#  Thermodynamic curvature of charged black holes with $AdS_{2}$ horizons

###### Abstract

Sign and magnitude of the thermodynamic curvature provides empirical
information about the nature of microstructures of a general thermodynamic
system. For charged black holes in AdS, thermodynamic curvature is positive
for large charge or chemical potential, and diverges for extremal black holes,
indicating strongly repulsive nature. We compute the thermodynamic curvature
at low temperatures, for charged black holes with AdS2 near horizon geometry,
and containing a zero temperature horizon radius $r_{h}$, in a spacetime which
asymptotically approaches $AdS_{D}$ (for $D>3$). In the semi-classical
analysis at low temperatures, the curvature shows a novel crossover from
negative to positive side, indicating the shift from attraction to repulsion
dominated regime near $T=0$, before diverging as $1/(\gamma T)$, where
$\gamma$ is the coefficient of leading low temperature correction to entropy.
Accounting for quantum fluctuations, the curvature computed in the canonical
ensemble is positive, whereas the one in the grand canonical ensemble,
continues to show a crossover from negative to positive side. Moreover, the
divergence of curvature at $T=0$ is cured irrespective of the ensemble used,
resulting in a universal constant.

##  1 Introduction

Thermodynamics of charged black holes in AdSD spacetimes has been well
studied, particularly with the motivation of understanding holographic field
theories at finite temperature [1]-[5]. The presence of an AdS2 factor in the
near horizon geometry which has the form AdS${}_{2}\times M_{d}$ (where
$M_{d}$ is a compact space with $D=d+2$) is expected to give universal
information about the low energy quantum theories for these black holes. The
study of low energy aspects based on the AdS2 factor has improved our
understanding with the advent of Sachdev-Ye-Kitaev models [7, 8, 9, 10] and
related studies in the low energy limit of string theories [6]-[41].

A universal property of black holes in AdSD with charge $Q$ (and also SYK
models) is that, there are interesting crossovers at low temperatures, with
various thermodynamic quantities receiving non-trivial classical and quantum
corrections [42, 43, 44, 45]. Energy and charge fluctuations of the low
temperature quantum theory lead to a Schwarzian action [24, 42]:

$\displaystyle I[f,\phi]$ $\displaystyle=$ $\displaystyle-
S_{0}(Q)+\frac{K}{2}\int_{0}^{1/T}d\tau(\partial_{\tau}\phi-i(2\pi\mathcal{E}T)\partial_{\tau}f)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\gamma}{4\pi^{2}}\int_{0}^{1/T}d\tau\,\\{\tan(\pi
Tf(\tau)),\tau\\},$

with the following notation for the Schwarzian

$\displaystyle\left\\{g(\tau),\tau\right\\}\,=\,\frac{g^{\prime\prime\prime}}{g^{\prime}}-\frac{3}{2}\,\left(\frac{g^{\prime\prime}}{g^{\prime}}\right)^{2}\,.$
(1.2)

The action is specified in terms of three parameters, namely $\gamma$, the
compressibility given as

$\displaystyle K=\frac{dQ}{d\mu}\Bigr{|}_{T=0}\,,$ (1.3)

and the electric field at the horizon $\mathcal{E}$, which can be computed as

$\displaystyle\frac{dS_{0}(Q)}{dQ}=2\pi\mathcal{E}\,.$ (1.4)

In particular, the dimensionless parameter $\mathcal{E}$ appeared first in the
works of Sen [6] while proposing an entropy function for general black holes
(see [8, 9] for its appearance in complex SYK models). Unlike the zero
temperature entropy $S_{0}$ and $\mathcal{E}$, the compressibility $K$ and the
coefficient $\gamma$ are not universal and depend upon the UV details of the
theory. There are further novel features to the above model with the inclusion
of quantum corrections and also including supersymmetric situations [43, 44,
45]

The aim of this paper is to attempt an understanding of the nature of
microstructures of such low temperature charged black holes in AdS following
recent developments [23, 42, 43, 44, 45, 46], using the methods of
thermodynamic geometry. The main idea of this approach is based on
thermodynamic fluctuation theory, which starts from writing the number of
microstates of a thermodynamic system as

$\Omega=e^{\frac{S}{k_{B}}}\,,$ (1.5)

where $k_{B}$ is the Boltzmann constant. One now considers a thermodynamic
system $I_{0}$ in equilibrium, with a sub-system $I$ in it, in addition to
having a couple of independent fluctuating variables, $x^{i}$ where $i=1,2$.
The number of microstates in eqn. (1.5) can now be related to the probability
$P(x^{1},x^{2})$ of locating the state of the system somewhere between
$(x^{1},x^{2})$ and $(x^{1}+dx^{1},x^{2}+dx^{2})$ . According to second law of
thermodynamics the pair $(x^{1},x^{2})$ picks the values that maximise the
entropy $S=S_{\text{max}}$. In effect, $(x^{1},x^{2})$ describe fluctuations
around the maximum and the probability around this maximum can be written as
[47]:

$P(x^{1},x^{2})\propto e^{-\frac{1}{2}\,\Delta l^{2}}\,,$ (1.6)

where the line element which measures thermodynamic distance between two
nearby fluctuation states is written as:

$\Delta l^{2}\,=-\frac{1}{k_{B}}\,\frac{\partial^{2}S}{\partial x^{i}\partial
x^{j}}\,\Delta x^{i}\Delta x^{j}\,.$ (1.7)

The distance is then shorter when the fluctuation between neighbouring states
is more probable. The thermodynamic curvature $R$ computed from the line
element in eqn. (2.8) contains much information, and has been studied for
various systems in nature, such as Ideal/van der Waals fluids, quantum gases
to other Bose/Fermi systems, including the Ising model and black holes
[48]-[62]. The current understanding based on the available empirical data is
that repulsive (attractive) interactions of a thermodynamic system turn out to
have positive (negative) value of $R$ [47]. $R$ typically diverges at the
phase transitions points and has zero crossings at the points where the
attractive and repulsive interactions are in balance or in a non-interacting
situation.

Thermodynamic geometry of charged black holes in AdS has been well studied in
the canonical, grand canonical and also other mixed ensembles in several works
(see e.g., [53, 54]). It has also been observed that the curvature $R$ is
generally positive for the large black hole branch above a certain critical
charge or chemical potential and diverges as $+1/T$ near $T=0$, i.e., in the
extremal limit. These computations are done for charged black holes in AdS,
far away from the horizon. In this work, out motivation is to study the
computation of $R$ for charged black holes in AdS, in a certain low
temperature regime where the dynamics is governed by corrections to the AdS2
near horizon geometry. The partition function has been computed recently in
both canonical and grand canonical ensembles, carefully taking into account
the quantum fluctuations [23, 42, 43, 44, 45, 46], resolving the mass-gap
puzzle [43, 44]. As noted above, the thermodynamic variables of black holes in
the low temperature limit show certain universal properties, but there are
also non-universal behaviours, which depend on the how the AdS2 is embedded in
the higher dimensional theory. Here, universality of thermodynamic quantities
implies independence of their characteristics on parameters of the geometry
far from the AdS2 near-horizon region. Indeed, the behaviour of $R$ in the low
temperature limit we find here, is due to the presence of a near horizon AdS2
geometry, and its nature differs far away from the horizon. More importantly,
$R$ diverges on the positive side at $T=0$, in the semi-classical limit, which
is similar to earlier calculations done in the full geometry [53, 54, 55]. In
addition, the curvature contains a novel crossover from the negative to
positive side, which objectively points towards the existence of attractive
interactions (bosonic in nature as per our conventions) which develop at low
temperatures, i.e., for $T\ll 1/r_{h}$, before becoming repulsive. Once the
quantum corrections to thermodynamic quantities are included, both the
curvatures computed in the canonical and grand canonical ensembles and
evaluated at $T=0$, turn out to be equal to a constant. The key take away
message is that the quantum corrections to the near horizon nearly AdS2
geometry cure divergence of the thermodynamic curvature at $T=0$. This
indicates that the microstructures are weakly repulsive in nature at $T=0$, as
opposed to the strongly repulsive nature anticipated from the analysis done
earlier in the full geometry [53, 54, 55].

The rest of the paper is as structured as follows. In section-(2), we set up
the notations and collect known details of thermodynamic geometry of charged
blackholes in AdS spacetimes. These computations will be for charged black
holes in D-dimensions far from the horizon. In particular, we note from
figure-(1), that the thermodynamic curvature is generally positive for large
black holes above a certain chemical potential and diverges at $T=0$. In
section-(3), we perform a computation of the thermodynamic curvature in semi-
classical as well as with the inclusion of quantum corrections to
thermodynamic quantities. Subsection-(3.1) contains the computation of
thermodynamic curvature in the semi-classical limit, where the corrections to
entropy come from corrections to the AdS2 geometry [42] and show that there is
a novel crossover. The $T=0$ behaviour is unmodified and matches with earlier
results [53, 54, 55]. In subsection-(3.2), we follow [43] and note down the
thermodynamic quantities in the low temperature limit, where the system can
also be described by an effective one dimensional Schwarzian action, based on
corrections to the near horizon AdS2 geometry. The quantum corrections to the
thermodynamic quantities in the canonical and grand canonical ensembles are
now known [43], and we use these to compute the curvature in both these cases.
In the canonical ensemble, the curvature is generally positive, with no
crossover and is a positive constant at $T=0$. In the grand canonical
ensemble, the curvature is a positive constant at $T=0$, but crosses over to
the negative side with slight increase of temperature, but still well below
$1/r_{h}$. At $T=0$, both the curvatures are finite as well as independent of
charge, giving a universal constant. We end with remarks in section-(4).

##  2 Charged black holes in AdSD

Let us start with the action of Einstein-Maxwell theory in AdSd+2 ($d>1$) in
the presence of a U(1) gauge field with the action [5].

$\displaystyle I=\int
d^{d+2}x\sqrt{g}\left[-\frac{1}{2\kappa^{2}}\left(\mathcal{R}_{d+2}+\frac{d(d+1)}{L^{2}}\right)+F^{2}\right],$
(2.1)

where we set $\kappa^{2}=8\pi$ and the gravitational constant $G_{N}=1$.
$\mathcal{R}_{d+2}$ is the Ricci scalar, with $L$ denoting the AdSd+2 radius.
The above theory is known to contain black hole solutions with the metric

$\displaystyle
ds^{2}=f(r)d\tau^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{d}^{2}\,,$ (2.2)

where $d\Omega_{d}^{2}$ stands for the metric of the $d$\- dimensional sphere,
and

$\displaystyle
f(r)=1+\frac{r^{2}}{L^{2}}+\frac{q^{2}}{r^{2d-2}}-\frac{m}{r^{d-1}}.$ (2.3)

As $r\rightarrow\infty$, the metric in Eq. (2.2) goes over to AdSd+2, whereas
near the horizon the geometry is AdS${}_{2}\times$Sd. The grand canonical
potential is given as

$\displaystyle\Omega(T,\mu)=\frac{\omega_{d}r_{0}(T,\mu)^{d-1}}{16\pi}\left(1-\frac{r_{0}(T,\mu)^{2}}{L^{2}}\right)-\frac{\omega_{d}(d-1)\mu^{2}r_{0}(T,\mu)^{d-1}}{2d}\,.$
(2.4)

from which the entropy and charge can be obtained to be

$\displaystyle
S(T,\mu)=\frac{\pi^{\frac{d+1}{2}}r_{0}^{d}}{2\Gamma\left(\frac{d+1}{2}\right)},$
(2.5) $\displaystyle Q(T,\mu)=\frac{(d-1)\pi^{\frac{d-1}{2}}\mu
r_{0}^{d-1}}{2\Gamma\left(\frac{d+1}{2}\right)}\,,$ (2.6)

where
$\omega_{d}=\frac{2\pi^{\frac{d+1}{2}}}{\Gamma\left(\frac{d+1}{2}\right)}$.
Focussing on the large black hole branch, the specific heat can be written as,

$\displaystyle
C_{\mu}=\frac{d^{2}\pi^{\frac{d+3}{2}}L^{2}T\left(\frac{\sqrt{dL^{2}\left(d^{3}\left(2\mu^{2}-1\right)-2d^{2}\mu^{2}+d\left(-2\mu^{2}+4\pi^{2}L^{2}T^{2}+1\right)+2\mu^{2}\right)}+2\pi
dL^{2}T}{d(d+1)}\right)^{d}}{\Gamma\left(\frac{d+1}{2}\right)\sqrt{dL^{2}\left(d^{3}\left(2\mu^{2}-1\right)-2d^{2}\mu^{2}+d\left(-2\mu^{2}+4\pi^{2}L^{2}T^{2}+1\right)+2\mu^{2}\right)}}\,,$
(2.7)

Black holes in the fixed potential ensemble are known to show distinct
behaviour for $\mu<1$ and $\mu>1$ [5] and our interest is in the later regime
where extremal limit can be taken. Using the grand potential, a thermodynamic
line element can be constructed as

$\Delta l^{2}\,=-\frac{1}{T}\,\frac{\partial^{2}\Omega(T,\mu)}{\partial
x^{i}\partial x^{j}}\,\Delta x^{i}\Delta x^{j},$ (2.8)

where the fluctuation coordinates $x^{i}$ are chosen as $(T,\mu)$. The above
line element is conformally related to the one in eqn. (1.7). The curvature
following from it can be computed straightforwardly [53, 54], though the
resulting expression is quite long to express here. We instead show the result
in figure-(1), which matches with the earlier computations [53, 54].

Figure 1: Thermodynamic curvature of charged black holes in AdS4 in the
$(T,\mu)$ plane) for $\mu>1$ (red curve) and $\mu<1$ (blue curve). The red
curve is for $\mu=1.05$ and the blue curve for $\mu=0.8$.

The low temperature behaviour for $\mu>1$ is given below which will be useful
later:

$\displaystyle
R=\frac{3\sqrt{3}\left(2\mu^{2}-1\right)}{4\pi^{2}\left(\mu^{2}-1\right)^{3/2}L^{3}T}-\frac{3\left(6\mu^{2}-1\right)}{4\left(\pi\left(\mu^{2}-1\right)^{2}L^{2}\right)}+\frac{\left(4\sqrt{3}\mu^{2}+\sqrt{3}\right)T}{2\left(\mu^{2}-1\right)^{5/2}L}+O\left(T^{2}\right)\,.$
(2.9)

We note that $R$ is positive and diverges as $T\rightarrow 0$, indicating
dominant repulsive type interactions of microstructures.

##  3 Low temperature nearly AdS2

At sufficiently low temperatures, non-constant modes on $S_{d}$ are not
excited and certain universal features can be obtained. The expressions in the
previous section were for general values of $T$ and $\mu$ and in this section,
following [42], the low temperature expressions (at fixed $\mu=\mu_{0}$) can
be obtained by writing $r_{0}$ as

$\displaystyle r_{0}=r_{h}(\mu_{0})+\frac{2\pi
L^{2}}{d+1}T+\mathcal{O}(T^{2})$ (3.1)

where

$\displaystyle r_{h}\equiv
L\left[\frac{(d-1)(\mu_{0}^{2}8\pi(d-1)-d)}{d(d+1)}\right]^{1/2}\,,$ (3.2)

is the zero temperature horizon radius and $\mu_{0}=\mu|_{T=0}$. The expansion
in eqn. (3.2), allows writing zero temperature entropy as

$\displaystyle S_{0}=\frac{2\pi\omega_{d}}{8\pi}\,r_{h}^{d}\,,$ (3.3)

where $\omega_{d}$ is the area of d-dimensional unit sphere. Note that the the
entropy $S_{0}$ is given terms of the area of the horizon in the d-dimensional
geometry. Compressibility can be written as

$\displaystyle
K=\frac{(d-1)\omega_{d}r_{h}^{d-3}\left[d(d+1)r_{h}^{2}+(d-1)^{2}L^{2}\right]}{(d+1)}\,.$
(3.4)

Zero temperature charge can be expressed in terms of $\mu_{0}$ as

$\displaystyle
Q=\frac{\left(d^{2}-1\right)\pi^{\frac{d-1}{2}}\mu_{0}\left(\frac{L\sqrt{d\left(d^{2}-1\right)\left(d\left(2\mu_{0}^{2}-1\right)-2\mu_{0}^{2}\right)}}{d(d+1)}\right){}^{d-1}}{2(d+1)\Gamma\left(\frac{d+1}{2}\right)}\,.$
(3.5)

As suggested in [42], the low temperature analysis can be done in the
canonical ensemble at constant $Q$.

###  3.1 Semi-classical analysis

The free energy in the fixed charge ensemble $F=\Omega+\mu\,Q$ gets
corrections to order $T^{2}$ [63, 42]:

$\displaystyle
F=\frac{\pi^{\frac{d-1}{2}}r_{h}^{d}\left(\frac{d^{2}r_{h}}{(d-1)L^{2}}+d\left(\frac{1}{r_{h}}-\frac{2\pi^{2}L^{2}r_{h}T^{2}}{(d-1)^{2}L^{2}+d(d+1)r_{h}^{2}}\right)-2\pi
T\right)}{4\Gamma\left(\frac{d+1}{2}\right)}\,.$ (3.6)

To this order, the entropy is given as

$\displaystyle S(Q,T\rightarrow 0)=S_{0}(Q)+\gamma\,T+\cdots\,,$ (3.7)

where $S_{0}(Q)$ is the zero temperature entropy and is non-zero. The
coefficient $\gamma$ (which also gives the gap scale $M^{-1}_{SL(2)}$) can be
deduced from the corrections to the near horizon AdS2 geometry[16, 17, 19, 10]
as

$\displaystyle\gamma=\frac{d\pi^{\frac{d+3}{2}}L^{2}r_{h}^{d+1}}{\Gamma\left(\frac{d+1}{2}\right)\left((d-1)^{2}L^{2}+d(d+1)r_{h}^{2}\right)}\,.$
(3.8)

Due to the above scaling of entropy with temperature as in eqn. (3.7), it was
believed that the statistical description breaks down at temperatures lower
than $1/\gamma$. The scale $M_{SL(2)}$ is now understood as the energy at
which the (approximate) near horizon conformal symmetry of AdS2 is broken.
This becomes evident once the quantum corrections are taken into account [43].
For now, we continue with the semi-classical analysis and return to consider
the quantum corrected thermodynamic quantities in the next subsection. The low
temperature behaviour of the chemical potential $\mu$ at leading order can be
written as [42]

$\displaystyle\mu(T)$ $\displaystyle=$
$\displaystyle\mu_{0}-2\pi\mathcal{E}T+\ldots\,,$ (3.9)

where

$\displaystyle 2\pi\mathcal{E}=\frac{\sqrt{2}\pi
Lr_{h}\sqrt{d\left((d-1)L^{2}+(d+1)r_{h}^{2}\right)}}{(d-1)^{2}L^{2}+d(d+1)r_{h}^{2}}\,.$
(3.10)

The second term in eqn. (3.9) is purely the contribution at the boundary of
AdS2 and its contribution at the AdSd+2 boundary can be computed in the
presence of time diffeomorphisms and gives the second term in eqn. (1). We
will see in the following section that eqn. (3.9) gets corrected further due
to quantum fluctuations and we will use the temperature dependent terms there
to compute the curvature. While using zero temperature relations, we will
continue to express the thermodynamic relations in terms of either $r_{h}$,
$Q$ or $\mu_{0}$ as per convenience as all these are constants, related to
each other. We can compute the thermodynamic curvature in both canonical and
grand canonical ensembles, but we prefer the later here, for comparison with
the results in the previous section. The particular length ratio $L/r_{h}$ is
kept fixed as the temperature is lowered in the limit $T\rightarrow 0$. We
will set $L=1$ in the plots.

The curvature can be computed straightforwardly by evaluating the metric from
eqn. (2.8), and the expression for the four dimensional case is

$\displaystyle
R=\frac{\sqrt{3}\left(3\left(1-2\mu_{0}^{2}\right)^{4}-4\pi^{2}\left(4\mu_{0}^{6}+8\mu_{0}^{4}-11\mu_{0}^{2}-1\right)L^{2}T^{2}\right)}{4\pi^{2}\left(\mu_{0}^{2}-1\right)^{3/2}\left(2\mu_{0}^{2}-1\right)^{3}L^{3}T}$
(3.11)

The above expression has been written in terms of $\mu_{0}$ for convenience,
and can be converted to the fluctuating variable $Q$ using eqn. (3.5). The
first few terms of eqn. (3.11) close to $T=0$ are

$\displaystyle
R=\frac{3\sqrt{3}\left(2\mu_{0}^{2}-1\right)}{4\pi^{2}\left(\mu_{0}^{2}-1\right)^{3/2}L^{3}T}-\frac{\sqrt{3}\left(4\mu_{0}^{6}+8\mu_{0}^{4}-11\mu_{0}^{2}-1\right)T}{\left(\mu_{0}^{2}-1\right)^{3/2}\left(2\mu_{0}^{2}-1\right)^{3}L}+O\left(T^{2}\right)$
(3.12)

Figure 2: Thermodynamic curvature for charged black holes with near horizon
AdS2 geometry at low temperature. The curvature crosses over from negative to
positive side at $T=0.282$ for $Q=0.2$ (or equivalently $\mu_{0}=1.052$).

The general result for curvature in figure-(1) is positive all along and
approaches zero at high temperatures. However, the low temperature curvature
in eqn. (3.11) behaves slightly differently, a seen in figure-(2). As the
temperature is lowered (and still much below the energy scale $1/r_{h}$), the
curvature can be negative, before resuming the positive behaviour and
subsequent divergence as $T\rightarrow 0$. The $T=0$ behaviour is universal.
That is, the first term of thermodynamic curvature in the general expression
in eqn. (1) and the low temperature computation in eqn. (3.12) agree exactly,
showing the typical $1/T$ divergence in the extremal limit. In fact, we note
that the coefficient of leading $1/T$ term in either of these low temperature
series for curvature is exactly $1/\gamma$.

Figure 3: Red curve represents the zero crossings of curvature (eqn. (3.13))
as a function of $\mu_{0}$. The black dotted and blue dashed curves represent
respectively $1/r_{h}$ and $1/(2\pi r_{h})$ plotted w.r.t. to $\mu_{0}$. $d=2$
for all the cases.

Based on the empirical understanding of thermodynamic curvature, the points of
crossover of $R$ indicate shift of the nature of collective interactions of
microstructures from attraction dominated to repulsion dominated at a
temperature, which is in the physical region of interest. In this same
temperature regime, the black hole also shows a crossover and the description
in terms of an effective action governed by the Schwarzian is feasible (see
figure-2 in [42]). The zero crossing of the thermodynamic curvature in eqn.
(3.11) occurs at

$\displaystyle T_{R=0}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}\left(1-2\mu_{0}^{2}\right)^{2}}{2\pi\sqrt{4\mu_{0}^{6}+8\mu_{0}^{4}-11\mu_{0}^{2}-1}L}\,$
(3.13)

as also seen in figure-(3), happening at temperatures much lower than
$1/r_{h}$. Of course, the crossing of $R$ shown in figure-(3) cannot be
trusted at temperature comparable to $1/r_{h}$.

###  3.2 Quantum corrections

In this subsection, we follow the analysis in [43], where the corrections to
thermodynamic quantities coming from quantum fluctuations were computed for
Reissner-Nordström near-extremal black holes at low temperatures. One of the
key findings of [43] is that in the fixed charge sector, the density of states
of the system does not show a gap, but rather a continuum of states, for the
case of non-supersymmetric black holes. The computed partition function in the
canonical and grand canonical ensembles shows that the low energy limit is
well captured by two dimensional Jackiw-Teitelboim gravity, which is coupled
to a $U(1)$ gauge field and other gauge fields coming from dimensional
reduction. We first compute the thermodynamic curvature in the canonical
ensemble and then in the grand canonical ensemble.

The partition function in the fixed charge ensemble was shown in [43] to be

$\displaystyle Z_{\rm RN}[T,Q]=\Big{(}\gamma\,T\Big{)}^{3/2}e^{\pi
r_{h}^{2}-M_{0}(Q)/T+2\pi^{2}\gamma T}.$ (3.14)

For our purposes, the quantum corrected entropy is

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{0}+\gamma
T+\frac{3}{2}\log\left(\gamma T\right)\,,$ (3.15)

where the various quantities appearing in the above equation can be found to
be [43]

$\displaystyle
Q^{2}=4\pi\left(r_{h}^{2}+\frac{3r_{h}^{4}}{L^{2}}\right),~{}~{}~{}~{}M_{0}=r_{h}\left(1+\frac{2r_{h}^{2}}{L^{2}}\right)\,.$
(3.16)

In the above equations, zero temperature entropy $S_{0}$ and $\gamma$ can be
obtained from equations (3.3) and (3.8) respectively, by setting $d=2$. The
specific heat following from eqn. (3.15) is:

$\displaystyle
C=\frac{1}{18}T\left(\frac{27}{T}+\frac{2\sqrt{6}\pi^{7/4}L^{5/2}\left(\sqrt{\pi
L^{2}+3Q^{2}}-\sqrt{\pi}L\right)^{3/2}}{\sqrt{\pi L^{2}+3Q^{2}}}\right)\,,$
(3.17)

with the chemical potential found to be

$\displaystyle\mu=-\frac{\sqrt{L^{2}+3r_{h}^{2}}}{8\sqrt{\pi}Lr_{h}\left(L^{2}+6r_{h}^{2}\right)^{3}}\left(24\pi^{2}L^{4}r_{h}^{3}T^{2}\left(L^{2}+2r_{h}^{2}\right)+L^{2}T\left(L^{2}+6r_{h}^{2}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left(L^{2}\left(8\pi
r_{h}^{2}+27\right)+6r_{h}^{2}\left(8\pi
r_{h}^{2}+9\right)\right)-4r_{h}\left(L^{2}+6r_{h}^{2}\right)^{3}\right)\,.$
(3.18)

The partition function in the fixed potential ensemble reads [43]

$\displaystyle Z_{\rm
RN}[\beta,\mu]=e^{\mu\frac{Q_{0}}{T}+S_{0}(Q_{0})-M_{0}(Q_{0})/T}\Big{(}\gamma
T\Big{)}^{3/2}e^{2\pi^{2}\gamma T}~{}Z_{U(1)},$ (3.19)

where $Z_{U(1)}$ is the contribution of the $U(1)$ mode, which we ignore for
now it does not induce any major contributions to the curvature to be
discussed below. With the above partition functions in both ensembles, one can
now do the computation of thermodynamic curvatures in both the canonical and
the grand canonical ensembles. For the canonical ensemble, one use the metric

$\displaystyle dl^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{T}\,\frac{\partial^{2}F}{\partial
T^{2}}(dT)^{2}+\frac{1}{T}\,\frac{\partial^{2}F}{\partial Q^{2}}(dQ)^{2}\,.$
(3.20)

whereas for the grand canonical, one can use the same metric noted earlier in
eqn. (2.8). For the grand canonical ensemble, one remembers that $Q$ is taken
to be function of only $\mu_{0}$, which is the zero temperature contribution
from eqn. (3.2). The thermodynamic curvature can be computed analytically as
such in both the ensembles as before, but the expressions are too long and
hence we present the results in figure-(4).

Figure 4: Red and blue curves represent respectively, the thermodynamic
curvatures in canonical and grand canonical ensembles in four dimensions for
$Q=2,L=1$. The value at $T=0$ is $R=1/3$.

There are major changes due to quantum corrections. First, the thermodynamic
curvature at $T=0$ does not diverge anymore, irrespective of the ensemble
used, and actually gives $R_{\rm T=0}=1/3$, which is a universal constant
independent of charge. This is in stark contrast to the behaviour of
thermodynamic curvature noted earlier for full AdSD background [51, 53, 54,
55] and also the semi-classical analysis in the nearly AdS2 near horizon
geometry found in the last section. From the existing results on Ruppeiner
geometry, one concludes that a small positive constant value of thermodynamic
curvature at $T=0$ implies the presence of weakly interacting repulsive nature
of degrees of freedom. In the grand canonical ensemble, as seen from
figure-(5), the thermodynamic curvature continues to have a crossover
behaviour (already seen in the semi-classical analysis in the last section),
which is below the scale $1/r_{h}$, and also below the conformal symmetry
breaking scale $M^{-1}_{SL(2)}$.

Figure 5: The dashed curve represents the temperature at which the curvature
in the grand canonical ensemble has zeroes as a function of charge $Q$ (or
equivalently the chemical potential $\mu_{0}$). The red and blue curves
represent the inverse of horizon radius and the conformal symmetry breaking
scale $M_{SL(2)}$ as a function of charge.

Due to the arguments in [43] for non-supersymmetric black holes, the
thermodynamic analysis in this limit is justified and one expects the
crossover to have a physical interpretation.

##  4 Conclusions

Charged black holes in AdS have long been used as holographic models for
strongly interacting quantum systems at finite density[39]. Close to the
boundary, the geometry asymptotes to AdSD, where the usual AdS/CFT relations
hold giving an understanding of bulk properties in terms of field theory in
$D-1$ spacetime dimensions. The low temperature correlations (for $D>3$)
however were linked to the presence of the near horizon AdS2 geometry alone
[29]. More recently, following examples of the SYK model, interesting one
dimensional Schwarzian action has been studied [19, 16, 17, 24, 23, 35, 36],
which has allowed novel computation of quantum properties, in comparison to
the AdSD approach. The effective one dimensional action has also been obtained
from the low energy limit of the Einstein-Maxwell theory which contains the
charged black holes in asymptotically AdSD spacetime [42, 43, 44, 45]. In
particular, at low temperatures, the black holes exhibit interesting
crossovers to the Schwarzian regime.The semi-classical analysis is now
supplemented by the quantum corrections, which play a crucial role in
resolving the mass gap puzzle in the non-supersymmetry near extremal black
holes [43].

An important byproduct of the recent developments in [42, 43, 44, 45] is that
the thermodynamic quantities of charged black holes at low temperatures
receive non-trivial corrections due to quantum fluctuations. Purely from
thermodynamics point of view, Ruppeiner geometry is known to give reliable
results on the nature of microscopic interactions of degrees of freedom, for a
wide range of systems, starting from black holes to quantum gases.
Thermodynamic geometry of charged black holes in AdSD studied earlier shows
that the curvature is generally positive and diverges at $T=0$[53, 54, 55]. In
this paper, we performed a computation of the curvature $R$ in the low
temperature regime, both in the semi-classical as well as the quantum regime,
based on the correction of thermodynamic quantities close to the near horizon
geometry giving novel behaviour. In the semi-classical limit, as the
temperature is lowered, keeping the charge fixed, $R$ shows crossover from
negative (bosonic) to positive (fermionic) side (before diverging as
$\frac{1}{\gamma T}$), in the region where the temperature is much less than
$1/r_{h}$. Once the quantum corrections are taken into account, the curvature
in the canonical ensemble is positive for all temperatures, but the one in the
grand canonical ensemble continues to show a crossover for a wide range of
charges and for $T\ll 1/r_{h}$, which is also the region where the black hole
undergoes interesting crossovers [42]. More importantly, both the curvatures
are finite at $T=0$, equalling to a universal constant, independent of charge
of black hole. This indicates that the interactions of microstructures are
weakly repulsive, as opposed to the strongly repulsive nature anticipated from
the analysis of thermodynamic curvature in the full AdSD geometry.

Some interesting questions which should be pursued are as follows. Black holes
in higher derivative gravity are known to give non-trivial corrections to the
entropy of black holes, both in supersymmetric and non-supersymmetric
situations. It is important to study their thermodynamic geometry in the near
horizon limit, with corrections to AdS2 geometry as done here. Second, it has
been suggested that the mass gap does exist for BPS black holes in
supersymmetric theories [44, 45] and it is interesting to check whether the
thermodynamic curvature in either of the ensembles has any non-trivial
behaviour beyond that found here.

## Acknowledgements

We thank Matthew Heydeman, Hao Geng and Sudipta Mukherji for helpful
correspondence and comments on the draft. A.S. wishes to thank the Council of
Scientific and Industrial Research (CSIR), Government of India, for financial
support. P.M. thanks IIT Bhubaneswar for Institute fellowship. C.B. thanks the
DST (SERB), Government of India, for financial support through the
Mathematical Research Impact Centric Support (MATRICS) grant no.
MTR/2020/000135 and the Institute of Physics of the Czech Academy of Sciences
& CEICO, Prague, for warm hospitality.

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