# Onset of Quantum Chaos in Random Field Theories

Micha Berkooz<EMAIL_ADDRESS>Adar Sharon
<EMAIL_ADDRESS>Navot Silberstein<EMAIL_ADDRESS>Erez Y. Urbach<EMAIL_ADDRESS>Department of Particle Physics and
Astrophysics,
Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

We study the quantum Lyapunov exponent $\lambda_{L}$ in theories with
spacetime-independent disorder. We first derive self-consistency equations for
the two- and four-point functions for products of $N$ models coupled by
disorder at large $N$, generalizing the equations appearing in SYK-like
models. We then study families of theories in which the disorder coupling is
an exactly marginal deformation, allowing us to follow $\lambda_{L}$ from weak
to strong coupling. We find interesting behaviors, including a discontinuous
transition into chaos, mimicking classical KAM theory.

##  I Introduction

Systems with disorder are ubiquitous in nature, and display a wide range of
interesting physical phenomena. Disorder can sometimes be modeled by
introducing random couplings, and averaging over these random couplings can
lead to simplifications which allow for exact computations, at least when the
number of degrees of freedom is parametrically large. A notable example is the
Sachdev–Ye–Kitaev (SYK) model, in which (nearly) conformal symmetry is
restored at low energies Sachdev and Ye (1993); Kitaev (2015a); Maldacena and
Stanford (2016); Kitaev and Suh (2018), and such averages allow for the
computation of the quantum chaos exponent $\lambda_{L}$. The latter is defined
by the fastest growing exponential mode in the double-commutator
$\langle[U(t),W(0)]^{2}\rangle_{\beta}\sim e^{\lambda_{L}t}$ for generic
operators $U,W$, over an appropriate time scale, with temperature $1/\beta$
Larkin and Ovchinnikov (1969).

The SYK model consists of $N$ free quantum mechanical fermions deformed by a
relevant all-to-all spacetime-independent disordered interaction.
Generalizations have appeared which consist of $N$ copies of other free
theories with similar interactions Murugan  _et al._ (2017); Fu  _et al._
(2017); Peng (2018); Bulycheva (2018); Chang  _et al._ (2021); Popov (2020);
Gross and Rosenhaus (2017); Lian  _et al._ (2019). In this letter we study
quantum chaos in a more general setting by studying $N$ copies of a general
core model $\mathcal{Q}$ (which can be a quantum field theory or a spin
system), deformed by a spacetime-independent all-to-all interaction. In
particular, we eventually focus on $Q$ being a conformal field theory (CFT) in
0+1 or in higher dimensions, and with an exactly marginal disorder
interaction. In such a setup we have better control over the space of
couplings over which we are averaging, eliminating the complications of the
renormalization group (RG) and without having to resort to strong coupling.

The theories we study are of the form

$\mathcal{Q}^{N}+\sum_{i_{1}\neq...\neq
i_{q}}^{N}J_{i_{1}...i_{q}}\mathcal{O}_{i_{1}}...\mathcal{O}_{i_{q}}\;,$ (1)

where $\mathcal{Q}^{N}$ denotes $N$ decoupled copies of the model
$\mathcal{Q}$, $\mathcal{O}_{i}$ with $i=1,...,N$ are the $N$ copies of a
local operator $\mathcal{O}$ in $\mathcal{Q}$, and $J_{i_{1}...i_{q}}$ are
Gaussian random variables with variance

$\langle J_{i_{1}...i_{q}}^{2}\rangle=\frac{J^{2}(q-1)!}{N^{q-1}}\;.$ (2)

In the case where $\mathcal{Q}$ is a CFT, we will take $\mathcal{O}$ to be a
primary operator of this core CFT, and equation (1) should be interpreted in
conformal perturbation theory in $J$. We call such theories disordered CFTs;
the simplest example of this setup is the SYK model itself, as a disordered
free fermion theory.

We will first derive self-consistency equations for the two- and four-point
functions of $\mathcal{O}_{i}$ at leading order in $1/N$ for general
disordered CFTs. These extend known results for disordered free theories like
the SYK model Maldacena and Stanford (2016); Kitaev and Suh (2018)), and for
spin systems Bray and Moore (1980). We will also discuss similar results for
the double-commutator (defined in (7) below). Although these self-consistency
equations are complicated, they are tractable in perturbation theory in $J$
and allow us to establish the existence of a kernel structure from which we
can extract the chaos exponent $\lambda_{L}$, which is the rate of growth of a
double commutator. The latter is given by the fastest growing eigenvector
(with eigenvalue 1) of a specific integral kernel $K_{R}$ (see equation (9)),
as long as this rate of growth is indeed positive Kitaev (2015b, a); Larkin
and Ovchinnikov (1969). We will denote by $\lambda_{L}^{\text{ker}}$ the
putative chaos exponent read from the diagonalization $K_{R}$.

Figure 1: The behaviors we find for the chaos exponent as a function of an
exactly marginal disorder deformation $J$: (a) continuous and (b)
discontinuous. $\lambda_{L}^{\text{ker}}$ corresponds to the dashed line and
$\lambda_{L}$ to the solid line.

We will be interested in computing the chaos exponent $\lambda_{L}$ as a
function of $J$, as it is varied from weak to strong coupling. The
diagonalization of $K_{R}$ is a difficult process, which can usually be done
only when conformal invariance is restored. Normally, the disorder is a
relevant deformation. To compute the chaos one first flow to the IR CFT (which
is equivalent to take $J=\infty$) and find the chaos exponent there Murugan
_et al._ (2017). In this work we focus on cases where the averaged interacting
theory is conformally invariant for all $J$. This can be done by demanding
that the disorder interaction itself be exactly marginal, in which case the
theory is conformally invariant for every realization of the couplings, and
the space of $J$’s forms a conformal manifold 111It is important that
$\mathcal{O}$ itself does not obtain an anomalous dimension as a function of
$J$, which will be the case in the examples in this letter.. The disorder
average is then simply a non-uniform average over this conformal manifold,
with no RG-related complications (for recent discussions of averages over
conformal manifolds see Maloney and Witten (2020); Afkhami-Jeddi  _et al._
(2021), and for the marginal case see Berkooz  _et al._ (2017)).

Surprisingly, we will find models where $\lambda_{L}^{\text{ker}}(J)$ is
negative at weak coupling. This signals a breakdown of the approximations
involved in the computation, and so in this case the true chaos exponent
simply vanishes, $\lambda_{L}=0$. In other words, we have

$\lambda_{L}=\max(0,\lambda_{L}^{\text{ker}})\;.$ (3)

As a result, there are two possible behaviors for the onset of chaos: either
the theory undergoes a continuous transition into chaos as in figure 1a or a
discontinuous transition as in figure 1b, corresponding to whether
$\lambda_{L}^{\text{ker}}$ is non-negative or negative at small enough $J$
respectively.222We implicitly assumed that the chaos exponent increases
monotonically with the strength of the disorder $J$. As we start with N
decoupled systems it is reasonable, but not necessarily true. For example, an
integrable deformation keeps the exponent at zero, but these are hard to come
by in this context.

The discontinuous transition into chaos is a surprising result, and it is
tempting to compare it to similar results in classical chaos, the most famous
one being the KAM theorem. In order to sharpen the comparison, we also discuss
what a single core CFT should obey in order for the transition into chaos to
be discontinuous. Similar works on the onset of quantum chaos include Stanford
(2016); de Mello Koch  _et al._ (2019); Chowdhury and Swingle (2017);
Steinberg and Swingle (2019); Maldacena and Stanford (2016); Peng (2018), and
Lian  _et al._ (2019); Hu and Lian (2021) are especially relevant.

We will apply our formalism to two classes of examples where the disorder
interaction is exactly marginal. The first class is disordered generalized
free fields in $1$d (following Gross and Rosenhaus (2017)) and in $2$d. The
second class is the disordered $\mathcal{N}=2$ supersymmetric (SUSY) $A_{q-1}$
minimal models. In practice, we will only discuss the simplest case of the
$A_{2}$ minimal model here. We will find a discontinuous transition in the
former and a continuous transition in the latter.

More details and discussions on the computations and results can be found in
the companion paper Berkooz  _et al._ (2021).

##  II Disorder around a nontrivial CFT

###  II.1 The kernel structure of the four-point function

We start by writing a self-consistency equation for the averaged two-point
function of (1),

$G(x)=\langle\mathcal{O}_{i}(x)\mathcal{O}_{i}(0)\rangle\;.$ (4)

Using the $G-\Sigma$ formalism Maldacena and Stanford (2016); Kitaev (2015a),
it can be shown that $G$ obeys a generalized Schwinger-Dyson (SD) equation at
leading order in $1/N$, which appears diagrammatically in figure 2a.

Figure 2: (a) The SD equations. $G$ insertions appear as red lines, and black
dots denote insertions of the deformation (1). (b) Examples of $n_{s}$, The
subtracted $n$-point functions. Dashed lines connect to the external points,
and numerical factors indicate symmetry factors. The blue numbers inside the
brackets denote possible ways of permuting the legs.

The equation includes subtracted $n$-point functions denoted by “$n_{s}$”,
which are combinations of the standard core CFT $n$-point functions with
additional theory-independent subtractions which can be derived order-by order
in $n$ Berkooz  _et al._ (2021). The first few subtracted $n$-point functions
(assuming $\mathcal{O}_{i}$ are real) are shown in figure 2b. Generalizations
to complex $\mathcal{O}_{i}$ exist, and also to superfields (in which case the
diagrams correspond to supergraphs).

The contributions to the averaged connected four-point function

$C=\frac{1}{N^{2}}\sum_{i,j}(\langle\mathcal{O}_{i}\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{j}\rangle-\langle\mathcal{O}_{i}\mathcal{O}_{i}\rangle\langle\mathcal{O}_{j}\mathcal{O}_{j}\rangle)$
(5)

also have a simple form, and obey an iterative ladder structure similar to the
case of disordered free fields:

$C=\sum_{n=0}^{\infty}K^{n}F_{0}\;,$ (6)

where the kernel $K$ and the initial diagram $F_{0}$ are defined in figure 3a.
The definition requires new subtracted $n$-point functions called
$n_{s}^{\prime}$, which are again theory-independent Berkooz  _et al._ (2021).
The first few $n_{s}^{\prime}$ for real $\mathcal{O}_{i}$ appear in figure 3b.

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

Some comments are in order. First, although the kernel $K$ is very
complicated, knowing that an iterative ladder structure exists for the four-
point function is already an important result since it allows for a systematic
computation of $\lambda_{L}$, as we now discuss. Second, although a full
solution of the two- and four-point functions requires knowing all $n$-point
functions of the core CFT, a solution to order $J^{2n}$ in perturbation theory
in $J$ only requires knowing the $2m$-point functions for $m\leq n+1$.

###  II.2 The double-commutator and chaos

A similar analysis also applies to the computation of the double-commutator:

$\begin{split}W_{R}(t_{1},t_{2})=\frac{1}{N^{2}}\sum_{i,j=1}^{N}&\langle[\mathcal{O}_{i}(\beta/2),\mathcal{O}_{j}(\beta/2+it_{2})]\\\
&\cdot[\mathcal{O}_{i}(0),\mathcal{O}_{j}(it_{1})]\rangle,\end{split}$ (7)

where we suppress the spatial positions. In chaotic theories, at large
Lorentzian times $t_{1},t_{2}$, the double commutator is expected to grow
exponentially:

$\lim_{t_{1},t_{2}\rightarrow\infty}W_{R}(t_{1},t_{2})\sim\frac{1}{N}\exp(+\lambda_{L}(t_{1}+t_{2})/2)\;.$
(8)

Since the double commutator can be written in terms of analytically continued
$4$-point functions (5), (7) satisfies a “retarded” version of the kernel
structure of figure 3a:

$W_{R}=\sum_{n=0}^{\infty}K_{R}^{n}F_{0}\ \Longrightarrow\
(1-K_{R})W_{R}=F_{0}\;,$ (9)

with $K_{R}$ the retarded kernel Kitaev (2015b). The retarded kernel is
composed of the same diagrams as in figure 3a, where one plugs in specific
analytic continuations in time of the $n$-point functions.

If $\lambda_{L}>0$, the ladder structure (however complicated) allows us to
compute it in cases where the averaged correlator has conformal symmetry (see
Murugan  _et al._ (2017) for a review). This is done by finding the largest
solution $\lambda_{L}^{\text{ker}}$ of the equation
$k_{R}(\lambda_{L}^{\text{ker}})=1$, where $k_{R}(\lambda)$ are the
eigenvalues of the retarded kernel $K_{R}$. If $\lambda_{L}^{\text{ker}}>0$
then we can identify $\lambda_{L}=\lambda_{L}^{\text{ker}}$, and otherwise we
learn that $\lambda_{L}=0$.

Importantly, if $J$ is exactly marginal, $k_{R}(\lambda_{L})$ can be found
perturbatively in $J$. As a result, one can compute
$\lambda_{L}^{\text{ker}}(J)$ in orders of $J$ by using finitely many core CFT
correlation functions at every order. The leading order of the equation is
explicitly

$\begin{split}&k_{R}(\lambda)=\frac{J^{2}}{4}\int
d^{2}x_{3}d^{2}x_{4}\exp(\lambda/2(t_{3}+t_{4}-t_{1}-t_{2}))\\\
&\cdot\langle[\mathcal{O}(\beta/2+it_{2}),\mathcal{O}(\beta/2+it_{4})][\mathcal{O}(it_{1}),\mathcal{O}(it_{3})]\rangle_{0}\\\
&\cdot\frac{G_{lr,\Delta(q-1)+\frac{\lambda}{2}}(3,4)}{G_{lr,\Delta+\frac{\lambda}{2}}(1,2)}+O(J^{4})\;,\end{split}$
(10)

where $\langle\cdot\rangle_{0}$ denotes an expectation value at $J=0$ (i.e. of
the core CFT). The integration range of the points $3,4$ is over the past
light-cone of the points $1,2$ respectively, and $G_{lr,\Delta}$ is the
analytically continued cylinder $2$-point function:

$G_{lr,\Delta}(1,2)=\frac{1}{\left(4\cosh\left(\frac{t_{12}-x_{12}}{2}\right)\cosh\left(\frac{t_{12}+x_{12}}{2}\right)\right)^{\Delta}}.$
(11)

The expansion in $J$ can also be used to determine whether the transition into
chaos would be continuous or discontinuous (see figure 1). This requires
finding the sign of $\lambda_{L}^{\text{ker}}$ in the limit $J=0^{+}$. Using
the leading contribution to $k_{R}$ in orders of $J^{2}$ (10), it is easy to
see that the exponent $\lambda_{L}^{\text{ker}}(J=0^{+})$ is given by the
maximal $\lambda$ for which the integral (10) diverges. If the integral
diverges at a positive (negative) value of $\lambda$, we have a continuous
(discontinuous) transition into chaos.

At $J=0$, the kernel vanishes, and so it is not clear that $\lambda_{L}(J=0)$
is related to chaos. Instead, at $J=0$ we find $N$ decoupled core CFTs, and we
expect

$\begin{split}\lim_{t_{1},t_{2}\rightarrow\infty}W_{R}(t_{1},t_{2})\mid_{J=0}\;\sim\frac{1}{N}\exp(+\lambda_{L}^{0}(t_{1}+t_{2})/2)\end{split}$
(12)

for some $\lambda_{L}^{0}$. Note that the CFTs are decoupled and so
$\lambda_{L}^{0}$ is a property of a single core CFT. However, we emphasize
that it is not a chaos exponent in a single core CFT (as we take $t_{1},t_{2}$
to be larger then any timescale of the core CFT); in fact, for unitary
theories in $2d$ it is always non-positive, $\lambda_{L}^{0}\leq 0$ Caron-Huot
(2017).

Surprisingly, under reasonable physical assumptions it can be shown that
$\lambda_{L}^{\text{ker}}(J=0^{+})$ and $\lambda_{L}^{0}$ are equal Berkooz
_et al._ (2021). This amounts to showing that the integral (10) diverges for
$\lambda\leq\lambda_{L}^{0}$ due to the large (negative) $t_{3},t_{4}$ regime
of the integrand. Putting these pieces together, we claim that
$\lambda_{L}^{\text{ker}}(J)$ satisfies

$\lambda_{L}^{\text{ker}}(J=0^{+})=\lambda_{L}^{0}.$ (13)

The LHS is calculated through the kernel equation $k_{R}(\lambda)=1$ in the
limit $J=0^{+}$, and the RHS is a property of the core CFT. We can interpret
the result by noticing that $\lambda_{L}^{0}$ describes the core CFT double-
commutator behavior at large time scales. At arbitrarily weak coupling this
behavior seems to control the $1\ll t\ll\log N$ behavior of the disordered
theory double-commutator. The result is striking: in order to determine the
type of transition into chaos, it is enough to find (the sign of)
$\lambda_{L}^{0}$ in a single core CFT. Below we give one example for each
type of transition into chaos. In both cases we find that our conjecture (13)
holds.

##  III Examples of the Onset of chaos

###  III.1 Disordered generalized free fields

We now discuss our first class of examples with exactly marginal chaos, which
are the disordered generalized free fields in $1$d and in $2$d. Generalized
free fields (GFFs) are non-local theories with no energy-momentum tensor, but
can be obtained by specific large-flavor limits of local theories, and are
good toy-models for more complicated theories. Conformal invariance allows us
to set the inverse temperature to be $\beta=2\pi$ in the following.

In $1$d, the generalized free fermion model is called the “cSYK” model and was
introduced in Gross and Rosenhaus (2017). Explicitly, the action is

$S=S_{0}+S_{\text{SYK}}$ (14)

with

$\begin{split}S_{0}&=-\Delta\sum_{i=1}^{n}\int
d\tau_{1}d\tau_{2}\chi_{i}\left(\tau_{1}\right)\frac{\operatorname{sgn}\left(\tau_{1}-\tau_{2}\right)}{\left|\tau_{1}-\tau_{2}\right|^{2-2\Delta}}\chi_{i}\left(\tau_{2}\right)\;,\\\
S_{\text{SYK}}&=\frac{i^{\frac{q}{2}}}{q!}\sum_{i_{1},\ldots,i_{q}=1}^{n}\int
d\tau J_{i_{1}i_{2}\ldots
i_{q}}\chi_{i_{1}}\chi_{i_{2}}\cdots\chi_{i_{q}}\;.\end{split}$ (15)

Here, $J_{i_{1}...i_{q}}$ are Gaussian random variables with variance $\langle
J_{i_{1}...i_{q}}^{2}\rangle=\frac{J^{2}(q-1)!}{N^{q-1}}$. Choosing
$\Delta=1/q$, the deformation becomes classically marginal, and it is argued
in Gross and Rosenhaus (2017) that it is exactly marginal at leading order in
$1/N$.

The two- and four-point functions of $\chi_{i}$ for this model were found in
Gross and Rosenhaus (2017). It is simple to extend the results also to the
double-commutator. One finds that the corresponding retarded kernel
$K_{R}^{\text{cSYK}}$ has eigenvalues

$k^{\text{cSYK}}_{R}(\lambda)=\left(\frac{\bar{b}(J)}{\bar{b}(J\to\infty)}\right)^{q}\frac{\Gamma(3-2\Delta)\Gamma(2\Delta+\lambda)}{\Gamma(1+2\Delta)\Gamma(2-2\Delta+\lambda)}\;,$
(16)

where $\bar{b}(J)$ solves the equation

$\frac{\bar{b}^{q}}{1-2\bar{b}}=\frac{1}{J^{2}\psi(1-\Delta)\psi(\Delta)},\;\psi(\Delta)\equiv
2\cos(\pi\Delta)\Gamma(1-2\Delta).$ (17)

$\lambda_{L}^{\text{ker}}$ is found by taking the largest solution to
$k^{\text{cSYK}}_{R}(\lambda)=1$. The result for $\Delta=1/4$ appears in
figure 4, and similar results apply for other $\Delta$.

Figure 4: The chaos exponent $\lambda_{L}(J)$ at $\Delta=1/4$ for disordered
GFFs in QM and for SUSY disordered GFFs in $2d$. Dashed lines represent
$\lambda_{L}^{\text{ker}}$ and solid lines represent $\lambda_{L}$.

At any $\Delta$, $\lambda_{L}^{\text{ker}}$ approaches the maximal value
$\lambda_{L}=1$ Maldacena  _et al._ (2016) at large $J$ as in the SYK model.
More relevant to this study, $\lambda_{L}^{\text{ker}}$ approaches $-2\Delta$
at $J=0$ (corresponding to the dashed red line), and so becomes negative at
small $J$ for any $\Delta>0$. As discussed above, we cannot identify
$\lambda_{L}^{\text{ker}}$ with $\lambda_{L}$ when the former is negative, but
we immediately learn that $\lambda_{L}=0$ in this regime. We thus conclude
that there is no chaos at small enough $J$, corresponding to $\lambda_{L}$
denoted by the solid red line. We thus find a discontinuous transition into
chaos, as in figure 1b. We also comment that the long time exponent
$\lambda_{L}^{0}$ of a single core CFT is equal to $\lambda_{L}^{0}=-2\Delta$,
which matches $\lambda_{L}^{\text{ker}}(J=0^{+})$, and so the conjecture (13)
is obeyed.

The same analysis can be done for disordered GFFs in 2d. In this case we
choose to work with $\mathcal{N}=2$ SUSY GFFs, since this results in an exact
conformal manifold even at finite $N$. The model consists of $N$ generalized
free chiral superfields $\Phi_{i}$ of dimension $\Delta=1/q$, coupled via the
superpotential

$W=\sum_{i_{1}\neq...\neq
i_{q}}^{N}J_{i_{1}...i_{q}}\Phi_{i_{1}}...\Phi_{i_{q}}\;.$ (18)

The computation is very similar, and the results appear in figure 4. At large
enough $J$, the chaos exponent for any $\Delta$ approaches the result in the
$2$d SUSY versions of the SYK model Murugan  _et al._ (2017); Bulycheva
(2018). In addition, we again find that $\lambda_{L}^{\text{ker}}$ approaches
$-2\Delta$ as $J\to 0^{+}$, and so for small enough $J$
$\lambda_{L}^{\text{ker}}$ is negative for any $\Delta>0$. As a result, we
again find a discontinuous transition into chaos for any $\Delta>0$. We also
find once again that the conjecture (13) is obeyed.

###  III.2 Disordered minimal models

We now discuss the disordered 2d $\mathcal{N}=2$ minimal models. A single core
CFT in this case consists of an $\mathcal{N}=2$ SUSY $A_{q-1}$ minimal model,
which can be represented by a single chiral superfield $\Phi$ and
superpotential $W=\Phi^{q}$. The full disordered theory is then

$W=\sum_{i=1}^{N}\Phi_{i}^{q}+\sum_{i_{1}\neq....\neq
i_{q}}J_{i_{1}...i_{q}}\tilde{\Phi}_{i_{1}}...\tilde{\Phi}_{i_{q}}\;.$ (19)

We emphasize that we interpret this equation in conformal perturbation theory
in $J$ around $N$ copies of the $A_{q-1}$ minimal model, where $\tilde{\Phi}$
is the chiral operator of dimension $1/q$ which appears in $A_{q-1}$. Since
the CFT at $J=0$ has no continuous non-R global symmetries, every classically
marginal operator is exactly marginal Green  _et al._ (2010); Kol (2002,
2010), and so each deformation $J_{i_{1}...i_{q}}$ for $i_{1}\neq...\neq
i_{q}$ is exactly marginal. Thus each realization of the model is conformal,
and as a result averaged correlators of $\tilde{\Phi}_{i}$ will also be
conformal (even at finite $N$).

We can now attempt to compute $\lambda_{L}(J)$ in perturbation theory. This
computation is generically difficult, and so we focus on the case $q=3$, where
a core CFT has central charge $c=1$ and so corresponds to the free compact
boson at a specific radius. We can identify the operators $\tilde{\Phi}_{i}$
in terms of vertex operators of the $c=1$ boson, and as a result we can read
off all $n$-point functions of the $\tilde{\Phi}_{i}$. Using this result to
compute the leading contribution to the retarded kernel at small $J$ given in
(10), we find that at small $J$ the exponent $\lambda_{L}^{\text{ker}}$
approaches zero. As a result, we expect a continuous transition into chaos in
this model, as in figure 1a. In particular, the value of $\lambda_{L}^{0}$ in
a single $A_{2}$ minimal model is also zero, so that the conjecture (13) is
again obeyed.

###### Acknowledgements.

We would like to thank O. Aharony, N. Brukner, C. Choi, Y. Jia, R.R. Kalloor,
B. Lian, J. Maldacena, O. Mamroud, M. Mezei, M. Rangamani, V. Rosenhaus, D.
Tong and M. Watanabe for useful conversations. This work was partly funded by
an Israel Science Foundation center for excellence grant (grant number
2289/18), 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. MB is the incumbent of the Charles and David Wolfson Professorial
Chair of Theoretical Physics.

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