Scalarized Einstein-Maxwell-scalar Black Holes in Anti-de Sitter Spacetime
SScalarized Einstein-Maxwell-scalar Black Holes in Anti-de Sitter Spacetime
Guangzhou Guo, ∗ Peng Wang, † Houwen Wu, ‡ and Haitang Yang § Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610064, China
In this paper, we study spontaneous scalarization of asymptotically anti-de Sitter charged blackholes in the Einstein-Maxwell-scalar model with a non-minimal coupling between the scalar andMaxwell fields. In this model, Reissner-Nordstr¨om-AdS (RNAdS) black holes are scalar-free blackhole solutions, and may induce scalarized black holes due to the presence of a tachyonic instability ofthe scalar field near the event horizon. For RNAdS and scalarized black hole solutions, we investigatethe domain of existence, perturbative stability against spherical perturbations and phase structure.In a micro-canonical ensemble, scalarized solutions are always thermodynamically preferred overRNAdS black holes. However, the system has much rich phase structure and phase transitions ina canonical ensemble. In particular, we report a RNAdS BH/scalarized BH/RNAdS BH reentrantphase transition, which is composed of a zeroth-order phase transition and a second-order one.
CONTENTS
I. Introduction 1II. EMS Model in AdS Space 2A. Equations of motion 3B. Asymptotic behavior 3C. Smarr relation 4D. Free energy 4III. Perturbations around Black Hole Solution 5A. Scalar perturbation around RNAdS black holes 5B. Time-dependent perturbation around scalarized black holes 6IV. Numerical Results 7A. Scalarized black holes 7B. Phase structure in a canonical ensemble 7V. Discussion and Conclusions 10Acknowledgments 11References 11
I. INTRODUCTION
The no-hair theorem states that a black hole can be uniquely determined via three parameters, its mass, electriccharge and angular momentum [1–3]. Although this theorem holds true in the Einstein-Maxwell field theory, it suffersfrom challenges due to the existence of hairy black holes possessing extra macroscopic degrees of freedom in othertheories. In fact, various black hole solutions, e.g., hairy black holes in the Einstein-Yang-Mills theory [4–6], blackholes with Skyrme hairs [7, 8] and black holes with dilaton hairs [9], have been discovered as counter-examples to theno-hair theorem. For a review, see [10].Spontaneous scalarization typically occurs in models with non-minimal coupling terms of scalar fields, which cansource the scalar fields to destabilize scalar-free black hole solutions and form scalarized hairy black holes. Thisphenomenon was first studied for neutron stars in scalar-tensor models [11] by coupling scalar fields to the Ricci ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ g r- q c ] F e b curvature. It demonstrated that there is a coexistence region for scalar-free and scalarized neutron star solutions,where the scalarized ones can be energetically preferred. Later, it was found that there also exists spontaneousscalarization of black holes in scalar-tensor models, provided that black holes are surrounded by non-conformallyinvariant matter [12, 13].Recently, the phenomenon of spontaneous scalarization has been studied in the extended Scalar-Tensor-Gauss-Bonnet (eSTGB) gravity [14–19]. In particular, asymptotically anti-de Sitter (AdS) scalarized black holes have beenstudied in a scalar-tensor model with non-minimally coupling the scaler field to the Ricci scalar and the Gauss-Bonnet term, as well as applications to holographic phase transitions [20]. In eSTGB models, the scalar field isnon-minimally coupled to the Gauss-Bonnet curvature correction of the gravitational sector, which leads to numericalchallenges for solving the evolution equations due to non-linear curvature terms. To better understand dynamicalevolution into scalarized black holes, a similar, but technically simpler, type of models, i.e., Einstein-Maxwell-scalar(EMS) models with non-minimal couplings between the scalar and Maxwell fields, have been put forward in [21],where fully non-linear numerical evolutions of spontaneous scalarization were presented. Subsequently, further studiesof spontaneous scalarization in the EMS models were discussed in the context of various non-minimal couplingfunctions [22, 23], dyons including magnetic charges [24], axionic-type couplings [25], massive and self-interacting scalarfields [26, 27], horizonless reflecting stars [28], stability analysis of scalarized black holes [29–33], higher dimensionalscalar-tensor models [34], quasinormal modes of scalarized black holes [35, 36], two U(1) fields [37], quasi-topologicalelectromagnetism [38], topology and spacetime structure influences [39] and the Einstein-Born-Infeld-scalar theory[40]. Besides the above asymptotically flat scalarized black holes, spontaneous scalarization was also discussed in theEMS model with a positive cosmological constant [41]. Additionally, spontaneous vectorization of electrically chargedblack holes was also proposed [42], analytic treatments were applied to study spontaneous scalarization in the EMSmodels [43–46], and an infinite family of exact topological charged hairy black hole solutions were constructed in theEMS gravity system [47].Studying thermodynamics of the EMS models not only provides evidence for endpoints of the dynamical evolutionof unstable scalar-free black holes, but also is interesting per se. Understanding the statistical mechanics of black holeshas been a subject of intensive study for several decades. In the pioneering work [48–50], Hawking and Bekensteinfound that black holes possess the temperature and the entropy. However, asymptotically flat black holes are oftenthermally unstable since they have negative specific heat. To make black holes thermally stable, appropriate boundaryconditions need to be imposed, e.g., putting the black holes in AdS space. Asymptotically AdS black holes becomethermally stable since the AdS boundary acts as a reflecting wall. The thermodynamic properties of AdS black holeswere first studied by Hawking and Page [51], who discovered the Hawking-Page phase transition between SchwarzschildAdS black holes and thermal AdS space. Later, motivated by AdS/CFT correspondence [52–54], there has been muchinterest in studying phase structure and transitions of AdS black holes [55–64]. In light of this, it is of great interestto study spontaneous scalarization of asymptotically AdS black holes and associated thermodynamic properties inthe EMS models with non-minimal couplings of the scalar and electromagnetic fields.The remainder of this paper is organized as follows. In section II, we introduce the EMS model with a negativecosmological constant and derive the free energy in a canonical ensemble. Section III is devoted to discussing linearperturbations in scalar-free and scalarized black hole solutions. In section IV, we present our main numerical results,including the domain of existence, entropic preference, effective potentials for radial perturbations, and phase structureand transitions in a canonical ensemble. We summarize our results with a brief discussion in section V. II. EMS MODEL IN ADS SPACE
In this section, we derive the equations of motion, asymptotic behavior, the Smarr relation and the Helmholtz freeenergy for asymptotically AdS scalarized black hole solutions in the EMS model. The action of the EMS model witha negative cosmological constant is S bulk = − π (cid:90) d x √− g (cid:104) R − − ∂φ ) − f ( φ ) F µν F µν (cid:105) , (1)where we take G = 1 for simplicity throughout this paper. In the action (1), the scalar field φ is minimally coupledto the metric g µν and non-minimally coupled to the gauge field A µ , F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic fieldstrength tensor, Λ = − /L is the cosmological constant with the AdS radius L , and f ( φ ) is the non-minimal couplingfunction of the scalar and the gauge fields. A. Equations of motion
Varying the action (1) with respect to the metric g µν , the scalar field φ and the gauge field A µ , one obtains theequations of motion, R µν − Rg µν − g µν L = 2 T µν , (cid:3) φ −
14 ˙ f ( φ ) F µν F µν = 0 , (2) ∂ µ (cid:0) √− gf ( φ ) F µν (cid:1) = 0 , where ˙ f ( φ ) ≡ df ( φ ) /dφ . The energy-momentum tensor T µν in eqn. (2) is given by T µν = ∂ µ φ∂ ν φ − g µν ( ∂φ ) + f ( φ ) (cid:18) F µρ F ρν − g µν F ρσ F ρσ (cid:19) . (3)In the following, we focus on the spherically symmetric ansatz for the metric, the electromagnetic field and the scalarfield, ds = − N ( r ) e − δ ( r ) dt + 1 N ( r ) dr + r (cid:0) dθ + sin θdϕ (cid:1) ,A µ dx µ = V ( r ) dt and φ = φ ( r ) . (4)Plugging the above ansatz into the equations of motion (2) yields N (cid:48) ( r ) = 1 − N ( r ) r − Q r f ( φ ( r )) − rN ( r ) φ (cid:48) ( r ) + 3 rL , (cid:0) r N ( r ) φ (cid:48) ( r ) (cid:1) (cid:48) = − ˙ f ( φ ( r )) Q f ( φ ) r − r N ( r ) φ (cid:48) ( r ) ,δ (cid:48) ( r ) = − rφ (cid:48) ( r ) , (5) V (cid:48) ( r ) = − Qr f ( φ ( r )) e − δ ( r ) , where primes denote the derivatives with respect to the radial coordinate r , and the integration constant Q canbe interpreted as the electric charge of the black hole solution. For later use, we introduce the Misner-Sharp massfunction m ( r ) by N ( r ) = 1 − m ( r ) /r + r /L . B. Asymptotic behavior
To obtain non-trivial hairy black hole solutions of the non-linear ordinary differential equations (5), one shouldimpose appropriate boundary conditions at the event horizon and the spatial infinity. Accordingly, in the vicinity ofthe event horizon at r = r + , we find that the solutions can be approximated as a power series expansion in terms of( r − r + ), m ( r ) = r + (cid:18) r L (cid:19) + m ( r − r + ) + · · · , δ ( r ) = δ + δ ( r − r + ) + · · · ,φ ( r ) = φ + φ ( r − r + ) + · · · , V ( r ) = v ( r − r + ) + · · · , (6)where m = Q r f ( φ ) , φ = − ˙ f ( φ ) Q (cid:2) f ( φ ) r − f ( φ ) r + Q + 3 f ( φ ) r /L (cid:3) , δ = − r + φ , v = − Qr f ( φ ) e − δ . (7)The two essential parameters, φ and δ , can be determined after matching the asymptotic expansion of the solutionsat the spatial infinity, m ( r ) = M − Q r + · · · , φ ( r ) = φ + r + · · · , δ ( r ) = 3 φ r + · · · , V ( r ) = Φ + Qr + · · · , (8)where f (0) = 1 is assumed, M is identified as the ADM mass, φ + can be interpreted as the expectation value of thedual operator in the conformal boundary theory from AdS/CFT correspondence, and Φ is the electrostatic potentialwith Φ = (cid:82) ∞ r + dre − δ ( r ) Q/ (cid:0) r f ( φ ( r )) (cid:1) . Note that the scalar field usually behaves as φ ( r ) ∼ φ − + φ + r in the asymptoticregion, and we set φ − = 0 in this paper, corresponding to the absence of the external source in the conformal boundarytheory. Consequently, we can use the shooting method to solve the non-linear differential equations (5) for solutionssatisfying the asymptotic expansions at the boundaries. It is also noteworthy that there is a scaling symmetry amongthe physical quantities, r → λr, M → λM, Q → λQ, L → λL, (9)which allows us to solve eqn. (5) numerically in terms of redefined dimensionless quantities. C. Smarr relation
The Smarr relation [65] can be used to test the accuracy of numerical scalarized black hole solutions, since itassociates the black hole mass with other physical quantities. The Smarr relation can be derived from computing theKomar integral for a time-like Killing vector K µ = (1 , , ,
0) in a manifold M . Integrating the identity ∇ µ ( ∇ ν K µ ) = K µ R µν over the time constant hypersurface Σ, whose boundary ∂ Σ consists of the event horizon r = r + and thespatial infinity r = + ∞ , one can use Gauss’s law to obtain (cid:90) ∂ Σ dS µν ∇ µ K ν = (cid:90) Σ dS µ K ν (cid:18) T µν − T g µν − g µν L (cid:19) , (10)where dS µν is the surface element on ∂ Σ, and dS µ is the volume element on Σ, accordingly. Making use of eqns. (3)and (5), we find that the Smarr relation is given by M = A H T Q Φ − e − δ r L + (cid:90) ∞ r + dre − δ ( r ) δ (cid:48) ( r ) r L , (11)where A H = 4 πr is the horizon area, and T = N (cid:48) ( r + ) e − δ ( r + ) / π is the Hawking temperature. For a RNAdS blackhole with δ ( r ) = 0, the Smarr relation (11) reduces to M = A H T Q Φ − r L , (12)where the last term is the P V term in the extended phase space of AdS black holes [66].
D. Free energy
Given a family of scalarized black holes, it is of interest to compute the Helmholtz free energy, which can be usedto investigate phase structure and transitions in a canonical ensemble with fixed charge Q and temperature T . Thefree energy, which is identified as the thermal partition function of black holes, can be derived via constructing theEuclidean path integral. In the semiclassical approximation, the partition function is evaluated by exponentiating theon-shell Euclidean action S E on-shell , Z ∼ e − S E on-shell , (13)where the on-shell action S E on-shell is obtained by substituting the classical solution into the action. However, the on-shell action S E on-shell normally diverges in asymptotically AdS spacetime. One then needs holographic renormalizationto remove divergences appearing in the asymptotic region. There are several methods to regularize S E on-shell , suchas the background-subtraction method [57] and the Kounterterms method [67–69]. Here, we adopt the countertermsubtraction method [70, 71] to regularize the action by adding a series of boundary terms to the bulk action.Specifically for the aforementioned bulk action S bulk in eqn. (1), the regularized action S R is supplied with threeboundary terms S R = S bulk + S GH + S ct + S surf , (14)where S GH is the Gibbon-Hawking boundary term to render the variational problem well-defined, S ct includes coun-terterms to eliminate divergences on asymptotic boundaries, and S surf is used to fix the charge rather than theelectrostatic potential when the action is varied [62, 72]. The three boundary terms are given by S GH = − π (cid:90) d x √− γ Θ ,S ct = 18 π (cid:90) d x √− γ (cid:18) L + L R (cid:19) , (15) S surf = − π (cid:90) d x √− γf ( φ ) F µν n µ A ν , where the integrals are performed on the hypersurface at the spatial infinity, γ is the determinant of the inducedmetric on the hypersurface, Θ is the trace of the extrinsic curvature, R is the scalar curvature of the induced metric γ , and n µ is the unit vector normal to the hypersurface. Using the equations of motion (2) and the asymptoticexpansion (8), we obtain the on-shell Euclidean version of S bulk , S GH , S ct and S surf , S E bulk, on-shell = 1 T (cid:32) e − δ ( r ) r N (cid:48) ( r ) − e − δ ( r ) r N ( r ) δ (cid:48) ( r )4 (cid:12)(cid:12)(cid:12)(cid:12) r =+ ∞ − T S − Q Φ (cid:33) ,S E GH, on-shell = − T (cid:20) e − δ ( r ) r N (cid:48) ( r ) − e − δ ( r ) r δ (cid:48) ( r ) N ( r )4 + e − δ ( r ) (cid:18) r − M + r L (cid:19)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) r =+ ∞ ,S E ct, on-shell = e − δ ( r ) T (cid:18) r L + r − M (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r =+ ∞ , (16) S E surf, on-shell = Q Φ T , where S = πr is the entropy of the black hole. Consequently, the regularized on-shell Euclidean action for the blackhole solution (4) is S E on-shell = S E bulk, on-shell + S E GH, on-shell + S E ct, on-shell + S E surf, on-shell = M − T ST . (17)The Helmholtz free energy F is related to the Euclidean action S E on-shell via F = − T ln Z = T S E on-shell , (18)which gives F = M − T S. (19)
III. PERTURBATIONS AROUND BLACK HOLE SOLUTION
In this section, we investigate linear perturbations around black hole solutions, which can help us understand thestability of the solutions.
A. Scalar perturbation around RNAdS black holes
We first examine a scalar perturbation δφ in a RNAdS black hole background. Note that if ˙ f (0) = 0 is imposed, aRNAdS black hole, which is described by N ( r ) = 1 − Mr + Q r + r L , A = Qr dt, δ ( r ) = 0 , φ ( r ) = 0 , (20)is manifestly a solution of the equations of motion (2). In this scalar-free solution background, we can linearize thescalar equation in eqn. (2) with a scalar perturbation δφ , (cid:0) (cid:3) − µ eff (cid:1) δφ = 0 , (21)where µ eff = − ¨ f (0) Q / (cid:0) r (cid:1) . In a (3 + 1)-dimensional asymptotically AdS spacetime of AdS radius L , a scalarfield can cause a tachyonic instability only if its mass-squared is less than the so-called Breitenlohner-Freedman (BF)bound µ BF = − / (cid:0) L (cid:1) [73]. For the scalar perturbation δφ , one always has µ eff > µ BF for large enough r , andhence, asymptotically, the RNAdS black hole is stable against the formation of the scalar field, which guaranteesthat scalarized black holes induced by the tachyonic instability of the scalar field are asymptotically AdS. However, if µ eff < µ BF in some region (e.g., near the event horizon), a RNAdS black hole may evolve to a scalarized black holeunder a scalar perturbation. Note that if ¨ f (0) <
0, one always has µ eff > µ BF , and hence a tachyonic instabilitycan not occur.To study how a scalarized black hole solution bifurcates from a scalar-free black hole solution, we calculate zeromodes of the scalar perturbation in the scalar-free black hole background. For simplicity, the scalar perturbation δφ is written as the decomposition with spherical harmonics functions, δφ = (cid:88) l,m Y lm ( θ, φ ) U l ( r ) . (22)With this decomposition, the scalar equation (21) then reduces to1 r ddr (cid:18) r N ( r ) dU l ( r ) dr (cid:19) − (cid:20) l ( l + 1) r + µ eff (cid:21) U l ( r ) = 0 . (23)Given the fixed values of l and ¨ f (0), requiring that the radial field U l ( r ) is regular at the event horizon and vanishesat the spatial infinity selects a family of discrete black hole solutions, which can be labelled by a non-negative integernode number n . In this paper, we focus on the l = 0 = n fundamental mode since it gives the smallest q of theblack hole solutions [21]. Due to the tachyonic instability, scalarized RNAdS black holes may emerge from these zeromodes, which composes bifurcation lines in the domain of existence for the scalarized black holes. B. Time-dependent perturbation around scalarized black holes
To investigate perturbative stability of the scalarized black hole solution (4), we then consider spherically symmetricand time-dependent linear perturbations. Specifically including the perturbations, the metric ansatz is written as [22] ds = − ˜ N ( r, t ) e − δ ( r,t ) dt + dr ˜ N ( r, t ) + r (cid:0) dθ + sin θdϕ (cid:1) , ˜ N ( r, t ) = N ( r ) + (cid:15) ˜ N ( r ) e − i Ω t , ˜ δ ( r, t ) = δ ( r ) + (cid:15) ˜ δ ( r ) e − i Ω t , (24)where the time dependence of the perturbations is assumed to be Fourier modes with frequency Ω. Similarly, theansatzes of the scalar and electromagnetic fields are given by˜ φ ( r, t ) = φ ( r ) + (cid:15)φ ( r ) e − i Ω t and ˜ V ( r, t ) = V ( r ) + (cid:15)V ( r ) e − i Ω t , (25)respectively. Solving eqn. (2) with the ansatzes (24) and (25), we can extract a Schrodinger-like equation for theperturbative scalar field φ ( r ), − d Ψ ( r ) dr ∗ + U Ω Ψ ( r ) = Ω Ψ ( r ) , (26)where Ψ ( r ) ≡ rφ ( r ), and the tortoise coordinate r ∗ is defined by dr ∗ /dr ≡ e δ ( r ) N − ( r ). Here, the effective potential U Ω is given by U Ω = e − δ Nr (cid:40) − N − r φ (cid:48) − Q r f ( φ ) (cid:32) − r φ (cid:48) + ¨ f ( φ )2 f ( φ ) + 2 rφ (cid:48) ˙ f ( φ ) f ( φ ) − ˙ f ( φ ) f ( φ ) (cid:33) + 3 r (cid:0) − r φ (cid:48) (cid:1) L (cid:41) . (27)One can show that the effective potential U Ω vanishes at the event horizon, whereas approaches positive infinity atthe spatial infinity. From quantum mechanics, the existence of an unstable mode with Ω < U Ω < U Ω ensures that scalarized black hole solutions arestable against the spherically symmetric perturbations. It is noteworthy that the appearance of a negative regionof U Ω cannot sufficiently guarantee the presence of an instability [26]. One can utilize other techniques, like the S -deformation method [74], to further discuss the stability of these solutions. IV. NUMERICAL RESULTS
In this section, we first study various properties of scalarized RNAdS black hole solutions and then investigate theirphase structure and transitions in a canonical ensemble. After the non-linear differential equations (5) are expressedin terms of a new dimensionless coordinate x = 1 − r + r with 0 ≤ x ≤
1, (28)they are numerically solved for scalarized black hole solutions using the NDSolve function in Wolfram Mathematica.In the remainder of this paper, we focus on the coupling function f ( φ ) = e αφ with α >
0. For this coupling function,one has f (0) = 1 and ˙ f (0) = 0, which ensures that RNAdS black holes are solutions of the EMS model, and ¨ f (0) > q = QM , a H = A H πM , ˜ T = T L, ˜ F = FL , ˜ r + = r + L , ˜ Q = QL , ˜ M = ML , (29)which are dimensionless and invariant under the scaling symmetry (9). To test the accuracy of our numerical method,we use the Smarr relation (11) and find that the numerical error can be maintained around the order of 10 − . A. Scalarized black holes
Here, we present the numerical results, e.g., the domain of existence, entropic preference and effective potentials, forscalarized black hole solutions, which are dynamically induced from RNAdS black holes. Without loss of generality, wefocus on ˜ Q = Q/L = 0 . µ eff < µ BF somewhere in the spacetime. Since the minimum value of µ eff occurs at the event horizon r = r + ,we only need to check µ eff < µ BF at r = r + . The region in the ( α, q ) parameter space of RNAdS black holes where µ eff < µ BF at r = r + is plotted in the upper left panel of Fig. 1. The distribution of values of ( µ eff − µ BF ) Q | r = r + is also displayed, which shows that the tachyonic instability region becomes larger as α increases, and the scalar fieldsuffers from a strong tachyonic instability when black holes are near-extremal. The bifurcation line is composed ofthe l = 0 = n zero modes of eqn. (23) for the scalar perturbation in RNAdS black holes, and represented by bluedashed lines in Fig. 1. When the tachyonic instability is strong enough, the scalar perturbation can lead to scalarizedblack holes with non-trivial scalar fields above the bifurcation line. In the upper right panel of Fig. 1, we display thedomain of existence for scalarized black holes, which is exhibited by a light blue region. The domain of existence isbounded by the bifurcation and critical lines, and resembles that of RN scalarized black holes [21]. On the criticalline, the mass and the charge of scalarized solutions remain finite, whereas its horizon radius vanishes. On the otherhand, the mass-to-charge ratio q of RNAdS black holes reaches the maximum in the extremal limit, which is shownby a horizontal dashed gray line. Moreover, there is a certain region bounded by the extremal and bifurcation lines,where scalarized and RNAdS black holes coexist.The reduced area a H as a function of reduced charge q is plotted for RNAdS and scalarized black holes in thelower left panel of FIG. 1, which demonstrates that scalarized black holes emerge from RNAdS black holes at thebifurcation points, marked by B , and eventually terminate on the critical line with zero a H . For a multiphase systemin a micro-canonical ensemble with conserved energy, the phase of maximum entropy is globally stable and will bepresent at equilibrium. Therefore, in the scalarized and RNAdS black holes coexisting region, our numerical resultsshow that scalarized solutions are entropically preferred over RNAdS black hole solutions, and hence are the globallystable phase in the micro-canonical ensemble.To study the stability of scalarized solutions, the effective potentials U Ω of scalarized solutions with α = 5 areplotted for several values of q in the lower right panel of FIG. 1, where solid and dashed colored lines correspond topotentials with and without negative regions, respectively. The scalarized solutions have positive effective potentialsfor a large enough value of q , and thus are free of radial instabilities. However, as q decreases towards the bifurcationline, there appear negative regions in the effective potentials, which means that radial instabilities cannot be excludednear the bifurcation line. B. Phase structure in a canonical ensemble
In this subsection, we consider phase structure and transitions of scalarized and RNAdS black holes in a canonicalensemble maintained at a given temperature of T and a given charge of Q . In a canonical ensemble, the globally FIG. 1. Plots of tachyonically unstable region for RNAdS black holes, and the domain of existence, entropic preference andeffective potentials for scalarized black holes. Here, we take
Q/L = 0 . Upper Left:
Density plot of (cid:0) µ eff − µ BF (cid:1) Q evaluated at the event horizon r = r + as a function of q and α for the scalar perturbation in RNAdS black holes. Weonly display the tachyonically unstable region, where µ eff < µ BF , in the α - q plane. The closer RNAdS black holes are tothe extremal limit, the more unstable the scalar field becomes. The blue dashed line represents the bifurcation line, wheretachyonic instabilities are strong enough to induce scalarized black holes. Upper Right:
Domain of existence for scalarizedRNAdS black holes in the α - q plane, which is highlighted by the light blue region and bounded by the critical and bifurcationlines. The critical line is depicted by a red solid line, on which the reduced horizon area a H vanishes. The horizontal dashedgray line denotes extremal RNAdS black holes, above which RNAdS black hole solutions do not exist. Lower Left:
Reducedhorizon area a H against q for RNAdS and scalarized black holes. The scalarized black hole solutions are always entropicallypreferred, which means that they are globally stable in a micro-canonical ensemble. Lower Right:
Effective potentials ofscalarized black holes with α = 5 for several values of q . Solid red lines denote positive definite effective potentials betweenthe event horizon and the spatial infinity, while dashed red lines represent those possessing negative regions. When q is largeenough, the scalarized black hole solutions are stable against radial perturbations. stable phase of a multiphase system, which exists at equilibrium, has the lowest possible Helmholtz free energy F ,which can be computed via eqn. (19). The rich phase structure of black holes usually comes from expressing thehorizon radius r + as a function of temperature T . If the function r + ( T ) is multivalued, there will be more than oneblack hole phase, corresponding to different branches of r + ( T ).To illustrate phase structure and transitions, we plot the reduced horizon radius ˜ r + and the free energy ˜ F asfunctions of reduced temperature ˜ T for scalarized and RNAdS black holes with three representative values of ˜ Q inFig. 2, where we have α = 5. In the left column of Fig. 2 with a small ˜ Q , the upper panel shows that threebranches of the RNAdS black hole solution coexist in some range of ˜ T , and are dubbed as large, intermediate andsmall RNAdS BHs, respectively, based on their values of horizon radius. At a high (low) enough temperature, onlythe large (small) RNAdS BH phase exists. On the other hand, there is only one phase for the scalarized black holesolution, which bifurcates from the RNAdS black hole solution at the bifurcation point B , and does not exist at a lowtemperature. The reduced free energy ˜ F is plotted against ˜ T for these four phases in the lower panel, which showsthat the scalarized black hole can not be the globally stable phase since there always exists some RNAdS black holephase of a lower free energy at a given ˜ T . In the coexisting region of the RNAdS black hole phases, a first-order phasetransition between large and small RNAdS BHs occurs at some point, where their free energies intersect each other. Scalarized BH RNAdS BH TL r + / L Q / L = B Scalarized BH RNAdS BH TL r + / L Q / L = B Scalarized BH RNAdS BH T min TL r + / L Q / L = B Scalarized BH RNAdS BH TL F / L Q / L = B B
Scalarized BH RNAdS BH TL F / L Q / L = B Scalarized BH RNAdS BH T min TL F / L Q / L = BB FIG. 2. Plots of the reduced horizon radius ˜ r + (the upper row) and the reduced free energy ˜ F (the lower row) versus thereduced temperature ˜ T for RNAdS (blue lines) and scalarized (green lines) black holes with several fixed values of the reducedcharge ˜ Q . Here, we focus on α = 5. Bifurcation points are labeled by B . When ˜ r + ( ˜ T ) is multivalued, black hole solutionshave more than one branch of different horizon radii in a canonical ensemble with fixed ˜ T and ˜ Q . Left column : There is aband of temperatures where three branches of RNAdS black hole solutions coexist, and a first-order phase transition occursbetween the large RNAdS BH phase (i.e., the branch with the largest horizon radius) and the small RNAdS BH phase (i.e.,the branch with the smallest horizon radius). Scalarized black holes emerge from the bifurcation point. Nevertheless, theyare not globally preferred since they always have a higher free energy than RNAdS black holes.
Center column : RNAdSblack hole solutions have only one branch, whose free energy is smaller than that of scalarized black holes. There is no phasetransition.
Right column : RNAdS black hole solutions have only one branch, whereas scalarized black hole solutions havetwo branches of different sizes. As ˜ T increases, the globally stable phase jumps from RNAdS black holes to scalarized blackholes (the branch with a larger horizon radius), corresponding to a zeroth-order phase transition at ˜ T min . Further increasing ˜ T ,there would be a second-order phase transition returning to RNAdS black holes at the bifurcation point B . Here, we observea RNAdS BH/scalarized BH/RNAdS BH reentrant phase transition. The coexisting region of the RNAdS black hole phases shrinks as ˜ Q increases until reaching a critical point, wherea second-order phase transition occurs between large and small RNAdS BHs. Beyond the critical point, large RNAdSBH is indistinguishable from small RNAdS BH, hence RNAdS black hole solutions have a single phase. In the centercolumn of Fig. 2, we depict ˜ r + ( ˜ T ) and ˜ F ( ˜ T ) for the case with a value of ˜ Q greater than the critical value. The upperpanel shows that both RNAdS and scalarized black holes have a single phase. Moreover, as displayed in the lowerpanel, the RNAdS black hole always has a smaller free energy than the scalarized black hole, and hence is the globallystable phase. Consequently, there is no phase transition in this case.However when ˜ Q is large enough, phase structure of scalarized and RNAdS black holes becomes much richer. Forexample, ˜ r + ( ˜ T ) and ˜ F ( ˜ T ) are plotted for scalarized and RNAdS black holes with a large enough ˜ Q in the rightcolumn of Fig. 2. It shows in the upper panel that the RNAdS black hole solution possesses a single phase, whereasthe scalarized solution can have two phases at some given ˜ T , namely large scalarized BH (i.e., the one with a largerhorizon radius) and small scalarized BH (i.e., the one with a smaller horizon radius). In fact, the scalarized blackhole solution has a minimum temperature ˜ T min , and large scalarized BH coexists with small scalarized BH between˜ T = ˜ T min and the bifurcation point B , where large scalarized BH and the RNAdS black hole merge. The lower panelexhibits the free energy as a function of ˜ T for the three phases. If we start increasing the temperature from ˜ T = 0,the system follows the blue line of the RNAdS black hole until ˜ T = ˜ T min , where the free energy has a discontinuityat its global minimum. Further increasing ˜ T , the inset shows that the system jumps to the lower green line of largescalarized BH, which corresponds to a zeroth-order phase transition between scalarized and RNAdS black holes. As˜ T continues to increase, the system follows the lower green line until it joins the blue line at the bifurcation point B , which corresponds to a second-order phase transition between scalarized and RNAdS black holes. Note that sincescalarized and RNAdS black holes have the same entropy at the bifurcation point, a phase transition between them atthe bifurcation point is second-order. In short, a RNAdS BH/scalarized BH/RNAdS BH reentrant phase transition0 C 0.5 1.0 1.50.050.100.150.200.250.300.35 Q / L TL RNAdS BH large scalarized BH l a r g e RNA d S BH s m a ll RNA d S BH FIG. 3. Phase diagram of RNAdS and scalarized black holes in a canonical ensemble with fixed temperature T and charge Q .Here, we take α = 5. The phase diagram exhibits the globally stable phases, which have the lowest free energy, and the phasetransitions between them. The light yellow and blue regions correspond to RNAdS and scalarized black holes, respectively. Afirst-order phase transition line (the purple line) separates the large RNAdS BH phase, which is above the line, and the smallRNAdS BH phase, which is below the line. The first-order phase transition line terminates at the critical point, labelled by C .The scalarized black hole phase is delimited by a zeroth-order phase transition line (the red line) and a second-order one (theblue dashed line), which coincides with the bifurcation line. is observed as ˜ T increases.In addition, it is interesting to consider the local thermodynamic stability of black hole phases against thermalfluctuations. In a canonical ensemble, the quantity of particular interest is the specific heat at constant charge, C Q = T (cid:18) ∂S∂T (cid:19) Q = 2 L π ˜ r + ˜ T ∂ ˜ r + ∂ ˜ T . (30)Since the entropy is proportional to the size of the black hole, a positive specific heat means that black holes radiateless when they are smaller. Thus, the thermal stability of a phase follows from C Q > ∂ ˜ r + /∂ ˜ T > ∂ ˜ r + /∂ ˜ T >
0. Inconsequence, the globally stable phases of scalarized and RNAdS black holes possess a positive C Q , and are thermallystable.To better illustrate the globally stable phases of lowest free energy and the associated phase transitions, we displaythe phase diagram of scalarized and RNAdS black holes in the ˜ Q - ˜ T plane in Fig. 3, where α = 5. There is a first-order phase transition line (the purple line) separating large and small RNAdS BHs for small ˜ Q , which terminatesat the critical point. This first-order phase transition is quite similar to the liquid/gas phase transition. When ˜ Q islarge enough, large scalarized BH (the light blue region) appears, and is bounded by zeroth-order (the red line) andsecond-order (the blue dashed line) phase transition lines. V. DISCUSSION AND CONCLUSIONS
In this paper, we investigated spontaneous scalarization of asymptotically AdS charged black holes in the EMSmodel, and studied phase structure of scalarized and RNAdS black holes in a canonical ensemble. We focused on anon-minimal coupling function f ( φ ) = e αφ , which leads to spontaneous scalarization due to the tachyonic instabilityof the scalar field near the event horizon. In practice, scalarized black holes can be induced from RNAdS black holeson the bifurcation line, which consists of zero modes of the scalar perturbation in RNAdS black holes. In the α - q plane with a fixed ˜ Q , the domain of existence for scalarized RNAdS black holes is bounded by the bifurcation andcritical lines, which resembles that of scalarized RN black holes very closely [21]. In a micro-canonical ensemble, wefound that scalarized black hole solutions are always entropically preferred over RNAdS black holes, and hence theglobally stable phase.On the other hand, the system has much richer phase structure in a canonical ensemble. After the Helmholtz freeenergy of the EMS model was computed, we obtained the phase structure of scalarized and RNAdS black holes. In the1small Q regime, scalarized black holes never globally minimize the free energy, and the corresponding phase diagramresembles that of the liquid/gas system closely. Nevertheless in the large Q regime, scalarized black holes can be theglobally stable phase in some parameter region. As the temperature increases at a given charge, the system undergoesa RNAdS BH/scalarized BH/RNAdS BH reentrant phase transition, which consists of zeroth-order and second-orderphase transitions.The phenomenon of reentrant phase transition was first observed in a nicotine/water mixture, and later discoveredin the context of black hole thermodynamics, e.g., Born-Infeld-AdS black holes [62, 75], higher dimensional singlyspinning Kerr-AdS black holes [76], AdS black holes in Lovelock gravity [77], AdS black holes in dRGT massivegravity [78], hairy AdS black holes [79]. For these black holes, reentrant phase transitions were found to includezeroth-order and first-order phase transitions. In this paper, we present an example of a reentrant phase transitionfor black holes, which is composed of zeroth-order and second-order phase transitions. Furthermore, the second-orderphase transition between scalarized and RNAdS black holes is of great interest since this implies that our results mayprovide an interesting model of holographic superconductors. We leave this for future work.AdS black holes can also been studied in the context of extended phase space thermodynamics, where the cosmo-logical constant is interpreted as a thermodynamic pressure P ≡ /L [60, 80]. In terms of P , the reduced quantitiesare expressed as ˜ T = T (cid:112) /P , ˜ F = F (cid:112) P/ , ˜ r + = r + (cid:112) P/ , ˜ Q = Q (cid:112) P/ , ˜ M = M (cid:112) P/ . (31)Note that M and F are identified as the enthalpy and the Gibbs free energy, respectively, in extended phase space.With eqn. (31), our results can be generalized to extended phase space thermodynamics. ACKNOWLEDGMENTS
We are grateful to Qingyu Gan and Feiyu Yao for useful discussions and valuable comments. This work is supportedin part by NSFC (Grant No. 11875196, 11375121, 11947225 and 11005016). [1] Werner Israel. Event horizons in static vacuum space-times.
Phys. Rev. , 164:1776–1779, 1967. doi:10.1103/PhysRev.164.1776 . I[2] B. Carter. Axisymmetric Black Hole Has Only Two Degrees of Freedom.
Phys. Rev. Lett. , 26:331–333, 1971. doi:10.1103/PhysRevLett.26.331 .[3] Remo Ruffini and John A. Wheeler. Introducing the black hole.
Phys. Today , 24(1):30, 1971. doi:10.1063/1.3022513 . I[4] M.S. Volkov and D.V. Galtsov. NonAbelian Einstein Yang-Mills black holes.
JETP Lett. , 50:346–350, 1989. I[5] P. Bizon. Colored black holes.
Phys. Rev. Lett. , 64:2844–2847, 1990. doi:10.1103/PhysRevLett.64.2844 .[6] Brian R. Greene, Samir D. Mathur, and Christopher M. O’Neill. Eluding the no hair conjecture: Black holes in sponta-neously broken gauge theories.
Phys. Rev. D , 47:2242–2259, 1993. arXiv:hep-th/9211007 , doi:10.1103/PhysRevD.47.2242 . I[7] Hugh Luckock and Ian Moss. BLACK HOLES HAVE SKYRMION HAIR. Phys. Lett. B , 176:341–345, 1986. doi:10.1016/0370-2693(86)90175-9 . I[8] Serge Droz, Markus Heusler, and Norbert Straumann. New black hole solutions with hair.
Phys. Lett. B , 268:371–376,1991. doi:10.1016/0370-2693(91)91592-J . I[9] P. Kanti, N.E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley. Dilatonic black holes in higher curvature stringgravity.
Phys. Rev. D , 54:5049–5058, 1996. arXiv:hep-th/9511071 , doi:10.1103/PhysRevD.54.5049 . I[10] Carlos A.R. Herdeiro and Eugen Radu. Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D ,24(09):1542014, 2015. arXiv:1504.08209 , doi:10.1142/S0218271815420146 . I[11] Thibault Damour and Gilles Esposito-Farese. Nonperturbative strong field effects in tensor - scalar theories of gravitation. Phys. Rev. Lett. , 70:2220–2223, 1993. doi:10.1103/PhysRevLett.70.2220 . I[12] Vitor Cardoso, Isabella P. Carucci, Paolo Pani, and Thomas P. Sotiriou. Matter around Kerr black holes in scalar-tensortheories: scalarization and superradiant instability.
Phys. Rev. D , 88:044056, 2013. arXiv:1305.6936 , doi:10.1103/PhysRevD.88.044056 . I[13] Vitor Cardoso, Isabella P. Carucci, Paolo Pani, and Thomas P. Sotiriou. Black holes with surrounding matter in scalar-tensor theories. Phys. Rev. Lett. , 111:111101, 2013. arXiv:1308.6587 , doi:10.1103/PhysRevLett.111.111101 . I[14] Daniela D. Doneva and Stoytcho S. Yazadjiev. New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization inExtended Scalar-Tensor Theories. Phys. Rev. Lett. , 120(13):131103, 2018. arXiv:1711.01187 , doi:10.1103/PhysRevLett.120.131103 . I[15] Hector O. Silva, Jeremy Sakstein, Leonardo Gualtieri, Thomas P. Sotiriou, and Emanuele Berti. Spontaneous scalarizationof black holes and compact stars from a Gauss-Bonnet coupling. Phys. Rev. Lett. , 120(13):131104, 2018. arXiv:1711.02080 , doi:10.1103/PhysRevLett.120.131104 . [16] G. Antoniou, A. Bakopoulos, and P. Kanti. Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-BonnetTheories. Phys. Rev. Lett. , 120(13):131102, 2018. arXiv:1711.03390 , doi:10.1103/PhysRevLett.120.131102 .[17] Pedro V.P. Cunha, Carlos A.R. Herdeiro, and Eugen Radu. Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor–Gauss-Bonnet Gravity. Phys. Rev. Lett. , 123(1):011101, 2019. arXiv:1904.09997 , doi:10.1103/PhysRevLett.123.011101 .[18] Carlos A. R. Herdeiro, Eugen Radu, Hector O. Silva, Thomas P. Sotiriou, and Nicol´as Yunes. Spin-induced scalarizedblack holes. Phys. Rev. Lett. , 126(1):011103, 2021. arXiv:2009.03904 , doi:10.1103/PhysRevLett.126.011103 .[19] Emanuele Berti, Lucas G. Collodel, Burkhard Kleihaus, and Jutta Kunz. Spin-induced black-hole scalarization in Einstein-scalar-Gauss-Bonnet theory. Phys. Rev. Lett. , 126(1):011104, 2021. arXiv:2009.03905 , doi:10.1103/PhysRevLett.126.011104 . I[20] Yves Brihaye, Betti Hartmann, Nath´alia Pio Aprile, and Jon Urrestilla. Scalarization of asymptotically anti–de Sitterblack holes with applications to holographic phase transitions. Phys. Rev. D , 101(12):124016, 2020. arXiv:1911.01950 , doi:10.1103/PhysRevD.101.124016 . I[21] Carlos A.R. Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, and Jos´e A. Font. Spontaneous Scalarization of Charged BlackHoles. Phys. Rev. Lett. , 121(10):101102, 2018. arXiv:1806.05190 , doi:10.1103/PhysRevLett.121.101102 . I, III A, IV A,V[22] Pedro G.S. Fernandes, Carlos A.R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas Sanchis-Gual. SpontaneousScalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features. Class. Quant. Grav. , 36(13):134002,2019. [Erratum: Class.Quant.Grav. 37, 049501 (2020)]. arXiv:1902.05079 , doi:10.1088/1361-6382/ab23a1 . I, III B[23] Jose Luis Bl´azquez-Salcedo, Carlos A.R. Herdeiro, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu. Einstein-Maxwell-scalar black holes: the hot, the cold and the bald. Phys. Lett. B , 806:135493, 2020. arXiv:2002.00963 , doi:10.1016/j.physletb.2020.135493 . I[24] D. Astefanesei, C. Herdeiro, A. Pombo, and E. Radu. Einstein-Maxwell-scalar black holes: classes of solutions, dyons andextremality. JHEP , 10:078, 2019. arXiv:1905.08304 , doi:10.1007/JHEP10(2019)078 . I[25] Pedro G.S. Fernandes, Carlos A.R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas Sanchis-Gual. Charged blackholes with axionic-type couplings: Classes of solutions and dynamical scalarization. Phys. Rev. D , 100(8):084045, 2019. arXiv:1908.00037 , doi:10.1103/PhysRevD.100.084045 . I[26] De-Cheng Zou and Yun Soo Myung. Scalarized charged black holes with scalar mass term. Phys. Rev. D , 100(12):124055,2019. arXiv:1909.11859 , doi:10.1103/PhysRevD.100.124055 . I, III B[27] Pedro G.S. Fernandes. Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair. Phys. Dark Univ. ,30:100716, 2020. arXiv:2003.01045 , doi:10.1016/j.dark.2020.100716 . I[28] Yan Peng. Scalarization of horizonless reflecting stars: neutral scalar fields non-minimally coupled to Maxwell fields. Phys.Lett. B , 804:135372, 2020. arXiv:1912.11989 , doi:10.1016/j.physletb.2020.135372 . I[29] Yun Soo Myung and De-Cheng Zou. Instability of Reissner–Nordstr¨om black hole in Einstein-Maxwell-scalar theory. Eur.Phys. J. C , 79(3):273, 2019. arXiv:1808.02609 , doi:10.1140/epjc/s10052-019-6792-6 . I[30] Yun Soo Myung and De-Cheng Zou. Stability of scalarized charged black holes in the Einstein–Maxwell–Scalar theory. Eur. Phys. J. C , 79(8):641, 2019. arXiv:1904.09864 , doi:10.1140/epjc/s10052-019-7176-7 .[31] De-Cheng Zou and Yun Soo Myung. Radial perturbations of the scalarized black holes in Einstein-Maxwell-conformallycoupled scalar theory. Phys. Rev. D , 102(6):064011, 2020. arXiv:2005.06677 , doi:10.1103/PhysRevD.102.064011 .[32] Yun Soo Myung and De-Cheng Zou. Onset of rotating scalarized black holes in Einstein-Chern-Simons-Scalar theory. Phys. Lett. B , 814:136081, 2021. arXiv:2012.02375 , doi:10.1016/j.physletb.2021.136081 .[33] Zhan-Feng Mai and Run-Qiu Yang. Stability analysis on charged black hole with non-linear complex scalar. 12 2020. arXiv:2101.00026 . I[34] Dumitru Astefanesei, Carlos Herdeiro, Jo˜ao Oliveira, and Eugen Radu. Higher dimensional black hole scalarization. JHEP ,09:186, 2020. arXiv:2007.04153 , doi:10.1007/JHEP09(2020)186 . I[35] Yun Soo Myung and De-Cheng Zou. Quasinormal modes of scalarized black holes in the Einstein–Maxwell–Scalar theory. Phys. Lett. B , 790:400–407, 2019. arXiv:1812.03604 , doi:10.1016/j.physletb.2019.01.046 . I[36] Jose Luis Bl´azquez-Salcedo, Carlos A.R. Herdeiro, Sarah Kahlen, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu.Quasinormal modes of hot, cold and bald Einstein-Maxwell-scalar black holes. 8 2020. arXiv:2008.11744 . I[37] Yun Soo Myung and De-Cheng Zou. Scalarized charged black holes in the Einstein-Maxwell-Scalar theory with two U(1)fields. Phys. Lett. B , 811:135905, 2020. arXiv:2009.05193 , doi:10.1016/j.physletb.2020.135905 . I[38] Yun Soo Myung and De-Cheng Zou. Scalarized black holes in the Einstein-Maxwell-scalar theory with a quasitopologicalterm. Phys. Rev. D , 103(2):024010, 2021. arXiv:2011.09665 , doi:10.1103/PhysRevD.103.024010 . I[39] Hong Guo, Xiao-Mei Kuang, Eleftherios Papantonopoulos, and Bin Wang. Topology and spacetime structure influenceson black hole scalarization. 12 2020. arXiv:2012.11844 . I[40] Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Einstein-Born-Infeld-scalar Black Holes. 12 2020. arXiv:2012.01066 . I[41] Yves Brihaye, Carlos Herdeiro, and Eugen Radu. Black Hole Spontaneous Scalarisation with a Positive CosmologicalConstant. Phys. Lett. B , 802:135269, 2020. arXiv:1910.05286 , doi:10.1016/j.physletb.2020.135269 . I[42] Jo˜ao M. S. Oliveira and Alexandre M. Pombo. Spontaneous vectorization of electrically charged black holes. Phys. Rev.D , 103(4):044004, 2021. arXiv:2012.07869 , doi:10.1103/PhysRevD.103.044004 . I[43] R.A. Konoplya and A. Zhidenko. Analytical representation for metrics of scalarized Einstein-Maxwell black holes and theirshadows. Phys. Rev. D , 100(4):044015, 2019. arXiv:1907.05551 , doi:10.1103/PhysRevD.100.044015 . I[44] Shahar Hod. Spontaneous scalarization of charged Reissner-Nordstr \ ”om black holes: Analytic treatment along the existence line. Phys. Lett. B , 798:135025, 2019. arXiv:2002.01948 .[45] Shahar Hod. Reissner-Nordstr¨om black holes supporting nonminimally coupled massive scalar field configurations.
Phys.Rev. D , 101(10):104025, 2020. arXiv:2005.10268 , doi:10.1103/PhysRevD.101.104025 .[46] Shahar Hod. Analytic treatment of near-extremal charged black holes supporting non-minimally coupled massless scalarclouds. Eur. Phys. J. C , 80(12):1150, 2020. doi:10.1140/epjc/s10052-020-08723-z . I[47] Subhash Mahapatra, Supragyan Priyadarshinee, Gosala Narasimha Reddy, and Bhaskar Shukla. Exact topological chargedhairy black holes in AdS Space in D -dimensions. Phys. Rev. D , 102(2):024042, 2020. arXiv:2004.00921 , doi:10.1103/PhysRevD.102.024042 . I[48] S. W. Hawking. Particle Creation by Black Holes. Commun. Math. Phys. , 43:199–220, 1975. [Erratum: Com-mun.Math.Phys. 46, 206 (1976)]. doi:10.1007/BF02345020 . I[49] Jacob D Bekenstein. Black holes and the second law.
Lett. Nuovo Cim. , 4(15):737–740, 1972. doi:10.1007/BF02757029 .[50] Jacob D. Bekenstein. Black holes and entropy.
Phys. Rev. D , 7:2333–2346, Apr 1973. URL: https://link.aps.org/doi/10.1103/PhysRevD.7.2333 , doi:10.1103/PhysRevD.7.2333 . I[51] S.W. Hawking and Don N. Page. Thermodynamics of Black Holes in anti-De Sitter Space. Commun. Math. Phys. , 87:577,1983. doi:10.1007/BF01208266 . I[52] Juan Martin Maldacena. The Large N limit of superconformal field theories and supergravity.
Int. J. Theor. Phys. ,38:1113–1133, 1999. arXiv:hep-th/9711200 , doi:10.1023/A:1026654312961 . I[53] S.S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. Gauge theory correlators from noncritical string theory. Phys.Lett. B , 428:105–114, 1998. arXiv:hep-th/9802109 , doi:10.1016/S0370-2693(98)00377-3 .[54] Edward Witten. Anti-de Sitter space and holography. Adv. Theor. Math. Phys. , 2:253–291, 1998. arXiv:hep-th/9802150 , doi:10.4310/ATMP.1998.v2.n2.a2 . I[55] Edward Witten. Anti-de Sitter space, thermal phase transition, and confinement in gauge theories. Adv. Theor. Math.Phys. , 2:505–532, 1998. arXiv:hep-th/9803131 , doi:10.4310/ATMP.1998.v2.n3.a3 . I[56] Andrew Chamblin, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. Charged AdS black holes and catas-trophic holography. Phys. Rev. D , 60:064018, Aug 1999. URL: https://link.aps.org/doi/10.1103/PhysRevD.60.064018 , arXiv:hep-th/9902170 , doi:10.1103/PhysRevD.60.064018 .[57] Andrew Chamblin, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. Holography, thermodynamics andfluctuations of charged AdS black holes. Phys. Rev. D , 60:104026, Oct 1999. URL: https://link.aps.org/doi/10.1103/PhysRevD.60.104026 , arXiv:hep-th/9904197 , doi:10.1103/PhysRevD.60.104026 . II D[58] Marco M. Caldarelli, Guido Cognola, and Dietmar Klemm. Thermodynamics of Kerr-Newman-AdS black holes andconformal field theories. Class. Quant. Grav. , 17:399–420, 2000. arXiv:hep-th/9908022 , doi:10.1088/0264-9381/17/2/310 .[59] Rong-Gen Cai. Gauss-Bonnet black holes in AdS spaces. Phys. Rev. D , 65:084014, 2002. arXiv:hep-th/0109133 , doi:10.1103/PhysRevD.65.084014 .[60] David Kubiznak and Robert B. Mann. P-V criticality of charged AdS black holes. JHEP , 07:033, 2012. arXiv:1205.0559 , doi:10.1007/JHEP07(2012)033 . V[61] Shao-Wen Wei and Yu-Xiao Liu. Insight into the Microscopic Structure of an AdS Black Hole from a ThermodynamicalPhase Transition. Phys. Rev. Lett. , 115(11):111302, 2015. [Erratum: Phys.Rev.Lett. 116, 169903 (2016)]. arXiv:1502.00386 , doi:10.1103/PhysRevLett.115.111302 .[62] Peng Wang, Houwen Wu, and Haitang Yang. Thermodynamics and Phase Transitions of Nonlinear Electrodynamics BlackHoles in an Extended Phase Space. JCAP , 04(04):052, 2019. arXiv:1808.04506 , doi:10.1088/1475-7516/2019/04/052 .II D, V[63] Shao-Wen Wei, Yu-Xiao Liu, and Robert B. Mann. Repulsive Interactions and Universal Properties of Charged Anti–deSitter Black Hole Microstructures. Phys. Rev. Lett. , 123(7):071103, 2019. arXiv:1906.10840 , doi:10.1103/PhysRevLett.123.071103 .[64] Peng Wang, Houwen Wu, and Haitang Yang. Thermodynamic Geometry of AdS Black Holes and Black Holes in a Cavity. Eur. Phys. J. C , 80(3):216, 2020. arXiv:1910.07874 , doi:10.1140/epjc/s10052-020-7776-2 . I[65] Larry Smarr. Mass formula for Kerr black holes. Phys. Rev. Lett. , 30:71–73, 1973. [Erratum: Phys.Rev.Lett. 30, 521–521(1973)]. doi:10.1103/PhysRevLett.30.71 . II C[66] David Kubiznak, Robert B. Mann, and Mae Teo. Black hole chemistry: thermodynamics with Lambda.
Class. Quant.Grav. , 34(6):063001, 2017. arXiv:1608.06147 , doi:10.1088/1361-6382/aa5c69 . II C[67] Rodrigo Olea. Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes. JHEP , 06:023,2005. arXiv:hep-th/0504233 , doi:10.1088/1126-6708/2005/06/023 . II D[68] Rodrigo Olea. Regularization of odd-dimensional AdS gravity: Kounterterms. JHEP , 04:073, 2007. arXiv:hep-th/0610230 , doi:10.1088/1126-6708/2007/04/073 .[69] Olivera Miskovic and Rodrigo Olea. Thermodynamics of Einstein-Born-Infeld black holes with negative cosmologicalconstant. Phys. Rev. D , 77:124048, 2008. arXiv:0802.2081 , doi:10.1103/PhysRevD.77.124048 . II D[70] Vijay Balasubramanian and Per Kraus. A Stress tensor for Anti-de Sitter gravity. Commun. Math. Phys. , 208:413–428,1999. arXiv:hep-th/9902121 , doi:10.1007/s002200050764 . II D[71] Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. Surface terms as counterterms in the AdS / CFT corre-spondence. Phys. Rev. D , 60:104001, 1999. arXiv:hep-th/9903238 , doi:10.1103/PhysRevD.60.104001 . II D[72] Bom Soo Kim. Holographic Renormalization of Einstein-Maxwell-Dilaton Theories. JHEP , 11:044, 2016. arXiv:1608.06252 , doi:10.1007/JHEP11(2016)044 . II D[73] Peter Breitenlohner and Daniel Z. Freedman. Stability in Gauged Extended Supergravity. Annals Phys. , 144:249, 1982. doi:10.1016/0003-4916(82)90116-6 . III A[74] Masashi Kimura. A simple test for stability of black hole by S -deformation. Class. Quant. Grav. , 34(23):235007, 2017. arXiv:1706.01447 , doi:10.1088/1361-6382/aa903f . III B[75] Sharmila Gunasekaran, Robert B. Mann, and David Kubiznak. Extended phase space thermodynamics for chargedand rotating black holes and Born-Infeld vacuum polarization. JHEP , 11:110, 2012. arXiv:1208.6251 , doi:10.1007/JHEP11(2012)110 . V[76] Natacha Altamirano, David Kubiznak, and Robert B. Mann. Reentrant phase transitions in rotating anti–de Sitter blackholes. Phys. Rev. D , 88(10):101502, 2013. arXiv:1306.5756 , doi:10.1103/PhysRevD.88.101502 . V[77] Antonia M. Frassino, David Kubiznak, Robert B. Mann, and Fil Simovic. Multiple Reentrant Phase Transitions and TriplePoints in Lovelock Thermodynamics. JHEP , 09:080, 2014. arXiv:1406.7015 , doi:10.1007/JHEP09(2014)080 . V[78] De-Cheng Zou, Ruihong Yue, and Ming Zhang. Reentrant phase transitions of higher-dimensional AdS black holes indRGT massive gravity. Eur. Phys. J. C , 77(4):256, 2017. arXiv:1612.08056 , doi:10.1140/epjc/s10052-017-4822-9 . V[79] Robie A. Hennigar and Robert B. Mann. Reentrant phase transitions and van der Waals behaviour for hairy black holes. Entropy , 17(12):8056–8072, 2015. arXiv:1509.06798 , doi:10.3390/e17127862 . V[80] Brian P. Dolan. Pressure and volume in the first law of black hole thermodynamics. Class. Quant. Grav. , 28:235017, 2011. arXiv:1106.6260 , doi:10.1088/0264-9381/28/23/235017doi:10.1088/0264-9381/28/23/235017