# Triple Points of Gravitational AdS Solitons and Black Holes

Constanza Quijada  Departamento de Física, Universidad de Concepción, Casilla,
160-C, Concepción, Chile  Andrés Anabalón  Departamento de Física, Universidad
de Concepción, Casilla, 160-C, Concepción, Chile  Instituto de Física Teórica,
UNESP-Universidade Estadual Paulista, R. Dr. Bento T. Ferraz 271, Bl. II, Sao
Paulo 01140-070, SP, Brazil.  Robert B. Mann  Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Julio
Oliva  Departamento de Física, Universidad de Concepción, Casilla, 160-C,
Concepción, Chile

###### Abstract

We present a triple point of a new kind for General Relativity, at which two
gravitational solitons can coexist with a planar black hole in anti de Sitter
space. Working in the context of non-linear electrodynamics, we obtain simple,
sensible spacetimes for which the thermodynamics can be studied in an analytic
manner. The spacetimes are charged under the non-linear electrodynamics
leading to an electric charge for black holes and a magnetic flux for
solitons. In the grand-canonical ensemble, we show that the phase space of the
theory is very rich, containing re-entrant phase transitions, as well as
triple points, for small values of the coupling controlling the non-linearity
of the electrodynamics Lagrangian.

##  I Introduction

Black holes in Anti-de Sitter (AdS) spacetime have thermal properties that may
be structurally different than those of their asymptotically flat
counterparts. Even in vacuum, the former may have positive specific heat,
leading to locally stable thermal objects, suitable for exploring phase
transitions. The Hawking-Page (HP) transition emerges in this manner, since
the difference in free energy between large black holes and thermal AdS
changes its sign at a given critical temperature $T_{c}$, above which the
black hole dominates, while for $T<T_{c}$ thermal AdS is the preferred phase
Hawking:1982dh . This phase transition is particularly relevant in holography
Maldacena:1997re , since it leads to a confinement/deconfinement transition in
the dual field theory Witten:1998zw . When gauge fields are introduced, it is
natural to consider charged solutions. In this context, in Chamblin:1999tk
the authors explored the structure of the phase space in both the canonical
and grand-canonical ensembles that accommodate charged, spherically symmetric
black holes, and showed that the effect of the electric potential on the phase
space structure is crucial. For large electric potential, there is a single
black hole that always dominates the canonical ensemble. If thermodynamic
pressure is taken into account, then black holes of fixed charge can undergo
Van der Waals phase transitions Kubiznak:2012wp .

Furthermore, General Relativity with a negative cosmological constant
naturally contains black holes with non-spherical horizons Mann:1997iz  that
can be planar or hyperbolic, leading to non-trivial topologies that propagate
up to the conformal boundary. Consequently these black holes are
asymptotically locally AdS. In holography, planar black holes are of special
relevance since they are dual to Conformal Field Theories (CFTs) at finite
temperature formulated in the flat conformal boundary
$\mathbb{R}_{t}\times\mathbb{R}^{d-1}$. In Horowitz:1998ha , the ground state
of the space of planar black holes was identified as a gravitational soliton,
dubbed the AdS soliton. The latter can be obtained from a double analytic
continuation of the planar black hole, and it therefore defines a new
configuration that competes as a possible dominant thermal phase. Phase
transitions between a planar black hole and the AdS soliton at finite
temperature were originally explored in Surya:2001vj , a study that was
extended for charged black holes in Banerjee:2007by , still within the context
of the neutral soliton, which is consistent in the grand-canonical ensemble.

Recently a new charged AdS-soliton configuration was discovered
Anabalon:2021tua . This solution of the Einstein-Maxwell system with a
negative cosmological constant can be obtained from a double analytic
continuation of the planar, electrically charged AdS black hole, which also
requires an analytic continuation of the electric charge, leading indeed to a
smooth, charged soliton. Even more, when this model is embedded in gauge
supergravity, both in dimensions four and five, the soliton turns out to be
supersymmetric111These studies were extended to $\mathcal{N}=4$, $SU(2)\times
SU(2)$ supergravity Freedman:1978ra , in Canfora:2021nca . for a particular
value of the charge Anabalon:2022aig . The magnetic charge of the soliton
leads to a magnetic flux, which enters as a control variable in a new
ensemble, extending the results of Banerjee:2007by , where only neutral
solitons were considered. It turns out that comparing the corresponding free
energy of different configurations at equal temperature, electric potential,
magnetic flux and period of a spatial cycle, leads to an interesting structure
of the phase space, since for large electric potential the black hole phase
dominates Anabalon:2022ksf .

It is well-known that for large electric fields, Maxwell’s equations receive
non-linear corrections coming from quantum effects induced by fermion loops,
as well as from the low energy limit of String Theory. Furthermore, non-linear
electrodynamics (NLED) naturally emerges inside materials with non-linear
constitutive relations. NLED was also employed to construct the first regular
black holes of Einstein’s theory coupled to a field theory Ayon-Beato:1998hmi
. With this motivation, here we commence an exploration of the effects of NLED
on the thermal stability of charged solitons.

There are different modifications of Maxwell theory, which are well-motivated,
depending on the particular setup one is considering Sorokin:2021tge . We
consider the most general form of NLED introduced by Gao Gao:2021kvr  that
leads to a simple correction of the gravitational and gauge potentials,
allowing analytic control in most of the study. Recently, the phase space of a
family of NLED Gao:2021kvr , was shown to posses multiple points at which four
and even five different black hole branches with the same free energy
Tavakoli:2022kmo  can coexist. Remarkably, we find that in this simple setup a
new sort of triple point can emerge, at which two solitons and one black hole
branch can coexist.

##  II Black holes and Solitons in NLED

We consider the framework defined by the theory Gao:2021kvr

$S=\frac{1}{2\kappa}\int_{\mathcal{M}}d^{4}x\ \sqrt{-g}\left(R-2\Lambda-
L_{\mathrm{EM}}\right)\ ,$ (1)

where the electromagnetic Lagrangian is given by Gao:2021kvr

$L_{\mathrm{EM}}=\sum_{i=1}^{\infty}\alpha_{i}\left(F^{2}\right)^{i}\ ,$ (2)

with $F^{2}=F_{\mu\nu}F^{\mu\nu}$ and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The $\alpha_{i}$ are
dimensionful coupling constants ($[\alpha_{i}]=\text{Length}^{2(i-1)}$), and
$A_{\mu}$ is the $U(1)$ Maxwell field.

By varying the action with respect to the metric and the gauge fields, the
following equations are obtained

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}$
$\displaystyle=$ $\displaystyle T_{\mu\nu}\ ,$ (3)
$\displaystyle\nabla_{\mu}\left(\frac{\partial L_{\mathrm{EM}}}{\partial
F^{2}}F^{\mu\nu}\right)$ $\displaystyle=$ $\displaystyle 0\ ,$ (4)

with

$T_{\mu\nu}=-\frac{1}{2}g_{\mu\nu}L_{\mathrm{EM}}+2\frac{\partial
L_{\mathrm{EM}}}{\partial F^{2}}F_{\mu\alpha}F_{\nu}^{\ \alpha}\ .$ (5)

Charged, static black hole spacetimes with planar horizons belong to the
following family of metrics and gauge potentials

$\displaystyle ds_{b}^{2}$ $\displaystyle=$
$\displaystyle-U(r)dt_{b}^{2}+\frac{dr^{2}}{U(r)}+\frac{r^{2}}{\ell^{2}}(d\varphi_{b}^{2}+dz^{2})\
,$ (6) $\displaystyle A_{b}$ $\displaystyle=$ $\displaystyle
A_{t}(r)dt_{b}+A_{\varphi}d\varphi_{b}\ ,$ (7)

where we have chosen to identify $\varphi_{b}$ via
$0\leq\varphi_{b}\leq\eta_{b}$, while $0<z<L$ and $A_{\varphi}$ is an
arbitrary constant.

Using this ansatz, the field equations (3) and (4) yield the relations

$\displaystyle
rU^{\prime\prime}+2U^{\prime}-\frac{6r}{\ell^{2}}+r\sum_{i=1}^{\infty}\alpha_{i}(-2)^{i}(A_{t}^{\prime})^{2i}$
$\displaystyle=$ $\displaystyle 0\ ,$ (8)
$\displaystyle\frac{r^{2}}{2}\sum_{i=1}^{\infty}i\alpha_{i}(-2)^{i}(A_{t}^{\prime})^{2i-1}-Q$
$\displaystyle=$ $\displaystyle 0\ ,$ (9)

where the prime denotes derivative with respect to the radial coordinate $r$
and $Q$ is an integration constant corresponding to the electric charge of the
black hole.

To solve this system of equations we employ the expansions

$A_{t}(r)=\sum_{i=1}^{\infty}b_{i}r^{-i}\quad\mathrm{and}\quad
U(r)=\frac{r^{2}}{\ell^{2}}+\sum_{i=1}^{\infty}c_{i}r^{-i}\ ,$ (10)

where $b_{i}$ and $c_{i}$ are constants. Inserting these expressions in (8)
and (9), and setting $\alpha_{1}=1$ (thereby recovering Maxwell theory when
$\alpha_{i\geq 2}=0$) yields

$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle Q\ ,$ (11)
$\displaystyle b_{5}$ $\displaystyle=$
$\displaystyle\frac{4}{5}Q^{3}\alpha_{2}\ ,$ (12) $\displaystyle b_{9}$
$\displaystyle=$
$\displaystyle\frac{4}{3}Q^{5}\left(4\alpha_{2}^{2}-\alpha_{3}\right)\ ,$ (13)
$\displaystyle b_{13}$ $\displaystyle=$
$\displaystyle\frac{32}{13}Q^{7}\left(24\alpha_{2}^{3}-12\alpha_{3}\alpha_{2}+\alpha_{4}\right)\
,$ (14) $\displaystyle\dots$ $\displaystyle=$ $\displaystyle\dots\ ,$ (15)

and

$c_{1}=-2M\quad c_{i}=\frac{4Q}{i+2}b_{i-1},\quad\mathrm{for}\quad i>1\ ,$
(16)

where $M$ corresponds to the mass of the black hole. All the non-listed
$b_{i}$ vanish.

The key point identified in Gao:2021kvr  is that there are infinitely many
choices of couplings $\alpha_{i}$ that yield a finite number of non-vanishing
$b_{i}$, and consequently $c_{i}$, and vice-versa. In such cases the simple
structure of the gauge potential and metric functions allows exploration of
the phase space in an analytic manner.

From (1), it is possible to obtain black holes solutions with 2, 3 or more
event horizons, setting the $b_{i}$ constants to zero for sufficiently large
$i$, consequently leading to relations between the $\alpha_{i}$. In the case
of the 2-horizon black hole, the values of $b_{i}$ are set to zero for $i\geq
5$, implying $\alpha_{i}=0$ for $i\geq 2$; thus we recover Einstein-Maxwell
theory with cosmological constant, and the black hole solution reduces to the
Reissner-Norstrom-AdS geometry.

The general form for a black hole solution is

$A_{t}=\sum_{i=1}^{\infty}\left(\frac{Q\tilde{b}_{i}}{r^{i}}-\frac{Q\tilde{b}_{i}}{r_{+}^{i}}\right)\quad
U=\frac{r^{2}}{\ell^{2}}-\frac{2M}{r}+\sum_{i=2}^{\infty}\frac{4Q^{2}\tilde{b}_{i-1}}{(i+2)r^{i}}$
(17)

assuming $U(r)$ has a root $r=r_{+}$, where ${b}_{i}=Q\tilde{b}_{i}$. The
Euclidean on-shell action yields a particularly simple expression for the free
energy (see section IV), in spite of the presence of an infinite series,
yielding

$\displaystyle M$
$\displaystyle=\frac{r_{+}^{3}}{2\ell^{2}}+\sum_{i=2}^{\infty}\frac{2Q^{2}\;\tilde{b}_{i-1}}{(i+2)r_{+}^{i-1}}$
(18) $\displaystyle T$
$\displaystyle=\frac{3r_{+}}{4\pi\ell^{2}}-\sum_{i=2}^{\infty}\frac{Q^{2}\;\tilde{b}_{i-1}(i-1)}{\pi(i+2)r_{+}^{i+1}}$
(19) $\displaystyle\phi_{b}$
$\displaystyle=\sum_{i=1}^{\infty}\frac{Q\;\tilde{b}_{i}}{r_{+}^{i}}$ (20)

for the mass, temperature, and electric potential of the black hole.

To obtain the general soliton solution, we apply a double Wick rotation to the
planar black hole metric (6) ($t_{b}\rightarrow i\varphi_{s}$ and
$\varphi_{b}\rightarrow it_{s}$) and an analytic continuation of the black
hole charge ($Q\rightarrow-iq$). This gives

$\displaystyle ds^{2}_{s}$ $\displaystyle=$
$\displaystyle-\frac{r^{2}}{\ell^{2}}dt_{s}^{2}+\frac{dr^{2}}{U(r)}+U(r)d\varphi_{s}^{2}+\frac{r^{2}}{\ell^{2}}dz^{2}$
(21) $\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle
A_{t}dt_{s}+A_{\varphi}(r)d\varphi_{s}$ (22)

where $0<z<L$, $0\leq\varphi_{s}\leq\eta_{s}$, identified, and

$A_{\varphi}=\sum_{i=1}^{\infty}\left(\frac{q\hat{b}_{i}}{r^{i}}-\frac{q\hat{b}_{i}}{r_{0}^{i}}\right)\quad
U=\frac{r^{2}}{\ell^{2}}-\frac{2\mu}{r}-\sum_{i=2}^{\infty}\frac{4q^{2}\hat{b}_{i-1}}{(i+2)r^{i}}$
(23)

where $\hat{b}_{i}(q)=\tilde{b}_{i}(iQ)$, and

$\displaystyle\mu$
$\displaystyle=\frac{r_{0}^{3}}{2\ell^{2}}-\sum_{i=2}^{\infty}\frac{2q^{2}\;\hat{b}_{i-1}}{(i+2)r_{0}^{i-1}}$
(24) $\displaystyle\Phi_{b}$
$\displaystyle=\sum_{i=1}^{\infty}\frac{Q\;\hat{b}_{i}}{r_{0}^{i}}\eta_{s}$
(25)

are the respective mass and magnetic flux of the soliton. The period
$\eta_{s}$ of the $\varphi_{s}$ coordinate of the soliton is fixed in terms of
$r_{0}$ and $q$ as

$\eta_{s}=\left[\frac{3r_{0}}{4\pi\ell^{2}}+\sum_{i=2}^{\infty}\frac{q^{2}\;\hat{b}_{i-1}(i-1)}{\pi(i+2)r_{0}^{i+1}}\right]^{-1}$
(26)

in order to avoid a potential conical singularity that may appear at the
origin of the soliton, located at $r=r_{0}$.

##  III Free energy and phase transitions

The on-shell Euclidean action receives three contributions

$I_{\textrm{Euc}}=-I_{\mathrm{bulk}}-I_{\mathrm{GH}}+I_{\mathrm{ct}}$ (27)

where

$\displaystyle I_{\mathrm{bulk}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa}\int_{\mathcal{M}}d^{4}x\sqrt{g}\left(R-2\Lambda-\sum_{i=1}^{\infty}\alpha_{i}\left(F^{2}\right)^{i}\right)\
,$ (28) $\displaystyle I_{\mathrm{GH}}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int_{\partial\mathcal{M}}d^{3}x\sqrt{h}\ K\ ,$
(29) $\displaystyle I_{\mathrm{ct}}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int_{\partial\mathcal{M}}d^{3}x\sqrt{h}\left(\frac{2}{\ell}-\frac{\ell}{2}\mathcal{R}(h)\right)\
.$ (30)

No further boundary term is required to compare the free energy at fixed
temperature, electric potential, spacelike period and magnetic flux. From (27)
we obtain

$\displaystyle G_{b}$ $\displaystyle=$
$\displaystyle-\frac{L\eta_{b}}{\kappa\ell^{2}}M$ (31) $\displaystyle G_{s}$
$\displaystyle=$ $\displaystyle-\frac{L\eta_{s}}{\kappa\ell^{2}}\mu$ (32)

for the free energy of the black hole and the soliton respectively.

(a) $\alpha_{2}=0.0$

(b) $\alpha_{2}=0.001$

Figure 1: Reduced free energy $G\ell^{2}/(\eta L)$ of the soliton branches as
function of the magnetic flux $\Phi$. We set the spacelike period $\eta=0.5$.
The green line is a new stable soliton branch.

To study phase transitions between the solutions, we need to match the
asymptotic geometries, implying

$\displaystyle\sqrt{g_{\tau_{b}\tau_{b}}(\rho\rightarrow\infty)}\beta_{b}$
$\displaystyle=$
$\displaystyle\sqrt{g_{\tau_{s}\tau_{s}}(\rho\rightarrow\infty)}\beta_{s}$
(33)
$\displaystyle\sqrt{g_{\varphi_{s}\varphi_{s}}(\rho\rightarrow\infty)}\eta_{s}$
$\displaystyle=$
$\displaystyle\sqrt{g_{\varphi_{b}\varphi_{b}}(\rho\rightarrow\infty)}\eta_{b}$
(34)

where $\rho$ is the cut-off in the radial direction. Since for each
configuration, the on-shell, regularized Euclidean actions provide a finite
result, we can directly take the leading term in the expansion of the cutoff,
leading to $\beta_{b}=\beta_{s}=\beta$ and $\eta_{s}=\eta_{b}=\eta$. We also
have to identify the electric potential of the black hole $\phi_{b}$ with that
of the soliton, and the magnetic flux of the soliton $\Phi_{s}$ with that of
the black hole.

In general phase transitions will occur between the planar black hole (17) and
the soliton (23) for a broad range of parameters and couplings. To illustrate
this, we consider the 3-horizon black hole, for which $b_{i\geq 5}=0$ implying

$\alpha_{i}=\frac{2^{i-1}(3i-3)!}{i!(2i-1)!}\alpha_{2}^{i-1}\quad\textrm{for
$i\geq 3$}$ (35)

In this case the theory is characterized by a single NLED coupling
$\alpha_{2}$, with the remaining infinite list of couplings given by (35). For
spherical black holes this choice of couplings also leads to very simple exact
solutions Gao:2021kvr , keeping analytic control over most of the thermal
properties. Once the solution is constructed, one could apply the methods of
Ayon-Beato:1999qin ; Ayon-Beato:2004ywd  to obtain an NLED that accommodates
such solutions.

Let us first analyze the many soliton branches that may appear for different
values of the magnetic flux. The upper panel of figure 1 reproduces the result
of Anabalon:2022ksf  in Maxwell electrodynamics for $\alpha_{2}=0$; the lower
panel shows that for small values of $\alpha_{2}$, a new locally stable
soliton branch emerges for a range of magnetic flux. The dotted segments
denote configurations for which the generalized specific heat is negative (the
free energy is not a convex function of the magnetic flux).

As $\alpha_{2}$ increases the swallow-tail shrinks and then disappears. A new
structure appears with negative free energy when $\alpha_{2}\sim 0.625$, as
shown in figure 2. We clearly see from the inset that when $\alpha_{2}=0.624$
a phase transition between solitons occurs, since for a given value of the the
magnetic flux, the dominant soliton phase switches from small to large. Again,
dotted segments represent configurations with non-convex free energy (negative
specific heat).

(a) $\alpha_{2}=0.6$

(b) $\alpha_{2}=0.624$

Figure 2: Reduced free energy $G\ell^{2}/(\eta L)$ of the small (blue) and
large (green) soliton branches as function of the magnetic flux $\Phi$ for
larger values of $\alpha_{2}$. We set the spacelike period $\eta=0.5$.

The new triple Point and a reentrant phase transition: Now that we have
explored the soliton phase space, it is instructive to take the black hole
phases into account. Figure 3 shows that for large values of the electric
potential $\phi$, there is a first order phase transition between the small
soliton and the black hole, whereas for small values of $\phi$ this transition
is between the black hole and the large soliton. Significantly, there is a
precise value of $\phi$ and the magnetic flux $\Phi$, at which the free energy
of the black hole (purple horizontal line), coincides with that of the two
locally stable solitons (blue and green lines). At such a point in phase space
the three configurations coexist, leading to a new type of triple point.

Figure 3: $G\ell^{2}/(\eta L)$ for solitons and black hole solutions as
function of the magnetic flux $\Phi$, for different, fixed values of the
electric potential $\phi$. Here $\alpha_{2}=0.624$, $T=0.6$ and $\eta=0.5$.

Notice also that there is a reentrant phase transition. For example at
$\phi=6.2$, the small soliton (blue curve) dominates for small $\Phi$; as
$\Phi$ increases there is a first order transition to the black hole. Further
increasing $\Phi$, there is another first order transition to the large
soliton (green curve). Finally, above a second critical magnetic flux, the
same black hole branch returns to being the dominant phase. The free energy of
the black hole is insensitive to variations of the magnetic flux, while the
free energy of the solitons is not sensitive to the electric potential. This
can be see from plots of the surfaces $G(\phi,\Phi)$, shown in Figure 4.

Figure 4: Reduced free energy $G\ell^{2}/(\eta L)$ for solitons (blue, orange
and green surfaces) and black hole (yellow surface) as function of the
magnetic flux $\Phi$ and the electric potential $\phi$. We set the nonlinear
coupling $\alpha_{2}=0.624$, the temperature $T=0.6$ and the period of the
spacelike cycle $\eta=0.5$.

A more thorough exploration of the parameter space for this system will be
provided elsewhere.

##  IV Discussion

We have shown the existence of a new type of triple point in General
Relativity in AdS, supported by NLED. This point in phase space allows for the
coexistence of two solitons and a black hole, at a given value of the
temperature, magnetic flux, spacelike period and electric potential. Even
though the NLED Lagrangian contains arbitrarily high powers of the Lorentz
invariant $F^{2}$, both the electric and gravitational potential of the planar
black hole are at most linear in the unique dimensionful coupling defining the
NLED, as already observed for spherically symmetric black holes by Gao
Gao:2021kvr .

We emphasize that the behaviour we have observed is generic for Einstein
gravity coupled to any NLED and not simply the 3-horizon case considered here.
In general multi-horizon black holes can be expected to exhibit multicritical
soliton black hole phase behaviour, contingent on the number of horizons and
the choice of parameters. This is because an increased number of horizons has
an increased number of NLED couplings, yielding a sufficient number of
thermodynamic variables to support multiple phases Wu:2022bdk .

Given the simple structure of the black holes in this theory, it would be
natural to explore the existence of non-stationary configurations supported by
this NLED, as recently done in Barrientos:2022bzm  for the case of MODMAX
theory Bandos:2020jsw , for accelerating black holes. We also expect that
multicritical points, similar to those observed for NLED black holes
Tavakoli:2022kmo , are also present for NLED solitons, and that perhaps
extensions to more complicated solitons Andrews:2019hvq  are possible. We
expect to return to these topics in the near future, as well a to provide a
more exhaustive exploration of the parameter space of the solitons and black
holes.

##  V Acknowledgments

We thank Patrick Concha, Octavio Fierro and Evelyn Rodriguez for enlightening
comments. This research has been supported in part by the Natural Sciences and
Engineering Research Council of Canada, and by FONDECYT grants 1221504,
1210635 and 1210500. This work was partially supported by ANID Fellowships
21191868 (C.Q.). The research of AA is supported in part by a visiting
researcher award of the FAPESP 2022/11765-7.

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# Analytic infinite derivative gravity, $R^{2}$-like inflation, quantum
gravity and CMB

Alexey S. Koshelev<EMAIL_ADDRESS>Departamento de Física, Centro de
Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, 6200
Covilhã, Portugal  K. Sravan Kumar
sravan.korumilli<EMAIL_ADDRESS>Van Swinderen Institute,
University of Groningen, 9747 AG, Groningen, The Netherlands  Alexei A.
Starobinsky<EMAIL_ADDRESS>L. D. Landau Institute for Theoretical
Physics RAS, Moscow 119334, Russian Federation

###### ?abstractname?

Emergence of $R^{2}$ inflation which is the best fit framework for CMB
observations till date comes from the attempts to attack the problem of
quantization of gravity which in turn have resulted in the trace anomaly
discovery. Further developments in trace anomaly and different frameworks
aiming to construct quantum gravity indicate an inevitability of non-locality
in fundamental physics at small time and length scales. A natural question
would be to employ the $R^{2}$ inflation as a probe for signatures of non-
locality in the early Universe physics. Recent advances of embedding $R^{2}$
inflation in a string theory inspired non-local gravity modification provides
very promising theoretical predictions connecting the non-local physics in the
early Universe and the forthcoming CMB observations.

This essay received an Honorable Mention in the 2020 Essay Competition of the
Gravity Research Foundation.

## Introducing the gravity modification

According to recent studies Krasnikov:1987yj ; Kuzmin:1989sp ;
Tomboulis:1997gg ; Biswas:2005qr ; Modesto:2011kw ; Biswas:2011ar ;
Biswas:2016egy ; Koshelev:2016xqb ; Koshelev:2017tvv ; Koshelev:2020foq  the
following higher derivative quadratic in curvatures gravity action captures
the long standing quests to build a much needed consistent theoretical and
phenomenological path towards quantum gravity:

$S=\int
d^{4}x\sqrt{-g}\left(\frac{M_{p}^{2}}{2}R+\frac{1}{2}\bigg{[}R\mathcal{F}_{R}\left(\square_{s}\right)R+W_{\mu\nu\rho\sigma}\mathcal{F}_{W}\left(\square_{s}\right)W^{\mu\nu\rho\sigma}\bigg{]}\right)\,.$
(1)

Here the first term with $M_{p}$ being the reduced Planck mass is the
canonical Einstein-Hilbert General Realtivity (GR) Lagrangian, $R$ as usual is
the Ricci scalar, while the part in brackets is the higher derivative
modification of GR. The crucial ingredient here is higher derivative
formfactors $\mathcal{F}$ which further turn out to be analytic non-
polynomial, i.e. essentially non-local, functions of the covariant
d’Alembertian $\square$. Hereafter we use
$\square_{s}={\square}/{\mathcal{M}_{s}^{2}}$ with $\mathcal{M}_{s}$ being the
scale of non-locality and $W_{\mu\nu\rho\sigma}$ denotes the Weyl tensor.

The above gravity action can be motivated and argued from different
perspectives. The most recent routine construction Biswas:2016egy ;
Koshelev:2017tvv  shows that such an action with analytic infinite derivative
(AID) form-factors appears as the most general action describing linear
fluctuations around maximally symmetric space-times. Analyticity of form-
factors is required by the existence of the low-energy GR limit while presence
of exactly infinite number of derivatives follows from imposing ghost-free
conditions.

The non-locality scale naturally bounded by the Planck mass from above and
also is significantly large and close to the Planck mass as it will follow
later. As such

$\mathcal{M}_{s}\lesssim M_{p}\,.$

Obviously, the non-locality scale can be associated with more fundamental
originating theories like string theory for example.

## From trace anomaly to AID gravity

Looking back we know that GR was extremely successful matching predictions of
several observations over the century including the very recent detection of
gravitational waves Wald:1984rg ; Abbott:2016blz . However, it is well known
from the very beginning that GR requires modifications at small distance and
time scales or at high energies which are assumed the energies to be
comparable with the Planck energy scales. Quantum Field Theory (QFT) would
obviously form an initial guiding path towards understanding rules for
modifying gravity Duff:1993wm . GR is a classical theory of geometry but it
cannot be quantized or made ultraviolet (UV) complete because it is non-
renormalizable. In this regard a different question was asked in the 1970’s
which is can quantum corrections to the geometry arise due to matter fields?
Answer to this question turned out to be one of the most important development
known as trace anomaly which led to significant developments in classical and
quantum gravity Duff:1993wm . The trace anomaly represents 1-loop quantum
corrections to the graviton propagator due to conformal fields leading to an
anomalous non-zero trace of energy momentum tensor which can in short be
written in the following form

$M_{p}^{2}R=g^{\mu\nu}\langle T_{\mu\nu}\rangle=aF+bG+c\square R\,,$ (2)

where one has the Weyl tensor square term
$F=W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$, the Gauss-Bonnet term
$G=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$ with
$R_{\mu\nu\rho\sigma}$ being Riemann tensor, $R_{\mu\nu}$ being Ricci tensor
and $a,b,c$ are dimensionless coefficients that depend on the number of
conformal fields. Trace anomaly has been widely studied and applied in various
fundamental theories including anticipated UV complete theories such as string
theory and supergravity (SUGRA) Duff:1993wm .

A first and very significant application of trace anomaly appeared in
cosmology and led to the discovery of a non-singular solution in the form of
Friedmann-Lemaître-Robertson-Walker (FLRW) space-time describing an epoch of
an exponential (quasi-de Sitter) expansion in the early Universe
Starobinsky:1980te ; Hawking:2000bb , the epoch well known as the cosmic
inflation. The paradigm of inflation does successfully resolves the standard
problems of the big bang cosmology Guth:1980zm ; Linde:1981mu . This
particular inflationary scenario is the Starobinsky inflation model happening
in a regime when the last term in (2) dominates over the rest and this effect
can be equivalently seen with the following effective action which we shortly
call as (local) $R^{2}$ gravity

$S_{R^{2}}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R+\frac{M_{p}^{2}}{12M^{2}}R^{2}\right]\,.$
(3)

where $M\ll M_{p}$ becomes the mass of a propagating scalar in this model,
named scalaron. During inflationary regime the scalaron degree of freedom
coincides with the Bardeen potential $\Psi$ Bardeen:1980kt  which parametrizes
metric fluctuations and whose two point correlation function gives the power
spectrum of curvature perturbations
$\mathcal{R}\approx-\frac{1}{\epsilon}\Psi$ which explains the density or
temperature fluctuations observed in the cosmic microwave background (CMB).
Here $\epsilon=-\frac{\dot{H}}{H^{2}}\approx\frac{M^{2}}{6H^{2}}\ll 1$ is the
so-called slow-roll parameter, $H,\,\dot{H}$ are the Hubble parameter and its
derivative with respect to the cosmic time respectively.

The only free parameter $M$ in this model is fixed to $M\approx 1.3\times
10^{-5}M_{p}$ by the observed Fourier power spectrum
$\mathcal{P}_{\mathcal{R}}=2.2\times 10^{-9}$ of primordial scalar
perturbations (or matter density perturbations) in the Universe corresponding
to $N=55$ number of $e$-foldings during inflation Akrami:2018odb . The key
predictions of $R^{2}$ inflation are the spectral index $n_{s}$ which is the
tilt of the power spectrum $P_{\mathcal{R}}$, the ratio of the tensor to the
scalar power spectra $r$ and the tensor tilt $n_{t}$ at the Hubble radius
exit. In the local $R^{2}$ model these parameters become:

$n_{s}=1-\frac{2}{N}\,,\quad r=\frac{12}{N^{2}}\,,\quad r=-8n_{t}\,.$ (4)

The current observational bounds read Akrami:2018odb  $n_{s}=0.9649\pm
0.0042\,\,\textrm{at}\,\,68\%\textrm{CL},r<0.064\,\,\textrm{at}\,\,95\%\,\textrm{CL}$
making $R^{2}$ inflation incredibly well consistent with the present CMB
measurements by PLANCK mission and the more recent BICEP2/Keck Array
experiments Akrami:2018odb . The last relation in (4) implies the negative
tensor tilt $n_{t}<0$ and this has been regarded as a definitive test
especially for $R^{2}$ inflation.

From the theoretical perspective $R^{2}$ gravity is proven to be
renormalizable by the addition of the Weyl tensor squared term, the idea
proposed and studied by Stelle Stelle:1976gc ; Stelle:1977ry  which makes the
theory UV complete in the scalar sector but the unitarity gets spoiled by
appearance of a spin-2 ghost. Despite there have been several attempts in the
last decades of realizing $R^{2}$ gravity within more fundamental UV complete
theories such as string theory and supergravity Linde:2014nna ;
Kehagias:2013mya , there are interesting advances beyond Stelle’s theory
having certain connections with a manifestly UV complete setup of string field
theory Witten:1985cc ; Witten:1986qs ; Zwiebach:1985uq ; Arefeva:2004odl  and
this is our main focus in the sequel.

Deeper developments of trace anomaly computations resulted in a non-local
effective theory of gravity Barvinsky:2015 ; Teixeira:2020kew . Such a non-
local gravity contains quadratic in curvatures terms with infinitely many
derivatives in a form of non-analytic functions such as
$\ln\frac{\square}{\mu^{2}}$ Barvinsky:2015  where $\mu$ is the theory cut-off
scale. These non-local gravitational theories could lead to ghost degrees of
freedom and moreover do not have a smooth low energy limit towards GR making
themselves not interesting as UV completion candidates of local $R^{2}$
gravity. Nevertheless, this provides an insight that a non-locality can play a
crucial role on the way to UV completion. Moreover, various approaches towards
quantum gravity including causal sets Belenchia:2014fda , non-commutative
theories Witten:1985cc ; Chu:2005nb , loop quantum gravity Hossenfelder:2007re
; Ashtekar:2007tv  and asymptotic safety Weinberg:2009wa ; Bonanno:2017pkg ;
Liu:2018hno ; Eichhorn:2018yfc ; Bosma:2019aiu  highlight the importance of
gravity being non-local near Planck scale. Since non-locality appears to be a
common feature in several frameworks aiming to construct quantum gravity,
building of a non-local gravity theory with viable inflationary scenario gives
us the best opportunity to look for its signatures in the CMB.111Signatures
for non-local nature of gravity has also been studied recently in the context
of astrophysical and gravitational wave observations Capozziello:2017xla ;
Calmet:2017rxl . In Capozziello:2017xla  gravitational energy-momentum pseudo
tensor is calculated in the context of non-local theories and its importance
with respect to the power carried by gravitational radiation from
astrophysical sources has been discussed. In Calmet:2017rxl , it was pointed
out that one can probe the parameters of the effective action of quantum
gravity with non-analytic form factors using a combination of gravitational
wave observations and pendulum type experiments searching deviations of
Newtonian potential.

In string field theory analytic non-locality does naturally appear because
interactions are intrinsically non-local. In this scenario infinite derivative
operators like $e^{{\square}/{\mathcal{M}_{s}^{2}}}$, with $\mathcal{M}_{s}$
being a string scale $\mathcal{M}_{s}\lesssim M_{p}$ enter the vertex terms of
space-time fields interactions.

Notably the AID gravity action in (1) joins at once several remarkable
properties: it manifestly inherits the non-local operator structure from the
string theory, this in turn targets questions of the UV completion by the very
structure of the action and also this proposal does solve the ghost problem of
Stelle gravity Biswas:2016egy . For all the aspects presence of AID form-
factors is the crux.

##  $R^{2}$-like inflation in AID gravity (1) and future CMB

It was proven that the local $R^{2}$ gravity inflationary solution that itself
satisfies $\square R=M^{2}R$ happens to be exact attractor solution of (1)
with very minimal conditions such that
$\mathcal{F}_{R}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=\frac{M_{p}^{2}}{6M^{2}}=\mathcal{F}_{1}$
and $\mathcal{F}_{R}^{(1)}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)=0$
(hereafter (1) means the derivative with respect to the argument). We recall
that $M$ is the scalaron mass. No condition arise on
$\mathcal{F}_{W}\left(\square_{s}\right)$ at the background level since Weyl
tensor is zero in FLRW space time. Study of scalar and tensor degrees of
freedom in quasi-de Sitter regime imply the following general structures of
form-factors

$\mathcal{F}_{R}\left(\square_{s}\right)=\mathcal{F}_{1}\frac{3e^{\gamma_{S}\left(\square_{s}\right)}\left(\square_{s}-\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)+\left(\frac{\bar{R}}{\mathcal{M}_{s}^{2}}+3\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)}{3\square_{s}+\frac{\bar{R}}{\mathcal{M}_{s}^{2}}}\,,\quad\mathcal{F}_{W}\left(\square_{s}\right)=\frac{\mathcal{F}_{1}\bar{R}}{\mathcal{M}_{s}^{2}}\frac{e^{\gamma_{T}\left(\square_{s}-\frac{2}{3}\frac{\bar{R}}{\mathcal{M}_{s}^{2}}\right)}-1}{\square_{s}-\frac{2\bar{R}}{3\mathcal{M}_{s}^{2}}}\,.$
(5)

where $\gamma_{S},\,\gamma_{T}$ are entire functions of the d’Alembertian
operator and $\bar{R}$ is the background curvature of the qusi-de Sitter
phase. The structure of form-factors is dictated by the absence of ghosts
requirement. The following heuristic hierarchy

$M^{2}\ll H_{\text{inf}}^{2}\lesssim\mathcal{M}_{s}^{2}\lesssim M_{p}^{2}\,,$
(6)

where $H_{\text{inf}}^{2}\approx\bar{R}/12$ is the Hubble parameter squared
during inflation is proven to keep the key well estimated observational
predictions of $\mathcal{P}_{\mathcal{R}},\,n_{s}$ to remain the same as in
local $R^{2}$ model Koshelev:2017tvv . Given the great observational precision
in this instances it is crucial to name new finding successful.

The two new predictions of inflation in AID theory (1) Koshelev:2016xqb ;
Koshelev:2017tvv ; Koshelev:2020foq  which can clearly discriminate local and
non-local theory, and many other setups of local $R^{2}$-like inflationary
scenarios Akrami:2018odb  are

$r=\frac{12}{N^{2}}e^{-2\gamma_{T}\left(\frac{-\bar{R}}{2\mathcal{M}_{s}^{2}}\right)}\Bigg{|}_{k=aH},\quad
n_{t}\approx-\frac{3}{2N^{2}}-\left(\frac{2}{N}+\frac{3}{2N^{2}}\right)\frac{\bar{R}}{2\mathcal{M}_{s}^{2}}\gamma_{T}^{(1)}\left(-\frac{\bar{R}}{2\mathcal{M}_{s}^{2}}\right)\Bigg{|}_{k=aH}\,,$
(7)

where $\bar{R}_{k=aH}\approx 4NM^{2}$ is the value of Ricci scalar at the
Hubble radius crossing. There are three important things to be noticed here:
tensor to scalar ratio can take any value depending on the form-factor of Weyl
tensor quadratic term; tensor tilt gets correction due to non-locality in the
model and a deviation from the standard consistency relation $r=-8n_{t}$ is
evident here; a possibility of a negative tensor tilt $n_{t}>0$ is not
excluded. This means that the non-local $R^{2}$-like inflation is manifestly
ready to be tested by future CMB observations Abazajian:2019eic  (see Fig. 1).

,

?figurename? 1: In the left panel, we depict the predictions of non-local
$R^{2}$-like inflation in the $(n_{s},r)$ plane of the latest CMB S-4 science
paper about future forecast of detecting $B$-modes Abazajian:2019eic . In the
right panel, we note that the predictions of non-local $R^{2}$-like inflation
can be anywhere within the likelihood projected $(n_{t},r)$ plane of latest
Planck 2018 Abazajian:2016yjj .

## Beyond two-point correlations

3-point scalar correlations that give non-Gaussian corrections provide another
test for the non-local $R^{2}$-like inflation Maldacena:2002vr ;
Meerburg:2019qqi ; Meerburg:2016zdz . In the Fourier space, the three-point
correlation function of curvature perturbations is defined through parameter
$f_{\text{NL}}$ which is called the reduced bi-spectrum. Physically this
measures the three-point tree-level interactions of curvature perturbation in
quasi-de Sitter regime. There are three triangular configurations which are
observationally measurable to constrain the non-Gaussianities in gravity
modifications. As per the latest CMB data with respect to Planck, BICEP2/Keck
Array and BK15 with TT,TE, EE+lowE+lensing Akrami:2019izv  we have

$f_{\text{NL}}^{sq}=-0.9\pm 5.1\,,\quad f_{\text{NL}}^{eq}=-26\pm 47\,,\quad
f_{\text{NL}}^{ortho}=-38\pm 24\,,\leavevmode\nobreak\
\textrm{at}\leavevmode\nobreak\ 68\%\leavevmode\nobreak\ \textrm{CL}.$ (8)

for the so-called squeezed, equilateral, and orthogonal configurations
respectively. Among these the squeezed limit which tells us about an
interaction between a long wavelength mode that has exited the Hubble radius
already with those modes which are still exiting the Hubble radius is what is
potentially measurable in near future and any detection of it away from
Maldacena’s single canonical field consistency relation would tell us more
about inflationary paradigm and UV completion Maldacena:2002vr ; Chen:2006nt ;
Arkani-Hamed:2015bza . This consistency relation is obeyed by local $R^{2}$
inflationary models and is read

$f_{\text{NL}}^{sq}=\frac{5}{12}\left(1-n_{s}\right)\,.$

In the case of non-local $R^{2}$-like inflation the result for the squeezed
limit of non-Gaussianities turns out to be Koshelev:2020foq

$\displaystyle
f_{NL}^{sq}\approx\frac{5}{12}\left(1-n_{s}\right)-23\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}-1\right]-\frac{4\bar{R}}{\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{(1)}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)\Bigg{|}_{k=aH}\,,$
(9)

which gets correction due to non-local tree level interactions of curvature
perturbation. The fact of getting a possibility of
$f_{\text{NL}}^{sq}\sim\mathcal{O}(1)$ (see Fig. 2) due to non-local effects
puts this non-local model within the scope of upcoming CMB and astrophysical
observations Meerburg:2019qqi ; Meerburg:2016zdz . Further, the other
configurations also get non-local corrections as follow Koshelev:2020foq

$\displaystyle f_{NL}^{eq}$
$\displaystyle\approx\frac{5}{12}\left(1-n_{s}\right)-49\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{S}^{2}}\right)}-1\right]-\frac{9\bar{R}}{\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{(1)}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)\Bigg{|}_{k=aH}\,,$
(10) $\displaystyle f_{NL}^{ortho}$
$\displaystyle\approx\frac{5}{12}\left(1-n_{s}\right)-43\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}-1\right]-\frac{22\bar{R}}{3\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{(1)}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)\Bigg{|}_{k=aH}\,.$

?figurename? 2: In the above plots, $f_{NL}$ versus the scale of non-locality
$\mathcal{M}_{s}$ (in the units of $M_{p}$) is depicted for squeezed (blue),
equilateral (red), and orthogonal (green) configurations for the polynomial
entire functions $\gamma_{S}$ given by eqs. (4.19) and (4.20) of
Koshelev:2020foq  and represented by solid and dashed lines respectively. Here
$N=55$ of $e$-foldings is assumed. In the limit $\mathcal{M}_{s}\to M_{p}$ the
predictions of the local $R^{2}$ model are recovered.

In conclusion, as the immense number of approaches to quantum gravity
indicates a non-local nature of gravity, realizing $R^{2}$-like inflation in a
non-local configuration given by AID gravity action (1) provides a unique hope
to test nature of gravity with CMB observations. Any detection of the
predictions of non-local $R^{2}$-like inflation would imply a giant leap
towards our understanding of quantum gravity.

###### Acknowledgements.

AK is supported by FCT Portugal investigator project IF/01607/2015. This
research work was supported by grants UID/MAT/00212/2019, COST Action CA15117
(CANTATA). KSK is supported by Netherlands Organization for Scientific
Research (NWO) grant number 680-91-119. AAS was partly supported by the
project number 0033-2019-0005 of the Russian Ministry of Science and Higher
Education.

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# Eigenstate Thermalization Hypothesis and Free Probability

Silvia Pappalardi<EMAIL_ADDRESS>Laboratoire de Physique de
l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université de Paris, F-75005 Paris, France  Laura Foini  IPhT, CNRS, CEA,
Université Paris Saclay, 91191 Gif-sur-Yvette, France  Jorge Kurchan
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France

(August 27, 2024)

###### Abstract

Quantum thermalization is well understood via the Eigenstate Thermalization
Hypothesis (ETH). The general form of ETH, describing all the relevant
correlations of matrix elements, may be derived on the basis of a ‘typicality’
argument of invariance with respect to local rotations involving nearby energy
levels. In this work, we uncover the close relation between this perspective
on ETH and Free Probability theory, as applied to a thermal ensemble or an
energy shell. This mathematical framework allows one to express in an
unambiguous way high order connected correlation functions (here identified as
free cumulants) in terms of standard correlation functions. This perspective
naturally incorporates the consistency property that local functions of ETH
operators also satisfy ETH. The present results open a direct connection
between quantum thermalization and the mathematical structure of Free
Probability, thus offering the basis for insightful analogies and new
developments.

Introduction \- The current framework for understanding the emergence of
thermal equilibrium in isolated quantum systems goes under the name of the
_Eigenstate Thermalization Hypothesis_ (ETH). Early works of Berry [1],
Deutsch [2] and Srednicki [3] recognized the importance of understanding the
eigenstates of chaotic systems a pseudo-random vectors that encode
microcanonical ensembles. Inspired by Random Matrix Theory (RMT), ETH was then
fully established by Srednicki in Ref.[4], incorporating some additional
structure required to account for non-trivial temperature or time dependences.
See Ref.[5] for a review. According to ETH, the matrix elements of local
observables $A$ in the energy eigenbasis $H|E_{i}\rangle=E_{i}|E_{i}\rangle$
are pseudo-random numbers, whose statistical properties are smooth
thermodynamic functions. In the original formulation, the average and variance
read

$\overline{A_{ii}}=\mathcal{A}(E_{i})\
,\quad\overline{A_{ij}A_{ji}}=F^{(2)}_{E_{ij}^{+}}(\omega_{ij})e^{-S(E_{ij}^{+})}\quad\text{for}\quad
i\neq j\ ,$ (1)

where $E^{+}_{ij}=(E_{i}+E_{j})/2$, $\omega_{ij}=E_{i}-E_{j}$ and $S(E)$ is
the thermodynamic entropy at energy $E$. While $\mathcal{A}$ represents the
microcanonical expectation value of $A$, $F_{E}^{(2)}(\omega)$ depends
implicitly on the observable $A$ ($|f_{A}(E,\omega)|^{2}$ with the standard
notations [5]) and it is associated to correlations on energy shell. In this
paper, we will refer to them as _on-shell correlations_. The ETH assumptions
(1) allow one to fully describe the local relaxation of observables to thermal
equilibrium as well as to characterize two-time dynamical correlation
functions [6; 5]. Since its formulation, ETH has motivated a considerable body
of numerical [7; 8; 9] and analytical work [9; 10], also in relation to
quantum entanglement [11; 12; 13].

However, despite its successes, there are still important open issues
concerning the foundations of ETH. In fact, while the validity of ETH has been
established for a few body semi-classical problems [14], the substantial
challenge is to demonstrate ETH in the context of many-body systems. Recently,
with the goal of describing thermal correlations of $q$ times, Ref.[15] argued
that it is necessary to relax any assumption of independence and gaussianity
on them, and introduced a general version of ETH which predicts the joint
statistical properties of $A_{ij}$. This was deduced from typicality arguments
[16; 17; 18; 19; 20], as applied to small rotations of nearby energy levels.
Such perspective and counting arguments inspired by RMT lead to the
understanding of the finite contribution of different matrix elements to
multi-time correlation functions directly from a diagrammatic approach. The
existence of matrix elements correlations beyond Eq.(1), recently confirmed
numerically [21; 22], has motivated discussions on the finer structure of the
ETH [23; 24; 25; 26].

In this paper, we identify the intimate relation between the general form of
the ETH and _Free Probability theory_. The latter can be thought of as the
generalization of classical probability to non-commutative random variables,
where the concept of “free independence” or “freeness” extends the one of
“independence”. Introduced by Voiculescu around 1985 in connection to the
theory of operator algebras [27], Free Probability theory turned out to have
important links with several branches of mathematics and physics [28; 29],
such as RMT [30] and combinatorics. In the ETH context addressed here, we are
interested in the combinatorial theory of free probability, which is based on
free cumulants and the concept of non-crossing partitions [31].
Our methodology is to use the properties of the ETH matrix elements and their
diagrammatic description to link it with the mathematical structure of free
probability. We will first recall the derivation of the general form of ETH,
based on invariance with respect to local rotations of nearby energy levels.
By discussing the ETH diagrams relevant to correlation functions, we will show
that they are in one-to-one correspondence to non-crossing partitions. Our
main result is an explicit expression for the thermal free cumulants –
connected correlation functions of order $q$ – in terms of sums of the matrix
elements over non-repeated indices: simple loops in the diagrammatic language,
discussed in [15]. Moreover, thanks to the results of Free Probability we
provide an expression for high order multi-time correlation functions in terms
of free cumulants. This leads to directly linking thermal free cumulants to
the Fourier transform of the ETH on-shell correlation of order $q$. As a
byproduct, these relations allow us to deduce bounds on the behaviour of on-
shell correlations in the frequency domain.

Figure 1: Impact of the local rotational invariance of $A_{ij}$ on the
correlations between three matrix elements. The operator $A$ in the energy
eigenbasis is depicted as a random matrix with a band structure. To each
matrix element $A_{ij}$ is associated with a “small” $U$ (box on the diagonal)
which acts as a pseudo-random unitary matrix. Matrix elements with different
indices (a) are characterized by different $U$ and their average vanishes.
When the indices are repeated on a loop (b) the $U$ appear in pairs and yield
a finite result.

General ETH -  The ETH in its enlarged formulation was discussed in Ref.[15]
to compute correlation functions of order $q$ depending on
$\vec{t}=(t_{1},t_{2},\dots,t_{q-1})$ times. The latter can be written in
terms of the product of $q$ matrix elements. The ETH amounts in the following
ansatz: the average of products with distinct indices $i_{1}\neq i_{2}\hbox
to10.00002pt{.\hss.\hss.}\neq i_{q}$ reads

$\overline{A_{i_{1}i_{2}}A_{i_{1}i_{2}}\hbox
to10.00002pt{.\hss.\hss.}A_{i_{q}i_{1}}}=e^{-(q-1)S(E^{+})}F_{E^{+}}^{(q)}(\omega_{i_{1}i_{2}},\hbox
to10.00002pt{.\hss.\hss.},\omega_{i_{q-1}i_{q}})$ (2)

and with repeated indices it factorizes in the large $N$ limit

$\overline{A_{i_{1}i_{2}}\hbox
to10.00002pt{.\hss.\hss.}A_{i_{l}i_{1}}A_{i_{1}i_{m}}\hbox
to10.00002pt{.\hss.\hss.}A_{i_{q}i_{1}}}=\overline{A_{i_{1}i_{2}}\hbox
to10.00002pt{.\hss.\hss.}A_{i_{l}i_{1}}}\;\overline{A_{i_{1}i_{m}}\hbox
to10.00002pt{.\hss.\hss.}A_{i_{q}i_{1}}}\ .$ (3)

In Eq.(2), $E^{+}=(E_{i_{1}}+\hbox to10.00002pt{.\hss.\hss.}+E_{i_{q}})/q$ is
the average energy,
$\vec{\omega}=(\omega_{i_{1}i_{2}},\dots,\omega_{i_{q-1}i_{q}})$ with
$\omega_{ij}=E_{i}-E_{j}$ are $q-1$ energy differences and
$F_{E^{+}}^{(q)}(\vec{\omega})$ is a smooth function of the energy density
$E^{+}/N$ and $\vec{\omega}$. Thanks to the explicit entropic factor,
$F^{(q)}_{E}(\vec{\omega})$ is of order one and thus contains Eq.(1) as a
particular case for $q=1,2$. We will refer to $F^{(q)}_{E}(\vec{\omega})$ as
the _on-energy shell correlations of order $q$._ This generalization of ETH,
which is necessary if matrix elements are considered to be not independent,
implies that correlation functions at order $q$ contain new information that
is not in principle encoded in lower moments.

The ETH ansatz in Eq.(2) can be derived using typicality arguments. The
central idea is to use local invariance of the $A_{ij}$, stemming from small
rotations that involve only nearby energy levels. The matrix elements are
evaluated by substituting the operator with a “locally” rotated one (see Fig.
1) $A^{u}=UAU^{\dagger}$, i.e.
$A_{ij}=\sum_{\bar{i}\bar{j}}U_{i\bar{i}}A_{\bar{i}\bar{j}}U^{*}_{\bar{j}j}$,
with $U_{i\bar{i}}=\langle E_{i}|E_{\bar{i}}\rangle$ and $|E_{\bar{i}}\rangle$
are the eigenstates of a slightly perturbed Hamiltonian [2]. By looking at a
sufficiently small energy range around $E_{i}$, the $U_{i\bar{i}}$ can be
thought as a pseudorandom unitary matrix. This is in analogy to Berry’s
conjecture, stating that the overlaps of chaotic eigenstates with a generic
basis can be thought as random Gaussian numbers. The size of this matrix has
to be “small” in order to keep intact the energy band structure of $A_{ij}$,
but it contains many level spacings. Hence the matrix elements are treated as
belonging to an ensemble of _local rotational invariances_. By averaging over
$U$, one can immediately deduce the finite contributions to any product of
matrix elements. Averages are non-zero only if the matrices $U$ appear at
least twice. For example, for $\overline{A_{ij}A_{lk}}$ the only finite
contribution comes from $\overline{A_{ij}A_{ji}}$ leading to Eq.(1). In the
same way, finite products of $q$ matrix elements necessarily have to be in a
loop (see Fig.1 for the pictorial example with $q=3$). When the indices are
different, this leads to Eq.(2). This scenario, complemented with some
entropic arguments, results also in the factorization of Eq.(3), see Ref.[15].

As a consequence, we remark that this approach also accounts for the validity
of the ETH ansatz between different operators, e.g.
$\overline{A_{ij}B_{ji}}=F^{(2)}_{E^{+},AB}(\omega)e^{-S(E^{+})}$, where we
make explicit the dependence on the operators. Clearly, the ensemble defined
by $U$ is the same for $A$ and $B$, since they come from changes in $H$ and
not from the operators themselves. Hence, the above argument can be applied to
any set of local observables.

Figure 2: ETH diagrams (a) and non-crossing partitions (b) for $q=4$. (a1-6)
Loop and cactus diagrams that contribute to ETH correlators. The arrow
indicate the presence of a time-dependence. With $\times[n]$ we indicate that
there are $n$ cyclic permutations. (a7) Cactus diagram. (b1-6) Non-crossing
partitions for $q=4$. Each of blocks contributes with a free cumulant $k_{n}$,
where $n$ is the number of $n$ points in that partition. For completeness, we
also represent the crossing partition after the dashed line.

Computing expectations: diagrammatic expansion - The ETH ansatz in Eq.(2)
allows one to compute multi-time thermal correlation functions of the form

$\displaystyle S_{\beta}^{(q)}(\vec{t})=\langle A(t_{1})A(t_{2})\dots
A(t_{q-1})A(0)\rangle_{\beta}$ (4)

where $\langle\bullet\rangle_{\beta}=\text{Tr}(\rho\,\bullet)$ and
$\rho=e^{-\beta H}/Z$ with ${Z=\text{Tr}(e^{-\beta H})}$. Here,
$A(t)=e^{iHt}Ae^{-iHt}$ is the observable at time $t$ in the Heisenberg
representation ($\hbar=1$).

As introduced in Ref.[15], one can determine the contribution of
$\overline{A_{i_{1}i_{2}}\dots A_{i_{q}i_{1}}}$ to the thermal correlation
$S_{\beta}^{(q)}(\vec{t})$ in a _diagrammatic fashion_ , see Fig.2a for $q=4$.
Let us briefly illustrate how to understand pictorially such ETH diagrams.
Products as $A_{i_{1}i_{2}}\dots A_{i_{q}i_{1}}$ are represented on a loop
with $q$ points. The matrix elements $A_{ij}$ live on the oriented edge
connecting two points $i$ and $j$. The arrows keep track of non-trivial time
dependences. The different diagrams correspond to all the different ways one
can contract the indices, i.e. identify two points. Such diagrams are
classified as:

$\bullet$ _loops_ : all distinct vertices lie on a single closed circle (e.g.
$A_{ij}A_{jk}A_{km}A_{mi}$ in Fig.2a1). Each loop with $n$ vertices
contributes with $\propto F^{(n)}e^{-(n-1)S}$;

$\bullet$ _cactus diagrams_ : trees of loops are joined to one another at
single vertex (e.g. $A_{ij}A_{ji}A_{ik}A_{ki}$ in Fig.2a2). Cactus with $p$
leafs contribute with $p$ products of the associated $F$. The example of a
two-leaf cactus in Fig.2a2 contributes with $\propto(F^{(2)}e^{-S})^{2}$.

The _non-cactus_ diagrams (e.g. $A_{ij}A_{ji}A_{ij}A_{ji}$ in Fig.2a7) have a
further constraint to indices with respect to the other diagrams, their
contribution is subleading for the correlation functions, as we now argue.

The thermal correlation $S^{(q)}_{\beta}(\vec{t})$ in Eq.(4) is given by the
sum over all indices (and correspondingly all the diagrams) with the proper
Boltzmann weight $e^{-\beta E_{i}}/Z$. The ETH ansatz (2)-(3) results in two
main outcomes: 1) all summations of elements with repeated indices (cactus
diagrams) factorize on results computed at the thermal energy
$E_{\beta}=\langle H\rangle_{\beta}$. This means that leaves may be severed.
2) The contribution of non-cactus diagrams is exponentially small with respect
to the other terms. These properties follow from the smoothness of the ETH
functions and the proper entropic counting. As an explicit example of 1), one
can compute the diagram (a6) of Fig.2, i.e.

$\displaystyle\frac{1}{Z}\sum_{i}{e^{-\beta E_{i}}}\overline{A_{ii}}^{4}$
$\displaystyle=\frac{1}{Z}\int dE{e^{-\beta
E+S(E)}}\mathcal{A}^{4}(E)=(S^{(1)}_{\beta})^{4}$ (5)

where we have substituted the ETH ansatz (1), summations with integrals
$\sum_{i}=\int dEe^{S(E)}$ and performed the integral in $E$ via saddle point
technique, which fixes the energy by the thermodynamic condition
$\beta=S^{\prime}(E_{\beta})$ and yields
$\mathcal{A}^{4}(E_{\beta})=(S^{(1)}_{\beta})^{4}$. On the other hand,
performing the same steps on the non-cactus diagram Fig.2(a7) and expanding
the entropies 111We substitute $\sum_{i\neq j}\to\int
dE_{1}dE_{2}e^{S(E_{1})+S(E_{2})}$ and then expand
$S(E_{1/2})=S(E^{+}\pm\omega/2)=S(E^{+})\pm
S(E^{+})\omega/2+S^{\prime\prime}(E^{+})\omega^{2}/8+\dots$. This leads to
$\int dE^{+}d\omega e^{2S(E^{+})}$, where we have neglected the second order
correction, since one can then identify $S^{\prime\prime}(E)=-\beta^{2}/C$,
where $C\propto N$ is the heat capacity at inverse temperature $\beta$, that
diverges in the thermodynamic limit. leads to

$\displaystyle\frac{1}{Z}$ $\displaystyle\sum_{i\neq j}{e^{-\beta
E_{i}}}\overline{|A_{ij}|^{2}}\,\overline{|A_{ij}|^{2}}$ (6)
$\displaystyle=\frac{1}{Z}\int dE^{+}d\omega e^{-\beta
E^{+}}e^{-\beta\omega/2}\left(F^{(2)}_{E^{+}}(\omega)\right)^{2}\sim\mathcal{O}(e^{-N})\
.$

In this paper, we rationalize that cactus diagrams correspond in fact to the
non-crossing partitions that play a role in free probability theory.

Hints of free probability theory -  We are interested in the combinatorial
aspects of free probability, which are based on non-crossing partitions and
free cumulants, as developed by Speicher [31]. A partition of a set
$\\{1,\dots q\\}$ is a decomposition in blocks that do not overlap and whose
union is the whole set. Partitions in which blocks do not “cross” are called
_non-crossing partitions_. The set of all non-crossing partitions of
$\\{1,\dots q\\}$ is denoted $NC(q)$. See the example in Fig.2b for the
partitions with $q=4$, with $\times[n]$ we denote the $n$ cycling
permutations. There are 14 non-crossing partitions of $q=4$ elements, and only
one is crossing, i.e. Fig.2b.a7.
Non-crossing partitions appear in the definition of free cumulants. Consider
some some normalized $\phi$ (i.e. $\phi(1)=1$), for example
$\phi(\bullet)=\text{Tr}(\bullet)/D$ for large $D\times D$ random matrices.
The _free cumulants_ $k_{q}$ are defined implicitly from the moment-cumulant
formula, that states that the moments of variables $A^{(i)}$ read

$\phi(A^{(1)}\dots A^{(q)})=\sum_{\pi\in NC(q)}k_{\pi}(A^{(1)}\dots A^{(q)})\
,$ (7)

where $\pi$ is a non-crossing partition. Here $k_{\pi}$ is a product of
cumulants, one term for each block of $\pi$. For instance for $A^{(i)}=A$
$\forall i$ and $k_{n}(A...A)=k_{n}$, one has $\phi(A)=k_{1}$,
$\phi(A^{2})=k_{2}+k_{1}^{2}$, $\phi(A^{3})=k_{3}+3\,k_{1}k_{2}+k_{1}^{3}$,
and $\phi(A^{4})=k_{4}+2k^{2}_{2}+4k_{3}k_{1}+6k_{2}k_{1}^{2}+k_{1}^{4}$, i.e.
Fig.2b. Classical cumulants are defined by a similar formula, where one sum
over all the possible partitions and not only on the non-crossing ones.
Notably, the relation between moments and cumulants differs in classical and
free probability only for $n\geq 4$. The vanishing of free cumulants
$k_{2}(AB)=0$ can be seen as an alternative definition of freeness between $A$
a $B$ (in the same way that classical mixed cumulants vanish for independent
variables). Another interesting property is that the third and higher-order
free cumulants of a Gaussian random matrix $A$ vanish, i.e. $k_{q\geq
3}(A...A)=0$ (exactly as for the classical cumulants of standard Gaussian
random variables). Notice that, at this level, Eq.(7) is a simple implicit
definition of free-cumulants in terms of moments, e.g. $k_{1}=\phi(A)$,
$k_{2}=\phi(A^{2})-\phi(A)^{2}$, etc..

ETH in these words - The relation between ETH and free cumulants is thus
clear: cactus diagrams correspond in fact to non-crossing partitions, on the
set identified by the matrices $A(t_{i})$. Likewise, non-cactus diagrams, not
being associated with any non-crossing partition, do not count. In an
alternative representation of Fig.S3b (see [33]) non-cactus diagrams are
equivalent to crossing partitions. As we argued, they do not contribute to the
ETH moments/free probability. Analogously to Eq.(7), we introduce an implicit
definition of the _thermal free cumulants_

$S_{\beta}^{(q)}(\vec{t})=\sum_{\pi\in
NC(q)}k^{\beta}_{\pi}(\,A(t_{1})A(t_{2})\dots A(0))\ .$ (8)

Also here, $k^{\beta}_{\pi}$ is a product of free cumulants, one for each
block of the partition $\pi$, as $k^{\beta}_{n}$ associated to $n$ operators.
Note that the cumulant $k^{\beta}_{n}(\vec{t})$ depends on the order in which
we consider different operators, and we make this implicit in its time
dependence. _The ETH ansatz ( 2) is then the precise statement that the
thermal free cumulants, may be substituted, for the purposes of computing time
correlations, by sums as_

$\displaystyle k^{\beta}_{q}(\vec{t})$ $\displaystyle=k^{ETH}_{q}(\vec{t})$
(9) $\displaystyle=\frac{1}{Z}\sum_{i_{1}\neq i_{2}\neq i_{q}}e^{-\beta
E_{i_{1}}}A(t_{1})_{i_{1}i_{2}}A(t_{2})_{i_{2}i_{3}}\dots A(0)_{i_{q}i_{1}}\
.$

_where all indices are different_. In other words, free cumulants are simply
given by the loops diagrams. This follows from two properties of the ETH
ansatz discussed above: a) that cactus diagrams factorize and b) that only
cactus diagrams (non-crossing partitions) matter. This first point is almost
trivial for $q=2$, where it is well known that via ETH one can compute

$\displaystyle k^{\beta}_{2}(t)$ $\displaystyle\equiv
S_{\beta}^{(2)}(t)-[S_{\beta}^{(1)}]^{2}=\langle
A(t)A(0)\rangle_{\beta}-\langle A\rangle^{2}_{\beta}$ (10)
$\displaystyle=\frac{1}{Z}\sum_{i\neq j}{e^{-\beta
E_{i}}}|A_{ij}|^{2}e^{i(E_{i}-E_{j})t}=k^{ETH}_{2}(t)$ (11)

where one uses that the diagonal ETH matrix element is a smooth function of
energy and therefore $\sum_{i}e^{-\beta E_{i}}A_{ii}^{2}\simeq\langle
A\rangle_{\beta}^{2}$ by saddle point integral, as in Eq.(5). One can show
that this factorization holds at all orders. For instance, for $q=4$ fixing
$k_{1}(A)=\langle A\rangle_{\beta}=0$, we obtain

$\displaystyle\begin{split}\langle
A(t_{1})&A(t_{2})A(t_{3})A(0)\rangle_{\beta}=k^{\beta}_{4}(t_{1},t_{2},t_{3})\\\
&+k^{\beta}_{2}(t_{1}-t_{2})\,k^{\beta}_{2}(t_{3})+k^{\beta}_{2}(t_{2}-t_{3})\,k^{\beta}(t_{1})\
,\end{split}$ (12)

where $k_{4}^{\beta}$ is the term coming from the simple loop in Fig.2a1 and
encodes all the correlations beyond the gaussian ansatz [25; 21]. See [33] for
the detailed calculation of Eq.(12).
In rotationally invariant random matrix ensembles, free cumulants are
associated with diagrams with all distinct indices, similarly to the ETH
result discussed here. Indeed, also in that case, one can show that only the
cactus diagrams matter [34].

Note that the structure of Free Probability incorporates the consistency
condition that products operators obeying ETH shall obey ETH [35; 4]. This can
be checked directly from the Free Probability non-crossing partitions, as
illustrated in the Supp.Mat.[33].

Free cumulants on-shell - The thermal free cumulants defined above in Eq.(9)
admit an extremely appealing expression in terms of the ETH ansatz (2). By
standard manipulations [33], one shows that

$k^{\beta}_{q}(\vec{t})=\int
d\vec{\omega}F^{(q)}_{E_{\beta}}(\vec{\omega})\,e^{i\vec{\omega}\cdot\vec{t}-\beta\vec{\omega}\cdot\vec{\ell}_{q}}\
,$ (13)

where we have introduced the _$q$ -th ladder shift_
$\vec{\ell}_{q}=\left(\frac{q-1}{q},\frac{q-2}{q},\dots,\frac{1}{q}\right)$.
This equation gives an important property: _ETH on-shell correlations of order
$q$, are related to the Fourier transform of the thermal free cumulants
$k^{\beta}_{q}$_, i.e.

$k^{\beta}_{q}(\vec{\omega})=F^{(q)}_{E_{\beta}}(\vec{\omega})e^{-\beta\vec{\omega}\cdot\vec{\ell}_{q}}$
(14)

This is very familiar for $q=2$, for which
$k^{\beta}_{2}(\omega)=F^{(2)}_{E_{\beta}}(\omega)e^{-\beta\omega/2}$, which
is the standard Kubo-Martin Schwinger (KMS) relation, leading to the
fluctuation-dissipation theorem. The presence of this thermal shift at each
$q$ – which only depends on temperature and the order of the correlation
function – shall be interpreted as a generalized KMS condition, see Ref.[36].
Eq.(14) naturally leads us to inspect the free cumulant expansion of the
shifted correlator $\overline{S}^{(q)}_{\beta}(\vec{t})\equiv
S^{(q)}_{\beta}(\vec{t}-i\beta\vec{\ell}_{q})$ given by

$\overline{S}^{(q)}_{\beta}(\vec{t})=\text{Tr}\left(\rho^{1/q}A(t_{1})\rho^{1/q}\dots
A(t_{q-1})\rho^{1/q}A(0)\right)\ ,$ (15)

which corresponds to a _regularized version_ of $S_{\beta}$. One can look at
the following connected correlation function of order $q$ of
$\overline{S}^{(q)}_{\beta}$, i.e.

$\bar{k}_{q}(\vec{t})=\frac{1}{Z}\sum_{i_{1}\neq i_{2}\neq
i_{q}}e^{-\frac{\beta}{q}(E_{i_{1}}+\dots
E_{i_{q}})}A(t_{1})_{i_{1}i_{2}}A(t_{2})_{i_{2}i_{3}}\dots
A(t_{q})_{i_{q}i_{1}}\ .$ (16)

Diagrammatically, it is associated to the loop with $q$ operators where the
thermal weight $\rho^{1/q}$ is equally split. Nicely, its Fourier transform
coincides with on shell correlations at the thermal energy $E_{\beta}$, i.e.
[15]

$\bar{k}_{q}^{\beta}(\vec{\omega})=F^{(q)}_{E_{\beta}}(\vec{\omega})\ .$ (17)

This allows accessing such correlations directly from the time-dependent
correlation functions in time and by taking their Fourier transform.

We now recall that correlation between matrix elements with large energy
differences should be small. This means that ETH correlation functions are
usually expected to decay fast at large frequencies $\omega\gg 1$ as

$F^{(q)}_{E}(\omega)\sim e^{-|\omega|/\omega^{(q)}_{max}}\ .$ (18)

The relations between free cumulants and $F^{(q)}_{E}(\vec{\omega})$ (14)
allow one to infer relevant properties of the latter. For instance, using
Eq.(13) and the fact that free cumulants at equal times shall be well defined,
in [33] we prove that on-shell correlations must decay at large frequencies in
all directions at least as

$F^{(q)}_{E_{\beta}}(\vec{\omega})\sim\exp\left(-\beta\frac{q-1}{q}|\omega_{i}|\right)$
(19)

$\forall i=1,\dots,q-1$. This gives the bound
$\omega^{(q-1)}_{max}\leq\frac{q-1}{q\beta}$, which generalizes the result for
$q=2,4$ of Ref.[25]. These kinds of constraints have been related to operator
growth or to time-scales of multi-time correlation functions (such as out-of-
time order correlators) [37; 25; 38; 39; 40], which have been proven to obey
strict bounds [41; 42; 43].

Conclusions - We have found that the Eigenstate Thermalization Hypothesis,
when generalized to all multi-point correlations in a way that is in the same
spirit as Berry, Deutsch and Srednicki, leads us directly to place it in the
realm of Free Probability. This is a branch of mathematics where many
developments have been made, and for which one may now turn to look for
connections and analogies.

There is, however, a fundamental new element: the ensembles of matrices we
need to consider are not homogeneously full, but rather have a band structure,
and a large, slowly varying diagonal. This structure exists on a specific
basis, the one where the Hamiltonian is diagonal and its eigenvalues are
ordered. The results are, likewise, always related to a specific energy shell,
and not the matrix as a whole. The moments that define equilibrium correlation
functions are then more complicated objects than those of a standard
rotationally-invariant Matrix Model. Nonetheless many results from these
appear to generalize to the ETH setting and call for a rigorous understanding.

The Eigenstate Thermalization Hypothesis is at its most interesting when it
fails, and integrability or many-body localization phenomena emerge. A more
global understanding of ETH may then lead to a finer understanding of these
effects.

###### Acknowledgements.

This paper has been submitted simultaneously with “Dynamics of Fluctuations in
the Open Quantum SSEP and Free Probability” by L. Hruza and D. Bernard [44],
which discusses the appearance of free cumulants in stochastic transport
models. The occurrence of free probability in both problems has a similar
origin: the coarse-graining at microscopic either spatial or energy scales,
and the unitary invariance at these microscopic scales. Thus the use of free
probability tools promises to be ubiquitous in chaotic or noisy many-body
quantum systems.
We thank A. Polkovnikov for useful suggestions. SP thanks C. Malvenuto for
discussions on non-crossing partitions. SP and JK are supported by the Simons
Foundation Grant No. 454943. This work is supported by ‘Investissements
d’Avenir’ LabEx PALM (ANR-10-LABX-0039-PALM) (EquiDystant project, L. Foini).

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——————————

Supplemental Material:
Eigenstate Thermalization Hypothesis and Free Probability

In this Supplementary Material, we provide additional analysis and background
calculations to support the results in the main text. In Sec.1 and Sec.2 we
discuss further properties of non-crossing partitions in relation to ETH, In
Sec.3, we derive the expression of the free cumulants within ETH. In Sec.4 we
discuss the large frequency dependence of on-shell correlations and prove
Eq.(19) of the main text. We conclude with Sec.5 with the detailed calculation
of the moment-cumulant calculation for $q=4$.

##  1 Cactus diagrams, non-crossing partitions and their duals

Figure S1: Relation between ETH diagrams (a), non-crossing partitions (b) and
their dual (c) for $q=4$.

Non-crossing partitions admit a complementary/dual representation, where the
partition is identified on the vertices rather than on the links. This is
illustrated in Fig.S1 where we represent the example for $q=4$. The relation
between the ETH diagrams (Fig.S1a) and the dual of the non-crossing partition
Fig.S1c is thus particularly simple: non-crossing diagrams are obtained by
pinching the blue regions that identify two or more indices. Furthermore, non-
cactus diagrams (a7) can be identified with the crossing partition (c7).

##  2 Consistency of ETH

Years ago, Srednicki asked the natural question about the consistency of ETH
under multiplication [35]. Because local functions of an observable share the
‘typicality’ properties assumed by the observable itself, this is quite
natural. However, it is instructive to see this within the framework of Free
Probability. Let us see this with an example: suppose we have
$B^{1}=A^{1}A^{2}$, $B^{2}=A^{3}A^{4}$. If $A^{1},\,A^{2},\,A^{3},\,A^{4}$
obey ETH, can we show that $B^{1},\,B^{2}$ automatically do too? Free
Probability diagrams tell us how to restate this question in an illuminating
way, as illustrated in Fig.S2.

We draw the lattice of all dual non-crossing diagrams of the $A^{\prime}s$
(Fig.S2d). With dual partition, we mean that points belonging to the same
block are identified and one shall think about the blue lines as $\delta$
functions, see Section above. Here, we represent it using the _lattice_
structure of non-crossing partitions, see e.g. Ref.[45]. An alternative way to
obtain these non-crossing partitions is to proceed to start from the
$B^{\prime}s$ in two steps: we first draw the non-crossing partitions for
$B^{(1)}$ and $B^{(2)}$ (Fig.S2a), where the blue lines identify the indices
in the products $B^{(1)}_{ij}B^{(2)}_{ji}$. Then, every $B$ is written as a
product of two $A$’s (Fig.S2b), where the hidden index is represented as an
empty dot. Each one of these diagrams, yields a sublattice of diagrams with
all new possible lines added (Fig.S2c), excluding the ones that link two
indices of the $B$’s, because they will be counted in another sublattice. All
these diagrams constitute a complete set of sums associated with the products
of the four $A$’s, i.e. (Fig.S2d). Just by inspection of each sublattice, we
recognize the development of the products of two $A$’s, as for example in
$\sum_{j\neq
i}A^{1}_{ij}A^{2}_{ji}+\sum_{j=i}A^{1}_{ij}A^{2}_{ji}=\sum_{j}A^{1}_{ij}A^{2}_{ji}=B_{ii}$.
Correspondingly, one can start from the $A^{\prime}s$, and – by proceeding in
the other sense – reconstruct the non-crossing partitions of the
$B^{\prime}s$.

The merit of this construction is that it relies exclusively on the
bookkeeping of Free Cumulants.

Figure S2: ETH consistency under multiplication and non-crossing partitions of
Free Probability. If the operators $A$ obeys ETH, what can we say about
$B=A^{2}$? The lattice non-crossing partitions associated to $A^{4}$ in (d)
can be divided in two sublattices in (c). Once summed over, this yield the
non-crossing partition of the operators $B$ (a, b).

##  3 Free cumulants within ETH

In this Section we compute free cumulants of Eq.9 of the main text within the
ETH ansatz [cf. Eq.2 of the main text]. One has

$\displaystyle k_{q}^{\beta}(\vec{t})$
$\displaystyle=\frac{1}{Z}\sum_{i_{1}\neq i_{2}\dots\neq i_{1}}e^{-\beta
E_{i_{1}}}e^{it_{1}(E_{i_{1}}-E_{i_{2}})+t_{2}(E_{i_{2}}-E_{i_{3}})+\dots
t_{q-1}(E_{i_{q-1}}-E_{i_{q}})}{A_{i_{1}i_{2}}A_{i_{2}i_{3}}\dots
A_{i_{q}i_{1}}}$ (S1) $\displaystyle=\frac{1}{Z}\sum_{i_{1}\neq i_{2}\dots\neq
i_{1}}e^{-\beta
E_{i_{1}}}e^{i\vec{t}\cdot\vec{\omega}}e^{-(q-1)S(E^{+})}F^{(q)}_{E^{+}}(\vec{\omega})$
(S2) $\displaystyle=\frac{1}{Z}\int dE_{1}\dots dE_{q}e^{-\beta
E_{1}}e^{i\vec{t}\cdot\vec{\omega}}e^{S(E_{1})+\dots
S(E_{q})-(q-1)S(E^{+})}\,F^{(q)}_{E^{+}}(\vec{\omega})$ (S3)

where from the first to the second line we have used
$\vec{\omega}=(\omega_{i_{1}i_{2}},\dots,\omega_{i_{q-1}i_{q}})$ with
$\omega_{ij}=E_{i}-E_{j}$ and substituted the ETH ansatz and from the second
to the third we have exchanged the summation with the integral
$\sum_{i_{1}}\to\int dE_{1}e^{S(E_{1})}$. We can thus Taylor expand the
entropies around energy $E^{+}$ as

$S(E_{i})=S(E^{+}+(E_{i}-E^{+}))=S(E^{+})+S^{\prime}(E^{+})(E_{i}-E^{+})+\frac{1}{2}S^{\prime\prime}(E^{+})(E_{i}-E^{+})^{2}+\dots\
.$ (S4)

Then, by summing over all the energies one obtains

$\sum_{i=1}^{q}S(E_{i})=qS(E^{+})+S^{\prime\prime}(E^{+})\sum_{i}(E_{i}-E^{+})^{2}+\dots\
,$ (S5)

where the linear term in $E_{i}-E^{+}$ vanishes (due to
$E^{+}=(E_{1}+E_{2}+\dots E_{q})/q$), while the quadratic term is subleading
to the thermodynamic property $S^{\prime\prime}(E^{+})=-\beta^{2}/C$ with
$C\propto N$ the heat capacity and $\beta=S^{\prime}(E^{+})$ the inverse
temperature at energy $E^{+}$. Since $E_{i}-E^{+}\propto\vec{\omega}$ and
$F_{E^{+}}(\vec{\omega})$ is a smooth function that decays decays fast at
large frequencies, we can neglect the second term in Eq.(S5). The free
cumulant then reads

$\displaystyle k_{q}^{\beta}(\vec{t})=\frac{1}{Z}\int dE_{1}e^{-\beta
E_{1}+S(E^{+})}\int dE_{2}\dots
dE_{E_{q}}e^{i\vec{t}\cdot\vec{\omega}}F^{(q)}_{E^{+}}(\vec{\omega})\ .$ (S6)

We can now rewrite

$\displaystyle E_{1}$ $\displaystyle=E^{+}+(E_{1}-\frac{E_{1}+E_{2}+\dots
E_{q}}{q})=E^{+}+\frac{q-1}{q}(E_{1}-E_{2})+\frac{q-2}{q}(E_{2}-E_{3})+\dots+\frac{1}{q}(E_{q-1}-E_{q})$
(S7) $\displaystyle=E^{+}+\vec{\ell}_{q}\cdot\vec{\omega}\ ,$ (S8)

where we have defined the ladder operator

$\vec{\ell}_{q}=\left(\frac{q-1}{q},\frac{q-2}{q}\dots,\frac{1}{q}\right)\ .$
(S9)

We substitute this into Eq.(S6) and change integration variables
$dE_{1}dE_{2}\dots dE_{1}=dE^{+}d\omega_{1}d\omega_{1}\dots d\omega_{q-1}$,
leading to

$\displaystyle k_{q}^{\beta}(\vec{t})=\frac{1}{Z}\int dE^{+}e^{-\beta
E^{+}+S(E^{+})}\int d\omega_{1}\dots
d\omega_{q-1}e^{i\vec{t}\cdot\vec{\omega}-\beta\vec{\ell}_{q}\cdot\vec{\omega}}F^{(q)}_{E^{+}}(\vec{\omega})\
.$ (S10)

Since $F^{(q)}_{E^{+}}(\vec{\omega})$ is a smooth function of $E^{+}$ of order
one, we can solve the integral over $E^{+}$ by saddle point, which simplifies
with the denominator and fixed the energy by the thermodynamic definition via
$S^{\prime}(E_{\beta})=\beta$. This immediately leads to Eq.13 of the main
text.

##  4 Large $\omega$ dependence of on-shell correlations

As shown in the previous Section, the thermal free cumulant can be written at
all times in terms of an on-shell correlation functions time a thermal weight
[cf. Eq.13 of the main text]. At times $t=0$ this leads to

$k_{q}^{\beta}(0)=\int
d\vec{\omega}\,\,F^{(q)}_{E_{\beta}}(\vec{\omega})e^{-\beta\vec{\ell}_{q}\cdot\vec{\omega}}=\int
d\vec{\omega}\,\,F^{(q)}_{E_{\beta}}(\vec{\omega})e^{-\frac{\beta}{q}((q-1)\omega_{1}+(q-2)\omega_{2}+\dots+\omega_{q-1})}\
.$ (S11)

This is just a combination of moments of the same operator at equal times and
therefore it should be well defined and finite. This imposes constraints on
the behaviour of $F^{(q)}(\vec{\omega})$ at large (negative) frequencies in
the direction of $\ell_{q}$. Furthermore, the function $F^{(q)}_{E^{+}}$ has
$(q-1)$ symmetries:

$F^{(q)}_{E^{+}}(\omega_{i_{1}i_{2}},\omega_{i_{2}i_{3}},\dots,\omega_{i_{q-1}i_{q}})=F^{(q)}_{E^{+}}(\omega_{i_{1}i_{2}}^{\prime},\omega_{i_{2}i_{3}}^{\prime},\dots,\omega_{i_{q-1}i_{q}}^{\prime}),$
(S12)

where
$(\omega_{i_{1}i_{2}}^{\prime},\omega_{i_{2}i_{3}}^{\prime},\dots,\omega_{i_{q-1}i_{q}}^{\prime})$
is obtained by putting
$\overline{\omega}=\omega_{i_{1}i_{2}},+\omega_{i_{2}i_{3}}+\dots+\omega_{i_{q-1}i_{q}}$,
permuting cyclically the set
$(\omega_{i_{1}i_{2}},\omega_{i_{2}i_{3}},\dots,\omega_{i_{q-1}i_{q}},-\overline{\omega})$,
omitting the new last term. This, together with Eq.(S11), implies that the
smooth function must fall at large frequencies in all directions at least as

$F^{(q)}_{E_{\beta}}(\omega_{1},\omega_{2},\dots,\omega_{q-1})\sim\exp(-\beta\frac{q-1}{q}|\omega_{i}|)\
,$ (S13)

which, for an exponentially decaying $F^{(q)}_{E}(\omega)\sim
e^{-|\omega|/\omega^{(q)}_{max}}$, yields the bound
$\omega_{max}^{(q)}\leq\frac{q-1}{q}\frac{1}{\beta}$.

##  5 Cumulant-moment calculation with $q=4$

Figure S3: Relation between dual non-crossing partitions, ETH diagrams and
thermal free cumulants for $q=4$ with $k_{1}=0$.

In this section we derive Eq.(12) of the main text. We compute

$\displaystyle S_{\beta}^{(4)}(t_{1},t_{2},t_{3})$
$\displaystyle=\text{Tr}\left(\frac{e^{-\beta
H}}{Z}A(t_{1})A(t_{2})A(t_{3})A(0)\right)=\frac{1}{Z}\sum_{ijkm}e^{-\beta
E_{i}}e^{i\omega_{ij}t_{1}+i\omega_{jk}t_{2}+i\omega_{km}t_{3}}\overline{A_{ij}A_{jk}A_{km}A_{mi}}$
(S14) $\displaystyle=\sum_{i\neq j\neq k\neq m}\frac{e^{-\beta
E_{i}}}{Z}e^{i\omega_{ij}t_{1}+i\omega_{jk}t_{2}+i\omega_{km}t_{3}}\overline{A_{ij}A_{jk}A_{km}A_{mi}}$
(S15) $\displaystyle\quad+\sum_{i\neq j\neq m}\frac{e^{-\beta
E_{i}}}{Z}e^{i\omega_{ij}(t_{1}-t_{2})+i\omega_{im}t_{3}}\overline{A_{ij}A_{ji}A_{im}A_{mi}}+\sum_{i\neq
j\neq k}\frac{e^{-\beta
E_{i}}}{Z}e^{i\omega_{ij}t_{1}+i\omega_{jk}(t_{2}-t_{3})}\overline{A_{ij}A_{jk}A_{kj}A_{ji}}\
,$ (S16)

where we set $A_{ii}=0$ without loss of generality. These decompositions can
be found by associating to each point on the loop and index and taking the
first three non-crossing partitions. Let us now compute with ETH each term.
The first coincides with the definition of free-cumulant in Eq.(9). Plugging
the ETH ansatz (2) into Eq.(S15), we have

$\displaystyle k_{4}^{\beta}(t_{1},t_{2},t_{3})$ $\displaystyle=\sum_{i\neq
j\neq k\neq m}\frac{e^{-\beta
E_{i}}}{Z}e^{i\omega_{ij}t_{1}+i\omega_{jk}t_{2}+i\omega_{km}t_{3}}F_{E^{+}}^{(4)}(\omega_{ij},\omega_{jk},\omega_{km})e^{-3S(E^{+})}$
(S17) $\displaystyle=\frac{1}{Z}\int
dE_{1}dE_{2}dE_{3}dE_{4}e^{S(E_{1})+S(E_{2})+S(E_{3})+S(E_{4})-3S(E^{+})}\,{e^{-\beta
E_{1}}}e^{i\omega_{12}t_{1}+i\omega_{23}t_{2}+i\omega_{34}t_{3}}F_{E^{+}}^{(4)}(\omega_{12},\omega_{23},\omega_{34})$
(S18) $\displaystyle=\frac{1}{Z}\int dE^{+}{e^{S(E^{+})-\beta E^{+}}}\int
d\omega_{12}d\omega_{23}d\omega_{34}e^{i\vec{\omega}\cdot\vec{t}}e^{-\beta\vec{\omega}\cdot\vec{\ell}_{4}}F_{E^{+}}^{(4)}(\omega_{12},\omega_{23},\omega_{34})$
(S19)

where from the first to the second line we have substituted summations with
integrals $\sum_{i}=\int dE_{1}e^{S(E_{1})}$. From the second to the third we
have expanded the individual entropies around the average one, i.e.

$S(E_{i})=S(E^{+}+(E_{i}-E^{+}))=S(E^{+})+S^{\prime}(E_{i}-E^{+})+\mathcal{O}(\omega^{2}S^{\prime\prime})\
.$ (S20)

When we sum over all of them, the term proportional to $S^{\prime}$ vanishes.
We have also re-written

$E_{1}=E^{+}+\frac{3}{4}\omega_{12}+\frac{2}{4}\omega_{23}+\frac{1}{4}\omega_{14}=E^{+}+\vec{\omega}\cdot\vec{\ell}_{4}\
,$ (S21)

where $\ell_{4}$ is defined in Eq.(S9) and changed integration variables
$dE_{1}dE_{2}dE_{3}dE_{4}=dE^{+}d\omega_{12}d\omega_{23}d\omega_{34}$. We thus
perform the saddle point integral over $E^{+}$, whose solution fixes the
$E_{\beta}$ energy from the canonical definition
$S^{\prime}(E_{\beta})=\beta$. This leads exactly to the result in Eq.(13) for
$q=4$. Let us now compute the other term as

$\displaystyle\sum_{i\neq j\neq k}$ $\displaystyle\frac{e^{-\beta
E_{i}}}{Z}\,\,e^{i\omega_{ij}(t_{1}-t_{2})+i\omega_{ik}t_{3}}e^{-S(E_{ij})-S(E_{jk})}F^{(2)}_{E_{ij}}(\omega_{ij})F^{(2)}_{E_{ik}}(\omega_{ik})$
(S22) $\displaystyle=\frac{1}{Z}\int dE_{1}{e^{S(E_{1})-\beta
E_{1}}}\left(\int
dE_{2}e^{i\omega_{12}(t_{1}-t_{2})}e^{S(E_{2})-S(E_{12})}F^{(2)}_{E_{12}}(\omega_{12})\right)\left(\int
dE_{3}e^{i\omega_{13}t_{3}}e^{S(E_{3})-S(E_{13})}F^{(2)}_{E_{13}}(\omega_{13})\right)$
(S23) $\displaystyle=\left(\int
d\omega_{12}e^{i\omega_{12}(t_{1}-t_{2})}e^{-\beta\frac{1}{2}\omega_{12}}F^{(2)}_{E_{\beta}}(\omega_{12})\right)\left(\int
d\omega_{13}e^{i\omega_{13}t_{3}}e^{-\beta\frac{1}{2}\omega_{13}}F^{(2)}_{E_{\beta}}(\omega_{13})\right)=k_{2}^{\beta}(t_{1}-t_{2})k_{2}^{\beta}(t_{3})\
.$ (S24)

Where from the second to the third line we have expanded the entropies

$S(E_{2})-S(E_{12})=S(E_{1}-\omega_{12})-S(E_{1}-\omega_{12}/2)=-S^{\prime}(E_{1})\frac{\omega_{12}}{2}+\mathcal{O}(\frac{\omega_{21}^{2}}{N})\
,$ (S25)

solved the integral over $E_{1}$ again by saddle point and changed the
integration from $E_{2/3}\to\omega_{12/3}$. We have thus found the nice
factorization between the free cumulants illustrated pictorially in Fig.S3. We
now evaluate the last term

$\displaystyle\sum_{i\neq j\neq k}$ $\displaystyle\frac{e^{-\beta
E_{i}}}{Z}e^{-i\omega_{ji}t_{1}+i\omega_{jk}(t_{2}-t_{3})}e^{-S(E_{jk})-S(E_{ij})}F^{(2)}_{E_{jk}}(\omega_{jk})F^{(2)}_{E_{ij}}(\omega_{ij})$
(S26) $\displaystyle=\frac{1}{Z}\int dE_{2}{e^{S(E_{2})}}\left(\int
dE_{1}e^{-i\omega_{21}t_{1}}e^{-\beta
E_{1}+S(E_{1})-S(E_{12})}F^{(2)}_{E_{12}}(\omega_{21})\right)\left(\int
dE_{3}e^{i\omega_{23}(t_{2}-t_{3})}e^{S(E_{3})-S(E_{23})}F^{(2)}_{E_{23}}(\omega_{23})\right)$
(S27) $\displaystyle=\frac{1}{Z}\int dE_{2}{e^{S(E_{2})-\beta
E_{2}}}\left(\int
dE_{1}e^{-i\omega_{21}t_{1}}e^{\beta\omega_{21}/2}F^{(2)}_{E_{12}}(\omega_{12})\right)\left(\int
dE_{3}e^{i\omega_{23}(t_{2}-t_{3})}e^{-\beta\omega_{23}/2}F^{(2)}_{E_{23}}(\omega_{23})\right)$
(S28) $\displaystyle=\left(\int
d{\omega_{12}}e^{i\omega_{12}t_{1}}e^{-\beta\omega_{12}/2}F^{(2)}_{E_{\beta}}(\omega_{12})\right)\left(\int
d\omega_{23}e^{i\omega_{23}(t_{2}-t_{3})}e^{-\beta\omega_{23}/2}F^{(2)}_{E_{\beta}}(\omega_{23})\right)=k^{\beta}(t_{1})k^{\beta}(t_{2}-t_{3})$
(S29)

where from the second to the third line we have expanded the entropies as in
Eq.(S25) and re-written the thermal weight as $e^{-\beta E_{1}}=e^{-\beta
E_{2}+\beta\omega_{21}}$. We can thus integrate over $E_{2}$ by saddle point
and obtain the result on the fourth line.

Putting it all together, we have

$S_{\beta}^{(4)}(t_{1},t_{2},t_{3})=k^{\beta}_{4}(t_{1},t_{2},t_{3})+k^{\beta}_{2}(t_{1}-t_{2})k^{\beta}_{2}(t_{3})+k^{\beta}_{2}(t_{1})k^{\beta}_{2}(t_{2}-t_{3})\
,$ (S30)

which is exactly the (free) cumulant-moment formula in Eq.(12).