# Extended phase space thermodynamics
for hairy black holes

Dumitru Astefanesei1111Email<EMAIL_ADDRESS>, Paulina
Cabrera1,2222Email<EMAIL_ADDRESS>, Robert B. Mann3,4333Email:
<EMAIL_ADDRESS>and Raúl Rojas5444Email<EMAIL_ADDRESS>

1Pontificia Universidad Católica de Valparaíso, Instituto de Física,
Av. Brasil 2950, Valparaíso, Chile

2Universidad Técnica Federico Santa María, Departamento de Física,
Av. España 1680, Valparaíso, Chile

3Department of Physics and Astronomy, University of Waterloo,
Waterloo, Ontario, N2L 3G1, Canada

4Perimeter Institute, 31 Caroline Street North,
Waterloo, ON, N2L 2Y5, Canada

5Departamento de Física, Universidad de Concepción,
Casilla 160-C, Concepción, Chile

###### Abstract

We expand our results in [1] to investigate a general class of exact hairy
black hole solutions in Einstein-Maxwell-dilaton gravity. The dilaton is
endowed with a potential that originates from an electromagnetic Fayet-
Iliopoulos term in $\mathcal{N}=2$ extended supergravity in four spacetime
dimensions. We present the usual thermodynamics by using the counterterm
method supplemented with boundary terms for a scalar field with mixed boundary
conditions. We then extend our analysis by considering a dynamical
cosmological constant and verify the isoperimetric inequality. We obtain a
very rich phase diagram and criticality in both the canonical and grand
canonical ensembles. Within string theory, the cosmological constant is
related to the radius of the external sphere (of the compactification) and can
be interpreted as a modulus. In this context, the existence of a critical
value hints to the fact that the thermodynamic properties of black holes in
lower dimensions depend on the size of the compactification.

###### Contents

  1. 1 Introduction
  2. 2 Hairy black hole chemistry framework
    1. 2.1 The canonical ensemble: Fixed electric charge
    2. 2.2 The grand canonical ensemble: Fixed conjugate potential
  3. 3 Extended phase space thermodynamics: the general case
    1. 3.1 Euclidean action and the usual thermodynamics
    2. 3.2 Smarr formula and the reverse isoperimetric inequality
    3. 3.3 Extended thermodynamics
      1. 3.3.1 The canonical ensemble
      2. 3.3.2 The grand canonical ensemble
    4. 3.4 Novel Phase Transitions
  4. 4 Conclusions
  5. A Novel Phase Behaviour for $\sigma=\infty$

##  1 Introduction

In the context of string theory, scalar fields arise naturally as moduli when
considering specific compactifications. These moduli will appear as fields in
the lower dimensional effective field theory (see, e.g., [2] and references
therein). One such modulus is the cosmological constant that appears when
considering gauged supergravity. Once embedded in string/M theory [3], the
radius of anti-de Sitter ($AdS$) spacetime is related to the radius of the
external sphere. Therefore, the cosmological constant becomes dynamical if the
radius of the sphere evolves in time [4].

In string theory, a sufficiently heavy compactified wrapped object will
effectively give rise to a lower dimensional black hole; examples in $AdS$
gravity include the well known $R$-charged black hole solutions [3, 5]. Here
we consider a different general class of exact hairy black hole solutions in
$AdS_{4}$ [6]555Similar or more general exact hairy black hole solutions in
$AdS$ were obtained in [7, 8, 9, 10, 11, 12, 13, 14]. It is also worth
mentioning that similar solutions [15], which are thermodynamically and
dynamically stable [16, 17, 18], exist in flat spacetime. that can be embedded
in a supergravity model with dyonic Fayet-Iliopoulos terms [19, 20] (see also
[21, 22, 23, 24, 25]). Interestingly, one can study the thermodynamics of
black holes in $AdS$ when the corresponding cosmological constant can vary
[26, 27] and is taken to be a thermodynamic variable similar to ‘pressure’
[28, 29, 30, 31]. In one of the pioneering works on this subject [32], it was
understood that there is a deep analogy between charged $AdS$ black holes and
Van der Waals fluids. What is important from a physical point of view is that,
since there is a critical value for the ‘pressure’, the thermodynamic
properties of the black holes in string theory depend on the size of the
compactification (external sphere) [4, 1]. It is also important to emphasize
that the cosmological constant represents a relevant thermodynamic quantity in
black hole thermodynamics, as follows from the fact that it is required for
the consistency of the Smarr formula, even as a fixed quantity [33, 34].

Previously we have provided a detailed analysis of the criticality phenomena
for a particular exact hairy black hole solution and compared its properties
with the Reissner-Nordström asymptotically anti-de Sitter (RN-$AdS$) black
hole [1]. Interestingly, the scalar field drastically changes its properties.
For example, in the grand canonical ensemble, the scalar field allows critical
phenomena and, unlike the RN-$AdS$ black hole, there is double criticality in
the canonical ensemble.

In this paper, we carry out a complete analysis of the criticality for a
general family of exact hairy black hole solutions [6]. In this case, the
potential of the scalar field contains two extra parameters besides the
cosmological constant that makes the thermodynamics in the extended phase
space even richer. In the canonical ensemble we previously observed the
particular interesting aspect of reentrant phase transitions [35]. However, in
this paper we shall not only provide the technical details and complete the
thermodynamic analysis in the canonical ensemble, but shall also investigate
the grand canonical ensemble, which also has interesting properties that are
distinct from the particular case presented in [1]. We use the counterterm
method developed in [36] (that is consistent with the Hamiltonian formalism
[37, 38]) for the scalar field to regularize the Euclidean action and
quasilocal stress tensor of Brown and York [39]. Armed with these results, we
present the usual thermodynamics of hairy black holes before considering the
extended phase space thermodynamics. We obtain the Smarr formula and provide a
concrete check of the isoperimetric relation and provide its physical
interpretation in this context. Unlike the RN-$AdS$ black hole, which has only
a single critical point in the canonical ensemble, and no interesting phase
behaviour in the grand canonical ensemble, for the class of hairy black holes
we consider, there exist two critical points in each ensemble, along with
reentrant phase transitions in some range of the electric charge and its
conjugate potential. For the grand canonical ensemble, one critical point
corresponds to the termination of a sequence of standard first order phase
transitions in which large black holes ‘condense’ to small ones. The other
corresponds to the beginning of a sequence of first order phase transitions
exhibiting novel behaviour, in which the specific volume increases in a large-
to-small phase transition instead of decreases. We shall consider these new
interesting thermodynamic properties in great detail.

The paper is organized as follows: as a set-up, in Section 2 we briefly review
the main results we have obtained in [1] for a particular charged hairy $AdS$
black hole solution. In Section 3, we present a detailed analysis of the usual
and, also, extended phase space thermodynamics for the general charged hairy
$AdS$ black hole solution in both, canonical and grand canonical, ensembles.
Particularly, we shall consider in detail the novel first order transitions
that appear above the second critical point. In the last section, we conclude
with a brief review of our results.

##  2 Hairy black hole chemistry framework

In this section, we review the thermodynamics of the exact asymptotically
$AdS$ charged hairy black hole solution found in [6], corresponding to the
limit $\gamma\rightarrow 1$ (when the ‘hair parameter666In the original
papers, the hair parameter is denoted by $\nu$. In order to avoid confusion
with the notation for the specific volume $v$, we shall use $\sigma$ instead
of $\nu$.’ $\sigma\rightarrow\infty$) for constant coupling in the exponential
between the scalar field $\phi$ and the Maxwell invariant $F^{2}\equiv
F_{\mu\nu}F^{\mu\nu}$, as shown in the gravitational action below. This is
done in the extended phase space where the (negative) cosmological constant
$\Lambda$ is a pressure term, allowing us to explore the thermodynamic
behaviour for the whole set of $AdS$ theories. We use this example as a set-up
for the complete analysis of the entire family that we shall consider in the
next section.

Let us consider the Einstein-Maxwell-scalar theory described by the action

$I=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\phi)^{2}-U(\phi)-e^{\phi}F^{2}\right]}$
(1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the gauge
field and $A_{\mu}$ the gauge potential, $\phi$ is the scalar field (dilaton)
and $(\partial\phi)^{2}\equiv g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$.
We adopt the unit system where the numerical values of the fundamental
constants are set to unity: $G=1$, $c=1$ (so that $\kappa=8\pi$), $\hbar=1$
and, for the electromagnetic sector, we fix $\mu_{0}=4\pi$.

This theory is known to support a spherically symmetric exact solution for the
following scalar field potential

$U(\phi)=2\alpha\left(2\phi+\phi\cosh\phi-3\sinh\phi\right)+\frac{2\Lambda}{3}\left(\cosh\phi+2\right)$
(2)

where $\alpha$ is an arbitrary dimensionful parameter that has its origin in
extended SUGRA [19, 20], and $\Lambda\equiv-3/\ell^{2}$ is the cosmological
constant, with $\ell$ being the $AdS$ radius. The self-interacting potential
(2), for small $\phi$, decays as
$U(\phi)=-{6}/{\ell^{2}}-{\phi^{2}}/{\ell^{2}}+\mathcal{O}(\phi^{4})$, as
expected for the $AdS$ asymptotics, and the solution to the corresponding
equations of motion is

$\displaystyle
ds^{2}=\Omega(x)\left[-f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{x^{2}f(x)}+d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right]\,,\quad
A_{\mu}=\left(-\frac{q}{x}+c\right)\delta^{t}_{\mu}\,,\quad\phi=\ln(x)$ (3)

where

$\displaystyle
f(x)=\alpha\left(\frac{x^{2}-1}{2x}-\ln{x}\right)+\frac{1}{\Omega(x)}\left(1-2q^{2}\frac{x-1}{x}\right)-\frac{\Lambda}{3}\,,\qquad\Omega(x)=\frac{x}{\eta^{2}(x-1)^{2}}$
(4)

are the metric functions, $\eta$ and $q$ are the two constants of integration
and $c$ in the gauge field is an additive constant that will be used to fixed
the gauge $A_{t}(x_{+})=0$, where $x_{+}$ is the location of the black hole
horizon, $f(x_{+})=0$. The radial $x$-coordinate has the range
$1<x\leq\infty$, where $x=\infty$ is the location of the central singularity
and $x\rightarrow 1$ is the boundary777This is known as the ‘positive branch’
of solutions. There is also a negative branch, which is not studied in this
paper.. The relation to the canonical (Schwarzschild-like) coordinate, at
least near the boundary, is given by $r=\sqrt{\Omega(x)}$.

For this solution, the conserved energy $E$, the Hawking temperature $T$, the
Hawking-Bekenstein entropy $S$, the electric charge $Q$ and its conjugate
potential $\Phi$, the pressure $P$ and thermodynamic volume $V$ are [6]

$E=\frac{q^{2}}{\eta}-\frac{\alpha}{12\eta^{3}}\,,\qquad
T=-\frac{x_{+}f^{\prime}(x_{+})}{4\pi\eta}=\frac{(x_{+}-1)^{2}}{8\pi\eta
x_{+}}\left[-\alpha-2\eta^{2}\left(\frac{x_{+}+1}{x_{+}-1}\right)+4\eta^{2}q^{2}\left(\frac{x_{+}+2}{x_{+}}\right)\right]$
(5) $S=\frac{\pi
x_{+}}{\eta^{2}(x_{+}-1)^{2}}\,,\qquad\Phi=\frac{q(x_{+}-1)}{x_{+}}\,,\qquad
Q=\frac{q}{\eta}\,,\qquad P=-\frac{\Lambda}{8\pi}\,,\qquad V=\frac{2\pi
x_{+}(x_{+}+1)}{3\eta^{3}(x_{+}-1)^{3}}$ (6)

and they satisfy the extended first law $dE=TdS+\Phi dQ+VdP$.

In order to work with dimensionless quantities, in the remainder of this
section we will consider the rescaled thermodynamic variables,

$\eta\rightarrow\sqrt{\alpha}\eta\,,\quad
E\rightarrow\frac{E}{\sqrt{\alpha}}\,,\quad T\rightarrow\sqrt{\alpha}T\,,\quad
S\rightarrow\frac{S}{\alpha}\,,\quad
Q\rightarrow\frac{Q}{\sqrt{\alpha}}\,,\quad P\rightarrow\alpha P\,,\quad
V\rightarrow\alpha^{-\frac{3}{2}}V$ (7)

This makes explicit the assumption that $\alpha>0$.

###  2.1 The canonical ensemble: Fixed electric charge

The ensemble with $T$ and $Q$ kept fixed is achieved by imposing the boundary
condition $\left.\delta(e^{\phi}\star F)\right|_{\partial\mathcal{M}}=0$. The
on-shell Euclidean action is now $\tilde{I}^{E}=I^{E}+I^{E}_{A}$, where
$I_{A}=(2/\kappa)\int{d^{3}x\sqrt{-h}e^{\phi}n_{\mu}F^{\mu\nu}A_{\nu}}$ is the
boundary term for the gauge field, and satisfies the quantum-statistical
relation $\beta^{-1}\tilde{I}^{E}=\mathcal{F}(T,Q)=E-TS$ [1].

We begin with the equation of state, given parametrically by the expressions

$P=\frac{3(x_{+}+1)^{2}}{8\pi
x_{+}}\left[\frac{2(x_{+}+1)^{2}}{x_{+}(x_{+}-1)}\frac{Q^{2}}{v^{4}}-\frac{1}{v^{2}}-\frac{x_{+}^{2}-1-2x_{+}\ln{x_{+}}}{2(x_{+}+1)^{2}}\right]$
(8) $T=\frac{(x_{+}+1)^{2}}{4\pi
x_{+}}\left[\frac{2(x_{+}+1)(x_{+}+2)}{(x_{+}-1)x_{+}}\frac{Q^{2}}{v^{3}}-\frac{1}{v}-\frac{v}{2}\frac{(x_{+}-1)^{3}}{(x_{+}+1)^{3}}\right]$
(9)

where $v\equiv 3V/2S$ is the specific volume that measures the thermodynamic
volume per degree of freedom [40]. It can be straightforwardly shown that, in
the limit $x_{+}\rightarrow 1$, these expressions reduce to the RN-$AdS$
equation of state, $P=T/v-1/(2\pi v^{2})+2Q^{2}/(\pi v^{4})+O(v^{-5})$, as
expected. In the opposite limit, $x_{+}\rightarrow\infty$, we obtain

$v=\frac{1}{\eta}+2\eta
r_{+}^{2}+\mathcal{O}(r_{+}^{4})\,\qquad\eta(r_{+}\rightarrow
0)=\frac{\left(1+\sqrt{1+4Q^{2}}\right)^{\frac{1}{2}}}{2Q}$ (10)

where $r_{+}\equiv\sqrt{\Omega(x_{+})}$. This indicates that in the limit
$x_{+}\rightarrow\infty$ ($S\rightarrow 0$), the specific volume tends to a
constant value, i.e., $V\propto S$.

Critical points in this ensemble satisfy the conditions

$\left(\frac{\partial P}{\partial
v}\right)_{T_{c},Q}=0\,,\qquad\left(\frac{\partial^{2}P}{\partial
v^{2}}\right)_{T_{c},Q}=0$ (11)

for different critical temperatures. We find two solutions to these equations,
corresponding to two critical points, for all fixed $|Q|>0$.

In Fig. 1, we depict the equation of state $P-v$ (fixed $T$) and free energy
$\mathcal{F}-T$ (fixed $P$) diagrams for $Q=1$. The first critical point
($c1$) is reminiscent of RN-$AdS$ criticality, whereas the second critical
point ($c2$) yields new features due to the scalar hair. This one is quite
novel. Below the critical temperature $T_{c2}$, $v(P)$ is a single-valued
function, whereas above this temperature it is multivalued; the critical point
corresponds to the point at which the local maximum and minimum of $P(v)$
become coincident, shown in the dashed line in the panel second from right in
Fig. 1. According to the $\mathcal{F}-T$ diagrams, both critical points are
associated with first-order phase transitions between thermally stable phases,
as follows from the fact that, for the coexisting phases, $C_{Q}\equiv
T(\partial S/\partial
T)_{Q}=-T(\partial^{2}\mathcal{F}/\partial{T}^{2})_{Q}>0$.

Figure 1: Equation of state and $\mathcal{F}-T$ diagram in the canonical
ensemble for $Q=1$. The left-hand (right-hand) panels show the behaviour
around the first (second) critical point. Critical isobars/isotherms are given
by dashed lines.

###  2.2 The grand canonical ensemble: Fixed conjugate potential

The thermodynamic ensemble with $T$ and $\Phi$ kept fixed is compatible with
the boundary condition $\left.\delta A_{\mu}\right|_{\partial\mathcal{M}}=0$.
The on-shell action computed in the Euclidean section, $I^{E}$, satisfies the
quantum-statistical relation $\beta^{-1}I^{E}=\mathcal{G}(T,\Phi)=E-TS-\Phi Q$
[1], where $\beta=T^{-1}$ is the periodicity in the Euclidean time and
$\mathcal{G}$ is the grand canonical thermodynamic potential.

Let us first consider the equation of state $P-T-v$. Parametrically, we have

$P(v,x_{+})=\frac{3(x_{+}+1)^{2}\Phi^{2}}{4\pi(x_{+}-1)v^{2}}+\frac{3(x_{+}+1)^{2}}{8\pi
x_{+}}\left[\frac{2x_{+}\ln{x_{+}}-x_{+}^{2}+1}{2(x_{+}+1)^{2}}-\frac{1}{v^{2}}\right]$
(12)
$T(v,x_{+})=\frac{(x_{+}+1)(x_{+}+2)\Phi^{2}}{2\pi(x_{+}-1)v}-\frac{(x_{+}-1)^{3}}{4\pi
x_{+}(x_{+}+1)v}\left[\left(\frac{x_{+}+1}{x_{+}-1}\right)^{3}+\frac{1}{2}v^{2}\right]$
(13)

where $1<x_{+}\leq\infty$. It is straightforward to show that, in the large
black hole limit $x_{+}\rightarrow 1$, the equation of state reduces to the
RN-$AdS$ equation of state, namely, $P={T}/{v}+(\Phi^{2}-1)/(2\pi
v^{2})+\mathcal{O}(1/v^{3})$, as expected.

While there is no critical phenomena for the RN-$AdS$ black hole in the grand
canonical ensemble, the situation here is more interesting. We have previously
reported one critical point in this ensemble [1]. Upon further investigating
this case we find at most two critical points, each satisfying the conditions

$\left(\frac{\partial P}{\partial
v}\right)_{T_{c},\Phi}=0\;,\qquad\left(\frac{\partial^{2}P}{\partial
v^{2}}\right)_{T_{c},\Phi}=0$ (14)

at different critical temperatures $T_{c}$. The equations in (14) have two
solutions if ${1/\sqrt{2}}<\Phi<1$, one solution if $\Phi>1$, and no solution
if $\Phi<1/\sqrt{2}$. This is illustrated in Fig. 2, where these three
situations are shown. We see that there are two kinds of critical isotherms
for the intermediate values of $\Phi$. One corresponds to the standard Van der
Waals case, where the $P-v$ curve has a point of inflection. The other has the
same novel features as in the canonical ensemble, corresponding to the
coincidence of the local maximum and minimum of $P(v)$. This novel point is
the only critical point for large values of $\Phi>1$.

Figure 2: Equation of state. The left-hand panel, for $\Phi=0.25<1/\sqrt{2}$,
shows no criticality. The middle panels, for $1/\sqrt{2}<\Phi=0.85<1$, show
two different critical isotherms, $T_{c1}$ and $T_{c2}>T_{c1}$, given by the
dashed lines. The right-hand panel, for $\Phi=1.25$, shows one critical
isotherm, again given by the dashed line.

The existence of critical isotherms is indicative of phase transitions, which
we examine by studying the thermodynamic potential $\mathcal{G}=E-TS-\Phi Q$.
In Fig. 3, we depict the thermodynamic potential for the three situations. It
is remarkable that all the critical points are associated with large-to-small
first order phase transitions between two thermally stable phases. Hawking-
Page phase transitions888The two phases involved in the first order Hawking-
Page transitions should be a large black hole and the ground state of the
theory. The fact that the solutions can be embedded in SUGRA is a sufficient
condition for the existence of a stable ground state of the theory. While
explicit construction of the ground state of the theory is outside the scope
of this paper, we would like to point out the reference [41] where exact hairy
soliton solutions were constructed in supergravity. (in which the hairy black
hole discharges to thermal AdS) only take place for $\Phi<1/\sqrt{2}$.

Figure 3: $\mathcal{G}-T$ diagrams. The left-hand panel, for
$\Phi=0.5<1/\sqrt{2}$, shows no criticality, but exhibits a Hawking-Page-type
phase transition at $\mathcal{G}=0$. The middle panels, for
$1/\sqrt{2}<\Phi=0.85<1$, show two different critical isobars, $P_{c1}$ and
$P_{c2}>P_{c1}$, given by the dashed lines. The right-hand panel, for
$\Phi=1.25$, shows one critical isobar, again given by the dashed line.

##  3 Extended phase space thermodynamics: the general case

In this section, we investigate the thermodynamics and critical behaviour of
the general class of solutions for which the scalar potential contains an
extra parameter, $\sigma$, that controls the coupling between the scalar and
gauge field, as well as the self-interaction scalar field potential. The
action is

$I=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\left[R-e^{\gamma\phi}F^{2}-\frac{1}{2}(\partial\phi)^{2}-U(\phi)\right]}$
(15)

where

$\gamma\equiv\left(\frac{\sigma+1}{\sigma-1}\right)^{\frac{1}{2}}$ (16)

and the scalar field potential is [6]

$\displaystyle U(\phi)$
$\displaystyle=\frac{2\alpha}{\sigma^{2}}\left[\frac{\sigma-1}{\sigma+2}\sinh\left(\frac{\sigma+1}{l_{\sigma}}\phi\right)-\frac{\sigma+1}{\sigma-2}\sinh\left(\frac{\sigma-1}{l_{\sigma}}\phi\right)+\frac{4\sigma^{2}-4}{\sigma^{2}-4}\sinh\left(\frac{\phi}{l_{\sigma}}\right)\right]$
$\displaystyle\quad+\frac{\sigma^{2}-4}{3\sigma^{2}}\left[\frac{\sigma-1}{\sigma+2}\exp\left(-\frac{\sigma+1}{l_{\sigma}}\phi\right)+\frac{\sigma+1}{\sigma-2}\exp\left(\frac{\sigma-1}{l_{\sigma}}\phi\right)+\frac{4\sigma^{2}-4}{\sigma^{2}-4}\exp\left(-\frac{\phi}{l_{\sigma}}\right)\right]\Lambda$
(17)

where $l_{\sigma}\equiv\sqrt{\sigma^{2}-1}$. We refer to the parameter
$\sigma$ as the ‘hair parameter’ in the sense that we recover the usual
RN-$AdS$ black hole for the specific value $\sigma=-1$. When $\sigma\leq-1$,
the coupling exponent is $0\leq\gamma<1$. We are interested in the cases
$\sigma>1$ for which the coupling is stronger, $\gamma>1$, and so the
contribution from the scalar field becomes non-trivial in its capacity for
inducing relevant changes on the thermodynamic properties we propose to
explore. The limit $\sigma=1$ corresponds to the Schwarzschild black hole and
the limit $\sigma\rightarrow\infty$ corresponds to the case studied in the
previous section999The limit $\sigma\rightarrow\infty$ should be carefully
taken [15]..

For small $\phi$, the potential decays in accordance with the $AdS$
asymptotics,

$U(\phi)=-\frac{6}{\ell^{2}}-\frac{\phi^{2}}{\ell^{2}}-\frac{1}{12}\left(\frac{\sigma^{2}-3}{\sigma^{2}-1}\right)\cdot\frac{\phi^{4}}{\ell^{2}}+\mathcal{O}(\phi^{5})$
(18)

where $\Lambda=-3/\ell^{2}$. As shown in Fig. 4, the potential is bounded from
below and has a global minimum at a finite value of $\phi$, for any (negative)
value of $\Lambda$, provided $\sigma>1$ and $\alpha>0$. We will assume
$\sigma>1$ and $\alpha>0$ from now on.

Figure 4: The scalar field potential $U(\phi)$ vs $\phi$. Left-hand panel:
$\sigma=\sqrt{3}$. Right-hand panel $\sigma=3$.

The equations of motion are

$R_{\mu\nu}-\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}U(\phi)-T_{\mu\nu}^{EM}=0\,,\quad\Box\phi-\gamma
e^{\gamma\phi}F^{2}-\frac{dU(\phi)}{d\phi}=0\,,\quad\nabla_{\mu}\left(e^{\gamma\phi}F^{\mu\nu}\right)=0$
(19)

where
$T_{\mu\nu}^{EM}=2e^{\gamma\phi}(F_{\mu\alpha}F_{\nu}{}^{\alpha}-\frac{1}{4}g_{\mu\nu}F^{2})$
is the energy-momentum tensor for the electromagnetic field. The exact
solution to the equations of motion, provided the potential (17), is

$ds^{2}=\Omega(x)\left[-f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{f(x)}+d\Sigma^{2}\right],\quad
A_{\mu}=\left(-\frac{q}{\sigma x^{\sigma}}+\frac{q}{\sigma
x_{+}^{\sigma}}\right)\delta_{\mu}^{t},\quad\phi=l_{\sigma}\ln(x)$ (20)

where $\eta$ and $q$ are the constants of integration related to the conserved
charges, namely the mass and electric charge of the black holes101010The
scalar field is ‘secondary hair’ that is present outside the horizon that has
no associated conserved charge., and $d\Sigma:=d\theta^{2}+\sin^{2}\theta
d\varphi^{2}$. The metric functions $f(x)$ and $\Omega(x)$ are

$f(x)=\frac{1}{\ell^{2}}+\alpha\left[\frac{1}{\sigma^{2}-4}-\frac{x^{2}}{\sigma^{2}}\left(1+\frac{x^{-\sigma}}{\sigma-2}-\frac{x^{\sigma}}{\sigma+2}\right)\right]+\frac{x}{\Omega(x)}\left[1-\frac{2q^{2}(x^{\sigma}-1)}{\sigma(\sigma-1)x^{\sigma}}\right]\,,\quad\Omega(x)=\frac{\sigma^{2}x^{\sigma-1}}{\eta^{2}\left(x^{\sigma}-1\right)^{2}}$
(21)

The black hole horizon is located at $x=x_{+}$, where $f(x_{+})=0$. There is
no loss in generality in assuming $\eta\geq 0$. The canonical radial
coordinate is given by the change $r^{2}=\Omega(x)$. In the asymptotic region
($x\rightarrow 1$),

$x=1+\frac{1}{\eta{r}}-\frac{\sigma^{2}-1}{24}\left(\frac{1}{\eta^{3}r^{3}}-\frac{1}{\eta^{4}r^{4}}+\cdots\right)$
(22)

Since the mass of the scalar field potential is $m^{2}=-2\ell^{-2}$, it should
be expected that the scalar field falls off as
$\phi(r)=\frac{A}{r}+\frac{B}{r^{2}}+\mathcal{O}(r^{-3})$. It turns out that,
indeed, $A={l_{\sigma}}{\eta}^{-1}$ and $B=-(1/2){l_{\sigma}}{\eta}^{-2}$. So
in this case, the scalar field obeys a mixed boundary condition since both
modes $A$ and $B$ are non vanishing. It is convenient to introduce a function
$W=W(A)$ that relates both $A$ and $B=B(A)$, by means of
$B(A)\equiv\frac{dW(A)}{dA}$. It follows straightforwardly that

$W(A)=-\frac{A^{3}}{6l_{\sigma}}$ (23)

This expression for $W(A)$ is relevant for the computation of the contribution
to the on-shell Euclidean action coming from the scalar field.

###  3.1 Euclidean action and the usual thermodynamics

In this section, we use counterterms consistent with the mixed boundary
condition of the scalar field [42, 36] to compute the on-shell Euclidean
action. We do the computation using the boundary condition $\left.\delta
A_{\mu}\right|_{\partial\mathcal{M}}=0$ for the gauge field. Once the action
is regularized, we use the Brown-York formalism [39] to compute the quasilocal
boundary stress tensor and the conserved energy. Finally, we verify the first
law of black hole thermodynamics and the quantum-statistical relation. We
would like to emphasize that the hair parameter $\sigma$ is kept arbitrary
and, therefore, this analysis is more general than the one presented in [1].

The full regularized action $I^{E}$ in the Euclidean section is composed of
the bulk part $I^{E}_{\text{bulk}}$ given by Eq. (15), the Gibbons-Hawking
boundary term $I^{E}_{GH}$ [43], the gravitational counterterm for
asymptotically $AdS$ spacetime $I^{E}_{g}$ [44], and the boundary term for the
scalar field $I^{E}_{\phi}$ [36],

$I^{E}=I^{E}_{\text{bulk}}-\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{-h}K}+\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{-h}}\left(\frac{2}{\ell}+\frac{\ell\mathcal{R}}{2}\right)+\frac{1}{2\kappa}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-h}\left[\frac{\phi^{2}}{2\ell}+\frac{W(A)}{\ell
A^{3}}\phi^{3}\right]$ (24)

respectively, where $\mathcal{R}$ is the Ricci scalar for the foliation
$x=x_{B}=const$, where $B$ stands for boundary, and $h_{ab}$ is the induced
metric on the boundary $\partial\mathcal{M}$. At the end, we shall consider
the limit $x_{B}\rightarrow 1$.

Let us proceed by computing the terms in (24) one by one. By using the
equations of motion, the bulk part of the action in the Euclidean section can
be reduced to
$I_{\text{bulk}}^{E}=-(1/4){\beta}{\eta}^{-1}[f(x_{B})\Omega^{\prime}(x_{B})-2\eta^{2}\int_{x_{+}}^{x_{B}}\Omega(x)\,dx]$.
Now, by expanding this first result in powers of $(x_{B}-1)$, we get

$I_{\text{bulk}}^{E}=\beta\left[\frac{1}{4\eta}+\frac{\alpha}{6\eta^{3}}-\frac{q^{2}}{\eta(\sigma-1)}+\frac{\sigma(x_{+}^{\sigma}+1)}{4\eta(x_{+}^{\sigma}-1)}-\frac{\sigma^{2}-1}{48\eta^{3}\ell^{2}}\right]+\frac{\beta}{2\eta^{3}\ell^{2}(x_{B}-1)^{3}}+\mathcal{O}\left(x_{B}-1\right)$
(25)

The next terms are computed on the hypersurface $x=x_{B}$, where
$h_{ab}dx^{a}dx^{b}=\Omega(x)\left[-f(x)dt^{2}+d\Sigma\right]$. For the
Gibbons-Hawking boundary term, let us first write down the extrinsic curvature
tensor111111The normal unit to the hypersurface $x=const$ is
$n_{\mu}=-\delta_{\mu}^{r}(g^{xx})^{-\frac{1}{2}}$, and
$K_{\mu\nu}=\nabla_{\mu}n_{\nu}$,
$K_{ab}=(\frac{dx^{\mu}}{dy^{a}})(\frac{dx^{\nu}}{dy^{b}})K_{\mu\nu}$. and its
trace,

$K_{ab}=\frac{\Omega^{\prime}}{2\eta}\left(\frac{f}{\Omega}\right)^{\frac{1}{2}}\left[\frac{(f\Omega)^{\prime}}{\Omega^{\prime}}\delta^{t}_{a}\delta^{t}_{b}-\delta^{\theta}_{a}\delta^{\theta}_{b}-\sin^{2}\theta\delta^{\phi}_{a}\delta^{\phi}_{b}\right]\,,\quad
K=-\frac{1}{2\eta}\left(\frac{f}{\Omega}\right)^{\frac{1}{2}}\left(\frac{f^{\prime}}{f}+\frac{3\Omega^{\prime}}{\Omega}\right)$
(26)

It follows that

$I_{GH}^{E}=\beta\left[{\frac{3q^{2}}{2\eta\left(\sigma-1\right)}}-\frac{3}{4\eta}-\frac{\alpha}{4\eta^{3}}+\frac{\sigma^{2}-1}{16\eta^{3}\ell^{2}}\right]-\frac{3\beta}{2\eta^{3}\ell^{2}(x_{B}-1)^{3}}-\frac{\beta}{\eta(x_{B}-1)}+\mathcal{O}\left(x_{B}-1\right)$
(27)

The Ricci scalar on the boundary is $\mathcal{R}=2/\Omega(x_{B})$ and, thus,
the gravitational counterterm is

$I_{g}^{E}=\beta\left[\frac{1}{2\eta}+\frac{\alpha}{6\eta^{3}}-\frac{q^{2}}{\eta(\sigma-1)}+\frac{\sigma^{2}-1}{8\eta^{3}\ell^{2}}\right]+\frac{\beta}{\eta^{3}\ell^{2}(x_{B}-1)^{3}}+\frac{\beta(\frac{1}{\eta}-\frac{\sigma^{2}-1}{8\eta^{3}\ell^{2}})}{x_{B}-1}+\mathcal{O}\left(x_{B}-1\right)$
(28)

For the boundary term of the scalar field, we consider the expression $W$
obtained in (23). The result is

$I_{\phi}^{E}=-\frac{\beta(\sigma^{2}-1)}{6\eta^{3}\ell^{2}}+\frac{\beta(\sigma^{2}-1)}{8\eta^{3}\ell^{2}(x_{B}-1)}+\mathcal{O}(x_{B}-1)$
(29)

Now, by adding up all the contributions, we find that the divergent terms
$\propto(x_{B}-1)^{-1}$ and $\propto(x_{B}-1)^{-3}$ cancel each other and, in
the limit $x_{B}\rightarrow 1$, the final result is finite

$I^{E}=I_{\text{bulk}}^{E}+I_{GH}^{E}+I_{g}^{E}+I_{\phi}^{E}=\beta\left[\frac{1}{4\eta}+\frac{\alpha}{12\eta^{3}}-\frac{q^{2}}{2\eta(\sigma-1)}+\frac{\sigma+1+(\sigma-1)x_{+}^{\sigma}}{4\eta(x_{+}^{\sigma}-1)}\right]$
(30)

The total action $I^{E}$ satisfies the quantum-statistical relation, as we
shall show. Let us compute the thermodynamic quantities for this solution. We
start by computing the conserved energy $E$. We use the Brown-York formalism
[39], which requires the quasilocal boundary stress tensor $\tau_{ab}$. For
the full action (24), we have

$\tau_{ab}\equiv-\frac{2}{\sqrt{-h}}\frac{\delta I}{\delta
h^{ab}}=-\frac{1}{\kappa}\left(K_{ab}-h_{ab}K+\frac{2}{l}h_{ab}-lG_{ab}\right)-\frac{h_{ab}}{2\kappa\ell}\left[\frac{\phi^{2}}{2}+\frac{W(A)}{A^{3}}\phi^{3}\right]$
(31)

and, according to the Brown-York formalism, the conserved energy is

$E=\oint_{s_{\infty}^{2}}{d^{2}x\sqrt{\Sigma}n^{a}\tau_{ab}\xi^{b}}$ (32)

where $\Sigma=\Omega^{2}(x_{B})\sin^{2}\theta$ is the determinant of the
metric on the 2-sphere $ds_{\Sigma}^{2}=\Omega(x_{B})d\Sigma$,
$\xi^{a}=\delta^{a}_{t}$ is the time-like Killing vector and $n_{a}$ the
normal unit to $t=const$, given by

$n_{a}=\frac{\delta_{a}^{t}}{\sqrt{-g^{tt}}}=\sqrt{\Omega f}\delta_{a}^{t}$
(33)

To compute the conserved energy, we only need the leading terms in powers of
$(x_{B}-1)$ of $\tau_{tt}$, that is,

$\tau_{tt}=\frac{1}{4\pi\ell}\left(\frac{q^{2}}{\sigma-1}-\frac{\alpha+3\eta^{2}}{6\eta^{2}}\right)(x_{B}-1)+\mathcal{O}\left[(x_{B}-1)^{2}\right]$
(34)

Using this result, the conserved energy of the system is

$E=\frac{q^{2}}{\eta(\sigma-1)}-\frac{\alpha+3\eta^{2}}{6\eta^{3}}$ (35)

Let us now compute the remaining thermodynamic quantities. The electric charge
$Q$ can be obtained, as usual, by using the Gauss law at the boundary

$Q=\frac{1}{4\pi}\oint_{s^{2}_{\infty}}{e^{\gamma\phi}\star
F}=\frac{q}{\eta}\,,\qquad\Phi=\frac{q}{\sigma
x_{+}^{\sigma}}\left(x_{+}^{\sigma}-1\right)$ (36)

where its conjugate potential is defined as $\Phi\equiv
A_{t}(x=x_{+})-A_{t}(x=1)$. The Hawking-Bekenstein entropy is $S=A/4$, where
$A=4\pi\Omega(x_{+})$ is the area of the event horizon, and the expression for
the Hawking temperature

$S=\frac{\pi\sigma^{2}x_{+}^{\sigma-1}}{\eta^{2}\left(x_{+}^{\sigma}-1\right)^{2}}\,,\qquad
T=-\frac{f^{\prime}(x_{+})}{4\pi\eta}=\frac{3E}{2S}-\frac{x_{+}^{-\frac{\sigma-1}{2}}}{4\sigma\sqrt{\pi{S}}}\left[\frac{4\pi\sigma
Q^{2}}{x_{+}S}+x_{+}^{\sigma}(\sigma-1)+\sigma+1\right]$ (37)

is obtained by removing the conical singularity in the Euclidean metric. It is
straightforward to show that first law for charged black holes $dE=TdS+\Phi
dQ$ is satisfied121212Since the scalar field is secondary hair, no scalar
charge appears in the first law [45].. Now, we can easily verify that the
Euclidean action, given by Eq. (30), satisfies the quantum-statistical
relation,

$\frac{I^{E}}{\beta}=E-TS-\Phi Q\equiv\mathcal{G}(T,\Phi)$ (38)

where $\mathcal{G}$ is the grand canonical thermodynamic potential. The first
law can be written as $d\mathcal{G}=-SdT-Qd\Phi$, from where it follows that
$\mathcal{G}=\mathcal{G}(T,\Phi)$.

###  3.2 Smarr formula and the reverse isoperimetric inequality

Let us now consider the extended phase space where the cosmological constant
represents the pressure of a perfect fluid of density $\rho=-P$, with
$P=-\Lambda/(8\pi)$. The extended first law is

$dE=TdS+VdP+\Phi dQ$ (39)

where $V$ is the thermodynamic volume given by

$V\equiv\left(\frac{\partial{E}}{\partial{P}}\right)_{Q,S}=\frac{2\pi\sigma^{2}}{3\eta^{3}}\frac{(\sigma+1)x_{+}^{2(\sigma-1)}+(\sigma-1)x_{+}^{\sigma-2}}{(x_{+}^{\sigma}-1)^{3}}$
(40)

The thermodynamic variables satisfy a simple relation known as the Smarr
formula. It can be obtained by scaling arguments, i.e., by looking at the
dimensions of the corresponding thermodynamic variables [34]. Since the theory
is given by two dimensionful constants ($\Lambda$ and $\alpha$), besides the
standard thermodynamic variables $(E,T,S,Q,\Phi)$, the Smarr formula
incorporates the extra pairs $PV$ and $\alpha\mathcal{B}$, and it reads

$E=2TS+\Phi Q-2PV-2\alpha\mathcal{B}$ (41)

where $\mathcal{B}\equiv\left({\partial E}/{\partial\alpha}\right)_{S,Q,P}$
measures how $E$ changes due to variations in $\alpha$. This suggests that the
first law can be further extended to

$dE=TdS+VdP+\Phi dQ+\mathcal{B}d\alpha$ (42)

provided we can find a concrete physical interpretation for $\alpha$.
Nevertheless, we treat the parameter $\alpha$ as a constant without variation
and so the quantity $\mathcal{B}$ is not going to be relevant in our analysis.

Returning to thermodynamic volume $V$, by using (22) we observe that $V=4\pi
r_{+}^{3}/3+\mathcal{O}(r_{+})$, or in other words the leading contribution is
the Euclidean volume, as could be expected from the fact that for large black
holes the scalar field and its self-interaction are negligible at the event
horizon. In general, the thermodynamic volume $V$ is conjectured to satisfy
the so-called Reverse Isoperimetric Inequality [4]:

$\mathcal{R}\equiv\left[\frac{(d-1)V}{\omega_{d-2}}\right]^{\frac{1}{d-1}}\left(\frac{\omega_{d-2}}{A}\right)^{\frac{1}{d-2}}\,\,\,\geq\,\,\,1$
(43)

where $d$ is the number of dimensions of the spacetime, $\omega_{d-2}$ is the
area of the unit cross section, and $A$ is the area of the black hole event
horizon. In our case, with $d=4$ and $\omega_{2}=4\pi$, we find that

$\mathcal{R}_{\sigma}(x_{+})={x_{+}^{-\frac{1}{6}(\sigma+1)}}\left[\frac{\sigma+1}{2\sigma}x_{+}^{\sigma}+\frac{\sigma-1}{2\sigma}\right]^{\frac{1}{3}}$
(44)

is the general expression131313Since the value of the scalar field on the
horizon is $\phi_{+}\equiv l_{\sigma}\ln(x_{+})$, the ratio $\mathcal{R}$
depends, alternatively, on $\sigma$ and $\phi_{+}$. for the ratio
$\mathcal{R}$. From the following considerations

$\lim_{x_{+}\rightarrow
1}\mathcal{R}_{\sigma}(x_{+})=1\,,\qquad\lim_{x_{+}\rightarrow\infty}\mathcal{R}_{\sigma}(x_{+})\rightarrow\infty\,,\qquad\frac{d\mathcal{R}_{\sigma}(x_{+})}{dx_{+}}=\frac{2^{\frac{2}{3}}x_{+}^{-\frac{1}{6}\sigma-\frac{7}{6}}(x_{+}^{\sigma}-1)(\sigma-1)}{12\sigma^{\frac{1}{3}}\left[x_{+}^{\sigma}(\sigma+1)+\sigma-1\right]^{\frac{2}{3}}}\geq
0$ (45)

it is straightforward to show that the inequality (43) is satisfied. The
physical interpretation is that, for a fixed thermodynamic volume, the charged
hairy $AdS$ black hole carries less entropy than its RN-$AdS$ counterpart, for
which $\mathcal{R}=1$. This is expected, because the remaining entropy is
carried by the scalar field (‘hairy’ degrees of freedom) outside the event
horizon.

Henceforth we shall rescale the thermodynamic quantities as in (7);
consequently $\alpha>0$ will not appear explicitly in any further expression.

###  3.3 Extended thermodynamics

With the on-shell Euclidean action properly computed, we have the tools to
study the thermodynamics. The hair parameter $\sigma$ introduces a new
phenomenon of reentrant phase transitions in both the canonical and grand
canonical ensembles, in addition to the novel transition behaviour seen in the
previous section for $\sigma\to\infty$.

####  3.3.1 The canonical ensemble

We first consider thermodynamics in the extended phase space with the electric
charge of the black hole being kept fixed. The boundary condition for the
gauge field is $\delta(e^{\gamma\phi}\star F)|_{\partial\mathcal{M}}=0$. The
thermodynamic potential in this ensemble is $\mathcal{F}(T,Q)=E-TS$ and can be
obtained by a Legendre transformation of the thermodynamic potential from the
grand canonical ensemble (38). This is equivalent to adding a boundary term
[46]

$I_{A}^{E}=-\frac{2}{\kappa}\int_{\partial\mathcal{M}}{d^{3}x\sqrt{h}e^{\gamma\phi}n_{\mu}F^{\mu\nu}A_{\nu}}=\beta\,Q\Phi$
(46)

to the action, yielding $\mathcal{F}(T,Q)=\beta^{-1}\tilde{I}^{E}=E-TS$, where
$\tilde{I}^{E}=I^{E}+I^{E}_{A}$.

Let us first study the equation of state, given parametrically by

$T=\frac{1}{4\pi\eta^{3}\Omega(x_{+})}\left[\frac{2\eta^{4}\left((\sigma+2)x_{+}^{\sigma}+2\sigma-2\right)Q^{2}}{\sigma(\sigma-1)x_{+}^{\sigma}}-1\right]-\frac{(\sigma+2)x_{+}^{\frac{1}{2}(\sigma+1)}+(\sigma-2)x_{+}^{-\frac{1}{2}(\sigma-1)}}{4\pi\sigma\sqrt{\Omega(x_{+})}}$
(47) $v=\frac{x_{+}^{\sigma}(\sigma+1)+\sigma-1}{\eta(x_{+}^{\sigma}-1)x_{+}}$
(48)

where $\eta=\eta(x_{+},P,Q)$ is obtained from141414Since $\eta>0$, Eq. (49)
has two solutions of interest. One of these solutions for $\eta$ is positive
only for $\sigma>2$, while the other one is positive only for $1<\sigma<2$.
For the particular case $\sigma=2$, note that  $\lim_{\sigma\rightarrow
2}\left[\frac{x_{+}^{2-\sigma}}{\sigma^{2}(\sigma-2)}-\frac{1}{\sigma^{2}-4}\right]=-\frac{1}{4}\left[\ln(x_{+})+\frac{3}{4}\right].$
$f(x_{+})=0$,

$\frac{2Q^{2}x_{+}^{2-2\sigma}(x_{+}^{\sigma}-1)^{3}\eta^{4}}{\sigma^{3}(\sigma-1)}-\frac{(x_{+}^{\sigma}-1)^{2}\eta^{2}}{\sigma^{2}x_{+}^{\sigma-2}}+\frac{x_{+}^{2}}{\sigma^{2}}\left(1+\frac{x_{+}^{-\sigma}}{\sigma-2}-\frac{x_{+}^{\sigma}}{\sigma+2}\right)-\frac{1}{\sigma^{2}-4}-\frac{8\pi
P}{3}=0$ (49)

For large black holes, the equation of state can be put in the form
$T=Pv+1/(2\pi v)-2Q^{2}/(\pi v^{3})+\mathcal{O}(v^{-5})$, regardless of the
value of the hair parameter $\sigma$. This is because the scalar field and its
self-interaction are negligible at the event horizon of a large black hole,
and thus the corrections to the equation of state due to the scalar are
subleading.

When the scalar field and its self-interaction become important near the event
horizon, which is the case for intermediate and small black holes, the
thermodynamic behaviour becomes more interesting. Two values for the electric
charge are relevant in our analysis: $Q_{\text{min}}$ and $Q_{0}$. Let us, for
concreteness, fix $\sigma=2$, for which $Q_{\text{min}}\approx 2.622$ and
$Q_{0}\approx 2.712$. In Fig. 5, we illustrate the equation of state for three
representative values of $Q$. For $Q<Q_{\text{min}}$ (left-hand panel in Fig.
5), there is no critical behaviour. For $Q>Q_{0}$ (right-hand panel in Fig.
5), the critical behaviour is qualitatively the same as that of the RN-$AdS$.
For $Q_{\text{min}}<Q<Q_{0}$ (middle plot in Fig. 5), two critical isotherms
are observed. These two critical isotherms are related to reentrant phase
behaviour.

Figure 5: Equation of state for $Q=1.50$ (left-hand panel), $Q=2.65$ (middle)
and $Q=2.90$ (right-hand panel), for the theory with $\sigma=2$. For
$Q_{min}<Q<Q_{0}$, two critical points are observed.

To appreciate better the nature of this double criticality within
$Q_{\text{min}}<Q<Q_{0}$, consider the $\mathcal{F}-T$ diagram, depicted in
Fig. 6. As pressure increases from small values (the bluer curve in Fig. 6),
an ‘inverted’ swallowtail appears. There is no first order phase transition in
this case because the curves that intersect themselves in the inverted
swallowtail are not at the global minimum of $\mathcal{F}$. However, as the
pressure further increases, the inverted swallowtail moves leftward with
respect to the lower part of the curve, eventually giving rise to a second
(standard) swallowtail. This standard swallowtail results in a first order
phase transition from large to small black holes in the direction of
decreasing temperature. For a tiny range for $P$, there is also a zeroth order
phase transition from a small black hole to a large one, characterized by a
jump discontinuity in $\mathcal{F}$, as well as in its first derivative
$(\partial\mathcal{F}/\partial T)_{P}$, as it is shown in the second panel of
Fig. 6.

Figure 6: $\mathcal{F}-T-P$ for $Q=2.65$ and $\sigma=2$. As pressure
increases, the inverted swallowtail moves leftward with respect to the
leftmost part of the curve, and a second (standard) swallowtail with a first
order phase transition appears in addition to a zeroth order phase transition.

This ‘reentrant phase behaviour’, reported also in [35], is new when compared
to the behaviour for hairy black holes in the theory $\sigma=\infty$. It
happens within the interval $Q_{min}<Q<Q_{0}$. It is also important to notice
that $\Lambda$ is not required to be considered a thermodynamic variable for
reentrant phase behaviour to occur, because the parameter that is being varied
is temperature. Therefore, once $Q$ is conveniently fixed (within
$Q_{\text{min}}<Q<Q_{0}$), it is only necessary to search for different
(fixed) values of $P$ for which reentrant phase transitions occur. For the
concrete case $Q=2.65$ in the theory $\sigma=2$, reentrant phase behaviour
exists approximately within the tiny interval $4.61\cdot 10^{-3}<P<4.65\cdot
10^{-3}$. These values of $P$ for which there is reentrant phase behavior were
obtained by solving numerically the equations $(\partial P/\partial
v)_{T,Q}=0$ and $(\partial^{2}P/\partial v^{2})_{T,Q}=0$.

####  3.3.2 The grand canonical ensemble

The parametric expressions for the equation of state in this ensemble are

$P=\frac{3}{8\pi\sigma^{2}}\left[\frac{x_{+}^{2-\sigma}}{\sigma-2}-\frac{x_{+}^{\sigma+2}}{\sigma+2}+x_{+}^{2}-\frac{\sigma^{2}}{\sigma^{2}-4}+\frac{\mathcal{X}_{1}^{2}}{x_{+}^{3\sigma-4}v^{2}}\left(\frac{\mathcal{X}_{2}-2\sigma+2}{\mathcal{X}_{2}-3\sigma}\frac{2\sigma\Phi^{2}}{\sigma-1}-1\right)\right]$
(50)
$T=\frac{x_{+}}{4\pi\sigma}\left[-\frac{(x_{+}^{\sigma}-1)^{3}}{\sigma\mathcal{X}_{1}}v+\frac{2\mathcal{X}_{1}\mathcal{X}_{2}}{(\sigma-1)x_{+}^{\sigma-2}(x_{+}^{\sigma}-1)}\frac{\Phi^{2}}{v}-\frac{\mathcal{X}_{1}(\mathcal{X}_{2}-\sigma)}{\sigma
x_{+}^{2\sigma-2}v}\right]$ (51)

where
$\mathcal{X}_{1}\equiv(\sigma+1)x_{+}^{2\sigma-2}+(\sigma-1)x_{+}^{\sigma-2}$
and $\mathcal{X}_{2}\equiv(\sigma+2)x_{+}^{\sigma}+2\sigma-2$ have been
defined for simplicity. The specific volume $v$ is obtained as usual,

$v\equiv\frac{3V}{2S}=\frac{x_{+}^{\sigma}(\sigma+1)+\sigma-1}{\eta(x_{+}^{\sigma}-1)x_{+}}$
(52)

In this ensemble, the value of $\Phi$ determines in a remarkable way the
thermodynamic behaviour of black holes. For
$\Phi<\Phi_{c}\equiv\sqrt{(\sigma-1)/(2\sigma)}$, there is no criticality,
i.e., the conditions for criticality given in (14) are not satisfied. For
$\Phi_{c}<\Phi<\Phi_{0}(\sigma)$, there is one critical point. The dependence
of $\Phi_{0}$ on $\sigma$ is shown in Fig. 7. Notice that
$\Phi_{0}(\sigma)<1/\sqrt{2}$. For $\Phi_{0}<\Phi<1/\sqrt{2}$, there are three
critical points. For $1/\sqrt{2}<\Phi<1$, there are two critical points and,
for $\Phi>1$, there is one critical point.

This rich behaviour is depicted in Fig. 8 for different values of $\Phi$. We
see from the upper left panel that for small
$\Phi<\sqrt{(\sigma-1)/(2\sigma)}$ there are no phase transitions, but once
$\Phi=\sqrt{(\sigma-1)/(2\sigma)}$ we see a new kind of criticality, in which
the local maximum and minimum of $P(v)$ are coincident, but $P(v)$ is not
single-valued, noted in the previous section. Above the critical point neither
$P(v)$ nor $v(P)$ are single-valued functions. For larger values of
$\Phi>\sqrt{(\sigma-1)/(2\sigma)}$ a novel phase transition is present (shown
in the upper central panel), whose behaviour we shall discuss in the next
section. As $\Phi$ crosses the next threshold at $\Phi=1/\sqrt{2}$, two new
critical points appear at very low pressures, shown in the inset of the upper
right panel in Fig. 8; this is the reentrant behaviour shown in the middle
diagram of Fig. 5. Over this range of values of $\Phi$, as temperature
increases, we will have the rentrant behaviour discussed in the previous
subsection, followed by a novel phase transition of the same type as for $\
\sqrt{(\sigma-1)/(2\sigma)}<\Phi<1/\sqrt{2}$. As $\Phi$ becomes larger than
$1/\sqrt{2}$, the middle critical point disappears, and only the smallest one
(corresponding to the standard Van der Waals case) and the largest one
(corresponding to the novel case) are present. For $\Phi>1$ the Van der Waals
critical point vanishes, and only the one corresponding to the novel case
remains.

Figure 7: The number of critical points depends on the value of $\Phi$. The
values $\Phi_{c}$ and $\Phi_{0}$ depend on $\sigma$.

\begin{overpic}[scale={0.38}]{plot23}\put(54.0,32.0){\includegraphics[scale={0.17}]{plot24}}
\end{overpic}

Figure 8: Equation of state $P-v$ for five characteristic values of the
conjugate potential, in the model $\sigma=3$. The panels show the cases:
$\Phi=0.50<\Phi_{c}$, $\Phi_{c}<\Phi=0.60<\Phi_{0}$,
$\Phi_{0}<\Phi=0.70<1/\sqrt{2}$, $1/\sqrt{2}<\Phi=0.85<1$, $\Phi>1$,
respectively. Dashed lines correspond to critical behaviour.

In order to elucidate the nature of the critical points, we study the
thermodynamic potential vs temperature. In parametric form we have

$\mathcal{G}=\frac{1}{12{\eta}^{3}}-\frac{\sigma}{4\eta(x_{+}^{\sigma}-1)^{2}}\left[x_{+}^{2\sigma}\left(1-\frac{2\sigma\Phi^{2}}{\sigma-1}\right)-1\right]$
(53)
$T=\frac{x_{+}}{4\pi\eta\sigma}\left[-\frac{(x_{+}^{\sigma}-1)^{2}}{\sigma
x_{+}^{\sigma}}+\frac{2\eta^{2}\mathcal{X}_{2}\Phi^{2}}{\sigma-1}-\frac{\eta^{2}(\mathcal{X}_{2}-\sigma)(x_{+}^{\sigma}-1)}{\sigma
x_{+}^{\sigma}}\right]$ (54)

where

$\eta=\sqrt{\frac{2\sigma^{2}(\sigma-1)x_{+}^{\sigma-2}}{(x_{+}^{\sigma}-1)\left[\mathcal{X}_{2}-\sigma
x_{+}^{\sigma}(3-4\Phi^{2})\right]}\left[\frac{8\pi
P}{3}+\frac{1}{\sigma^{2}-4}-\frac{x_{+}^{2}}{\sigma^{2}}\left(1+\frac{x_{+}^{-\sigma}}{\sigma-2}-\frac{x_{+}^{\sigma}}{\sigma+2}\right)\right]}$
(55)

with $\mathcal{X}_{2}$ given in the line below (51). By studying the
thermodynamic potential, we find that for almost every critical behaviour
observed in $P-v$ diagrams, there is a standard swallowtail typical for large-
to-small first order phase transitions between stable phases. However this
swallowtail does not signify a standard Van der Waals transition, but rather
novel phase behaviour that we shall discuss in more detail in the next
subsection. The only exception occurs for the interval
$\Phi_{0}<\Phi<1/\sqrt{2}$, where there is reentrant phase behaviour in
addition to the novel behaviour. All the different situations are depicted in
Fig. 9 and the exceptional case for the reentrant phase behaviour within
$\Phi_{0}<\Phi<1/\sqrt{2}$ is depicted in Fig. 10. In this case, there is a
reentrant phase transition in the sense that there is a range of (fixed values
of) $P$ for which the system can go from large to small to large black hole in
the direction of decreasing temperature.

\begin{overpic}[scale={0.42}]{plot30}\put(25.0,24.0){\includegraphics[scale={0.24}]{plot31}}
\end{overpic}

Figure 9: $\mathcal{G}$ vs $T$ diagram for five characteristic values of
$\Phi$, in the model $\sigma=3$. The panels show the cases:
$\Phi=0.50<\Phi_{c}$, $\Phi_{c}<\Phi=0.60<\Phi_{0}$,
$\Phi_{0}<\Phi=0.70<1/\sqrt{2}$, $1/\sqrt{2}<\Phi=0.85<1$, $\Phi>1$,
respectively. Dashed lines correspond to critical behaviour. These diagrams
are consistent with the equation of state depicted in Fig. 8

Figure 10: $\mathcal{G}-T$ for the special case $\Phi_{c}<\Phi<\Phi_{0}$. We
take $\Phi=0.70$ in the model $\sigma=3$. For this case, there are three
critical isobars. Here we show only the reentrant phase behaviour. As pressure
increases from small values, an inverted swallowtail appears (at the first
critical point). Then the branch of large stable black holes intersects the
inverted swallowtail and a second swallowtail (a standard one) forms. This is
when a reentrant phase transition takes place, as detailed in the second
panel. The standard swallowtail shrinks to zero at the second critical point.

One (dimensionless) quantity that provides information about the criticality
properties of the system is the ‘critical compressibility factor’, defined as

$z_{c}\equiv\frac{P_{c}v_{c}}{T_{c}}$ (56)

For both a Van der Waals fluid and the RN-$AdS$ black hole, $z_{c}$ turns out
to be exactly $3/8$. Unlike the critical exponents, which are expected to be
universal, $z_{c}$ is known to differ from one substance to another in
ordinary chemistry [47, 48, 49]. In this case, the critical compressibility
factor depends on $\Phi$, as depicted in Fig. 11, for $\sigma=3$. The
dependence of $z_{c}$ on $\Phi$ indicates that the conjugate potential plays
the role of a fluid parameter that characterizes the nature the dual conformal
field theory. Also in Fig. 11, we have depicted the critical pressures for the
whole range of $\Phi$. Notice that in the limit $\Phi\rightarrow\Phi_{c}^{+}$,
$P\rightarrow\infty$, and in the other hand, in the limit
$\Phi\rightarrow\infty$, $P\rightarrow 0.0768$.

\begin{overpic}[scale={0.20}]{plot38}
\put(48.0,48.0){\includegraphics[scale={0.09}]{plot39}} \end{overpic}

Figure 11: Left-hand panel: Critical compressibility factor $z_{c}$ vs $\Phi$,
for $\sigma=3$, and $\Phi_{c}\approx 0.5774$, $\Phi_{0}\approx 0.6957$. Right-
hand panel: Critical pressures vs $\Phi$.

###  3.4 Novel Phase Transitions

Several of the phase transitions that appear in both the canonical and grand
canonical ensembles have rather unusual features. First, the free-energy
diagrams exhibit swallowtails whose size increases with increasing pressure,
as is clear from the rightmost diagram in Fig. 1, the two rightmost diagrams
in Fig. 3, and the second and fourth diagrams in Fig. 9. This kind of
phenomenon has been seen before in Lovelock gravity [50], and is referred to
as a reverse Van der Waals phase transition. It denotes a situation where
condensation of large black holes into small ones takes place at increasingly
higher temperatures and pressures above a critical point, instead decreasing
values of these quantities, as holds for the standard case [31]; an example
for the class of hairy black holes we are considering is shown in the middle
diagram in Fig. 5, with the re-entrant behaviour shown in the right panel for
Fig. 6.

However the $P-v$ diagrams that correspond to the swallowtails in the third
diagram in Fig. 1, the two rightmost diagrams in Fig. 2, and all but the two
leftmost diagrams in Fig. 8 indicate that something quite different is going
on in these cases. The phase transition for these novel cases takes place at
the temperature $T_{c2}$. There is a ‘subcritical temperature’
$T_{sc}(<T_{c2})$ at which

$\left(\frac{\partial v}{\partial
P}\right)_{T_{sc}}=0\,,\qquad\left(\frac{\partial^{2}v}{\partial
P^{2}}\right)_{T_{sc}}=0$ (57)

For $T\leq T_{sc}$, $P$ is a single-valued function of $v$, while for
$T>T_{sc}$ there is a region in $v$ where $P$ is triple-valued.

For concreteness, we focus on the situation $1/\sqrt{2}<\Phi<1$ in the grand
canonical ensemble, for which there are two critical points. We fix
$\Phi=0.85$ and $\sigma=3$. The critical points are

$c1:\quad(P=5.13\cdot 10^{-6},v=42.67,T=7.09\cdot 10^{-4})\,,\qquad
c2:\quad(P=0.1347,v=3.8361,T=0.4789)$ (58)

This case is interesting because the critical point $c1$ taking place at the
lower pressure is similar to the Van der Waals one, whereas the second
critical point, $c2$, at higher pressure, has no analogue with standard
thermodynamic systems. In Fig. 11 we depict in detail the critical isotherms
for this case. Near the critical point $c1$ we observe standard Van der Waals
behaviour, but around the critical point $c2$, the pressure is multi-valued:
for a given $(T,v)$, there are at most three possible values of $P$, as is
clear from the right diagram in Fig. 11.

\begin{overpic}[scale={0.38}]{plot41}
\put(60.0,42.0){\includegraphics[scale={0.14}]{plot42}} \end{overpic}

Figure 12: Equation of state for $\Phi=0.85$ ($\sigma=3$). There are two
critical isotherms (black dashed curves), one corresponding to the critical
point $c1$ (left), the other to the critical point $c2$ (right).

In order to understand the distinction between these two kinds of phase
transitions, let us begin by analyzing the phase transition for the more
familiar critical point $c1$. Fig. 13 depicts the $\mathcal{G}-T$,
$\mathcal{G}-v$ and $\mathcal{G}-P$ diagrams151515In the $\mathcal{G}-v$
diagram –with $T$ kept fixed– the issue is that pressure is not held fixed
along a given isotherm, but this is precisely the case for black hole
chemistry.. From $\mathcal{G}-T$, we know that the large-to-small first order
black hole phase transition is discontinuous in the entropy
$S=-(\partial\mathcal{G}/\partial{T})_{P}$ in the direction of decreasing
temperature. From the $\mathcal{G}-P$ diagram, we know that it is
discontinuous in $V=(\partial\mathcal{G}/\partial{P})_{T}$ in the direction of
increasing pressure, which is quite intuitive. Essentially, for an ensemble of
large black holes, as the thermodynamic volume decreases the pressure
increases, until a point is reached where ‘condensation’ begins, with the
large black holes changing into small ones with no further increase in
pressure or free energy, as is clear from the central panel in Fig. 13. The
pressure at which this occurs is given by Maxwell’s equal area law. As volume
is further decreased, more and more large black holes in the ensemble will
condense into small ones, until the entire ensemble consists of small black
holes. As the volume decreases even more, the pressure significantly
increases, since further condensation is impossible. The situation is fully
analogous to a gas condensing into a liquid at a given temperature as the
volume of the system decreases.

Figure 13: $\mathcal{G}-T$, $\mathcal{G}-v$ and $\mathcal{G}-P$ diagrams for
the first critical point. The (first order) phase transition is discontinuous
in $S=-(\partial\mathcal{G}/\partial{T})_{P}$ and
$V=(\partial\mathcal{G}/\partial{P})_{T}$.

Figure 14: $\mathcal{G}-T$, $\mathcal{G}-v$ and $\mathcal{G}-P$ diagrams for
the second critical point. The first order phase transition is discontinuous
in $S=-(\partial\mathcal{G}/\partial{T})_{P}$ and
$V=(\partial\mathcal{G}/\partial{P})_{T}$, like in the previous case.

Now, consider the second critical point, for which the corresponding diagrams
are depicted in Fig. 14. Here the swallowtails in the $\mathcal{G}-T$ and
$\mathcal{G}-P$ planes exhibit opposite behaviour compared to the previous
case, growing as $P$ increases (left diagram) and as $T$ increases (right
diagram). The central panel is most instructive – we see that neither
$\mathcal{G}(v)$ nor $v(\mathcal{G})$ are single-valued functions above
$T_{c2}$. For these temperatures the first order transition corresponds to
what appears to be a form of ‘reverse condensation’, in which black holes of
smaller specific volume condense into black holes of larger specific volume!

We can understand this behaviour by considering the equation of state and its
corresponding $\mathcal{G}-P$ diagram, both illustrated in Fig. 15, with
$T=1.62$. For this choice of $T$, the transition takes place at $P=1$,
corresponding to the intersection point of the swallowtail. The arrows
indicate the novel transition from large-to-small specific volumes. Consider
the system at point A, corresponding to a large black hole of negative free-
energy. As the specific volume $v$ decreases, the free-energy and pressure
both increase until point B is reached. At this point the system undergoes
‘reverse condensation’, moving from B to D to F at constant $P$, with
small-$v$ black holes condensing into larger-$v$ ones. This all takes place at
the swallowtail crossover. After this, the system is at point F, corresponding
to a larger value of $v$. The equation of state then indicates that as $P$
increases, $v$ will again decrease (and the free energy will increase). The
net effect is a small-$v$-to-large-$v$ first order transition, despite the
fact that increasing pressure corresponds to decreasing $v$. Note that points
C and E correspond to the cusps in the swallowtail; the system does not
actually transit through these points.

We emphasize that ‘reverse condensation’ is a property of the specific volume
$v$; the thermodynamic volume $V$ decreases at the first-order transition, as
is clear from central and right panels of Fig. 15. Note that the equal-area
law can be applied here, despite the fact that $P$ is not a single-valued
function of $V$ and vice-versa.

This peculiar form of transition takes place because neither $v$ nor $V$ are
monotonically increasing functions of the horizon size $1/x_{+}$ as shown in
Fig. 16. An inspection of the central and right panels indicates the
distinction between the two: the transition point B is smaller than the local
maximum of $v(1/x_{+})$ but larger than the local maximum of $V(1/x_{+})$.
Hence during the transition the thermodynamic volume $V$ and the horizon
radius of the black hole consistently decrease, as is clear from the left
panel, whereas the specific volume increases. During condensation, the black
hole gets smaller in size, but larger in specific volume, shown in the central
panel. Although it may seem counterintuitive that $V$ decreases while $v$
increases, this occurs because the entropy decreases considerably, and hence
$v=3V/(2S)$ has a larger local maximum.

Figure 15: Equation of state $P-v$ and $P-V$, and the $\mathcal{G}-P$ diagram
for $\Phi=0.85$ and $T=1.62$ in the model $\sigma=3$. The trajectory of points
is compared side by side for the phase transition that occurs at $P\approx 1$.
Point A corresponds to the large-$v$ and large-$V$ phase, respectively. Point
B (and F) corresponds to the intersection point of the swallowtail. Points C
and E correspond to the local maximum and minimum of $P(v)$ (and $P(V)$),
respectively. Point G corresponds to the small-$v$ and small-$V$ phase,
respectively.

Figure 16: $P-x_{+}^{-1}$, $v-x_{+}^{-1}$ and $V-x_{+}^{-1}$ for $T=1.62$ and
$\Phi=0.85$, in the model $\sigma=3$. The limit $x_{+}^{-1}\rightarrow 1$
corresponds to the large black hole limit.

The case $\sigma\to\infty$ also exhibits similar behaviour, but has a few
special features that we discuss in the Appendix.

##  4 Conclusions

We have investigated the thermodynamics of a four-dimensional asymptotically
$AdS$ family of exact hairy black hole solutions [6], whose scalar field has a
non-trivial self-interacting potential that can be embedded in supergravity
[19, 20]. The Euclidean action was regularized by using the counterterm method
in the presence of a scalar field satisfying mixed boundary conditions [36]
and the conserved energy was obtained by using the quasilocal formalism of
Brown and York [39]. We have explicitly shown that the first law is satisfied
and the Smarr relation holds as long as the two parameters of the theory
($\Lambda$ and $\alpha$) are included.

By considering the cosmological constant as a pressure term, we have analyzed
the thermodynamics in the extended phase space. After revisiting the study of
the special theory $\sigma\rightarrow\infty$ in Section 2, we studied the
general case in Section 3. We have proved that the thermodynamic volume
satisfies the Reverse Isoperimetric Inequality for the set of the hair
parameter $\sigma>1$. One of the main new results is the existence of
reentrant phase transition in both the canonical and grand canonical ensemble
for suitable values of $Q$ and $\Phi$, respectively. As pointed out in [51],
for a phase transition to be reentrant, it must involve the transformation of
a system from one state into a macroscopically similar state via at least two
phase transitions through the variation of a single thermodynamic parameter.
In this case, this parameter is the temperature, while the pressure is held
fixed. As $P$ is not required to be variable in order to have reentrant phase
transition, these results do not restrict to the black hole chemistry
approach.

For the case $\sigma\rightarrow\infty$, $P$ is double-valued in both,
canonical and grand canonical, ensembles and no reentrant phase transition was
observed. For finite $\sigma$, there is a region of the extended phase space
where the pressure becomes triple-valued only in the grand canonical ensemble.
This region, for which $\Phi>\Phi_{c}\equiv\sqrt{(\sigma-1)/(2\sigma)}$, is
where the second critical point belongs to. We have analyzed in detail this
multi-valuedness in pressure and the associated phase transition and the
results are summarized in Fig. 15. We would like to point out that there also
exist examples in ordinary chemistry systems where $P$ is a multi-valued
function [52].

The results presented in this paper, i.e., the multiple critical points, the
reentrant phase behaviours in both ensamble, and multi-valuedness of the
pressure and its associated novel phase transition, are new features of
charged hairy black holes in $d=4$ spacetime dimensions. They do not exist
when either the scalar field or its self-interaction is turned off. Therefore,
this study offers concrete evidence that a self-interacting scalar field can
drastically change and enrich the thermodynamic behaviour of black holes.

In the context of string theory, it has recently been arguedw that variations
of the cosmological constant can be interpreted as variations in the volume of
the sphere of compactification [4, 1]. Therefore, in accordance with our
results, namely, the existence of several critical isobars, the size of the
sphere of compactification leads to a different thermodynamic behaviour of
black holes.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. DA was supported during this work by the Fondecyt
grant 1200986. The work of RR was supported by the FONDECYT grant 3220663. The
work of PC was supported by ANID Grant No. 21182145 and UTFSM’s grant No.
072/2021 thanks to the DPP’s support.

##  Appendix A Novel Phase Behaviour for $\sigma=\infty$

For $\sigma=\infty$, as long as $\Phi>1/\sqrt{2}$ (values of $\Phi$ for which
the system exhibits critical phenomena), the specific volume

$v=\sqrt{2}\sqrt{2\Phi^{2}-1}-\frac{\sqrt{2}\left(2\sqrt{2}\pi\sqrt{2\Phi^{2}-1}T-8\Phi^{2}+3\right)}{\sqrt{2\Phi^{2}-1}x_{+}}+\mathcal{O}(x_{+}^{-2})$
(59)

goes to a constant value as $x_{+}\rightarrow\infty$. The equation of state
consequently develops a branch characterized by an almost completely vertical
line in $P-v$. One way to see this is that in the very tiny black hole limit,
the entropy and the thermodynamic volume becomes proportional. Indeed, both
decays in the same order of $x_{+}$,

$S=\frac{2\pi(2\Phi^{2}-1)}{x_{+}}+\mathcal{O}(x_{+}^{,2})\,,\qquad
V=\frac{4\sqrt{2}\pi(2\Phi^{2}-1)^{3/2}}{3x_{+}}+\mathcal{O}(x_{+}^{-2})$ (60)

Another way to see this proportionality is by writing
$x_{+}=1/(\eta^{2}r_{+}^{2})+2-\eta^{2}r_{+}^{2}+\mathcal{O}(r_{+}^{4})$,
obtained from $\Omega(x_{+})=r_{+}^{2}$ in the limit $x_{+}\rightarrow\infty$,
in the expression for $V$ given in (6). The entropy is simply $S=\pi
r_{+}^{2}$ and the volume is

$V=\frac{2\pi r_{+}^{2}}{3\eta}+\mathcal{O}(r_{+}^{4})$ (61)

In any case, $V/S=2/(3\eta)$, where
$\eta(x_{+}\rightarrow\infty)=1/(\sqrt{2}\sqrt{2\Phi^{2}-1})$ can be directly
obtained by taking the limit $x_{+}\rightarrow\infty$ in the expression for
$\eta$ obtained from the horizon equation $f(x_{+})=0$. The equation of state
near the second critical point and the $\mathcal{G}-P$ diagram are depicted in
Fig. 17 for $\Phi=0.85$. For this value of $\Phi$,
$v(x_{+}\rightarrow\infty)\approx 0.9434$. Apart from the fact that the
specific volume tends to a constant value in the limit
$x_{+}\rightarrow\infty$, the phase transition follows the same general
features as for finite $\sigma$.

Figure 17: Equation of state $P-v$, $P-V$ and the corresponding
$\mathcal{G}-P$ diagram for $\Phi=0.85$, $T=1.62$ in the model
$\sigma=\infty$. For this isotherm, the phase transition occurs at $P\approx
3.7$.

For completeness, we have plotted $P-(x_{+})^{-1}$, $v-(x_{+})^{-1}$ and
$V-(x_{+})^{-1}$.

Figure 18: $P-x_{+}^{-1}$, $v-x_{+}^{-1}$ and $V-x_{+}^{-1}$ for $T=1.62$ and
$\Phi=0.85$, in the model $\sigma=3$. The limit $x_{+}^{-1}\rightarrow 1$
corresponds to the large black hole limit.

A subtle difference in this case, compared with the finite $\sigma$ case, is
observed in the second panel of Fig. 18. After the transition is taking place,
from B to F, the specific volume $v$ still increases a little bit when moving
from F to G. It follows from (59) that, in the limit $x_{+}\rightarrow\infty$,
$(\partial v/\partial x_{+})_{T}$ is positive only provided

$T>\frac{8\Phi^{2}-3}{2\sqrt{2}\pi\sqrt{2\Phi^{2}-1}}$ (62)

For $\Phi=0.85$, this inequality is $T>0.47$, which is fulfilled since
$T_{c2}\approx 0.82>0.47$. Therefore, the first order phase transition
associated with the second critical point $c2$ has the peculiarity that, for
the small$-S$ phase, $v$ slightly increases as $S$ decreases, contrary to the
case of finite $\sigma$.

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