Thermal fluctuations of black holes with non-linear electrodynamics and
charged Renyi entropy
Gabriel Arenas-Henriquez☕, Felipe Diaz🚲, and Yerko Novoa\Bat

☕ Centre for Particle Theory, Department of Mathematical Sciences,
Durham University, South Road, Durham, DH1 3LE, UK
🚲Departamento de Ciencias Físicas, Universidad Andrés Bello,
Sazié 2212, Santiago de Chile
\BatDepartamento de Filosofía, Universidad de Santiago de Chile,
Avenida Libertador Bernardo O’Higgins 3677, Santiago, Chile
Abstract

We extend the charged Renyi entropy to a more general holographic scenario.
Coupling an arbitrary non-linear electrodynamics Lagrangian density to AdS
gravity, we analyse the thermodynamic features of non-linearly charged
hyperbolic black holes and the thermal fluctuations in the grand canonical
ensemble. We provide a general form for the relevant holographic quantities
that describes a CFT with a global $U(1)$ symmetry in terms of horizon data
and we compute the first thermal fluctuation of the charged Renyi entropy. We
demonstrate the validity of the formulae through an analytic example; the
Coulomb source in $2+1$ dimensions. We propose this model to be dual to
charged free bosons in $1+1$ dimensions. The corrections generates a
subleading logarithmic divergence in the entanglement entropy which appear in
some Condensed Matter systems with spontaneous symmetry breaking due to IR
effects in the ground state. We comment on the possibility of interpreting
these results in terms of holography beyond the saddle point approximation.

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###### Contents

  1. 1 Introduction
  2. 2 Holographic charged Renyi Entropy
  3. 3 AdS Gravity and non-linear Electrodynamics
    1. 3.1 Holographic Quantities
  4. 4 Thermal fluctuations and stability
  5. 5 Holographic charged free boson in 1+1
  6. 6 Summary and Discussion
  7. A Power-law Maxwell electrodynamics

##  1 Introduction

The AdS/CFT correspondence [1, 2, 3] maps the gravity partition function in
anti-de Sitter (AdS) space to a conformal field theory (CFT) in one dimension
lower. The original proposal matches on-shell highly symmetric strongly
coupled gauge theories at large $N$ with low energy limits of String theory,
implying a UV/IR correspondence [4, 5]. After a long and successful path, it
has been extended to large class of quantum field theories that are described
in terms of IR saddles of quantum gravity in hyperbolic spaces, showing a
surprising relation between black hole physics and critical many-body quantum
systems, see for instance [6] and reference therein.

The String theory partition function contains higher-curvature terms in the IR
limit such that considering Kaluza–Klein reductions can lead to non-linear
electrodynamics (NED) coupling to the Einstein sector [7]. Moreover, non-
linear kinetic terms for the vector fields also appear in quantisation of
String actions [8]. Therefore, NED Lagrangians, originally introduced as an
alternative to Maxwell electrodynamics to solve problems in quantum
electrodynamics, can be considered in the duality and allows to extend the
possible holographic field theories. For a nice review on NED and its
applications see [9].

An important matter is to see how the correspondence is extended beyond the
classical saddle in the gravity side, such that quantum corrections of the
String partition function can modify the properties of the dual CFT. An
interesting holographic quantity that allows to explore this issue is the
entanglement entropy [10, 11] which has lead to understand properties of
spacetime itself [12]. The quantum corrections to the holographic entanglement
entropy have been proposed in [13, 14] where the extra term appearing are of
zero order in the gravity coupling.

In [15], an extension of holographic entanglement was introduced when the
quantum field theory has a global charge. For a spherical entangling surface,
the bulk dual corresponds to charged static topological black holes in the
grand canonical ensemble. The charged entanglement has been also studied for
the case when the bulk is coupled to Born–Infeld electrodynamics [16] and
higher-derivative Einstein–Maxwell electrodynamics [17].

The aim of this paper is twofold; first we generalise the charge Renyi entropy
to the case where the bulk theory contains non-linear sources, and to treat
the issue of Gaussian fluctuations in the thermal partition function evaluated
on the hyperbolic charged static solution in order to study corrections to the
holographic charged Renyi entropy proposed in [15]. The fluctuations corrects
the black hole entropy adding a new subleading term that has been extensively
study in the literature, and whose coefficient will depend on the approach to
quantum gravity, see for instance [18] and reference therein. This new
contribution modifies the IR structure of the Renyi entropy in a very specific
way while keeping the UV entanglement unmodified. Interestingly, this behavior
has been observed in field theories with spontaneous symmetry breaking (SSB)
of a continuous symmetry due to low energy excitations on the ground state
[19, 20, 21, 22, 23, 24]. This means that, holographically, this contribution
cannot be extracted from the classical saddle point of the dual gravitational
theory [25, 26]. While the matter fields does not play a relevant role in the
particular structure of the corrected Renyi entropy, the consideration of non-
linear sources allows us to explore different holographic models beyond the
examples of charged AdS black holes known in the literature so far. In
particular, we focus in the holographic dual of two-dimensional charged free
bosons that we found to match the Renyi entropy and twist operators obtained
using Einstein AdS gravity coupled to conformal electrodynamics in the bulk.

The present work is organised as follows:

#### $\diamond$

In section 2, we give a brief review of the standard Renyi entropy and its
generalisation to theories with global symmetries. Using the proposal of [27,
15] for the holographic dual of (charged) Renyi entropy for spherical
entangling region, we introduce the main concepts that are going to be used
extensively through this manuscript.

#### $\diamond$

In section 3, we extend the gravity dual theory to include non-linearly
charged black holes by considering a general NED Lagrangian on top of the
Einstein–Hilbert action in asymptotically AdS spacetimes. We analyse, in a
general framework, the thermodynamic description and we compute relevant
quantities such as the vacuum energy, temperature, and the specific heat.
Then, in subsection 3.1 we obtain the desire holographic data in presence of
the NED Lagrangian, such as the conformal weights of the twist operators, the
magnetic response, and the charged Renyi entropy.

#### $\diamond$

In section 4, we review the method of [28] to compute thermal fluctuations in
the gravity partition function in the grand canonical ensemble, and discuss
how the new terms modify the Renyi entropy, as proposed in [29], and the twist
operators conformal weights in general NED.

#### $\diamond$

In section 5, we present a specific example in three-dimensional gravity in
which the NED theory is conformal and the black hole solution is
asymptotically AdS. This last feature differs with the static charged BTZ [30]
which contains a logarithmic term in the metric that breaks the asymptotic
symmetries of the solution. The resulting Renyi entropy and conformal weights
reproduce the result of a charged free boson in two-dimensional CFT, which is
dual to the Einstein–Maxwell theory in higher dimensions, but not in $d=2$ as
can be seen in [15]. Furthermore, we compute the thermal fluctuations of
system generating the extra logarithmic term in the thermal entropy. This new
term generates a subleading logarithmic divergence in the Renyi and
entanglement entropy, which resembles the entanglement entropy of SSB systems.
Therefore, the coefficient appearing in the logarithmic corrections to the
black hole thermal entropy can be interpreted as the number of Goldstone modes
in the dual CFT appearing in a SSB of a continuous symmetry.

We conclude with some discussion of the main results presented in this paper
and further directions.

##  2 Holographic charged Renyi Entropy

Renyi entropy corresponds to a quantity that generalises various notions of
entropy while preserving additivity for non-dependant events, and has shown to
be an important quantity in several areas of physics. Particularly, Renyi
entropy gives information of the entangled structure in many-body quantum
ground states. Considering a system subdivided in $A$ and $B$ by the
entangling region $\Sigma$, tracing over the states in $B$ defines the reduced
density matrix $\rho_{A}={\rm Tr}_{B}\rho$ , where $\rho$ corresponds to the
normalised density matrix of the full system. Then, the Renyi entropy of the
configuration is

$\displaystyle S_{n}=\frac{1}{1-n}\log{\rm
Tr}_{A}\rho_{A}^{n}\leavevmode\nobreak\ ,$ (2.1)

and satisfies the following inequalities [31, 32]

$\displaystyle\partial_{n}S_{n}\leq 0\leavevmode\nobreak\
,\qquad\partial_{n}\left(\frac{n-1}{n}S_{n}\right)\geq 0\leavevmode\nobreak\
,\qquad\partial_{n}((n-1)S_{n})\geq 0\leavevmode\nobreak\
,\qquad\partial_{n}^{2}((n-1)S_{n})\leq 0\leavevmode\nobreak\ .$ (2.2)

Different limits of the Renyi entropy give useful information about the
reduced density matrix; the $n\to 1$ limit gives the Von Neumann entropy

$\displaystyle S_{\rm E}:=\lim_{n\to 1}S_{n}=-{\rm
Tr}_{A}\rho_{A}\log\rho_{A}\leavevmode\nobreak\ ,$ (2.3)

that defines the entropy of entanglement between the degrees of freedom in
both regions. Other two interesting limits are

$\displaystyle S_{0}:=\lim_{n\to 0}S_{n}=\log{\mathbf{d}}\leavevmode\nobreak\
,\qquad S_{\infty}:=\lim_{n\to\infty}S_{n}=-\log{\lambda_{\rm
max}}\leavevmode\nobreak\ ,$ (2.4)

where $\mathbf{d}$ is the number of non-zero eigenvalues of $\rho_{A}$, that
is, the dimension of the subspace of the Hilbert space in the region $A$ that
participates in the entanglement, and $\lambda_{\rm max}$ is the largest
eigenvalue of $\rho_{A}$. An interesting fact is that the second Renyi
entropy, $S_{2}:=\lim_{n\to 2}S_{n}$ has been measured experimentally in [33]
corresponding to the bipartition entanglement between two identical systems,
and the proposal to extend this experiment for the inclusion of global
symmetries can be found in [34].

Furthermore, the Renyi entropy has a thermal interpretation by considering a
field theory at temperature $T_{0}$ and with free energy functional
$F(T_{0})$. For a Gibbs state at temperature $T=T_{0}/n$, the Renyi entropy
can be defined as [35]

$\displaystyle
S_{n}=\left(1-\frac{1}{n}\right)^{-1}\frac{1}{T_{0}}\left[F(T_{0})-F(T_{0}/n)\right]\leavevmode\nobreak\
,$ (2.5)

and can easily be related to the thermal entropy of the system by means of
standard thermodynamic relations, viz.

$\displaystyle
S_{n}=\left(1-\frac{1}{n}\right)^{-1}\frac{1}{T_{0}}\int^{T_{0}}_{T_{0}/n}S_{\rm
thermal}(T)dT,$ (2.6)

such that the Renyi parameter is associated with the decreasing of the
temperature.

In practice, obtaining the Renyi entropy is rather complicated but, for
certain field theories, it can be done through the Replica trick [36]. This
allows to replace the computation of the $n$-th power of the reduced density
matrix, $\rho_{A}^{n}$ , with obtaining the partition function of the original
field theory on the replicated manifold, $Z_{n}$ , which contains $n$-copies
of the original manifold glued cyclically along the $n$-fold branch-cuts
introduced in the entangling region, i.e.,

$\displaystyle{\rm Tr}_{A}\rho_{A}^{n}=\frac{\int\left[\prod_{i=1}^{n}{\cal
D}\phi_{i}\right]\prod_{j}\sigma_{j,n}(\omega_{k}){\rm
exp}\\{-\sum_{k=1}^{n}I_{k}[\phi]\\}}{\int\left[\prod_{i=1}^{n}{\cal
D}\phi_{i}\right]\leavevmode\nobreak\ {\rm
exp}\\{-\sum_{k=1}^{n}I_{k}[\phi]\\}}=\frac{Z_{n}}{Z_{1}^{n}}\leavevmode\nobreak\
,$ (2.7)

with $\sigma_{n}$, usually referred as twist operators, located at the end-
points $\omega_{k}$ of the $n$-fold branch-cuts [36, 27, 37]. Then, the Renyi
entropy (2.3) becomes

$\displaystyle S_{n}=\frac{1}{1-n}(n\log Z_{1}-\log Z_{n})\leavevmode\nobreak\
.$ (2.8)

The computation of the Euclidean path integral on a complicated Riemann
surface also correspond to compute correlation function of twist operators,
which is indeed something achievable in quantum field theories. For a two-
dimensional CFT, the twist operators are primaries with scaling dimension

$\displaystyle
h_{n}=\frac{c}{12}\left(n-\frac{1}{n}\right)\leavevmode\nobreak\ ,$ (2.9)

where $c$ is the central charge of the theory, and in higher-dimensional CFT a
power expansion of the conformal weights $h_{n}$ in the Renyi index $n$ around
$n=1$ found to be proportional to the structure coefficients of the theory
[27, 38, 39, 37].

Holographically, Ryu and Takayanagi [10, 11] proposed, and later proved by
Lewkowycz and Maldacena [40], that the entanglement entropy of a field theory
is dual to extremising the area of a co-dimension 2 surface whose boundary
corresponds to be conformal to the entangling region $\Sigma$. This idea
generalises the Bekenstein–Hawking area law for black hole entropy in the case
in which the minimal surface equals the black hole horizon.

For the case of a zero-temperature $d$-dimensional CFT in flat space with
spherical entangling region, Casini, Huerta, and Myers [41] showed that the
density matrix can be mapped to a thermal density matrix on a CFT in a
hyperbolic cylinder $\mathbb{R}\times{\mathbb{H}}^{d-1}$ of radius $R$, which
corresponds to a slicing of AdS space, at temperature111See [42] for the
extension of the CHM map to CFTs with planar boundaries or planar defects.

$\displaystyle T_{0}=\frac{1}{2\pi R}\leavevmode\nobreak\ ,$ (2.10)

such that the entanglement entropy of the non-thermal CFT is mapped to the
thermal entropy of the CFT on the hyperbolic cylinder at temperature $T_{0}$.
This is usually known as the CHM map, and corresponds to a unitary
transformation of the density matrix such that, after taking traces, the
quantum information of the system remains invariant. From the bulk point of
view, one can use the CHM map to relate the entanglement entropy of a non-
thermal CFT with spherical entangling region to the Bekenstein–Hawking entropy
of a static topological black hole with Hawking temperature $T_{0}$.
Therefore, this map in the gravity side corresponds to a coordinate
transformation which renders the black hole horizon into the Ryu–Takayangi
minimal surface.

The holographic Renyi entropies are usually mapped to gravity duals on non-
smooth geometries [43] due to the Replica trick. These geometries contain
conical singularities [44] that must be included in the gravity partition
function. The CHM map reduces the computation to expressing the formula (2.6)
in terms of the black hole entropy and temperature defined by the horizon
radius $r_{h}$. Therefore, the equation (2.6) from the point of view of the
bulk, becomes

$\displaystyle S_{n}={}$
$\displaystyle\left(1-\frac{1}{n}\right)^{-1}\frac{1}{T_{0}}\int^{x_{1}}_{x_{n}}S(x)\partial_{x}T(x)dx\leavevmode\nobreak\
,$ (2.11) $\displaystyle={}$
$\displaystyle\left(1-\frac{1}{n}\right)^{-1}\frac{1}{T_{0}}\left[S(x)T(x)\rvert_{x_{n}}^{x_{1}}-\int_{x_{n}}^{x_{1}}T(x)\partial_{x}S(x)dx\right],$
(2.12)

where $x:=r_{h}/L$ , with $L$ the AdS radius, $S(x)$ and $T(x)$ are the black
hole entropy and temperature respectively, and $x_{n}$ represents the change
of the horizon radius in terms of the change of temperature $T(x_{n})=T_{0}/n$
.

One can be a bit more curious and wonder what happens with the above
definitions in a CFT with a continuous global symmetry. This question has been
already answered in [39] for the case of a Wilson line that encircles the
entangling surface. The conserved global charge adds a fixed chemical
potential $\mu$ to the free energy without modifying the form of (2.11) and
the twist operators get a dependence on the chemical potential due to the
magnetic flux carried by the entangling surface. Their conformal dimension
$h_{n}(\mu)$ is extracted from the leading order behaviour of the correlators

$\displaystyle\langle
T_{ab}\sigma_{n}(\mu)\rangle=-\frac{h_{n}}{2\pi}\frac{\delta_{ab}}{y^{d}}\leavevmode\nobreak\
,\qquad\langle
T_{ij}\sigma_{n}(\mu)\rangle=-\frac{h_{n}}{2\pi}\frac{dn_{i}n_{j}-(d-1)\delta_{ij}}{y^{2}}\leavevmode\nobreak\
,$ (2.13)

where $n_{i}$ are unit vectors orthogonal to the insertion of the twist
operators, $y$ is the perpendicular distance between the insertion of the
stress tensor and the twist operator, and the indices $(a,b)$ denote
tangential directions to the twist operators, and indices $(i,j)$ denote
normal directions. In fact, the conformal weights $h_{n}(\mu)$ can be computed
in terms of the energy density ${\cal{E}}(T,\mu)$ of the CFT [39] as

$\displaystyle h_{n}(\mu)=\frac{2\pi
n}{d-1}R^{d}\left({\cal{E}}(T_{0},0)-{\cal{E}}(T_{0}/n,\mu)\right)\leavevmode\nobreak\
.$ (2.14)

In the bulk, the difference of energy densities is extracted from the
$(t,t)$-component holographic stress tensor [45] which in the case of the
topological black hole leads to

$\displaystyle h_{n}(\mu)=-\pi n\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}m(x_{n},\mu)\leavevmode\nobreak\ ,$ (2.15)

where $m$ is a parameter related with the mass of the black hole written in
terms of the chemical potential and $x_{n}$. The last expression holds
independently of the gauge sector of the bulk theory. The gauge field
$A_{\mu}$ in the bulk is dual to a current $J_{i}$ at the boundary from which
the leading order divergence of the correlator

$\displaystyle\langle
J_{i}\sigma_{n}(\mu)\rangle=-\frac{ik_{n}}{2\pi}\frac{\epsilon_{ij}n^{j}}{y^{d-1}}\leavevmode\nobreak\
,$ (2.16)

where $\epsilon_{ij}$ is the volume form of the space transverse to the
entangling region, and the structure coefficient $k_{n}(\mu)$ is referred as
the magnetic response, given by

$k_{n}(\mu)=2\pi nR^{d}\rho(x_{n},\mu)\leavevmode\nobreak\ ,$ (2.17)

where $\rho(n,\mu)$ corresponds to the charge density of the theory appearing
in the first law of thermodynamics

$\displaystyle d{\cal E}=TdS+\frac{\mu}{2\pi R}d\rho\Rightarrow\frac{\mu}{2\pi
R}=\left(\frac{\partial{\cal E}}{\partial\rho}\right)_{S}\leavevmode\nobreak\
.$ (2.18)

Additionally, the coefficient $C_{T}$ in the leading singularity of the stress
tensor two-point function, which encodes information about the central charges
of the theory, can be expressed as [27, 38],

$\displaystyle
C_{T}=\frac{d-1}{2\pi^{\frac{d-2}{2}}\Gamma(d/2)}h_{1,0}\leavevmode\nobreak\
,$ (2.19)

where $\Gamma(d)$ is the Gamma function and $h_{1,0}$ is a coefficient in an
expansion of the conformal weight around $n=1$ and $\mu=0$,

$\displaystyle
h_{n}(\mu)=\sum_{a,b}\frac{1}{a!b!}h_{a,b}(n-1)^{a}\mu^{b}\leavevmode\nobreak\
,\qquad
h_{a,b}=\partial_{n}^{a}\partial_{\mu}^{b}h_{n}(\mu)\Big{\rvert}_{{\begin{subarray}{c}n=1\\\
\mu=0\end{subarray}}}\leavevmode\nobreak\ .$ (2.20)

Thus, the coefficients of the most singular order of the correlators, which
encodes most of the crucial information about the CFT, can be extracted by
expansions of the conformal weights of twist operators. In the next section we
will describe the bulk theory and the holographic setup.

##  3 AdS Gravity and non-linear Electrodynamics

For the bulk theory, we consider the Einstein–Hilbert AdS action coupled to an
arbitrary NED density ${\cal L}({\cal S})$ that depends on the Lorentz
invariant scalar ${\cal{S}}=F_{\mu\nu}F^{\mu\nu}$ , where $F_{\mu\nu}$ is the
field strength of the $U(1)$ gauge field $A_{\mu}$ ,

$\displaystyle I=\int d^{d+1}x\sqrt{g}\left(\frac{1}{2\ell_{\rm
P}^{d-1}}\left({\cal R}+\frac{d(d-1)}{L^{2}}\right)+{\cal
L}({\cal{S}})\right)\leavevmode\nobreak\ ,$ (3.1)

where ${\cal R}={\cal R}^{\mu\nu}g_{\mu\nu}$ is the Ricci scalar, and $L$ is
the characteristic AdS radius. The equations of motion reads

$\displaystyle E^{\mu}_{\nu}\equiv{}$ $\displaystyle{\cal
R}^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu}({\cal R}-2\Lambda)-\ell_{\rm
P}^{d-1}T^{\mu}_{\nu}=0\leavevmode\nobreak\ ,$ (3.2) $\displaystyle
E^{\mu}\equiv{}$ $\displaystyle\nabla_{\nu}\left(F^{\mu\nu}\frac{\partial{\cal
L}}{\partial{\cal S}}\right)=0\leavevmode\nobreak\ ,$ (3.3)

where the NED energy-momentum tensor is

$\displaystyle T^{\mu}_{\nu}=\delta^{\mu}_{\nu}{\cal L}-4\frac{\partial{\cal
L}}{\partial{\cal S}}F^{\mu\alpha}F_{\nu\alpha}\leavevmode\nobreak\ .$ (3.4)

Solving the field equations for a static ansatz222The normalisation chosen is
such that for $k=-1$ [27] the boundary line element becomes conformally
equivalent to $ds^{2}=-dt^{2}+R^{2}dH_{d-1}^{2}$ corresponding to a hyperbolic
cylinder of radius $R$ where $dH^{2}_{d-1}$ is the volume element of the
$(d-1)$ Lobachevsky disc.

$\displaystyle
ds^{2}=-f(r)\frac{L^{2}}{R^{2}}dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\gamma_{mn}dy^{m}dy^{n}\leavevmode\nobreak\
,$ (3.5)

where the curvature of the horizon is labeled by the topological parameter
$k=0,1,-1$ for flat, spherical, or hyperbolic case, respectively. The gauge
field ansatz

$\displaystyle A_{\mu}=\left(\frac{L}{R}\phi(r)-\frac{\mu}{2\pi
R}\right)\delta^{t}_{\mu}\leavevmode\nobreak\ ,\qquad\mu\equiv 2\pi
L\phi(r_{h})\leavevmode\nobreak\ ,$ (3.6)

where $\mu$ is the chemical potential chosen such that the gauge potential
vanishes at the horizon, generates an electric field
$E(r)=-\partial_{r}\phi(r)$ and the associated field strength

$\displaystyle
F_{\mu\nu}=\frac{L}{R}E(r)\left(\delta^{t}_{\mu}\delta^{r}_{\nu}-\delta^{t}_{\nu}\delta^{r}_{\mu}\right)\leavevmode\nobreak\
,$ (3.7)

such that

$\displaystyle{\cal{S}}=-2E^{2}\leavevmode\nobreak\ .$ (3.8)

We can plug the metric function $f(r)$ into the equations of motion and using
the fact that $T^{t}_{t}=T^{r}_{r}$, we get

$\displaystyle
E^{t}_{t}=E^{r}_{r}=\frac{d-1}{2r^{2}}\left(rf^{\prime}(r)+(d-2)(f(r)-k)-d\frac{r^{2}}{L^{2}}\right)-\ell_{\rm
P}^{d-1}T^{r}_{r}=0,$ (3.9)

that is solved for

$\displaystyle f(r)=\frac{r^{2}}{L^{2}}+k-\frac{m}{r^{d-2}}+\frac{2\ell_{\rm
P}^{d-1}}{d-1}\frac{{\cal Q}}{r^{d-2}}\leavevmode\nobreak\ ,$ (3.10)

where

$\displaystyle{\cal Q}(r):=\int_{\infty}^{r}du\leavevmode\nobreak\
u^{d-1}T^{r}_{r}=\int_{\infty}^{r}du\left(u^{d-1}{\cal L}({\cal
S})-4{\tilde{q}}E\right)\leavevmode\nobreak\ .$ (3.11)

The parameter $m$ is an integration constant that can be related to the black
hole mass through the Noether-Wald formalism. The electric charge of the black
hole can be obtained as the Noether charge associated to the $U(1)$ gauge
symmetry [46]. The black hole horizon is given by the root of $f(r_{h})=0$. We
can use this to write the mass as a function of the horizon,

$\displaystyle
m=\frac{r_{h}^{d-2}}{L^{2}}\left(r_{h}^{2}+kL^{2}\right)+\left(\frac{2\ell_{\rm
P}^{d-1}}{d-1}\right)\frac{{\cal Q}(r_{h})}{r_{h}^{d-2}}$ (3.12)

and thus, express the metric function as

$\displaystyle f(r)=\frac{r^{2}}{L^{2}}+k+\frac{2\ell_{\rm
P}^{d-1}}{d-1}\frac{1}{r^{d-2}}{\cal
Q}(r)-\left(\frac{r_{h}}{r}\right)^{d-2}\left(\frac{r^{2}_{h}}{L^{2}}+k+\frac{2\ell_{\rm
P}^{d-1}}{d-1}\frac{1}{r^{d-2}_{h}}{\cal Q}(r_{h})\right)\leavevmode\nobreak\
.$ (3.13)

The black hole temperature is

$\displaystyle
T=\frac{T_{0}}{2}Lf^{\prime}(r_{h})=\frac{T_{0}}{2}\frac{L}{r}\left[d\frac{r_{h}^{2}}{L^{2}}+(d-2)k+\frac{2\ell_{\rm
P}^{d-1}}{d-1}r_{h}^{2}T^{r}_{r}({r_{h}})\right]\leavevmode\nobreak\ ,$ (3.14)

such that extremality occurs at

$\displaystyle T_{r}^{r}(r_{h})=-\frac{d-1}{2\ell_{\rm
P}^{d-1}r_{h}^{2}}\left(d\frac{r^{2}_{h}}{L^{2}}+k(d-2)\right)\leavevmode\nobreak\
,$ (3.15)

and the Wald entropy corresponds to

$\displaystyle
S^{(0)}={\mathtt{V}}\left(\frac{r_{h}}{L}\right)^{d-1}\leavevmode\nobreak\
,\qquad{\mathtt{V}}\equiv{2\pi}\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}V_{\Sigma}\leavevmode\nobreak\ ,$ (3.16)

where $V_{\Sigma}$ is the volume of the horizon that for the hyperbolic case
($k=-1)$

$\displaystyle
V_{\mathbb{H}^{d-1}}=\int_{\mathbb{H}^{d-1}}d\Sigma_{d-1}\sim\frac{\Omega_{d-2}}{d-2}\frac{R^{d-2}}{\delta^{d-2}}+\dots\leavevmode\nobreak\
,\qquad\Omega_{d-2}\equiv\frac{\pi^{\frac{d-2}{2}}}{\Gamma(d/2)}\leavevmode\nobreak\
,$ (3.17)

is divergent, regularised by inserting a short-distance cutoff $\delta$, such
that the Renyi entropy universal divergences are already encoded in this
quantity [41]. For $d=2$ , the volume logarithmically diverges

$\displaystyle
V_{\mathbb{H}^{1}}=2\log\left(\frac{2R}{\delta}\right)\leavevmode\nobreak\ ,$
(3.18)

which encapture the logarithmic divergences in two-dimensional CFT
entanglement entropy [36]. We can integrate the gauge field equation (3.3) to
obtain a generalised Gauss law

$\displaystyle E^{t}=0=\partial_{r}\left(r^{d-1}E(r)\frac{\partial{\cal
L}}{\partial{\cal S}}\right)\Rightarrow E(r)\frac{\partial{\cal
L}}{\partial{\cal S}}\Bigg{\rvert}_{{\cal
S}=-2E^{2}}=-\frac{\tilde{q}}{{r^{d-1}}}\leavevmode\nobreak\ ,$ (3.19)

where ${\tilde{q}}$ can be related with the physical electric charge of the
solution.

It is known that for black holes with a non-linear electrodynamic source, the
Smarr relation is not satified, see [47] for a discussion. Nonetheless, in
[48] was noticed that instead of relying on the Smarr relation to derive the
first law of black hole thermodynamics, it is possible to get the quantum
statistical relation from the regularised Euclidean action

$\displaystyle G\equiv TI={\cal E}-TS-\frac{\mu}{2\pi
R}\rho\leavevmode\nobreak\ .$ (3.20)

It follows that the first law of black hole thermodynamics is

$\displaystyle d{\cal E}=TdS+\frac{\mu}{2\pi R}d\rho\leavevmode\nobreak\ ,$
(3.21)

where the total energy of the system ${\cal E}=M+\frac{1}{2}(1+(-1)^{d})E_{\rm
vac}$ , with $M$ as the ADM mass, gets a contribution from the Casimir energy
of the dual field theory when $d$ is even [49],

$\displaystyle E_{\rm
vac}=-\frac{2V_{\Sigma}}{\sqrt{\pi}}\frac{\Gamma((d+1)/2)}{\Gamma(d/2)}\frac{(-L)^{\frac{d}{2}-1}}{\ell_{\rm
P}^{d-1}}\lim_{r\to\infty}\int_{0}^{1}dy\left(f-\frac{r}{2}f^{\prime}\right)\left(y^{2}\left(\frac{r^{2}}{L^{2}}-f\right)+k\right)^{\frac{d}{2}-1}\leavevmode\nobreak\
,$ (3.22)

that corresponds to the vacuum energy of global AdS spaces in odd dimensions.
Assuming finiteness of the total energy, the Casimir energy is

$\displaystyle E_{\rm
vac}=(-k)^{\frac{d}{2}}\frac{L^{\frac{d}{2}-1}}{\ell_{\rm
P}^{d-1}}V_{\Sigma}=\frac{(\pi L)^{\frac{d}{2}-1}}{(d-2)\ell_{\rm
P}^{d-1}\Gamma(d/2)}\frac{R^{d-2}}{\delta^{d-2}}+\dots\leavevmode\nobreak\ ,$
(3.23)

where the second equality corresponds to the hyperbolic case $k=-1$. We will
use the generalised quantum statistical relation to see how the thermal
fluctuations modify the known holographic quantities such as the conformal
weight of the twist operators.

The black hole specific heat

$\displaystyle{\cal{C}}=T\left(\frac{\partial}{\partial
T}S^{(0)}\right)=T(d-1){\mathtt{V}}x^{d-2}\frac{\partial x}{\partial
T}\leavevmode\nobreak\ ,$ (3.24)

where

$\displaystyle\frac{\partial x}{\partial
T}=\frac{2}{T_{0}}\left[d-\frac{k(d-2)}{x^{2}}+\frac{2\ell_{\rm
P}^{d-1}L^{2}}{d-1}\left(T^{r}_{r}+x\partial_{x}T^{r}_{r}\right)\right]^{-1},$
(3.25)

is expressed in terms of the NED energy-momentum tensor evaluated at the
horizon $x$. In other words, the specific heat is

$\displaystyle{\cal{C}}={}$
$\displaystyle\frac{2(d-1){\mathtt{V}}x^{d}T}{T_{0}\left[dx^{2}-k(d-2)+\frac{2\ell_{\rm
P}^{d-1}L^{2}}{d-1}x^{2}\left(T^{r}_{r}+x\partial_{x}T^{r}_{r}\right)\right]}\leavevmode\nobreak\
,$ (3.26)

that by using the entropy formula (3.16) is cast in the same footing as it is
presented in the rest of the literature for particular NED theories (see for
instance [50, 51]),

$\displaystyle{\cal
C}=(d-1)S^{(0)}\left[\frac{dx^{2}+(d-2)k+\frac{2L^{2}\ell_{\rm
P}^{d-1}}{d-1}x^{2}T^{r}_{r}}{dx^{2}-(d-2)k+\frac{2L^{2}\ell_{\rm
P}^{d-1}}{d-1}x^{2}\left(T^{r}_{r}+x\partial_{x}T_{r}^{r}\right)}\right]\leavevmode\nobreak\
.$ (3.27)

Requiring positive heat capacity implies that,

$\displaystyle
T^{r}_{r}(x)+x\partial_{x}T^{r}_{r}(x)\geq-\frac{d-1}{2\ell_{\rm
P}^{d-1}L^{2}}\left(d-\frac{k(d-2)}{x^{2}}\right)\leavevmode\nobreak\ ,$
(3.28)

where all terms are evaluated at ${\cal S}=-2E^{2}$ . Additionally, the weak
energy condition $\rho=-T_{\mu\nu}u^{\mu}u^{\nu}\geq 0$ puts an extra
constraint in the NED energy-momentum tensor,

$\displaystyle T^{r}_{r}=T^{t}_{t}={\cal L}+4\frac{\partial{\cal
L}}{\partial{\cal S}}E^{2}\geq 0\leavevmode\nobreak\ .$ (3.29)

Then

$\displaystyle\partial_{x}T^{r}_{r}(x)\geq-\frac{d-1}{2\ell_{\rm
P}^{d-1}L^{2}x}\left(d-\frac{k(d-2)}{x^{2}}+\frac{2\ell_{\rm
P}^{d-1}L^{2}}{d-1}T^{r}_{r}\right)\leavevmode\nobreak\ ,$ (3.30)

which for the topological black hole with hyperbolic horizon $k=-1$, the
r.h.s. of the inequality is always negative. As we saw in section 2, the
charged Renyi entropy for a CFT is defined in the grand canonical ensemble,
i.e., the chemical potential is fixed. Therefore, in order to obtain the
logarithmic corrections of the charged Renyi entropy, we are going to analyse
thermal fluctuations in the grand canonical ensemble where the stability
condition is modified [52], and the value of the heat capacity will play an
important role when particularising the result to different black hole
solutions.

###  3.1 Holographic Quantities

Having defined the relevant bulk expressions, we proceed with the dual
boundary description. We follow to list the relevant holographic quantities,
which in fact, the NED contributions to the boundary data can be expressed in
terms of the $rr$-component of the energy-momentum tensor evaluated at the
horizon.

The conformal weights of the twist operators

$\displaystyle h_{n}(\mu)=\pi n\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}\left(x_{n}^{d-2}L^{d-2}(1-x^{2}_{n})-\left(\frac{2\ell_{\rm
P}^{d-1}}{d-1}\right)\frac{{\cal
Q}(x_{n},\mu)}{\left(Lx_{n}\right)^{d-2}}\right)\leavevmode\nobreak\ ,$ (3.31)

and the magnetic response (2.17) can be obtained with the generalised charge
density [48]

$\displaystyle\rho(x_{n},\mu)=-4\left(\frac{L}{R}\right)^{d-2}x_{n}^{d-1}\left(E\frac{\partial{\cal
L}}{\partial{\cal S}}\right)\Bigg{\rvert}_{{\cal
S}=-2E^{2},\leavevmode\nobreak\ x=x_{n}}\leavevmode\nobreak\ ,$ (3.32)

and the central charge of the dual theory can be obtained by taking a series
expansion of the structure coefficients around $n=1$ . All the holographic
quantities depend on $x_{n}$ which corresponds to the largest solution to
$T(x_{n},\mu)=T_{0}/n$, viz.

$\displaystyle nx_{n}^{2}\left(d+\frac{2L^{2}\ell_{\rm
P}^{d-1}}{(d-1)}T^{r}_{r}(x_{n},\mu)\right)-2x_{n}+kn(d-2)=0\leavevmode\nobreak\
,$ (3.33)

with $k=-1$. The last equation cannot be always solved with analytic methods
for the horizon radius $x_{n}$. As a matter of fact, for most of the
lagrangians ${\cal L}({\cal{S}})$, (3.33) is solved numerically. The same
occurs for the chemical potential; for each NED the chemical potential can be
obtained numerically in terms of the electric charge and then be replaced in
the desire quantity333See for instance [16] for Born–Infeld electrodynamics.

Finally, the holographic charged Renyi entropy is obtained by evaluating
(2.11) leading to

$\displaystyle S_{n}={}$
$\displaystyle\frac{n}{n-1}\frac{{\mathtt{V}}}{2}\left\\{x^{d-2}+x^{d}+\frac{2\ell_{\rm
P}^{d-1}L^{2}}{d-1}\left({\cal Q}(x,\mu)+\int
x^{d}\partial_{x}T^{r}_{r}(x,\mu)dx\right)\right\\}\Bigg{\rvert}_{x_{n}}^{x_{1}}\leavevmode\nobreak\
,$ (3.34)

and the Renyi inequalities (2.2) can be checked using

$\displaystyle\partial_{n}S_{n}=-\frac{\partial x_{n}}{\partial
n}\left[(d-2)x^{d-3}_{n}+dx^{d-1}_{n}+\frac{2\ell_{\rm
P}^{d-1}L^{2}}{d-1}x_{n}^{d-1}\left(T^{r}_{r}(x_{n})+x_{n}\partial_{x_{n}}T^{r}_{r}(x_{n})\right)\right]\leavevmode\nobreak\
,$ (3.35)

where

$\displaystyle\frac{\partial x_{n}}{\partial
n}=\frac{x^{2}_{n}\left(d+\frac{2\ell_{\rm
P}^{d-1}}{d-1}T^{r}_{r}(x_{n})\right)-(d-2)}{2-4nx_{n}\left[d+\frac{2\ell_{\rm
P}^{d-1}}{d-1}\left(T^{r}_{r}(x_{n})+\frac{x_{n}}{2}\partial_{x_{n}}T^{r}_{r}(x_{n})\right)\right]}\leavevmode\nobreak\
.$ (3.36)

Finally, the Renyi index can be related with the black hole parameters as

$\displaystyle
n=\frac{2L}{r_{h}}\left[\frac{dr_{h}^{2}}{L^{2}}-(d-2)+\frac{2\ell_{\rm
P}^{d-1}}{d-1}r^{2}_{h}T_{r}^{r}(r_{h})\right]^{-1}\leavevmode\nobreak\ ,$
(3.37)

which can be used to check phase transitions in the dual theory [15, 53].

In the following section, we will include thermal fluctuations to black hole
entropy and use the holographic dictionary to interpret the correction from
the CFT perspective.

##  4 Thermal fluctuations and stability

Thermal fluctuations induces logarithmic corrections in the black hole
entropy. In the grand canonical ensemble with $K$ fixed chemical potentials
$\mu_{i}$, with $i=1,\dots,K$ , the standard procedure to obtain the
corrections requires to consider the partition function as a discrete sum over
energy states and take the continuum limit [54]. The entropy acquires a
correction [28] of the type

$\displaystyle S=S^{(0)}+\frac{1}{2}\log{\rm det}\leavevmode\nobreak\
D_{AB}+\log J\leavevmode\nobreak\ ,$ (4.1)

where $J$ is a Jacobian [54] that appears in due to the energy spectrum
transformation. To compute the new contributions, one assumes that the charges
and energy are linear in their respective quantum numbers, and that the
symmetric matrix $D_{AB}$ is given by

$\displaystyle
D_{AB}=\frac{\partial}{\partial\chi^{A}}\frac{\partial}{\partial\chi^{B}}\log
Z\leavevmode\nobreak\ ,$ (4.2)

where $\chi^{A}=\\{-\beta\mu_{i},\beta\\}$ with $\beta=T^{-1}$, are the
thermodynamic quantities considered in the fluctuations of the thermal
partition function $Z(\beta,\mu_{i}/T)$. The logarithmic corrections only
makes sense if ${\rm det}D_{AB}\geq 0$. This imposes conditions on the
thermodynamic quantities such as the heat capacity or Maxwell relations [55].
For instance, considering only one chemical potential [52], the determinant is

$\displaystyle{\rm
det}D_{AB}=T^{3}{\cal{C}}\kappa-T^{3}\mu-T^{4}\alpha^{2}\leavevmode\nobreak\
,$ (4.3)

where

$\displaystyle{\cal C}=T\left(\frac{\partial S}{\partial
T}\right)_{\mu}\leavevmode\nobreak\
,\qquad\kappa=-\frac{1}{T}\left(\frac{\partial
N}{\partial\lambda}\right)_{T}\leavevmode\nobreak\
,\qquad\alpha=\left(\frac{\partial N}{\partial
T}\right)_{\mu}\leavevmode\nobreak\ ,$ (4.4)

with $\lambda=-\mu/T$ and $N$ corresponding to the thermodynamic charge
conjugated to the chemical potential (electric charge in our case). Then, the
condition on the determinant leads to

$\displaystyle{\cal{C}}\geq
T\alpha\left(\frac{\alpha}{\kappa}+\frac{\mu}{T}\right)\leavevmode\nobreak\ .$
(4.5)

This expression defines thermal local stability of the system. In the
canonical ensemble, this condition simplifies to

$\displaystyle
S=S^{(0)}+\frac{1}{2}\log\left({\cal{C}}T^{2}\right)\leavevmode\nobreak\ ,$
(4.6)

in agreement with [18].

Consideration of thermal or quantum fluctuations have shown to modify the
entropy as

$\displaystyle S=S^{(0)}-{\mathtt{C}}\log S^{(0)}+\dots\leavevmode\nobreak\ ,$
(4.7)

and the value of the ${\mathtt{C}}$ coefficient depends on the particular
black hole solution and the ensemble under consideration, but changes
depending on the approach to quantum gravity and allows to test holographic
theories beyond the saddle approximation, see for instance [18, 56, 57, 54,
58, 59, 60, 61, 62, 63, 55]. We will see how the ${\mathtt{C}}$ coefficient
can be interpreted in an holographic context by considering thermal
fluctuations on NED solutions and check how affect the holographic quantities
computed in subsection 3.1.

The thermal fluctuation back reacts into the Renyi entropy [29, 64] and the
charged Renyi entropy that becomes

$\displaystyle S_{n}={}$
$\displaystyle\frac{n}{n-1}\frac{1}{T_{0}}\int^{x_{1}}_{x_{n}}\left(S^{(0)}-{\mathtt{C}}\log\left({\mathtt{V}}x^{d-1}\right)\right)\partial_{x}T(x,\mu)dx$
(4.8) $\displaystyle={}$
$\displaystyle\frac{n}{n-1}\left\\{\frac{{\mathtt{V}}}{2}\left(x^{d-2}+x^{d}\right)-\frac{{\mathtt{C}}}{2}\left[\frac{(2-d(1-x^{2}))(d-1)+(2-d(1+x^{2}))\log{\mathtt{V}}x^{d-1}}{x}\right]\right\\}\Bigg{\rvert}_{x_{n}}^{x_{1}}$
$\displaystyle+{}$ $\displaystyle\frac{n}{n-1}\left(\frac{\ell_{\rm
P}^{d-1}L^{2}}{d-1}\right)\int^{x_{1}}_{x_{n}}\left({\mathtt{V}}x^{d-1}-{\mathtt{C}}\log{\mathtt{V}}x^{d-1}\right)\left(T^{r}_{r}(x,\mu)+x\partial_{x}T^{r}_{r}(x,\mu)\right)dx\leavevmode\nobreak\
,$

where all the quantities are evaluated at fixed chemical potential prior to
integration. The first line of (4.8) corresponds to the universal term coming
from the Einstein–Hilbert action, while the second line contains the
contribution of the matter sector. Notice that turning off the gauge field and
taking ${\mathtt{C}}=3/2$, (4.8) reduces to the result obtained in [29]. It is
also important to notice that the last expression has a smooth $\mu\to 0$
limit. In fact, we can see from the first line that there is no explicit
dependence on the chemical potential, being relevant just for the value of
$x$. This means that the structure of the next to leading order contribution
in the Renyi entropy is fixed by the gravity solution, and not by the matter
source. Later on, we will examine this in detail for a concrete example where
we will see that this structure is present in systems where there has been
some spontaneous symmetry breaking of a spatial continuous symmetry.

The Renyi inequalities (2.2) are satisfied if the Wald entropy is positive
[65] as can be seen from

$\displaystyle\partial_{n}\left(\frac{n-1}{n}S_{n}(\mu)\right)=\frac{1}{n^{2}}S(T_{0}/n,\mu)\leavevmode\nobreak\
\geq 0\leavevmode\nobreak\ .$ (4.9)

Therefore, the Renyi inequalities can be violated if
${\mathtt{C}}\geq\frac{S^{(0)}}{\log S^{(0)}}$. In the grand canonical
ensemble ${\mathtt{C}}$ can take positive or negative values, but in the
canonical ensemble ${\mathtt{C}}$ is indeed positive [18], opening the
possibility for negative entropy. Nonetheless, as the value of ${\mathtt{C}}$
is usually small compared with the value of the entropy, the bound can be
violated only for microscopic black holes [66]. One can think of this coming
from an evaporation process of a black hole where at late times becomes so
small that the thermal fluctuations start dominating over the classical saddle
causing a “negative” entropy. However, this description is not accurate in
such scenario. In fact, these microscopic black holes appear to be thermally
unstable [67], which prevents us from describing the thermal fluctuations with
this method. A more precise description would require to consider how the
geometry itself reacts due to quantum effects modifying the thermodynamic
quantities in the regime where the solution is locally stable [68]. Therefore,
the Renyi inequalities of the dual theory can be understood as the thermal
stability condition of the black hole [69] even when logarithmic corrections
are considered.

Let us consider, for a moment, only the gravity sector, i.e. $\mu=0$ . We can
compute the conformal weight of the twist operator by integrating the first
law [27, 16]. This leads to

$\displaystyle h_{n}(\mu=0)={}$
$\displaystyle\frac{n}{(d-1)V_{\mathbb{H}^{d-1}}}\int^{x_{1}}_{x_{n}}T(x)\partial_{x}S(x)dx$
$\displaystyle={}$ $\displaystyle\pi n\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}\left(x_{n}^{d-2}-x_{n}^{d}\right)-\frac{n{\mathtt{C}}}{2V_{\mathbb{H}^{d-1}}}\frac{(1-x_{n})(dx_{n}+d-2)}{x_{n}}\leavevmode\nobreak\
,$ (4.10)

where

$\displaystyle
x_{n}=\frac{1}{dn}\left(1+\sqrt{1-2dn^{2}+d^{2}n^{2}}\right)\leavevmode\nobreak\
.$ (4.11)

Thus,

$\displaystyle\partial_{n}h_{n}\rvert_{n=1}=\frac{1}{d-1}\left(2\pi\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}-\frac{{\mathtt{C}}}{V_{\mathbb{H}^{d-1}}}\right)\leavevmode\nobreak\
.$ (4.12)

The second term is identified as next to the leading order contribution in the
saddle point approximation of the gravity dual [4]. From the quantum field
theory perspective, it corresponds to an IR correction of the central charge,
which is in agreement with the $c$\- and $F$-theorem in $d=2$ and $d=4$,
respectively [70, 71]. The latter is true if and only if, the coefficient
${\mathtt{C}}$ is positive, which is indeed required for the black hole to
satisfy the generalised second law [18].

Extending the result to nontrivial chemical potential we get

$\displaystyle h_{n}(\mu)={}$
$\displaystyle\frac{n}{(d-1)V_{\mathbb{H}^{d-1}}}\left(\int_{x_{n}}^{x_{1}}T(x,\mu)\partial_{x}S(x,\mu)dx-\frac{V_{\mathbb{H}^{d-1}}\mu}{2\pi
R}\rho(x_{n},\mu)\right)$ (4.13) $\displaystyle={}$ $\displaystyle\pi
n\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}\left(x_{n}^{d-2}-x_{n}^{d}\right)-\frac{n{\mathtt{C}}}{2V_{\mathbb{H}^{d-1}}}\frac{(1-x_{n})(dx_{n}+d-2)}{x_{n}}-\frac{n\mu}{2\pi
R}\frac{\rho(x_{n},\mu)}{d-1}\leavevmode\nobreak\ ,$ (4.14)

where $x_{n}=x_{n}(\mu)$ . In the grand canonical ensemble, the logarithmic
correction can have a negative ${\mathtt{C}}$ coefficient, such that the
entropy is always positive [72, 73, 28].

In the next section, as an analytic example, we particularise the results
obtained for the charged Renyi entropy and its logarithmic corrections for
three-dimensional AdS gravity coupled to Coulomb source. We will show how the
bulk seems to describe charged free bosons in $1+1$ dimensions, and how the
extra subleading logarithmic term in the entanglement resembles the
entanglement entropy of systems with SSB. For additional examples, see the
Appendix A.

##  5 Holographic charged free boson in 1+1

In the usual three-dimensional AdS gravity with Maxwell electrodynamics, the
BTZ black hole [30] is charged by

$\displaystyle{\cal Q}(r)=-\frac{q^{2}}{4\ell_{\rm
P}}\log\left(\frac{r}{\sqrt{mL^{2}}}\right)\leavevmode\nobreak\ .$ (5.1)

Clearly, this solution does not asymptotes to global AdS. From the dual point
of view, it can be interpreted as a behaviour that breaks conformal invariance
of the boundary field theory [74, 75, 76]. We can avoid this issue by
considering a particular NED theory whose Lagrangian corresponds to444From the
Power-law Lagrangian (A.1) studied in Appendix A, this corresponds to the
conformal invariant case $p=3/8$ in three dimensions, and
$\gamma=-2^{-3/4}\lambda$ .

$\displaystyle{\cal L}({\cal S})=-2^{\frac{1}{4}}\lambda|{\cal
S}|^{\frac{3}{4}}\leavevmode\nobreak\ ,$ (5.2)

which is indeed conformal in three dimensions [77]. The (traceless) energy-
momentum tensor

$\displaystyle T^{\mu}_{\nu}=\lambda|{\cal
S}|^{\frac{3}{4}}\left(\delta^{\mu}_{\nu}-\frac{3F^{\mu\alpha}F_{\nu\alpha}}{{\cal
S}}\right)\leavevmode\nobreak\ ,$ (5.3)

produces a Coulomb-like electric field

$\displaystyle E(r)=\frac{{\tilde{q}}}{r^{2}}\leavevmode\nobreak\ ,$ (5.4)

where it has been chosen $\lambda=-|{\tilde{q}}|^{1/2}/2\ell_{\rm P}$ and
there is a black hole solution [78] described by the metric function

$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}-m+\frac{2{\tilde{q}}^{2}}{r}\leavevmode\nobreak\ ,$
(5.5)

which indeed, is asymptotically AdS avoiding the aforementioned problem of the
charged BTZ black hole. The solution has a proper spacetime singularity at
$r=0$ that can be seen from the quadratic invariant

$\displaystyle
R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}=\frac{12}{L^{4}}+\frac{12{\tilde{q}}^{4}}{r^{6}}\leavevmode\nobreak\
.$ (5.6)

Therefore, this solution shares more similarities with a Reissner–Nordström-
AdS black hole than the usual charged BTZ black hole. The temperature of the
solution

$\displaystyle
T(x,\mu)=T_{0}\left(x-\tilde{\mu}^{2}\right)\leavevmode\nobreak\
,\qquad\tilde{\mu}^{2}:=\left(\frac{\mu}{2\pi}\right)^{2}\frac{1}{L}\leavevmode\nobreak\
.$ (5.7)

From above, it is easy to see that the temperature is positive if
$x\geq\tilde{\mu}^{2}$ (for real chemical potential) and the specific heat at
fixed chemical potential is

$\displaystyle{\cal
C}=S^{(0)}-{\mathtt{V}}\tilde{\mu}^{2}=\frac{T}{T_{0}}{\mathtt{V}}\leavevmode\nobreak\
.$ (5.8)

For vanishing charge, it reproduces the standard result $S^{(0)}>0$ that
guarantees local stability in the canonical ensemble. The horizon radius can
be computed analytically,

$\displaystyle x_{n}=\frac{1}{n}+\tilde{\mu}^{2}\leavevmode\nobreak\ .$ (5.9)

In fact, the solution for the horizon radius is always real giving freedom to
the value of $\mu$. This adds the possibility of Wick rotate the chemical
potential $\mu^{2}_{\rm E}=(i\mu)^{2}$ in the dual theory. Nonetheless, if we
ask for a positive horizon radius, we have that $(\tilde{\mu}_{E})^{2}\leq
n^{-1}$ for fixed $n$ . In the dual CFT, there is an upper bound for the
imaginary chemical potential due to the non-analyticity of the free energy
[39].

In order to highlight some features of this model, let us consider for a
moment the difference of free energies, at same temperature, between thermal
AdS3, $I^{E}_{AdS}$, and the solution $I^{E}_{\rm C}$,

$\displaystyle\Delta I^{E}(\mu)\equiv I^{E}_{\rm C}-I^{E}_{AdS}=-\frac{\pi
Lx}{\ell_{\rm
P}}\frac{T_{0}}{T}\left(2x+\tilde{\mu}^{2}\right)V_{\mathbb{H}^{1}}\leavevmode\nobreak\
,$ (5.10)

showing no Hawking–Page phase transition for a real chemical potential.
Instead, for imaginary chemical potential, we get

$\displaystyle\Delta I^{E}(\mu_{\rm E})\equiv I^{E}_{\rm
C}-I^{E}_{AdS}=T_{0}S^{(0)}\left(\frac{\tilde{\mu}_{\rm
E}^{2}-2x}{\tilde{\mu}_{\rm E}^{2}+x}\right)\leavevmode\nobreak\ .$ (5.11)

It clearly suggest that there could be a first order phase transition between
thermal AdS and the black hole at critical temperature
$T_{c}/T_{0}=\frac{3}{2}\tilde{\mu}_{\rm E}^{2}$ . This last point is
radically different from the BTZ scenario where the Hawking–Page phase
transition does not occur due to mass gaps between both configurations555In
this case, it is possible to produce a Hawking–Page phase transition by
considering contributions of conical deficits in the partition function [79,
80, 81]. [82]. Nonetheless, phase transitions in three-dimensional gravity
coupled to NED seems to be dependant of the thermodynamical ensemble [83], and
the solution (5.5) in the canonical ensemble does not shows phase transitions
for real electric charge [84].

As the solution is asymptotically AdS, we can compute the holographic
quantities defined in the previous sections. Let us start with the magnetic
response given in (2.17),

$\displaystyle k_{n}(\mu)=\frac{3R}{L}n\mu x_{n}\leavevmode\nobreak\ ,$ (5.12)

and the conformal weight of the holographic twisted operators from (3.31) for
imaginary chemical potential

$\displaystyle h_{n}(\mu)=\frac{\pi L}{\ell_{\rm
P}}\left[\left(n-\frac{1}{n}\right)+n\left(\frac{\mu_{\rm
E}}{2\pi}\right)^{4}\right]\leavevmode\nobreak\ .$ (5.13)

We can expand the conformal weights as done in (2.20), from which we read off
the central charge $c$ by noting that

$\displaystyle\lim_{n\to 1}\lim_{\mu\to 0}\partial_{n}h_{n}(\mu)=\frac{2\pi
L}{\ell_{\rm P}}\leavevmode\nobreak\ .$ (5.14)

The black hole entropy computed from the Wald formula is given simply by
$S=\frac{\pi L}{\ell_{\rm P}}V_{\mathbb{H}^{1}}x={\mathtt{V}}x$ , and the
Renyi entropy for an imaginary chemical potential is

$\displaystyle S_{n}=\frac{\pi L}{\ell_{\rm
P}}\left(\frac{1}{2}\left(1+\frac{1}{n}\right)-\left(\frac{\mu_{\rm
E}}{2\pi}\right)^{2}\right)V_{\mathbb{H}^{1}}\leavevmode\nobreak\ .$ (5.15)

This result can be matched with the CFT computation of the Renyi entropy for
free charged bosons in $d=2$ [39] by identifying $\left(\frac{\mu_{\rm
E}}{2\pi}\right)^{2}=\frac{|\mu_{\rm C}|}{2\pi}$ where $\mu_{\rm C}$ is the
imaginary chemical potential of the conformal bosons. In this way, the rate in
which the Renyi entropy decreases is dictated by the absolute value of the
imaginary chemical potential in both theories. The generalised twist operators
in a bosonic two-dimensional charged conformal field theory have conformal
dimensions $h_{n}$ of the form $(\tilde{m}/n+\mu_{\rm
E}/2\pi+\ell_{\tilde{m}})^{2}$ where $\tilde{m}$ and $\ell_{\tilde{m}}$ are
constants. The linear term in $\mu_{E}$ can be eliminated by the $U(1)$ gauge
freedom [15], and the form of the conformal weights and Renyi entropy are
exactly as (5.13) and (5.25) for small values of $\mu_{\rm C}$. The Renyi
entropy limits are

$\displaystyle S_{0}=\frac{{\mathtt{V}}}{2n}\leavevmode\nobreak\ ,\quad S_{\rm
E}={\mathtt{V}}(1+\tilde{\mu}^{2})\leavevmode\nobreak\ ,\quad
S_{\infty}=\frac{{\mathtt{V}}}{2}\left(1+2\tilde{\mu}^{2}\right)\leavevmode\nobreak\
,\quad
S_{2}={\mathtt{V}}\left(\frac{3}{4}+\tilde{\mu}^{2}\right)\leavevmode\nobreak\
,$ (5.16)

and for large chemical potential

$\displaystyle\lim_{\mu\to\infty}S_{n}(\mu)={\mathtt{V}}\tilde{\mu}^{2}+\frac{{\mathtt{V}}}{2n}(n+1)\leavevmode\nobreak\
,$ (5.17)

that is, indeed, $n$-independant at the leading order. The Renyi entropy
inequalities (2.2) can be checked analytically

$\displaystyle\partial_{n}S_{n}={}$
$\displaystyle-\frac{{\mathtt{V}}}{2n}\leavevmode\nobreak\ ,$
$\displaystyle\partial_{n}\left(\frac{n-1}{n}S_{n}\right)=\frac{{\mathtt{V}}(1+n\tilde{\mu}^{2})}{n^{3}}\leavevmode\nobreak\
,$ (5.18) $\displaystyle\partial_{n}\left((n-1)S_{n}\right)={}$
$\displaystyle\frac{{\mathtt{V}}}{2}\left(1+\frac{1}{n^{2}}+2\tilde{\mu}^{2}\right)\leavevmode\nobreak\
,$ $\displaystyle\leavevmode\nobreak\
\partial_{n}^{2}\left((n-1)S_{n}\right)=-\frac{{\mathtt{V}}}{n^{3}}\leavevmode\nobreak\
,$ (5.19)

which are satisfied for all real values of $\mu$. Also notice,
$\mu\partial_{\mu}S_{n}=2{\mathtt{V}}\tilde{\mu}^{2}$ , which is positive for
real $\mu$ . Despite this, the inequalities restricts the value of $\mu^{2}$
at fixed $n$ when the chemical potential is imaginary while the positive
horizon radius $\tilde{\mu}_{\rm E}\leq n^{-1}$ ensures the inequalities to be
satisfied.

The thermal fluctuations are obtained using the prescription given in section
4 in terms of the determinant of the matrix $D_{AB}$ and the Jacobian $J$. In
particular, the determinant,

$\displaystyle{\rm det}D_{AB}=\frac{L((S^{(0)})^{2}-\ell_{\rm
P}L{\mathtt{V}}^{2}\mu^{2})^{2}\left(\pi R(S^{(0)}-\ell_{\rm
P}L{\mathtt{V}}\mu^{2})((S^{(0)})^{2}-\ell_{\rm
P}L{\mathtt{V}}^{2}\mu^{2})-L{\mathtt{V}}^{2}\right)}{2\pi^{2}R^{2}(S^{(0)})^{4}{\mathtt{V}}^{2}}\leavevmode\nobreak\
,$ (5.20)

that for the zero chemical potential goes as $3\log S^{(0)}$ coinciding with
the value obtained by Carlip [56] from considering thermal fluctuations on the
torus partition function of a left- and right-moving CFT . The Jacobian is

$\displaystyle J=\frac{S^{(0)}{\mathtt{V}}}{(S^{(0)})^{2}-\ell_{\rm
P}L{\mathtt{V}}^{2}\mu^{2}}\leavevmode\nobreak\ ,$ (5.21)

and then, the corrected entropy becomes

$\displaystyle S={}$ $\displaystyle S^{(0)}-3\log
S^{(0)}+\log\left[\left((S^{(0)})^{2}-\ell_{\rm
P}L{\mathtt{V}}^{2}\mu^{2}\right)\left(((S^{(0)})-\ell_{\rm
P}L{\mathtt{V}}\mu^{2})((S^{(0)})^{2}-\ell_{\rm
P}L{\mathtt{V}}^{2}\mu^{2})\right)\right]\leavevmode\nobreak\ ,$ (5.22)

plus terms that are independent of the entropy of the system. As we can see,
the corrections contain a leading contribution of the form $-{\mathtt{C}}\log
S^{(0)}$ where we identify ${\mathtt{C}}=3$ . This kind of complicated
expression for the fluctuation appear to be common for higher-dimensional
black holes in open ensembles [55], and are valid in the stability regime

$\displaystyle{\cal C}\geq\frac{2\mu^{2}}{T}\frac{{\mathtt{V}}L}{2\pi
RS^{(0)}}\leavevmode\nobreak\ .$ (5.23)

This last inequality imposes a bound on the chemical potential given by

$\displaystyle\tilde{\mu}^{4}-2\tilde{\mu}^{2}\left(\frac{S^{(0)}}{{\mathtt{V}}}+\frac{L}{S^{(0)}}\right)+\left(\frac{S^{(0)}}{{\mathtt{V}}}\right)^{2}\geq
0\leavevmode\nobreak\ ,$ (5.24)

and (5.23) recovers the standard $S^{(0)}\geq 0$ restriction for vanishing
chemical potential.

Considering only the first correction, the Renyi entropy renders

$\displaystyle
S_{n}=\frac{n}{n-1}\frac{{\mathtt{V}}}{2}\left(x_{1}^{2}-x_{n}^{2}\right)+\frac{n}{n-1}{\mathtt{C}}\left(x_{1}-x_{n}-x_{1}\log{\mathtt{V}}x_{1}+x_{n}\log{\mathtt{V}}x_{n}\right)\leavevmode\nobreak\
,$ (5.25)

with ${\mathtt{C}}=3$ . Again, the corrected charged Renyi entropy has the
following limits

$\displaystyle S_{0}={}$
$\displaystyle\frac{{\mathtt{V}}}{2n}-{\mathtt{C}}\log\frac{{\mathtt{V}}}{n}\leavevmode\nobreak\
,$ (5.26) $\displaystyle S_{\rm E}={}$
$\displaystyle{\mathtt{V}}(1+\tilde{\mu}^{2})-{\mathtt{C}}\log\left({\mathtt{V}}(1+\tilde{\mu}^{2})\right)\leavevmode\nobreak\
,$ (5.27) $\displaystyle S_{\infty}={}$
$\displaystyle\frac{1}{2}(x_{1}-x_{\infty})(2{\mathtt{C}}+{\mathtt{V}}(x_{1}+x_{\infty}))-{\mathtt{C}}x_{1}\log{\mathtt{V}}x_{1}+{\mathtt{C}}x_{\infty}\log{\mathtt{V}}x_{\infty}\leavevmode\nobreak\
,$ (5.28) $\displaystyle S_{2}={}$
$\displaystyle{\mathtt{V}}\left(\frac{3}{4}-\tilde{\mu}^{2}\right)+{\mathtt{C}}+{\mathtt{C}}\left(1-2\tilde{\mu}^{2}\right)\log\left({\mathtt{V}}\left(\frac{1}{2}-\tilde{\mu}^{2}\right)\right)-2{\mathtt{C}}(1-\tilde{\mu}^{2})\log\left({\mathtt{V}}(1-\tilde{\mu}^{2})\right)\leavevmode\nobreak\
,$ (5.29)

where $x_{1}=1+\tilde{\mu}^{2}$ and $x_{\infty}=\tilde{\mu}^{2}$ . The large
$\mu$ limit yields to

$\displaystyle\lim_{\mu\to\infty}S_{n}(\mu)={\mathtt{V}}\tilde{\mu}^{2}-{\mathtt{C}}\log\left({\mathtt{V}}\tilde{\mu}^{2}\right)\leavevmode\nobreak\
,$ (5.30)

which is again independent of $n$ at the leading order.

The thermal fluctuations generate a new subleading logarithmic divergence
$\sim\log{\mathtt{V}}\tilde{\mu}^{2}$ in the Renyi (5.25) as well the
entanglement entropy (5.27), and holds even in higher-dimensional Maxwell
electrodynamics (see Appendix A). As anticipated from (4.8), this term is
determined by the fact that the Bekenstein–Hawking entropy of the hyperbolic
black hole acquires a logarithmic contribution due to fluctuations. This
entanglement entropy behavior has been observed in systems with non-smooth
entangling regions [85, 86] or systems with SSB of a continuous symmetry due
to low energy excitations of the ground state [19, 20, 21, 22, 23, 24]. The
term associated to the SSB diverges logarithmically with the system size, just
as in (5.27), and the coefficient in front is related to the number of
Goldstones modes [20]. In [20], the derivation of the subleading contributions
was done by implementing a direct wave function calculation, and was proposed
to have the form $\sim\log(L^{d-1}\alpha)$ where $L$ is the size of the
system, and $\alpha$ gives information regarding the IR scales in the energy
gaps of the theory which depends on intrinsic physical properties of the
system, such as the total spin and spin-wave velocity. Then, we can associate
the ${\mathtt{C}}$ coefficient with number of Goldstone modes appearing in the
dual theory, and $\alpha$ will depend on $\tilde{\mu}$ and coefficients in
${\mathtt{V}}$ as can be seen from (5.27). We further comment on this on
section 6.

The derivatives of the modified Renyi entropy are

$\displaystyle\partial_{n}S_{n}={}$
$\displaystyle-\frac{(n-1)(2{\mathtt{C}}n+{\mathtt{V}}(n-1))+2{\mathtt{C}}(1+\tilde{\mu}^{2})\log\left(\frac{n}{n-1}(1+\tilde{\mu}^{2})\right)}{2n^{2}(n-1)^{2}}\leavevmode\nobreak\
,$ (5.31) $\displaystyle\partial_{n}\left(\frac{n-1}{n}S_{n}\right)={}$
$\displaystyle\frac{{\mathtt{V}}}{n^{3}}(1+n\tilde{\mu}^{2})-\frac{{\mathtt{C}}}{n^{3}}\log\left(\frac{{\mathtt{V}}}{n}\left(1+n\tilde{\mu}^{2}\right)\right)\leavevmode\nobreak\
,$ (5.32) $\displaystyle\partial_{n}((n-1)S_{n})={}$
$\displaystyle\frac{{\mathtt{V}}}{2}\left(1+\frac{1}{n^{2}}+2\tilde{\mu}^{2}\right)+{\mathtt{C}}\left[\frac{n-1}{n}-\log\left({\mathtt{V}}(1+\tilde{\mu}^{2})\right)+\tilde{\mu}^{2}\log\left(\frac{n^{-1}+\tilde{\mu}^{2}}{1+\mu^{2}}\right)\right]\leavevmode\nobreak\
,$ (5.33) $\displaystyle\partial_{n}^{2}((n-1)S_{n}))={}$
$\displaystyle-\frac{1}{n^{3}}\left({\mathtt{V}}-\frac{n{\mathtt{C}}}{1+n\tilde{\mu}^{2}}\right)\leavevmode\nobreak\
,$ (5.34) $\displaystyle\mu\partial_{\mu}S_{n}={}$ $\displaystyle
2\tilde{\mu}^{2}\left({\mathtt{V}}+\frac{{\mathtt{C}}}{n-1}\log\frac{1+n\tilde{\mu}^{2}}{n+n\tilde{\mu}^{2}}\right)\leavevmode\nobreak\
,$ (5.35)

which satisfy the Renyi inequalities (2.2) when the coefficient ${\mathtt{C}}$
smaller than ${\mathtt{V}}$ as can be see from comparing Figure 1 and Figure
2. The equation (5.35) shows that $\mu\partial_{\mu}S_{n}$ remains positive
for ${\mathtt{C}}\geq 0$ but now is $n$ dependant. As we mentioned earlier,
the Renyi entropy for this solution shares many properties with the higher-
dimensional Einstein-Maxwell case, even at the level of logarithmic
corrections; for instance, in Figure 1 the Renyi entropy approaches an
asymptotic value that depends on the chemical potential. In Figure 3, we can
see the growth of entangled pairs as $\tilde{\mu}$ gets larger, and in Figure
4, the Renyi entropy decreases dramatically for imaginary chemical potential,
and shows a maximum value in which the entropy is well-defined. Again, this
resembles the free boson calculation where the analytic continuation is only
valid within a finite window of values for the chemical potential. Finally,
the conformal weights receive the correction

$\displaystyle h_{n}(\mu)=\frac{\pi L}{\ell_{\rm
P}}\left[\left(n-\frac{1}{n}\right)+n\left(\frac{\mu_{\rm
E}}{2\pi}\right)^{4}\right]-\frac{{\mathtt{C}}}{V_{\mathbb{H}^{1}}}(1-x_{n})n\leavevmode\nobreak\
,\quad\partial_{n}h_{n}(\mu)\big{\rvert}_{n\to 1,\mu\to 0}=\frac{2\pi
L}{\ell_{\rm P}}-\frac{{\mathtt{C}}}{V_{\mathbb{H}^{1}}}\leavevmode\nobreak\
,$ (5.36)

decreasing the value of the dual central charge.

Figure 1: Corrected Renyi entropy for three-dimensional gravity coupled to
Coulomb sources with logarithmic corrections as a function of $n$ normalised
by $S_{1}(0)$ on the left panel, and
$\frac{n}{n-1}\frac{S_{n}(\mu)}{S_{1}(0)}$ on the right panel, for different
values of $\tilde{\mu}$ with a UV cutoff so that ${\mathtt{V}}=10^{40}$ . From
bottom to top, the curves corresponds to $\tilde{\mu}=0,0.2,0.4,0.6,0.8,1$.

Figure 2: Corrected Renyi entropy for three-dimensional gravity coupled to
Coulomb sources with logarithmic corrections as a function of $n$ normalised
by $S_{1}(0)$ on the left panel, and
$\frac{n}{n-1}\frac{S_{n}(\mu)}{S_{1}(0)}$ on the right panel, for different
values of $\tilde{\mu}$ with a UV cutoff so that ${\mathtt{V}}=1$ . From top
to bottom, the curves corresponds to $\tilde{\mu}=0,0.2,0.4,0.6,0.8,1$.

Figure 3: Corrected Renyi entropy for three-dimensional gravity coupled to
Coulomb sources with logarithmic corrections as a function of $\tilde{\mu}$
normalised by $S_{1}(0)$ on the left panel, and normalised by $S_{1}(\mu)$ on
the right panel, for different values of $\tilde{\mu}$ with a UV cutoff such
that ${\mathtt{V}}=10^{40}$ . From top to bottom, the curves corresponds to
$n=1,2,3,4,5$

Figure 4: Corrected Renyi entropy with imaginary chemical potential
$\tilde{\mu}_{\rm E}=i\tilde{\mu}$ for Coulomb sources with logarithmic
corrections normalised by the entanglement entropy with (non-)zero chemical
potential in the (right) left panel. The UV cutoff is chosen such that
${\mathtt{V}}=10^{40}$ . From bottom to top, the curves corresponds to
$n=1,2,3,4,5$.

##  6 Summary and Discussion

In this paper, we extended the holographic charged Renyi entropy to a bulk
theory with general non-linear electrodynamics. Supplementing Einstein–Hilbert
term with a Lagrangian density that depends on the Lorentz invariant ${\cal
S}=F_{\mu\nu}F^{\mu\nu}$ , allowed us to treat the thermodynamics of non-
linearly charged black holes in a generic way without the need of specifying
the model under consideration. With this, we constructed the holographic
ingredients to compute the charged Renyi entropy: generalised charge (3.32),
magnetic response (2.17), and the twist operators and their conformal weights
(3.31). We then considered thermal fluctuations in the grand canonical
ensemble. The local stability of the system constraints the range of
parameters in which the method to obtain the logarithmic corrections [28] is
valid. The condition (4.5) is in fact, translated in terms of the energy-
momentum tensor of the matter fields evaluated over the black hole horizon.
The new corrected Renyi entropy (3.34) can be easily separated between the
ones coming from the Einstein–Hilbert term and the ones given by the gauge
sector. This comes in handy to recover the uncharged case [29] given by
turning off the chemical potential. The Renyi inequalities (2.2) are satisfied
as long as the thermal entropy of the system is positive, constraining the
value of the coefficient ${\mathtt{C}}$ in terms of the entropy.

We interpret these new contributions by studying specific examples; the
Coulomb source in 2+1 dimensions section 5 and Maxwell theory in $d\geq 3$ in
Appendix A. In both cases, the structure of the subleading contributions to
the Renyi entropy have the flavour of coming from spontaneous symmetry
breaking of some continuous symmetry in the CFT. If we follow the holographic
dictionary, the duality matches UV excitation in one side to IR effects in the
other. This means that the corrections that we have found, from the gravity
side, corresponds to high energy fluctuations beyond the classical saddle
which from the boundary field theory point of view, are low energy
excitations. In fact, the SSB contribution to the entanglement entropy does
not modifies its UV structure [20], and therefore it cannot be seen
holographically from the classical saddle in the AdS partition function [25,
26]. A proposal for quantum corrections of holographic entanglement entropy
was given in [13], were it is claimed to be dual to the relative entropy
between bulk regions disconnected by the Ryu–Takayanagi surface. In the
Klebanov–Strassler theory at large $N$, the corrections, which are zero order
in the gravity coupling, are associated to a spontaneously broken global
symmetry due to bulk massless excitations, and have the same subleading
logarithmic structure as the one presented in section 4. Similar corrections
can be also found in other holographic theories, as in [87], by considering
1-loop contributions to the gravity partition function.

Considering thermal fluctuations leads to new terms with the same structure
which allow us to identify the ${\mathtt{C}}$ coefficient with the number of
Goldstone modes in the dual theory. This matching, gives the possibility to
study quantum correlations in theories with SSB and Maxwell fields by means of
holography beyond the saddle point as pointed out in [24]. It is important to
remark that the Renyi inequalities ensure unitarity and stability of the
boundary theory constraining the holographic dictionary [69]. This restricts
the possible geometries to be considered in the partition function, see [88]
for a discussion in three-dimensions. Furthermore, the consideration of extra
Maxwell fields in a system with SSB leads to a double logarithmic divergence
$\sim\log\log L$ in the quantum mutual information, which only depends on the
system length [24, 89]. The mutual information is a quantity measuring
correlation between subsystems, and is constructed in terms of the
entanglement entropy in such a way that is independent of renormalisation
schemes. In our case, a double logarithmic structure in the entanglement
entropy appears for odd dimensional bulk due to the subleading contributions
in the divergent volume ${\mathtt{V}}$ [90] which are of the form $\sim\log L$
, which also holds in Maxwell theory in any dimensions as can be seen in
(A.28) . This indicates the possibility of obtaining the double logarithmic
structure in the quantum mutual information by considering fluctuations
generating a logarithmic correction in the black hole entropy. We plan to
refine and present these ideas in a separate work.

The breaking of continuous symmetries should also influence the gravity
theory. As a matter of fact, this should be seen as some “quantum”
backreaction in the geometry due to Gaussian fluctuations in the thermal
partition function [91]. The backreaction can be modeled by adding the
Barvinsky–Vilkovisky–de Witt non-local effective action [92, 93, 94] that,
indeed, produces the logarithmic corrections on the black hole entropy [95,
96, 97, 68]. The deformations can break the original black hole symmetries
generating terms of the form of (5.1), as can be seen in [97]. This could lead
to model breaking of conformal invariance in the AdS boundary when the quantum
corrections dominate over the saddle, as could be for final stages of black
hole due to evaporation. Indeed, the backreaction of the energy fluctuations
in the geometry has been shown to break local symmetries of the solution, such
as the spherical symmetry of the Schwarzschild black hole [91]. This breaking
of local symmetries in the bulk should correspond to breaking the CFT global
spherical symmetry being a concrete realisation of our proposal. We plan to
refine this idea in a future work.

We also noticed that the fluctuations modify the conformal weights of the
twist operators, which leads to a subleading modification of the numerical
value of the central charge. This is in contrast with the magnetic response
that does not suffer a direct adjustment due to the presence of the
${\mathtt{C}}$, leaving the structure coefficients associated with the
$\langle JJ\rangle$ and $\langle TJJ\rangle$ correlators unmodified, which
encaptures universal features of the CFT [15, 98], but modifying the ones of
the stress-tensor two point function. In order to see modifications of the
magnetic response one would have to consider the aforementioned modifications
of the black hole geometry in order to modify the thermodynamic quantities
such as the electric charge.

Another interesting result was obtained in section 5 for a particular case of
NED in three-dimensional gravity. In the standard description of 2+1 gravity
coupled to Maxwell, the BTZ black holes acquires a logarithmic contribution to
the metric due to the presence of the electric charge. This logarithmic term
modifies the asymptotic behavior of the solution being not asymptotically AdS
any longer. Then, the holographic charged Renyi entropy does not reproduce the
expected free theory in the bulk as in dimensions. Instead, to avoid this
problem, we considered the Coulomb source term that produces a charged black
hole with global AdS in the asymptotic region [78]. In this way, the dual
field theory is conformally invariant which allows to properly interpret the
bulk quantities using the holographic dictionary. We showed that the structure
for the conformal weights of the twist operators, and the charged Renyi
entropy, is the same as the one found in [15] for charged free bosons in 1+1
dimensions. This offers the possibility of a complete correspondence between
these two models. However, it also brings a few points that are need to be
investigated in order to have a concrete statement about the duality:

#### $\diamond$

From a classical perspective, the solution presented should be geodesically
complete. This has not been clarified in the previous literature, and it
corresponds to an important issue when understanding the isometries of the
boundary theory.

#### $\diamond$

The asymptotic behavior of the bulk fields would give insight about the
sources and vacuum expectation values of the boundary operators. For this, we
require to apply holographic renormalisation techniques [99] that also would
allow us to write down the Euclidean action and holographic stress tensor, in
order to fully characterise the boundary field theory.

#### $\diamond$

It would be interesting to see if the three-dimensional NED theory arises as a
sector of String theory. There are two possible ways of how to obtain this
Lagrangian: As coming from a conformal limit of an effective description of a
theory that interacts with matter, or as a Kaluza–Klein reduction of higher-
curvature gravity that would correspond to IR limits of String/M theory [7].

We hope that these questions and issues can be examine with more detail in the
future.

## Acknowledgments

We would like to thank A. Das, A. Donos, S. Hosseini, V.G. Matyas, R. Olea,
and P. Sundell for enlightening discussions and comments on various topics
related to this manuscript. The work of Gah is funded by Becas Chile (ANID)
Scholarship N${}^{\rm o}\leavevmode\nobreak\ 72200271$. The work of FD is
supported by Beca Doctorado nacional (ANID) 2021 Scholarship N${}^{\rm
o}\leavevmode\nobreak\ 21211335$ and ANID/ACT210100 Anillo Grant “Holography
and its applications to High Energy Physics, Quantum Gravity and Condense
Matter Systems”.

##  Appendix A Power-law Maxwell electrodynamics

Consider the non-linear Lagrangian666See [100] and reference therein for
holographic applications of the Power-law Maxwell Lagrangian.,

$\displaystyle{\cal L}=-2\gamma F^{2p}\leavevmode\nobreak\ ,$ (A.1)

where $\gamma$ is a real constant that, in principle, can depend on the
electric charge of the theory, but we will consider in this Appendix both
quantities to be independent. The energy-momentum tensor

$\displaystyle
T^{\mu}_{\nu}=-2\gamma\left(\delta^{\mu}_{\nu}-4p\frac{F^{\mu\alpha}F_{\nu\alpha}}{F^{2}}\right)F^{2p}\leavevmode\nobreak\
,$ (A.2)

generates an electric field and potential

$\displaystyle E(r)=\frac{Q}{r^{\delta}}\leavevmode\nobreak\
,\qquad\phi(r)=\frac{Q}{(1-\delta)r^{\delta-1}}\leavevmode\nobreak\ ,$ (A.3)

where

$\displaystyle\delta=\frac{d-1}{2p-1}\leavevmode\nobreak\ ,\qquad
Q=\left(\frac{(-1)^{p+1}{\tilde{q}}}{2^{p}p\gamma}\right)^{\frac{1}{2p-1}}\leavevmode\nobreak\
,\qquad\frac{\mu_{\rm p}}{2\pi
L}=\frac{Q}{(1-\delta)r_{h}^{\delta-1}}\leavevmode\nobreak\ .$ (A.4)

Notice that when $p=\frac{d+1}{4}$ , the electric field always has the Maxwell
form and the Lagrangian becomes conformal leaving the energy-momentum
traceless [77]. The $rr$-component of the energy-momentum tensor is

$\displaystyle T_{r}^{r}=(-1)^{p}(2p-1)\gamma 2^{p+1}\frac{Q^{2p}}{r^{2\delta
p}}\leavevmode\nobreak\ ,$ (A.5)

and that evaluated at the horizon gives

$\displaystyle T^{r}_{r}(x,\mu_{\rm
p})=\gamma(-1)^{p}\left(\frac{\tilde{\mu}_{\rm
p}}{xL}\right)^{2p}\leavevmode\nobreak\
,\qquad\tilde{\mu}^{2p}\equiv(2p-1)2^{p+1}\left(\frac{(1-\delta)\mu_{\rm
p}}{2\pi L}\right)^{2p}\leavevmode\nobreak\ .$ (A.6)

Its derivative

$\displaystyle\partial_{x}T_{r}^{r}=\gamma(-1)^{p+1}\left(\frac{2p}{x}\right)\left(\frac{\tilde{\mu}_{\rm
p}}{xL}\right)^{2p}\leavevmode\nobreak\ ,$ (A.7)

is positive for $p$ odd and satisfies the stability condition in the canonical
ensemble . In the case of $p$ even, must be analysed to check the temperature
regime of stability, see [50] for a discussion. In the case of even $p=2m$ ,
we get an upper bound for the chemical potential

$\displaystyle d+\frac{d-2}{x^{2}}\geq\frac{2\ell_{\rm
P}^{d-1}L^{2}}{\delta}\gamma\left(\frac{\tilde{\mu}_{\rm
p}}{Lx}\right)^{4m}\leavevmode\nobreak\ .$ (A.8)

The charge dependent sector of the metric function becomes

$\displaystyle{\cal
Q}(r)=-\frac{2\gamma(d-1)(-2)^{p}Q^{2p}}{\delta(\delta-1)}\frac{1}{r^{\delta-1}}\leavevmode\nobreak\
,$ (A.9)

such that, in terms of $\tilde{\mu}_{\rm p}$, the specific heat at fixed
chemical potential

$\displaystyle{\cal
C}=(d-1)S^{(0)}\left[\frac{dx^{2}+k(d-2)+\frac{2(-1)^{p}\gamma\ell_{\rm
P}^{d-1}L^{2}x^{2}}{d-1}\left(\frac{\tilde{\mu}_{\rm
p}}{xL}\right)^{2p}}{dx^{2}-k(d-2)+\frac{2(2p-1)(-1)^{p}\gamma\ell_{\rm
P}^{d-1}L^{2}x^{2}}{d-1}\left(\frac{\tilde{\mu}_{\rm
p}}{xL}\right)^{2p}}\right]\leavevmode\nobreak\ ,$ (A.10)

and temperature

$\displaystyle
T=\frac{T_{0}}{2x_{n}}\left[dx_{n}^{2}-(d-2)+(-1)^{p}\gamma\frac{2\ell_{\rm
P}^{d-1}}{d-1}\left(\frac{\tilde{\mu}_{\rm
p}}{L}\right)^{2p}\frac{1}{x_{n}^{2p-2}}\right]\leavevmode\nobreak\ ,$ (A.11)

where $x_{n}$ is the largest solution to

$\displaystyle
dnx_{n}^{2p}-n(d-2)x_{n}^{2p-2}-2x_{n}^{2p-1}+n(-1)^{p}\gamma\frac{2\ell_{\rm
P}^{d-1}}{d-1}\left(\frac{\tilde{\mu}_{\rm
p}}{L}\right)^{2p}=0\leavevmode\nobreak\ .$ (A.12)

Analytic solutions for $x_{n}$ can be found for several values of $p$ .
Particularly, for the Maxwell case, $p=1$, the equations is quadratic, and
when the theory becomes conformal the equation becomes

$\displaystyle
dnx^{\frac{d+1}{2}}-2x_{n}^{\frac{d-1}{2}}-n(d-2)x_{n}^{\frac{d-3}{2}}+(d-2)\tilde{\mu}^{\frac{d+1}{2}}=0\leavevmode\nobreak\
,$ (A.13)

which is quadratic in $d=3$ and corresponds to Maxwell theory .

Considering the pure Maxwell, which corresponds to $p=1$ with
$\gamma=\frac{\ell_{\ast}^{2}}{16\ell_{\rm P}^{d-1}}$ , and following the
conventions of [39], the Lagrangian becomes

$\displaystyle{\cal L}=-\frac{\ell_{\ast}^{2}}{8\ell_{\rm
P}^{d-1}}F_{\mu\nu}F^{\mu\nu}\leavevmode\nobreak\ ,$ (A.14)

and the theory admits a charged solution with a hyperbolic horizon when $d>2$
with metric function

$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}-1-\frac{m}{r^{d-2}}+\frac{q^{2}}{r^{2(d-2)}}\leavevmode\nobreak\
,$ (A.15)

and corresponding gauge field

$\displaystyle
A=\left(\sqrt{\frac{2(d-1)}{(d-2)}}\frac{Lq}{R\ell_{\ast}r^{d-2}}-\frac{\mu_{\rm
M}}{2\pi R}\right)dt\leavevmode\nobreak\ .$ (A.16)

where $\mu_{\rm M}$ is chosen such that the gauge field vanishes at the
horizon, namely

$\displaystyle\mu_{\rm
M}=2\pi\sqrt{\frac{2(d-1)}{(d-2)}}\frac{Lq}{\ell_{\ast}r^{d-2}_{h}}\leavevmode\nobreak\
.$ (A.17)

There is a black hole horizon at $r_{h}$ and the mass parameter can be written
as a function of the horizon,

$\displaystyle
m=\frac{r_{h}^{d-2}}{L^{2}}(r^{2}_{h}-L^{2})+\frac{q^{2}}{r_{h}^{d-2}}\leavevmode\nobreak\
,$ (A.18)

As we mentioned in sec the full solution is parametrised by the AdS radius
$L$, the electric charge $q$ and the horizon $r_{h}$,

$\displaystyle
f(r)=\frac{r^{2}}{L^{2}}-1+\frac{q^{2}}{r^{2d-4}}-\left(\frac{r_{h}}{r}\right)^{d-2}\left(\frac{r_{h}^{2}}{L^{2}}-1+\frac{q^{2}}{r_{h}^{2d-4}}\right)\leavevmode\nobreak\
.$ (A.19)

The black hole has an associated temperature given by

$\displaystyle
T=\frac{T_{0}}{2}Lf^{\prime}(r_{h})=\frac{T_{0}}{2}\left[d\frac{r_{h}}{L}-(d-2)\frac{L}{r_{h}}\left(1+\tilde{\mu}_{\rm
M}^{2}\right)\right]\leavevmode\nobreak\
,\qquad\tilde{\mu}_{\rm{M}}\equiv\sqrt{\frac{d-2}{2(d-1)}}\left(\frac{\mu_{\rm
M}\ell_{\ast}}{2\pi L}\right),$ (A.20)

and entropy given by the Wald formula evaluated over the horizon,

$\displaystyle
S^{(0)}={\mathtt{V}}\left(\frac{r_{h}}{L}\right)^{d-1}\leavevmode\nobreak\ .$
(A.21)

The horizon radius in terms of the Renyi parameter is

$\displaystyle x_{n}=\frac{1}{dn}\left(1+\sqrt{1+d(d-2)(1+\tilde{\mu}_{\rm
M}^{2})n^{2}}\right)\leavevmode\nobreak\ .$ (A.22)

The heat capacity has been computed in [101], in agreement with (3.26), and
shown to be always stable. Considering the first order corrections in (4.6)
can be used and gives

$\displaystyle S=S^{(0)}-{\mathtt{C}}_{\rm M}\log S^{(0)}+{\rm
const.}\leavevmode\nobreak\ ,\qquad{\mathtt{C}}_{\rm M}=1\leavevmode\nobreak\
,$ (A.23)

and the coefficient ${\mathtt{C}}_{\rm M}$ has been found in [54] for
arbitrary dimensional AdS RN in the (grand) canonical ensemble. Using the
result into the Renyi entropy

$\displaystyle S_{n}=S_{n}^{(0)}+S_{n}^{(1)}\leavevmode\nobreak\ ,\ $ (A.24)

where the first order $S^{(0)}_{n}$ was found in [39],

$\displaystyle
S^{(0)}_{n}=\frac{{\mathtt{V}}}{2}\frac{n}{n-1}\left[\left(1+\tilde{\mu}^{2}\right)(x_{1}^{d-2}-x_{n}^{d-2})+x_{1}^{d}-x_{n}^{d}\right]\leavevmode\nobreak\
,$ (A.25)

and the first correction

$\displaystyle S^{(1)}_{n}={}$ $\displaystyle-\frac{{\mathtt{C}}_{\rm
M}}{2}\left(\frac{n}{n-1}\right)\int^{x_{1}}_{x_{n}}\left(d+(d-2)(1+\tilde{\mu}_{\rm
M}^{2})x^{-2}\right)\log\left({\mathtt{V}}x^{d-1}\right)dx$ $\displaystyle={}$
$\displaystyle\frac{n}{n-1}\frac{{\mathtt{C}}_{\rm
M}}{2}(d-2)(1+\tilde{\mu}_{\rm
M}^{2})\left(\frac{\log\left({\mathtt{V}}x_{1}^{d-1}\right)+(d-1)}{x_{1}}-\frac{\log\left({\mathtt{V}}x_{n}^{d-1}\right)+(d-1)}{x_{n}}\right)\leavevmode\nobreak\
.$ (A.26)

The resulting Renyi entropy (A.24) has limits

$\displaystyle S_{0}{}$
$\displaystyle=\frac{{\mathtt{V}}}{2}\left(\frac{2}{d}\right)^{d}\frac{1}{n^{d-1}}-{\mathtt{C}}_{\rm
M}\left[\log\left(\left(\frac{2}{dn}\right)^{d-1}{\mathtt{V}}\right)-(d-1)\right]\leavevmode\nobreak\
,$ (A.27) $\displaystyle S_{\rm E}{}$
$\displaystyle=\frac{{\mathtt{V}}}{2}\frac{(d-2)x^{d-2}_{1}}{dx_{1}-1}\left(\frac{dx_{1}^{2}}{d-2}+1+\tilde{\mu}_{\rm
M}^{2}\right)-{\mathtt{C}}_{\rm
M}\log\left({\mathtt{V}}x_{1}^{d-1}\right)\leavevmode\nobreak\ ,$ (A.28)
$\displaystyle S_{\infty}{}$
$\displaystyle=\frac{{\mathtt{V}}}{2}\left[\left(1+\tilde{\mu}_{\rm
M}^{2}\right)\left(x_{1}^{d-2}-x_{\infty}^{d-2}\right)+x_{1}^{d}-x_{\infty}^{d}\right]$
$\displaystyle+\frac{{\mathtt{C}}_{\rm M}}{2}(d-2)(1+\tilde{\mu}_{\rm
M}^{2})\left(\frac{\log\left({\mathtt{V}}x_{1}^{d-1}\right)+(d-1)}{x_{1}}-\frac{\log\left({\mathtt{V}}x_{\infty}^{d-1}\right)+(d-1)}{x_{\infty}}\right)\leavevmode\nobreak\
,$ (A.29) $\displaystyle S_{2}{}$
$\displaystyle={\mathtt{V}}\left[\left(1+\tilde{\mu}_{\rm
M}^{2}\right)\left(x_{1}^{d-2}-x_{2}^{d-2}\right)+x_{1}^{d}-x_{2}^{d}\right]$
$\displaystyle+{\mathtt{C}}_{\rm M}(d-2)(1+\tilde{\mu}_{\rm
M}^{2})\left(\frac{\log\left({\mathtt{V}}x_{1}^{d-1}\right)+(d-1)}{x_{1}}-\frac{\log\left({\mathtt{V}}x_{2}^{d-1}\right)+(d-1)}{x_{2}}\right)\leavevmode\nobreak\
.$ (A.30)

An interesting feature is that, to lading order in $\mu$, the Renyi entropy is
$n$ independent [39] (maximal disorder), and this remains true when the
logarithmic corrections are considered

$\displaystyle\lim_{\mu\to\infty}S_{n}(\mu)={\mathtt{V}}\left(\frac{d-2}{d}\right)^{\frac{d-1}{2}}\tilde{\mu}_{\rm
M}^{d-1}-{\mathtt{C}}_{\rm
M}\log\left[{\mathtt{V}}\left(\frac{d-2}{d}\right)^{\frac{d-1}{2}}\tilde{\mu}_{\rm
M}^{d-1}\right]\leavevmode\nobreak\ ,$ (A.31)

As can be checked by the numerical results plotted in Figure 5, the Renyi
inequalities (2.2) hold when the coefficient ${\mathtt{C}}$ is not comparable
with the horizon volume. As can be seen from Figure 6, the growth in the
entangled pairs created with the increasing of real chemical potential holds
also as in [39].

Figure 5: Charged Renyi entropy for Maxwell fields in $d=4$ normalized by the
entanglement entropy with zero chemical potential on the left panel, and
$\frac{n-1}{n}\frac{S_{n}(\mu)}{S_{1}(0)}$ on the right panel. The UV cutoff
has been chosen such that ${\mathtt{V}}=10^{40}$ . The curves corresponds to
$\tilde{\mu}_{\rm M}=0,0.5,1,1.5,2$ from bottom to top.

Figure 6: Charged Renyi entropy for Maxwell fields in $d=4$ normalized by the
entanglement entropy with zero chemical potential on the left panel, and non-
zero chemical potential on the right panel. The UV cutoff has been chosen such
that ${\mathtt{V}}=10^{40}$ . The curves corresponds to $n=1,2,3,4,5$ from top
to bottom.

Finally, the conformal weights of the dual twist operators are

$\displaystyle h_{n}(\mu)={}$ $\displaystyle\pi n\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}\left(x_{n}^{d-2}-x_{n}^{d}\right)-\frac{n{\mathtt{C}}}{2V_{\mathbb{H}^{d-1}}}\frac{(1-x_{n})(dx_{n}+d-2)}{x_{n}}-\frac{nx_{n}^{d-2}}{2\pi
R^{d}}\frac{L^{d-2}}{\ell_{\rm P}^{d-1}}\tilde{\mu}_{\rm
M}^{2}\leavevmode\nobreak\ ,$ (A.32)
$\displaystyle\partial_{n}h_{n}\Big{\rvert}_{n=1,\mu=0}={}$
$\displaystyle\frac{2\pi}{d-1}\left(\frac{L}{\ell_{\rm
P}}\right)^{d-1}-\frac{{\mathtt{C}}}{V_{\mathbb{H}^{d-1}}}\leavevmode\nobreak\
,$ (A.33)

Then, the extra subleading logarithmic divergence in the entanglement entropy
(A.28), which has been discussed in section 6 to be related to systems with
SSB, also appearin higher-dimensional Maxwell electrodynamics. This remains
present in the zero charge limit.

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