# Scalarization of planar anti-de Sitter charged black holes in Einstein-
Maxwell-scalar theory

Chatchai Promsiri<EMAIL_ADDRESS>Faculty of Science, King
Mongkut’s University of Technology Thonburi, 126 Prachauthid Road., Bangkok
10140, Thailand  Takol Tangphati<EMAIL_ADDRESS>Faculty of Science, King
Mongkut’s University of Technology Thonburi, 126 Prachauthid Road., Bangkok
10140, Thailand  Ekapong Hirunsirisawat<EMAIL_ADDRESS>Theoretical
and Computational Physics (TCP); Theoretical and Computational Science Center
(TaCS), Faculty of Science, King Mongkut’s University of Technology Thonburi
(KMUTT), Pracha Uthit Road, Bangkok, 10140, Thailand  Learning Institute, King
Mongkut’s University of Technology Thonburi (KMUTT), Pracha Uthit Road,
Bangkok, 10140, Thailand  Supakchai Ponglertsakul<EMAIL_ADDRESS>Strong Gravity Group, Department of Physics, Faculty of Science, Silpakorn
University, Nakhon Pathom 73000, Thailand

###### Abstract

We construct scalarized planar charged black holes in Einstein-Maxwell-scalar
(EMS) theory with the presence of a negative cosmological constant. Domains of
existence of black hole solutions are given in term of nonminimally coupling
constant $\alpha$. Perturbative stability of a scalarized black hole is
investigated by calculating its quasinormal modes. Thermodynamic properties of
the scalarized planar solution are also discussed. We observe no evidence of
instability of the scalarized black holes. Moreover, we find that scalarized
planar charged AdS black holes are thermodynamically preferred over scalar-
free solutions in grand canonical and canonical ensembles. The transition
between scalar-free solutions and scalarized solutions is found to be the
thermal second order phase transition. The transition of these solutions
shares some similar features with conductor-superconductor phase transition.

##  I Introduction

It is widely accepted that black holes in general relativity (GR) cannot
express other physical quantities rather than mass, electric charge and
angular momentum or as John Wheeler simply puts it as a black hole can have no
hair [1]. The no-hair conjecture states that any stationary black hole
solutions of Einstein-Maxwell theory (i.e., electrovacuum) must belong to the
Kerr-Newmann family. For instance, Bekenstein shows noneexistence of
asymptotically flat black holes with scalar, vector, and spin-2 meson hair [2,
*Bekenstein:1972ky, *Bekenstein:1972ny, *PhysRevD.51.R6608]. However, it is
found that a black hole can possess macroscopic external degrees of freedom or
hairs in several other setups. These include when GR couples with other types
of matter fields, asymptotic structure of spacetime is modified, and
gravitational theories beyond Einstein-Maxwell theory, etc. Interestingly,
these hairy black holes have much richer physics than their bald counterpart.
There are several studies providing a counterexample of the no-hair
conjecture, e.g., black holes with Yang-Mills hair and its variants [6,
*Aichelburg:1992st, *Donets:1992zb, *Kleihaus:1997rb, *Winstanley:1998sn,
*PhysRevD.93.064064, *PhysRevD.47.2242, *Ponglertsakul:2016fxj], hairy black
holes in a boxlike boundary [14, *PhysRevLett.116.141101,
*PhysRevLett.116.141102, *Sanchis-Gual:2016tcm, *Basu:2016srp, *Peng:2017squ,
*Dias:2018yey], and hairy black holes in modified gravity theories [21,
*BenAchour:2018dap, *Creminelli:2020lxn, *Lee:2021uis, *Erices:2021uyu]. For a
nice review on this subject, see [26, 27].

A bald black hole is able to dynamically develop into a black hole with scalar
hair via spontaneous scalarization. This usually occurs in the models with
nonminimally coupled scalar field. The coupling term of scalar field tends to
make the scalar-free black hole solution unstable and leads to the formation
of a black hole with nontrivial scalar field profile outside its horizon, or a
scalarized black hole (SC BH). Spontaneous scalarization is originally
considered in scalar tensor theory for neutron star where scalar field is
nonminimally coupled to Ricci curvature [28] and later extends to rapidly
rotating neutron star [29]. It is found that the scalarized neutron star is
energetically favored over the scalar-free solution. Despite this, it is shown
in [30, 31] that black holes in scalar tensor theory do not differ from GR,
however, scalarized black holes in scalar tensor theory are made possible by
surrounding black holes with nonconformally invariant matter [32, 33].

Beyond scalar tensor theory, spontaneous scalarization is also found in
extended scalar tensor Gauss-Bonnet (eSTGB) gravity where scalar field is
generally coupled to the Gauss-Bonnet curvature term [34,
*PhysRevLett.120.131104, *PhysRevLett.120.131102, *PhysRevLett.123.011101,
*PhysRevLett.123.011101, *PhysRevLett.126.011103, *PhysRevLett.126.011104].
Despite the fact that many studies have been devoted to explore spontaneous
scalararization in the eSTGB gravity, however, one of the remaining tasks is
to determine the endpoint of instability of the scalar free solutions and
dynamical evolution of scalarized solutions. Nonlinear higher curvature terms
in the eSTGB gravity render these problems to be challenging. There is,
however, a considerably simpler model that also allows for the spontaneous
scalarization, i.e., Einstein-Maxwell-scalar (EMS) theory with nonminimally
coupling function between scalar field and Maxwell field. A fully nonlinear
dynamical evolution from Reissner-Nordström (RN) black hole into scalarized
black hole in the EMS theory is investigated in [40]. In this model,
scalarized black holes depend on the coupling function between the scalar
field and the Maxwell field. In [41], the dependence of various coupling
functions and dynamical features of scalarized black holes are discussed.
Moreover, stability and quasinormal modes of scalarized black holes are also
investigated by several studies [42, 43, 44, 45]. More interestingly,
spontaneous scalarization in the EMS theory in asymptotically de Sitter (dS)
[46], anti de Sitter (AdS) [47, *Zhang:2021etr, *Guo:2021zed] and in a cavity
[50] are proposed and studied.

The seminal papers of Bekenstein [51] and Hawking [52] suggest that black
holes could have entropy and nonzero temperature. Since then, black hole
thermodynamics has become one of the most interesting topics to black hole
physics communities. Thermodynamics of BHs in asymptotically flat spacetime
has been often studied, however, there are some unsettled issues about thermal
equilibrium configurations. By putting the BHs in AdS space, Hawking and Page
[53] find that BHs can be in thermal equilibrium with its surroundings since
the AdS boundary acts as a reflecting wall. The thermodynamics and phase
transition of Reissner-Nordström black hole in AdS (RN-AdS) are studied in
[54, *Chamblin:1999hg, *Burikham:2014gwa]. Recently, by investigating
thermodynamics of asymptotically AdS BHs in the EMS theory in a normal phase
space, where the cosmological constant $\Lambda$ is fixed, scalarized
solutions are found to be thermodynamically preferred over the RN-AdS BHs in
the microcanonical ensemble [49]. Interestingly, this system exhibits a
reentrant phase transition, which consists of zeroth order and second order
types of phase transitions between RN-AdS and SC BHs in some range of
parameters. However, one may consider $\Lambda$ as a thermodynamic variable
analogous to pressure $P$ in the first law of BH thermodynamics. This
framework is called an extended phase space approach [57, *Dolan:2012jh,
*Kubiznak:2012wp, *Gunasekaran:2012dq, *Kubiznak:2016qmn,
*Altamirano:2014tva]. In this way, phase structure of BHs in the EMS model has
been studied in both canonical and grand canonical ensembles [63].

It is known that the BHs in asymptotically AdS spacetime can have three
different geometries of event horizon with positive, zero and negative
curvature constants. These are called the spherical, planar and hyperbolic
BHs, respectively. The thermodynamics and phase structure of these topological
BHs were studied in various theories of gravity, for example see [64,
*Emparan:1999gf, *Cai:2001dz, *Cai:2014znn, *PhysRevD.102.024042,
*Priyadarshinee:2021rch, *KordZangeneh:2020qeg]. Intriguingly, thermal
properties of AdS BHs and an area law of BH’s entropy lead to the development
of gauge/gravity duality, which states that the thermodynamics of BHs in a
higher dimensional bulk AdS space is holographically dual to a thermal state
of gauge theories that living into the AdS boundary [71, *Witten:1998zw,
*Gubser:1998bc, *Aharony:1999ti]. By means of gauge/gravity duality, planar BH
has received more attention in nongravitational physics communities, such as,
condensed matter physics [75, *Gubser:2005ih, *PhysRevLett.101.031601,
*Hartnoll:2008kx, 79, 80, 81, 82], hydrodynamics [83, *Son:2007vk,
*Bhattacharyya:2007vjd, *Bhattacharyya:2008mz, *Burikham:2016roo,
*Baggioli:2020ljz, *Ahn:2022azl, *Baggioli:2022pyb] and quantum information
[91, *Ryu:2006ef, *Ogawa:2011bz, *Wu:2014xva, *Momeni:2016yts, *Chen:2021lnq].
In recent years, holographic superconductor in a probe limit with nonminimally
coupled EMS theory are considered [97, *Mohammadi:2022buy]

In the present paper, we extend the previous studies of [49, 63] by
considering spontaneous scalarization of four dimensional AdS BH with planar
horizon in EMS gravity with nonminimal coupling of the scalar and Maxwell
fields. We establish domain of existence, perturbative stability of scalarized
solutions. The law of BH mechanics demonstrates the mathematical analogy
between the dynamics of BH and the law of thermodynamics [99]. Thus, it is
interesting to investigate the spontaneous scalarization in the context of BH
thermodynamcis. The differences between spherical and planar horizon of BHs on
the scalarization mechanism is also discussed in both mechanical and
thermodynamical perspectives.

This paper is organized as follows. We begin Sec. II with the action for the
Einstein-Maxwell-scalar model in AdS spacetime and present its field equations
along with the boundary conditions for the numerical method. Moreover, the
bifurcation line that presents the emergence of scalarized planar AdS charged
black hole is studied. In Sec. III, the numerical results of the scalarized
planar AdS charged black hole are demonstrated with the analysis of the
stability of the scalarized solutions. In Sec. IV, the study of the
thermodynamics properties of the scalarized planar AdS charged black hole is
presented in the grand canonical ensemble (fixed potential) and canonical
(fixed charge) ensemble. We also present the Euclidean regularized action
technique for eliminating the divergences at the boundary and from the
electromagnetic (EM) field. Finally, we summarize the results and discuss the
novel findings in the conclusions.

##  II Einstein-Maxwell-Scalar model in AdS spacetime

In this section, we discuss basics equations involving the EMS-AdS theory. We
will derive equations of motion and boundary conditions that allow an
existence of scalarized solutions. Then we explore the origin of instability
that leads to scalarization.

###  II.1 The model

We consider the model such that massless scalar field ($\varphi$) is minimally
coupled to gravity but nonminimally coupled to gauge field $A_{\mu}$. The
action is given by

$\displaystyle S$ $\displaystyle=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\bigg{[}R-2\Lambda-2\nabla_{\mu}\varphi\nabla^{\mu}\varphi-\mathcal{G}(\varphi)F_{\rho\sigma}F^{\rho\sigma}\bigg{]},$
(1)

where $F_{\rho\sigma}=\nabla_{\rho}A_{\sigma}-\nabla_{\sigma}A_{\rho}$ and
coupling between the scalar field and the gauge field is denoted by
$\mathcal{G}(\varphi)$. The cosmological constant relates to the AdS radius
$L$ by $\Lambda=-3/L^{2}$.

Varying (1) with respect to $g^{\mu\nu},A^{\mu}$ and $\varphi$, we obtain the
following equations of motion

$\displaystyle R_{\mu\nu}-\frac{g_{\mu\nu}R}{2}+\Lambda g_{\mu\nu}$
$\displaystyle=2T_{\mu\nu},$ (2)
$\displaystyle\nabla_{\mu}\left(\mathcal{G}F^{\mu\nu}\right)$
$\displaystyle=0,$ (3) $\displaystyle\nabla^{\mu}\nabla_{\mu}\varphi$
$\displaystyle=\frac{1}{4}\frac{d\mathcal{G}}{d\varphi}F_{\rho\sigma}F^{\rho\sigma},$
(4)

where the energy-momentum tensor is defined by

$\displaystyle T_{\mu\nu}$
$\displaystyle=\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}\partial^{\rho}\varphi\partial_{\rho}\varphi+\mathcal{G}\left(g^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\right).$
(5)

Remark that, the nonminimally coupling $\mathcal{G}$ cannot take an arbitrary
form. Instead, we must choose this coupling function that allows an existence
of the scalar free solution. This means when $\varphi=0$, the action (1)
admits Reissner-Nordström-AdS solution. This puts a condition on function
$\mathcal{G}$ i.e., $\mathcal{G}(0)=1$ and
$\frac{d\mathcal{G}(0)}{d\varphi}=0$. In the absence of the gauge field, the
action (1) admits black hole solutions with scalar hair [100,
*Winstanley:2002jt, *Sudarsky:2002mk].

###  II.2 Equations of motion

Here we consider spacetime metric with planar symmetry. The spacetime metric,
the gauge field and the scalar field take the following forms

$\displaystyle ds^{2}$
$\displaystyle=-N(r)e^{-2\delta(r)}dt^{2}+N(r)^{-1}dr^{2}+\frac{r^{2}}{L^{2}}\left(dx^{2}+dy^{2}\right),$
(6) $\displaystyle A_{\mu}$ $\displaystyle=\\{V(r),0,0,0\\},$ (7)
$\displaystyle\varphi$ $\displaystyle=\phi(r).$ (8)

We define mass function $m(r)$ as

$\displaystyle N(r)$
$\displaystyle\equiv-\frac{2m(r)}{r}+\frac{r^{2}}{L^{2}}.$ (9)

Putting these into equations of motion (2)–(4), we obtain

$\displaystyle m^{\prime}$
$\displaystyle=\left(\frac{r^{3}}{2L^{2}}-m\right)r\phi^{\prime
2}+\frac{Q^{2}}{2r^{2}\mathcal{G}},$ (10) $\displaystyle\delta^{\prime}$
$\displaystyle=-r\phi^{\prime 2},$ (11) $\displaystyle V^{\prime}$
$\displaystyle=-\frac{Q\text{e}^{-\delta}}{r^{2}\mathcal{G}},$ (12)
$\displaystyle 0$
$\displaystyle=\left(\frac{r^{4}}{L^{2}}-2rm\right)\phi^{\prime\prime}+\left(\frac{4r^{3}}{L^{2}}-2m-\frac{Q^{2}}{r\mathcal{G}}\right)\phi^{\prime}+\frac{Q^{2}}{2r^{2}\mathcal{G}^{2}}\frac{d\mathcal{G}}{d\phi}.$
(13)

We denote derivative with respect to radial coordinate $r$ by $\prime$ e.g.,
$m^{\prime}=\frac{dm}{dr}$. $Q$ is constant of integration which can be
associated to black hole’s charge (see AppendixA). When $\phi=0$, the RN-AdS
with planar symmetry is the solution to these equations. To obtain a
scalarized planar AdS charged black hole, we must solve these nonlinear
differential equations (10) with appropriated boundary conditions.

###  II.3 Boundary conditions

We assume that there exists regular event horizon located at $r=r_{+}$ i.e.,
$N(r_{+})=0$. The field functions $\\{m,\delta,V,\phi\\}$ are expected to be
finite near the event horizon and at spatial infinity. Therefore, we expand
the field functions near the event horizon accordingly

$\displaystyle m$ $\displaystyle=m_{0}+m_{1}(r-r_{+})+...,$ (14)
$\displaystyle\delta$ $\displaystyle=\delta_{0}+\delta_{1}(r-r_{+})+...,$ (15)
$\displaystyle V$ $\displaystyle=V_{0}+V_{1}(r-r_{+})+...,$ (16)
$\displaystyle\phi$ $\displaystyle=\phi_{0}+\phi_{1}(r-r_{+})+...$ (17)

where equation of motions (10)–(13) allow us to determine the following

$\displaystyle m_{0}$ $\displaystyle=\frac{r_{+}^{3}}{2L^{2}},\quad
m_{1}=\frac{Q^{2}}{2r_{+}^{2}\mathcal{G}(\phi_{0})},$ (18)
$\displaystyle\delta_{1}$ $\displaystyle=-r_{+}\phi_{1}^{2},$ (19)
$\displaystyle V_{1}$
$\displaystyle=-\frac{e^{-\delta_{0}}Q}{r_{+}^{2}\mathcal{G}(\phi_{0})},$ (20)
$\displaystyle\phi_{1}$
$\displaystyle=\frac{Q^{2}L^{2}}{2r_{+}\mathcal{G}(\phi_{0})\left(Q^{2}L^{2}-3r_{+}^{4}\mathcal{G}(\phi_{0})\right)}\frac{d\mathcal{G}(\phi_{0})}{d\phi}.$
(21)

Here, we use the gauge freedom to set $V_{0}=0$. At spatial infinity, we
obtain

$\displaystyle m$ $\displaystyle=M-\frac{Q^{2}}{2r}+...,$ (22)
$\displaystyle\phi$ $\displaystyle=\frac{\phi_{f}}{r^{3}}+...,$ (23)
$\displaystyle\delta$ $\displaystyle=\frac{3\phi_{f}^{2}}{2r^{6}}+...,$ (24)
$\displaystyle V$ $\displaystyle=\Phi+\frac{Q}{r}+...,$ (25)

Here, the parameter $M$ is related to the Komar mass $E$ of black hole (see
Appendix A). The leading order of scalar field at spatial infinity is denoted
by constant $\phi_{f}$. The electrostatic potential $\Phi$ is defined as the
difference between the gauge field at infinity and event horizon

$\displaystyle\Phi=A_{t}(\infty)-A_{t}(r_{+})=V(\infty)-V(r_{+}).$ (26)

Since our equations of motion (10)–(13) has shifted symmetry in $V$, therefore
we can add arbitrary constant such that $V(r_{+})=V_{0}=0$. Thus the electric
potential of black hole solution becomes

$\displaystyle\Phi=A_{t}(\infty)=V(\infty).$ (27)

Remark that one obtain an explicit form of $\Phi$ by integrating (12)

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle-\int_{r_{+}}^{\infty}dr\frac{Qe^{-\delta}}{\mathcal{G}(\phi)r^{2}}.$
(28)

This equation serves as a good check on our numerical results.

###  II.4 Tachyonic instability

Before discussing on scalarized solution, it is useful to understand what
drives bald planar AdS charged black hole away from its stability. Thus in
this subsection, we consider a linear scalar perturbation on fixed background
of planar AdS charged black holes. We obtain such a solution by choosing

$\displaystyle\phi(r)=0,\quad\delta(r)=0,\quad
N(r)=-\frac{2M}{r}+\frac{r^{2}}{L^{2}}+\frac{Q^{2}}{r^{2}},\quad
A_{t}=\frac{Q}{r}.$ (29)

Equation governing scalar perturbation on curved background can be obtained by
linearization of (4) with a small perturbation $\delta\varphi$

$\displaystyle\left(\square-\mu_{\text{eff}}^{2}\right)\delta\varphi$
$\displaystyle=0,$ (30)

where

$\displaystyle\mu_{\text{eff}}^{2}=-\frac{d^{2}\mathcal{G}(0)}{d\phi^{2}}\frac{Q^{2}}{2r^{4}}.$
(31)

It turns out that the second derivative of the nonminimally coupling function
plays a role as the effective mass of scalar field. In asymptotically flat
spacetime, tachyonic instability arises if $\mu_{eff}^{2}<0$. For
asymptotically AdS spacetime, if $\mu_{eff}^{2}<\mu_{BF}^{2}$ where
$\mu_{BF}^{2}=-\frac{9}{4L^{3}}$ is the Breitenlohner-Freedman bound [103],
the tachyonic instatilbity occurs. We therefore choose the nonminimally
coupling such that $\frac{d^{2}\mathcal{G}(0)}{d\phi^{2}}>0$. Throughout this
work, we particularly consider the coupling in the following form

$\displaystyle\mathcal{G}$ $\displaystyle=e^{\alpha\phi^{2}},$ (32)

where $\alpha$ is a positive constant. Thus the effective mass squared is
$\mu_{\text{eff}}^{2}=-\frac{\alpha Q^{2}}{r^{4}}$. This form of coupling
function also satisfies the requirements discussed in the Sec. II.1. For other
forms of the coupling functions, for instance, hyperbolic, power law, and
fraction are investigated thoroughly in [41].

We expand the small perturbation $\delta\varphi$ as

$\displaystyle\delta\varphi$
$\displaystyle=U(r)e^{i\left(k_{1}x+k_{2}y\right)},$ (33)

where $k_{1},k_{2}$ can be considered as wave vector in planar direction.
Therefore Eq. (30) can be expressed in the form

$\displaystyle\left(r^{2}NU^{\prime}\right)^{\prime}-\left[\frac{\alpha
Q^{2}}{r^{2}}-L^{2}\vec{k}^{2}\right]U=0,$ (34)

where $\vec{k}^{2}\equiv k_{1}^{2}+k_{2}^{2}$. The above equation is solved by
assuming that the radial function $U$ is regular at the event horizon and
smoothly vanishing at spatial infinity. For a given, $Q=0.4,L=4,\vec{k}=0$, we
solve this equation for each $\alpha$. As a results, we obtain a bifurcation
line which marks the location where scalarized solutions bifurcate from planar
AdS charged black hole. See Fig.2 below for an example plot. We remark that in
Einstein-scalar-Gauss-Bonnet-AdS theory, a scalar perturbation on scalar-free
spherical BH is comparatively easier leading to tachyonic instability than
those with planar horizon [104, *Kiorpelidi:2022kuo].

##  III Scalarized planar AdS charged solutions

To obtain scalarized solution, we solve (10)–(13) with boundary conditions as
discussed in the previous section. We apply numerical shooting method where
$\phi_{0}$ and $\delta_{0}$ are chosen such that the boundary conditions at
spatial infinity are satisfied. For a given value of the event horizon
$r_{+}$, black hole charge $Q$, AdS radius $L$ and coupling constant $\alpha$,
we numerically integrate (10)–(13) from $(r_{+}+\epsilon)$ to some certain
distance ($r_{\infty}$) where $\epsilon$ is set to $10^{-9}$. We search for
$\phi_{0}$ and $\delta_{0}$ that make $\\{m,\delta,V,\phi\\}$ behave
asymptotically as (22)–(25). We obtain $M$ and $\Phi$ by identifying
$M=m(r_{\infty}),\Phi=V(r_{\infty})$. We find that the numerical value of
$\Phi$ agrees very well with (28).

To illustrate its behaviors, the plots of scalarized planar AdS charged black
holes as the functions of the radius $r$ are displayed with setting
$Q=0.4,~{}L=4$. In Fig 1, we display example plots of scalarized planar AdS
charged black holes. In the left panel, the free parameters are found to be
$\phi_{0}=0.622,~{}\delta_{0}=0.318$. We find that the black hole mass and
electrostatic potential are $0.092$ and $-0.314$ respectively. It is obvious
that the scalar field profile develops in the exterior region of the black
hole horizon and decreases rapidly as $r$ is larger . The scalar field
profiles can be characterized by number of node $n$ in the exterior region of
black hole. These are shown in the middle panel. For the rest of this work, we
shall particularly focus on the $n=0$ mode. Lastly, the right panel indicates
that there is no essential singularly anywhere outside the black hole’s
horizon since the Kretschmann scalar $R_{abcd}R^{abcd}$ is always finite for
$r>r_{+}$.

Figure 1:  Left: Example plots of the field functions for $\alpha$=9.3 and
$M$=0.092, $q$=4.35. Middle: The profiles of scalar field for $n$=0, $n$=1,
$n$=2 modes with $\alpha$=10 and $q$=3.60, 3.62 and 3.91 respectively. Right:
Plot the Kretschmann scalar as a function of r for scalarized black hole
geometry with $\alpha$=18.7 and $M$=0.152, $q$=2.63.

Solution space of scalarized solution is illustrated in Fig 2. The solution
space is displayed in $\alpha-q$ plane where $q\equiv\frac{Q}{M}$ is defined
as the reduced charge. In this plot, we only express the region where
tachyonic instability occurs i.e., $\mu_{eff}^{2}<\mu_{BF}^{2}$. Here we only
evaluate $\mu_{eff}^{2}$ at event horizon since it takes the smallest possible
value. The domain of existence of scalarized solution is bounded by the
bifurcation line (red dashed line) and critical line (solid blue line). The
bifurcation line indicates the location in phase space where scalarized
solutions bifurcates from bald black holes. The maximum value of $q$ for each
$\alpha$ form the critical line where scalarized black hole ceases to exist.
We find no solution above the critical line. Remark that, the scalarized black
holes and planar AdS black holes coexist in the region bounded by the
bifurcation line and the extremal line. On the extremal line the black holes
have zero Hawking temperature. In this plot, the extremal line is located at
$q=3.604$ or $M=0.11098$. This also implies that above the extremal line the
mass of scalarized solutions are lower than those of planar AdS charged black
holes. From this plot the area of existence increases with coupling constant
$\alpha$. We remark that the domain of existence plot of planar black hole is
qualitatively similar to those of spherically symmetric solution [49]. The
difference is that in spherical symmetric setup the extremal line locates at
$q=1$.

Figure 2: Domain of existence of scalarized planar AdS charged black hole. The
red dashed line is the bifurcation line and the dashed black line is the black
hole extremal line. The shaded area above the bifurcation line is the domain
of existence of scalarized solutions. The upper limit of this domain is
critical line (solid blue line). The colored area below the bifurcation line
is the area where planar AdS charged black hole exists. All the shaded area is
displayed only when $\mu_{eff}^{2}<\mu_{BF}^{2}$.  Figure 3:  The reduced
horizon areas $a_{H}$ against the reduced charge $q$ for scalarized planar AdS
BHs in the EMS gravity is plotted with $\alpha=5,10,15$ and $20$ corresponding
to dashed pink, dash-dotted black, dashed green and solid red, respectively,
compared with the case of the bald AdS charged BH (solid blue). Vertical
(solid blue) line represents the extremal point in this case.

In thermodynamics, it is known that the most preferred state is the state with
the largest entropy. Thus, if two states have different entropy, the state
with higher entropy is preferred over the other. According to Bekenstein [51],
entropy of BHs is proportional to its surface area at horizon radius, i.e.,
$S_{\text{BH}}\sim A_{H}$, where $S_{\text{BH}}$ is the Bekenstein-Hawking
entropy and the event horizon area of planar BH from the metric tensor is
given by $A_{H}=\mathcal{V}_{2}r^{2}_{+}/L^{2}$. Here $\mathcal{V}_{2}$
denotes the spatial extension in the $xy$ plane. In this case, we introduce
the reduced horizon area $a_{H}$, which measures the event horizon area with
respect to the area at $r=2M$, i.e., $a_{H}=r^{2}_{+}/4M^{2}$. Thus we display
the reduced horizon area as a function of $q$ in Fig 3. At small $q$, only
planar AdS charged black holes exists. Beyond the extremal point $q=3.604$
(marked by the vertical line), the planar AdS charged black holes do not
exist. When $q$ reaches certain number, scalarized planar black hole
bifurcates from the bald one. The bifurcation points for
$\alpha=5,~{}10,~{}15,\text{ and }20$ are at $q=3.438,~{}3.065,~{}2.735,\text{
and }2.466$, respectively. Remarkably, we observe that there exists a region
where planar AdS black holes co-exist with the scalarized solutions. It is
clear that in the co-exist region scalarized solutions are entropically
favored over planar AdS charged black holes since their reduced horizon areas
are relatively larger than those of planar AdS charged black holes. This
agrees with spherical symmetric situation where scalarized black holes are
found to be globally stable in microcanonical ensemble [49].

###  III.1 Stability of scalarized solutions

In this subsection, we explore linear stability of scalarized black holes. We
consider time dependent linear perturbations in EMS theory. Therefore, we
expand the field functions $\\{N,\delta,V,\phi\\}$

$\displaystyle N(t,r)$
$\displaystyle=N_{0}(r)+\epsilon_{1}N_{1}(r)e^{-i\bar{\omega}t},$ (35)
$\displaystyle\delta(t,r)$
$\displaystyle=\delta_{0}(r)+\epsilon_{1}\delta_{1}(r)e^{-i\bar{\omega}t},$
(36) $\displaystyle V(t,r)$
$\displaystyle=V_{0}(r)+\epsilon_{1}V_{1}(r)e^{-i\bar{\omega}t},$ (37)
$\displaystyle\varphi(t,r)$
$\displaystyle=\phi_{0}(r)+\epsilon_{1}\phi_{1}(r)e^{-i\bar{\omega}t},$ (38)

with frequency $\bar{\omega}$ and $\epsilon_{1}$ is order parameter. Here
$\delta_{0}$ and $\phi_{0}$ are not free parameters but rather represent
metric function and scalar function at equilibrium. Thus at first order (in
$\epsilon_{1}$), (2)–(3) imply the following,

$\displaystyle N_{1}$ $\displaystyle=-2rN_{0}\phi_{0}^{\prime}\phi_{1},$ (39)
$\displaystyle\delta_{1}^{\prime}$
$\displaystyle=-2r\phi_{0}^{\prime}\phi_{1}^{\prime},$ (40) $\displaystyle
V_{1}^{\prime}$
$\displaystyle=\frac{Qe^{-\delta_{0}}}{r^{2}\mathcal{G}(\phi_{0})^{2}}\left(\mathcal{G}(\phi_{0})\delta_{1}+\frac{d\mathcal{G}(\phi_{0})}{d\phi_{0}}\phi_{1}\right).$
(41)

We redefine $\phi_{1}\equiv\psi/r$ and define tortoise coordinate
$dr_{\ast}=e^{\delta_{0}}N_{0}^{-1}dr$. Therefore, the perturbed Klein-Gordon
equation can be expressed in the Schrödinger-like form

$\displaystyle-\frac{d^{2}\psi}{dr_{\ast}^{2}}+\left(V_{\bar{\omega}}-\bar{\omega}^{2}\right)\psi$
$\displaystyle=0,$ (42)

where

$\displaystyle V_{\bar{\omega}}$
$\displaystyle=N_{0}e^{-2\delta_{0}}\left[\frac{3-6r^{2}\phi_{0}^{\prime
2}}{L^{2}}-\frac{Q^{2}\left(1-2r^{2}\phi_{0}^{\prime
2}\right)}{r^{4}\mathcal{G}}-\frac{N_{0}}{r^{2}}-\frac{Q^{2}}{2r^{4}\mathcal{G}^{2}}\left(4r\phi_{0}^{\prime}\frac{d\mathcal{G}}{d\phi_{0}}+\frac{d^{2}\mathcal{G}}{d\phi_{0}^{2}}\right)\right].$
(43)

Clearly, the perturbation potential $V_{\bar{\omega}}$ vanishes at event
horizon and positively diverge as we approach spatial infinity. This is
illustrated in Fig 4 where the potential is plotted for various value of
$\alpha$. Equation (42) will have no bound state if $V_{\bar{\omega}}$ is
positive everywhere. The existence of bound state leads to unstable mode with
$\bar{\omega}^{2}<0$. From Fig 4, there exists a region where
$V_{\bar{\omega}}<0$ at small $\alpha$. However, as we increase $\alpha$, the
potential exhibits no negative region anywhere outside the event horizon.
Nevertheless, the existence of negative potential region does not sufficiently
leads to development of instability [41, 106]. Stability of scalarized
solution can be analyzed using the S-deformation technique [107,
*Kimura:2018whv].

Figure 4: The plot of perturbation potential against radial coordinate $r$
with $Q=0.4$ and $L=4$ for various value of $\alpha$. The event horizon
locates at $r_{+}=0.862$.

Figure 5:  Imaginary part of lowest-lying quasinormal frequency
($\bar{\omega}_{I}$) plots against charge $q$ (Left) and coupling constant
$\alpha$ (Right). The colored solid lines represent $\bar{\omega}_{I}$ of
scalarized black holes where dashed lines are planar AdS charged black holes.
The red vertical line in the left figure is charge at extremal limit
$q=3.604$.

Furthermore, stability of scalarized solution can be determined by calculating
quasinormal modes (QNMs) [109, 110]. We investigate whether the solutions are
stable against the linear scalar perturbation (42). For asymptotically AdS
spacetime, QNMs are defined to be the modes that satisfy purely ingoing wave
at the horizon and vanishing at infinity i.e.,

$\displaystyle\psi(r\to r_{h})\sim
e^{-i\bar{\omega}r_{\ast}},~{}~{}~{}~{}~{}\psi(r\to\infty)\sim 0.$ (44)

The frequency $\bar{\omega}$ corresponding to these modes are quasinormal
frequencies. As a consequence of the boundary conditions, the frequencies are
complex and discrete. Imaginary part of $\bar{\omega}$ will reveal whether the
perturbations are stable (exponentially decay) or unstable (exponentially
growth). Thus we numerically solve scalar perturbation (42) equation with
background numerical nodeless solutions (as mentioned earlier in this section)
and imposing boundary conditions (44). The aim is to obtain the quasinormal
frequencies as a function of charge $q$ and nonminimally coupling constant
$\alpha$. This is done by using built-in function $NDSolve$ in Wolfram’s
Mathematica. In addition, we implement psuedospectral method in order to
obtain the quasinormal frequencies. We refer the readers to [111] for a nice
introduction and Mathematica’s package of this method.

In Fig. 5, we illustrate an imaginary part of quasinormal frequencies
$\bar{\omega}_{I}$ as a function of $q$ and $\alpha$. In these plots,
$\bar{\omega}_{I}$ of scalarized black holes are displayed by the solid curves
where those of RN-AdS are shown by the dashed curves. For each fixed value of
$\alpha$, we plot scalarized solutions from the smallest possible $q$ i.e.,
$q$ at the bifurcation line to the largest possible value ($q$ at the critical
line). We observe that the quasinormal frequencies are always negative.
Negativity of $\bar{\omega}_{I}$ implies that these scalarized solutions are
linearly stable. The larger the $\alpha$, the more stable these solutions are.
More interestingly, $\bar{\omega}_{I}$ develops nontrivial behavior as a
function of $q$. At small $q$, it appears that the solutions become more
stable as $q$ is increased. However, beyond a certain value of $q$, the
solutions are less stable as $q$ is increased. We note that similar trends are
also observed in [49, 112] for spherically symmetric solutions. Despite this,
we find no evidence of unstable modes of scalarized black holes. Moreover,
QNMs of RN-AdS are also computed. We find that as $q$ increases, the RN-AdS
becomes more unstable as expected. The RN-AdS becomes most unstable when we
reach extremal limit denoted by red vertical line. In addition, the right
figure of Fig. 5, we plot $\bar{\omega}_{I}$ as a function of $\alpha$ for
various values of $q$. We notice that when $q$ is increases the
$\bar{\omega}_{I}$ becomes more positive. While increasing $\alpha$, the
scalarized solution become more stable. Similar to the previous case, we find
that the RN-AdS solutions are unstable against scalar perturbation.

##  IV Thermodynamics

In this section, we investigate the thermodynamics and phase structure of
scalarized black holes with the planar horizon in the grand canonical ensemble
and canonical ensemble [49, 63].

###  IV.1 Action calculation

To analyze the thermodynamics behaviors of black holes in the grand canonical
ensemble with fixed electric potential $\Phi$ or canonical ensemble with fixed
electric charge $\tilde{Q}=\frac{\mathcal{V}_{2}Q}{4\pi L^{2}}$ (see Appendix
A). We study the Euclidean action of the solution in an imaginary time
$t\rightarrow i\tau$ by identifying the period
$\beta_{\text{H}}=1/T_{\text{H}}$ with the Hawking temperature $T_{\text{H}}$
[113]. In the semiclassical approximation, the thermal partition function is
given by

$\displaystyle\mathcal{Z}\sim\text{e}^{-S^{E}_{\text{on-shell}}},$ (45)

where $S^{E}_{\text{on-shell}}$ defined as the on-shell Euclidean action.
Nevertheless, the action (1) diverges as spacetime volume increases. Thus we
must renormalize the action to eliminate the divergences from the asymptotic
AdS spacetime.

We introduce the Euclidean regularized action $S^{E}_{R}$ including the
Euclidean bulk action $S^{E}_{\text{bulk}}$ from (1), Gibbons-Hawking boundary
term $S^{E}_{\text{GH}}$ for eliminating the divergence at the boundary, the
counterterm $S^{E}_{\text{ct}}$ for eliminating the divergence from the AdS
asymptotic at the boundary, and $S^{E}_{\text{surf}}$ for eliminating the
divergence of the EM field,

$\displaystyle
S^{E}_{R}=S^{E}_{\text{bulk}}+S^{E}_{\text{GH}}+S^{E}_{\text{ct}}+S^{E}_{\text{surf}}.$
(46)

Note that in the above equation, we have changed the time coordinate $t$ into
the Euclidean time $\tau=it$. For convenience, in the Euclidean spacetime, we
will denote $d^{4}x=d\tau drdxdy$. The Euclidean bulk and the boundary terms
are given by

$\displaystyle S^{E}_{\text{bulk}}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int
d^{4}x\sqrt{g}\bigg{[}R+\frac{6}{L^{2}}-2\nabla_{\mu}\varphi\nabla^{\mu}\varphi-\mathcal{G}(\varphi)F^{2}\bigg{]},$
(47) $\displaystyle S^{E}_{\text{GH}}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\int d^{3}x\sqrt{\gamma^{(3)}}\Theta,$ (48)
$\displaystyle S^{E}_{\text{ct}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int
d^{3}x\sqrt{\gamma^{(3)}}\left(\frac{2}{L}+\frac{L}{2}\mathcal{R}\right),$
(49) $\displaystyle S^{E}_{\text{surf}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int
d^{3}x\sqrt{\gamma^{(3)}}\mathcal{G}(\varphi)F^{\mu\nu}n_{\mu}A_{\nu},$ (50)

where $\gamma^{(3)}$ is the determinant of the induced metric on the
hypersurface at $r\rightarrow\infty$, $\Theta$ is the trace of the extrinsic
curvature ${\Theta^{\mu}}_{\mu}$, $\mathcal{R}$ is the scalar curvature of the
induced metric $\gamma_{ij}^{(3)}$, and $n_{\mu}$ is the unit normal vector on
the hypersurface.

Now we consider the Euclidean bulk term $S^{E}_{\text{bulk}}$ along with the
Einstein field equation in (2) and its trace. The Euclidean bulk term can be
written as

$\displaystyle S^{E}_{\text{bulk}}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int
d^{4}x\sqrt{g}\left[\frac{6}{L^{2}}+\mathcal{G}(\varphi)F^{2}\right].$ (51)

With the $\tau\tau$ component of the field equation in Eq. (2), we obtain the
identity as follows,

$\displaystyle R_{\tau\tau}+\frac{3}{L^{2}}Ne^{-2\delta}$ $\displaystyle=$
$\displaystyle-\mathcal{G}(\varphi)NV^{\prime 2}.$ (52)

The field strength tensor square is given by

$\displaystyle F^{2}=-2e^{2\delta}V^{\prime 2}.$ (53)

Plugging Eqs. (52) and (53) into (51), the Euclidean bulk term becomes

$\displaystyle S^{E}_{\text{bulk}}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int
d^{4}x\sqrt{g}\left[4e^{2\delta}\mathcal{G}(\varphi)V^{\prime
2}+\frac{2e^{2\delta}}{N}R_{\tau\tau}\right],$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int^{\beta_{H}}_{0}d\tau\int
dxdy\int^{\infty}_{r_{+}}dr\sqrt{e^{-2\delta}\frac{r^{4}}{L^{4}}}\left[4e^{2\delta}\mathcal{G}(\varphi)V^{\prime
2}+\frac{2e^{2\delta}}{N}R_{\tau\tau}\right],$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{V}_{2}}{4\pi
T_{H}}\int^{\infty}_{r_{+}}dr\frac{r^{2}}{L^{2}}\left[e^{\delta}\mathcal{G}(\varphi)V^{\prime
2}+\frac{1}{2N}e^{\delta}R_{\tau\tau}\right].$

We substitute the $\tau\tau$ component of the Ricci tensor, which directly
calculated from the metric tensor in Eq. (6), into the above equation. One can
integrate out the coordinate $r$ to obtain

$\displaystyle S^{E}_{\text{bulk}}$ $\displaystyle=$
$\displaystyle\frac{1}{T_{\text{H}}}\left[\frac{\mathcal{V}_{2}}{16\pi
L^{2}}r^{2}e^{-\delta}\left(N^{\prime}-2N\delta^{\prime}\right)\big{|}_{r\rightarrow\infty}-T_{\text{H}}S_{\text{BH}}-\tilde{Q}\Phi\right].$
(54)

Here, the Hawking temperature of the scalarized black hole is

$\displaystyle
T_{\text{H}}=\frac{1}{4\pi}N^{\prime}(r_{+})e^{-\delta(r_{+})}.$ (55)

The Bekenstein-Hawking entropy is given in the form of the area law as

$\displaystyle
S_{\text{BH}}=\frac{A_{\mathcal{H}}}{4}=\frac{\mathcal{V}_{2}r_{+}^{2}}{4L^{2}}.$
(56)

Note that the horizon area can be calculated from

$\displaystyle A_{\mathcal{H}}=\int
d^{2}x\sqrt{\gamma^{(2)}}=\frac{\mathcal{V}_{2}r^{2}_{+}}{L^{2}},$ (57)

where $\gamma^{(2)}$ is a determinant of the induced metric associated to the
horizon surface and $\mathcal{V}_{2}=\int dxdy$ is the spatial extension of
black hole in $\mathbb{R}^{2}$.

The first term in (54) diverges at $r\rightarrow\infty$. To remove this
divergence, we make use of the Gibbons-Hawking term to cancel them out. The
extrinsic curvature is defined by

$\displaystyle\Theta_{\mu\nu}\equiv\frac{1}{2}{h^{\alpha}}_{\mu}{h^{\beta}}_{\nu}\left(\nabla_{\alpha}\sigma_{\beta}+\nabla_{\beta}\sigma_{\alpha}\
\right).$ (58)

The metric on the hypersurface $h_{\mu\nu}$ is defined as follows,

$\displaystyle h_{\mu\nu}\equiv g_{\mu\nu}-\sigma_{\mu}\sigma_{\nu},$ (59)

where $\sigma_{\mu}=(0,1/N^{1/2},0,0)$ is the unit normal vector pointing
outward from the hypersurface. The trace of the extrinsic curvature can be
expressed explicitly

$\displaystyle\Theta=\frac{N^{\prime}}{2\sqrt{N}}-\delta^{\prime}\sqrt{N}+\frac{2\sqrt{N}}{r}.$
(60)

Now the Gibbons-Hawking boundary term is,

$\displaystyle S^{E}_{\text{GH}}$ $\displaystyle=$
$\displaystyle-\frac{1}{8\pi}\int d^{3}x\sqrt{\gamma^{(3)}}\Theta,$ (61)
$\displaystyle=$
$\displaystyle-\frac{1}{T_{H}}\left[\frac{\mathcal{V}_{2}}{16\pi
L^{2}}r^{2}e^{-\delta}\left(N^{\prime}-2N\delta^{\prime}\right)+\frac{\mathcal{V}_{2}}{4\pi
L^{2}}e^{-\delta}\left(-2M+\frac{r^{3}}{L^{2}}\right)\right]_{r\rightarrow\infty}.$

The first term in the above equation cancels out the divergence in the bulk
action (54). The second term in the above equation leads to another
divergence. However, one finds that the counterterm $S^{E}_{\text{ct}}$ yields

$\displaystyle S^{E}_{\text{ct}}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{V}_{2}}{4\pi
L^{2}T_{\text{H}}}e^{-\delta}\left(\frac{r^{3}}{L^{2}}-M\right)\Bigg{|}_{r\rightarrow\infty},$
(62)

where $\mathcal{R}$ vanishes with planar symmetry. The surface boundary action
$S_{\text{surf}}^{E}$ vanishes when the potential $\Phi$ is fixed at the
boundary since the field strength of the gauge field is zero on this surface.
Therefore the grand potential is given by

$\displaystyle\Omega=T_{H}S^{E}_{R}=E-T_{H}S_{\text{BH}}-\tilde{Q}\Phi.$ (63)

On the other hand, if we fix the electric charge $Q$ of the black hole
instead, the surface term now becomes nonvanishing, i.e.,

$\displaystyle S^{E}_{\text{surf}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\int
d^{3}x\sqrt{\gamma^{(3)}}\mathcal{G}(\varphi)F^{\mu\nu}n_{\mu}A_{\nu},$ (64)
$\displaystyle=$ $\displaystyle\frac{\mathcal{V}_{2}}{4\pi
T_{\text{H}}L^{2}}Q\Phi,$ $\displaystyle=$
$\displaystyle\frac{1}{T_{\text{H}}}\tilde{Q}\Phi.$

Therefore, it contributes to the on-shell action in Eq.(46). In this case, the
free energy is the Helmholtz free energy which can be written in the following
form

$\displaystyle F=T_{\text{H}}S_{\text{R}}^{E}=E-T_{\text{H}}S_{\text{BH}}.$
(65)

###  IV.2 Fixed potential

The behaviors of the reduced temperature $TL$ with reduced horizon radius
$r_{+}/L$ are illustrated in Fig. 6. In these plots, we fix the electric
potential $|\Phi|=0.5$ and $1.2$ in the left and right panels, respectively.
At low temperature, there are two branches of black holes with planar horizon:
RN-AdS BH (solid blue curve) and SC BH (solid red, dashed green and dot-dashed
black curves which correspond to coupling constant $\alpha=5,10$ and $15$
respectively). In both BH’s configurations, the Hawking temperature is
proportional to BH’s horizon radius, however, the SC BH has higher temperature
than the RN-AdS BH until its temperature increases to $T_{B}$, where $T_{B}$
represents the temperature at the bifurcation point. For $T>T_{B}$, SC BH
disappears and there is only one BH branch which is RN-AdS.

Figure 6: Reduced temperature $TL$ plots against reduced horizon radius
$r_{+}/L$ with $L=1$. Left: for fixed $|\Phi|=0.5$ Right: for fixed
$|\Phi|=1.2$. The solid blue and red lines are RN-AdS and scalarized solution
with $\alpha=5$. The dashed green and dot-dashed black lines are scalarized
solution with $\alpha=10$ and $15$ respectively.

In Fig. 7, we plot the reduced heat capacity $C/L^{2}$ as a function of the
reduced temperature $TL$ with corresponding to the temperature profiles in
Fig. 6 for the SC BH branch (red, dashed green and dot-dashed black curves)
compared with the RN-AdS BH (blue curve). The heat capacity of these two
branches are both positive and increased when the increasing temperature, so
these two branches are locally in thermal equilibrium against microscopic
fluctuations. The behavior of the heat capacity of these two branches are
significantly different, namely, the heat capacity of the SC BH is larger than
the RN-AdS BH and reaches its maximum value at $T_{B}$ where $T>T_{B}$ SC BHs
are no longer exist. On the other hand, the heat capacity of the RN-AdS BH
increases monotonically with when $T>0$.

As shown in Fig. 6, when $r_{+}<r_{e}$, where $r_{e}$ denotes the horizon
radius of extremal RN-AdS BH, there exists only SC BH while the existence of
the RN-AdS BH in this region would violate the cosmic censorship conjecture
[114]. Note that the SC BH approaches the extremal limit with zero temperature
comparatively slower than the RN-AdS BH since its heat capacity is greater
than RN-AdS BH. In other words, the SC BH radiates more energy to reduce one
unit of the temperature. Remarkably, in the planar horizon case, the SC BH
exists at near zero temperature up to $T_{B}$. In contrast to spherical
horizon case, the SC BH exists at the lowest (nonzero) temperature
$T_{\text{min}}$ up to arbitrarily high temperature [63].

In order to investigate the global stability of BH in EMS system with fixed
$|\Phi|$ ensemble, one needs to consider the grand potential. For planar black
hole, the extensive quantities such as mass, entropy, charge and free energy
are diverge since $\mathcal{V}_{2}$ has an infinite extension. Therefore it is
more appropriate to use the density of these quantities as $x_{i}=4\pi
L^{2}X_{i}/\mathcal{V}_{2}$ where $X_{i}$ represents the extensive quantities.
In this way, the grand potential in Eq. (63) can be written in the form

$\displaystyle\Omega=\frac{\mathcal{V}_{2}}{4\pi L^{2}}\omega,$ (66)

where

$\displaystyle\omega=M-T_{\text{H}}s_{\text{BH}}-Q\Phi,$ (67)

defined as the grand potential density. Here $M$, $Q$ and $s_{\text{BH}}=\pi
r^{2}_{+}$ are the mass, charge and entropy density, respectively. In Fig. 8,
we present the reduced grand potential density $\omega/L$ as the function of
reduced temperature $TL$. These graphs show that there are two branches of
black hole emerged at zero temperature with different values of the grand
potential: RN-AdS BH and SC BH. For the solid blue, solid red, dashed green
and dot-dashed black curves correspond to RN-AdS BH and SC BH with different
coupling constant $\alpha=5,10$ and $15$, respectively. At low temperature
$T<T_{B}$, the thermodynamically preferred state is the SC BH since it has
lower grand potential than the RN-AdS BH.

At $T=T_{B}$, the SC BH and RN-AdS BH coalesce, while RN-AdS BH persists for
all high temperature. Since the heat capacity, which associated to the second
derivative of the free energy, i.e., $\displaystyle
C=-T\left(\frac{\partial^{2}\Omega}{\partial T^{2}}\right)_{\Phi}$, is
discontinuous. Thus the SC - RN-AdS BH phase transition is the second order
type of phase transition. In the left panel in Fig. 8, we explore the
influence of $\alpha$ on the phase structures of BH in EMS system by varying
$\alpha=5,10,15$ for fixed $|\Phi|=0.85$. We find that the grand potential of
SC BH are more negative value when the $\alpha$ increases, so the SC BH with
larger $\alpha$ are more thermodynamically favored than the BH with smaller
$\alpha$. In the right panel in Fig. 8, we fixed $\alpha=15$ and vary
$|\Phi|=0.5,0.85$ and $1.2$ respectively. The grand potential curves of SC BH
and RN-AdS BH are both decreased from top to bottom when $|\Phi|$ is
increasing.

Figure 7: Show the heat capacity $C/L^{2}$ plot against the reduced
temperature $TL$ with $L=1$. Left: for fixed $|\Phi|=0.5$ Right: for fixed
$|\Phi|=1.2$. The solid blue and red lines are RN-AdS and scalarized solution
with $\alpha=5$. The dashed green and dot-dashed black lines are scalarized
solution with $\alpha=10$ and $15$ respectively.

Figure 8: Grand potential density $\omega$ plots against the reduced
temperature $TL$ with $L=1$. Left: for fixed $|\Phi|=0.85$ and varying
$\alpha=5,10,15$ Right: for fixed $\alpha=15$ and varying
$|\Phi|=0.5,0.85,1.2$. Figure 9: Temperature of black hole solutions plot
against electrostatic potential for $\alpha=10$. The dashed black line
indicates the temperature at the bifurcation points. The lower region is
dominated by scalarized solution while the upper region belongs to RN-AdS.

Figure 9 displays the phase diagram of the grand canonical ensemble of BH in
EMS system with the lowest grand potential at fixed $\alpha=10$ in the
$TL-|\Phi|$ plane. The dashed black curve represents the line of coexistence
of the SC BH and the RN-AdS BH, where the transition across this line is of
the second order type. This occurs when the grand potential of these
configurations are degenerated at $T=T_{B}$. In the upper region, there is
only RN-AdS BH; however, when the temperature decreases to $T_{B}$, the RN-AdS
BH develops a scalar hair near the event horizon to form the SC BH, which
represents in the lower region of the coexistence line. In this region, the SC
BH can be the globally preferred thermal state.

###  IV.3 Fixed charge

To study the BH thermodynamics of the EMS gravity in the canonical ensemble,
we fix the bulk electric charge $Q$ of the BH solutions instead of fixing the
electric potential $\Phi$. Fig 10 shows the behavior of $TL$ as a function of
$r_{+}/L$ with three different values of $Q/L$. For low temperature $T<T_{B}$
region, there are two branches of BH solutions of different horizon radii. We
find that these two BH solutions with small and large horizon radii correspond
to the SC BH and RN-AdS BH respectively.

Figure 10: Plots of the reduced Hawking temperature $TL$ against the reduced
horizon radius $r_{+}/L$ of two BH solutions in the EMS gravity with three
different values of the reduced charge $Q/L=0.1,0.5,\text{ and }1.5$ in the
left, middle and right panels, respectively. The RN-AdS BH is identified by
the solid blue curve, while the SC BHs with different $\alpha=5,10\
\text{and}\ 15$ are represent in solid red, dashed green and dot-dashed black
curves, respectively.

In Figs 10, 11, and 12, the RN-AdS BH is identified by the solid blue curve,
while the SC BHs with different $\alpha=5,10\ \text{and}\ 15$ are represented
in solid red, dashed green and dot-dashed black curves, respectively. When the
temperature increases to $T_{B}$, these two BHs are degenerate at the same
horizon radius and SC BH transits to RN-AdS BH. Consequently, the RN-AdS BH is
the only phase that exists at any high temperature regime. By varying
$Q/L=0.1,0.5$ and $1.5$ from the left to the right panels in Fig 10, we find
that $T_{B}$ and its corresponding horizon radius increase when $Q/L$
increases. At each value of $Q/L$, we also compare the effect of the strength
of the coupling function between the massless scalar field and the Maxwell
gauge field on the thermal behavior of SC BHs by varying $\alpha$. These
graphs indicate that $T_{B}$ and its corresponding horizon radius increase
with $\alpha$.

Figure 11: Plot of the reduced heat capacity $C/L^{2}$ as the function of the
reduced temperature $TL$ for planar SC and RN-AdS BHs with the variation of
$\alpha=5,10,\text{ and }15$ and $Q/L=0.1,0.5,\text{ and }1.5$.

To investigate the local stability of two BH solutions of the EMS gravity in
the canonical ensemble, we plot the reduced heat capacity $C/L^{2}$ against
the reduced temperature $TL$ in Fig 11 for different reduced charge $Q/L$.
These graphs illustrate that the heat capacity of these two BH backgrounds are
both positive, so they can be in thermal equilibrium with heat reservoir at
fixed temperature. However, SC BH only exists in the range $0\leq T\leq T_{B}$
with higher heat capacity than the RN-AdS BH and consequently disappear for
$T>T_{B}$ region. On the other hand, RN-AdS BH always exists at any
temperature. For fix $\alpha$ and vary BH’s charge, $T_{B}$ is shifted to the
right when the bulk charge of the black hole increases. Moreover, when we fix
$Q$ of the black hole and vary $\alpha$, the results indicate that $T_{B}$
move to the right when $\alpha$ is larger.

Figure 12: The Helmholtz free energy against the temperature of the phase
transition between the scalarized planar black hole and the RN-AdS planar
black hole with the variation of $\alpha=5,10,\text{ and }15$ and
$Q/L=0.1,0.5,\text{ and }1.5$.

A free energy corresponding to the canonical ensemble is the Helmholtz free
energy, which is denoted by $F$. For planar BH, it diverges in the same way as
the grand potential of the fix $\Phi$ ensemble. Thus, we use the Helmholtz
free energy density $\mathcal{F}$ instead of $F$ to investigate the global
stability of BHs in the EMS gravity. The Helmholtz free energy density is
defined as

$\displaystyle F=\frac{\mathcal{V}_{2}}{4\pi L^{2}}\mathcal{F},$ (68)

where

$\displaystyle\mathcal{F}=M-T_{\text{H}}s_{\text{BH}}.$ (69)

Figure 12 displays the Helmholtz free energy as a function of temperature. In
high temperature regime $T>T_{B}$, we find the planar RN-Ads BH is the most
globally preferred phase in the AdS space. As one lower temperature to $T_{B}$
and keep $Q/L$ to be constant, the RN-AdS BH becomes unstable and develops an
atmosphere in the form of massless scalar with finite peak near the event
horizon to form the SC BH. The transition between these two BH configurations
can be classified to the second order phase transition since $C/L^{2}$ shows a
discontinuity and $\mathcal{F}/L$ is continuous and smooth function at
$T=T_{B}$ as shown in Fig 11 and Fig 12, respectively. For low temperature
$T<T_{B}$, the SC BH can be the globally preferred state. As we increase
$Q/L$, both BH configurations move to lower $\mathcal{F}/L$, while $T_{B}$ of
the system increases. By varying $\alpha=5,10$ and $15$ and fixed
$Q/L=0.1,0.5$ and $1.5$, respectively. we find $\mathcal{F}/L$ of SC BHs are
lower from the top to bottom.

For the canonical ensemble, Fig 13 demonstrates the phase diagram of RN-AdS
and SC BHs with the lowest $\mathcal{F}/L$ in the $TL-Q/L$ plane when
$\alpha=10$ . The dashed black line represents the second order phase
transition dividing the RN-AdS BH (upper region) from the SC BH (lower
region).

Figure 13: The plot of the temperature against the charge where the phase
transition of the scalarized planar black hole emerging from the RN planar
black hole with the parameter set as follows: $\alpha=10$, $Q/L=0.1$, and
$Q\in[0.0,1.0]$.

##  V Conclusions

In this work, the spontaneous scalarization of asymptotically AdS charged
black holes of planar event horizon in the Einstein-Maxwell-calar theory with
nonminimally coupling between scalar field and Maxwell field is investigated.
It is tachyonic instability that drives scalar-free charged black holes away
from their stability. When the condition $\mu^{2}_{eff}<\mu^{2}_{BF}$ holds,
scalarized planar charged black holes emerge when charge $q$ and the coupling
constant $\alpha$ are sufficiently large as illustrated in Fig. 2. Moreover,
in the region where RN-AdS and SC BHs co-exist, we find that SC BHs are
entropically preferred over RN-AdS. From our investigation, we observe that
the SC BHs with low $\alpha$ exhibit a possibility of instability while
perturbation equations of SC solutions with higher $\alpha$ do not develop any
bound states. By computing quasinormal frequencies, we find that all
scalarized black holes investigated in this work are linearly stable against
scalar perturbations. In fact, the scalarized solutions become more stable as
$\alpha$ is increased. In contrast, we show that the RN-AdS black holes do
suffer from tachyonic instability. Despite that, comparing to the spherically
symmetric case, topological scalarized solutions are more difficult to obtain.
However, we find that their domains of existence, the reduced event horizon
areas of black hole and stability behaviors share several similarities.

In terms of thermodynamic behaviors of planar BH solutions in the EMS gravity,
we find two stable branches of BHs: RN-AdS BH and SC BH in both grand
canonical and canonical ensembles. There is only planar RN-AdS BH branch at
high temperature regime $T>T_{B}$. By decreasing temperature, planar RN-AdS BH
becomes unstable below $T_{B}$ and develops nonzero finite value of real
scalar field near its event horizon, i.e., $\phi_{0}\neq 0$. The phase
transition between RN-AdS BH to SC BH is the second order phase transition
since the behavior of heat capacity shows a discontinuity at $T=T_{B}$.
Remarkably, SC BH are thermodynamically favored than RN-AdS BH in $T<T_{B}$
region because they have smaller free energy than RN-AdS BH. The phase diagram
in Fig 9 and Fig 13 display a region where RN-AdS BH and SC BH exist with the
lowest free energy in grand canonical and canonical ensemble, respectively.
Unlike the spherical SC BHs that have no extremal limit since they only exist
from nonzero temperature up to arbitrary temperature, the planar SC BHs emerge
from the bifurcation point at $T_{B}$ which is thermodynamically preferred
than RN-AdS BH. Below $T_{B}$, planar SC BH approaches toward the extremal
limit. The spherical BHs in the EMS model has richer thermodynamic phase
structure and phase transition than the planar horizon case. In the spherical
case, the BH configuration exhibits a phase transition in a similar way as the
liquid-gas one at small $Q$ regime, while the reentrant phase transition can
occurs in some range of parameters in the large $Q$ limit. Nevertheless, the
thermal second order phase transition between planar RN-AdS BH and SC BH in
the EMS model is reminiscent of normal conductor-superconductor phase
transition. This similarity may demonstrate as follows. In the gauge/gravity
duality, one can interpret the radial coordinate $r$ of AdS space as
geometrical view of the renormalisation group flow. In other words, running
the coordinate $r$ from the boundary to the interior of the bulk in AdS
corresponds to flowing down the energy scale from high energy (UV) to low
energy (IR) of the field theory. In the left panel of Fig. 1 shows a scalar
field profile in the bulk spacetime. The scalar field is concentrated in the
deep interior of spacetime and monotonically decreases toward zero amplitude
near the boundary. In thermodynamics perspective, SC BH with $\phi_{0}\neq 0$
will develop in the low temperature limit $T<T_{B}$, however, for $T>T_{B}$
the scalar field’s amplitude vanishes $\phi_{0}=0$, at this point the SC BH
transits to scalar-free BH. Such mechanism implies a spontaneous symmetry
breaking of $U(1)$ gauge symmetry in the IR limit and the symmetry is restored
at the UV physics. This suggests that SC BH and RN-AdS BH in the EMS model
might behave as a superconducting phase and normal conducting phase,
respectively, as in the Abelian-Higgs model of superconductivity [115].

As a general extension of this work, it is interesting to explore the
dependence of these scalarized black holes on various form of coupling
functions where the similar work is done with spherical horizon [41]. A
nonlinear dynamical evolution of scalarized spherical black hole is studied in
[116]. Similar should be done for the planar case where it should offer more
insight into perturbative stability of scalarized planar black holes. Given,
thermodyanmics behaviors and phase structure of our scalarized solutions, it
would be extremely interesting to explore these solutions in the context of
holographic superconductor. We leave these for future works.

###### Acknowledgements.

The authors would like to thank Carlos A. R. Herdeiro, Alexandre M. Pombo and
Napat Poovuttikul for very useful discussions. This work (Grant No. RGNS
64-217) was supported by Office of the Permanent Secretary, Ministry of Higher
Education, Science, Research and Innovation (OPS MHESI), Thailand Science
Research and Innovation (TSRI) and Silpakorn university. T. Tangphati was
supported by King Mongkut’s University of Technology Thonburi’s Post-doctoral
Fellowship. E. Hirunsirisawat acknowledges the financial support provided by
the Center of Excellence in Theoretical and Computational Science (TaCS-CoE),
KMUTT. Moreover, this research project is supported by Thailand Science
Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2023 under
project number FRB660073/0164.

##  Appendix A MASS AND CHARGE OF PLANAR EMS SOLUTION

In this appendix, we shall derive mass and electric charged of planar BH
solutions in the EMS gravity via the Komar integral, which we have used them
to derived some thermodynamic quantities of BHs. Let’s first consider, the
Maxwell’s equation with an external current $J^{\mu}$. Thus Eq. (3) becomes

$\displaystyle\nabla_{\nu}\left(\mathcal{G}F^{\mu\nu}\right)=J^{\mu},$ (70)

where $\mathcal{G}=e^{\alpha\phi^{2}}$. The electric charge $\tilde{Q}$
passing through a spacelike hypersurface $\Sigma$ can be defined by an
integral over spatial coordinates $x^{i}$ on $\Sigma$

$\displaystyle\tilde{Q}=-\frac{1}{4\pi}\int_{\Sigma}d^{3}x\sqrt{\gamma^{(3)}}n_{\mu}J^{\mu}=-\frac{1}{4\pi}\int_{\Sigma}d^{3}x\sqrt{\gamma^{(3)}}n_{\mu}\nabla_{\nu}\left(\mathcal{G}F^{\mu\nu}\right),$
(71)

where $\gamma^{(3)}_{ij}$ is the induced metric and $n_{\mu}$ is the unit
normal vector associated to $\Sigma$. By using the Stokes’s theorem, we can
rewrite the volume integral in the above relation as a surface integral

$\displaystyle\tilde{Q}=-\frac{1}{4\pi}\int_{\partial\Sigma}d^{2}x\sqrt{\gamma^{(2)}}n_{\mu}\sigma_{\nu}\mathcal{G}F^{\mu\nu},$
(72)

where $\gamma^{(2)}_{ij}$ is the induced metric and $\sigma_{\mu}$ is the unit
normal vector associated to $\partial\Sigma$. For planar EMS solutions in Eq.
(6), the surfaces $\Sigma$ and $\partial\Sigma$ are the constant-time
hypersurface and $\mathbb{R}^{2}$ surface at spatial infinity, respectively.
Therefore the unit normal vectors associated to $\Sigma$ and $\partial\Sigma$
are

$\displaystyle n_{\mu}=\left(-N^{1/2}(r)e^{-\delta(r)},0,0,0\right),\ \ \
\text{and}\ \ \ \sigma_{\mu}=\left(0,N^{-1/2}(r),0,0\right),$ (73)

respectively. The line element on surface $\partial\Sigma$ is

$\displaystyle\gamma^{(2)}_{ij}dx^{i}dx^{j}=\frac{r^{2}}{L^{2}}\left(dx^{2}+dy^{2}\right),$
(74)

and the volume element takes the form

$\displaystyle d^{2}x\sqrt{\gamma^{(2)}}=\frac{r^{2}}{L^{2}}dxdy.$ (75)

From definition of normal vectors in Eq. (73), one can compute

$\displaystyle
n_{\mu}\sigma_{\nu}\mathcal{G}F^{\mu\nu}=e^{-\delta(r)}\mathcal{G}V^{\prime}(r).$
(76)

Substituting Eq. (75) and Eq. (76) into Eq. (72) gives

$\displaystyle\tilde{Q}=-\frac{\mathcal{V}_{2}}{4\pi
L^{2}}\left(r^{2}e^{-\delta(r)}\mathcal{G}V^{\prime}(r)\right)_{r\rightarrow\infty},$
(77)

where $\mathcal{V}_{2}$ denotes spatial area of constant $r$ surface. Remark
that the electric charge $\tilde{Q}$ is evaluated at spatial infinity. By
using Eqs. (23)-(25), we obtain

$\displaystyle\tilde{Q}=\frac{\mathcal{V}_{2}Q}{4\pi L^{2}}.$ (78)

In an asymptotically AdS space, the physical mass can be measured with respect
to a reference AdS background. In this way, the Komar mass can be written in
the following form

$\displaystyle
E=\frac{1}{4\pi}\int_{\partial\Sigma}d^{2}x\sqrt{\gamma^{(2)}}n_{\mu}\sigma_{\nu}\nabla^{\mu}K^{\nu}-E_{\text{AdS}}.$
(79)

We define the time-translation Killing vector $K^{\nu}\equiv(1,0,0,0)$,
$n_{\mu}$ and $\sigma_{\mu}$ are defined in Eq. (73). The Komar integral of
vacuum AdS space $(E_{\text{AdS}})$ is given by

$\displaystyle E_{\text{AdS}}=\frac{\mathcal{V}_{2}r^{3}}{4\pi L^{4}}.$ (80)

Therefore, we have

$\displaystyle
n_{\mu}\sigma_{\nu}\nabla^{\mu}K^{\nu}=\frac{1}{2}e^{-\delta(r)}N^{\prime}(r)\left(1-2\delta^{\prime}(r)\right).$
(81)

Thus the first term of (79) becomes

$\displaystyle\frac{1}{4\pi}\int_{\partial\Sigma}d^{2}x\sqrt{\gamma^{(2)}}n_{\mu}\sigma_{\nu}\nabla^{\mu}K^{\nu}$
$\displaystyle=$
$\displaystyle\frac{\mathcal{V}_{2}}{4\pi}\left[\frac{r^{2}}{2L^{2}}e^{-\delta(r)}N^{\prime}(r)\left(1-2\delta^{\prime}(r)\right)\right],$
(82) $\displaystyle=$
$\displaystyle\frac{\mathcal{V}_{2}}{4\pi}\left[\frac{M}{L^{2}}-\frac{Q^{2}}{2L^{2}r}+\frac{r^{3}}{L^{4}}+\mathcal{O}(r^{-2})\right].$

Evaluating at the boundary ($r\rightarrow\infty$) with $E_{\text{AdS}}$, we
obtain the Komar mass of planar BH solutions as

$\displaystyle E=\frac{\mathcal{V}_{2}M}{4\pi L^{2}}.$ (83)

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