# More on chaos at weak coupling

Rohit R. Kalloor  Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot, Israel  Adar Sharon  Simons Center for Geometry
and Physics, SUNY, Stony Brook, NY 11794, U.S.A.

###### Abstract

We discuss aspects of the quantum Lyapunov exponent $\lambda_{L}$ in theories
with an exactly marginal SYK-like random interaction, where $\lambda_{L}$ can
be computed as a continuous function of the interaction strength
$\mathcal{J}$. In $1d$, we prove a conjecture from [1] which states that at
small $\mathcal{J}$, $\lambda_{L}$ can be found by considering a specific
limit of the four-point function in the decoupled theory. We then provide
additional evidence for the $2d$ version of this conjecture by discussing new
examples of Lyapunov exponents which can be computed at weak coupling.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Four-point function and double-commutator
    2. 2.2 Chaos
    3. 2.3 The continuity conjecture
  3. 3 Proving the continuity conjecture in QM
  4. 4 The chiral SYK model
  5. 5 The disordered $\mathcal{N}=2$ $\mathcal{A}_{3}$ minimal model
    1. 5.1 Details of duality to free fields
    2. 5.2 Correlation functions
    3. 5.3 Chaos
  6. 6 Conclusions
  7. A  Superappendix
    1. A.1 The supersymmetry algebra
    2. A.2 The $\mathcal{A}_{3}$ minimal model
  8. B Simplification of $1d$ integral

##  1 Introduction

The quantum Lyapunov exponent $\lambda_{L}$ is an intriguing and complicated
observable in quantum field theories. Of particular interest are theories
where $\lambda_{L}$ saturates an upper bound [2], since this might indicate
that they have semiclassical gravity duals. However, in this paper we will be
interested in the opposite limit, and would like to study how the chaos
exponent behaves as we turn on some small coupling $\mathcal{J}$. The
classical version of this question has been studied extensively, and many
interesting behaviors have been found for chaos at weak coupling. For example,
the KAM theorem schematically states that an integrable system deformed by a
small (integrability-breaking) deformation remains non-chaotic even for a
finite but small enough deformation. We will ask an analogous question in a
quantum setting: starting with $N$ decoupled theories and slowly increasing
the strength of a random interaction between them, how does $\lambda_{L}$
behave?

Our setup is the following. The basic building block is a conformal field
theory (CFT) $\mathcal{C}$, called the “core CFT”, which contains a primary
$\Phi$. Next, we take $N$ copies of the core CFT, and deform it by a random
interaction:

$\mathcal{C}^{N}+J_{i_{1}...i_{q}}\int d^{d}x\Phi_{i_{1}}...\Phi_{i_{q}}\;.$
(1.1)

The couplings $J_{i_{1}...i_{q}}$ are random with Gaussian measure and
variance $\langle
J_{i_{1}...i_{q}}^{2}\rangle=(q-1)!\frac{\mathcal{J}^{2}}{N^{q-1}}$ (with no
sum over repeated indices).111We emphasize that our disorder is spacetime-
independent. The deformation should be understood in terms of conformal
perturbation theory around $N$ copies of the CFT $\mathcal{C}$, and we will be
studying these theories in the large-$N$ limit. We will call such theories
“disordered CFTs”.

Disordered CFTs are a generalization of ideas appearing in disordered free
theories, which can be obtained by setting $\mathcal{C}$ to be a free field
theory, and setting $\Phi$ to be the corresponding free field. A famous
example of disordered free theories is the SYK model [3, 4], and some
additional examples can be found in [5, 6, 7, 8, 9, 10, 11, 12, 13]. In
general, the random interactions allow for some exact computations in the IR
for disordered free fields, assuming the theory flows to an interacting CFT
[3, 14, 15]. This was generalized to arbitrary core CFTs in [1, 16]. In these
theories, $\lambda_{L}$ can be read off from a certain out-of-time-ordered
four-point function (or alternatively, the double-commutator) [4, 17, 18, 2].
This is a difficult computation in general, but it is aided by the large-$N$
limit and conformal invariance in the IR. In particular, we must also restrict
ourselves to dimensions $d\leq 2$, since we will be interested in the theory
at finite temperature, but in small dimensions a conformal transformation can
map such theories to flat space.

An especially interesting subsector of disordered CFTs can be obtained by
demanding that the interaction term in (1.1) is exactly marginal, such that
$\mathcal{J}$ parametrizes a line of CFTs. This behavior is not generic, but
can be obtained by using supersymmetry or by considering chiral theories, both
of which we will discuss in this paper.222Similar behavior can be obtained in
QM without these assumptions [11]. Having a line of CFTs allows us to follow
interesting observables as we continuously move between different CFTs, and in
particular will allow us to study $\lambda_{L}$ as a function of
$\mathcal{J}$.

In [1], the chaos exponent was studied in the weak-coupling limit
$\mathcal{J}\to 0$ in disordered CFTs where $\mathcal{J}$ is exactly marginal.
In particular, in the examples that were discussed it was observed the chaos
exponent in the limit $\mathcal{J}\to 0$ is also given by the leading
exponential behavior of the double-commutator (DC) in a single core CFT. In
other words, if the DC of a single core CFT behaves at large times as
$\exp(\lambda_{L}^{0}t)$, then the chaos exponent of the interacting theory in
the limit $\mathcal{J}\to 0$ is given by

$\lambda_{L}(J\to 0)=\lambda_{L}^{0}\;.$ (1.2)

It was conjectured that this result is general; we will call it the continuity
conjecture.

The equality (1.2) is surprising for two main reasons. First,
$\lambda_{L}^{0}$ cannot be interpreted as a chaos exponent in a single core
CFT (since some form of a large-$N$ limit is required for this
interpretation), and yet in the interacting theory it dictates the behavior of
the chaos exponent at weak coupling. Second, it is possible for
$\lambda_{L}^{0}$ to be _negative_. A negative chaos exponent seems counter-
intuitive, and indeed the result cannot be trusted in the standard method of
computing $\lambda_{L}$ due to some assumptions that are required. A more
precise version of the continuity conjecture is then

$\lambda_{L}(J\to 0)=\max(\lambda_{L}^{0},0)\;.$ (1.3)

Despite this fact, if $\lambda_{L}^{0}$ is negative then one can still
determine that the chaos exponent of the theory is at most zero for a finite
range of values of $\mathcal{J}$. As a result, it is enough to show that in a
single core CFT $\lambda_{L}^{0}<0$ in order to find a discontinuous
transition into chaos, see figure 1. This is reminiscent of classical KAM
theory.

(a)

(b)

Figure 1: Two types of behaviors for the dependence of $\lambda_{L}$ on the
exactly marginal interaction $\mathcal{J}$: (a) continuous and (b)
discontinuous.

This paper includes two main results. First we will prove the continuity
conjecture (1.3) in quantum mechanics (QM). We then discuss two examples in
$2d$ and show that they obey the continuity conjecture, providing further
evidence for the $2d$ version of the conjecture. The first example is the
chiral SYK model discussed in [12], where the chaos exponent was already found
for all $\mathcal{J}$. We compare the result at small $\mathcal{J}$ to
$\lambda_{L}^{0}$ and find agreement. Next we discuss the disordered
$\mathcal{A}_{3}$ minimal model. Since this theory is dual to a free CFT, we
can compute all $n$-point functions for its primaries, and we use this result
to compute the chaos exponent at small $\mathcal{J}$. Comparing the result to
$\lambda_{L}^{0}$ we again find agreement.

In the examples discussed in this paper, $\lambda_{L}^{0}$ always turns out to
be non-negative, so that the physical picture is that of figure 1(a). We will
discuss ideas for how to generate cases with a negative value and a
discontinuous transition into chaos.

This paper is organized as follows. In section 2 we introduce the theories we
consider in this paper and the methods for computing the chaos exponent. We
then prove the continuity conjecture in QM in section 3. In section 4 we
discuss our first $2d$ example, the chiral SYK model, and show that the
continuity conjecture is obeyed. In section 5 we discuss our second example,
the disordered $\mathcal{A}_{3}$ minimal model. We compute the chaos exponent
at small $\mathcal{J}$ and again show that the continuity conjecture is
obeyed. We discuss some future directions in section 6.

##  2 Background

In this section we review the basic method discussed in [1] for obtaining the
chaos exponent $\lambda_{L}$ at weak coupling for a general disordered CFT.

###  2.1 Four-point function and double-commutator

In disordered free theories, it is possible to write down self-consistency
equations for the two- and four-point functions of the fundamental field,
which can then be solved using a conformal ansatz when the conformal symmetry
is restored in the IR. In [1] it was shown that similar equations can be
written down for general disordered CFT for the operator $\Phi$ in (1.1),
which can in principle be solved for any value of $\mathcal{J}$ when it is an
exactly marginal deformation. However, solving these equations requires
knowledge of all $n$-point functions of $\Phi$ in the core CFT, and so is very
difficult in general. In this section we will review the derivation in [1],
omitting some details.

In the following we will only use the four-point function, and so we only
discuss the self-consistency equation for the four-point function and for the
DC. The discussion can also be immediately generalized to SUSY theories, in
which case the diagrams discussed should be understood as supergraphs.

We would first like to compute the connected four-point function

$C=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\langle\Phi_{i}\Phi_{i}\Phi_{j}\Phi_{j}\rangle_{conn}$
(2.1)

in the deformed theory (1.1). The diagrams contributing to $C$ have an
iterative ladder structure at large $N$, so that the full result for the four-
point function is given by

$C=\sum_{n=0}^{\infty}K^{n}F_{0}=\frac{F_{0}}{1-K}\;,$ (2.2)

where $F_{0},K$ are defined in figure 2.

Figure 2: The kernel $K$ and initial diagram $F_{0}$ for general disordered
CFTs. Red lines denote full propagators $G$, and black dots denote insertions
of the disorder interaction, with $q-2$ red propagators between each pair.

The computation of $K$ and $F_{0}$ requires knowing all “subtracted $n$-point
functions”, denoted by $n_{s}^{\prime}$. These are given by the standard
$n$-point functions but with some specific subtractions of lower-order
correlators, and are explicitly defined in [1]. Examples of some subtracted
$n$-point functions appear in figure 3. From these it is in principle possible
to compute the full four-point function $C$. One can also perform perturbation
theory in $\mathcal{J}$; at leading order only the first diagram in the
expression for $K$ in figure 2 contributes, which is determined by the CFT
four-point function and two-point function, and takes the form

$K(1,2;3,4)=(q-1)J^{2}\langle\Phi_{1}\bar{\Phi}_{2}\Phi_{3}\bar{\Phi}_{4}\rangle_{s}^{\prime}G(3,4)^{q-2}\;,$
(2.3)

with $G(3,4)$ the $\Phi$ propagator between the points $x_{3},x_{4}$.

Next we discuss the computation of the double-commutator, defined as

$\begin{split}W_{R}(t_{1},t_{2})&=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\left\langle[\Phi_{i}(\beta/2),\Phi_{j}(\beta/2+it_{2})][\Phi_{i}(0),\Phi_{j}(it_{1})]\right\rangle\\\
&=\lim_{\varepsilon\rightarrow
0}\frac{1}{N^{2}}\sum_{i,j=1}^{N}\left\langle\left(\Phi_{i}\left(\varepsilon\right)-\Phi_{i}\left(-\varepsilon\right)\right)\left(\Phi_{i}\left(\beta/2+\varepsilon\right)-\Phi_{i}\left(\beta/2-\varepsilon\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\left.\Phi_{j}\left(it_{1}\right)\Phi_{j}\left(\beta/2+it_{2}\right)\right\rangle\;.\end{split}$
(2.4)

We have suppressed the spatial coordinates, since they will be unimportant,
and we have kept the (real and imaginary) time coordinates. By
$\langle...\rangle$ we mean the Euclidean time-ordered thermal trace. Note
that (2.4) is just a combination of analytically-continued Euclidean four-
point functions on the cylinder, which in a $d=2$ CFT is an analytically-
continued flat-space correlator (2.1).

Figure 3: Examples of correlation functions $n_{s}^{\prime}$. Dashed lines
corresponds to external points, while solid lines are connected via $\Sigma$’s
in figure 2.

The diagrams contributing to the double-commutator also obey an iterative
ladder structure. The corresponding kernel $K_{R}$ and initial diagram
$F_{0,R}$ have the same diagrammatics as the four-point function (appearing in
figure 2), but the correlators are analytically-continued versions of the ones
appearing above. The details appear in [1], and we will write down explicitly
only the leading contribution in $\mathcal{J}$, which comes from the four-
point function:

$K_{R}(t_{1},t_{2},t_{3},t_{4})=\mathcal{J}^{2}\left\langle\Delta\mathcal{O}\left(it_{1}\right)\Delta\mathcal{O}\left(\frac{\beta}{2}+it_{2}\right)\mathcal{O}\left(it_{3}\right)\mathcal{O}\left(\frac{\beta}{2}+it_{4}\right)\right\rangle^{\prime}_{s}G^{q-2}_{lr,\Delta}(3,4)+O(\mathcal{J}^{4})\;.$
(2.5)

Here we have denoted
$\Delta\mathcal{O}(z)=\mathcal{O}(z+\varepsilon)-\mathcal{O}(z-\varepsilon)$
where we eventually take the limit $\epsilon\to 0$. The integration range in
principle for all points is over the complex time contour appearing in figure
4, although various cancellations as we take $\epsilon\to 0$ lead to the
result above. In particular, we have used the expression for the thermal two-
point function for a scalar operator of dimension $\Delta$ between points from
different “rails” (see figure 4):

$G_{lr,\Delta}(1,2)=\frac{1}{\left(4\cosh(\frac{t_{12}-x_{12}}{2})\cosh(\frac{t_{12}+x_{12}}{2})\right)^{\Delta}}\;.$
(2.6)

Figure 4: The complex time contour chosen for the computation of the DC. Red
dots denote insertion points of operators. We call the two excursions from the
real axis the “left rail” and the “right rail”.

###  2.2 Chaos

Having written down the ladder diagrams which contribute to the DC, we can now
compute the chaos exponent. In principle, this is done by computing the DC
explicitly, and then taking the large-time limit $t_{1},t_{2}=t\to\infty$,
which should lead to exponentially growing behavior,

$W_{R}(t_{1},t_{2})\sim\exp(\frac{\lambda_{L}}{2}(t_{1}+t_{2}))f(t_{1}-t_{2})\;.$
(2.7)

where from now on we set $\beta=2\pi$. However, computing the full DC is
difficult, and instead the existence of an iterative ladder structure offers
us a shortcut. The ladder structure leads to the self-consistency equation

$W_{R}=F_{0,R}+K_{R}W_{R}\;.$ (2.8)

At large times, we assume that the $F_{0,R}$ term is negligible, and so
$W_{R}$ obeys the equation

$W_{R}=K_{R}W_{R}\;,$ (2.9)

which is just an eigenvalue equation for $K_{R}$. As a result, the
exponentially-growing solution for $W_{R}$ must be an eigenfunction of the
retarded kernel $K_{R}$ with eigenvalue $1$. Thus, the chaos exponent is found
by guessing solutions of the form (2.7) and finding their eigenvalue
$k_{R}(\lambda_{L})$ under $K_{R}$. The largest $\lambda_{L}$ for which
$k_{R}(\lambda_{L})=1$ is the chaos exponent.

The precise form of the eigenvalue is constrained due to conformal invariance,
and takes the form of a two-point function on the cylinder. In $2d$ we have
the ansatz

$W_{\lambda}(1,2)=\frac{\exp(-\frac{h+\tilde{h}}{2}(t_{1}+t_{2})+\frac{h-\tilde{h}}{2}(x_{1}+x_{2}))}{(2\cosh(\frac{t_{12}-x_{12}}{2}))^{\Delta-h}(2\cosh(\frac{t_{12}+x_{12}}{2}))^{\Delta-\tilde{h}}}\;.$
(2.10)

In general we require
$h=-\frac{\lambda}{2}+ip,\tilde{h}=-\frac{\lambda}{2}-ip$ for real
$\lambda,p$, but in practice in the examples appearing here the maximal chaos
exponent will have $p=0$.

Finding the chaos exponent now amounts to computing the eigenvalue of the
eigenfunction $W_{\lambda}$ under $K_{R}$. Since the theories we consider are
conformally invariant for any value of $\mathcal{J}$, we can perform this
computation in a perturbative expansion in $\mathcal{J}$. Generally the
eigenvalue is given by

$k_{R}(\lambda,\mathcal{J})=\frac{\int d^{2}x_{3}d^{2}x_{4}K_{R}\cdot
W}{W}\;,$ (2.11)

and plugging in the leading-order result for $K_{R}$ we find that in $2d$ and
at leading order in $\mathcal{J}$,

$\begin{split}k_{R}(\lambda,\mathcal{J})=&\mathcal{J}^{2}\int
d^{2}x_{3}d^{2}x_{4}\left\langle\Delta\mathcal{O}\left(it_{1}\right)\Delta\mathcal{O}\left(\beta/2+it_{2}\right)\mathcal{O}\left(it_{3}\right)\mathcal{O}\left(\beta/2+it_{4}\right)\right\rangle^{\prime}_{s}\\\
&\cdot\frac{G_{lr,\Delta+\frac{\lambda}{2}}(3,4)}{G_{lr,\Delta+\frac{\lambda}{2}}(1,2)}\exp\left(\frac{\lambda}{2}(t_{3}+t_{4}-t_{1}-t_{2})\right)\cdot
G_{lr,2\Delta(q-2)}(3,4)+O(J^{4})\\\
=&\frac{\mathcal{J}^{2}}{4}\frac{\exp(-\frac{\lambda}{2}(t_{1}+t_{2}))}{G_{lr,\lambda/2}(1,2)}\frac{\int^{\infty}_{u_{1}}du_{3}\int_{-\infty}^{u_{2}}du_{4}}{u_{34}^{2+\frac{\lambda}{2}}}\frac{\int_{v_{1}}^{\infty}dv_{3}\int_{-\infty}^{v_{2}}dv_{4}}{v_{34}^{2+\frac{\lambda}{2}}}\mathcal{G}_{R}(\chi,\bar{\chi})+O(\mathcal{J}^{4})\;.\end{split}$
(2.12)

With $\chi,\bar{\chi}$ the conformal cross-ratios. Here we have performed the
change of variables

$\begin{split}u_{3}=e^{x_{3}-t_{3}}\;,&\quad v_{3}=e^{-x_{3}-t_{3}}\;,\\\
u_{4}=-e^{x_{4}-t_{4}}\;,&\quad v_{4}=-e^{-x_{4}-t_{4}}\;.\end{split}$ (2.13)

$\mathcal{G}_{R}$ is the retarded normalized 4-point of the undeformed CFT at
$\mathcal{J}=0$,

$\begin{split}\mathcal{G}_{R}(\chi,\bar{\chi})&=\frac{\left\langle[O(\beta/2+it_{2}),O(\beta/2+it_{4})][O(it_{1}),O(it_{3})]\right\rangle_{s}^{\prime}}{G_{lr,\Delta}(1,2)G_{lr,\Delta}(3,4)}\\\
&=\lim_{\varepsilon_{1},\varepsilon_{2}\rightarrow
0}\mathcal{G}_{++}-\mathcal{G}_{+-}-\mathcal{G}_{-+}+\mathcal{G}_{--}\end{split}$
(2.14)

with the normalized four-point
$\mathcal{G}_{\pm_{1},\pm_{2}}=\mathcal{G}(u_{1}e^{\pm
i\varepsilon_{1}},v_{1}e^{\pm i\varepsilon_{1}},u_{2}e^{\pm
i\varepsilon_{2}},v_{1}e^{\pm i\varepsilon_{2}},u_{3},v_{3},u_{4},v_{4})$.

####  2.2.1 Consistency of the perturbative expansion

We now discuss perturbative corrections to the chaos exponent which are
subleading in the $\mathcal{J}\to 0$ limit. The computation of the chaos
exponent in perturbation theory in $\mathcal{J}$ requires several assumptions.
Expanding the eigenvalues of the retarded kernel, we expect to see

$k_{R}(\lambda,\mathcal{J})=\mathcal{J}^{2}f_{2}(\lambda)+\mathcal{J}^{4}f_{4}(\lambda)+...\;,$
(2.15)

First, we would like to understand when it is enough to find leading term
$f_{2}(\lambda)$ discussed above in order to find the leading value of the
chaos exponent in the limit $\mathcal{J}\to 0$. The chaos exponent is found by
setting $k_{R}=1$, and so if we keep only the $\mathcal{J}^{2}$ term,
$\lambda_{L}$ is found by analyzing at which values of $\lambda$ the function
$f_{2}(\lambda)$ diverges as $1/\mathcal{J}^{2}$. The largest such $\lambda$
can then be identified with the chaos exponent. In order for this procedure to
be consistent it is enough to require two conditions on the higher-order
terms:

  1. 1.

If the largest value of $\lambda$ at which $f_{2}$ diverges is $\lambda_{0}$,
then all other $f_{n}$ diverge at values $\lambda\leq\lambda_{0}$.

  2. 2.

If $f_{2}$ diverges as $\frac{1}{(\lambda-\lambda_{0})^{\alpha}}$, then other
$f_{n}$ diverge as $\frac{1}{(\lambda-\lambda_{0})^{\beta_{n}}}$ for
$\beta_{n}\leq n\alpha/2$.

In this case the leading value of $\lambda_{L}$ is indeed $\lambda_{0}$. All
examples discussed in [1] were shown to obey these conditions, and we will
show that the examples discussed here also obey them).

Next, it is natural to ask when it is consistent to perform a perturbative
expansion around $\lambda_{0}$ in the limit of small $\mathcal{J}$ in order to
obtain the chaos exponent in a series in $\mathcal{J}$ of the form

$\lambda_{L}=\lambda_{0}+\mathcal{J}^{2}\lambda_{1}+...$ (2.16)

This requires a strict inequality in the second condition above, i.e. we
require

$\beta_{n}<n\alpha/2\;.$ (2.17)

To see why this is the case, assume an expansion of the form

$k_{R}=\sum a_{n}\frac{\mathcal{J}^{2n}}{\lambda^{2n}}\;,$ (2.18)

so that $\alpha=2$ and $\beta_{n}=n=n\alpha/2$ and the inequality is exactly
saturated. As a result, the expansion is not in terms of $\mathcal{J}$, but in
terms of $\mathcal{J}/\lambda$. We can now try to compute $\lambda_{L}$ in
perturbation theory. Write

$\lambda_{L}=\lambda_{0}+\mathcal{J}^{2}\lambda_{1}+...$ (2.19)

Plugging in this expansion, we immediately learn that the leading order is
given by $\lambda_{0}=0$, as discussed above. However, in order to find the
value of the subleading term $\lambda_{1}$, we must take into account all of
the terms in the expansion, since all terms they are all of the same order in
$\mathcal{J}$. As a result, while we can still compute $\lambda_{0}$ in such
theories, we cannot perform perturbation theory around it without knowing
$k_{R}$ completely (or at least to all orders in some double-scaling limit).
So a perturbative calculation of $\lambda_{L}$ beyond the leading order using
this method fails. We will see this happen in the chiral SYK example discussed
in section 4.

On the other hand, for any theory where there is a strict inequality
$\beta_{n}<n\alpha/2$, a perturbative expansion in $\mathcal{J}$ should be
possible. This is the result in e.g. the generalized free fields examples in
[1].

###  2.3 The continuity conjecture

We now discuss the behavior of the chaos exponent as $\mathcal{J}\to 0$. Note
that in equation (2.12), the prefactor $\mathcal{J}^{2}$ vanishes in this
limit, and so in order for us to be able to solve for
$k_{R}(\lambda_{L},\mathcal{J})=1$, we must find values of $\lambda$ for which
the integral in (2.12) diverges as $1/\mathcal{J}^{2}$. The chaos exponent in
this limit is then given by the largest value of $\lambda$ for which this
divergence occurs. Assuming that the retarded four-point function of the
theory at a single core CFT behaves at large times $t_{1},t_{2}$ as333In
$d\leq 2$ this is related to the Regge limit.

$\mathcal{G}_{R}(t_{1},t_{2})\sim\exp(\lambda_{L}^{0}(t_{1}+t_{2})/2)\;,$
(2.20)

it is easily seen that the divergence occurs precisely at
$\lambda=\lambda_{L}^{0}$. As a result, it was conjectured in [1] that the
chaos exponent in the limit $\mathcal{J}\to 0$ is given precisely by the
leading exponential behavior of the double-commutator in the free theory at
$\mathcal{J}=0$. Thus the computation of the leading behavior of $\lambda_{L}$
is particularly simple, and is a property of a single core CFT.444Note that
$\lambda_{L}^{0}$ does not have any interpretation in terms of a chaos
exponent in a single core CFT, since this requires some form of a large-$N$
limit or another weak-coupling expansion. Several examples were discussed in
[1] which were shown to obey this conjecture.

##  3 Proving the continuity conjecture in QM

The continuity conjecture discussed in 2.3 relates the chaos exponent in
disordered CFTs at small $\mathcal{J}$ to the late-time behavior of the DC in
a single core CFT. It states that assuming that $\mathcal{G}_{R}$ defined in
(2.14) (or equivalently, the double-commutator) behaves at large times as
$\exp(\lambda_{L}^{0}t)$, then the chaos exponent as $\mathcal{J}\to 0$
approaches $\lambda_{L}^{0}$. We will now prove this statement under the
assumption that perturbation theory in $\mathcal{J}$ is valid for the
eigenvalues of the retarded kernel, so that the leading order is obtained by
considering just the contribution of the four-point function to the retarded
kernel, see section 2.2.

Consider the leading contribution to $k_{R}$, which comes from the four-point
function diagram in figure 2. In $1d$ this contribution simplifies to (see
Appendix B for a derivation):

$k_{R}=\frac{1}{|z_{12}|^{2\Delta}}\int^{\infty}_{z_{1}}dz_{3}\int^{z_{2}}_{-\infty}dz_{4}\frac{\mathcal{G}_{R}(\chi)}{|z_{34}|^{2+\lambda}}\;.$
(3.1)

Here, we performed the change of variables $z=e^{-t}$ on the “left rail”
(points $1,3$) and $z=-e^{-t}$ on the “right rail” (points $2,4$), see e.g.
[5] for details. The conformal cross-ratio is

$\chi=\frac{z_{12}z_{34}}{z_{13}z_{24}}\;.$ (3.2)

Next we perform another change of coordinates to the coordinates
$\kappa,\chi$, where $\kappa$ is defined as $z_{3}=\kappa/\chi$. The integral
becomes is

$k_{R}(\lambda)=\int_{-\infty}^{0}d\chi\frac{\mathcal{G}_{R}(\chi)}{\chi^{1-\lambda}}\int_{-\infty}^{\chi-1}\frac{d\kappa}{\kappa^{1+\lambda}(\kappa-\chi)^{1+\lambda}(\kappa-(\chi-1))^{-\lambda}}\;.$
(3.3)

The chaos exponent $\lambda_{L}$ is now given by the largest value of
$\lambda$ for which the integral diverges in the limit
$t_{3},t_{4}\to-\infty$, which in these coordinates corresponds to
$\chi\rightarrow 0$. Note that the $\kappa$ integral is finite as long as
$\lambda>-1$ (corresponding to $\lambda_{L}^{0}>-1$). At $\chi=0$ the $\kappa$
integral is smooth; it does not vanish or diverge. Therefore we can expand
$\mathcal{G}_{R}(\chi)$ around $\chi=0$ inside the integral. Let us denote the
leading order term by
$\mathcal{G}_{R}(\chi)=c_{0}\chi^{-\lambda_{L}^{0}}+...$. Note that since at
large times we have $\chi\sim e^{-t}$, we can identify $\lambda_{L}^{0}$ with
the rate of growth of the DC of a single core CFT as in (2.20). Plugging in
this expansion, we see that the integral converges only for
$\lambda>\lambda_{L}^{0}$, which means that the chaos exponent at small
$\mathcal{J}$ is $\lambda_{L}^{0}$. We have thus proved the continuity
conjecture in $1d$.

##  4 The chiral SYK model

The chiral SYK model was introduced in [12] (see also [19]). The theory is a
disordered free theory in $2d$, where the core CFT is a free chiral Majorana
fermion. The theory was shown to have a line of fixed points characterized by
$\mathcal{J}$, so that the chaos exponent can be found as a function of
$\mathcal{J}$. Since the theory is chiral, instead of the standard chaos
exponent it is more interesting to consider the velocity-dependent chaos
exponent, given by considering a large time and large distance limit with the
ratio $v=x/t$ kept constant. The corresponding chaos exponent is denoted
$\lambda_{v}$. It was found that the chaos exponent always starts at zero as
$\mathcal{J}\to 0$, and rises as we increase $\mathcal{J}$. In particular,
choosing $v$ such that $\lambda_{v}$ is maximal, one finds that at infinite
$\mathcal{J}$ the chaos exponent $\lambda_{v}$ saturates the bound on chaos
[2]. We will be interested in the weak-coupling limit, where $\mathcal{J}$ is
close to zero. We will show that the continuity conjecture is obeyed for the
velocity-dependent chaos exponent, and discuss corrections in $\mathcal{J}$.

The velocity-dependent chaos exponent at weak coupling can be easily extracted
from Appendix B of [12] (see equation (B.6)), and in the limit $\mathcal{J}\to
0$ one finds

$\lambda_{v}(\mathcal{J})=\begin{cases}\frac{2\pi}{\beta}\eta(v-1)+O(\mathcal{J}),&u_{-}<v<u_{+}\\\
0,&\text{else}\end{cases}$ (4.1)

where $u_{\pm}=1\pm\frac{\mathcal{J}}{2\pi}$ and

$\eta=\frac{\sqrt{3}}{\sqrt{1-\mathcal{J}^{2}}}\frac{1-v}{\sqrt{\mathcal{J}^{2}-(1-v)^{2}}}\;.$
(4.2)

Note that since $u_{-}<v<u_{+}$, $\eta$ is finite in the limit $\mathcal{J}\to
0$.

Since the theory is chiral, it is not surprising that the chaos exponent
vanishes outside of a cone around the speed of light. Taking the strict
$\mathcal{J}\to 0$ limit, we find that $\lambda_{v}(\mathcal{J}\to 0)$
vanishes trivially unless $v=1$ (since $u_{-}=u_{+}=1$), but at $v=1$ it turns
out to vanish as well. Thus we find $\lambda_{v}(\mathcal{J}\to 0)=0$.

We would like to compare this to $\lambda_{L}^{0}$, which describes the large-
time behavior of the DC in a single copy of the free core CFT. The DC in the
core CFT is given by $G_{R}(13)G_{R}(24)$, where $G_{R}(ij)$ is the retarded
propagator between spacetime points $i,j$ and is given by

$G_{R}(t,x)=\frac{1}{\beta\sqrt{u_{+}u_{-}}}\frac{\Theta\left(t-u_{+}^{-1}x\right)\Theta\left(u_{-}^{-1}x-t\right)}{\sqrt{\sinh\left[\frac{\pi}{\beta}\left(t-u_{+}^{-1}x\right)\right]\sinh\left[\frac{\pi}{\beta}\left(u_{-}^{-1}x-t\right)\right]}}\;.$
(4.3)

Next we take the large-time limit where $t=t_{1}=t_{2}$ and $x=x_{1}=x_{2}$
are large while $v=x/t$ is kept constant. We find (up to unimportant
constants)

$DC\propto\begin{cases}\exp\left(-\frac{\pi}{\beta}(u_{-}^{-1}-u_{+}^{-1})vt\right),&u_{-}<v<u_{+}\\\
0,&\text{else}\end{cases}$ (4.4)

Plugging in $\mathcal{J}=0$ we find as a result that $\lambda_{v}^{0}=0$
always. Thus we have found

$\lambda_{v}^{0}=\lambda_{v}(\mathcal{J}\to 0)=0\;,$ (4.5)

and so the continuity conjecture is obeyed.

We can also try to understand subleading corrections to the chaos exponent as
discussed in section 2.2.1. The eigenvalues of the chiral SYK model were
computed in [12], and we can expand the result in $\mathcal{J}$:

$k_{R}=\frac{3\mathcal{J}^{2}}{\lambda^{2}}-\frac{6\mathcal{J}^{3}}{\lambda^{3}}+O\left(\mathcal{J}^{4}\right)\;.$
(4.6)

Expanding to higher orders one finds $\alpha=2$ and $\beta_{n}=n=n\alpha/2$ in
the notation of section 2.2.1. As a result, the expansion is not in terms of
$\mathcal{J}$, but in terms of $\mathcal{J}/\lambda$, and so a perturbative
computation of $\lambda_{L}$ will fail at higher orders in $\mathcal{J}$.

##  5 The disordered $\mathcal{N}=2$ $\mathcal{A}_{3}$ minimal model

We now discuss the case where the core CFT is the $\mathcal{N}=2$
supersymmetric $\mathcal{A}_{3}$ minimal model. This model can be constructed
using a single chiral superfield $X$ with superpotential

$W=X^{4}\;.$ (5.1)

The model has central charge $c=3/2$ and its spectrum includes a chiral
primary of dimension $1/4$ which we will call $\Phi$. This theory can be
identified with the theory of a free boson $H$ an a free fermion $\chi$, which
combine into a single free $\mathcal{N}=1$ chiral multiplet [20]. The core CFT
thus has a free field representation.

The disordered $\mathcal{A}_{3}$ minimal model with $q=4$ can then be
constructed as

$(\mathcal{A}_{3})^{N}+\sum_{i_{1}\neq...\neq i_{4}}J_{i_{1}...i_{4}}\int
d^{2}xd^{2}\theta\Phi_{i_{1}}...\Phi_{i_{4}}\;.$ (5.2)

In particular, this deformation is marginal. In fact, as discussed in [1], it
can be shown using standard arguments [21, 22, 23] that any realization of
this theory is a CFT, so that the interaction is exactly marginal (even
without averaging).

In order to compute $\lambda_{L}$ we need to know $n$-point functions of
$\Phi$. Since the $\mathcal{A}_{3}$ minimal model has a free field realization
in terms of $\mathcal{N}=1$ superfields, we should be able to identify the
components of $\Phi$ with products of operators from the free boson and free
fermion CFT. In practice, the various components will be mapped to products of
vertex operators and fermionic twist fields, whose $n$-point functions are
known. Plugging the results into the retarded kernel, we will be able to read
off $\lambda_{L}$. Following the discussion above, we will focus on the four-
point function which gives the leading contribution to $\lambda_{L}$ at small
$\mathcal{J}$.

Our notations appear in appendix A. In particular, we use lightcone
coordinates $x^{\pm}$ so that a two-point function takes the form
$(x^{+}x^{-})^{\Delta}\equiv|x|^{2\Delta}$.

###  5.1 Details of duality to free fields

The free field representation of the $\mathcal{A}_{3}$ minimal model consists
of a free Majorana fermion $\chi$ and a free compact boson $H$ at the self-
dual radius $R=1/\sqrt{2}$ (see for instance [20] or [24]). The SUSY algebra
is generated by the operators

$\displaystyle G_{\pm}$
$\displaystyle=\chi_{\pm}\exp\left(i\sqrt{2}H_{\pm}\right)\;,$
$\displaystyle\bar{G}_{\pm}$
$\displaystyle=\chi_{\pm}\exp\left(-i\sqrt{2}H_{\pm}\right)\;,$ $\displaystyle
j_{\pm}^{(R)}$ $\displaystyle=\frac{i}{\sqrt{2}}\partial_{\pm}H\;,$ (5.3)

where $\pm$ stand for the left/right moving parts of the fields and
operators.555The theory actually has $\mathcal{N}=3$ SUSY, but we will only
need the $\mathcal{N}=2$ subalgebra above for our purposes.

The fundamental superfield $\Phi$ has scaling dimensions
$\left(\tfrac{1}{8},\tfrac{1}{8}\right)$, and there are five superprimaries in
the NS sector: $\Phi$, $\Phi^{2}$, $\bar{\Phi}$, $\bar{\Phi}^{2}$,
$\bar{\Phi}\Phi$. Their bottom components map to the following free theory
operators:

$\displaystyle\phi=\sigma\exp\left(i\frac{H}{2\sqrt{2}}\right)\;,$
$\displaystyle\qquad\bar{\phi}=\sigma\exp\left(-i\frac{H}{2\sqrt{2}}\right)\;,$
$\displaystyle\phi^{2}=\exp\left(i\frac{H}{\sqrt{2}}\right)\;,$
$\displaystyle\qquad\bar{\phi}^{2}=\exp\left(-i\frac{H}{\sqrt{2}}\right)\;,$
$\displaystyle\bar{\phi}\phi$ $\displaystyle=\epsilon=\chi_{+}\chi_{-}\;,$
(5.4)

where $\sigma$ and $\mu$ (which will be of use later), with
$(h,\bar{h})=(1/16,1/16)$, are the twist fields of the fermion theory. The
other components of these superfields may be worked out via free field OPEs
(see Appendix A). For convenience, we list in Table 1 the components of the
basic superfield:

$\displaystyle\Phi(y^{+},y^{-})=\phi(y^{+},y^{-})+\theta^{+}\psi_{+}(y^{+},y^{-})+\theta^{-}\psi_{-}(y^{+},y^{-})+\theta^{+}\theta^{-}F(y^{+},y^{-})$
(5.5)

along with their dimensions and R-charges.

Table 1: The operators in the multiplet of $\Phi$ in the $\mathcal{A}_{3}$ minimal model. $\mathcal{O}$ | $\mathcal{O}_{\text{free}}$ | $(h,\bar{h})$ | $q_{R}$
---|---|---|---
$\phi$ | $\sigma\ e^{i\frac{H}{2\sqrt{2}}}$ | $\left(\frac{1}{8},\frac{1}{8}\right)$ | $\left(\frac{1}{4},\frac{1}{4}\right)$
$\psi_{+}$ | $\mu\ e^{i\frac{-3H_{+}+H_{-}}{2\sqrt{2}}}$ | $\left(\frac{5}{8},\frac{1}{8}\right)$ | $\left(-\frac{3}{4},\frac{1}{4}\right)$
$\psi_{-}$ | $\mu\ e^{i\frac{H_{+}-3H_{-}}{2\sqrt{2}}}$ | $\left(\frac{1}{8},\frac{5}{8}\right)$ | $\left(\frac{1}{4},-\frac{3}{4}\right)$
$F$ | $\sigma\ e^{-i\frac{3H}{2\sqrt{2}}}$ | $\left(\frac{5}{8},\frac{5}{8}\right)$ | $\left(-\frac{3}{4},-\frac{3}{4}\right)$

###  5.2 Correlation functions

Having identified the components of the superfield $\Phi$ with operators from
free field theories, we can now compute all $n$-point functions of the field
$\Phi$.

####  5.2.1 Two-point function

Superconformal symmetry fixes the form of the two-point function to be

$\langle\Phi\bar{\Phi}\rangle=\frac{1}{|\langle 12\rangle|^{2\Delta}}$ (5.6)

where the relevant value for our theory is $\Delta=1/4$. By expanding the
result in the superspace coordinates on both sides, we can identify two-point
functions of the various components of $\Phi$, which allows us to set their
normalization. We find

$\begin{split}\langle\phi\bar{\phi}\rangle=\frac{1}{|x_{12}|^{2\Delta}}\;,&\qquad\langle
F\bar{F}\rangle=\frac{4\Delta^{2}}{|x_{12}|^{2(\Delta+1)}}\;,\\\
\langle\psi^{+}\bar{\psi}^{+}\rangle=-\frac{2\Delta}{|x_{12}|^{2\Delta}x_{12}^{+}}\;,&\qquad\langle\psi^{-}\bar{\psi}^{-}\rangle=-\frac{2\Delta}{|x_{12}|^{2\Delta}x_{12}^{-}}\;.\end{split}$
(5.7)

####  5.2.2 Four-point function

We move on to the computation of the four-point function. Superconformal
symmetry fixes its form to be

$\displaystyle\langle\Phi(x_{1})\bar{\Phi}(x_{2})\Phi(x_{3})\bar{\Phi}(x_{4})\rangle=\frac{1}{|\langle
12\rangle\langle 34\rangle|^{\frac{1}{2}}}f(\chi_{s}^{+},\chi_{s}^{-})\;,$
(5.8)

where $f$ is an arbitrary function of the superconformal cross ratios

$\chi_{s}^{\pm}=\frac{\langle 12\rangle^{\pm}\langle 34\rangle^{\pm}}{\langle
14\rangle^{\pm}\langle 32\rangle^{\pm}}\;.$ (5.9)

Note that the bottom component of $\chi_{s}^{\pm}$ is the usual non-
supersymmetric cross-ratio
$\chi^{\pm}=\frac{x_{12}^{\pm}x_{34}^{\pm}}{x_{14}^{\pm}x_{32}^{\pm}}$.

We start by calculating the bottom component of this four point function.
Using the identification (5.1), this is equivalent to computing the product of
a four-point function of vertex operators and of twist operators $\sigma$. The
results are well known, and we find

$\langle\phi(x_{1})\bar{\phi}(x_{2})\phi(x_{3})\bar{\phi}(x_{4})\rangle=\left|\frac{1}{x_{12}x_{34}}\right|^{\frac{1}{2}}\frac{1}{\sqrt{2}}\sqrt{1+|\chi|+|\chi-1|}\;.$
(5.10)

Since there is a single superconformal cross-ratio of each chirality, it is
simple to uplift this result to the full supermultiplet; it suffices to
replace $\chi^{\pm}\to\chi_{S}^{\pm}$ and the prefactor of
$\left|\frac{1}{x_{12}x_{34}}\right|^{\frac{1}{2}}$ with its supersymmetric
analog $\frac{1}{|\langle 12\rangle\langle 34\rangle|^{\frac{1}{2}}}$. The
result is thus

$\displaystyle\langle\Phi(x_{1})\bar{\Phi}(x_{2})\Phi(x_{3})\bar{\Phi}(x_{4})\rangle=\frac{1}{|\langle
12\rangle\langle
34\rangle|^{\frac{1}{2}}}\frac{1}{\sqrt{2}}\sqrt{1+|\chi_{s}|+|\chi_{s}-1|}\;.$
(5.11)

As a consistency check, we have checked that expanding both sides in the
superspace coordinates gives the expected result for four-point functions of
other components of $\Phi$.

###  5.3 Chaos

We can now compute the chaos exponent using the procedure outlined in section
2.2. This requires diagonalizing the retarded kernel. We perform this
diagonalization by first writing down the kernel for the standard four-point
function, and then performing the analytic continuation required to obtain the
retarded kernel. In the following we will focus on the bottom component of all
four-point functions, assuming that they produce that leading behavior at long
times, as was the case in similar models [5, 6, 1].

####  5.3.1 The chaos exponent at weak coupling

We start with the eigenvalues of the standard kernel. The eigenvalues of the
bosonic kernel are given by

$k(h,\mathcal{J})=\frac{\int d^{2}X_{3}d^{2}\bar{X}_{4}K\cdot W}{W}\;,$ (5.12)

where $d^{2}X=d^{2}xd^{2}\theta$ and $d^{2}\bar{X}=d^{2}xd^{2}\bar{\theta}$.
At leading order in $\mathcal{J}$ the kernel $K$ is given by the leading
(super-)diagram in figure 2 and $W$ is given by (2.10). Plugging in the form
of the four-point function (5.11) we find666We are ignoring the subtractions
here. They are necessary to compute the eigenvalues $k$, but will not
contribute to the eigenvalues of the retarded kernel which are our main
interest.

$\int KW=\int
d^{2}X_{3}d^{2}\bar{X}_{4}\langle\Phi_{1}\bar{\Phi}_{2}\Phi_{3}\bar{\Phi}_{4}\rangle\frac{1}{|\langle
34\rangle|^{3/2-2h}}=\frac{1}{\sqrt{2}|\langle 12\rangle|^{1/2}}\int
d^{2}X_{3}d^{2}\bar{X}_{4}\frac{\sqrt{1+|\chi_{s}|+|\chi_{s}-1|}}{|\langle
34\rangle|^{2-2h}}\;.$ (5.13)

We will focus on the bosonic part of this integral. Using

$\begin{split}\langle 12\rangle^{\pm}=x_{12}^{\pm}\;,\qquad&\langle
34\rangle^{\pm}=x_{34}^{\pm}-2\theta_{3}^{\pm}\bar{\theta}^{\pm}_{4}\;,\\\
\chi_{s}^{\pm}=\frac{x_{12}^{\pm}(x_{34}^{\pm}-2\theta_{3}^{\pm}\bar{\theta}^{\pm}_{4})}{x_{14}^{\pm}x_{32}^{\pm}}&=\chi^{\pm}(1-\frac{2\theta_{3}^{\pm}\bar{\theta}^{\pm}_{4}}{x_{34}^{\pm}})\;.\end{split}$
(5.14)

we can perform the Grassman integrals to obtain

$k=\frac{1}{|z_{12}|^{1/2}}\int
d^{2}x_{3}d^{2}x_{4}\frac{(x_{34}^{-})^{h-2}({x}_{34}^{+})^{h-2}}{4\sqrt{2}}\mathcal{I}(\chi^{\pm})$
(5.15)

where

$\begin{split}\mathcal{I}=&\frac{1}{\left(|\chi-1|+|\chi|+1\right)^{3/2}}\left[\frac{|\chi|\left(4h\left(|\chi-1|-\chi^{+}+1\right)-2|\chi-1|+3\chi^{+}-2\right)+\chi^{-}\left(7|\chi|-2\chi^{+}+4\right)}{|\chi-1|}\right.\\\
&+\frac{\chi^{-}\left(4h\left(2\chi^{+}|\chi-1|+2\chi^{+}|\chi|-|\chi-1|-2|\chi|+\chi^{+}-1\right)-6\chi^{+}\left(|\chi-1|+|\chi|\right)+4|\chi-1|\right)}{|\chi-1|}\\\
&\left.-4(h-1)\left(|\chi-1|+|\chi|+1\right)\left(\frac{|\chi|-\chi^{+}\left(|\chi-1|+|\chi|\right)}{\chi^{+}-1}-4(h-1)\left(|\chi-1|+|\chi|+1\right)\right)\right]\;.\end{split}$
(5.16)

We can now use analytic continuation to find the eigenvalues of the retarded
kernel at finite temperature. More precisely, first we need to go to the
cylinder, the do the analytic continuation. The mapping to the cylinder is
given by [5]

$\begin{array}[]{lll}u=e^{x-t},&v=e^{-x-t},&\text{ (left rail) }\\\
u=-e^{x-t},&v=-e^{-x-t},&\text{ (right rail) }\end{array}$ (5.17)

with the rails identified in figure 4. The analytic continuation is done by
taking the following times:

$\tau_{1}=\beta/2\pm\epsilon_{1}+it_{1},\quad\tau_{2}=\pm\epsilon_{2}+it_{2},\quad\tau_{3}=\beta/2+it_{3},\quad\tau_{4}=it_{4}\;.$
(5.18)

The procedure is then to compute the four-point function for each of the four
choices of signs for the $\epsilon$’s, and then add them up where each term
gets a sign which is positive if both epsilons have the same sign and negative
otherwise. This gives the double-commutator

$\left\langle[\Phi_{i}\left(\beta/2+it_{1}\right),\Phi_{j}\left(\beta/2+it_{3}\right)][\overline{\Phi}_{i}\left(it_{2}\right),\overline{\Phi}_{j}\left(it_{4}\right)]\right\rangle$
(5.19)

which can be written as

$\left\langle\left(\Phi_{i}\left(\beta/2+\epsilon_{1}+it_{1}\right)-\Phi_{i}\left(\beta/2-\epsilon_{1}+it_{1}\right)\right)\left(\overline{\Phi}_{i}\left(\epsilon_{2}+it_{2}\right)-\overline{\Phi}_{i}\left(-\epsilon_{2}+it_{2}\right)\right)\Phi_{j}\left(\beta/2+it_{3}\right)\overline{\Phi}_{j}\left(it_{4}\right)\right\rangle\;,$
(5.20)

where we eventually take the limit $\epsilon_{i}\to 0$. Note that for most
combinations of $\tau$’s, the $\epsilon$-dependence drops out in this limit.
For example, in the combination $\tau_{1}-\tau_{4}$ the real part is always
dominated by $\beta$ and not by the $\epsilon$ term, and so the sign of
$\epsilon$ does not affect the result when we take $\epsilon\to 0$. As a
result, $\chi^{\pm}(\epsilon)=\chi^{\pm}$ does not depend on $\epsilon$:

$\chi^{\pm}=\frac{\sinh\frac{x_{12}\pm i\tau_{12}}{2}\sinh\frac{x_{34}\pm
i\tau_{34}}{2}}{\sinh\frac{x_{14}\pm i\tau_{14}}{2}\sinh\frac{x_{32}\pm
i\tau_{32}}{2}}\;.$ (5.21)

However, $|\chi-1|$ does depend on $\epsilon$. Following this procedure for
$|\chi-1|$ we find that the analytic continuation takes

$|\chi-1|=\sqrt{(\chi^{-}-1)(\chi^{+}-1)}\to-4\sqrt{|(\chi^{-}-1)(\chi^{+}-1)|}\;,$
(5.22)

assuming $t_{13}>\left|x_{13}\right|$ and $t_{24}>\left|x_{24}\right|$, and
the expression vanishes otherwise.

Up to overall factors which will not be important, the integral then becomes

$\begin{split}k_{R}=\int(KW)_{R}=\int
du_{3}dv_{3}du_{4}dv_{4}\frac{1}{u_{34}^{2-h}v_{34}^{2-h}}\mathcal{I}_{R}\end{split}$
(5.23)

where $\mathcal{I}_{R}$ is given by taking $\mathcal{I}$ and subtracting
$\mathcal{I}$ but where each $|\chi-1|$ is replaced by $-|\chi-1|$. In these
new variables, we have

$\chi=\frac{u_{12}u_{34}}{u_{14}u_{32}},\qquad\bar{\chi}=\frac{v_{12}v_{34}}{v_{14}v_{32}}\;,$
(5.24)

and the integration region is
$u_{3}>u_{1},\;u_{2}>u_{4},\;v_{3}>v_{1},\;v_{2}>v_{4}$.

As discussed in 2.2, the chaos exponent as $\mathcal{J}\to 0$ is found by
looking for divergences, which should appear at large $|u_{i}|,|v_{i}|$; the
chaos exponent is $\lambda_{0}=-2h_{*}$ where $h_{*}$ is the value for which
the integral diverges. We can find this analytically, since we expect the
divergence to come from taking large $u_{3},v_{3},u_{4},v_{4}$ (and they are
all of the same order of magnitude), which means we are taking
$\chi,\bar{\chi}\to 0$. In this limit the integrand behaves as

$\frac{1}{u_{34}^{2-h}v_{34}^{2-h}}$ (5.25)

which diverges for $h\geq 0$. So the chaos exponent at weak coupling is

$\lambda_{L}(\mathcal{J}\to 0)=0\;.$ (5.26)

####  5.3.2 Contributions from higher $n$-point functions

As discussed in section 2.2, we must make sure that contributions to the
kernel from higher $n$-point functions don’t diverge at lower values of
$\lambda$ than $\lambda_{L}(\mathcal{J}\to 0)=0$, otherwise the approximations
made above are invalid. We know all of the higher $n$-point functions since we
have mapped the theory to a free theory, and so it is a matter of plugging the
results into the kernel.777As in [1], we can ignore the contribution of
subtractions in this calculation.

As an example, we consider the contribution from the bottom component of the
$2n$-point function for any $n$ to the kernel in figure 2. Using the free
field realization, we find that any $2n$-point function of the bottom
component $\phi$ of the superfield $\Phi$ takes the form

$\langle\phi(x_{1})...\phi(x_{n})\bar{\phi}(y_{1})...\bar{\phi}(y_{n})\rangle=\frac{1}{2^{n/2}}\sqrt{\sum_{\begin{subarray}{c}\epsilon_{i}^{x}=\pm
1,\;\;\epsilon_{i}^{y}=\pm 1,\\\
\sum\epsilon_{i}^{x}+\epsilon_{i}^{y}=0\end{subarray}}\frac{\prod_{i<j}|x_{ij}|^{\left(\epsilon_{i}^{x}\epsilon_{j}^{x}+1\right)/2}|y_{ij}|^{\left(\epsilon_{i}^{y}\epsilon_{j}^{y}+1\right)/2}}{\prod_{i,j}|x_{i}-y_{j}|^{\left(1-\epsilon_{i}^{x}\epsilon_{j}^{y}\right)/2}}}\;.$
(5.27)

Combined with the propagators (5.7), we can find the contributions to the
kernel at any order and look for divergences as
$x^{+}_{3}\to\infty,x^{+}_{4}\to\\-\infty$ and similarly for $x_{3,4}^{-}$. In
this limit, the contribution from the six point function behaves as
$\frac{1}{|x_{34}|^{4-2h}}$, which matches the behavior found in the
contribution from the four-point function. The divergence is thus also at
$h=0$, and so our approximation is consistent at this level. One can repeat
the analysis for other components of the 2n-point function which contribute to
$k_{R}$, and as a result the approximations discussed above are consistent.

####  5.3.3 Continuity conjecture

We now compute $\lambda_{L}^{0}$ and compare to $\lambda_{L}(\mathcal{J}\to
0)$ in order to show that the continuity conjecture is obeyed. This amounts to
taking the bosonic part of the analytically-continued four-point function of a
single core CFT (5.11) and studying its behavior in the large-time limit
$t_{3}\sim t_{4}=t\to\infty$.

First we must do the analytic continuation discussed above. The bosonic part
of the 4-pt function is

$\frac{1}{\sqrt{2}}\frac{\sqrt{1+|\chi|+|\chi-1|}}{|z_{12}z_{34}|^{\frac{1}{2}}}\;.$
(5.28)

Mapping to the cylinder and performing the analytic continuation discussed
above, we find

$\frac{\sqrt{1+|\chi|+|\chi-1|}}{|z_{12}z_{34}|^{\frac{1}{2}}}\to
2\theta(t_{13}-|x_{13}|)\theta(t_{24}-|x_{24}|)\frac{\sqrt{1+|\chi|+|\chi-1|}-\sqrt{1+|\chi|-|\chi-1|}}{|z_{12}z_{34}|^{\frac{1}{2}}}\;.$
(5.29)

Now we can take the large-time limit, which amounts to taking
$\chi,\bar{\chi}\sim e^{-t}$ for $t$ large. We find

$\sqrt{1+|\chi|+|\chi-1|}-\sqrt{1+|\chi|-|\chi-1|}\approx\sqrt{2}-\sqrt{2}e^{-t/2}+...$
(5.30)

and so the analytically-continued four-point function is finite in the
large-$t$ limit, since it is not exponentially vanishing in $t$. As a result,
$\lambda_{L}^{0}=0$ since there is no exponential behavior. We thus find that
$\lambda_{L}^{0}=\lambda(\mathcal{J}\to 0)=0$ and so the continuity conjecture
is obeyed.

##  6 Conclusions

In this paper we have proven the continuity conjecture in QM and given
additional evidence for it in $2d$. We also studied the disordered
$\mathcal{A}_{3}$ minimal model at leading order in the random coupling
$\mathcal{J}$.

There are many interesting questions left for future work. For example, in
both of the $2d$ examples given here (the chiral SYK model and the disordered
$\mathcal{A}_{3}$ model), the chaos exponent at small coupling was
$\lambda_{L}^{0}=0$. In [1], the example of the disordered $\mathcal{A}_{2}$
minimal model was also shown to have $\lambda_{L}^{0}=0$. These examples
should be compared to the case of disordered generalized free fields where
$\lambda_{L}^{0}<0$, leading to a discontinuous transition into chaos as
$\mathcal{J}$ increases [1].888This behavior is not related to the non-
locality of the generalized free field theory; for example, a negative
$\lambda_{L}^{0}$ was found in [25], although this corresponds to the chaos
exponent on thermal Rindler space and not flat space. We can ask why this
interesting behavior did not occur here, and an obvious conjecture is that
$\lambda_{L}^{0}$ can only be non-zero when the core CFT does not have a free
theory realization. Unfortunately, this means that performing computations in
theories with $\lambda_{L}^{0}<0$ will be difficult, since the computations
require knowledge of exact $n$-point functions of the CFT. One possible
direction would be to perform these computations in $\mathcal{N}=2$ minimal
models with higher central charges, where the four-point functions are known
exactly, but performing the necessary integrals over them is a very
complicated technical task which we leave to future work.

Another future direction would be to prove the continuity conjecture in $2d$.
This becomes more complicated due to the appearance of the butterfly velocity.
The divergence which leads to the chaos exponent at weak coupling appears at
large times, but can appear at any value of the ratio $v=t/x$, and an
additional integral must be performed over these values. The final result
depends nontrivially on $v$, and the result must be studied carefully in order
for the chaos exponent to be read off. However, the chiral example discussed
above is evidence that even with a nontrivial butterfly velocity, the
continuity conjecture is obeyed. It is yet to be seen whether it is obeyed
only for the velocity which leads to the maximal chaos exponent, or whether it
is obeyed for any velocity.

It would also be interesting to extend calculations of
$\lambda_{L}(\mathcal{J})$ for a continuous range of $\mathcal{J}$ to higher
dimensions, building on [9]. For example, considering $N$ free bosons in $3d$,
the deformation $((\varphi_{i})^{2})^{3}$ is exactly marginal at leading order
in $1/N$ [26]. Similarly for $N$ free matter multiplets in a $3d$
$\mathcal{N}=1$ supersymmetric theory, the superpotential deformation
$(|\Phi_{i}|^{2})^{2}$ is also exactly marginal at leading order in $1/N$
[27]. One could perform a computation of the chaos exponent analogous to that
of the $O(N)$ model [28] as a function of the exactly marginal
deformation.999A better understanding of whether it is enough to be exactly
marginal at leading order in $1/N$ is required in order to perform these
computations. The chaos exponent is expected to be non-positive since the
theory is close to being free, and it would be interesting to see if it is
exactly zero.

## Acknowledgements

The authors would like to thank E. Y. Urbach and N. Silberstein for
collaborating on this work in its early stages, and B. Lian for collaboration
on related topics. The authors would also like to thank Micha Berkooz and
Doron Gepner for enlightening discussions. The work of RRK was supported in
part by an Israel Science Foundation (ISF) center for excellence grant (grant
number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant 994296
(Simons Collaboration on Confinement and QCD Strings), by grant no. 2018068
from the United States-Israel Binational Science Foundation (BSF), by the
Minerva foundation with funding from the Federal German Ministry for Education
and Research, by the German Research Foundation through a German-Israeli
Project Cooperation (DIP) grant “Holography and the Swampland”, and by a
research grant from Martin Eisenstein.

##  Appendix A  Superappendix

###  A.1 The supersymmetry algebra

$\mathcal{N}=(2,2)$ SUSY in two dimensions is closely related to
$\mathcal{N}=1$ in four dimensions. There are two real spinors’ worth of
supercharges: $Q^{(a)}_{\alpha}$, where $\alpha$ is a Lorentz (spinor) index
and $a=1,...,\mathcal{N}$ is a separate (R-symmetry) index. The SUSY algebra
is

$\displaystyle\\{Q^{a}_{\alpha},Q^{b}_{\beta}\\}$
$\displaystyle=2\gamma^{\mu}_{\alpha\beta}P_{\mu}\delta^{ab}\;,$ (A.1)

and the gamma matrices are:

$\displaystyle\gamma^{0}_{\alpha\beta}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}\;,\qquad\gamma^{1}_{\alpha\beta}=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}\;.$

We can break the spinor representation into the left/right-moving sectors,
which are one-dimensional and transform by a phase under the Lorentz group:

$\displaystyle Q^{a}_{\pm}$ $\displaystyle\rightarrow
e^{\pm\frac{1}{2}\eta}Q^{a}_{\pm}$ (A.2)

In Lorentzian signature, the supercharges are real:

$\displaystyle\overline{Q^{a}_{\pm}}$
$\displaystyle=\left(Q^{a}_{\pm}\right)^{*}=Q^{a}_{\pm}$ (A.3)

####  A.1.1  $\mathcal{N}=(1,1)$

A real $\mathcal{N}=(1,1)$ scalar superfield $\mathcal{A}(x,\theta)$ consists
of a real scalar $\phi(x)$ and its superpartners: a Majorana spinor
$\psi_{\alpha}(x)$, and another scalar $F(x)$.

$\displaystyle\mathcal{A}(x,\theta)$
$\displaystyle=e^{\theta^{\alpha}Q_{\alpha}}\phi(x)$ (A.4)
$\displaystyle=\phi(x)+\theta^{\alpha}\psi_{\alpha}(x)+\theta^{\alpha}\theta^{\beta}\epsilon_{\alpha\beta}F(x)+\dots$
$\displaystyle=\phi(x)+\theta^{+}\psi_{+}(x)+\theta^{-}\psi_{-}(x)+\theta^{+}\theta^{-}F(x)+\dots$
(A.5)

where $\theta^{\alpha}$ is a real Grassman
spinor,101010$\left(\theta^{\pm}\right)^{2}=0$. and is given a Lorentz
transformation so that the superfield is a scalar. We’ve broken up
$\psi_{\alpha}(x)$ into Lorentz irreps, and the equations of motion will tell
us that $\psi_{\pm}$ are left/right moving (holomorphic/antiholomorphic in
Euclidean signature). Higher components are descendants since the supercharge
squares to a derivative. There is a discrete R-symmetry

$\displaystyle Q_{\alpha}$ $\displaystyle\rightarrow-Q_{\alpha}\;,$ (A.6)

that leaves the SUSY algebra invariant. This gives the following
transformations for the fields:

$\displaystyle\phi(x)$ $\displaystyle\rightarrow(-1)^{\gamma}\phi(x)\;,$
$\displaystyle\psi_{\alpha}(x)$
$\displaystyle\rightarrow(-1)^{\gamma+1}\psi_{\alpha}(x)\;,$ $\displaystyle
F(x)$ $\displaystyle\rightarrow(-1)^{\gamma+2}F(x)\;.$ (A.7)

With $\theta^{\alpha}\rightarrow-\theta^{\alpha}$, the superfield
$\mathcal{A}(x,\theta)$ has the same R-charge as its bottom component (here,
the scalar; see (A.4)). The Lagrangian (free-field plus superpotential) in
terms of superfields may be written as

$\displaystyle\mathcal{L}$ $\displaystyle=\int
d^{2}\theta(\frac{1}{2}\mathcal{D}\mathcal{A}\bar{\mathcal{D}}\mathcal{A}+W(\mathcal{A}))\;.$
(A.8)

####  A.1.2  $\mathcal{N}=(2,2)$

The $\mathcal{N}=(2,2)$ algebra has two Majorana supercharges, and we need two
real Grassman spinors $\theta^{\alpha}_{a}$. A complex superfield
$\mathcal{X}$ takes the form

$\displaystyle\mathcal{X}(x,\theta^{a})$
$\displaystyle=e^{\theta_{a}^{\alpha}Q^{a}_{\alpha}}\phi(x)=\phi+\theta_{a}^{\alpha}\psi^{a}_{\alpha}+...$
$\displaystyle=\phi(x)+\theta^{+}_{a}\psi^{a}_{+}(x)+\theta^{-}_{a}\psi^{a}_{-}(x)+...$
$\displaystyle\bar{\mathcal{X}}(x,\theta^{a})$
$\displaystyle=\bar{\phi}+\theta_{a}^{\alpha}\bar{\psi}^{a}_{\alpha}+...$
$\displaystyle=\bar{\phi}(x)+\theta^{+}_{a}\bar{\psi}^{a}_{+}(x)+\theta^{-}_{a}\bar{\psi}^{a}_{-}(x)+...$
(A.9)

The bar stands for complex conjugation (see (A.3)). It is convenient to use
complex combinations of the supercharges:

$\displaystyle
Q_{\pm}=Q^{1}_{\pm}+iQ^{2}_{\pm}\;,\qquad\overline{Q}_{\pm}=Q^{1}_{\pm}-iQ^{2}_{\pm}\;,\qquad\overline{(Q_{\pm})}=\overline{Q}_{\pm}\;.$
(A.10)

The $\mathcal{N}=(2,2)$ SUSY algebra may be written as:

$\displaystyle\\{Q_{\pm},\overline{Q}_{\pm}\\}$ $\displaystyle=2P_{\pm}=2(\pm
P_{0}+P_{1})\;.$ (A.11)

The Grassman variables are now:

$\displaystyle\theta^{\pm}=\theta_{1}^{\pm}+i\theta_{2}^{\pm}\;,\qquad\overline{\theta}^{\pm}=\theta_{1}^{\pm}-i\theta_{2}^{\pm}\;.$
(A.12)

The superfield takes the form

$\displaystyle\mathcal{X}(x,\theta,\bar{\theta})$
$\displaystyle=e^{\bar{\theta}^{\alpha}Q_{\alpha}+\theta^{\alpha}\bar{Q}_{\alpha}}\phi(x)$
(A.13)
$\displaystyle=\phi(x)+\bar{\theta}^{+}\tilde{\psi}_{+}(x)+\theta^{+}\psi_{+}(x)+\bar{\theta}^{-}\tilde{\psi}_{-}(x)+\theta^{-}\psi_{-}(x)+...$
$\displaystyle\bar{\mathcal{X}}(x,\theta,\bar{\theta})$
$\displaystyle=\bar{\phi}(x)+\bar{\theta}^{+}\bar{\psi}_{+}(x)+\theta^{+}\bar{\tilde{\psi}}_{+}(x)+\bar{\theta}^{-}\bar{\psi}_{-}(x)+\theta^{-}\bar{\tilde{\psi}}(x)+...$
(A.14)

with

$\displaystyle\psi_{\pm}=\psi_{\pm}^{1}-i\psi_{\pm}^{2}\;,\qquad\tilde{\psi}_{\pm}=\psi_{\pm}^{1}+i\psi_{\pm}^{2}\;.$
(A.15)

The $\mathfrak{u}(1)_{R}$ symmetry rotates $(Q,\bar{Q})$ in opposite
directions:

$\displaystyle Q_{\pm}\rightarrow
e^{i\alpha}Q_{\pm}\;,\qquad\bar{Q}_{\pm}\rightarrow
e^{-i\alpha}\bar{Q}_{\pm}\;,$ $\displaystyle\theta^{\pm}\rightarrow
e^{i\alpha}\theta^{\pm}\;,\qquad\bar{\theta}^{\pm}\rightarrow
e^{-i\alpha}\bar{\theta}^{\pm}\;.$ (A.16)

There is also an axial $\mathfrak{u}(1)_{\tilde{R}}$ which rotates the $\pm$
components with opposite phases.

##### Chirality

The superfield (A.13) is in a reducible representation. Imposing

$\displaystyle Q(\phi(x))$ $\displaystyle=\bar{Q}(\bar{\phi}(x))=0\;,$

we have

$\displaystyle\Phi(x,\theta,\bar{\theta})$
$\displaystyle=e^{(\bar{\theta}^{\alpha}Q_{\alpha}+\theta^{\alpha}\bar{Q}_{\alpha})}\phi(x)=e^{(\theta^{\alpha}\bar{Q}_{\alpha}+i\theta^{\alpha}\gamma^{\mu}\bar{\theta}P_{\mu})}\cancel{e^{\bar{\theta}^{\alpha}Q_{\alpha}}}\phi(x)$
$\displaystyle=\phi(x)+\theta^{+}\psi_{+}(x)+\theta^{-}\psi_{-}(x)+\theta^{+}\theta^{-}F(x)+...$
$\displaystyle\bar{\Phi}(x,\theta,\bar{\theta})$
$\displaystyle=\bar{\phi}(x)+\bar{\theta}^{+}\bar{\psi}_{+}(x)+\bar{\theta}^{-}\bar{\psi}_{-}(x)+\bar{\theta}^{+}\bar{\theta}^{-}\bar{F}(x)+...$
(A.17)

This makes $\Phi(x,\theta,\bar{\theta})$ a chiral superfield. An
$\mathcal{N}=(2,2)$ chiral (scalar) multiplet contains only one complex
fermion. It is easy to write down interactions for such superfields:

$\displaystyle\mathcal{L}$ $\displaystyle=\int d^{2}\theta
d^{2}\bar{\theta}\bar{\Phi}\Phi+\int d^{2}\theta W(\Phi)+\int
d^{2}\bar{\theta}\bar{W}(\bar{\theta})\;.$ (A.18)

Of course, these are not the only possible interactions.

###  A.2 The $\mathcal{A}_{3}$ minimal model

We are interested in the following $\mathcal{N}=(2,2)$ theory:

$\displaystyle\mathcal{L}$ $\displaystyle=\int d^{2}\theta
d^{2}\bar{\theta}\bar{X}X+\int d^{2}\theta X^{4}+\int
d^{2}\bar{\theta}\bar{X}^{4}\;,$ (A.19)

with $X$ a chiral superfield.

Table 2: Operators from the Ising model. The last entry $\chi$ is the fermion. $\sigma$ and $\mu$ are the spin and disorder operators; these aren’t mutually local. $\mathcal{O}$ | $\Delta$ | $\ell$
---|---|---
$\sigma$ | $\frac{1}{8}$ | 0
$\mu$ | $\frac{1}{8}$ | 0
$\epsilon$ | $1$ | 0
$\chi_{\pm}$ | $\tfrac{1}{2}$ | $\tfrac{1}{2}$

It’s easily seen that the interactions preserve $\mathfrak{u}(1)_{R}$ with
$q_{X}=1/2$; this blocks the superpotential from picking up other terms under
RG flow. The theory in the IR is a Landau-Ginsburg minimal model with central
charge $c=\tfrac{3}{2}$ that may also be obtained by putting together a free
scalar and the Ising model. In the following, $H$ is a free boson, and Table 2
are the operators we need from the Ising model.

Firstly, we identify the SUSY generators111111Recall that the SUSY currents
are operators with dimensions $(\tfrac{3}{2},0)$ or $(0,\tfrac{3}{2})$
($G_{\pm}$ respectively). and R symmetry current:

$\displaystyle G_{\pm}$
$\displaystyle=\chi_{\pm}\exp\left(i\sqrt{2}H_{\pm}\right)\;,$
$\displaystyle\bar{G}_{\pm}$
$\displaystyle=\chi_{\pm}\exp\left(-i\sqrt{2}H_{\pm}\right)\;,$ $\displaystyle
j_{\pm}^{(R)}$ $\displaystyle=\frac{i}{\sqrt{2}}\partial_{\pm}H\;.$ (A.20)

$H_{\pm}$ are the left/right moving parts of the scalar field $H$. The bottom
components of the superconformal primaries are given by:

$\displaystyle\phi=\sigma\exp\left(i\frac{H}{2\sqrt{2}}\right)\;,$
$\displaystyle\qquad\bar{\phi}=\sigma\exp\left(-i\frac{H}{2\sqrt{2}}\right)\;,$
$\displaystyle\phi^{2}=\exp\left(i\frac{H}{\sqrt{2}}\right)\;,$
$\displaystyle\qquad\bar{\phi}^{2}=\exp\left(-i\frac{H}{\sqrt{2}}\right)\;,$
$\displaystyle\bar{\phi}\phi$ $\displaystyle=\epsilon\;.$ (A.21)

We can then find the free field expressions for their SUSY descendants using
(A.20) and the OPE’s:

$\displaystyle\chi_{\pm}(x^{\pm})\times\sigma(x)\sim\frac{1}{\sqrt{x^{\pm}}}\mu\;,\qquad\chi_{\pm}(x^{\pm})\times\mu(x)\sim\frac{1}{\sqrt{x^{\pm}}}\sigma\;.$
(A.22)

This leads to:

$\displaystyle\bar{G}_{+}(x^{+})\times\phi$ $\displaystyle\sim\frac{1}{x^{+}}\
\mu\exp\left(\frac{i}{2\sqrt{2}}\left(-3H_{+}+H_{-}\right)\right)=\frac{1}{x^{+}}\psi_{+}\;,$
$\displaystyle\bar{G}_{-}(x^{-})\times\phi$ $\displaystyle\sim\frac{1}{x^{-}}\
\mu\exp\left(\frac{i}{2\sqrt{2}}\left(H_{+}-3H_{-}\right)\right)=\frac{1}{x^{-}}\psi_{-}\;,$
$\displaystyle\bar{G}_{+}(x^{+})\times\psi_{-}$
$\displaystyle\sim\frac{1}{x^{+}}\
\sigma\exp\left(-i\frac{3H}{2\sqrt{2}}\right)=\frac{1}{x^{+}}F\;,$
$\displaystyle\bar{G}_{-}(x^{-})\times\psi_{+}$
$\displaystyle\sim\frac{1}{x^{-}}\
\sigma\exp\left(-i\frac{3H}{2\sqrt{2}}\right)=\frac{1}{x^{-}}F\;.$ (A.23)

We summarise these results in Table 3.

Table 3: The operators in the multiplet of $X$ in the $\mathcal{A}_{3}$ minimal model. $\mathcal{O}$ | $\mathcal{O}_{\text{free}}$ | $(h,\bar{h})$ | $q_{R}$
---|---|---|---
$\phi$ | $\sigma\ e^{i\frac{H}{2\sqrt{2}}}$ | $\left(\frac{1}{8},\frac{1}{8}\right)$ | $\left(\frac{1}{4},\frac{1}{4}\right)$
$\psi_{+}$ | $\mu\ e^{i\frac{-3H_{+}+H_{-}}{2\sqrt{2}}}$ | $\left(\frac{5}{8},\frac{1}{8}\right)$ | $\left(-\frac{3}{4},\frac{1}{4}\right)$
$\psi_{-}$ | $\mu\ e^{i\frac{H_{+}-3H_{-}}{2\sqrt{2}}}$ | $\left(\frac{1}{8},\frac{5}{8}\right)$ | $\left(\frac{1}{4},-\frac{3}{4}\right)$
$F$ | $\sigma\ e^{-i\frac{3H}{2\sqrt{2}}}$ | $\left(\frac{5}{8},\frac{5}{8}\right)$ | $\left(-\frac{3}{4},-\frac{3}{4}\right)$

##### Mutual locality of the operators

Firstly, $G$, $\bar{G}$, and $j^{R}$ are mutually local (of course, the
fermionic operators are mutually local only in the fermionic sense – they pick
up a $-1$) since the bosonic pieces are the conserved currents of the
$\mathfrak{su}(2)_{1}$ model. Now, all the operators within the $X$ and
$\bar{X}$ multiplet are obtained by operator products with the currents. So to
show that these are mutually local, we just need to show that the bottom
components don’t see branch cuts with respect to each other or the currents.

The OPE of two vertex operators (analytically continued to Euclidean space)
take the form:

$\displaystyle e^{ik_{1}.H}(z)\times e^{ik_{2}.H}(0)$ $\displaystyle\sim
z^{k_{1}^{R}k_{2}^{R}}\bar{z}^{k_{1}^{L}k_{2}^{L}}e^{i(k_{1}+k_{2}).H}(0)$
$\displaystyle=(z\bar{z})^{\tfrac{1}{2}k_{1}.k_{2}}\left(\frac{z}{\bar{z}}\right)^{-\tfrac{1}{2}(k_{1}^{L}k_{2}^{L}-k_{1}^{R}k_{2}^{R})}e^{i(k_{1}+k_{2}).H}(0)$
(A.24)

where $k_{1,2}.H=k_{1,2}^{R}H_{+}+k_{1,2}^{L}H_{-}$ and
$k_{1}.k_{2}=k_{1}^{R}k_{2}^{R}+k_{1}^{L}k_{2}^{L}$. When one operator goes
around the other, the right-hand side picks up $e^{2\pi
i(k_{1}^{L}k_{2}^{L}-k_{1}^{R}k_{2}^{R})}$. Setting this phase to 1 gives the
Narain condition:

$\displaystyle k_{1}\odot
k_{2}=k_{1}^{L}k_{2}^{L}-k_{1}^{R}k_{2}^{R}\in\mathbbm{Z}$ (A.25)

This will be used to check locality between vertex operators. The R-current
plays well with all vertex operators, so we won’t have to bother with it.

Coming back to our operators, the Ising parts of $\phi$ and $\bar{\phi}$ are
mutually local. So we just need to examine the vertex operators, which are
easily seen to satisfy (A.25). Actually it quite easy to see that all the
bottom components listed in (A.21) are mutually local. The first nontrivial
check is between $(G,\bar{G})$ and $\phi$. But we can easily verify from
(A.23) that there are no branch cuts; the same conclusion holds for
$(G,\bar{G})$ and $\bar{\phi}$ via the conjugate of (A.23). $\bar{\phi}\phi$
and the currents are also easily kosher. To work out $\phi^{2}$ case, we need
only focus on the vertex operators. The only non-zero $\odot$ products are
$(\pm\sqrt{2},0)\odot(\tfrac{1}{\sqrt{2}},0)=\pm 1$.

##  Appendix B Simplification of $1d$ integral

Using the iterative ladder structure, we can write the full retarded kernel as

$K_{R}(1,2)=K_{0}(1,2;3,4)G_{lr}^{q-2}(3,4)\;.$ (B.1)

At leading order, $K_{0}$ is just the subtracted 4-pt function. It is more
convenient to work with the rescaled expression

$K_{R}^{\prime}=e^{\Delta\left(t_{1}+t_{2}-t_{3}-t_{4}\right)}K_{R}\;.$ (B.2)

Next, we perform the change of variables $z=e^{-t}$ on the left rail (points
1,3) and $z=-e^{-t}$ on the right (points 2,4), see figure 4, leading to

$K_{R}^{\prime}dt_{3}dt_{4}=K_{R}^{\prime}\frac{dz_{3}}{z_{3}}\frac{dz_{4}}{z_{4}}\;.$
(B.3)

The two-points are fixed up to a constant that we will ignore:

$\begin{split}G_{lr}&=\frac{1}{\left(2\cosh\frac{t_{3}-t_{4}}{2}\right)^{2\Delta}}=\frac{z_{3}^{\Delta}z_{4}^{\Delta}}{(z_{34})^{2\Delta}}\;,\\\
G_{R}\left(1,3\right)&=\theta\left(t_{1}-t_{3}\right)\frac{2\cos(\pi\Delta)b}{\left(2\sinh\frac{t_{1}-t_{3}}{2}\right)^{2\Delta}}=\theta(z_{3}-z_{1})\frac{2\cos(\pi\Delta)bz_{1}^{\Delta}z_{3}^{\Delta}}{(z_{13})^{2\Delta}}\;.\end{split}$
(B.4)

Now we use the fact that $K_{0}$ is a (sum of) four-point functions, so we can
write it as

$K_{0}^{\prime}=e^{\Delta\left(t_{1}+t_{2}-t_{3}-t_{4}\right)}G_{lr}(1,2)G_{lr}(3,4)\mathcal{G}(\chi)$
(B.5)

with $\chi$ the conformal cross-ratio and $G$ some function. Putting this
together and setting $\Delta=1/q$ we find

$K_{R}^{\prime}dt_{3}dt_{4}=\frac{\mathcal{G}_{R}(\chi)dz_{3}dz_{4}}{(z_{12})^{2\Delta}(z_{34})^{2\Delta\left(q-1\right)}}$
(B.6)

Finally, applying $K_{R}^{\prime}$ to the eigenfunction
$W_{+}^{\prime}=\frac{1}{|z_{34}|^{2\Delta+\lambda}}$ we find

$K_{R}^{\prime}W_{+}^{\prime}dt_{3}dt_{4}=\frac{\mathcal{G}_{R}(\chi)dz_{3}dz_{4}}{|z_{12}|^{2\Delta}|z_{34}|^{2+\lambda}}\;.$
(B.7)

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