tilde{\chi}_{\beta}(\omega^{\prime})\rangle=\frac{A_{\alpha\beta}(\omega)}{1+e^{-\beta\omega}}\,2\pi\delta(\omega+\omega^{\prime}).$
(4.8)

The thermal state $\rho$ restricted to this subalgebra will be denoted by
$\varrho$. In most cases, $\varrho$ and $\rho$ can be used interchangeably;
for example,

$\varrho^{-s}\tilde{\chi}_{\alpha}(\omega)\varrho^{s}=\rho^{-s}\tilde{\chi}_{\alpha}(\omega)\rho^{s}=e^{s\beta
H}\tilde{\chi}_{\alpha}(\omega)e^{-s\beta
H}=e^{-s\beta\omega}\tilde{\chi}_{\alpha}(\omega).$ (4.9)

The distinction between $\varrho$ and $\rho$ will be important in the next
section, when we will consider perturbations.

The next piece of formalism is useful to describe the structure of the
perturbed TFD. For this purpose, it is convenient to represent
$\chi_{\alpha}(it)$ as a delayed signal coming from the “past horizon”, i.e. a
standard heat bath with a flat spectral function. More exactly (see appendix A
for detail),

$\tilde{\chi}_{\alpha}(\omega)=\sum_{\beta}L_{\alpha\beta}^{\mathrm{R}}(\omega)\tilde{\psi}_{\beta}(\omega),$
(4.10)

where
$\\{\tilde{\psi}_{\alpha}(\omega),\tilde{\psi}_{\beta}(\omega^{\prime})\\}=\delta_{\alpha\beta}\,2\pi\delta(\omega+\omega^{\prime})$.
The last condition is equivalent to the equation
$L^{\mathrm{R}}(\omega)L^{\mathrm{R}}(\omega)^{\dagger}=A(\omega)$, and we
also require that $L^{\mathrm{R}}(\omega)$ admits an analytic continuation to
the upper half-plane and has reasonable asymptotic behavior at
$\omega\to\infty$. Due to the analyticity condition,
$\tilde{\psi}_{\alpha}(\omega)$ is defined for complex $\omega$ with positive
imaginary part by inverting the transformation (4.10) and using the
regularization
$\tilde{\chi}_{\alpha}(\omega)\approx\int_{-\infty}^{+\infty}\chi_{\alpha}(it)e^{i\omega
t}u(t)\,dt$, where $u$ is a broad Gaussian. But if
$\operatorname{Im}\omega<0$, then $\tilde{\psi}_{\alpha}(\omega)$ is hard to
express in terms of the original operators of the model. Nonetheless, we will
use it as a formal construct in equations (4.28), (4.29) for quantum states.
This is legitimate because by definition, a state is a linear functional on
observables. If one uses this definition directly, as in (LABEL:coh_part0),
(4.27), the past horizon representation is not necessary, though it helps
avoid carrying the spectral function around. In the derivation of our main
result, we will not use it at all.

The canonical annihilation operators for the double system are defined as
follows:

$\displaystyle\tilde{a}_{\alpha}(\omega)=\frac{\Gamma\left(\frac{1}{2}-i\frac{\beta\omega}{2\pi}\right)}{\sqrt{2\pi}}\Bigl{(}{\underbrace{\varrho^{\frac{1}{4}}\tilde{\psi}_{\alpha}(\omega)\varrho^{-\frac{1}{4}}}_{e^{\beta\omega/4}\tilde{\psi}_{\alpha}(\omega)}}\boldsymbol{\cdot}I+iI\boldsymbol{\cdot}{\underbrace{\varrho^{-\frac{1}{4}}\tilde{\psi}_{\alpha}(\omega)\varrho^{\frac{1}{4}}}_{e^{-\beta\omega/4}\tilde{\psi}_{\alpha}(\omega)}}\Bigr{)},$
(4.11)
$\displaystyle\tilde{a}_{\alpha}(\omega)|\varrho^{\frac{1}{2}}\rangle=0,\quad\tilde{a}_{\alpha}(\omega^{*})^{\dagger}|\varrho^{\frac{1}{2}}\rangle=\frac{\sqrt{2\pi}}{\Gamma\left(\frac{1}{2}-i\frac{\beta\omega}{2\pi}\right)}\,\big{|}\varrho^{\frac{1}{4}}\tilde{\psi}_{\alpha}(-\omega)\varrho^{\frac{1}{4}}\bigr{\rangle},$
(4.12)

where the overall factor is chosen such that
$\\{\tilde{a}_{\alpha}(\omega),\tilde{a}_{\beta}(\omega^{\prime})^{\dagger}\\}=\delta_{\alpha\beta}\,2\pi\delta(\omega-\omega^{\prime})$
when $\omega$, $\omega^{\prime}$ are real. In the OTOC analysis, we will use a
variant of this equation:

$\bigl{\\{}\tilde{a}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)},\,\tilde{a}_{\beta}\bigl{(}\omega^{\prime}-i\tfrac{\varkappa}{2}\bigr{)}^{\dagger}\bigr{\\}}=\delta_{\alpha\beta}\,2\pi\delta(\omega-\omega^{\prime})\qquad\text{for
}\,\omega,\omega^{\prime}\in\mathbb{R}.$ (4.13)

###  4.2 Retarded vertex function as a deformation of the TFD

Let us again set $\beta=2\pi$ and consider fermionic operators $X_{j}$ at
complex times $\theta_{j}=\tau_{j}+it_{j}$ with

$\tau_{1}=\pi,\quad\tau_{3}=\frac{\pi}{2},\quad\tau_{2}=0,\quad\tau_{4}=-\frac{\pi}{2},\qquad
t_{1}\approx t_{2}>t_{3}\approx t_{4}.$ (4.14)

This configuration is shown in Fig. 1b. It contains a pair of operators in the
future (namely, $X_{1}$ and $X_{2}$) and another pair in the past ($X_{3}$ and
$X_{4}$). We may interpret either pair as a source that perturbs the
thermofield double $|\rho^{1/2}\rangle$, whereas the other pair measures the
resulting effect. Here, we choose to regard $X_{3}$ and $X_{4}$ as sources and
interpret the OTOC as a matrix element of a combination of $X_{1}$, $X_{2}$
acting on the double system:

$\operatorname{OTOC}_{X_{1},X_{2},X_{3},X_{4}}(\theta_{1},\theta_{2},\theta_{3},\theta_{4})=\bigl{\langle}\rho^{1/4}X_{4}(it_{4})^{\dagger}\rho^{1/4}\big{|}-iX_{1}(it_{1})\boldsymbol{\cdot}X_{2}(it_{2})\big{|}\rho^{1/4}X_{3}(it_{3})\rho^{1/4}\bigr{\rangle}.$
(4.15)

We assume that $t_{1}\approx t_{2}\approx 0$, whereas $t_{3},t_{4}$ are large
and negative. Thus, we focus on the perturbation effect while pushing the
sources to distant past (or distant future in other cases). For simplicity,
let $X_{4}=X_{3}^{\dagger}$ and $t_{4}=t_{3}$; then the above expression
becomes (up to an overall factor) the expectation value of
$-iX_{1}(it_{1})\boldsymbol{\cdot}X_{2}(it_{2})$ on the pure state
$|\Psi\rangle\langle\Psi|$, where

$|\Psi\rangle=b^{-1/2}\big{|}\rho^{1/4}X_{3}(it_{3})\rho^{1/4}\bigr{\rangle},\qquad
b=\bigl{\langle}\rho^{1/4}X_{3}(it_{3})\rho^{1/4}\big{|}\rho^{1/4}X_{3}(it_{3})\rho^{1/4}\bigr{\rangle}=\langle{X_{3}(\theta_{3})X_{4}(\theta_{4})}\rangle.$
(4.16)

If $X_{1}=\chi_{\alpha}$ and $X_{2}=\chi_{\beta}$ are elementary Majorana
operators, we may restrict $|\Psi\rangle\langle\Psi|$ to the previously
defined subalgebra, i.e. trace out the other degrees of freedom. (This amounts
to using the “naive model” when detecting the TFD deformation.) Thus, we
obtain a mixed state $R$ of a system of free fermions, and the OTOC is
expressed as follows:

$b^{-1}\operatorname{OTOC}_{\chi_{\alpha},\chi_{\beta},\cdots,\cdots}(\pi+it_{1},it_{2},\cdots,\cdots)=\operatorname{Tr}\bigl{(}(-i\chi_{\alpha}(it_{1})\boldsymbol{\cdot}\chi_{\beta}(it_{2}))\,R\bigr{)}.$
(4.17)

The expression on the right-hand side is a perturbed version of the Wightman
function,

$\langle{\chi_{\alpha}(\pi+it_{1})\chi_{\beta}(it_{2})}\rangle=\operatorname{Tr}\bigl{(}(-i\chi_{\alpha}(it_{1})\cdot\chi_{\beta}(it_{2}))\,|\varrho^{1/2}\rangle\langle\varrho^{1/2}|\bigr{)}.$
(4.18)

In this section, we study the deformation of the TFD in the linear order.
Thus, the OTOC is given by equation (3.1), which can be simplified as follows:

$b^{-1}\operatorname{OTOC}_{\chi_{\alpha},\chi_{\beta},\cdots,\cdots}(\pi+it_{1},it_{2},\cdots,\cdots)=\langle{\chi_{\alpha}(\pi+it_{1})\chi_{\beta}(it_{2})}\rangle-c\,e^{\varkappa\frac{t_{1}+t_{2}}{2}}\Upsilon^{\mathrm{R}}_{\alpha\beta}(\pi+i(t_{1}-t_{2})).$
(4.19)

The coefficient $c$ depends on the deformation source, i.e. on $X_{3}$ and
$t_{3}$, but we may assume that $c>0$ by fixing the normalization of
$\Upsilon^{\mathrm{R}}$. The Fourier transform of this function,
$\widetilde{\Upsilon}^{\mathrm{R}}(\omega)=\int_{-\infty}^{+\infty}\Upsilon^{\mathrm{R}}(\pi+it)e^{i\omega
t}dt$, can be shown to satisfy the following relations:

$\widetilde{\Upsilon}^{\mathrm{R}}(\omega)=\widetilde{\Upsilon}^{\mathrm{R}}(-\omega)^{T}=\widetilde{\Upsilon}^{\mathrm{R}}(-\omega)^{*}=\widetilde{\Upsilon}^{\mathrm{R}}(\omega)^{\dagger},\qquad\widetilde{\Upsilon}^{\mathrm{R}}(\omega)\geqslant
0.$ (4.20)

The last inequality can be derived from the positivity of the incoherent
deformation part, defined below, which is proportional to
$\cos\frac{\pi\varkappa}{2}$ and related to commutator OTOCs.

The last term in (4.19) characterizes the deformation of the free-fermion
vacuum:

$\delta\langle{-i\chi_{\alpha}(it_{1})\boldsymbol{\cdot}\chi_{\beta}(t_{2})}\rangle:=\operatorname{Tr}\bigl{(}(-i\chi_{\alpha}(t_{1})\boldsymbol{\cdot}\chi_{\beta}(t_{2}))\,\delta
R\bigr{)}=-c\,e^{\varkappa\frac{t_{1}+t_{2}}{2}}\Upsilon^{\mathrm{R}}_{\alpha\beta}(\pi+i(t_{1}-t_{2})),$
(4.21)

where

$\delta R=R-|\varrho^{1/2}\rangle\langle\varrho^{1/2}|.$ (4.22)

An arbitrary infinitesimal deformation of a pure state can be decomposed into
a “coherent” and an “incoherent” part:

$\delta
R=|\xi\rangle\langle\varrho^{1/2}|+|\varrho^{1/2}\rangle\langle\xi|+\sum_{j}\bigl{(}|\eta_{j}\rangle\langle\eta_{j}|-\langle\eta_{j}|\eta_{j}\rangle|\varrho^{1/2}\rangle\langle\varrho^{1/2}|\bigr{)},\qquad|\xi\rangle,|\eta_{j}\rangle\perp|\varrho^{1/2}\rangle.$
(4.23)

In the case of a free-fermion system, $|\xi\rangle$ is a linear combination of
$a^{\dagger}_{\alpha}a^{\dagger}_{\beta}|\varrho^{1/2}\rangle$, whereas
$|\eta_{j}\rangle$ is a combination of
$a^{\dagger}_{\alpha}|\varrho^{1/2}\rangle$. To compute $|\xi\rangle$ and
$|\eta_{j}\rangle$, we will consider
$\delta\langle\tilde{a}_{\alpha}(\omega+i\frac{\varkappa}{2})\,\tilde{a}_{\beta}(\omega^{\prime}+i\frac{\varkappa}{2})\rangle$
and
$\delta\langle\tilde{a}_{\alpha}(\omega+i\frac{\varkappa}{2})^{\dagger}\tilde{a}_{\beta}(\omega^{\prime}+i\frac{\varkappa}{2})\rangle$,
respectively.

Since the annihilation and creation operators are defined in terms of the
“past horizon” operators $\tilde{\psi}_{\beta}(\omega)$ (see Eq. (4.10)), the
first calculation step is to pass from
$\delta\langle{-i\chi_{\alpha}(\pi+it_{1})\boldsymbol{\cdot}\chi_{\beta}(t_{2})}\rangle$
in (4.21) to

$\delta\bigl{\langle}{\tilde{\psi}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}\boldsymbol{\cdot}\tilde{\psi}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2}\bigr{)}}\bigr{\rangle}=-ic\,\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(\omega)\,\delta(\omega+\omega^{\prime}),$
(4.24)

where

$\widetilde{\Xi}^{\mathrm{R}}(\omega)=L^{\mathrm{R}}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}^{-1}\,\widetilde{\Upsilon}^{\mathrm{R}}(\omega)\left(L^{\mathrm{R}}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}^{-1}\right)^{\dagger}.$
(4.25)

The function $\widetilde{\Xi}^{\mathrm{R}}(\omega)$ has properties similar to
(4.20).

The next step is to express the products of $a$ and $a^{\dagger}$ using the
definition (4.11). Each such expression has four terms, but it is sufficient
to consider only the terms with $\tilde{\psi}_{\alpha}$ and
$\tilde{\psi}_{\beta}$ on different sides. (Indeed, the expectation value of
any one-sided observable is not affected by the deformation.) Thus,

$\displaystyle\delta\bigl{\langle}{\tilde{a}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}\tilde{a}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2}\bigr{)}}\bigr{\rangle}$
(4.26)
$\displaystyle=\frac{\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega)\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega^{\prime})}{2\pi}\begin{aligned}
\Bigl{(}\;&ie^{\pi(\omega-\omega^{\prime})/2}\,\delta\bigl{\langle}{\tilde{\psi}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}\boldsymbol{\cdot}\tilde{\psi}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2})}\bigr{\rangle}\\\
{}-{}&ie^{\pi(\omega^{\prime}-\omega)/2}\,\delta\bigl{\langle}{\tilde{\psi}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2}\bigr{)}\boldsymbol{\cdot}\tilde{\psi}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}}\bigr{\rangle}\Bigr{)}\end{aligned}$
$\displaystyle=c\,\bigl{|}\Gamma\bigl{(}\tfrac{1}{2}+\tfrac{\varkappa}{2}-i\omega\bigr{)}\bigr{|}^{2}\,\delta(\omega+\omega^{\prime})\,2\sinh(\pi\omega)\,\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(\omega),$

and similarly,

$\displaystyle\delta\bigl{\langle}{\tilde{a}_{\alpha}\bigl{(}\omega+i\tfrac{\varkappa}{2}\bigr{)}^{\dagger}\tilde{a}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2}\bigr{)}}\bigr{\rangle}$
(4.27)
$\displaystyle=\frac{\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega)\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega^{\prime})}{2\pi}\begin{aligned}
\Bigl{(}\;&ie^{\pi(\omega-\omega^{\prime}-i\varkappa)/2}\,\delta\bigl{\langle}{\tilde{\psi}_{\alpha}\bigl{(}-\omega+i\tfrac{\varkappa}{2}\bigr{)}\boldsymbol{\cdot}\tilde{\psi}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2})}\bigr{\rangle}\\\
{}+{}&ie^{\pi(\omega^{\prime}-\omega+i\varkappa)/2}\,\delta\bigl{\langle}{\tilde{\psi}_{\beta}\bigl{(}\omega^{\prime}+i\tfrac{\varkappa}{2}\bigr{)}\boldsymbol{\cdot}\tilde{\psi}_{\alpha}\bigl{(}-\omega+i\tfrac{\varkappa}{2}\bigr{)}}\bigr{\rangle}\Bigr{)}\end{aligned}$
$\displaystyle=c\,\bigl{|}\Gamma\bigl{(}\tfrac{1}{2}+\tfrac{\varkappa}{2}-i\omega\bigr{)}\bigr{|}^{2}\,\delta(\omega-\omega^{\prime})\,2\cos\bigl{(}\tfrac{\pi\varkappa}{2}\bigr{)}\,\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(-\omega).$

From this we obtain the expressions for the coherent part,

$\displaystyle|\xi\rangle$
$\displaystyle=c\int\frac{d\omega}{2\pi}\,\frac{\bigl{|}\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega)\bigr{|}^{2}}{2\pi}\sinh(\pi\omega)\sum_{\alpha,\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(\omega)\,\tilde{a}_{\beta}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}^{\dagger}\tilde{a}_{\alpha}\bigl{(}\omega-i\tfrac{\varkappa}{2})^{\dagger}\,|\varrho^{1/2}\rangle$
(4.28)
$\displaystyle=c\int\frac{d\omega}{2\pi}\,\sinh(\pi\omega)\sum_{\alpha,\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(\omega)\,\big{|}\varrho^{1/4}\tilde{\psi}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}\tilde{\psi}_{\alpha}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\varrho^{1/4}\bigr{\rangle},$

and for the incoherent part,

$\displaystyle\sum_{j}|\eta_{j}\rangle\langle\eta_{j}|$ $\displaystyle=c\cdot
2\cos\bigl{(}\tfrac{\pi\varkappa}{2}\bigr{)}\int\frac{d\omega}{2\pi}\,\frac{\bigl{|}\Gamma(\frac{1}{2}+\frac{\varkappa}{2}-i\omega)\bigr{|}^{2}}{2\pi}\sum_{\alpha,\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(-\omega)\,\tilde{a}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}^{\dagger}|\varrho^{1/2}\rangle\langle\varrho^{1/2}|\tilde{a}_{\alpha}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}$
(4.29) $\displaystyle=c\cdot
2\cos\bigl{(}\tfrac{\pi\varkappa}{2}\bigr{)}\int\frac{d\omega}{2\pi}\sum_{\alpha,\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(-\omega)\,\big{|}\varrho^{1/4}\tilde{\psi}_{\beta}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\varrho^{1/4}\bigr{\rangle}\bigl{\langle}\varrho^{1/4}\tilde{\psi}_{\alpha}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}\varrho^{1/4}\big{|}.$

Note that the last equation implies that
$\widetilde{\Xi}^{\mathrm{R}}(-\omega)\geqslant 0$.

As a curiosity, let us also represent $\delta R$ as
$cT|\varrho^{1/2}\rangle\langle\varrho^{1/2}|$, where $T$ is a superoperator
acting on the left (physical) subsystem:444In the discussion of $T$, the
notation $X\cdot Y$ is understood as the left-right action on
$|\varrho^{1/2}\rangle\langle\varrho^{1/2}|$, namely, $X$ acts on the ket and
$Y$ acts on the bra. This is in contrast with the previous usage of
$X\boldsymbol{\cdot}Y$, where $X$ acts on the left subsystem and $Y$ acts on
the right subsystem. As we have clarified, $T$ acts on the left subsystem
only.

$T=i\sin\bigl{(}\tfrac{\pi\varkappa}{2}\bigr{)}(P\cdot I-I\cdot
P)+2\cos\bigl{(}\tfrac{\pi\varkappa}{2}\bigr{)}\mathcal{L},$ (4.30)

where $P$ is a Hermitian operator,

$P=\int\frac{d\omega}{2\pi}\,\sinh(\pi\omega)\sum_{\alpha\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}(\omega)\,\tilde{\psi}_{\alpha}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\tilde{\psi}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}$
(4.31)

and $\mathcal{L}$ is a Lindbladian,

$\mathcal{L}=\int\frac{d\omega}{2\pi}e^{\pi\omega}\sum_{\alpha,\beta}\widetilde{\Xi}^{\mathrm{R}}_{\alpha\beta}\begin{aligned}
&(\omega)\Bigl{(}-i\tilde{\psi}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}\cdot\tilde{\psi}_{\alpha}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\\\
&-\tfrac{1}{2}\tilde{\psi}_{\alpha}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\tilde{\psi}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2})\cdot
I-\tfrac{1}{2}I\cdot\tilde{\psi}_{\alpha}\bigl{(}-\omega-i\tfrac{\varkappa}{2}\bigr{)}\tilde{\psi}_{\beta}\bigl{(}\omega-i\tfrac{\varkappa}{2}\bigr{)}\Bigr{)}.\end{aligned}$
(4.32)

An interesting observation is that the Lindbladian part in $T$ comes with the
prefactor of $\cos\frac{\pi\varkappa}{2}$, which is called the decoherence
factor in [12]. In the limit of maximal chaos, $\varkappa\rightarrow 1$, it
tends to zero.555This does not mean that the incoherent part in $\delta R$
vanishes, as the coefficient $c$ may diverge while the combination
$c\cos\frac{\pi\varkappa}{2}$ approaching a finite limit.

###  4.3 Size operator

A “size operator” is any positive-semidefinite operator that measures the
magnitude of the TFD deformation. Since we consider a one-parameter family of
quantum states, and only to the first order in the parameter $c$ (as defined
by Eq. (4.21)), there are many ways to measure. This is based on an underlying
assumption: the measurement takes place around a certain time $t_{0}$, say,
$t_{0}=0$, whereas the deformation is produced by a relatively weak source at
a distant time. Within the naive model, we may use the operator
$Q_{\mathrm{R}}^{(\text{naive})}=\int\frac{d\omega}{2\pi}\frac{d\omega^{\prime}}{2\pi}\sum_{\alpha,\beta}v_{\alpha\beta}(\omega,\omega^{\prime})\,\tilde{a}_{\alpha}\bigl{(}\omega+i\frac{\varkappa}{2}\bigr{)}^{\dagger}\tilde{a}_{\beta}\bigl{(}\omega^{\prime}+i\frac{\varkappa}{2}\bigr{)}$
with some positive-semidefinite $v$. To assure convergence (in view of the
exponential factors in the definition (4.11) of $\tilde{a}_{\alpha}(\omega)$),
we may require that $v_{\alpha\beta}(\omega,\omega^{\prime})$ decay
sufficiently fast as $\omega,\omega^{\prime}\to\infty$. In the time domain,
the previous equation becomes

$Q_{\mathrm{R}}^{(\text{naive})}=\int
dt\,dt^{\prime}\sum_{\alpha,\beta}u_{\alpha\beta}(t,t^{\prime})\,e^{-\varkappa(t+t^{\prime})/2}a(\chi_{\alpha}(it))^{{\dagger}}a(\chi_{\beta}(it^{\prime})),\qquad
u\geqslant 0,$ (4.33)

where, according to the definition (4.5),

$a(\chi_{\alpha}(it))=\chi_{\alpha}\bigl{(}-\tfrac{\pi}{2}+it\bigr{)}\boldsymbol{\cdot}I+iI\boldsymbol{\cdot}\chi_{\alpha}\bigl{(}\tfrac{\pi}{2}+it\bigr{)}.$
(4.34)

The “window function” $u$ can be, for example, Gaussian to eliminate any
potential divergence. (In practice, it is not an issue.) In the SYK case, the
indices $\alpha,\beta$ are redundant because the naive model consists of a
single field, $\chi(it)=\chi_{j}(it)$ for some fixed $j$.

In the actual SYK model, the TFD deformation affects all Majorana modes
equally, and therefore, the size operator may be averaged over the modes.
Changing the overall normalization, we arrive at the following
definition:666For infinite temperature and
$u_{\mathrm{R}}(t,t^{\prime})=\delta(t)\delta(t^{\prime})$, we get
$Q_{\mathrm{R}}=\sum_{j}a(\chi_{j})^{{\dagger}}a(\chi_{j})$, where
$a(\chi_{j})=\chi_{j}\boldsymbol{\cdot}I+iI\boldsymbol{\cdot}\chi_{j}$. In
this case, $\langle\mathcal{O}|Q_{\mathrm{R}}|\mathcal{O}\rangle$ is the
“size” of the operator $\mathcal{O}$ in the sense of Ref. [17]. See Ref. [27,
28, 29] for more proposals of operator size at finite temperature.

$Q_{\mathrm{R}}=\sum_{j=1}^{N}\int
dt\,dt^{\prime}\,u_{\mathrm{R}}(t,t^{\prime})\,e^{-\varkappa(t+t^{\prime})/2}a(\chi_{j}(it))^{{\dagger}}a(\chi_{j}(it^{\prime})),\qquad
u_{\mathrm{R}}\geqslant 0.$ (4.35)

By construction, $Q_{\mathrm{R}}$ is positive-semidefinite, annihilates
$|\rho^{\frac{1}{2}}\rangle$, and measures the magnitude of the forward-
propagating scrambling mode. The definition of the operator $Q_{\mathrm{A}}$
measuring the backward-propagating scrambling mode is similar but involves the
coefficient function
$u_{\mathrm{A}}(t,t^{\prime})e^{\varkappa(t+t^{\prime})/2}$.

To elaborate a bit, the “magnitude” is understood as the coefficient $c$.
However, it is convenient to normalize $Q_{\mathrm{R}}$ by the condition

$\delta\bigl{\langle}{Q_{\mathrm{R}}}\bigr{\rangle}=Cc.$ (4.36)

Let us express it more explicitly. For each filed $\chi=\chi_{j}$, we have